Paper Title: Evolution of H$α$ Equivalent Widths from $z \sim 0.4-2.2$: implications for star formation and legacy surveys with Roman and Euclid.




firstpage–lastpage
Full classification of Pauli Lie algebras
    [black]https://orcid.org/0000-0003-1626-2761Lennart Bittel
    August 5, 2024
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§ ABSTRACT  
Past studies have investigated the evolution in specific star formation rate (sSFR) and its observational proxy ( equivalent width; EW) up to z ∼ 6; however, such measurements are prone to selection biases that may overestimate the typical sSFR/EW at a given redshift. We investigate the `intrinsic' (corrected for selection and observational effects)  EW distributions of z ∼ 0.4 - 2.2 narrowband-selected  samples from HiZELS and DAWN using a forward modeling approach where we assume an `intrinsic' exponential EW distribution, apply selection and filter effects, and compare with observed  EW distributions. We find an `intrinsic' EW – stellar mass anti-correlation, M^γ, with steepening slopes γ = -0.18±0.03 to -0.24^+0.06_-0.08 at z ∼ 0.4 and z∼ 2.2, respectively. Typical rest-frame  EW increases as (1+z)^1.78^+0.22_-0.23 for a 10^10  emitter from 15^+2.4_-2.3Å (z ∼ 0.4) to 67.7^+10.4_-10.0Å (z ∼ 2.2). The  EW redshift evolution becomes steeper with decreasing stellar mass highlighting the high EW nature of low-mass systems at high-z. We model this redshift evolving EW–stellar mass anti-correlation, , and find it produces  luminosity functions and SFR functions strongly consistent with observations validating the model. This allows us to use our best-fit  model to investigate the relative contribution of  emitters towards cosmic star formation at a given epoch. We find  >200Å emitters (potentially bursty systems) contribute significantly to overall cosmic star formation activity at z ∼ 1.5 - 2 making up ∼ 40% of total cosmic star formation consistent with sSFR > 10^-8.5 yr^-1 (mass doubling times of ≲ 300 Myr) which we find makes up ∼ 45 - 55% of cosmic star formation at z ∼ 2. Overall, this shows the importance of high EW systems in the high-z Universe. Our best-fit  model also reproduces the cosmic sSFR evolution found in simulations and observations (coupled with selection limits typically found in narrowband surveys), such that tension between the two can simply arise from selection effects in observations. Lastly, we forecast Roman and Euclid grism surveys using our best-fit  model and include the effects of limiting resolution and observational efficiency. We predict number counts of ∼ 24000 and ∼ 30000 0.5 < z < 1.9  emitters per deg^-2, respectively, down to >5×10^-17  including 10^7.2 - 8  galaxies at z > 1 with  >1000Å. Both Roman and Euclid will enable us to observe with unprecedented detail some of the most bursty/high EW, low-mass star-forming galaxies near cosmic noon.
 







§ INTRODUCTION


Constraining the star-formation histories of galaxies is crucial in understanding their stellar mass buildup over cosmic time and how they become the large, present-day structures in the local Universe. Star-formation history is generally smooth with galaxies residing on a linear correlation between their star-formation rates (SFRs) and stellar mass (e.g., `Main Sequence'; ). However, a unique subset of galaxies reside above this correlation with elevated SFR at a given stellar mass and are known as starbursts given their enhanced burst of star formation activity. Understanding how bursty star-formation activity occurs, how dominant or minor this mode of star formation was at various epochs, and what its overall contribution to total cosmic star-formation activity was is important in understanding the evolutionary path of galaxies into the present structures we see in the local Universe.


One commonly used tracer to identify sources that may have recent bursty star-formation activity is the /UV ratio where  traces instantaneous star-formation (∼ 3 - 10 Myr; ) via bright, massive, short-lived O and B type stars and UV continuum light tracing longer periods of SF activity (∼ 100 – 200 Myr; ). An elevated  emission in respect to UV continuum flux would, in principle, signify a star-forming galaxy undergoing some recent, bursty star-formation activity in the past ∼ 10 Myr. This approach has been used to study burstiness of galaxies in the local (e.g., ), z < 2  (e.g., ), and z > 2 Universe  (e.g., ). However an elevated /UV ratio can also result from a varying IMF (e.g., ), underestimation (overestimation) of UV () dust correction, and variations in the nebular-stellar dust reddening (e.g., ).  emission can also be affected by the escape fraction of ionizing photons (e.g., ). Furthermore, binary stellar populations may increase the lifetime of massive O and B type stars raising the SF timescale of  emission to be comparable with UV  (e.g., ). 


While /UV has been used for estimating recent star formation and potential burstiness, a recent study by <cit.> used a sample of 310 z ∼ 2 star-forming galaxies with Keck/MOSFIRE and LRIS coverage and high-resolution, rest-frame UV HST and found no correlation between /UV and SFR surface density (Σ_SFR). <cit.> also finds a lack of elevated Civ and Siiv emission and P Cygni features with increasing /UV ratio which would have been indicative of the presence of massive stars and stellar driven winds and concludes that /UV ratio may not be a reliable tracer of bursty star-formation activity. We therefore need a more reliable indicator of burstiness, especially for high-z star-forming galaxies.

The term `burstiness' is used commonly within the literature but with differences in regards to its meaning. In the context of this work, we refer to `burstiness' and `bursty star formation' as a galaxy that has undergone a rapid and intense period of star-formation activity relative to its past star formation. This is also traced by both the instantaneous sSFR and emisison line EW which is defined as the ratio of recent to past star-formation activity. Other definitions of `burstiness' include the variations of SFR factoring in both the rise and fall of star formation activity encompassing its full SFH. This should not be confused with our definition as mentioned above.

 equivalent width (EW) is an observational alternative to investigate the recent star-forming histories of galaxies and has several advantages. First,  EW is by definition an observational proxy of specific star formation rate (sSFR) and, therefore, traces the doubling mass timescale from recent star-formation activity (e.g., ). Second, it is even more sensitive to recent star-formation activity as it is the ratio between  emission and rest-frame R continuum (traces even older stellar populations; timescales > 100Myr; ). Lastly,  EW is, essentially, independent of dust correction if one assumes a nebular-stellar reddening of unity (f = E(B -V)_stellar/E(B-V)_nebular, f ∼ 1) although assuming different f values will result in systematically different   with factor 10^0.4 k(6563Å) E(B-V)_stellar [ 1/f  - 1], where k(6563Å) is the reddening curve at 6563Å. Overall,  equivalent width has several key advantages over /UV as a tracer of bursty star-formation activity.

The distribution of  EW in a sample of galaxies would qualitatively show the range of recent star-formation history from normal to bursty star-forming galaxies. Past studies of  EW distributions find an anti-correlation with stellar mass with a redshift evolution where low-mass galaxies tend to have higher EWs and the typical EWs  increase with redshift as (1+z)^1.8 up to z ∼ 2 and (1+z)^1.3 from z ∼ 2 - 6 (e.g., ). This suggests for wider  EW distributions and increasing number of bursty star-forming galaxies with increasing redshift and at lower stellar masses. This implies that bursty star-forming galaxies may be an important contributor to overall star-formation activity in the high-z Universe. For example, <cit.> used a sample of 1000 0.3 < z < 2  emitters with HST grism spectroscopy and found that rest-frame EW>100Å sources contributed ∼ 34% to the overall z ∼ 1  - 2 cosmic star-formation rate density. Recent JWST observations are also finding a ubiquitous population of high  EW systems ). Interpreting  EW as a tracer of burstiness, these results may suggest rapid intense periods of star formation activity may be a common mode of star formation activity in the high-z Universe. <cit.> suggest that a bursty phase of star-formation in the high-z Universe would result in high EW systems and such a feature would decrease with decreasing redshift as the stellar continuum increases with the overall stellar mass build-up. 


However, past studies of typical  EWs, its anti-correlation with stellar mass, and its redshift evolution may be overestimated simply due to selection biases not being taken into account. We demonstrated this in <cit.> where we used the z = 0.47  narrowband-selected LAGER sample <cit.> with a forward modeling approach also used in this study (see <ref>) to investigate how selection effects could affect the underlying measurements derived from  EW distributions. In that study, we found an intrinsic (selection-corrected) EW_0() ∝ M^-0.16±0.03 with >5σ significance from a null correlation such that an intrinsic anti-correlation between  EW and stellar mass exists. This intrinsic correlation can also recover the  luminosity function coupled with a stellar mass function. However, not correcting for selection effects results with a steeper correlation of EW_0() ∝ M^-0.25±0.04 in strong agreement with past studies (e.g., ) and does not recover the  luminosity function. 



In order to properly gauge the  EW redshift evolution and anti-correlation with stellar mass, we need to carefully take into account and correct for selection effects. In this study, we extend our work from <cit.> by applying our intrinsic  EW modeling via our forward modeling approach on the narrowband-selected High-z Emission Line Survey (HiZELS; ) and the Deep And Wide Narrowband survey (DAWN; ). Both surveys are currently the largest near-IR narrowband surveys available and have several key advantages for the purpose of this work. The samples are selected based on emission line contribution within the narrowband filter and have measurement for emission line flux, continuum flux density, and equivalent width. The selection functions are also very simple to model, typically a combination of line flux and EW-limits as demonstrated in <cit.>. The emission line-selection also limits redshift uncertainties to the narrowband FWHM and eliminates any internal redshift evolution in the sample (e.g., a sample that has a wide redshift bin width). Lastly, the combination of HiZELS and DAWN covers from z ∼ 0.4 to ∼ 2.2 allowing for a comprehensive overview of the changing star-formation histories of star-forming galaxies from high-z to low-z. Specifically, we would address the role of bursty star-forming galaxies at different cosmic times in regards to cosmic star-formation activity. Lastly, we will use our modeling approach to also present predictions for the range of  EW planned legacy surveys with Roman and Euclid can observe given survey conditions (e.g., flux limits, grism resolution, survey area, etc.) 

The paper is structured as follows: the samples are defined in <ref>. We then provide an overview of our forward modeling approach in <ref> and refer the reader to <cit.> for more details. <ref> highlights our results in the study where we present measurements of the intrinsic, selection-corrected  EW – stellar mass anti-correlation and its redshift evolution, show how it can reproduce both the  luminosity function and SFR function, how different types of star-forming galaxies contribute to overall cosmic star-formation activity at different cosmic times, and how selection effects can augment measurements of the cosmic  EW and sSFR evolutions causing disagreement with simulations. In <ref> we predict the number counts and range of  EW that can be observed with  Roman and Euclid. In <ref>, we present discussions on the implications our work has on survey planning, how starburst galaxies may be important sources in the high-z Universe, what our results imply regarding the potential of high  EW emitters to be potential sources of reionizing the Universe at z > 6, and how our constraints on the  EW distribution can help in assessing low-z contaminants/interlopers in high-z  emitter-selected samples. Lastly, we present our main conclusions in <ref>.

Throughout this paper, unless otherwise stated, we assume ΛCDM cosmology (H_0 = 70 km s^-1 Mpc^-1, Ω_m = 0.3, Ω_Λ = 0.7) and a <cit.> initial mass function (IMF). Magnitudes assume the AB magnitude system.




§ SAMPLES




In this work, we use samples derived from two of the largest, ground-based near-IR narrowband surveys described below. Survey parameters and sample sizes are listed in Table <ref>.






 §.§ HiZELS



The High-z Emission Line Galaxy Survey (HiZELS; ) is a large 2 deg^2 narrowband survey covering COSMOS and UDS/SXDS using four narrowband filters on Subaru/SuprimeCam and UKIRT/WFCAM. The four narrowband filters were used to select  emitters at z ∼ 0.4 - 2.2 <cit.>, as well as   and  emitters at z ∼ 0.84 - 3.2 and z ∼ 1.5 - 4.7, respectively <cit.>, in 4 discrete redshift slices of Δ z ∼ 0.01 - 0.03 in width. We use the  samples as selected by <cit.> for this work and refer the reader to the specific study for details regarding the survey design and sample selection.

Briefly, the sample selection used in HiZELS is uniform throughout all four narrowband samples. The selection function is threefold: 1) a 3σ narrowband magnitude (line flux) limit, 2) a rest-frame EW cut of 25Å, and 3) a color significance cutof Σ > 3. The narrowband magnitude limit is imposed to remove sources with poor S/N. The equivalent width cut is set to remove any sources that could potentially mimic an emission-line within the narrowband filter profile (e.g., continuum features). The last selection limit is the color significance cut which removes sources where the nebular color excess (broadband – narrowband) is not significant in respect to photometric errors. This essentially removes sources that may have emission lines present within the narrowband filter, but are not statistically significant above a threshold (e.g., Σ > 3 can be thought of as 3σ significance). 

The color significance, Σ, is the ratio of the broadband minus the narrowband flux densities (proxy for emission line flux) versus the associated photometric errors in both bands added in quadrature. Essentially it is a signal-to-noise measurement typically used in narrowband surveys where a Σ > 3 cut would translate as a S/N > 3 limit in the emission line flux uncertainies. This ensures a clean sample and the removal of sources mimicing emission line galaxies simply due to photometric uncertainties in the narrowband and/or broadband.

Selection is done on a pointing-to-pointing basis to take into account varying depths per chip (see Figure <ref> for HiZELS/NBH). We refer the reader to Table 2 of <cit.> for further details on exposure times and 3σ limiting magnitudes for each pointing. The NB921 imaging was done using Subaru/SuprimeCam and is homogeneous in depth for the 4 pointings covering COSMOS, while the central pointing in UDS is 0.6 mag fainter compared to the other 4 pointings. NB921 UDS is also ∼ 2 mag deeper than the corresponding COSMOS imaging. The other 3 HiZELS narrowband samples (NBJ, NBH, and NBK) were observed using UKIRT/WFCAM which consists of 4 chips per pointing. Figure <ref> shows the 3σ NBH depth inhomogeneity which can vary by ∼ 0.5 mag. Each pointing/chip can also have varying effective comoving volumes in respect to other pointings/chips due to masked regions. All of these factors need to be taken into account when we generate mock samples that will be used in assessing the intrinsic EW distributions for each sample.




 §.§ DAWN

The Deep And Wide Narrowband (DAWN) survey is a near-IR narrowband imaging survey of COSMOS, UDS, and EGS using a single, customized NB1066 filter (λ_c = 10660Å, FWHM= 35Å) on NEWFIRM installed on the 4 m KPNO/Mayall telescope. <cit.> was first to investigate the z = 0.62  properties using a single, deep 0.22 deg^2 pointing in COSMOS selecting 116  emitters reaching a 5σ  line flux limit of ∼ 1 × 10^-17 erg s^-1 cm^-2. <cit.> extended this work by incorporating 8 shallow, flanking fields around the deep COSMOS pointing increasing the total areal coverage to ∼ 1.5 deg^2. The 5σ depth of the deep pointing is 23.6 mag while the surrounding flanking fields have 5σ depths varying between 20.1 and 22.1 mag corresponding to  line flux limits of 5 - 30 × 10^-17 erg s^-1 cm^-2. We refer the reader to Table 1 and Figure 1 of <cit.> for more details regarding the survey design and pointing properties. A total of 241 z = 0.62  emitters were selected by <cit.> using the following selection: rest-frame EW_0 > 11Å (observer-frame EW> 18 Å), 5σ limiting magnitude cuts, and Σ > 3. 




§ FORWARD MODELING OF EW DISTRIBUTIONS


In this section, we outline the methodology used in measuring intrinsic  EW distributions following the approach outlined in <cit.> and refer the reader to that study for further details.



 §.§ Brief Overview of Methodology


We use the approach outlined in <cit.> which was used for measuring the intrinsic  EW distribution of z = 0.47  emitters from the LAGER survey <cit.>. The approach models  emitters by randomly sampling from an assumed intrinsic EW distribution and stellar mass function to generate an intrinsic  sample with rest-frame EW, stellar mass, and  luminosity, where the latter is inferred from the EW and stellar mass (assuming a mass-to-continuum luminosity model). <cit.> found that an exponential rest-frame EW profile best represents observations of  EW distributions and is defined as:


    p( EW|W_0) = 1/W_0 e^-( EW/W_0)

where  is the characteristic EW and describes both the average EW for a sample (EW= [0 - ∞)Å) and the width of the exponential distribution. Higher  corresponds to wider EW distributions and includes higher rest-frame EWs. We also assume an exponential distribution profile in this study. 

We generate a mock sample by first assigning an intrinsic, rest-frame EW and stellar mass, where the former is randomly assigned by sampling the intrinsic, exponential rest-frame EW distribution, as defined in Equation <ref> with an assumed , and the latter by randomly sampling from an assumed stellar mass function (see <ref>). Rest-frame R-band continuum luminosity, L_R, is determined by combining the assigned stellar mass with our redshift-dependent M/L_R calibration and assigning L_R with a 0.1 - 0.2 dex scatter (depending on the redshift) to take into account variations in the IMF and star-formation histories at a given stellar mass (see <ref>). The combination of L_R and rest-frame EW determines the line flux/luminosity and the contribution of  is determined by using an empirical, observationally-driven calibration (see <ref>). We then incorporate the narrowband filter transmission curve effects by convolving the modeled line flux and continuum flux densities with the narrowband and broadband filters to get the observed magnitudes (see <ref>). The mock photometry will be used as part of the selection process to measure line flux, rest-frame EW, and continuum flux densities using the same techniques applied in typical narrowband surveys (e.g., assuming top-hat approximations, see <ref>).


With each mock sample, we convolve the selection functions corresponding to the observed samples, as described in <ref>. These selection functions include an observed + line flux limit cut (narrowband magnitude limit), observer-frame + EW cut, and a color significance cut. This results in mock samples having incompleteness due to selection that best represent the observed samples which will be used in this study to constrain intrinsic EW distributions. The observer-frame + EW cut places a selection limit that primarily affects bright continuum (high-mass) galaxies. The + line flux and color significance limits dominate at faint continuum (low-mass) galaxies and translate to an increasing EW cut with decreasing stellar mass/continuum luminosity (inversely proportional to EW). This can arbitrarily form an EW – stellar mass anti-correlation due to selection effects alone, although we show in <cit.> that a shallower, but statistically significant (> 5σ) anti-correlation is present when said effects are taken into account.

The mock samples described above are used to compare with the rest-frame EW distributions from the observed samples described in <ref>. This is done by first subdividing both the observed and mock samples in bins of rest-frame R-band luminosity, L_R. We then compare the rest-frame EW distributions of the mocks with the distributions from the observed samples per each L_R bin using a χ^2 minimization approach where  is augmented to reach a minimum χ^2 and the observed errors are Poisson. The best-fit  is essentially the e-scaling factor that defines the shape/width of the intrinsic EW distribution before incompleteness from selection effects. The intrinsic  also represents the mean EW of the sample at a given L_R.

In the following sections, we describe in detail the varying components and assumptions made in our approach.












 §.§ Stellar Mass Function Assumption



One of the main starting points of our approach is populating our mock samples by randomly drawing  emitters from an assumed stellar mass function (SMF). These are then converted to rest-frame R-band luminosity using an empirical model described in <ref>, which is an observable in narrowband surveys. It is crucial that the assumed SMF used in generating the mock samples be consistent with the observations. 

We follow the same approach as in <cit.> by assuming the z ∼ 0.4 - 2.2  HiZELS SMFs of <cit.>. These SMFs are based directly on -selected emitters and represent the underlying population of star-forming galaxies. The observed samples we use in this study are also drawn primarily from HiZELS (see <ref>) and follow the same selection function and thin redshift slices. 






 §.§  contamination correction



Narrowband filters are powerful observational tools to capture  emission lines and other strong nebular emission lines in blind surveys. However, the resolving power is not high enough to differentiate between  and the neighboring 6541,6583Å emission lines. This is also true for blind slitless grism surveys (e.g., ) and for strong EW systems selected using nebular broadband excess (e.g., ). Although assuming a constant / ratio to correct for  contamination would work within first order, studies of the BPT diagnostic clearly show / ratios varying by a dex for star-forming galaxies (e.g., ). Starburst galaxies are reported to have low / ratios indicative of low gas-phase metallicities in comparison to `normal' star-forming galaxies (e.g., ) and the typical / ratios of galaxies decreases with increasing redshift (e.g., ). 

In this study, we provide results based on both + and  EWs, where the latter takes the  contamination into account. We adopt the <cit.> EW calibration which is a simplification of the SDSS-based calibration <cit.>. The latter is a 5th degree polynomial fit to the / line ratios of SDSS-selected  emitters as dependent on rest-frame + EW and is defined as:

    log_10/ =     -0.924 + 4.802x - 8.892x^2 
        +6.701x^3-2.27x^4+0.279x^5

where x = log_10EW_0(+). This calibration and comparable polynomial fit calibrations have been used extensively in  narrowband surveys (e.g., ). 

A simplification of this calibration was presented by <cit.> which used median stacked / ratios of z = 0.84  emitters with Keck/MOSFIRE and Subaru/FMOS spectra in rest-frame + EW bins. They found the median line ratios were consistent with the local SDSS calibration with minimally elevated / ratios at EW(+)∼ 25Å, although still consistent within 1σ. The simplified, linear calibration is defined as:

    f(/) = -0.296 log_10EW_0(+) +0.8

for rest-frame EW(+) between ∼ 15Å and ∼ 400Å where f(/) is the / flux ratio. We take into account this limited range in our mock samples by setting the  contribution being constant at the bounds (e.g., EW< 15Å mock emitters have / ratios fixed at 2.8 consistent with EW= 15Å using Equation <ref>). The key advantage of using Equation <ref> is the reduced number of free parameters in comparison to Equation <ref> and the focus on observables being the + EW. 


We implement the <cit.> calibration in our mock samples by using the intrinsic rest-frame EW drawn from Equation <ref> and determine the / contribution as being f_ = / +. In the case where we are modeling the intrinsic  EW, we apply a 1 + f_ factor to determine the intrinsic + EW. Note that narrowband surveys do not have the resolving resolution to deblend the two emission lines such that line fluxes and EWs from narrowband photometry are the combination of  and . Therefore, we need to incorporate an / prescription to simulate narrowband and broadband photometry in our forward modeling approach.




 §.§ Mass-to-Continuum Luminosity



One of the observables in narrowband surveys is the rest-frame R-band continuum flux density centered at 6563Å, which is sensitive to both young and old stellar populations and traces the bulk of stellar mass in a galaxy (e.g, ). Therefore, the rest-frame R-band continuum observed in  narrowband surveys can be used as a proxy for stellar mass.

In <cit.>, we formed an empirical, single power law calibration to convert stellar mass to rest-frame R-band luminosity, L_R. This was done using COSMOS2015 stellar mass measurements <cit.> along with L_R measured from CTIO/NB964 and DECam z photometry for all sources in LAGER with a stellar mass coverage between 10^7 and 10^11  (see Figure 2 of ).

We follow a similar approach using the COSMOS2015 stellar mass catalog, cross match it with our sample catalogs, and measure the R-band luminosity – stellar mass correlation. Since we are interested in the continuum luminosity and given the well-constrained photometric redshifts from the COSMOS2015 catalog, we augment the selection to include sources that have photometric redshifts within the selection range of the sample (e.g., HiZELS/NB921 selected sources with 0.35 < z_phot < 0.45) but without the rest-frame + EW> 25Å (11Å for DAWN) and Σ > 3 criteria. This way our calibration is based on galaxies within a specific redshift range where the continuum flux density is consistent with rest-frame R-band centered at 6563Å based on the photo-z regardless if emission lines are above the selection threshold or not.




Figure <ref> shows the L_R – stellar mass correlation for all our samples. The HiZELS/NB921 observations cover the widest range in stellar mass and can be characterized by a single power law up to 10^10   with a slope of -0.6 consistent with our z = 0.47 measurement <cit.>. However, we find a shallower trend at >10^10  up to the highest stellar masses in the sample. We take this feature into account by modeling the L_R – stellar mass correlation in the form of a double power law defined as:

    L_R = 2 L_R,10y^γ/1.+y^-0.6

where y = M/(10^10), γ is the high-mass end slope, and L_R,10 is the rest-frame continuum luminosity at 10^10 . The low-mass end is fixed to a slope of -0.6 constrained by the HiZELS/NB921 and LAGER/NB964 <cit.> samples. This is due to the lack of coverage in the z > 1 samples. 



We use Equation <ref> along with the best-fit parameters shown in Table <ref> to convert stellar mass to rest-frame L_R when generating mock  emitters in our forward modeling approach. We randomly perturb the assigned L_R per each generated  emitter by drawing from a normal distribution centered at 0 and with 1σ set to the scatter in L_R at fixed stellar mass. The scatter takes into account variations in star-formation histories and IMFs per galaxy. 





 §.§ Filter Profile Bias and Modeled Photometry



Measurements of line flux and EW in narrowband surveys assume the filter is a top-hat with 100 percent transmission and width corresponding to the narrowband FWHM. Given that narrowband filters are not perfect top-hats, this introduces a bias in the final line flux and EW measurement where intrinsically bright sources detected towards the filter wings will appear faint with low EWs. This also introduces a probed volume bias where faint galaxies will only be detected towards the filter center where transmission is at its highest resulting in low comoving volumes probed compared to intrinsically bright galaxies that can be observed even towards the filter wings (e.g., ).

To take this effect into account, we randomly assign a redshift to each mock emitter that covers the full wavelength coverage of the narrowband filter. We then convolve the emission line flux (measured from assigned EW_0 and stellar mass with mass-to-L_R conversion) and continuum flux density (measured from assigned stellar mass and Equation <ref>) with the narrowband and broadband transmission curves and measure the observed flux densities and magnitudes. The latter will be used in <ref> when applying selection criteria. We then follow the top-hat assumption and measure the mock line fluxes, EW, and continuum flux density by assuming:

    f_NB =     f_c + F_L/ΔNB
    
    	f_BB =     f_c + F_L/ΔBB

where f_NB and f_NB are the flux densities observed in the narrowband and broadband filters with their corresponding FWHM defined as ΔNB and ΔBB, respectively. Both flux densities are a combination of the observed continuum flux density, f_c, and line flux F_L. We use this F_L and rest-frame EW (F_L/(f_c (1+z))) affected by the top-hat assumption to compare with observations and constrain EW distributions.




 §.§ Constraining EW distributions



The final step to form a mock  sample is to apply the selection criteria corresponding to the observed sample that we want to compare to and constrain the EW distributions. This is done by using the mock photometry generated using the mock line flux, continuum flux density, and EW convolved with the narrowband and broadband transmission curves as described in <ref>. The selection criteria for each of our observed samples is described in <ref> and is primarily a combination of an EW cut, line flux cut, and color significance cut. Since the narrowband surveys in this work also have inhomogeneity in depth and survey area as shown in Figure <ref>, we apply the selection criteria on a pointing-to-pointing basis.

We use the mock  samples to constrain the intrinsic EW distributions by subdividing the sample in bins of stellar mass and comparing the underlying EW distributions of both the observed and mock samples (with selection effects applied to mimic the observations). Since the mock samples are dependent on the intrinsic  that was assigned to the sample, we know then based on our assumptions what the intrinsic EW distribution should be to reproduce the EW distributions in observations when selection biases are included. 

We start by normalizing both distributions (observed sample and mock) to unity, where the binning scheme is based on the EW distribution from the observed sample. We then follow a χ^2 minimization approach:

    χ^2 = ∑_i=0^i=N (n_o,i - n_m,i()/σ_o,i)^2

where n_o,i and n_m,i are the normalized number of emitters in the observed and mock samples for the i^th bin, respectively, and σ_o,i is the Poisson error for the observed sample in the i^th bin. We report the assigned intrinsic  that minimizes χ^2 as the best-fit that describes the intrinsic  distribution that best matches the EW distribution in observations when selection biases are applied.




§ RESULTS





 §.§ Hα Equivalent Width and Continuum/Stellar Mass










Past studies of star-forming galaxies have reported a  EW – stellar mass anti-correlation with the typical EW increasing as (1+z)^1.8 up to z ∼ 2 at fixed stellar mass (e.g., ). However, we showed in <cit.> that selection limits can severely bias the underlying correlation by increasing the EW – stellar mass slope by a factor of ∼ 1.6 based on the z ∼ 0.5  LAGER. In this section, we extend that work to investigate the  EW – continuum R-band luminosity/stellar mass anti-correlation where we follow the methodology outlined in <ref> to measure the `intrinsic' anti-correlation (selection/sample bias corrected).


Figure <ref> shows EW_0(+) and EW_0() in terms of rest-frame L_R continuum luminosity centered at 6563Å. We find a clear EW – L_R anti-correlation at all observed redshifts with the slope becoming steeper with increasing redshift. At fixed L_R, the typical EW increases with redshift highlight a strong redshift evolution. For each sample, we fit a simple power law of the form:

    (L_R) = W_0,40(L_R/10^40erg s^-1cm^-2Å^-1)^β [Å]

where  is the typical rest-frame EW at a L_R, β is the correlation slope, W_0,40 is the typical rest-frame EW at L_R = 10^40 erg s^-1 cm^-2 Å^-1. The best-fits are overlaid in Figure <ref> with the 1σ and 2σ confidence regions highlighted as dark and light colored regions. The best-fit parameters are also shown in Table <ref>.

The HiZELS samples show an EW(+) – L_R anti-correlation with statistical significance >4σ from a null hypothesis (no correlation), while the DAWN z = 0.62 sample shows ∼ 3σ significance which is mostly due to the smaller sample size in comparison to the HiZELS samples. The slope is found to become steeper with increasing redshift from β(z=0.4) = -0.19 to β(z = 2.23) = -0.55 corresponding to a factor of ∼ 3 increase. The  EW – L_R anti-correlation also shows a strong statistical significance of ∼ 4 - 8σ from a null hypothesis with a steeper slope from β(z = 0.4) = -0.25 to β(z = 2.23) = -0.51 (a factor of ∼ 2 increase). The shallower increase in the slope compared to the EW(+) – L_R anti-correlation is attributed to the contribution of  at low EWs where / is higher.

The strong increase in the slope for both EW(+) and EW() anti-correlations with L_R is somewhat driven by the changing M/L_R with redshift as shown in <ref> and Figure <ref>. We take this into account by repeating the same measurement and analysis by converting L_R to stellar mass using our M/L_R calibration from Equation <ref>. Figure <ref> shows the anti-correlation with stellar mass where we fit simple power laws of the form:

    (M) = W_0,10(M/10^10M_⊙)^γ [Å]

where W_0,10 is the typical EW for an emitter at 10^10  and γ is the anti-correlation slope. The best-fit power laws are shown in Table <ref> where we find that the slopes become shallower in comparison to the EW–L_R for both EW(+) and EW(). We find that the anti-correlation between EW and stellar mass for both emission lines has >3.5σ significance. The slope becomes steeper with increasing redshift although shallower in comparison to the EW–L_R anti-correlation. This implies that the previous steepness is primarily driven by changes in M/L_R over cosmic time; however, a steeper slope in the EW–stellar mass anti-correlation is still present. This suggests that the EW distributions of low-mass  emitter populations become wider and consist of even higher EW systems with increasing redshifts. It also implies that the contrast in EW distributions between low-mass and high-mass  systems becomes even stronger with increasing redshift as low-mass systems have wider EW distributions than high-mass systems by cosmic noon.




  §.§.§ Redshift Evolution of Typical  Equivalent Widths


The typical  EW for 10^10   emitters is also seen to increase with redshift rising from ∼ 15Å at z ∼ 0.4 to ∼70Å by z ∼ 2 suggesting a redshift evolution where  emitters at high-z have significantly higher EWs compared to their low-z counterparts. Figure <ref> shows that at any fixed stellar mass the typical EW increases with increasing redshift but also shows that the redshift evolution is mass dependent where low-mass  emitters have a stronger increase in  with increasing redshift.

We model this mass-dependent redshift evolution using a power law of the form:

    W_0(M,z) = W_0,10(1+z)^α_1(M/10^10 M_⊙)^γ_0 + α_2(1+z) [Å]

which is a combination of two main properties: 1) the evolution of  highlighted by W_0,10 (typical EW of a z = 0  emitter with stellar mass of 10^10 ) and a redshift evolution defined by the slope, α_1; and, 2) the evolution in the anti-correlation slope described by γ_0 (power law slope at z = 0) that evolves with redshift with a slope, α_2. We simultaneously fit the EW–stellar mass anti-correlation and redshift evolution using all our samples and (M) measurements. 

Figure <ref> shows the best-fits of our redshift and stellar mass dependent EW model with the parameters highlighted in Table <ref>. We find that the typical  EW at 10^10  evolves as ∼ (1+z)^1.8, similar to previous measurements up to z ∼ 2 (e.g. ). At z = 0, the + EW – stellar mass anti-correlation is nearly flat with γ_0 = - 0.03^+0.07_-0.06 while the  EW anti-correlation has a slope of -0.10^+0.07_-0.07, which is probably due to an increase in  contribution at the high-mass end causing a shallower slope. However, both the EW(+) and EW() show a steepening slope where γ(z ∼ 2) is ∼ -0.25 and ∼ -0.35 for + and , respectively. This implies that  emitters not only have increasingly higher typical EWs with increasing redshift, but also their low-mass populations show increasingly wider EW distributions in comparison to their high-mass counterparts. In the scope of star-formation activity, our results suggest that populations of low-mass  emitters have a wide, diverse range of star-forming galaxies from significantly high EWs corresponding to high sSFR (active, bursty star-forming systems) to low EW, steady star-formation histories.






  §.§.§ A Biased View of the  EW Evolution


In our previous work <cit.>, we showed how selection biases can result in a steeper EW – stellar mass correlation for the z = 0.47  sample drawn from the narrowband LAGER survey. Figure <ref> shows the redshift evolution of typical  rest-frame EW at a fixed stellar mass range of 10^9.5 - 10  for different selection limit cases. We find that typical  EW corrected for selection bias (black solid line) increases by nearly an order of magnitude from z ∼ 0 to z ∼ 2 and is consistent with a (1+z)^1.8 evolution found in previous studies (e.g., ). However, we find observations to have systematically higher measurements of  EWs up to z ∼ 2. 

We test how selection limits alone can resolve the discrepancy between past observations and our intrinsic measurement of the  EW redshift evolution. In a typical narrowband survey, two major selection limits are applied to identify potential emission line galaxy candidates. The first is an EW cut to remove bright narrowband sources that have strong continuum features mimicking emission lines in narrowband photometry. The second is a line flux cut (e.g., narrowband magnitude limit) to remove sources that are close to a certain depth in the narrowband images.

Figure <ref> shows the typical evolution for varying rest-frame EW cuts and we find an increasing typical EW measurement with increasing selection limit at fixed redshift. The effect is more pronounced at lower redshift and this is caused by the narrower EW distributions at such regimes (e.g., low ). Applying a line flux limit of 10^-16.5 erg s^-1 cm^-2 in conjunction with an EW_0 > 10Å limit strongly matches with the (1+z)^1.8 evolution of <cit.> and observational measurements. This does not imply each survey has these specific selection limits as other combinations of line flux and EW_0 limits can match with the observations. The main point is that it clearly shows not correcting for selection limits will result in an inflated typical  EW measurement for any stellar mass range. This highlights the importance of measuring the intrinsic  EW distributions, its anti-correlation with stellar mass, and its redshift evolution. 







 §.§ Equivalent Width and  luminosity functions


Three major statistical properties of an emission line galaxy population are the stellar mass function (integrated past star-formation history), equivalent width distribution (the ratio between instantaneous and past star formation activity), and the luminosity function (instantaneous star formation activity). Our forward modeling approach described in <ref> assumes a stellar mass function and M/L_R to populate our samples with  emitters with varying luminosities and EWs. We constrain the `intrinsic' EW distribution and show that it depends strongly on both redshift and stellar mass (see Figures <ref> and <ref>). In this section, we test if our (M,z) model can also reproduce  luminosity functions which would provide strong evidence supporting our model and approach.

