Paper Title: Different behaviour of the gas-phase and stellar metallicity in the central part of MaNGA galaxies.






       

Faculty of Physics, Ludwig-Maximilians-Universität, Scheinerstr. 1, 81679 Munich, Germany 
Main Astronomical Observatory, National Academy of Sciences of Ukraine, 
27 Akademika Zabolotnoho St., 03143, Kyiv, Ukraine
Instituto de Astrofísica de Andalucía (CSIC), Apartado 3004, 18080 Granada, Spain  



We quantified the disparity between gas-phase and stellar metallicity in a large galaxy sample obtained from the MaNGA DR17 survey. We found that the gas metallicity is on average closely aligned with the stellar metallicity in the centers of intermediate-mass galaxies. Conversely, the difference is notably larger within the center of massive galaxies. It reaches about -0.18 dex on average for the most massive galaxies,
while for low-mass galaxies, the gas metallicity exhibits a slightly lower value than the stellar metallicity.
Moreover, the most prominent instances of a reduced gas-phase metallicity in relation to stellar metallicity were observed within the centers of massive red galaxies with low specific star formation rates. Because of the absence of a correlation between the integral mass fraction of neutral gas and the disparity between gas and stellar metallicity, we suggest that the diminished gas-phase metallicity in the centers of massive galaxies might be attributed to the replenishment of gas-depleted central regions through processes such as radial gas flows or accretion from the circumgalactic medium rather than gas infall from the intergalactic medium. 







Gas-phase and stellar metallicity in the central part of MaNGA galaxies
Zinchenko & Vílchez
Different behaviour of the gas-phase and stellar metallicity in the central part of MaNGA galaxies
    
        I. A. Zinchenko<ref>,<ref> J. M. Vílchez<ref>
       
    
==================================================================================================







§ INTRODUCTION


The study of the chemical composition of gas and stars in galaxies is crucial for understanding the fundamental principles of galaxy formation. The chemical composition of the gas in galaxies is constantly shaped by the complex interplay of various physical processes, such as feedback from stars and active galactic nuclei (AGN), gas inflow, outflow, and radial gas flows, while the metallicity of stars usually remains equal to the metallicity of the gas from which a star was formed. 

One of the most renowned fundamental relations is the connection between the stellar mass and metallicity of a galaxy. Generally, more massive galaxies exhibit higher metallicities  in their gas component <cit.> and in their stellar component <cit.>. The mass-metallicity relation (MZR) provides constraints and insights into the interplay among gas inflows, outflows, star formation rates, and chemical enrichment processes. Additionally, the MZR provides valuable insights into the role of feedback mechanisms in regulating the metal content of galaxies, thereby shaping their diverse properties. Nevertheless, it can be challenging to interpret the MZR because the chemical evolution models involving both outflows and inflows are degenerate.

Conversely, <cit.> demonstrated that the stellar metallicity exhibits greater sensitivity to the rate of the gas outflow than to the gas inflow. Consequently, a simultaneous analysis of the gas-phase and stellar metallicities can alleviate the degeneracy and unravel the balance of gas inflow and outflow in star-forming galaxies. 
Nonetheless, properties such as the metallicity, the star formation rate, the gas mass fraction, and the inflow and outflow rates can vary with the distance from the center of a galaxy. An examination of the gas-phase metallicity in an extensive sample of galaxies over the past decade has unveiled that the gas-phase metallicity profile within galaxies can exhibit a more complex shape than previously assumed and can deviate from the traditional exponential decline <cit.>. 

Furthermore, a decline in luminosity-weighted stellar metallicity has been observed in the nuclei of NGC 628 <cit.> and in the Milky Way <cit.>. The relation between the gas-phase oxygen abundance and stellar metallicity in a galaxy was studied by <cit.> for a sample of galaxies from the Sloan Digital Sky Survey Data Release 2 (SDSS DR2). They concluded that the two quantities appear to be correlated because the stellar metallicity always was lower than the gas-phase metallicity, but showed a large scatter. This was interpreted as showing the effect of gas inflows or outflows. This relevant result was affected by the limitation of the spatial resolution natural to the SDSS spectral aperture (3 arcsec fiber diameter) projected onto each galaxy. Some recent studies specifically addressed this question. Notably, <cit.> studied the mass-metallicity relation of the gas and the stellar component in a sample of local SDSS (DR12) galaxies and found that the difference in metallicity increases for galaxies with lower masses and can reach between 0.4–0.8 dex at 10^9 M. An extension of this study using a selection of the Mapping Nearby Galaxies at Apache Point Observatory Data Release 14 galaxies <cit.> has found negative stellar metallicity gradients, which appear to be steeper for more massive galaxies, whereas the gas abundance gradients do not depend on mass. 
Moreover, <cit.> studied a sample of galaxies selected from the SAMI survey, for which stellar metallicity and gas chemical abundances data are available. Their work confirmed previous results overall and showed that the ratio of the gas to stellar metallicity appears to be systematically below the predictions of the simple closed-box model. This reinforced the idea that a major role is played by gas inflows and enriched outflows. Notwithstanding the above, however, some limitations persisted for a spatially resolved analysis because the spectra that were analyzed collected integrated light within one effective radius for each galaxy.

Hence, it is clear that the analysis of spatially resolved information of the chemical abundances of gas and stars offers valuable insights into the study of the mechanisms that have governed the growth and assembly of galaxies <cit.>. 
In this work, we investigate the distributions of the gas-phase and stellar metallicity in the central regions of an extensive sample of nearby galaxies from the MaNGA survey. The MaNGA survey is a part of the Sloan Digital Sky Survey IV and employs integral field spectroscopy (IFS). This technique enables the acquisition of spatially resolved information about the gas and unresolved stellar populations and circumvents potential aperture biases that are attributed to the diverse sizes and redshifts of galaxies. In this way, it allows comparable physical regions to be sampled consistently.

The paper is structured as follows. 
In Sect. <ref> we provide a detailed description of the data and the criteria we employed for the sample selection. 
Section <ref> describes the methods with which we obtained the gas-phase and stellar metallicities. 
In Sect. <ref> we discuss potential scenarios for the formation of the difference between the metallcity of the gas and stellar components. 
Finally, Sect. <ref> presents a summary of the main findings.





§ DATA




For this study, we prepared a sample of galaxies from the MaNGA SDSS DR17 survey <cit.>.
We analyzed the MaNGA spectra following the prescriptions described in <cit.>.
In brief, the stellar background in all spaxels was fit using the public version of the code STARLIGHT <cit.>, which was
adapted for the parallel processing of the datacubes.
To fit the stellar spectra, we used simple stellar population (SSP) spectra from the evolutionary synthesis models by <cit.>. The resulting stellar spectrum was subtracted from the observed spectrum to obtain a pure gas spectrum. To analyze the physical parameters of the stellar populations, we used data from the MaNGA Firefly Value-Added Catalog <cit.>.

To fit the emission lines, we used our code ELF3D in the optical spectra. This code is able to fit complex line profiles such as blends or profiles with broad and narrow components. The recent ELF3D version is based on the LMfit library and shared-memory parallelization. This approach reduces the number of dependencies and improves the portability of the code. We fit each emission line with a single-Gaussian profile. Thus, for each spectrum, we measured the fluxes of the
[OII]λ λ3727,3729,
Hβ,
[OIII]λ4959,
[OIII]λ5007,
[NII]λ6548,
Hα,
[NII]λ6584, and
[SII]λ6717,6731 lines. The line fluxes were corrected for interstellar reddening using the analytical approximation of the Whitford interstellar reddening law <cit.>, assuming a Balmer line ratio of Hα/Hβ = 2.86. When the measured value of Hα/Hβ was lower than 2.86, the reddening was set to zero.

To select spaxels associated with HII regions, we applied the log([OIII]λ5007/Hβ) – log([NII]λ6584/Hα) diagram <cit.>. For the further investigation, we selected only spectra whose main ionization source were massive stars according to the dividing line proposed by <cit.>.
We selected only spectra with a signal-to-noise ratio S/N > 5 in all the [OII]λ λ3727,3729, Hβ, [OIII]λ5007, Hα, [NII]λ6584, and [SII]λ6717,6731 lines.

The stellar masses, effective radii, and colors for the MaNGA sample were taken from the NASA-Sloan Atlas (NSA) catalog[http://nsatlas.orgnsatlas.org].
The NSA catalog lists parameters derived by different methods. We chose stellar masses that were derived from the K-correction fit for elliptical Petrosian fluxes using the <cit.> initial mass function and simple stellar population models from <cit.>.
Effective radii (R_e) were selected to correspond to the 

Sersic 50% light radius along the major axis in the r band.
The inclination and position angle of the major axis of the galaxies in the NSA catalog were obtained from the Sersic fit to the surface brightness profile in the r band that is available in the NSA catalog.






§ GAS-PHASE AND STELLAR METALLICITIES




We compare the stellar and gas-phase metallicities within the central region (R < 0.5 R_e) and beyond (R_e < R < 1.5 R_e) here to explore potential differences in the behavior in the central area and in the galaxy disk.

We assumed the stellar metallicities of MaNGA galaxies from the Firefly MaNGA Value-Added Catalogue <cit.>, which were built using the full-spectrum fitting code Firefly <cit.> on the MaStar stellar library <cit.>. The calibration used spectral templates computed using the ATLAS9 model-atmosphere grid <cit.>, which employs the updated solar abundances from <cit.>. The FF VAC presents two variants corresponding to the stellar population models MaStar <cit.> and M11-MILES <cit.>. <cit.> showed in their Fig. 6 that MILES and MaStar provide similar luminosity-weighted metallicities, whereas the FF VAC mass weighted metallicities from MaStar are higher on average than those in MILES, in particular toward the lower metallicity range.

We derived stellar metallicities within each radial bin as the median of the light-weighted metallicities in Voronoi cells, with cell centers falling within the respective bin. Because the MaNGA Firefly VAC expresses the metallicity as Z_star = [Z/H], it does not need to be rescaled to the solar metallicity. We compared stellar and gas-phase metallicities and therefore adopted luminosity-weighted stellar metallicities for this investigation. In this context, the luminosity-weighted metallicity proves more representative than the mass-weighted metallicity because it captures the metallicity of younger stellar populations, while the mass-weighted metallicity reflects that of older stars better.

The MaNGA Firefly VAC exists in two versions. One version employs stellar population models by <cit.>, based on the MILES stellar library <cit.>. The other version employs the MaStar models described in <cit.>. We considered stellar metallicities obtained using both sets of stellar population models. This approach facilitates the identification of potential systematic uncertainties caused by the application of distinct stellar population models.

The gas-phase metallicity was traced through the gas-phase oxygen abundance, denoted as 12 + log(O/H). The calculation of this abundance employed the empirical R calibration <cit.>. Subsequently, the oxygen abundance was converted into the gas-phase metallicity, denoted as Z_gas = [O/H], by scaling the measured O/H ratio to the oxygen abundance of the Sun, 12 + log(O/H)_⊙ = 8.69 <cit.>. While this conversion from the oxygen abundance to the total metallicity might be influenced by [O/Fe] variations across the galactic disk, <cit.> quantified these variations and found that they remain below 0.05 dex on average along the galactic radius. It is known that metallicities calculated by different calibrations may show some scatter. However, it has been shown that the oxygen abundances derived by R calibration agree with the oxygen abundances derived by the direct T_e method, which is considered to provide an accurate gas-phase metallicity <cit.>. 


For the further analysis, we selected only galaxies with available measurements of the gas and stellar metallicity. Therefore, our final sample consists of 3659 galaxies.




















§ DISCUSSION




According to the closed-box model of galaxy evolution, the metallicity is expected to increase over time, and stars are assumed to inherit the metallicity of the gas that is present at the time of their formation. Thus, the closed-box model posits that the stellar metallicity cannot surpass the gas metallicity. However, the dynamic processes of gas inflow and outflow have the potential to disrupt this simple relation.

In Fig. <ref> we depict the difference between gas-phase metallicity and light-weighted stellar metallicity, denoted as Z_gas - Z_star within the inner R < 0.5 R_e and the intermediate R_e < R < 1.5 R_e regions of our MaNGA galaxy sample. This difference is plotted as a function of stellar mass. We found that intermediate-mass galaxies exhibit a comparable gas phase and stellar metallicities in the central and intermediate part of the disk on average. This reinforced our analysis. Conversely, massive galaxies, characterized by log(M/M_) ⩾ 10, exhibit a systematically lower gas-phase metallicity compared to the stellar metallicity within the central region of a galaxy, as was reported for some galaxies of the SAMI survey in <cit.>. In other words, the value of Z_gas - Z_star is negative and reaches -0.18 dex on average for the most massive galaxies. 
This demonstrates a monotonous diminishing trend with increasing stellar mass. Importantly, this trend remains consistent regardless of whether the stellar metallicity was derived based on MaStar or MILES stellar population models. While the Z_gas - Z_star within the intermediate disk region is negative on average, the deviation is subtle and is nearly constant within the scatter for galaxies with log(M/M_) ⩾ 9.25. 
Therefore, the Z_gas - Z_star dip shown by some massive galaxies in the SAMI sample <cit.> for an integral metallicity within R_e would be exptected to form in the very central part of the galaxies and become mild at R_e. For intermediate-disk regions, a very slight reduction in Z_gas - Z_star is also apparent at the lower stellar mass end.
We concentrate our investigation below on the chemical abundance within the central regions of MaNGA galaxies.




 §.§ Systematics and selection effects


In addition to the inflow of low-metallicity gas and the outflow of enriched gas, three mechanisms have the potential to cause a decrease in the observed gas-phase metallicity: i) the nonlinear relation between the oxygen abundance and Z_gas, ii) the selection effect at the highest metallicity, and iii) the depletion of certain chemical elements in dust.
Consequently, our initial evaluation focused on assessing the influence of these mechanisms on our data. This assessment aims to ensure that the observed negative Z_gas - Z_star in our sample is free from observational biases.

We define the stellar metallicity as the abundance of all heavy chemical elements (in relation to the solar chemical abundance). For the gas phase, we adopted O/H (relative to solar O/H) as a proxy for the metallicity. However, it is well known that the abundance ratios of certain chemical elements can change with metallicity. One prominent example is the O/H - N/O relation <cit.>. This phenomenon can lead to a slight underestimation of the total gas-phase metallicity in the high-metallicity range, where N/H increases more rapidly than O/H. Consequently, Z_gas - Z_star is expected to correlate with the N/O ratio. If this effect plays a significant role in our case, Z_gas - Z_star is expected to correlate with the N/O ratio for massive galaxies. In Fig. <ref>, however, the median N/O ratio is color-coded in bins of different masses and Z_gas - Z_star does not show a correlation with the N/O ratio in massive galaxies where Z_gas - Z_star is negative.

The MaStar stellar library uses a fixed solar [α/Fe] ratio. <cit.> noted that these models are “qualitatively – [α/Fe]-enhanced” below solar metallicity, but are scaled solar from Z_ and above. We can investigate the effects of the departures from solar partition in the stellar metallicity derivation in the following way. The total metallicity can be evaluated from the iron abundance through a direct formulation when [O/Fe] is taken into account, for instance, [Z/H] = [Fe/H] + 0.93[O/Fe] <cit.>, because the total metallicity is driven by the oxygen abundance, the [α/Fe] ratio can be expressed as [O/Fe]. Since O and Mg follow the same general enhancement and similar distributions of [O/Fe] and [Mg/Fe], we can make a sampling of [Mg/Fe] derivations. In a recent paper by  <cit.>, the derived [Mg/Fe] values range from 0.06 to 0.22 dex for a sample of star-forming nearby galaxies with stellar masses 8 ⩽log(M/M_) ⩽ 10. For more massive local galaxies with 10 ⩽log(M/M_) ⩽ 11.6, the derived [Mg/Fe] appears to increase from 0 to ∼ 0.3 dex <cit.>. The [Mg/Fe] derivations in the Milky Way show comparably large enhancements associated with the bulge and the thick disk, but the [Mg/Fe] values in the thin disk are always much lower and close to zero from the solar metallicity and above <cit.>.
Therefore, we assumed a face value of ∼ 0.2 dex for an [Mg/Fe]  
average in massive galaxies, which, according to the formula for [Z/H] above, would imply ∼ 0.2 dex to be added to stellar [Fe/H]. This might increase the absolute value of the negative Z_gas - Z_star difference shown by the highest-mass galaxies.


It is also possible that our sample lacks galaxies with the highest gas-phase metallicity because these galaxies are very gas-poor. Consequently, the emission lines that are needed to estimate the oxygen abundance might be too faint to detect. In this scenario, the average Z_gas - Z_star for massive galaxies might be expected to increase when the signal-to-noise ratio (S/N) threshold for emission lines in the sample were lowered. To investigate this, we considered three samples of galaxies with emission lines with an S/N > 5, S/N > 3, and S/N > 10, and we compared the average Z_gas - Z_star for massive galaxies. We observed no substantial differences in the median values of Z_gas - Z_star for massive galaxies with various S/N thresholds. Thus, we conclude that this selection effect does not play a significant role in our case.

Another effect that might account for the decrease in the gas-phase metallicity is the depletion of oxygen and other heavy elements into dust. At solar metallicity, the oxygen depletion into dust could reach up to ≈ 0.12 dex and is expected to be weaker for galaxies with lower metallicity <cit.>. Consequently, Z_gas - Z_star might be underestimated by up to ∼ 0.1 dex. 
In this case, star-forming regions with enhanced depletion rates should be linked to higher amounts of dust, leading to higher extinction or color excess E(B-V) in the light from HII regions. Depletion does not impact stellar metallicity because dust grains are destroyed during star formation, releasing all heavy elements into the stellar atmosphere. Therefore, if depletion significantly contributes to the decrease in Z_gas - Z_star, an anticorrelation between E(B-V) and Z_gas - Z_star is expected. However, in Fig. <ref>, our sample does not show any significant correlation between Z_gas - Z_star and E(B-V) in the central regions of the massive and low-mass galaxies, where the decrease in averaged Z_gas - Z_star is most pronounced. This fact argues against a significant influence of dust depletion on Z_gas - Z_star, although it does not rule out a 0.1 dex offset in Z_gas - Z_star.




 §.§ Connection with the physical properties of a galaxy


The inflow of low-metallicity gas can also contribute to the decrease in gas-phase metallicity while increasing the gas mass fraction. The integral amount of neutral hydrogen (HI) for some MaNGA galaxies in the HI-MaNGA program is available as estimates, which involves 21 cm radio observations with the Green Bank Telescope and data from the Arecibo Legacy Fast ALFA (ALFALFA) catalog <cit.>. Assuming that the mass fraction of helium and heavy elements in the interstellar medium (ISM) is 25%, the gas fraction can be defined as

    μ = 1.25 M_ HI/(1.25 M_ HI + M_ star),

where M_ HI is the mass of HI, and M_ star is the stellar mass. 

In Fig. <ref>, we illustrate the relation between Z_gas - Z_star, M_ star, and μ. Our findings show no significant enhancement of μ at low Z_gas - Z_star values. Instead, μ tends to be slightly lower for massive galaxies with lower Z_gas - Z_star. This observation argues against attributing Z_gas - Z_star solely to the massive infall of low-metallicity gas from regions outside the galaxy. It also implies that the total amount of HI in the galaxy is not the key factor for lowering the gas metallicity compared to the stellar component. However, this does not rule out the possibility of a gas redistribution within the ISM and circumgalactic medium (CGM).

Nevertheless, for galaxies with log(M/M_) > 9.75, a correlation exists between the specific star formation rate (sSFR) and Z_gas - Z_star, where galaxies with a higher sSFR tend to exhibit higher Z_gas - Z_star values in the center, as shown in Fig.<ref>. Simultaneously, Fig.<ref> illustrates that the global u-r color index of a galaxy increases as Z_gas - Z_star decreases for galaxies with log(M/M_) > 9.75.

Comparing Fig.<ref> and Fig.<ref>, we can assume an anticorrelation between sSFR and E(B-V) for galaxies with stellar masses of about log(M/M_) = 10. However, the values of E(B-V) and sSFR in these figures are derived in bins of a given Z_gas - Z_star and log(M/M_). To test whether this anticorrelation between E(B-V) and sSFR is physical or just a result of the projection in multidimentional parameter space, we add Fig.<ref>, which does not involve Z_gas - Z_star and compares E(B-V, sSFR, and log(M/M_) directly. Figure <ref> does not show a global anticorrelation between E(B-V) and sSFR. However, the correlation between E(B-V) and sSFR in a narrow stellar mass range around log(M/M_) = 10 is only mild.


To summarize the trends observed in both figures, it becomes evident that Z_gas - Z_star is diminished in the central regions of galaxies exhibiting redder colors and lower sSFR at the current epoch. These galaxies have experienced mild star formation over the previous ∼ 1 Gyr, resulting in their reddened color. One plausible explanation for a negative Z_gas - Z_star in this case is that high-metallicity gas has been consumed during active star formation in the center first due to the inside-out growth of the galaxy. Additionally, it might have been expelled from the centers of massive galaxies through stellar or AGN feedback. Consequently, radial gas flow or accretion from the CGM could have refilled this central cavity with gas of lower metallicity.

The process of radial gas flow of gas toward the center could lead to a reduced gas-phase metallicity in the center due to the negative radial metallicity gradient typical for disk galaxies. In the case of infall from the CGM, low-metallicity gas should be distributed throughout the galactic radius. However, the resulting metallicity at a given radius would depend on the quantity of higher-metallicity gas at the specific galactocentric radius where the infalling gas is diluted. As a result, the centers of massive galaxies, characterized by a lower HI amount, might maintain the low metallicity of the infall gas better than less massive galaxies and the outer regions of the disk, where the amount of higher metallicity in-situ gas is higher.

This scenario finds support from the analytical chemical evolution model proposed by <cit.>. They found that a model without gas outflow but with gas inflow could produce a negative Z_gas - Z_star, reaching up to ∼ 0.3 dex at 
a high stellar-to-gas mass ratio, typical for the centers of massive galaxies, for a model with a gas mass accretion factor ω = 1. This result might initially seem contradictory to our previous conclusion that Z_gas - Z_star is not dependent on μ in our galaxy sample. However, it is important to note that the gas inflow in this context may originate from the CGM or from the outer disk, meaning that the total gas mass within the galaxy is conserved.







§ SUMMARY AND CONCLUSIONS




We analyzed the sample of galaxies from the MaNGA DR17 survey in order to measured the difference between the gas-phase and stellar metallicity, denoted as Z_gas - Z_star. Our final sample of galaxies with available measurements of the gas and stellar metallicity comprised 3659 galaxies.
The gas-phase metallicity was derived using the R calibration, and the luminosity-weighted stellar metallicity was obtained from the MaNGA Firefly VAC. The main results of our analysis are listed below.




  * In our galaxy sample, Z_gas is on average approximately equal to Z_star in the centers of intermediate-mass galaxies. Conversely, Z_gas - Z_star becomes slightly negative for low-mass galaxies and significantly negative in the centers of massive galaxies, with an average difference of up to -0.2 dex.


  * Beyond the effective radius, a mild decrease in Z_gas - Z_star with stellar mass is observed for intermediate- and high-mass galaxies. This trend becomes significant for low-mass galaxies, with an average difference of up to -0.3 dex.


  * The most prominent instances of negative Z_gas - Z_star in the centers are prevalent in massive red galaxies with a low sSFR.



Since we did not find a correlation between integral mass fraction of neutral gas and Z_gas - Z_star, we propose that the observed negative Z_gas - Z_star trend in the centers of massive galaxies could be attributed to the replenishment of gas-depleted central regions through processes such as gas radial flow or accretion from the CGM. 





We are grateful to the referee for his/her constructive comments. 

The authors gratefully acknowledge the computational and data resources 
provided by the Leibniz Supercomputing Centre (www.lrz.de). 

SDSS-IV acknowledges support and resources from the Center for High-Performance
Computing at the University of Utah.
The SDSS web site is https://www.sdss.org/www.sdss.org. 

