Paper Title: The frequency of transiting planetary systems around polluted white dwarfs.



firstpage–lastpage
Ge-based Clinopyroxene series: first principles and  experimental local probe study
    João P. Araújo
    August 5, 2024
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§ ABSTRACT 
This paper investigates the frequency of transiting planetary systems around metal-polluted white dwarfs using high-cadence photometry from ULTRACAM and ULTRASPEC on the ground, and space-based observations with TESS.  Within a sample of 313 metal-polluted white dwarfs with available TESS light curves, two systems known to have irregular transits are blindly recovered by box-least-squares and Lomb-Scargle analyses, with no new detections, yielding a transit fraction of 0.8_-0.4^+0.6 per cent. Planet detection sensitivities are determined using simulated transit injection and recovery for all light curves, producing upper limit occurrences over radii from dwarf to Kronian planets, with periods from 1 h to 27 d.  The dearth of short-period, transiting planets orbiting polluted white dwarfs is consistent with engulfment during the giant phases of stellar evolution, and modestly constrains dynamical re-injection of planets to the shortest orbital periods. Based on simple predictions of transit probability, where (R_* + R_ p)/a≃0.01, the findings here are nominally consistent with a model where 100 per cent of polluted white dwarfs have circumstellar debris near the Roche limit;  however, the small sample size precludes statistical confidence in this result.
Single transits are also ruled out in all light curves using a search for correlated outliers,  providing weak constraints on the role of Oort-like comet clouds in white dwarf pollution.



    circumstellar matter – 
    planetary systems – 
    white dwarfs










§ INTRODUCTION


Transit photometry is the leading method of detecting extrasolar planets, and is responsible for several thousand discoveries that constitute the bulk of confirmed detections <cit.>. From a geometric standpoint, the transit method favours the detection of planets on orbits close to their host stars, whereas sensitivity favours larger transit depths, such as those caused by giant and dwarf gas planets.

Relative to main-sequence stars, white dwarfs provide two potential advantages in transit searches: their small radii and the ubiquitous presence of closely orbiting planetary debris. Compared to the sun, while a typical white dwarf radius R_*=0.013 R_⊙ decreases the nominal transit probability by two orders of magnitude, it produces a 50 per cent transit depth for an Earth-size planet, and thus for similar photometric precision yields a sensitivity gain of nearly four orders of magnitude. Unlike main-sequence stars, there is comprehensive evidence of planetesimal debris discs orbiting within roughly 1 R_⊙ of metal-polluted white dwarfs <cit.>. At face value, this predicts around 1 per cent of polluted white dwarfs may exhibit transits. 

At least 25 per cent of white dwarfs exhibit signatures of planetary systems, which include emission from circumstellar dust and gas <cit.>, photospheric metals that require ongoing replenishment <cit.>, and transits <cit.>.  In support of these observations, there are theoretical predictions for planets closely orbiting white dwarfs, possibly resulting from debris disc evolution <cit.>, or post-main sequence dynamics <cit.>.

Transits provide a distinct view of white dwarf planetary systems, with a modest but growing number of discoveries <cit.>. These have revealed new empirical insights, and confirmed inferences based on prior work.  The rings of debris can have semimajor axes significantly larger than a solar radius, and thus potentially be highly eccentric <cit.>, where some exhibit line of sight gas absorption <cit.>.  The debris discs are highly dynamic, with large scale heights <cit.>, and therefore distinct from the canonical picture based on the rings of Saturn <cit.>. White dwarfs with transits generally do not have infrared excesses, suggesting their detection results from favourable viewing geometry and not enhanced dust production <cit.>. 

Currently, white dwarf transits fall into two categories: those with regular transits from self-gravitating bodies <cit.>, and irregular transits consistent with circumstellar debris fields, where metals have been detected when high-resolution spectroscopy is available <cit.>.  While all white dwarfs with infrared dust emission exhibit photospheric metals <cit.>, some that exhibit transits do not, or currently lack decisive data <cit.>.

Given that white dwarf pollution requires the ongoing or recent deposition of circumstellar debris, it may be inevitable that some exhibit transits. However, despite key forerunner investigations of transiting planets around white dwarfs <cit.>, the transit frequency toward polluted white dwarfs is currently unconstrained, and in particular for transiting debris discs. This work aims to constrain the frequency of planetary systems around metal-polluted white dwarfs, and utilizes standard methodology of box-least-squares and simulated injection and recovery for high-cadence photometric observations of 33 targets from the ground, and 313 objects from space.  The observations are described in Section 2, and the transit signal search and sensitivity are detailed in Section 3.  The results are discussed in Section 4, including a determination of transit frequency and comparison to related surveys, with a brief summary and outlook in Section 5.




§ OBSERVATIONS AND DATA


The data used in this paper are based on two programs targeting polluted white dwarfs: the first is a ground-based transit search for a sample of 33 stars using the high-speed photometry cameras ULTRACAM and ULTRASPEC <cit.>, and the second analyzes Transiting Exoplanet Survey Satellite (TESS; ) pdcsap light curves for 313 objects. 

The latter targets are a bright subsample that represent around 20 per cent of all known polluted white dwarfs, and were identified by cross-matching literature sources with the TESS Input Catalogue (TIC v8.2; ).






 §.§ ULTRACAM and ULTRASPEC


High-speed photometry for 19 polluted white dwarfs was obtained with ULTRACAM, a triple-beam imaging camera that enables simultaneous acquisition of data in wavelength regions covering the u, g, and riz bands, and facilitates rapid capture of the field of view via frame transfer technology. The observations were carried out using super-SDSS filters u_ sg_ sr_ s (with higher throughput compared to the conventional filters; ), between 2017 Nov 17 and 2021 Aug 20 at La Silla Observatory, where ULTRACAM is mounted on the 3.5-m New Technology Telescope.

These data were reduced using the HiPERCAM pipeline software <cit.>.  Differential photometry was performed on the source and at least one comparison star using a variable aperture (the number of field stars available for normalisation ranged from one to five depending on the field). The frames were corrected for bias and flat-fielded using normalized sky flats obtained in a continuous spiral (to remove stars) during twilight. To maximise the signal-to-noise ratio and the sampling cadence in the final light curves used here for transit detection, only the g and r bands are used in the subsequent analysis. These data were then co-added to further improve the fidelity of the final light curves, with a median scatter in the individual g and r bands of 1.2 and 1.7 per cent, respectively.

A further 14 polluted white dwarfs were observed with ULTRASPEC, which is mounted on the 2.4-m Thai National Telescope at Doi Inthanon. ULTRASPEC operates similarly to ULTRACAM, but instead utilises a single channel.  The data were obtained with a Schott KG5 filter to maximise light throughput, where the transmission approximates the wavelengths spanned by ugr combined. The ULTRASPEC observations were taken between 2015 Nov 30 and 2020 Jan 09, and the data were reduced as for ULTRACAM. The median scatter in ULTRASPEC target star light curves is 2.5 per cent. The time stamps for light curves from both instruments were converted to Barycentric Julian Day using Barycentric Dynamical Time.  All ground-based target observations are summarized in Table <ref>.




 §.§ TESS Data


TESS has surveyed the entire sky for transiting planets since its launch in 2018. During science operations, its four cameras take nearly continuous exposures, from which full-frame images and target pixel files of selected targets are produced by a dedicated data pipeline. All available light curves from Sectors 1–65 with 120 s-cadence data were downloaded via the Science Processing Operations Center pipeline. In addition, 214 of these stars had 20 s-cadence light curves, which were downloaded and analyzed in the same fashion as the 120 s data. This sample had mean TESS light curve baselines of 2.34 and 1.72 sectors for the 120 and 20 s cadences, respectively.  When downloading the light curves, only data with pipeline quality flags of 0 or 512 (impulsive outlier flag) were accepted. Despite typically indicating anomalous data, the impulsive outlier flag was accepted, based on the short durations expected for white dwarf transits (less than 2 min for a transiting Earth analogue, and thus only one or two in-transit measurements for the 120 s-cadence data). 

The source photometry obtained from TESS is susceptible to crowding owing to the 21 arcsec size and response function of the camera pixels. For exoplanet transits, there will be a reduction in the observed depths in blended light curves, leading to underestimated planet radii. This problem is worse for white dwarfs as they are intrinsically faint, where the TESS estimation of crowding, encoded by the crowdsap parameter, ranges from 0.011 to 0.998 (median and standard deviation 0.53±0.32) for stars in this study.  Despite the correction of any flux dilution in the pdcsap light curves <cit.>, crowding often results in false positive detections of variability and transit-like signals (see Section <ref>).

All long-term trends were removed from the light curves using a Tukey bi-weight slider from the wōtan package <cit.>, where a kernel size of 0.2 d was used for most of the data; this is approximately a factor of three larger than the maximum expected transit duration for a planet with radius 2 R_ Jup and period 27 d.  In special cases where the light curve displayed rapid variability (owing to contamination from nearby sources), a smaller, tailored kernel size was adopted.  The de-trended TESS light curves have significantly larger scatter than for the ground-based observations, with a median of 12.5 per cent, illustrated in Fig. <ref>.



§ METHODS AND DATA ANALYSIS


The following section details the algorithms used to search for new transiting systems, and the procedures 
to determine (upper limit) occurrences for planetary systems, in the case of either real or null detections. First, a transit search was executed for all light curves using the box-least-squares algorithm <cit.>, which identifies exoplanet transit signals by fitting boxcar functions to light curves phase folded on trial periods. Second, transit injection and recovery was used to determine the sensitivities of the respective instruments to planetary transit signals. 








 §.§ The transit search algorithm


For the ULTRACAM and ULTRASPEC light curves, where the median light curve duration is 4.1 h, BLS was used to search for transit signals in a period range spanning 15 min to 7 h, with a grid spacing of 3.6 s, where the upper period bound is slightly less than twice the median, ground-based observation baseline.  The simulated period range for the ground-based data encompasses orbits from the inner edge of a debris disc to just outside the Roche limit. For the TESS sample, periods from 1 h to 27 d, spaced by 60 s, were searched for both the 120 and 20 s cadence light curves. The lower bound of 1 h is interior to the Roche limit for a white dwarf, and the upper bound of 27 d corresponds to the approximate baseline of a single TESS sector. In both cases, the period grid is sufficiently fine to detect short-duration transits, while also enabling calculation of the BLS periodogram for each light curve at reasonable computational expense. From the periodogram, three candidate periods that pass the detection threshold are selected. The first candidate period corresponds to the signal with the highest power in the BLS spectrum, and likewise the second and third candidate periods correspond to the next two respective maxima in the periodogram, with the condition that they are neither harmonics of the best candidate period nor of each other. 



Owing to the irregular dimming events in the light curves of white dwarfs with transiting debris, the Lomb-Scargle algorithm was used in addition to BLS, to facilitate the detection of any possible periodic signals. Both algorithms were used to perform a blind search in both the ground-based and TESS samples. For all light curves, the Lomb-Scargle periodogram was computed for the raw data, and BLS was run after de-trending. 

The detection thresholds for all light curves are individually determined using a bootstrap randomisation technique <cit.>. In short, for a given light curve, the fluxes are randomly shuffled while preserving the original time stamps of the observations. The periodogram is subsequently re-computed for each randomised light curve to extract the signal with the highest peak in the power spectrum. This randomisation process is repeated a total of 1000 times per light curve, from which the 99.9^ th percentile power is calculated, and this value is adopted as the detection threshold for that light curve.  The median signal detection efficiencies (defined in eq. 6 of ) and standard deviations of the detection thresholds are 10.8±2.9 for the TESS light curves, and 12.1±3.7 for the ground-based data. 

All TESS targets were inspected for contamination owing to aperture crowding using tess-localize <cit.>, which determines the source of variability in the sky for a given target pixel file and set of frequencies. The input frequencies were informed by the output of the Lomb-Scargle periodogram for each light curve. All periodograms and phase-folded light curves for each candidate period were visually examined for each target, with particular attention given to stars that were determined to be variable by tess-localize. Visual inspection may introduce uncertainties in the characterisation of light curve variability, but is unlikely to have a significant effect on the overall results. There were 32 instances where signals in the Lomb-Scargle periodogram passed the associated false alarm threshold, but that were subsequently determined to be spurious by tess-localize. This corresponds to a 9.6 per cent contaminant rate for variability detections in the space-based sample. 




 §.§ Transit injection and recovery


In an ideal scenario, the planetary occurrence is simply the ratio of the number of detected systems to the total sample size N. However, only a proportion Np_ transit = N · (R_* + R_ p)/a of planetary systems transit the stellar disc, and within this reduced subset, only a fraction S(R_ p, P) – the detection sensitivity – can be found with a given instrument. Both of these factors hinder the detection of transiting systems in a photometric survey, and in the case of a null detection, yield only upper limits on their actual frequency. To account for both of these, an effective sample size N' can be used instead <cit.>, defined as 

    N'(R_ p, P) = N · p_ transit(R_ p, P) · S(R_ p, P)   ,

and this approach is used to determine all transit occurrences hereafter. 

The detection sensitivity encodes the probability that an instrument can detect planets of a given period and radius, and can be computed via the method of injection and recovery. This process injects a simulated transit signal into a light curve, for which a BLS periodogram is computed using the injected data. Because the period and radius of the synthetic transit are known quantities, if the correct period or a harmonic is identified by BLS, then a real signal (i.e. regular transits from a self-gravitating body or irregular dimming events from debris) of the same period and radius should have also been detected. The sensitivity is then simply the ratio of the number of recovered injections to the total number of simulations. The Python software package batman <cit.> was used to inject transit signals into the de-trended light curves where no pre-existing transit features were present (32 ground-based, 311 and 213 TESS 120 s- and 20 s-cadence light curves, respectively). Because transits were injected following light curve de-trending, it may be possible that the resulting detection sensitivities are marginally higher than otherwise. All sample stars were assumed to have R_*=0.013 R_⊙ and M_*=0.65 M_⊙, with uniform limb darkening, and the impact parameter of the synthetic planet was drawn from a uniform distribution spanning 0 to (R_ p + R_*)/R_* (i.e. grazing transits are permitted). The injected transits had zero eccentricity: on the one hand, tides are theorised to circularise orbits within a few Myr for planets that survive until the white dwarf phase <cit.>; on the other hand, the data provide no actual constraints on eccentricity in the absence of detections. 

The model integration time was chosen to match the exposure times for the ground-based observations and TESS cadences, respectively. 

To initialise the injection and recovery process, a period-radius grid was defined using eight logarithmically-spaced radii from 0.0625 to 8.00 R_⊕ (corresponding to dwarf planets like Ceres up to the radius of Saturn). The injected period range for each sample was identical to the period grid used in the respective transit searches in Section <ref>. For each injection, a light curve was randomly selected from the sample, and values for the period and radius of the synthetic transit signal were randomly drawn from a uniform distribution. A synthetic transit signal was considered to be recovered by the search algorithm if the best-fit and injected periods were within a tolerance of 2 per cent. The simulations were run until the recovered sensitivity converged to within 0.5 per cent of the local median value within each period-radius cell, with a total of 37 800 injections performed for each of the ground-based and TESS 120 s cadence light curves, and 20 400 injections for the TESS 20 s subset (fewer in the latter case, to reduce computational expense). 




 §.§ Occurrence constraints


If transits are detected, the occurrence in radius-period phase space is the number of detections n divided by the effective sample size N' <cit.>. In the case that no transits are found, upper limit occurrences can be calculated as a function of radius and period. Assuming a binomial distribution describes the probability of finding n planetary systems in an effective sample of N' white dwarfs <cit.>, the rate of occurrence for a null detection is given by <cit.>


    f_ max = 1 - (1-C)^1/1+N' ,


where f_ max is the maximum occurrence for a null detection at an arbitrary confidence level C. 




The uncertainty in effective sample size is 

    σ_N' = √((p_ transit× N ×σ_S)^2 + (S × N ×σ_ transit)^2) ,


where σ_S and σ_ transit are the uncertainties on the detection sensitivity and transit probability, respectively. The uncertainty on the maximum occurrence f_ max for a null detection is therefore

    σ_f_ max = | ∂ f_ max/∂ N' σ_N'| = | ln(1-C) · (1-C)^1+N'/(1+N')^2 σ_N'| .

As a compromise between under- and over-predicting the transiting planet occurrence, a value of C = 0.95 (i.e. a 2 confidence interval) is adopted for all rates and uncertainties presented and discussed hereafter. Because the 20 s-cadence data have a higher detection sensitivity than the 120 s data, the optimum effective sample size is a combination of data from both cadences. The combined uncertainty can be computed via error propagation, and both of these steps were executed following eqs. 8 and 9 in <cit.>.




 §.§ Single transit detection


The non-periodic nature of single transits requires an alternative detection strategy. To search for such signals in the TESS light curves, the data were combed for outlier flux points, using a sliding filter of size 8 h (a factor of five greater than the maximum expected duration for a Jupiter-size planet with a 200 d period). Fluxes more than four standard deviations below the filter median were flagged as candidate single transits. This threshold represents a good balance between sensitivity to shallow transits and a manageable number of spurious detections. These were subsequently filtered by only accepting correlated changes in flux, with a minimum number of four and 12 data points in the event for 120 and 20 s cadence light curves, respectively. This minimum-case signal was chosen to be representative of a transiting Mars-size planet on a 45 d orbit. 

Adjacent arrays were then compared to reduce the possibility of long-term systematics or correlated noise being flagged as candidate events. This detection routine was tested on a sample of 200 TESS light curves with injected single transit signals, and produced a 90 per cent true positive rate with zero false negatives for (synthetic) transit depths larger than 4 per cent. The single transit detection algorithm was then run for the ground-based data and TESS light curves. No single transits, consistent with a planetary body, were confidently detected in either the ground-based or TESS data. However, two false positives, WD 0145+234 and WD 2253+803, were flagged in this search, and are shown in Fig. <ref>. The double-dip feature present in the Sector 43 light curve of WD 0145+234 is caused by a spurious increase in the sky estimate in the pdcsap pipeline aperture photometry, which is likely caused by a passing asteroid. As is the case with WD 2230–125 (see Fig. <ref>), the estimated speed of the passing objects are consistent with the orbits of asteroids, which are slower than the earth (25 km/s) owing to wider orbits (although the relative speed to the TESS satellite can be slightly smaller or larger). The speeds of these objects are not consistent with satellites. The single light curve dip apparent for WD 2253+803 in Sector 58 was caused by an instrumental pointing excursion lasting approximately 30 min, during which fluxes were re-distributed into adjacent pixels, resulting in corresponding dimming and brightening events in different parts of the array. Although these features are not true dimming events in the target star light curves, their detection underscores the ability of this method to recover single transits.








§ RESULTS AND DISCUSSION


This section outlines the results of the transit search, and derives upper limits for transiting planet occurrence based on their null detection. The TESS results are compared with related studies and simple predictions based on transit probability, applied to polluted white dwarfs.



 §.§ Recovered transiting and variable systems


The search blindly recovered two of three targets with known irregular transits from debris clouds via their TESS light curves; ZTF J0328–1219 <cit.> and WD 1054–226 <cit.>, but not WD 1145+017 <cit.>.  Normalized BLS periodograms computed for the pdcsap light curves of both stars with detected transits are shown in Fig. <ref>. Of the remaining, metal-rich white dwarfs with known transiting debris, SDSS J0107+2107, ZTF J0139+5245 and SBSS 1232+563 are too faint to be included in the TIC and are not part of the target sample. WD 1856+534, which has regular transits from a candidate giant planet <cit.>, does not meet the sample selection criteria because it is not metal-polluted. Thus, a nominal fraction of 0.8_-0.4^+0.6 per cent of polluted white dwarfs exhibit detectable transits in TESS, where the median value and limits (16^ th and 84^ th percentiles) are derived using a beta distribution with a flat prior and a binomial likelihood.

The non-detection of WD 1145+017 is notable, as it has been observed to exhibit transit depths of 40 to 60 per cent in observations spanning at least a few years <cit.>.  Despite having T=17.3 mag, it has an excellent crowdsap of 0.93, hence variations of several per cent should be detectable in TESS data (; see Fig. <ref>). Nevertheless, neither BLS nor Lomb-Scargle reveal any significant periodicities, and the null detection suggests a change in the circumstellar activity of the prototype transiting white dwarf. This is corroborated by recently published, ground-based observations that show the transits are greatly diminished in depth and frequency in 2022 and 2023, and notably correlated with long-term, flux-calibrated photometry <cit.>.  Had WD 1145+017 been detected, the transiting fraction would have been 1.2^+0.7_-0.5 per cent, and thus not substantially changed.


The Lomb-Scargle search recovered two known ZZ Ceti stars that passed the tess-localize validation process: ZZ Psc (G29-38; ) and KX Dra (PG 1541+651; ). Notably, the BLS algorithm did not detect either of these pulsators, where the light curve variations are increases from the baseline flux.  Fig. <ref> plots the detected amplitudes of the variable polluted stars (both transiting and pulsating), together with the detection thresholds for non-variable pdcsap light curves, each computed using Lomb-Scargle. Only stars with crowdsap >0.5 are included in the plot, because photometric contamination from nearby field stars can superficially augment the variability threshold.  For the 180 stars plotted, the figure demonstrates excellent sensitivity to variable signals with amplitudes smaller than 1 per cent in most cases.








 §.§ Results of the injection and recovery


The resulting detection sensitivities, transit occurrences and upper limits for all samples are displayed in Fig. <ref>. The errors for the detection sensitivities were calculated using bootstrap re-sampling (this procedure is outlined in Section <ref>). The results of the injection and recoveries were re-sampled a total of 10 000 times per period-radius bin, and the standard deviation of the re-sampled distribution encodes the error that arises owing to a finite number of injections.  The ground- and space-based results have some common trends: for synthetic planets with larger radii and shorter orbital periods, the transit depths are larger and there are more transits in a given light curve, thus enabling higher detection sensitivity, while the opposite is true for simulated planets with smaller radii and longer periods. Because transit probability is also higher for closer orbits, Eq. <ref> provides stronger constraints for such planets. The shorter light curve duration (median 4.1 h) and sample size for the ground-based observations result in weak upper limits for transits with periods longer than around 2 h.  This effect can be seen by comparing the strongest occurrence constraints for the ground-based data (31.7 per cent for P<1 h) with a similar period-radius bin for the TESS sample (10.6 per cent for 1 h < P < 3 h). The visual inspection process was not applied to the synthetic transits, hence there may be potential imperfections in the recovery, and thus detection sensitivity. Because self-gravitating objects such as planets and brown dwarfs cannot have orbits interior to the Roche limit, the ground-based results presented in Fig. <ref> are better suited to describe the occurrence of circumstellar debris around polluted white dwarfs. However, for orbits longer than several hours, the space-based results are applicable to the occurrence of debris clouds and self-gravitating planetary bodies.






 §.§ Comparison with prior work


The first effort to constrain the occurrence of transiting planets around white dwarfs used data from the Wide-Angle Search for Planets survey (WASP) where the analysis consisted of 194 white dwarfs with 12 mag <V<15 mag that were selected without regard to spectral type, mass, or multiplicity <cit.>.  Assuming the sample are all products of single star evolution, the WASP data most strongly constrain the frequency of brown dwarf and giant planetary companions to no more than 10 per cent for periods less than 0.2 d. The results presented here for TESS provide similar, focused limits for polluted white dwarfs in corresponding period-radius bins, but also provide modest constraints for planets at greater distances, and significantly smaller radii.


A search for planetary transits around white dwarfs was also performed using data from the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; ). The size of the white dwarf sample in this study is significantly uncertain, with 661 strong candidates identified by reduced proper motion, but the bulk of the 1718 candidates selected using color-color cuts that likely include contaminants.  Nevertheless, the study found no transit candidates for light curves with a few 10^3 epochs per star, with four to eight consecutive 240 s exposures per night.  The study reports an upper limit of 0.4 per cent for transiting planets with R_ p≳ 2  R_⊕ and semimajor axes smaller than 0.02 au. Keeping in mind that this upper limit frequency may be significantly underestimated based on the uncertain sample size of bona fide, single white dwarfs, TESS compares more favorably in the bulk of the parameter space in Fig. <ref>, especially for planets smaller than Earth at any orbital distance.

A search for transiting substellar objects around 1148 high-probability white dwarfs in K2 recovered nine eclipsing binaries, and the transits towards WD 1145+017 <cit.>. An upper limit frequency of 82 per cent was placed for planets larger R_ p ≈ 0.5  R_⊕ with orbital periods less than a few days, and is comparable to 84 per cent found with the TESS sample of single, polluted white dwarfs.  However, K2 is more sensitive to planets with P < 15.6 h and R_ p >  R_⊕), assuming a significant fraction of the sample white dwarfs are intrinsically single. With the crude assumption that the values reported in each K2 period-radius bin are representative of the entire bin range, then orbits between 10 and 27 d are roughly equally-well constrained by TESS, with weak constraints on smaller planetary radii than K2.   


Using both forward modelling and hierarchical Bayesian inference, it has recently been predicted that transiting rocky planets around white dwarfs are exceedingly rare <cit.>. While consistent with the results presented here for transits of self-gravitating bodies, the prediction does not seem to account for engulfment, which is primarily responsible for their absence.  In fact, direct engulfment is expected to clear all planetary bodies out to several au following both the first-ascent and asymptotic giant branches <cit.>. Thus, the candidate transiting planet WD 1856+534b is indeed a surprise, but likely accounted for by Lidov-Kozai migration <cit.>.  Furthermore, the prediction does not capture the necessarily frequent orbital injection of rocky planetesimals that are required for the observed debris discs and white dwarf photospheric pollution. This planetary material must have an origin further out to avoid engulfment during prior stellar evolution.  If the two stars with irregular transits recovered in this work are indicative of a 100 per cent occurrence of closely-orbiting planetary material, then a similar fraction of polluted white dwarfs are host to distant planets that are dynamically responsible for the star-grazing orbits required to approach the Roche limit.



 §.§ Debris transits consistent with frequent occurrence


Because all white dwarfs in this study are metal-polluted, the accretion of planetary material is either ongoing or recent, and thus most or all targets are expected to have circumstellar debris, where two stars were detected to have associated transits. If all polluted white dwarfs have debris discs orbiting near their Roche limits (1.2 R_⊙), and these are sufficiently opaque and vertically extended to marginally result in detectable transits (p_ transit=1.4 per cent), then 4.4 events are expected in a sample of 313.  These predictions are within the error bars of the 0.8^+0.6_-0.4 per cent transit fraction and 2.0 events recovered in this study.  The results are broadly compatible with a model where all polluted white dwarfs have debris orbiting near the Roche limit, and thus their detections are the result of favourable viewing geometry.

The bulk of occurrences presented in Fig. <ref> are upper limits relevant to null detections, with the exception of the two bins where transiting debris is identified (outlined in white). Owing to the modest detection sensitivity and small transit probability, both bins with detected transits have n/N' that saturates at 100 per cent. Alternatively, using a modified Eq. 2 to compute the upper limit occurrence given a single detection also yields unity. Despite a small effective sample size at these radii and periods, these occurrence values are consistent with circumstellar material orbiting all polluted white dwarfs, and in agreement with the prediction of transit probability above.




 §.§ Constraints on exocomets


There are several models that have considered Oort-like comet clouds to account for metal pollution in white dwarfs, where the first preceded the discovery of their debris discs <cit.>.  All studies agree that a modest fraction of distant comets should remain bound, even in the case of asymmetric mass loss during stellar evolution <cit.>. Given the potential numbers, exocomets might significantly contribute to metal pollution, where simulations demonstrate they can be perturbed from several thousand au into the Roche limit of a white dwarf, either by a natal kick <cit.>, or from Galactic tides <cit.>. Extrasolar comets are apparent toward young stars like  Pic via time-series spectroscopy <cit.>, and more recently via transits in that system <cit.>, and others <cit.>.  It thus seems plausible and worthwhile to consider their survival and possible transits near periastron around polluted white dwarfs, particularly because cometary comae are typically Earth-sized or larger, and so their transits should be straightforward to detect in white dwarf light curves.  

The probability that a transit is detected within a sample of N=313 white dwarfs is 


    p_ detect = p_ transit× N × m × f_ orbit× f_ coma,


where m is the number of exocomets hosted by each star, f_ orbit is the fraction of the orbital period observed (e.g. by TESS), and f_ coma is the fraction of the orbit over which a coma (i.e. outgassing) is present and detectable. To estimate this overall probability, the following parameters are adopted: a semimajor axis of 3000 au <cit.>, periastron at 1 R_⊙, p_ transit = (R_* + R_⊕)/a/(1-e^2) for Earth-sized comae, f_ orbit = 8.5 × 10^-7 for the median observing coverage of 2.3 TESS sectors in this study, and f_ coma = 5.5 × 10^-8 for activity that begins around 150 K (i.e. interior to 0.15 au) for a white dwarf luminosity of 2×10^-3 L_⊙. Under the assumption that each exocomet orbit approaches the Roche limit, m = 2 × 10^12 comets are needed to achieve p_ detect=0.5, and thus a reasonable chance of detecting one in transit.  This ad hoc scenario would require all polluted white dwarfs to have bountiful Oort-like clouds, each dominated by star-grazing orbits. While unrealistic, this calculation nevertheless provides a detection benchmark, where it is notable that a recent model predicts distant planetesimal belts can be excited to high eccentricities following white dwarf natal kicks <cit.>.

There are models that specifically calculate the rate at which Oort-cloud analogs are delivered to the Roche limit of white dwarfs, including the effects of Galactic tides, as well as the mitigating influences of companions and stellar evolution <cit.>.  The most optimistic of these predict that comets may reach the innermost regions at rates as high as ∼2×10^4 Myr^-1, in the absence of a companion and without substantial loss of comets during post-main sequence evolution (each of which can reduce this number by an order of magnitude).  During the median observing coverage of 2.3 TESS sectors for the 313 white dwarfs in the sample, this rate predicts each star could have up to 0.0035 comets on such steeply infalling orbits, where a fraction of 0.011 of these would transit.  Overall, there would have been a 1 per cent chance at best to detect an exocomet transit in the space-based sample, and this probability is likely less favourable based on the aforementioned model considerations. Thus, the lack of a cometary transit here does not significantly constrain the prevalence of extrasolar Oort clouds, but future samples that are orders of magnitude larger should be more informative.




§ CONCLUSION AND OUTLOOK


This study searched for transiting planetary systems around polluted white dwarfs using ground-based observations with ULTRACAM and ULTRASPEC, and space-based data from TESS. Both the BLS and Lomb-Scargle algorithms were used to search for periodic signals in the target light curves, and two systems known to have periodic dimming events associated with debris discs were blindly recovered. The calculated detection sensitivities provide upper limits for the occurrence of planets and planetesimals ranging from dwarf to Kronian planetary radii.  The nominal fraction of polluted white dwarfs with detectable transits is 0.8_-0.4^+0.6 per cent.

The 40 pc sample of white dwarfs contains 1076 stars with 99.3 per cent spectroscopic completeness <cit.>. Assuming a pollution fraction of 30 per cent <cit.>, and the transit incidence found here, then 2.6_-1.6^+2.0 metal-rich white dwarfs within this volume are expected to exhibit transits. Currently, WD 1054–226 is the only polluted white dwarf with transits within 40 pc (ZTF J0328–1219 is just outside this volume), and it is therefore possible that one or two additional nearby, polluted white dwarfs will have transiting debris discs. 

While models for white dwarf pollution require major planets to dynamically perturb asteroids to the Roche limit, these are expected to have semimajor axes of several to tens of au <cit.>.  However, planets might be born of recycled disc material, as a result of viscous spreading beyond the Roche limit for sufficiently massive debris discs <cit.>, or achieve temporary, star-grazing orbits due to unpacking of previously stable, multi-planet systems <cit.>. While none are detected here, the planetary constraints are modest outside of orbital periods of a few days.

There is a basic necessity for planets in polluted white dwarf systems, as the tidally-disrupted planetesimals responsible for debris discs and pollution require dynamical perturbations.  Moreover, these orbits have to be sufficiently distant to escape engulfment during stellar evolution, and will likely have expanded further owing to stellar mass loss <cit.>.  Such widely-bound planets may be directly imaged by JWST if they are at least as massive as Jupiter at favorable angular separations <cit.>.  However, the requisite number of observations to achieve a detection will be challenging. 

It is likely that the most promising avenue for discovering planets orbiting (polluted) white dwarfs will be astrometric detection with Gaia. The fraction of white dwarfs that host planets detectable with Gaia astrometry is predicted to be as high as 0.1 per cent <cit.>, yielding an order of ten planets on au-scale orbits. There are currently three white dwarfs with candidate planets that have been inferred through a combination of Gaia and Hipparcos astrometry as proper motion anomalies <cit.>. Furthermore, Gaia DR4 should provide epoch photometry for millions of stars, and increase the number of white dwarfs with space-based light curves by over two orders of magnitude compared to TESS.  While the cadence and photometric precision of Gaia will remain inferior to TESS, the sheer numbers should result in new transiting systems.  Rubin-LSST will obtain light curves for up to 10^8 white dwarfs, but the observing cadence and baseline may complicate immediate characterization of any transits.  Nevertheless, if the 0.8 per cent fraction determined here is correct to an order of magnitude, then several thousands of transiting systems await detection. Even if only a tiny fraction of these are sufficiently bright for follow up, this can transform the field from the study of individual stars to statistical analyses of a population.



§ ACKNOWLEDGEMENTS


The authors thank E. M. Bryant for useful discussions about injection and recovery simulations.  A. Robert acknowledges support from a Science and Technology Facilities Council studentship. This work was supported by Naresuan University (NU), and National Science, Research and Innovation Fund (NSRF) (grant No. R2566B021). This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101020057).The authors acknowledge the European Southern Observatory for the award of telescope time via programs 0100.C-0546, 0102.C-0700, 0103.C-0247, and 105.209J. This work has made use of data obtained at the Thai National Observatory on Doi Inthanon, operated by NARIT.  ULTRACAM and ULTRASPEC are funded by the Science and Technology Facilities Council (grant ST/V000853/1). This paper utilizes data collected by the TESS mission, which is funded by the NASA Explorer Program, including targets from Guest Investigator programs: G011113, G011179, G011203, G011288, G011294, G022017, G022057, G022065, G022077, G022158, G022176, G022217, G03046, G03124, G03126, G03207, G03221, G03252, G03274, G04036, G04056, G04091, G04106, G04123, G04137, G04171, G04177, G04200, G04209, G04211, G04219, G04236, G04240, G05003, G05057, G05071, G05081, G05084, G05115, G05148. 





§ DATA AVAILABILITY

ULTRACAM and ULTRASPEC data are available on reasonable request to the authors. TESS data are available through the Mikulski Archive for Space Telescopes. 





mnras






















§ GROUND-BASED OBSERVATION SUMMARY











 Paper Title: Magnetic field diagnostics of prominences with the Mg II k line: 3D Stokes inversions versus traditional methods.

 



Prominence diagnostics with Mg ii k



Štěpán et al.


Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165 Ondřejov, Czech Republic
Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
Consejo Superior de Investigaciones Científicas, Spain



 

The Mg ii k resonance line is commonly used for diagnosing the solar chromosphere. We theoretically investigated its intensity and polarization in solar prominences, taking 3D radiative transfer and Hanle and Zeeman effects into account. We used an optically thick 3D model representative of a solar prominence and applied several inversion methods to the synthetic Stokes profiles, clarifying their pros and cons for inferring prominence magnetic fields. We conclude that the self-consistent 3D inversion with radiative transfer is necessary to determine the magnetic field vector, although its geometry cannot be inferred with full fidelity. We also demonstrate that more traditional methods, such as those based on the weak field approximation or the constant-property slab assumption, can offer useful information under certain conditions.




Magnetic field diagnostics of prominences with the Mgii k line
 3D Stokes inversions versus traditional methods
    
Jiří Štěpán1
Tanausú del Pino Alemán2,3
Javier Trujillo Bueno2,3,4

    Received XXXX; accepted XXXX
===============================================================================================================







§ INTRODUCTION


Among the various spectral lines used in the diagnostic of the solar chromosphere and prominences, we have the  h and k lines, located at 280.3 nm and 279.6 nm, respectively. This ultraviolet (UV) resonance doublet is known for its strong sensitivity to the physical conditions of the plasma, making it a valuable tool for probing.

Recently, there has been increased interest in these UV lines thanks to the spectroscopic observations obtained by the Interface Region Imaging Spectrograph <cit.> and the spectropolarimetric data collected by the two Chromospheric LAyer SpectroPolarimeter missions, CLASP2 <cit.> and CLASP2.1 <cit.>. These suborbital space experiments have provided unprecedented observations of  h and k lines, confirming previous theoretical predictions and providing new insights into the structure and dynamics of the upper solar atmosphere <cit.>.

Building on the success of these missions, the scientific community is now planning the Chromospheric Magnetism Explorer (CMEx) space telescope <cit.>, which will focus on the spectropolarimetry of the spectral region of the  h & k doublet. This new mission is expected to provide detailed information on the magnetic field and dynamics of the plasma in the chromosphere and prominences.

For the last 40 years, many studies have attempted to understand the formation of the Mg ii h & k lines in prominences to explain the available prominence observations. The first 2D freestanding slab models of prominences were developed by <cit.> using the complete frequency redistribution (CRD) approximation for calculating the  k line intensity. Later, <cit.> found similar results for the core of the h and k lines with both the CRD approximation and accounting for partial frequency redistribution (PRD) effects in prominences. <cit.> conducted a thorough analysis using 1D prominence slab models and confirmed that the results obtained using a two-level  atomic model without a continuum agree with those found using a multilevel plus continuum model. Their investigation indicated that prominences are generally optically thick in the k line, with line-center thicknesses reaching up to 10^3 or 10^4. <cit.> compared 1D models with IRIS data of prominences and find that the integrated line intensities require optically thick plasma.

The magnetic field strength in quiescent solar prominences presumably ranges from a few to several tens of gauss, although it can occasionally reach significantly higher values <cit.>. While in the chromosphere the h & k doublet exhibits remarkable linear polarization signals in the profile wings due to J-state interference and magneto-optical effects <cit.>, only the line core is visible in prominences. Therefore, our study focused on the core of the  k line. This line is sensitive to both the Zeeman and Hanle effects. Its circular polarization degree due to the Zeeman effect scales linearly with ℛ=1.5× 10^-4B <cit.>, allowing for the determination of the line-of-sight (LOS) component of the magnetic field, , while the transversal component in the plane of the sky (which scales with ℛ^2) would be very difficult (or impossible) to determine via the Zeeman effect at the expected magnetic field strengths. The linear polarization of the  k line arises dominantly from the scattering of anisotropic radiation and the Hanle effect. The critical Hanle field of the line is B_ H≈ 22  G <cit.> and is therefore suitable for quiescent prominence diagnostics. However, diagnosing magnetic fields becomes challenging when the field is not constant along the LOS and when radiation transfer needs to be considered because the prominence plasma is optically thick (τ>1) and not in local thermodynamic equilibrium (LTE), that is, when it is in the so-called non-LTE regime.

The main aim of this paper is to theoretically investigate the Stokes inversion problem of the  k line in solar prominences, taking the effects of 3D radiative transfer (RT) into account. To this end, we used an optically thick 3D prominence model with a magnetic field that varies spatially along the model's loop-like structure. Because the model's physical properties vary along the three spatial directions, the axial symmetry of the incident radiation field can break at each point within the medium without the need of a magnetic field. Such nonmagnetic causes of symmetry breaking can have an important impact on the linear polarization signals caused by the scattering of anisotropic radiation <cit.>,  which at the line center are sensitive to the presence of magnetic fields via the Hanle effect. In addition to anisotropic radiation pumping and the Hanle effect, our study also includes the Zeeman effect, which dominates the line's circular polarization. Given that in prominences the  k line does not show extended wings, where the effects of PRD and J-state interference are very important <cit.>, in this investigation we solved the 3D non-LTE RT problem assuming CRD without J-state interference. Our approach to the 3D Stokes spectral synthesis and inversion problem without assuming LTE can be found in <cit.>.

Here we focused on investigating the suitability of the  k line for spectropolarimetric diagnostics of optically thick prominences, which requires understanding the magnetic field's impact on the polarization of the emergent spectral line radiation. In particular, we ask if our 3D Stokes inversion, which consistently accounts for RT without assuming LTE, can uncover the global magnetic field geometry of an optically thick prominence using the  k line. Section <ref> describes the prominence model and the spectral synthesis of the emergent Stokes profiles, taking the effects of RT  into account in the 3D model. In Sect. <ref> we apply the weak field approximation <cit.>, a Bayesian approach based on the constant-property slab model, and the full 3D Stokes inversion to the synthetic data, assessing the goodness of the inference. Finally, Sect. <ref>  outlines our conclusions.





§ PROMINENCE MODEL AND SYNTHESIS OF THE STOKES PROFILES 
























Figure <ref> illustrates the prominence model we used to investigate the performance of the three inference methods we applied to the synthetic Stokes profiles. Although it is an academic prominence model with a relatively simple magnetic field geometry, it features the main ingredients that can affect the performance of the inversion, namely a nontrivial variation of the physical quantities both across the field of view (FOV) and along the LOS, and a relatively large optical thickness to manifest the effects of RT in 3D geometry.

For simplicity, we assumed that all Mg atoms are in the Mg ii ionization stage. At chromospheric temperatures the Mg ii ion is indeed the dominant species <cit.>, and at about 15–20 kK the Mg ii/Mg iii fraction quickly decreases and Mg iii becomes the dominant species <cit.>. We modeled the Mg ii k resonance line using a two-level atom model and in the limit of CRD. These two approximations can be justified because we can directly specify the total population in the lower and upper levels of the k line of Mg ii (avoiding the need to account for an equation of state and for ionization and recombination processes) and because the line center, dubbed k_3, is not strongly affected by PRD effects <cit.>.[While strong velocity gradients can make the line center sensitive to PRD effects due to the Doppler shifts <cit.>, the lack of bright wings in prominence profiles may diminish this effect.] Finally, we assumed an abundance of 7.544 for magnesium. Given the academic nature of our 3D prominence model, we assumed that hydrogen, which is the most important donor of electrons under the considered thermodynamic conditions, is fully ionized. We could then estimate the electron volume density directly from the volume density of magnesium and its abundance. The inelastic collisional rates in the  k line transition have been calculated using the data from <cit.>.

The spatial domain of our prominence model is 50×50×50 Mm^3; it contains a loop-like structure whose legs start at the top of the chromosphere and extends up to about 40 Mm above it; the spatial domain sits on top of the C model of , hereafter the FAL-C model, at about 2.2 Mm above the visible solar surface. The Mg ii number density is prescribed and decreases outward from the inner part of the prominence body. The optical thickness is maximum at the central part of the apex of the loop-like structure and exceeds 100 at the  k line center. The prominence is isothermal, with a temperature of 10 kK, with a microturbulent velocity of 5 km/s <cit.>. For simplicity, we do not include bulk velocities in this model. Finally, the magnetic field is such that the field lines follow the loop-like structure (see Fig. <ref>), with a strength reaching about 100 G close to the chromosphere and decreasing to about 40 G at the apex. These relatively strong magnetic fields have been chosen to produce a circular polarization signal above the considered noise level.

The prominence is illuminated by the underlying chromosphere, not limited to the computational domain. This illumination is axially symmetric and only depends on the angle between the propagation direction and the local vertical. The spectral profiles and their center-to-limb variation (see Fig. <ref>) have been calculated with HanleRT-TIC[The 1D non-LTE code is publicly available at <https://gitlab.com/TdPA/hanlert-tic>.] <cit.> using the FAL-C model. We note that in our solution we did not approximate the geometry of the solar chromosphere by an infinite plane, but we took the curvature of the solar surface into account.

We solved the non-LTE RT problem in 3D by applying the spectral synthesis mode of our code <cit.>, obtaining the J^K_Q radiation field tensor components everywhere in the domain, and the emergent Stokes profiles for the chosen LOS (see Fig. <ref>). The resulting FOV is shown in Fig. <ref>. The linear polarization is due to the scattering of anisotropic radiation and the Hanle effect, while the circular polarization is caused by the Zeeman effect. For the application of the inference methods presented in Sect. <ref>, we selected two particular positions in the FOV (black and orange dots in Fig. <ref>). The spectra corresponding to these two locations, after adding Gaussian polarimetric noise with σ=5× 10^-11 erg s^-1 cm^-2 Hz^-1 sr^-1, on the order of 5×10^-4 of the line-center intensity, are shown in Fig. <ref>. The material behind the black dot LOS has an optical thickness at the line center of about 160, showing the typical self-reversal in its intensity profile, while the orange dot LOS has a more modest optical thickness of about 2.5 at the line center, showing no self-reversal. In Fig. <ref> we show how  and the line center opacity change with the optical thickness at the line center, τ, along the LOS for these two selected positions.





§ MAGNETIC FIELD INFERENCE



In this section we apply three methods to infer the magnetic field in the prominence model described in Sect. <ref>. In Sects. <ref> and <ref> we apply two commonly used inference techniques, both of them based on strong assumptions for the RT. These two methods are applied pixel by pixel, and for the presentation of the results we chose two particular positions in the FOV (see the black and orange dots in Fig. <ref>), which are representative of a LOS with significant optical thickness at the line center (>100) and a LOS with a modest optical thickness at the line center (∼2.5). In Sect. <ref> we apply our 3D Stokes inversion to infer the magnetic field vector in the whole prominence body and its surroundings. In particular, we are interested in assessing the goodness of the fits in the different methods and discussing their advantages and disadvantages.




 §.§ Weak field approximation (WFA)



When the Zeeman splitting of the line's levels produced by the magnetic field is much smaller than the spectral line width, and under other certain assumptions, it is possible to find a closed and simple expression for the circular polarization, the so-called WFA <cit.>,

    V(λ) = -Cλ^2_0B_ LOSg_ eff∂ I/∂λ(λ),

where C = 4.6686× 10^-13 G^-1Å^-1, λ_0 is the wavelength of the spectral line in Å, g_ eff is the effective Landé factor (7/6 for the  k line), and  is the longitudinal component of the magnetic field in gauss.

The first assumption that must be fulfilled for the applicability of the WFA is that the Zeeman effect must be the only mechanism contributing to the polarization. While in solar prominences the linear polarization of the Mg ii k line is produced by the scattering of anisotropic radiation and the Hanle effect, the circular polarization is dominated by the Zeeman effect. Consequently, in this subsection we focus only on the circular polarization profiles.