We follow a similar approach as discussed in <ref> with the difference being that we assign EW to our samples based on the best-fit (M,z) in Equation <ref> and Table <ref>.  luminosity is measured using the rest-frame EW and rest-frame R-band luminosity. We do not assume any prescription for dust in this case such that  luminosities reported here are based on observered (not corrected for dust) luminosities. We also apply a constant EW cut consistent with HiZELS and DAWN when making our predicted  LFs to compare with observations. This is because the observed LFs are drawn from EW-limited samples.

Figure <ref> shows our predicted  LFs based on assuming the <cit.> SMF along with the intrinsic  (M,z) model that was constrained using the HiZELS and DAWN  EW distributions. We find a strong agreement between our predicted LFs and the observed LFs from <cit.> and <cit.>. This suggests that our observationally-constrained (M,z) model can recover the  LF based on simple assumptions. However, despite this strong agreement, there are some minor discrepancies. For example, the predicted z = 0.40 LF is slightly underestimated at L_ < 10^40.75 erg s^-1. We also find the predicted z = 2.23  LF somewhat overestimates in the faint-end although still within 1σ. The strong agreement with observed LFs is an indication that our `intrinsic' EW–stellar mass–z model can robustly describe all three major statistical properties of emission line galaxies. In our previous work, we found similar agreement in <cit.> where the `intrinsic' –stellar mass anti-correlation was able to recover the  LF while not correcting for selection biases resulted in an underestimation of the bright-end.  We also find that our predicted  LFs are in strong agreement with the z = 0.4  LF of <cit.>, which is based on a 16 deg^2 survey using Subaru/HSC. We also find a strong agreement with the  LFs from the HST grism survey, WISP <cit.>, suggesting that our approach can also describe  populations regardless of selection. 

Overall, the agreement between our predicted  LFs and the observed LFs from the literature gives strong evidence that our intrinsic (M,z) model empirically constrained by the HiZELS and DAWN samples can robustly describe the general  emission line galaxy population up to cosmic noon. It also allows us to investigate global  population properties at different equivalent widths (sSFRs) which, in turn, allows us to study the variety of star-formation histories and how it contributes to the total cosmic star-formation activity at various epochs.




 §.§  Star Formation Rate Functions


Following the same approach used in <ref>, we use our best-fit  model to investigate how well it can recover the dust-corrected  star formation rate functions (SFRFs). We begin by generating mock  samples using our best-fit  model with observed  luminosity measured by using the rest-frame EW and R-band continuum luminosity. This is then converted to star formation rates using the <cit.> calibration

    SFR() = 4.4×10^42 L_  yr^-1

where we assume a <cit.> IMF. Dust corrections are then applied to each mock  emitter by using the <cit.> –stellar mass prescription, which is calibrated on SDSS DR7 but has been shown to work up to z ∼ 2 (e.g., ). For each mock emitter we assign  using a normal distribution, 𝒩 (μ=(M),σ=0.28 mag), which is centered on  defined by the calibration at a given stellar mass and perturbed by a random number drawn from a normal distribution with σ = 0.28 mag to incorporate the observational scatter in the dust correction <cit.>.

Figure <ref> shows the SFRFs assuming our  model compared with the  star formation rate functions (SFRFs) from <cit.> and <cit.>. We find a strong agreement between our predicted SFRFs and those from the literature at all redshift slices, specifically for the z = 0.4 and z = 2.23 SFRFs where our predicted SFRFs are in near perfect agreement with observations. Our z = 1.47 SFRF slighlty underestimates the number densities <40  yr^-1 although it is still within 1σ uncertainty. The main difference occurs with the z = 0.84 SFRF where we find our predicted SFRF underestimates the observed number densities at <6  yr^-1. Overall, our  model shows that we can describe both the observed  LF and the dust-corrected  SFRF up to z ∼ 2.2 by using information regarding the intrinsic  equivalent width distributions and an assumption of the stellar mass function for star-forming galaxies. 







 §.§ Cosmic Star Formation Rates: Which types of galaxies contribute the most?







Given that equivalent width is a ratio between emission line flux (instantaneous SF) and continuum flux density (average SF over longer timescales), it can be used as a proxy in understanding star-formation histories of galaxies. We have shown in <ref> and <ref> how our  model can produce  LFs and SFRFs consistent with observations. This allows us to use our model to investigate what types of galaxies contribute the most to overall cosmic star formation activity at different epochs in galaxy evolution by measuring the relative SFR contribution of star-forming galaxies at different cosmic times in varying bins of  EW and threshold limits, stellar mass, and sSFR(). For each subdivision, we generate a mock  sample using our  model and measure its corresponding SFRF. We then integrate the full SFRF to measure star-formation rate densities (SFRDs).

Figures <ref> and <ref> show our SFRDs for the full sample as a light blue, dashed line along with comparisons to other  measurements and the  SFRD evolution of <cit.> (constrained up to z ∼ 5). The light blue circles are SFRDs for the full  population at specific redshifts corresponding to the HiZELS and DAWN samples using the (M) power-law fit described in Table <ref> and Equation <ref>. We find that our  SFRDs from both  and (M) models are in strong agreement with the observed  SFRDs <cit.> suggesting that both approaches are tracing the observed cosmic SFRD evolution. However, we do note that the resulting SFRD at z ∼ 0.84 and 1.47 is underestimated in respect to the HiZELS measurement and may be linked to the underestimation in the faint-end shown in the SFRFs (Figure <ref>). Overall, our model does predict the cosmic star formation history for the full population of  emitters up to z ∼ 2.2.

We first investigate the relative contribution of  emitters based on their rest-frame EWs as shown in Figure <ref> and Table <ref>. We find that at z > 1.5,  emitters with rest-frame EW >200Å contribute ∼ 40 percent of the total cosmic star-formation activity. This is somewhat expected given that the  EW distributions become wider (higher ) with increasing redshift. However, if we consider  EW as a proxy for comparing instantaneous-to-past star formation activity, then this would suggest bursty, star-forming galaxies play a major role at the peak of cosmic star-formation activity. 

Furthermore, at z > 1.5 the main star-formation contribution is found to reside in EW > 100Å  emitters (∼ 60 - 65 percent); however, the role of such high EW systems decreases with decreasing redshift. Between z ∼ 0.4 and 1.5, half of cosmic star-formation activity is found within  emitters with rest-frame EW between 25Å and 100Å. This suggests that the bulk of star formation activity shifts from extreme emission line galaxies to intermediate systems (e.g., `normal'/main-sequence galaxies) signifying a population shift.

By z ∼ 0.4, cosmic star-formation activity is primarily subdivided between low EW systems (0Å < EW_0 < 25 Å) and intermediate systems (25Å < EW_0 < 100Å), with star formation in z < 0.4 SF-galaxies found in low EW systems. This would signify that the main contributor of star-formation activity at z < 1.5 is primarily from increasingly less active, low EW and SFR  systems while at z > 1.5 the main contribution is from  emitters exhibiting bursty, episodic SF activity producing high EWs. The left panel of Figure <ref> also shows how  emitters with varying EW limits contribute to overall star-formation activity and shows the importance of high EW sources at z > 1.5 while low EW systems have a more significant role at lower redshifts.

Indeed if we subdivide our sample in sSFR, we also see a similar result as shown in the left panel of Figure <ref>. At z > 1.5, we find  emitters with 10^-8.5 < sSFR/yr^-1 < 10^-7.0 contribute half of cosmic star-formation activity. This suggests the main contribution to cosmic SFRD at z ∼ 2 is from highly active star-forming galaxies that have mass-doubling timescales of ∼ 10 - 300 Myr and are most likely systems with bursty, episodic star-formation histories. Between z ∼ 0.6 and z ∼ 1.5, this shifts to  emitters with sSFR between 10^-9.5 yr^-1 and 10^-8.5 yr^-1, which represent `normal' star-forming galaxies with mass-doubling times of ∼ 300 Myr to 3 Gyr. At z < 0.6, > 50 percent of star-formation activity comes from  emitters with 10^-11. < sSFR/yr^-1 < 10^-9.5 which represent star-forming galaxies that fall below the main sequence (e.g., galaxies in the process of quenching).

Figure <ref> also shows the relative contribution of  emitters to cosmic star formation activity for varying stellar masses. At all redshifts up to z ∼ 2, we find that massive, star-forming galaxies (10. < log_10 M/ < 11.5) have the highest contribution to cosmic SFRD. By z∼ 0.4 the relative contribution of M ∼ 10^10.5 - 11  galaxies is 48 percent and drops to ∼ 34 percent by z ∼ 2. We find the relative contribution of intermediate (10^9.0 - 10.0 ) and high (>10^10 ) galaxies to decrease with increasing redshift while low-mass (10^8.0 - 9.0 ) galaxies increasingly contribute to overall cosmic SF activity with increasing redshift. The bulk of star-formation at z > 1 is found within <10^10  systems. This highlights the importance of low-mass galaxies in the high-z Universe which will also tend to be the sources with elevated EW and sSFR (potential starburst galaxies). 

Overall, our  model indicates that high EW/sSFR systems are the main contributors to cosmic star-formation activity at z > 1.5 primarily in low-mass galaxies.. This suggests low mass systems with high EW/sSFR at z > 1.5 (population of extreme emission line galaxies undergoing bursty, episodic star formation activity) are the main contributors to cosmic SFRD. At z < 1.5, we notice a population shift where intermediate EW/sSFR and massive systems (e.g., `normal' main-sequence galaxies) start to dominate cosmic SF activity. This continues to z ∼ 0.4 where we find the final transition where the bulk of star-formation in the Universe is found in low EW/sSFR and massive  emitters. This highlights key aspects of where star-formation activity primarily occurred at varying periods in cosmic time. 















 §.§ sSFR Evolution: The role of selection bias




Observationally, the typical sSFR at a stellar mass of ∼ 10^9.5 - 10  is found to evolve as (1+z)^2.4 - 3.0 at z < 2.2 <cit.> and (1+z)^1.1 - 1.5 up to z ∼ 7 <cit.>. Figure <ref> shows the cosmic sSFR evolution where observations find more than an order-of-magnitude increase from the local Universe to cosmic noon. 

Large hydronamical simulations such as EAGLE and Illustris <cit.> are able to predict the shape of this redshift evolution. However, there is a factor of ∼ 2 - 3 discrepancy in the typical sSFR (normalization of the trend) between observations and simulations. We explore the cause of this discrepancy by using our observationally-constrained  model to investigate the cosmic evolution of sSFR and how selection biases could affect the underlying measurement of typical sSFR at fixed stellar mass. 

We measure sSFR using the assigned stellar mass for each  emitter and its corresponding  SFR which is determined by converting stellar mass to  luminosity via the assigned EW and applying the <cit.> calibration assuming Chabrier IMF. The typical sSFR at a given redshift and stellar mass range is determined by taking the median sSFR of the mock sample for that respective redshift and stellar mass range. We find that the typical sSFR from our  model can be parameterized as:

    sSFR(z,M)    =   sSFR_10(M/10^10)^β_0 (1+z)^ξ(M)
    ξ(M)    =   ξ_10 + βlog_10(M/10^10)

where sSFR_10 is the typical sSFR at z = 0 and stellar mass of 10^10 , β_0 is the sSFR – stellar mass correlation slope at z = 0, ξ_10 is the redshift-dependent sSFR slope for 10^10  star-forming galaxies, and β defines how the redshift evolution slope changes with stellar mass. Essentially, Equation <ref> is a combination of the sSFR – stellar mass correlation and cosmic sSFR evolution similar to how we model the – stellar mass correlation and redshift evolution. 

Figure <ref> shows the typical sSFR within the range of 10^9.5 - 10   with a median stellar mass of ∼ 10^9.8  for three limiting rest-frame EW limits: 0Å, 25Å, and 100Å. In all three cases we do not consider any  limiting line flux; however, we do include the cosmic sSFR evolution parameterization in Table <ref> for all cases including EW and  line fluxes limits. The `intrinsic' sSFR evolution is described as the case where EW_0 > 0Å and no  line flux limits are applied which we show as a black solid line in Figure <ref>. We find that our  model shows an increasing sSFR with increasing redshift where we find close to a two order-of-magnitude increase by z ∼ 2.2 and described by a (1+z)^3.36 evolution. This is somewhat steeper compared to what is found in observations <cit.>.

We include observational measurements of the typical sSFR within the same stellar mass range of our measurements with a comparable median stellar mass of ⟨ M ⟩∼ 10^9.8  in Figure <ref>. <cit.>. We find that our `intrinsic' sSFR evolution measurement is consistently below observations by a factor of ∼ 2 - 3.  The only exception is the z ∼ 0 measurement of <cit.> which has a limiting  EW_0 > 3Å  and is also a factor of ∼ 3 below the z ∼ 0 <cit.> and <cit.> measurements. Comparing to simulations, we find that at z > 1.5 we are in agreement with Illustris <cit.> while at z < 1.5 we are in agreement with the L25 and L100 simulations from EAGLE <cit.>. 

Overall, our `intrinsic' model better matches simulations while the disagreement with observations is due to selection bias effects in the latter. For example, the z ∼ 0 <cit.> measurement is in agreement with our `intrinsic' sSFR evolution with the reason being that embedded in this measurement is a low limiting EW_0 > 3Å limit which is close to the EW_0 > 0Å assumed in our `intrinsic' model. <cit.> is also not in agreement with the z ∼ 0 measurements of <cit.> and <cit.>. We also see a scatter of ∼ 0.3 dex in the literature measurements which is primarily arising from different sample selections and SF calibrations used. 

Figure <ref> also shows the cosmic sSFR evolution for the case of a >25Å and >100Å limiting rest-frame EW case as a solid green and red line, respectively. We find that the > 25Å case is in agreement with observations and has a (1+z)^2.38 evolution consistent with past studies <cit.>. This highlights how applying a selection bias to our mock  samples can reconcile the discrepancy between our `intrinsic' sSFR evolution (in a agreement with simulations) and observations. It also further suggests how selection biases can play an important role in how we measure the cosmic sSFR evolution.

Interestingly, applying an EW_0 > 100Å cut results in a (1+z)^1.94 evolution somewhat consistent with the (1+z)^1.1 - 1.5 evolution measured at z > 2.2  <cit.>. Although our models are observationally-constrained up to z = 2.23, it is interesting to note that the + EWs measured via Spitzer IRAC color excess measured at z > 2 are primarily biased towards high EWs such that the shallower  (1+z)^1.1  - 1.5 evolution is probably due to changing selection limits with increasing redshift. Overall, we find that the major culprit in the discrepancy between observations and simulations when it comes to the cosmic sSFR evolution is the result of selection limits causing elevated median sSFR. 






§ FORECASTING NEXT-GENERATION SURVEYS: ROMAN AND  EUCLID


Future slitless grism surveys using Nancy Grace Roman Space Telescope and Euclid will be able to observe sizeable samples of star-forming galaxies that would enhance our understanding of galaxy evolution physics. Accurate estimation of the number counts for planned surveys using these state-of-the-art facilities is crucial in regards to survey designing and planning of science goals and objectives. A simple, back-of-the-envelope estimation of the expected number counts and redshift distribution can be determined by using reported line luminosity functions in combination with a limiting line flux threshold. However, slitless grism surveys have a resolution limit that can be interpreted as a minimum limiting EW threshold. 

For example, HST/WFC3 G141 has a resolving power of R ∼ 130 for point-sources which acts also as an observed EW limiting threshold of 85Å to 130Å at 1.1μm and 1.7μm, respectively. For z ∼ 0.7 – 1.6  emitters, this corresponds to a rest-frame EW threshold of ∼ 50Å. Using our best-fit  model, we find the typical EW ranges from ∼ 23Å to ∼ 90Å for 10^10   emitters from z ∼ 0.7 to 1.6, respectively. Given that  corresponds to 50 percent of the total exponential EW distributions, it implies that a HST/WFC3 G141 survey of  emitters could miss 88 and 43 percent of the total  EW distribution at z ∼ 0.7 and 1.6, respectively. The low-z end is especially affected given that EW distributions become narrower (lower ) with decreasing redshift such that the 50Å rest-frame EW limit becomes an important factor towards low-z. 

This effect of limiting EW weakens with decreasing stellar mass. For example, a similar grism survey that observes 10^8   emitters will miss 27 percent and 14 percent of the total  EW distribution at z ∼ 0.7 and 1.66, respectively, which is due to wider distributions (higher ) at lower stellar masses. This highlights the importance of incorporating EW distributions in survey number count forecasting for grism surveys. 

In this section, we present our predicted number counts and EW properties for large grism surveys planned with Nancy Grace Roman Space Telescope and Euclid. We use our best-fit  model to generate mock samples of  emitters corresponding to the redshift range associated with the planned grism surveys. We also convolve the grism throughputs with the intrinsic (not corrected for dust) + luminosities in our mock samples to measure the expected + line fluxes that would be observed in such surveys. For each telescope, we consider three different + line flux limits: >2 × 10^-16 erg s^-1 cm^-2 (“shallow”), >1 × 10^-16 erg s^-1 cm^-2 (“intermediate”), and >5 × 10^-17 erg s^-1 cm^-2 (“deep”). The limiting observed EW thresholds are also included to take into account missing low EW sources due to the changing EW distributions.








 §.§ Nancy Grace Roman Space Telescope


The  Nancy Grace Roman Space Telescope is a 2.4-meter space-based observatory planned for launch in mid-2027 <cit.>. The primary extragalactic astrophysics instrument on-board will be its Wide Field Instrument (WFI) which will cover 0.28 deg^2 per exposure and a wavelength coverage from 4800Å to 23000Å with 8 broadband imaging filters. Roman/WFI will also include a single grism, G150, centered at 1.465 and covering from 1 to 1.93 with a limiting resolution of R∼ 461 at 1 (2 pix). The G150 wavelength coverage corresponds to 0.5 < z < 1.9  emitters; however, the limiting resolution sets a minimum observed + equivalent width of ∼ 22Å. This corresponds to a rest-frame EW limit of ∼ 15Å and ∼ 8Å  at z ∼ 0.5 and 1.9, respectively. Given that  at z ∼ 0.5 is ∼ 22Å for 10^10   emitters, the expected number counts can be significantly reduced in the low-z, high-mass end in comparison to relying solely on past reported luminosity functions to forecast number counts.

Figure <ref> and Table <ref> show our expected redshift distributions for a Roman/G150 survey using our  model and applying selection limits based on an observed + emission line flux cut and the limiting resolution/EW threshold. We find our approach predicts a redshift distribution peaked at z ∼ 0.7 - 0.8 for our shallow survey and increases to z ∼ 0.9 - 1.0 with the deep survey. In comparison to the simulation of <cit.>, we are in agreement at the peak of the redshift distirbution. The discrepancy at other redshifts is primarily due to incorporating the grism throughput and limiting EW thresholds, which is evident in the drop seen at z ∼ 0.6 and z > 1. We also compare to the <cit.> simulation results which assumes three different dust prescriptions <cit.> used to uncorrect their intrinsic sample. All three predict higher number counts in comparison to our results and <cit.>. This is primarily due to the simulated <cit.> LFs overestimating the bright-end compared to observations by almost an order-of-magnitude (depending on the assumed dust prescription; see their Figure 9). <cit.> also used simulated LFs that are more consistent with observations although there are redshift ranges for which the bright-end is overestimated primarily towards low-z (see their Figure 1). However, the main discrepancy arises from our inclusion of limiting EW thresholds and the G150 throughput which are not included in the <cit.> measurements.

For our shallow survey, we find a significant drop in the number densities with increasing redshift which is due to (1) bright ELGs are fainter at high-z compared to low-z and (2) decreasing grism sensitivity at redder wavelengths (increasing redshift). As shown in Table <ref>, we predict that Roman/G150 observations will observe ∼ 1600 and ∼ 720 + emitters per deg^-2 at 0.5 < z < 1 and 1 < z < 1.9, respectively, for a bright survey. In comparison, we predict a deep survey will observe ∼ 11300 and ∼ 12700 + emitters per deg^-2 at 0.5 < z < 1 and 1 < z < 1.9, respectively, highlighting how decreasing the line flux limit would help in observing larger numbers of high-z emitters. A shallow, wide survey would need to observe an area 10 times larger than a small, deep survey to achieve comparable number counts albeit for a brighter population of  emitters.

Figure <ref> shows the redshift distribution and range in rest-frame + EW and stellar mass that would be observed in a 10 deg^2 Roman survey for the case of an intermediate and deep survey. An observed EW> 22Å cut is applied consistent with the G150 limiting resolution. We find that the intermediate survey probes down to ∼ 10^8  and a maximum + EW_0 ∼ 1000Å while the deep survey pushes the stellar mass limit down to ∼ 10^7.2  with a maximum  + EW_0 ∼ 2000Å. The line flux limit selection effect is more prevalent in the intermediate survey as it removes low EW systems above the limiting EW threshold at fixed stellar mass with increasing redshift in the high-mass end (> 10^10 ). By z> 1.5 the minimum EW_0 is ∼ 30Å for 10^11  emitters corresponding to an observed EW ∼ 75Å (z ∼ 1.5). The line flux limit does affect the deep survey such that low-EW systems will also be missed with increasing redshift at fixed stellar mass. However, we do predict that >10^11  systems will still be detectable at the minimum observed EW∼ 22Å at the highest redshifts probed.

For a ∼ 2000 deg^2 and a line flux limit of 10^-16  comparable to the High Latitude Spectroscopic Survey (HLSS; ), we predict a total of ∼ 17.6 million + emitters at 0.5 < z < 1.9 and ∼ 7.5 million + emitters at 1 < z < 1.9 would be observed. This is a factor 1.6 lower than the predictions of <cit.> which predict ∼ 12 million 1 < z < 2 + emitters at 10^-16  (6.5σ) and is based on semi-analytical modeling (Galacticus; ) coupled with an N-body simulation (UNIT; ) and calibrated to best match the number counts from WISP <cit.> and HiZELS <cit.>. If we remove the limiting EW and inclusion of the grism throughput, then our predictions are ∼ 12 million 1 < z < 1.9 + emitters fully consistent with <cit.>. This highlights the importance of taking into account limiting resolution and observational efficiency in number count predictions. Based on the line flux limit, HLSS will potentially observe ≳ 0.2 L^⋆(z = 0.5) and ≳ 1.2 L^⋆(z = 1.9)  emitters which will strongly constrain not only the bright-end of the luminosity function, but also part of the faint-end at lower redshifts given the large sample size. Besides star-forming galaxies, HLSS will also pickup potential AGNs with the fraction rising to ∼ 15 percent at the highest redshift end <cit.>. Overall, our predictions are consistent in that HLSS will have a statistically significant sample of + emitters that are primarily star-forming and would constrain key statistical properties near cosmic noon.




 §.§ Euclid


Euclid is a 2.3-meter space observatory that launched in July 2023 and will be able to observe ∼ 0.55 deg^2 in a single pointing with its two main instruments: Visual Imaging Channel (VIS) and Near-Infrared Spectrometer and Photometer (NISP). The latter includes a slitless spectroscopy mode (NISP-S) with two main grism configurations: a `blue' grism with a single grating (BGS) and three `red' grisms (RGS) that cover the same wavelength range but configured to have three different dispersion directions (0^∘, 180^∘, and 270^∘) to decrease contamination from overlapping spectra. The `blue' grism covers 0.92 - 1.25 with an effective wavelength of ∼ 1.1 and the `red' grisms will cover 1.25 - 1.85 with an effective wavelength of ∼ 1.5. This corresponds to  emitters at 0.4 < z < 0.9 and 0.9 < z < 1.8 for the blue and red grisms, respectively. Both grisms will have resolution of R ∼ 450 for a 0.5” diameter source corresponding to an observed EW threshold of ∼ 27Å. 

Figure <ref> shows the predicted redshift distribution using the Blue and Red grism and tabulatated in Tables <ref> and <ref>, respectively. We predict that Euclid/Blue will be able to observe a total of ∼ 2250, ∼ 6000, and ∼ 12000 0.4 < z < 1  emitters per deg^2 in a bright, intermediate, and deep survey, respectively, and Euclid/Red will be able to observe ∼ 1350, ∼ 6200, and ∼ 18500 0.9 < z < 1.9 + emitters per deg^2. Figure <ref> shows that both Euclid/Blue and Red grism surveys can cover a wider stellar mass and EW range with the deep pointing capturing more high EW, low-mass systems compared to the intermediate pointing. Especially towards high-z in the Red grism, we predict in a 10 deg^2 field Euclid would be able to capture numerous >1000Å  emitters for dwarf-like systems (<10^8 ) while also capturing massive star-forming galaxies (>10^10 ) with rest-frame + EW up to ∼ 400Å.

The Euclid Wide Survey <cit.> is one of the main planned surveys which will include NISP Red grism observations over ∼ 14500 deg^2 area within a six year time frame and down to an observed flux limit of 2×10^-16 . Based on our models, we predict that  Euclid Wide will be able to observe ∼ 1350  emitters per deg^2 resulting in a sample size of 19.5 million 0.9 < z < 1.9  emitters. Given the bright emission line flux limit, Euclid Wide will be sensitive to ∼ L^⋆  and ∼ 2.5 L^⋆  emitters at z ∼ 1 and ∼ 2, respectively. However, the large area coverage will provide one of the strongest constraints on the bright-end of the  luminosity function and will potentially have ∼ 15  - 25 percent of the sample as AGN <cit.>.





§ DISCUSSION





 §.§ Implications for Survey Planning


Our  model in conjunction with a set of observing parameters (e.g., grism throughput, redshift range, limiting EW, and line flux threshold) can provide an assessment of estimated number counts and redshift distributions. Our approach also provides information regarding the potential range of stellar masses,  line fluxes/luminosities, and  EW that could be observed in a given assumed survey. This extra amount of detail not only provides accurate number count predictions by taking into account factors such as a limiting EW threshold in grism surveys, but also gauges the range of star-forming galaxy properties that could be potentially observed in a given planned survey.

 For example, Roman and Euclid show great promise in selecting a wide range of  emitters within a 10 deg^2 survey as shown in Figure <ref> reaching down to 10^7.5  galaxies at z ∼ 1.5 with rest-frame  EW >1000Å. The planned wide field surveys with both observatories will be able to capture orders-of-magnitude more low-mass dwarf galaxy systems near cosmic noon especially numerous cases of high EW systems which may signify bursty, episodic star-formation histories. Such systems could be analogs of potentially ionizing sources in the z > 6 Universe (e.g., ).

Another science case that can be planned using our models is the search for high mass, high EW systems which are rare and require large areas to probe. For example, Figure <ref> shows that at 10 deg^2 we predict a survey would observe many 10^11  galaxies exhibiting  EW > 100Å. Although not as extreme to low-mass galaxies, but in respect to the typical  EW at 10^11  and the corresponding redshift, these sources are indeed `extreme'. However, such sources may not appear within a ∼ 1 deg^2 where the comoving volume probed is significantly reduced. Using our models, one could gauge what survey areas are needed to ensure that such sources could potentially be observed.

The two science cases defined above are examples of how our models can be used to aid in survey planning and defining science cases. However, we do caution readers that our predictions assume that the  EW distribution is best described by an exponential distribution which, as we also found in <cit.>, is not entirely true at extreme EW (>1000s EW) where outliers from the exponential distribution are found. These outliers will also have uncertain EWs given the poorly constrained faint continuum and contribute a small fraction (e.g., <1% for the z = 0.47 LAGER sample; ). Overall, our  model makes a great addition to survey planning, presents accurate number count predictions by taking into account the limiting EW threshold in grism surveys, and includes added information regarding the range of galaxy properties that may be observed.




 §.§ The Major Role of Starburst Galaxies in cosmic star formation history


In <ref>, we show that z > 1.5  emitters with EW_0 > 200Å contribute ∼ 40 percent of the overall cosmic star-formation activity and  emitters with sSFR ∼ 3 - 100 Gyr^-1 also dominate the overall cosmic star formation activity at z > 1.5. The rapid evolution of the cosmic SFRD for >200Å  systems is consistent with past studies that have found such high EW systems to be rare in the local/low-z Universe (e.g., ) but are numerous in the z ≳ 1 Universe (e.g., ). It also provides supporting evidence of how bursty star-formation activity may be a dominate mode of star-formation in the high-z Universe.


Past studies report varying results on the relative contribution of starburst galaxies to cosmic SFRD. <cit.> reports ∼ 10% contribution from >10^10  starburst galaxies selected at 1.5 < z < 2.5 using Herscel/PACS. <cit.> used ∼ 2000 Hershcel/SPIRE-selected 0 < z < 4 galaxies and finds an upper limit of ∼ 28% and ∼ 61% at 0 < z < 0.5 and 2 < z < 4, respectively, with a >10^10  stellar mass limit. We report ∼ 34% contribution from 10^10 - 11.5  z ∼ 2.2  emitters with no EW cut as shown in Figure <ref> and Table <ref>. At z ∼ 2, the equivalent width distributions range between ∼ 30 - 70Å for 10^10 - 11.5  galaxies with >100Å (>200Å) corresponding to ∼ 20% (5%) of the total >10^10   population and a cosmic SFRD contribution of ∼ 7% (2%) in respect to all star-forming galaxies. The higher fractions observed in Herschel studies could be primarily due to selection of dusty star-forming galaxies which are either missed or have their dust corrections underestimated in narrowband-selected  surveys. This may also be due to a volume-effect where high EW, massive galaxies are missed due to their rarity (e.g., `Ando effect'; ). Overall, the main consensus at z < 2 based on past studies and the results in this paper is that massive, starburst galaxies play a minor role in overall cosmic star formation activity.

The vast majority of z > 1.5  emitters with high rest-frame EW are found within the low-mass regime which is expected given the EW_0 – Stellar Mass anticorrelation (see <ref>). <cit.> used HST grism spectroscopy of ∼ 1000  emitters at 0.3 < z < 2 and report a SFRD contribution of 13%, 18%, and 34% for EW_0 > 300, 200, and 100Å, respectively, for M ∼ 10^8.2 - 10  and SFR > 2 (10)  yr^-1 at z ∼ 1 (∼ 2). We find > 100Å and > 200Å  emitters contribute 65.7% and 39.4%, respectively, at z ∼ 2; however, limiting this to the same range as <cit.> results in ∼ 31.2% and 18.6%, respectively, which is in strong agreement with <cit.>. This also highlights how the majority of SFRD contribution at high EW is coming from <10^10  emitters where the contribution at z ∼ 2.2 is ∼ 43% and  26% for EW_0 > 100Å and 200Å, respectively, and dominated by low-mass dwarf galaxies which will have higher number densities and  equivalent widths in comparison to high mass systems.

In the z > 2 Universe, studies suggest an even higher contribution. <cit.> find >50% contribution of starbursts to cosmic SFRD at 3.9 < z < 4.9 measured using Spitzer/IRAC 3.6 excess consistent with strong  emission. <cit.> finds that the majority of z ∼ 4.5 galaxies, especially at <10^10, have undergone a major bursty phase within the last 50 Myr period with a factor >5 enhancement in their SFR. <cit.> also find similar results for 2.5 < z < 4 EELGs suggesting a recent ∼ 50 Myr burst SF phase. Spectroscopic observations of low mass, high EW systems also find the presence of violent bursts (e.g., ) supported by high gas fractions (e.g., ). <cit.> suggests that strong EWs and a bursty phase is quite common in the high-z Universe and is common evolutionary path. Galaxies undergo an initial phase of bursty star formation activity resulting in faint continuum and high EWs. The continuum brightness increases due to the build-up of stellar mass after each subsequent burst resulting in a decreasing EW with time. This could explain the decrease in  EW with decreasing redshift that we also find in our study. Overall, our results suggest that starburst galaxies (interpreted as high EW/sSFR systems) are important contributors to cosmic SF activity in the high-z Universe dominated by low-mass, dwarf galaxies and that such a SF mode may be the dominant path of star-formation in the z > 2 Universe.





 §.§ High EW Hα Emitters may Reionize Universe


Our results show an increase in the  EW up to z ∼ 2.2 with an emphasis on low-mass, high EW systems contributing a significant portion of cosmic SF activity around cosmic noon. The elevated level of star-formation suggests that such galaxies contain significant populations of massive, luminous, short-lived stars (e.g., O and B type stars) which would be pumping ionizing radiation into the surrounding medium. <cit.> show a correlation at z ∼ 2 where increasing  EW corresponds to an increase in the ionizing photon production efficiency, . This has also been observed by other studies at z ∼ 2 <cit.> and z ∼ 4.5 <cit.>. Recently, <cit.> used HST grism spectroscopy and UV imaging of 0.7 < z < 1.5  emitters and found elevated /UV ratios consistent with bursty star-formation activity (see  for caveats of /UV as tracer of burstiness) primarily in <10^8.5  emitters. These sources also had increasing EW with increasing /UV and decreasing stellar mass. <cit.> also reported a correlation between  and  EW such that low-mass, faint  emitting star-forming galaxies may be probable sources that reionized the Universe at the Epoch of Reionization. Our results would then suggest that the typical  of star-forming galaxies would be high, especially for low-mass, high  EW starburst galaxies. Given that low-mass galaxies are more abundant than high-mass galaxies and the increasing EW with redshift at a given stellar mass, it is then probable that such galaxies are likely sources that play an important role in reionizing the Universe.



 §.§ Implications for low-z interlopers in Lyα Samples at z > 6


Investigating how galaxies formed and evolved to the present-day structures we see in the local Universe requires us to understand the high-z Universe, especially at the Epoch of Reionization, where we expect the first generation of galaxies to have formed. There are several approaches to probe this cosmic regime with one popular method being the selection of  emitting galaxies via narrowband-surveys such as LALA <cit.>, SXDS <cit.>, HiZELS <cit.>, CF-HiZELS <cit.>, CALYMHA <cit.>, SILVERRUSH <cit.>, SC4K <cit.>, J-PLUS <cit.>, LAGER <cit.>, and other surveys (e.g., ). The technique relies on both deep narrowband and broadband photometry to differentiate between high-z  emitters and low-z foreground ELGs, such as  emitters and include a minimum EW threshold. However, low-z foreground contaminants are always an issue plaguing not only narrowband-selected  samples, but -selected samples in general. 

One efficient way to gauge the level of contamination that would arise in narrowband-selected  samples is by assessing the number of high EW foreground emitters that could mimic low EW high-z  emitters. For example, let's assume two different z ∼ 6.6 narrowband  surveys with the condition that Survey A has ()> 30Å and Survey B has ()> 10Å. Now let's assume that there maybe  foreground emitters misidentified as  emitters. The corresponding (+) of these contaminants would be 163Å and 54Å for Survey A and B, respectively. Assuming that the median stellar mass of the  contaminants is ∼ 10^8  (e.g., ) and using Equation <ref> with the best-fit + parameters in Table <ref>, then the probability of having  emitters with >163Å and >54Å is ∼ 1.2% and ∼ 23%. This implies that a low EW threshold survey, such as Survey B in our example, would be able to capture a larger number of  emitters given the lower EW threshold, but would also significantly increase the risk of foreground  contamination.

We can also assess the contamination of higher redshift  emitters using our  model. For example, let's assume a z ∼ 9  narrowband survey with the condition that Survey A has  ()> 50Å and Survey B has ()> 25Å. This would correspond to an (+) threshold of 240Å and 121Å, respectively. Assuming a median stellar mass of ∼ 10^9.5  (e.g., ), this would correspond to a probability of having an  emitter above these thresholds as 0.2% and 4.1%, respectively. However, depending on the survey design, if the median stellar mass is ∼ 10^8.5 , then the probabilities are 1.2% and 10.7%, respectively. 