SDSS-IV is managed by the Astrophysical Research Consortium for the 
Participating Institutions of the SDSS Collaboration including the 
Brazilian Participation Group, the Carnegie Institution for Science, 
Carnegie Mellon University, the Chilean Participation Group,
the French Participation Group, Harvard-Smithsonian Center for Astrophysics, 
Instituto de Astrofísica de Canarias, The Johns Hopkins University, 
Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / 
University of Tokyo, Lawrence Berkeley National Laboratory, 
Leibniz Institut für Astrophysik Potsdam (AIP),  
Max-Planck-Institut für Astronomie (MPIA Heidelberg), 
Max-Planck-Institut für Astrophysik (MPA Garching), 
Max-Planck-Institut für Extraterrestrische Physik (MPE), 
National Astronomical Observatories of China, New Mexico State University, 
New York University, University of Notre Dame, 
Observatório Nacional / MCTI, The Ohio State University, 
Pennsylvania State University, Shanghai Astronomical Observatory, 
United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, 
University of Colorado Boulder, University of Oxford, University of Portsmouth, 
University of Utah, University of Virginia, University of Washington,
University of Wisconsin, Vanderbilt University, and Yale University. 







 Paper Title: First measurement of the triaxiality of the inner dark matter halo of the Milky Way.

 


















   Kapteyn Astronomical Institute, University of Groningen, Landleven 12, NL-9747 AD Groningen, the Netherlands,

   




Stellar streams are particularly sensitive probes of the mass distribution of galaxies.
  
   In this work, we focus on the Helmi Streams, the remnants of an accreted dwarf galaxy orbiting the inner Milky Way. We examine in depth their peculiar dynamical properties, and use these to provide tight constraints on the Galactic potential, and specifically on its dark matter halo in the inner 20 kpc.
  
   We extract 6D phase-space information for the Helmi Streams from Gaia DR3, and confirm that the Streams split up into two clumps in angular momentum space, and that these depict different degrees of phase-mixing. To explain these characteristics we explore a range of Galactic potential models with a triaxial NFW halo, further constrained by rotation curve data.
  
  We find that a Galactic potential with a mildly triaxial dark matter halo, having p=1.013^+0.006_-0.006, q=1.204^+0.032_-0.036, M_discs=4.65^+0.47_-0.57·10^10 M_⊙ and M_DM(< 15 kpc)=1.14^+0.11_-0.10·10^11 M_⊙, is required to form two clumps in angular momentum space over time. Their formation is driven by the fact that the clumps are on different orbital families and close to an orbital resonance.  This resonance also explains the different degrees of mixing observed, as well as the presence of a dynamically cold subclump (also known as S2).
   
   This first and very precise measurement of the triaxiality of the inner dark matter halo of the Galaxy uniquely reveals the high sensitivity of phase-mixed streams to the exact form of the gravitational potential.







H.C. Woudenberg and A. Helmi
A first measurement of the Milky Way's inner dark matter halo's triaxial shape

First measurement of the triaxiality of the inner dark matter halo of the Milky Way
    Hanneke C. Woudenberg 
          Amina Helmi
          
    Received ..... .., ....; accepted ......, ....
===================================================================================



§ INTRODUCTION
 

Structure formation is hierarchical in the ΛCDM cosmological model, with cold dark matter (CDM) halos predominantly growing via mergers <cit.>. For Milky Way-like galaxies, this usually involves a small number of major mergers at early times and many minor mergers throughout their history <cit.>. While in CDM only simulations DM halos have triaxial shapes <cit.>, in hydrodynamical simulations including baryonic physics the DM halos tend to be rounder due to presence of baryons  <cit.>. In general DM halos are more spherical in the inner regions and are found to have radially varying shapes <cit.>. Further, the evolution of a DM halo's shape is influenced by its merger history and its mode of accretion <cit.>, and also by the fundamental nature of the DM particle <cit.>.

The merger history of the Milky Way has been studied in detail over the past decades, starting with the discovery of the Sagittarius dwarf galaxy <cit.> and its tidal tails <cit.>, and the discovery of the nearby Helmi Streams <cit.>. By now, we have established a picture of the Milky Way's merger history which seems to have been rather quiescent. The advent of Gaia DR2 revealed that the last major merger of the MW happened about 10 Gyr ago with the massive dwarf galaxy Gaia-Enceladus-Sausage <cit.>. The debris of a range of smaller dwarf galaxies has also been found in the inner Galactic halo <cit.>. These accreted dwarf galaxies also bring in their own globular clusters and possibly satellite galaxies <cit.>. Furthermore, more than 100 thin stellar streams have been discovered so far <cit.>. 

Such narrow distant stellar streams have been used to study the mass distribution of our Galaxy and particularly the shape of its DM halo. Examples include the modelling of GD-1 <cit.>, the disrupting Palomar 5 <cit.> and the Sagittarius Stream <cit.>. In recent years it has become clear that the determination of the shape of our DM halo is further complicated by the perturbations from, for example, the Large Magellanic Cloud <cit.>. 

Some authors have recently pointed out that non-integrability of the Galactic potential can also leave imprints on the morphology and dynamics of streams, especially on streams near or on separatrices, the transitions between orbit families <cit.>. Such streams provide us with a direct probe of the location of resonances, chaotic regions and the orbital families that are present in a gravitational system. These depend both on the characteristic parameters of the gravitational potential, such as the halo flattening, halo scale radius or disc mass <cit.>, and also on the distribution function <cit.>.

In this work, we use the detailed dynamics of the Helmi Streams to constrain the Milky Way's gravitational potential, and in particular the shape of its DM halo. The Helmi Streams, the first ever substructure identified to occupy the inner Galactic halo <cit.>, are the remnants of an accreted dwarf galaxy. Follow-up studies over the past 25 years, starting with <cit.>, have created a picture of the history and characteristics of the Streams <cit.>. The parent dwarf galaxy is thought to have a mass of about 10^8   M_⊙ and to have been accreted 5 to 10 Gyr ago <cit.>. The Helmi Streams' stars show abundance patterns that are distinct from other substructures <cit.>. Between 7 to 15 GC are thought to be associated to it <cit.>. As was noted by <cit.>, it seems curious that such a low mass galaxy would have been accreted to the inner halo at z < 1 and is now orbiting the inner Milky Way. Possible mechanisms could involve group infall, but as we shall demonstrate below, this might not be necessary and a purely dynamical explanation is likely. 

<cit.> found that the Helmi Streams split into two clumps in angular momentum space which are chemically indistinguishable, supporting a common origin. One of the clumps was found to exhibit a kinematically cold subclump <cit.> Later, the two clumps were recovered using a single-linkage clustering algorithm in Gaia EDR3 <cit.> and Gaia DR3 data <cit.>, with a gap identified in between as an underdensity. <cit.> used the existence of the gap to constrain the flattening of the DM halo density to be q_ρ = 1.2. In such a potential, the gap is long-lasting as the effective gravitational potential (provided by the sum of mass all components) is close to being spherical, and one of the clumps is on an orbital resonance. This however does not explain how the two clumps were formed, which is the question this paper will address. We do this by inspecting the orbital structure in the region of phase space occupied by the streams for a range of Galactic potential models with a triaxial DM halo. 

This paper is structured as follows.  
Sect. <ref> discusses the data and the methods used in this paper, which involves the application of a chaos indicator, the study of orbit families and orbital frequency analysis. 
In Sect. <ref> we present our analysis of the orbital properties of the Helmi Streams in a range of triaxial potentials. This leads to a method to constrain the Galactic potential in the region probed by the streams, and in particular the shape of the inner DM halo. 
In Sect. <ref>  we discuss the results of this analysis. 
We end with a general discussion in Sect. <ref> and present our conclusions in Sect. <ref>. 


§ DATA AND METHOD
 



 §.§ Generalities



We use the Helmi Streams (HS) sample of <cit.>, which was constructed by applying a single-linkage clustering algorithm <cit.> on the Gaia DR3 <cit.> RVS sample for a selection of halo stars in a local volume of 2.5 kpc. This sample consists of 319 members with distance errors < 20%. The distances to the stars were obtained by inverting their parallaxes after correcting these for a zeropoint offset following <cit.>. 

To convert to Galactocentric cartesian and cylindrical coordinates[We use a right-handed Galactocentric cartesian coordinate system and denote the coordinates as (x, y, z, v_x, v_y, v_z). The GC is at its centre, x increases in the disc plane away from the location of the Sun, meaning the Sun is located at negative x, y points in the direction of Galactic rotation and z points towards the North Galactic Pole. Galactocentric cylindrical coordinates are denoted as (R, ϕ, z, v_R, v_ϕ, v_z).], we follow the assumptions of <cit.> and <cit.>, as we will use their rotation curve data later in this work, which requires consistency. Hence, we assume  a height above the midplane of z_⊙ = 0.025   kpc <cit.> and a distance from the Sun to the Galactic Centre of R_⊙ = 8.122   kpc <cit.>, which is consistent with more recent measurements of orbits around Sgr A^* <cit.> and other dynamical measurements <cit.>. To correct for the motion of we Sun, we use the proper motion of Sgr A^* by <cit.>, μ_l,Sgr   A^* = -6.379 ± 0.026 mas yr^-1, μ_b,Sgr   A^* = -0.202 ± 0.019 mas yr^-1, and the relation that 


    v_j = 4.74057  km   s^-1( μ_jmas   yr^-1) ( dkpc)


where j = (l,b) in this case. This gives v_y, ⊙ = 245.6 km   s^-1 and W_⊙ = v_z, ⊙ = 7.8  km   s^-1. We assume U_⊙ = v_x, ⊙ = 11.1  km   s^-1 <cit.>, which is consistent with a range of more recent estimates <cit.>. The z component of the angular momentum vector, L_z, is defined such that it is positive for prograde orbits, i.e. its sign is flipped. 

We use  <cit.> for orbit integration and to compute dynamical quantities. Our fiducial MW model follows the <cit.> potential with some modifications. The <cit.> axisymmetric potential model consists of a bulge, an exponential thin and thick disc, an HI gas disc, a molecular gas disc and a spherical NFW DM halo. Instead of the 3 kpc of the original model, in this work, we  set a thick disc scale length of R_thick = 2 kpc[A value of R_thick= 2 kpc is in agreement with the findings of <cit.> and <cit.> and references therein. Such a thick disc thus has a smaller scale length than the thin disc, R_thin = 2.5 kpc, as we could expect from their formation histories <cit.>.]. We also assume a prolate DM halo with a flattening in the density of q_ρ = 1.2, following <cit.>. To constrain the mass of the stellar discs, the DM halo scale radius and density in this new fiducial model, we use recent rotation curve data with associated uncertainties derived using axisymmetric Jeans equations by <cit.> and <cit.>. The procedure followed to determine the values of the characteristic parameters is described in detail in Appendix <ref>. The resulting rotation curve and a comparison to other models from literature is shown in Fig. <ref>. Our fiducial model has a total mass within 20 kpc of M_tot(r < 20  kpc) = 2.2^+0.1_-0.1· 10^11 M_⊙, compatible with other estimates <cit.>, and v_c(R_⊙) = 233.2 ± 2.6  km   s^-1. Consequently, our V_⊙ = v_y, ⊙ - v_LSR = 12.4  km   s^-1, well in agreement with for example the work by <cit.>, <cit.> or the commonly used value found by <cit.>. 





 §.§ Review of dynamics



Integrals of motion (IoM) are quantities that only depend on a body's phase-space coordinates and are constant along an orbit. In a time independent potential, the total energy E = E_kin + E_pot is an IoM. In a spherical potential, all components of the angular momentum vector, L⃗ = r⃗p⃗ = (L_x, L_y, L_z), are IoM, and hence so is the perpendicular angular momentum vector component, L_^2 = L_x^2 + L_y^2. In an axisymmetric potential, symmetry with the polar angle θ is broken and therefore only L_z = R v_ϕ remains an IoM, though L_ is sometimes used to characterise orbits (as a proxy for a third integral). In a triaxial potential, also symmetry with respect to the azimuthal angle ϕ is broken and none of the components of the angular momentum vector are IoM. 

A regular orbit is quasi-periodic, and the time series of its phase space coordinates w(t) = (𝐱(t), 𝐯(t)) can approximated by the sum


    w(t) = ∑_k^∞ a_k e^i ω_k t,


where ω_k are orbital frequencies and a_k the associated amplitudes. Therefore, a Fourier transform of the orbit will give us a spectrum with peaks at ω_k of amplitude a_k. A regular orbit can be described by its three independent fundamental frequencies, meaning that all other peaks ω_k in the spectrum are a linear combination of those,  ω_k = n⃗_⃗k⃗·Ω⃗, with n⃗ is a vector of three integers. If the fundamental frequencies are commensurable, i.e. one is a ratio of small integers of the other, n⃗·Ω⃗ = 0, the orbit is said to be resonant. A singly-resonant orbit is confined to a two-dimensional surface, while a doubly-resonant orbit traces a closed one-dimensional curve. A sufficiently tight ensemble of particles can be resonantly trapped by a resonant orbit, meaning that their orbits have a finite libration amplitude around the resonance <cit.>. Depending on how strongly trapped the particles are, and thus how small the libration amplitude is, they will occupy a subspace of phase-space around the resonant orbit. While an ensemble of particles on regular non-resonant orbits phase-mixes at a rate ∝ t^-3, where t is time, an ensemble of particles trapped by a resonant orbit phase-mixes slower, at a rate ∝ t^-2 for a singly-resonant orbit and at a rate ∝ t^-1 for a doubly-resonant orbit <cit.>. 



At the edge of a family of resonant orbits we find a separatrix, which denotes a transition between orbit families. A separatrix is a stochastic region[Stochastic regions are found in near- or non-integrable potentials, and generally realistic galactic potential models are of this type.] hosting chaotic orbits. Such orbits are not quasi-periodic and do not have three IoM. Substructures on chaotic orbits therefore diffuse and mix faster than substructures on regular orbits <cit.>. Moreover, stellar streams on separatrices can exhibit unusual morphologies and similarly undergo faster diffusion <cit.>. Naturally, triaxial potentials host a wider range of orbits, and consequently more separatrices and more chaotic orbits than an axisymmetric potential <cit.>. However, also axisymmetric potentials can host chaotic orbits <cit.>, including simple models like the Miyamoto Nagai potential <cit.>. The location of chaotic regions, resonances, separatrices and different orbit families can be analysed using e.g. the orbital frequencies <cit.>, and these will of course depend on a potential's characteristics <cit.>.




 §.§ Characterising the Helmi Streams



Figure <ref> shows the distribution of the HS stars in IoM (E, L_z, L_), velocity and configuration space. The top left panel shows that the HS separate in two clumps in (L_z, L_) space, as first identified by <cit.>. We empirically separate the sample into the hiL clump, having larger L_, and loL clump, having lower L_, and classify a star with a given L_, * and L_z, * as follows


    L_, *{[ > 0.35 · L_z, * + 1525  kpc   km   s^-1                                       →; < 0.35 · L_z, * + 1525  kpc   km   s^-1                                      →, ].


see also Fig. <ref>. There are 191 and 128 stars in the hiL and loL clumps (respectively)[The reason that there are more hiL stars than loL stars, a ratio of ∼ 2:3, could be caused by the fact that the loL clump is located closer to the disc in integrals of motion space, which results in a stronger background, possibly making it harder for the clustering algorithm to pick up the structure.]. The median position of the hiL stars is (L_z, L_) = (1234, 2207)   kpc   km   s^-1, while the median position of the loL stars is  (L_z, L_) = (1436, 1791)   kpc   km   s^-1. We confirm that these two clumps have consistent stellar populations and metallicity distributions, with a median metallicity of -1.45 dex and standard deviation of 0.42 dex using LAMOST LRS DR7 metallicity estimates <cit.>.

The lower left panel of Fig. <ref>, displaying the (E, L_) distribution, shows that on average the hiL stars are less bound and the loL stars span a larger range in energy. The hiL stars have larger apocentres and reach higher above the midplane, see also Fig. <ref>. This is robust for different Galactic potential models, though of course the exact apo- and pericentre distribution changes.  



The two middle panels of Fig. <ref> show the velocity distribution of the two clumps. Interestingly, the ratio of stars with positive v_z, to  negative v_z differs between the two clumps. For the hiL clump this ratio is N_hiL(v_z^+)N_hiL(v_z^-) = 13178∼ 0.07, while for the loL clump this ratio is N_loL(v_z^+)N_loL(v_z^-)= 4979∼ 0.62. In earlier work, the bimodality in v_z of the entire HS has been used to obtain a rough estimate for the time of accretion <cit.>, since it is an indication of a structure's degree of phase-mixing. The hiL clump not only appears to be less phase-mixed, but it forms a tighter, more coherent structure in velocity space than the loL clump stars. The hiL clump even exhibits a kinematically cold subclump (in black in Fig. <ref> and Fig. <ref>), and defines a stream-like structure in configuration space, as the two right panels of Fig. <ref> show. 

We select the subclump as 1320 ≤ L_z ≤ 1410   kpc   km   s^-1, 2330 ≤ L_≤ 2500   kpc   km   s^-1 and x > - 9.2 kpc and find 9 members. Four of these stars have LAMOST LRS DR7 metallicity estimates <cit.>, which are [Fe/H] = -1.77, -2.04, -1.48 and -2.19 dex, with typical uncertainties of 0.1 dex <cit.> to ∼ 0.3 dex at the metal-poor end <cit.>. Given these uncertainties and the metallicity distribution of the HS, the subclump is indistinguishable from the HS in its stellar populations. Although this may apparently contrast <cit.>' independent identification of the subclump in Gaia DR2, naming it S2, subsequent high-resolution spectroscopic follow-up of S2 by <cit.> also support the association. This has been convincingly demonstrated by <cit.> who studied the abundance patterns of a sample of HS stars and homogeneously reanalysed the spectra of three <cit.> stars and concluded that their abundance patterns are fully consistent.  Similarly, <cit.> identified and considers the subclump as a part of the HS. Given its much tighter distribution, the subclump must have evolved dynamically more slowly than the rest of the substructure, as it is significantly less phase-mixed. This could happen if the dynamical evolution of the subclump is affected by a resonance.




 §.§ Orbital characterisation: frequency analysis and chaos indicators


In cylindrical coordinates, we typically use three orbital frequencies (Ω_R, Ω_ϕ,Ω_z) which are related to the orbit's oscillation in the radial, vertical and azimuthal directions. For a regular orbit, the components of its frequency spectrum are linear integer combinations of the three orbital frequencies (Ω_R, Ω_ϕ, Ω_z)[This set of three orbital frequencies is not necessarily equal to the three fundamental orbital frequencies discussed in Sect. <ref>.]. For a resonant orbit, two or more frequencies are commensurate, while for a chaotic orbit the orbital frequencies vary with time. To determine the orbital frequencies corresponding to a star's orbit, we use a modified version of  <cit.>, as described in detail in Appendix <ref>.

In contrast to regular orbits, a chaotic orbit 
has less than 3 integrals of motion, and as a consequence it undergoes orbital diffusion. Therefore, if the MW's potential hosts a stochastic region in between the two HS clumps, this could possibly lead to a depletion of stars in that region. To identify chaotic behaviour, we make use of the Lyapunov exponent Λ. Take an orbit x⃗(t) and an orbit infinitely close to it, x⃗(t) + w⃗(t). Here, w⃗(t) is the so-called deviation vector, which grows as a power-law in time for a regular orbit, but exponential in time for a chaotic orbit. The Lyapunov exponent Λ measures the time variation of w⃗(t),


    Λ≡lim_t→∞ln | w⃗|t


To compute Λ, we resort to its finite-time estimate. For a period of time where the orbit is regular, ln | w⃗| / t fluctuates around a constant  value, and in that case Λ is set to zero. When an orbit is chaotic, Λ is estimated as the average value of  ln | w⃗| / t over the period of exponential growth. Thus, the larger Λ is, the more chaotic the orbit is. The Lyapunov exponent also gives an indication of the timescale of chaoticity. In this work, the finite-time estimate of Λ is computed using  over an integration time of 100 Gyr.





§ ANALYSIS
 



 §.§ The HS' orbits in a triaxial potential


To investigate how the dynamical properties of the HS can be explained, we study their orbits in a range of mildly triaxial potentials, and we explore the orbital structure in the region of phase space occupied by the HS. We explore triaxial potentials as,  on the basis of cosmological simulations, it is likely that DM halos have a triaxial shape, particularly at larger radii <cit.>. Moreover, there is evidence that Milky Way's halo is being perturbed and deformed by the infall of the Large Magellanic Cloud <cit.>.

We modify the fiducial MW model based on the <cit.> potential described in Sect. <ref> (see also Appendix <ref>) by replacing its NFW halo with a triaxial halo:


    ρ_NFW(x, y, z) = ρ_0/r̃r_s( 1 + r̃r_s)^2,


with


    r̃ = √( x^2 + y^2p^2 + z^2q^2)


and where


    p^2 = b^2a^2,     q^2 = c^2a^2   .


We make sure that the circular velocity at the position of the Sun is equal to v_c(R_⊙) = 233.2  km   s^-1 when varying (p, q)[To this end, we update the initial density ρ_0,i of a DM halo with a given  p, q to ρ_0, new, while we keep the other components of the potential fixed. We use that v_c(R_⊙)^2 = v_c, baryons(R_⊙)^2 + v_c, halo(R_⊙)^2, where v_c, baryons(R_⊙) is the contribution of all baryonic components to v_c(R_⊙), which is fixed, and v_c, halo(R_⊙) is the contribution of the DM halo. Since v_c, halo(R_⊙) follows the proportionality v_c, halo(R_⊙)^2 ∝ρ_0, we compute the density ρ_0, new = 233.2^2 - v_c, baryons(R_⊙)^2v_c, halo^2(R_⊙)ρ_0,i.]. We explore the range 1.05 ≤ q ≤ 1.35 and 0.90 ≤ p ≤ 1.10, which is motivated by disc stability constraints <cit.> and the requirement that to maintain the gap we need an effective potential that is roughly spherical <cit.>. To study how varying p and q influences HS' orbits, we integrate these for 100 Gyr and compute (1) the finite-time estimate of the  Lyapunov exponent to study chaoticity, (2) we quantify the variation in L_z and L_y for each orbit to distinguish different types of orbit families, and (3) we determine the orbital frequencies Ω_R, Ω_ϕ and Ω_z as outlined in Appendix <ref> to identify possible orbital resonances. The average/median values of these quantities for the loL stars, hiL stars and subclump stars are shown in the left, central and right columns of Fig. <ref> respectively. We discuss the behaviour of each dynamical quantity in the following paragraphs.



  §.§.§ Chaoticity: the Lyapunov exponent


The average Lyapunov exponent of the hiL stars, loL stars and subclump are shown separately in the form of a map in the top row of Fig. <ref>. There is a band of chaoticity (with light green-yellow corresponding to Λ≳ 0.04) in the region q ∼ 1.13, p ∼ 0.90, till q ∼ 1.25, p ∼ 0.97 for the loL stars, while for hiL stars the chaoticity band occupies a  narrower region from q ∼ 1.05, p ∼ 0.90  till  q ∼ 1.15, p ∼ 0.97, and an even narrower region with chaotic behaviour is present for q ∼ 1.05, p ∼ 0.90,  till  q ∼ 1.13, p ∼ 0.97 for the subclump stars. The band of chaoticity is thus located at larger q for the loL stars, and it also appears broader. This is due to the loL stars occupying a larger (and different) volume in phase space than the hiL stars, see Fig. <ref>. Interestingly, there is no analogous chaotic region for potentials with p > 1.



  §.§.§ y-tube and z-tube orbits and variations in L_z and L_y


We can get more insight into what causes the band of chaoticity by inspecting the orbits of the HS stars in the different potentials, of which examples are shown in Fig. <ref> (see also Appendix <ref>). The tube orbit family comprises regular orbits which fill a doughnut-like shaped volume in configuration space. Tube orbits have a sense of rotation around the principal axes of the potential, and depending on which axis they are called short- or inner/outer long axis tube orbits <cit.>. As we vary p and q, the long and short axis swap orientation[For example, in a simple axisymmetric case, z is the short axis if q < 1, while z is the long axis if q > 1.]. Therefore, we choose a more general naming convention and define the z-axis tube orbits (z-tube orbits) as rotating around the z-axis, while the y-axis tube orbits (y-tube orbits) are defined as rotating around the y-axis. Tube orbits are centrophobic, i.e. they do not pass through the centre of the potential, which results in a non-zero time-averaged angular momentum about whichever axis they rotate. This can also be appreciated from Fig. <ref> and <ref>, which shows that the z-tube orbit rotates around the z-axis, keeping its L_z roughly conserved in time, while its L_y varies by a large amount with a time-average equal to zero. On the other hand, the y-tube orbit revolves around the y-axis, keeping its L_y roughly conserved, while its L_z varies by a large amount with a time-average equal to zero.