The second assumption for the applicability of the WFA is that  must be constant along the LOS. While this condition seems really restrictive and close to impossible to fulfill, in practice it means that  must be approximately constant along the LOS in those regions actually contributing to the emergent profiles. It is also clear from Eq. (<ref>), which relates Stokes V and the derivative of the intensity I, that when the profiles are formed in extensive regions, the contributions to both Stokes parameters must come from the same regions along the LOS.

If we assume uncorrelated and Gaussian noise, we can use Eq. (<ref>) to calculate  as follows <cit.>,

    B_ LOS = -1/C∑_j V(λ_j)I'(λ_j)/∑_j(I'(λ_j))^2±σ/C√(∑_j(I'(λ_j))^2),

where I'(λ_j) = λ^2_0g_ eff∂ I/∂λ(λ_j),
λ_j are the observed wavelengths, σ is the standard deviation of the Gaussian distribution of the noise in Stokes V, and the error is computed from the covariance matrix assuming a wavelength-independent standard deviation and a confidence level of 68.3 % (1σ).

In Fig. <ref> we show the WFA fit to the Mg ii k circular polarization profiles shown in Fig. <ref>, corresponding to the black and orange dots in Fig. <ref>, by applying Eq. (<ref>). The fits are quite good and the corresponding  are -34.9±2.3 and -36.6±3.6 G. From Fig. <ref>, we see that the retrieved  correspond approximately to the magnetic field around the maximum of the line-center opacity. We note that the τ scale in Fig. <ref> corresponds to the line center wavelength and that the opacity quickly decreases for the line wings. Therefore, even if at the line core we cannot “see” the regions with τ∼ 80 in Fig. <ref>, at the near wing wavelengths we can.

The WFA has the advantage of providing an estimation of  with negligible computing effort given that in our 3D model the required conditions are satisfied. However, the WFA returns a single number and an uncertainty that only accounts for the noise; there is thus no information about the magnetic field geometry and stratification, nor about other sources of uncertainty in the inferred values. For the particular cases we studied in this work, it turns out that  is relatively constant along the formation region of the profiles, keeping the same polarity along the whole LOS, and thus the inferred  are rather good estimates. However, for more complex magnetic field geometries, with potential cancellation effects (different polarities along the LOS) or strong source function gradients, the WFA does not guarantee a good estimation of .




 §.§ Constant-property slab Bayesian inversion



Most of the magnetic field inference in prominences (and filaments) over the last two decades relies on the modeling of spectropolarimetric observations in the He i triplet at 10830 Å and D_3 triplet at 5877 Å assuming a constant-property slab illuminated by the solar radiation of the underlying quiet Sun disk <cit.>. Several inference methods based on this model can be found in the literature, such as look-up tables based on principal component analysis <cit.>, minimization methods such as that implemented in the Hanle and Zeeman light (HAZEL) code <cit.>, or through Bayesian statistical approaches <cit.>.

One of the main assumptions of this modeling approach is that the radiation pumping within the slab is fully dominated by the cylindrical symmetric illumination from the underlying solar disk (i.e., that the excitation of the atoms within the slab is not affected by RT within the slab). In order for this approximation to be reasonable, the optical thickness of the slab plasma should be small 
enough <cit.>. The model then assumes that all properties of the plasma within the slab are constant along the LOS. Typically, a single slab is assumed in the modeling, but several components – both side by side <cit.> and one after the other along the LOS <cit.> – have been considered.

When the modeling assumptions are satisfied, these inference methods can provide estimations of the magnetic field vector while allowing for the study of ambiguities and uncertainties. However, when the optical depth is on the order of, or larger than, the unity along any direction within the prominence (or filament), or if the plasma properties are not constant in the region along the LOS where the line forms, the accuracy of the inference can be severely compromised. Moreover, the inference methods based on Bayesian statistics, while providing a clear picture of the uncertainties and ambiguities, are computationally heavy, especially when considering more than a single slab.

In prominences, the Mg ii k line investigated in this work typically shows larger optical thickness than the He i triplet lines  <cit.>. Consequently, an unsuitable performance of the constant-property slab model is, a priori, expected. At the pixel marked with a black dot in Fig. <ref> the plasma of our 3D model is very optically thick at the k-line center, with an optical depth of over one hundred. Its intensity profile (black curve in Fig. <ref>) shows a clear self-reversal, which cannot be reproduced assuming a single constant-property slab. On the contrary, at the orange dot pixel in Fig. <ref> the plasma of our 3D model has a total optical depth of about 2.5 along the LOS. This pixel is near the prominence “edge” and it can thus “see” most of the underlying chromosphere. We note, however, that the prominence body blocks some of the chromospheric radiation, so the assumption of cylindrically symmetric illumination is not fully valid. The  is approximately constant along the LOS for the orange dot pixel (see the bottom panel of Fig. <ref>), but the inclination of the magnetic field vector changes along the LOS.

We calculated the Bayesian posterior distribution for the case of a single-component constant-property slab model inversion of the emergent Stokes profiles at the location of the orange dot pixel. The parameters of our model are (i) , (ii) the cosine of the polar angle of the magnetic field inclination with respect to the solar radius (μ_B), (iii) its azimuth in the plane normal to the radius with respect to the projection of the LOS on such a plane (χ_B), (iv) the thermal width of the line (Δ v_D), and (v) and the line-center optical thickness (τ). We used a Jeffreys prior[We performed identical calculations using a uniform prior for τ and the results we have obtained are very similar. Nevertheless, our numerical experiment shows that the uniform prior for τ leads to slight overestimation of both τ and .] for τ and uniform priors for the rest of the parameters, between 0 and 1000 G for , between -1 and 1 for μ_B, between 0 and 2π for χ_B, and between 0.1 and 10 km/s for Δ v_D.

Figure <ref> shows the marginalized posterior distributions for , μ_B, and χ_B. Although the physical properties of the 3D model at the selected pixel do not fulfill the applicability conditions, because the optical depth is larger than unity, the illumination is not cylindrically symmetric, and μ_B changes along the LOS, the inferred  turns out to be as good as with the WFA (see Sect. <ref>) and, moreover, the constant-property slab approach is capable of finding χ_B up to the ambiguities. This inversion method is much slower than the WFA, but in exchange it provides additional physical information.

Even though the  inference is rather good, the magnetic field strength is overestimated by about a factor of 2 (hence the magnetic field energy density by a factor of 4) due to the significant uncertainty in μ_B. Due to the symmetry assumed in the 3D model, there are ambiguous solutions for both μ_B and χ_B.

For an optically thin prominence with a not-too-complex magnetic field geometry, approaches based on this constant-property slab model (Bayesian inference, principal component analysis, etc.) seem to be optimal since they can provide maximum information on uncertainties even when there are no self-consistent RT constraints.




 §.§ 3D Stokes inversion



The third inference method we applied to the synthetic Stokes profiles that we calculated by solving the non-LTE RT problem in our 3D prominence model is the 3D Stokes inversion approach described in <cit.>. This method approaches the inverse problem by finding the physical quantities in the whole spatial domain of the 3D model simultaneously. Different regions of the model's spatial domain are coupled by the transfer of polarized radiation and, in addition, the solution can have additional constraints such as those from the magneto-hydrodynamic equations. As described in detail in the aforementioned paper, this mesh-free method does not rely on calculating a sequence of self-consistent forward models that lead to the minimum of a merit function. Instead, it follows an unconstrained minimization method in which unphysical solutions are allowed but penalized via regularization terms in the merit function. This allows relatively accurate solutions to be obtained within a significantly shorter computing time. This method does not only provide a solenoidal magnetic field vector B⃗ everywhere in the model's spatial domain, but for our particular case also other thermodynamic quantities such as the atomic number density.

We solved the inversion problem using 480 CPU cores of the OASA computer of the Astronomical Institute in Ondřejov. The solution shown here was reached in about 20 hours or 10^4 CPU hours. As mentioned above, we penalized unphysical (non-self-consistent) solutions and magnetic field vector distributions not fulfilling ∇·B⃗=0. The initial state of the magnetic field vector components has been chosen so that all the amplitudes of the basis functions were randomly sampled from a normal distribution with zero mean and a standard deviation of 20 G. The initial guess of the atomic number density has been a constant function equal to log_10N=1 in the units of [N]=cm^-3.

To evaluate the goodness of the inversion, we quantified the error in the inferred magnetic field with

    e = B⃗_ inv-B⃗/B⃗ ,

where B⃗ is the true magnetic field vector (see Fig. <ref>) and B⃗_ inv is magnetic field vector resulting for the inversion. Figure <ref> shows the histogram of this quantity evaluated at 10^5 randomly located points in the whole 3D domain (blue histogram) and with the same number of random points within the prominence body (orange histogram). As the Stokes parameters of the  k line give us information about the “visible” surface of the prominence, whose size is comparable to that of the model's domain, the two histograms are relatively close.

We find a typical relative error in the inferred magnetic field vector of about 20–30 %. However, it is important to emphasize that using this relative error as a measure of the quality of the inference can be misleading. For instance, a small spatial displacement of a magnetic loop, for example at position (X,Z)=(0,-25)  Mm, can lead to a very significant relative error. In Fig. <ref> we show a cut in the X–Z plane of the original and inferred spatial distribution of the magnetic field vector and of the atomic number density, demonstrating that the inversion does a pretty good job in recovering the overall physical model.

Given the available CPU time, we performed about two dozen inversions with different initializations of the model and different setups of the inversion algorithm. This is not enough to make any quantitative conclusions about the ability of different setups to perform the inversion. But overall, we have found that the model has (i) converged to a solution similar to the one presented in this section, (ii) ended up in a local minimum of the merit function, or (iii) has diverged. This last situation has occurred in cases where the inversion algorithm setup can be considered too aggressive (i.e., too large iteration steps, etc.). Importantly, the solution did never converge to any solution fundamentally different and ambiguous from the one presented here. This is indicative that, at least for this 3D academic prominence model, the global consistency imposes constraints strong enough to remove the ambiguities present in approaches assuming either no coupling between the FOV pixels or unrealistic symmetric conditions. Consequently, the nontrivial spatial coupling through RT seems to impose strong constraints on the solutions that are possible, leading to robust results.

Concerning the linear polarization signals caused by scattering processes, it is important to emphasize that they are sensitive mostly to regions not much deeper than optical depth unity along the LOS, where the anisotropy of the radiation can be substantial so as to produce observationally relevant linear polarization signals. Given that the Mg ii k line can be very optically thick in this prominence model, the inversion cannot be expected to perfectly recover the magnetic field in these optically inaccessible regions for which the Stokes profiles do not give us enough information. Nevertheless, the 3D inversion provides a sufficiently good estimate of the global structure and strength of the magnetic field.





§ DISCUSSION AND CONCLUSIONS



We solved the non-LTE problem of the generation and transfer of polarized radiation in the Mg ii k line in an academic 3D prominence model, where the magnetic field lines follow an optically thick loop-like structure. We chose this relatively simple geometry to facilitate the comparison between different magnetic field inference approaches. We have found that, despite 3D RT effects, for this relatively simple prominence model both the WFA and the constant-property slab approaches can give good estimations of the longitudinal component of the magnetic field from the observed Stokes V profile. To some degree, the constant-property slab approach is able to estimate the magnetic field transversal component from the linear polarization. To recover the full 3D picture, a full Stokes inversion method that includes the effects of RT in 3D is necessary, such as the one we applied in this study.

We have found that the WFA provides a fast estimation of   that is suitable at least for a simple structure such as our prominence model. For profiles with small enough optical thickness along the LOS, methods based on the constant-property slab model seem to be the best approach in terms of recovered information and computational time required, if the prominence has a simple enough geometry. When the optical thickness exceeds unity, we can take advantage of the RT coupling by applying our full 3D RT approach and inferring the global structure of the prominence.

Due to the significant optical thickness in our 3D model, also found in actual prominences <cit.>, the Stokes Q and U profiles are mostly sensitive to the outermost layers of the prominence (in the direction toward the observer). This is more critical in prominences because, unlike on-disk observations, the wings are not observed. In contrast with the Hei lines more popularly used in prominence diagnostics, the  k line is a strong resonance line with a spectral structure. This entails that the chromospheric radiation illuminating the prominence plasma is sensitive to velocities, introducing frequency shifts between the absorption profile and the incoming illumination spectrum, and to variations in the chromospheric surface that produce significant changes in the intensity of this line.

The significant optical thickness can be even more problematic if the real physical scenario is that of many small-scale threads with their own prominence-corona transition regions <cit.>. In this case, the inversion can become extremely challenging. Therefore, spectral lines with smaller optical thickness in prominences, which provide more spatially averaged information on the magnetic field, may provide very valuable additional information.

The scientific importance of developing a space telescope that enables routine spectropolarimetric observations in the near-UV region of the  h and k lines cannot be overestimated, because the polarization signals that the combined action of scattering processes and the Hanle and Zeeman effects introduce in this spectral region encode a wealth of information on the magnetism and geometry of chromospheric and prominence plasmas. Equally important is the development of advanced plasma diagnostic techniques capable of providing reliable information on the magnetic field vector.      





J.Š. acknowledges the financial support from project  of the Astronomical Institute of the Czech Academy of Sciences.
T.P.A.'s participation in the publication is part of Project RYC2021-034006-I, funded by MICIN/AEI/10.13039/501100011033, and the European Union
“NextGenerationEU”RTRP. T.P.A. and J.T.B. acknowledge support from the Agencia Estatal de Investigación del Ministerio de Ciencia, Innovación y Universidades (MCIU/AEI) under grant “Polarimetric Inference of Magnetic Fields” and the European
Regional Development Fund (ERDF) with reference
PID2022-136563NB-I00/10.13039/501100011033.


aa




 Paper Title: Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem.



Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem]Effects of Variable Mass, Disk-Like Structure, and Radiation Pressure on the Dynamics of Circular Restricted Three-Body Problem













1]L. B. Putra

[2]I. Nurul Hudaibnu.nurul.huda@brin.go.id

1]H. S. Ramadhan

1,2]M. B. Saputra

3]T. Hidayat

[1]Departemen Fisika, FMIPA, Universitas Indonesia,  Depok, 16424,  Indonesia

*[2]Research Center for Computing, National Research and Innovation Agency,  Bogor,  Indonesia

[3]Department of Astronomy and Bosscha Observatory, Institut Teknologi Bandung, Bandung, Indonesia





In this paper, we intend to investigate the dynamics of the Circular Restricted Three-Body Problem. Here we assumed the primaries as the source of radiation and have variable mass. The gravitational perturbation from disk-like structure are also considered in this study. There exist five equilibrium points in this system. By considering the combined effect from disk-like structure and the mass transfer, we found that the classical collinear equilibrium points depart from x-axis. Meanwhile, this combined effect also breaks the symmetry of tringular equlibrium point positions. We noted that the quasi-equilibrium points are unstable whereas the triangular equilibrium points are stable if the mass ratio μ smaller than critical mass μ_c. It shows that the stability of triangular equilibrium points depends on time.






















[
    [
    
=====






§ INTRODUCTION



Circular Restricted Three Body Problem (CRTBP) consists of the movement of the third body with respect to the two primaries. The primaries move in a circular orbit and the third body is influenced by but not influences the primaries. In the classical case, the primaries and the third body are assumed as the point mass <cit.>. There exist five equilibrium points which are divided into two categories named collinear equilibrium points L_1, L_2, and L_3 and triangular equilibrium points L_4 and L_5. The collinear equilibrium points are always unstable while the triangular equilibrium points are stable if the mass ratio μ < μ_c = 0.038520896504551.

The complexity of nature has made the CRTBP not suitable for some cases. Therefore, some authors have tried to develop the CRTBP by incorporating various effects. For instance, <cit.> and <cit.> have considered the effect of photogravitation in the CRTBP for mimicking the stellar objects . More recently, the influence of disk-like structure has been incorporated in the CRTBP <cit.>. There are several studies that combined various additional effects in CRTBP. For instance, <cit.> has studied CRTBP by assuming both primaries are radiating and oblate bodies, together with the effect of disk-like structure. <cit.> combined the effect of photogravitational and disk-like structure, with addition of oblateness and finite-straight segment for the primaries, to study the stability of equlibrium points in CRTBP.

Several close binary star systems have been discovered <cit.>. It is already studied that some of them have planets that have mass much less compared to the binary <cit.>. Meanwhile, previous studies also suggest that there is also possibility that an asteroid belt like structure also exist in the binary system <cit.>. In certain cases, the transfer of mass between binary stars is unavoidable <cit.>. However, accurately predicting how mass moves between close-orbiting stars is still a major challenge. In the case of CRTBP, the transfer mass between star in binary sytem can be modelled by the variability of mass of each primary. The study of variable mass in the restricted three-body problem was done by <cit.> in 1930s. More recently, <cit.> studied the CRTBP system which the primaries have variable masses but the sum of their masses remains constant. <cit.> consider the variation of mass of primaries in accordance with the combined Meshcherskii law. Several studies also consider the variable mass of the third body <cit.>.



In this study, we investigate the possible movement of the infinitesimal mass in the close binary star system. We used a framework of CRTBP where the binaries are primaries. We assumed that the stars emit radiation and there is a mass transfer between primaries. We also considered a disk-like structure surrounding this three-body system, mimicking the Kuiper or asteroid belt structure. 

This paper is outlined as follows. In the Section <ref>, we give a detail about the equation of motion of the system. The detail about the equilibrium points is given in Section <ref>. Section <ref> describes the stability of the system. Finally, the conclusion is given in Section <ref>. Here we used Mathematica software to conduct a numerical calculation or algebraic manipulation.




§ EQUATION OF MOTION



Let the mass of the first and second primaries are m_1 and m_2 respectively. The mass ratio between primaries is represented by μ = m_2/(m_1 + m_2), where 0 < μ < 1. Hence we represent the mass of the primaries by 1-μ and μ. For simplifying the problem, we consider the system in a two dimensional rotational coordinate Oxy and the primaries always lays on x-axis. The origin of the coordinate system is located in the positon of m_1. We take the distance between primaries as the unit of length and the unit of time is chosen in such a way so that the gravitational constant is unity. Let (x, y) be the position of the third body. We follow <cit.> for describing the CRTBP where the primaries have variable mass. However, we also consider the effect of the radiation pressure on the primaries and the disk-like structure surrounding the three-body system. 

The radiation force (F_p) has an opposite direction with respect to the gravitational force (F_g). In order to consider the radiation pressure in the CRTBP, we defined the mass reduction factor q = 1 - (F_p/F_g), where 0 < 1 - q ≪ 1. Meanwhile, the disk-like structure effect can be modelled by following <cit.>. The potential of disk-like structure for planar version is given as

    V(x,y) = M_b/√(r^2 + T^2)

where M_b is the total mass of the disk-like structure and r^2 = x^2 + y^2 is the radial distance of the infinitesimal mass. The mass parameter of the disk-like structure is M_b and is assumed to be small compared to the total mass of the primaries (M_b<<1). Here T = a + b means the dust belt's density profile, where a and b are the flatness and core parameters of the disc respectively. 



Figure <ref> shows a graphic representation of the system. We follow <cit.> to model the mass transfer between the primaries. It has a center of inertia (A) as its center. This center is a static point. There is also center of mass (D) that moves around point A. It has to be noted that in our case the distance between A and D is so small. For simplicity, we shall consider conservative linear mass transfer law between the primaries,

    μ(t)=m_2(t)/m_1(t)+m_2(t)=kt.

Here t means time and 0<t<1/k. It has to be noted that the sum of mass m_1(t) and m_2(t) is constant. We assume that the rate of transfer k is much slower compared to the orbital period of the primaries, i.e. k<<1/n where n is the mean motion of the two body system,

    n^2=1+2M_br_c/(r_c^2+T^2)^3/2.

The reference radius of the disk-like structure is given by r_c^2=1-μ+μ^2 as in <cit.>. Assuming that the transfer mass between primaries is very slow and that the dominant order in the expansion is the first order, we have

    μ (t)≈μ (t_0)+μ(t_0)(t-t_0),

where we could define μ(t_0)≡ k and μ(t_0)≡μ_0, so that

    μ (t)=μ_0+k(t-t_0).

Note that for μ_0=kt_0, <ref> reverts back to <ref>. In the case of μ_0=kt_0, the domain for t is (t_0-μ_0/k)<t<(t_0+1-μ_0/k).

The equation of motion of the system is given as follows

    x   -2ny=W_x, 
    y   +2nx=W_y,

where W_x and W_y mean the partial derivative of W with respect to x and y respectively. The pseudo potential is given by

    W=1/2n^2(x^2+y^2)-μ n^2 x+(1-μ)q_1/r_1+μ q_2/r_2+M_b/(ℛ^2+T^2)^1/2.

The first derivatives of the pseudo potential with respect to the third body position is given by

    W_x   =n^2x-μ n^2-(1-μ)q_1x/r_1^3-μ q_2(x-1)/r_2^3-M_b(x-μ)/(ℛ^2+T^2)^3/2, 
    
        W_y   =n^2y-(1-μ)q_1y/r_1^3-μ q_2y/r_2^3-M_b(y-2k/n)/(ℛ^2+T^2)^3/2.

Here q_1 and q_2 are the radiation pressure factor for m_1 and m_2. If we do not consider the radiation and disk-like structure effects, <ref> will be similar to the equation of motion in <cit.>. We consider the same coordinate system in <cit.> where the origin of the rotational coordinate is the position of m_1, hence

    r_1^2   =x^2+y^2, 
    
        r_2^2   =(x-1)^2+y^2.

Since the center of the disk-like structure is the point C, i.e. the point around which the primaries barycenter orbits, the radial distance of the infinitesimal mass becomes

    ℛ^2=(x-μ)^2+(y-2k/n)^2.





§ EQUILIBRIUM POINTS





 §.§ Quasi-Collinear Points


The collinear points L_1, L_2, and L_3 are the solution located in the interval 1 < x < ∞, 0 < x < 1, and -∞ < x < 0, respectively. The position of collinear points are found by considering x = y = ẋ = ẏ = y = 0 into <ref>. We have

    (x-μ)(n^2-M_b/((x-μ)^2+4k^2/n^2+T^2)^3/2)-(1-μ)q_1x/x^3-μ q_2(x-1)/(x-1)^3=0, 
    2M_bk/n((x-μ)^2+4k^2/n^2+T^2)^3/2=0.

It is clear from <ref> that the collinear equilibrium points only exist if M_b = 0 or k = 0. Nevertheless, we searched a possible equilibrium points near x-axis when M_b ≠ 0 and k ≠ 0 by calculating the equilibrium points numerically. The numerical values are obtained using a numerical algorithm in Mathematica. Here we consider μ_0=0.3 and t_0=0. The resulting time dependence graph can be seen in <ref>. As expected, we found that the equilibrium points are quasi-collinear. They shifted slightly towards the +y axis, due to the existence of mass variation and the disk like structure, which has the point C (that is not on the barycenter, nor is it anywhere in the x axis) as its center.

According to <ref>, the position of L_1, L_2, L_3 have affected by M_b. Higher M_b makes L_1, L_2, and L_3 position further away from x-axis. Meanwhile, higher k means that the position of L_1, L_2, and L_3 are shifted higher with respect to the original position as time increase (see <ref>). In contrast, as shown in <ref> the value of q_1 and q_2 has contributed lower compared to k in determining the shift of L_1, L_2, and L_3 as time increase.





 §.§ Triangular Points


In order to find the position of equilibrium points, we have to solve <ref> by considering x = y = ẋ = ẏ = 0. The position of  triangular equilibrium points can be calculated by considering y ≠ 0. Hence we get

    (x-μ)[n^2-M_b/(ℛ^2+T^2)^3/2]-(1-μ)q_1x/r_1^3-μ q_2(x-1)/r_2^3 = 0, 
    
        y [n^2-M_b/(ℛ^2+T^2)^3/2]   -(1-μ)q_1y/r_1^3-μ q_2y/r_2^3+2kM_b/n(ℛ^2+T^2)^3/2 = 0.

We assume that the position of triangular points in the modified CRTBP is the perturbed version of classical case (r_1 = 1; r_2 = 1), i.e.

    r_1   =1+ϵ_1, 
    
        r_2   =1+ϵ_2,

where ϵ_1,2≪ 1. By substituting <ref> to <ref>, neglecting higher order of ϵ_1,2, and solving for x and y, we have position of triangular equilibrium points as follows

    x    = 1/2+ϵ_1-ϵ_2, 
    
            y    = √(3/4+ϵ_1+ϵ_2),

Following <cit.>, we consider <ref> and <ref> into <ref>. Hence, with additional expansion of k to the first order, we obtain

    ϵ_1   =-1-q_1/3+M_b(1-2r_c)/3(r_c^2+T^2)^3/2, 
    ϵ_2   =-1-q_2/3+M_b(1-2r_c)/3(r_c^2+T^2)^3/2.

Substituting <ref> to <ref> yields the triangular points L_4 and L_5

    x=1/2-q_2-q_1/3

and

    y=±√(3)/2(1-2/9(2-q_1-q_2)+4/9M_b(1-2r_c)/(r_c^2+T^2)^3/2).

It can be seen that the triangular points for this system are identical (to the first order) with the constant primary mass counterpart, albeit with a (slow) time dependence.

<Ref> show the effect of M_b in the position of triangular points. We observe that the position of L_4 and L_5 is not symmetric due to the combination of disk-like structure and mass transfer. This asymmetric is larger when M_b is higher. We noted also that the position of L_4 and L5 is closer to the primaries  with increasing M_b. In <Ref>, it is clear that the decreasing value of k makes L_4 closer to the primaries, in contrast with L_5. From <Ref> we observe that radiation pressure has the impact on the location of triangular equilibrium points. The location of L_4 and L_5 are closer to the source of radiation pressure when the radiation pressure getting stronger, either for m_1 or m_2. 


















§ LINEAR STABILITY






The stability of equilibrium points are studied by introducing the perturbation in the equilibrium point (x_0,y_0), hence we define

    x   =x_0+α, 
    
        y   =y_0+β,

where α and β is small displacements with respect to the equilibrium points. By substituting <ref> to <ref> and expand the equation, we get

    α   -2nβ=W_xx^0α+W_xy^0β, 
    β   +2nα=W_yx^0α+W_yy^0β,

where

    W_xx   =n^2 + (1-μ)q_1/r_1^3(-1+3x^2/r_1^2) + μ q_2/r_2^3(-1+3(x-1)^2/r_2^2) + M_b/(R^2+T^2)^3/2(-1+3(x-μ)^2/(R^2+T^2)), 
    
        W_yy   =n^2 + (1-μ)q_1/r_1^3(-1+3y^2/r_1^2) + μ q_2/r_2^3(-1+3y^2/r_2^2) + M_b/(R^2+T^2)^3/2(-1+3(y-2k/n)^2/(R^2+T^2)), 
    
        W_xy   =W_yx=3(1-μ)q_1xy/r_1^5 + 3μ q_2(x-1)y/r_2^5 + 3M_b(x-μ)(y-2k/n)/(R^2+T^2)^5/2.

The characteristic equation is given by

    λ^4+(4n^2-W_xx^0-W_yy^0)λ^2+W_xx^0W_yy^0-(W_xy^0)^2=0.

The solution of this equation is given as follows 

    λ_i = ±√((-b ±√(b^2 - 4c))/2);    i=1,2,3,4.

where b = 4n^2-W_xx^0-W_yy^0 and c = W_xx^0W_yy^0-(W_xy^0)^2. The stability of equilibrium points can be achieved when all λ_i are purely imaginary, otherwise we have an unstable equilibrium point. 

Table <ref> shows the characteristic roots (λ_i) of the collinear equilibrium points by considering several configuration of perturbing parameters. All λ_1 have the form real which signify instability. In the range of mass parameter 0 < μ < 1 we found that b^2-4c > 0 for L_1, L_2, and L_3. Consequently we have at least one positive real for the solution of characteristic equation. Hence, the collinear equilibrium points are always unstable.

In the case of triangular equilibrium points, the stability is achieved when 0<μ<μ_c, where μ_c means the critical mass. Following <cit.>, the critical mass is given as follows

    μ_c   =1/2(1-√(23/27))-22-q_1-q_2/27√(69)
       +(3/2+(76-8r_c)(r_c^2+T^2)/27√(69)-83+12r_c^2/6√(69))M_b/(r_c^2+T^2)^5/2.

Table <ref> shows the examples of characteristic roots in the stability of triangular equilibrium points. All case have stable equilibrium points during t=0 and unstable in t=0.2 and t=0.3. We noted that λ_1 and λ_3 in L_4 are similar to L_5 for the case 1 and case 2. However, due to the combination of disk-like structure and mass transfer effects, this similarity is not sound for case 3 and case 4.

Since, in our case, μ depends on time, besides μ_c there exists also a so called critical time (t_c) as follows

    t_c=t_0+(μ_c-μ_0)/k,

where t>t_c means unstable. <ref> shows the effect of perturbing parameters M_b, q_1, and T on t_c for the case of k=0.1, μ_0=0.3, and t_0=3. We noted that t_c becomes shorter when M_b and 1-q_1 increase. In contrast, t_c is longer if T increases.













§ CONCLUSIONS


We have studied the influence of mass transfer, disk-like structure, and radiation pressure, on the position and stability of CRTBP equilibrium points. In this system, we found there are five equilibrium points, where two of them are triangular equilibrium points, and the others are quasi-collinear equlibrium points. Unlike the classical collinear equilibrium points, we noted that L1, L2, and L3 are slightly departed from x-axis since there exist the effects from disk-like structure and mass transfer. Moreover, the symmetry of L_4 and L_5 is broken when we consider the mass transfer and disk-like structure together. Furthermore, we found that the quasi-collinear equilibrium points remain unstable. The stability of triangular points depend on the initial mass parameter μ_0 as well as the time. We found there exist critical time for achieving the stability of triangular points.



§ ACKNOWLEDGMENT

We thank the reviewers for their insightful comments and suggestions on the manuscript.This research has been supported by RIIM LPDP-BRIN 2023-2025 and UI research grant No. PKS-026/UN2.F3.D/PPM.00.02/2023. 








 Paper Title: Searching for New Cataclysmic Variables in the Chandra Source Catalog.

 

   

   
   Kazan Federal University, Kremlevskaya Str.18, 420008, Kazan, Russia

     IlhIGaliullin@kpfu.ru
         Department of Astronomy, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA, 91125, USA
        Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA
        Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
        Anton Pannekoek Institute for Astronomy, University of Amsterdam, 1090 GE Amsterdam, The Netherlands
        IPAC, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA
        Division of Physics, Mathematics, and Astronomy,
            California Institute of Technology, Pasadena, CA 91125, USA 
             
             
    Ilkham Galiullin, Antonio C. Rodriguez et al.
    
   



 
  
  
   
     Cataclysmic variables (CVs) are compact binary systems in which a white dwarf accretes matter from a Roche-lobe-filling companion star. In this study, we searched for new CVs in the Milky Way in the Chandra Source Catalog v2.0, cross-matched with Gaia Data Release 3 (DR3). 
  
      We identified new CV candidates by combining X-ray and optical data in a color-color diagram called the “X-ray Main Sequence".  We used two different cuts in this diagram to compile pure and optically variable samples of CV candidates. We undertook optical spectroscopic follow-up observations with the Keck and Palomar Observatories to confirm the nature of these sources.   
  
      We assembled a sample of 25,887 Galactic X-ray sources and found 14 new CV candidates. Seven objects show X-ray and/or optical variability. All sources show X-ray luminosity in the 10^29-10^32 erg s^-1 range, and their X-ray spectra can be approximated by a power-law model with photon indices in the Γ∼ 1-3 range or an optically thin thermal emission model in the kT ∼ 1-70 keV range. We spectroscopically confirmed four CVs, discovering two new polars, one low accretion rate polar and a WZ Sge-like low accretion rate CV. X-ray and optical properties of the other 9 objects suggest that they are also CVs (likely magnetic or dwarf novae), and one other object could be an eclipsing binary, but revealing their true nature requires further observations.  
  
    These results show that a joint X-ray and optical analysis can be a powerful tool for finding new CVs in large X-ray and optical catalogs. X-ray observations such as those by Chandra are particularly efficient at discovering magnetic and low accretion rate CVs, which could be missed by purely optical surveys. 

   

   Searching for New Cataclysmic Variables in the Chandra Source Catalog
    Ilkham Galiullin1,
          Antonio C. Rodriguez2,
          Kareem El-Badry2,
          Paula Szkody3,
          Abhijeet Anand4,
          Jan van Roestel5,
          Askar Sibgatullin1,
          Vladislav Dodon1,
          Nikita Tyrin1,
          Ilaria Caiazzo2,
          Matthew J. Graham2,
          Russ R. Laher6,
          Shrinivas R. Kulkarni2,
          Thomas A. Prince7,
          Reed Riddle2,
          Zachary P. Vanderbosch2
          Avery Wold 2
          
    Received XXX; accepted XXX
====================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================










§ INTRODUCTION


Cataclysmic variables (CVs) are compact semi-detached binaries in which a white dwarf (WD) accretes matter from a Roche-lobe filling companion star <cit.>. CVs have orbital periods, typically spanning 66 min – 10 hr, except for ultra-compact AM Canum Venaticorum (AM CVn) stars, which possess ≲ 1 hr periods <cit.>. In magnetic CVs,  the magnetic field of the WD (B≳ 1 MG) significantly influences the accretion onto the surface of the WD. Magnetic CVs are split into intermediate polars (IP) with truncated disks, and polars, where the strong field (B ∼ 10-250 MG) prevents disk formation. Studies of CVs augment our understanding of various astrophysical phenomena, including their role as potential progenitors for Type Ia supernovae <cit.>. They also present optimal non-relativistic laboratories to study accretion processes <cit.>. Furthermore, due to their short orbital periods, some CVs and many ultra-compact AM CVn systems are classified as prospective "verification binaries” for future planned space-based gravitational wave observatories such as LISA <cit.>.

From the theoretical standpoint, CVs are predicted to form as a result of Common Envelope Evolution (CEE) <cit.> that first brings the stars closer together. Angular momentum loss (AML) is the key factor in the long-timescale evolution of CVs. As AML occurs, the separation between the stars decreases, leading to a reduction in the orbital period. The evolutionary model considers two mechanisms for AML: Magnetic wind Braking (MB) <cit.> and Gravitational-wave Radiation (GR) <cit.>. Although the model initially aligned well with the observational data of that era, subsequent findings revealed numerous discrepancies between theory and observations <cit.>. Consequently, empirical revisions were proposed to rectify these inconsistencies <cit.>.

CVs exhibit a broad range of observational features, therefore detailed multiwavelength studies are necessary to draw accurate conclusions about their nature and population. X-ray emission analysis is a powerful tool for searching for and studying CVs. The first X-ray observations of CVs took place with HEAO-1 <cit.> and Einstein <cit.>. Early X-ray surveys led to the discovery of a unique, previously unidentified population of magnetic CVs <cit.>. The all-sky X-ray survey conducted by the eROSITA telescope aboard the Spektrum-Roentgen-Gamma (SRG) mission <cit.> is proving to be groundbreaking by cataloging millions of sources <cit.> and has already been successful in discovering a collection of new and rare CVs due to its ability to reach much fainter fluxes <cit.>.


Despite not being an all-sky X-ray survey, the Chandra observatory provides unique sub-arcsecond on-axis resolution which is comparable with optical surveys <cit.>. The Chandra Source Catalog v2.0 (hereafter, CSC2)  provides the uniformly calibrated, science-ready data products compiled from the archival Chandra data <cit.>. Such data collection enables the study of properties of X-ray sources with spectral and timing analyses and classifying new objects in the catalog. In addition to new X-ray surveys and catalogs, all-sky optical photometric surveys provide a wealth of new data. The Zwicky Transient Facility (ZTF) is the deepest Northern sky time-domain photometric survey, conducted at Palomar Observatory on the Samuel Oschin 48-inch telescope. The 47 deg^2 field-of-view camera images the sky roughly every two days down to ∼20.5 mag in g, r, and i filters <cit.>. 

In this paper, we aim to search for and study new CVs in the CSC2, cross-matched with the  third  Gaia Data Release (hereafter, Gaia DR3). In Section <ref>, we describe the construction steps of a sample of Galactic X-ray sources based on the CSC2–Gaia DR3 catalog. In Section <ref>, we describe the selection of new CV candidates in the CSC2–Gaia DR3 catalog using two different cuts in the X-ray and optical color-color diagram, known as the “X-ray Main Sequence". We compile pure and optically variable samples of CV candidates. In Section <ref>, we present the follow-up observations conducted with the 5m Hale Telescope at Palomar Observatory and the 10m Keck I Telescope on Mauna Kea. We also present detailed X-ray timing and spectroscopic analyses. In Section <ref>, we present the sample of 14 CV candidates found in the CSC2–Gaia DR3 catalog. We separately discuss the nature of four spectroscopically confirmed CVs and four variable CV candidates. Our results are summarized in Section <ref>.





§ SAMPLE OF GALACTIC X-RAY SOURCES



We began by constructing a sample of Galactic X-ray sources from the publicly available CSC2 <cit.>. We cross-matched CSC2 with the Gaia DR3  catalog <cit.>, keeping only objects with significant parallaxes and proper motions. We identified the known sources in our sample by cross-matching with the Simbad database and other publicly available CV catalogs. All sample creation steps are presented in detail below.






 §.§ Chandra Source Catalog

As a first step, we used the master source table from the CSC2 with a total of about ∼ 317,000 unique source detections. We set the following criteria[For detailed flag descriptions see <cit.> and CSC2 web-page: <https://cxc.cfa.harvard.edu/csc/columns/flags.html>] to select the point-like X-ray sources with measured parameters:


    
  * extent_flag=False; only point-like sources.
    
  * conf_flag=False; source regions do NOT overlap.
    
  * pileup_flag = False; ACIS pile-up fraction does NOT exceed ∼ 10% in all observations.
    
  * sat_src_flag=False; the source is NOT saturated in all observations.
    
  * streak_src_flag=False; the source is NOT located on an ACIS readout streak in all observations.
    
  * dither_warning_flag = False; highest statistically significant peak in the power spectrum of the source region count rate does NOT occur at the dither frequency of the observation.
    
  * man_add_flag=False and man_inc_flag=False; the source was NOT manually added nor included in the catalog via human review.
    
  * man_match_flag=False; source detections were NOT manually matched between overlapping stacked observations via human review.
    
  * man_pos_flag=False; best-fit source position was NOT manually modified via human review.
    
  * man_reg_flag=False; source region parameters were NOT manually modified via human review.
    


Using these criteria, we selected 271,438 unique objects from CSC2.




 §.§ Cross-match with Gaia


For the next step, we cross-matched 271,438 objects with the Gaia DR3 catalog <cit.> to select only Galactic sources. As a search radius for each object, we used the individual X-ray positional error (with a search radius equal to err_ellipse_r0, the semi-major axis of the 95% confidence ellipse). We only kept sources meeting the following criteria:

    
  * RUWE < 1.4 <cit.>; good fit to the astrometric observations.
    
  * Only ONE Gaia object must be within the search radius.
    
  * Parallax and proper motion are measured with a signal-to-noise ratio S/N ≥ 3.


These selection steps reduced the number of sources to 26,079. To remove any potential  high proper motion stars, we applied an additional filtering criterion:


    
  * pm×Δ t ≤ err_ellipse_r0;  The angular motion of the Gaia object (caused by a proper motion (pm)) during the time interval Δ t does not exceed the search radius (err_ellipse_r0).   The time interval Δ t is an epoch difference between Gaia DR3 (Epoch 2016) and CSC2 (Epoch 2000) catalogs.


This filter finally resulted in 25,887 objects in the CSC2–Gaia DR3 catalog.




 §.§ Identification of Known Sources



To identify known sources in the CSC2–Gaia DR3 catalog, we cross-matched our objects with the Simbad database <cit.> using Gaia coordinates and a 3 search radius. We grouped our objects based on the Simbad hierarchical structure[<http://simbad.cds.unistra.fr/guide/otypes.htx>], making seven general types: STARS (single and binaries), ACCRETING BINARIES (X-ray binaries, CVs, Nova and Symbiotic Stars), GALAXY, ISM, PROPERTIES, BLENDS (not well-defined objects) and UNKNOWN (object is not presented in Simbad database). In Appendix <ref>, we present a detailed description of our general object types compiled from the Simbad hierarchical structure. We found Simbad classifications for 12,181[While creating the catalog of known systems, we had multiple matches (more than one) within the search radius for 810 of the 12,181 objects, giving about 7% of possible spurious associations in the catalog. However, for most objects with multiple matches a Simbad database provided a similar main object type (as defined in Table <ref>). We adopted the Simbad object type from the nearest source for those objects.]  of the 25,887 objects. 32 of these 12,181 were known CVs or CV candidates. 

To check that we did not miss any known CVs, we independently cross-matched our 25,887 objects with the following catalogs using a 2 search radius: i) Ritter and Kolb catalog <cit.> (Final version; December 31, 2015); ii) Cataclysmic Variables Catalog (2006 edition; <cit.>); iii) The Open Cataclysmic Variable catalog <cit.> (only confirmed CVs are kept in the list). We combined the information from the Simbad database and these three CV catalogs to make the final list of known CVs. Out of 25,887 sources, we finally compiled a sample of 40 known CVs. This sample of 40 consists of the CVs from Simbad plus those from the three catalogs mentioned above.

We note that 52 objects (out of 25,887) were classified as GALAXY. These objects are primarily quasars and galaxies hosting active galactic nuclei that show significant parallax and proper motion. The high proper motions and parallaxes of these extragalactic objects could be due to variability-induced motion <cit.>. Detailed investigation of these objects is beyond the scope of this paper. 





§ CV SELECTION WITH THE X-RAY MAIN SEQUENCE







 
We combined X-ray and optical data to search for new CV candidates in the CSC2–Gaia DR3 catalog. We used the “X-ray Main Sequence", a phase space of  X-ray-to-optical flux ratio (F_X/F_opt) and a Gaia optical BP–RP color <cit.>, to select CV candidates and create a pure sample of these objects. The empirical cut in the “X-ray Main Sequence" is given by the following functional form,

    log(F_X/F_opt)_cut= A + B × (BP-RP),

where A= -3.5 and B=1 are linear parameters from <cit.>. We computed the F_X/F_opt ratio based on X-ray and Gaia optical fluxes. The X-ray fluxes were calculated in the 0.5–7 keV energy band from CSC2. The aperture-corrected (90% enclosed counts fraction) energy flux is calculated using the power-law model with a photon-index of Γ=2 and the individual Galactic hydrogen column density N_H(Gal)  in the direction of the source. Gaia G band magnitudes were converted to optical fluxes using the AB magnitude system and a central wavelength of 6,000 Å. Out of 25,887 sources in the CSC2–Gaia DR3 catalog, we selected only sources with measured X-ray fluxes and optical magnitudes:


  * flux_powlaw_aper90_b/error ≥ 2; Source X-ray flux in the 0.5–7 keV energy band is measured with signal-to-noise ratio more than S/N ≥ 2.

  * Gaia G band, BP and RP magnitudes are measured having signal-to-noise ratio more than S/N ≥ 2.

These selection steps finally resulted in 19,665 sources (with 36 known CVs).



 §.§ First Selection: A pure sample of CV candidates 



To compile a pure sample of CV candidates and avoid contamination from stellar objects, we searched for optimal values for linear parameters A and B in Eq. <ref>. We created two samples of known sources classified based on the Simbad hierarchical structure: one consisting of CVs (36 sources) and the other of stellar objects (STARS, 4781 sources). For the STARS sample, we included only objects with sub-types classified as "YSO", "Binaries", "Massive Stars", "Main-sequence Stars", or "Evolved Stars" (see Table <ref>). The left panel of Figure <ref> shows an F_X/F_opt and Gaia BP–RP color diagram for known CVs (red squares) and STARS (blue dots). It's evident that known CVs are primarily located in the upper left corner of the diagram and could be visually separated from STARS. We computed the purity and completeness using a sample of these known CVs and STARS (see Appendix <ref> for more details). Based on our completeness/purity analysis we selected A = -1.5, and B=0.7 that gives the highest purity (≈100 %) and ≈52% completeness.
  
We only searched for new CV candidates among objects without precise classification from the Simbad database. Out of 19,665 objects, we selected 13,956 sources classified as one of the following object types: i) UNKNOWN (9,868 sources); ii) PROPERTIES (355 sources); iii) STARS, sub-type "Star" (3,720 sources); iv) STARS, sub-type "WD" (10 sources); v) STARS, sub-type "BD" (3 sources). The right panel of Figure <ref> shows the F_X/F_opt and Gaia BP–RP color diagram for these objects. We found 11 new CV candidates (out of 13,956 objects) with our selection based on our most optimal values for  A = -1.5 and B=0.7 (see objects above the solid red line in the right panel of Figure <ref>).

We note that we independently found three objects (recently confirmed as CVs) as CV candidates using the F_X/F_opt and Gaia (BP–RP) color diagram. During the writing of this manuscript, we learned that these systems had been previously discovered. However, these objects have only been mentioned in one or two papers, so we included those objects in our analysis and discussed them individually in Appendix <ref>. 




 §.§ Second Selection: An optically variable sample of CV candidates with ZTF photometry



Many CVs appear as periodic sources in optical light curves. Furthermore, accretion can lead to outbursts, which can also be observed in their optical light curves, particularly in non-magnetic CVs. Therefore, even after compiling a sample of 11 CV candidates based purely on an F_X/F_opt and Gaia BP–RP color diagram, we also searched for possible variability in a larger sample of sources. We used the empirical cut on the F_X/F_opt and Gaia BP–RP color diagram from <cit.> (A = -3.5 and B=1) and searched for possible variability in 375 sources (see objects above the dashed black line in the right panel of Figure <ref>).

We searched for periodic and outbursting-type variability in sources based on ZTF optical photometry for both  g and r light curves, considering only data with a signal-to-noise ratio greater than 5, SNR_ZTF > 5. We retrieved ZTF light curves for our objects and used the barycentric corrected modified Julian date (MJD) in our analysis. Light curves have a photometric precision of ∼0.015 mag at the bright limit of 14 mag, decreasing to a precision of ∼0.1 mag at the faint limit of 20.5 mag. We searched for periodic variability using the Lomb-Scargle periodogram, as implemented in the  tool to search for periods ranging from 5 min to 10 days. Once the period search was performed, we excluded all sources with orbital periods that are multiples or fractions of the sidereal day. To first order, the periods are reported up to a precision of 30 sec, which is determined by the exposure time. To calculate the significance of a period, we first computed the median absolute deviation (MAD) of the periodogram. Then, we took the periodogram value at the highest peak and divided it by the MAD. This ratio served as our significance metric. We plotted the distribution of significance metrics and kept only those sources in the upper 80^th percentile of the distribution. To search for outbursting sources, we looked for ZTF light curves with more than 5 outburst points Δ mag≥ 1, with each point more than 3 standard deviations from the median.