We do note two main caveats with this approach. The first is that our  model assumes that  EW distributions are best-defined by an exponential distribution; however, there will be high EW outliers although the fraction is not significantly high (e.g., ) and is also not well-constrained given the faint broadband photometry used to constrain the continuum flux density. The second caveat is that this approach measures the possible foreground contamination only from  emitters. There will also be  and  foreground contaminants in  samples. Overall, we have shown our   model can be used to assess the level of  foreground contamination in -selected samples based on the assumed () threshold and a range of possible stellar masses of the contaminants.




§ CONCLUSIONS



In this paper, we have measured and constrained the intrinsic  EW distributions from z ∼ 0.4 to 2.2, modeled them using our best-fit  model, and explored the implications of our results in terms of the overall star-formation properties of  emitters and its implications for future space-based grism surveys. Our main results are:

	
  * We use the forward-modeling approach that was primarily developed in our first paper <cit.> but now expanded to the HiZELS and DAWN samples where we measure the intrinsic  EW distributions from z ∼ 0.4 to 2.2  in a uniform manner that takes into account selection and filter profile effects.
	
  * An intrinsic EW – L_R  and EW – stellar mass anti-correlation is found at all redshift slices where low-mass/faint continuum  emitters are found to have higher typical  EWs compared to high-mass/bright continuum emitters. We find the slopes become steeper with increasing redshift signifying how low-mass galaxies experience a stronger, rapid evolution in their typical  EWs compared to high-mass galaxies.
	
  * We develop a combined double power-law model, , that is used to constrain both the typical  EW redshift evolution and its anti-correlation with stellar mass. We find that for a 10^10   emitter, the typical  at z = 0 is 9.26^+1.71_-1.53 Å and increases as (1+z)^1.78^+0.22_-0.23 such that by z ∼ 2.2 the typical  is ∼ 73Å. The stellar mass anti-correlation slope at z = 0 is -0.10^+0.07_-0.07 and becomes steeper with redshift as -0.05(1+z) such that by z ∼ 2.2 the anti-correlation slope is -0.26 signifying the importance of low-mass emitters and their  EW evolution.
	
  * Our intrinsic  model shows typical  lower than past observations but we find the main reason is due to selection effects resulting in elevated typical EWs in past measurements. We demonstrated how applying a combination of  and line flux limits can result in  redshift evolutions consistent with past observations. We also demonstrate a similar effect in the cosmic sSFR evolution where we use our intrinsic  model and find our measurements are more consistent with simulations. However, incorporating selection effects, we find elevated sSFR that is consistent with observations. This suggests that the tension between observations and simulations may be rooted from selection effects.
	
  * Our best-fit  model predicts  luminosity functions and star-formation rate functions that are consistent with past narrowband and slitless grism surveys highlighting our model can describe both the  equivalent width distributions and luminosity functions coupled with an assumed stellar mass function.
	
  * Given that we can predict accurate LFs and SFRFs, we extended our work to investigate the relative contribution of  emitters for varying EWs, stellar masses, and sSFRs. We find that in the z > 1.5 Universe, cosmic SF activity is primarily arising from galaxies with  >100Å and sSFRs>10^-8.5 yr^-1 contributing 50 - 60 percent of the total cosmic SFRD. Contribution from such sources rapidly decreases with decreasing redshift with the 0.5 < z < 1.5 Universe primarily dominated by 25 <  < 100Å and 10^-9.5 < sSFR < 10^-8.5 yr^-1 systems and by z < 0.5 it is dominated by < 25Å and 10^-9.5 yr^-1 systems. Overall, we find that low-mass galaxies with high EW and sSFR are important contributors to cosmic star-formation activity at z > 1.5 and most likely extend as such into the high-z Universe. Such sources can potentially be analogs of high-z ionizing sources responsible for the reionization of the Universe.
	
  * We present number count slitless grism survey predictions for Roman and Euclid incorporating the grism throughput and limiting resolution, where the latter acts as a limiting EW threshold. We find that both Roman and Euclid will be able to find numerous  emitters over a wide range of stellar masses and EWs. With the current survey parameters of Roman/HLSS, we can expect to observe ∼ 17.6 million 0.5 < z < 1.9 + emitters. The current Euclid Wide Survey will also ∼ 19.5 million 0.9 < z < 1.9  emitters with the main difference being the Euclid Wide Survey will be limited to >L^⋆  emitters essentially constraining the bright-end while Roman/HLSS will reach as low as 0.2 L^⋆ at the lowest redshift. 


Overall, our  model has demonstrated two major points. First, it suggests past studies may be overestimating the typical  EWs and sSFRs simply due to selection biases and other observational effects not taken into account that also cause disagreement with simulations. Second, corroborating previous results, our model suggests that low mass, high EW  systems are indeed important contributors to overall star-formation activity around cosmic noon and can be important analogs of sources that reionized the Universe at z > 6 given their high EWs. Both Roman and Euclid will be able to observe these sources in large numbers such that the next few years will be an exciting time in studying bursty star-formation activity in unprecedented detail.




§ ACKNOWLEDGMENTS

AAK acknowledges this work was supported by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center, administered by the Universities Space Research Association (USRA) contracted through NASA. 



§ DATA AVAILABILITY

Data from the public HiZELS narrowband survey catalogs are available on https://cdsarc.cds.unistra.fr/viz-bin/cat/J/MNRAS/428/1128VizieR. Raw data for the DAWN survey are also publicly accessible via the https://astroarchive.noirlab.edu/portal/search/NOIRLab Astro Data Archive by searching for programs 2013A-0287, 2013B-0528, and 2013B-0236.







§ SURVEY PREDICTION NUMBER COUNTS


Below are the number counts based on our  model discussed in <ref> and shown in Figure <ref>. 
















 Paper Title: Ram-pressure stripped radio tails detected in the dynamically active environment of the Shapley Supercluster.



firstpage–lastpage
Spatial Transformer Network YOLO Model for Agricultural Object Detection


 



    Yash Vivek Zambre^1, Ekdev Rajkitkul^2, Akshatha Mohan^1, Joshua Peeples^1
 ^1Department of Electrical and Computer Engineering, Texas A&M University, College Station, TX, USA

	^2Department of Computer Science and Engineering, Texas A&M University, College Station, TX, USA

		yashzambre@tamu.edu, ekdev@tamu.edu, akshatha.mohan@tamu.edu, jpeeples@tamu.edu
    
=========================================================================================================================================================================================================================================================================================================================================================================




§ ABSTRACT 
We study the radio continuum emission of four galaxies experiencing
ram-pressure stripping in four clusters of the Shapley supercluster at
redshift z∼ 0.05. Multi-band (235-1367 MHz) radio data,
complemented by integral-field spectroscopy, allow us to detect and
analyse in detail the non-thermal component both in the galaxy discs
and the radio continuum tails. Three galaxies present radio continuum
tails which are tens of kiloparsecs long.  By deriving the radio
spectral index in the inner and outer tails and comparing our findings
with the distribution of the extraplanar ionised gas and the results
of N-body/hydrodynamical simulations, we demonstrate that these
tails are caused by the ram pressure which, together with the ionised
gas, sweeps the magnetic field from the galaxy discs. We suggest that
the radio continuum emission in these tails can be differently powered
by (i) in situ star formation; (ii) relativistic
electrons stripped from the disc; (iii) shock excitation or a
combination of them. All the ram-pressure stripped galaxies are found
in environments where cluster-cluster interactions occurred and/or are
ongoing thus strongly supporting the thesis that cluster and group
collisions and mergers may locally increase the ram pressure and
trigger hydrodynamical interactions between the intracluster medium
and the interstellar medium of galaxies.






galaxies: evolution – radio continuum: galaxies – galaxies: ISM – galaxies: clusters: individual: Shapley Supercluster








§ INTRODUCTION


Galaxies orbiting in the cluster environment experience the
hydrodynamical interaction of their cold interstellar medium (ISM)
with the hot diffuse intracluster medium (ICM). This interplay may
result in the ISM evaporation <cit.> and `starvation'
<cit.> or may remove the cold gas supply by means of the
external pressure exerted by the ICM on the ISM <cit.>.  The
primary strength of these processes consists of explaining the lower
star formation rates <cit.> and the redder
colours <cit.> as well as the HI deficiency
<cit.> seen in cluster galaxies with respect to
the field population. At present, galaxy evolution studies support the
ram-pressure stripping (RPS) as the dominant hydrodynamical mechanism
contributing to the evolution of cluster galaxies <cit.>.

In their pioneering work <cit.> modelled the RPS process by
means of an analytic expression where the ram pressure opposes the
gravitational restoring force per unit area - a decreasing function of
the disc radius. It results that in the dense environment of the
cluster cores the ram pressure can effectively remove the cooler ISM
in the galaxy starting from outside and thus quenching star formation
in the ram-pressure stripped regions. Subsequent studies based on
observations <cit.>
and simulations <cit.> showed that the RPS
can occur, or start at least, also in less dense environments and/or
out of the cluster cores differently affecting high- and low-mass
galaxies. The time-scales for RPS is about one cluster crossing time
(∼10^9 yr).

The characteristic signatures of RPS are the presence of i)
extraplanar ISM stripped from the galaxy disc in the shape of one-side
tail; ii) distortion and ultimate truncation of the gaseous disc
without corresponding distortion of the old stellar component
<cit.>.  Besides, the ISM of late-type galaxies has a
multi-phase nature whose components (warm diffuse ionised hydrogen,
cold and warm neutral hydrogen, dense molecular gas, interstellar
dust), having different surface densities, are differently affected by
the ram pressure
<cit.>. Moreover,
as observed in nearby clusters
<cit.>,
also the non-thermal component can be stripped from the galaxy discs
forming cometary tails. Actually, the first case of late-type galaxies
experiencing RPS was detected in the radio continuum
<cit.>.

It follows that multi-band observations of both the tail and the
galaxy are crucial to distinguish among the different components in
the tail, and high-resolution observations are required in order to
resolve the physics of the gas, the stars and the plasma in the
galaxy. Moreover, multi-band studies of individual galaxies affected
by RPS are often complemented with dedicated numerical simulations
<cit.>

The effectiveness of such cluster-related mechanism in transforming
galaxies encouraged an extensive investigation of RPS supported by the
capabilities of the new observing facilities and the availability of
focused and highly resolved simulations down to scales of few tens of
parsecs. The spectacular appearance of an ongoing RPS also plays a
role in the general interest to study the RPS process and contributed
to boost systematic searches of RPS candidates, mainly based on
visible photometry <cit.>, but recently also using
radio continuum <cit.>. These works enabled to extend
the census of ongoing RPS in nearby clusters and to carry out detailed
studies of individual galaxies exploiting the now available
instrumental capabilities of the integral-field spectrographs
<cit.>.

The advent of the new generation radio telescopes has given new
impetus to the study of galaxy tails in radio continuum allowing their
detection at sensitivities and resolutions which were inaccessible to
wide-field radio surveys until now. Through 120-168 MHz radio
continuum images from the LOFAR two-meter Sky Survey,
<cit.> identified star-forming galaxies
(∼ 150) with extended asymmetric radio continuum tails likely
experiencing RPS in low redshift (z< 0.05) clusters and
groups. <cit.> carried out a 1.4 GHz continuum and HI
emission survey in the Coma cluster to investigate the radio
properties of RPS galaxies. They detected radio continuum tails in
50% of the targets demonstrating the widespread presence of
relativistic electrons and magnetic fields in the stripped
tails. <cit.> extended the study of the Coma cluster using
uGMRT detecting 19 RPS radio continuum tails out of the observed 24
extended radio sources which are cluster members. With the collected
uGMRT observations, they were able to derive the spectral structure in
all the tailed radio sources. Thanks to the new instrumentation
facilities, we can now address the open issues about the nature and
origins of the radio tail in RPS galaxies and how the radio continuum
emission in the tails correlates with the emission in other bands and
with the galaxy environment. Sophisticated simulations including the
effects of either intra-cluster or galaxy magnetic fields are also
available <cit.>.



In this framework, the role of the cluster and/or group interactions
in triggering the RPS mechanism is debated
<cit.>. Deep and
high-resolution radio continuum imaging of merging clusters provides,
on one side, detailed descriptions of the cluster merger features
tracing the intra-cluster diffuse non-thermal emission, on the other
side, it reveals the radio continuum tail of individual galaxies. This
allows us to investigate the cause-effect relationship, i.e. the link
between the large-scale mass assembly and the involved galaxies.

Superclusters are the most massive structures in the Universe
(10^15-10^16 M_⊙). In such dynamically active and locally
dense structures one can potentially observe the effects of dynamical
processes such as cluster-cluster collisions and group-cluster mergers
and sample different environments from cluster cores to filaments and
fields. In the harsh environment of the Shapley supercluster, we have
undertaken a search of supercluster members showing radio continuum
tails with the aim to investigate if these features are linked with
the ongoing or past cluster-cluster interaction.





We report on four radio continuum tails (see Fig. <ref>)
associated to RPS galaxies which are supercluster members and
investigate if and to which extent ram-pressure affects the
non-thermal component of these galaxies and discuss our results in the
light of the ongoing dynamical activity in the Shapley
supercluster. The paper is organised as follows.  In
Section <ref> we provide a short description of the selection
and properties of the RPS galaxies in our sample. The radio
observations and data reduction are described in
Section <ref> where the radio properties of the galaxies are
also derived. In Section <ref>, complementing the radio
observations with multi-band data, we analyse the radio tails in the
framework of the RPS scenario.  We investigate the origin of the
electrons powering the radio tails and highlight the role of the
environment in Section <ref> and in Section <ref> the main
results are summarised.


Throughout the paper we adopt a cosmology with Ω_M=0.3,
Ω_Λ= 0.7, and H_0=70 km s^-1Mpc^-1. According
to this cosmology 1 arcsec corresponds to 0.96 kpc at the median
redshift (z ∼ 0.048) of the Shapley supercluster. The radio
spectral energy distribution is modelled as a power law S = A ν^α.





§ RPS GALAXIES IN THE SHAPLEY SUPERCLUSTER



The Shapley supercluster <cit.> is the most massive
supercluster in the local Universe. It encompasses at least 25 Abell
clusters in the redshift range 0.035<z<0.06 over a ∼
15×10 deg^2 region
<cit.>. At the heart of the Shapley
supercluster is the high-density Shapley Supercluster Core (SSC, at
median redshift z=0.048), comprising three Abell clusters and two
poor clusters, forming a continuous filamentary structure 2 deg
(∼7 Mpc) in extent filled with hot gas
<cit.>. Evidence of cluster interaction and merging in
the SSC were extensively provided by dynamical studies
<cit.>, X-ray observations
<cit.> and radio continuum analysis
<cit.>. Over
the past years, the Shapley Supercluster Survey
<cit.> traced the filaments feeding the SSC, and
revealed how the surrounding clusters are connected to each other and
to the SSC <cit.>.

The four galaxies analysed in this work have been identified in the
region of the Shapley supercluster covered by the ShaSS by
<cit.>. From a spectroscopic catalogue, which is 95
per cent complete at magnitude i < 18, the RPS candidates were
selected in the deep (down to magnitude r∼ 24 at 5σ)
images collected at the ESO VLT Survey Telescope (VST) according the
criteria introduced by <cit.>: (i) disturbed morphology
indicative of a unilateral external force; (ii) brightness knots and
colour gradients suggesting bursts of SF; (iii) evidence of tails. The
final sample of RPS candidates turned out to constitute the 0.5 per
cent of the supercluster galaxies in agreement with the 0.6 per cent
found by <cit.>. Among the 13 selected galaxies 9 actually
presented extraplanar ionised gas and only 4 turned out to be
significantly affected by ram pressure. This does not imply that these
are the only galaxies affected by ram pressure across ShaSS, but that
the visual selection of the RPS candidates in the ShaSS imaging
allowed the identification of these cases <cit.>. All
four galaxies are members of the supercluster although associated with
different clusters.

Ongoing RPS has been already ascertained for three galaxies
(SOS 114372, SOS 90630, SOS 61086) thanks to the observations
carried out with the Wide-Field Spectrograph
<cit.>, then complemented by
N-body/hydrodynamical simulations <cit.>.
The fourth galaxy, ShaSS 421015419 (hereafter ShaSS 421) has been
observed with MUSE - Multi-Unit Spectroscopic Explorer
<cit.>. The analysis of this particular object is in progress
and it will be published in detail elsewhere (Merluzzi et al., in
preparation), however there is evidence that this galaxy is also
affected by RPS as shown by the preliminary results presented
here. Therefore for sake of completeness, we include this object in
the sample.

In Table <ref> the main global properties of each galaxy are
listed. In Fig. <ref> the positions of the four galaxies
across the ShaSS are indicated together with those of the clusters of
galaxies.

The radio tail of SOS 61086 has been already studied in
<cit.>, however, in Section <ref> we recall the
main results for a more comprehensive discussion.





§ RADIO DATA



The galaxies SOS 114372, SOS 90630 and SOS 61086 are covered by
several radio observations of the SSC over a frequency range from
235 MHz to 1367 MHz, carried out with a variety of radio
interferometers. In particular, the data were collected with i)
GMRT <cit.> and uGMRT <cit.> at 235 MHz,
325 MHz, 400 MHz (Band 3) and 610 MHz; ii) MeerKAT
<cit.> at 1.28 GHz; and iii) the Australian Square
Kilometre Array Pathfinder <cit.> at 887 MHz (EMU
Evolutionary Map of the Universe, <cit.>, Early Science
Project ESP 20, <cit.>), 943 MHz and 1367 MHz as part
of the POSSUM <cit.> Pilot 2 survey. On the other hand, for
the galaxy ShaSS 421 only ASKAP imaging at 887 MHz (Early Science
Project ESP 20) is available at present. The summary of the
observations used in this paper is presented in
Table <ref>. For projects 22_039, 30_024 and 42_019 the
parameters of the full resolution images are reported.

The observing details and data reduction strategy concerning projects
30_024, 22_039, ESP 20 and MGCLS <cit.> were provided in <cit.>
where the radio tail of galaxy SOS 61086 was also studied. MGCLS
images at full resolution (see Table <ref>) and at the
resolution of 15^''×15^'' have been
used.  Beyond the total intensity images, the MGCLS data products
provide in-band spectral index (in the range 908-1656 MHz) and full
Stokes information.

Project 42_019 consists of 5 uGMRT pointings in Band 3
(250-550 MHz) which cover the SSC from west of the cluster A 3558 to
east of A 3562. It was designed to study the diffuse emission
features in this region (Trehaeven et al. in prep). The two datasets
which include SOS 90630 and SOS 114372 are centred at RA_J2000 =
13^h30^m0^s and DEC_J2000 =
-31^∘39^'00^'' and RA_J2000 =
13^h27^m58^s and DEC_J2000 =
-31^∘30^'30^'' for the two sources
respectively.  The data were recorded with 8192 spectral channels and
an integration time of 4 sec in full Stokes mode. The sources 3C 48
and 3C 236 were used as primary calibrators, and 1311–222 was used
as phase calibrator. The data were processed by means of SPAM
<cit.>. The
full band was split into 6 sub-bands each 33.3 MHz wide and the
sub-band were then imaged together using WSClean (v2.10) at the common
frequency of 400 MHz. Residual amplitude errors are of the order of
8%.  The final deep images were produced with weighting=`Briggs' using robust=-0.5 and a set of tapering to
properly image the features on different angular scales. Primary beam
correction was performed using the task PBCOR in the NRAO Astronomical
Image Processing System (AIPS).

All galaxies discussed in this paper (SOS 61086, SOS 90630,
SOS 114372 and ShaSS 421) are included in the ∼ 6×6
deg^2 field imaged by ASKAP at 887 MHz as part of the EMU Early
Science Programme <cit.> ESP 20 and already
presented in <cit.>, where observation and data reduction
details are given.  In brief, the telescope was configured to produce
36 electronically formed beams arranged on the sky in a 6×6
square grid; each individual beam covers an area of ∼ 1 deg^2
for a total instantaneous field of view of ∼ 31 deg^2.  The
observations were performed with 35 out of the 36 antennas of the
array, at 887 MHz, with bandwidth Δν=288 MHz (288 channels
each of bandwidth of 1 MHz). The final image is centred at RA_
  J2000=13^h25^m50^s, DEC_
  J2000=-31^∘03^'05^''.  Residual amplitude
calibration errors are of the order of 5%.

Finally, the galaxies SOS 61086, SOS 90630, SOS 114372 were
observed by ASKAP as part of the POSSUM Pilot 2 survey (POSSUM2),
whose goal is to study the magnetic fields of the Universe in
different environments.  The pointing centres for Band 1
(800-1088 MHz) and Band 2 (1152-1440 MHz) are respectively
RA_J2000 = 13^h29^m47^s and DEC_J2000 =
-30^∘17^'10^'' and RA_J2000 =
13^h28^m51^s and DEC_J2000 =
-29^∘43^'04^''. The Band 1 data was taken
in the closepack36 PAF beam configuration and Band 2 was in the
square_6×6 configuration. Due to the different frequencies,
the final image in Band 2 covers a slightly smaller sky area, each of
the electronically formed beam being ∼ 30% smaller than in
Band 1. The data reduction of the POSSUM2 observations is the same as
the procedure followed for the Rapid ASKAP Continuum Survey (RACS),
detailed in <cit.>. Residual amplitude calibration errors
are of the order of 5%. The datasets in both bands provide total
intensity images, polarisation information through Rotation Measure
(RM) synthesis, and spectral index information (obtained dividing the
Taylor 1 by the Taylor 0 images). The final full resolution images of
the galaxies SOS 90630 and SOS 114372 at all radio frequencies are
reported in the Appendix <ref>. The analysis of the polarisation
will be presented in a forthcoming paper.





 §.§ Radio images and fluxes



Our aim is to investigate the nature of the extraplanar radio
continuum emission, i.e. to identify the mechanisms responsible of
such a feature as well as which processes are powering the electrons
outside the galaxy disk. The multi-band radio data with their
resolution and sensitivity enable us to derive the radio properties in
different regions - discs and tails. However, the different angular
resolutions in our dataset (see Table <ref>) need to be
considered in defining the galaxy regions. For SOS 90630 and
SOS 114372, we can distinguish two regions: one comprehending the
galaxy and the inner part of the tail (hereafter `gal+tail1') and the
other covering the outer part of the tail (tail2). This choice was
made to account for the different angular resolutions of our radio
images.

For SOS 90630 in Fig. <ref> we notice that in the full
resolution images the length of the tail is similar (∼ 43 kpc)
at four frequencies, while at 235, 610 and 1367 MHz the sensitivities
do not (or only barely) allow the detection of the radio tail outside
the visible galaxy (see Table <ref>).

In Fig. <ref> the tail of SOS 114372 shows, at the
sensitivity of our images, different extensions in projection at
different frequencies, e.g. ∼ 72 kpc at 943 MHz and 29 kpc at
1367 MHz. We notice that the ASKAP-POSSUM2 image at 943 MHz has the
second better sensitivity in the dataset and traces the extraplanar
emission further out the galaxy disk.

To account for the different angular resolutions in our datasets,
which range from 7.66^''×7.28^'' of the
full-resolution MeerKAT images to
18^''×18^'' for ASKAP-POSSUM Band 1,
and ensure a consistent spectral analysis, we convolved all images to
the same angular resolution of
18^''×18^'', with the exception of GMRT
235 MHz for SOS 90630 (whose beam is larger) and GMRT 610 MHz
(which is not used in the analysis, see below). The convolved images
are shown in Figs. <ref> and <ref>.

At each frequency, for SOS 90630 and SOS 114372 we derived i)
the integrated flux density; ii) the flux density of gal+tail1;
iii) the flux density of the remaining outer part of the tail
(tail2). At the adopted resolution the gal+tail1 component is all
included within the beam of the image, and S_ gal+tail1 has
been obtained fitting the peak in the image with a Gaussian component
(see Fig. <ref>). The total flux, S_ tot, was derived
by integration over the whole area taking the images at 943 MHz as
reference. The flux of tail2 is obtained subtracting S_
  gal+tail1 from S_ tot.  These values are listed in
Table <ref>.

For each galaxy and each component we estimated the uncertainty
Δ S on the flux density S using the relation

  
    Δ S=√((σ×√(N_ beam))^2+(ξ_ cal× S)^2)


where σ is the local noise in the image, N_ beam is the
extent of the source (or component) in number of beams, ξ_
  cal is the calibration uncertainty.

For ShaSS 421 we derived S_ tot=5.16±0.26 mJy.  All the
other flux density values are reported in Table <ref>.

In the following we provide a description of the radio morphology for
each galaxy and a zero-order analysis of the integrated spectrum of
the region encompassing the galaxy and the beginning of the radio tail
(gal+tail1), and of the outer tail region (tail2). In the description
of the morphologyy we benefit of the high-resolution MeerKAT images,
although the quantitative analysis is based on the convolved
images.



SOS 90630. This galaxy shows a radio continuum tail oriented
westwards with respect to the position of the optical galaxy centre,
coincident with the radio peak. At 1.28 GHz it extends for 45 arcsec
(∼43 kpc) with hints that the radio tail forks in two branches
perpendicular to the direction of the tail at 40 arcsec west from the
centre, as seen in the MeerKAT 1283 MHz image at full resolution
(Fig. <ref>). We point out that the
westernmost knot is most likely associated with the background galaxy
visible in Fig. <ref> and for this reason we
do not consider it as part of the tail. The total radio power of
SOS 90630 at 1367 MHz is P_ 1367 MHz=1.49×10^22
W Hz^-1.

In order to check for spectral steepening along the tail, we separated
the flux density of the radio emission in the two components as
explained above.  Usings the flux density obtained from the
18^''× 18^'' images for each component
we fitted the spectra of these two components with a power law (the
610 MHz flux density values were not included in the fit as they are
clearly overestimated). Despite the scatter of our measurements along
the tail, mainly due to the different radio interferometers used in
this analysis, our data show an unambiguous steepening, with
α_ 325 MHz^ 1367 MHz(gal+tail1)=-0.63±0.11 and
α_ 325 MHz^ 1367 MHz(tail2)=-1.03±0.12. The
integrated spectrum of the radio emission has a spectral index
α_ 235 MHz^ 1367 MHz(tot)=-0.62±0.09.



SOS 114372. The radio emission in this galaxy follows the
optical disc (but see Section <ref>), for a total size of
∼ 30 kpc, and is characterised by a radio tail roughly
perpendicular in projection to the optical disc in the north-west
direction, as shown in Fig. <ref>. The
radio tail is resolved along its minor axis, and shows two ridges of
emission extending ∼40 arcsec (∼38 kpc) north of the disc,
clearly visible at all frequencies in the higher resolution
images. The total radio power at 1367 MHz for this source is P_
  1367 MHz=6.35×10^22 W Hz^-1.

As in the case of SOS 90630, using the flux density values derived
from the 18^''×18^'' images a single
power law fit shows a clear spectral steepening in tail2 compared to
the combination of the galaxy and beginning of the tail.  In
particular, we measured α_ 325 MHz^
  1367 MHz(galaxy+tail1)=-0.72±0.10 and α_
  325 MHz^ 1367 MHz(tail2)=-1.49±0.12. The integrated
spectrum of the radio emission is dominated by the galaxy and has a
spectral index α_ 325 MHz^
  1367 MHz(tot)=-0.70±0.10.

ShaSS 421. Only the 887 MHz ASKAP observation is available for
ShaSS 421. No radio tail is detected at the sensitivity of the image,
whose local noise is ∼ 50 μJy b^-1, significantly
higher than for SOS 90630 and SOS 114372. The radio emission extends
for ∼ 62 kpc along the optical disc, as shown in
Fig. <ref>. The radio morphology has a bipolar
shape. The radio continuum emission detected by ASKAP peaks in the
centre of the galaxy stellar disc. Moving away from the centre along
the galaxy major axes in both sides of the disc, the radio emission
shows recollimation, to broaden again in a `wrapped candy'
morphology. The radio power of the emission is P_
  887 MHz=3.45×10^22 W Hz^-1. The radio morphology is
actually suggestive of a radio galaxy of nuclear origin. We extracted
the 1.4 GHz flux density from the NRAO VLA Sky Survey
<cit.> and obtained a total spectral index
α_ 887 MHz^ 1400 MHz=-1.04±0.18. Further studies
on the origin of the radio emission are ongoing.





§ RADIO TAILS OF RPS GALAXIES



In this section we analyse the observed radio properties of
SOS 90630, SOS 114372 and ShaSS 421 in the framework of the RPS
scenario as supported for these objects by multi-band studies and
hydrodynamical simulations. We also recall the main results obtained
by <cit.> for SOS 61086.  In
Figs. <ref>,
<ref>, <ref> and
<ref> we show the VST r-band image of each
galaxy with superimposed the radio continuum emission at 1.28 MHz
(red contours) and the Hα emission (cyan contours).



 §.§ SOS 90630



The galaxy SOS 90630 is a member of the poor cluster
SC 1327-312. The VST r-band image
Fig. <ref> shows the highly asymmetric light
distribution with a western arm extending well out of the disc. The
deep MeerKAT radio continuum emission at 1.28 MHz peaks at the galaxy
centre and extends west in the same direction of the asymmetric
emission observed in the visible image.

To estimate the equipartion magnetic field for the galaxy and inner
tail (gal+tail1) and for the outer part of the tail (tail2) we used
the formulae in <cit.>. For both regions we assumed a
cylindrical volume, a ratio between electrons and protons k=1 and a
filling factor Φ=1, obtaining H_ eq∼ 1.8 μG and
∼ 0.9 μG, respectively.



In order to estimate the age of the radiating electrons in the two
regions under considerations, we derived the break frequency ν_
  br by fitting the synchrotron spectrum with the software Synage
<cit.>. The best fit is obtained assuming a JP model
<cit.> and an injection spectral index α_
  inj=-0.5. The fit provides a break frequency ν_ br=
10327 MHz for gal+tail1, and ν_ br=1994 MHz for tail2.  The
fit of the radio emissions are shown in Fig. <ref>,
upper row.  The corresponding radiative ages, computed following Eq. 1
in <cit.> which accounts for the equivalent magnetic field
strength of the CMB radiation, are t_ rad∼ 4.0×10^7 yr
for the galaxy and inner tail and t_ rad∼ 7.7×10^7 yr
for the outer tail.

The MeerKAT inband spectral index in the frequency range 908-1656 MHz
could be derived only for the nuclear component and inner part of the
tail.  The surface brightness in the region corresponding to tail2 is
too low and falls below the sensitivity limit of each frequency
channel used to derive the inband spectral
index. Fig. <ref> clearly shows the steepening from
the nuclear region, where the spectral index is α∼ -0.8, out
to the edge of tail1 where α∼ -1.8.

In Fig. <ref> we also show the distribution
of the Hα emission from WiFeS IFS (cyan contours) as analysed
by <cit.>. The gas out of the disc extends ∼ 4 kpc in
projection in the NW side, and along a more than 40 kpc-long tail to
the west. The radio tail extends in projection with almost the same
length. The ionised gas disc is also truncated in the South-East
side. Due to the different resolution it is not clear if the
non-thermal emission of the galaxy is characterised by a similar
truncation.  Moreover, the moderate disc truncation observed in the
ionised gas distribution could be related to the galaxy inclination
angle whose impact on the amount of gas stripped from a disc is
significant when the wind is close to edge-on as was shown with
hydrodynamical and numerical simulations
<cit.>. In this case much less gas is removed
from the galaxy reducing the truncation effect.

<cit.> found that the star formation across SOS 90630 is
highly asymmetric with a `crown' of Hii regions tracing the
eastern disc and other star-forming regions in the western disc
elongated toward the tail in a spiral arm <cit.>.

A single burst of star formation over the last 200 Myr that is still
ongoing is responsible of the young stellar population detected in the
centre and in the eastern edge of the galaxy disc, which is likely
induced by the compression exerted by the ram pressure. Due to their
ageing as traced by the inband spectral index in
Fig. <ref>, the electrons in the inner tail can be
originated by this ongoing star formation.



N-body/hydrodynamical simulations <cit.> enabled to estimate the parameters of
the hydrodynamical interaction between the gas disc of the galaxy and
the ICM of host cluster SC1327-312 indicating a ram pressure acting
close to edge-on, a wind velocity V_ICM= 500  km s^-1 and
the time of the ram-pressure onset of about 120 Myr.

Together with the similar length of the radio tail and the ionised gas
tail, the consistency among i) the epoch of the onset of
stripping (120 Myr); ii) the age of the youngest stellar
population (<200 Myr) and iii) the age of the radio tail
(<100 Myr) supports a scenario where the radio emitting plasma and
the warm gas tails are related to the same event, i.e. both affected
by the ram pressure with similar efficiency <cit.>.



 §.§ SOS 114372



The bright star-forming spiral galaxy SOS 114372 is the only galaxy
of the sample which is member of a rich cluster, A 3558.  Extraplanar
ionised gas has been detected along the full extent of the galaxy disc
out to 13 kpc in projection from it (cyan contours in
Fig. <ref>). Running
N-body/hydrodynamical simulations of RPS for this galaxy (see
Appendix <ref>), <cit.> reproduced the observed
distribution and surface brightness of the ionised gas. In this case,
the wind angle turns out to be intermediate between face-on and
edge-on, the estimated wind velocity is ∼ 1400 km s^-1 and
the derived time of onset of the RPS is t ∼ 60 Myr.

MeerKAT observations reveal a widespread emission out of the galaxy
disc whose extension in projection largely overcomes (38 kpc long at
1.28 MHz) the ionised gas tail. In this case the truncated ionised
gas disc observed in the NE edge of the galaxy corresponds to a clear
truncation also in the radio continuum (see
Fig.  <ref>). We point out that the IFS
observations do not cover the whole galaxy, for the SW edge of the
galaxy IFS data are not available <cit.>.  Therefore, it is misleading to compare radio and
IFS observations in this part of the galaxy.



We computed the equipartition magnetic field and estimated the
radiative age of the electrons in the gal+tail1 and in the outer part
of the tail (tail2) under the same assumptions used for SOS 90630. We
obtained H_ eq∼ 2.56 μG and 0.86 μG
respectively. We fitted the spectra of the two regions as for
SOS 90630. Again the best fit was provided by a JP model (see
Fig. <ref>, lower row). We derived ν_
  br=6851 MHz and ν_ br=1428 MHz for gal+tail1 and tail2,
respectively, with a corresponding age for the radiating electrons of
t_ rad∼ 4.8×10^7 yr and t_ rad∼
9.3×10^7 yr.

The MeerKAT inband spectral index, shown in
Fig. <ref> with radio continuum and Hα
contours overlaid, covers only the disc of the galaxy. It is flat in
the internal disc and consistent with the estimate of the spectral
index of the integrated spectrum throughout the whole extent of the
disc. We notice, however a marginal steepening at the north-eastern
edge of the disc and a region characterised by the higher values of
the spectral index SW in the disc. Unfortunately, due to sensitivity
limits, the inband spectral index along the tail could not be derived.

In a region SW from the centre <cit.> identified a
starburst. The agreement between the age (∼100 Myr) of the
youngest stellar population in this region and the age (started ∼
60 Myr ago) of the gas stripping process inferred by the
N-body/hydrodynamical simulations led to ascribe this starburst to
the compression of the ISM caused by the ram pressure - one of the
first detections of ram-pressure induced star formation. On the other
hand, in the NE disc a region of star formation significantly lower
than in the rest of the disc was also identified. The full spectral
modelling and the line strengths provided evidence for the detection
of a starburst which lasted for ∼ 0.3 Gyr and which has been
just shut down. It is remarkable that these two regions correspond to
the regions where the extreme values of the radio spectral index are
measured in the disc as shown in Fig. <ref>,
i.e. the spectral index steepens in the post-starburst NE region and
flattens in the SW starburst region of the disc. This shows how the
1.28 MHz MeerKAT observations are able to resolve the star formation
in the Shapley Supercluster galaxies.