A way to differentiate between these two orbit families is thus by inspecting the orbit's variation in L_z and L_y, which we define as


    Δ L_i = max (L_i) - min (L_i)


where i refers to the component of the angular momentum, in this case y or z, and Δ L_i is calculated over at least one period in angular momentum space (e.g. the period of the z-tube orbit shown in Fig. <ref> is about 25 Gyr). Figure <ref> illustrates the expected range of variation in L_y and L_z: the z-tube orbit has Δ L_z ≈ 500   kpc   km   s^-1 and Δ L_y ≈ 4500   kpc   km   s^-1, while the y-tube orbit has Δ L_z ≈ 3000   kpc   km   s^-1 and Δ L_y ≈ 500   kpc   km   s^-1. The bottom row of Fig. <ref> shows a chaotic orbit without a third IoM which has Δ L_z ≈Δ L_y ≈ 4500   kpc   km   s^-1.




The second row of Fig <ref> shows a map of the (Δ L_z) for the hiL stars and loL stars for different p and q, while the third row of Fig. <ref> shows the (Δ L_y). These maps clearly show the regions occupied by the z-tube and y-tube orbits. A comparison of the top three rows of Fig. <ref> reveals that the chaotic band coincides with the transition between these two orbit families. In this chaotic band, (Δ L_z) is larger, as expected. The maps also show that potentials with DM halo shapes p >1 and p <1 have a roughly similar behaviour and largely mirror each other, but for p > 1 there is a region with a lower (Δ L_z) that is not present in p < 1. This region corresponds to an orbital resonance that traps orbits, as we discuss in the next section.

Summarising, Fig. <ref> shows that there is a part of (p, q) parameter space where the hiL stars are on y-tube orbits, while the loL stars are on z-tube orbits. This holds roughly from (p, q) = (0.9, 1.10) till (p, q)= (0.98, 1.17) and from (p, q) = (1.01, 1.20) till (p, q)=(1.10, 1.05). In such a configuration, we would expect that the hiL clump stars slowly change their L_z over time, while the loL clump's L_z stays roughly conserved, leading to a separation of the two clumps over time. Such behaviour could therefore  possibly explain the formation of two (the hiL and loL)  clumps.




  §.§.§ Orbital frequencies and resonances



To investigate further the transition in orbit family at these specific p and q, we inspect the behaviour of the orbital frequencies Ω_R , Ω_ϕ, and Ω_z. The bottom row of Fig. <ref> shows a map of the ratio of Ω_ϕ:Ω_z for the values of p and q explored in the other panels. This reveals that the change in orbit family seen in the two middle rows is related to the Ω_ϕ : Ω_z = 1:1 resonance. For p < 1, the chaotic band neatly overlaps with the white dotted
line indicating the resonance. Therefore, the chaotic behaviour seen in the top row is due to the stochastic layer around the 1:1 resonance. This happens for p < 1 because the short and long axis of the potential are exchanged: for an oblate effective potential, z is the short axis, while for p <1 and a prolate effective potential, the short axis is y. Because of this, the 1:1 resonance works as a separatrix: stars on this resonance do not know whether to rotate around the z-axis or y-axis, as the chaotic orbit in Fig. <ref> shows. This orbit seems to jump between a y-tube orbit (0-10 Gyr), z-tube orbit (10-20 Gyr) and a resonantly trapped orbit (20-25 Gyr).



On the other hand, for p >1, the region that has a lower (Δ L_z) overlaps with the 1:1 resonance, and in this case the resonance actually stabilises the orbits and resonantly traps them. These resonantly trapped orbits have angular momenta both around the z and y-directions, and orbit in a plane in (y, z), see also Fig. <ref> and Appendix <ref>. Their L_y, L_z and L_ are almost constant in time, as seen in Fig. <ref>, indicating that in this region the effective potential is roughly spherical.

Given these findings, a picture emerges where the loL and hiL clump could be on different orbital families. This would require a potential with (p, q) values such that the hiL clump is on the y-tube orbits and the loL clump on the z-tube orbits. Furthermore, the presence of the kinematically cold subclump in the hiL clump suggests that it is associated with a stabilising orbital resonance, as present for p >1, as this will slow down phase mixing. From these considerations, it naturally follows that the subclump could be on the Ω_ϕ : Ω_z = 1:1 resonance, while (part of) the hiL clump would be resonantly trapped by that same resonance. The loL clump would not be on the 1:1 resonance, and would phase-mix at a normal rate as this would explain the asymmetries reported in Sec. <ref> regarding the number of stars in the streams with positive and negative z-velocities. The opposite effect would occur for p<1, as the orbits are chaotic close to the Ω_ϕ : Ω_z = 1:1 resonance and undergo quick orbital diffusion, and a kinematically cold subclump could not be sustained. This is why p < 1 is disfavoured.





  §.§.§ Resonance trapping



We study the effect of resonance trapping by the Ω_ϕ : Ω_z = 1:1 resonance in more detail in Fig. <ref>. We integrate a set of orbits in a potential with a triaxial halo with p = 1.02, q = 1.19, as in this potential the subclump stars are strongly resonantly trapped (see for example Appendix <ref>). The orbits' initial conditions are generated by varying (L_z, L_) on a grid and using the energy and 3D position of a subclump star. We inspect the variation in angular momentum,  Δ L_y, z = √(Δ L_y^2 + Δ L_z^2), to quantify how strongly trapped an orbit is. Orbits on the Ω_ϕ : Ω_z = 1:1 resonance have a roughly conserved L_y and L_z, while z-tube orbits and y-tube orbits do not conserve their L_y and L_z, respectively (see Fig. <ref>). Hence, a small value of Δ L_y, z indicates that an orbit is resonantly trapped, allowing us to probe the extent of the resonance, as we show in Fig. <ref>. 

Figure <ref> shows that the resonance covers a region that is as large as the hiL clump, though not each orbit is as strongly trapped as the other. The smaller the variation in angular momentum Δ L_y, z is, the more strongly resonantly trapped the orbit is, as can be seen by comparing the left and right panels of Fig. <ref>. The right panel shows the (y, z) projection of the orbits over an integration time of 20 Gyr, and reveals that resonant orbits define a 2D planar structure in (y, z) space. 





  §.§.§ The hiL and loL clump on different orbit families



To understand in which potential the individual hiL and loL clump stars are on the y-tube, z-tube or resonantly trapped orbits, we compute Δ L_z and Δ L_y for each individual HS star in the potentials for the range  1.00 < p < 1.04 and 1.13 < q < 1.23. Based on visual inspection of a large range of orbits, we employ the following empirical criteria based on the orbit's Δ L_z and Δ L_y (see Fig. <ref> and Sect. <ref>) to call them:


    [ ;  ]
    = {[ if   [Δ L_y  > 1200    kpc   km   s^-1]  &;                           [Δ L_z  < Δ L_y]; if   [Δ L_z  > 1200    kpc   km   s^-1]  &;                           [Δ L_z  > Δ L_y];  if   [Δ L_y  < 400    kpc   km   s^-1]  &;          [Δ L_z  < 400    kpc   km   s^-1];  if   [Δ L_y  < 750    kpc   km   s^-1]  &;          [Δ L_z  < 750    kpc   km   s^-1]; if   [Δ L_y  < 1200    kpc   km   s^-1]  &;         [Δ L_z  < 1200    kpc   km   s^-1];                                            ].


Figure <ref> shows the result. For axisymmetric potentials (i.e. p =1), Δ L_z is equal to zero for all stars for, as expected, but for p > 1 the stars occupy different regions of the (Δ L_z, Δ L_y) space, meaning they are on different orbit families. While the loL stars are mainly on z-tube orbits, i.e. they largely conserve their L_z, the hiL stars are mainly resonantly trapped or are on y-tube orbits. The subclump stars are naturally also either resonantly trapped or on y-tube orbits. Hence, this figure illustrates that it is possible to have the HS stars on different orbit families in the same Galactic potential, even though they belong to the same accreted parent satellite (see also Appendix <ref>).

Following our findings, we require the hiL stars to be on y-tube orbits, or possibly being resonantly trapped, we require the loL stars to be on the z-tube orbits, and we require the subclump to be strongly resonantly trapped on the Ω_ϕ : Ω_z = 1:1 resonance. The percentages in Fig. <ref> indicate the percentage of the hiL, loL and subclump stars that are on the desired orbit family. The range of p and q for which the percentage of hiL stars on y-tube orbits, the percentage of loL stars on z-tube orbits and the percentage of subclump stars on resonantly trapped orbits is larger than 50% is for (p, q) = (1.01, 1.17-1.21) and (p, q) = (1.02, 1.18-1.20). For even larger values of q, an increasing number of loL stars end up on y-tube orbits, while for smaller q than this range a decreasing number of subclump stars are resonantly trapped. For p larger than this range, either the majority of subclump stars are not resonantly trapped or the majority of loL stars are on y-tube orbits.



 §.§ A simulated Helmi Streams-like progenitor in a mildly triaxial potential



We now investigate whether a simulated HS-like progenitor on a HS-like orbit would develop the structure seen (the hiL, loL clump) over a reasonable timescale in Galactic potentials for a range of p and q. To obtain realistic simulated HS-like phase-space positions that resemble the observations, we use  of the set of four simulated HS-like dwarf galaxies by <cit.>. This progenitor has two components: stars and dark matter. The star particles follow a Hernquist profile with a stellar mass of 10^8 M_⊙ and a scale radius of 0.585 kpc. The DM halo particles follow a truncated NFW profile with a total mass of 3.62 · 10^9 M_⊙.

To allow a good comparison to observations, we randomly select 319 star particles (the same number as stars observed in the HS) that belong to the core of  , which we define as being the 73% most bound star particles. We place the particles on a HS-like orbit by re-centring them to the position and velocity of a central HS star, which has (E, L_z, L_) = (-116246   km^2   s^-2, 1272   kpc   km   s^-1, 1871   kpc   km   s^-1) and thus lies in the middle of the HS's IoM distribution. The particles have an initial (L_z, L_) distribution that is positively correlated, meaning that particles with a larger L_z generally have a larger L_, and there is of course no gap present. We treat the particles as test particles and integrate their orbits in potentials with a range of  1.13 < q < 1.23 and 1.00 < p < 1.04, motivated by the results reported in the previous sections, over a timescale of 0 - 8 Gyr, where the upper limit is motivated by literature estimates for the accretion time of the HS <cit.>, though these estimates need to be taken with caution given our findings (see Sect. <ref> for a more in-depth discussion).

We inspect the behaviour of the particles in (L_z, L_) space over time. Of course, the particles' orbits depend on exactly how the progenitor was re-centred, but the behaviour remains qualitatively similar for re-centrings to different central HS stars. Figure <ref> shows the distribution in (L_z, L_) for q = 1.18, p = 1.02 for different snapshots and illustrates how the two clumps develop and a gap is formed over time. We see how particles on y-tube orbits separate themselves from the particles on z-tube orbits as their Δ L_z is large. Particles that are resonantly trapped (in green), which have Δ L_y,z < 1200    kpc   km   s^-1, are located at (L_z, L_) values that are roughly consistent with those of the subclump. The final (L_z, L_) distribution at 8 Gyr resembles the observed HS distribution, showing two separated clumps of stars whose relative orientation is also roughly reproduced. This separation remains if we convolve the distribution with the expected observational errors.

In Fig. <ref> we show different distributions after 8 Gyr of integration time for a few combinations of p and q (and Fig. <ref> in Appendix <ref> shows the distributions after 8 Gyr for the entire range 1.13 < q < 1.23 and 1.00 < p < 1.04). For p=1, the particles remain as a single clump, since in an axisymmetric potential they are all on z-tube orbits which have constant L_z. Instead, for p>1, we find (L_z, L_) distributions that resemble the HS' observations, with the cases of 1.18 < q < 1.21 and p ∼ 1.02 matching the most convincingly, showing two separate clumps of stars. For p > 1.02, two clumps can also be created, but the extent in  (L_z, L_) is larger than the range covered by the HS' observations, and this extent grows as p gets larger. This is because L_z changes faster in a more triaxial potential. This means that there is a degeneracy between the accretion time and the value of p, as for a more recent (earlier) accretion time, a higher (lower) value of p gives similar (L_z, L_) distributions. 

The characteristics of the progenitor determine its extent in IoM space. If we re-do our test-particle experiment using the smaller and lighter  by <cit.>, we similarly find that particles can be on different orbit families, but the extent in IoM space of the HS is not reproduced being too small. This confirms that  provides a better description of the HS. 

In summary, this experiment shows that a HS-like progenitor can evolve from a single clump in (L_z, L_) space to two clumps over time because of the local orbital structure of the Galactic potential. Consequently, this should allow us to place constraints on the inner DM halo's shape of the Milky Way using the observed HS' dynamics.







§ A CONSTRAINT ON THE MILKY WAY'S GALACTIC POTENTIAL



We can strongly constrain the effective Galactic potential, and thus the potential's characteristic parameters, in the region of phase-space probed by the HS, as the observable properties of the Streams require a specific local orbital structure. That is what we set out to do in this section.

In Sect. <ref> we found that in our Galactic potential models with (p, q) = (1.01, 1.17-1.21) and (p, q) = (1.02, 1.18-1.20) the HS stars are on orbits such that the formation of the HS clumps may be explained. However, this range of (p,q) values was obtained for a potential whose parameters were all fixed except for the shape of the DM halo. To provide a more robust estimate including uncertainties and to explore the influence of degeneracies, we now proceed to vary the disc mass, halo scale radius, halo density and p and q while fitting the rotation curve data by <cit.> and <cit.>, following the method outlined in Appendix <ref>. We simultaneously maximise the number of subclump stars that are strongly resonantly trapped by the Ω_ϕ:Ω_z = 1:1 resonance, which we translate into the requirement Δ L_y, sub  < 750   kpc   km   s^-1 and Δ L_z, sub  < 750   kpc   km   s^-1. We focus on this resonance as it is pivotal in setting the local orbital structure, and it is required to explain the existence of the kinematically cold subclump. We run a Monte Carlo Markov Chain (MCMC) using  <cit.> with 40 walkers and 4000 steps. By performing the MCMC, we can estimate how much the location of the resonance shifts by varying parameters of the potential. Moreover, the MCMC can sample the parameter space more finely than we have done so far. Our likelihood has the form


    ℒ(θ⃗) = - 101 + 100 ·N_sub,   strongly   trappedN_sub,   tot  - 12 N_RC∑_i=1^N_RC(  v_c, i^d - v_c^mσ_v_c, i)^2


where the superscript d indicates the data and m the model. The two likelihood terms have a relative importance of about 10:1, respectively. We set a simple flat prior and allow


    P(θ⃗) = {[                                                                                                                                                                                                         1 if{[                         1.0 ≤ p ≤ 1.1;                         0.7 ≤ q ≤ 1.4;                     10 ≤ r_h≤ 30  kpc;         10^5≤ρ_0, h≤ 10^8  M_⊙ kpc^-3; 3.5 · 10^10≤ M_discs≤ 5.5 · 10^10 M_⊙ ].;                                                                                                                                                                                                         0                                                                                                                                                                                                otherwise. ].


and take θ⃗_i = [p_i, q_i, r_h,i/  kpc,  ρ_0,h,i/(M_⊙  kpc^-3),   M_discs,i/M_⊙] = [1.02, 1.20, 15.7, 10 · 10^6, 4.5 · 10^10] as an initial guess, motivated by our findings and the parameters of the fiducial potential. Recall that we have fixed the discs scale-lengths and relative density near the Sun (see Appendix <ref>), which is why we consider the sum of the masses of the thin and thick discs, M_discs, as the free parameter.






 §.§ Results



The MCMC converged, though with a relatively low acceptance fraction of ∼0.27. We have removed the first 1000 steps as burn-in and steps of walkers that got stuck. Figure <ref> shows the posterior parameter distribution for p, q, M_discs and M_DM(<15kpc) in blue, where M_DM(<15kpc) is computed by integrating Eq. <ref> for a given r_h, ρ_0,h, p and q within an ellipsoid which has a minor axis of 15 kpc and axis ratios set by p and q. We constrain ourselves to 15kpc as this is the limit of our rotation curve data, and thus show M_DM(<15kpc). In this context, the degeneracy between M_discs and M_DM(<15kpc) seen in Fig. <ref> is not surprising. The other degeneracies between the parameters can easily be explained as well. The degeneracy between the disc mass and q reflects the fact that the required local orbital structure constrains the total effective potential flattening. For a larger disc mass, a larger q is required to compensate for the flattening introduced by the disc. Following a similar reasoning, a more massive DM halo implies a lower q, though this degeneracy is weaker. The degeneracy between p and q reflects the fact that the Ω_ϕ : Ω_z = 1:1 resonance is located on a diagonal line in p and q, see Fig. <ref> and Fig. <ref>. In almost all unique potentials of the chain, ∼ 99.6 %, all nine subclump stars are strongly resonantly trapped. In more than 22% of the potentials, 5 or more subclump stars are on the resonant orbit while the rest is strongly resonantly trapped.

Though this posterior parameter distribution beautifully reflects where the subclump is resonantly trapped, it does not contain information about the orbits of the hiL and the loL stars. We find that in all potentials, the hiL stars are either resonantly trapped or on the y-tube orbits, as desired, but in a majority of the potentials, especially for larger p, most loL stars are on y-tube orbits as well. Therefore, we analyse the orbits of the loL stars in each sampled potential, and add the constraint that at least 40% of the loL stars are on z-tube orbits. This constrained posterior parameter distribution is shown in black in  Fig. <ref> and shows that a low p is required, in concordance with Fig. <ref> (though in that case only p and q were varied).

We proceed now to determine our best-fit model. An inspection of Fig. <ref> shows that the posterior distribution of p is skewed and that parameters are correlated. Hence, the median or average values of the marginalised distributions do not reflect the peak of the posterior parameter distribution. Therefore, to obtain the Galactic potential model that satisfies the likelihood best, we select the global peak value of the posterior distribution. Next, we use  to obtain a kernel-density estimate of the posterior distribution, where we use the default settings but with a bandwidth (which scales the width of the kernel) equal to 2/3  to capture the finer details of the distribution[The obtained peak values are consistent when the default bandwidth  or  is used, which are determined using Scott's Rule and Silverman’s Rule, respectively. For a bandwidth equal to 1/2 , a local maximum instead of the global maximum is obtained.]. The peak values are those corresponding to the highest density and are shown with solid green lines in Fig.<ref>. Given the peak values, we then take the 31.73% (68.27%)  percentile of the data lower (higher) than the peak value as the lower (upper) 1σ uncertainty. These are indicated with dashed green lines in Fig.<ref>. Hence, we find p=1.013^+0.006_-0.006, q=1.204^+0.032_-0.036, M_discs=4.65^+0.47_-0.57·10^10 M_⊙ and M_DM(< 15 kpc)=1.14^+0.11_-0.10·10^11 M_⊙. In the best-fit potential, 5 subclump stars are on the resonance, while 4 are strongly resonantly trapped. All hiL stars are either on the resonance (9%), resonantly trapped (57%) or on y-tube orbits (21%), and 80% of the loL stars are on z-tube orbits. This is in agreement with our findings of Fig. <ref>.


§ DISCUSSION
 




 §.§ Comparison to the literature


In this work, we have constrained the Milky Way's DM halo shape and characteristic parameters, and the mass of its discs. This is the first constraint on the degree of triaxiality of the DM inner halo and it is  based on the dynamics of phase-mixed streams. Here, we compare the Galactic potential characteristics that we found to the literature. 

We constrained the discs' total mass M_discs=4.65^+0.47_-0.57·10^10 M_⊙. Given that we assumed a fixed local density ratio and fixed parameters for the scale length and scale height, this implies M_thin = 3.12^+0.31_-0.38· 10^10 M_⊙ and M_ thick = 1.53^+0.15_-0.19· 10^10 M_⊙. This is consistent with <cit.> and references therein. We also constrained the total DM halo mass within 15 kpc to be M_DM(< 15 kpc)=1.14^+0.11_-0.10·10^11 M_⊙. This corresponds to a local DM density of ρ_0, h(R_⊙, z_⊙) = 8.3^+1.1_-0.8· 10^-3 M_⊙ pc^-3 or 0.32^+0.04_-0.03 GeV cm^-3, which is consistent with the local DM densities of the models of <cit.> and <cit.>, but also with the compilation of estimates presented in <cit.> and <cit.>. The total mass within 15 kpc is M_tot(<15  kpc) = 1.80^+0.08_-0.07· 10^11 M_⊙.

The posterior parameter distribution for the DM halo scale radius r_h reveals that a value below the lower bound of 10 kpc would be preferred. This could indicate that a different DM profile is more suitable, possibly a contracted NFW <cit.>. We note that <cit.> have suggested in their fit to recent rotation curve data that a cored Einasto profile provides a better fit over a (generalised) NFW profile. Despite being cored, their Einasto profile is relatively steep with an r_-2 = 9.17 kpc, which is very comparable to the value we obtain (as r_h = r_-2 for an NFW profile).

Similar to the fiducial potential presented earlier (and discussed in Appendix <ref>), the circular velocity at the position of the Sun is 233.3^+2.8_-2.1 km   s^-1. Also the model's rotation curve, shown in Fig. <ref>, agrees with the rotation curve of the fiducial potential. Lastly, we confirm that our model agrees with the determination of the vertical force 1.1 kpc from the Galactic plane as a function of radius by <cit.>.

The HS constrain the shape and mass distribution of the effective Galactic potential, which is obtained by adding the contributions of all Galactic components. Fig. <ref> shows the isopotential contours of the effective potential of our best-fit model in different planes. To place our findings in context, we compare our results to the effective flattening of Galactic potential models from the literature. We evaluate the shape of all these models by approximating the effective isopotential at different radii by a triaxial ellipsoid and determining the axis ratios. Fig. <ref> shows q_Φ and p_Φ as a function of Galactocentric coordinate x, while Fig. <ref> shows q_Φ at a fixed value, x = 15 kpc. The comparison in the innermost regions, x < 5 kpc, is not very meaningful as the curves differ due to different assumptions on the discs, bulge and halo profile (e.g. <cit.> assumes a contracted DM halo, and <cit.> assumes a heavier bulge). The constraints obtained from fitting spatially coherent streams result most often in oblate q_ϕ-values that agree within uncertainty with each other. For x > 10 kpc most models assume a spherical DM halo, and as a consequence 
the resulting effective potentials are oblate in the region occupied by the HS. Instead, the effective flattening of our model turns prolate for x ≳ 13 kpc. Our findings agree with the work by <cit.> and <cit.>, and also <cit.>'s models turn prolate at x ∼ 18 kpc.

As the HS constrain the effective potential, we cannot distinguish with certainty whether the measured triaxiality is due to the DM halo's shape or to the effect of a perturbation such as the LMC. We can however neglect the triaxiality induced by the Galactic bar and spiral arms, as these do not strongly affect the dynamics of the HS since they orbit high above the plane (see Fig. <ref>). To investigate the effect of the LMC in the region probed by the HS, we compare p_Φ of our model to the ℒ3, ℳ11 and ℒ2, ℳ11 time-dependent Galactic potential models by <cit.>. In these two models, the LMC is on its second passage around the Milky Way. Both models have a MW halo with a virial mass of M_vir, MW = 11 · 10^11 M_⊙, and have M_LMC = 3 · 10^11 M_⊙ and M_LMC = 2 · 10^11 M_⊙, respectively <cit.>. The right panel of Fig. <ref> shows that the present-day perturbation of the LMC exceeds the triaxiality required by the HS, with 1 < p_Φ < 1.06. However, at earlier times, the perturbation is of the same order as the required triaxiality. Although beyond the scope of this work, it could be worthwhile to investigate the behaviour of the HS orbits in a potential including both the LMC and a prolate DM halo. The <cit.> models assume a spherical NFW DM halo as a starting point, and as a consequence, all HS stars are placed on z-tube orbits and in (L_z, L_) space the Helmi Streams' clumps do not remain separated in time. 