In total, we had 178 (out of 375) sources with ZTF photometry. Using our selection criterion described above, we found five periodic sources (see Figures <ref>, <ref>, <ref> and <ref>). With ZTF photometry, only one outbursting source (2CXO J024131.0+593630) was identified, which we discuss in Appendix <ref>. Based on the analysis of optical ZTF light curves, we extended our final sample of CV candidates to include a total of 14 candidates[Two periodic sources were already presented in the sample of 11 new CV candidates (selected by F_X/F_opt and Gaia BP–RP color). Therefore, we added only three variable sources (out of five) in our final sample to include 14 CV candidates in total.].  

Out of 14 sources, 2 objects were marked as variables with a Gaia flag (phot_variable_flag=VARIABLE). These objects show outbursting-type variability. Figure <ref> shows optical light curves based on Gaia photometry (SNR_ Gaia > 3). Table <ref> (column 9) summarizes the variability properties of the sources in the final sample. We note that the number of spurious matches in our final sample is effectively zero due to the ∼1–2" localization of X-ray sources made possible by Chandra. Furthermore, our candidates that show optical variability are even less likely to be false matches, since optical variable sources, particularly accreting compact objects, are typically associated with X-ray emission (e.g., see Figure 2 of <cit.>). 














§ DATA AND ANALYSIS



































 §.§ Optical Spectra



We performed optical spectroscopic follow-up observations for four sources in our sample, focusing on those located in the Northern Hemisphere that were accessible to our telescopes and exhibited strong optical variability and/or a high F_X/F_opt ratio. We used both the Double Spectrograph <cit.> on the 5m Hale Telescope at Palomar Observatory and the Low-Resolution Imaging Spectrometer <cit.> at the 10m Keck I Telescope on Mauna Kea. During all DBSP observations, we used the 600/4000 grism on the blue side and the 316/7500 grating on the red side. A 1.5 slit was used, and the seeing on all occasions was 1.5 – 2.0. All P200/DBSP data were reduced with [<https://dbsp-drp.readthedocs.io/en/stable/index.html>], a Python-based pipeline built upon the existing  package <cit.> and optimized for reducing spectral data. During all LRIS observations, we used the 600/4000 grism on the blue side and the 600/7500 grating on the red side. We used a 1.0 slit, and the seeing on all occasions was around 0.7. LRIS data were reduced with , an IDL-based pipeline optimized for LRIS long slit spectroscopy <cit.>. All data were flat fielded and sky-subtracted using standard techniques, wavelength-calibrated with internal arc lamps, and flux-calibrated using a standard star. Table <ref> shows all data acquired for the four CV candidates. Single spectra were taken at a time for each object (no coadds), with typical exposure times of 600 seconds for Keck I/LRIS and 900 seconds for P200/DBSP.

Figure <ref> shows Chandra X-ray images and optical images from Pan-STARRS <cit.> of the same sky region for four spectroscopically confirmed CVs in our sample. Figures <ref> - <ref> show the optical spectra of these sources. We identified prominent emission lines and computed their equivalent widths (EWs) by fitting Gaussian profiles to them. Table <ref> shows the EWs of selected lines and the He II(4686) / Hβ ratio[2CXO J044048.3+292434 was observed twice, and both observations show equivalent width measurements which are in agreement to within 1σ. Therefore, in Table <ref>, we only report the values from the first exposure.]. 

We conducted an extensive cross-match analysis between our 14 objects and publicly available optical data from the Sloan Digital Sky Survey (SDSS) and the Dark Energy Spectroscopic Instrument (DESI) collaboration <cit.>. With 2 search radius, we found no matches between SDSS/APOGEE catalogs (APOGEE DR17 StarHorse VAC, <cit.>; APOGEE DR17 astroNN VAC, <cit.>; APOGEE DR17 DistMass VAC, <cit.>; APOGEE Net VAC <cit.>; MaStar Stellar Parameters VAC, <cit.>) and our CV sample. We cross-matched our objects with the early data release (EDR) of DESI collaboration <cit.>, which contains spectra of more than approximately 1 million objects <cit.>. We found no identification of our objects with this catalog.









 §.§ X-ray spectra and luminosities



 
        




For each of the 14 sources in our sample, we queried the archival Chandra data using the observation identifier (obsid) from source observation summary results in the CSCview application[<https://cxc.cfa.harvard.edu/csc2.0/gui/intro.html>]. For some objects, we additionally searched for new Chandra observations within a 10 search radius using the Chaser[<https://cda.harvard.edu/chaser/>] webtool and included them in our analysis. The Chandra archival observations were processed by following standard Chandra Interactive Analysis of Observations (CIAO) <cit.>  threads[<http://cxc.harvard.edu/ciao/index.html>] (CIAO version 4.16.0 and CALDB version 4.11.0). The chandra_repro tool was used to perform the initial calibration, reprocess the data, and create a level 2 event file. To extract the source and background spectra, we used the specextract tool in CIAO. A point-source aperture correction was applied for each source spectra by setting the parameter correctpsf=yes. We computed the radii of the Chandra point spread function (PSF) from the PSF map, which was constructed with mkpsfmap tool for 90% of encircled counts fraction (ECF) and the 2.3 keV energy (parameters ecf=0.9 and energy=2.3 respectively). Source and background regions were centered at source positions from CSC2. The circle with the radius equal to the Chandra PSF radius was used for the source regions, and annuli with the inner and outer radii equal to two and four times the Chandra PSF radius were used for background regions. 

We performed the spectral fit using  the  v12.13 package <cit.> in the 0.5–7 keV energy band. We binned the X-ray spectra to have at least three counts per bin and used C-statistics <cit.> to get best-fit results. We used the error command in XSPEC to compute 1σ confidence intervals for the parameters. For some sources in our sample, we estimated only the 1σ lower limit (or upper limit) of the parameters. Each spectrum was approximated using two spectral models: the power-law model ( in ) and the optically thin thermal plasma emission model ( in ). To take into account interstellar absorption, we used the Tuebingen-Boulder ISM absorption model ( in ) and the solar elemental abundances from <cit.>.

All sources in our sample are located at different distances, leading to various Galactic hydrogen column densities. We computed the distances of the sources from Gaia parallaxes, and their errors were calculated using the standard error propagation. Using the source distances, we adopted the E(B-V) values from the Bayestar dust map <cit.>. If data from the Bayestar dust map were not available, we utilized the Galactic plane reddening map <cit.>. We computed the hydrogen column density by assuming Cardelli's extinction law (R_V≈ 3.1) and the relation N_H (map)≈ R_V× E(B-V) × 2.21×10^21 cm^-2 <cit.>. For sources having more than 100 counts in their X-ray spectra, we computed hydrogen column density (N_H(fit)) and metal abundance directly from the fit. We found no significant deviation between the best-fit results of N_H(fit) and N_H(map) computed from the Bayestar dust map. For other sources, we approximated their spectra with a fixed N_H(map) and assuming solar metallicity. In the cases when N_H(map) was not available at the specified distance of the source or when the distance fell outside the interval classified as reliable, we fixed the hydrogen column density to the Galactic value <cit.>.


The best-fit results obtained from approximating the X-ray spectra with  and  models are presented in Appendix <ref> (see Table <ref> and Figure <ref>). For sources with more than one Chandra observation, we analyzed the X-ray spectra for each observation separately. However, we found no significant parameter variations between these separate observations subsequently performed a joint fit of the spectra in XSPEC by combining the parameters across multiple observations. We computed absorption corrected fluxes in the 0.5–7 keV energy band for the power-law model using the cflux command and converted them to luminosities. The approximate errors of the unabsorbed X-ray luminosities were computed from the uncertainties in the distances and fluxes using standard error propagation. Table <ref> (columns 10 and 11) shows the distances and unabsorbed X-ray luminosities in the 0.5–7 keV energy band.




 §.§ X-ray variability and timing analysis



We searched our sample of 14 CV candidates for X-ray variable sources using the CSC2 variability indices in the 0.5–7 keV energy band: var_intra_index_b ≥ 8 (variability within a single observation) and/or var_inter_index_b ≥ 8 (variability across multiple observations). We found only one source matching these criteria (2CXO J063805.1–801854). We also checked the significance of the difference between the maximum and minimum fluxes between observations using the ratio R=(F_max-F_min)/(σ_F_max^2 + σ_F_min^2)^1/2 and found only one source having R ≥ 3 (2CXO J123727.5+655211).

Out of the 14 CV candidates in our sample, we performed an X-ray timing analysis for three sources with more than 300 source counts in their spectra (see Table <ref>). If sources had more than one archival Chandra observation, we used the observation with the longest exposure time for the timing analysis. We extracted the background-subtracted X-ray light curves with the dmextract tool in CIAO, using the same source and background regions as those used for the X-ray spectral analysis. Barycentric corrections were applied for the event files by using axbary tool in CIAO. We used the XRONOS sub-package in FTOOLS[<http://heasarc.gsfc.nasa.gov/ftools>] for timing analysis <cit.>. First, we calculated the power spectrum density using a Fast Fourier Transform (powspec command in the XRONOS package) to search for periodic signal. The epoch-folding (EF) technique was then applied to constrain the period of the sources <cit.> using the efsearch command in the XRONOS package. We estimated the best-fit period from the EF method by fitting a Gaussian function to the χ^2 vs period diagram. The period error was estimated using the bootstrap technique similarly as described in <cit.>. To determine the significance of the detected period, we generated 1000 random Poisson, non-periodic light curves with mean source count rates and applied the powspec and efsearch tools on these randomly generated light curves. We computed the maximum value of the power spectrum density and the χ^2 (as defined in the EF method) for each generated light curve. A 3σ upper confidence level was calculated for the distributions of power spectrum densities and χ^2 values. A strong signal was considered present in the observed X-ray light curves if the results from the powspec and efsearch tools exceeded the 3σ confidence level determined from the simulations. We then used the efold tool to construct X-ray light-curves folded with the best-fit periods.

We found strong periodic X-ray signals only in one source out of three (see Table <ref>). Figure <ref> shows the X-ray pulse profile normalized by the mean count rate of that source. The pulse fraction (PF) was computed using maximum and minimum count rates, where PF=(F_max-F_min)/(F_max+F_min).  We discuss the nature of the X-ray periodic source in Section <ref>.




 §.§  Hertzsprung–Russell and X-ray luminosity – orbital period diagrams








To compute extinction-corrected absolute G band magnitudes for CV candidates, we directly adopted the interstellar extinction A_G  (at 6000 Å) from Bayestar or the Galactic plane reddening maps when available. Otherwise, we converted the hydrogen column densities presented in Table  <ref>  for the mekal model (see Section <ref>)  to interstellar extinction A_G  using the <cit.> extinction function. Figure <ref> shows the location of the CV candidates on the Hertzsprung–Russell (HR) diagram along with Gaia objects within 100 pc that have significantly measured parallaxes (> 3). 

We constructed an X-ray luminosity – orbital period diagram for six objects in our sample with periods computed from the optical and X-ray data (see Figure <ref>). To compare our objects with other known CVs, we cross-matched the CV catalog of Ritter and Kolb (Final Edition, frozen on 31 December 2015, <cit.>) with the the second ROSAT all-sky survey (2RXS) catalogue <cit.>, keeping only sources with a single match within 20. We then cross-matched with Gaia DR3 <cit.>, keeping only sources with significant parallaxes and proper motions (5σ), which yielded 259 sources. Next, we used the WebPimms[<https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3pimms/w3pimms.pl>] tool to convert ROSAT source count rates (0.1–2.4 keV) to unabsorbed X-ray luminosities in the 0.5–7 keV energy band, assuming a power-law model with a photon index of 1.7 and hydrogen column density of 3× 10^20 cm^-2. To show different CV types in Figure <ref>, we used the classification scheme of the Ritter and Kolb catalog, which categorizes them as into polar (AM), intermediate polar (DQ), or otherwise as non-magnetic. In Figure <ref>, we show different period minimum values for CVs from <cit.> (≈82 min) and <cit.> (≈76 min) as well as CV orbital period gap values from <cit.> (≈ 2.5-3.2 hr) and <cit.> (≈ 2.2-3.2 hr).







§ RESULTS AND DISCUSSION



Table <ref> shows the parameters of the 14 CV candidates found in the CSC2–Gaia DR3 catalog using  the F_X/F_opt and the Gaia BP–RP color diagram and the period search with ZTF optical photometry (see Section <ref>).  Figure <ref> shows that most of our CV candidates are located between the main sequence and the WD regions, which is consistent with the distribution of known CVs in the Gaia HR diagram <cit.>. The X-ray luminosities of all our objects vary in the ≈ 10^29 - 10^32 erg s^-1 range, which matches with the range of X-ray luminosities of some known CVs (see Figure <ref>). X-ray spectra of our CV candidates can be approximated by a power-law model with photon indices in the Γ≈ 1-3 range (or kT ∼ 1-70 keV for the optically thin thermal emission model). Such X-ray spectra are seen in some magnetic and non-magnetic CVs <cit.>. 

Our variability search indicates that seven sources out of these 14 CV candidates are variable in nature. The X-ray luminosity – orbital period diagram shows that most of our variable sources have X-ray luminosities and periods typically seen for magnetic and non-magnetic CVs (see Figure <ref>). Only two sources (2CXO J182117.2–131405 and  2CXO J185603.2+021259) are located in a distinct region of the phase space compared to the main sample of CVs, and exhibit low X-ray luminosity. These two objects might be a low-accretion rate  CV and an eclipsing binary[Such a binary would be an “active binary", like RS CVn systems, which have orbital periods from as low as 5–6 hours to ∼10 days. In these systems, X-rays originate from the hot corona of one or both stellar components <cit.>.]. We spectroscopically confirm four objects in our sample as new CVs. Their optical spectra show prominent emission lines or cyclotron humps, features typically observed in CVs (see Figures <ref> - <ref>). Below, we provide a detailed discussion of these CVs and other variable objects in our sample.


We note that some objects in our sample were previously suggested to be CV or quiescent low-mass X-ray binary (qLMXB) candidates using a multiwavelength machine-learning method, however, the results have low confidence <cit.>. Objects having Gaia optical light curves were also classified as possible CV candidates using a machine learning variability classifier in Gaia DR3 <cit.>. In Appendix <ref>, we discuss three recently discovered CVs, which we independently selected using the methods described in this paper. These examples demonstrate that the F_X/F_opt and Gaia BP–RP color diagram, as suggested by <cit.> can be a powerful method for searching and independently selecting accreting binary candidates. Without optical follow-up, it is challenging to distinguish between CVs and qLMXBs, as they display similar F_X/F_opt ratios and have overlapping X-ray luminosity ranges. Further spectroscopic follow-up to place constraints on the mass of the accreting object is required for the rest of the objects in our sample to clarify their nature.  






 §.§ Spectroscopically Confirmed CVs



Here, we discuss the nature of four objects from our final sample of CV candidates, for which we obtained optical spectra using Keck I/LRIS or Palomar/DBSP.




  §.§.§ 2CXO J044048.3+292434, a new polar



We present the optical spectrum of J044048.3+292434 (hereafter J04404) in the right panel of Figure <ref>. It shows Hydrogen Balmer and Helium emission lines along with a high-excitation He II 4686Å emission line, yielding a high line ratio  of He II / Hβ≈ 0.64, which is typically seen in polars (see Table <ref>).

J04404 shows a flux ratio of about F_X/F_opt≈ 0.6, and an X-ray luminosity L_X ≈ 7.1× 10^30 erg s^-1 (see Table <ref>). The X-ray spectrum can be approximated by a power-law model with an unusually shallow photon index Γ≈ 0.5 (or kT  55.5 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>), a characteristic observed in some magnetic CVs <cit.>. J04404 is located close to the period minimum of CVs in the X-ray luminosity–period diagram (see Figure <ref>). Its optical and X-ray properties indicate that  J04404 is a polar near the CV period minimum.

In the left panels of Figure <ref>, we present the long-term ZTF light curve of  J04404 and the phase-folded light curve at the best period (1.30 h = 78 min). It is evident that J04404 underwent a state change, commonly observed in polars, around MJD = 59000, where the average optical r filter magnitude increased from r∼21 to r∼19 (see top left panel of Figure <ref>). 
 
In some cases, polars can show two peaks in their optical light curve within a single orbital period due to cyclotron beaming[A two-component cyclotron model has been used to model the optical and near-infrared light curves of polars, such as in <cit.>. A two-component cyclotron model leads to two peaks in a single orbital period, and can be due to any of the following: two magnetic poles of different field strength, optical depth, and/or viewing angles.]. The phase-folded optical light curve of J04404 shows ∼ 2 mag peak-to-peak modulation, which is characteristic of cyclotron beaming following a state change in polars (see the bottom left panel of Figure <ref>). In order to determine if the orbital period of J04404 is 1.30 hr or 2.60 hr, we utilized the fact that the Hydrogen Balmer and He II emission lines in polars typically vary by ≈1000 km/s within a single orbital period <cit.>. In the case of J04404, two spectra taken 1.4 hr apart do not show a significant radial velocity shift. In the bottom right panel of Figure <ref>, we show both spectra of J04404, taken 1.4 hr apart. We also show a ≈1000 km/s shift, which would be expected if 1.4 hr were approximately half of the orbital period. However, we note that if the system were viewed nearly perpendicular to the accretion stream, such a radial velocity shift would not be observed. Time-resolved spectroscopy for 2.6 hr is necessary to confirm the orbital period of this system, leading us to currently label the orbital period as either 1.3 or 2.6 hr.





  §.§.§ 2CXO J044147.9-015145, a low accretion rate non-magnetic CV



In Figure <ref> (left panel), we present the long-term ZTF optical light curve of 2CXO J044147.9-015145 (hereafter J04414). A preliminary Lomb-Scargle period analysis reveals no periodicity in the ZTF  data, indicating that further data are required to determine the possible orbital period of this system. The optical spectrum of J04414 is shown in Figure  <ref> (right panel). Hydrogen and Helium emission lines are present, with no detectable He II 4686Å emission line, giving a 3σ upper limit for the line ratio He II / Hβ < 0.05 (see Table <ref>). J04414 shows a flux ratio of F_X/F_opt≈ 0.3 and an X-ray luminosity L_X ≈ 3.1× 10^29  erg s^-1. The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.8 (or kT ≈ 4.5 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). The optical and X-ray properties suggest that J04414 is a low accretion rate non-magnetic CV (WZ Sge type). This classification is based on the apparent WD Balmer absorption lines and weakly double-peaked central emission lines seen in the center of its  optical spectrum. There is no sign of the donor signature in the optical spectrum. The Gaia HR diagram (see Figure <ref>) also supports this classification, as J04414 is located in the WD region, where WZ Sge type systems are concentrated  <cit.>.






  §.§.§ 2CXO J123727.5+655211, a new polar



In Figure <ref> (top panels), we present the long-term ZTF light curve and the phase-folded light curve at the best period for 2CXO J123727.5+655211 (hereafter J12372). The ZTF light curve reveals a significant period at 2.12 hr. The color dependence of cyclotron beaming is evident, with the ZTF i filter showing the highest modulation (peak-to-peak ∼ 2 mag) and the ZTF g filter showing the lowest (peak-to-peak ∼ 1 mag) (see the top left panel of Figure <ref>). 
 
We present the optical spectrum of J12372 in Figure <ref> (bottom panel). The spectrum shows Hydrogen, Helium, and high excitation He II 4686Å emission lines, yielding a line ratio of He II / Hβ≈ 0.37 (see Table <ref>).

J12372 shows a flux ratio F_X/F_opt≈ 1.1 and an X-ray luminosity L_X ≈ 4.4× 10^31 erg s^-1 (see Table <ref>). The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.4 (or kT ≈  38.3 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). X-ray timing analysis revealed no significant period in the longest exposure Chandra observation (see Table <ref>). J12372 shows significant X-ray flux variability (R≈ 6, see Section <ref>), with a factor of ≈ 2 between the Chandra observations, but no statistically significant variability between the spectral parameters was found. J12372 lies close to the CV period gap in the X-ray luminosity–period diagram (see Figure <ref>). Together with the optical light curves and spectrum, this indicates that J12372 is a polar.








  §.§.§ 2CXO J182117.2-131405, low accretion rate polar




Figure <ref> (top panels) shows the optical long-term ZTF light curve of 2CXO J182117.2–131405 (hereafter J18211) and the phase folded light curve. We determined  its best period to be 2.01 hr using the Lomb-Scargle period analysis. The phase-folded optical light curve shows cyclotron beaming  near phase 0.15 and ≈ 1 mag peak-to-peak modulation (see top right panel of Figure <ref>). This modulation could be either due to cyclotron beaming or an eclipse of the WD by the donor star. High-speed photometry is needed to further investigate these scenarios.

J18211 has a low number of X-ray counts, and its spectrum can be approximately fitted by a power-law model, resulting in a large uncertainty for a photon index  Γ∼ 1.7 (or kT  1.6 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). J18211 shows a flux ratio F_X/F_opt≈ 0.15 and a lower X-ray luminosity (L_X ≈ 6× 10^28 erg s^-1) than typically seen in CVs (see Table <ref>). J18211 is located near the edge of the CV period gap in the X-ray luminosity–period diagram (see Figure <ref>). In this part of the diagram, the canonical picture of CV evolution predicts that detached systems begin to undergo accretion again. Hence, J18211 could be a low accretion rate polar (previously referred to as a pre-polar).

We present the optical spectrum of J18211 in Figure <ref> (bottom panel). The spectrum is dominated by broad humps corresponding to cyclotron harmonics with no significant Hβ and He II 4686Å emission lines detected. Using the equation for the electron gyration frequency around a magnetic field, we can define the wavelength of cyclotron harmonics as the following:


    λ_n = 10710/n(100  MG/B)sinθ


With the spacing of two harmonics, it is possible to constrain the magnetic field strength, as the spacing must always be the ratio of integers. We must assume a cyclotron beaming angle, which should be close to 90^∘ since the amplitude of the cyclotron harmonics is so large (An angle θ > 60^∘ is a good approximation for cyclotron beaming factors greater than 2 <cit.>). Detailed cyclotron modeling is beyond the scope of this paper. In this work, we fixed the magnetic field at different values and approximately matched the location of the cyclotron humps in the spectrum to different cyclotron harmonics (see Figure <ref>). This combination of the cyclotron wavelengths observed in the spectrum of J18211 and the large beaming amplitude constrains the magnetic field to be B≈54-63 MG. This magnetic field strength is typical of low accretion rate polars <cit.>.






 §.§ Variable CV candidates



Here, we discuss four CV candidates classified based on their X-ray and optical variability, but for which optical spectra are not yet available.






  §.§.§  2CXO J063805.1–801854: A magnetic CV?


2CXO J063805.1–801854 (hereafter J06380) shows F_X/F_opt≈ 0.8 and X-ray luminosity L_X ≈ 2.4× 10^31 erg s^-1 (see Table <ref>). Its spectrum can be approximated by a power-law model with an unusually shallow photon index Γ≈ 0.2 (or kT  75 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). Such a shallow photon index is observed in the X-ray spectra of some magnetic CVs, such as polars or IPs <cit.>. X-ray timing analysis revealed a ≈ 13,200 sec periodicity (PF=61%), which is consistent with the period previously found by <cit.> in the same Chandra data (see Table <ref> and Figure <ref>). There is no available optical ZTF photometry for J06380. The Gaia optical light curve shows an outbursting-type variability with ≈ 3 mag outburst lasting around ≈ 100 days and a possible recurrence time of approximately ≈ 500 days (see Figure <ref>). Figure <ref> shows that J06380 lies within main sample of known CVs. The X-ray periodicity (in the range of CV orbital periods), along with a high X-ray luminosity and hard spectrum, suggests that J06380 could be a magnetic CV, likely an IP.




  §.§.§   2CXO J165219.0–441401: A possible dwarf nova?


2CXO J165219.0–441401 (hereafter J16521) shows F_X/F_opt≈ 1.2 and a high X-ray luminosity L_X ≈ 3× 10^32 erg s^-1 (see Table <ref>). The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.5 (or kT ≈ 5.8 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). There is no ZTF photometry for J16521 and the Gaia optical light curve shows a ≈ 3 mag outburst with a possible ≈ 500 days recurrent time (see Figure <ref>). This suggests that J16521 could be a dwarf nova.




  §.§.§ 2CXO J173118.6+224248: A possible non-magnetic CV?


2CXO J173118.6+224248 (hereafter J17311) shows F_X/F_opt≈ 0.3 and an X-ray luminosity L_X ≈ 2.4× 10^30 erg s^-1 (see Table <ref>). The optical ZTF light curves show ≈0.5 mag variability with a period of about ≈1.8 hr (see Figure <ref>). However, the actual period may be twice this value, possibly due to ellipsoidal modulation. The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 2.2 (or kT ≈ 3.0 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>), a characteristic observed in non-magnetic CVs. J17311 is located close to the main sequence of known CVs in the X-ray luminosity–orbital period diagram (see Figure <ref>). The X-ray properties and optical variability suggest that J17311 is a non-magnetic CV.





  §.§.§ 2CXO J185603.2+021259: A possible eclipsing binary?


2CXO J185603.2+021259 (hereafter J18560) shows F_X/F_opt≈ 0.05 and a lower X-ray luminosity (L_X ≈ 2.5× 10^29 erg s^-1), than observed in CVs (see Table <ref>). The ZTF optical light curve shows deep eclipses of approximately ≈ 1 mag in the r filter (or approximately ≈ 1.5 mag in the g filter) with a period of about 5.5 hr (see Figure <ref>).  Out of the eclipse sinusoidal variability with ≈ 0.1 mag is seen, possibly caused by the contribution from the donor. The X-ray spectrum can be approximated by a power-law model having a photon index Γ≈ 2.8 (or kT ≈ 1 keV for an optically thin thermal plasma model) (see Table <ref> and Figure <ref>). Figure <ref> shows that J18560 is located below the main CV sample, similar to J18211. These observed properties suggest that J18560 could be an eclipsing binary.






§ CONCLUSIONS



We searched for new Galactic CV candidates among 25,887 sources in the CSC2, cross-matched these with Gaia DR3, and studied their optical and X-ray properties. We summarize our findings below:




  – We used a modern concept suggested by <cit.> to classify new CVs among stellar objects based on a combination of F_X/F_opt and Gaia BP–RP color (so called the “X-ray Main Sequence") (see Figure <ref>). We further defined a new threshold cut to compile a pure sample of CV candidates and minimize contamination from stellar objects (see Eq.<ref> and Section <ref>).




  –  We excluded known Galactic stellar objects by cross-matching the CSC2 – Gaia DR3 catalog with the Simbad database and other publicly available CV catalogs (see Section <ref>). Out of 13,956 unclassified Galactic objects, we identified 14 new CV candidates using two different cuts on the “X-ray Main Sequence"  (see Table <ref> and Section <ref>).




  – All objects show X-ray luminosities in the range L_X ≈ 10^29-10^32 erg s^-1. Their X-ray spectra can be approximated by a power-law model with photon indices in the Γ∼ 1-3 range or by an optically thin thermal emission model in the kT ∼ 1-70 keV range (see Figure <ref> and Table <ref>). These X-ray properties are mostly typical of CVs (both magnetic and non-magnetic), but some objects might also be qLMXBs.





  – We performed optical follow-up observations with Keck I/LRIS and P200/DBSP and spectroscopically confirmed four objects as new CVs. Their optical spectra show prominent emission lines or cyclotron humps, typically seen in CVs (see Tables <ref> and <ref>). We determined orbital periods for three of these objects using ZTF photometry. Among these, we discovered two polars, one low accretion rate polar and one non-magnetic CV (WZ Sge type) (see Section <ref>). 




  – The X-ray and optical properties of four variable CV candidates suggest that  three are potential magnetic systems or dwarf novae, while one may be an eclipsing binary (see Section <ref>). Further spectroscopic follow-up is required to confirm the nature of these objects.





Our analysis shows that a multiwavelength approach is a powerful tool for discovering new CVs and studying their properties. We illustrated that the “X-ray Main Sequence" presented in <cit.> allows us to compile a sample of CV candidates for a detailed study. This approach can also be applied to other X-ray catalogs or upcoming all-sky SRG/eROSITA data to search for accreting binary systems. Further investigation of selection effects is necessary to use the “X-ray Main Sequence" to compile volume-limited samples of CVs and study their population.




This research has made use of data obtained from the Chandra Data Archive and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.

Based on observations obtained with the Samuel Oschin Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part of the Zwicky Transient Facility project. ZTF is supported by the National Science Foundation under Grants No. AST-1440341 and AST-2034437 and a collaboration including current partners Caltech, IPAC, the Weizmann Institute of Science, the Oskar Klein Center at Stockholm University, the University of Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity College Dublin, Lawrence Livermore National Laboratories, IN2P3, University of Warwick, Ruhr University Bochum, Northwestern University and former partners the University of Washington, Los Alamos National Laboratories, and Lawrence Berkeley National Laboratories. Operations are conducted by COO, IPAC, and UW. The ZTF forced-photometry service was funded under the Heising-Simons Foundation grant No.12540303 (PI: Graham).

We are grateful to the staffs of the Palomar and Keck Observatories for their work in helping us carry out our observations. 

This work has made use of data from the European Space Agency (ESA) mission Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia Data Processing and Analysis Consortium (DPAC, <https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.


This research made use of matplotlib, a Python library for publication quality graphics <cit.>; NumPy <cit.>; Astroquery <cit.>; Astropy, a community-developed core Python package for Astronomy <cit.>; and the VizieR catalogue access tool, CDS, Strasbourg, France. 

IG, AS, VD, NT acknowledge support from Kazan Federal University. ACR acknowledges support from the National Science Foundation via an NSF Graduate Research Fellowship. The authors thank the anonymous referee for useful and inspiring suggestions which helped to improve the manuscript.






 



§ GENERAL OBJECT TYPES WITH A SIMBAD AND F_X/F_OPT-(BP-RP) SELECTION CUT 









Table <ref> shows our general object types based on the Simbad hierarchical structure. The columns are: (1) our main object type; (2) object sub-type; (3) title of Simbad hierarchical structure used to define the main object type and its sub-type; (4) number of sources per each object sub-type, including known sources and their candidates. No source in our sample had an object type from the sixth Simbad's hierarchy ("Gravitation"); therefore, it's not shown in Table <ref>. 

We investigated purity and completeness to construct a pure sample of new CV candidates and avoid contamination from stellar objects. Based on the Simbad hierarchical structure, we created two samples of known CVs (36 sources) and STARS (4781 sources) (see Section <ref>).

<cit.> computed purity and completeness for the red and blue galaxies classified by optical color. We applied a similar approach to define the purity and completeness as:





  * True positive (TP) = known CVs classified as CVs based on a given threshold (i.e., known CVs with an observed X-ray-to-optical ratio exceeding the threshold cut, (F_X/F_opt)_obs≥ (F_X/F_opt)_cut).


  * False positive (FP) = known STARS classified as CVs based on a given threshold (i.e., known STARS with an observed X-ray-to-optical ratio exceeding the threshold cut, (F_X/F_opt)_obs≥ (F_X/F_opt)_cut).


  * False negative (FN) = known CVs classified as STARS (i.e., known CVs with an observed X-ray-to-optical ratio below the threshold cut, (F_X/F_opt)_obs < (F_X/F_opt)_cut).



Purity (P) is computed as the fraction of true identification to all identifications, 

    P = TP / (TP + FP),

and the completeness (C) is calculated as the fraction of true detections to all that should have been classified, 

    C = TP / (TP + FN).

To find the optimal combination of linear parameters (A, B) for Eq.<ref>, we created a grid of parameters with the 0.1 steps in the range 0  A 3 and -8  B 3 and computed the purity and completeness. 

Figure <ref> shows the purity and completeness as a function of linear parameters (A, B). For illustrative purposes, we show the purity and completeness only for three B parameters. The B=0 case corresponds to the canonical selection cut based only on the X-ray-to-optical flux ratio without additional information from Gaia colors. For example, the cut F_X/F_opt = 0.1 (where A = -1 and B=0) gives ≈27% purity and ≈48% completeness. The drop of the purity above the A  -0.2 is caused by a low number of sources and is not statistically significant. The <cit.> empirical cut (A = -3.5 and B=1) shows ≈25% purity and ≈78% completeness. In our study, we chose to use A = -1.5 and B=0.7, which gives ≈100 % purity and ≈52% completeness.

We note that our threshold is defined based only on a small sample of confirmed CVs (36 objects). To study the CV population and create a volume-limited sample, the F_X/F_opt and Gaia BP–RP color diagram should be further investigated for selection effects with a large sample of confirmed CVs and stellar objects. A detailed investigation is beyond the scope of this paper.








§ RECENTLY DISCOVERED CVS

















We selected three objects from the CSC2–Gaia DR3 catalog using the X-ray main sequence as CV candidates (see Figure <ref> and Table <ref>).  In the course of this study, two of these objects (2CXO J024131.0+593630 and 2CXO J143435.3+334049) were recently discovered as CVs with SDSS spectra and ZTF light curves <cit.>. Another object, 2CXO J174133.7–284034[We included this system in our analysis because it appears as an X-ray source ("X") in Simbad, not explicitly as a CV. ], was confirmed as a CV via optical spectroscopy by <cit.>. However, we included this object in our analyses to search for possible X-ray variability through timing and spectral analyses. Here, we individually discuss the X-ray and optical properties of these objects.




 §.§ 2CXO J024131.0+593630, a non-magnetic CV




In the left panel of Figure <ref>, we present the long-term optical ZTF light curve of 2CXO J024131.0+593630 (hereafter J02413), which shows two significant (≈ 2 mag in r filter and ≈ 3 mag in g filter) short outbursts. A preliminary Lomb-Scargle period analysis reveals no periodicity in the ZTF and Gaia data, indicating that further observations are needed to determine the possible orbital period of this system.

We observed J02413 with P200/DBSP on 6 December 2023 for 900s. The optical spectrum of J02413 is presented in the right panel of Figure <ref>. Single-peaked Hydrogen Balmer emission lines are clearly visible, and a weak He II 4686Å emission line is also present, giving a low line ratio He II / Hβ≈0.12  (see Table <ref>). However, as noted in Table <ref>, the detection of He II 4686Å does not pass a visual inspection, so the value of the ratio should be interpreted as an upper limit. We also note that the detection (or non-detection) of He II 4686Å does not affect our final interpretation of this object. 

J02413 shows a high flux ratio F_X/F_opt≈ 1.7 and an X-ray luminosity L_X ≈ 1.5× 10^31 erg s^-1 (see Table <ref>). The X-ray spectrum of J02413 can be approximated by a power-law model with Γ≈ 1.8 (or kT ≈ 5.5 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). We found no significant period for J02413 from the X-ray timing analysis of the longest exposure Chandra observation (ObsID = 7152). J02413 also does not show X-ray flux variability and or changes in spectral parameters between observations. Our analysis, using optical and X-ray properties, suggests that J02413 is a CV, likely non-magnetic. This agrees with <cit.> conclusion, where J02413 is identified as a new non-magnetic CV (U Gem type). The spectrum of this object is not shown in <cit.>, making our spectrum a useful addition to the literature.




 §.§ 2CXO J143435.3+334049, low-accretion rate CV



In Figure <ref> (left panel), we present the long-term ZTF light curve of 2CXO J143435.3+334049 (hereafter J14343). A preliminary Lomb-Scargle period analysis reveals no periodicity in the ZTF  data, indicating that further observations are needed to determine the possible orbital period of this system.

We observed J14343 with P200/DBSP on 21 January 2023 for 900s. The optical spectrum of J14343 is presented in Figure <ref> (right panel). The WD Balmer absorption lines are seen with double-peaked emission lines at the center, though the peaks appear to be asymmetric in height and width for both the Hα and Hβ lines. We do not detect the high-excitation He II 4686Å emission line providing an upper limit for a line ratio  He II / Hβ < 0.15 (see Table <ref>). There is no sign of the donor signature in the optical spectrum. The spectrum of this object in Figure <ref> is different from the one shown in <cit.>, with the Balmer emission lines here being noticeably asymmetric. This could warrant future study of this system to further understand the nature of this source.

J14343 shows a flux ratio F_X/F_opt≈ 0.3  and a lower X-ray luminosity (L_X ≈ 4.8× 10^29 erg s^-1) than typically seen in CVs (see Table <ref>). We have not detected statistically significant flux and spectral parameter variability between Chandra observations. The X-ray spectrum can be approximated by a power-law model with a photon index Γ≈ 1.7 (or kT  5.2 keV for optically thin thermal emission model) (see Table <ref> and Figure <ref>). X-ray and optical properties suggest that J14343 is a low-accretion rate CV (WZ Sge type), similar to what <cit.> concluded in their analysis. 




 §.§ 2CXO J174133.7–284034, a non-magnetic CV


2CXO J174133.7–284034 (hereafter J17413 shows a flux ratio F_X/F_opt≈ 1.0 and an X-ray luminosity L_X ≈ 4.4× 10^31 erg s^-1 (see Table <ref>). The X-ray spectrum of J17413 can be approximated by a power-law model with Γ≈ 1.8 (or kT ≈ 5.6 keV for an optically thin thermal emission model) (see Table <ref> and Figure <ref>). We have not detected statistically significant flux and spectral parameter variability between Chandra observations. We found no significant period for J17413 with the X-ray timing analysis for the longest exposure Chandra observation (ObsID = 9565). <cit.> observed optical variability (≈ 1 mag) with no periodicity. <cit.> observed prominent emission lines and confirmed via optical spectroscopy that J17413 is a CV. Our X-ray analysis supports  <cit.> and <cit.>  conclusions and also suggests that J17413 is likely a non-magnetic system.





 




§ X-RAY SPECTRA FIGURES



Figure <ref> shows Chandra X-ray spectra of all objects from Tables <ref> and <ref>. Table <ref> shows the results of the approximation of X-ray spectra with two spectral models: power-law ( in  with photon index parameter, Γ) and  optically thin thermal plasma emission model ( in  with temperature kT, and metal abundance [Z] parameters).





























aa







 Paper Title: TESS Giants Transiting Giants. VI. Newly Discovered Hot Jupiters Provide Evidence for Efficient Obliquity Damping after the Main Sequence.






0000-0003-2657-3889]Nicholas Saunders
NSF Graduate Research Fellow
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0003-4976-9980]Samuel K. Grunblatt
Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA

0000-0003-2657-3889]Ashley Chontos
Henry Norris Russell Fellow
Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA

0000-0003-2657-3889]Fei Dai
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0001-8832-4488]Daniel Huber
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

UTF8gbsn
0000-0002-2696-2406]Jingwen Zhang (张婧雯)
NASA FINESST Fellow
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0001-7409-5688]Guðmundur Stefánsson
Anton Pannekoek Institute for Astronomy, 904 Science Park, University of Amsterdam, Amsterdam, 1098 XH

0000-0002-4284-8638]Jennifer L. van Saders
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0002-4265-047X]Joshua N. Winn
Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA

0000-0003-3244-5357]Daniel Hey
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA



0000-0001-8638-0320]Andrew W. Howard
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0003-3504-5316]Benjamin Fulton
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E. California Boulevard, Pasadena, CA 91125, USA

0000-0002-0531-1073]Howard Isaacson
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA
Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD, Australia

0000-0001-7708-2364]Corey Beard
NASA FINESST Fellow
Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA

0000-0002-8965-3969]Steven Giacalone
NSF Astronomy and Astrophysics Postdoctoral Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0002-4290-6826]Judah van Zandt
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA

0000-0001-8898-8284]Joseph M. Akana Murphey
NSF Graduate Research Fellow
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA

0000-0002-7670-670X]Malena Rice
Department of Astronomy, Yale University, New Haven, CT 06511, USA

0000-0002-3199-2888]Sarah Blunt
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
CIERA, Northwestern University, Evanston IL 60201, USA

0000-0002-1845-2617]Emma Turtelboom
Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA

0000-0002-4297-5506]Paul A. Dalba
Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA

0000-0001-8342-7736]Jack Lubin
Department of Physics & Astronomy, University of California Irvine, Irvine, CA 92697, USA

0000-0002-4480-310X]Casey Brinkman
Institute for Astronomy, University of Hawaiʻi at Mānoa, 2680 Woodlawn Drive, Honolulu, HI 96822, USA

0000-0003-3179-5320]Emma M. Louden
Department of Astronomy, Yale University, New Haven, CT 06511, USA

0000-0002-3221-3874]Emma Page
Department of Physics, Lehigh University, 16 Memorial Drive East, Bethlehem, PA 18015, USA




0000-0001-8621-6731]Cristilyn N. Watkins
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0001-6588-9574]Karen A. Collins
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0003-2163-1437]Chris Stockdale
Hazelwood Observatory, Australia


0000-0001-5603-6895]Thiam-Guan Tan
Perth Exoplanet Survey Telescope, Perth, Western Australia


0000-0001-8227-1020]Richard P. Schwarz
Center for Astrophysics  Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA


0000-0001-8879-7138]Bob Massey
Villa '39 Observatory, Landers, CA 92285, USA

0000-0002-2532-2853]Steve B. Howell
NASA Ames Research Center, Moffett Field, CA 94035, USA

0000-0001-7246-5438]Andrew Vanderburg
Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA


Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA

0000-0002-4715-9460]Jon M. Jenkins
NASA Ames Research Center, Moffett Field, CA, 94035








0000-0002-6892-6948]Sara Seager
Department of Physics, and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA

0000-0002-8035-4778]Jessie L. Christiansen
NASA Exoplanet Science Institute, IPAC, MS 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA

0000-0002-6939-9211]Tansu Daylan
Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA





Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA



0009-0008-9808-0411]Max Brodheim
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0009-0004-4454-6053]Steven R. Gibson
Caltech Optical Observatories, Pasadena, CA, 91125, USA

0000-0002-7648-9119]Grant M. Hill
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0000-0002-6153-3076]Bradford Holden
UCO/Lick Observatory, Department of Astronomy and Astrophysics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA

0000-0002-5812-3236]Aaron Householder
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA


Caltech Optical Observatories, California Institute of Technology, Pasadena, CA 91125, USA

0000-0003-2451-5482]Russ R. Laher
NASA Exoplanet Science Institute/Caltech-IPAC, MC 314-6, 1200 E California Blvd, Pasadena, CA 91125, USA

0009-0004-0592-1850]Kyle Lanclos
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0000-0003-0967-2893]Erik A. Petigura
Department of Physics & Astronomy, University of California Los Angeles, Los Angeles, CA 90095, USA

0000-0001-8127-5775]Arpita Roy
Astrophysics & Space Institute, Schmidt Sciences, New York, NY, 10011, USA

0000-0003-3856-3143]Ryan A. Rubenzahl
NSF Graduate Research Fellow
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0000-0002-4046-987X]Christian Schwab
School of Mathematical and Physical Sciences, Macquarie University, Balaclava Road, North Ryde, NSW 2109, Australia

0000-0003-3133-6837]Abby P. Shaum
Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA

0009-0007-8555-8060]Martin M. Sirk
Space Sciences Laboratory, University of California, 7 Gauss Way, Berkeley, CA 94720, USA


Space Sciences Laboratory, University of California, 7 Gauss Way, Berkeley, CA 94720, USA

0000-0002-6092-8295]Josh Walawender
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743

0000-0002-4037-3114]Sherry Yeh
W. M. Keck Observatory, 65-1120 Mamalahoa Hwy., Kamuela, HI 96743








§ ABSTRACT 

    The degree of alignment between a star's spin axis and the orbital plane of its planets (the stellar obliquity) is related to interesting and poorly understood processes that occur during planet formation and evolution. Hot Jupiters orbiting hot stars (≳6250 K) display a wide range of obliquities, while similar planets orbiting cool stars are preferentially aligned. Tidal dissipation is expected to be more rapid in stars with thick convective envelopes, potentially explaining this trend. Evolved stars provide an opportunity to test the damping hypothesis, particularly stars that were hot on the main sequence and have since cooled and developed deep convective envelopes. We present the first systematic study of the obliquities of hot Jupiters orbiting subgiants that recently developed convective envelopes using Rossiter-McLaughlin observations. Our sample includes two newly discovered systems in the Giants Transiting Giants Survey (, ). We find that the orbits of hot Jupiters orbiting subgiants that have cooled below ∼6250 K are aligned or nearly aligned with the spin-axis of their host stars, indicating rapid tidal realignment after the emergence of a stellar convective envelope. We place an upper limit for the timescale of realignment for hot Jupiters orbiting subgiants at ∼500 Myr. Comparison with a simplified tidal evolution model shows that obliquity damping needs to be ∼4 orders of magnitude more efficient than orbital period decay to damp the obliquity without destroying the planet, which is consistent with recent predictions for tidal dissipation from inertial waves excited by hot Jupiters on misaligned orbits.






§ INTRODUCTION
 





Giant planets on close-in orbits (hot Jupiters) have been observed in a wide variety of stellar environments, but it remains unclear how they arrive at such small orbital separations from their host stars <cit.>. Two popular scenarios are disk-driven migration and high-eccentricity migration <cit.>. In both theories, giant planets form at larger semi-major axes, before losing angular momentum to either the protoplanetary disk (in disk-driven migration; e.g. ) or to another planet or star (in high-eccentricity migration; e.g. ). In the latter case, the planet ends up on an eccentric orbit that gradually shrinks and circularizes as the eccentricity is damped by tidal interactions with the star. 

The stellar obliquity—the angle between the spin axis of the star and the planet's orbital axis—is a key piece of evidence to distinguish between these pathways as each scenario produces a distinct distribution of spin-orbit angles. Disk migration results in aligned populations at early times, while high-eccentricity migration predicts early scattering events that might alter the obliquity <cit.>. Observations of hot Jupiters around main-sequence stars have shown that the stellar obliquity distribution has a strong dependence on the star's effective temperature <cit.>, with hot hosts displaying a wide range of obliquities, and cool hosts being preferentially aligned. The critical separation is approximately at the Kraft break <cit.>, around 6250 K, corresponding to the break between stars with primarily radiative envelopes (>6250 K) and stars with thick convective envelopes (<6250 K). The distinct distributions of obliquities separated by the Kraft Break suggest two possible scenarios:

    
  * Hot Jupiters have unique formation pathways around hot and cool stars, resulting in aligned orbits around cool stars and a broad range of obliquities around hot stars.
    