The flatness of the spectral index towards the SW starburst region is
consistent with the decreasing trend of the spectral index with
increasing SFR found in nearby galaxies covering a wide range in
morphology, SFR, stellar mass, and environment (field vs. interacting
galaxies) studied with both global <cit.> and
spatially resolved measurements <cit.>. This
suggests that the star-formation process could affect the energetics
of the cosmic-ray electron population in a galaxy by increasing the
number of younger and more energetic relativistic particles in galaxy
regions with higher star-formation activity (e.g. starburst).

Also for this object the consistency among i) the epoch of the
onset of stripping (∼ 60 Myr); ii) the age of the youngest
stellar population (<100 Myr) and iii) the age of the radio
tail (<100 Myr) suggest that the relativistic electrons and the
ionised gas are both stripped from the disc by the same
mechanism. However, unlike SOS 90630, the length of the radio and
optical tails are significantly different.





 §.§ ShaSS 421


ShaSS 421 is the brightest and most massive galaxy of the sample. A
detailed study of this object, based on MUSE observations, is in
preparation. Here we briefly mention the main features supporting the
ongoing RPS scenario.

Fig. <ref> shows evidence of extraplanar ionised
gas extending NE from the disc. Within a radius ∼ 250 kpc there
are not other cluster members with signs of disturbed morphology,
therefore excluding galaxy interaction as the cause of the observed
extraplanar emission. Further evidence in favour of the RPS scenario
is given by the orientation of the ShaSS 421 tail which is opposite
to the peak of galaxy density as derived in the ShaSS
<cit.>. The galaxy is located ∼ 385 kpc in
projection (0.3r_200) from the centre of AS 0726 as shown in
Fig. <ref>.

The galaxy was targeted with MUSE. Two contiguous pointings slightly
overlapped in the centre covered an area of ∼ 1^'×
2^' including the galaxy and its extraplanar region. In
Fig. <ref> the ionised gas emission traced by
Hα is shown. The extraplanar gas extends NW up to 15 kpc out
of the disc and along all the disc length. The one-sided extraplanar
gas emission as well as the truncation of the gas disc along the SW
edge of the galaxy support an ongoing RPS event.

In this galaxy, we do not observe a radio tail.  The analysis of the
MUSE spectra showed that the galaxy host an AGN which could be
responsible of the radio emission intensity and shape. However, the
extent of the H_α emission is larger than the radio emission,
which does not reach either the outer edges of the galaxy or the
south-eastern extent above the galaxy disc. It seems that the RPS
mechanism is not capable to affect the non-thermal component of this
galaxy.



 §.§ SOS 61086



The radio tail of SOS 61086 has been studied in
<cit.>. Here we recall the main features of this object
and the proposed interpretation in order to compare them with those of
the other galaxies in the sample.

SOS 61086 is a member of the poor cluster
SC 1329-313. <cit.> showed that this galaxy is characterised
by a tail of ionised gas extending up to ∼ 30 kpc in
projection. They demonstrated that the galaxy is affected by RPS and
estimated the time of the onset (`age') of RPS of about 250 Myr in
agreement with the age of the young stellar population (< 200 Myr),
which likely suggests a ram-pressure induced star formation.

<cit.> reported on the radio tail revealed at 1.28 MHz
with MeerKAT. The radio-emitting plasma and the ionised gas show equal
extent in projection from the galaxy disc and similar truncation in
the gas disc itself (see Fig. <ref>). Taking
into account the steepening of the radio spectral index along the
tail, the age of the relativistic electrons and their velocity, which
is found to be consistent with the wind velocity inferred from the
N-body/hydrodynamical simulation, <cit.> concluded that
the radio tail revealed by MeerKAT observations in SOS 61086 is due
to the ongoing RPS and it is fuelled by relativistic electrons
stripped from the galaxy.



§ DISCUSSION



In Section <ref>, we provide evidence that the ionised gas tail and
that traced by the relativistic electrons are due to the same
mechanism - the hydrodynamical interaction between the ICM and the
ISM. A step further is to understand if the radio continuum emission
in the tail is powered either by in situ star formation
<cit.> or by relativistic electrons stripped from the
galaxy by ram pressure and possibly re-accelerated
<cit.>.

Among the four RPS galaxies showing ionised gas tails, three present
consistent synchrotron tails revealed by multi-frequency radio
observations, while the 887 MHz ASKAP observations do not detect
radio continuum extraplanar emission in the most massive galaxy of the
sample, ShaSS 421. Beyond the different instrumental characteristics
(sensitivity and spatial resolution), which are crucial for the
detection of the faint extraplanar emission, the galaxy properties and
orbits as well as their own environment may play an important role in
the process of sweeping away the ISM from the disc.

In the following we investigate the origin of the electrons in the
tail and describe the different environments of the RPS galaxies in
our sample also in order to possibly distinguish the effects of the
magnetic field in the RPS process.



 §.§ Origin of the relativistic electrons in the tail of SOS 90630


The relative lengths of the radio continuum emission and the Hα
tails can provide a piece of information to disentangle the origin of
the relativistic electrons in the tail. Radio tails either longer
<cit.> or shorter <cit.> than
Hα ones have been observed. However, it is difficult to compare
the lengths of the tails because differences in extent may be due on
differences in sensitivity and projection effects.

The velocity of the plasma containing the relativistic electrons
stripped from the galaxy can be estimated from the tail length and the
radiative age. The age estimated for the relativistic electrons (∼
77 Myr) in tail2 is actually an average value throughout the extent
of the tail which can be associated to the average distance from the
centre of the outer tail, i.e. 30 kpc (see Fig. <ref>). The
tail length should be then corrected for the projection effect using
the angle between the ram-pressure wind velocity, as derived by the
simulations, and the line of sight. In this case this angle is ϕ∼ 67 deg.

Under these assumptions, we obtain a velocity of the stripped
non-thermal component of ∼ 420 km s^-1, which is consistent
with the wind velocity (V_ICM= 500 km s^-1) resulting from
the N-body/hydrodynamical simulations of RPS affecting
SOS 90630. The result indicates that the radio tail is produced by
the gas stripped from the disc.

The other piece of information is given by the radio spectral index
across the galaxy and the tail. Also in this case, opposite behaviours
have been reported in the literature for different
objects. <cit.> did not observe a steepening of the
spectral index in the extraplanar radio emission of NGC 4438, while
other studies <cit.> reported a rapid
ageing of the electrons with increasing distance from the galaxy
disc. For SOS 90630 we actually observe a steepening in the inband
spectral index (Fig. <ref>) which denotes an electron
ageing from the galaxy disc to the inner tail thus supporting that the
relativistic electrons are, at least in the inner tail, stripped from
the galaxy disc. Also considering the spectra of gal+tail1 and tail2
(see Section <ref>) we measure a clear steepening of the
spectral index in tail2 although corresponding to only a modest
ageing.

However, we notice that the line diagnostics analysis shows that the
excitation of the gas both within the main body of the galaxy and
within its extended tail is due to Hii regions
<cit.>. Therefore, even if we tend to interpret the radio
emission of the tail as due to electrons stripped from the galaxy
disc, in the outer tail we cannot exclude a contribution due to in situ star formation, which would account for the modest ageing
observed in this region.



 §.§ Origin of the relativistic electrons in the tail of SOS 114372


The spectra of gal+tail1 and tail2 (see Section <ref>) show
a steepening of the spectral index in tail2 which suggests that the
relativistic electrons origin from the galaxy disc. The ageing
observed in tail2 is nevertheless not dramatic. <cit.> showed
that the emission in some regions of the galaxy and in the whole gas
outflow presents a significant contribution from shock excitation,
together with emission powered by star formation. The distribution of
the regions with higher velocity dispersion and shock-ionised gas
across the galaxy shows that the ram pressure compresses and strips
the gas out of SOS 114372, forming a tail of turbulent shock-ionised
gas and dust <cit.>. Shock due to ram pressure may similarly
affect and accelerate also the relativistic electrons
<cit.> and explain their modest ageing in the outer
tail.

Although the difference in length of the radio and Hα tails may
depend to difference in sensitivity, the almost 3× longer radio
tail of SOS 114372 deserves further investigations.

A more extended tail in radio than Hα could be consistent with
similar results found in galaxies and explained in terms of different
time-scales involving star formation traced by the two emissions
<cit.>. While radio emission traces star formation
on time-scales >10 Myr, Hα emission traces star formation on
smaller time-scales. <cit.> found that the older star
formation traced by the 1.4 GHz emission is more extended than the
younger one traced by the Hα emission on a sample of
star-forming galaxies. Since our highest radio frequencies
(1283-1367 MHz, see Table <ref>) are very close to the
canonical radio tracer of star formation at 1.4 GHz, the finding of
<cit.> could be in line with a more extended tail in radio
than Hα in SOS 114372 (and to a lesser extent in
SOS 90630). Following their work, an older stellar population could
be more easily detected and more extended in the radio because it
takes time for the radio emission to be produced in supernovae and for
the cosmic rays to be transported far away from the galaxy disc along
the tails. The galaxy was also observed with MUSE by
<cit.>. They observed a ionised gas tail consistent with
the radio tail length at 1283 MHz, but still much shorter than the
radio tail observed at 943 MHz.

Also for this galaxy, we tentatively estimate the velocity of the
stripped plasma. The radiative age (∼ 90 Myr) in tail2 can be
associated to a tail length of 38 kpc (see
Fig. <ref>). After correction for the projection effects,
i.e. accounting for an angle ϕ∼ 54 deg between the wind
velocity and the line of sight provided by the simulations, we obtain
a velocity of stripped non-thermal component of ∼
510 km s^-1 which is 2.7× lower than the wind velocity
(V_ICM=1400 km s^-1) inferred from the
N-body/hydrodynamical simulations of RPS run for SOS 114372.
However, we notice that the simulations ran by <cit.> did not
include galactic magnetic fields. If the ram pressure differently
affects the gas and the non-thermal component in the galaxy, it is
also possible that the radio continuum and the ionised gas tails
present different morphology which cannot be foreseen by the
simulations now available for SOS 114372.



 §.§ The environment of the RPS galaxies




  §.§.§ SOS 90630 in the poor cluster SC 1327-312


The cluster SC 1327-312 has a well defined centre with the peak of
the X-ray emission coinciding with a bright galaxy. The velocity
dispersion of the cluster is 535±17 km s^-1
<cit.>. The cluster lies almost entirely within the virial
radius of A 3558 and it remains hard to disentangle the two
clusters. In particular, SOS 90630 is located in the outskirt of
SC 1327-312, 0.2r_200 SW with respect to the centre.

<cit.> analysed the X-ray gas profile of SC 1327-312
showing that it is rather symmetric. However, the whole SSC shows a
continuous filamentary structure 2 degrees in extent, that is filled
with hot gas as seen by both Planck and XMM-Newton satellite
<cit.>. <cit.> identified two clear
overdensities of blue supercluster galaxies in the western region of
the poor cluster SC 1329-313, and a linear structure bisecting the
clusters A 3558 and SC 1327-312. This is actually the region where
the clusters A3562 and A3558 are both experiencing merging events,
which may suggest that this merging has recently triggered star
formation in these blue galaxies, possibly through shock fronts
produced when the ICMs of merging clusters collide. Alternatively,
these overdensities could reflect infall regions along the filament
connecting the clusters A3558 and A3562. All this places SOS 90639 in
an environment where cluster interactions occurred and/or are ongoing.

The role of the magnetic field in RPS events has been investigated by
<cit.> who ran magnetohydrodynamical simulations of RPS
including a galactic magnetic field. Their work showed that the total
amount of gas stripped in the magnetised galaxies does not
significantly differ from that removed in unmagnetised ones.  They do
not find any dramatic difference in the morphology of the tails with
and without magnetic field, although the magnetic fields in the disc
lead to larger unmixed structures in the tail. Actually, we observe a
fragmented ionised gas tail in SOS 90630.

<cit.> simulated the interaction of disc galaxies orbiting in
a cluster and exposed to a uniformly magnetised wind including
radiative cooling and self-gravity of the gas. Such an interaction
produces a very filamentary structure in the tails which then appear
bifurcated in the plane of the sky. These features could be
distinguished in the tail of SOS 90630 suggesting that the galaxy is
exposed to a uniformly magnetised wind.

We should mention that if the RPS explains the truncation of the gas
disc as well as other features (gas distribution and star formation in
the galaxy disc), the length of the ionised gas tail needs the help of
another force to be reproduced. <cit.> suggested that both
RPS and the tidal interaction with a massive companion whose external
edge is visible in the lower right corner of
Fig. <ref>) contribute to the formation of
the long tail, as observed in other galaxies
<cit.>. We notice that the simulations of
<cit.> assumed that a constant ram pressure is exerted on
the galaxy disc. On the other hand, <cit.> showed that varying
the ram pressure strength along the galaxy orbit leads to
significantly different density and velocity structure in the
tail. This new evidence given by the simulations implies that we
cannot exclude a priori that SOS 90630 is actually affected by
RPS alone.



  §.§.§ SOS 114372 in the rich cluster A 3558


The bright and massive (M_⋆ = 7.7× 10^11 M_⊙)
galaxy SOS 114372 is a member of the rich cluster A 3558 having a
velocity dispersion of 1007±25 km s^-1 <cit.>. The
galaxy is located ∼1 Mpc from the cluster centre which
corresponds to 0.4r_200, in an environment characterised by a
lower density with respect to the very cluster core. SOS 114372
demonstrates that RPS can affect bright (L>L^⋆) and massive
cluster galaxies both within and beyond the cluster core
<cit.> and thus showing that the ICM is
neither static nor homogeneous.

Actually, a non-homogeneous ICM is expected for non-virialized merging
and post-merging clusters as those in the SSC. From the analysis of
XMM-Newton and Chandra observations, <cit.>
identified a cold front in the A 3558 ICM probably caused by the
sloshing of the core induced by the perturbation of the gravitational
potential associated with a past merger. This cold front is expanding,
perturbing the density and the temperature of the ICM. From their
Fig. 3 it is clear that the cold front is moving in the direction of
SOS 114372 and that it is ∼ 5 arcmin (≃ 300 kpc in
projection) from the galaxy. Unfortunately, the X-ray observations do
not include the region of SOS 114372.



  §.§.§ ShaSS 421: a massive galaxy member of AS 0726


ShaSS 421 is a peculiar case of a massive galaxy affected by RPS in a
low-density environment. In fact, the parent clusters of ShaSS 421,
AS 0726, presents an extremely low X-ray luminosity
L_X,bol=6.9×10^42 erg s^-1 and gas temperature
T_X=0.90 keV <cit.>. On the other hand,
thanks to the newly acquired redshifts in this area, <cit.>
measured a velocity dispersion of 603±48 km s^-1 and the
corresponding dynamical mass of M_200=2.4×
10^14 M_⊙, which suggests a hotter and denser ICM with
respect to the ROSAT detection. Beside this, the dynamical analysis
revealed that AS 0726 is a system of 2-3 groups with ShaSS 421
associated to the most prominent peak in the galaxy spatial
distribution. Therefore, the complex and unrelaxed dynamics of
AS 0726 could contribute to the ongoing RPS observed in
ShaSS 421. The group-group merger would either trigger a shock front
in the ICM, or allow the high encounter velocity between ICM and
galaxy <cit.>.

A datailed analysis of this galaxy is ongoing by means of deep MUSE
data.



  §.§.§ SOS 61086 in the poor cluster SC 1329-313
 

SOS 61086 is a member of the poor cluster SC 1329-313. The galaxy
shows both Hα and radio tails extending out to 30 kpc north of
the disc. The tail orientation, as projected in the sky, is almost
tangential to the cluster outskirt suggesting that the galaxy orbit is
not radial.

<cit.> detected an inter-cluster radio bridge at GHz
frequencies which extends between the cluster A 3562 and the group
SC 1329-313. This bridge takes the form of an arc which connects the
cluster and the group from north, and remarkably follows the X-ray
emission in this region. The authors interpreted the observed low
brightness intercluster diffuse radio emission as a trace of
turbulence injected in the ICM by the flyby of SC 1329-313 north of
A 3562 into the supercluster core
<cit.>. Because of its position, the trajectory
of SC 1329-313 falling into the SSC, and the tail orientation,
SOS 61086 is the textbook case of a galaxy where the RPS is triggered
by the perturbed ICM during a minor merger event.



§ SUMMARY AND CONCLUSIONS



In this work we study a sample of four galaxies undergoing
ram-pressure stripping in different environments of the Shapley
supercluster. The radio continuum emission of three of them
(SOS 90630, SOS 114372 and ShaSS 421) is analysed here for the
first time. Radio observations collected with various interferometers
and spanning the frequency range from 235 to 1367 MHz are available
for the galaxies SOS 90630, SOS 114372 and SOS 61086, while only
887 MHz ASKAP observations cover ShaSS 421.

SOS 90630 and SOS 114372 both show radio continuum extraplanar
emission in different radio bands. At 1.283 GHz MeerKAT radio
continuum tails extend ∼ 40 kpc in projection out of the disc,
while we do not detect a radio continuum tail in the ASKAP data of
ShaSS 421.

Together with the total flux, in the tailed galaxies we are able to
disentangle the radio emission of the galaxy and the inner tail
(gal+tail1) from that of the outer tail (tail2). By means of the
multi-frequency dataset, we derive the spectral index along the tails,
which in both galaxies steepens in the outer tail as it was also
observed in the fourth galaxy <cit.>.
Furthermore, we derive the inband spectral index across the disc of
SOS 114372 and the disc and inner tail of SOS 90630 from the
1.28 GHz MeerKAT data. In both galaxies the tail of relativistic
electrons is superimposed to the tail of ionised gas as traced by
Hα emission. However, at 1.28 GHz the radio tail extends in
projection with almost the same length of the optical tail for
SOS 90630 (as for SOS 61086), while for SOS 114372 the length of
radio tail significantly exceeds that of the optical one.

We interpret our findings in the framework of the RPS scenario, which
was ascertained for these galaxies by the analysis of IFS
observations, multi-band data and N-body/hydrodynamical simulations
<cit.>.

In SOS 90630 and SOS 114372 the consistency among i) the epoch
of the onset of stripping (inferred by the hydrodynamical
simulations); ii) the age of the youngest stellar population
(derived by the IFS data) and iii) the radiative age of the
electrons in the tail (obtained from the synchrotron spectrum)
supports a scenario where the radio emitting plasma and the warm gas
tails are related to the same event, i.e. by the ram pressure, with
similar efficiency. This was also the case of SOS 61086
<cit.>.

SOS 90630. Taking into account the steepening of the spectral
index, the ageing of the electrons and the source (Hii regions)
of the gas excitation in the tail, we suggest that the radio emission
of the inner tail is due to electrons stripped from the galaxy disc,
while in the outer tail we cannot exclude a contribution due to in situ star formation, which would account for the modest ageing
observed in this region.

SOS 114372. The steepening of the spectral index indicates that
also in this galaxy the relativistic electrons origin from the disc,
however the Hα emission in the tail is mainly powered by shock
excitation which may accelerate also the relativistic electrons and
explain their modest ageing in the outer tail.

Considering also SOS 61086, the three detected radio tails are all
fuelled by relativistic electrons stripped with the plasma from the
disc. However, while the electrons in the tail of SOS 61086 do not
show any signs of re-acceleration or rejuvenation, those in the tails
of the other two galaxies are likely either accelerated by shock
excitation (SOS 114372) or further powered by in situ star
formation (SOS 90630).

It is remarkable that all the galaxies of our sample are located in
environments where cluster-cluster interaction occurred and/or is
ongoing. We underline again that the galaxies considered in this work
are the most dramatically affected by RPS across the ShaSS region.

The correlation between clusters and groups merger and RPS events has
been suggested by <cit.> who identified three RPS galaxies
showing active star formation in their tails in Abell 2744. Being
these objects found in close proximity to the merger-related feature,
the authors proposed that the observed star formation has been
triggered by the rapid increase in pressure during an interaction with
the shock front. <cit.> conducted a systematic search
for RPS galaxies in 63 MACS clusters at intermediate redshifts
identifying 53 RPS candidates. Their analysis of a three-dimensional
modelling of the orbits of the galaxies suggests that the most extreme
events are primarily triggered in massive cluster mergers where the
RPS may initiate far from the cluster core because of the large
relative (galaxies-ICM) velocities. This could be actually the case of
SOS 114372. In addition, <cit.> suggested that extreme
RPS can also occur in mergers of poor clusters and groups. This is
likey what we observe for ShaSS 421 in the unrelaxed poor cluster
AS 0726.

A first theoretical study of RPS in the environment of galaxy cluster
merges was presented by <cit.> considering the A901/2
system where two clusters and two groups are simultaneously
merging. They concluded that the galaxies experiencing RPS are
preferentially located near a boundary where the gas moving with the
cluster/subcluster and the gas from the remaining of the system
collide. This recalls the case of SOS 61086 affected by RPS along the
trajectory of the poor cluster SC 1329-313 falling into the SSC.

If cluster-cluster interaction plays a role in favouring the RPS, as
it seems, then the detection of RPS events can be used as a diagnostic
tool to constrain the geometry of cluster collision as proposed by
<cit.>. Actually, the orientation of the tail of SOS 61086
supports and reinforce the scenario of the minor merger of
SC 1329-313 and A 3562 in the SSC, as recently traced by the
intercluster diffuse emission detected at 1.28 GHz by MeerKAT
<cit.>.

The question now is why our sample is limited to four galaxies
experiencing RPS across the harsh environment of the Shapley
supercluster. These ascertained cases of ongoing RPS have been
selected from the optical images, thus biased to objects showing star
formation and/or shock excitation of the gas into the tail. Different
biases, both observational and intrinsic to the mechanism (e.g. the
short duration of the stripping event), make it difficult to identify
RPS candidates and deep/telescope time expensive IFS observations are
needed to trace low brightness extraplanar ionised gas. Therefore,
even if the information provided by IFS observations will be always
essential, the current available and forthcoming radio facilities
(MeerKAT, LOFAR, uGMRT, ASKAP) will allow to map the atomic gas
content and detect extra-planar atomic hydrogen as well as to detect
tails of non thermal component by means of blind surveys of large sky
regions <cit.>. We aim to use this approach to
make a census of RPS galaxies and possibly trace their radio continuum
and Hi tails in the Shapley supercluster.



§ ACKNOWLEDGEMENTS


The authors thank the referee for helping them to improve their work.
PM, TV, VC and GB acknowledge INAF Mini Grant 2022 ShaSEE -
  Shapley Supercluster Exploration and Exploitation. Basic research
in radio astronomy at the Naval Research Laboratory is supported by
6.1 Base funding. This work was supported in part by the Italian
Ministry of Foreign Affairs and International Cooperation, grant
number ZA23GR03. We thank the staff of the GMRT that made these
observations possible. GMRT is run by the National Centre for Radio
Astrophysics of the Tata Institute of Fundamental Research. This
scientific work uses data obtained from Inyarrimanha Ilgari Bundara /
the Murchison Radio-astronomy Observatory. We acknowledge the Wajarri
Yamaji People as the Traditional Owners and native title holders of
the Observatory site. CSIRO ASKAP radio telescope is part of the
Australia Telescope National Facility
(https://ror.org/05qajvd42). Operation of ASKAP is funded by the
Australian Government with support from the National Collaborative
Research Infrastructure Strategy. ASKAP uses the resources of the
Pawsey Supercomputing Research Centre. Establishment of ASKAP,
Inyarrimanha Ilgari Bundara, the CSIRO Murchison Radio-astronomy
Observatory and the Pawsey Supercomputing Research Centre are
initiatives of the Australian Government, with support from the
Government of Western Australia and the Science and Industry Endowment
Fund. The MeerKAT telescope is operated by the South African Radio
Astronomy Observatory, which is a facility of the National Research
Foundation, an agency of the Department of Science and
Innovation. Based on observations collected at the European
Organisation for Astronomical Research in the Southern Hemisphere
under ESO programmes 097.B-0917(A) and 099.B-0016(A).




§ DATA AVAILABILITY


In this work we exploit the radio continuum observations of the
Australian Square Kilometre Array Pathfinder
<cit.>. In particular, at 887 MHz we used the data
of the survey EMU Evolutionary Map of the Universe
<cit.> together with the Early
Science Project ESP 20 <cit.>, and at 943 MHz and
1367 MHz the data of the POSSUM <cit.>
Pilot 2 survey. We also benefit of one pointing of the MeerKAT Galaxy
Cluster Legacy Survey <cit.> at 1283 MHz. The
VST r-band images of the galaxies have been collected and analysed
in the Shapley Supercluster Survey <cit.>.  The raw
images are available through the ESO archive
(http://archive.eso.org/cms.html), ESO Programmes 096.A-0129
091.A-0050, 089.A-0095, 090.B-0414. The reduced r-band images can be
provided by the authors upon request. The ESO MUSE observations of
ShaSS 421 (ESO Programme 097.B-0917) will be published by Merluzzi et
al. (in preparation).








mnras
 


















§ FULL RESOLUTION AND CONVOLVED RADIO IMAGES OF THE GALAXIES SOS 90630 AND
  SOS 114372











In this section we show the radio images of the two galaxies newly
studied in this work and for which multi-frequency radio observations
are available (see Table <ref> for the angular resolution
and rms of each image). The full resolution images are shown in
Figs. <ref> and <ref>, while the images
convolved to 18×18 arcsec^2 are shown in
Figs. <ref> and <ref>.



§ SPECTRAL FITS



In Table <ref> we give the fluxes measured from the
radio images convolved to the resolution of 18×18 arcsec^2
except GMRT 235 MHz for SOS 90630.  In Fig. <ref>
we plot the spectral fits to the data in Table <ref> for
SOS 90630 and SOS 114372 obtained with the program Synage
<cit.>. The parameters used for the fit are detailed in
Sections <ref> and <ref>.





§ N-BODY/HYDRODYNAMICAL SIMULATIONS OF THE RPS GALAXIES



The main goal of the simulations is to understand to what extent ram
pressure might explain the main observed features of the gas: the
truncation of the disc, the shape and extension of the tail and the
kinematics. We ran N-body/hydrodynamical simulations for the galaxies
SOS 114372, SOS 90630 and SOS 61086 in <cit.> to
which we refer for a complete description. The aim of the simulations
was to reproduce as closely as possible the observed velocity field of
the gas and thus constrain the ICM wind angle and velocity and the
time of the onset of ram pressure. The simulations were run from wind
angles close to edge-on to close to face-on at intervals of 10^∘
and the wind velocity range was set taking into account as the lower
limit the galaxy line-of-sight velocity relative to the cluster
systemic velocity and as the upper limit the escape velocity (derived
from the Hernquist model of each cluster). The ICM densities were
derived from the X-ray profile of each cluster <cit.>.

For the three galaxies the simulations were performed adopting a model
galaxy resembling the pre-interaction properties of each galaxy. The
initial conditions are calculated according to <cit.>
based on the work of <cit.>. Our model galaxy includes a dark
matter halo, a stellar disc, a gaseous disc and a stellar bulge. The
disc scale length of the galaxy is directly related to the angular
momentum of the disc with the spin as a free parameter. The stellar
and gaseous disc have exponential surface density profiles. For the
stellar bulge and the dark matter halo a <cit.> profile
is adopted.

The simulations were done with the cosmological N-body/hydrodynamic
code GADGET-2 <cit.> for SOS 114372 while the
cosmological moving-mesh code AREPO <cit.> is used for
SOS 90630 and SOS 61086 <cit.>.













 Paper Title: VEGAS-SSS: Tracing Globular Cluster Populations in the Interacting NGC3640 Galaxy Group.

 


   

   



INAF Osservatorio Astr. d’Abruzzo, Via Maggini, 64100 Teramo, Italy

              marco.mirabile@inaf.it
         Gran Sasso Science Institute,  Viale Francesco Crispi 7, 67100 L’Aquila, Italy
         University of Naples “Federico II”, C.U. Monte Sant’Angelo, Via Cinthia, 80126 Naples, Italy
        INAF − Astronomical Observatory of Capodimonte, Salita Moiariello 16, I-80131, Naples, Italy
        SRON Netherlands Institute for Space Research, Landleven 12, 9747 AD Groningen, The Netherlands
        Kapteyn Astronomical Institute, University of Groningen, Postbus 800, 9700 AV Groningen, The Netherlands
        European Southern Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile
        European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany


    
   


  
  
   Globular clusters (GCs) are among the oldest stellar systems in the universe. As such, GCs population are valuable fossil tracers of galaxy formation and interaction history. This paper is part of the VEGAS-SSS series, which focuses on studying the properties of small stellar systems (SSSs) in and around bright galaxies.
  
   We use the multi-band wide-field images obtained with the VST to study the properties of the globular cluster (GC) population in an interacting pair of galaxies.
  
   We derived ugri photometry over 1.5× 1.5 sq. degrees centered on the galaxy group composed by two elliptical galaxies: NGC 3640 and its fainter companion NGC 3641. We studied the GC system properties from both the ugri and gri matched catalogs. GC candidates were identified based on a combination of photometric properties (colors, magnitudes) and morphometric criteria (concentration index, elongation, FWHM, etc.), using sources with well-defined classifications from spectroscopic or imaging data available in the literature and numerical simulations as references. The selection criteria were also applied to empty fields to determine a statistical background correction for the number of identified GC candidates.
   
  
    The 2D density maps of GCs appear to align with the diffuse light patches resulting from merging events of the galaxies. The highest density peak of GCs is observed to be on NGC 3641 rather than NGC 3640, despite the latter being the more massive galaxy. The azimuthal averaged radial density profiles in both galaxies reveal that the GC population extends beyond the galaxy light profile and indicate the likely presence of an intra-group GC component. A color bimodality in (u-r) and (g-i) is observed for NGC 3641, whereas NGC 3640 shows a broad unimodal distribution. Analysis of the GC Luminosity Function indicates that both galaxies are roughly located at the same distance (∼ 27 Mpc). We provide an estimate of the total number of GCs, and determine the specific frequency for NGC 3640, S_ N =2.0±0.6, which aligns with expectations, while for NGC 3641 we find a large S_ N = 4.5±1.6. 
  
   


VEGAS-SSS: Tracing Globular Cluster Populations in the Interacting NGC 3640 Galaxy Group 
    Marco Mirabile1,2,3https://orcid.org/0009-0007-6055-3933
          Michele Cantiello1https://orcid.org/0000-0003-2072-384X
          Pratik Lonare1 https://orcid.org/0009-0000-0028-0493
        Rossella Ragusa4
        Maurizio Paolillo1,4
        Nandini Hazra1,2
        Antonio La Marca5,6https://orcid.org/0000-0002-7217-5120
        Enrichetta Iodice4
        Marilena Spavone4
        Steffen Mieske7https://orcid.org/0000-0003-4197-4621
        Marina Rejkuba8https://orcid.org/0000-0002-6577-2787
        Michael Hilker8
        Gabriele Riccio1https://orcid.org/0000-0002-6399-2129
        Rebecca A. Habas1https://orcid.org/0000-0002-4033-3841
        Enzo Brocato1
        Pietro Schipani4
        Aniello Grado4
        Luca Limatola4https://orcid.org/0000-0002-1896-8605
          
    
==============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================





§ INTRODUCTION

A globular cluster (GC) is among the simplest astrophysical stellar systems in the universe. It is a very dense system of old stars with a typical age t ≥ 10 Gyr. Its half-light radius is typically  ∼2.5pc, and it exhibits a mean absolute magnitude in the visual passband of M_V∼ -7.5mag, and with stellar masses between ∼ M_⊙^4-M_⊙^6 <cit.>. 

Initially, GCs were thought to host single-age and single-metallicity stellar populations. However, studies of the Milky Way (MW) GCs and old star clusters in the Magellanic Clouds and other nearby galaxies have revealed the ubiquitous presence of chemical inhomogeneities indicative of quite complex chemical evolution characterised by multiple stellar populations as recently reviewed by <cit.>. Although they are more than originally believed, GCs generally exhibit simpler stellar populations compared to their host galaxies in terms of metallicity and age distribution, allowing for more precise analysis of their properties due to their less intricate star-formation histories. Additionally, the old age and high luminosity of GCs make them valuable tools for extragalactic studies. The old age makes GC systems relics that trace the formation and evolution of a galaxy and its surrounding environment. Indeed, they are tracers of the main star formation events in the early universe and later trace the dynamical evolution of their host galaxy <cit.>. Thanks to their brightness, GCs can be observed at large distances, especially in spheroidal galaxies where GCs appear as bright point sources against the smooth light profile of the galaxy. Systematic investigations of GC systems in external galaxies have provided valuable insights into their properties such as: the luminosity function, spatial distribution, color profiles and kinematics. These properties serve as effective indicators of the past formation and evolution of galaxies <cit.>.
Despite their attractiveness as tracers of galaxy evolution, much remains to be understood <cit.>.

The GC systems in external galaxies typically exhibit a universal luminosity function that is well approximated by a Gaussian, characterized by a peak – known as the turnover magnitude (TOM) –at a nearly constant absolute magnitude of M_V^TOM∼ -7.5 mag <cit.>. The width of the Gaussian distribution depends on the mass (luminosity) of the galaxy <cit.> with more massive galaxies showing wider luminosity/mass distribution function. The near-universality of the GC luminosity function (GCLF) has led to its use as a standard candle, serving as a secondary distance indicator <cit.>. 

With the advent of deep wide-field surveys, studies of the spatial distributions of GCs have revealed the existence of rich GC substructures within galaxy groups and clusters <cit.>.
Thanks to their intrinsic luminosity, GCs can be studied out to very large galactocentric radii (far beyond 10 R_e[R_e is the effective radius of the galaxy, defined as the distance within which half of the galaxy's light is concentrated.]), where the galaxy's surface brightness fades away  <cit.>. The association of GCs with the host cluster/group allows for a direct exploration of the substructures therein, such as their distribution and density. These findings can then be used to validate predictions from galaxy evolution models.

In the last decade, the VEGAS (VST Survey Telescope Early-type GAlaxy Survey) survey has been collecting deep multi-band optical imaging data for galaxy groups and clusters <cit.>. Its aim is to study the faintest low-surface brightness features in the group/cluster and to characterize the GC population around the observed systems <cit.>.

In this study, we present an analysis of the GC system in the interacting pair dominated by NGC 3640. NGC 3640 and its companion NGC 3641 were classified as an E3 and an E galaxy, respectively, by <cit.>. They are ∼27 Mpc away based on surface brightness fluctuation (SBF) distance estimates <cit.>. NGC 3640 is part of a loose group composed of at least eight galaxies <cit.>. This group is considered to be dynamically young, as evidenced by the absence of X-ray emissions <cit.>.

<cit.>  used both photometric and spectroscopic data to investigate the morphology and kinematics of NGC 3640. Through the analysis of the surface brightness profile, they identified signatures indicative of past merging events, primarily manifested as shells. Additionally, P88 observed a likely dust lane along the minor axis of the galaxy (from north to south), pointing toward NGC 3641. With spectroscopic data, they determined that NGC 3640 is a fast rotator along its major axis (from east to west) with a V/σ ratio of 1.5. P88 proposed that NGC 3640 is in an advanced stage of merging, although not yet completed due to the absence of strong nuclear radio emission, which is a typical post-merger sign. They suggest that NGC 3640 likely merged with a gas-poor disk galaxy.