 §.§ Limitations, uncertainties and systematics


A number of simplifying assumptions have been made throughout this work. To start, we assumed that the DM halo follows an NFW profile with a triaxial shape whose axes are aligned with those defined by the MW disc. Recent work by <cit.> has however suggested that the Milky Way's DM halo might be titled. Besides this, there is evidence that the Milky Way's DM halo could be contracted <cit.> and follows a mass distribution whose shape may vary with radius <cit.>. Next, in all analyses performed in this work we have assumed a static potential. However, the effect of the LMC's perturbation could prove important, as discussed in the previous section[The MW is also being perturbed by the ongoing merger with the Sagittarius dwarf galaxy <cit.>. However, we expect that it is unlikely that this led to the formation of the two clumps, as all stars will have felt the perturbation equally independent of their orbital phase. Interestingly, though, the HS and Sagittarius have similar orbital frequency ratios, possibly suggesting a connection, such as group infall <cit.>.]. It would also be interesting to explore the local orbital structure for non-parametrised forms of the Galactic potential,  using for example (low-order) basis function expansions <cit.>. 

The peculiar motion of the Sun also introduces a systematic uncertainty on the estimated values of p and q. A larger (smaller) value or the velocity of the Sun could lead to an overcorrection (undercorrection) when transforming from observables to Galactocentric coordinates. This means that we add (subtract) velocity to (from) the true motion of the stars, which in turn will make them reach farther (less far) out. In general, the farther away one is from the Galactic Centre, the more the effective potential becomes dominated by the DM halo (see also Fig. <ref> in Appendix <ref>). Hence, if all HS stars would have larger apocentres, the DM halo's value of q that is needed to reach an effective flattening of q_Φ = 1 would be smaller. We use the subclump stars to obtain an order of magnitude estimate of this effect. For these stars, a difference in v_LSR and thus V_⊙ of ∼ 7   km   s^-1 leads to a shift in apocentre of ∼ 2 kpc, while the pericentres remain similar. This in turns shifts the location of the Ω_ϕ :Ω_z = 1:1 resonance in q by about ∼0.03. Hence, the systematic uncertainty associated to this effect is σ_sys∼ 0.03, which is very small. 
 


 §.§ Accretion time estimate



In the literature, the asymmetry between the numbers of stars in the HS with positive and negative v_z has been used to estimate the time of accretion time  <cit.>.  However, such estimates will be biased towards more recent accretion times as the hiL clump, which we have established is on or close to a resonance, will phase-mix more slowly. Therefore, to estimate the time of accretion, one should use the HS' loL clump only.  The loL clump's ratio, N_ loL(v_z^+)N_ loL(v_z^-)∼ 0.62, is significantly different from the ratio we obtain for all HS stars,  N_ HS(v_z^+)N_ HS(v_z^-)∼ 0.24. 

To illustrate how a resonance affects phase-mixing time-scales, we set up a test-particle simulation experiment. We use  by <cit.> in a potential with p = 1.02, q=1.19 and explore two cases. Firstly, we integrate the orbit of a subclump star, which is on the Ω_ϕ:Ω_z = 1:1 resonance, to an apocentre 8 Gyr ago and re-centre  to that 6D phase-space coordinate. We then integrate the orbits of all particles forward to the present day. We select particles that are in a local volume, i.e. d < 2.5 kpc, and inspect their positions and velocities. 
The result is shown in the top two rows of Fig. <ref>. A large part of the  particles is resonantly trapped, as is evident from the flattened particle distribution in the (y,z) plane. We clearly see that  is not yet fully phase-mixed and its streams can still be identified as coherent structures in configuration space. Moreover, the local volume particles have a tight distribution in velocity space and a low N(v_z^+)N(v_z^-)∼ 0.01 ratio. 
Next, we follow the same procedure but use a loL star, which is on a z-tube orbit, for re-centring. The result is shown in the bottom two rows of Fig. <ref>. The particles appear more mixed, as they show less substructure in configuration space, have a more diffuse velocity distribution and a significantly higher N(v_z^+)N(v_z^-)∼ 0.7 ratio. We find that the velocity distribution and the v_z ratio of the particles match the loL observations roughly after at least 5 Gyr of evolution, implying that, according to this experiment, the HS were likely accreted at least 5 Gyr ago. This accretion time estimate is robust to recentrings to different loL-like orbits and to forward or backward integration, as is expected for a structure on a regular, non-resonant orbit. This experiment emphasises that the degree of asymmetry in the number of stars associated to streams that have undergone some amount of phase-mixing cannot be straightforwardly used to estimate accretion times when a substructure is located close to or on a resonance (a similar conclusion seems to apply to the streams from Gaia-Enceladus, see , although in this case, due to resonances with the Galactic bar).

An independent measure of the accretion time of the HS can be derived from the rate of variation in L_z of the y-tube orbits. Fig. <ref> shows that the timescale over which L_z varies is linked to the value of p: a larger value of p implies a faster change in L_z. Fig. <ref> shows the (L_z, L_) distribution of the particles after 8 Gyr of integration time, and p > 1.02 results in distributions that span a too large range in L_z compared to the observations. However, the same figure but for an integration time of 4 or 6 Gyr shows two clumps in the (L_z, L_) distribution with an extent resembling the HS observations for p = 1.03 and p = 1.04, respectively. What exact value of p is preferred in these test-particle simulations thus depends on the accretion time of the HS. Since our MCMC results, based on the observed HS stars, indicate that p ≲ 1.02, this allows us to tentatively constrain the Helmi Streams' accretion time to about at least 6 Gyr ago, confirming our earlier estimate based on the degree of phase-mixing seen for the loL streams. 






§ CONCLUSION



We have studied the dynamical properties of the (phase-mixed) Helmi Streams using Gaia DR3 data and have used it to constrain the Galactic potential within 20 kpc, the region probed by the streams stars' orbits. The dynamics of the Helmi Streams are peculiar, as their stars separate into two clumps in angular momentum space <cit.>, of which one has a larger total angular momentum (hiL) than the other (loL). The phase-space distribution of the Helmi Streams indicate that the hiL clump is less phase-mixed than the loL clump, as is apparent from their v_z distribution. We have also confirmed that the hiL clump contains substructure in the form of a kinematically cold subclump (known as S2, see ).

To explain these characteristics, we have studied the orbital structure in the volume of phase-space probed by the Helmi Streams in a range of Galactic potentials models with a triaxial NFW halo, parametrised by the (density) axis ratios p and q. We have used the Lyapunov exponent as chaos indicator, the variation in L_z and L_y to classify orbits, and orbital frequencies (Ω_R, Ω_ϕ, Ω_z) to identify resonances. We 
have found that for triaxial DM halos with flattening q ∼ 1.2, the Ω_ϕ:Ω_z = 1:1 resonance acts as a separatrix between z-tube and y-tube orbits, which are orbits that rotate around the z- and y-axis, respectively. In the case of p<1, this resonance causes chaoticity, while for p>1, this resonance stabilises orbits and resonantly traps them. This means p>1 is favoured, as it is able to sustain the kinematically cold subclump. The location of the resonance in relation to the hiL and loL clumps, and therefore also the transition in orbit family, varies with (p, q).  Consequently, there is a region of (p,q) parameter space where the hiL clump is on a different orbit family than the loL clump, providing an explaination of the longevity and separation of the clumps.    Specifically, we have found that the hiL clump can be partially resonantly trapped, while the kinematically cold subclump is strongly resonantly trapped and the loL clump is located off-resonance, and this in turn explains their different degrees of phase-mixing. In such Galactic potential models, the hiL and loL clump with the correct dynamical characteristics can be formed over time from a single progenitor, which we have demonstrated by integrating and inspecting the orbits of simulated star particles of a Helmi Streams-like progenitor.

We have taken these findings a step further and used them to constrain the characteristics of the Galactic potential. We have performed a Monte Carlo Markov Chain sampling of Galactic potential models in which we vary p, q, the total disc (thin and thick) mass, DM halo scale radius and density, and we require the kinematically cold subclump to be resonantly trapped on the Ω_ϕ:Ω_z = 1:1 resonance. We also require the model to fit recent rotation curve data <cit.>. Additionally, we require that at least 40% of the loL stars is on z-tube orbits. We find  p=1.013^+0.006_-0.006, q=1.204^+0.032_-0.036, M_discs=4.65^+0.47_-0.57·10^10 M_⊙ and M_DM(< 15 kpc)=1.14^+0.11_-0.10·10^11 M_⊙. Our best-fit model has v_c(R_⊙)=233.3^+2.8_-2.1km   s^-1. In the best-fit potential, 5 subclump stars are on the resonance, while 4 are strongly resonantly trapped. All hiL stars are either on the resonance (9%), resonantly trapped (57%) or on y-tube orbits (21%), and 80% of the loL stars are on z-tube orbits. Our constraint on the shape of the DM halo is extremely strong. This can be can be explained by the fact that we combine precise rotation curve data, which strongly constrains the mass enclosed on the midplane of the Galaxy, with a constraint on the local orbital structure of the potential, which depends on the effective shape and characteristics of the potential. 

Our findings suggest that the orbital structure of the potential can strongly influence the dynamical properties of substructures in the Galaxy, especially if these are located near a separatrix. Among the possible consequences, it may complicate the identification of debris due to chaos which accelerates phase-mixing <cit.>. Furthermore the determination of the accretion time may also affected due to the presence of resonances which in contrast slows down the process of phase-mixing. On the other hand, it gives us a powerful and exciting new tool to probe the orbital structure of the potential, and hence its characteristic parameters <cit.>, 
as is shown in this work. 






This research has been supported by a Spinoza Grant from the Dutch Research Council (NWO). HCW thanks Lorenzo Posti and Paul McMillan for providing the chains of their MCMC runs.

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.



Throughout this work, we've made use of the following packages:  <cit.>,
           <cit.>, 
           <cit.>, 
           <cit.>,
           <cit.>,
           <cit.>,
           <cit.>,
           <cit.>, 
           <cit.>,
           <cit.> and Jupyter Notebooks <cit.>.






         






aa 








§ OUR FIDUCIAL GALACTIC POTENTIAL MODEL



Throughout this work, we use a modified version of the <cit.> potential. The <cit.> potential is an axisymmetric potential model consisting of a bulge, an exponential thin and thick disc, a HI gas disc, a molecular gas disc and a spherical NFW DM halo. Since recent work has shown that this model might be slightly too heavy <cit.>, we perform a simple Monte Carlo Markov Chain using  <cit.> with 40 walkers and a 1000 steps to optimise the scale radius of the halo r_h, the density of the halo ρ_0, h and M_discs, the sum of the thin and thick disc mass, assuming a thick disc scale length of R_thick = 2 kpc <cit.> and a fixed local density normalisation f_d, ⊙ = ρ_thick(R_⊙, z_⊙)/ρ_thin(R_⊙, z_⊙) = 0.12 <cit.>. Furthermore, following <cit.>, we assume an axisymmetric halo with a density flattening q = 1.2. We use rotation curve data with associated uncertainties derived using axisymmetric Jeans equations by <cit.>[In the case of <cit.>, we take the mean of the measured uncertainties and add the systematic uncertainties (extracted from their Fig. 4), assuming a systematic uncertainty of 12% for the five bins largest in R, quadratrically, such that σ_v_c, tot^2 = σ_meas^2 + σ_syst^2. We treat the different contributions to the total systematic uncertainties as independent and add them quadratically.], who used Gaia DR2 data, and <cit.>[In the case of <cit.>, we similarly add the systematic uncertainties (extracted from their Fig. 12) quadratically and also treat the different contributions to the total systematic uncertainty as independent by summing them quadratically. ], who used APOGEE, LAMOST, 2MASS and Gaia EDR3 data and employed supervised machine-learning to obtain distances[We use these two datasets as they use consistent values for R_⊙, v_⊙, and z_⊙. Alternative works on rotation curves make different assumptions on their solar motion and position. For example, <cit.> assumes R_⊙ = 8.34 kpc and z_⊙ = 0.027 kpc, and <cit.> assumes R_⊙ = 8.178 kpc and z_⊙ = 0.0208 kpc. This in turn leads to different adopted values of the solar motion.]. 
We restrict ourselves conservatively to rotation curve data with R < 15 kpc, because beyond this radius systematic errors associated to assumptions behind the methods used to derive the rotation curve become important <cit.>. However, we checked that if we take data within R <20 kpc, our conclusions remain similar. We set a simple flat prior and allow 


    P(θ⃗) = {[                                                                                                                     1 if{[                   10 ≤ r_h≤ 30  kpc;       10^5≤ρ_0, h≤ 10^8  M_⊙ kpc^-3; 3 · 10^10≤ M_discs≤ 5.5 · 10^10 M_⊙ ].;                                                                                                                     0                                                                                                            otherwise. ].


We find v_c(R_⊙) = 233.2 ± 2.6   km   s^-1 and median and percentiles  r_h = 15.7^+8.8_-4.4 kpc, ρ_h, 0 = 10^+8_-5· 10^6   M_⊙ kpc^-3 and M_discs = 4.5^+0.6_-0.7· 10^10 M_⊙, and use these median values throughout this work to set our fiducial potential. The contributions as a function of radius of all components of the potential are shown in Fig. <ref>. The total mass within 20 kpc, the region probed by the Helmi Streams, of this model is M_tot(r < 20  kpc) = 2.2^+0.1_-0.1· 10^11 M_⊙, compatible with other estimates <cit.>.





§ ORBITAL FREQUENCIES DETERMINATION






To recover the fundamental orbital frequencies corresponding to a star's orbit, we use a modified version of  <cit.>.  is an implementation of the Numerical Analysis of Fundamental Frequencies (NAFF) method <cit.>. It performs a Fast Fourier Transform of a complex time series w(t) + i v_w(t), where w are the coordinates, for example Poincaré symplectic coordinates <cit.> or Cartesian coordinates. If a 3D time series is inputted,  stores the three frequency spectra (frequency and amplitude) of each of the time series and sorts them by amplitude. Next, the frequency with the highest amplitude is chosen as the first leading frequency. It then determines the second leading frequency by picking the frequency with the second highest amplitude, requiring it to be of a different coordinate and requiring a minimum difference in frequency between the first two leading frequencies (this difference is by default set to 10^-6 but can be changed). In a similar way, the third leading frequency is found. This set of three leading frequencies typically corresponds to the set of three fundamental frequencies, but this does not necessarily need to be the case, something that was noted by <cit.> and <cit.> as well.

To make the fundamental frequency determination more robust, we proceed as follows. We integrate orbits for 100 Gyr, as the longer an orbit is integrated, the more reliable the frequency determination becomes <cit.>. We output each 10 Myr and construct a set of three time series in Poincaré symplectic polar coordinates for each orbit[Poincaré's symplectic polar coordinates, a variation on cylindrical polar coordinates, are used as input for  throughout this work, and thus also in the case of mildly triaxial potentials. This choice of coordinates ensures relatively good results and gives more information about the orbital properties than for example the orbital frequencies in Cartesian coordinates do.]. This is used as input for , which determines the frequency spectra with settings  = 15 and . [To a certain extent, the frequencies recovered by the algorithm are dependent on the settings in the code, as also shown by <cit.>.]. We start with the z coordinate, and from its frequency spectrum define the frequency with the highest amplitude to be Ω_z. Next, we filter the frequency spectra of R and ϕ such that frequencies that are a multiple of Ω_z are removed. This is done by requiring a minimum difference of 10^-4 between Ω_ϕ and Ω_z and Ω_R. Of course, this presents issues if the orbit in question is a resonant orbit. Therefore, we step wise make the required minimum difference smaller (10^-5, 10^-6, 10^-7, 10^-8) if the peak that is picked results in an unexpected frequency ratio. For the Helmi Streams, this translates to the requirement (Ω_ϕ: Ω_z - 1) < 0.15, which effectively means that we assume that all Helmi Stream stars have orbital frequencies that are drawn from a continuous distribution, i.e. they have values that are within a certain range. Next, the frequency with the highest amplitude in the ϕ spectrum is chosen as Ω_ϕ. Then, the remaining frequency spectrum of R is filtered such that the R frequencies that are a multiples of Ω_ϕ are removed. Finally, the frequency with highest amplitude that is independent is chosen as Ω_R. However, as linear integer combinations of this peak with Ω_z or Ω_ϕ may appear, one needs to be careful to choose the peak that corresponds to the independent Ω_R. For the Helmi Streams, this is done by requiring that (Ω_R: Ω_z - 0.7) < 0.15). The fact that multiples of the peaks in Ω_z, Ω_ϕ and Ω_R appear in all three spectra is indicative of the fact that the motion in R, ϕ and z is coupled. The procedure to determine the orbital frequencies is illustrated in Fig. <ref> and solves the problem of the ficticious Ω_z : Ω_R = 1:2 branch reported by <cit.>.




§ ORBIT MAPS OF A HELMI STREAMS'  SUBCLUMP STAR



We randomly picked a subclump star and integrated its orbit for 20 Gyr in the fiducial potential with a triaxial NFW halo for a range of 0.95 < p < 1.05 and 1.10 < q < 1.30 to create orbit maps. Figure <ref> shows the map in (R, z), Fig. <ref> shows the map in (x, y), Fig. <ref> shows the map in (x,z) and Fig. <ref> shows the map in (y,z). By inspecting these maps, different orbit families can be recognised. The resonantly trapping Ω_ϕ: Ω_z is 1:1 resonance clearly stands out for p>1 in Fig. <ref>, while such a region is not apparent for p <1. The y-tubes and z-tubes can be recognised easily from  Fig. <ref> and Fig. <ref>, as on these families the star orbits rotate around the y and z axis, respectively.

















§ ORBITAL FREQUENCIES OF THE HELMI STREAMS



Figure <ref> shows the distribution of orbital frequencies of the Helmi Streams stars in the potential where we have replaced the DM halo with a triaxial halo for a range of p and q. If stars are on the Ω_ϕ : Ω_z resonance, they fall on the dotted line in the figure. Using Δ L_y and Δ L_z to classify the orbits, we checked and found that the orbital frequencies of z tube orbits and truly resonant orbit are determined reliably. However, the algorithm sometimes has some problems to  determine the orbital frequencies of y tube orbits, which are either classified as resonant orbits or placed on a wrong branch, which both stand out as line features in Fig. <ref>. This can be attributed to the fact that the cylindrical coordinates (R, ϕ, z) are not the most suitable coordinates to describe the motion of y tube orbits. However, the frequency determination is good enough for our purposes, which is to use the median of the orbital frequencies of the hiL, loL and subclump stars to map out where orbital resonances reside (see Fig. <ref> of the main text). It is also interesting to see the effect of the Ω_ϕ : Ω_z 1:1 resonance for p >1, clearly trapping orbits, and p < 1, resulting in a chaotic and thus underpopulated region in frequency space.






§  IN POTENTIALS WITH DIFFERENT (P, Q)



In Sect. <ref> we explore the (L_z, L_) distributions of  particles after they have have been re-centred to the 6D phase-space coordinates of a central HS star 8 Gyr ago and integrated forward to the present day. Figure <ref> shows  the  (L_z, L_) distribution after 8 Gyr of integration time for the range 1.13 < q < 1.23 and 1.00 < p < 1.04. For the range 1.18 < q < 1.21 and 1.02 < p < 1.03 we find distributions that match the HS' observations most convincingly, showing two separated clumps of stars, driven by the fact that the stars in the two clumps are on different orbital families.










 Paper Title: Search for extraplanar radio emission in a sample of edge-on starburst galaxies.

























Example

  
  

  







2022
Prospects for low-frequency radio astronomy in S. A.



 









1Facultad de Ciencias Astronómicas y Geofísicas,
  UNLP, Paseo del Bosque S/N, 1900, La Plata,
  Buenos Aires, Argentina.

2Instituto Argentino de Radioastronomía,
  Berazategui, Buenos Aires, Argentina. 


Galante et al.

Extraplanar radio emission in starburst galaxies


C. A. Galante, J. Saponara, G. E. Romero, & P. Benaglia


Galante, C. A.
Saponara, J.
Romero, G. E.
Benaglia, P.



The intense star formation in starburst galaxies results in the formation of intense winds that sweep matter up to several kpc out of the galactic plane. These winds can be detected in many bands of the electromagnetic spectrum.
In this opportunity, we present our project to study three edge-on galaxies, candidates to show extraplanar emission at radio wavelengths. We will describe the criteria followed in selecting the sample and the software developed to detect the presence of the wind. Besides, we will discuss the next steps of this project.




La intensa formación estelar que presentan las galaxias starburst resulta en la formación de fuertes vientos que arrastran materia hasta varios kpc por fuera del plano galáctico, emitiendo radiación en varias bandas del espectro electromagnético. En esta oportunidad, presentamos nuestro proyecto para estudiar tres galaxias vistas de canto, candidatas a presentar emisión por fuera del plano galáctico en ondas de radio. Describiremos el criterio utilizado para selecionar la muestra de galaxias y el software creado para detectar la presencia de los vientos. Además, discutiremos los próximos pasos a seguir de este proyecto.




Galaxies: halos
Galaxies: starburst
Radio continuum: galaxies
Radiation mechanisms: synchrotron







Search for extraplanar radio emission in a sample of edge-on starburst galaxies
    
  C. A. Galante,1,2 
  J. Saponara,2
  G. E. Romero,1,2
  and P. Benaglia2
    
===============================================================================






§ GENERAL



Starburst galaxies exhibit a remarkable level of star formation activity. The combined effect of intense stellar winds generated by hot stars and a high rate of supernova explosions leads to gas ejection from the star-forming region into the surrounding environment, creating powerful winds <cit.>. These winds play an important role in shaping the formation and evolution of galaxies. They can inhibit star formation by depleting the central gas reservoirs <cit.>. As the gas escapes, star formation decreases and the outflow stops. However, the arrival of fresh gas can reignite the starburst. This influx of fresh gas is often triggered by gravitational interactions with neighboring galaxies, or when previously ejected gas that remains gravitationally bound falls back into the galaxy as high-speed clouds <cit.>. However, some of the gas may not be able to return to the galactic disk because it is no longer gravitationally bound. Consequently, this gas enriches the intergalactic medium as it is carried away by the outflow, supplementing it with heavy elements.

Different phases make up the outflow, and the relative positions of each phase provide important clues to understanding the three-dimensional picture. The hot gas in the cavity emits X-rays, the diffuse ionized gas surrounding the hot gas emits H_α, and the atomic gas in a cooler shell pulled up by a thick disk emits at the 21-cm line <cit.>. In addition, the presence of magnetic fields and the relativistic particles that emit synchrotron radiation can be studied by detecting the extra-planar radio continuum emission <cit.>. The best example is the starburst galaxy NGC 253, located at a distance of 3.9 Mpc in the Sculptor group of galaxies. The nuclear starburst is thought to be forming stars at a high rate and producing a superwind that transports material into the halo. The galaxy and its extra-planar gas have been well studied in recent years at various wavelengths <cit.>. 


We are carrying out a project to detect and characterize the extra-planar non-thermal emission in three edge-on galaxies, to determine the non-thermal contribution, to measure the magnetic field and its distribution, and to detect embedded sources. In the following sections, we describe the sample of galaxies, the method implemented to detect the continuum emission and the current status of this project.








§ THE GALAXY SAMPLE




From the Imperial IRAS-FSC Redshift Catalog <cit.>, we selected the galaxies according to the following criteria: 1. They should have an
inclination higher than 75^∘ to allow for a clear separation of the halo emission from the disk emission. 2. They must fulfill that log(L_IR/L_)>10 to guarantee the intense star formation activity. 3. The ratio of infrared fluxes at 60 and 100 μm should be S_60/S_100 > 0.4 to increase the chance of presenting disk-halo interactions <cit.>. 4. The angular sizes (θ) and distances (D) must satisfy θ > 3 arcmin and D < 20 Mpc, to ensure their detection at radio wavelengths and to allow us to resolve the emission in z-direction.

The following galaxies constitute the galaxy sample. We list the main properties of the galaxies, i inclination, D distance, θ angular size, the L_IR, SFR, S_60/S_100 and the flux density at 1.4 GHz (S_1.4) in Table <ref>.