  * Hot Jupiters are formed by the same mechanism around hot and cool stars, and the spin-orbit angle is realigned during the main sequence by tidal dissipation in stars with convective envelopes.


The correlation between spin-orbit alignment and the presence of a convective envelope has driven theories for realignment through the tidal damping of obliquity <cit.>. During their main sequence lifetimes, low mass stars (≲1.2 M_⊙) have surface convection zones that deepen as the star cools and evolves into a subgiant. However, it is challenging to constrain the timescales of damping mechanisms using a main sequence population in part due to uncertain age estimates. More massive stars ≳1.2 M_⊙) have radiative envelopes on the main sequence, which cool and become convective as the stars drop below the Kraft break during the subgiant or red giant branch phase. As these stars only develop substantial convective envelopes after leaving the main sequence, they provide a unique avenue to test the timescales of tidal damping. If convection zones efficiently damp obliquities, as required by high-eccentricity migration, subgiants that were formerly 'hot' and had a broad range of obliquities should be found to have low obliquities due to rapid obliquity damping.

Studies have suggested that the presence of a convective envelope drives efficient damping of orbital period, eccentricity, and obliquity <cit.>. Diminishing orbital period has been unambiguously observed in only a small handful of systems <cit.>, though these measurements provide useful benchmarks for the timescales and efficiencies of mechanisms for orbital evolution. Orbital decay is driven by the transfer of orbital angular momentum to stellar rotation <cit.>. As an orbiting planet raises a tidal bulge in the outer envelope of its host star, turbulent viscosity <cit.> and the excitation of waves through the stellar interior <cit.> sap energy from the orbit. This orbital evolution can take place over human timescales <cit.>, and the changing orbital period can be observed. 











The sky-projected obliquity can be constrained by obtaining high-resolution spectra during a planet's transit and modeling the Rossiter-McLaughlin (RM) effect <cit.>. As a planet crosses in front of its host star, it blocks regions of the stellar surface that are red-shifted or blue-shifted relative to the observer due to the stellar rotation. These red- and blue-shifts imprint an additional signal onto the traditional radial velocity curve caused by the planet's orbit, the morphology and amplitude of which depend on the relative alignment of the planet's orbital path across the rotating star, and can be used to constrain the sky-projected obliquity of the system. The nearly all-sky coverage of NASA's Transiting Exoplanet Survey Satellite (; ) has enabled the discovery of a large sample of hot Jupiters orbiting bright subgiants which are amenable to measurements of the RM effect <cit.>, allowing us measure spin-orbit alignment and understand late-stage planetary orbit evolution. 

In this paper we present observations of the RM effect for three hot Jupiters orbiting subgiants observed by : (R_ p = R_ J, P_ orb = days, R_⋆ = 1.89±0.18 R_⊙, T_ eff = 6106±100 K), (R_ p = R_ J, P_ orb = days, R_⋆ = 2.23±0.05 R_⊙, T_ eff = 6223±100 K), and (R_ p = R_ J, P_ orb = days, R_⋆ = 1.82±0.25 R_⊙, T_ eff = 6020±100 K). is a previously confirmed system <cit.>, while and are newly-identified planets confirmed by our work as part of the Giants Transiting Giants Survey. We find that all three systems have a low sky-projected obliquity (|λ| < 15^∘), indicating that tidal realignment is efficient after the post-main-sequence emergence of a convective envelope. We then test models for realignment, and find that enhanced efficiency of obliquity damping after the main sequence can lead to realignment without inspiral and engulfment. 















§ OBSERVATIONS
 



 §.§ Photometry
 



and were discovered as part of our Giants Transiting Giants survey (GTG; ). data are available for in 30-minute cadence Full-Frame Images (FFIs) for Sectors 17, 18, and 24, spanning from October 7, 2019 - May 13, 2020, and in 200-second cadence FFIs for Sector 58 from October 29 - November 26, 2022. It was flagged by our survey as a planet candidate in 2021, and was selected for 2-minute cadence observations in Sector 58. data are available for in 30-minute FFIs in Sector 12 (April 22 - May 21, 2019), 10-minute cadence in Sector 39 (May 26 - June 24, 2021), and 200-second cadence in Sector 66 (June 2  - July 1, 2023). We identified as a planet candidate in 2021, and it received 2-minute cadence observations in Sectors 39 and 66. 

was first identified by the MIT Quick Look Pipeline (QLP; ) in 2019, and data are available in 30-minute cadence FFIs for Sectors 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, and 26, spanning from July 18, 2019 - July 4, 2020. data are available in 10-minute cadence FFIs for Sectors 40, 41, 47, 48, 49, 50, 51, 52, 53, 54, 55, spanning from June 24, 2021 - September 1, 2022. In 200-second cadence FFIs, it has so far been observed in Sectors 56, 57, 58, 59, and 60, covering September 1, 2022 - January 18, 2023. was scheduled for 2-minute cadence observations beginning on April 16, 2020 with Sector 24, and was selected for 2-minute cadence data through Sector 60. The planet was confirmed by <cit.>, and was followed up as part of the -Keck Survey (TKS; ). 

All photometry data used are the Presearch Data Conditioning simple aperture photometry (PDC_SAP; ) 2-minute light curves produced by the Science Processing Operations Center (SPOC; ) at NASA Ames Research Center downloaded from the Mikulski Archive for Space Telescopes (MAST).

We searched for additional transiting planets within these systems in the photometry by masking the known transits and performing a Box-fitting Least Squares (BLS) periodogram analysis with the  implementation of the  BLS tools. We did not identify additional transiting planet signals.

The Science Processing Operations Center (SPOC; ) performed multi-transiting planet search in the subsequent 2-min cadence observations and detected the signatures of TOIs 1181 b, 6029 b and 4379 b using a noise-compensating matched filter <cit.>. The transit signatures were fitted with an initial limb-darkened transit model <cit.> and were subjected to a number of diagnostic tests <cit.>, including the difference image centroiding test, which located the host stars within 0.409 ± 2.6, 0.78 ± 2.5, and 2.6 ± 2.6. The SPOC searches likewise failed to identify additional transiting planet signals.



 §.§ Radial Velocity Follow-up
 

High precision RVs were taken using the High Resolution Echelle Spectrograph (HIRES) on the 10-meter Keck I telescope on Maunakea, Hawaiʻi <cit.>. For , 87 RV measurements were taken between December 2, 2019 and May 16, 2022. For , 46 RV measurements were taken between December 25, 2020 and October 6, 2022. For , 12 RV measurements were taken between September 21, 2021 and July 16, 2022. RV observations were taken in the HIRESr configuration with an iodine cell in the light path for wavelength calibration. The C2 decker was used for all observations, providing a spectral resolution of R≈60,000. RVs were reduced with the California Planet Search (CPS; ) pipeline. Table <ref> in Appendix <ref> contains all RV observations and corresponding uncertainties used in this analysis.



 §.§ Rossiter-McLaughlin Observations
 

In addition to RVs over the full phase of the planets' orbits, we obtained RV observations during each planet's transit to measure the Rossiter-McLaughlin (RM) effect. We obtained high resolution spectra during transit events with HIRES for and , and with the Keck Planet Finder (KPF; ) for . 

The HIRES observing configuration during the RM observations matches that described in <ref>. We obtained 40 HIRES RV observations during the transit of on September 8, 2020. The typical counts for each ∼10-minute exposure were 20,000 e^- with an average RV uncertainty of ∼3 m s^-1. We obtained full coverage of the transit of , with ∼1 hour before ingress and ∼1.5 hours after egress to establish an RV baseline. For , we obtained 34 HIRES RVs on September 18, 2022 with typical counts of 5,000 e^- over ∼13-minute exposures and average RV uncertainty of ∼5 m s^-1. Poor weather prevented observation of the full transit egress, and we were unable to continue the observations after the transit event concluded.

We obtained 30 observations of on June 6, 2023 using KPF, with 9-minute exposures. KPF is a fiber-fed spectrograph with a resolution of R≈98,000. Observations were taken with simultaneous wavelength calibration from an etalon lamp fed through a dedicated calibration fiber. In addition to simultaneous calibrations, we took dedicated etalon calibration exposures (so-called “slew-cals") each hour to monitor RV stability. Due to its southerly declination (-32^∘), the window to observe the transit of was tight, and the star was not visible above the Keck telescope dome during ingress. RVs were measured using two distinct approaches—first, using a cross-correlation function (CCF) on a set of spectral lines produced by the public KPF data reduction pipeline[https://github.com/Keck-DataReductionPipelines/KPF-Pipeline/https://github.com/Keck-DataReductionPipelines/KPF-Pipeline/]. This reduction method resulted in a typical RV uncertainty of ∼7 m s^-1. We also reduced the RVs by analyzing the spectra using an adapted version of the SpEctrum Radial Velocity Analyser code (SERVAL; ), which measures shifts in the observed spectra relative to a matched template spectrum, but found that the CCF reduction resulted in RVs with higher precision and smaller scatter. The KPF data reduction pipeline is being actively developed, and future improvements to the measured RV precision are likely to be implemented.



 §.§ Ground-based Imaging
 

We obtained high-contrast imaging of and to search for close stellar companions which might have biased the measurement of the amplitude of the transit signal observed by . was observed by the NIRC2 near-infrared imager in the Kp bandpass on the 10-meter Keck II telescope on June 26, 2023.

We first pre-processed the data using the Vortex Imaging Processing (VIP) software package <cit.>. We performed flat-fielding, bad-pixel removal, and correct for geometric distortions by applying the solution in <cit.>. Since the data were taken in vertical angle mode, we de-rotated each image according to the parallactic angles and then stacked the individual images into a combined image. To register the eight frames, we identify the star's position by fitting a 2D Gaussian to the stellar point spread function (PSF) in each frame.

A companion to was identified with a separation of 763 ± 20 milliarcseconds at a position angle of 110.2 ± 0.5^∘. We measured a difference in magnitude in the Ks bandpass of Δ m=4.44 ± 0.05. The reduced and derotated image can be found in Figure <ref> in Appendix <ref>.

After obtaining the pre-processed cubes, we extracted the astrometry of the companion applying a least-squares minimization code. Specifically, we fit the PSF of the companion using the unsaturated PSF of the primary star in the same images (see details in ).

was observed by the 8-meter Gemini South telescope on Cerro Pachon using the speckle interferometric instrument Zorro <cit.> on May 18, 2022 UT. Three sets of 1000 × 60 ms images were obtained simultaneously at 562 and 832 nm, the data was reduced using our standard software pipeline <cit.> and the limiting magnitude contrast was computed for each filter as a function of the angular separation from TOI-4379. The 5-σ contrast curves reached depths of 5 to 6.5 magnitudes in the two filters over the angular range of 0.1 to 1.2 arcsec. Within these contrast levels and angular limits, no close companion was detected. The raw and reduced data, including the contrast curves and reconstructed images at each wavelength, are available on the  website[exofop.ipac.caltech.edu/tess/target.php?id=97766057exofop.ipac.caltech.edu/tess/target.php?id=97766057].



 §.§ Ground-based Photometry


The TESS pixel scale is ∼ 21 pixel^-1, often causing multiple stars to blend in the TESS photometric aperture. To confirm the true source of the TESS detection, we acquired ground-based follow-up photometry of the fields around TOI-6029 and TOI-4379 as part of the TESS Follow-up Observing Program <cit.>[tess.mit.edu/followup]. We used the , which is a customized version of the  software package <cit.>, to schedule our transit observations.



  §.§.§ LCOGT


We observed an ingress window of in Sloan i' band on UTC 2023 August 03 from the Las Cumbres Observatory Global Telescope <cit.> 1 m network node at McDonald Observatory near Fort Davis, Texas, United States (McD). The 1 m telescopes are equipped with 4096×4096 SINISTRO cameras having an image scale of 0389 per pixel, resulting in a 26×26 field of view. The images were calibrated by the standard LCOGT  pipeline <cit.>, and differential photometric data were extracted using  <cit.>. The transit is detected within the 35 follow-up photometric aperture which excludes the flux from the nearest known neighbor in the Gaia DR3 catalog (Gaia DR3 430269136521931776) that is ∼17 north of .

We also observed a full transit window of in Sloan i' band on May 15, 2022 (UTC) from the LCOGT 1 m network node at Cerro Tololo Inter-American Observatory in Chile (CTIO). The transit is detected within the 27 follow-up photometric aperture, which excludes most of the flux from the nearest known neighbor in the Gaia DR3 catalog (Gaia DR3 6027376203516633728), which is ∼3 west of . 



  §.§.§ PEST

We observed a full transit window of in Sloan g' band on May 27, 2023 (UTC) using the Perth Exoplanet Survey Telescope (PEST), located near Perth, Australia. The 0.3 m telescope is equipped with a 5544×3694 QHY183M camera with an image scale of 07 pixel^-1, resulting in a 32×21 field of view. A custom pipeline based on [c-munipack.sourceforge.net] was used to calibrate the images and extract the differential photometry. The transit is detected within the 64 follow-up photometric aperture, which includes the flux from the nearest known neighbor in the Gaia DR3 catalog. Figure <ref> shows the ground-based transit observations of and with best-fit transit models. All light curve data are available on the  website[exofop.ipac.caltech.edu/tess/target.php?id=419523962exofop.ipac.caltech.edu/tess/target.php?id=419523962], but are not included in the joint model of this work. 



§ HOST STAR CHARACTERIZATION
 



 §.§ High-resolution Spectroscopy
 

We obtained out-of-transit, iodine-free spectra with Keck/HIRES for each of our host stars to extract precise stellar parameters. Stellar properties were computed using SpecMatch empirical (SpecMatch-Emp; ) and synthetic (SpecMatch-Synth; ) methodologies. All of our host stars have effective temperatures T_ eff > 5000 K, and we therefore adopt values from SpecMatch-Synth. Our adopted values can be found in Table <ref>.



 §.§ Isochrone Fitting
 

We used the [https://github.com/danxhuber/isoclassifyhttps://github.com/danxhuber/isoclassify] Python package <cit.> to compute additional stellar properties. We ran the code in grid mode where we pass input observables taken from SpecMatch-Synth and position and parallax distance from Gaia DR3 <cit.> and K-band magnitude adopted from the Input Catalog (TIC;  v8.2.  uses the stellar model grid to compute stellar mass (M_⋆), stellar radius (R_⋆), surface gravity (logg), distance (d), luminosity (L), and stellar age. We used the Combined19 allsky dust map from  <cit.>. 

We account for the companion to described in <ref> by running  in binary mode, including the correction for the measured companion magnitude. For the companion to , we obtain a mass of 0.44±0.02 M_⊙, radius of 0.45±0.02 R_⊙, and temperature of 3528±40 K.

The  outputs and associated uncertainties can be found in Table <ref>. All host stars are found to be evolved (∼2 R_⊙) stars with masses between 1.2 and 1.6 M_⊙ and metallicities between 0.1 and 0.4 dex.



 §.§ Asteroseismology
 

Finally, to improve the characterisation of the host stars, we searched for signatures of stellar p-mode oscillations in the light curve.  To do this, we first remove transits and apply a simple high pass-filter to remove further low-frequency variability. We use the PySYD pipeline <cit.> to search for variability in the host stars. We find no evidence of oscillations in TOI-6029 and TOI-4379. In TOI-1181, however, we identify a weak signal associated with the global oscillation power excess, commonly referred to as ν_ max, at 737±69 μHz at an SNR of 4.55. This value is slightly lower than the expected ν_ max of 930 μHz based on asteroseismic scaling relations (cf. ). As a result of this, and non-detections in the other host stars, we do not use asteroseismology to constrain the stellar parameters.







§ PLANET MODELING
 



 §.§ Joint Transit and RV Fit
 

We used the  Python package <cit.> to simultaneously model the planet transit, radial velocity curve, and RM effect. All photometry data used are 2-minute cadence observations reduced by the Science Processing Operations Center (SPOC; ) pipeline. We used the 2-minute cadence data as it provides the best constraint on the ingress and egress times, which are important for characterizing the shape of the Rossiter-McLaughlin signal. These data are also reduced with a uniform approach which accounts for contamination by nearby stars, providing the most accurate constraint on the transit depth and planet radius. To enable longer sampling within reasonable computation time (≲ 24 hours), we select only one Sector of 2-minute cadence data for each target (Sectors 73, 58, 66 for , , and , respectively). Including additional Sectors did not significantly impact the measured transit depth, period, or epoch.

We fit for orbital period, P, transit epoch, t_0, ratio of planetary to stellar radius, R_p/R_⋆, cosine of orbital inclination, cosi, RV semi-amplitude, K, sky-projected spin-orbit angle, λ, and the eccentricity and argument of periastron, parameterized by the expressions √(e)cosω and √(e)sinω. For each instrument, we also fit for two quadratic limb darkening coefficients, described by q_1, TESS, q_2, TESS, q_1, HIRES, q_2, HIRES, q_1, KPF, and q_2, KPF. We include an offset parameter for the RV instruments, which captures small global offsets for all RVs taken by a given instrument. 



For each target, we tested (a) constant instrumental offset shared between all observations for a given instrument and (b) instrumental offset with a linear slope to capture long term trends in the RV signal. The linear offset improved the fit by reducing structure in the residuals for , while the fits to and preferred the constant intrumental offset. We predicted the RV signal due to the resolved companion for to test whether it could cause the observed trend, assuming the companion was observed at maximum separation on a circular orbit. We found that the maximum slope caused by a companion of mass M=0.44 M_⊙ at the observed angular separation is 0.001 m s^-1 day^-1, a factor of 10 smaller than the observed slope of ∼0.01 m s^-1 day^-1. We conclude that the outer companion is unlikely to be the source of the RV slope and may indicate the presence of an unresolved stellar or planetary-mass companion. Further monitoring of RV and astrometry for this system is required to determine the source of the RV trend.



We also included the RM effect in our  model (see  for details). The amplitude of the RM effect depends on the ratio of the planet's radius (R_ p) to the star's radius (R_⋆), the projected rotation velocity of the star (vsini), and impact parameter (b), given by <cit.> as:

    A_ RM≈2/3(R_ p/R_⋆)^2 vsini√(1-b^2).

As stars evolve off of the main sequence, they become larger, more luminous, and slower in their rotation. These changes have three primary effects that make RM observations challenging—first, the larger radius causes the transit duration to increase, limiting available observation windows; second, the increased stellar radius also causes the ratio R_ p/R_⋆ to decrease, reducing the RM amplitude; finally, the slowed stellar rotation (vsini_⋆) also reduces the RM amplitude. Additionally, granulation for subgiants can introduce RV jitter with an amplitude of ∼5 m s^-1 on timescales of hours <cit.>. 

By modeling the RM signal jointly with the photometric and RV observations, the constraints on the obliquity are strengthened. The RVs provide additional information about the out-of-transit RV baseline, and the deviation from that baseline informs the spin-orbit angle. The photometric transit gives a constraint on the planet radius, which plays a strong role in the amplitude of the RM signal. In turn, the amplitude of the RM signal sheds additional light on the planet radius. Because we have an independent constraint on the planet radius, the projected stellar rotation velocity, vsini_⋆, is also well-constrained by the amplitude of the RM signal. 



All priors and inferred values for our model parameters can be found in Table <ref>. For each planet, we additionally derive planet mass, M_p, planet radius, R_p, and eccentricity, e from the outputs of our model. Figures <ref>, <ref>, and <ref> show the transit, RV, and RM fits, respectively. For , a significant long-term RV trend was detected, shown in Figure <ref>. 



For each target, we sample the posterior distribution using affine-invariant Markov Chain Monte Carlo (MCMC). We initiate 100 walkers with 50,000 sampling steps and 5,000 burn-in steps per walker, ensuring that the chains ran at least 30 times the autocorrelation length and successfully converged. The residual scatter in the RVs is consistent with what we would expect for a subgiant (10-15 m s^-1), and there is no remaining structure in the residual photometry.













 §.§ Doppler Shadow
 

As an independent measure of the obliquity, we analyzed the spectral timeseries for each of our systems, and identified a Doppler shadow signal in the HIRES observations for and (Figures <ref> and <ref>, respectively). We selected lines in the non-Iodine portion of the HIRES spectra (4000-5000 Å) to search for variations in the line profile during transit. We performed 5-σ outlier clipping and removed the continuum and blaze function with a polynomial fit. Each spectrum was then cross-correlated with the SpecMatch template spectrum and the globabally-averaged line profile was subtracted off to isolate line profile variations. Our approach follows <cit.>. Analysis of the KPF data for did not result in the Doppler shadow detection, due to low signal-to-noise ratio of the observations, and the fact that the line profile is dominated by turbulence rather than rotational modulation.







§ RESULTS
 



 §.§ TOI-1181
 

is an intermediate-mass (M_p= M_J) hot Jupiter on a circular orbit. The planet radius is typical for a hot Jupiter of this mass (R_p= R_J). At a T_ eff of 6054 K, its subgiant host star has cooled after its main sequence lifetime and is likely to have developed a non-negligible surface convective zone. The RM effect of has the highest amplitude in our sample, due in part to the high projected rotation velocity of its host star (vsini_⋆ = 10.6±1.0 km s^-1), providing the most precise measurement of the spin-orbit angle. The posterior from our model gives the constraint on the sky-projected obliquity of TOI-1181 of λ = ^∘. For , we measure an obliquity from the Doppler shadow analysis of 7.5±17.2^∘, which is consistent with the measurement from the RM analysis. 



 §.§ TOI-6029
 

is the most massive planet in our sample, with M_p= M_J, as well as the smallest (R_p= R_J), indicating a high density relative to other hot Jupiters. Its host star is the hottest, most massive, and most luminous in the sample. TOI-6029 sits near the Kraft break at T_ eff = 6223 K, indicating that the system has had a substantial surface convective envelope (≳10% of stellar radius) for only a relatively brief window of its lifetime. The out-of-transit residuals to our RV model have larger scatter than those of the other two systems, and we searched for additional non-transiting planets. We performed a Lomb-Scargle periodogram analysis on the residuals as well as attempting a two-planet fit to the observations, neither of which returned conclusive evidence of additional planetary companions. The lack of evidence for additional non-transiting planets indicates that the residual scatter is likely caused by stellar jitter. The spin-orbit alignment of the TOI-6029 system is similar to that measured for TOI-1181, with λ = ^∘. The Doppler shadow measurement for provides an obliquity constraint of 22.3±8.6^∘, which is consistent with the degree of misalignment provided by the RM measurement. We note that the sign of the obliquity from the different methods is flipped for due to the degeneracy of the orientation of the transit chord.



 §.§ TOI-4379
 

is similar to Jupiter in mass (M_p= M_J) but has a larger radius (R_p= R_J). Its host star has the lowest projected stellar surface rotation velocity in our sample (vsini_⋆ = 5.9±1.0 km s^-1). The lack of sufficient RV baseline observations caused the measured sky-projected obliquity of the system to have larger uncertainty than the previously discussed measurements. The spin-orbit angle of the TOI-4379 system is consistent with being fully aligned (λ = ^∘).



 §.§ Population Comparison
 

We now compare the radii, masses, and incident flux of our planet sample to the known population of exoplanets. Literature values were downloaded from the NASA Exoplanet Science Institute (NExScI) Exoplanet Archive[https://exoplanetarchive.ipac.caltech.eduexoplanetarchive.ipac.caltech.edu] on January 3, 2024. We limited the sample to exoplanets with both measured radii and masses, and removed planets for which the fractional uncertainty on either mass or radius exceeded 20%. Figure <ref> shows the well established relationship between incident flux, planet radius, and planet mass. We estimate incident flux with the relation from <cit.> given by

    F_ p/F_⊕ = (R_⋆/R_⊕)^2 ( T_ eff,⋆/ T_ eff,⊙)^4 (a/ AU)^-2.

, , and show inflation that is consistent with known planets of similar mass. The planets presented in this work constitute a small representative sample of hot Jupiters. We refer to <cit.> for a population-level comparison of masses and radii for planets orbiting main-sequence and subgiant stars.


















§ TIDAL REALIGNMENT
 



 §.§ Comparison to Main Sequence Obliquities
 

Figure <ref> shows a Hertzsprung-Russell (H-R) diagram of host stars for systems with measured sky-projected obliquities. To populate this figure, we downloaded obliquity measurements from the  catalog <cit.> and included additional measurements reported in <cit.>. We then performed cuts to select hot Jupiters systems (M_P>0.3 M_J, P_ orb < 10 days, a/R_⋆≤8) and removed obliquity measurements with uncertainties >30^∘.



To estimate the evolution of our host stars, we fit evolutionary tracks from  Isochrones & Stellar Tracks (; ). We adopt the  mass and interpolate between values for metallicity in order to identify isochrone models that pass through our host stars near our stellar age estimates. The resulting tracks are shown as dotted lines in Figure <ref>. The evolutionary tracks show that these stars have indeed dropped below 6250 K during their post-main sequence evolution. In Figure <ref> we highlight two additional hot Jupiter systems for which the host star is likely somewhat evolved: WASP-71 b at 6180 K <cit.> and HAT-P-7 b at 6310 K <cit.>.

Near the estimated main-sequence positions of our targets, hot Jupiter hosts are observed with a wide range of obliquities. We test the likelihood that our three aligned systems are randomly drawn from the previously-aligned population in the observed distribution using a bootstrapping technique: we select a range of effective temperature between 6300 K and 6700 K which encompasses the zero-age main sequence (ZAMS) positions of our host stars. Next, we perform 100,000 random draws of three measured obliquities and calculate the fraction of draws in which all three systems were aligned (<20^∘), and find that we retrieve the observed measurements ∼10% of the time. If we include the literature measurement of WASP-71 b, the fraction drops to ∼4%. This indicates that it is unlikely for our measured obliquities to be randomly sampled, however a larger sample would improve the confidence of this constraint.

Using a sample of subgiants which have decreased in temperature and crossed the Kraft break, we can test whether systems that were part of a frequently misaligned population become realigned after developing a deep surface convective envelope. Main sequence stars hotter than the Kraft break—similar to the H-R positions of our sample at the ZAMS—show a wide range of obliquities. If spin-orbit misalignment persists over the full lifetime of planetary systems, hot Jupiters orbiting stars which are hotter than the Kraft break should still be misaligned. In contrast, if the presence of a convective envelope causes spin-orbit realignment, the population of stars which gained deep convective envelopes after the main sequence should host hot Jupiters which are aligned or in the process of being aligned.







 §.§ Evolution of the Convective Envelope
 

Figure <ref> qualitatively suggests that misaligned hot Jupiters can be realigned as their host stars cross the Kraft break on the subgiant branch, implying efficient obliquity damping due to a convective envelope. To quantify whether obliquity damping can indeed act over subgiant evolution timescales, we examine the evolution of the surface convective envelope. We performed stellar modeling with the Modules for Experiments in Stellar Astrophysics code (; ) to trace the emergence of surface convective envelopes and estimate the time each star spent cooler than the Kraft break. Our models used initial elemental abundances from <cit.> and an atmospheric temperature structure following an Eddington T() relation with fixed opacity. We assume no diffusion and no core or envelope overshoot. 



We used the best-fit stellar properties for each star (given in Table <ref>) as initial properties, and evolved our stellar models from the pre-main sequence until the base of the red giant branch. In Figure <ref>, we show the depth of the surface convection zone boundary in units of fractional stellar radius R_ CZ/R_⋆ (where 0 is the center and 1 is the surface) as a function of stellar age.





Near the ZAMS, the convective boundary is near R_ CZ/R_⋆=1, indicating a thin surface convection zone. As the stars evolve into subgiants, the convective boundary moves deeper into the stellar interior. While the mass fraction (M_ CZ/M_⋆) contained in the convective envelope remains small (≲0.01%), the deepening surface convection zone may lead to enhanced tidal damping. We note that at the age when these stars drop below 6250 K, the outer ∼10% of the stellar envelope is convective, but there is no clear feature to indicate a substantial deepening of the convective envelope over a short timescale. However, the observed change in the distribution of obliquities corresponds to this temperature, and we therefore select 6250 K as the threshold for enhanced damping. Realignment may require a minimum threshold of material for efficient damping, or it may correspond to the disappearance of the convective core. Further stellar modeling will be required to examine this threshold with more detail in the future.

If we assume that efficient realignment begins when a star decreases in temperature and drops below the Kraft break, we can roughly estimate the timescale for realignment. Because all systems in our sample are already aligned, we can only estimate an upper limit to this timescale by calculating the time between when the host star crossed the Kraft break and the estimated age at which it is observed today. This results in an upper limit on the realignment timescale of ∼1.5 Gyr for , and ∼0.5 Gyr for and . Using and as references, we can place a global upper limit on the timescale for realignment at ∼ 500 Myr. 



 §.§ Realignment Model
 

Stellar tides have been suggested to effectively damp out the obliquity of hot Jupiter systems, however they also damp orbital periods on similar or shorter timescales <cit.>. This leads us to an important question—how can a system realign without the planet facing runaway inspiral and engulfment? 

To test this, we model the evolution of orbital period and obliquity under the influence of tidal damping. We construct a toy model for orbital evolution, using a single stellar evolution model that is characteristic of stars in our sample, with M_⋆=1.4 M_⊙, [Fe/H]=0.3 dex, and M_p=1.2 M_J. For simplicity, we assume a circular orbit for the planet. We use our toy model to explore the damping efficiencies required to reproduce a post-main sequence distribution similar to what we observe. 

We assume exponential decay prescribed by ȧ = - a / _a for the semi-major axis, a, and λ̇ = - sin(λ) / _λ for the obliquity, λ.








The coefficients _a and _λ are the damping timescales, defined by <cit.> as

    _a ≈ 1.28 ( Q'_⋆/10^7) (M_⋆/10^3 M_ p) (ρ̅_⋆/ρ̅_⊙)^5/3(P_ orb/1 d)^13/3 Gyr

and 

    _λ≈ 1.13 _a (κ/0.1) (M_⋆/10^3 M_ p) (ρ̅_⊙/ρ̅_⋆)^2/3(10 d/P_ s) (1 d/P_ orb)^1/3

where Q'_⋆ is the tidal quality factor of the star, ρ̅_⋆ is the mean stellar density, and κ is related to the stellar moment of inertia I_⋆ by I_⋆ = κ M R^2.

The calculation of _λ in Equation <ref> requires knowledge of the stellar rotation period. We therefore produced our stellar models with a modified version of which calculates and stores parameters necessary to trace the rotational evolution of the star (see  for details). We assume standard Skumanich-like spindown over the main sequence lifetime. This is a reasonable assumption for stars in this mass range, as they are not expected to undergo significant magnetic braking <cit.>. 

The choice of tidal quality factor Q'_⋆ has a significant effect on the dynamical evolution. However, values for Q'_⋆ range over multiple orders of magnitude for subgiants <cit.>. In Figure <ref>, we show the impact of this choice on our dynamical models. If Q'_⋆ is assumed to be the same for the damping of P and λ, the planet enters a phase of runaway inspiral and is engulfed before it can realign (panel a). If we instead define the damping timescales _a and _λ with different effective tidal quality factors, it is possible to produce a model in which the planet realigns before it can be engulfed. Because the obliquity distribution appears distinct on either side of the Kraft break, we assume that _a and _λ share the same tidal quality factor until the model drops below 6250 K, at which point we invoke a deviation in Q'_⋆, described by the order of magnitude difference δlogQ'_⋆. Figure <ref>b shows that a difference in effective tidal quality factor δlogQ'_⋆≳4 is required to efficiently realign hot Jupiters on orbits between ∼3-8 days without entering a phase of runaway inspiral. 



Figure <ref> shows the sample from Figure <ref> but instead plotting obliquity versus effective temperature, highlighting the correlation between alignment and stellar effective temperature. To examine the change in λ as the host star evolves, we take the model in Figure <ref> b and plot it in λ versus T_ eff space. We observe that the toy model indeed reproduces an evolutionary track in which a deepening convective envelope during subgiant evolution causes obliquity damping.



The toy model shown in Figures <ref> and <ref> make several assumptions. First, we assume that the tidal quality factor Q'_⋆ does not evolve smoothly in time, and instead drops precipitously in the case of λ-damping after the star crosses the Kraft break. It is unlikely that this fully captures the behavior seen in nature, as Q'_⋆ depends on the interior stellar structure, which evolves smoothly over main sequence lifetimes. However, main sequence internal structure changes are small, and we choose to fix the values of Q'_⋆ on the main sequence, as well as after the threshold is crossed. We also assume that the value of Q'_⋆ applied to the damping of period is held fixed for the entire evolution of the star. As the stellar structure evolves after the star leaves the main sequence, its Q'_⋆ should change, though the relationship between Q'_⋆ and stellar radius and density is uncertain. We again choose the simple case of holding the value fixed, though future studies of how Q'_⋆ relates to stellar interior evolution may allow us to improve this model in the future. There are promising pathways to better characterize Q'_⋆ through eclipsing binary system evolution <cit.> and asteroseismology <cit.>.



We now consider what physical mechanisms could lead to a large difference in the efficiencies of the orbital period and obliquity damping timescales. <cit.> invoke sustained core-envelope decoupling as a potential source of enhanced obliquity damping. In this scenario, the planet is able to exert a torque on the outer envelope of the star without rapidly losing orbital angular momentum. They note, however, that this explanation is challenged by evidence that the core and envelope of main sequence stars appear to be well-coupled <cit.>. It is unclear whether this expectation applies to subgiant stars, and the theory of sustained core-envelope decoupling could be tested with asteroseismic measurements of radial differential rotation or core-envelope misalignments in evolved stars <cit.>.

Another explanation, posed by <cit.>, suggests that in the presence of a convective envelope, a hot Jupiter on a misaligned orbit can excite inertial waves which dissipate angular momentum as they propagate through the stellar interior. <cit.> examined tidal dissipation in stars with convective envelopes in the context of hydrodynamical theory, and found that these inertial waves can result in the value of Q'_⋆ being reduced by up to 4 orders of magnitude. Because these inertial waves are static in the inertial frame, they do not contribute to orbital decay <cit.>, and only impact the efficiency of obliquity damping. This suggested order of magnitude difference is consistent with our toy model in Figures <ref> and <ref> following the prescription for tidal damping laid out in <cit.>.

A recent study by <cit.> showed that hot Jupiter orbits can resonantly lock to gravity mode pulsations in the radiative interior of cool stars and efficiently damp the system's obliquity (similar to semi-major axis damping described in ). This mechanism is able to damp the stellar obliquity without causing the planet to inspiral into its host star. The interior structure of our subgiant hosts is amenable to gravity mode pulsations, providing another possible mechanism for the observed obliquity damping.





 §.§ Predicting Orbital Evolution of Main Sequence Systems
 

To investigate period and obliquity damping over a range of parameters, we calculate our toy model over a grid of main sequence orbital period, P_ ZAMS, planet mass, M_ p, and δlogQ'_⋆. We consider a system “engulfed” if the orbital period is reduced to zero before the star reaches the base of the red giant branch, “realigned” if the obliquity drops below 10^∘ and the orbital period is greater than zero for the full track, and “not aligned” if the planet is not engulfed but retains an obliquity greater than 10^∘. This threshold corresponds to the typical uncertainty on a sky-projected obliquity measurement (see Figure <ref>).

Figure <ref> shows the parameter space where realignment is expected for a range of values for δlogQ'_⋆ as a function of main sequence orbital period, P_ ZAMS, and planet mass, M_ p. The regions inside the contours are expected to realign without being engulfed before the star reaches the base of the red giant branch. As the orbital period increases, a larger M_ p is required to efficiently damp the obliquity. However, if the planet mass is too high, the damping of orbital period will be efficient enough to catalyze runaway inspiral. Figure <ref> can be used as a rough guide to predict whether a hot Jupiter orbiting a hot main sequence star will realign after the main sequence.







§ CONCLUSIONS


The obliquities of hot Jupiter systems have been observed to show a strong dependence on the effective temperature of the host star, with cool stars (<6250 K) hosting mostly aligned planets and hot stars (≥ 6250 K) hosting mostly misaligned planets. Theories have suggested that a surface convective envelope can damp the obliquity of a hot Jupiter's orbit.

To test this theory, we have measured the sky-projected spin-orbit angles for three subgiants hosting hot Jupiters. Each of the host stars in our sample crossed the Kraft break and gained deep surface convective envelopes during their post-main sequence evolution. Our evolved hot Jupiter systems have low sky-projected obliquities, indicating efficient realignment as stars develop convective envelopes. 

We summarize our main conclusions as follows:

    
  * We confirm two newly identified hot Jupiters orbiting subgiant stars: (R_ p = R_ J, M_ p = M_ J, P_ orb = days), and (R_ p = R_ J, M_ p = M_ J, P_ orb = days).
    
  * We measured the RM effect for three hot Jupiters orbiting subgiants that have recently developed convective envelopes. All three are aligned (|λ| < 15^∘), which is consistent with the picture that convective envelopes can efficiently damp obliquities. We place an upper limit on the timescale for tidal realignment of ∼500 Myr.
    
  * A simple model incorporating tidal realignment to explain these observations requires the effective tidal quality factor for obliquity damping to be at least four orders of magnitude lower than that for orbital decay (δlog Q'_⋆≳ 4). This is consistent with angular momentum dissipation through the driving and damping of inertial waves in the stellar convective envelope or coupling to gravity modes in the radiative interior.



















Further work is required to place tighter constraints on the timescales for obliquity damping. A larger sample of obliquity measurements for planets orbiting subgiant stars similar to those reported here would provide additional benchmarks for the dynamical evolution, and potentially distinguish between mechanisms for enhanced λ-damping efficiency. One approach to better understand the relationship between λ and T_ eff would be to measure the stellar obliquity for massive subgiants which have not yet crossed the Kraft break. By observing systems at a variety of evolutionary stages, we will better constrain the efficiency of obliquity damping to test high-eccentricity migration as the mechanism for hot Jupiter formation.



§ ACKNOWLEDGEMENTS


N.S. acknowledges support by the National Science Foundation Graduate Research Fellowship Program under Grant Numbers 1842402 & 2236415. A.C. & D.H. acknowledge support from the National Aeronautics and Space Administration (80NSSC21K0652). D.H. acknowledges support from the Alfred P. Sloan Foundation and the Australian Research Council (FT200100871). M.R. is supported by Heising-Simons grant #2023-4478. K.A.C. and C.N.W. acknowledge support from the mission via subaward s3449 from MIT. T.D. acknowledges support by the McDonnell Center for the Space Sciences at Washington University in St. Louis. J.M.A.M. is supported by the National Science Foundation (NSF) Graduate Research Fellowship Program (GRFP) under Grant No. DGE-1842400. 



This paper made use of data collected by the mission and are publicly available from the Mikulski Archive for Space Telescopes (MAST) operated by the Space Telescope Science Institute (STScI). Funding for the mission is provided by NASA's Science Mission Directorate. We acknowledge the use of public data from pipelines at the Science Office and at the Science Processing Operations Center. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products. Funding for the mission is provided by NASA's Science Mission Directorate. 

Some of the data presented herein were obtained at Keck Observatory, which is a private 501(c)3 non-profit organization operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. 

The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the Native Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.



This work makes use of observations from the LCOGT network. Part of the LCOGT telescope time was granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is funded by NSF.

Some of the observations in this paper made use of the High-Resolution Imaging instrument Zorro and were obtained under Gemini LLP Proposal Number: GN/S-2021A-LP-105. Zorro was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Zorro was mounted on the Gemini South telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigación y Desarrollo (Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina), Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea).



This research has made use of the Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5) website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program.

 <cit.>,  <cit.>,  <cit.>

tange2018

aasjournal





§ GROUND-BASED IMAGING FIGURES
 

Figure <ref> shows the high-contrast image of obtained with the Keck/NIRC2 near-infrared imager (for details, see <ref>). The corrected and derotated image, taken in the Kp bandpass, shows a companion with a separation of 763±20 milliarcseconds at a position angle of 110.2±0.5^∘.

Figure <ref> shows photometric time-series observations of the transits of and . These data were obtained with the ground-based LCOGT and PEST telescopes (for details, see <ref>), and they display transit depths and ephemerides consistent with our predictions. 



































§ RADIAL VELOCITY MEASUREMENTS
 

Table <ref> lists RV measurements obtained with Keck/HIRES and Keck/KPF for each of our targets. A machine-readable version of this table containing all observations is available.


l l l l l
    Target     Instrument     Time (BJD)     RV (m/s)     RV Uncertainty (m/s) 
 
    TOI-1181     Keck/HIRES     2459000.040402     -67.993     3.735 

TOI-1181     Keck/HIRES     2459003.045446     14.927     3.907 

TOI-1181     Keck/HIRES     2459004.050001     72.699     3.539 

TOI-1181     Keck/HIRES     2459026.093511     -50.794     4.016 

TOI-1181     Keck/HIRES     2459029.024667     118.585     3.540 



















































































⋮                

TOI-6029     Keck/HIRES     2459423.093788     84.436     6.597 

TOI-6029     Keck/HIRES     2459442.102513     -136.860     6.772 

TOI-6029     Keck/HIRES     2459447.07665     -26.309     7.122 

TOI-6029     Keck/HIRES     2459449.108516     -30.814     6.492 

TOI-6029     Keck/HIRES     2459457.094269     111.598     6.563 










































⋮                

TOI-4379     Keck/HIRES     2459478.753368     -38.193     8.845 

TOI-4379     Keck/HIRES     2459482.716898     104.904     5.204 

TOI-4379     Keck/HIRES     2459633.119397     63.349     4.963 

TOI-4379     Keck/HIRES     2459694.041773     79.915     5.086 

TOI-4379     Keck/HIRES     2459710.004739     24.236     4.605 

⋮                








TOI-4379     Keck/KPF     2460101.830527187     68.210     15.086 

TOI-4379     Keck/KPF     2460101.837867083     49.998     9.299 

TOI-4379     Keck/KPF     2460101.844402326     37.009     7.578 

TOI-4379     Keck/KPF     2460101.85159022     40.815     9.384 

TOI-4379     Keck/KPF     2460101.858644097     48.221     7.549 


























⋮                

Radial velocity measurements for TOI-1181, TOI-6029, and TOI-4379 used in our analysis. A full machine-readable table can be accessed online.
    






 Paper Title: Effect of area divergence and frequency on damping of slow magnetoacoustic waves propagating along umbral fan loops.



firstpage–lastpage
Of Mice and Machetes
    Christopher Henney-Turner and Philip Welch
    August 5, 2024
==============================================




§ ABSTRACT 
Waves play an important role in the heating of solar atmosphere, however, observations of wave propagation and damping from the solar photosphere to corona through chromosphere and transition region are very rare. Recent observations have shown propagation of 3-min slow magnetoacoustic waves (SMAWs) along fan loops from the solar photosphere to corona. In this work, we investigate the role of area divergence and frequencies on the damping of SMAWs propagating from the photosphere to the corona along several fan loops rooted in the sunspot umbra. We study the Fourier power spectra of oscillations along fan loops at each atmospheric height which showed significant enhancements in 1–2 min, 2.3–3.6 min and 4.2–6 min period bands. Amplitude of intensity oscillations in different period bands and heights are extracted after normalizing the filtered light curves with low-frequency background. We find damping of SMAW energy flux propagating along the fan loop 6 with damping lengths ≈170 km and ≈208 km for 1.5-min and 3-min period bands. We also show the decay of total wave energy content with height after incorporating area divergence effect, and present actual damping of SMAWs from photosphere to corona. Actual damping lengths in this case increases to ≈172 km and ≈303 km for 1.5-min and 3-min period bands. All the fan loops show such increase in actual damping lengths, and thus highlight the importance of area divergence effect. Results also show some frequency-dependent damping of SMAW energy fluxes with height where high-frequency waves are damped faster than low-frequency waves. 



waves – Sun: chromosphere – Sun: corona – sunspots – Sun: transition region –  Sun: UV radiation









§ INTRODUCTION

 

Chromospheric and coronal energy losses in the solar active regions are approximately 2×10^7 and 10^7 erg cm^-2 s^-1, respectively <cit.>. To maintain the hot atmosphere of the Sun, magnetohydrodynamic (MHD) waves is one of the proposed mechanisms to transfer energy into the upper solar atmosphere <cit.>. These waves are excited by the interaction of magnetic field lines and convection motions at the photosphere. The available energy flux provided by the photosphere is about 1.2×10^9 erg cm^-2 s^-1 for a mean field strength of 500 G <cit.>. The transfer and conversion of this mechanical energy flux into heat in the upper atmosphere <cit.> is still not completely understood and is commonly called the coronal heating problem. Henceforth, the photosphere can provide sufficient energy to replenish the atmospheric losses and waves propagating upward may carry these energy fluxes with them into the upper solar atmosphere. Nature of waves i.e., either longitudinal or transverse, frequency, magnetic topology, wave amplitude, propagation speed and other parameters such as plasma density, field strength, etc. may decide upon the amount of energy fluxes carried by them. 

One of the outward propagating waves is slow magnetoacoustic waves (SMAWs) which are easily detected in the imaging observations due to their compressive nature <cit.>. <cit.> provide comprehensive reviews on the propagation of slow magnetoacoustic waves along open structures. Slow magnetoacoustic waves are subjected to damping while propagating in the solar corona <cit.>. Propagating waves develop into shocks due to amplitude steepening at heights where a sharp change in density or temperature occurs (such as chromosphere and transition region) which can be observed as saw-tooth patterns in the velocity oscillations <cit.>. <cit.> found that slow magnetoacoustic waves steepen non-linearly while propagating into the corona, leading to wave-enhanced dissipation. However, slow waves shock dissipation within the sunspot umbra do not contribute to the umbral atmospheric heating <cit.>. Although damping of slow magnetoacoustic waves in the solar atmosphere is very well studied theoretically, calculating damping flux along the solar atmosphere observationally through wave propagation is very complex. This is due to the fact that the dynamics of the photosphere, chromosphere and corona are very different. Furthermore, in the lower atmosphere, effects due to opacity and area divergence are difficult to evaluate <cit.>.

Damping of these waves are studied in various atmospheric structures including magnetic pores <cit.>, umbra <cit.>, coronal holes <cit.>, coronal loops <cit.> etc. These waves are also important for seismological studies and provide estimates on important plasma parameters that are otherwise difficult to calculate in the solar atmosphere such as adiabatic index, compressive viscosity, thermal conductivity, etc. <cit.>. 