<cit.> conducted an analysis on a sample of E and S0 galaxies, estimating heuristic merger ages based on UBV colors for each galaxy. In this work, NGC 3640 is identified as a young elliptical candidate, indicating formation through mergers less than ∼ 7 Gyr ago.
Near-IR color studies of NGC 3640 carried out by <cit.> argue against a major starburst occurring within the last 3 Gyr. Additionally, <cit.> and <cit.> investigated the stellar populations of NGC 3640 using long slit spectra. <cit.> reported an age of the center of 2.5 Gyr, whereas <cit.> found an older (4 Gyr) and less metal-rich center when examining the same region. Moreover, <cit.> concluded that NGC 3640 is consistent with having undergone a dissipational merger up to ∼7.5 Gyr ago.
Consistent with that, <cit.> observed no HI emission from NGC 3640, but detected it from an inclined low surface brightness (LSB) dwarf spiral galaxy located at coordinates 11h21m51.95s, +03°24'17” (J2000), with a radial velocity consistent with that of NGC 3640[<cit.> used a radial velocity of V_opt=1314  km s^-1 for NGC 3640, resulting in Δ V=62  km s^-1.] (V_ H=1180  km s^-1).

<cit.> conducted the first and most detailed study, to the best of our knowledge, about the GC population of NGC 3640. They observed NGC 3640 with the Hubble Space Telescope (HST) and estimated a lower limit for the total GC population of ∼50 GCs. The GC candidates showed a color distribution peaked at V-I=1.08^+0.080_-0.035 mag.

Despite the extensive literature on NGC 3640, none of the analyses have definitively confirmed or ruled out any interactions with the less massive companion galaxy, NGC 3641, which is at a projected angular distance of only ∼ 2.5 arcmin (∼20 kpc) from NGC 3640. The two galaxies show a quite high relative radial velocity of ∼500 km/s which may suggest that if they interacted, it should have been a high speed encounter.




In this study, we investigate the GC population to gain further insights into the history of this system of galaxies.
The paper is structured as follows. In Sect. <ref>, we outline the data used, detailing the procedures for source detection, photometric calibration, and completeness estimation. Section <ref> focuses on the selection criteria for GC candidates based on their morphological and photometric properties. We also provide the final GC candidates catalogs as well as the single candidates in each observed passbands. In Sect. <ref>, we present the analysis conducted using the identified GC candidates. Section <ref> delves into the investigation of GC content in dwarf galaxies across the observed field. A brief summary of our conclusions is
presented in Sect. <ref>.






§ OBSERVATIONS AND DATA ANALYSIS



In this section, we provide a brief overview of the imaging data used in this work and the image processing procedure.




 §.§ Observations and image processing 




The observations used in this work are part of the VST Elliptical GAlaxy Survey (VEGAS, P.I. E.Iodice, ). The survey was performed with the INAF VLT Survey Telescope (VST), which
is a 2.6 m diameter optical telescope located at Cerro
Paranal, Chile <cit.>. The imaging is in the u, g, r
and i-band using the 1×1 square degree field of view camera
OmegaCAM <cit.>. 
A detailed description of the survey and procedures adopted for data acquisition and reduction can be found in <cit.>, <cit.>, and <cit.>. Here, we only provide a brief overview. 
The data were processed with VST-tube <cit.>, a pipeline specialized for the data reduction of VST-OmegaCAM, executing pre-reduction (bias subtraction, flat field normalization), illumination and fringe corrections (for the i-band), photometric and astrometric calibrations. The final co-added images were normalized to a Zero Point (ZP) of 30.00 mag, calibrated using the PSF magnitude within 40 (19.04 pixel) circular aperture of reference standard stars observed with the Sloan Digital Sky Survey <cit.>. Table <ref> summarizes the main characteristics of the observations. Figure <ref> shows a color composite image obtained with the python package make_lupton_rgb <cit.> of the observed field. The figure shows the two studied galaxies at its center and the two other galaxies in the field, NGC 3643 and NGC 3630, along with the dwarfs identified in this work (white boxes, see Sect. <ref>).
















 §.§ Photometry and photometric calibration


In order to detect point-like sources close to the centre of the galaxies, where the background surface brightness is high, we first modeled and subtracted the light profile of all the brightest galaxies in the observed field. This is because the steepness of the galaxy surface brightness profile is reflected as drastic changes in the background, reducing the efficiency of detection for faint sources in these regions. The luminosity profiles of NGC 3640, NGC 3641, NGC 3630, and NGC 3643 were independently derived in all available passbands. To model the light distribution of the galaxies we used the python Elliptical Isophote Analysis package[<https://photutils.readthedocs.io/en/stable/isophote.html.>]. A description of how the model is generated can be found in <cit.>. Figure <ref> shows an example of the original image and the residual (original minus model), for NGC 3640 and NGC 3641 in the u, g and r-band. When examining the residuals in the u and g wavelength range, we did not observe any dust lane, contrary to the observations made by P88. After modelling and subtracting the light of
the bright galaxy in the field, NGC 3640, and its companion, NGC 3641, the field clearly shows the underlying GCs as point sources, as well as a rich web of low-surface brightness features. The system of shells previously reported by P88 is
strongly enhanced in the galaxies-subtracted images (Fig. <ref>, right panels), but there are no obvious signs of localized dust patches/lanes.

To obtain photometry of the sources in the field, we used SExtractor <cit.> on the galaxy-subtracted images for each filter independently. For compact sources, we chose as reference the aperture magnitude within an eight-pixel diameter (∼ 168 at OmegaCAM pixel scale). This choice relies on maximizing the signal vs. noise within the aperture. The aperture correction (AC), required to take into account the missing flux beyond such aperture, was determined by selecting ∼100 bright, isolated point-like sources, then measuring the magnitude difference between our reference aperture (8 pixels) and the pipeline calibration diameter (i.e. 19.04 pixels). 
We investigated potential spatial variations of the AC value across the four bands by dividing the observed fields into four regions. Estimates of the AC value for each region were obtained and found to be consistent with each other within the error.
Table <ref> presents the aperture corrections derived for each passband along with their uncertainties (estimated using the median absolute deviation MAD). The same  bright and compact sources were also used to estimate the median FWHM (Full Width at Half Maximum), reported in Table <ref>.


To give a rough idea of the difference in depth between the available passbands, we note that the number of detections ranged between ∼50000, in the u-band, and ∼320000 in the r-band catalog above 1σ the background level. 

The catalogs were then matched using a matching radius of 10. The matching radius was chosen after several tests with various radii, balancing between completeness (most affected in the shallower u-band) and contamination (most affected in our deepest and richest image, the r-band). We produced both ugri- and gri-matched catalogs, because of the much shallower u-band data. On these catalogs, we applied the following color corrections provided by the VST-Tube pipeline <cit.>:






    m_u_new    =m_u +(m_u-m_g)×0.025(± 0.002) 
    
    m_g_new   =m_g +(m_g-m_i)×0.023(± 0.001)  
    
    m_r_new   =m_r +(m_r-m_i)×0.054 (± 0.001)  
    
    m_i_new   =m_i -(m_g-m_i)×0.003(± 0.001)


As sanity check, we matched and compared our photometry with the SDSS PSF-magnitude measurements <cit.>. The two photometric systems are equivalent, and we identified ∼18000 matching sources in the ugri catalog (within a 10 matching radius). Table <ref> reports the median magnitude offset, Δ mag=m_VST-m_SDSS,  of the matched source together with the rms_MAD, showing good photometric consistency within errors in all bands.

For the foreground extinction correction, we obtained the extinction values from the IRSA (NASA/IPAC Infra-Red Science Archive) dust query module of Astropy, which provides E_B-V at the position of the source. The adopted values to multiply E_B-V to obtain the extinction are  4.239, 3.303, 2.285, 1.583, for ugri, taken from <cit.>.
The complete single band catalogs, as well as the ugri and gri matched catalogs, are available on the VEGAS portal webpage[<http://old.na.astro.it/vegas/VEGAS/Welcome.html>], and on the CDS.

















 §.§ Completeness


Part of the analysis that we perform in the forthcoming sections requires correcting the number counts of faint sources by the detection completeness function, which we derived using the procedure outlined in this section. This function provides the fraction of detected sources over the injected ones (f≡ N_ detected/N_ injected) as a function of magnitude. We have developed a Python code to estimate the completeness function, in four main steps:

    
  * Point Spread Function modeling;
    
  * Injection of artificial sources;
    
  * Detection of sources and matching with the catalog of injected ones;
    
  * Estimation of the completeness ratios, and fitting of the completeness functions. 






In more detail, we first derived a PSF model from ∼10 bright, compact and isolated sources in the image. A cutout of the selected PSF-model candidates was extracted from the image using the 'extract stars' routine in Astropy. The PSF stars were given as input into the EPSFBuilder module of the photutils package[<https://photutils.readthedocs.io/en/stable/epsf.html>] to generate an effective PSF of 43×43 pixel^2 (∼9×9 arcsec^2). 

The magnitudes of ∼2000 artificial stars to be injected were generated using a probability-density distribution (PDF) that matches with the observed total luminosity function of the sources in the image. The PDF is populated independently for each available passband using a random generator[One should note that the total luminosity function also contains the non-point sources component. However, we are not interested in the completeness fractions for extended objects, and the impact of ignoring the two components (compact and extended) is negligible for our purposes.], and the stars are then injected into the image of interest along an equally spaced grid, with a spacing of ∼ 150. The simulations we run are, by construction, independent of the source color. This assumption is made with the intent of keeping the passbands separate, so as to avoid reducing the quality of our data to that of the image with the least depth in the matched sample (see also Sect. <ref> and footnote <ref>).

We run our detection and photometry procedures on the simulated image as described in the previous Sect. <ref>, adopting the same parameters that were used to generate the catalogs. 

The catalogs of detected and injected sources were matched within 10 radius, and cleaned for spurious matches. For the cleaning process we inspected the broadening of the Δ mag=m_ injected-m_ detected function versus the detected magnitude. We adopt a magnitude dependent, iterative sigma-clipping approach. Briefly, we first divide our sample into bins of 0.5 mag width  and calculate the median Δ mag and its rms_ MAD within each bin. For bright sources we considered as matched the sources within a fixed Δ mag interval. For faint sources, instead, we consider the sources to be matched when Δ mag≤± 2 rms_ MAD.
The bright/faint separation cut is set to the point where the rms_MAD 0.1 mag. This approach is depicted in Fig. <ref> for one of the r-band simulations. In the figure, the gray shaded area at magnitudes brighter/fainter than m_r∼23 mag is the region of selection for bright/faint sources. 

Finally, the ratio of the number of detected sources which passed the cleaning stage was normalized to the number of injected sources in the same magnitude bin, f=N_detected/N_injected,  providing us with the discrete completeness ratio behaviour as a function of magnitude. 

To reduce sampling effects, we repeated the simulations 150 times, during which the positions and magnitudes of the injected sources varied. Ultimately, the completeness function is obtained as the median of all the completeness functions generated.

We fitted the completeness ratios adopting  a modified Fermi function <cit.>:


    f_F(m)=1+C·exp[b(m-m_50)]/1+exp[a(m-m_50)]



where m_50 is the magnitude of the 50% completeness limit, a regulates the steepness of the cutoff, b (which must be < a) influences the point at which the deviation above unity starts, and C determines the amplitude of the deviation.

The completeness function in each band was determined by using the procedure described above, injecting the artificial stars on cutouts of the images with ∼ 6×6 arcmin^2 size (see Fig. <ref>). We considered both fields centered on NGC 3640/NGC 3641, and off-galaxy fields. The size of the cutouts are big enough to contain ∼5 R_e of the galaxies. Table <ref> reports the fitted m_50, as well as the interpolated magnitude of the 80% completeness limits (m_80) for the on-galaxy and off-galaxy fields. For the off-galaxy fields, we also give the galactocentric distance, with respect to NGC 3640 measured from the center of the field. Figure <ref> shows the completeness functions derived in three off-galaxy regions and on-galaxy for all four filters. In the figure, we also reported the expected position of the turn-over magnitude of the GCLF  (see Sect. <ref>). In all filters, but the u-band, the data are 80% complete or more at the TOM level. The u-band data has 0% completeness level at the turn-over magnitude, meaning that the ugri matched catalog is limited to the very bright end of the GCLF.
As expected, the off-galaxy regions are more complete than the on-galaxy region at fixed magnitude, with the exception of the region closer to the edge of the image (R_gal>>30 arcmin, see below).

We also derived the completeness functions for 26 other 6× 6 arcmin^2 off-galaxy regions at various galactocentric radii, R_gal. The resulting 50% and 80% magnitude limits for the r-band are reported in Table <ref>. The full sample of fitted functions and the m_80 vs. R_gal plots are also reported in Fig. <ref>. In the left panel of the figure all functions derived in boxes with R_gal≤ 30' are shown with gray dashed curves, while the functions at larger radii are plotted with red dot-dashed lines. The right panel shows the m_80 vs. R_gal behavior.

A small but non negligible depth variation with R_gal exists, a result which is expected because of the specific observing strategy adopted, as described in <cit.>. The images were acquired using a step-dither observing strategy. This result will be later used to select the areas suitable for estimating the population of contaminants (mostly foreground MW stars, but also background galaxies) for the purpose of analyzing the GC population (see Sect. <ref>).











 §.§ Testing the PSF



To obtain a reliable completeness function, we need to have a well-characterized PSF. To test the robustness of the PSF models used for deriving the completeness functions, we compared the magnitudes of unsaturated, bright and compact sources obtained using PSF photometry[We used the PSFPhotometry module from the photutils Python package (for more details, see the link https://photutils.readthedocs.io/en/stable/psf.htmlhere.)] to the magnitudes obtained from SExtractor. The median Δ phot=mag_SExtractor-mag_PSF and the corresponding rms_MAD are reported in Table <ref>. In all cases, but the u-band, the
offset is within the errors. Even in the u-band, the effect is negligible for the specific purpose of deriving a completeness function, especially if one considers that the u image quality is the worst among the available images. 

 Using the same PSFPhotometry, we  checked the quality of the PSF model using the PSF residuals images, where stars are modeled and subtracted using this task. As an independent proof of the reliability of our PSF-modeling procedure, the PSF subtracted images have generally very flat residuals.













§ GC SELECTION




In this section, we will present the procedures to identify  the GCs from the ugri and gri matched catalogs. For reference, the typical radius of a GC in the MW is ∼ 2.5 pc <cit.>. At the adopted distance for NGC 3640 (see Table <ref>), one OmegaCAm pixel (0.21 arcsec) corresponds to ∼27.5 pc, meaning that at the image quality of our data, GCs are point-like sources. 
Hence, we choose to identify the GC candidates using their shape (i.e. their morphometry) and their photometric properties. in particular we  select the GCs based on their  magnitude, compactness and color.










 §.§ Magnitude selection



The GC system in a galaxy exhibits a universal Gaussian GCLF, which can be used to define the magnitude range expected for GCs in the target galaxy, once the distance to the host is known. The width of the GCLF scales with the total galaxy luminosity. 

Using Eq. (5) from <cit.>, and adopting a total galaxy magnitude M_z,gal∼-22.4 mag (see Table <ref>), we evaluated σ_g^GCLF= 1.1 ± 0.2 mag.
Adopting a turn-over magnitude of M_g^TOM∼ -7.5 mag[Derived assuming a mean TOM for Fornax galaxies of ∼24.0 mag from <cit.>, and a distance modulus to this cluster of m-M=31.51±0.15 from <cit.>.], and the distance modulus to NGC 3640 of 32.14 from <cit.>, the expected GCLF peak would be m_g^TOM∼ 24.7 mag. Assuming GC median color (g-r)∼ 0.6 mag, (g-i)∼ 1.0 mag and (u-r)∼ 2.0 mag from <cit.>, we obtained the TOM in the other observed filters:  m_u^TOM∼ 26.1 mag,  m_r^TOM∼ 24.1 mag, m_i^TOM∼ 23.7 mag. Additionally, assuming identical GCLF width in the r and g bands[As reported in Table <ref> the GCLF width does not change much (Δσ=0.02±0.3 mag) from the g- to the z-band, allowing us to assume the value of σ_g in the r-band as well.], we select all sources in the matched catalog within ±3σ^GCLF around the m_r^TOM, namely 20.8 mag≤ m_r≤ 27.4 mag, as GC candidates. As reported in Table <ref>, we adopted a more generous cut at the bright limit, to avoid any possible bias from the adopted distance to the galaxy. The faint detection limit is, in practice, much brighter than the expected m_TOM+3σ because of the incompleteness. We applied the magnitude cut only from the r-band, as we consider this as our reference passband, both because of its fainter depth relative to the TOM (see Fig. <ref>) and for the compactness of its point-like sources (see Table <ref> and Sect. <ref>).





 






 §.§ Morphometric selection



To identify GC candidates, we also adopt the FWHM, flux radius, and elongation (major-to-minor axis ratio) properties, all derived using SExtractor. We decided not to use the star classifier provided by SExtractor due to its lower accuracy in classifying stars and galaxies in the fainter regime. Although there is some degree of redundancy between these indicators, this approach helps in culling the catalog more robustly than by using just one indicator. In addition, for each source, we measured the magnitude concentration index, CI, defined as the difference in magnitude measured at two radial apertures <cit.>. This parameter, has proven to be very effective in separating compact sources from extended ones. After  several tests, we adopted  as reference the  8- and 4-pixel aperture radii: CI=m_8 pix-m_4 pix. For compact sources, after aperture correction is applied to the magnitudes at both radii, the CI should be statistically consistent with zero. However, we chose to not apply any aperture correction to the magnitudes used to estimate the CI, hence our CI sequence of compact sources would simply be offset with respect to zero, by an amount that corresponds to the AC difference for magnitudes at 8 and 4 pixel. Figure <ref> shows the CI versus magnitude plot in r,  which is our reference selection passband thanks to its image quality. This provides the best separation diagnostic between point sources and slightly resolved background ones.


To define selection ranges for identifying compact sources, we independently estimated the mean properties of point-like, bright sources in each passband using the matched ugri catalog. The results, along with the rms_MAD, are presented in Table <ref>.
  
When dealing with the selection of faint sources, two factors require consideration: the increase in photometric error, and the blending with brighter sources. The former introduces broadening in the distribution of the selection parameter considered as the magnitude gets fainter, while the latter results in fainter sources exhibiting larger radii due to the contamination from neighbouring sources which increases the local background. In order to take into account these effects and taking CI as reference parameter, we applied a selection procedure similar to the one adopted for the cleaning of the injected-to-detected catalog for the study of the completeness (Sect. <ref>). The mean locus of the accepted sources is fixed to the median CI of the bright sample, while the broadening is taken to be ± 5 σ_mag. bright for sources brighter than m_r∼ 23 mag, and it is set to ± 3σ_mag for sources fainter than m_r=23 mag. Here, σ_mag. bright is the rms of the CI_r reported in Table <ref>, while σ_mag is the rms  determined in the each magnitude bin fainter than m_r=23 mag.
The results from this approach are shown in Fig. <ref>, where the r-band CI for the matched ugri catalog is reported, together with the curves 

 





adopted for culling the sample: the accepted compact sources are within black dashed lines.


After the CI_r cleaning, we also inspected the other morphometric parameters, and further clean the sample from possible outliers, as shown in Fig. <ref>. For the FWHM, elongation and FLUX_RADIUS parameters, we decided to adopt upper/lower limits to identify bona-fide GCs. Table <ref> contains the parameters values we adopted for the final catalog of GC candidates. To choose these limits,  we made several tests to balance the effects of completeness and contamination, allowing the selection parameters vary from narrower intervals (lower completeness and contamination) to broader (higher completeness and contamination). For these tests, which were run iteratively with the color-color selection (see section <ref>) and included a comparison of the GC population properties  (i.e. radial profile, color distribution and density map, see Sect. <ref>) for each adopted selection range, we observed consistency in the results over a wide range of adopted selection intervals. This result suggest that, globally, the selection is driven by the CI parameter. In conclusion, we decided to adopt the wide  selection ranges of the FWHM, elongation and flux radius as reported in Table <ref>.









 §.§ Color selection




The selection of bona-fide GC candidates can be improved taking advantage of the multi-band coverage of the present dataset. With an approach similar to the one presented in <cit.> and relying on the same GC master catalog adopted there (composed mainly of spectroscopically confirmed GC in Fornax), we used color-color plots to further refine the catalog of pre-selected GC candidates.

The color-color plots for the ugri and gri catalogs are reported in Fig. <ref>, as Hess diagrams. 
Additionally, we included isodensity contours of the GCs from the master catalog obtained from the Fornax Deep Survey <cit.>. The contour level containing 85% (80%) of the sources in the catalog is highlighted with a thick blue (magenta) line for the ugri (gri) catalog and serves as our reference contour[The isodensity levels of the master catalog are obtained using a kernel density estimator[<https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gaussian_kde.html>] (KDE). The adopted parameters were:  bw_method='scott' and bw_adjust=0.7.].

Ideally, one would like to have a GC candidate sample with low contamination and high completeness. However in order to choose which contour to use for color-color selection, we need to balance two effects. Narrow color-color density contours imply low contamination but also a fairly incomplete GC sample; wide  contours imply higher completeness, but also high contamination. To search for the contour level that is the best compromise between the two effects we made several tests. In each test, we adopted different contour levels, and inspected the properties of the resulting catalog (2D-map, radial and color distributions, etc.). Finally, we chose the contour levels containing the 85%[The choice to use two different contour levels is consequence of the higher contamination presented in the gri catalog.] (80%) of  the sources of the master catalog as our reference for the ugri (gri) matched catalog because narrower contours resulted in too sparse final sample of GC candidates, which  is prone to statistical sampling effects, while larger contours were too much contaminated by the background sources and spurious over/under densities in the regions of the mosaic characterised by higher/lower S/N.




The color-color selection algorithm was independently run on the ugri  and on the gri catalog. The former is expected to provide a final GC-sample with lower contamination compared to the gri, thanks to the use of two independent colors, u-r and g-i, and the wider wavelength range covered (more details will be given in Sect. <ref>) . However, it is also much less complete than the final gri GC-sample, because of the much shallower u-band depth compared to the other passbands (Sect. <ref>).






 §.§ Final catalog


The final sample of GC candidates derived from the ugri (gri) catalog contains 769 (4944) sources. The basic parameters of this sample are reported in Table <ref>. 
The  catalog contains: i) the source ID and its RA and Dec coordinates; (ii) the automatically estimated magnitudes by SExtractor, the aperture corrected AB magnitudes and their error in all available bands;  (iii) the morphometric parameters (FWHM, Concentration Index, Flux Radius, elongation) in the reference passband (i.e. r-band), and the E_B-V reddening at the position of the source.

This catalog still contains point source contaminants matching 
the properties of the GC population, especially at the faint magnitude level. Nevertheless, thanks to the
large covered area by VST images, and assuming that any population of contaminants
is uniform over the inspected area, the GCs in the region can be analyzed using background subtraction methods  <cit.>.






























§ GC POPULATION: SAMPLE ANALYSIS 


In this section, we study the spatial, color and luminosity distribution of the selected GC samples. We expect that the final GC-list still contains some contamination from fore- and back-ground sources. With the only exception of the 2D surface density maps, to reduce the effect of such contamination we take advantage of the large observed field, and use a statistical background decontamination method, as described in our previous works <cit.>. This technique involves statistically characterizing the GC contamination levels by inspecting regions far from the galaxies, where no GCs are expected to be. The underlying assumption is that the background sources identified as contaminants are evenly distributed across the field. Therefore, by using the final catalogs of GC candidates, the on-galaxy population is decontaminated from spurious sources using the catalog of off-galaxy candidates as a reference.
 






 §.§ Surface density map of GCs over NGC 3640 field



Thanks to the large format of the VST images, under the assumption that the level of contamination is roughly constant across the observed field, we can study the 2D map of the GC-candidates, without the need of subtracting the background. We inspected the GC distribution maps centered on NGC 3640, applying a kernel density estimator technique[We used the kdeplot packaged of seaborn v.11.2 (link https://seaborn.pydata.org/generated/seaborn.kdeplot.htmlhere) with the following parameters: bw_method='scott' and bw_adjust=0.1. ]. The distribution is inspected using both the ugri and the gri catalogs. A zoom on the central region, over a galactocentric radius of ∼0.3 degrees, is shown in Fig. <ref>. 





In the figure, the global GC population, centered on the pair of galaxies NGC 3640 and NGC 3641, stretches from southeast to northwest with an inclination nearly identical
to the line connecting the two galaxies. A key result from our analysis is that among the two galaxies, NGC 3641, which is also  fainter/less massive, is the one showing the largest central density of GCs. Indeed, the peak of the density appears on NGC 3641.
In the 2D map, an over-density of GC candidates on the other two bright galaxies in the group, NGC 3630 and NGC 3643, is also observed.

To further investigate the 2D distribution of GCs, the iso-density contours from the  surface density map in Fig. <ref> are shown in Fig. <ref> overlaid on the galaxy-subtracted image. 
The southeastern tail of the contour levels appears to be spatially connected to an LSB feature labeled "A" in Fig. <ref>. In the northeast extension of the density contours, we note two such connections. First, the GC contours follow the shape of what appears to be a complex structure of streams and merging features north on NGC 3640 (label B in the figure). Secondly, a bridge of GCs seems to connect the distribution on the galaxy pair along the direction of a disturbed dwarf in the field (Dwarf 12, see Sect. <ref>). We inspected the contour levels in the background region (between 25 and 30 arcmin, see next section) to ensure that the bridge is a real feature and not an artifact. Observations of the background contour levels reveal more rounded structures, with no evidence of any distinct patterns. Inspecting the contour levels in the background regions, we find them to host a lower density population and to be less extended compared to the asymmetric structure of the GC bridge and the other substructures mentioned here.


As a sanity check, we also inspected the maps from the ugri catalog. The general behaviour of the features just described (south-east/north-west stretch, highest density peak on NGC 3641, broader distribution in the region of complex streams, etc.) is preserved, as seen in Fig. <ref> (right panel).
  
We also investigated the overdensity of sources with color 0.15≤(g-i)≤0.65 and 1≤(u-r)≤ 1.6, present in the left panel in Fig. <ref>, to understand if these sources are foreground MW stars or young GCs. We inspected the 2D density map to understand the nature of these sources. We observed a U-shape overdenstity features extending from south-est to south-west respect to the galaxy pair. This overdensity might be associated to a  intermediate GC population formed after merging event of NGC 3640. However, spectroscopic datasets are required to confirm or rule out this scenario.





 §.§ Radial GC number density profile




To study the radial distributions of the GC systems of NGC 3640 and NGC 3641, we investigated the azimuthally averaged surface density profiles centered on each galaxy. Because of the small angular separation between the two galaxies,
we need to adopt some criterion to separate our sample of GC candidates into two subgroups: one belonging to NGC 3640 and another one to NGC 3641[Ideally one would require spectroscopy, which would be very effective in this system thanks to the radial velocity offset between the two (see Table <ref>).].

Figure <ref> shows the criterion we adopted. The separation criterion relies on the effective radius of the galaxies, which is directly linked to the radial extent of the two GC populations  <cit.>. We split the sample using the tangent line between the two circles shown in Fig. <ref>. These circles are centered on the galaxies and have a radius of α effective radii, α R_e. The value of α that makes the two circles tangent is α∼3.9. 



Then, all GC candidates  above the tangent (yellow line  in Fig. <ref>) are associated to NGC 3640, the ones below the tangent are associated to NGC 3641. For the sake of clarity, it is essential to emphasize that the described criterion has been fine-tuned specifically for this case. We also inspected the surface brightness (SB) level of the two galaxies, and found that the point where the two SB profiles reach equality, is at about the same location as the tangent point (within a few arcsec). For the motivation detailed below, we decided to use the criterion based on the effective radii.

Before inspecting the radial density profile on the single galaxies, we analysed the radial surface density profile for the global GCs population (i.e. the combined GCs systems of NGC 3640 and NGC 3641 without any distinction), assuming the center to be the tangent point where the two circles meet in Fig. <ref>.

We considered concentric circular annuli, 1.5 arcmin wide, and counted the GCs candidates within each annulus. For this profile, we masked the regions around saturated pixels, bright stars, image artefacts, etc. The area of the annuli is corrected for such masking.
Figure <ref> shows the radial density distribution of the bulk GCs population using the gri catalog.
We mainly used this global GC profile to identify the region where the background density becomes constant. Since this catalog is richer than the ones we get splitting the sample in two, it allows to constrain more robustly the background contamination level. The gray-shaded region in Fig. <ref> indicates the spatial regime, between 25 and 30 arcmin, that we adopted to estimate the background level (shown as a red-shaded region in the figure). We assumed that all GC candidates  in this region are interlopers (either background galaxies or foreground MW stars) that pass our photometric and morphological selections. We chose this region because interlopers should have a uniform distribution around the group, leading to a flat radial distribution. The region also needs to be away from the outer image area, which typically has lower signal-to-noise ratio because of the observational strategy. At group-centric radii beyond 30 arcmin the distribution approaches to the edges of the images, resulting in regions with lower completeness fractions (Sect. <ref>). This generates the drop seen in the density profile at R>30 arcmin.
The profile in Fig. <ref> shows a monotonic decrease within R≤10 arcmin, followed by a flat density up to 20 arcmin. We rejected the flat region between 10 and 20 arcmin as a possible background area because it has a GC candidate density a factor of ∼2 higher than the 25-30 arcmin region, which cannot be attributed solely to differences in image depth (see Fig. <ref>, right). Additionally, the color distribution of sources in this region shows a blue-density peak compared to the distribution within 25-30 arcmin (see Fig. <ref>), as expected in the case of a residual GC component. As discussed below, we associate this flat region with a possible population of intra-group GC candidates, which drops off after ∼25 arcmin.



We inspected the same radial profile for the ugri catalog and, although it has many fewer GC candidates, the general behaviour (in terms of trend and background identification) is confirmed. 








 

 





 §.§ Individual GC Radial profile




As anticipated, we also analyzed the GCs radial density profiles considering all GC candidates above the separation limit (tangent yellow line in Fig. <ref>) associated with NGC 3640, and the ones in the remaining area that are associated with NGC 3641. We used circular concentric annuli, adopting a constant annular width of 1.5 arcmin for NGC 3640. For NGC 3641, we used radial binning with a width of 0.15 arcmin up to 6 arcmin, followed by an annular width of 1.5 arcmin. This adjustment was necessary to provide a more precise characterization of the inner region of the galaxy, where the density of GCs is relatively high.

Figure <ref> shows the  radial profiles for NGC 3640 (left panel) and NGC 3641 (right). The light-red horizontal line is the background level from the global profile, estimated as shown in Fig. <ref> with a thickness corresponding to  1σ level. Both profiles show decreasing trend outward from the galaxy center until R_gal∼10 arcmin for NGC 3640 and to ∼5 arcmin for NGC 3641. The density reaches a plateau at  ∼20 arcmin before dropping to the background level. The background levels for both galaxies agree within errors with the global value. Table <ref> reports the estimated  mean and the rms of the global background (Fig. <ref>, between 25-30 arcmin), along with the values estimated for the individual galaxies  within the gray shaded regions in Fig. <ref>. 

The two radial profiles in Fig. <ref>, highlight a feature already discussed in the previous section: NGC 3641 shows substantially larger GC density in the inner regions compared to NGC 3640, despite it being ∼ 3 mag fainter (see Table <ref>).








We also inspected the background subtracted radial profiles, shown in Figure <ref>, which we fitted with a R^1/4 de Vaucouleurs profile, log(ρ_GC)=a_0 + a_1r^1/4, out to the transition radius between the central radial trend and the plateau (R_t∼9 arcmin for NGC 3640, abd R_t∼5 arcmin for NGC 3641). In the figure we also show the galaxy SB profiles in the g-band (with an arbitrary shift; R. Ragusa, priv. communication). The preliminary galaxy light profile reaches a depth of μ_g∼ 28 mag/arcsec^2 at a distance of ∼ 5 arcmin from NGC 3640 and ∼ 1 arcmin from NGC 3641. 

For both galaxies we observe a good match between the SB profiles and the radial GC densities with the GCs profile reaching larger radii, a result consistent with expectations <cit.>. 

In a previous work from the VEGAS series, <cit.>, taking advantage of the same wide-field used here, analyzed the LSB features in the intra-group space within this field. 
Focusing on the intra-group light (IGL), the authors found that the light fraction of this component compared to the total luminosity of the pair is ∼8% in both g and r bands, with average colors g-r ∼ 0.7 mag and g-i ∼ 0.9 mag, which are in full agreement with theoretical predictions <cit.>. These results were achieved using the well-tested method in the literature of multi-component decomposition of the averaged azimuthally surface brightness profile of the brightest central galaxy in the group or cluster, applied in this case to NGC 3640. The results of this analysis (Ragusa et al. , in prep.) indicated a transition radius of ∼ 1.5 arcmin. In the context of IGL/ICL studies, the transition radius refers to the separation radius between the brighter and inner component of the galaxy, gravitationally bound to it (which, in the case of NGC 3640, follows a Sersic profile), and the outer exponential component, representing the diffuse stellar envelope of the galaxy plus the IGL component. This transition radius, related to the behavior of the core/envelope stars in the galaxy, should not be confused with the GC transition radius described above, which is about a factor of 7 times larger. Although we expect that in a virialized system the two transition radii would be correlated with each other, here the galaxy system is actively merging. It will be of interest to analyze this correlation using a comprehensive study based on multiple systems –both interacting and non-interacting– and on numerical simulations of galaxies that include proper GC systems modeling.

The radial density profile of NGC 3640 shown in Fig. <ref> (left panel) shows a plateau of GCs density between 10≤ R_gal [arcmin] ≤ 20 which may be associated with an extended population of intra-group GCs, possibly generated or enriched by the recent merging activity that characterizes this group. For the profile of NGC 3641 (right panel) at the galactocentric distance of NGC 3630 and NGC 3643, we notice two relatively small peaks in the distribution over the plateau over-density, corresponding to the galactocentric radii of the GC systems in these two galaxies (also visible in the 2D maps Fig. <ref>).

We also inspected the radial trends of the red and blue GC sub-populations in both the ugri and gri catalog for NGC 3641. We identified the blue population as all the sources with (g-i)≤0.95 and the red ones with (g-i)>0.95 (see Sect. <ref>). In both cases galaxies the red population is more concentrated than the blue one and contributes less to the region where the intra-group population is located.
 
All these results are also confirmed from the analysis of the GC candidates using the ugri catalog. Additionally, these results are also confirmed when adopting a different criterion for radial binning. As a test, we made radial bins chosen to have an equal number of GC candidates (thus the same statistical error) this leads to the same results illustrated above.







 §.§ Color distribution



As discussed previously, studying the colors of GCs provides valuable information that enables us to trace mainly the metallicity distribution of the GC population, thereby revealing insights into the formation history of the host galaxy.

We examine the color distribution of the GC candidates associated with NGC 3640 and NGC 3641 using both the ugri and gri GCs candidates catalogs. We take the first catalog as reference in this section, as the analysis of colors benefits from the broader wavelength coverage (and lower contamination) of the ugri sample, despite its shallower depth due to the brighter magnitude cut of the u-band data.



As discussed in Sect. <ref>, we initially associate the GC candidates based on their galactocentric distance from the galaxies. Here, we further narrowed down the candidates analysed by imposing that their distances is: R≤ R_t (see Sect. <ref> and Table <ref>).

To correct for background contamination, we  used the background region defined in Sect. <ref>, to characterize the color distribution of the background sources. 

Figs. <ref>-<ref> show the color histograms for (g-i) and (u-r). The gray-filled bars represent the corrected color distribution (total minus background distribution) for the sample of GC candidates in NGC 3640 and NGC 3641, respectively. The error bars are computed assuming Poisson errors in the number of contaminating sources in each color bin. 