 NGC 1055: This galaxy shows evidence of a minor merger observed at the low-angular resolution of the VLA and in Chandra X-ray observations (with low integration time).

NGC 4527: This galaxy was detected in X-rays by Chandra. Low-angular resolution observations carried out with the VLA suggest the presence of a point-like source embedded in the diffuse radio continuum emission.

NGC 1421: This galaxy was previously observed with the VLA. <cit.> suggest evidence of extra-planar emission and possible embedded sources.




§ METHODOLOGY





We developed a software to identify and characterize the presence of extra-planar emission. 
The software splits the galaxy image into multiple strips perpendicular to the galactic disk (in z-direction) and averages them together to create mean intensity profiles, hereafter z-profiles. The width of this z-profile is given by the combined effect of the telescope resolution, the galaxy inclination, the intrinsic height of the galactic plane, and the possible presence of a radio halo. The software models these components by assuming that the galactic plane is an infinitesimally thin disk. When it is inclined and projected onto the sky, its profile along the z-direction can be described either by a Gaussian or an exponential function. Each function is convolved with a Gaussian-shaped beam to model the effect introduced by the telescope resolution. If a single function is not enough to fit the z-profile, a second exponential component is added to model the emission from the halo. 

We tested the software using artificial galaxies. These were created considering different combinations of disk diameters and heights, besides an inclination range between 80^∘ and 90^∘. Our software could find the presence of the halos for artificial galaxies with inclinations greater than ∼83^∘. We found no dependence of the halo scale heights with the inclination. In Fig. <ref>, we show the results obtained for an artificial galaxy created with an inclination of 84^∘ and a halo with 10 arcseconds FWHM. The best fit was found for a two-component function with an exponential disk, properly identifying the halo.





§ CURRENT STATUS







We found that the software we developed to detect and characterize the extra-planar emission works correctly by testing it using the artificial galaxies described in Sect. <ref>. The three galaxies of the sample were observed with the Giant Metrewave Radio Telescope (GMRT) at band-4 (550 MHz) and band-5 (1420 MHz). In order to attain 10μJy beam^-1 rms and considering a fudge factor of 2.5 following ETC recommendations, we observed each source for 3 and 3.5 hours, at bands 5 and 4 respectively, including overhead. Because NGC 5690 is dimmer and the smallest, this galaxy was observed during 4 and 3.5 hours at bands 5 and 4. Currently, we are calibrating and processing the data. A very preliminary image of the galaxy NGC 4527 with synthesised beam of 25”× 25” and a noise level of 1 mJy beam^-1, is shown in Figure <ref>. 





Acknowledgements:  GER acknowledges financial support from the State Agency for Research of the Spanish Ministry of Science and Innovation
under grants PID2019-105510GB-C31AEI/10.13039/501100011033/ and
PID2022-136828NB-C41/AEI/10.13039/501100011033/, and by“ERDF A way of
making Europ”, by the “European Unio”, and through the “Unit of
Excellence María de Maeztu 2020-2023” award to the Institute of
Cosmos Sciences (CEX2019-000918-M). Additional support came from PIP 0554 (CONICET). 






[Bauer et al.(2007)]Bauer Bauer, M., Pietsch, W., Trinchieri, G., et al.  2007, , 467, 979

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[Dahlem et al.(2001)]Dahlem Dahlem, M., Lazendic, J. S., Haynes, R. F., Ehle, M., Lisenfeld, U.  2001, , 374, 42

[del Valle et al.(2018)]delValle del Valle, M. V., Müller, A. L., Romero, G. E.  2018, , 475, 4298

[Heesen et al.(2009a)]heesena Heesen, V., Beck, R., Krause, M., Dettmar, R. J.  2009a, , 494, 563

[Heesen et al.(2009b)]heesenb Heesen, V., Krause, M., Beck, R., Dettmar, R. J.  2009b, , 506, 1123

[Irwin et al.(1999)]Irwin Irwin, J. A., English, J., Sorathia, B. 1999, , 117, 2102

[Krause et al.(2020)]Krause Krause, M., Irwin, J., Schmidt, P., et al.  2020, , 639, 112

[Romero et al.(2018)]Romero Romero, G. E., Müller, A. L. & Roth, M.  2018, , 616, A57



[Veilleux et al.(2005)]Veilleux Veilleux , S., Cecil, G., Bland-Hawthorn, J. 2005,  , 43, 769

[Wang & Rowan-Robinson(2009)]Wang Wang, L. & Rowan-Robinson, M.  2009, , 398, 109







 Paper Title: North Polar Spur: gaseous plume(s) from star-forming regions at $\sim$3-5 kpc from Galactic Center?.





 


Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany 
Space Research Institute (IKI), Profsoyuznaya 84/32, Moscow 117997, Russia
Universitäts-Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany
Ioffe Institute, Politekhnicheskaya st. 26, Saint Petersburg 194021, Russia
Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskaya str., Moscow 119017, Russia


NRC `Kurchatov Institute', acad. Kurchatov Square 1, Moscow 123182, Russia
Institute of Applied Physics of the Russian Academy of Sciences, 46 Ul'yanov str., Nizhny Novgorod 603950, Russia






















We argue that the North Polar Spur (NPS) and many less prominent structures are formed by gaseous metal-rich plumes associated with star-forming regions (SFRs). The SFRs located at the tangent to the 3-5 kpc rings might be particularly relevant to NPS.  A multi-temperature mixture of gaseous components and cosmic rays rises above the Galactic disk under the action of their initial momentum and buoyancy. 
Eventually, the plume velocity becomes equal to that of the ambient gas, which rotates with different angular speed than the stars in the disk.  As a result, the plumes acquire characteristic bent shapes. 
An ad hoc model of plumes' trajectories shows an interesting 
resemblance to the morphology of structures seen in the radio continuum and X-rays. 


North Polar Spur


    
North Polar Spur : gaseous plume(s) from star-forming regions at ∼3-5 kpc from Galactic Center?
    
Eugene Churazov 1,2 
Ildar I. Khabibullin 3,1,2 
Andrei M. Bykov 4 
Nikolai N. Chugai 5 
Rashid A. Sunyaev 2,1 
Victor P. Utrobin 5,6 
Igor I. Zinchenko 7

    August 5, 2024
================================================================================================================================================================






§ INTRODUCTION











North Polar Spur <cit.> is an elongated bright structure first discovered in all-sky radio images along with other similar structures dubbed Loops <cit.>. Unlike other Loops, NPS (the northern part of Loop I) is visible in soft X-rays across its full extent <cit.>.    

Given the NPS's large angular size, ≳90, the first proposed explanation <cit.> was a shell of a very nearby, d≲100 pc, supernova remnant. Alternatively, a scenario in which NPS represents a distant, d≳10 kpc, Galaxy-scale structure has been put forward, with the possible energetic outburst in the Galactic center being considered as the primary source behind it <cit.>. The latter might be caused by an intense star-formation episode in the vicinity of the Galactic Center or by the activity of the supermassive black hole Sgr A* <cit.>. Both scenarios suggest that NPS is powered by a shock wave propagating in the hot medium (either interstellar or circumgalactic). This shock is responsible for the acceleration of radio-emitting relativistic electrons, as well as the heating and compression of the X-ray-emitting hot gas. 


Although these two scenarios involve very different physical sizes, time scales and required energetics, it turned out to be very difficult to unequivocally distinguish between them based on the available data, with contradicting conclusions having been reached from, e.g., the Faraday Rotation and X-ray absorption measurements <cit.>. Recent discoveries of the giant structures in gamma-rays <cit.> and soft X-rays in the southern Galactic sky <cit.>, the "bubbles" appear to be more symmetric with respect to the Galactic center, strengthening the case for the central energy release as a viable solution <cit.>.  

In this Letter, we consider another scenario for NPS formation motivated by the morphological and spectral properties of the soft X-ray emission measured in the course of the eROSITA all-sky survey <cit.>. We propose that NPS is produced by a break-out of the massive star formation regions associated with the end of the Galactic bar, which locates its base at d∼5 kpc from us <cit.>. Flows of enriched hot gas move upward from the Galactic disk and get entrained in the relative rotation of the circumgalactic medium above the disk, resulting in the spiral-like structure, appearing like a loop in the sky. In this model, it is the advection of the enriched gas and relativistic particles, rather than a shock, that is responsible for the appearance of NPS, predicting the high metal abundance of X-ray emitting gas and lack of evolutionary signatures in the direction perpendicular to the NPS edge.















§ X-RAY PICTURE




For X-ray analysis, we use the data accumulated by eROSITA <cit.> telescope onboard the SRG  X-ray observatory <cit.> in the course of its all-sky survey in the Eastern Galactic hemisphere (i.e. the Galactic longitude range 0^∘<l<180^∘). Details on the data preparation and analysis are given in Appendix <ref>. Images extracted from the eROSITA all-sky survey in the 0.52-0.61 keV and 0.7-0.96 keV bands are shown in Fig. <ref> (left and middle panels, respectively). The former band includes the triplet of He-like oxygen O VII near 574 eV, while the bright lines of helium-like neon Ne IX (at 905 and 921 eV) and neon-like iron Fe XVII (between 725 and 826 eV) fall into the latter. For plasma in collisional ionization equilibrium (CIE), the peak line emissivities in these bands are at temperatures ∼ 0.2  keV and  ∼ 0.6  keV, respectively.

Rather complex X-ray morphology of NPS is obvious from these images and the comparison between them. One can readily see that the correlation length of X-ray structures is larger in the direction along NPS for both bands but a simple combination of concentric spherical shells is not a good description of NPS.
For example, the surface brightness in the 0.52-0.61 keV band dominated by the O VII line, has a sharp outer edge but a rather flat surface brightness across the inner region, unlike an edge-brightened shell. Similarly, the 0.7-0.96 keV surface brightness is composed of a fainter diffuse emission surrounding a bright central "filament". 

More quantitatively this is illustrated via profiles of the X-ray emission in several narrow bands. The white dashed lines in Fig. <ref> outline a wedge that is aligned with the NPS "outer" boundary. Radial profiles extracted from this wedge are shown in Fig. <ref>. The steep rise, corresponding to the "edge" of NPS, is visible in all bands. However, further inside NPS, the profiles do not follow a typical edge-brightened behavior characteristic for a spherical shock (see, e.g., profiles at energies below 0.61 keV). This suggests that in 3D the volume occupied by the plasma emitting at these energies resembles a flattened sheet rather than a spherical shell. In harder bands, the profiles feature a steep jump, followed by a gradual increase of intensity before reaching a peak at x≈ 35  degrees in this plot. The center of the wedge is at (l,b)=(338^∘,32^∘) in Galactic coordinates. 
  

An alternative to the local SNR or GC-related scenarios, which associate the steep rise with a shock, is the assumption that NPS is a multi-temperature gas, where a hotter (>0.3 keV) component, responsible for the emission in iron lines above 0.7 keV, coexists with a warm (e.g., 0.15-0.2 keV) plasma, producing the bulk of the emission below 0.7 keV and most prominently in oxygen lines. 

In what follows, we consider a scenario in which such multi-temperature gas is venting from the active star-forming regions in the central parts of the Galactic disk. 
























§ MORPHOLOGICAL MODEL



Here we consider a simple morphological model that is based on a broad-brush picture of the evolution of massive star-forming region in the Galactic disk <cit.>. The first shock waves of supernova explosions originating from the most massive newborn stars in such a region produce hot and dense gas strongly enriched with metals of the ejecta. Initially confined by the surrounding dense and cold gas, the hot gas manages to break through and form an (adiabatically cooled) plume that moves away from the disk and interacts with the interstellar (ISM) and circumgalactic (CGM) media forming some sort of chimneys and plumes (see Appendix <ref>). The trajectories of rising plumes might be complicated, especially when the gas is multi-phase and magnetized. They are also affected by the pressure, magnetic fields, and velocities in the ambient ISM and/or CGM <cit.>.   



For illustration purposes, we consider an ad hoc model that encapsulates all the complicated physics in three parameters. Namely, the initial (vertical) velocity of the plume _z,0 and two effective scale heights z_ϕ and z_z. Here z_ϕ describes how the plume that initially moves together with stars and gas close to the disk plane, decelerates and joins the pressure-supported hot halo gas, which is either non-rotating or rotating slower than the stellar disk. In turn, the z_z parameter controls the decline of the vertical (perpendicular to the disk) velocity component. As a result, the two velocity components have the following dependence on z:     


    _z(z)=   _z,0  e^-z/z_z
    _ϕ(z)=   _ rot  [1- e^-z/z_ϕ ].




    _z(z)=_z,0  e^-z/z_z    and  _ϕ(z)=_ rot  [1- e^-z/z_ϕ ].


A flat rotation curve with _ rot = 220   km s^-1 is adopted for the sake of simplicity, the effective scale heights are set to z_ϕ=2  kpc and z_z=5  kpc, and _z,0= 200   km s^-1. The choice of these parameters is rather arbitrary, but it suffices for illustration purposes and the final answer is not dramatically sensitive to it.

For the source of the plume, e.g. a star-forming region, we assume a circular trajectory in the disk plane with the velocity _ϕ, src. In our illustrative model, the following two versions are considered:


    _ϕ, src=   _ rot    ( a) 
    _ϕ, src=   Ω R     ( b),



    ( a)   _ϕ, src=_ rot   and    (b)  _ϕ, src=Ω R  ,

where R is the distance from the Galactic Center. Version (a) corresponds to the source moving with the same velocity as stars, while version (b) mimics a pattern motion. The former case is relevant when the active phase of a given star-forming region associated with the dense gas is very long and the active region always moves together with the gas (with the velocity _ rot). The latter case corresponds to a situation when the spatial distribution of the plume sources is linked to the Galactic bar and/or spiral arms. For illustration of this case, we adopted the value Ω=33  km s^-1 kpc^-1 derived for the bar pattern speed <cit.>. 

As a further simplification, we assume that none of these parameters depend on the position of the plume within the galaxy. 

The above relations (<ref>-<ref>) can be trivially integrated to derive a position of the gas lump released some time t ago relative to the current position of the "source". By design, plume trajectories appear in 3D as spirals rising from the disk from the current location of the plume source (see Appendix <ref> and Fig. <ref>). 
We note here that for a strongly underdense lump, the buoyancy force is solely set by pressure gradients in the ambient gas. These gradients themselves depend on the rotation velocity of the halo gas and can have both radial and vertical components. In the above model, we explicitly neglect the former component, although in real systems it can force the plume to move radially. In particular, for slow (fast) rotation of the halo gas, underdense lumps can move to larger (smaller) radii (see Fig. <ref>).

As a final step, we project the trajectory onto the sky as seen from the position of the Sun. This modifies the appearance of the plume due to different distances from the Sun to various segments of the plume 
and the Sun's motion relative to the source. An example of plume trajectories for a set of "sources" chosen by hand is shown in Fig <ref>. A table with the positions of sources is given in the Appendix <ref>. The trajectories have been integrated for ∼ 140  Myr, which is approximately one rotation period around the Galactic Center near the Sun for the adopted value _ rot.

Figure <ref> illustrates the key morphological properties of the plumes. Naturally, the trajectories remain confined to the regions defined by the radial distance to the source from the Galactic Center, while the notable left-right asymmetry is imposed by the rotation direction of the Galaxy. All trajectories have rising parts close to the Galactic Plane, with the direction of the curvature set by the mutual position of the source and the Sun and the direction of the Galaxy rotation. This is clearly illustrated by the family of trajectories for sources located at the 5 kpc ring around GC. The morphological difference is even more pronounced for sources located at larger distances from GC (cases marked at Cygnus X and Vela in Table <ref>). For the Cygnus X region, the plume's apparent trajectory rises to the Galactic Poles before forming a spiral. For the Vela region, the plume is bent much earlier and follows an almost horizontal line at |b|∼ 20-25 degrees.  

If NPS is indeed a gaseous plume, its base should be associated with the star formation regions that are currently at Galactic longitudes l∼ 20^∘-40^∘ (see Fig. <ref>). Incidentally, this range of longitudes corresponds to the tangential direction to the 3-5 kpc rings around the Galactic Center and indeed hosts the most active star-forming regions in the Galaxy, including a prototypical mini-starburst complex W43 (see Appendix <ref>). Below we discuss this scenario in more detail.



§ DISCUSSION



Motivated by the morphological model presented above, we assume that the leading edge of NPS is located at a tangent to the 5 kpc ring. In this case, the distance of the NPS base from the Sun is D_ Sun,NPS≈ 6.6  kpc. Of course, some parts of the plumes can be closer to the Sun, while some further away. 

Given the apparent size of NPS on the sky, one can assume that its physical size (the "depth" along the line of sight) is of the same order as the distance from the Sun, i.e. l∼ D_ Sun,NPS. Based on that, we can estimate the hot gas density from the observed peak surface brightness I_X in the 0.7-1.05 keV band. This yields a proton number density 

    n_p∼ 7× 10^-4( Z/Z_⊙ )^-1/2( l/6  kpc )^-1/2  cm^-3,

where Z is the metal abundance with respect to the Solar value <cit.>. In this derivation, we assumed that the gas is in collisional ionization equilibrium with temperature 0.6  keV and used  model <cit.> to predict the emissivity. The metallicity dependence in the above expression is valid for Z/Z_⊙≳ 0.1.   The corresponding cooling time of the gas with temperature kT≈ 0.6  keV can be estimated using the cooling function of <cit.> as t_ cool∼  2× 10^9( n/10^-3  cm^-3 )^-1( Z/Z_⊙ )^-1/2  yr.




While the 3D geometry of NPS is uncertain, we assume that its volume is ≈ l^3 (i.e. its line-of-sight extension is comparable to its transverse size) to get an estimate of the total energy (enthalpy) E∼ l^3 (n_p+n_e) 5/2 kT∼ 2× 10^55  erg



for l=6  kpc.
Assuming that the integrated energy release associated with the formation of 1 solar mass of stars is ∼ 10^49  erg, it needs ∼ 2× 10^6 M_⊙ of gas to be converted into stars. Assuming the star-formation rate of 0.1  M_⊙  yr^-1, it takes ∼ 20 Myr to generate enough energy for powering the entire NPS. The long cooling time and preferential accumulation of trajectories in the NPS region imply that it could be a result of cumulative contributions from many sources.  


In the simplest version of the above scenario, the gas in NPS is in pressure equilibrium with the ambient medium. Assuming that the temperature of the ambient medium is ∼ 0.15  keV, the regions filled with ∼ 0.6  keV plasma should have ∼ 4 times lower density. Yet, NPS appears much brighter than the diffuse emission of the Galaxy. There are two plausible reasons for that. One is that for lines of Fe XVII, Ne IX, and Ne X, the gas in the halo is simply too cool to produce a bright emission. The second reason is that the plumes are plausibly much more metal-rich than the halo gas. This is particularly important for O VII lines, which are present in the spectrum of the halo. The gas in the plumes could easily be a factor of 10 or more metal-rich and even in the absence of shock, it can shine prominently (see Fig. <ref>). In Appendix <ref> we argue that the plume metallicity is not much larger than the Solar one. This implicitly suggests that the abundance in the halo is ≲ 0.1. 



An important implication of the model is the global asymmetry of the Milky Way diffuse X-ray and radio emission. The asymmetry is present even if the distribution of plumes' sources is itself symmetric. In 3D, the plumes can form a symmetric pattern, but the direction of the Galaxy rotation produces the apparent asymmetry when viewed from the Sun's position. Some extra asymmetry could come from global gas motions in the Galactic halo <cit.>, e.g. caused by interaction with the satellite galaxies, but it comes on top of the "natural" East-West asymmetry. The North-South asymmetry, which is clearly visible in both radio and X-ray sky, in this model is attributed to the asymmetry of the primary sources of the plume. When the hot gas finds its way of least resistance through the dense cold gas in the disk, it may preferentially go to one side rather than forming a symmetric structure on both sides.




The morphological model considered above, treated trajectories of individual gas plumes as independent, neglecting the hydrodynamic nature of the flows and, in particular, possible gas mixing instabilities that would naturally arise in such a situation. The sharpness of the NPS's outer edge as well as its overall filamentary appearance might be a manifestation of the suppression of these instabilities by the well-ordered and sufficiently strong magnetic field. 

Indeed, the synchrotron radio emission from the NPS region is known to be polarized <cit.>. 
The polarization implies that the magnetic field is ordered. In the model discussed above, these structures can be especially prominent plumes located at different distances from the Galactic Center. The ordering is caused by stretching the field lines during certain phases of plume evolution and/or by the halo gas motions rather than by shocks (as illustrated in Appendix <ref>). 

On a more speculative side, we note that in this model the gas vented from the disk tends to accumulate high above the plane in general direction of the Galactic Center. From this point of view, a question arises whether structures like the eROSITA bubbles <cit.> could be produced by the same mechanism. In this case, these structures can evolve on a longer time scale than implied by the shock-driven scenario. In fact, different regions across NPS could come from gas lumps having different ages (and different star-forming regions) and, therefore, can have different properties.  

In the proposed scenario, the NPS might be a Galactic analog of the magnetic structures directly observed to reach significant distances above the disks in many nearby galaxies, e.g. NGC 4217 <cit.>. On the other hand, NPS might also be a case similar to the off-disk parts of the so-called anomalous spiral arms in NGC 4258, which are believed to be powered by the interaction of the relativistic jet with the galactic disk <cit.>. In all these cases, we might witness signatures of the disk-halo interaction via intense feedback episodes resulting in complex multi-phase and magnetized interface region <cit.>.



One observational test that can falsify the plume scenario is the measurements of the non-equilibrium ionization (NEI)  signatures in the gas, which are pertinent to the shock scenario <cit.>. Indeed, for the gas electron density ∼ 10^-3  cm^-3 and the downstream plasma velocity ∼ 200   km s^-1, the ionization parameter τ=n_e× t∼ 1.5× 10^11  cm^-3 s is expected at a distance of 1  kpc downstream of the shock. Such scales are easily resolved for any distance to NPS (for the "nearby" shock scenario the ionization parameter is even lower at the same angular distance from the shock) and can be derived from the spectra. The complication here is that NPS does not have a simple "layered" structure that allows for clean separation of shells at different 3D distances from the edge. As a result, the lines of O VII, O VIII, Fe XVII, Ne IX, and Ne X are always present in a proportion that is difficult to predict. We discuss the NEI scenario in a forthcoming publication.    





If the model outlined above is correct, NPS and similar structures offer a possibility to probe the rotation pattern of the hot gas in the Milky Way. Future large grasp microcalorimetric missions like Line Emission Mapper <cit.> capable of mapping the entire NPS region in the course of its all-sky survey <cit.> will be instrumental in measuring abundances and velocities using emission lines of elements from carbon to iron across the 0.2-2 keV energy band to narrow the range of plausible models. 






§ CONCLUSIONS

While the North Polar Spur is usually attributed to a shock front associated with the activity of our Galactic Center or a local SNR, we discuss an alternative scenario. It posits that metal-enriched plumes rise above the disk from active star-forming regions. Interactions with the hot halo gas give these plumes the appearance of bent spirals. Their shapes sensitively depend on the rotation pattern of the hot gas above the disk. These plumes are mostly hotter than the ambient gas and can be metal-rich compared to the hot gas in the Milky Way halo. 
In this model, the NPS itself is associated with the star formation within a 3-5 kpc distance from the Galactic Center. Fainter plumes might be associated with other star-forming regions in the Galaxy. 







§ ACKNOWLEDGMENTS


This work is partly based on observations with the eROSITA telescope onboard SRG space observatory. The SRG observatory was built by Roskosmos in the interests of the Russian Academy of Sciences represented by its Space Research Institute (IKI) in the framework of the Russian Federal Space Program, with the participation of the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The eROSITA X-ray telescope was built by a consortium of German Institutes led by MPE, and supported by DLR. The SRG spacecraft was designed, built, launched, and is operated by the Lavochkin Association and its subcontractors. The science data are downlinked via the Deep Space Network Antennae in Bear Lakes, Ussurijsk, and Baikonur, funded by Roskosmos. 