The power spectra of intensity and velocity oscillations in the umbral atmosphere reveal the presence of enhanced powers in various period bands <cit.>. In sunspots at the photosphere, the power of 5-min oscillations dominates over 3-min oscillations <cit.>. At heights above the photosphere, 3-min oscillations dominate due to the acoustic cut-off of 5-min oscillations <cit.>. The presence of 3-min oscillations at specific locations in the umbral photosphere was reported by <cit.>, and their propagation from photosphere to corona along fan loops was studied in detail by <cit.>. Moreover, low period oscillations (≈ 1-min) at some atmospheric heights in the umbra are also reported <cit.>. <cit.> reported low period oscillations (≈ 1.5 min) at all the heights above the umbra from imaging observations. <cit.> reported low-period oscillations in the corona at the location of light-bridge in sunspot. While propagating, these waves show evidence of damping. Damping of slow magnetoacoustic waves of various periods along the integrated umbral atmosphere from photosphere to transition region was studied by <cit.> and along open structures in corona were studied by <cit.> etc. All these results indicate some frequency-dependent damping of slow magnetoacoustic waves. Moreover, recently <cit.> reported different damping lengths for 3-min slow waves propagating along the same fan loop observed from different instruments with non-parallel lines of sight.




To balance the total horizontal pressure, the cross-sectional area of the magnetic flux tube increases with height in the solar atmosphere. This leads to the damping of slow magnetoacoustic waves due to area divergence and is related to the conservation of wave energy within the flux tube. Area divergence causes the wave amplitude to decay as 1/√((A(s))), where A(s) is the cross-sectional area of coronal loop with height <cit.>. Area divergence does not contribute to heating as the wave energy redistributes itself to conserve wave energy within the flux tube which is purely a geometric effect. These expanding fields also enhance the transmission of waves to the corona by decreasing the reflection from chromosphere and transition region <cit.>. Therefore, it is important to take into account the effect of area divergence for an accurate estimate of wave damping. 
 
Till now most of the work on slow magnetoacoustic wave energy propagation in the umbral atmosphere is carried out either over the whole integrated umbra or at some unidentified location within the umbra <cit.>. In this work, our focus is to study the damping of slow magnetoacoustic waves propagating along the already identified fan loops rooted in the sunspot umbra <cit.>. Here, we are exploring the effect of area divergence and frequency on the damping of these waves while propagating from photosphere to corona along fan loops. We describe observations in Section <ref>, data analysis and results in Section <ref>, and finally summarise and discuss our results in Section <ref>.



§ OBSERVATIONS



To investigate the damping of slow magnetoacoustic waves in the solar atmosphere, we utilize multi-wavelength observations of fan loops within the active region NOAA AR 12553 observed on June 16, 2016 by Atmospheric Imaging Assembly <cit.>, Helioseismic and Magnetic Imager <cit.> both onboard Solar Dynamics Observatory <cit.>, and Interface Region Imaging Spectrograph <cit.>. We obtained 4 hours of simultaneous data starting from 07:19:13 UT. The same dataset was previously utilized by <cit.> to trace the source region of 3-min slow magnetoacoustic waves propagating along the fan loops rooted in the umbral region.

AIA/SDO provides full-disc images of the Sun in seven EUV and three UV-visible passbands covering the upper and lower solar atmospheres, respectively. HMI/SDO provides full-disc images of the photospheric Sun in the intensity continuum, Dopplergram and magnetogram derived from Fe I 6173 Å spectral line. All the SDO images were calibrated, co-aligned, and rescaled to a common 0.6 pixel^-1 resolution and 12-s temporal resolution using the robust SDO library of Rob Rutten[https://robrutten.nl/rridl/00-README/sdo-manual.html] which incorporates spline interpolation to rescale the time series. Images obtained from IRIS-SJI 2796 Å passband have an exposure time of 2 s with an effective cadence of 6.88 s and 0.166 pixel^-1 resolution. IRIS has a field of view (FOV) of 60×65 which is centered around the sunspot umbra. Details on data preparation are provided in <cit.>. Fig. <ref> shows the images of the sunspot and fan loops as obtained from AIA, IRIS and HMI passbands as labeled. 

The selected sunspot is slightly off the disc center (heliocentric coordinates X ≈ -25, Y ≈ -125), the angle between local vertical and line-of-sight (LOS) is ≈ 7.7 ^o, which leads to μ = cos θ≈ 0.99. Therefore, any projection effect with respect to the disc center will be negligible. 



 §.§ IRIS spectroscopic observations



IRIS also observed in spectroscopic mode and recorded the time series in 2-step raster mode. Both slit width and pixel size along the slit direction are 0.166/pixel. Each raster step has an exposure time of 2 s and slit length of 399 pixels. We have extracted the near-ultraviolet spectra of Mg II at 2796 Å (formation temperature of 10,000 K) from one slit location of the raster to generate time series. We averaged over the spectra for a region marked with a white line (11-pixels) in Fig. <ref> and further binned over two time-frames to improve upon the signal. The extracted region is representative of the quiet umbral atmosphere. After all binning, the effective cadence becomes 13.76 s. We then fitted average spectral profiles with a single Gaussian function and constant background and further extracted the Doppler shifts and peak intensity of the profiles with time (for more details see Appendix <ref>). The final Doppler velocity oscillations are plotted in the right panel of Fig. <ref> with the blue box showing an enlarged view of the oscillatory saw-tooth pattern with time.






§ DATA ANALYSIS AND RESULTS



Fig. <ref> shows the analyzed fan loop structures in AIA 171 Å passband. Overplotted contour represents the umbral boundary of a sunspot at 9000 DN, as obtained from the HMI continuum image. We have identified eight fan loops emanating from the sunspot umbra for our study purpose which are labeled accordingly. In AIA 171 Å image asterisks (*) represent the coronal foot-points of fan loops from where the loops are visible in corona. Locations of all the loops at different atmospheric heights from the transition region to the photosphere are already identified <cit.> and marked with small circles (o) in Fig. <ref>. We estimate the inclination angle of the traced loops from the photosphere to corona to be ≈ 10^o-30^o  as calculated from the spatial offsets between the loop locations identified at the photosphere (HMI continuum) and at the corona (AIA 171 Å). Here, we are presenting results from loop 6 as a representative example and present results from all the loops in Section <ref>.

We carried out analysis for three different cases. Firstly, analysis is carried out for the loop locations identified at different atmospheric heights denoted by small circles (o) and asterisks (*) in Fig. <ref>. These are the locations where the correlation value obtained between light curves from two atmospheric heights is maximum and represents the center of the loop. Secondly, we also carried out analysis over all the pixels that fall within the contour area where the correlation value between two atmospheric heights becomes ≈95% of the maximum correlation value. This area can be considered as the upper limit on the size of the loop i.e., loop cross-section. The area within this correlation threshold decreases as we move into the lower atmosphere (see Appendix <ref> for details). This is as per the expectations from the theory of flux tube expansion with height. Details on technique and methodology are described in <cit.>. Thirdly, analysis is also carried out over the whole integrated umbra as marked by the overplotted contours in Fig. <ref> for comparison with previous results <cit.>.



 §.§ Fourier powers and oscillation amplitudes






In Fig. <ref>, we show fast Fourier transform (FFT) power spectrum obtained from the background subtracted light curves of loop location 6 from different passbands as labeled <cit.>. Background light curves are obtained by taking 16-min running average of original light curves at those atmospheric heights. The FFT power spectrum and confidence levels are obtained using standard IDL routine fft_powerspectrum. Plots clearly show a wide distribution of power peaks in different period ranges. 1–2 min (16.67–8.33 mHz), 2.3–3.6 min (7.25–4.63 mHz) and 4.2–6.0 min (3.97–2.78 mHz) ranges show enhanced powers at different atmospheric heights. Here, we term these three dominant period ranges as 1.5-min, 3-min and 5-min period bands respectively. It should also be noted that these period bands have different period (frequency) widths. These period bands are shaded with green, blue and red colors, respectively. We will be following the same color code to represent different period bands further in the paper. We finally sum over the FFT powers within their respective bands. Fourier power in 3-min period band is present in all the passbands which suggests the propagation of 3-min waves from the photosphere to corona along the fan loops as reported in <cit.>. Enhanced FFT powers in the 1.5-min period band are present in AIA 1700, IRIS 2796, AIA 304 and AIA 171 Å passbands. The lack of Fourier powers in this band at photospheric heights (HMI continuum and Dopplergram) could be either due to their real absence at these heights or due to poor cadence of HMI observations which is 45 s. Enhanced Fourier powers in the 5-min band are present only at the photospheric heights (HMI continuum and Dopplergram) and are absent at higher atmospheric heights which could be due to atmospheric cutoff. 

 Since we have found enhanced powers in the 1.5-min and 3-min period bands in almost all the passbands, and thus at all the atmospheric heights, we will be utilizing these period bands for further analysis of wave damping. We apply bandpass filters over these three period ranges on the original light curves (with 3×3 pixel binning for SDO and 5×5 pixel binning for IRIS). Since the absolute wave amplitudes of filtered light curves in the given period ranges can not be compared directly, these filtered light curves are normalized by background light curves present at that atmospheric heights. These normalized filtered light curves represent relative wave amplitudes with time are shown in Fig. <ref> for 1.5-min, 3-min and 5-min period bands at all atmospheric heights. These curves show clean intensity oscillations in all three period bands. However, due to the presence of several nearby power peaks in the different period bands, we notice oscillations in the form of unclean wave packets. The bottom panel of Fig. <ref> shows the filtered velocity oscillations obtained from HMI Dopplergram which provides the velocity amplitude variations of 1.5-min, 3-min and 5-min period bands with time. From the plots in Figs. <ref> and <ref>, we notice that 5-min oscillation dominates at the photosphere and 3-min oscillation dominates over 1.5-min oscillation at the higher heights where 5-min oscillation is absent. The root mean square (RMS) relative amplitudes are obtained using the formula √(1/n∑(I'/I_o)^2) for each period band, where I_o is background light curves and I' is filtered light curves obtained in the ranges shown by shaded region in Fig. <ref>. The standard errors in the RMS values are obtained using the formula σ/√(N), where σ is the standard deviation and N is the number of data points <cit.>. The obtained RMS amplitudes and standard errors at each height are multiplied with √(2) to determine the relative intensity amplitudes and associated error-bars. In the left panel of Fig. <ref>, we plot relative amplitude of intensity oscillations for different period bands present at various atmospheric heights.  
 Furthermore, we determine the relative intensity amplitude at all the pixels within the loop contours and then average them to determine the relative amplitudes of these waves within the loop cross-sections along the loop. 
 
 Relative intensity amplitude variations with height are determined using different methods as mentioned earlier and are plotted in the left panel of Figs. <ref> and <ref> for single maximum correlated pixel and loop cross-sectional area respectively. Variations in amplitudes with height are similar for both the methods. Relative intensity amplitude variation with height initially shows amplitude steepening due to rapid fall of density. The relative intensity amplitude is maximum at chromospheric height (IRIS 2796 Å) which makes the waves non-linear. This can also be seen in the saw-tooth pattern of velocity oscillations obtained at chromospheric height using Mg II 2796 Å line as shown in the right panel of Fig. <ref>, which eventually develops into shocks <cit.>. High-period (5-min) shows atmospheric cutoff as the amplitude of these waves decreases as they move from photospheric height to chromospheric height. As seen in the FFT power spectrum, enhanced power in 1.5-min period band is absent only at photosphere whereas enhanced power in 5-min period band is present only at photosphere. Therefore, the relative intensity amplitudes of 5-min period at heights above photosphere and 1.5-min period at photosphere are unreliable. Henceforth, all these unreliable data points are connected using dashed lines in Fig. <ref> and further for representation purpose only. 
 
Velocity amplitudes of 1.5-min, 3-min and 5-min oscillations are determined assuming linear approximation in transition region (AIA 304 Å) and corona (AIA 171 Å) using formula

    I'/2I_o = v'/v_s,

where v' is the velocity amplitude and v_s is the speed of sound <cit.>. We determined the amplitude of velocity oscillations at the photosphere using HMI Dopplergram data for 3-min and 5-min to be ≈ 33 m s^-1 and 73 m s^-1, respectively as shown in Fig. <ref>. For chromospheric height, velocity amplitude of 1 km s^-1 was found for 3-min oscillations from Mg II 2796 Å line (see details in Appendix <ref>). Using the velocity amplitudes of photospheric and chromospheric oscillations, we linearly interpolated and extracted the velocity amplitude of 3-min oscillations for temperature minimum region.

These derived velocity amplitudes of 3-min oscillations will be utilized to calculate wave energy flux. Since we have relative intensity amplitude of 1.5-min, 3-min and 5-min intensity oscillations, so to calculate velocity amplitudes of 1.5-min and 5-min oscillations for temperature minimum and chromospheric heights, we scaled their relative intensity amplitudes with relative intensity and velocity amplitudes of 3-min oscillations at respective atmospheric heights. Finally, the obtained velocity amplitude variations with height are shown in the right panel of Figs. <ref> and <ref> for maximum correlated pixel and loop cross-sectional area respectively. Moreover, inclination of loop from photosphere to corona will change the velocity perturbations by inverse factor of cos(10^o-30^o) ≈ 0.98-0.87 (see Section <ref>). This will lead to changes of only about 2-15% and are within the estimated error bars of velocity amplitude.



 §.§ Energy flux of waves
 









To calculate the wave energy flux (F), we are using the WKB approximation, F= ρδ v^2 v_p where ρ, δ v, v_p, are mass density, velocity amplitude, and slow wave propagation speed respectively. However, under a typical coronal condition of low-beta plasma, sound speed (v_s) is a good approximation to the phase speed of slow magnetoacoustic waves <cit.>. Since μ≈ 0.99 for our analyzed sunspot, we can assume v_p equals v_s and also any projection effect on velocity perturbations will also be negligible and within the error bars. Sound speed is calculated using the formula v_s = √(γ R T/μ), where γ= 5/3 is the polytropic index, R =8.314×10^7 erg K^-1 mol^-1 is the gas constant, μ is the mean molecular weight, and T is the temperature <cit.>. Further, we are utilizing density and temperature from the sunspot atmospheric model of <cit.> and assume 20% errors in density and temperature, see details in Appendix <ref>.  

The estimated wave energy fluxes are decreasing from ≈ 10^7 erg cm^-2 s^-1 at the photosphere to ≈ 2× 10^3 erg cm^-2 s^-1 at the low corona for 3-min waves. For 1.5 min waves, energy flux is ≈ 3× 10^5 erg cm^-2 s^-1 at the temperature minimum which decreases to ≈ 400 erg cm^-2 s^-1 at the low corona. Wave energy flux of 1.5-min waves at the photosphere may be unreliable due to the poor cadence of the HMI continuum. For 5-min waves, energy flux at photosphere is ≈ 6× 10^7 erg cm^-2 s^-1 which decays rapidly in the atmosphere. These variations in wave energy fluxes are plotted in Fig. <ref> for loop center locations at different heights and also for average wave energy fluxes within the loop cross-sectional areas at 95% contour level. Plots show consistent decrease in wave energy fluxes with height even though the oscillation amplitudes are increasing with height in the lower atmosphere due to rapid fall in density. Decrease in wave energy fluxes with height indicates the damping of slow magnetoacoustic waves across these atmospheric layers. 

For comparison purpose, we also provide amplitude of oscillations and average wave energy fluxes over the whole integrated umbra with height in Appendix <ref>. In this case also obtained amplitudes and wave energy fluxes at different atmospheric heights are of similar order of magnitude as those obtained for previous cases.  




 §.§ Dependence of wave damping on frequency and area divergence



The wave energy flux at different heights in the solar atmosphere are estimated for different cases as mentioned in Section <ref> and plotted in Fig. <ref>. To determine the dependence of frequency and area divergence on the damping of these waves, we fitted the obtained energy fluxes using exponential decay function. In Fig. <ref>, solid lines represent the best fits with the data following a function F=F_0e^-h/L_d, where F is the wave energy flux with height h (see  Appendix <ref>), L_d is the damping length, and F_0 is appropriate constant. Fits are performed using MPFITFUN routine <cit.>. The exponential decay function appears to be consistent with the data. To improve the chi-square of fit, points in black color cross as shown in Fig. <ref> are excluded from the fit. Damping lengths (L_d) obtained from the maximum correlated pixel are ≈ 171±10 km and 208±07 km whereas those from loop cross-sectional area are 169±09 km and 208±06 km for 1.5-min and 3-min period waves respectively. Damping lengths are also printed in Fig. <ref>. The damping length of 1.5-min period wave is less than 3-min period wave. This indicates some frequency-dependent damping of slow waves where high-frequency waves are damped faster than low-frequency waves. Also, damping lengths obtained from maximum correlated pixels and loop areas are similar. This may indicate that wave energy fluxes are uniformly distributed across the loop cross-sections.

Since loops are expanding with the height as reported in <cit.>, the wave energy is redistributed over the whole loop cross-section to conserve the wave energy within the flux tube. This effect will lead to the decay of wave energy flux with height which is purely a geometric effect without any actual physical damping of waves. To take this geometric effect into account so as to estimate the actual damping of slow magnetoacoustic waves, we multiplied the wave energy flux with the cross-sectional area of the loop at that respective height. The cross-sectional area of the loop at ≈ 95 %  contour level is provided in Appendix <ref>. The obtained variation of total wave energy content at different atmospheric heights for different periods are shown in Fig. <ref>. The plot clearly demonstrates the actual decay of total wave energy content carried by slow magnetoacoustic waves propagating from photosphere to corona along the fan loop. We again fitted the total wave energy curves with exponential decay functions to obtain the actual damping lengths. Obtained actual damping lengths (L_d) in this case are ≈ 172±03 km and 303±10 km for 1.5-min and 3-min period waves respectively. These damping lengths are for actual damping of total wave energy content with height which are larger than the damping lengths obtained from the wave energy fluxes with height. Findings indicate that area divergence does play a role in the damping of slow magnetoacoustic waves. 

We performed the same analysis on the other seven loops identified in Fig. <ref>. Here also all the loops show the presence of 1.5-min, 3-min and 5-min period waves. Similar variations of relative intensity and velocity amplitudes along the solar atmosphere are noted. Damping lengths of wave energy fluxes for 1.5-min period are less than the damping lengths of 3-min period for all the loops. Damping lengths obtained from identified loop locations and loop cross-sectional areas are listed in Table <ref>. Findings confirm almost uniform distribution of wave energy fluxes across the loop cross-sections. However, the longer damping length of the total wave energy content propagating along the different loops is due to area expansion of the loops, and thus indicates area-dependent damping of waves. Table <ref> also shows different damping lengths for 1.5-min and 3-min period bands as found above. Results provide a clear dependence of frequency and loop area divergence on the damping of slow waves. 














§ DISCUSSION AND SUMMARY



In this work, we utilized FFT power spectra of fan loops rooted in sunspot umbra and found waves with different periods. We noticed enhanced powers in three period bands such as 1–-2 min (16.67–8.33 mHz), 2.3–-3.6 min (7.25–4.63 mHz) and 4.2–6.0 min (3.97–2.78 mHz) at different atmospheric heights. 1–2 min period band is present above the photosphere, 2.3–3.6 min period band is present at all the atmospheric heights whereas 4.2–6 min period band is present only at the photosphere. The 1.5-min period band is also reported by <cit.> at different atmospheric heights within the umbra. The absence of 1.5-min period at the photosphere could be either due to poor cadence of HMI data or due to their real absence in our observations. However, it should be noted that <cit.> also did not find any enhancement in 1.5-min period band at the photospheric height despite having better cadence data.  <cit.> detected short period (≈1-min) oscillations at light-bridge locations within sunspot umbra. Additionally, we could not find any significant power enhancement in the 3-min period band in the power spectrum of the integrated umbra at the photosphere, though significant enhancements were observed at heights above the photosphere (see Appendix <ref>). This is similar to the previous reports <cit.>. This could be due to the fact that 3-min enhancements are found only at the specific locations in the umbral photosphere potentially at the sites of umbral dots <cit.> and also at the foot-points of fan loops <cit.>. 

We estimated wave energy fluxes for 3-min and 5-min period bands at the photosphere which are ≈ 10^7 and ≈ 6× 10^7 erg cm^-2 s^-1 respectively whereas that for 1.5-min period at temperature minimum region is ≈ 5× 10^5 erg cm^-2 s^-1 which are similar to the previous reports. <cit.> estimated the slow magnetoacoustic waves energy ﬂuxes for 6-10 mHz (1.7-2.7 min) band at the photosphere as 2×10^7 erg cm^-2 s^-1 and <cit.> reported energy ﬂuxes to be ≈ 10^7 erg cm^-2 s^-1 for 3-min period waves. For 5-min period band, <cit.> reported slow wave energy flux to be ≈ 3 × 10^7 erg cm^-2 s^-1 at an atmospheric height of 100 km inside the pore. As waves propagate upward these energy fluxes decay with height and provide evidence of wave damping as demonstrated in Fig. <ref>. The sharp drop in energy fluxes from photosphere to chromosphere can be due to leaky wave characteristic <cit.>, strong shocks dissipation and radiative cooling <cit.>. These waves propagating along loops can also get damped due to various non-ideal MHD effects depending on the different physical conditions of the loop <cit.>. There are several damping mechanisms proposed for slow magnetoacoustic waves such as compressive viscosity <cit.>, thermal conduction <cit.>, area divergence <cit.>, mode coupling <cit.>, shocks <cit.> etc. 

In this work, we obtained the damping length of ≈208 km for 3-min slow wave energy flux propagating along fan loops from photosphere to corona and ≈ 170 km for 1.5-min period waves propagating from temperature minimum region to corona. <cit.> reported damping lengths of ≈ 149 km and 119 km from decay of relative intensity amplitude of slow wave propagation in integrated umbra from chromosphere to transition region for periods ≈ 2.5-min and 1.3-min respectively. Both the results indicate some frequency-dependent damping of slow magnetoacoustic waves in the lower atmosphere. Frequency-dependent damping lengths are also reported in the corona along open structures <cit.>. We also noted that wave energy flux at the center of the loop and average wave energy flux obtained within the loop cross-sections (at ≈ 95% contour level) are of similar order (see Fig. <ref>). Fluxes are similar because velocity amplitudes at all the pixels within the loop cross-sections are almost similar.
This may indicate that wave energy fluxes are uniformly distributed across the fan loop cross-sections and also a similar damping mechanism is operating throughout the umbra. Therefore, damping of slow magnetoacoustic waves propagating in different regions of the solar atmosphere shows some kind of frequency-dependent damping even though different damping mechanisms might be operating at different heights.  

One of the dominant damping mechanisms of slow magnetoacoustic waves is area divergence which is very well studied in the solar corona <cit.>. However, its influence in the lower atmosphere is still unexplored. In this work, we utilized the cross-sectional area of the fan loops obtained at different heights in the lower atmosphere <cit.>. We explored the effect of area expansion on wave damping, by multiplying the loop cross-sectional area with the energy flux of slow waves at that atmospheric height. The obtained total slow wave energy propagating along fan loop in unit time at different heights is plotted in Fig. <ref> and represents actual damping of slow waves. 
Here damping lengths are 172±03 km and 303±10 km for 1.5-min and 3-min periods respectively. After incorporating the effect of area divergence, the actual damping lengths obtained are larger than those found from the damping of wave energy fluxes. This indicates that the waves are losing energy faster (i.e., damping length is short) when the area divergence effects are not considered. This is due to the fact that wave energy is redistributed across the loop cross-sections due to area divergence. This clearly explains the importance of area divergence in the damping of waves. 

Moreover, it should be noted that calculating the actual damping of wave energy flux along the solar atmosphere is a very complex problem as the dynamics of photosphere, chromosphere and corona are very different. Furthermore, the umbral photosphere itself is a complex region due to the presence of plasma-β= 1 layer which affects the propagation of wave modes <cit.>. Additionally while calculating the energy fluxes, we are assuming waves to be linear. Calculated energy fluxes for 3-min waves at photosphere, chromosphere and corona are ≈ 10^7, ≈ 10^4 and ≈ 10^3 erg cm^-2 s^-1. However, how much of this wave energy flux is damped, how much is getting reflected, and how much is propagating upwards still needs detailed investigation <cit.>. Detailed MHD simulations with non-local thermodynamic equilibrium modeling will be required to understand the wave dynamical processes at various atmospheric heights <cit.>.

In summary, we studied the damping of slow magnetoacoustic waves propagating along fan loops in the umbral atmosphere and showed oscillations in the period bands 1.5-min, 3-min and 5-min. We estimated slow magnetoacoustic wave energy fluxes propagating along the fan loops at different heights and provided the evidence of wave damping with damping lengths of ≈ 170 km and ≈ 208 km for 1.5-min and 3-min periods respectively for loop 6. We showed the decay of total wave energy content within the loop cross-section with height and provided evidence of actual damping of slow magnetoacoustic waves from the photosphere to corona with damping lengths of  ≈ 172 km and ≈ 303 km for 1.5-min and 3-min periods respectively for the same loop. Results showed some frequency-dependent damping of slow magnetoacoustic wave energy flux with height where high-frequency waves are damped faster than low-frequency waves. We have also demonstrated the importance of the role of area divergence in the damping of slow magnetoacoustic waves. 
 


§ ACKNOWLEDGMENTS

We thank the referee for helpful comments and suggestions that improved the quality of presentation. The research work at the Physical Research Laboratory (PRL) is funded by the Department of Space, Government of India. AR thanks PRL for her PhD research fellowship. AIA and HMI data are courtesy of SDO (NASA). IRIS is a NASA small explorer mission developed and operated by LMSAL with mission operations executed at NASA Ames Research Center and major contributions to downlink communications funded by the Norwegian Space Center (NSC, Norway) through an ESA PRODEX contract.





§ DATA AVAILABILITY


The observational data utilized in this study from AIA and HMI on-board SDO are available at   <http://jsoc.stanford.edu/ajax/lookdata.html> and data from IRIS mission are available at <https://www.lmsal.com/hek/hcr?cmd=view-recent-events&instrument=iris>.












mnras
 







§ APPENDIX






 §.§ Area of loop cross-sections



  Variation of cross-sectional areas of loop 6 at different atmospheric heights (at ≈95% of the maximum correlation values) is shown in Fig. <ref>. Here, we see flux tube expansion with height. Details on the technique and methodology for deriving loop cross-sections are described in <cit.>.
  Since we are tracing the same loop, we can assume the filling factor to be one from photosphere to corona. However, at the photosphere, loop cross-section is limited to 1-pixel due to limitations on resolution. We have assumed 1- and 2-pixel errors in the width of the loops calculated using SDO and IRIS passbands respectively. 1-pixel of SDO and 2-pixel of IRIS passbands correspond to 435 km and 241 km on the Sun respectively.
 
  



 §.§ Average umbral Mg II 2796 Å line profile


We have extracted the average near-ultraviolet spectra of Mg II 2796 Å in umbral region marked with a white line (11-pixels) in Fig. <ref>. 
We then fitted average spectral profiles with the Gaussian function and constant background as shown in Fig. <ref>. The fitted parameters were used to extract Doppler shifts and thus Doppler velocity oscillations as shown in the right panel of Fig. <ref>. 

The fitted parameters were also used to extract peak intensity with time and thus relative intensity amplitude is obtained from spectroscopic data. We then determined the relative intensity amplitude near the slit from imaging data (assuming similar intensity and velocity oscillations) and found that the intensity amplitude obtained from spectroscopic data is 1.25 times larger than the intensity amplitude obtained from imaging data. Therefore, to obtain velocity amplitudes from imaging data (IV_amp), we are using formula  

    SI_amp/SV_amp = II_amp ×1.25/IV_amp
, where II_amp and SI_amp are relative intensity amplitude obtained from imaging and spectroscopic data respectively, and SV_amp is velocity amplitude obtained from spectroscopic data  for 3-min oscillations. 



 



 §.§ Temperature and density of loop from photosphere to corona



 To determine the formation heights for AIA, IRIS and HMI passbands in the sunspot umbra, we utilize the sunspot model of <cit.>.  Temperature and density variations with respect to the formation height are shown in Fig. <ref>. We associate temperature at the peak of response functions as passbands temperature and obtain the corresponding formation heights. However, all these passbands will also have some cooler and hotter temperature contributions as well <cit.>. Simultaneously at those formation heights, we determine electron (N_e) and hydrogen (N_h) number density separately. The mass density is given by ρ= (N_e+N_h)m_p, where m_p is the proton mass (m_p = 1.67×10^-24 g) <cit.>. The left axis shows the temperature and the right axis shows the total mass density (ρ). Furthermore, we have assumed 20% error in density and temperature in all our calculations <cit.>. These values play a key role in determining the wave energy fluxes with height. 
 
 





 §.§ Results from integrated umbra


Previously, <cit.> and <cit.> calculated the wave amplitudes and energy fluxes in the umbral atmosphere over the entire integrated umbra. For comparison with previous results, we also carried out a similar analysis on the integrated umbra (see HMI continuum image of Fig. <ref>). We determined the average FFT power spectra of the whole umbra at all the atmospheric heights, and obtained the presence of various periods as shown in Fig. <ref>. We performed a similar analysis as described earlier to derive relative intensity amplitude, velocity amplitude and wave energy flux in the umbra for three period bands and plot them in Fig. <ref>. Due to lack of significant enhancement in the FFT power of 3-min period band at photosphere, we are considering it as an unreliable data point. The damping lengths of wave energy flux for 1.5-min and 3-min period bands obtained from temperature minimum to corona are printed at the top right corner of Fig. <ref>. 















 Paper Title: Flare Accelerated Electrons in Kappa-Distribution from X-Ray Spectra with Warm-Target Model.





Yingjie Luo
Yingjie.Luo@glasgow.ac.uk

0000-0002-5431-545X]Yingjie Luo (骆英杰) School
of Physics & Astronomy, University of Glasgow, G12 8QQ, Glasgow, UK


0000-0002-8078-0902]Eduard P. Kontar
School of Physics & Astronomy, University of Glasgow, G12 8QQ,
Glasgow, UK

0000-0002-2651-5120]Debesh Bhattacharjee
School of Physics & Astronomy, University of Glasgow, G12 8QQ,
Glasgow, UK

§ ABSTRACT 
X-ray observations provide essential and valuable insights into the acceleration
and propagation of non-thermal electrons during solar flares. Improved X-ray
spectral analysis requires a deeper understanding of the dynamics of energetic
electrons. Previous studies have demonstrated that the dynamics of accelerated
electrons of a few thermal speeds are more complex. To better describe
the energetic electrons after injection, a model considering energy diffusion
and thermalization effects in flare conditions (warm-target model) has recently
been developed for Hard X-ray spectral analysis. This model has demonstrated how
the low-energy cut-off, which can hardly be constrained in cold-target
modeling, can be determined. However, the power-law form may not be the most
suitable representation of injected electrons. The kappa distribution, which is
proposed as a physical consequence of electron acceleration, has shown
successful application in RHESSI spectral analysis. In this study, we employ the
kappa-form injected electrons in the warm-target model to analyze two M-class
flares, observed by RHESSI and STIX, respectively. The best-fit results show
that the kappa-form energetic electron spectrum generates lower non-thermal
energy when producing a similar photon spectrum in the fit range compared to the
power-law form. We also demonstrated that the fit parameters associated with
kappa-form electron spectrum can be well determined with small fit
uncertainty. Further, the kappa distribution, which covers the entire electron energy range, enables the determination of key electron
properties such as total electron number density and average energy in the flare
site, providing valuable information on electron acceleration processes.







§ INTRODUCTION
 

Solar flares are explosive energy-release events of which the emissions span
over a wide electromagnetic wavelengths ranging from radio to γ-ray. In
order to understand the acceleration mechanism and the transport of energetic
electrons, it is crucial to know where and how the energy is released
<cit.>. Energetic electrons from solar flares can be
indirectly observed through X-ray and radio emissions. Due to the unique
advantages, Hard X-ray (HXR) diagnostics is essential for determining the
electron properties (see <cit.> for
reviews). HXR emission observed from solar flares is dominantly due to
electron-ion bremsstrahlung. The intensity of HXR emission  is linearly
proportional to the non-thermal electron density, so the observed HXR photon
spectrum provides a straightforward relationship with the energetic electrons.
Furthermore, HXR emission is less affected by propagation effects than radio
emissions, resulting in minimal modification of the original electron properties
during observations.

Over the past two decades, advanced X-ray instruments, such as the Reuven Ramaty
High-Energy Solar Spectroscopic Imager <cit.>
and the Spectrometer/ Telescope for Imaging X-rays
<cit.>, have facilitated the collection of
high-quality spectra and spectroscopic imaging. The frequently observed X-ray
sources in the corona and chromosphere footpoints
<cit.>
strongly supports the 'thick-target' interpretation
<cit.>, where flare-accelerated
electrons propagate down to the dense chromosphere and deposit all their energy
within the chromosphere. According to the thick target model, the observed X-ray
photon spectrum is determined by the injected electron spectrum, bremsstrahlung
cross-section, and the energy loss-rate within the target
<cit.>. When the energy of electrons is high compared to
the target, i.e. E ≫ k_BT (k_B is the Boltzmann constant,
T is the temperature), the model is referred to as a cold target. In
cold-target conditions, the kinetic energy loss is predominantly due to Coulomb
collisions <cit.>, and while the other
effects, such as return current Ohmic losses
<cit.>,
could affect the results. While such a cold target model is widely utilized for
the basis of HXR spectral analysis, observations necessitate that the coronal
plasma during flares can reach high temperatures over 10^7 K, highlighting
the need to improve the model. Studies have demonstrated that the behavior of
electrons in the warm-target condition (E∼ k_BT) is more complex and
significantly different from what is predicted by the cold-target model.
<cit.> highlighted a rapid drop in the energy loss rate in
warm-target conditions, indicating a substantial overestimation of electron flux
by the cold-target model in generating observed photon spectra. Additionally,
collisional energy diffusion and thermalization <cit.> have
a significant impact on the dynamics of injected electrons. Furthermore, a
physically self-consistent model requires consideration of both energy and
spatial diffusions during electron transportation <cit.>.
Taking into account these advances, <cit.> developed a warm-target model that provides a physically
sound treatment of electron dynamics in the warm-target condition (referred to
as the warm-target model in the following). This model includes collisional
energy diffusion, suggesting that injected electrons thermalize within the
target rather than losing all their energy and disappearing. The HXR spectrum
generated by the warm-target model includes both the contributions from the the
electrons in the cold-target chromosphere as well as the the component from the
corona.

The warm-target model has been utilized to analyze RHESSI flares to more
precisely fit the characteristics of non-thermal electrons
<cit.>. One
significant challenge in inferring non-thermal electron properties is that the
non-thermal emission is typically masked by the thermal emissions below
20-30 keV <cit.>. This makes it difficult to determine the
low-energy cut-off, E_c, which is essential in non-thermal electron energetics
<cit.>.  
This parameter, along with the closely-related non-thermal power are the key
parameters to characterize the acceleration mechanism. The cold-target condition
E ≫ k_BT is typically applicable in the range over 20 keV, providing
solely an upper limit for the low-energy cut-off E_c. On the other hand, the
warm-target model considers the thermalization of injected electrons, which
enables effectively constraining electron properties, as the spectra of
thermalized electrons are proportional to the injected electrons.
<cit.> have demonstrated that using the warm-target model
with a power-law distribution can significantly enhance the accuracy of
determining the low-energy cut-off E_c compared to the cold-target model.

However, it is important to recognize that the power-law form of the injected
electron spectrum is not unique and not always the most suitable representation
in many cases <cit.>. The power-law distribution requires a
low energy cut-off (E_c) to prevent infinite electron number. The presence of
the low energy cut-off in the injected electron spectrum, 
while suggested to have physical significance in terms
of the critical velocity of run-away and free acceleration
<cit.>, 
results in a non-physical discontinuous electron distribution of
Maxwellian plus the power law. 
Around the low-energy cut-off E_c, where positive distribution slope (df/dv>0) exists, Langmuir waves are expected to generate and grow <cit.>. The interaction between the waves will result in a flattened distribution around
the low-energy cut-off <cit.>, suggesting the injected (accelerated) 
electron distribution should be nonincreasing function 
for all energies. On the other hand, the kappa distribution, consisting of a Maxwellian core and smoothly merged power-law tail, has been studied to represent accelerated electrons. 
The kappa distribution is often used
to model the entire energetic electron spectrum from in-situ observations
<cit.>
and has emerged as a promising alternative for describing the accelerated
electrons in HXR spectral fitting
<cit.>.
Previous studies have shown that while spatially integrated spectra cannot be
well-fitted only with the electrons obeying a kappa distribution, kappa-form
accelerated electrons with a prominent thermal component yields promising
results. 

In addition to not requiring an arbitrary energy cut-off E_c, the adoption of
a kappa-form injection offers several advantages over the power-law
distribution. First, the kappa-form electron spectrum is believed to result from
acceleration out of thermal equilibrium, and its parameters can provide crucial
information on electron acceleration. Various studies have delved into the
origin of the kappa distribution from different perspectives, including
collisional relaxation along with wave-particle interaction
<cit.>, beam-plasma interactions
<cit.>. Moreover,
originating from the Fokker-Planck equation,  has
pointed out that the kappa electron distribution results from stochastic
acceleration in the presence of Coulomb collisions and velocity diffusion,
specifying insight into electron kinetics during solar flares. Similarly,
simulations of electron acceleration during magnetic reconnection lead to a
distribution resembling a kappa-distribution <cit.>.
Another advantage of the electron spectrum in kappa distribution over the
power-law distribution is its entire electron kinetic energy coverage. Kappa
distribution can provide electron information at low energy (∼keV), which
is not sensitive to X-ray instruments but can be observed through other
diagnostics such as EUV observations. This allows for a more comprehensive study
of electron distribution through multi-wavelength observations, providing
additional constraints <cit.>.
Furthermore,  due to its whole electron energy coverage and convergent nature,
the kappa distribution allows for the determination of parameters such as
electron number density and average electron energy, which is not achievable
with the power-law distribution. The kappa distribution is closely related to
the thermal properties of the ambient plasma. Parameters such as electron number
density and average electron energy can be directly compared with the
corresponding thermal properties of the pre-accelerated plasma, serving as an
important constraint and cross-check.


The paper investigates the spectrum of accelerated (injected) electrons in solar
flares. Section <ref> discusses relationship between kappa-form
injected electron spectrum and the mean electron flux based on the warm-target
model. We utilize the warm-target model with kappa form injected electrons to
analyze solar flares observed by RHESSI on February 24, 2011, and by STIX on
March 28, 2022. The best-fit results and a comparison between the kappa and
power-law form injection is provided in Section <ref>. Section
<ref> discusses the uncertainty of fitting parameters related to
the kappa distribution with different approaches. The resulting characteristics
such as electron number density and average electron energy are also shown. The
findings and the physical significance of the derived kappa parameters are
summarized and discussed in Section <ref>.



§ SOURCE-INTEGRATED SPECTRUM FROM KAPPA-FORM INJECTED ELECTRONS



<cit.> highlighted the concept of the source-integrated
density-weighted mean electron flux ⟨ nVF ⟩(E)  [electrons 
cm^-2s^-1keV^-1] and expressed the observed HXR spectrum
I(ε) [photons  cm^-2s^-1keV^-1] as:

    I(ε)=1/4π R^2∫_ε^∞ Q(ε,E)⟨ nVF ⟩(E)  dE ,


where R is the distance between the observer and the Sun, and
Q(ε,E) is angle-averaged bremsstrahlung cross-section (see
discussion in <cit.> for RHESSI). Determination of the mean
electron flux from the observed photon spectrum requires only the well-studied
bremsstrahlung cross-section <cit.>. To obtain the injected
electron rate spectrum Ṅ(E)  [electrons  s^-1keV^-1] from the
obtained mean electron flux, a model for the dynamics of injected electrons is
necessary. The commonly used cold-target model establishes the relationship
between mean electron flux and injected electron rate as 

    ⟨ nVF ⟩(E) = E/K∫_E^∞Ṅ(E_0) dE_0 ,

were K = 2 π e^4 ln(Λ) is collision parameter, e  [esu] is
the elementary charge and ln(Λ) is the Coulomb logarithm
<cit.>. To account for the electron dynamics in the
warm-target corona and cold chromosphere, <cit.> included
the energy diffusion, transport and thermalization of the accelerated electrons,
rewriting Equation (<ref>) 

    ⟨ nVF ⟩(E) = 1/2K E  e^-E/k_BT ∫_E_min^Ee^E'/k_BT/E'G(√(E'/k_BT)) dE' ×∫_E'^∞Ṅ(E_0) dE_0 .


<cit.> further simplify equation (<ref>) and
re-write as:

    ⟨ nVF ⟩(E) ≈Δ EM √(8/π m_e)E/(k_B T)^3/2 e^-E/k_BT + E/K∫_E^∞Ṅ(E_0) dE_0 ,

where Δ EM, referred as 'thermalized emission measure', is due to the
thermalization of accelerated electrons, can be explicitly written as 

    Δ EM ≈π/K√(m_e/8)(k_B T)^2Ṅ_0/E_min^1/2 ,
            E_min≈ 3 k_BT(5λ/L)^4 ,

and Ṅ_0  [electrons  s^-1]= ∫_0^∞Ṅ(E)  dE
is the total electron injection (acceleration) rate,
λ=(k_BT)^2/2Kn is the collisional mean free path. The half loop
length L, number density n, and temperature T are the parameters
associated with  the coronal part of the loop. Equation <ref> consists
of two parts, refer as the thermalized and non-thermal components, corresponding
to the warm- (coronal) and cold-target conditions, respectively. It is worth
noting that the thermalized component is proportional to the total injected
electron rate Ṅ_0, which is why this model is effective in constraining
the nonthermal component when fitting the observed X-ray spectrum. This
warm-target model 'f_thick_warm', which utilizes Equations <ref> and <ref>, is implemented in Solar Software (SSW) and
 package <cit.>. Function
'f_thick_warm' utilizes the commonly employed broken power-law form
of injected electrons, and it is worth noting that the warm-target model
(Equations <ref> and <ref>) can accept any electron
injection rate form. As mentioned earlier, the use of a kappa distribution for
injection is a promising alternative and has been applied to fit the RHESSI
spectra.


 derives the kappa distribution as a result of
stochastic acceleration in collisional plasma:

    f_k(v)=n_k/π^3/2v_te^3κ^3/2Γ(κ)/Γ(κ-3/2)(1+v^2/κ v_te^2)^-κ ,

where v_te is the thermal speed (1/2 m_e v_te^2=k_B
T_κ), and κ is the kappa index, which importantly represents the
ratio between the acceleration τ_acc(v) and collisional deceleration
τ_c(v) time scales κ=τ_acc(v)/2τ_c(v).
While different formats of the kappa distribution exist <cit.>, they are identical in essence
and only differ in their format and use of parameters. Furthermore, the physical
interpretation of κ vary when additional factors are are taken into
account. In this study, we have chosen to adopt Equation <ref>
for the kappa distribution.


The relationship between injected electron rate spectrum and electron
distribution for velocity is Ṅ(E)dE=A v f_k(v) d^3v, A is the
injection area. For isotropic electron distribution (d^3v=4π v^2dv), the
injected electron rate spectrum becomes (see red curves in Figure <ref>):

    Ṅ(E)=An_kΓ(κ)/Γ(κ-3/2)κ^3/2√(8/π m_ek_BT_κ)E/k_BT_κ/(1+E/κ k_BT_κ)^κ ,

where n_k [cm^-3]= ∫f_k(v) d^3v is the total electron number
density, Γ(x) is the gamma function. The total injection rate becomes:

    Ṅ_0= ∫_0^∞Ṅ(E)  dE = 2A n_k √(2k_BT_κ/m_e)κ^1/2/(κ-2)B(κ-3/2,1/2),

where B(x,y) is the beta function.  
Equation <ref> can be expressed as:

    Ṅ(E)=Ṅ_0/k_BT_κ(κ-1)(κ-2)/κ^2E/k_BT_κ/(1+E/κ k_BT_κ)^κ.


According to Equation <ref>, the mean electron flux from a kappa-form
injected electrons is given by <cit.>:


    ⟨ nVF ⟩(E) ≈Δ EM √(8/π m_e)E/(k_B T)^3/2 e^-E/k_BT + Ṅ_0E/K1+(1-1/κ)E/k_BT_κ/(1+E/κ k_B Tκ)^κ-1.


For high energy electrons, the kappa  distribution can be well approximated by a
power-law distribution with a spectral index κ-1. Figure <ref>
shows that the two distributions have an identical injection rate near 100 keV.
However, in the energetically important deka-keV range, kappa distribution is
distinct from a power-law distribution, and thus, a different photon spectrum
emerges. Figure <ref> illustrates the electron injection rate (top
panels, red and black for kappa- and power-law injection ,respectively) and the
resulting photon spectra (bottom panels) generated by the warm-target model in a
typical flare coronal condition (L_loop = 30  Mm, n_loop
=  5× 10^10 cm^-3, and k_B T_loop=1.5
 keV). 



§ APPLICATION OF THE WARM-TARGET MODEL TO OBSERVATIONS
 

In this section, we apply the warm-target model to analyze two well-observed
flare event. One on 2011 February 24, observed by the RHESSI and one on 2022
March 28 observed by the STIX, both events are situated in the limb and
providing spectroscopic images with two footpoint and looptop sources. The
warm-target model with kappa-form electron injection (function
f_thick_warm_kappa, based on Equation <ref>)
encompasses parameters of kappa distribution (total electron injection rate
Ṅ_0, kappa temperature T_κ, and kappa index κ) and thermal
properties of the target coronal loop (the half loop length L, number density
n_loop, and temperature T_loop). 

In this study, the thermal properties of the target plasma are determined by the
HXR observations of the thermal component. Following the approach outlined in
<cit.>, the thermal component of the X-ray spectra prior to
the HXR burst can provide the plasma temperature (T) and emission measure
(EM). Moreover, the HXR spectroscopic imaging can estimate the coronal loop
length (L) and coronal X-ray loop source size for determining the electron
number density (n_th). While this method is straightforward, we recognize
that other methods, such as differential emission measure analysis <cit.> from EUV
flare context or combining X-ray data with EUV data
<cit.>, can yield thermal properties
<cit.>. 



 §.§ 2011 February 24 RHESSI flare
 









The first flare of study is a GOES-class M3.5 flare located on the eastern limb
from the Earth's perspective. The flare occurred on February 24, 2011 with GOES
Soft X-ray flux peaked at ∼07:35 UT <cit.>. Our
attention is drawn to the first HXR peak at approximately 07:30 UT (red shadowed
time range in Figure <ref> left panel). The HXR spectroscopic
imaging of the burst displays a 'classical' configuration of the loop top source
and two HXR footpoints (Figure <ref> right panel). The X-ray
observations from the RHESSI of the selected HXR burst benefits from the
low/negligible pile-up effect. Furthermore, RHESSI was in optimal working
condition during the flare, providing more functional detectors. 