To study the detailed properties of the color distribution (e.g the presence of bimodality), the probability distribution functions obtained from the background-subtracted color distributions were randomly populated with ∼ 300 points. The probability function was derived by subtracting the normalized color distribution of sources within the background region (see Sect. <ref>) from the corresponding profile of on-galaxy sources, which was obtained using all GC candidates within R_t. The negative density bins (ρ<0) were set to zero. Then, the resulting distribution was inspected using Gaussian Mixture Models <cit.>, which uses a likelihood-ratio test to compare the goodness of fit for double (or multiple) Gaussian versus a single Gaussian. GMM outputs the best-fit Gaussian parameters:  mean (μ), standard deviation (σ) and the fraction of sources attributed to each Gaussian. 
Additionally, it also provides statistical tests to assess the confidence level of the multi-modal fit. These tests include the kurtosis, which measures the skewness of the distribution, and D, a parameter that is based on the separation of the means
relative to their widths expected to be >2 for bimodal distributions. A unimodal distribution would exhibit a strong peak, resulting in positive kurtosis value and D<2. For NGC 3640 GC population, GMM confirms the preference for a unimodal distribution; whereas for NGC 3641, a color bimodality is preferred in both colors.

The numerical experiment of repopulating the distribution functions was repeated ten times. Table <ref> reports the median and rms_MAD of the parameters from the Gaussian best fit to the distributions for NGC 3640 and NGC 3641. Figures <ref>, <ref> show the background-subtracted distribution with the fitted Gaussians. 

In order to verify if our results are independent from the adopted choice we conducted several tests, including: i) varying the distance up to which we considered the GC candidates belonging to the galaxies[Among the several distances we tested, we also inspect the GC color distribution adopting the sources with R_gal≤3.9R_e (i.e. all the sources within the green circle in Fig. <ref>). Even with only a few tens of objects we confirm the observed behavior.]; ii)  adopting the GC candidates associated with the plateau as background region. All the tests yielded results consistent with our reference GC selection. The GC population on NGC 3640 exhibited a unimodal color distribution, while NGC 3641 consistently showed a bimodal distribution. However, the GMM runs provide statistically different fractions 
for the red population, ∼ 26% (∼9%) of the total background-corrected GC population for g-i (u-r) color. The two colors have different sensitivities to metallicity, with (u-r) being the more sensitive of the two, and this difference might be an expression of this effect. To further inspect this behavior, we made the same analysis for the (g-i) color distribution, using the more contaminated gri matched catalog. The analysis with the gri confirms that NGC 3640 has a preference for a unimodal skewed blue distribution, while NGC 3641 confirms its bimodality with a fraction of red GCs of the order of 40%. Table <ref> reports the GMM parameters for the (g-i) color for NGC 3641.
With the aim of inspecting the mean positions of red/blue GC-candidates around NGC 3641, we divided the GC sample into two sub-populations: a blue one with (g-i)≤0.95 mag, and a red one with  (g-i)>0.95 mag. The adopted separation value is determined as the point where the two color distributions intersect shown in Fig. <ref>. The mean galactocentric distance for red GCs is ∼1 arcmin, while for the blue GCs we find ∼2 arcmin. The higher central concentration of the red population relative to the blue one is a feature already observed in massive galaxies <cit.>.  We obtain consistent results even if we assume the blue/red cut using (u-r)≈2.25 mag, or if the gri catalog is used. Actually, using the (u-r) color cut, we observe a red sub-population slightly more concentrated (∼0.6 arcmin) toward the central region than the one observed using the (g-i) cut.  












 §.§ GC Luminosity function and specific frequency



In this section, we focus on the analysis of the GCLF for both NGC 3640 and NGC 3641 to estimate the total number of GCs (N_GC) and constrain the relative distance between the two galaxies by comparing their GCLFs. We adopted a single-band data, the g- and r-bands, discarding the  u- and i-bands. The g and r catalogs are derived independently from the respective images as described in Sect. <ref>. The u-band data are too shallow, failing to reach the turnover magnitude of the GCLF (see Fig. <ref>). The i-band, instead, is characterized by a high fraction of spurious detections close to the galaxy core, making them difficult to reject based solely on a single-band analysis. 
The decision to adopt a single-band approach arises from the lack of a completeness function estimated for the multi-band matched catalog, which requires a procedure that accounts for the color of the sources[To determine the completeness fraction of a GC-candidates catalog derived from multi-band cross-matches, we would need to simulate source injection using the expected GCLF, color distribution, and galaxy background contamination in all used bands. The function itself would then be derived from a selection based on the same criteria adopted for identifying GC-candidates. However, such a procedure would be excessive for the present work. Moreover, the expected outcome would, in most cases, be dominated by the worst passband in the matched sample. Ultimately, this approach would not benefit from the advantage of the large format of VST data.].  






















Single-band catalogs exhibit higher contamination but also greater completeness compared to the multi-band ones. As also discussed in Section <ref>, we take advantage of the wide sky coverage of the VST imager to efficiently characterize the population of fore/back-ground interlopers. This allows us to clean the contaminants from the luminosity function by subtracting the density function derived in background areas from the on-galaxy luminosity function, thereby revealing the luminosity function of GCs.
For the r-band, which is our reference image for morphometry  (FWHM, CI, Flux radius, elongation), we adopted the same magnitude, morphometric and CI selection strategy described in Sect. <ref>. 
For the g-band data, where selection parameters were not required except for colors in the multi-band selection, we obtained selection parameters independently from the r data.  We applied the CI cuts using the same basic procedure outlined in in Sect. <ref>.

Additionally, for both bands, we applied a lower threshold to the FWHM, set at the median FWHM value minus 3×σ_FWHM, to exclude sources with a FWHM much smaller than the value expected for point-like sources [The additional cut was not necessary for the ugri and gri catalogs because, thanks to the matching procedure, we were able to discard these detected sources –likely fake detections– from our catalogs.]. The final selection parameters are given in Table <ref>. 




To characterize the GCLF, all sources within 3.9 R_e from the galaxy center were included in the analysis (see Sect. <ref> and Fig. <ref>). For the background region, we adopted the same area as described in Sect. <ref>, specifically, group-centric radii between 25 and 30 arcmin (measured from the tangency point of the two circles in Fig. <ref>). Then, for the on-galaxy and off-galaxy regions, we derived the LFs.
To ensure a reliable GCLF, completeness corrections for undetected sources are mandatory, for both the LF on-galaxy and the background LFs.

We calculate the radial completeness fraction within the adopted region, using the same strategy outlined in Sect. <ref>. For NGC 3640, we retrieved the completeness function in two regions: one within 1.2 arcmin (∼2R_e) and another one between 1.2 and 2 arcmin (∼3.9R_e) from the center of the galaxy. For NGC 3641, we only retrieved the completeness function within the 3.9R_e, as analyzing a smaller region would suffer from sampling effects. 
However, in both cases, we also take in to account the shape selection  criteria in deriving the completeness (see Table <ref>). For the background region, the completeness is obtained from the average functions estimated in 4 different regions across the adopted radial annulus[The completeness is indeed very stable across the background annulus, with an rms of 0.01 (1%) across the four different regions.]. Figure <ref> and <ref> present the completeness functions derived in the g- and r-band, respectively. Once more highlighting the expected observation that completeness on-galaxy is lower than that of the background/off-galaxy. 


The gray shaded bars in Figs. <ref>-<ref> show the completeness corrected and background subtracted LF for both the galaxies in the g- and r-bands, respectively. The uncertainty on each magnitude bin was calculated assuming Poisson errors on the number of contaminating objects, and a conservative 5% error on the completeness corrections.

Each distribution was then fitted with a Gaussian function. Initially, we allowed the fit to estimate the dispersion of the distribution, and obtained values consistent within the errors with the expected σ^GCLF (see Table <ref>). However, given the very large uncertainties on σ_GCLF we chose to fix the value of the GCLF width to the empirical expectation <cit.>. We repeated this procedure ∼10 times, varying the binning width and the phasing of the histogram. The final adopted values for the peaks and uncertainties are the mean and the rms of the obtained fitting parameters.

The positions of the peaks of the Gaussian functions shown in Figs. <ref>-<ref> (red lines) are reported in Table <ref>. 
In this table we also report the Δμ between the GCLF peaks. Taking advantage of the GCLF as a distance indicator, we notice that in both bands the two galaxies are consistent with being basically at the same distance. Possibly, the g-band estimates reveal NGC 3640 being ∼10% farther than its fainter companion. However, given the higher quality of our r-band, we prefer the scenario in which the two galaxies are roughly at the same distance which is also the result from SBF distances of the two <cit.>.


Taking advantage of the fitted Gaussian parameters,  we estimated the total number of GCs, N_GC, as follows. First, we integrated the GCLF up to its peak and then double the GC number obtained. This provides the total number of GCs within 3.9R_e. 
Recent literature quotes a range of 3-5 R_e as the half-number radius for the GC spatial distribution <cit.>. Adopting these results and considering our area coverage, to estimate N_GC we doubled once more the GC number from the GCLF integration. 

To estimate the uncertainty of N_GC, we derived alternative values for N_GC by varying the selection parameters for GC identification and the half-number radius within reasonable intervals. As a result, we observed that N_GC changes by ∼30%, which we assume as uncertainty. 
The final N_GC values are reported in Table <ref>. Despite the fact that the g- and r-band N_GC estimates are derived using independent catalogs, with non negligible difference in terms of image quality, we note that the total population of GCs in the two passbands is in good agreement. 

For NGC 3640, the only value available in the literature is N_GC∼50 from <cit.>. However, this value represents only a lower limit, as it was obtained by counting the selected sources over the HST area.







Adopting as M_V the value reported in Table <ref>, we evaluated the specific frequency (S_ N) of the two galaxies, taking as reference the weighted average between the two N_GC estimates. 
The specific frequency provides an estimate of the number of GCs per unit of V-band luminosity <cit.> and is derived as:

    S_ N=N_GC×10^0.4 × (M_V+15)


The S_ N values we obtain are: S_ N =2.0±0.6 for NGC 3640, and S_ N = 4.5±1.6 for NGC 3641.








§ DWARF GALAXIES IN THE FIELD AND THEIR GCS CONTENT



We used the wide-field coverage of the VST data to investigate the GC population in dwarf galaxy candidates across the observed field. The investigation of the GC populations in dwarf galaxies is currently a strongly discussed topic with a number of studies focusing on the universality of the GCLF and their dark matter content <cit.>, because these low-mass galaxies provide some of the most critical constraints for cosmology <cit.>. 


As a first steps we visually inspected the g and r images independently, and then cross-matched the identified dwarf candidates from both fields (the visual inspection was independently made by the first and second authors of this work). We searched for objects that showed a diffuse brightness profile, with no evidence of the typical features found in bright galaxies (spiral arms, shells, bars), with a number of point-like GC candidates centered around the dwarf (although we did not find any GC-rich dwarf). Then, after inspecting the brightness profiles, we also rejected objects with bright central cores. This yielded to a final sample of 27 galaxies, all listed in Table <ref>. Among the selected dwarfs, we identify as Dwarf 8 (upper middle panel in Fig. <ref>)  the edge-on dwarf spiral galaxy observed by <cit.> to exhibit HI emission. Indeed, it is one of the bluest galaxies in the sample. Figure <ref> shows the position of the dwarfs identified across the observed field. 
 
We also checked the literature and found a sample of dwarfs identified in the MATLAS Survey <cit.>. In <cit.>, they characterized the properties of 19 dwarfs found over ∼ 1×1 sq. degrees around NGC 3640, 10 of which are in common with our list of 27 identified dwarfs. <cit.> classified the Dwarf 12 as an Ultra Diffuse Galaxy (UDG, with μ_0,g≥ 24mag/arcsec^2 and R_e≥1.5kpc), a result confirmed by our analysis. <cit.> analysed MUSE data of a sample of dwarf galaxies in the MATLAS sample, including three dwarfs identified around NGC 3640. They confirmed the membership of these dwarfs to the galaxy group. Additionally, from the MATLAS website none of the dwarfs identified in the field around NGC 3640 show evidence of GCs overdensity, with some exceptions discussed below. In Table <ref>, we add a column to include information regarding the dwarfs we have in common with MATLAS. In conclusion, adding up our sample of 17 newly detected dwarfs with the total sample identified by the MATLAS survey, brings the total number of dwarf galaxies in the ∼1.5×1.5 deg^2 area around NGC 3640 to 26.


To analyze the properties of the galaxies in this work, we developed a code that integrates the AutoProf pipeline <cit.>. In brief, AutoProf automatically fits elliptical isophotes to galaxy images and extracts accurate flux measurements along them. It also estimates the local background value for each object and masks out any sources (e.g., stars, background galaxies, GCs) that could affect surface photometry measurements. In our code, we enhanced the masking procedure by incorporating additional sources to mask –particularly focusing on the inner region of the galaxy– and added a module to derive the Sersic parameters <cit.> to the isophotal profiles produced by AutoProf. The procedure was run on cutout images with size of ∼9×9 arcmin^2 around each of our candidate dwarfs, both in the g- and r-bands, adopting as reference image the g, and then running forced photometry on the r-band frame. For each dwarf candidate we derived both structural (effective radius R_e, and Sersic index n) and photometric (e.g., effective surface brightness μ_eff) properties. The resulting parameters are reported in Table <ref>, together with their uncertainties, which we decided to conservatively round-off to 0.1 on all the fitted parameters. Due to their extremely faint surface brightness, structural and photometric properties could not be retrieved in four of the 27 identified dwarfs. 
Along with these properties, we also estimated the total integrated magnitude and the (g-r) color within one effective radius, adopting the mean  R_e of the g- and r-bands as radius. All magnitudes and colors are corrected for Galactic extinction using values from <cit.>. Then, we inspected the color-magnitude diagram and compare the properties of the dwarfs selected in this work with the relation retrieved by <cit.>: (g-r) = -0.0314× M_r + 0.145, σ=0.15. We found that the 80% of our sample is within 1σ from the relation and the full catalog is included within 2σ, suggesting that the identified dwarfs are located at roughly the same distance of the group.
To derive the total magnitude, m_T, we decided to adopt the following procedure. First, we derived m_X,T (with X being g or r) by using the asymptotic value of the growth curve derived from the fitted isophotes. We also derive m_T using the equation m_eq.,T=μ_e-2.5log_10(2π R_eff^2) from <cit.> and <cit.>, where μ_e  is the effective surface brightness and R_eff is the effective radius. Finally, as a third option, we integrate the light profile up to the radius corresponding to a surface brightness value of ∼ 27 mag/arcsec^2 which we considered as our photometry limiting radius, beyond which we are contaminated by the background. The reason for having three different approaches is due to the peculiarities of the profiles for some dwarfs, which did not allow using the simple curve of growth approach straightforwardly. 


For galaxies exhibiting a flat asymptotic curve of growth, we assumed the asymptotic value as the total magnitude m_T (Case A). In cases where the curve of growth was affected by the presence of bright contaminants (stars or galaxies), we  assumed an upper limit to the magnitude  m_eq.,T from the equation above (Case B). Finally, in cases where the behavior of the curve of growth was ambiguous, and the results from m_eq.,T were substantially different from the expected magnitudes based on the broad asymptotic value of the curve of growth (e.g. due to  the presence of a plateau region immediately followed by an increase of the flux) we opted for the magnitude estimated using the third method (Case C). For Case A galaxies the uncertainty was determined by taking the semi-difference between the maximum and minimum magnitude estimates among the three methods used. For Case B, we only have upper limits, while for Case C we conservatively assumed an uncertainty of 0.5 mag.

In Figs. <ref>-<ref>, we show the cutout from the VST r-band image centered on the dwarf galaxies (left panel), the surface brightness profile in the r-band (middle panel) and the color profile for each of the galaxies (right panel). 



We also analyzed the GC content in the identified galaxies by examining both the g- and r-catalogs, and applying to the same selection criteria outlined in Sect. <ref>. 

Additionally, we selected sources based on their distance from the dwarf galaxy candidate. We considered sources associated with the galaxy to be those located within 4R_e <cit.>.

For background estimation, we assessed the local background of GC-candidates surrounding each galaxy using three annuli, with radii that are multiples of the galaxy's R_e. Specifically, the adopted radii were: 6R_e, 7R_e, 8R_e, and 9R_e. For each galaxy, the adopted radius was the mean R_e calculated between the g- and r-band estimates.
We estimated the number of GCs associated with each galaxy using the formula N_GC^dw= (ρ_gal-ρ_background)× A_gal. 
Here, ρ_gal is the on-galaxy GC-candidates density and ρ_background is the background density, both measured in number of sources per square arcmin. The circular area containing the sources associated with the galaxies (π R^2, with R=4 R_e) was used as A_gal.  
We repeated this process for all three background regions and adopted the middle value among the three estimates as the final result. None of the galaxies we selected showed a statistically significant overdensity of sources: they either had no GC-candidates or had a number consistent with zero within the uncertainty. Upon a visual inspection, however, we note that the dwarfs with ID=7  and 12 in Table <ref> exhibit a central concentration of compact sources, consistent with a non-zero GC population. These results are basically consistent with what is observed by the MATLAS team, who also find no GC population in ten of the dwarfs we have in common. Additionally, MATLAS data also suggest that Dwarf 12 hosts a population of GCs <cit.>, as well as Dwarf 13 <cit.> which in our data appears to be nucleated (see Appendix <ref> and Fig. <ref>).

Regarding Dwarf 12, this galaxy distinctly exhibits a tidal interaction pattern (as depicted in the lower right panel of Fig. <ref>), and it is located in close in projection to a GC-overdensity in the 2D maps, as anticipated in Sect. <ref>,.




§ DISCUSSION






The analysis of the GC system around the two bright galaxies, NGC 3640 and its close-in-projection companion NGC 3641, confirms known properties of the pair and expected characteristics of GC systems around bright galaxies. However, we also identified several new features.








From the analysis of the GCs' 2D density maps, we observe that the GC population appears to follow the diffuse light pattern originating from the past merging events involving NGC 3640. Indeed, we notice a stretching in the distribution of GCs from southwest to northeast, aligning with the observed light pattern of merging features, including what appears being a dwarf galaxy with tidal features (Dwarf 12, Fig. <ref>, lower right panel, and lower panel in Fig <ref>). 

The highest GC density peak is on NGC 3641, despite it being less massive than NGC 3640. Furthermore, there is no compelling evidence of a spatial displacement between the GCs density peak and the location of the galaxy photocenter. 

For both of the brightest galaxies in the field, the radial density profile of GCs candidates follows that of the galaxy light (see Fig. <ref>). 
We find an over-density in radial density profile, that we associate to an intra-group GC component, possibly linked to the extended merging activity  of NGC 3640. This GCs population could now be part of the dark matter halo of the galaxy, and gradually fades into the background  at galactocentric radius R_gal∼ 30 arcmin.


When studying the GCs color distributions, NGC 3641 presents a bimodal distribution, quite typical in normal elliptical galaxies <cit.>, while NGC 3640 shows a broad unimodal color distribution. 

There are known cases that deviate from the typical bimodal distribution, and this is  generally attributed to specific evolutionary paths, like merging activity <cit.>, or low accretion fractions <cit.>. The unimodal g-i distribution of GCs host in NGC 3640 could possibly arise from an interaction that happened 3-5 Gyr ago, as pointed out by the shells around NGC 3640, and which could have been accompanied by a star formation event. However, NGC 3640 does not exhibit obvious evidence of recent intense star-formation activity, which could have led to the formation of a population of intermediate-color, relatively young (3-5 Gyr old) massive star clusters. Indeed, simple stellar population models predict (g-i)>0.9 for stellar populations with ages t≥3 Gyr and [Fe/H]≥0.0 <cit.>. Therefore, even such a population of intermediate age GCs should not be the main driver of the unimodality. Only stellar populations with t≤2 Gyr reach the blue colors required to fill the gap in the bimodal GC distribution. Additionally, unlike the field around NGC 1316 <cit.>, we do not observe substantial diffuse dust patterns in our images <cit.>. 
One alternative explanation could be the dry merging of two GC systems with different color distributions, which, when combined, generate the broad color distribution we observe. 

Furthermore, one may notice that the (u-r) color distribution reaches approximately the same red/metal-rich border for both galaxies. However, the (g-i) color distribution of NGC 3641 reveals the presence of a GC population (g-i)∼0.2 mag redder than that in NGC 3640. This is a partly unexpected result, given that more massive galaxies tend to host relatively redder GC systems compared to lower mass ones <cit.>.  This is only specific for the ugri matched catalog since, as mentioned above, using the gri catalog we observe a broad unimodal (g-i) color distribution which consistently reaches the same red limit for the two galaxies.  
An explanation for this behavior might arise from a combination of effects. On the observational side, we must note that the (u-r) color is a more efficient tracer of metallicity than the (g-i) because of the wider wavelength coverage, which spans a magnitude interval over a factor of two larger. 
Moreover, there is an additional observational bias due to the spatial segregation of red GCs and the different image depth at smaller galactocentric radii. Red GCs tend be concentrated around the core of the galaxy <cit.>, hence in our dataset these have a lower detection efficiency compared to the blue GCs, because of the brighter detection limit around the center of NGC 3640 compared to the smaller and fainter companion NGC 3641 (see Fig. <ref>).
Finally, the bluer bands in our observational dataset, namely the u and g, are both shallower than the r-band relative to the TOM, hence the efficiency of detection for red GCs is relatively lower than for blue ones. 
The combination of such effects, plus the peculiar color distribution on NGC 3640 as highlighted by its color unimodality, could conspire toward the observed bluer (g-i) color in the more massive NGC3640 relative to NGC 3641. 
Given the degeneracy between the age and metallicity for the optical colors, which is further complicated by the interaction history of the system <cit.>, additional near-IR photometry or spectroscopic data may be necessary to fully explain the peculiar behavior of the GC system in NGC 3640.

Analyzing the GCLF of both galaxies, we found that the turn-over magnitudes of the distribution on NGC 3640 and NGC 3641 are consistent with each other, suggesting that the galaxies lie roughly at the same distance (∼ 27 Mpc). This result is consistent with existing distance estimates from  SBF measurements.

Furthermore, through the study of the GCLF, we estimated the total number of GC populations and the specific frequency of the galaxies. Despite the mentioned merging activity, NGC 3640 exhibits a specific frequency aligned with values for galaxies of the same mass (S_ N=2.0±0.6), a result that likely support the dry-merging scenario. NGC 3641 shows a specific frequency S_ N=4.5±1.6, about four times higher than expected for galaxies of its same magnitude <cit.>, indicating a relatively rich GC population. Examining the behavior of the specific frequency as a function of absolute magnitude in <cit.>, we found other galaxies, such as NGC 1336, NGC 4352, NGC 1172, NGC 1419, NGC 1380A/B with -18≤ M_V ≤-20 and S_N≥3. By inspecting the main properties (i.e. R_e, dynamical masses and GC mass)  of galaxies with S_N≤2 and with S_N≥3, we found no evidence of any differences between these objects, as all the inspected properties are consistent each other within the estimated uncertainties. We noted that all of the galaxies, but NGC 1380A/B, live in isolated environments (i.e. distant from all the massive galaxies of the group/cluster). 
Hence, although exceptional, NGC 3641 is not unique in terms of S_ N.
Summarizing, the high S_ N of NGC 3641 could be explained if this old, relatively compact elliptical (see Fig. <ref>), experienced an intensive early starburst with rapid enrichment and high cluster formation efficiency (CFE), which also boosted its S_ N and resulted in a more concentrated GC population.
The relatively low S_ N of NGC 3640 could be a result of the increased brightness of the galaxy due to the presence of an intermediate-age population, similar to the peculiar brightest galaxy in Fornax, NGC 1316 <cit.>. Hence, compared to its less massive companion, NGC 3640 might have had a more gentle star formation history and efficiency, with a lower CFE, explaining its lower S_ N value.



Finally, we also examined the content of GCs in dwarf galaxy candidates across the field. The results of this analysis indicate that there is no evidence for an over-density of GCs for these galaxies. Nevertheless, we find a couple of dwarf galaxies, particularly one that appears to be a tidally distorted dwarf (Dwarf 12, Fig. <ref>), exhibiting a concentration of compact sources in their cores.. Further studies will be necessary to investigate in more detail the true origin of such sources.

Combined together, the observed 30% larger recession velocity of NGC 3641 compared to its more massive companion, the lack of any obvious evidence of merging features around NGC 3641 and its high density of GCs seem to suggest that we are observing a not-too-close and fast encounter between the bright and loose galaxy NGC 3640 and the denser and fainter companion, NGC 3641, moving along our line of sight. The combination of the specific geometry of the encounter and dynamical friction effects might be the cause of the increased GC density (by a projection effect) and the lack of evidence for typical morphological irregularities in NGC 3641. The high S_N might further indicate that the GCs system has possibly experienced few external interaction events, similar to the case of NGC 1336 in Fornax <cit.>.




§ SUMMARY


In this work, we presented a thorough analysis of the GC population around the interacting galaxy NGC 3640 and its companion NGC 3641. The analysis takes advantage of the multi-band (ugri) wide ∼ 1.5× 1.5 sq. degrees area coverage of the field from the VEGAS survey carried out with the VST telescope.
Our main results can be summarized as follows:




    
  * The GC 2D density map seems to follow the galaxy's light patches resulting from past merging event; 
    
  * The highest over-density peak of GCs is observed to be on NGC 3641 rather than NGC 3640, the more massive galaxy among the two;
    
  * The surface density profiles of the GCs seem to show the presence of an intra-group GC component,  consistent with the scenario of ongoing merging; 
    
  * The color distribution of NGC 3641 shows a bimodal distribution  as expected for a normal elliptical galaxy of this magnitude, while a broad unimodal distribution is observed for the GCs in  NGC 3640, possibly due to the past merging events;
    
  * From the study of the GCLF, the two galaxies appear to be roughly located at the same distance (∼ 27 Mpc); 
    
  * The value of the specific frequency for NGC 3640 aligns with expectations, while NGC 3641 appears to host a very rich GC population.
    
  * The identification of 17 new dwarf galaxies around NGC 3640.
    


In the future, thanks to the upcoming facilities such as the Large Synoptic Survey Telescope (LSST), we anticipate significant advancements in our ability to further characterize GC systems like the one we have observed. The LSST, with its wide-field survey capabilities and unprecedented depth, will provide invaluable data that will enable us to delve deeper into the properties  of these systems. This enhanced observational capacity will offer new insights into the formation, evolution, and interactions of galaxies and their associated GC populations.
We are actively seeking new optical and IFU data for objects like NGC 3641, where we observe extremely high concentrations of GCs. Such data are particularly interesting and ideal for the characterization of the peculiarly high S_ N as observed in this galaxy.




§ ACKNOWLEDGEMENTS



We thank the anonymous referee for her/his comments and
constructive suggestions. This work is based on visitor mode observations collected at the European Southern Observatory (ESO) La Silla Paranal Observatory within the VST Guaranteed Time Observations, Programme ID: 097.B-0806(A). 
MM acknowledges financial support from the INAF-OAAb, INFN institute of Naples, the University of Naples Federico II and the VST project (P.I. P. Schipani). 
EI, MS, MC acknowledges support by Italian Ministry for Education University and Research (MIUR) grant PRIN 2022 2022383WFT “SUNRISE”, CUP C53D23000850006, and by the VST funds. EI, MS, MC and RH acknowledge funding from the Italian National Institute of Astrophysics (INAF) through large grant PRIN 12-2022 "INAF-EDGE" (PI L. Hunt). 

This research has made use of the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology. We also acknowledge the usage of the Extragalactic Distance Database (EDD, <https://edd.ifa.hawaii.edu/>). We made extensive use of the softwares of SExtractor <cit.> and Topcat <cit.>). This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France <cit.>. The original description  of the VizieR service was published in <cit.>. This research has made use of the SDSS database <www.sdss.org>. Funding for the Sloan Digital Sky Survey V has been provided by the Alfred P. Sloan Foundation, the Heising Simons Foundation, the National Science Foundation, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. SDSS telescopes are located at Apache Point Observatory, funded by the Astrophysical Research Consortium and operated by New Mexico State University, and at Las Campanas Observatory, operated by the Carnegie Institution for Science. This work made use of Astropy (<http://www.astropy.org>) a community-developed core Python package and an ecosystem of tools and resources for astronomy <cit.>.



aa.bst






    





§ GC CANDIDATES

We present an excerpt from the table of GC candidates selected as described in Sect. <ref>.









§ COMPLETENESS TEST

We present the table with the magnitude limits obtained from the completeness function for 26 off-galaxy regions.










§ DWARFS MEASURED PROPERTIES










§ DWARF GALAXY CANDIDATES IN THE FIELD



In this appendix we show the LSB dwarf candidates identified (see Sect. <ref>). The error bars are estimated by propagating the error on the isophotal flux, background value, zeropoint and extinction. As uncertainty on the background we assumed the 30% of the fluctuation estimated by Autoprof[ Autoprof estimates the sky noise as
the 68.3 percentile of flux values below the sky level. For more detail see <cit.>.]. Judging from the images and radial profiles, Dwarf 12 and 13 might be nucleated.


       

























 Paper Title: The AGORA high-resolution galaxy simulations comparison project: CosmoRun data release.




















































0000-0002-6299-152X]Santi Roca-Fàbrega
Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, SE-221 00 Lund, Sweden
Departamento de Física de la Tierra y Astrofísica, Facultad de Ciencias Físicas, Plaza Ciencias, 1, 28040 Madrid, Spain; mailto:santi.roca_fabrega@fysik.lu.sesanti.roca_fabrega@fysik.lu.se

0000-0003-4464-1160]Ji-hoon Kim
Institute for Data Innovation in Science, Seoul National University, Seoul 08826, Korea; mailto:mornkr@snu.ac.krmornkr@snu.ac.kr
Center for Theoretical Physics, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea
Seoul National University Astronomy Research Center, Seoul 08826, Korea

0000-0001-5091-5098]Joel R. Primack
Department of Physics, University of California at Santa Cruz, Santa Cruz, CA 95064, USA; mailto:joel@ucsc.edujoel@ucsc.edu



Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748, Garching, Germany

0000-0002-9144-1383]Minyong Jung
Center for Theoretical Physics, Department of Physics and Astronomy, Seoul National University, Seoul 08826, Korea


DiSAT, Università degli Studi dell'Insubria, via Valleggio 11, I-22100 Como, Italy
Dipartimento di Fisica “G. Occhialini”, Università degli Studi di Milano-Bicocca, I-20126 Milano, Italy

0000-0001-7457-8487]Kentaro Nagamine
Department of Earth and Space Science, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
Kavli IPMU (WPI), University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japan
Department of Physics & Astronomy, University of Nevada Las Vegas, Las Vegas, NV 89154, USA

0000-0002-3764-2395]Johnny W. Powell
Department of Physics, Reed College, Portland, OR 97202, USA

0000-0001-5510-2803]Thomas R. Quinn
Department of Astronomy, University of Washington, Seattle, WA 98195, USA


Institute of Physics, Laboratoire d'Astrophysique, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland


Shikoku Gakuin University, 3-2-1 Bunkyocho, Zentsuji, Kagawa, 765-8505, Japan


Instituto de Astronomía, Universidad Nacional Autónoma de México, A.P. 70-264, 04510, Mexico, D.F., Mexico


<http://www.AGORAsimulations.org>






§ ABSTRACT 
The AGORA Cosmorun <cit.> is a set of hydrodynamical cosmological “zoom-in” simulations carried out within the AGORA High-resolution Galaxy Simulations Comparison Project <cit.>. These simulations show the formation and evolution of a Milky Way-sized galaxy using eight of the most widely used numerical codes in the community (Art-I, Enzo, Ramses, Changa, Gadget-3, Gear, Gizmo and Arepo.). In this short report, we describe the public release of the raw output data from all of these simulations at z = 8, 7, 6, 5, 4, 3, 2 (plus at z=1, 0 when available), and several metadata files containing the halo centers, virial quantities, and merger trees. The data from even thinner timesteps will be released as soon as the upcoming collaboration papers (VII-IX) are submitted and accepted.





























§ THE AGORA INITIATIVE:  PAST, PRESENT, AND FUTURE
 

Since its launch in 2012, the AGORA High-resolution Galaxy Simulations Comparison Project (Assembling Galaxies of Resolved Anatomy) has taken aim at carefully comparing high-resolution galaxy simulations on multiple code platforms widely used in the contemporary galaxy formation research.[See the Project website at <http://www.AGORAsimulations.org/> for more information about the AGORA Collaboration. ]  
The main goal of this initiative has been to ensure that physical assumptions are responsible for any success in galaxy formation simulations, rather than manifestations of particular numerical implementations, and by doing so, to collectively raise the predictive power of numerical galaxy formation studies. 
Over 160 individuals from over 60 different academic institutions worldwide are participating or participated in the collaborative effort of the Project.

The first result by the AGORA Collaboration <cit.> was our flagship paper which explained the philosophy behind the initiative and detailed the publicly available Project infrastructure we have assembled. 
Also described was the proof-of-concept test, in which we field-tested our infrastructure with a dark matter-only cosmological zoom-in simulation using the common initial condition <cit.>, finding a robust convergence amongst participating codes. 
In the second paper of the Project <cit.>, we focused on the evolution of an isolated Milky Way-mass galaxy. 
All participating codes shared the common initial condition <cit.>, common physics models <cit.>, and common analysis platform <cit.>. 
Subgrid physics models such as Jeans pressure floor, star formation, supernova feedback energy, and metal production were carefully constrained across code platforms.
With a spatial resolution of 80 pc that resolves the scale height of the disk, we find that any intrinsic inter-code difference is small compared to the variations in input physics such as supernovae feedback. 
In the third paper <cit.> we presented the Cosmorun project and the calibration process we designed to reduce the number of variables to account for when comparing results from the different code groups. For these simulations, we adopted most of the subgrid physics and simulation strategies developed for Paper II but including the most recent version of the Grackle library, leaving only to the decision of each code group the stellar feedback strategy to be used. 
Following Paper III publication, the Cosmorun data has been used to study many aspects of the formation and evolution of MW-size galaxies focusing on the differences between the participant codes and their stellar feedback strategies. In particular, we presented the analysis of changes in the merger history, the stellar disk properties and the stellar metallicity distribution in <cit.>,  the satellites number counts in <cit.> and the circumgalactic medium (CGM) in <cit.>. In Table <ref> we include a summary of the AGORA papers published until August 2024. Currently, many new sub-projects are using the Cosmorun to study the properties and formation of stellar disks, the quenching of satellites, and major merger properties, among many others.
After twelve years of workshops and teleconferences, and with common infrastructures built together, the AGORA Collaboration has become a successful open forum where users of different simulation codes can talk to and learn from one another, promoting collaborative and reproducible research essential in any scientific community.    





§ PUBLIC RELEASE OF COSMORUN DATA
 

All code groups started their simulations from a common initial condition (IC) generated with Music <cit.>,[The website is <https://www-n.oca.eu/ohahn/MUSIC/>.] and run by keeping physics prescriptions the same for all codes (e.g., gas cooling and heating, star formation parameters), although some variations were made in each code <cit.>. Only the decision concerning the stellar feedback prescription and metal production to be used was left to each code group. 
Code groups were asked to use a prescription close to the most widely-used practice in each code community. 
Spatial resolution was ∼80 physical pc at z = 4 to resolve the internal structure of a target halo, and to make our physics prescriptions less reliant on platform-specific models.  


Here we provide the simulation snapshots used in the analysis of Paper III plus the one at z = 3, 2 and 1, 0 when available (see Appendix <ref> for a detailed description of the released data). We also provide the metadata of the centers, several virial quantities, and the results of applying the rockstar halo finder to each one of the datasets.
The cohort of widely-used, state-of-the-art galaxy simulation codes who contributed to this release includes:  the Lagrangian smoothed particle hydrodynamics codes Changa <cit.>, Gadget-3 <cit.>, and Gear <cit.>, the Eulerian adaptive mesh refinement codes Art-I <cit.>, Enzo <cit.> and Ramses <cit.>, the moving-mesh code Arepo <cit.> and the mesh-free finite-volume Godunov code Gizmo <cit.>. 