The development and construction of the eROSITA X-ray instrument was led by MPE, with contributions from the Dr. Karl Remeis Observatory Bamberg & ECAP (FAU Erlangen-Nuernberg), the University of Hamburg Observatory, the Leibniz Institute for Astrophysics Potsdam (AIP), and the Institute for Astronomy and Astrophysics of the University of Tübingen, with the support of DLR and the Max Planck Society. The Argelander Institute for Astronomy of the University of Bonn and the Ludwig Maximilians Universität Munich also participated in the science preparation for eROSITA. The eROSITA data were processed using the eSASS/NRTA software system developed by the German eROSITA consortium and analyzed using proprietary data reduction software developed by the Russian eROSITA Consortium.


IK acknowledges support by the COMPLEX project from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program grant agreement ERC-2019-AdG 882679.


aa

 






















§ X-RAY DATA ANALYSIS



 Data from all four consecutive scans are combined together after filtering for the periods of enhanced solar activity, which causes a strongly elevated level of the instrumental background. For the imaging analysis, the data taken with all seven Telescope Modules (TMs) are combined, while for the spectral analysis, only five TMs protected by the on-chip filter (i.e. TMs 1-4, and 6) are used. Data reduction, filtering, vignetting-correction, and background subtraction are performed in the same way as was done in the previous studies exploring Galactic diffuse X-ray sources <cit.>.


In particular, the energy-dependent contribution of the instrumental background is modeled and subtracted based on the calibration data accumulated via observations with the "Closed Filter Wheel" configuration, while corrections for exposure time and vignetting are conducted so that the data are characterized by Field-of-View averaged response matrices. 








§ PLUMES IN 3D


In this section, we outline two simple scenarios for plume trajectories. In the first scenario, a plume is produced by a continuous source. The plume initially moves together with stars/gas in the disk and eventually joins the motion of the halo gas. In the second scenario, a trajectory of a massless buoyant bubble is considered.



 §.§ Plumes from a moving source

The first case corresponds to the model described by equations <ref> and <ref>. Fig. <ref> illustrates the shapes of the "plumes" in a stationary 3D frame. The blue curve shows the case when the source of the plume moves with the same velocity as the gas in the disk. As the plume rises, it gradually joints the slowly rotating gas in the disk. The red curve shows modest modifications that appear when the source of the plume moves slower than the gas and stars in the disk. By design, the rising plumes retain the same distance from the Galactic Center in projection to the disk.   












 §.§ Trajectory of a massless buoyant bubble

Here we consider the trajectory of a single (massless) bubble moving in a stratified atmosphere of the Galaxy. 
Similarly to galaxy clusters <cit.>, the velocity of the bubble is set by the balance of pressure gradients and the drag force acting on the bubble, i.e.

    u^2u⃗/u1/2ρ A C_d=-∇ P V,

where u⃗ is the 3D velocity vector of the bubble velocity with respect to the halo gas (u= | u⃗ |), ρ and P are the halo gas density and pressure, respectively, A and V are the area and volume of the bubble, and C_d is the appropriate drag coefficient <cit.>. The bubble size is assumed to be smaller than the pressure scale height and the velocity remains subsonic. For an atmosphere in hydrostatic equilibrium, 1/ρ∇ P=-∇ϕ and, therefore,   u⃗ can be directly calculated from the gravitational potential ϕ, which in our case can include a contribution from halo gas rotation. 
To this end, we use the approximation of the MW potential from <cit.> to which we add a centrifugal term (-V_h^2ln R) for a cylindrical rotation, where R is the distance from the MW center in the disk plane and V_h is the gas rotation velocity. The gas rotation velocity of the gas is added to u⃗ to get the bubble velocity in the non-rotating frame. The effects of rotation on the bubble trajectory are illustrated in Fig. <ref>. This figure shows that different configurations are possible depending on the gas rotation pattern. We note here, that unlike Fig. <ref>, the trajectories of individual "bubbles" are shown rather than the "plumes" that are formed by many bubbles released at different times.











§ ENHANCED METALLICITY OF PLUMES




The enhanced metallicity of the NPS hot material enriched by supernovae is important for the proposed scenario. 
The thermal energy of the NPS E_ th∼ 2 × 10^55 erg (see <ref>) suggests the explosion of N_ sn∼ 2× 10^4 
  core-collapse supernovae (CCSN) with a typical kinetic energy of 10^51 erg (lifetime of a star-forming region is short for SNe Ia to explode). 
We rely on two sets of oxygen nucleosynthesis calculations  for CCSN 
 progenitors in the range of 11 - 40M_⊙ by <cit.>
 (WW95) and 
<cit.> 
  (K06).
In K06, the  data on 11M_⊙ progenitors are lacking. We, therefore, adopt the ejected mass of oxygen from WW95 for this particular progenitor.  The abundance of other elements relative to oxygen is assumed to follow (approximately) the solar composition. 

 The average mass of ejected oxygen per CCSN (m_ O) is inferred assuming Salpeter initial mass function   dN/dm ∝ m^-2.35 in the mass range of 0.1 – 100M_⊙. We find comparable values of m_ O, 2.2M_⊙ and 2.6M_⊙,  
  for WW95 and K06 data, respectively.
The average m_ O=2.4M_⊙ multiplied by the CCSN number N_sn  provides us with the total amount of oxygen synthesized by supernovae of the star-forming region M_ O≈ 5× 10^4M_⊙.
The synthesized oxygen mixed with the gas of the solar composition (solar abundance X()_⊙ = 0.01) produces enhanced overall metallicity. 
The total mass of the mixture (M_ hot) is fixed by the thermal energy of the X-ray-emitting gas 
 E_ th = (5/2)kT_ xM_ hot/(μ m_p). 
For the average molecular weight μ = 0.61 and temperature kT_ x =0.6 keV one obtains  
 M_ hot = 4×10^6M_⊙. 
The expected oxygen abundance of the NPS hot gas is then X() = X()_⊙ + M_ O/M_ hot≈ 0.02, twice the solar value.
  



§ THE CHIMNEY MODEL





The clustering of core-collapse supernovae produced at the end of the evolution of massive stars in OB associations and compact clusters has a profound effect on the interstellar medium 
and the galactic ecology <cit.>. Multiple correlated supernovae and the powerful winds of OB stars create superbubbles and eventually (for powerful enough systems) produce the chimney-type structures <cit.>. We consider such superbubbles as potential sources of the plumes. 
 To study the effect of the galactic magnetic field on the superbubble breakthrough from the disk to halo <cit.> performed 3D MHD simulations for different assumptions on the large-scale magnetic field structure. The magnetic field with the strength of ∼ 5 μG with a broad ∼ kpc scale height can confine a superbubble of a modest kinetic luminosity 3× 10^37  for about 20 Myr within the scale height of | z |∼ 300 pc from the disk, while in a model with the field that follows the scaling B ∝ρ^1/2 (with the mid-plane field magnitude of 5 μG) the superbubble blows out to the halo. The evolution of superbubbles in density stratified disks that blow out into galactic halo may be a subject of Rayleigh-Taylor instabilities <cit.>. 
 A parsec-resolution multi-phase simulation of the local star-forming galactic disk with the account for SNe feedback effects revealed that the hot galactic outflows (with gas temperatures above 10^6 K) may carry about 10%-20% of the energy and 30%-60% of the metal mass injected by supernovae <cit.>. These values are broadly consistent with the energy requirements needed to produce the X-ray plumes discussed in the text.  
 
 Non-thermal particles accelerated by shocks from SNe and stellar winds at the active phase of the evolution of superbubble which is about 10 Myrs <cit.>. Then relativistic particles will be blown out to the low halo with the frozen-in magnetic fields of the chimney-type plasma outflow. Relativistic electrons of energy ∼ 100 GeV would have in the halo a lifetime of about 10 Myrs due to synchrotron - Compton losses and will radiate in magnetic fields of a few μG at frequencies of  50 GHz. The synchrotron-Compton cooling time for the relativistic electron of energy E_ GeV (measured in GeV) in the galactic halo can be estimated as 
 
    t_ syn∼  10^9E_ GeV^-1/ 1 + 0.1 B_μ G^2  yr.


 

























§ THE ROLE OF W43


Below, we discuss in brief the mini-starburst complex W 43 as a prototype 
powerhouse located in the Galactic molecular ring assuming that an ensemble of such complexes 
may supply the plumes over a long period of ∼ hundred million years. 
The molecular complex W 43 with the estimated 
mass ∼ 7 × 10^6 is presumably located at the connecting point
of the Scutum-Centaurus Galactic arm and the Galactic bar <cit.>. W 43 excited a giant H II region dubbed NGC 3603. The equivalent diameter of the coherent complex of molecular clouds is ∼ 140 pc and it is surrounded by an atomic gas envelope of about two times larger diameter. <cit.> noted that the transition from circular to elliptical orbits
in the spiral arm and bar potentials might cause high-velocity streams and thus efficient star formation episodes of current rate at least ∼ 0.01 yr^-1 as estimated from 8 μm luminosity measurements but it is likely increasing to ∼ 0.1 yr^-1. On longer time scales ∼ 100 Myrs of interest for this paper multiple episodes of high star formation rate can be expected from colliding molecular clouds entrained by the large scale streams.   

The giant HII region NGC 3603 which is associated with the complex produces the Lyman continuum photon rate of about  4 × 10^51 photons s^-1 is the most powerful HII region in the Galaxy <cit.>. A compact dense cluster of young massive stars HD 97950 (also known as W 43 cluster)  powering NGC 3603 has an estimated dynamic stellar mass of 18,000  including   
more than 70 O and 3 WR stars. The distance to the powerful starburst region NGC 3603 
of 7.2 ± 0.1 kpc was estimated by <cit.> from GAIA DR2 stellar parallax measurements (see also <cit.> for a recent discussion). From the VLBA trigonometric parallax measurements of masers  toward W43 <cit.> derived a distance of 5.49_-0.34^+0.39 kpc to the complex <cit.>.  The origin and ages of 
molecular clumps and stellar population in NGC 3603 are still the subject of debate <cit.>. ALMA observations revealed that the core mass function in W43-MM1 is much shallower in the high-mass range than the standard initial mass function <cit.>.  

<cit.> discussed the formation of cluster W 43  due to a collision of two molecular clouds about 1 Myr ago. This age is consistent with that of the highest mass stellar population in the cluster. 
The suggestion by <cit.> about the origin of the star-forming complex W 43/NGC 3603 due to the collision of molecular clouds following their circular orbits in the Scutum-Centaurus arm and the clouds at the elliptical orbits of the Galactic bar makes the complex W 43 as well as other complexes along the Molecular Ring where the Galactic arms meet the bar to be favorable locations for the intense star formation episodes in the past.   
 
Very high energy emission source HESS J1848-018 <cit.> can possibly be associated with W 43. From  Fermi-LAT detection of extended high energy gamma-ray source source in the direction of W 43 <cit.> estimated the total cosmic ray energy within the source to explain the detected flux to be (2.3 ± 0.3) × 10^48 erg.

















 Paper Title: A closer look at dwarf galaxies exhibiting MIR variability: AGN confirmation and comparison with non-variable dwarf galaxies.





Archana Aravindan
aarav005@ucr.edu

0000-0001-7578-2412]Archana Aravindan
Department of Physics and Astronomy, University of California, Riverside, 900 University Ave, Riverside CA 92521, USA

0000-0003-4693-6157]Gabriela Canalizo
Department of Physics and Astronomy, University of California, Riverside, 900 University Ave, Riverside CA 92521, USA

0000-0002-4902-8077]Nathan Secrest
U.S. Naval Observatory, 3450 Massachusetts Avenue NW, Washington, DC 20392-5420, USA

0000-0003-2277-2354]Shobita Satyapal
George Mason University, Department of Physics and Astronomy, MS3F3, 4400 University Drive, Fairfax, VA 22030, USA

0000-0002-4375-254X]Thomas Bohn
Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan




































































§ ABSTRACT 
Detecting active black holes in dwarf galaxies has proven to be a challenge due to their small size and weak electromagnetic signatures. Mid-infrared variability has emerged as a promising tool that can be used to detect active low-mass black holes in dwarf galaxies. We analyzed 10.4 years of photometry from the ALLWISE/NEOWISE multi-epoch catalogs, identifying 25 objects with AGN-like variability. Independent confirmation of AGN activity was found in 68% of these objects using optical and near-infrared diagnostics. Notably, we discovered a near-infrared coronal line [S9] λ 1.252 μm in J1205, the galaxy with the lowest stellar mass (log M_* = 7.5 M_⊙) and low metallicity (12 + log(O/H) = 7.46) in our sample. Additionally, we found broad Paα potentially from the BLR in two targets, and their implied black hole masses are consistent with black hole-stellar mass relations. Comparing non-variable galaxies with similar stellar masses and WISE W1-W2 colors, we found no clear trends between variability and large-scale galaxy properties. However, we found that AGN activity likely causes redder W1-W2 colors in variable targets, while for the non-variable galaxies, the contribution stems from strong star formation activity. A high incidence of optical broad lines was also observed in variable targets. Our results suggest that mid-infrared variability is an effective method for detecting AGN activity in low-mass galaxies and can help uncover a larger sample of active low-mass (< 10^6 M_⊙) black holes in the universe.



























§ INTRODUCTION
 

The formation of the first black holes (BHs) in the universe is still not well understood <cit.>. There are two most commonly discussed formation scenarios for black hole seeds, which eventually evolved into supermassive black holes (SMBHs) at the centers of nearly every massive galaxy. In one scenario, the first BHs were formed by the collapse of early Population III stars. The BHs formed by this method are likely to have masses in the range 100 - 1000 M_⊙ <cit.> and should undergo phases of accretion at super-Eddington rates in order to reach the masses of the SMBH that are currently present in the universe <cit.>. In an alternate scenario, the first BHs were formed by the direct collapse of gas in the early universe <cit.>, and have heavier masses of about 10^4 -10^5 M_⊙ <cit.>. Other commonly suggested formation mechanisms include runaway collisions in dense star clusters <cit.>, which can produce BH seeds with masses 10^3 - 10^5 M_⊙.

Direct observational evidence for the nature of the earliest BHs is challenging to obtain. However, observing BHs in dwarf galaxies in the nearby universe could help narrow down the formation scenarios <cit.>. If dwarf galaxies have a higher occupation fraction (percentage of galaxies with BHs) of low-mass (< 10^3 M⊙) BHs, it is likely that the BH seeds were formed from the collapse of Population III stars <cit.>. Conversely, if they have a lower occupation fraction of high-mass (> 10^5 M⊙) BHs, the direct collapse scenario is favored <cit.>.

Detecting active black holes, or active galactic nuclei (AGN), which are actively accreting material, is one of the best methods to determine the presence of BHs in galaxies. Although AGN detections cannot provide a complete census of the BH occupation fraction in dwarf galaxies (since only a small fraction of BHs are likely to be actively accreting and visible as AGN), identifying AGN can still offer constraints on the occupation fraction. Moreover, AGN detections can guide follow-up spectroscopic observations to determine BH masses using methods such as broad-line measurements, further refining our understanding of BH formation scenarios.

With advancements in telescope technology and the advent of large-scale surveys, it has become clear that a sizeable number of dwarf galaxies are likely to host AGN <cit.>. Results from JWST indicate the presence of BH in low-metallicity galaxies in the early universe <cit.> and local dwarfs can be used as laboratories in which to study similar BH, and their effects on the host galaxy in much greater detail than is possible at high z <cit.>. Additionally, the detection of AGN in dwarf galaxies can be used to constrain the BH masses and occupation fractions <cit.>.

It is challenging to identify active BH in dwarf galaxies using traditional diagnostic methods such as the Baldwin, Phillips & Terlevich (BPT) diagram <cit.> or Veilleux-Osterbrock (VO) diagrams <cit.> as dwarf galaxies are more likely to have low gas-phase metallicities <cit.>. <cit.> show that, as the BH mass decreases, the hardening of the spectral energy distribution from the accretion disk changes the ionization structure of the nebula. The enhanced high-energy emission from IMBHs results in a more extended partially ionized zone compared with models for higher-mass BH. This effect produces a net decrease in the predicted [O3]/Hβ and [N2]/Hα emission line ratios, and the positions of galaxies can shift from the AGN region to the star-forming (SF) region. While diagrams that use [S2] and [O1] fluxes are useful for recovering AGN across a range of BH masses and galaxy metallicities  <cit.>, these diagrams still miss a large fraction of AGN <cit.>, most likely due to the strong contribution from star formation to the line ratios. The low-mass BH in dwarf galaxies also have a small gravitational sphere of influence compared to more massive BH, making them impossible to detect dynamically in galaxies apart from the nearby ones <cit.>. A low-mass BH, accreting at the Eddington luminosity, will be of comparable luminosity to that produced by stars in the galaxy, also making it challenging to accurately detect AGNs in dwarf galaxies.

A promising method to locate AGNs in dwarf galaxies is by making use of variability. AGNs are known to be variable across the entire electromagnetic spectrum and in different time scales, ranging from months to days depending on the mass of the galaxy <cit.>. This variability is often attributed to instabilities in the accretion disk or surface temperature fluctuations <cit.>. <cit.> made use of optical variability using the Sloan Digital Sky Survey <cit.> and Palomar Transient Factory <cit.> to identify AGNs in dwarf galaxies. They found that nearly 75% of low-mass galaxies in their sample with AGN-like variability have narrow emission lines dominated by star formation and would be completely missed by the BPT diagram. Similarly <cit.> used the High Cadence Transit Survey  <cit.> to look for optical variability and found that nearly 95% of variable AGNs in their sample do not exhibit any AGN-like optical emission line ratios. <cit.> found 44 out of 25,714 dwarf galaxies had optically variable AGN candidates using optical photometry from the Zwicky Transient Facility <cit.>. A large sample of optically variable AGN in dwarf galaxies was also obtained from the photometric analysis of various deep field surveys such as Dark Energy Survey <cit.> and COSMOS <cit.>, as well as predictions for possibly uncovering a larger sample from future surveys such as the Rubin Observatory <cit.>.

However, high levels of extinction can severely reduce the effectiveness of using optical variability to uncover AGN in low-mass galaxies.  Mid-infrared (MIR) variability, on the other hand, is less sensitive to optically obscured and Compton thick AGN, which form a significant fraction of the low luminosity AGN population <cit.>. Ultraviolet and optical emission from the accretion disk is reprocessed by the dust surrounding it and is re-radiated in the MIR. Thus, any fluctuations from the disk would be manifested in the MIR as well, leading to variability in the MIR <cit.>.

<cit.> found a low incidence of MIR variable dwarf galaxies (0.02%) using 8.4 years of photometry from the AllWISE/NEOWISE multi-epoch catalogs and concluded that high cadence data is necessary for MIR selection of AGN in dwarf galaxies. <cit.> used forward–modeled MIR photometry of time-resolved WISE coadded images to uncover 148 MIR variable galaxies out of a sample of 79,879 dwarf galaxies. They also found that spectroscopic approaches to AGN identification would have missed 69% of their WISE MIR variable dwarfs.

While AGNs are variable in the MIR, it is important to realize that there are several other transients that are capable of varying in the infrared as well. Although MIR variability selection is less affected by supernovae (SN) contamination than optical variability selection due to the fainter SEDs of SN <cit.>, there has been evidence of infrared variability caused by SN and variable stars <cit.>. There have also been instances of infrared transients from interactions between stars and planets <cit.>, or from mergers between a pair of compact objects such as neutron stars or white dwarfs <cit.>. Although analyzing the evolution of the light curves with time can eliminate non-AGN sources such as rapid, one-off transient events, studies have shown that intermediate-mass black holes are capable of varying over timescales ranging from a few hours to a few days, making it difficult to distinguish them from more rapid transients <cit.>.

Thus it is important to determine if MIR variability is accurately detecting AGN, specifically in dwarf galaxies. In this work, we aim to determine the effectiveness of MIR variability in tracing AGN activity by selecting a sample of dwarf galaxies that are variable in the MIR from Wide-field Infrared Survey Explorer <cit.> light curves. We then obtained NIR spectroscopic observations of the subset of targets that do not show signs of AGN activity in the optical in order to look for other AGN indicators. These include the presence of NIR coronal lines which have proved to be a useful method to identify the presence of AGN in dwarf galaxies <cit.> or broad lines from the broad line region (BLR) that might be too obscured to detect in the optical. We also analyzed how galaxies with variability compare to non-variable galaxies with similar WISE colors to study the differences, if any, between the two samples of galaxies. We do this to understand the effects of any biases in host galaxy properties on the observed MIR variability to indicate if MIR variability is preferentially detected in certain types of galaxies. 

The paper is organized as follows: In Section <ref>, we describe the variable sample selection, present details of the targets, and describe the observations and data reduction.  In Section <ref>, we present results of evidence of AGN activity in a subsample of MIR variable dwarfs. In Section <ref>, we compare the sample of variable dwarfs to a control sample of non-variable dwarf galaxies and analyze the differences between the properties of the two samples. We then summarize our conclusions in Section <ref>. Throughout the paper, we assume ΛCDM cosmology, with h = 0.73.



§ DATA AND OBSERVATIONS




 §.§ Sample Selection
  

As in <cit.>, we created a sample of dwarf galaxies from the NASA-Sloan Atlas (NSA), version . We started by matching the full NSA catalog to AllWISE with 10, which returned a unique match for 637168 out of 641409 (99%) of objects. We required that the elliptical Petrosian masses and Sérsic masses differ by no more than 0.5 dex, which <cit.> found to effectively remove galaxies cut off in their SDSS images, leaving out 8739 objects.

Because we use catalog AllWISE Multiepoch and NEOWISE-R PSF photometry for the light curves, we require that the galaxies not be extended or convolved with other sources in WISE, respectively, requiring , . This left 459000 galaxies. Finally, to place our sample on the WISE W1-W2, W2-W3 color-color diagram, we require that all objects have valid AllWISE photometry in the first three bands, with corresponding  equal to zero, leaving 316715 galaxies. <cit.> found that making a cut of AllWISE W2 < 14.5 mag removes the Eddington bias from the light curves, leaving 224969 systems. Of these, 6226 galaxies have elliptical Petrosian stellar masses less than 2×10^9 h^-2 M_, which corresponds to log(M_⋆/M_) < 9.6 using h=0.73. 


ccccccccccc

Summary of AGN confirmations in variable dwarf galaxies
Target     Stellar Mass     8cAGN indicators other than MIR variability     Observed with NIRES

3-10
        Optical BL      BPT     He II     Optical variability     WISE     NIR Coronal lines     NIR BL     NIR Diagnostics     
J000011.72+052317.4     9.29     (a)     ⋯      ⋯     ⋯          ⋯     ⋯     ⋯     ⋯

        J082912.67+500652.3     9.53      (b,c)      (e)      ⋯      (g)      (i)      ⋯     ⋯     ⋯     ⋯  

        J084702.07+035202.0^     9.57     ⋯     ⋯     ⋯      ⋯      (h)     ⋯     ⋯           

        J085431.18+173730.5     9.29     (d)     ⋯     ⋯     ⋯     ⋯     ⋯     ⋯     ⋯     ⋯

        J090613.76+561015.3     9.30     ⋯     (e)          ⋯          (j)     ⋯     ⋯      ⋯  

        J092810.51+150228.0^     9.47     ⋯     ⋯     ⋯     ⋯     (h)    ⋯             

        J093608.61+061525.9     8.29     ⋯     ⋯     ⋯    ⋯     ⋯    ⋯     ⋯      ⋯     †

        J095438.79+403204.4     9.52     (a, b)     ⋯    ⋯     ⋯          ⋯    ⋯    ⋯     ⋯ 

        J102315.19+122227.9     9.01     ⋯     ⋯          ⋯     ⋯     ⋯     ⋯      ⋯     †

        J103206.01+225921.6^     8.87     ⋯     ⋯     ⋯     ⋯     (h)        ⋯           

        J114433.55+072643.0     9.51     ⋯     ⋯     ⋯    ⋯          ⋯                 

        J120325.67+330846.1     9.10     ⋯     (e)         ⋯     (i)      ⋯     ⋯     ⋯     ⋯ 

        J120503.54+455151.0^     7.52     ⋯     ⋯     ⋯      ⋯     (h)         ⋯     ⋯     

        J122608.78-025226.2     8.79     ⋯     ⋯     ⋯     ⋯      ⋯      ⋯      ⋯     ⋯     †

        J125305.96-031258.7     8.61     ⋯    ⋯     (f)     ⋯     ⋯    ⋯     ⋯   ⋯     ⋯  

        J130819.11+434525.6     9.57     ⋯        (f)     ⋯         ⋯     ⋯     ⋯     ⋯  

        J132053.67+215510.8^∗     8.62     ⋯     (e)    ⋯     ⋯        ⋯    ⋯    ⋯     

        J135844.73+123546.9     9.34     ⋯    ⋯         ⋯      ⋯     ⋯     ⋯     ⋯     

        J140630.09-001939.3     8.93     (c)     ⋯     ⋯     ⋯        ⋯     ⋯    ⋯     ⋯ 

        J142835.84+570834.3     9.42     ⋯     ⋯     ⋯     ⋯     ⋯     ⋯     ⋯    ⋯     

        J143744.58+524333.3     8.51     ⋯     ⋯     (f)     ⋯     ⋯     ⋯    ⋯     ⋯     †

        J152637.36+065941.6     9.39     ⋯     ⋯     (f)     ⋯         ⋯     ⋯     ⋯     ⋯  

        J154404.29+275334.0     9.14         ⋯     ⋯     (g)     ⋯     ⋯     ⋯     ⋯     ⋯  

        J161243.20+124505.2     9.26     ⋯     ⋯    ⋯    ⋯     ⋯     ⋯    ⋯     ⋯     

        J233245.03-005845.8     9.45     (a, b)     ⋯     ⋯     ⋯         ⋯     ⋯     ⋯     ⋯
Column(1): SDSS name of the target. Column (2): Stellar Mass of the target in units of  logM_. Columns(3-10) AGN indicators other than MIR variability. Black checkmarks indicate confirmation in literature (references indicated in parenthesis), while blue checkmarks indicate confirmation based on the analysis carried out in this work based on the SDSS and NIRES spectrum. Column(3): Presence of broad optical Balmer lines  (a)<cit.>, (b)<cit.>, (c)<cit.>, (d)<cit.>. Column(4): Line ratios indicative of AGN activity based on the optical BPT diagram (e)<cit.> Column(5): Line ratios indicative of AGN activity based on He II λ 4686 Ådiagnostic diagrams (f)<cit.> Column (6): Presence of optical variability (g)<cit.> Column(7): WISE colors indicative of AGN activity (h)<cit.>, (i)<cit.> Column(8): Presence of NIR coronal lines (j)<cit.> Column(9): Presence of broad NIR Paschen lines Column(10): Line ratios indicative of AGN activity based on NIR diagnostic diagrams. Column(11): Whether the target was observed with NIRES. Only targets that had no other confirmation of AGN in literature (black checkmarks) were observed (see exceptions below)

: Targets that only had WISE colors indicative of AGN activity and no other AGN indicator at the time of proposing for observations.