For spectral analysis, we first gather the thermal properties of the target
coronal loops according to the images of the burst (07:30:00 to 07:30:44 UT,
Figure <ref> left panel red shadowed time range) and spectra before
the interval analysed (short time interval closely prior to the burst analysed,
refer as 'pre-burst' in the following, in this case 07:29:52 to 07:30:00 UT,
Figure <ref> left panel black shadowed time range). In the analysis
of the pre-burst X-ray spectra, we utilize functions f_vth
(iso-thermal) and f_thick2 (cold-thick-target). In this stage, we
focus solely on the thermal plasma parameters and can overlook the constraints
associated with using the cold-target model. f_vth from the best-fit
result of the pre-burst spectra provides us with the plasma temperature
(k_BT_0=1.30  keV) and emission measure (EM_0=0.127×10^49 cm^-3) before the injection/acceleration of nonthermal electrons.
Additionally, we use X-ray spectroscopic imaging (Figure <ref>
right panel) to estimate the size of the X-ray source and target coronal loop.
We determine the source size in various dimensions (see pink and red solid lines
in Figure <ref> right panel, r=l/2=3.7 Mm and d=15.3  Mm, respectively) by considering the 50% contour of X-ray loop top
sources. The cross-area can be calculated as A=π r^2=4.21×10^10 cm^2, and the source volume can be obtained through the cylindrical
formula: V=A× d=π r^2 d=6.46×10^26 cm^3. Thus the
thermal electron number density can be estimated using
n_loop=√(EM_0/V)=4.4×10^10 cm^-3. To obtain the
coronal loop length, we construct a semi-circular loop (see cyan dashed line in
Figure <ref> right panel) that connects two footpoints and passes
through the loop top source. Consequently, we determine the thermal properties:
loop temperature (k_B T_loop=k_BT_0=1.30  keV),
loop number density (n_loop=√(EM_0/V)=4.4×10^10 cm^-3), and the the half-loop length (L=15.8  Mm).


Once the thermal properties are determined, we proceed to fit the spectra of the
burst time interval (07:30:00 to 07:30:44 UT) in 6–100 keV using functions
f_vth and f_thick_warm_kappa. In function
f_vth, the temperature and emission measure are fixed according to
the pre-burst spectra (T=T_0, EM=EM_0). The warm-target algorithm propose
that the emission measure during the burst interval is a combination of the
emission measure from the pre-burst and the emission measure of thermalized
injected electrons (EM=EM_0+Δ EM, photon spectrum associated with EM_0
and Δ EM are plotted as the red solid curve and dashed curves in Figure
<ref> right panel). Since the plasma is slowly changing during
the 40 sec interval (electron diffusion time along the loop ∼56 seconds,
use <cit.> Equation 11), the parameters in
f_thick_warm_kappa related to thermal properties of the loop are
fixed in the fit (the half loop length L, number density n_loop,
temperature T_loop, and the relative elemental abundances, default to
be 1 <cit.>). The parameters of kappa-form injection
spectrum (total electron injection rate Ṅ_0, kappa temperature
T_κ, and kappa index κ, except the high energy cut_off
E_high fix to 10^4  keV) are determined from the fit. Moreover,
fit with a single power-law electron injection in the warm-target model
'f_thick_warm' was performed for comparison: the total injection rate
Ṅ_0, power-law index δ, and low-energy cut_off E_c are free,
the break-energy E_break and high energy cut_off E_high is
fixed to 10^4  keV. The best-fit parameters (see Table <ref>) and generated photon spectra are plotted in Figure <ref> (blue and green curves for warm-target model with kappa and
power-law form injection, respectively, same color representation for the
normalized residuals in the bottom panel).
 

The fit using the warm-target model, with kappa and power-law forms of electron
injection, both produces favorable fit results with reasonable parameters. With
the systematic uncertainty of 0.02, which is default for RHESSI in
, the reduced χ^2 is 1.93 and 2.11 for the fit with kappa and
power-law forms of electron injection, respectively. From Figure <ref> left bottom panel, the significant normalized residuals remain
below 12 keV, where thermal bremsstrahlung dominates. This may arise from the
use of the iso-thermal assumption f_vth for the thermal emission
<cit.>.
From the photon spectrum made by individual component (Figure <ref> right panel), we found that for this HXR burst, the emission
measure contributed by the thermalized injected electron (Δ EM, dashed
curves) is of comparable magnitude to the iso-thermal component (EM, red solid
curve). Here, we note that although the ratio of Δ EM/EM is affected by
the pre-burst timing chosen, all reasonable selections of pre-burst timing can
yield acceptable fit results. The different ratios only arise from the
identification of pre-accelerated plasma. 

In Figure <ref> left panel, we plot the injected electron rate
spectrum (blue and green curves represent the best-fit kappa and power-law form
injection, respectively). We noticed that the power-law and kappa forms electron
injection exhibit similarly between 40-100 keV but behave differently
approximately below 30 keV. At the power-law low energy cut-off E_c, the
electron injection rate Ṅ(E_c) of the kappa distribution is almost one
order of magnitude smaller (1.11 vs. 7.88 ×10^35 electrons sec^-1 keV^-1 for kappa and power-law form injection,
respectively). Additionally, the fitted injected electron spectrum in the kappa
distribution decreases more rapidly (κ>δ+1) at higher energy.
Although the best-fit electron spectrum in kappa and power-law form have a
similar total injection rate (Ṅ_0 = 22.5 vs. 20.3 ×10^35 electrons sec^-1 for kappa and power-law form injection,
respectively), the power-law electron spectrum produces a more substantial
nonthermal power than the kappa form electron spectrum (see Table <ref>, 3.62 vs 2.23 ×10^28 erg/s,
P_κ/P_power-law≈ 0.61). Moreover, for kappa form
electron spectrum, the nonthermal power produced by the electrons above
power-law low energy cut_off E_c is 1.41 ×10^28 erg/s. The
equations used for calculating the power are listed below, for power-law
distribution:



    P=∫_E_c^∞ E'N(E')  dE'=δ-1/δ-2E_CṄ_0.


For kappa distribution:


    P     = ∫_E_1^E_2 E'N(E')  dE' = Ṅ_0/(k_BT_κκ)^2(κ-3)(k_BT_κκ+E”)×
        (E”/k_BT_κκ+1)^-κ[2(k_BT_κκ)^2+2k_BT_κ(κ-1)κ E”+(κ^2-3κ+2)E”^2]|_E_1^E_2.


Thus


    P=∫_0^∞ E'N(E')  dE'= 2Ṅ_0k_BT_κκ/κ-3;



    P     = ∫_E_c^∞ E'N(E')  dE'= Ṅ_0/(k_BT_κκ)^2(κ-3)(k_BT_κκ+E_c)×
        (E_c/k_BT_κκ+1)^-κ[2(k_BT_κκ)^2+2k_BT_κ(κ-1) κ E_c+(κ^2-3κ+2)E_c^2].


We also include a distribution plot (f_k(E), f_k(E)dE=⟨
f_k(v) ⟩ d^3v) of the accelerated kappa-form electrons (blue curve in
Figure <ref> right panel, n_k=1.67×10^9 cm^-3, calculated from Equation <ref>, here A=4.21×10^10 cm^2). The
Maxwellian thermal distribution, which is take from the iso-thermal component
f_vth (total electron number density n_k=n_th=4.4×10^10 cm^-3, k_BT=1.30 keV), is also shown in red curve for
comparison. We will provide a more detailed discussion of the parameters
obtained in the following section.






 §.§ 2022 March 28 STIX flare
 









We conducted the same spectral analysis to the other well-observed GOES class
M4.0 flare on March 28, 2022. GOES Soft X-ray flux peaks at ∼ 11:29 UT; the
flare-associated HXR bursts were detected by STIX onboard the Solar Orbiter
<cit.>. The STIX temporal profile is shown in
Figure <ref> left panel, with the time difference of 335.7 seconds
between the Sol-O and the Earth has been applied to the STIX data. The flare
site is situated in the eastern limb from Sol-O's perspective. Extreme
Ultraviolet Imager <cit.>/Full Sun Imager (FSI)
provides EUV context from the same viewpoint (Figure <ref> right
panel). This flare comprises several HXR bursts, and our focus is on the burst
occurring at approximately 11:21 UT (red shadowed time range in Figure <ref> left panel). 


The selected burst and pre-burst time ranges for this flare are
11:21:24–11:21:44 UT and 11:21:12–11:21:17 UT (Figure <ref> left
panel red and black shadowed regions), respectively. For this particular event,
bottom pixels have a noticeable larger counts than the top pixels. Thus,
spectrum for fitting and spectroscopic imaging relies solely on the data from
the bottom pixels. The fit range is set to be 6 – 50 keV. The pre-burst spectra
yield emission measure and temperature values of k_BT_0=1.67  keV
and EM_0=0.0532×10^49 cm^-3. Figure <ref> right
panel displays X-ray spectroscopic images using the CLEAN algorithm
<cit.> and are overlaid on EUI/FUI 174 Å flare context.
Two looptop sources (red contours, 8-12 keV), of which the southern one existed
before the flare <cit.>, and two footpoint sources (blue contours, 20-50
keV) are observed. According to the 50% contour of the two loop top sources, we
obtain the source size in different dimensions (r_1=l_1/2=3.4  Mm,
r_2=l_2/2=3.2  Mm, pink lines, and d_1=12.1  Mm, d_1=9.6
 Mm, red lines). The cross-area (using northern source, A=π
r_1^2=3.64×10^10 cm^2) and source volume (V=π r_1^2 d_1+π
r_2^2 d_2=7.59×10^26 cm^3) can be calculated accordingly. The
constructed semi-elliptical coronal loop (cyan dashed line in Figure <ref> right panel), which rooted at two HXR footpoint sources and passing
through the loop top sources, features a half-loop length of L=18.4  Mm.
As a result, we can determine the target loop's thermal properties (k_B
T_loop=k_BT_0=1.67  keV and
n_loop=√(EM_0/V)=2.4×10^10 cm^-3).

The best-fit results using f_vth and
f_thick_warm_kappa/f_thick_warm are shown in Figure
<ref> and Table <ref>. To account for STIX
calibration uncertainty, which is still unknown, 6% systematic error was added
in  following  
<cit.>. From the Figure <ref> left panel,
distinct difference between the kappa and power-law form injection electron
spectrum was noted below 30 keV, consistent with the findings from the 2011
February 24 flare. For this event, low energy cut_off E_c obtained from
f_thick_warm is 12.46 keV. For comparison, the injection rate
Ṅ(E=E_c) is 0.641 vs. 4.21 ×10^35 electrons sec^-1 keV^-1 for kappa and power-law form injection,
respectively. The kappa form electron distribution, same as 2011 February 24
flare, will decrease more rapidly at higher energy (κ>δ+1). The
nonthermal power generated by the injected electron spectrum (3.66 vs 2.95
 ×10^28 erg/s, for electrons in power-law and kappa form
distribution, respectively) also exhibits significant dissimilarities. The kappa
form electrons generate less nonthermal power P_κ/P_power-law≈ 0.81 despite larger total injection rates: 20.5 vs. 13.6  [10^35 electrons sec^-1]. Figure <ref> also
illustrates a comparison between the distribution of accelerated kappa-form
electrons (blue curve, total electron number density n_k=1.1×10^9 cm^-3) and ambient thermal Maxwellian distribution (red curve, total
electron number density n_th=2.4×10^10 cm^-3,
k_BT=1.67 keV) in the right panel.






§ ANALYSIS TO THE FIT PARAMETERS AND FIT RESULTS
  In
this section, we will further investigate how well the three free kappa
parameters (Ṅ_0, κ, and T_κ) can be constrained by the fit
with the warm-target model. We will provide the uncertainty of the fit
parameters using different methods. Additionally, we will compare the total
electron density n_k and average electron energy E_avg derived from
the fit kappa-form spectrum with the corresponding thermal properties.





 §.§ 2011 February 24 RHESSI flare
 







In the previous section, we show that fitting using the functions
f_vth and f_thick_warm_kappa involves three free
parameters: Ṅ_0, κ, and T_κ, all associated with the kappa
distribution. To estimate the uncertainty of each parameter at different
confidence levels, we apply the method used in <cit.>. We
began by assessing the uncertainty of total injection rate Ṅ_0. We carry
out spectral fits with Ṅ_0 set at a fixed value while setting two other
kappa parameters (κ, and T_κ) for free. By doing this, we can
obtain the best-fit full χ^2 as well as the corresponding kappa index
κ and kappa temperature T_κ at this fixed Ṅ_0.
Subsequently, we repeat these fits multiple times with different fixed
Ṅ_0 values and record the corresponding full χ^2 values. Further,
we define Δχ^2=χ^2-min(χ^2), where min(χ^2) is the minimum
of all the obtained full χ^2 values. Δχ^2 reaches 0 at the
Ṅ_0 value of 22.5 ×10^35 electrons sec^-1
(Figure <ref> panel a), consistent with the best-fit results
shown in Table <ref>. The Δχ^2–Ṅ_0 curve
(Figure <ref> panel a) displays a prominent minimum, and the
Δχ^2 value changes rapidly in the vicinity of this minimum. The
Ṅ_0 within the 1-σ uncertainty level (corresponding to a 68%
confidence level, red horizontal dashed line in Figure <ref>
panel a) ranges 22.5^+0.8_-0.6 ×10^35 electrons sec^-1 (red vertical dashed lines in Figure <ref>
panel a), and the Ṅ_0 within the 3-σ uncertainty level
(corresponding to a 99% confidence level, green horizontal dashed line in
Figure <ref> panel a) ranges 22.5^+2.4_-1.8
×10^35 electrons sec^-1 (green vertical dashed lines in
Figure <ref> panel a). The Δχ^2–Ṅ_0 curve
and the estimated uncertainty suggest that the Ṅ_0 can be effectively
constrained by the fit with the warm-target model.


The same approach is used to analyze two other kappa parameters, κ, and
T_κ, for uncertainty. Both the Δχ^2–T_κ (Figure
<ref> panel c) and Δχ^2–κ (Figure
<ref> panel d) curves reveal a distinct minimum, similar to
the Δχ^2–Ṅ_0 curve. The range of k_BT_κ for
1-σ and 3-σ uncertainty levels (red and green horizontal dashed
lines in Figure <ref> panel c) is 1.29^+0.04_-0.04 keV
and 1.29^+0.13_-0.12 keV (red and green vertical dashed lines in Figure
<ref> panel c), respectively. The range of κ for
1-σ and 3-σ uncertainty levels (red and green horizontal dashed
line in Figure <ref> panel d) is 5.14^+0.04_-0.04 and
5.14^+0.12_-0.12 (red and green vertical dashed lines in Figure <ref> panel d), respectively.  The small fit uncertainties for all
three kappa parameters indicate that all parameters can be accurately determined
using the warm-target model.

In this study, we utilize the 3-σ uncertainty level to define the
acceptable range of the fit parameters and investigate the physical properties
within this acceptable range. One valuable property we examine is the average
energy of the electrons E_avg. Here, we first adopt the acceptable
range of Ṅ_0, which falls between 20.3 and 24.9 ×10^35 electrons sec^-1. Each given Ṅ_0 within the acceptable
range corresponds to a fit kappa-form electron spectrum. The average electron
energy for the kappa-form electron spectrum E_avg is given by:

    E_avg  =∫_0^∞ Ef_k(E)  dE/∫_0^∞ f_k(E)  dE 
    =3/2κ/κ-2.5k_B T_κ.

For comparison, the average electron energy of the Maxwellian electron spectrum
is E_avg=3/2k_B T. The obtained
E_avg–Ṅ_0 curve is depicted by the black curve in Figure
<ref> panel b. It is worth noting that, within the
acceptable range (green vertical dashed lines in Figure <ref> panel b), the average electron energies derived from the
kappa-form electron spectrum E_avg-κ are consistently higher than the
average electron energy of the thermal plasma in the target loop
(E_avg-loop=3/2k_B T_loop=1.95 keV, black horizontal
dashed lines in Figure <ref> panel b). According to Equation
<ref>, the total electron number density n_k can be
determined once the injection area A is known. We use the cross-area of the
loop top X-ray coronal source (A=π r^2=4.21×10^10 cm^2,
red outlier contour in Figure <ref> right panel) as the injection
area A to calculate the total electron number density n_k. The
n_k–Ṅ_0 curve is shown in blue curve in Figure <ref> panel b. Within the acceptable range of Ṅ_0, the total
electron number density n_k is found to be lower than the thermal electron
density of the target coronal loop n_loop (blue horizontal dashed lines
in Figure <ref> panel b, 4.4×10^10 cm^-3).
While the electron injection area cannot be precisely determined, and it may be
smaller than the observed loop cross area. n_k is unlikely to be higher than
the loop density n_loop, which requires a extremely small injection
area A<1.60×10^9 cm^2. The results indicate that the values
of E_avg-κ and n_k are physically reasonable when compared with the
ambient thermal plasma. The fitted kappa-form electron spectrum and associated
properties (E_avg-κ and n_k) within the acceptable range for
T_κ and κ closely resembles those obtained within the acceptable
range for Ṅ_0, and consistent with the previous findings. Consequently,
we have opted not to display them in the plot.


In order to further evaluate how well the fit parameters can be determined and
cross-check their uncertainty, we utilized the Monte Carlo analysis provided by
. Detailed information on this method is available in the 'Fit
Parameter Uncertainty Analysis' section from <cit.>. In
addition, <cit.> discussed commonly used methods, including
the Monte Carlo method, for estimating the constraint of fit parameters. The
Monte Carlo analysis requires a predetermined best-fit result, which is
available in Table <ref>. The Monte Carlo sampling explores
the multidimensional parameter space (Ṅ_0, κ, and T_κ)
surrounding the best-fit results by assuming a Gaussian probability function for
each parameter. Figure <ref> shows the posterior probability
density function for the three free parameters obtained from the Monte Carlo
analysis. The final fit results of the Monte Carlo analysis, as well as the
predetermined best-fit results from the forward fit, are also depicted in red
and blue dots, respectively. The coroner plot diagonal panels show the
one-dimensional projection of the probability density function. For the 2011
February 24 M3.5 flare, each parameter exhibits a prominent peak and a
relatively narrow width, indicating that they are well-constrained around the
best-fit results. The 1-σ uncertainty level range for each parameter is
shown in Figure <ref>: Ṅ_0=22.5^+0.6_-0.6
×10^35 electrons sec^-1,
κ=5.14^+0.03_-0.03, and
k_BT_κ=1.29^+0.04_-0.03 keV, comparable with the uncertainty
obtained from the previous method. The non-diagonal panels display the
probability density function projected for different parameter pairs, which
illustrate the correlations between different parameters. We observed that
κ and T_κ are positively correlated, while Ṅ_0 is
anti-correlated with κ and T_κ. The photon spectrum and normalized
residuals generated by Monte Carlo sampling, Monte Carlo final fit result, and
best forward fit result are shown in gray, red, and blue curves in Figure
<ref>.





 §.§ 2022 March 28 STIX flare
 







We conducted the same analysis on the 2022 March 28 M4.0 flare observed by STIX.
Similar to the 2011 February 24 M3.5 flare, the Δχ^2–Ṅ_0,
Δχ^2–T_κ, and Δχ^2–κ curves (black curves in
Figure <ref> panel a, c, and d, respectively) all exhibit a
prominent minimum. The ranges for each parameter at 1-σ and 3-σ
uncertainty levels are listed below: for Ṅ_0, 20.5^+1.3_-1.2
×10^35 electrons sec^-1, and 20.5^+3.9_-3.8
×10^35 electrons sec^-1; for k_BT_κ,
2.97^+0.17_-0.16 keV, and 2.97^+0.55_-0.45 keV; for κ,
8.85^+0.41_-0.38, and 8.85^+1.45_-1.01. The spectral fit for the
2022 March 28 M4.0 flare, due to a narrow fit range (6–50 keV) and a 6%
systematic error from STIX (2% for RHESSI), results in relatively larger
uncertainties, particularly for T_κ and κ. However, these
uncertainties still demonstrate that the warm target fit effectively constrains
the fit parameters. We adopt the 3-σ uncertainty level as the acceptable
range, consistent with our approach for the 2011 February 24 M3.5 flare. The
derived average electron energy E_avg and total electron density n_k
both satisfy E_avg-κ>E_avg-loop and n_k<n_loop within the
acceptable range (Figure <ref> panel b). Additionally, the
Monte Carlo analysis results are displayed in Figures <ref> and
<ref>. The 1-σ uncertainty level range for each
parameter is: Ṅ_0=20.5^+0.2_-0.2 ×10^35 electrons sec^-1, κ=8.85^+0.17_-0.14, and
k_BT_κ=2.97^+0.06_-0.05 keV. The correlations between these
parameters are consistent with the findings from the 2011 February 24 M3.5 flare
(Figure <ref> diagonal panels).






§ SUMMARY
  

In this study, we utilized the warm-target model to analyze two GOES M-class limb
flares. One M3.5 class flare occurred on February 24, 2011, and is observed by
RHESSI, while STIX observed the other M4.0 class flare on March 28, 2022. We
used the warm-target model due to its ability to accurately determine nonthermal
electron properties, such as nonthermal power, which can hardly be constrained
by the commonly used cold-target model. Unlike previous investigations, 
the kappa distribution is used to
characterize the accelerated/injected electrons. 
Unlike the power-law distribution, the kappa distribution is finite
as the electron speed approaches zero, 
thereby effectively covering the entire electron energy range without the
requirement of an arbitrary low energy cutoff. 
Consequently, the fitted
kappa-form electron spectrum can characterize electrons of which the kinetic
energy falls below the X-ray instrument's sensitive range. 
The results of the fitting and subsequent analysis are summarized below:




  * The best-fit results using the warm-target model in kappa-form injected
electrons feature reasonable electron spectrum. For comparison, we also employed
the warm-target model in the power-law electron distribution to fit the HXR
spectrum. We found that the obtained power-law and kappa-form electron spectrum
generate similar photon spectrum within the fitting range. However, the
kappa-form electron spectrum produces less nonthermal power despite having a
comparable or higher total injection rate Ṅ_0. This energy difference
is due to the behavior of the electron spectrum. The obtained
kappa-form and power-law-form electron spectra exhibited similar behavior at the
deka-keV level, while the kappa-form spectrum displayed a significantly lower
flux below ∼30 keV. Moreover, the kappa-form electron spectrum decreased
more rapidly than the power-law spectrum (κ>δ+1) at higher energies
(typically above 100 keV, out of the fit range).


  * The fit with the warm-target model in kappa-form electron injection
involves three free parameters (Ṅ_0, κ, and T_κ). 
Δχ^2-Ṅ_0 curve reveals a prominent minimum, 
suggesting an accurate determination of Ṅ_0 within a narrow range. 
For the M3.5 flare on February 24, 2011, Ṅ_0 within 1-σ 
uncertainty level ranges 
22.5^+0.8_-0.6 ×10^35 electrons sec^-1, and the
3-σ uncertainty level range is 22.5^+2.4_-1.8 ×10^35 electrons sec^-1. For the M4.0 flare on March 28, 2022,
Ṅ_0 within 1-σ uncertainty level ranges 20.5^+1.3_-1.2
×10^35 electrons sec^-1, and the 3-σ
uncertainty level is 20.5^+3.9_-3.8 ×10^35 electrons sec^-1. We apply the same analyses to the other two fit parameters,
κ and T_κ, for their uncertainty. 
For the M3.5 flare on February 24, 2011, κ
within 1-σ and 3-σ uncertainty level range is
5.14^+0.04_-0.04 and 5.14^+0.12_-0.12, respectively.
k_BT_κ within 1-σ and 3-σ uncertainty level range is
1.29^+0.04_-0.04 keV and 1.29^+0.13_-0.12 keV, respectively. For the
M4.0 flare on March 28, 2022, κ within 1-σ and 3-σ
uncertainty level range is 8.85^+0.41_-0.38 and 8.85^+1.45_-1.01,
respectively. k_BT_κ within 1-σ and 3-σ uncertainty
level range is 2.97^+0.17_-0.16 keV and 2.97^+0.55_-0.45 keV,
respectively. All the three parameters, Ṅ_0, T_κ and κ are
tightly constrained within a narrow range. Additionally, we carry out a Monte
Carlo analysis to further confirm the effectiveness in determining the fit
parameters with the warm-target model. The resulting posterior probability
density function for all three kappa parameters demonstrates distinct peaks and
relatively narrow widths, indicating well-constrained fit parameters. The Monte
Carlo analysis also provide a 1-σ level uncertainty for each parameter:
for 2011 February 24 M3.5 flare, Ṅ_0=22.5^+0.6_-0.6 ×10^35 electrons sec^-1, κ=5.14^+0.03_-0.03, and
k_BT_κ=1.29^+0.04_-0.03 keV; for 2022 March 28 M4.0 flare,
Ṅ_0=20.5^+0.2_-0.2 ×10^35 electrons sec^-1, κ=8.85^+0.17_-0.14, and
k_BT_κ=2.97^+0.06_-0.05 keV.



  * Unlike the power-law distribution, which requires a low energy cut-off
E_c, the kappa-form electron spectrum works for all energies, anchored at zero
speed, and extend to the speed of light. Therefore, the kappa distribution
facilitates the determination of the accelerated electron number density n_k
(with the information of the injection area A) and the average electron energy
E_avg. For both flares in this study, we find that, the derived total electron energy density is lower
than the thermal number density of the target loop (n_k<n_loop,
1.7×10^9 cm^-3<4.4×10^10 cm^-3 and
1.1×10^9 cm^-3<2.4×10^10 cm^-3 for 2011
February 24 and 2022 March 28 flare, respectively), while the average electron
energy surpasses that of the ambient Maxwellian thermal plasma (
E_avg-κ>E_avg-loop, 3.77 keV > 1.95 keV and 6.21 keV > 2.51 keV
for 2011 February 24 and 2022 March 28 flare, respectively).



In this study, we have demonstrated that the fit with the warm-target model
provides a convincing kappa-form accelerated electron spectrum. The associated
kappa distribution parameters are valuable for understanding electron
acceleration and transport. Here, we follow the model proposed by
<cit.>, which uses the Fokker-Planck equation to analyze
the evolution of flare-associated electrons. 
proposed that the kappa distribution arises from stochastic acceleration in the
presence of Coulomb collisions and velocity diffusion. The kappa index κ
is given by κ=τ_acc(v)/2τ_c(v). The collisional
deceleration time can be estimated as: τ_c(v)≈v^3/Γ, where
the collisional parameter Γ=2Kn/m_e^2. Here, we take the parameters
from the 2011 February 24 M3.5 flare event for study. For the upper limit of
electron energy E_l enabling the complete stop within the target, we have
E_l=√(2KnL)≈19.0 keV. At E_l, we have the electron speed
v_l=8.0×10^9 cm/s and the collisional deceleration time
τ_c(v_l)≈0.71 seconds. Thus the acceleration time can be
determined by τ_acc(v_l)=2κτ_c(v_l)≈7.33
seconds. The collisional diffusion time
τ_d(v_l)=2v_l^5/Γ v_te^2≈19.77 seconds, where
thermal speed v_te=√(2k_BT_κ)/m_e≈2.1×10^9
cm/s.

The ratio between the density of accelerated nonthermal electrons
(n_nth) and ambient thermal electrons (n_th) can provide
valuable insights into electron acceleration during solar flares. Previous
studies have shown that the ratio of nonthermal electrons to protons
(n_p=n_nth+n_th in the hydrogen plasma) is on the order
of approximately 1% <cit.>. In this study, we determined
the density of accelerated nonthermal electrons in corona by integrating the
kappa electron spectrum over 10 keV(n_nth[>10 ]∼1.2×10^8 cm^-3, and 2.0×10^8 cm^-3 for 2011 February 24 M3.5 and 2022 March 28 M4.0 flare,
respectively). For the thermal electron density in the flare site, We take the
loop electron density obtained from the X-ray thermal component
(n_th=n_loop). The ratio between the density of coronal
accelerated nonthermal electrons and ambient protons is found to be less than
1% (n_nth/n_p∼0.003, and 0.008, for 2011 February 24
M3.5 and 2022 March 28 flare, respectively), consistent with recent studies
<cit.>.

Here, we note the electron number density is calculated using Equation <ref> and is dependent on the injection area A. We estimated the
injection area A using 50% of the loop top X-ray source contour. The
resulting total injected electron density is approximately an order of magnitude
lower than the estimated coronal loop thermal electron density (1.7×10^9 cm^-3<4.4×10^10 cm^-3 and 1.1×10^9 cm^-3<2.4×10^10 cm^-3 for 2011 February 24 and 2022
March 28 flare, respectively). 
It is important to note that the electron injection
in the coronal region may be non-uniform in height.
According to <cit.>, 
the actual injection might be smaller.
In this study, we are utilizing the spatially integrated spectrum for analysis. At present, the signal-to-noise ratio does not sufficiently support precise spatially resolved spectral diagnostics at the arcsecond scale. Once we obtain the spatially resolved spectral information from the improved X-ray observations, the electron density derived from the kappa distribution will be more persuasive.

Since both the RHESSI and Sol-O/STIX are not sensitive to photons below 
∼3-6 keV, the thermal properties obtained from
RHESSI X-ray spectral diagnostics could differ from those obtained from SDO/AIA EUV
diagnostics <cit.>. Furthermore, thermal properties 
inferred from the observed X-ray spectrum are likely to be height 
dependent <cit.>. The obtained kappa-form energetic electrons 
allows for a more comprehensive combined study of the X-ray and EUV diagnostics. 
Potential future applications of the derived kappa distribution also include studying 
acceleration mechanisms based on the average electron energy and nonthermal power. 


The work was supported via the STFC/UKRI grant ST/T000422/1.



aasjournal




 Paper Title: Structural properties of subgroups of stars associated with open clusters.



firstpage–lastpage
Quantum repeater protocol for deterministic distribution of macroscopic entanglement
    Tim Byrnes
    August 5, 2024
====================================================================================




§ ABSTRACT 
Recent studies have identified star clusters with multiple components based on accurate spatial distributions and/or proper motions from Gaia DR3, utilising diverse diagnostics to improve our understanding of subgroup evolution. These findings motivated us to search for subgroups among the objects examined in our previous work, which employed fractal statistics. The present study considers seven open clusters that exhibit significant dispersion in age and/or proper motion distributions, suggesting that they would not be single clusters.
For characterizing the stellar groups, we calculate the membership probability using Bayesian multi-dimensional analysis by fitting the observed proper motion distribution of the candidates.  A probability distribution  is also used to determine the distance of the cluster, which is obtained from the mean value of the distance 
modes. 
The photometry from Gaia DR3 is compared with evolutionary models to estimate the cluster age and total mass.
 In our sample, double components are found only for Markarian 38 and NGC 2659. The other five clusters are confirmed as being single. The structural parameters, such as  
 𝒬,  Λ_ MSR and  Σ_ LDR
 are compared with results from N-body simulations 
to investigate how the morphology of the stellar clustering evolves.
The new results, for a more complete sample of cluster members, provide a better definition of the distribution type (central concentration or substructured region) inferred from the  m  -  s  plot.

 




stars: pre-main sequence – 
ISM: clouds – 
open clusters and associations: general.





 


§ INTRODUCTION



In the studies of stellar groupings, mass segregation and high surface density of sources are the properties typically expected in young star clusters. However, many stellar groups showing more or less concentrated clustering have been observed  probably due to different conditions for star formation and dynamical evolution,
 such as NGC 2264 <cit.>, NGC 2548 <cit.>,  NGC 6231 <cit.>, NGC 3105 <cit.>, and  NGC 346 <cit.>, among other examples.
The formation of bound clusters or dispersed stellar groups depends on the initial physical condition of turbulence effects in a molecular cloud <cit.>. For instance, recently formed clusters tend to follow the fractal structure of the clouds where they are found embedded <cit.>. 

Exploring the relationship between stellar groups and their natal star-forming regions hints at the hierarchical structure of star distributions and cluster evolution. 
For instance, to systematically search for reliable small-scale structures in star-forming regions that could be related to the star formation process, <cit.> developed a robust procedure to statistically analyse different regions having a varied sample of initial conditions. Their method was tested for synthetic and observed clusters, presenting successful results for regions with high degree of structures, where significant small-scale substructures were detected. For concentrated regions, they find a main structure surrounded by smaller ones. <cit.>  argue that multi-scale analysis is needed to disentangle the complexity of the region. 
In this work, we are particularly interested in clusters showing a low degree of structure, which could be evidence that structural characteristics may not change within the early stages of cluster evolution. 

<cit.> explored the fractal structure of a large sample of open clusters  noticing some objects that seem to belong to larger groups due to the possible presence of separate components having slightly different values of parallax and proper motion. 
NGC 2659 is one of those considered in the literature as a single cluster  <cit.>. However, in a study of the 
interaction of adjacent open clusters found in the Galaxy, which have spatial projected separation lower than 50 pc,   <cit.> suggest 
NGC 2659 is a binary cluster.


Among the open clusters studied by HGH19, there are several objects for which it was suggested the possible presence of more than a single component.
This hypothesis is due to the large deviations in the mean values of astrometric and dynamical parameters,
as well as differences when comparing with results from the literature, meaning that subgroups appeared mixed and considered a single cluster. 
To explore the changes in structural parameters, we  have selected a sample of clusters to be analysed in light of the possible presence of subgroups. 

The paper is organized as follows. In Sections 2 and 3, we respectively present some of the literature results related to our sample and describe the data used to revisit previous studies.
The analysis presented in Sect. <ref> is based on characterizing stellar groups that depend on identifying the cluster membership, mode of distance, 
mass, and age. Section <ref> is dedicated to the statistical analysis of the surface density distribution. In  Sects. <ref> and <ref>, we respectively present a comparison with previous results and summarize our conclusions. Finally, two appendices are included to present additional results (supplementary material available only in digital form).

 


§ SAMPLE SELECTION



For the revisiting analysis proposed here, we selected seven clusters from HGH19 with a larger number of members, 
which are  listed in Table <ref>. We have searched the recent literature for the parameters to compare with our previous results (HGH19). 
Most of the parameters are presented by Cantat-Gaudin and collaborators, whose works are 
dedicated to studying the structure and history of the Milky Way by characterising open clusters and studying the Galactic 
disc using the distance, age, and interstellar reddening for stellar clusters identified with Gaia astrometry 
<cit.>.





Our sample is also included in the list of clusters revisited by  <cit.>, for which updated parameters are presented 
based on an isochrone fitting code <cit.> to the Gaia DR2 photometry, taking into account the nominal errors
to derive distance, age, and extinction of each cluster. 
<cit.>  also used the method of isochrone fitting to estimate visual extinction and age for part of our sample.

 <cit.>  used Gaia EDR3 data to map the segments of the nearest spiral arms in the Milky Way, 
based on the overdensity distribution of young upper main sequence stars, open clusters, and classic Cepheids. 
They used a list of open clusters from <cit.>, but their analysis was restricted to the 
intrinsically bright objects, selecting the open clusters with more than five members 
with absolute magnitude M_G ≤ 0. 
We notice that four of the objects in our sample (Collinder 205, 
NGC 2659, NGC 2168, NGC 6494) coincide with the bright clusters present in the list of  <cit.>.

A machine learning code (SytarGO) was used by <cit.> to study the hierarchical star formation process by analysing
the three-dimensional morphology of young clusters, which revealed spatial and kinematic substructures. Two of our clusters appear in the sample studied by <cit.>: IC 2602 and NCG 3532; however, the morphology is indicated only for the second one, the “halo" type.




 §.§ Previous results


In this section, we summarize individual comments for our sample to complement the information provided in Table 1. Similar information could not be included for all clusters due to varying availability in the literature searched.

∙ Collinder 205: The distance that was previously estimated by us for Collinder 205 (Col205 hereafter) is about 2 kpc (HGH19) in agreement with the value found by <cit.>, but is considerable  larger than d ∼ 1.4 kpc found by different works <cit.> and smaller than d ∼ 2.4 kpc presented by <cit.>. There are also some discrepancies in our previous result of cluster age that tends to be younger than the ages found by other works <cit.>.
Col205 was included in the Galactic mapping survey of chemical abundances  of open clusters  <cit.>. As expected for clusters near the Sun, the metallicity ([Fe/H] = -0.07 ± 0.19) and all other elements are within the normal of solar neighborhood cluster mean abundances. This result is consistent with [Fe/H] = -0.169 found by <cit.>.

∙  IC 2602:  Density maps were obtained by <cit.> with the method of fitting  Gaussian functions to kernel density estimates, aiming to test in the solar vicinity the structural and dynamical early evolution theories,
IC 2602 is one of the studied objects, which was considered a benchmark in this kind 
of test because it is one of the nearest clusters with ages ∼ 50 Myr. 
<cit.> used the Gaia-ESO Survey <cit.>, products to estimate the gravity index, the lithium 
 equivalent width, and the metallicity to identify candidate members for IC 2602. They also used the radial velocities to derive the cluster membership
  probabilities and the intrinsic velocity dispersion that changes from ∼0.48 km s^-1 to ∼0.20  km s^-1  depending on the number of members considered. 
    
∙  Markarian 38: The estimates of proper motion and distance of the cluster Markarian 38 (hereafter Mrk38) presented by <cit.> are in excellent agreement with those from other works <cit.>. An estimation of the morphological aspect is given by <cit.>, indicating that the ellipticity ∼ 0.13) is measured for both the core and the overall shape of the cluster. According to the study of morphological evolution of a sample of open clusters presented by <cit.>, the overall shape of clusters becomes more elliptical as they grow older, while their core remains circular. This seems not to be the case for Mrk 38, whose core tends to be more elliptical.

∙  NGC 2659: <cit.> has explored the spatial distribution of Gaia sources aiming to identify possible double and multiple groups of star clusters, such as their “Group 18” that contains 8 open clusters. Four of these clusters were previously identified as a group by <cit.>, including  NGC 2659, which was suggested to constitute a pair with UBC 482. According to <cit.>, this pair has similar values of proper motion and parallax but are 1 apart in the space distribution, corresponding to ∼ 50 pc of projected separation. However, their distances exceed 250 pc <cit.>. For 82 members in NGC 2659 and 79 in UBC 482, <cit.> found ages of  44 Myr and  27 Myr, respectively.

∙  NGC 3532: The adopted age in the literature for NGC 3532 is ∼ 300 Myr <cit.>,
but it was two orders of magnitude larger than the value estimated by HGH19. This cluster was included in the three-dimensional kinematic 
study by  <cit.> to estimate the membership probabilities of open clusters. The morphology of NGC 3532  was studied by <cit.>  
giving the shape parameters corresponding to the ellipticity of the core e_core = 0.074 ± 0.031 and of the cluster overall e_all = 0.120 ± 0.045. 
<cit.> identified the high mass ratio of binaries found in a sample of 23 open clusters to estimate their fraction and trace their radial segregation. Their results show that NGC 3532 is one of the clusters with the lowest fraction of high mass-ratio binaries compared to the rest of 
the population. These results agree with <cit.>, arguing that NGC 3532 is not a binary-rich cluster, and its binary mass ratio follows a nearly uniform distribution.

∙  NGC 6494: <cit.>  infer a core radius r_c = 1.8 pc and A_V = 1.28 mag considering 1273 members in NGC 6494.  
<cit.> analysed a sample of 870 members and obtained
r_c = 2.55 pc, where the core has low eccentricity, while the halo has an elongated distribution, such 
as a tail-like structure with semi-major axis = 16.5 pc and a semi-minor axis =  6.29 pc. A comparable 
result was found by <cit.>, estimating for NGC 6494 a value of 0.15 for the ellipticity of the core.
<cit.> and <cit.> have adopted extinction correction, respectively E(B-V) = 0.27 and A_V = 0.85 mag that coincide with the values used by us. The age estimated by HGH19 is lower than the range of  380 Myr to 479 Myr  <cit.>.


The examples above are the primary sources used for parameter comparison (see Table <ref>). 
Our criteria for sample selection were based on the large differences found among our previous results and the literature, suggesting that discrepancies in the parameters and/or large uncertainties could indicate more than one structure in these selected clusters, possibly not noticed by HGH19.  
The data used in our analysis are described as follows.

 


§ DATA SETS






 §.§ Optical data



The astrometric and photometric data was obtained  from Gaia DR3 <cit.> by adopting 
query ranges defined on the basis of parameters estimated by HGH19: J2000 equatorial coordinates (α, δ),
parallax (ϖ), and proper motion (μ_α^⋆, μ_δ), where μ_α^⋆≡μ_αcosδ.  
Table  <ref> gives the adopted position and size of the clusters, compared with other results from the literature. 

Using the positions given in Table <ref>, we searched for all the sources in the area defined by a radius 10 percent larger than the cluster radius (HGH19). 
To avoid Gaia sources showing low quality of  the astrometric solution,  the selection was  restricted to objects having
RUWE[Re-normalised unit weight error (see details in the technical 

note GAIA-C3-TN-LU-LL-124-01).] < 1.4. 

To estimate individual mass and age of the cluster members, we constructed Colour-Magnitude Diagrams (see Sect. <ref>) with  photometric data at bands 
G (∼ 639 nm), G_BP (∼ 518 nm), and G_RP (∼ 782 nm) from Gaia DR3. 
The observed apparent magnitudes were corrected for extinction based on reddening E(G_BP-G_RP)  that is derived from A_0,  the 
line-of-sight extinction available for some of the Gaia sources. A_0 is one of the stellar parameters inferred by the algorithm Aeneas that
 is part of the package GSP-Phot (General Stellar Parametrizer from Photometry)[Details are found in the Gaia DR3 documentation 
 (Sect. 11.3.3)  at https://gea.esac.esa.int/archive/documentation/GDR3/ .].  The method performed by Aeneas consists of a simultaneous fitting of 
 the observed parallax, BP/RP spectra, and apparent magnitude (Gaia G band) using the technique of Bayesian posterior maximisation. 

In the case of  sources with no reddening information given by the Gaia catalogue, we estimate the colour excess E(B-V) 
based on the colour-magnitude diagram by fitting the observed unreddened magnitudes to the distribution of magnitudes that were corrected by the algorithm 
Aeneas. The value of  E(B-V)  that gives the best fitting was used to infer a mean value for visual extinction (A_V) by adopting R=A_V/E(B-V)=3.1 <cit.>. 
We checked the validity of the mean values of  A_V  by inspecting the extinction maps provided by <cit.>,  discussed in Sect. <ref>.
The conversion between  A_V and the expected extinction at the Gaia photometric bands was adopted
from <cit.>.









 §.§ Infrared photometry and mapping



We checked  in our sample the occurrence of circumstellar dust emission to identify the 
cluster members 
that are disc-bearing stars. Large infrared excess is expected for Young Stellar Objects (YSOs) that still 
are embedded in their natal cloud (Class 0 objects), as well as for pre-Main Sequence  stars with large amounts of 
circumstellar dust (Class I and Class II). 
Considering that protostellar discs only survive up to a few tens of Myr, members of older 
clusters are expected to be Class III objects (without discs). The classification based on the emission at near- and mid-infrared 
confirms the evolutive status of the clusters and could help identify the presence of mixed 
populations (different ages) in the sample. In Appendix <ref> (supplementary material), we describe the search for disc-bearing sources performed by fetching from public catalogues the infrared counterparts of the Gaia sources.
A cross-matching between the position of the sources was performed on the 2MASS <cit.>, 
and All-WISE <cit.> catalogues. The observed magnitudes at  2MASS (JHK) and WISE (W1: 3.35 μm; W2: 4.6 μm, W3: 11.6  μm) bands were fetched only for the sources showing flags indicating good photometric quality  
and data that are unaffected by known artifacts. 

Aiming to inspect the dust distribution in the direction of the clusters, we searched for the continuum images at  70-500 μm obtained by the Herschel Infrared Galactic Plane Survey <cit.>. We found only two   ∼ 2 strip tiles covering objects of our sample: Fields 264 and 269,  respectively, containing the clusters NGC 2659 and Collinder 205. 
A third cluster, Mrk38, is found near the southeast corner of Hi-GAL Field 11 but outside the image (see Fig. <ref>,  supplementary material). In this case, it is not possible to directly infer the local
 infrared emission. However, we could verify the lack of dust emission at this corner of Field 11 and around its neighborhood (Hi-GAL Field 13). Inspecting the far-infrared maps gives us some clues that the dust distribution is in good agreement with the visual extinction maps, as discussed in the following.


 



 §.§ Visual extinction maps



The dust distribution in the Galaxy is related to the A_V maps and traces the gas distribution in molecular clouds. 
 
Among the existing dust-based maps, there are two  of our 
 particular interest: the visual extinction maps from <cit.>, and the column density mapping
 derived from the Hi-GAL survey <cit.> using the point process mapping 
 (PPMAP)[https://www.astro.cf.ac.uk/research/ViaLactea/ .] 
 described by <cit.>. In this case, the  Herschel images of dust continuum emission are used to produce image cubes of differential column density as a function of dust temperature and position.
 
In this work we use the A_V maps[https://darkclouds.u-gakugei.ac.jp.] produced from the optical DSS[Digitized Sky Survey - STScI/NASA.] 
 images <cit.>, in spite of their lower resolution (∼  6 pixel^-1) than the extinction maps based on 2MASS data <cit.>. This choice was made to avoid unreal (A_V < 0) values that could occur in some regions mapped by using near-infrared photometry, which is not the case in the DSS maps. 

Since we are interested in a rough estimation 
 of A_V, to be used to check the reddening correction (see Sect. <ref>), we 
 can consider the comparison  between dust extinction maps produced from different data sets as a 
 confirmation that the position of the clusters coincides (or not) with regions showing low levels of dust 
 concentrations. 

Aiming to illustrate  the use of the A_V maps as a diagnosis of the presence of dust concentration, 
in Fig. <ref> we compare the maps from <cit.> with the Herschel PPMAP, presented in the form of 2D map of integrated line-of-sight column density of 
hydrogen molecules (in units of 10^20  cm^-2). 
In Table <ref>, we present the range of extinction values inferred by visual inspection of the 
A_V maps. The values of E(B-V) used to correct the reddening (as discussed in Sect. <ref>) 
are also presented in Table <ref>, which are in good agreement with the estimates from 
the A_V maps, excepting NGC 2659, which shows E(B-V) that is 0.19 mag larger than the value expected from the extinction map.