This release is intended to encourage the community to re-run the Cosmorun ICs both after applying the provided calibration procedure or not and using their favourite stellar feedback. Our goal is to enlarge the library of Cosmorun models thus enhancing our insight on how variations in the baryonic physics processes affect the formation of MW size galaxies. The coordination of the Collaboration and its members are at their disposal to help with the calibration process and with running new models.

The simulation data and several of the common analysis scripts in yt are available through the Project website[<http://www.AGORAsimulations.org/> or <http://sites.google.com/site/santacruzcomparisonproject/blogs/quicklinks/>]. Additionally, the authors of the published AGORA papers (see Table <ref>) will be happy to provide help to users interested in analyzing the released data.
Also available in the same link are isolated and cosmological initial conditions generated by the AGORA Collaboration for galaxy simulations, and the links to the key software.  
We encourage numerical galaxy formation community members to use these resources freely for their research.






§ DETAILS OF THE RELEASED DATA

In Table <ref> we provide the information on the snapshot number at each redshift for each one of the participant codes. We also provide the link to the data for each code, below the table.


In the metadata folder[https://users.flatironinstitute.org/ chayward/agora_public_release/PaperIV/Metadata/https://users.flatironinstitute.org/∼chayward/agora_public_release/PaperIV/Metadata/] you will find three different kind of files and a subfolder (https://users.flatironinstitute.org/ chayward/agora_public_release/PaperIV/Metadata/MergerTrees_rockstar/MergerTrees_rockstar) which contains the merger trees generated using rockstar <cit.> and consistent-trees <cit.>. All filenames contain the name of the corresponding code except for Changa which is labeled as  CHANGA_6.0e51erg_NSB for the Changa-T and CHANGA_3.5e51erg_SB for the fiducial Changa. Here we describe the content of each file and of the merger-trees folder:



  * centers_CODE_final_noID.txt: This file contains the centers of the main halo inside the simulated cosmological box in physical kiloparsecs. The data is structured as follows: redshift, snapnum (see Table <ref>), [x,y,z].


  * Rvir_cent_CODE.txt: This file contains similar information to the previous one but in a simpler format, including information on the R_200c. The data is structured as follows: redshift, snapnum (see Table <ref>), x, y, z, R_200c (in physical kpc).


  * Rvir_cent_others_CODE.txt: This file contains information about many general properties of the central halo. The data is structured as follows: redshift, snapnum (see Table <ref>), R_200c [kpc], M_200c [M_⊙], M_*(R<R_200c) [M_⊙], M_*,gal(R<0.15R_200c) [M_⊙], R_1/2 [kpc], M_tot(R<R_1/2) [M_⊙], Z_* [Z_⊙], R_1/2*,gal [kpc], M_*,gal(R<R_1/2*) [M_⊙], Z_*,gal [Z_⊙].


  * MergerTrees_rockstar: In this folder you will find a sub-folder containing the merger for each one of the participant codes except for the fiducial Changa. The files can be read and analysed using the  ytree[https://ytree.readthedocs.io/en/latest/https://ytree.readthedocs.io/en/latest/] tool provided by the yt[https://yt-project.orghttps://yt-project.org] community.  















 Paper Title: CEERS Key Paper. IX. Identifying Galaxy Mergers in CEERS NIRCam Images Using Random Forests and Convolutional Neural Networks.





0000-0002-8018-3219]Caitlin Rose
0000-0001-9187-3605]Jeyhan S. Kartaltepe
Laboratory for Multiwavelength Astrophysics, School of Physics and Astronomy, Rochester Institute of Technology, 84 Lomb Memorial Drive, Rochester, NY 14623, USA


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

0000-0002-1416-8483]Marc Huertas-Company
Instituto de Astrofísica de Canarias, La Laguna, Tenerife, Spain
Universidad de la Laguna, La Laguna, Tenerife, Spain
Université Paris-Cité, LERMA - Observatoire de Paris, PSL, Paris, France

0000-0003-3466-035X]L. Y. Aaron Yung
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA

0000-0002-7959-8783]Pablo Arrabal Haro
NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA

0000-0002-9921-9218]Micaela B. Bagley
Department of Astronomy, The University of Texas at Austin, Austin, TX, USA

0000-0003-0492-4924]Laura Bisigello
INAF, Istituto di Radioastronomia, Via Piero Gobetti 101, 40129 Bologna, Italy
Dipartimento di Fisica e Astronomia "G.Galilei", Universitá di Padova, Via Marzolo 8, I-35131 Padova, Italy

0000-0003-2536-1614]Antonello Calabrò 
INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy

0000-0001-7151-009X]Nikko J. Cleri
Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA

0000-0001-5414-5131]Mark Dickinson
NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA

0000-0001-7113-2738]Henry C. Ferguson
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA

0000-0001-8519-1130]Steven L. Finkelstein
Department of Astronomy, The University of Texas at Austin, Austin, TX, USA

0000-0003-3820-2823]Adriano Fontana
INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy

0000-0002-5688-0663]Andrea Grazian
INAF – Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122, Padova, Italy

0000-0001-9440-8872]Norman A. Grogin
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA

0000-0002-4884-6756]Benne W. Holwerda
Physics & Astronomy Department, University of Louisville, 40292 KY, Louisville, USA

0000-0001-9298-3523]Kartheik G. Iyer
Dunlap Institute for Astronomy & Astrophysics, University of Toronto, Toronto, ON M5S 3H4, Canada

0000-0001-8152-3943]Lisa J. Kewley
Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

0000-0002-5537-8110]Allison Kirkpatrick
Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA

0000-0002-8360-3880]Dale D. Kocevski
Department of Physics and Astronomy, Colby College, Waterville, ME 04901, USA

0000-0002-6610-2048]Anton M. Koekemoer
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA

0000-0003-3130-5643]Jennifer M. Lotz
Gemini Observatory/NSF's National Optical-Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719, USA

0000-0003-1581-7825]Ray A. Lucas
Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA

0000-0002-8951-4408]Lorenzo Napolitano
INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy
Dipartimento di Fisica, Università di Roma Sapienza, Città Universitaria di Roma - Sapienza, Piazzale Aldo Moro, 2, 00185, Roma, Italy

0000-0001-7503-8482]Casey Papovich
Department of Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA
George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX, 77843-4242 USA

0000-0001-8940-6768]Laura Pentericci
INAF – Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone, Italy

0000-0003-4528-5639]Pablo G. Pérez-González
Centro de Astrobiología (CAB), CSIC-INTA, Ctra. de Ajalvir km 4, Torrejón de Ardoz, E-28850, Madrid, Spain

0000-0003-3382-5941]Nor Pirzkal
ESA/AURA Space Telescope Science Institute

0000-0002-5269-6527]Swara Ravindranath
Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA
Center for Research and Exploration in Space Science and Technology II, Department of Physics, Catholic University of America, 620 Michigan Ave N.E., Washington DC 20064, USA

0000-0002-6748-6821]Rachel S. Somerville
Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA

0000-0002-4772-7878]Amber N. Straughn
Astrophysics Science Division, NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA

0000-0002-1410-0470]Jonathan R. Trump
Department of Physics, 196 Auditorium Road, Unit 3046, University of Connecticut, Storrs, CT 06269, USA

0000-0003-3903-6935]Stephen M. Wilkins
Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QH, UK
Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta

0000-0001-8835-7722]Guang Yang
Nanjing Institute of Astronomical Optics and Technology, Nanjing 210042, China



§ ABSTRACT 

A crucial yet challenging task in galaxy evolution studies is the identification of distant merging galaxies, a task which suffers from a variety of issues ranging from telescope sensitivities and limitations to the inherently chaotic morphologies of young galaxies. In this paper, we use random forests and convolutional neural networks to identify high-redshift JWST CEERS galaxy mergers. We train these algorithms on simulated 3<z<5 CEERS galaxies created from the IllustrisTNG subhalo morphologies and the Santa Cruz SAM lightcone. We apply our models to observed CEERS galaxies at 3<z<5. We find that our models correctly classify ∼60-70% of simulated merging and non-merging galaxies; better performance on the merger class comes at the expense of misclassifying more non-mergers. We could achieve more accurate classifications, as well as test for the dependency on physical parameters such as gas fraction, mass ratio, and relative orbits, by curating larger training sets. When applied to real CEERS galaxies using visual classifications as ground truth, the random forests correctly classified 40-60% of mergers and non-mergers at 3<z<4, but tended to classify most objects as non-mergers at 4<z<5 (misclassifying ∼70% of visually-classified mergers). On the other hand, the CNNs tended to classify most objects as mergers across all redshifts (misclassifying 80-90% of visually-classified non-mergers). We investigate what features the models find most useful, as well as characteristics of false positives and false negatives, and also calculate merger rates derived from the identifications made by the models.





§ INTRODUCTION
 

Galaxy merger studies are important for understanding the evolution of galaxies over cosmic time, as mergers can affect multiple physical phenomena and processes such as star formation and AGN activity <cit.>, the morphological evolution of spiral galaxies to elliptical galaxies <cit.>, and the build up of massive structures <cit.>. To better understand galaxy evolution, it is necessary to be able to accurately identify merging galaxies over a wide range of mass ratios <cit.> and redshifts.

However, at high redshift, merger identification can be difficult. The limited sensitivity of modern telescopes can result in samples biased toward the most massive galaxies and major mergers <cit.>. Cosmological surface brightness dimming and poor angular resolution can result in the loss of faint, detailed structures. As rest-frame optical light is shifted towards the infrared, infrared telescopes like JWST are needed to observe the stellar population. Finally, high redshift galaxies can inherently have irregular morphologies that are difficult to discern from merger signatures <cit.>.

Classic methods of identifying mergers – visual classifications <cit.>, the close pair method <cit.>, and quantitative morphology parameters <cit.> – all have limitations. The close pair method requires that both companions have accurate redshifts in order to identify pairs that are physically close in space, and is most sensitive to galaxies that have not yet merged. Visual classifications are qualitative and subjective, as well as time consuming. Finally, individual morphology parameters are each sensitive to different merger phases as well as noise <cit.>. Recently, studies have begun investigating machine learning methods for merger identification tasks, as they show promise for exploiting complex data (whether from multi-band images or from pre-computed morphology parameters) to better detect merger signatures in high redshift images than classic methods <cit.>. Furthermore, the use of deep, high-resolution near-infrared JWST imaging will also help in the detection of merger features at high redshifts. One such JWST survey is the Cosmic Evolution Early Release Science Survey <cit.>, which surveyed galaxies at 0.5 < z < 13 over 100 arcmin^2 in the Extended Groth Strip HST legacy field <cit.>. 

In <cit.>, we trained random forests to identify merging galaxies at 0.5 < z < 4 in simulated CEERS images from IllustrisTNG, using a suite of morphology parameters measured using  <cit.> and  <cit.>, as well as merger history information from IllustrisTNG. We found that the forests correctly classified ∼60% of mock CEERS mergers and non-mergers across all redshift bins. In this follow-up study, we investigate using convolutional neural networks trained directly on the mock images for the task of merger identification at high redshift, as well as test our trained models on real galaxies from CEERS (described in <ref>).

In recent years, many studies have trained neural networks to classify galaxies by morphology in general <cit.> or to identify mergers specifically <cit.>. However, most studies regarding merger identification train and test on galaxy samples at low redshift. At high redshift, <cit.>, <cit.>, and <cit.> each demonstrate that convolutional neural networks show promise for accurately identifying high redshift mergers. In particular, <cit.> apply their convolutional neural network (“DeepMerge") to z=2 simulated Illustris-1 HST F814W and F160W images created in <cit.> and achieve an accuracy of 76% on the merger class for noisy data.

In this paper, we train DeepMerge[https://github.com/deepskies/deepmerge-publichttps://github.com/deepskies/deepmerge-public] to identify galaxy mergers from simulated CEERS images from IllustrisTNG. Then we explore using our trained random forest from <cit.> and DeepMerge algorithms to identify merging galaxies in real CEERS imaging. In <ref>, we describe both the simulated JWST CEERS images and observed images, and how our datasets were curated. In <ref>, we describe training DeepMerge, as well as measuring quantitative parameters for observed galaxies. In <ref> and <ref>, we present our DeepMerge results on simulated data, then show our attempts to identify observed mergers using both DeepMerge and random forests. We summarize and conclude in <ref>.



§ DATA
 



 §.§ CEERS Observations and Data Reduction


The Cosmic Evolution Early Release Science Survey <cit.> is a JWST Early Release Science program (Proposal ID #1345) that observed the EGS deep field in 10 NIRCam pointings. The first four NIRCam pointings – hereafter CEERS1, CEERS2, CEERS3, and CEERS6 – were taken on June 21, 2022 in seven filters (F115W, F150W, F200W, F277W, F356W, F410M, and F444W) with a typical exposure time of 2835 seconds per exposure. We use these four pointings in this work. A full description of CEERS and the data reduction steps can be found in <cit.> and <cit.>, respectively. To summarize, the CEERS NIRCam images were reduced with version 1.7.2 of the JWST Calibration Pipeline[jwst-pipeline.readthedocs.iojwst-pipeline.readthedocs.io] <cit.> with custom modifications and with current NIRCam reference files[jwst-crds.stsci.edujwst-crds.stsci.edu, jwst 0989.pmap, jwst nircam 0232.imap]. The images were processed through Stages 1 and 2 of the pipeline, where reduction steps included detector-level corrections; removal of 1/f noise, wisps, snowballs; and astrometric calibration. The images were distortion-corrected and combined into final mosaics in Stage 3 of the JWST pipeline using the drizzle algorithm with inverse variance map weighting <cit.>. The final mosaics for each pointing in each filter are background-subtracted and have pixel scales of 003/pixel. The publicly released images are available at https://ceers.github.io/releases.htmlhttps://ceers.github.io/releases.html and on MAST via doi: [10.17909/z7p0-8481]https://dx.doi.org/10.17909/z7p0-8481.

In this work, we use the sample of 850 galaxies at z>3, from the four CEERS pointings, that were selected by <cit.> as well as their visual classifications (described in <ref>).



 §.§ Simulated CEERS Images
 

The simulated mock CEERS images used in this work were created by <cit.>. The images were constructed from IllustrisTNG100-1 <cit.> and the Santa Cruz SAM lightcone <cit.>. We refer the reader to <cit.> for further details on the creation of the pristine, noiseless images. The images are available in six filters (F115W, F150W, F200W, F277W, F356W, F444W) and cover ∼ 100 square arcmin (with pixel scales of 003/pixel) and contain over 100,000 galaxies, to mimic the size of the complete CEERS mosaic of the EGS field. The images were convolved with model PSFs, then noise was added on top to mimic CEERS noise <cit.>. The mock CEERS images are publicly available at https://ceers.github.io/ancillary_data.htmlhttps://ceers.github.io/ancillary_data.html.

These images are accompanied by catalogs with simulation information such as redshift, star formation rate, and stellar mass. Additionally, the merger history catalogs for IllustrisTNG galaxies <cit.> were also available. These catalogs give the IllustrisTNG snapshot numbers for each galaxy’s most recent past merger and next future merger (both major and minor). This merger history information makes IllustrisTNG an ideal dataset for training and testing machine learning algorithms to identify galaxy mergers at different stages. From the full simulated mosaic, we obtained morphology measurements for 5438 galaxies with z > 3. Figure <ref> shows the redshift and mass distribution for these objects in gray. This sample also peaks at z ∼ 3 and extends out to z ∼ 9. This sample also contains more low-mass galaxies than in the observed CEERS sample. While <cit.> note that the IllustrisTNG galaxy stellar mass function (GSMF) is in better agreement with observations than the Illustris GSMF, expecially at z≤1, IllustrisTNG may still overproduce low-mass galaxies at higher redshifts. Stellar masses for the observed CEERS galaxies were computed by <cit.> using the EGS photometry <cit.> from CANDELS <cit.> and estimated redshifts <cit.>.





§ METHODS
 



 §.§ Visual Classifications of Observed Galaxies
 

Visual classifications were reported by <cit.>, and were carried out by members of the CEERS team. Images of the galaxies in our sample were hosted on the Zooniverse project builder[https://www.zooniverse.org/lab https://www.zooniverse.org/lab]. Volunteer classifiers viewed stamps of the galaxies in seven JWST NIRCam filters and three HST  ACS/WFC3 filters, as well as RGB stamps and JWST segmentation map stamps <cit.> from  <cit.>. Each object was classified by three volunteer classifiers who followed a morphology classification scheme based on that used in <cit.>. The five tasks in this scheme ask volunteer classifiers to select main morphology classes, interaction classes, structural and quality flags, and, if the classifier wishes, to leave comments about the object. The main morphology class options were: Disk, Spheroid, Irregular / Peculiar, Point Source / Unresolved, and Unclassifiable / Junk. There were four interaction class options. 

(1) Merger: Galaxies (that are single objects) that appear to have experienced a merger due to the presence of structures like tidal tails, double nuclei, or other asymmetries. All mergers should have Irregular/Peculiar selected as their main classification, but not all galaxies classified as irregulars are mergers.

(2) Interaction within  segmentation map: Two or more galaxies that appear to be interacting within the same segmentation map. There must be clear evidence of an interaction (tidal arms, bridges, etc.).

(3) Interaction beyond  segmentation map: Two or more galaxies that appear to be interacting that have their own distinct segmentation maps. 

(4) Non-interacting companion: The main galaxy has a nearby companion, yet has no evidence of tidal interaction.

The structural and quality flags question included flags to mark galaxies with merger features such as tidal tails and double nuclei. Figure <ref> shows examples of galaxies from the different interaction classes.





 §.§ Morphology Parameters for Observed Galaxies
 

 <cit.> was used to detect z>3 galaxies in each of the four pointings. Weighted F150W+F200W+F277W+F356W images were used as the detection images. We ran  in both “cold" and “hot" mode, in order to deblend overlapping galaxies and detect faint galaxies, respectively. The “cold" and “hot" detections were then combined following the logic of the  program <cit.>.

 (Galaxy Analysis over Large Areas: Parameter Assessment by -ting Objects from ) is an IDL program that runs the multiwavelength Sérsic fitting program  <cit.> on large survey images. Information from the output  catalog was used as initial guesses for the Sérsic fits from . Empirical point-spread functions (PSFs) made from stacking stars <cit.> were used for these fits. We ran  on all seven filters simultaneously to obtain Sérsic indices and related size and magnitude information. In addition to the output Sérsic model catalog,  also returns input image, model, and Sérsic residual stamps for each galaxy in the catalog.

We also ran the non-parametric morphology program  <cit.> separately on each filter for each of the four pointings, for the z>3 objects in the  catalog, in order to obtain morphology parameters such as concentration (C), asymmetry (A), and clumpiness (S) <cit.>, and the Gini coefficient (G) and the moment of light (M_20) <cit.>. The output  segmentation map was used as input to  to aid in calculating these parameters, as well as the PSFs from <cit.>.

Finally, it has been shown that asymmetric or unusual features indicative of mergers can be more prominent in Sérsic residuals <cit.>. Therefore, we also run  on the Sérsic residuals output by . To do so, we add an offset of +1 to every pixel of all residual images, since  requires that input images have positive flux which is not necessarily true for residual images.

We begin with the catalog of visual classifications for 850 objects at z>3 from <cit.>. After combining all of the above catalogs (matching by CANDELS ID number or by RA and Dec), we create a sample of 798 observed z>3 CEERS galaxies that have successful morphology measurements from both the  and  catalogs. 



 §.§ Training Random Forests on Simulated Galaxies
 

In <cit.>, we trained and tested the random forest algorithm <cit.> on the simulated images described in <ref>. To do so, we ran  and  on the mock CEERS data in a similar fashion as described in <ref>. These morphology parameters were used as the input “features" for the forests, which were the single Sérsic index n, the two-component Sérsic indices n_bulge and n_disk, and various  parameters <cit.>, in all six filters, for both science and residual images. From the merger history catalogs, we convert the IllustrisTNG snapshot numbers to ages of the Universe in Gyr, then calculate the difference in time between the galaxy of interest and the last and next merger events. The resulting time since the last merger (both major and minor) and the time until the next (both major and minor) are used as the “labels" for the random forests. “Mergers" are objects that had, or will have, a merging event within ± 250 Myr, in accordance with <cit.>, as merger signatures from outside that time frame will have likely vanished. “Non-mergers" are objects which either have a merging event outside that time frame, or have not had merging events at all. In <cit.>, we restrict the mock CEERS dataset to have  signal-to-noise per pixel (S/N_F115W) > 3 and _ =2.0 (which indicates successful Galfit measurements). Then, we split the final sample of 40,391 mock galaxies into seven redshift bins from 0.5 < z < 4 and trained a separate forest for each bin, using  from Python's  package as the random forest algorithm and  <cit.> for hyperparameter optimization.

Following the same procedure in <cit.>, we additionally train a random forest for z = 4 - 5. In <ref>, we describe the outcome of training and testing random forests for merger identification on the mock CEERS dataset. In <ref>, we apply the 3 < z < 3.5, 3.5 < z < 4, and 4 < z < 5 random forests to observed CEERS galaxies.

Additionally, we compute the Spearman correlations between the morphology parameters and the merger timescales using  from the  package in order to understand which features might be most important for the random forests. The Spearman correlation describes the relationship between variables, where +1 or -1 indicates perfect positive or negative correlations, respectively, and 0 indicates no correlation. For each of the redshift bins studied in this work (3 < z < 3.5, 3.5 < z < 4, and 4 < z < 5), the strongest correlations ranged from ∼ |0.1| - |0.2|. For example, the most correlated features for the 4 < z < 5 bin were: F356W concentration with a correlation coefficient of -0.21, F200W residual flux (within an elliptical aperture) with a correlation coefficient of 0.11, F444W asymmetry with a correlation coefficient of 0.27, and F356W residual flux (within an elliptical aperture) with a correlation coefficient of 0.07.



 §.§ Training the DeepMerge Algorithm on Simulated Galaxies
 
An alternative to using segmentation maps and morphology parameters is to train a machine learning algorithm directly on the image data. Here, we explore using the DeepMerge[https://github.com/deepskies/deepmerge-publichttps://github.com/deepskies/deepmerge-public] convolutional neural network <cit.> to identify mock CEERS mergers. Convolutional neural networks have special “convolutional" layers which convolve the input image with a kernel, in order to extract features from the input image while preserving spatial information from the image. DeepMerge is a relatively simple network containing only three convolutional layers.



As with the random forest analysis, we use the same mock CEERS sample (with S/N_F115W > 3 and _ =2.0). This time, we restrict our mock CEERS sample to three redshift bins at 3 < z  <3.5, 3.5 < z  <4, and 4 < z  <5 (see Table <ref>). We limit ourselves to these three redshift bins, since we will apply our networks to our observed CEERS galaxies for which visual classifications are only available for z>3. We cut out stamps of size 75 by 75 pixels of each galaxy. The time since last merger and time until next merger were again used to determine the “labels" for DeepMerge.

We use stratified sampling to split our data into training, validation, and test sets. The ratio of the train/validation and test split was 0.18, which was smaller than the split used for the random forests in <cit.>, in order to divert more examples to the training set. The training and validation split was 0.3. Due to the low numbers of objects in each redshift bin, we employ augmentation to create a larger training set. This included flipping and rotating the images, as well as zooming in and zooming out while maintaining a consistent stamp size of 75 by 75 pixels. For each object, we provide DeepMerge with all six filters (F115W, F150W, F200W, F277W, F356W, F444W). We also normalize all images based on one of the six filters and then stretch the images using an arcsinh stretch to emphasize low surface brightness features. Figure <ref> shows an example of normalizing, stretching, and augmenting a 3 < z  <3.5 galaxy merger. Our data sets are imbalanced since there are more non-mergers than mergers. In order to help prevent our classifier from becoming biased toward the non-merger class, we maintain balanced classes during training by randomly undersampling the non-merger class, which decreases the size of the non-merger class. The validation and test sets are left unbalanced to better represent proportions of real-world data.



We start with the architecture of DeepMerge, and train using our data. We employ multiple strategies in order to properly train the DeepMerge networks. In addition to normalizing, augmenting, and balancing the training set, we also test:


  * Using different ratios for the split between the training, validation, and test sets.

  * Using different subsets of augmentations (e.g., only rotations).

  * Using only three filters, or only one rest-frame filter, rather than all six available filters.

  * Reducing the number of layers within the network, in order to reduce the complexity of the network.

  * Shifting the location of the dropout layers within the network. Dropout layers mask contributions from some neurons to the next layer in an effort to prevent overfitting.

  * Using rectified linear unit (ReLU) activation rather than sigmoid activation in the final layer. These activation functions determine how the result of the last layer will be mapped to a class probability.

  * Using different patience values for early stopping[https://keras.io/api/callbacks/early_stopping/https://keras.io/api/callbacks/early_stopping/], which is a method that stops training once a given metric (in this case, validation loss) has stopped improving in order to combat overfitting <cit.>.

  * Tracking the difference between training and validation accuracy, rather than validation loss, for early stopping.

Most of the above strategies resulted in networks that performed very well in one class but very poorly in the other. We also attempted to narrow our merger definition to include only major mergers, but the small number of labeled mergers in the final data sets (with the 3 < z < 3.5 bin having only 376 mergers) resulted in networks that classified all objects as non-mergers. We also attempted to include objects with any _ (about ∼10 objects in each redshift bin), as this flag only pertains to the quality of the  measurements which are not relevant here, but the inclusion of these objects also did not improve the performance of the networks. The reason for this may be that these objects were typically found in very crowded regions which may have caused DeepMerge to focus on the surrounding galaxies.

For all redshift bins, normalizing and arcsinh stretching the images resulted in improvements. We use all available filters. For the 3.5 < z < 4 and 4 < z < 5 redshift bins, we reduce the number of layers in DeepMerge to two sets of Convolution2D, BatchNormalization, MaxPooling2D, and Dropout layers rather than three sets. For the 3.0 < z < 3.5 redshift bin, we also reduce the number of layers to two sets of Convolution2D, BatchNormalization, MaxPooling2D; additionally we use only one Dropout layer which we shifted to just before DeepMerge's Dense layers. We train for 500 epochs with a batch size of 32. For the 3 < z < 3.5 redshift bin, we normalize with respect to the F115W filter; for the 3.5 < z < 4 redshift bin, we normalize with respect to the F150W filter; and for the 4 < z < 5 redshift bin, we again normalize with respect to the F150W filter. For each redshift bin, we perform the following augmentations on the training set: flip up-down, flip left-right, flip up-down and left-right, rotate at angles from 45^∘ to 270^∘, zoom-in by a factor of 1.5, and zoom-out by a factor of 0.75. For the 3 < z < 3.5 bin, for early stopping, we set patience to 50 and monitor the validation loss; for the 3.5 < z < 4 redshift bin, we set patience to 5 and monitor the ratio between validation and training accuracy so that training stops once this ratio is less than 0.8; for the 4 < z < 5 redshift bin, we set patience to 50 and monitor the validation loss.

In <ref>, we describe the best outcome of training and testing the DeepMerge network for merger identification on the mock CEERS dataset at 3 < z < 5. In <ref>, we apply the DeepMerge to observed CEERS galaxies.



§ ANALYSIS
 



 §.§ Visual Classification Results for Observed CEERS Galaxies
 

We construct four possible “merger" definitions based on the visual classifications. As described in <ref>, classifiers can select a main morphology class, an interaction class, and structure and quality flags. Our different merger definitions are motivated by these different categories:

Group 1 (G1): at least two out of the three classifiers assigned the Merger class or Interaction within segmentation map class or Interaction beyond segmentation map class.

Group 2 (G2): at least two out of the three classifiers assigned the Irregular main morphology class, since many merging galaxies appear irregular, although not all irregular galaxies are mergers.

Group 3 (G3): at least two out of the three classifiers assigned the tidal tails or double nuclei structure flags, which are merger signatures.

Group 4 (G4): one out of the three classifiers assigned the Merger class or Interaction within segmentation map class or Interaction beyond segmentation map class.

For groups 1, 2, and 3, we examine objects for which at least two out of three classifiers indicated that the object was potentially a merger, which means classifiers were overall more confident that these objects actually are mergers. For group 4, we examine objects where only only one classifier indicated that a given object had merger signatures. All objects from Group 1 will also be in this group, as well as additional objects for which classifiers were less confident about their merger status.

We start with the 798 objects that result from combining our morphology catalogs in <ref>. Since we restricted our mock CEERS dataset to have S/N_F115W > 3 and _ =2.0, we do the same to our real CEERS dataset. Doing so results in a  final sample of 369 galaxies at 3 < z < 5. Table <ref> shows the total number of objects from this restricted real CEERS data set, as well as the number of mergers from each group, in our three redshift bins. In this paper, we focus on merger definitions using Groups 1, 2, and 4, since we found that most Group 3 objects are also in the other groups. Only two objects were assigned the tidal tails structure flag without also being assigned an interaction class by at least one person.







 §.§ Classical Techniques: Performance on Observed CEERS Galaxies
 
Past studies have often used classical techniques such as G vs. M_20 <cit.> to identify mergers. In such plots, the merger discriminating line is 

    G > -0.14 M_20 + 0.33,

defined by <cit.>. 
The first panel of Figure <ref> shows G vs. M_20 for the observed CEERS dataset, color-coded by visual classification (Groups 1, 2, and 4 have been abbreviated as G1, G2, and G4 in the legend). G vs. M_20 does not appear to effectively separate visually classified mergers and non-mergers, which is to be expected since G-M_20 is not sensitive to all stages of a merger. The fraction of correctly classified mergers in each group, based on the merger discriminating line, is only 27.6% for Group 1, 21.7% for Group 2, and 19.2% for Group 4. Of the objects in the merger region above the merger discriminating line, only 35.6% are actually Group 1 mergers, 57.6% are actually Group 2 mergers, and 62.7% are actually Group 4 mergers. Most objects tend to fall in the late-type region of the plot. This performance of the G-M_20 method is similar to that in Figure 6 of <cit.> for the simulated CEERS data, where only 19-23% of the simulated mergers were correctly classified, and only 4-50% of predicted mergers were actually mergers.

G-M_20 has been shown to be unable to select a pure sample of mergers in other studies. For example, <cit.> measured G and M_20 for visually-classified HST UDF galaxies in several redshift bins from 0.4 < z < 3 and find that, besides peculiars/mergers, most other galaxy types were actually also found above the merger discriminating line. In their last redshift bin of 2.5 < z < 3, they find that ∼93% of peculiars are found in the merger region, and of all objects in the merger region, ∼67% are peculiars. <cit.> plot G-M_20 for 0.002 < z < 0.15 KiDS-GAMA galaxies with visual classifications from Galaxy Zoo. They report no specific numbers for the fraction of mergers selected by G-M_20, but find that most mergers and many non-mergers appear in the merger region. However, they then also include visual classifications from <cit.>, and find that most <cit.> mergers lie in the non-merger region. <cit.> plot G-M_20 for 0.01 < z < 3.5 visually-classified COSMOS galaxies and also do not report specific numbers, but find that most (major and minor) mergers appear in the non-merger region. They explored several automated methods besides G-M_20 and find that no technique is sensitive to all merger stages. <cit.> determine G-M_20 for simulated Illustris-1 galaxies, with mergers defined by a timescale of ±250 Myr, and also do not report specific numbers, but found that diagnostics including G-M_20 select a highly incomplete sample of mergers, as most mergers appeared in the non-merger region.

The second and third panels show the merger classifications predicted by the random forest (<ref>) and the DeepMerge network (<ref>), respectively. These panels are both color-coded by the color bar at the far right, which represents the merger probability output by either the random forest or neural network. Probabilities higher than 0.5 mean the object was classified as a merger (and colored purple or light purple). Probabilities lower than 0.5 mean the object was classified as a non-merger (and colored orange or yellow). See <ref> for further discussion of these two panels.



 §.§ Random Forests: Performance on Simulated CEERS Galaxies
 

Here we summarize the results from training random forests on mock CEERS galaxies from IllustrisTNG in <cit.>. Across all redshift bins, the forests accurately classified about ∼60% of merging and non-merging galaxies (according to our merger definition of ±250 Myr). The forests performed better than both a random classifier and classical techniques. For lower redshift bins, rest-frame asymmetry features were more important and for higher redshift bins, rest-frame bulge and clump features were more important to the forests for merger identification. The feature importances were calculated by the random forest and accessed via the  attribute. The forests misclassified non-mergers that had segmentation maps contaminated with emission from nearby or background galaxies or had merger signature persisting from outside the chosen merger time frame. The forests also misclassified mergers that appeared visually undisturbed. Finally, the merger fraction and merger rate calculated using random-forest-selected mergers were underestimated compared to previous works that used Illustris-1.



In this work, we additionally trained a random forest for z = 4 - 5. This forest correctly classified 59% of non-merging galaxies and 67% of merging galaxies. The confusion matrix and receiver operating characteristic (ROC) curve are shown in Figure <ref>, which indicate similar performance to the forests in <cit.>.



 §.§ DeepMerge: Performance on Simulated CEERS Galaxies
 



Figure <ref> shows the confusion matrices and ROC curves resulting from applying our trained DeepMerge models to the simulated CEERS test set. Our DeepMerge networks correctly classify ∼ 60 - 70% of merging and non-merging galaxies. The ROC curves show that the networks perform better than a random classifier. As in <cit.>, we categorize the test set results into four classes:


  * True Positives (TP): the number of true mergers correctly classified by the random forest.

  * False Positives (FP): the number of non-mergers incorrectly classified as mergers.

  * True Negatives (TN): the number of correctly classified non-mergers.

  * False Negatives (FN): the number of true mergers incorrectly classified as non-mergers.


There are several metrics used to quantify the performance of the neural networks:


  * True Positive Rate (TPR) – also known as recall or completeness:
    
    TPR = recall = TP/TP + FN.


  * False Positive Rate (FPR) – also known as fall out:
    
    FPR = FP/FP + TN.


  * Positive Predictive Value (PPV) – also known as precision:
    
    PPV = precision = TP/TP + FP.


  * F1 Score – the harmonic mean of precision (P) and recall (R):
    
    F_1 = 2P × R/P + R = TP/TP + 1/2 (FP + FN).


Since our simulated CEERS datasets are imbalanced, the F1 score is a more useful metric than accuracy for comparing the performance of our models. Table <ref> compares the F1 scores of the random forests and DeepMerge networks for the simulated data. Based on the F1 scores, the DeepMerge networks technically performed better than the random forests in each redshift bin, particularly the 3.5 < z < 4 bin.



As in <cit.>, we investigate the use of Gradient-weighted Class Activation Mapping (Grad-CAM) maps <cit.>. These mappings, calculated from the gradient of the predicted class of the input image with respect to the activations in the last convolutional layer, allow us to validate that the network is “looking" at the correct regions in the input image. However, for our complex data with six filters, the Grad-CAMs were not directly interpretable and showed a six-by-six pattern of activations all around the input image. Therefore, we investigate Grad-CAMs from one of the previous tests described in <ref>, where we use only one rest-frame filter rather than all six. Figure <ref> shows some examples of Grad-CAMs from the z=4-5 bin trained using only the F356W filter. For this test, the network correctly classified 62% of simulated CEERS non-mergers and 53% of simulated CEERS mergers. As shown in Figure <ref>, some Grad-CAMs from this F356W filter test show that the network does in fact activate around the edges of the central galaxies and around the edges of neighbors as we would expect, but other Grad-CAMs show only activations in a few random pixels. The distributions of the F356W magnitude and signal-to-noise per pixel indicate that objects with these “bad" Grad-CAMs are not necessarily more likely to have a fainter magnitude and lower S/N. The peak magnitude for objects with “bad" Grad-CAMs is 28.8 ± 1.2, while the peak magnitude for objects with “good" Grad-CAMs is 28.1 ± 1.5. Of objects with “bad" Grad-CAMs, 55.7% were correctly classified (either true positives or true negatives) and 44.4% were incorrectly classified (either false positive or false negative). Of the objects with “good" Grad-CAMs, 57.6% were correctly classified and 42.4% were incorrectly classified.