∗ J1320 was classified as an SN candidate in spite of being classified as BPT AGN. <cit.>.

† Targets planned for observations but could not be observed due to poor weather. 
Note: These are the reported findings from the literature for the above targets. It is possible that further targeted searches across different wavelengths could reveal additional AGN indicators.



ccccccccc[ht!]
Observing Log 

Galaxy     UT Date Obs.     Redshift     Stellar Mass      Exp. Timea     Slit PA     Airmass     
Telluric     Flux variationb
 [-0.1cm]
(SDSS Name)     (YYYY MM DD)          (logM_)     (s)     (degrees)              (%)
J084702.07+035202.0     2023 04 07     0.059     9.57      8×240     306     1.05     HD 85504     15
 
        J092810.51+150228.0     2023 04 08     0.078     9.47     4×240     45     1.50     HD 89239     25

        J103206.01+225921.6     2023 04 08     0.058     8.87      4×240     0     1.50     HD 101060     50
 
        J114433.55+072643.0     2023 04 07     0.066     9.51     4×240     322     1.30     HD 95126     15
 
        J120503.54+455151.0 c     2023 04 07     0.065     7.52     6×240     0     1.17     HD 95126     15
 
        J132053.67+215510.8     2023 05 08     0.022     8.62      3×240     30     1.28     HD 116960     <10
 
        J135844.73+123546.9     2023 05 08     0.024     9.34      4×240     90     1.24     HD 122945     <10
 
        J142835.84+570834.3     2023 05 08     0.059     9.42      6×240     344     1.28     HD 116405     <10
 
        J161243.20+124505.2     2023 05 08     0.034     9.26      3×240     4     1.46     HD 210501     <10
aExposures were typically done in ABBA nodding.
bVariation in flux between individual exposures.
cIndicates galaxies with coronal line detections. See Section <ref>







Single-epoch photometry from the AllWISE Multiepoch Photometry (MEP) and NEOWISE tables for these 6226 dwarf galaxies were subsequently queried using the AllWISE source positions, using zero position offset for MEP and a 3 tolerance for the NEOWISE positions. The single-epoch photometry was likewise cleaned, requiring valid measurements in W1 and W2, allowing a maximum deviation of ±1 mag between the single-epoch W2 and AllWISE W2 corresponding to the limit set by the Eddington bias,  and  for uncomplicated PSF photometry, W1 and W2  and  =0,  and . This removed 11% of single-epoch measurements, retaining cleaned data for all 6226 dwarf galaxies. On average, there are 276 single-epoch photometric measurements for each source, from which variability metrics were calculated as in <cit.>. In that work, we found that Pearson r between W1 and W2 is the most useful initial discriminant for genuine variability, with r > 0.4 being optimal for WISE data. This produced an initial sample of 60 objects, allowing manual inspection of their light curves. We further classified the light curves that exhibit clear variability into those with rapid AGN-like variability and those with supernova-like flares or slow declines in apparent magnitude. 

Although we started with a sample of dwarf galaxies, cutting on objects with AGN-like photometric variability creates a strong selection bias on the otherwise small fraction of contaminating background quasars and galaxies with incorrect spectroscopic redshifts. To mitigate this, we uploaded the WISE coordinates of the variable dwarf galaxy candidates to the SDSS thumbnail server and removed clear instances of nearby background quasars or objects with spectra not matching their catalog redshifts. After removing these contaminants, we were left with a final sample of 25 dwarf galaxies with bona fide, AGN-like, MIR variability (see Fig. <ref> for an example). As a check, we estimated the number of background quasars that could be contaminating our sample of 25 objects. Using the SDSS/BOSS quasar catalog <cit.>, 98% of which have WISE photometry (738088 objects), we find that 114119 have W2 < 14.5 mag like our sample. As this catalog covers ∼28% of the sky, the corresponding sky density of SDSS/BOSS quasars is ∼10 deg^-2. However, because our AGN selection criteria is mid-IR based, we correct for the obscured population that may have been largely missed by the SDSS/BOSS surveys. We conservatively assume that about half of the mid-IR AGN-selected sources are heavily obscured <cit.>, giving a sky density of 20 deg^-2. The final sample of 25 objects has a maximum NSA/AllWISE position offset of 44, of which 6023 of the 6226 dwarf galaxies satisfy. Then, the expected number of contaminant background QSOs is ∼0.6. Thus, even if mid-IR variability selects all background QSOs, it is likely that there is at most one contaminating background QSO in our final sample of 25 dwarf galaxies. The final sample is presented in Table <ref>. 

The stellar masses listed in Table <ref> were obtained from the NSA catalog <cit.>.  In order to estimate the uncertainty in these masses, we compared the masses listed by NSA with masses reported in other catalogs. We found an average deviation of 0.01 dex between the NSA masses and those reported in the MPA-JHU catalog <cit.>. This falls in the range of the expected deviations (-0.2 dex - 0.2 dex) between the MPA-JHU and NSA stellar masses determined by <cit.>. We also determined an average deviation of 0.3 dex between the NSA masses and those reported by the Portsmouth group <cit.> for galaxies with masses determined in all three catalogs. These values provide a measure of the average uncertainty in the masses of our sample. Thus, the stellar masses that we use fall within the range of expected masses of dwarf galaxies even with the predicted uncertainty. Also see Fig. <ref> for their positions on the optical BPT diagram.

We then extensively searched to see if any variable dwarf galaxies had already been identified as AGN in the literature. Twelve of the 25 (48%) dwarf galaxies were already identified as AGN by a combination of other optical methods, such as the presence of broad Balmer lines, strong He2 emission, optical variability, and emission line ratios indicative of AGN from the BPT diagram (see Table <ref> for references).

Although the MIR variable galaxy J1320 has BPT line ratios indicative of AGN, it was also classified as a potential SN candidate in the literature <cit.>, so we do not include it among the targets for which AGN activity was previously confirmed. Additionally, only one of the 25 (J1526) variable galaxies was classified as AGN from X-ray observations <cit.>. Apart from the references mentioned in the caption of Table <ref>, we also cross-referenced the variable targets in this work with <cit.> for BPT AGN, <cit.> for optical broad lines, and <cit.> for X-ray data, but we did not find any matches.



 §.§ Observations and Reductions
 



We selected a subsample of variable dwarf galaxies for observations in the NIR. The 13 galaxies in the subsample include nine galaxies with no confirmations of AGN by any method (including J1320) and four galaxies that only had WISE colors indicative of AGN activity. These were the only galaxies in the sample exhibiting MIR colors indicative of AGN without any other indicators across the criteria examined. We included the latter four galaxies because young starburst galaxies can often mimic the mid-infrared colors of AGN galaxies <cit.> and may not truly be AGNs. Probing the NIR spectra of the variable targets could uncover evidence of AGN activity, such as broad Paschen lines from the BLR or coronal lines indicative of a hard radiation field. 

NIR spectroscopy was obtained on three separate dates (UT 2023 April 7, UT 2023 April 8 and UT 2023 May 8) with the Keck II Near-Infrared Echellette Spectrometer <cit.>. NIRES is a NIR echellette spectrograph with a wavelength coverage across five bands (Z, Y, J, H, K) from 0.94 to 2.45 μm. There is a small gap in coverage between 1.85 to 1.88 μm, but this region is dominated by water absorption from the atmosphere and thus has low atmospheric transmission. The average spectral resolution of the five orders ranges between 84 and 89 km s^-1 (R = 3400). Observations on UT 2023 April 7 and UT 2023 April 8 were taken under variable and heavy cloud cover, while observations on UT 2023 May 8  were taken through light cirrus. Due to unfavorable weather conditions, we could get data for only nine of the 13 targets. Individual exposures for each target were taken for 4 minutes using the standard ABBA nodding, amounting to total exposure times ranging from 12 to 32 minutes, depending on the brightness of the target. A0 spectral class stars were used as telluric standard stars, with measured magnitudes in J, H, and K bands. Observations of the telluric stars were taken before or after each science target to correct for the atmospheric absorption features. A summary of the observations is shown in Table <ref>. 



The data were reduced using two modified pipelines. The flat fielding and background subtraction were done by the first pipeline using techniques described in <cit.> and <cit.>. Rectification, telluric correction, wavelength calibration, and extraction were done with a slightly modified version of REDSPEC [<https://www2.keck.hawaii.edu/inst/nirspec/redspec.html>]. The telluric correction was done by dividing each science target spectrum by the spectrum of its corresponding telluric star and multiplying it by a blackbody curve of the same temperature. Wavelength calibration was done using strong OH lines, and the 1D spectra were then median combined. Flux calibration of individual targets was done by converting the magnitude of the telluric star to the associated flux in that band [<https://irsa.ipac.caltech.edu/data/SPITZER/docs/dataanalysistools/tools/pet/magtojy/>]. A minor corrective factor (<5%) was introduced due to the differences between the center of the NIR bands and the center of the wavelength coverage. Because we observed through clouds, we found 15 - 50 % variations in the flux levels between individual exposures for the first two nights and less than 10 % variation in flux on the final night (Flux variations for individual objects are listed in Table <ref>). This introduced uncertainty in the flux calibration, as the latter was done relative to telluric standards observed immediately before or after each science target.  We estimated that the uncertainties are of the order of the flux variations, and we included these added uncertainties in the measurements of fluxes.



 §.§ Spectral fitting


All NIR spectra were fit using the Bayesian AGN Decomposition Analysis for SDSS Spectra 
<cit.> code, which was modified to cover the NIR wavelength regime. A narrow component for each emission line was fit simultaneously with a second-order polynomial for the continuum. Absorption features were fit using the penalized pixel fitting method (pPXF, <cit.>) with templates from the eMILES <cit.> stellar library. 

The error (noise) in the fit was determined by calculating the standard deviation of the continuum flux, represented by the grey-shaded region in the lower panel of Figure <ref>
(b). The σ_residuals, also indicated in the residual plot (lower panel) of Figure <ref>(b) denotes the standard deviation of the residuals obtained by subtracting the modeled spectrum from the observed data.



§ EVIDENCE OF AGN ACTIVITY




 §.§ Detection of coronal lines in J1205
 
Coronal lines (CL) are high ionization fine-structure lines that arise from collisionally excited forbidden fine-structure transitions in highly ionized species, with ionization potentials that extend well beyond the Lyman limit to several hundred electron volts (eV). They are typically attributed to AGN activity and have been successfully detected in dwarf galaxies hosting AGN <cit.>
We looked for NIR coronal lines in the nine galaxies observed with NIRES.

The NIR coronal line [S9] λ1.252 μm (I.P. 328.2 eV and critical density log (n_e)=9.4 cm^-3) was detected in J1205. The fit to the spectrum showing the [S9] is shown in Fig. <ref>. The [S9] line is a 5σ detection, and the measured flux of the line is 2.11 ± 0.52 × 10^-16 erg cm^-2 s^-1. The full width at half maximum (FWHM) of the line is 138 ± 10 km s^-1, which is greater than the spectral resolution of NIRES in that order (85 km s^-1) and thus is unlikely to be a noise spike or a cosmic ray signal. The errors in the flux and FWHM are given by the standard deviation of the posterior distribution from the MCMC fitting of the lines. Based on the commonly observed NIR emission lines in galaxies <cit.>, no other emission lines are likely to be found at that wavelength.

The detection of [S9] in J1205 is tantalizing evidence that it likely hosts an AGN. This is the first instance of a NIR coronal line being discovered in J1205, providing strong evidence that J1205 has an active black hole. J1205 has the lowest stellar mass and metallicity among all the targets <cit.>. It was classified as a compact starforming galaxy with extremely high [O3]/[O2] flux ratios by <cit.> and shown to have a [Ne5] λ3426 Å detection and evidence of hard ionization by <cit.>. Spectroscopic fits to the Hα emission line in spectra obtained from the Large Binocular Telescope (LBT) indicate the presence of a broad line feature, which is attributed to expansion motions of supernova remnants <cit.>.  <cit.> classified J1205 as a green pea galaxy possibly hosting an AGN based on MIR colors and variability from WISE. The low metallicity and stellar mass of J1205 would have made it impossible to detect the presence of the AGN by traditional methods such as the BPT diagram. This indicates that MIR variability can be a promising tool and, sometimes, the only method to detect AGN in low-mass galaxies.



While the presence of [S9] in J1205 is likely to indicate AGN activity, it could also possibly be the result of a tidal disruption event <cit.>. This can be determined by follow-up observations to check if the coronal line persists or declines after an interval of time. However, a TDE would still indicate the presence of a black hole in J1205.

We did not detect any other coronal lines apart from [S9] in J1205. It is possible that circumnuclear stellar populations dominate the NIR continuum and thus drown out the  CL emission, making fainter lines more difficult to detect <cit.>, especially since J1205 has been classified as a compact star-forming galaxy with very young ages (< 3 Myr) for star formation bursts. There is a feature near the noise limit that is suggestive of [Si6] (see Fig. <ref>), but the feature did not have sufficient signal to be fit using BADASS. We independently fit a Gaussian to this feature and determine the flux to be 1.78 ± 0.88 × 10^-17 erg cm^-2 s^-1, which is below the 3σ limit (see Fig. <ref>). Thus, this feature does not appear to be a 3σ detection, but it is possible that with deeper observations, this feature can be detected with greater confidence.

While the detection of the coronal line [S9] alone and the absence of [Si6], the most commonly observed coronal line in galaxies <cit.>, may seem surprising, such occurrences are not unprecedented. There have been a few instances of galaxies that have a [S9] detection, but only upper limits to the [Si VI] flux are reported <cit.>. Comparing the [Si6] to [S9] flux ratios recorded in the literature <cit.>, which range from 0.1 to 8 (more than half the galaxies have ratios less than 3.5), our measured ratio of 0.25 (assuming the 3σ flux limit for [Si6]) falls within this observed range. It is possible that we simply did not obtain sufficiently deep observations to detect the [Si6] line in J1205.





No coronal lines were detected in the remaining eight targets observed with NIRES, potentially due to the relatively low S/N of the data. To investigate this, we plotted the upper limits to the [Si6] flux for each target versus the WISE W2 (4.6 μm) flux in Fig. <ref>. We determined the upper limits to the [Si6] fluxes in the non-detections by estimating the 1σ noise levels in the region around [Si6] flux and multiplying it by three to obtain the 3σ upper limit. The solid line in Fig. <ref>indicates the best-fit relation, which was derived based on observations from <cit.> (indicated as black dots) by <cit.>. The plot indicates that J1428 is the only target for which [Si6] emission is highly unlikely to be present, as its upper limits are comfortably below the scatter of the relation, and it would have been detected if it were present. For the rest of the targets, we would likely require deeper observations in order to detect [Si6].  The absence of CLs in these targets is unlikely to be due to high extinction, as J1205 (the target with CL detection) does not have significantly lower extinction (calculated from the Hα over Hβ Balmer decrement) than the rest of the observed targets.

While the presence of CLs in a galaxy's spectrum confirms the existence of a black hole, the absence of CLs does not mean the contrary. <cit.> analyzed a sample of 9 dwarf galaxies with optical and NIR line ratios indicative of  AGN and found NIR CLs only in 5 of them (55 %). Several other works in the literature have found less than 100% CL detections in samples of Sy1 and Sy2 galaxies with similar spectral windows: 66% (36 out of 54, <cit.>), 25%, (5 out of 20, <cit.>), and 43% (44 out of 102, <cit.>). 

The relatively low detection rates in all of these samples could be an observational effect, as ground-based observations are always hampered by telluric absorption from the atmosphere. Depending on a target's redshift, the CLs can be shifted into regions of strong telluric absorption and/or in regions dominated by strong OH skylines, making it difficult to detect weak CLs. Other factors include a dominant contribution from the stellar population to the NIR continuum and the presence of stellar absorption features.


 


 §.§ Detection of broad Paα lines
 

Two of the variable targets (J0928 and J1144) show a broad line component in Paα at 1.876 μm. We justify the need for a second component based on the F-test. The F-test determines the significance of the fit between a complex (higher-order) and a simple (lower-order) model in order to justify the number of Gaussian components we include in our fit. To do this, we calculated the standard deviation of the residuals of the fit (given by σ). We performed the F- test, given as F = (σ_lower-order)^2/(σ_higher-order)^2, for one versus two components for the Gaussian fits to the Paα line. The F-test was calculated over a wavelength range of 1.86–1.89 μm, which includes the Paα λ 1.876 μm. An F value greater than 3 indicates that the higher-order fit (i.e., the fit using more components) is justifiable. Using this method, we determined that the Paα emission lines in J0928 and J1144 both required a second Gaussian component, which we allowed to freely vary to fit the broad feature present in the emission lines using BADASS. The fits to the spectrum can be seen in Fig. <ref>.

Both J0928 and J1144 show a broad component to Paα while not showing a broad component in the optical Balmer lines (at least at the spatial resolution of the SDSS spectrum, observed with a fiber that includes significant contaminating/diluting stellar light). In galaxies with strong dust obscuration, it is possible that the broad line region (BLR) does not manifest itself in optical emission lines. However, NIR wavelengths are less affected by dust extinction, making it easier to observe broad components that would be obscured at optical wavelengths. This method has been used to uncover optically hidden AGN in massive galaxies <cit.>. There were no indicators of a broad component being present in any of the forbidden lines in the NIR for the two targets. Combined with the fact that the SDSS spectrum of the two targets also did not have a significant broad component in the forbidden [O3] lines, it is unlikely that the broad component is produced by an outflow.



Assuming the broad lines measured are from the BLR, we used the FWHM and luminosity of the broad Paα component to estimate black hole masses. Using virial mass estimators from <cit.>, we determined the black hole mass (Table <ref>) from the following equation:

    M/M_⊙ = 10^7.07( L_Pa α/10^42 erg s^-1)^0.49×( FWHM/10^3 km s^-1)^2




The errors given in Table <ref> are the standard deviation of the posterior distribution from the MCMC fitting of the lines. The measured black hole masses agree with the predicted black hole masses based on the stellar masses of J1144 and J0928 (Fig. <ref>) from <cit.>.





 §.§ NIR diagnostic diagrams
 

Optical diagnostic diagrams like the BPT diagram have been well studied and used extensively to distinguish between AGN and SF regions in galaxies. However, similar diagnostic diagrams in the NIR are scarce, chiefly due to a lack of large spectroscopic surveys in the infrared. Nevertheless, there have been several attempts to recreate similar diagnostic diagrams in the infrared that can be used to distinguish between AGN and SF regions <cit.>. In particular, <cit.> used CLOUDY photoionization models to identify AGN - SF diagnostics based on the ratio of bright near-infrared emission lines such as  [S3] 9530 Å, [C1] 9850 Å, [P2] 11880 Å, [Fe2] 12570 Å, and [Fe2] 16470 Å to the Paschen lines. They used a sample of 130 AGN and SF galaxies with redshifts 0<z<3 observed using JWST-NIRSpec <cit.> as part of the Cosmic Evolution Early Release Science Survey <cit.> and found that this diagram is able to successfully distinguish between AGN and SF galaxies. Furthermore, they found 60% more AGN using NIR diagnostic diagrams than when using optical BPT diagrams. Thus NIR diagnostic diagrams are able to uncover optically `hidden' AGNs, thus highlighting the importance of using NIR wavelengths to potentially locate AGN in dusty galaxies. Since optical emission lines are significantly affected by extinction, it is possible to overlook AGN indicators from optical emission lines due to extinction. On the other hand, NIR emission lines are less sensitive to dust and thus could prove to be more helpful in identifying AGN activity in dusty galaxies.

Most of the MIR variable targets in our sample fall in the SF region of the optical BPT diagram as seen in Fig. <ref>. We plotted them in the various NIR diagnostic diagrams in Fig. <ref>. We primarily used the ratios of [S3] λ9530 Å/Paγ and [S3] λ9530 Å/Paβ to [Fe2]  λ16470 Å/Paα and [Fe2] λ12570 Å/Paβ, obtained from NIRES. For four of the targets (marked as filled circles in Fig. <ref>), these were the lines with signal-to-noise ratios (S/N) > 3 in our sample, and their flux values could be estimated with minimum uncertainties. For the remaining targets, we could only determine one of the ratios as at least one of the emission lines on the x-axis ([Fe2] 1.64μm, [Fe2] 1.25μm, Paα or Paβ) fell in the telluric region or were in a noisy part of the spectrum, and thus we could not measure the fluxes. We instead find the upper limits (if the [Fe2] lines were undetermined) or lower limits (if the Paschen lines were undetermined) to the emission line ratios and showed the targets as open circles with arrows to indicate upper or lower limits (both if neither of the line fluxes used in the ratio could be determined) to the ratios in Fig. <ref>. The upper limits for J1032, J1428, and J1320 did not exclude the possibility that these targets are AGN.

We found that four of the variable targets (J0847, J0928, J1144, and J1358) fell in the composite and AGN regions across all four NIR diagnostic diagrams. J0928, which has a broad Paα line, moved from the SF region in the optical to the AGN region of two of the NIR diagnostic diagrams. The other broad line target, J1144, also moved into the composite region of all four NIR diagnostic diagrams. 

While NIR diagnostic diagrams appear to be more effective in some cases at identifying obscured AGN in dwarf galaxies, they are still limited by the gas-phase metallicities of the galaxies. J1205, with a coronal line indicating the presence of a black hole, still lies firmly in the SF region of all the NIR diagnostic diagrams.