§ CHARACTERIZATION OF STELLAR GROUPS



The optical data extracted from the Gaia DR3 catalog (see Sect. <ref>) were used for three 
purposes: to analyze the vector-point diagram (VPD)  
constructed using proper motion (μ_α^⋆, μ_δ) of the stars to identify the members of different groups; using the parallax (ϖ) measured for the
members to estimate the  distance mode; and 
determining cluster age and individual stellar masses from isochrones fitting in the optical colour-magnitude diagram.  
The following sections are dedicated to describing the adopted methodology for identifying structures, the characterization of the clusters, 
and their respective stellar content.






 §.§ Membership probability



In the literature, the search for substructures has been conducted using methods and techniques based on spatial and kinematic analyses to identify discrete structures <cit.>, spatial stellar associations <cit.>,  or dynamical analyses  <cit.>, among other approaches. 
Young star clusters that are no longer embedded in their natal molecular cloud <cit.> are not expected to exhibit overdensities in their spatial distribution, which requires different criteria to identify which stars are physically related members confidently. Common velocities and distance are the main characteristics that stars inherit from the original cloud, giving the cluster a coeval movement until the tidal disruption that destroys the cluster <cit.>.

In this work, we adopt the method from  <cit.> for identifying cluster members by their proper motion modeled according to a 
Probability Distribution Function  (PDF), following the formalism presented by <cit.>. It was adopted  as a segregation 
procedure based on a mixed bivariate density function model 
for proper motions, considering that the cluster and the field are independent, as suggested by <cit.>. 

We adopted a PDF model that considers a sum of normal distributions, whose number of components can be two (cluster+field) or three (2 subclusters+field). The choice depends on visual inspection of proper motion VPD. Considering that candidates are projected against the same area, they do not present large differences in spatial distributions that could help distinguish subclusters. By this way, the only other parameter considered is parallax. The resulting PDFs are defined by the mean, standard deviation, and correlations for proper motion and parallax.

In summary, the 
membership probability is defined by using the maximum likelihood method that depends on the contrast between cluster members and 
field-stars. 
Accurate membership probabilities can be achieved by using  Bayesian 
 multi-dimensional analysis and performing a global optimization procedure  based on the  Cross-Entropy technique 
 <cit.>, to fit the observed distribution of proper motions and to obtain the probability of a given star 
belonging or not to the cluster. We also used a genetic algorithm to improve the first guess of the parameters set.  HGH19 gives details on the membership probability calculation. 





Our analysis is restricted to the samples of stars with 50 percent  or more of membership probability, which will be hereafter referred to as P_50 members or simply members. The mean values for the parameters estimated for the clusters are presented in Table <ref>.
Among the objects we studied, only for Mrk38 and NGC 2659 we found a second group in the same studied region. In these cases of double groups, the main cluster is denoted by index a and the secondary group by b.  

The distribution of members in the proper motion VPD and their position in the plot of equatorial coordinates are shown in Fig. <ref> for 
Collinder 205 (hereafter Col205), a representative of single clusters[The figures corresponding to the other clusters of the sample are given in Appendix <ref>  (supplementary material)].,  and the two targets that were found bimodal. The main criteria for choosing the single or two-component
clusters are based on the separation of groups showing different proper motions. This separation is also evidenced by the mean values of proper 
motion and distances shown for each subgroup in  Table <ref>. 
Despite the clear separation on proper motion, the spatial distribution of the members of the two groups associated with NGC 2659 roughly covers the same projected area, with a slight trend of subgroup  a being more concentrated in the SW region, while subgroup  b tends to 
the NE region.  On the other hand, both components of Mrk38 occupy the same projected area without any separation in the plot of equatorial 
coordinates.


Another criterion confirming the cluster membership is based on the mean value of the parallax (ϖ).  In  Fig. <ref> (top panel), we present the distribution of 
ϖ as a function of apparent magnitude in band G, using different colour grades to indicate membership of Col205. 
It can be noted that the most 
 probable members show a narrow distribution around ϖ∼ 0.5 mas, mainly for the brighter sources (G < 17 mag). 
 Fainter objects, for which the membership probability is lower, show a dispersion around the mean value of ϖ. 
 To infer the distances of the clusters, we adopted a statistical methodology based on a parallax distribution, which is described below.











 §.§ The most probable distance



We discuss here the procedure to improve the distance determination based on Gaia parallax of objects at kilo-parsec scale distances. 
In this case, the simple method of inversion of ϖ does not give accurate results, mainly for objects having high fractional parallax uncertainty, 
 defined as f = σ_ϖ / ϖ. 
 According to <cit.>,  the estimation by the inversion of the parallax tends to overestimate the distance modulus, which is a poor approximation due to systematics and correlations in the Gaia astrometric solution.
 
For instance,   the use of model fitting to obtain the PDF by an automated Bayesian approach (BASE-9) was adopted by <cit.> to infer 
the distance modulus and estimate the age for 268 open clusters. They found a median offset of -0.11 mag for the difference between the distance moduli derived from the analysis 
of BASE-9 results and the inversion of the median parallax. 

<cit.> also used a Bayesian inference method to estimate the distance to the W3 complex (d = 2.14 kpc). They derived the PDF by adopting the exponentially 
decreasing space density suggested by  <cit.>. The distances found for W3  and its substructures, considering the median value of the PDF, are about 5 percent different 
from those obtained by the simple inversion of parallaxes. For a more extreme case, the massive stellar cluster Westerlund 1 (Wd 1), <cit.> 
revisited the distance estimation by using Gaia EDR3 and considering new cluster members.  They inferred an improved distance of 4.06 kpc for Wd 1, adopting the parallax 
method suggested by  <cit.>  when dealing with cluster members with large uncertainties on the individual parallaxes. In these cases, the distance is assumed to be the same for all the cluster members and is calculated by following a maximum-likelihood procedure <cit.>.

In Fig. <ref> (bottom panel), we show the plot of  f as a function of G magnitude, aiming to evaluate, among the objects of our sample, the occurrence of 
observed parallaxes with large fractional parallax uncertainty. We verified that only the faint (G > 18 mag) members of our clusters may present high 
parallax uncertainty  (f > 0.3), which occurs for less than 20 percent of the members, excepting for Mrk38a and Mrk38b that have 
more than 50 percent of their members showing f > 0.3.


Considering that most objects in our sample do not show extreme cases of high fractional parallax uncertainty, we calculate the
distance for each cluster member  by using a simple PDF to estimate the true distance d = 1/ ϖ_ true based on the observed 
parallax (ϖ). For the individual distance estimation, we adopt  the probability distribution suggested by  <cit.>, assuming that the observed parallax is normally distributed around the true parallax:

    p(ρ | ϖ_ true)  =  1/(2 π ρ^4  σ_ϖ^2)^0.5  exp( -(1/ρ - ϖ_ true)^2/2  σ_ϖ^2),

where ρ = 1/ϖ and σ_ϖ is the uncertainty on the individual observed parallax.  The mode of the 
distribution was obtained for each member of the cluster, and finally, we calculated the mean value of the modes that
is adopted as the distance of the whole cluster. 
Figure <ref> displays histograms of the distribution of distance modes, whose mean values are presented in Table <ref>. Only the members with low fractional parallax uncertainty (f < 0.3) were considered in the calculation
of the mean value of distance modes.




 §.§ Colour-Magnitude diagrams



We used the photometric data from Gaia DR3 to compare the observed colours and magnitudes of the cluster members 
with theoretical isochrones that give us an estimation of stellar mass and age.
The isochrones were adopted from 
PARSEC[Version v1.2S+COLIBRI PR16 of PARSEC models available 
on http://stev.oapd.inaf.it/cgi-bin/cmd.] evolutive models <cit.>, 
which were plotted 
 in the (M_G)_0 × [G_BP - G_RP]_0 diagram, where the absolute magnitude was estimated by adopting for all the stars the same value of distance inferred for the cluster (see Sect. <ref>). The observed apparent magnitudes were corrected for extinction following the procedure described in 
 Sect. <ref>. Figure  <ref> shows the distribution of Gaia sources in the colour-magnitude diagram, compared with the PARSEC isochrone 
 that provides a good fitting of the observed data, which was adopted as the cluster age.  
 The uncertainty on the age is indicated by the grey area shown in Fig. <ref>, which is defined by lower and upper isochrones encompassing the distribution of data points. 
 
 



 
Stellar mass for each cluster member was estimated by adopting a simple interpolation method based on algebraic fitting inside the regions delimitated by the 
intersection between the theoretical lines (isochrones and evolutionary tracks). By localizing the observed position of a given star on the colour-magnitude diagram, the method 
infers the value of the mass by interpolating the theoretical values from the two nearest evolutionary tracks. 
The sum of the individual masses obtains the total observed mass of the cluster.


To validate the results derived from the colour-magnitude diagrams, in Fig. <ref>, we plot the individual masses (estimated by us) 
 as a function of effective
 temperature (provided by Gaia) for Col205. Fig. <ref>   (supplementary material) shows the plots for the other single clusters of our sample.  
 As an illustration,   the theoretical lines corresponding to three examples of isochrones from stellar models are also plotted. 
It can be noted a good mass-temperature correlation of low-mass stars (< 2 M_⊙) following a main sequence, while some massive stars having low temperature coincide with the Red Giants region, as shown by the isochrones of 119 Myr (two members of NGC 2168), and 300 Myr (NGC 6494).

The good agreement among the parameters, obtained from different databases (Gaia, PARSEC) and the individual masses estimated by us, gives confidence on the observed mass of the cluster. 

A similar confirmation is obtained when comparing the isochrone corresponding to the cluster age with the distribution of observed G magnitude as a function of the effective 
temperature (T_ eff) shown in Fig. <ref>. It is interesting to note the expected dispersion due to the presence of binary stars. In the case of NGC 3532, the separation of 
binaries is very clear and can be fitted by the same isochrone, but using an offset of Δ G = 0.75 mag. According to the simulations performed by  <cit.>  using 
evolutive models that include unresolved binaries in the Main Sequence population, this offset best fits the secondary distribution in the G × [BP-RP] 
diagram.
























 §.§ Mass function



The results for the sample containing only P_50 members give accurate mean values of astrometric parameters, distance, and age of the cluster. However, the incompleteness of the sample has an impact on the correct estimation of parameters depending on total mass, such as crossing time and dynamical age, as well as  for  the cluster size and core radius derived from the fitting of the surface density distribution (see Sect. <ref>).

Part of the problem can be solved by enlarging the sample at least to counting with a similar number of members found in the published catalogues (see Table <ref>). For some clusters (Mrk38, NGC 2659, and NGC 6494), the counting of P_50 members are close to the cataloged values. For the other clusters, we increased the list of considered members by choosing lower limits of membership probability. The adopted limit was P= 31 percent for Col205 and IC 2602, P=20 percent for NGC 2168, and P=4.5 percent for NGC 3532,
which defined the sample containing the total number of stars that we consider as observed members (N_ obs). These observed members were included in determining the individual mass based on the colour-magnitude diagram.

Observational bias due to Gaia detection limit must also be taken into account when discussing the sample 
completeness mainly for clusters at large distances <cit.>. 
We derived the mass function of the clusters to estimate the potential contribution of faint low-mass stars that may not have been detected by
 Gaia. The histogram of observed mass distribution was used in 
the fitting of the mass function  ξ (m) ∝ m^- (1+ χ) adopted from <cit.>. Following the method described by <cit.>, we use the slope of the observed mass distribution 
(χ given in Table <ref>)
 to obtain  the number of lacking faint stars (N_MF), below the limit of detection.  N_MF is estimated by integrating the Initial Mass Function suggested by Kroupa in the range of low-mass stars, assuming 
 χ= 0.3 ± 0.5 for m < 0.5 M_⊙. Table <ref> gives the total number of stars
  N_ tot = N_ obs + N_MF and the total mass of the cluster 
  M_ tot = M_ obs + M_MF, where  
  M_ obs is the sum of individual mass derived from the colour-magnitude diagram  of the observed stars,
  and M_MF was estimated by integrating the mass function in the range of low masses.





 §.§ Comparison with previous results


When increasing the lists of sources with more objects that have membership probability lower than 50 percent, the samples are comparable to the maximum number of members reported in the literature (see Table  <ref>), showing completeness that varies from 72 percent to 100 percent for Col205, IC 2602, NGC 2168, NGC 3532. 
For the lists that remained with P_50 members only, the number of objects is between the minimum and the maximum values reported in the literature, varying in the range of 42 -  60 percent of the maximum values, respectively, for NGC2659a and NGC6494. On the other hand, for Mrk38a, the list of P_50 objects is 0.62 larger than the maximum value found in the literature.

Concerning the age estimation obtained from the isochrone fitting, which strongly depends on the massive and intermediate-mass stars, the inclusion of more objects (mainly low-mass stars) does not change the age obtained for the P50 members. Comparing the results given in Table <ref> with those of Table <ref>, we find a good agreement (more than 0.75 of the maximum value from the literature) for the clusters IC 2602, Mrk38a, NGC 2659a, and NGC 3532. On the other hand, for NGC 2168 and NGC 6494, we found ages more compatible with results from <cit.> and <cit.> that reported lower values when compared with <cit.>. In the case of Col205, our estimation (42 Myr) is almost 5 times larger than the previous results.

As discussed in Sect. <ref>, an enlarged list of possible members (P > 10 percent) was analysed for NGC 2659a, but there was no change in the parameters that were determined for the constrained sample (P_50 members). The main impact of the incompleteness is on the estimation of total mass (see Sect. <ref>). However, since the cluster mass is not provided in the public catalogues we analysed (Table  <ref>), it cannot be compared with our derivations.








§ SPATIAL DISTRIBUTION



In the literature, several methods exploring the surface density distribution have been adopted 
to better understand the formation and evolution of 
substructures in the distribution of young stars, investigating if they are linked (or not) to the mass segregation 
in their original clouds. Recently, statistical tests have been performed to quantify the kinematic 
substructures of star-forming regions based on N-body simulations of artificial data points 
distributions  <cit.>.

Different examples of works are found covering structures ranging from large-scale distributions to the smallest ones.  
For instance, <cit.>  used the ALMA 1.3 mm and 3 mm continuum images to investigate the relation of core distributions and mass segregation with 
the density and kinematics of the gas of star-forming clouds. 
In nearby clusters, such as NGC 1333, the method to estimate mass segregation was applied to investigate the distribution of very low-mass stars to understand the origin of planetary-mass stars <cit.>.
Considering more distant regions, the estimation of parameters related to the fractal statistics have been explored for extragalactic objects, for instance, a large sample of star clusters 
of the Magellanic Clouds  that were studied by numerical simulations <cit.>  and based on observational  VISCACHA data[VISCACHA: VIsible Soar photometry of 
star Clusters in tApii and Coxi HuguA’ Survey <cit.>.] <cit.>. 

Here, we infer the geometric structure of our sample by analysing the stars' spatial distribution that hints at how the stellar clustering's morphology evolves. 
This section is dedicated to the morphology diagnosis that uses the density profile fitting to estimate the size of the cluster and its core radius;  the 
parameters related to the fractal statistics; local surface density, and mass segregation.  

The methodology for fitting the surface density profile and calculating fractal parameters is adopted from HGH19, whose description is summarized below.






 §.§ Cluster size and core radius


The mean values of equatorial coordinates  and respective standard deviations 
(α±σ_α, δ±σ_δ) given in Table <ref> roughly correspond to the center of the 2D projected spatial distribution of the cluster members.  
As indicated by the different values of  σ_α and σ_δ, some clusters may display an elongated 
distribution, which area is better represented by a convex hull. Figure <ref> shows an example of this geometric distribution that defines the minimal spanning tree (MST), 
the smallest network of lines connecting the set of data points without forming closed loops. The sum of the edge lengths in the MST is minimized, and the area of the convex hull contains all 
points projected on the cluster plane.

The cluster size was estimated using two different methods based on different lists of objects that provide minimum and maximum values for the cluster radius.

First, we studied only the distribution of the most probable members (P_50).
Following the methodology proposed by <cit.>, we constructed the MST adopting 
the algorithm from <cit.>.  The lower value for the radius (R_P50)  given in Table <ref> is defined by the radius of the circle having the same area as the convex hull. The standard deviation of R_ P50 was estimated using the Bootstrap method <cit.>. 

A second method was adopted to estimate the total radius of the cluster (R_ tot) that is achieved
 by fitting the density profile of the region containing the projected distribution of both cluster members and field stars.
The observed density profile is determined by counting the number of stars as a function of their distance to the center of the cluster. We adopted the  King-like radial density profile (RDP)  model suggested by <cit.> that is adapted from  the surface brightness profiles proposed by <cit.>: 

    σ (r) = σ_ bg + σ_0/1+(r/r_c)^2 ,


where σ_0 is the density at the center of the cluster and  σ_ bg is the background  density. The radius of the core (r_c) is derived from  σ (r_c) = σ_0 / 2. 

The  Levenberg–Marquardt method <cit.> was adopted for the RDP fitting based on maximum-likelihood statistics, with 
goodness-of-fitting function given by  χ^2.

The maximum value for the cluster radius (R_ tot) is defined by the point where the cluster stellar density reaches the background density (see Table <ref>). 

Based on the parameters derived from the RDP fitting, we 
also calculate the density-contrast (δ_c) parameter that quantifies how compact the cluster is. 
According <cit.>, this parameter is given by 

    δ_c  = 1+(σ_0/σ_bg) ,


where compact clusters are expected to have 7 < δ_c < 23. Table <ref> gives the values of density-contrast calculated for our clusters. According to this criterion, only  NGC 2659a is considered compact.















 §.§ Crossing time and dynamical age



The radius  obtained from the fitting of the King's profile  corresponds to an area larger than the distribution of P_50 members and possibly encompasses low-mass field stars. 
The same can be said about the estimation of M_ tot  that is considered an upper limit for the total mass of 
the cluster. Due to the differences in the lists of objects, we calculated two values for crossing time and dynamical 
age, giving a range of expected values. These ranges are presented in Table <ref>, where the first value 
corresponds to the calculation using the minimum estimation for mass and radius (M_P50, R_P50) and 
the second value is a result from the use of the maximum parameters (M_ tot, R_ tot).

Based on the mass M and radius R adopted for the cluster, we used  the expression
T_ cr = 10(R^3/GM)^1/2 to estimate the crossing time. The ratio of cluster age to crossing time is used to quantify the dynamical age expressed by the parameter Π. According to  <cit.>, the expanding objects have Π<1 and can be distinguished from bound star clusters with Π >1.

Five objects of our sample 
show Π < 1 that corresponds to unbound clusters (Col205, IC 2602, Mrk38a, NGC 2168, and NGC 2659a), in agreement with previous results (HGH19). On 
the other side, the new results for NGC 3532 and NGC 6494 indicate Π >1, leading to a new 
classification that suggests these are bound star clusters. As discussed in Sect. <ref>, the 
main difference from previous results is the larger number of members that are considered in the
 present work, which increased the total mass of the cluster, as well as the new isochrone fitting that gives an older age for these objects. 
  
The range of Π values presented by Mark38b and NGC 2659b indicates they are  unbound clusters. Since HGH19 did not 
individually analyse these subgroups, we compared them with their main companions. Both cases show considerable differences in 
dynamical age, suggesting that Mark38b and NGC 2659b are
stellar groups respectively distinguished from Mrk38a and NGG 2659a.







	




 §.§ Fractal statistics



The fact that most young clusters are found in a concentrated hierarchy of clusters within clusters makes their geometric distributions well-represented by fractals. In fractal star clusters, the level of substructures can be inferred using statistical analysis and measuring the 𝒬-parameter for observed clusters or simulations of artificial data point distributions  <cit.>.

The method for determining the fractal parameters m, s, and 𝒬 was first proposed by <cit.>, 
aiming to describe the geometrical structure of points distribution and statistically quantifies the substructures. Measurements on the MST allow us to obtain
𝒬 = m/s, where  m is the mean edge length that is related to the 
surface density of the points distribution,  and s is the mean separation of the points. 

Generally, an initially fractal star-forming region is expected to evolve to become smoother and more centrally concentrated. However, different initial conditions or differences in establishing the borders of the considered region will cause significant changes in the position of the pair of parameters  m - s in a plot that is often used to separate smooth distributions from substructured regions <cit.>.

Inferences of m  and s  depend on the total number of considered points N, which correspond in this work to the number of P_50 members, and are normalised by A_ N, the area of the convex hull  <cit.>.

These fractal parameters are useful to distinguishing fragmented (𝒬 <  0.8)  from smooth distributions  (𝒬 > 0.8),
small-scale fractal subclustering can be quantitatively distinguished from distributions with large-scale radial clustering. 

 Table <ref> gives the fractal parameters obtained for our sample, which are discussed in Sect. <ref> based on the comparative analysis with previous results. In this case, the study considers the expected distribution in the m-s plot indicating different types of structures, according to numerical simulations <cit.>. 

Our new results better indicate the cluster type distribution, probably due to the larger number of members considered here compared to HGH19.  The results are not well defined only in the case of  NGC 2659b, 
but the value  𝒬 = 0.87 indicates a radial concentration, while
Mrk38a (𝒬 = 0.73) is near the homogeneous boundary between fractals and radial profiles.


 An additional analysis of the cluster type distribution was performed by calculating the fractal dimension for the clusters that have 
 𝒬 < 0.8.  Following <cit.>, for instance, we adopted a simple definition of the Box-Counting dimension from <cit.>. 
 In summary, this method considers several cubes
(N_δ (F))  in a F ∈δ^n set with a δ size. If N_δ (F) intersects F, the box-counting dimension D_b 
is defined by 
the slope of  log(N_δ (F)) versus  - log (δ).
We found fractal box dimension D_b  > 2.6 for IC 2602, Mrk38a, and NGC 3532, indicating they do not show high levels of substructures, tending to smooth distributions as expected for regions having fractal dimension 
D ∼ 3. For Mrk38b, which has 𝒬∼ 0.8, the type of distribution remains undefined.

For clusters with 𝒬 > 0.8, we adopted the radial distribution  n ∝ r^-α  in order to estimate α
through the fitting of star counts in intervals of radius r+dr <cit.>. In this case, we found 
1.2 < α < 2.4 corresponding to intermediary 
distributions, in between uniform density profile (α = 0) and centrally concentrated distribution  (α = 3).




 §.§ Distribution of massive stars



Analysing the properties of the massive star as a function of age, surface density, and structure is useful in 
order to understand better the origin and evolution of mass segregation of stellar clusters 
<cit.>

In this section, we discuss two forms to quantify if massive stars are concentrated (or not) in a projected distribution by calculating 
 the mass segregation ratio (Λ_ MSR) and plotting  the local surface density (Σ) as a function of mass.

The parameter  Λ_ MSR <cit.> is used to determine if 
massive stars are closer to each other. It is defined by: 

    Λ_ MSR = ⟨ l_ average⟩/l_ subset,


where ⟨ l_ average⟩ is the average (median) length of the MST measured for sets of N_ MST random stars,  
and l_ subset is the same measure made for  the subset of  N_ MST most massive stars.  
In this work, we use a subset of N_ MST = 10.
We adopted the standard deviation from the dispersion associated with the roughly Gaussian distribution around ⟨ l_ average⟩ to represent the uncertainty on the estimate of the mass segregation ratio. The cases of  
 mass segregation are indicated by Λ_ MSR > 1.  

Another parameter to be considered is Σ_ LDR, commonly used to indicate if the massive stars are in regions of higher 
surface density than those where low-mass stars are found. 
The calculation of the  local stellar surface density  <cit.> for an individual star i is expressed by:

    Σ_i = N-1/π r^2_i,N,


where r_i,N is the distance between a given star and  its N^th nearest neighbour.  
In this work, we set N=10.
Following <cit.>, we adopted the local density ratio:


    Σ_ LDR = Σ̃_ subset/Σ̃_ all,


where Σ̃_ subset is the average measured for a given subset of stars, and  Σ̃_ all corresponds to
the average calculated for all the stars of the cluster. If the subset contains massive stars concentrated in dense regions, its local
surface density is expected to be higher than the whole sample. In this case, the mass segregation is indicated by 
Σ_ LDR > 1. With this method, it is possible to quantify if massive stars are found in dense regions, which is  different from the mass segregation  measured by 
Λ_ MSR <cit.>. 

Following the same method  to infer  the standard deviation of R_ P50 (see Sect. <ref>), the 
Σ_ LDR uncertainties were estimated using the bootstrap technique <cit.>. The method creates a simulated data set considering a variation of each parameter (positions) within a confidence level corresponding to a given distribution. Then, the standard deviation of this set of values is calculated. 

An example of the results of  Λ_ MSR  and Σ compared with the mass 
distribution is shown in  Fig. <ref>. The parameters related to the fractal statistics and the geometric 
structure of the clusters are given in Table <ref>.
The resulting Σ_ LDR, obtained from the ratio of the values corresponding to the horizontal lines in 
Fig. <ref> (right panel) shows that our clusters tend not to have massive stars concentrated  in regions
of high surface density, which is indicated by Σ_ LDR∼ 1.

Aiming to quantify the significance of the deviation from the median for all stars, we calculate the p-values using a Kolmogorov-Smirnov test 
for the cumulative distribution of the radial distances from the center of all stars and the 10 most massive stars. Following <cit.>, we adopted a p-value < 0.1 as the significance threshold. 
If Σ_ LDR∼ 1 and p-value > 0.1, the distributions show no significant difference, indicating that the most massive stars are not 
mass segregated. This is confirmed for Mrk38a,b and NGC 2659a,b. However, Col205 and NGC 3532 are too close to the adopted threshold, making it difficult to gauge whether the differences are significant. On the other hand, for IC 2602, NGC 2168, and NGC 6494, the parameters 
Σ_ LDR > 2 and p-value < 0.1 are clear signature of mass segregation.

In particular for NGC 6494, <cit.> found mass segregation ratio  Λ_ MST = 1.37 that is lower than our result but 
still compatible (see Table <ref> and Fig. <ref>).






§ COMPARATIVE ANALYSIS






 §.§ Astrometric results



The mean values of position found for the cluster members (see Table <ref>) were compared with  previous results 
from the literature (see Table <ref>) by calculating the offset on equatorial coordinates (α, δ) with respect the values obtained in this work. 
The conversion of angular measurements to linear dimensions, which gives  Δ pos (position offset in parsec), was made using the mean value of the distance mode.

The comparison of proper motion was made by calculating the offset of tangential  velocity 
Δμ = (Δμ_α^⋆^2 + Δμ_δ^2)^0.5,  given in mas yr^-1 that was converted 
into km s^-1 in the same way we used for the position offset.

In the case of double clusters, we adopted the results for Mrk38a and NGC 2659a as a reference, meaning that these main groups and the single clusters present offset equal zero in this comparative analysis. 










The estimated ranges quantifying the differences and similarities between our results and those in the literature are shown in Table <ref>. The table provides the minimum and maximum values of position and velocity offsets, along with their respective references.
Figure <ref> displays the offsets Δ pos and  Δμ  for the entire sample compared with the literature, identifying individual works by coloured bars.

First, we discuss the single clusters. Compared with our new results, the offsets are very low, excepting the comparison with <cit.> data  (green bars) for NGC 2168 and NGC 6494, which are based on UCAC4[The fourth U.S. Naval Observatory CCD Astrograph Catalogue.]. A possible explanation for 
these higher velocity offsets  is the difference in the proper motion used from a different database,  which in some 
cases may present larger uncertainties when compared with the Gaia data. Another possible cause of discrepancies between our results and the literature is related to different estimates of the 
cluster distance. This problem is avoided in the  left panel of Fig. <ref>  that displays Δ pos as a function 
of  Δμ, both given in angular measurements.
Again, the points showing large offset in velocity are NGC 2168 and NGC 6494, due to the values adopted from UCAC4.


Proceeding with the same evaluation for double clusters, we found larger discrepancies than expected. For Mrk38, the separation is larger on 
proper motion. 
On the other hand, the position we found for Mrk38a is in good 
agreement with the literature, while Mrk38b shows  negligible Δ pos in comparison with its pair, corresponding to minimal differences on the projected distribution.

Noticeable discrepancies are also found for NGC 2659 compared with HGH19, which results are roughly in between the values obtained for both groups in the present work, mainly for NGC2659b that has the 
largest distance from its pair.
The discrepancies of both components of NGC 2659 in comparison with
HGH19 indicate that our previous results were obtained from a mixing of members of both groups. It can be also noted that our present results for NGC 2659a are in better 
agreement with literature <cit.>.

We have checked the possible coincidence of NGC 2659b with other open clusters (OCs) previously suggested as part of the same group. Figure <ref> compares results only for NGC 2659 and other OCs that could have similar properties. 
These OCs were reported in the literature as candidates to possible companions of NGC 2659, which could constitute a double or multiple cluster.
This plot shows the offsets in position as a function of differences in cluster distance (Δ dist) when compared with our present results.
In this case, offsets with positive values indicate more distant objects (background), while negative values are used for lower distances (foreground). 
The symbols have different colours representing the offset of tangential velocity. The blue symbols indicate similarity in velocity compared with NGC2659a, 
as it is noted for the results from the literature <cit.>, while red symbols correspond to velocities similar to the results for NGC2659b (Δμ > 2  mas  yr^-1, where ⋆ indicates this work, and the red square is from HGH19).

The OCs <cit.> are represented by blue circles indicating similarity in velocity compared with NGC2659a. However, three of them have larger distances (Δ dist> 100 pc):   Gull5, UBC 482, and Cas61. Other five OCs have lower distances (Δ dist< -70 pc): LP58; Pismis 8; SAI92; Rupr71;  and NGC2645. The last candidate, Cas28, 
has projected position larger than 1 degree in the sky, which is also observed for all the other clusters. Due to these large discrepancies, it is more conservative not considering they belong to the same group.  






Finally, UBC 246, also named Pismis 9, was suggested to form a pair with NGC 2659a  <cit.>, which has a projected position and tangential velocity similar to NGC2659b. However, its distance is larger at more than 150 pc <cit.>. 
Even when considering the uncertainty in estimation, the extent of their separation remains too significant to align with the same subgroup.
It must be kept in mind that our selection of 96 members is more restrictive than other works that consider samples with at least three times more objects scattered around the mean spatial position, which naturally span in larger ranges of distances. To be more conclusive about the similarities and differences between NGC 2659b and UBC 246, it is necessary to include additional objects, candidates showing membership lower than 50 percent, to have the same basis of comparison with other works.  

 <cit.> reported a  list of 270 members for UBC 246. To achieve a similar sample, for NGC 2659b, we selected the objects that have 
membership probability P>0.1 percent and  fractional parallax uncertainty f <0.5. 
The mean values (and respective standard deviation)[(α,δ)=(130.712(0.006), -44.935(0.054)) deg, ϖ=0.509(0.052) mas, μ_α^⋆= -4.379(0.101) mas yr^-1, 
μ_δ = 3.015(0.134) mas yr^-1, d = 1916(218) pc.]  for this enlarged 
sample are in good agreement with the literature for UBC 246 <cit.>, except for the slightly different distance: d = 2120 pc <cit.> or d = 2025 pc <cit.>, but still within the uncertainties. This confirms that NGC 2659b is indeed the same cluster previously cataloged as UBC 246.

When comparing NGC 2659b with NGC 2659a (using a sample of 154 objects that have P>10 percent, f<0.5), the differences between 
the mean values for each parameter (presented above) remain comparable with those obtained for the samples restricted to P_50 members. The
new results show less than 0.5 percent for differences in position and proper motion, while the standard deviations, as expected, are almost 50 
percent larger than the previous results (P_50 members). The main difference, but not significant, occurs for the new values of distance mode, 
which change the offset 
from Δ dist = 2066 -1979 = 87 pc (measured for samples of P_50 members) to Δ dist = 1980 - 1916 = 64 pc.
An illustration of these differences is shown by dotted lines linked to the star points representing NGC 2659a,b in Fig. <ref>.
These  results indicate that enlarging the sample (by a factor of ∼3 in the case of NGC2659b) does not give different parameters that were 
found for  the list of the most probable members. Despite the similarity in the projected distribution of both groups, NGC 2659b tends to be in a 
region  more than 0.05 deg to the NE away from NGC 2659a. There is a difference of more than 50 pc in distances, where NGC 2659b is in the 
foreground, and significant differences in proper motion and age confirm that this cluster is not a subgroup of NGC 2659a.






 §.§ Structure



The early evolution of star clusters and their original structure can be explored by comparing the fractal 
parameters obtained from the observed surface density distribution with theoretical simulations.  In a 
recent study of the NGC 2264 star-forming region, <cit.> used the 𝒬-parameter to 
quantify the spatial distribution of stars for two subclusters centered around the stars S Mon and IRS 1/2.
Both groups have 𝒬∼ 0.8, meaning they have neither a substructured nor a centrally 
concentrated distribution. According to different age estimates in the literature, the star formation 
activity seems to have started first in the S Mon region (∼ 5 Myr) and more recently (∼2 Myr) 
for the group IRS 1/2 <cit.>. This is one example of studies using the𝒬-parameter, comparing it with the mass segregation ratio and the local density, similar to the analysis performed in the present work.

Here, we aim to investigate the variation in the fractal parameters if single or multiple structures are considered 
in the distribution of  our cluster sample. Figure <ref> shows the locus of different regions defined by numerical 
simulations from <cit.> using convex hull area normalization for synthetic star-forming regions that 
contain 300 points. We adapted ellipses covering the span of points from the results for each adopted 
geometry (fractal dimension or radial density profile) to display the loci of six parameters used in the simulations
<cit.>. Overimposed on these ellipses,  we 
plot the values of m  and s  obtained in this work compared with previous results from HGH19.
Above the line of  𝒬 = 0.8 is found the loci of simulations with RDP described by n ∝ r^-α, 
where α = 0 indicates uniform density profile, 
while α = 2.9 corresponds to centrally concentrated distributions. On the other hand, the regions with 
 𝒬 < 0.8 correspond to simulations  that adopt geometries varying from very substructured fractal 
 (fractal dimension D = 1.6) to smoother  distributions (D = 3).

 It can be noted in Fig. <ref> the improvement we obtained in the accuracy of the fractal parameters,  
 probably due to the larger number of members
 considered in this work for most of our clusters, compared with previous results (HGH19). 
 This means that the position of the clusters in the  m - s  plot coincides with well-defined 
 regions representing RDP distributions in the case of Col205, 
 NGC 6494 (α = 1), NGC 2659a (α = 2), and  NGC 2168 (α = 2.5). On the other hand,
 IC 2602 and NGC 3532  have 𝒬 < 0.8, coinciding with the simulated fractal region F2.6. 
 As discussed in Sect. <ref>, the legend of Fig. <ref> also shows the results from calculating 
 the fractal box-dimension (D_b) or the slope (α) of the RDP, respectively labeled by “f" or “r" aiming to compare with 
 the simulated regions directly.

A last diagnosis of our analysis is obtained from the plots of the  𝒬-parameter against the 
mass segregation ratio and the local density shown in Fig. <ref>. In these plots, the 
areas corresponding to the N-body simulations from <cit.> are displayed, representing results 
for a highly substructured star-forming region under subvirial[ α_ vir = 0.3, where α_ vir is the ratio of  potential energy to kinetic energy.] initial conditions. 
Previous results (HGH19) are also plotted for comparison, with principal differences noted for Λ_ MSR tending to be larger in the present work.
Σ_ LDR seems to show lower values, but for all the objects of our sample, both parameters 
remain in the 1.0 to 2.5 range. These values coincide with the interval expected by the simulations
corresponding to the early evolution of the clusters (age = 2 and 5 Myr) but
 do not reach the maximum values found by  <cit.> for these adopted initial conditions. No significant changes were noted in the 
𝒬 range, since our sample coincides with the range expected by 
the simulations, except for Mrk38b, which shows inverse mass segregation.






 Comparing 𝒬, Λ_ MSR and Σ_ LDR obtained from simulations for the 
 initial conditions adopted by <cit.>, it can be concluded that the dynamical evolution erases the initial 
 structure (see in Fig.  <ref> the area corresponding to the distribution of points 
at 0 Myr). After 2 Myr, their levels of substructure decrease significantly, and their levels of mass 
 segregation increase. Although the ages of our sample are much older, their  values of  𝒬 
 and mass segregation suggest that most of our clusters would be consistent with those initial conditions. 
 However, as discussed in HGH19, they would also be consistent with other conditions. For instance, we also 
 compare our results with simulations of synthetic regions from <cit.> 
 using α_ vir = 1.5 and D = 3. 
 Fig.  <ref>   shows a shaded area corresponding to the smoothed span of 
 the points resulting from these simulations at three ages (0, 2 and 5 Myr). In this case, the choice of initial 
 conditions is related to the simulated distribution of points coinciding with the same range of 
 parameters found for our clusters.








§ DISCUSSION


 
We selected seven OCs from a sample previously studied by us to investigate the possible presence of double- or multiple structures. The choice of the objects was based on the large dispersion of their ages, which were estimated by using near-infrared photometry (HGH19). In the present work, the characterization of the cluster members was revisited in the light of Gaia DR3 astrometric and photometric data that allowed a significant increase in the list of studied stars, corresponding to a more complete population when compared with our previous results. Near- to mid-infrared data and visual extinction maps were used to inspect the circumstellar and interstellar environments that could be related to dust emission from a protoplanetary disc or the presence of a surrounding molecular cloud. However, our sample does not show large amounts of dust emission, considering that only a few members of the younger clusters seem to be disc-bearing stars, and the A_V levels in the direction of the clusters are lower than 1 mag for most of them.

We considered members of the clusters the stars with a membership larger than 50 percent (P_50), estimated by fitting a Probability Distribution Function to the observed proper motion. Only members presenting low fractional uncertainty of parallax (f < 0.3) were used to estimate the mean values of the astrometric parameters, as well as in the calculation of the most probable distance of the cluster that is determined from the distribution of the distance mode of the members.
In total, we analysed 9 stellar groups, five of which are single clusters: Col205, IC 2602, NGC 2168, NGC 3532, and NGC 6494, which do not show any subgroup in their spatial distribution neither in proper motion. The other four groups are related to two candidates that possibly are pairs of clusters Mrk 38a,b and NGC 2659a,b, whose components are mainly distinguished when comparing the proper motion and the distance of the subgroups, while their position overlaps in the same projected region.

The colour-magnitude diagram constructed with the Gaia photometry was corrected from reddening using the extinction automatically 
estimated by the code Aeneas of the GSP-Phot package. For the Gaia sources lacking this correction, we have adopted a mean value 
of E(B-V) 
that agrees with the A_V maps and provides a good fit for the isochrone corresponding to the cluster age. Individual stellar masses, determined from interpolation between evolutive tracks, 
were compared with the effective temperature, confirming that the mass estimates are reliable mass. 
A comparison of the astrometric results with the literature was performed by measuring offsets in position (Δ pos) and proper motion calculated in the form of tangential velocity
 (Δμ), where our present results were adopted as references (that means zero points). 
We found low values for these offsets in the case of the single clusters that show Δ pos < 1 pc and Δμ < 1 km s^-1 suggesting an excellent agreement with the literature.


The surface stellar distribution of subgroup Mrk38b entirely coincides with the main cluster Mrk 38a (Δ pos∼ 0.1 deg), but the offset on proper motion corresponds to more than 10 km s^-1. This subgroup is about 500 pc more distant than the distance we found for the main cluster. These differences suggest Mrk38b is a distinguished stellar group, which is older (250 Myr) and more scarce (n = 2.4 stars pc^-2) than  Mrk38a (age= 30 Myr, n = 14 pc^-2) that is the main group previously known as a single cluster.
Due to unusual nature of Mrk38b, as a second cluster in the direct line of sight, we examined the possibility that it is instead an asterism caused by a combination of the Gaia missions limited ability to detect (fainter) stars more distant than Mrk38b and/or large parallax uncertainties of identified members. We performed a test of eliminating sources with a parallax uncertainty f>0.2 and re-calculated the membership probabilities. In this case, the number of sources is insufficient for the algorithm to discern two groups. Only one group is found, showing an elongated distribution in the considered parameter space. However, through visual inspection, it is still possible to identify a few members coinciding with the ranges of proper motion and parallax found for Mrk38b.

The largest offsets are found for NGC 2659b, separated from the main cluster by 3.3 pc and  tangential velocity offset > 20 km s^ -1.  In Fig. <ref> (right panel), we compared the main cluster NGC 2659a with other OCs from the literature that were suggested to be part of the same group due to their similar proper motion. However, all of these OCs have shown large differences in distance (> 100 pc) or projected position (> 1 degree), which are too distant from each other to be considered components of the same stellar group. NGC2659b shows differences that indicate it is not a substructure of NGC 2659a. NGC 2659b   has projected position and proper motion coinciding with the cluster UBC 246, but the distance we estimated is lower than the results from the literature. 

The calculation of the structural parameters was based on the surface density distribution by constructing the MST that is used to estimate the size of the cluster and fractal parameters, which were compared with N-body simulations from the literature. For instance, the 
m - s plot (see Fig. <ref>) indicates that clusters with 
𝒬 > 0.8 (corresponding to radial distribution): Col 205, NGC 2168, NGC 2659a, and NGC 6494 occupy the areas coinciding with simulations that define the locus of centrally concentrated regions (R = 1 to 2.5). NGC 2659b also has 𝒬 > 0.8, but its 
m and  s parameters do not coincide with the loci predicted by the simulations, just appearing near the region of smooth uniform distribution (R=0).
Only  IC 2602 and NGC 3532 do not show radial distribution, but low levels of substructures are found since they coincide with simulated fractal regions (F=2.6). Mrk38a and Mrk38b appear out of the simulated regions, but Mrk38a (𝒬 < 0.8) seems to be tending to the region of smooth fractal distribution (F=3), while Mrk38b (𝒬∼ 0.8) has undefined distribution.
These results are consistent with those calculated in Sect. <ref>.

The total mass of the clusters was used to estimate the crossing time and the dynamical age (Π) that indicates if the cluster is gravitationally bound. Half of our clusters are considered unbound since they have Π < 1. 
A possible exception is Mrk38a, which has Π∼ 1 and could be a bound young cluster whose age is near its crossing time.
Larger values of dynamical ages were found for the oldest clusters of our sample: Mrk38b, NGC2659b, NGC 3532, and NGC 6494. 
There is a Λ_ MSR > 1 trend for most of the clusters, except for the subgroup Mrk38b. However, the mass segregation is only evident for half of the sample, which has 
Λ_ MSR≳ 2. The other clusters are consistent with Λ_ MSR∼ 1 (within the errors).
On the other hand, based on the values of  Σ_ LDR∼ 1 we conclude that massive stars tend not to be concentrated in regions of high 
surface density for most of our clusters, excepting IC 2602 and NGC 6530 that show mass segregation signature 
(Σ_ LDR > 2 and  p-value < 0.1).

The diagram comparing the mass segregation parameters as a function of 𝒬 also displays the results from N-body simulations considering different initial conditions. These simulations indicate the data points' distribution during the artificial clusters' early evolution. The plots in Fig. <ref> show that most of our clusters present the same distribution found for simulations at 2 and 5 Myr. Only Col205, NGC 2168, and NGC2659a appear out of the areas predicted by the simulations. 
Their high values of 𝒬 indicate central concentration that is confirmed by their position in the 
m  - s  plot tending to RDP distributions (R2 to R2.5 regions in Fig. <ref>). The comparison with the simulations (highlighted areas in Fig. <ref>) indicates that most of our clusters tend to smooth distribution (𝒬∼ 0.8) or a radial concentration   (𝒬 > 0.8) as they are evolving. One possibility is that these clusters could have lost their original substructures if they had been formed in highly substructured regions. Or, on the contrary, they retained their original geometry that possibly had low levels of substructures. 




§ ACKNOWLEDGEMENTS

We warmly thank Professor Vera Jatenco-Pereira for the valuable suggestions on the text of this manuscript.
We thank the anonymous referee for the constructive comments and suggestions.
We acknowledge support from FAPESP (2020/15245-2; 2023/08726-2).
This work has made use of data from the European Space Agency (ESA) mission
Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
<https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.
This research has made use of the SIMBAD astronomical database <cit.>, and
the VizieR catalogue access tool <cit.>
operated at CDS, Strasbourg, France.
This publication makes use data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the
Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. 




§ DATA AVAILABILITY


The data underlying this article are available in machine-readable form at the CDS (VizieR On-line Data Catalogue),
which correspond to the tables containing individual members' parameters and results. 














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§ CIRCUMSTELLAR EMISSION



The identification of cluster members showing  infrared (IR) emission is based on the WISE  colours diagram using photometry at 3.4 μm, 4.6 μm, and 12 μm that is commonly adopted to 
distinguish between disc-bearing stars and pre-main sequence (PMS) stars without discs. 
 <cit.> proposed limits on the [3.4 μm - 4.6 μm] × [4.6 μm-12 μm] diagram defining the expected locus for Class I and Class II objects, which are PMS with the presence of protoplanetary disc, as indicated by their significant IR-excess. On the other hand,  field stars and Class III PMS objects lacking dust discs do not show IR emission and are well separated in the WISE colours diagram.

Following the methodology described by <cit.>,  we extracted from the AllWISE catalogue only the sources with good photometry quality at band W3 (12 μm) and avoiding contamination from fake detections, according with the filters proposed by <cit.>: 0.45 < W3_rχ^2 <1.15 and W3_snr > 5, where rχ^2 and snr correspond to the photometric error and signal-to-noise ratio, respectively.

In Fig. <ref> (top panel), we show the WISE colours diagram for all the cluster members
 that were selected from the AllWISE catalogue.  As expected for the older clusters, most objects do not exhibit an IR-excess and are located within the region of Class III or field stars, indicating the lack of protoplanetary disc.
 
Only 13 objects belonging to the youngest clusters of our sample appear in the region of disc-bearing stars (Class I and Class II). Nine of them are members of Col205, which corresponds to a significant fraction of disc-bearing stars (11 percent), considering the number of members we studied. For young clusters at this age (42 Myr), it is observed a lower fraction of disc-bearing stars <cit.>. In the case of NGC2659a (30 Myr old), we found only 3 Class II objects, meaning a fraction of 3 percent of stars with disc, more compatible with other clusters of the same age.  
The other Class II object is member of NGC 2168, whose low fraction (0.1 percent) of disc-bearing stars is in agreement with the age 119 Myr found for this cluster

The bottom panel of Fig. <ref>  displays a similar diagram showing the [H-K] colour from 2MASS as a function of [3.4 μm - 4.6 μm], which confirms the low-level of IR emission for most members of the studied clusters.
Once the diagrams in Fig.  <ref> are constructed for cluster members, we suggest 
 that the lacking-disc stars of our sample are Class III objects. That means, once they have a high membership probability, they can be distinguished from MS field stars.