 §.§ Random Forests: Performance on Observed CEERS Galaxies
 

After training on the simulated images, we apply the random forest algorithm to observed CEERS galaxies, selected in <ref>, at the following redshift bins: 3 < z < 3.5, 3.5 < z < 4, and 4 < z < 5. Figure <ref> shows the confusion matrix for the 3 < z < 3.5 redshift bin for Group 1 (Figure <ref> in the Appendix shows all the Group 1, 2, and 4 confusion matrices for each redshift bin). Here, we use the visual classifications as the true merger labels for the comparison. Since the visual classifications are subjective and volunteer classifiers may misclassify objects or disagree on classifications, the true merger labels may not be correct. In summary, the forests generally did not perform as well on the observed galaxies as they did on the simulated galaxies in the test sets. The only dataset where the random forest performed better than on the test set was Group 1 at 3 < z < 3.5, where the forest correctly classified 63% of non-mergers and 66% of mergers. The forests performed the worst in the highest redshift bin, where only 25% - 30% of mergers were correctly classified. Table <ref> shows how the precision, recall, and F1-score change as a function of redshift for these groups.









 §.§ DeepMerge: Performance on Observed CEERS Galaxies
 

After training on the simulated images, we apply our DeepMerge networks to observed CEERS galaxies at the same redshift bins of 3 < z < 3.5, 3.5 < z < 4, and 4 < z < 5. Figure <ref> shows the confusion matrix for the 3 < z < 3.5 redshift bin for Group 1 (Figure <ref> in the Appendix shows all the Group 1, 2, and 4 confusion matrices for each redshift bin). In summary, the networks did not perform as well on the observed galaxies as they did on the simulated galaxies in the test sets. At 3 < z < 3.5 and 4 < z < 5 the networks classified ∼ 15-20% of non-mergers correctly and ∼ 70-80% of mergers correctly across all visual classification groups. The networks performed the worst at 3.5 < z < 4, with only 6-9% of non-mergers correctly classified and ∼85% of mergers correctly classified for all visual classification groups. Table <ref> shows how the precision, recall, and F1-score change as a function of redshift for these groups.

Based on the confusion matrices, the DeepMerge networks tend to incorrectly classify more objects as mergers than the random forests. In addition to the strategies employed in <ref>, we additionally tried the following strategies during training to prevent underfitting and improve performance on the observed CEERS data set:


  * Giving DeepMerge only one rest-frame filter to train on, or summing the six filters together into one image, in order to reduce the complexity of the problem.

  * Reducing the number of layers in the network, in order to reduce the complexity of the network.

  * Adjusting the probability threshold for merger classification (default 0.5), such that only objects with merger probabilities higher than 0.55, 0.6, or 0.65 were identified as mergers.

The results of these tests did not significantly improve performance on the observed CEERS data set. While some tests did result in confusion matrices that were slightly more balanced, the networks still either classified almost all objects as mergers or almost all objects as non-mergers.



§ DISCUSSION
 



 §.§ Classical Techniques: Observed
CEERS Galaxies with Machine Learning Classifications


The second and third panels of Figure <ref> show G vs. M_20 for the observed CEERS dataset, color-coded by the merger probabilities predicted by the random forests and the DeepMerge network, respectively. Objects were classified as mergers if the merger probability was greater than 0.5, and classified as non-merger if the merger probability was less than 0.5. As we saw in <ref>, the fraction of correctly classified mergers in each group, based on the merger discriminating line, ranged from only 19.2 - 27.6%. Thus, the machine learning algorithms were able to correctly classify more mergers than the G-M_20 method, and Figure <ref> shows that many of these mergers are below the G-M_20 merger discriminating line as expected given that G-M_20 only selects mergers at a narrow portion of the merger phase. Therefore the algorithms are able to identify mergers that the G-M_20 method misses due G-M_20's limited sensitivity. A visual inspection of the second panel of Figure <ref> reveals a potential trend where objects with very low probabilities from the random forest are far below the merger discriminating line while objects with higher probabilities are closer to or above the line. A visual inspection of the third panel of Figure <ref> shows that most galaxies were classified as mergers with relatively high probabilities, and there appears to be no clear trend in relation to the merger discriminating line.



 §.§ Feature Importance


In <cit.>, we calculated the relative importance of the features (morphology parameters) given to the random forests, which shows how useful a given feature is for identifying simulated mergers. We found that at lower redshift, bluer filters and asymmetry features were more useful, while at higher redshift, redder filters and bulge and clump features were more useful. Here we calculate feature importance for the 4 < z < 5 random forest for simulated galaxies, which are (in order of importance):

    
  * F277W multimode (M)
    
  * F444W moment of light (M_20)
    
  * F356W Gini (G)
    
  * F444W concentration (C)
    
  * F356W concentration (C)

This continues the trend seen in Figure 11 in <cit.> where redder filter and bulge and clump features are more important at higher redshift.

Of the most correlated features identified by the Spearman correlation in <ref>, we see that only one of those features was present in the top five most important features calculated by the random forest, indicating that the Spearman correlation is not necessarily a good predictor of the random forest results.

It is important to note that morphology parameters are known to be sensitive to different merger stages. For example, <cit.> find that, when simulating a galaxy as observed with HST from z ∼ 3 to z ∼ 1 (with a merger at z = 1.6), the G-M_20 merger statistic is sensitive to the early phases of the merger, while the MID statistics are more sensitive after coalescence. Future studies with larger datasets could benefit from investigating random forest feature importance as a function of merger stage, in addition to redshift.



 §.§ Galaxy Categorization Results


Figure <ref> shows the categorization of the mock CEERS test set at 3.5 < z < 4 into true positives, false positives, true negatives, and false negatives as output by DeepMerge. The top left hand corner of each stamp gives the probability of the object being a merger as determined by the DeepMerge network. The stamps are arranged in order of decreasing
probability, so the horizontal orange line effectively represents the 0.5 probability threshold between merger and non-merger classifications. In each redshift bin, the probabilities range from ∼ 0.3 - 0.7, which is similar to the range of probabilities output by the random forest.

In Figure <ref>, the bottom left hand corner of each stamp shows the merger timescale for each galaxy, which includes both major and minor mergers. A positive timescale indicates the time since a past merger, and a negative timescale indicates the time until a future merger. Galaxies can have both past and future mergers, so whichever timescale is smallest is shown here. True positives and false negatives will have merger timescales < 0.25 Gyr, while false positives and true negatives will have merger timescales > 0.25 Gyr. Finally, the segmentation map outlines are color-coded by whether the given timescale corresponds to a major or minor merger. 





Like the false positives in <cit.>, the false positives in Figure <ref> tend to have irregular segmentation maps due to neighboring or background galaxies, or asymmetric features that are possibly due to a merging event outside of our merger timescale of ± 250 Myr. The average past and future timescales of true negatives are 0.72 ± 0.30 Gyr and 0.62 ± 0.24 Gyr, respectively. The average past and future timescales of false positives are 0.49 ± 0.17 Gyr and 0.41 ± 0.19 Gyr, respectively. This indicates that false positives are slightly more likely to be closer in time to a merging event than other non-mergers. On the other hand, the false negatives appear less disturbed, possibly due to minor mergers that would have a minor effect on morphology. The fraction of minor mergers among true positives and false negatives in the simulated CEERS test set is 42.9% and 52.4%, respectively, which indicates that false negatives are slightly more likely to be minor mergers. This contrasts with the random forest results in <cit.> where false positives were not more likely to be closer in time to a merging event and false negatives were not more likely to be minor mergers.

As with the simulated CEERS images, we also categorize the observed CEERS images into true positives, false positives, true negatives, and false negatives. Figure <ref> shows this categorization resulting from the random forest and from the DeepMerge network. In each case, the “true" label is based on Group 1 visual classification (whether or not two or more people said the object was a merger or interaction). As with Figure <ref>, the probability of the object being a merger as determined by each algorithm is given in the upper left hand corner of each stamp, and stamps are arranged in decreasing probability.

One reason for the poorer performance of these algorithms on the observed data is that the random forest and the DeepMerge network were both trained on data where the ground truth labels were determined by the Illustris past or future merger timescales within ± 250 Myr, while the ground truth labels for the observed CEERS galaxies were based on visual classifications. It is possible that if the algorithms could have been trained on simulated data with labels based on visual classifications, or applied to observed data where the labels were somehow determined by merger timescales, then the algorithms may perform better. Another reason is that the observed galaxies' visual classifications are subjective and were determined by only three classifiers. In some cases, it appears that a different visual classification could be appropriate. For example, the random forest's second false positive in Figure <ref> (or DeepMerge's first false positive in Figure <ref>) could in fact be a merger. Likewise, DeepMerge's second false negative in Figure <ref> could be a non-merger. If these objects had been assigned the opposite visual classification, then the algorithms would have correctly classified them.



 §.§ Merger Rates
 



Figure <ref> shows the simulated CEERS and observed CEERS merger rates. In each panel, the black line is the theoretical Illustris merger rate derived by <cit.> assuming a merger timescale of 0.5 Gyr, for comparison. The second and third panels show the simulated CEERS merger rate as determined by the random forests (<ref>) and by the DeepMerge network (<ref>), respectively, and are in good agreement with each other.

For the DeepMerge-selected simulated CEERS mergers, we calculate the merger rate in each of the three redshift bins. The merger rate was calculated following the same procedure as in <cit.>, first using the following equation to calculate the merger fraction: 

    f_merger(NN) = N_NN/NPPV/TPR <M/N> .
 
Then we divide the merger fraction by our merger timescale of 0.5 Gyr to obtain the merger rate. Here, N_NN is the number of mergers determined by the DeepMerge network and N is the total number of objects in our sample in each redshift bin. PPV is the positive predictive value, or precision, of the classifier. TPR is the true positive rate, or recall, of the classifier. PPV/TPR is meant to correct for the classifier's known incompleteness and purity based on the training set. <M/N> is the average number of merging events per true merger, and accounts for the possibility that mergers can experience more than one merging event within the ± 250 Myr timescale used to determine the labels for the neural network. The DeepMerge simulated CEERS merger rate is shown as the green line in the right panel of Figure <ref>, and is in good agreement with the random forest simulated CEERS merger rate.

The first panel of Figure <ref> shows the observed CEERS merger rate for each visual classification group. This was calculated by first determining the merger fraction using simply:

    f_merger(obs) = N_obs/N.
 
We then divide the merger fraction by our timescale of 0.5 Gyr used for the simulated data. Here, N_obs is the number of mergers based on our visual classification groups, and N is the total number of objects in our observed CEERS sample in each redshift bin. The second and third panels also give the observed CEERS merger rate based on the machine learning outputs. These merger rates were calculated as in Equation <ref>, where PPV/TPR is from the respective machine learning algorithm and <M/N> is calculated from the simulated data. Like the simulated merger rates, the observed merger rates in the first and second panel are underestimated compared to the theoretical Illustris merger rate. However, the DeepMerge observed merger rate is higher than the simulated CEERS merger rates, the theoretical Illustris merger rate, and the observed merger rates in the first panel due to the fact that DeepMerge overestimated the number of mergers. The random forest observed merger rate is in better agreement with the observed merger rates in the first panel. In addition, the observed merger rates appear relatively flat while the simulated mergers rates increase at z > 4.



 §.§ Comparison with Previous Works
 
 
As discussed in <ref>, few studies apply convolutional neural networks to the task of merger identification at high redshift. We compare our work to <cit.>, <cit.>, and <cit.>. The DeepMerge network is introduced in <cit.> to identify z=2 simulated Illustris-1 HST mergers. They have access to four different “camera” perspectives, which they use to create a sample of 8930 unique galaxies. They then augment the merger class to create a more balanced sample of 8120 mergers and 7306 non-merger before they split their data into train and test sets. Their merger class consists of mergers with a stellar mass ratio of at least 0.1 and within a time frame of ± 250 Myr. They find that DeepMerge achieves an accuracy of 76% when training and testing on noisy data. They report correctly classifying 78% of mergers and 73% of non-mergers in their noisy test set. In <cit.>, they use domain adaption techniques to apply their trained network to real data; however, this is done at 0.005 < z < 0.1 using data from the Galaxy Zoo project <cit.>.

<cit.> use simulated IllustrisTNG300-1 CANDELS major mergers (up to z∼3) within ± 0.3 Gyr and non-mergers. Additionally, they impose a limit on the separation between each pair of galaxies. With these criteria, they curate a sample of ∼30,000 distinct major merger candidates. For the non-mergers, they impose a timescale limit of ± 0.5 Gyr (i.e., the galaxy must have a merger timescale outside this time frame). Using these criteria, they build up a random sample of non-mergers that equals the number of mergers in each redshift bin. They train two classifiers – one to identify mergers vs. non-mergers, and a second to further subdivide mergers into pre- and post-mergers – and are able to correctly classify 87% and 94% or pre- and post-mergers, respectively. They apply their methodology to a sample of 3759 real CANDELS galaxies with visual classifications given by <cit.>, but do not report an accuracy for this test. They do note that since their validation sample is balanced while the CANDELS sample is not, the performance of the model on simulated data will not translate to the same performance of the model on real data.

Finally, <cit.> both train and test their convolutional neural network on observed CANDELS galaxies up to z < 4, in the F814W, F125W, and F160W bands, using visual classifications from <cit.>. They construct training samples of 694 objects in each class in each of their five redshift bins from 0 < z < 4. They also augment their images to create a larger training set. They report a final accuracy of 81.8% after testing on the full redshift range of 0 < z < 4.

Each of these studies reports better performance than we report in this work, which could be due to a number of differences. First, <cit.> and <cit.> use Illustris-1 and IllustrisTNG300-1, respectively, while we use IllustrisTNG100-1. <cit.> impose additional criteria to define their merger sample and non-merger samples. Perhaps most importantly, both <cit.> and <cit.> are able to curate a large training set, even before augmentation. All three studies also use balanced training, validation, and test sets, whereas our validation and test sets are left unbalanced since real data will be unbalanced. Additionally, <cit.> augment their data prior to splitting their data into training, validation, and test sets, and it is unclear if they ensured that all augmentations of a given galaxy end up in the same subset. Otherwise, the training, validation, and test sets would be correlated which could inflate the accuracy of the network. <cit.> do not specifically comment on the performance of their first classifier, which separates mergers from non-mergers. <cit.> do not specifically report performance on their non-merger class. Thus these papers may have focused on optimizing performance to correctly identify most mergers, rather than focusing on the purity of the merger and non-merger classifications, similarly to <cit.>. Finally, <cit.> and <cit.> study galaxies at z < 3 with HST images while we push to 3 < z < 5 with JWST. <cit.> do extend out to z = 4, but their training set consists for data from the full range of 0 < z < 4.



§ CONCLUSIONS
 

In this work, we investigate using the DeepMerge convolutional neural network algorithm to identify merging galaxies at 3 < z < 5 in simulated CEERS images constructed from IllustrisTNG and the Santa Cruz SAM. Additionally, we calculate morphology parameters for observed CEERS galaxies using the morphology programs  and . We apply the trained random forests from <cit.>, a 4 < z 
 < 5 random forest trained in this work, and the DeepMerge neural network trained in this work to observed CEERS galaxies, and verify using visual classifications from <cit.>. We find the following results:


  * The 4 < z < 5 random forest performed similarly to those in <cit.>, correctly classifying 59% of non-merging galaxies and 67% of merging galaxies.

  * The DeepMerge network correctly classified ∼60 - 70% of simulated merging and non-merging galaxies. Although Grad-CAMs from our tests using six filters were not clearly interpretable, Grad-CAMs from our tests using one filter showed that the networks usually activated around the edges of galaxies as expected.

  * The random forests and neural networks did not perform as well when applied to real CEERS galaxies. In particular, the neural network tended to classify most objects (60 - 80%) as mergers. This could be due to a number of reasons, such as the different methods of defining mergers for the simulated data and the observed data.

  * As in <cit.>, DeepMerge false positives tend to appear to have close neighbors or asymmetric features, and false negatives appear less disturbed. The false positives appear to be more likely to be closer in time to merging events than the true negatives and false negatives appear more likely to be minor mergers than true positives. The categorization of observed CEERS galaxies shows that positive classifications do look more irregular and asymmetric and negative classifications do look more undisturbed, but also that visual classifications are subjective.

  * Merger rates from (a) observed visual classifications only, (b) RF-selected observed merger classifications, and (c) DeepMerge-selected simulated merger classifications generally agree with each other and with the RF-selected simulated merger rate from <cit.>. The merger rate calculated using DeepMerge-selected observed merger classifications is overestimated comparatively, due to the fact that DeepMerge misclassified almost all observed non-mergers.




 §.§ Future Work


One clear area of improvement is curating a larger sample of simulated mergers in order to increase the number of objects in the training set which could help the network better learn merger signatures. This could be done by remaking the pristine simulated images using the larger IllustrisTNG300-1 simulation <cit.> or using another large volume, high resolution simulation. A larger sample would also enable the performance as a function of other physical parameters that can impact the observed merger signatures, such as the gas fraction of the galaxies, their merger ratio, and their relative orbits. Another area of improvement would be investigating the use of transfer learning techniques to improve performance on the observed simulated CEERS data. <cit.> show that using transfer learning techniques improves the performance of their simulation-trained networks on observed data by up to ∼ 20%. Finally, it would be interesting to train and test machine learning techniques directly on observed CEERS galaxies <cit.> once more visual classifications become available. It would also be interesting to compare the Grad-CAM maps to the regions of the images for which the morphology parameters are sensitive, in order to better understand the features that the neural network has learned.


Support for this work was provided by NASA through grants JWST-ERS-01345.015-A and HST-AR-15802.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. This research is based in part on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. This work is based in part on observations made with the NASA/ESA/CSA James Webb Space Telescope. The data were obtained from the Mikulski Archive for Space Telescopes at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127 for JWST. These observations are associated with program #1345.

The authors acknowledge Research Computing at the Rochester Institute of Technology for providing computational resources and support that have contributed to the research results reported in this publication <cit.>

The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper (http://www.tacc.utexas.eduhttp://www.tacc.utexas.edu).





JWST (NIRCam)

Source Extractor <cit.>, Galapagos-2 <cit.>, statmorph <cit.>, DeepMerge <cit.>





§ CONFUSION MATRICES FOR OBSERVED CEERS GALAXIES

Figure <ref> shows the results of applying the trained random forests to observed CEERS galaxies in <ref>. Figure <ref> shows the results of applying the trained DeepMerge neural networks to observed CEERS galaxies in <ref>.






aasjournal



 Paper Title: Captured molecules could make a Bose star visible.



v.flambaum@unsw.edu.au
igor.samsonov@unsw.edu.au
School of Physics, University of New South Wales, Sydney 2052, Australia

§ ABSTRACT 
A Bose star passing through cold molecular clouds may capture atoms, molecules and dust particles. The observational signature of such an event would be a relatively small amount of matter that is gravitationally bound. This binding may actually be provided by invisible dark matter forming the Bose star. 
We may expect a relative excess of heavier atoms, molecules, and solid dust compared to the content of giant cold molecular clouds since the velocity of heavy particles at a given temperature is lower and it may be small compared to the escape velocity, v_rms = √(3k_B T/m_gas)≪ v_esc=√(2GM/R). Finally, the velocity of this captured matter cloud may correlate with the expected velocity of free dark matter particles (e.g. expected  axion wind velocity relative to Earth).



Captured molecules could  make a Bose star visible
    I. B. Samsonov
    August 5, 2024
==================================================






§ INTRODUCTION

Bose stars  are assumed to be formed from cold dark matter fields (scalars, axions) which may form Bose condensate and solitons bound mainly by the gravitational  attraction <cit.>. 
Bose condensation temperature T_B ∝ n^2/3/m, so in thermal equilibrium axions (or scalars) form Bose condensate for sufficiently small mass m  and large  density n.  

A simple estimate, similar to that for hydrogen atom,  makes relation between mass M≈ N m and size R  of Bose star, where N is the number of bosons and m is their mass. 
Estimating kinetic energy of the ground state using uncertainty relation for minimal momentum p  ∼ħ/r and estimating gravitational energy we obtain equation for the non-relativistic energy of Bose star: 

    E=a_1 Nħ^2/mr^2 - a_2 N(N-1)/2Gm^2/r,

 where a_1, a_2 are constants ∼ 1. Minimizing energy by varying  size r or using virial theorem (U=- 2E_k, E=-E_k) we obtain size of gravitationally bound  Bose star   

    R=4 a_1  ħ^2/a_2 (N-1)G m^3,

 binding energy

    E=N (N-1)^2  a_2^2 G^2 m^5/16 a_1ħ^2,

and number density 

    n  ∼N/V∼ 0.004 N (N-1)^3  G^3 m^9/ħ^6,

where the star volume is V ∼ 4 π R ^3/3.
We see very strong dependence on the boson mass m. We also see that the density n in   the ground state is proportional to N^4 so any, even small non-gravitational self interaction between bosons may become important  for large N.
We may add the self-interaction  contribution of the self-interaction term λϕ^4 to the Bose star energy 

    E=a_1 Nħ^2/mr^2 - a_2 N(N-1)/2Gm^2/r+a_3  λN(N-1)/2 m^2 r^3  ,

where a_1, a_2 a_3 are constants ∼ 1. Using dE/dr=0,  this expression  gives quadratic equation for r with two solutions. For the QCD axion, the selfinteraction constant λ=- m^2/(24 f_a^2) is expressed via the axion decay constant f_a (see e.g. <cit.>).  In the case of λ<0 the energy tends to minus infinity for r → 0, so the Bose star is metastable. Solution with larger r corresponds to a minimum whereas the smaller r solution is maximum. More accurate analytical calculations using variational approach will be given in the next section.

Distribution of ordinary matter is given by Boltzmann distribution in the gravitational field of Bose star. Velocity distribution is given by Maxwell distribution. We can  add molecular matter terms E_m to the energy E:

    E_m=   a_4 N_m 5/2 kT(r) - a_5 N_M N Gm_m m/r
       - a_6 N_m(N_m-1)/2Gm_m^2/r .

If the total mass of the molecular cloud bound to the Bose star, M_m=N_m m_m, is relatively small as compared with the mass of the Bose star, N_m m_m ≪ Nm, we may assume that in the leading approximation molecular distribution does not affect Bose star parameters. Then we may include gravitational effect of molecules using iterations. 

Time of formation of Bose stars very strongly depends on the Boson  mass m. For m >  10^-22 eV the formation time is smaller than the lifetime of the Universe 13.7 billion years (see e.g. <cit.>).

Total mass of the Bose star M is strongly model dependent and depends on time. There are estimates of a  typical scale for expected mass M. The QCD axion dark matter field with axion mass m ∼ 10^-5  eV makes  Bose star with mass  M ∼  10^-15 M_⊙  <cit.>. Fuzzy dark matter with particle mass m ∼ 10^-22  eV may compose a Bose star with mass M ∼  10^8 M_⊙  <cit.>. Bose star accumulating  certain critical mass becomes unstable, collapses  and decays to relativistic jets <cit.>.
 
Axions and scalars are neutral, so in zero approximation Bose star is transparent and is hardly observable by telescopes.  In principle, Bose stars may be observed using gravitational lensing of passing light and axion decays to photons. We discuss a different possibility: Bose star may  capture atoms, molecules and dust. Radiation and absorption of light by this matter could make Bose stars visible. 




§ VARIATIONAL APPROACH


The variational approach allows one to get an approximate analytic solution to the particle density which minimizes total energy of the system of interacting particles. We consider N bosonic DM particles with position vectors r⃗_i and momenta p⃗_i. Given that each particle has mass m, they can interact gravitationally. The corresponding non-relativistic Hamiltonian is

    H  = -ħ^2/2m∑_i=1^N ∇_k^2 - ∑_i<j^N G m^2/r_ij ,

where G is the gravitational constant and r_ij≡ |r⃗_i - r⃗_j|. 

In the ground state, all particles are in the same state with the wave function ψ(r⃗_i), and the full system is described by a product of these wave functions, 

    Ψ = ∏_i=1^N ψ(r⃗_i) .

Below we consider two approximate analytic solutions to the function ψ(r⃗).



 §.§ Exponential ansatz


The ground state wave function may be searched for within the ansatz

    ψ(r) = 1/√(π) R^-3/2e^-r/R ,

where R is a free parameter. The function ψ(r) is the 1s solution of the stationary Schrödinger equation with the potential V = -ħ^2/m Rr and energy E_0 = -ħ^2/2 mR^2.

Making use of the function (<ref>) we find the expectation value of the Hamiltonian (<ref>):

    E = ⟨Ψ|H|Ψ⟩ 
        =N/2[
        ħ^2/m R^2 - 5/8Gm^2/R (N-1)
        ].

This energy reaches its minimum value 

    E_min = -25/512G^2 m^5/ħ^2 N(N-1)^2

at

    R = 16/5ħ^2/Gm^3 (N-1) .


The energy of the solution (<ref>) agrees at 90% level with the corresponding numerical result in Ref. <cit.>.
The root-mean-squared (RMS) radius of this solution is

    r_rms = (⟨Ψ | r_i^2 | Ψ⟩)^1/2
        =√(3) R .


As an example, consider a Bose star with mass of Bennu asteroid, M=7× 10^10 kg composed of QCD axions with mass m=10^-5 eV. It consists of N=4× 10^51 particles. Its size is

    r_rms = 4× 10^10 .

Note that the diameter of the Bennu asteroid is 0.5 km.

The RMS speed:

    v_rms = 1/m(
        ⟨Ψ |p⃗_i^2| Ψ⟩)^1/2 =ħ/m R .

For the axion star considered above, it is v≈ 8× 10^-16c. Thus, the non-relativistic approximation is justified. However, for solutions with boson  mass m ∼ 10^-5 eV and total mass of order the Solar mass, a relativistic treatment is needed.





 §.§ Gaussian ansatz


The simple exponential ansatz (<ref>) for the wave function provides a rather low accuracy for the energy. A better accuracy may be achieved within the Gaussian ansatz <cit.>

    ψ(r) = ( 2/π R^2)^3/4
        e^-r^2/R^2 ,

where R is the parameter to be found. Making use of this wave function, we find the expectation value of the Hamiltonian (<ref>):

    E = ⟨ H ⟩ = 3Nħ^2/2m R^2 - N(N-1)Gm^2/√(π)R .

This energy reaches its minimum value 

    E_min = -N(N-1)^2/6πG^2 m^5/ħ^2

at

    R = 3√(π)ħ^2/(N-1)Gm^3 .


The energy (<ref>) agrees at 98% level with the numerical solution presented in Ref. <cit.>. Therefore, we will consider only the Gaussian ansatz in what follows.

The RMS radius calculated with the wave function (<ref>) is 

    r_rms = √(3) R/2 = 3√(3π)ħ^2/2(N-1)Gm^3 ,

and the RMS speed is

    v_rms
        =√(3)ħ/m R .


The non-relativistic approximation is applicable for  v_rms≲ 0.1 c. Making use of equations (<ref>) and (<ref>), this relation translates into the constraint on the particle number and mass,

    Nm^2<0.1 
        ħ c√(3π)/G≈ 1.5 × 10^-10 g^2 .






 §.§ Gravitational potential


The Bose star solution is given by the product of N identical wave functions (<ref>) each describing a particle with mass m. They create a gravitational potential V(r) obeying the Poisson equation

    ∇^2 V(r) = -4π G ρ(r) ,

where the mass density is

    ρ = N m ψ^†ψ
        =N m( 2/π R^2)^3/2
        e^-2r^2/R^2 .

The solution to Eq. (<ref>) reads

    V(r)    = - G N m erf (√(2) r/R)/r
       ≈ -GNm{[ 2√(2)/√(π)R
        -4√(2) r^2/3√(π)R^3                            (r≪ R) ,;                                 1/r                            (r≫ R) . ].


On the boundary r=r_rms, the gravitational potential is

    V(r_rms) = -2G Nm/√(3) Rerf( √(3/2))
        ≈ - 0.2 N^2 G^2 m^4/ħ^2 .

Thus, the escape velocity on the boundary is

    v_esc = √(2|V(r_rms)|)≈ 0.6 NGm^2/ħ .


Consider, for example, the QCD axion with mass m=10^-5 eV and the total mass comparable with the mass of the Earth, Nm = M_⊕. The size of this object is about r_rms≈ 0.4 m. The corresponding escape velocity at the boundary is v_esc≈ 0.14 c.

If the DM particle mass is m=10^-9 eV, the corresponding RMS radius and escape velocity are r_rms≈ 6.3 R_⊕, v_esc≈ 4.3 km/s.



 §.§ Adding λϕ^4 selfinteraction


The scalar field ϕ is related to the wave function (<ref>) as

    ϕ = 1/√(m)Ψ .

Making use of the Gaussian ansatz (<ref>), we find

    λ∫ϕ^4 d^3r = λ/π^3/2 m^2 R^3 .

In the contribution to the energy, this expression should be multiplied by N(N-1)/2 which accounts for the pairwise particle interaction. Thus, the total energy of this field configuration reads



    E    = E_kin + E_grav + E_λ ,
    
        E_kin   = 3Nħ^2/2m R^2,
    
        E_grav   = - N(N-1)Gm^2/√(π)R ,
    
        E_λ   =  N(N-1)/2λħ^3/π^3/2 c m^2 R^3 .


This expression reaches its extrema at

    R_± = 3√(π)ħ^2/2G m^3(N-1)( 
            1±√(1+2λ G m^2(N-1)^2/3π^2ħ c)).


The case λ<0 corresponds to unstable solution because the energy (<ref>) is unbounded from below in this case. However, there is a potential barrier separating energy minimum at R_+ in Eq. (<ref>) from the collapse area close to r=0, with the barrier maximum at R_-. In the case of small λ, we obtain minimum of energy

    E_min = -N(N-1)^2/6πG^2 m^5/ħ^2 + O(λ)

at

    R = 3√(π)ħ^2/G m^3(N-1)
        +λħ(N-1)/2π^3/2mc+O(λ^2) ,

and maximum at small R ∝λ,

    R = -λħ(N-1)/2π^3/2mc + O(λ^2) .

In the case of small λ this barrier will be sufficiently  high to make  the   decay time of the  metastable Bose star  exponentially  long, making this kind of Bose star potentially observable.  

For positive selfinteraction constant, λ>0, only “+” solution in Eq. (<ref>) is physical. If λ is sufficiently small, the leading contribution has the same form as in Eq. (<ref>).
For strong selfinteraction, 2λ G m^2(N-1)^2/3π^2ħ c≫1,
the parameter R is independent of N,

    R = ħ^3/2/m^2√(3λ/2π cG) .






§ IDEAL GAS DISTRIBUTION AROUND THE BOSE STAR


Let us consider an ideal gas with particles of mass m_gas in the gravitational potential V(r)<0 at temperature T. Assuming that the total mass of the gas is sufficiently small so that the gravitational potential is not changed significantly, we can write the density distribution in the gas cloud,

    ρ(r) = ρ_0 exp( m_gas V(r)/k_BT) .

The total mass of the gas cloud is

    M_gas = 4πρ_0 ∫_0^∞exp( m_gas V(r)/k_BT) r^2 dr .


The gas at every point should follow the Maxwell-Boltzmann distribution

    f(v) = 4π v^2 ( m_gas/2π k_BT)^3/2 e^-m_gas v^2/2 k_B T .

This distribution, however, should be truncated at v=v_esc. Note that v_rms = √(3k_B T/m_gas).

At the Earth's surface, the temperature is about T=300 K. Empirically, we find the condition for the gas molecules to be captured by the gravitational field of the Earth:

    v_rms≲ 0.1v_esc .

This corresponds to the escape probability suppressed by the  exponential factor in the distribution (<ref>) exp(-3/2v_esc^2/v_rms^2)≲exp(-150). For the Earth, this condition implies that the gas molecules with mass number A<6 escape from the gravitational attraction. This is confirmed by the fact that the fractions of hydrogen and helium gases in the Earth's atmosphere are very low. 

In the case of Bose stars, making use of Eq. (<ref>) the above condition may be cast in the form

    m_gas >830k_BTħ^2/N^2 G^2 m^4 ,

Representing the gas particle mass as m_gas = A m_p, with m_p the proton mass, we find the condition on the atomic number A of gas particles captured near the surface of the Bose star,

    A>   830k_BTħ^2/N^2 G^2 m^4 m_p
       = 1.35 · 10^-29(M_⊙/M)^2(eV/m)^2
        T/10 K .




The temperature in cold molecular clouds in our Galaxy is typically in the range from 10 to 20 K. Assuming T=10 K, we estimate the possibility of capture of gases in the fuzzy dark matter with m=10^-22 eV and total mass of the object Nm=10^7M_⊙:

    A>13 .

Thus, this object can accumulate only heavy atoms and molecules, as well as dust particles. 

In Fig. <ref>, we present plots of mass (and radius) of the Bose star as a function of scalar particle mass m which can capture molecules with atomic mass numbers A=1, 10, and 100.

If radius R of Bose star is small, there  will be too little matter inside the star and impossible to observe it. However, matter may be captured to distant orbits, but this will require bigger mass M  of the star, since the escape velocity will be smaller; for matter at distance r, the  required star mass is M(r)=M_0 R/r, where M_0 is the star mass required to capture matter inside. 

Cold molecular gas and dust may also be captured by large axion clusters.

If matter capture conditions  are  fulfilled inside Bose star for all A, there will be violation of such conditions outside Bose star at some distance r from the centre. Firstly, this condition will be violated for A=1, going further, for higher A.  Therefore, there will be separation of heavy and light elements outside the Bose star. This is important for Bose star of large mass and small radius.




§ SUMMARY


We applied the analytic Gaussian solution for Bose star density to study the distribution of interstellar gas particles within the gravitational field of a Bose star, assuming the gas does not significantly perturb the star's gravitational field. We examined the conditions under which interstellar gas particles with a given atomic number A could escape the gravitational attraction of a Bose star with a total mass M composed of scalar dark matter particles of mass m. Our findings indicate that, for a range of values of m and M, the Bose star can capture atoms, molecules, and dust particles.

A small cloud of interstellar gas cannot form a gravitationally bound state without an additional gravitational source. Thus, the observation of such a gravitationally bound gas cloud would suggest it is held by the gravitational field of a Bose star or a dark matter cluster. Due to differences in escape velocities between light and heavy molecules, molecular clouds captured by a Bose star may exhibit higher metallicity compared to typical giant molecular clouds. The discovery of small, gravitationally bound molecular clouds with high metallicity in the interstellar medium would provide compelling evidence supporting this dark matter model.




Acknowledgements.— 
We are grateful to John Webb for interest in this work and useful discussions. The work was supported by the Australian Research Council Grants No. DP230101058 and No. DP200100150.










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 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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Song F., Esamdin A., Hu Q., Zhang M. 2022, , 666, A75

[Spina et al.2021]Spina
Spina L., Ting Y. -S., De Silva G. M. et al., 2021, , 503, 3279

[Tarricq et al.2022]tarricq22
Tarricq Y., Soubiran C., Casamiquela L. et al., 2022, , 659, A59

[Uribe & Brieva1994]Uribe94
Uribe A., Brieva E., 1994, , 214, 171

[Vicente et al.2016]belen16
Vicente B., Sánchez N., Alfaro E. J., 2016, , 461, 2519

[Wenger et al.2000]Simbad
Wenger M., Ochsenbein F., Egret D. et al., 2000, A&AS, 143, 9

[Wright et al.2010]Wright10
Wright E.L., Eisenhardt P.R.M., Mainzer A.K. et al., 2010, , 140, 1868

[Yen et al.2018]Yen18
Yen S.X., Reffert S., Schilbach E., Röser S., Karchenko N.V., Piskunov A.E.,  2018, , 615, A12















§ 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






