§ ADDITIONAL FIGURES






































 Paper Title: Chemical abundance gradients of organic molecules within a protostellar disk.




















































Levi G. Walls
levigwal@umich.edu

0000-0001-6253-1630]Levi G. Walls
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107

0000-0002-2555-9869]Merel L.R. van 't Hoff
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107



0000-0003-4179-6394]Edwin A. Bergin
Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1107















§ ABSTRACT 
Observations of low-mass protostellar systems show evidence of rich complex organic chemistry. Their low luminosity, however, makes determining abundance distributions of complex organic molecules (COMs) within the water snowline challenging. However, the excitation conditions sampled by  differing molecular distributions may  produce substantive changes in the resulting emission.  Thus, molecular excitation may recover spatial information from spatially unresolved data. By analyzing spatially-unresolved NOrthern Extended Millimeter Array (NOEMA) observations of CH_3OH and CH_3CN, we aim to determine if CH_3OH and CH_3CN are distributed differently in the protostellar disk around HOPS-370, a highly-luminous intermediate mass protostar. Rotational diagram analysis of CH_3OH and CH_3CN yields rotational temperatures of 198 ± 1.2 K and 448 ± 19 K, respectively, suggesting the two molecules have different spatial distributions. Source-specific 3D LTE radiative transfer models are used to constrain the spatial distribution of CH_3OH and CH_3CN within the disk. A uniform distribution with an abundance of 4×10^-8 reproduces the CH_3OH observations. In contrast, the spatial distribution of CH_3CN needs to be either more compact (within ∼120 au versus ∼240 au for CH_3OH) or exhibiting a factor of ≳ 15 increase in abundance in the inner ∼55 au.  A possible explanation for the difference in spatial abundance distributions of CH_3OH and CH_3CN is carbon-grain sublimation.























 



§ INTRODUCTION

Complex organic molecules (COMs) are carbon-bearing molecules with 6 or more atoms <cit.>. Many COMs are formed in ice mantles on dust grains. For example, hydrogenation of CO ice can lead to the formation of simple oxygen-bearing COMs, such as methanol (CH_3OH), up to biologically relevant molecules like glycerol <cit.>. Whether a COM will remain in the ice or transitions into the gas depends on energetics and the local temperature <cit.>. The binding energies of many COMs are similar to that of water, resulting in the transition between gas and ice to occur roughly around the water snowline (∼ 150 K for pressures P ∼ 10^-12 bar).  For example, the binding energy is 5640 K for amorphous solid water <cit.>, 5728 K for methanol <cit.>, and 4954 K for methyl cyanide  <cit.>, as measured for carbonaceous grains.   

Based on this picture, the spatial distribution of COMs in circumstellar material is expected to be uniform post-sublimation within the snowline in the absence of gas phase chemical processing. However, for some species, for example, the nitrogen-bearing COM methyl cyanide, CH_3CN, (additional) gas phase formation routes have been proposed <cit.>. In addition, chemical processing in hot (≳300-500 K) gas may be possible, potentially following the destruction of carbonaceous grains <cit.> or through high-temperature pathways that become active <cit.>. 

For low-mass sources, the snowline lies within a few au from the star in disks up to a few tens of au from the star in protostellar envelopes, making resolving the snowline location technically challenging. As such, most studies of low-mass protostellar sources have focused on the detection of COMs <cit.> or the chemical inventory of individual sources <cit.>, rather than their spatial distribution (see reviews by e.g., ). 

Evidence for chemical changes in hot gas have been found toward high-mass protostars. <cit.> is one of the first studies of the high mass star forming region Orion KL to indicate the observed kinematic and spatial difference in N-COM and O-COM abundances could be due to hot gas phase chemistry. More recent studies on the Orion KL region also show abundance enhancements of N-COMs relative to O-COMs <cit.>.  
<cit.> finds the ratio of column densities in hot gas relative to warm gas for N-COMs is approximately 5 times higher than for O-COMs. While all of these studies toward high-mass systems indicate resolved chemical segregation <cit.> no systematic study of the spatial distribution of COMs within the snowline of low-mass sources has been conducted. 


The disk around the highly-luminous protostar HOPS-370 <cit.> also displays evidence of hot-gas chemistry based on observations with the NOrthern Extended Millimeter Array (NOEMA; van 't Hoff et al. accepted). Being an intermediate mass protostar <cit.> with a luminosity of ∼314 L_⊙ <cit.>, this system is warm, which pushes the water snowline out to ∼225 au. This source has been well-studied and well-modeled: ALMA observations resolved a disk structure in the dust continuum emission (of 62 au) and rotation in multiple molecular lines perpendicular to the outflow direction, suggesting the presence of a near edge-on (inclination of 74) gas disk <cit.> that subtends a radial extent of ∼ 05. As such, HOPS-370 is a good target to study the spatial distribution of COMs within the snowline.

In this paper, we use spatially unresolved NOEMA observations of  and  in the disk around HOPS-370 to probe their spatial abundance distributions within the snowline using molecular excitation and source-specific radiative transfer modeling.
In  <ref>, we show how  can be and is used as a temperature probe. 
In  <ref> and  <ref>, we present the  and  observations toward HOPS-370 and our analysis that indicates differences in the spatial distributions of these species within the NOEMA beam.  These results are discussed in  <ref> and the main conclusions are summarized in  <ref>.




§  AS A TEMPERATURE PROBE
 
Symmetric-top molecules like  are excellent probes for the gas temperature. These molecules break symmetry along the rotational axis in one dimension.  Thus their energy states are characterized by the two quantum numbers J (total angular momentum) and K (the projection of J onto the axis of symmetry).  
Radiative transitions for symmetric-tops have ΔJ = 1 and ΔK = 0, while ΔK≠ 1 transitions are forbidden. 
The selection rules for dipole transitions imply collisions dominate over radiative processes when relative populations are explored across K ladders (same J - different K), and thus probe the gas temperature <cit.>.  To further enhance this capability, the rotational emissions of symmetric top molecules for transitions with the same J but different K are found closely spaced in frequency. Thus a single observation might capture multiple energy states at once.


In general, the excitation temperature, T_exc, defines the relative population of upper level u to lower level l in the Boltzmann distribution:


    N_u/N_l = g_u/g_lexp( -E_u - E_l/k_B T_exc).


If molecular emission is in LTE then the excitation temperature is equal to the kinetic temperature (T_k).  For complex molecules, with multiple emission lines sampling a range of excitation states, it is common to use the rotation diagram method <cit.> to measure the rotational temperature, T_rot.  Here the rotational temperature is a relative measure of the excitation temperature.  Below we explore the relation of T_rot and T_k for the J=14-13 transition of CH_3CN using a series of statistical equilibrium calculations to lay the ground work for our subsequent observational analysis.  

The methodology laid out by <cit.> states

    ln( N_u/g_u) = ln( N_T/Q(T_rot)) - E_u/T_rot,

where N_u is the upper level column density of the molecule, g_u the statistical weight of that level and E_u is the upper energy level in units of K. The total column density of the molecular gas is given by N_T and the partition function is Q(T_rot). 

For optically thin transitions, ln (N_u / g_u) is related to the observed total integrated flux, W in units of K km s^-1, via

    N_u/g_u = 8π k_B/h c^3ν^2 W/A_ul g_u,

where k_B is the Boltzmann constant, G is the gravitational constant, h is Planck's constant, c is the speed of light, A_ul is the Einstein coefficient for spontaneous emission, and ν is the transition frequency.  

With Eq. <ref> presented as such, a linear regression can be fit in semi-log space to the data, ln(N_u/g_u), as a function of upper energy level, E_u. The intercept is then proportional to the total column density and the negative slope yields the rotational temperature. For a fixed abundance, lowering the rotational temperature, T_rot, steepens the slope.

We use   slab model radiative transfer simulations <cit.> of  J=14_K - 13_K, K=0-12 transitions for different kinetic (gas) temperatures, H_2 density, and total CH_3CN column densities.  The spectroscopic data and collisional rate coefficients we adopt in these simulations are provided by <cit.>. For a given set of physical conditions  calculates the line emission, from which upper level column densities and rotational temperatures are derived using the method outlined above.  Fig. <ref> displays the resulting rotational temperatures versus gas density for different kinetic (gas) temperatures, and total   column densities.   Based on this analysis, LTE (in this case T_rot = T_k) is reached near densities of n_H_2∼ 10^6 cm^-3.  Furthermore, LTE conditions are retrieved (within a factor of 2) when n_H_2 < 10^6 cm^-3 provided N_T >10^16 cm^-2 and the emission is optically thin. 

Additionally, a statistical equilibrium argument illustrates that LTE is appropriate for realistic disk temperatures and densities such as those described above (see  <ref> for our disk model). Ignoring radiative processes, a system is in LTE when densities exceed the critical density, i.e. n_H_2≫ n_cr≡ A_ul/C_ul, where C_ul is the collisional rate coefficient from the upper level to the lower level. <cit.> show updated collisional rate coefficients for E- (K = 3n ± 1 for n ≥ 0) which are lower by at most a factor of 5 as compared with the rates from <cit.>; A- (K = 3n for n ≥ 0) is even closer to agreement. We thus approximate the most-up-to-date E- collisional rate cofficients of <cit.> at 140 K (roughly corresponding to the lowest temperature in our disk model). Calculating the critical density in this most conservative case, we find values do not exceed ∼3.4×10^7 cm^-3 (occurring for K≤ 2 transitions). This is approximately an order of magnitude lower than 2.2×10^8 cm^-3 (corresponding to the lowest gas number density in our disk model). Other locations in the disk are increasingly dense and hot and, thus, would only further solidify the fact the disk is in LTE. This effectively validates the use of CH_3CN to probe the gas temperature as demonstrated previously <cit.>. 


In our observational analysis we will explore, and contrast, the rotational temperatures of CH_3OH to CH_3CN.  
CH_3OH is an asymmetric top molecule and its rotational temperature relationship to the gas kinetic temperature is more complex.  However, if its emission is in LTE then a comparison between measured rotational temperatures may be revealing.
For analysis of observations in the PErseus ALMA CHEmistry Survey (PEACHES), <cit.> performs 1D, isothermal non-LTE radiative transfer calculations of . They find that for the ( 5_1,  4) A^- - ( 4_1,  3) A^-  transition (ν_t = 0) at (243915.788 MHz) transition, the excitation temperature approached the kinetic temperature to within 10-30% for gas density estimates appropriate for this type of sources (between ∼4×10^9 and 10^12 cm^-3). A similar analysis is done by <cit.> in their study of IRAS 16293, albeit for a different frequency range, but with a similar conclusion. Furthermore, models of <cit.> show LTE is appropriate for  at densities around ∼10^9 cm^-3, albeit for a low kinetic temperature, T_k = 50 K. We target the same frequency range as the PEACHES sources and the density for the HOPS-370 disk falls within the same range (see  <ref>), suggesting LTE is a valid assumption for our modeling of the optically thin emission spectrum of .    




 

§ OBSERVATIONS
 
HOPS-370 was observed by the Northern Extended Millimeter Array (NOEMA) as a part of the Chemistry And Temperature: Searching with NOEMA for Destruction Of carbon GrainS project (CATS-N-DOGS; HOPS-370 project code: W20AF) four times from December 2020 to March 2021 (van 't Hoff et al. accepted). In this work, we use the first three observations, which were at 1 mm spanning the frequency ranges of 240.9–248.6 GHz and 256.4–264.1 GHz each with 2 MHz (Δ v = 2.3 km s^-1 spectral resolution) and spent 3.0, 1.9, and 2.6 hours, on source in configuration 11C, 11C, and 10A, respectively.  

The data are calibrated using the CLIC package of the  software[<https://www.iram.fr/IRAMFR/GILDAS>], using the most recent  version available at the time of data reduction (July 2021). For each observation the bandpass calibrator was 3C84. The phase calibration was done using J0542-0913 and J0458-020 for each observation. The flux was calibrated using LKHA101 for all observations except for the second, in which MWC349 was used. 

Imaging and cleaning was done using the  package of . Due to the line richness of the spectrum, the data was imaged without continuum subtraction and no self-calibration was performed. The resulting beam size is  3.37× 0.6 (9^∘ ). The gas disk of HOPS-370 is ∼0.5in size (∼200 au in diameter at ∼400 pc; ), and thus unresolved in these observations. The noise level has been measured in a region off source in the image cube and is 16 mK in 2 MHz channels.

After extracting spectra toward the continuum peak position, the continuum was subtracted using a statistical method as, for example, used by <cit.> for the line-rich spectrum of the protostellar binary IRAS16293. This method entails the formation of a density distribution of flux values for the spectrum of each sideband. In the case of only continuum emission, the density distribution would represent a symmetric Gaussian centered at the continuum level. In the case of continuum and line emission, the Gaussian will contain an exponential tail toward higher values. The continuum level is then determined by fitting a skewed Gaussian to the distribution, and subtracted from the spectrum. The continuum-subtracted   J = 14_K - 13_K spectrum is shown in Fig. <ref>. 

We securely detect 12 transitions of the 14_K - 13_K ladder of  shown in Fig. <ref> (see also Table <ref>). The 16 mK noise level results in a weak detection of the K=11  transition. There are multiple other weak lines present near the K=12 frequency, which makes a detection of the K=12 transition very uncertain. We therefore do not report a detection of the K=12 transition. The K=5  transition at 257403.585 MHz coincides with the (18_3, 16) A^+ - (18_2, 17) A^-  transition (ν_t = 0) at 257402.086 MHz. We thus report a detection of the K=5  transition, but do not include it (nor the  transition) in the observed rotation diagram analysis. 






§ ANALYSIS
 
In this section, we model the emission of  and  via full 3D  radiative transfer calculations. The disk around HOPS-370 has a well-characterized physical structure <cit.>. We use this structure for 3D radiative transfer modeling with the LIne Modeling Engine <cit.> which includes both gas and dust emission and their respective optical depth effects. We then convolve the predicted emission to the NOEMA beam (at the source distance) to match the spatially unresolved observations. Using the strategy outlined in Sect. <ref>, we derive the rotational temperature for each molecular species.  A focus of our efforts will be to use the observed molecular excitation to set constraints on the overall spatial distribution of  and . In  <ref> we first describe the physical model in detail, and in  <ref> and  <ref> - <ref> we present the results of our modeling efforts for  and , respectively.


  



 §.§ Physical structure and model setup
  
Following the approach of <cit.>, we adopt the gas disk structure  from their modelling results of  emission, where 
the gas surface density, Σ_g, of the disk is assumed to be a power-law with an exponential cut-off <cit.>

    Σ_g(r) = Σ_0 ( r/r_c)^-γexp[-( r/r_c)^2 - γ],

where Σ_0 = 51.25 g cm^-2 is the gas surface density normalization, r is the cylindrical radius, r_c = 110 au is the disk critical radius, and γ = 0.89 is the surface density power-law index. 

The vertical structure of the disk is assumed to be in hydrostatic equilibrium, such that:

    h_g(r) = (k_B r^3 T_g(r)/G M_* μ_m m_H)^1/2

where k_B is the Boltzmann constant, G is the gravitational constant, M_* = 2.37 M_⊙ is the mass of the central protostar, μ_m = 2.37 is the mean molecular weight <cit.>, and m_H is the mass of the hydrogen atom. A radial power-law is adopted for the temperature profile, T_g(r), 

    T_g(r) = T_0 ( r/au)^-q

where T_0 = 998.7 K is the temperature at 1 au <cit.>. The temperature power-law index q=0.35 was chosen based on results for other protostellar disks, e.g., L1527 <cit.>. 
With this temperature profile, as shown in the right panel of Fig. <ref>, the 150 K water snowline is located at 225 au. 
See  <ref> for more discussion on the effect temperature profile choice has on our modelling work. 
 

The gas (mass) density then is found by integrating Eq. <ref>:

    ρ_g(r) = 1/√(2π)Σ_g(r)/h_g(r)exp[-1/2(z/h_g(r))^2],

where Σ_g(r) is given by Eq. <ref>, h_g(r) is given by Eq. <ref>, and z is the cylindrical height above the midplane. The gas number density, n_H_2, shown in the left panel of Fig. <ref> is found by dividing the gas mass density (Eq. <ref>) by the mean molecular weight.

The dust density is assumed to be 1% of the gas density <cit.>. Furthermore, the dust size distribution is assumed to follow n(a) ∝ a^-3.5 <cit.>, where a is the radius of the dust, which is assumed to be between 0.005 and 1 μm. Dust grains are assumed to be mixtures of water ice (mass fraction: 1.19×10^-3), iron  (mass fraction: 2.53×10^-4), olivine (mass fraction: 2.51×10^-3), refractory and volatile organics (mass fraction: 4.13×10^-3), orthopyroxene (mass fraction: 7.33×10^-4), and troilite (mass fraction: 5.69×10^-4). Following the results of <cit.>, we model the mass-weighted dust opacities presented in <cit.> using  <cit.>.  






The main focus of this work is investigating the effects of varying the abundance distribution (e.g., constant abundance uniformly distributed over the entire disk) of  and . The physical structure outlined above is used to perform LTE radiative transfer calculations for different transitions of  and  with  <cit.>.  solves the radiative transfer equation on a grid of Voronoi cells constructed from the Delauney triangulation of randomly chosen points, whose acceptance can be weighted based on, for example, a provided molecular abundance distribution. The result is an image cube containing the intensity for a prescribed spectral resolution. 

For both  and  we provide as input a Keplerian velocity profile and thermal Doppler broadening parameter in addition to the gas density (e.g. Eq. <ref>) and temperature structure (e.g. Eq. <ref>) as modeled by <cit.>. Additionally we adopt in  the distance <cit.> and inclination angle <cit.> for the disk around HOPS-370. The molecular abundance and its distribution for each molecular species is also provided and is the main quantity we vary.  

The spectroscopic information for both  and  (i.e. ν, g_u, E_u, A_ul) is present in  <cit.> and provided in a file conforming to the  format <cit.>. The spectroscopic data for  is provided 
by <cit.> (see also Table <ref>). 

Since  calculates the partition function internally using the energy levels in the provided molecular data file, we calculate the partition function in the same way for the analysis of the observations using 

    Q(T) = ∑_i g_i exp(-E_i/k_B T),

where g_i and E_i is the statistical weight and energy of level, i, respectively. In calculating the total column density for each species, the partition function is evaluated at the rotational temperature. This is justified given LTE conditions have been established within the disk (see  <ref>).

Given the temperatures considered here, however, higher  vibrational states can contribute to its partition function. We determine the multiplicative vibrational correction factor at the rotational temperature (a factor ≲2) by interpolating the correction factors provided in CDMS. Importantly, the derived rotational temperature does not depend on the partition function.

We include in our models the K=0-12 lines of the J=14_K-13_K transition, each at a velocity resolution of 0.16 km s^-1. While we include the  K=0 and 1 lines in our observational and model rotation diagrams, we do not include them when performing the linear regression as these transitions are blended (see Table <ref> for more information on what transitions are used in constructing the rotation diagrams in Figs. <ref>, <ref>, and <ref>).

The spectroscopic data for  is provided by <cit.> (see also Table <ref>).
From the  entry for , which consists of  torsionally-excited ν_t ≤ 2 states up to J=40 with A- and E-symmetry, we construct a molecular data file containing 38,643 total transitions up to 3 THz. The Einstein A_ul values are calculated as detailed by <cit.>, and we calculate the partition function from the line list as detailed above. Each line is modeled with a velocity resolution of 1 km s^-1.  


Each model yields an image cube containing the intensity in Jy/pixel for each line considered per molecule. For each line, the continuum is determined as the mean intensity in the first three velocity channels devoid of line emission. This is then subtracted from the rest of the velocity channels per line per molecule. This data is then converted to brightness temperature via the Rayleigh-Jeans law <cit.>:

    T_b = 1.222×10^3I/ν^2 θ_min θ_max,

where I is the spectral intensity in mJy, ν is the line frequency in GHz, and θ_min = 06 and θ_max=337 are the beam minor and major axes, respectively. After extracting the spectrum from the data cube, these modeled fluxes are then convolved with a 1D Gaussian assuming the velocity resolution of the NOEMA observations, i.e. Δ v = 2.3 km s^-1 and integrated with respect to velocity. The models, in which abundances and spatial distributions for  and  are varied, are analyzed with rotation diagram analysis and, in this way, compared with observations. 



 §.§ 
 
Fig. <ref> shows the rotation diagram for the  observations of HOPS-370 (see also van 't Hoff et al. accepted). A fit to the data using least-squares minimization yields a rotational temperature of T_rot = 198 ± 1.2 K and a total column density of N_T = (3.42 ± 0.04)×10^16 cm^-2. Given the higher number of  transitions with E_u ≳ 400 K relative to lower energy transitions, it is the higher energy transitions which strongly influence the derived rotational temperature. 

To model the  emission, we adopt a constant abundance of X(CH_3OH) = 4×10^-8 distributed uniformly throughout the 240 au disk, based on the results by <cit.> towards HOPS-370. This model is also shown in Fig. <ref> and results in a rotational temperature of T_rot = 197 K. 
Fitting the ν_t = 0 and 1 torsional states or the A- and E-symmetries separately results in very similar results. This is in agreement with the observed value.







 §.§ : constant abundance model
 
The rotation diagram for the observed K-ladder of J=14_K-13_K   transition is shown in Fig. <ref>. A least squares minimization results in an observed  rotational temperature of T_rot = 448 ± 19 K and a total column density of N_T = (8.28 ± 0.13)×10^14 cm^-2, given the discussion in  <ref>. The temperature, which is unaffected by our treatment of the partition function, is approximately 250 K higher than observed for  and well above the estimated uncertainties. This difference likely indicates different spatial distributions for both COMs. 

To study the spatial distribution of , we first explore a constant abundance distributed uniformly throughout the 240 au disk, as done for . The  abundance is varied to find the abundance that reproduces the observed peak K=4 flux to within 1σ (16 mK). We choose the K=4 line as it has an intermediate upper-level energy (E_u = 207 K) and large signal-to-noise (S/N ∼ 70). The K= 0 - 2 lines show some degree of blending, obfuscating the true fluxes of each line and the K=3 line (and K=6, 9, ...), while having high signal-to-noise, has a different statistical weight  due to the threefold symmetry of the methyl group and nuclear spin statistics. The K=5 transition is blended with a  line. As such, the line K=4 has the highest signal-to-noise of unblended transitions, is unaffected by statistical weights, and has an upper energy level that is readily observable.

The rotation diagram for the full J=14-13 ladder using the resulting constant abundance X(CH_3CN) = 5.2×10^-10 is shown in Fig. <ref>. This model severely underpredicts high K fluxes. This results in a rotational temperature of only T_rot = 225 K, which is ∼220 K cooler than observed. To obtain a higher rotational temperature, a higher  abundance is required in the warmer gas closer to the star. However, simply increasing the  abundance, results in low-K transitions being overpredicted.
Thus, a constant abundance  distribution is unable to reproduce the NOEMA observations, in contrast to . 






 §.§ : compact abundance models
 
The simplest way to increase the calculated rotational temperature while ensuring simulated fluxes agree with observations is by limiting the distribution of  to warmer gas. In these models, we assume a constant  abundance inside a given radius, r_out, and zero outside. Beginning with a radius, r_out∼ 25 au, we first systematically increase r_out in steps of ∼ 50 au. Based on the effect increasing r_out (that is, increasing the relative amount of warm gas we are introducing) has on the T_rot and K=4 line flux, we refine the step size such that a better agreement between model and observations is found. At each step, we determine whether the model and observed rotation diagram agree. This is accomplished when the K=4 line flux agrees with observations to within 1σ (16 mK) and the rotational temperature agrees to within 1σ (19.0 K). We limit the  abundance to be below 10^-4, which corresponds to the canonical CO abundance. Table <ref> provides the results of a set of models that met one or both of the above listed criteria.


cccccccc 

0pt 
Summary of compact model results with varying outer extent of emission

 r_out [au]    T_g(r_out) [K]     X_in(CH_3CN)     Peak T_b,K=4     Fits?^a     T_rot [K]     Fits?^b    N_T×10^14 [cm^-2]  

110     193     6.4× 10^-9     1.727     √     531                 12.15
 
110     193     4.0× 10^-9     1.518                 444     √     8.56 
 
115     190     4.2× 10^-9     1.726     √     436     √     9.39 
 
120     187     5.0× 10^-9     1.985                 452     √     11.17
 
120     187     3.1× 10^-9     1.713     √     386                 8.02 
 
167     167     9.0× 10^-10     1.731     √     257                5.13 
 
The observed values are: Peak T_b,K=4 = 1.725 ± 0.016 K, T_rot = 448 ± 19, N_T = (8.28 ± 0.13)×10^14 cm^-2. 
aIf a checkmark is present, this model reproduces the observed peak K=4 flux to within 1σ (16 mK).
bIf a checkmark is present, this model reproduces the observed rotational temperature to within 1σ (19.0 K).


When r_out≲ 25 au, the emitting area is too small, regardless of abundance; even with an abundance of 10^-4, the peak K=4 flux is not reached. Systematically increasing r_out, and thereby the emitting area, the smallest r_out for which the K=4 line flux can be reproduced is ∼ 100 au. However, for these models, the rotational temperature is generally ≳ 1σ larger than observed. 
A model with r_out = 110 au and abundance of X(CH_3CN) = 4.0×10^-9 (see Fig. <ref>, panel a) is able to reproduce the observed rotational temperature, but yields a K=4 line flux that is >1σ lower than observed. Increasing the abundance slightly to X(CH_3CN) = 6.4×10^-9 reproduces the K=4 flux, but results in a too high rotational temperature, as was the case for the model with r_out = 100 au. 
The smallest r_out that matches both criteria is 115 au (Fig. <ref>, panel b). This model has an abundance of X(CH_3CN) = 4.2×10^-9.  
Thus, the inner boundary for a compact model is ∼110 au.

Moving r_out further out quickly produces results comparable to the models with r_out = 110 au. As shown in Fig. <ref> (panel c), when r_out = 120 au, a model with X(CH_3CN) = 3.1×10^-9 is able to reproduce the observed K=4 line flux, but not the rotational temperature. On the other hand, a model with a higher abundance of X(CH_3CN) = 5.0×10^-9 reproduces the rotational temperature (to within 1σ), but does not reproduce the K=4 line flux. The outer boundary for a compact model is thus ∼120 au. In summary,  can be distributed as a compact abundance inside ∼ 110 - 120 au, while the observed rotational temperature and K=4 line flux cannot be simultaneously reproduced (to within 1σ) for models outside this radial range.








 §.§ : jump abundance models
 
The compact models illustrate the sensitivity of  to warm gas or more strictly the need for its emission to have significant contributions from gas closer to the star.  However, these models are somewhat inconsistent with chemical expectations.  That is, nearly the entire disk lies within the  sublimation point.  Thus  should have an abundance that is distributed throughout the disk, albeit with a strong emission component that samples the hot gas closer to the star, but also the warm gas that lies just inside the water snowline. Another way to increase the contribution of hot gas compared to a constant or compact abundance distribution is by including a non-zero abundance outside r_out that is lower than the inner abundance. This is accomplished with a jump abundance model <cit.>. 

We systematically increase r_jump and the relative inner and outer  abundances such that again the K=4 flux and rotational temperature are within 1σ of observed values.  As before, we only consider abundances less than the canonical CO abundance; that is, X_in(CH_3CN) < 10^-4. 
For realistic lower limits on the  abundance in the outer disk, we adopt an abundance ratio X(CH_3CN)/X(CH_3OH) of 0.1% based on the weighted-average column density ratio presented by <cit.> using observations of high-mass protostars in the ALMAGAL sample and literature values toward low- and intermediate-mass protostars. This ratio is ∼10 times lower than found by, for example, <cit.> (CALYPSO survey) and <cit.> (PEACHES survey), so the lowest outer abundance we consider (X_out(CH_3CN) ∼ 10^-11) may be conservative. 



Table <ref> shows the result of varying the jump location and corresponding inner/outer abundances required to reproduce the observations. In each case, there is a  enhancement of ≳ 15 inside the jump radius relative to outside. Fig. <ref> shows rotation diagrams for a range of these models.

ccccccc 

0pt 

Summary of jump-abundance models that can reproduce the observations.

 r_jump [au]     T_g(r_jump) [K]     X_in(CH_3CN)     X_out(CH_3CN)     Peak T_b,K=4 [K]     T_rot [K]     N_T×10^14 [cm^-2]

35      288     9.60× 10^-7     4.90× 10^-10     1.721     444     8.00 

55      246     1.90× 10^-8     4.60× 10^-10     1.727     447     8.64 

90      207     6.10× 10^-9     2.70× 10^-10     1.726     445     9.66 

100     199     5.30× 10^-9     1.70× 10^-10     1.712     448     9.70 

105     196     4.95× 10^-9     1.15× 10^-10     1.715     448     9.72 

The observed values are: Peak T_b,K=4 = 1.725 ± 0.016 K, T_rot = 448 ± 19, N_T = (8.28 ± 0.13)×10^14 cm^-2. 





While not shown in Fig. <ref>, when the jump location is located inside 35 au (where the disk temperature is ≳ 290 K), levels with K≤4 are generally well-populated, while higher K-levels are not. For K ≥ 6, peak line fluxes are ∼45% lower than observed on average. This ultimately results in a derived rotational temperature about 200 K lower than observed, even with an inner  abundance in excess of the canonical CO abundance.
This is primarily due to having too small an emitting area as well as optically thick dust obscuring the emission from the inner most region.  However, a jump at 35 au with an abundances of X_in(CH_3CN) = 9.60× 10^-7  and X_out(CH_3CN) = 4.90× 10^-10 provides a good match to the  observations. 

Good agreement between model and observation is found for models with jump radii out to 105 au (Fig. <ref> and Table <ref>). As r_jump is increased we also see that while inner/outer abundances decrease (as expected and also seen in the compact abundance distributions), the inner abundance is consistently ≳15 larger than the outer abundance.  Furthermore, we see that despite retrieving the observed rotational temperature and K=4 peak flux, the total column density increases as the jump radius is moved outward. This is expected since more material is being added as the radius increases. Our most concentrated jump model, with r_jump = 35 au has a total column ∼2σ lower than observed. In contrast, our best-fit compact abundance model with r_out=115 au, results in a total column that is ≳ 11σ larger than observed. 
This is to say, the observed total column necessitates a jump abundance model with  being distributed inside ∼55 au, much more concentrated than the best-fit compact abundance model. 

When the jump location is shifted outward of 105 au (where the disk gas temperature is <196 K), the jump model stops being as effective as, for example, the compact emission model with r_out=110 au at reproducing observed values. Outside 105 au, the contribution from the warm rather than hot gas begins to dominate, and it becomes difficult to reproduce emission from the high K-levels. 

To summarize, models with 35 < r_jump < 105 au are able to simultaneously reproduce the observed rotational temperature as well as the peak K=4 flux. For models with r_jump < 35 au, the emitting area is too small, and, as such, the rotational temperature cannot be reproduced despite using an abundance of 10^-4. For models where r_jump is larger than 105 au, we cannot distinguish whether  is distributed as a jump model or compact abundance. Furthermore, when the total column density is considered, models indicate  needs to have a compact component with the jump radius being r_jump≲ 55 au.   




§ DISCUSSION
 



We observe  to have a rotational temperature of T_rot = 448 ± 19 K, which is approximately 250 K warmer than observed for . Models of their spatial distribution indicate that  must indeed be different from , with  shown to be enhanced in the warm gas relative to . Nominally this requires a  production mechanism in the hot gas or a destruction mechanism preferentially diminishing  in the warm gas. Below we explore how detailed excitation might influence our results and discuss potential mechanisms that could enhance  in the hot gas or deplete it in the warm gas.




 §.§ Effect of temperature profile choice
 
The choice of temperature structure could have a significant impact on the outcome of our modelling work <cit.>. A potentially influential choice in our model is the assumption of a vertically isothermal temperature profile. <cit.> and <cit.> illustrated that the temperature structure for an evolved disk is not simply vertically isothermal. However, for deeply embedded protostars, the addition of an envelope (and its opacity) acts to intercept and reprocess the incident starlight <cit.>. Ultimately, this results in only a weak vertical temperature gradient compared to more evolved disks. This is confirmed for HOPS-370 specifically by 2D  calculations <cit.>, as shown in Fig. <ref>; the temperature does not vary greatly as a function of polar angle (height), especially between 1 and 100 au (demarcated by the gray arrows). In other words, the assumption of a vertically isothermal structure is reasonable for HOPS-370. 


Assuming the temperature is vertically isothermal implies that a 1D temperature profile can be used to describe the entire 2D disk temperature. We adopt the power-law temperature profile from <cit.>, who model the molecular line emission and dust continuum, albeit separately. This is done because calculating the dust temperature self-consistently using the radiative transfer calculations of  is computationally expensive. This quickly becomes intractable when full dust thermal calculations are to be done for each model with a different spatial distribution for multiple molecular species. To assess the validity of this simplified temperature profile, Fig. <ref> shows the difference between the dust temperature calculated self-consistently with  and the power-law temperature profile adopted in  <ref> based on the NOEMA  observations.


A power-law fit to the 2D  data yields:

    T_g+d(r) = 1470K (r/au)^-0.47

whereas in  <ref> we find that the observed  rotation diagram can be described with the following power law as modelled by <cit.>:

    T_g(r) = 998.7K (r/au)^-0.35.

With a larger T_0 and steeper power-law index, q, the fit to the  calculated temperature profile is hotter at 1 au but colder further out in the disk (> 10 au) than the power-law we assume (i.e. Eq. <ref>). This could affect our constraint on the jump location. However, even at 240 au, where there is the largest deviation between the two temperature profiles, we see there is at most a difference of ∼35 K between models. Any difference between the temperature structures where the jumps were placed is less than a factor of 2, which is less than ∼50 K. Even though a slightly different radial temperature profile may influence the exact location of the outer radius in the compact models or the jump radius in the jump-abundance models, it will not affect the conclusion that  is differently distributed than  within the snowline of HOPS-370.


    






 §.§ Pebble drift
  
One potential explanation for the existence of an increase in  in the inner disk could be pebble drift. From the perspective of dynamics, dust of different sizes interacts differently with the gas. The small dust is dynamically well-coupled to the gas, while large dust experiences maximum inward radial drift. The presence of a jump abundance distribution could in principle be explained by the inward drift of large dust, causing an enhanced abundance in the inner parts of the disk where the ice sublimates <cit.>. 
 


However, if radial drift were occurring and giving rise to the jump abundance seen in , we would also expect to a jump abundance distribution for  unless there is some additional ice chemistry serving to keep  enhanced outside the  outer radius or jump location in the outer disk. Since the simpler constant abundance model reproduces the  fluxes observed by NOEMA, a jump model is unnecessary. Thus, radial drift is unlikely to be the reason for the difference in abundance distributions between  and .




 §.§ Destroying CH_3CN in warm gas
 
Models of their spatial distribution indicate inner  abundances are heightened by a factor of ∼ 15 relative to outer within ∼105 au, where warm gas (150 K≲ T_g ≲ 300 K) resides. This enhancement could be due to a chemical mechanism reducing the  in the warm gas. 

Two such destruction mechanisms could be gas-phase reactions between He^+ or H_3^+ for both  and . The reaction between  He^+ and  occurs on a timescale similar to that for He^+ and .  However,  the reaction between  H_3^+ and  is approximately a factor of three faster than for  <cit.>. After ice sublimation this would lead to quicker re-equilibration towards chemical equilibrium for  with a timescale that is a factor of three shorter.   This could lead to a shift in the abundance of  relative to  in the outer disk gas, but any model would have to be fine-tuned in timing to capture this change within the short period of time as timescales are < 10^5 yrs at these high densities <cit.>.
Thus, the elevated  abundance is difficult to explain by a gas-phase destruction mechanism. 






 §.§ Carbon-grain sublimation
 
We see  is characterized by hot gas, where there is a relatively concentrated  abundance enhanced relative to . This could indicate the presence of hot gas phase chemistry taking place, which could be the result of carbon grain sublimation. Carbon-grain sublimation is the process by which molecules in refractory carbon grains are released to the gas. This occurs at ≳ 300 K; the so-called “soot line" <cit.>. 


When dust grains are destroyed at the soot line, more carbon and nitrogen will be observed in the hot gas relative to oxygen, since the cometary carbon and nitrogen budget is higher for dust grains than for ices, while the reverse is true for the cometary oxygen budget <cit.>.  After the initial release of carbon- and nitrogen-bearing species in the hot gas,  could then form there via reactions between CH_3^+ and HCN, the product of which, CH_3CNH^+, rapidly equilibrates to  via electronic dissociative recombination <cit.>. The conditions in HOPS-370 could facilitate these gas phase reactions. 


<cit.> provide three potential signatures of carbon grain sublimation and top-down chemistry to occur within young disks: 


  * An increased abundance of  N-COMs e.g.,  inside the soot line,

  * N-COMs being distributed within a smaller spatial area than O-COMs e.g., , and therefore

  * N-COMs exhibiting higher excitation temperatures than O-COMs. 

Potential evidence for carbon-grain sublimation has been observed by <cit.> for high mass protostars in the ALMAGAL survey. However, we also see potential evidence in HOPS-370.


The excitation temperature for the 1 mm NOEMA observations of  (see also van 't Hoff et al. accepted) is T_rot = 448 ± 19 K, approximately 250 K warmer than observed for  (T_rot = 198 ± 1.2 K). 

While we find a compact  abundance out to 115 au is able to reproduce the observations, so do several jump abundance models. All of these jump abundance models have the jump occurring between 35 ≲ r_jump≲ 105 au, corresponding to 196 < T_g < 288 K. Models with smaller jump radii (e.g. r_jump≲ 55 au) are favored based on the total column density calculated being most similar to observations. 
While the 300 K line is expected to correspond to the soot line, our analysis is agnostic of this.  However, we clearly see an enhanced  abundance inside of the expected soot line location. Thus, criteria 1 and 3 are satisfied.


The rotation diagram analysis we perform significantly constrains the location where any hot gas chemistry might be taking place, despite the size of the NOEMA beam. Regardless, future observations with high angular resolution could spatially resolve the location of the water snowline (∼225 au or 095) and maybe even the putative soot line (∼31 au or 013 assuming the soot line occurs at 300 K for HOPS-370).   Regardless, there is clear evidence for chemical shifts in this system that hint at changes that can be resolved.




§ CONCLUSION
 
With spatially-unresolved 1 mm NOEMA observations of  and  in the protostellar disk around HOPS-370, 
we aim to determine if  and  are distributed differently. 
We use the radiative transfer modelling software  in combination with a source-specific physical model to calculate spectra for each of these molecules for different parameterized spatial distributions. Rotation diagrams are used to analyze the observed and modeled spectra. Our main findings are as follows:


  *  has an observed rotational temperature of T_rot = 198 ± 1.2 K, while  is observed to be significantly warmer with T_rot = 448 ± 19 K.

  * The  observations can be reproduced with a constant abundance model out to 240 au.

  * The  observations cannot be reproduced by a uniform distribution similar to . Instead, the  emission is either described by a more compact abundance distribution within ∼115 au or by a jump-abundance distribution with the increase in abundance happening for radii between ∼35 and ∼105 au. In the inner region, the  is enhanced by a factor of ∼15 compared to the outer region.  






These changes suggest chemical gradients are present inside the water snowline within hot gas.
HOPS-370 is a unique laboratory for characterization of chemistry in hot gas which is needed to understand the formation of inner solar/stellar system planets like our own.  

L.G.W. acknowledges support from the Rackham Merit Fellowship, under which a majority of this work was funded. M.L.R.H. acknowledges support from the Michigan Society of Fellows. 




§ MOLECULAR DATA

Tables <ref> and <ref> provide spectroscopic data for observed and modelled transitions of  and , respectively (see also  <ref>). For , we provide in Fig. <ref> the observed transitions (with quantum numbers as they appear in the  database), along with the resultant Gaussian fits calculated with .

cccccccc 

0pt 
Spectroscopic data for  obtained from 
Upper level     Lower level     Frequency     E_u     g_u     A_ul     Observed flux     Flux error 
 
(J_K_a, K_c)      (J_K_a, K_c)     [MHz]     [K]          [s^-1]     [K km s^-1]     [K km s^-1] 
 

            v_t = 0                  


(25_3, 22) A^-     (25_2, 23) A^+     241588.758     803.7     204     8.19e-05     7.4304     0.56  
 
( 5_0,  5) A^+     ( 4_0,  4) A^+     241791.352      34.8      44     6.05e-05     25.436     1.17  
 
(24_3, 21) A^-     (24_2, 22) A^+     242490.245     745.7     196     8.19e-05     9.3002     1.33  
 
(23_3, 20) A^-     (23_2, 21) A^+     243412.610     690.1     188     8.20e-05      9.499     0.193 
 
( 5_1,  4) A^-     ( 4_1,  3) A^-     243915.788      49.7      44     5.97e-05     26.813     0.714 
 
(21_3, 18) A^-     (21_2, 19) A^+     245223.019     585.7     172     8.23e-05     13.751     0.977 
 
(20_3, 17) A^-     (20_2, 18) A^+     246074.605     537.0     164     8.25e-05     17.744     0.978 
 
(19_3, 16) A^-     (19_2, 17) A^+     246873.301     490.7     156     8.27e-05     17.737     0.979 
 
( 4_2,  2) A^+     ( 5_1,  5) A^+     247228.587      60.9      36     2.12e-05     21.876     1.44  
 
(18_3, 15) A^-     (18_2, 16) A^+     247610.918     446.6     148     8.29e-05     19.583     1.39  
 
(17_3, 14) A^-     (17_2, 15) A^+     248282.424     404.8     140     8.30e-05     20.338     0.258 
 
(19_3, 17) A^+     (19_2, 18) A^-     258780.248     490.6     156     9.01e-05     13.311     0.451 
 
(20_3, 18) A^+     (20_2, 19) A^-     260381.463     536.9     164     9.14e-05     15.384     1.22  
 
(21_3, 19) A^+     (21_2, 20) A^-     262223.872     585.6     172     9.28e-05     11.785     0.879 
 
 
(14_1, 14) E_2     (13_2, 11) E_2     242446.084     248.9     116     2.29e-05     21.325     1.32  
 
(16_2, 15) E_1     (15_3, 13) E_1     247161.950     338.1     132     2.57e-05     19.189     1.45  
 
(23_1, 22) E_1     (23_0, 23) E_1     247968.119     661.4     188     4.44e-05     8.6042     0.934 
 
(24_1, 23) E_1     (24_0, 24) E_1     259581.398     717.0     196     4.91e-05     6.8224     1.38  
 
(21_4, 18) E_2     (20_5, 16) E_2     261061.320     623.6     172     3.02e-05     7.3807     1.09  
 
( 2_1,  1) E_1     ( 1_0,  1) E_1     261805.675      28.0      20     5.57e-05     19.046     0.315 
 
            v_t = 1                  


( 5_0,  5) A^+     ( 4_0,  4) A^+     241267.862     458.4      44     6.00e-05     10.786     0.621 

( 5_1,  4) A^-     ( 4_1,  3) A^-     241441.270     360.0      44     5.79e-05     13.471     0.643 

(17_2, 15) A^-     (16_1, 15) A^-     259273.686     652.7     140     5.58e-05     8.2852     0.74  

 
( 5_1,  4) E_2     ( 4_1,  3) E_2     241238.144     448.1      44     5.75e-05     11.515     0.654 
 
(12_2, 11) E_2     (13_3, 11) E_2     247840.050     545.1     100     6.28e-05     12.673     0.696 
 
The quantum numbers (QNs) for each level of the transition are listed as (J_K_a, K_c), where J is the total rotational angular momentum, K_a is the projection of J onto the A and C inertial axes, respectively. Following the parentheses in each entry is the symmetry A or E. For transitions with A-symmetry, the parity is denoted with + or -. For transitions with E-symmetry, the subscript 1 indicates K_a ≥ 0, while the subscript 2 indicates K_a < 0 



    


cccrrccc 

0pt 
Spectroscopic data for  obtained from LAMDA
Upper level     Lower level     Frequency     E_u     g_u^a     A_ul      Observed flux      Flux error  

J_K     J_K     [MHz]     [K]          [s^-1]     [K km s^-1]     [K km s^-1]

                v_t = 0             

14_12     13_12     256817.163     1118.7     29.0     3.89e-4     —        —   
 
14_11     13_11     256930.140      955.2     14.5     5.62e-4     1.5713     0.266 
 
14_10     13_10     257033.444      805.8     14.5     7.20e-4     4.6887     1.05  
 
14_9     13_9      257127.036      670.5     29.0     8.63e-4     9.5756     0.78  
 
14_8     13_8      257210.878      549.4     14.5     9.91e-4     7.317      1.2   
 
14_7     13_7      257284.935      442.4     14.5     1.11e-3     9.6442     1.2   
 
14_6     13_6      257349.180      349.7     29.0     1.20e-3     21.106     1.2   
 
14_5     13_5      257403.585      271.2     14.5     1.29e-3     —        —   
 
14_4     13_4      257448.128      207.0     14.5     1.36e-3     21.196     0.301 
 
14_3     13_3      257482.792      157.0     29.0     1.41e-3     31.866     0.269 
 
14_2     13_2      257507.562      121.3     14.5     1.45e-3     28.774     0.305 
 
14_1     13_1      257522.428       99.8     14.5     1.47e-3     30.202     0.198 
 
14_0     13_0      257527.384       92.7     14.5     1.48e-3     23.013     0.338 
 
aThe statistical weights presented in the  entry for  have the contribution from the nuclear spin factored out. However, those present in the  file (and used in our calculation of the total column density) have the nuclear spin factor included. See, for example, <cit.> and <cit.> for discussion of the statistical weights for molecules with C_3ν symmetry, like .
While not used in constructing the observed rotation diagram, we include the K=5 and 12 lines in our models and the linear regressions. In both observed and model rotation diagrams, we do not include the K=0 and 1 lines in the rotation diagram analysis as these transitions are blended. 



 
 

 
  
 
 

 
 














NOEMA






 <cit.>,
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        , <cit.>,
        astropy <cit.>, <cit.>, 
        SciPy <cit.>,
        Matplotlib <cit.>
          











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