 §.§ Ruling out supernova contribution to variability
 
Although the nine galaxies that were observed using NIRES are determined to exhibit AGN-like variability from the light curves, our selection method does not completely rule out contributions from SN. Five of the galaxies have alternate evidence of AGN activity based on the presence of coronal lines, broad Paα lines, and emission line ratios indicative of AGN, which makes it unlikely that there is a significant contribution from SN to the variability.  However, <cit.> showed that in the optical, broad Hα lines could be caused by SN activity. The ideal method to rule out the possibility of broad lines being caused by SN is to obtain multi-epoch spectroscopy that would show whether the broad emission lines persist. Although we do not have multi-epoch spectra and cannot check for line variability, we did inspect all the light curves by eye, and determined that they do not show the rapid increase and decline in their curves that is characteristic of supernovae. Furthermore, we inspected the NIR spectrum for the common NIR emission lines associated with SN <cit.> and did not find any features that indicated the presence of SN.



 §.§ Summary of results




Dwarf galaxies with active black holes have been found to not display AGN signatures uniformly across different detection methods. If we have a dwarf galaxy that is variable in the MIR, how reliably can we determine that it is actually hosting an active black hole? In our work, we found that out of 25 dwarf galaxies that exhibit MIR variability, we have independent evidence that 17 of them (68%) host an AGN, strongly suggesting that MIR variability is due to the presence of an AGN in these targets. Twelve of these were classified as optical AGN, and the remaining five had NIR signatures of hosting an AGN. Four did not have any optical indicators of AGN, but we were not able to obtain NIR observations for them, so the true fraction of independently confirmed AGN in the sample of MIR variable AGN can be higher than 68% (see Fig. <ref> for a summary). 

Our results showed that if a galaxy is variable in the MIR, there is a high probability that it is actually hosting an active black hole. Using TESS light curves, <cit.> identified a similar fraction (62%) of optically variable galaxies that were either previously classified as AGNs in the literature or confirmed as AGNs based on a combination of emission-line diagnostics, mid-IR colors, or X-ray luminosity in their analysis. Thus, variability is a good indicator of AGN activity in dwarf galaxies and can be used to recover a larger fraction of AGN that is missed by other methods.

Signs of active black holes in dwarf galaxies are often difficult to distinguish from those originating from SF activity. The low luminosity of the AGN can lead to the possibility that SF activity capable of producing similar luminosities often act as interlopers. In the infrared in particular, <cit.> found that most of the BPT-confirmed AGNs from <cit.> were not selected as WISE AGN, which could be due to the host galaxy light dominating in the mid-infrared color space. Conversely, star-forming dwarf galaxies are capable of heating dust in such a way that mimics the infrared colors of more luminous AGNs. That is why W1-W2 colors alone should not be used to select for AGN in dwarf galaxies.




§ COMPARISON WITH NON-VARIABLE DWARF GALAXIES



Given the potential contamination from SF processes at infrared wavelengths, it is crucial to test whether MIR variability is accurately tracing AGN activity or if it is being caused by SF interlopers in dwarf galaxies. We have found that MIR variability is frequently accompanied by alternate indicators of AGN (see Sec. <ref> and Sec. <ref>), making it less likely that it is caused by stellar processes. Additionally, we compared dwarf galaxies that exhibit MIR variability with similar dwarf galaxies that do not exhibit MIR variability. We did this to recover differences in optical and infrared galaxy properties, if any, between the two samples of galaxies and understand the effects of any potential biases caused by host galaxy properties on the observed MIR variability.




 §.§ Control sample selection

We selected a control sample of non-variable dwarf galaxies, which were matched in WISE W1-W2 colors and stellar mass with the sample of variable dwarf galaxies. We obtained five control galaxies on average for each of the 25 target galaxies that we initially classified as variable and thus created a control sample of 136 non-variable dwarf galaxies. A two-sample Kolmogorov-Smirnov (K-S) test between the W1-W2 colors and stellar masses of the variable galaxies and non-variable control galaxies gives p=0.99 and p=0.56, respectively, indicating that the control galaxies do not have systematically different WISE colors or stellar masses (see Fig. <ref>). None of the galaxies in the control sample passed the variability tests based on our initial constraints (see Section 2.1), and thus can be safely assumed to be non-variable. The two samples have similar redshift distributions, ranging from 0.01 to 0.1.



We then obtained emission line flux values for [O2] λλ3727,3729 Å, He2 λ4686, Hβ λ4861 Å, [O3] λ 5007 Å, [O1] λ 6300 Å, [N2] λ6584 Å,  Hα λ6563 Å and [S2] λ6731 Å from the MPA-JHU catalog <cit.>. We measured extinction values for individual galaxies from the Hα/Hβ Balmer decrement. We then obtained extinction corrected [O3] λ5007 Å luminosity values for all the galaxies in the target and the control samples. We also measured the star-formation rates (SFRs) for the galaxies using a number of methods. We directly obtained the SFRs for the galaxies from the MPA-JHU catalog, which were calculated based on the methods used in <cit.>. We independently estimated SFRs from the extinction-corrected Hα luminosity values based on the relations by <cit.> as well as from the extinction-corrected [O2] fluxes using Equation 11 in <cit.>. The SFRs calculated by these different optical methods agreed with each other. We determined the optical colors (u-r) from SDSS for the different galaxies. 





 §.§ Large-scale properties of variable and non-variable dwarf galaxies
 

Comparing various properties of variable and non-variable targets with similar W1-W2 colors, we did not find a significant difference between the extinction values, [O3] luminosities, optical colors, and SFRs between both samples (Fig. <ref>). 

Using the emission line values obtained from the MPA-JHU catalog, we plotted the BPT diagrams of both samples (see Fig. <ref>). We found that 3/25 (12%) of the variable targets lie above the <cit.> demarcation line for the AGN region, while 8/136 (0.05%) of the non-variable controls lie in the AGN region.

We obtained He2 λ4686 emission line fluxes for eight of the variable galaxies and eighty-two of the non-variable galaxies from the MPA-JHU catalog. No significant difference was found between the positions of the variable and non-variable galaxies in the He2λ4686/Hβ versus [N2]λ6584/Hα diagnostic diagram presented in <cit.>. While some of the variable galaxies were previously identified as AGNs using this diagnostic, we identified four additional variable galaxies that meet the AGN criteria. These galaxies are marked with blue checkmarks in Column 5 in Table <ref>.

Determining gas-phase metallicities for the targets is challenging as the traditional methods that depend on various emission lines such as [O3] and [O2] often do not account for the contamination due to AGN activity. Moreover, studies have shown that there is an ambiguity in determining metallicities using methods such as the ratio of [O2]λλ 3726,3729 + [O3]λλ4959,5007 over Hβ  <cit.>, particularly in the range of metallicities (7.5 < 12 + log(O/H) < 8.5) that we would expect these dwarf galaxies to have <cit.>. Thus, to comment on the metallicities of the targets, we looked solely at the [N2]/Hα line ratios from the BPT diagram, which can be used as a proxy for metallicity <cit.>. 

We found that, on average, variable targets have higher metallicities (log [N2]/Hα > -1) as compared to the non-variable targets. Only 2/25 (8%) targets have [N2]/Hα lower than -1 as compared to 59/136 (43%) for the control sample, as seen in Fig. <ref>.  BPT-selected AGNs in dwarf galaxies are expected to be biased towards higher metallicities <cit.>. However, we adopted a completely independent method of selecting AGN in dwarf galaxies, and we still found that variable dwarf galaxies (tracing the AGN) have higher metallicities than non-variable dwarfs. Since AGN contribution can also lead to an increase in the [N2]/Hα ratio, a large fraction of variable targets with high [N2]/Hα ratios may be a direct consequence of the presence of AGNs in the variable targets.



 §.§ Optical broad lines and W2-W3 colors



We obtained the SDSS spectra of all the galaxies in the target and the control samples. We then performed spectral fits for the entire sample in the regions that include the prominent Balmer lines (4400 - 7000 Å) and fit simultaneously for  Hβ λ 4861 Å, [O3] λλ 4963, 5007 Å, Hα λ 4861 Å, and [N2]  λλ 6548, 6583 Å. We found that variable targets have broader Balmer lines compared to non-variable targets (Fig. <ref>): 
6/25 (24%) targets have Hα FWHM > 500 km s^-1 while only 2/136 (0.015%) of the controls have Hα FWHM > 500 km s^-1. Broad Balmer lines could indicate the presence of AGN activity, but there could be potential contamination from star formation. We did not find any broad outflowing component in the forbidden [O3] lines of both samples, indicating that neither sample has strong outflows in ionized gas.

This higher incidence of optical broad lines in the sample of galaxies exhibiting MIR variability as compared to non-variable galaxies is surprising. The presence of broad lines in the spectrum of AGN is typically attributed to an orientation effect. In type 1 AGNs, the nucleus is oriented so that we can observe the BLR and narrow line regions (NLR). MIR variability, on the other hand, should have no dependence on orientation, as the variability is believed to originate from the dust surrounding the accretion disk and is expected to be isotropic. 
However, <cit.> found that low-luminosity type 2 AGNs tend to have lower variability amplitudes than their type 1 counterparts. They attribute this to the type 2 AGN having distinctive central structures due to their low luminosity or their MIR brightness being contaminated by emission from the cold dust in the host galaxy. On the other hand, recent observations of extended (tens to hundreds of parsecs) MIR emission from the polar regions of AGNs suggest that MIR emission may be dominated by dust from the polar regions rather than from the torus <cit.>. This interpretation possibly suggests that we can simultaneously observe the BLR as well as MIR emission. Our observations of a large number of Type 1 AGN that are variable in MIR can be explained with either of these scenarios.



We also obtained WISE W1, W2, W3, and W4 fluxes for all the galaxies. Examining the infrared colors of both samples (Fig. <ref>), we found that non-variable targets have higher values of W2-W3 (redder W2-W3 colors) on average than variable targets (3.92 vs 3.18). Redder W2-W3 colors indicate activity caused by starburst events <cit.>. A p-value of 0.027 (< 0.05) indicates that there is a statistically significant difference in W2-W3 between the variable and control samples. This suggests that variable targets have bluer W2-W3 colors, indicating that there is likely lesser contribution from dust heated by star formation in the MIR for the variable targets. This difference in the contribution of SF activity is not apparent from the optical, where both samples have similar SFRs (see Fig. <ref>). Even with similar W1-W2 colors, variable dwarfs still have bluer W2-W3 colors. Thus, it is likely that AGN activity causes redder W1-W2 colors in variable targets, while for the non-variable galaxies, the contribution stems from strong star formation activity.



§ CONCLUSIONS
 
In this work, we selected a sample of twenty-five dwarf galaxies that are identified to be variable in the MIR from the light curves from the most recent release of the AllWISE/NEOWISE catalog. We determined that 12/25 (48%) of these galaxies had been confirmed as AGN in the literature by several optical diagnostic methods, such as the BPT diagram, presence of broad lines or strong He II emission, indicating that MIR variability accurately traces black hole activity in these small dwarf galaxies. For a subsample of the variable dwarf galaxies that did not have any optical confirmations of AGN activity, we obtained NIR observations using Keck/NIRES. Our main results can be summarized as follows (also see Fig. <ref>):


    
  * We present the first detection of a NIR coronal line ([S IX]) in J1205, which has the lowest mass (log(M_*) = 7.5 M_⊙) among all the objects in our sample. Most commonly used AGN diagnostics, such as BPT diagrams and He II emission lines, fail to detect the presence of an AGN in J1205, thus highlighting that  MIR variability can be a powerful and sometimes the only tool to detect AGNs in low-mass galaxies.
    
  * We find evidence of a broad Paα line potentially from the BLR in two of the variable targets (J0928 and J1144). These targets lie in the SF region of the optical BPT diagram, but they lie in the composite or AGN regions of the NIR diagnostic diagrams. The black hole masses implied by their broad lines are consistent with the black hole to stellar mass relation from RV15.
    
  * Using NIR diagnostic diagrams, which make use of NIR emission line ratios in order to differentiate between AGN and SF regions in galaxies, we find that several of the variable targets (4/9) that we observed moved into the AGN/composite region of the diagnostic diagram compared to their location in the SF regions of the optical BPT diagram. 
    
  * Comparing a sample of variable and non-variable dwarf galaxies with similar stellar mass and WISE W1-W2 colors, we find that variable dwarfs have bluer W2-W3 colors than non-variable dwarfs. The redder W1-W2 colors of non-variable dwarfs are likely because of high star formation. Thus, variable dwarfs accurately trace AGN activity independent of mid-infrared colors.
    
  * Variable dwarfs have a higher fraction of galaxies that have broad (> 500 km s^-1) Balmer (Hβ and Hα) lines than the non-variable dwarfs (24% compared to 0.015%).  While this could simply indicate a higher incidence of AGN, it is remarkable that such a large fraction would be type 1 AGN. 

We find independent confirmation of AGN activity in at least 68 % out of 25 MIR variable dwarf galaxies, indicating that MIR variability can be used to accurately detect AGN in dwarf galaxies. Improving methods to detect low-mass active black holes is the first step in constraining the black hole occupation fraction in dwarf galaxies, thus determining the evolutionary pathways for black hole seeds in the early universe. Our study shows that using indicators in the infrared is crucial to obtaining a more complete census of black holes in dwarf galaxies.  With the increasing amount of infrared data being available from JWST and the upcoming infrared transient survey telescopes such as the Palomar Gattini-IR <cit.>, harnessing the power of MIR variability will likely be fruitful in uncovering large numbers of active black holes in dwarf galaxies.


We thank the anonymous referee for their thoughtful feedback and constructive comments that greatly helped to improve this paper. Partial support for this project was provided by the National Science Foundation under grant No. AST 1817233. The data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Maunakea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain. 

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, and NEOWISE, which is a project of the Jet Propulsion Laboratory/ California Institute of Technology. WISE and NEOWISE are funded by the National Aeronautics and Space Administration.

SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. 


WISE, NEOWISE, KCWI/Keck, SDSS

astropy <cit.>,
          BADASS <cit.>,
          pPXF, <cit.>,
          REDSPEC (<https://www2.keck.hawaii.edu/inst/nirspec/redspec.html>),
          SciencePlots <cit.>






The non-uniform detection of coronal lines across galaxies is likely due to insufficient observation times in order to detect the typically faint coronal lines. We present the upper limits for the [Si6] flux for the nine galaxies observed with NIRES in this paper (Table <ref>). The upper limits to the [Si6] fluxes were determined by estimating the 1σ noise levels in the region around [Si6] flux and multiplying it by three to obtain the 3σ upper limit indicated in Table <ref>. None of the observed galaxies had [Si6] detections.





















aasjournal












 Paper Title: Charting the Lyman-alpha escape fraction in the range 2.9<z<6.7 and consequences for the LAE reionisation contribution.

 

   
   
   
 I. Goovaerts & T. T. Thai 
   Charting the escape fraction

 
   
   
          

   Aix Marseille Université, CNRS, CNES, LAM (Laboratoire d’Astrophysique de Marseille), UMR 7326, 13388 Marseille, France

              ilias.goovaerts@lam.fr; tranthai.physics@gmail.com
         Department of Astrophysics, Vietnam National Space Center, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
         Graduate University of Science and Technology, VAST, 18 Hoang Quoc Viet, Cau Giay, Vietnam
         Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria
         Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
         Leibniz Institut für Astrophysik (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
         
        
        
         

   
 
  
  
   The escape of photons at redshifts greater than two is an ongoing subject of study and an important quantity to further understanding of emitters (LAEs), the transmission of photons through the interstellar medium and intergalactic medium, and the impact these LAEs have on cosmic reionisation.
  
   This study aims to assess the escape fraction, , over the redshift range 2.9<z<6.7, focusing on  Very Large Telescope/Multi Unit Spectroscopic Explorer (VLT/MUSE) selected, gravitationally lensed, intrinsically faint LAEs. These galaxies are of particular interest as the potential drivers of cosmic reionisation.
  
   We assessed in two ways: through an individual study of 96 LAEs behind the A2744 lensing cluster, with James Webb Space Telescope/Near-Infrared Camera (JWST/NIRCam) and HST data, and through a study of the global evolution of using the state-of-the-art luminosity functions for LAEs and the UV-selected `parent' population (dust-corrected). We compared these studies to those in the literature based on brighter samples. 
  
   We find a negligible redshift evolution of for our individual galaxies; it is likely that it was washed out by significant intrinsic scatter. We observed a more significant evolution towards higher escape fractions with decreasing UV magnitude and fit this relation. When comparing the two luminosity functions to derive in a global sense, we saw agreement with previous literature when integrating the luminosity functions to a bright limit. However, when integrating using a faint limit equivalent to the observational limits of our samples, we observed enhanced values of , particularly around z∼6, where becomes consistent with 100% escape. This indicates for the faint regimes we sampled that galaxies towards reionisation tend to allow very large fractions of photons to escape. We interpret this as evidence of a lack of any significant dust in these populations; our sample is likely dominated by young, highly star-forming chemically unevolved galaxies. Finally, we assessed the contribution of the LAE population to reionisation using our latest values for and the LAE luminosity density. The dependence on the escape fraction of Lyman continuum photons is strong, but for values similar to those observed recently in z∼3 LAEs and high-redshift analogues, LAEs could provide all the ionising emissivity necessary for reionisation.   
  
  
   
   
   
   Charting the escape fraction in the range 2.9<z<6.7 and consequences for the LAE reionisation contribution
    I. Goovaerts1T. T. Thai1,2,3R. Pello1P. Tuan-Anh2,3 N. Laporte1 J. Matthee4 T. Nanayakkara5 J. Pharo6
    Received 9th July 2024; accepted 13th July 2024
==========================================================================================================






§ INTRODUCTION


For more than 20 years, the study of high-redshift galaxies has been facilitated by the line <cit.>. Galaxies for which this line is observable, emitters (LAEs; equivalent width, >25Å), have formed an essential part of the drive to characterise star-forming galaxies (SFGs) at increasingly higher redshifts (the most recent examples include, , see also  and references therein). The line, while intrinsically very strong and therefore an attractive target for observations, undergoes a very complicated process escaping a galaxy due to the resonant nature of the line <cit.> as well as dust within the galaxy <cit.>, inflows and outflows <cit.>, and the intergalactic medium (IGM) <cit.>. Understanding the escape of these photons is of paramount importance to the characterisation of these high-redshift SFGs as well as to understanding their impact on extragalactic processes such as cosmic reionisation. 

The escape of photons is also expected to be connected to the escape of ionising Lyman continuum photons (λ_LyC<912 Å) <cit.>, although any relation between these two photon escapes tends to be reported with significant scatter between individual galaxies. Since the escape of Lyman continuum photons is extremely challenging to observe directly at redshifts higher than z∼3 due to the intervening IGM, the escape of photons can be an important proxy.
 
Many efforts have been made towards a better understanding, both on the theoretical and observational front, and a better quantification of the escape fraction (henceforth ). The well-known relations in <cit.> quantify the basic relationship between luminosity and the star-formation rate (SFR) expected to produce it, subject to assumptions on the initial mass function (IMF), stellar metallicity, and star-formation history (SFH). However, the resonance of the line means that photons will scatter in a galaxy's neutral hydrogen. During this process, dust may absorb these photons ( and references therein), so the escape fraction is expected to decrease with increasing dust content <cit.>. The size and stellar mass of galaxies may also play a role, with generally increasing with lower stellar mass <cit.>.

The increasing neutrality of the IGM at redshifts above z∼5-6 <cit.> is expected to play a significant role in visibility. Neutral hydrogen around the galaxy and along the line of sight absorbs emission, meaning that the number density of LAEs decreases around these redshifts <cit.>. However, this effect is subject to significant uncertainties, and for individual cases, it is highly dependant on the physical conditions and environment of the galaxy in question <cit.>. 

Despite these challenges, strides have been made in recent years towards quantifying and its evolution with redshift. <cit.>, henceforth , studied the evolution of dust properties and across a sample of ∼100 blank-field LAEs in the range 1.9<z<3.8 and compared star-formation rate densities (SFRDs) derived from LAE and UV luminosity functions (henceforth LFs) in order to quantify the evolution of with redshift. They found a sample median of ∼ 29% when considering their LAEs individually and a negligible redshift evolution. 

<cit.> found a significantly lower , around 5% at z=2.2 using a blind narrow-band survey searching for Hα and . The authors made use of the Hα line to estimate un-obscured star formation and compared this to the star formation derived from . The Hα line originates from the same process as the line and is less sensitive to dust attenuation; therefore, it is often a more reliable indicator of star formation.

One general agreement in the literature is that increases with redshift. Low values such as 0-5% are found in the aforementioned study as well as <cit.>, 
 <cit.>, <cit.>, <cit.>, and <cit.>. Larger values such as 30-40% and even higher have been found at redshifts towards the epoch of reionisation (;  ).

Another general agreement found in the literature relates to sample selection. Samples that are selected by their emission typically display higher values of when compared to samples selected by Hα <cit.>.

By contrast, significant scatter in is expected in all samples, likely due to the large number of factors that affect photon escape and the differing conditions present in any sample of galaxies. This can be seen in all studies on , from low redshift <cit.> to the higher redshift, such as , <cit.>, <cit.>, and <cit.>.

Currently, the most powerful estimators for determining in an indirect manner include <cit.> and line separation <cit.>. The correlation with line separation is likely due to the presence of outflows, to which the line shape is sensitive <cit.>. These outflows produce channels of low HI column density in the interstellar medium (ISM) and circumgalactic medium (CGM), which allows for more escape <cit.>. 

We can also constrain in a global way by comparing the SFRDs derived from the LAE and general UV-selected populations (with SFR derived from the UV emission). The ratio of these gives the escape fraction of photons over a specific survey volume. Employing this method, <cit.>,  and <cit.> found a gentle evolution over their redshift ranges that could be fitted with a power law of the form: = C× (1+z)^ξ, where C=(4.79^+5.68_-0.69) × 10^-4, ξ=3.38^+0.10_-0.37 at z∼ 2.2-6.7 <cit.>, C= 3.98^+2.32_-1.46× 10^-3, ξ=2.2±0.3 at z=1.9-3.8  and C=5.0 × 10^-4, ξ=2.8 in z=0-6 <cit.>.

<cit.> mentioned that may reach unity at z∼ 11 ^+0.8_-0.6, while <cit.> argued that would reach unity around a redshift of six. In general, such research has been done with a few LAEs fainter than log  L_Lyα [erg/s] < 40, hence without the ability to fully constrain the faint part of the LAE LF and its impact on the SFRD.

Any difference in the evolution of may come from the integration limits during the estimation of the and UV luminosity densities. Hence, data on the fainter luminosity regime is necessary in order to integrate to fainter luminosity limits and better understand these faint populations' effect on the evolution of the LF as well as with redshift.

Fortunately, the recent VLT/Multi Unit Spectroscopic Explorer (MUSE) observations of lensing clusters <cit.> and blank fields <cit.> have allowed us to observe  down to log L_Lyα [erg/s]∼ 39 ,  39.5, respectively.    

In this paper, we seek to combine two state-of-the-art LFs of intrinsically faint lensed galaxies, the latest UV LF from <cit.> and the latest LAE LF from <cit.>, allowing us to compare the SFRD derived from emission, SFRD_Lyα, to the total SFR density, SFRD_total, and hence track the redshift evolution of over a wide redshift range: 2.9<z<6.7. In <cit.>, the independent LF comparison is advocated as a method to calculate , though it is subject to cosmic-variance related errors. We can significantly reduce the uncertainties related to cosmic variance by using six lensing clusters for the UV LF and 17 lensing clusters for the LAE LF. The LFs we used provide this study with a unique opportunity to explore the faint galaxy regime (M_UV<-13 and log L_Lyα [erg/s]>39) without the need for extrapolation in the LFs.

We also calculated on an object-per-object basis in order to compare galaxy properties as well as the global evolution, based on spectral energy distribution (SED) fitting of HST and JWST/NIRCam data of ∼100 LAEs in the A2744 lensing cluster. The NIRCam data, extending to ∼5 μ m, allowed us to access, for the first time, the rest-frame optical emission of these LAEs and greatly improves the reliability of the properties extracted with SED fitting. The NIRCam imaging notably also covers the entire MUSE field of view, meaning the full sample size of MUSE LAEs can be explored. This would not be the case with calculations of SFR from Hα and dust attenuation from the Balmer decrement obtained from spectroscopy, for example from JWST/NIRSpec's MSA mode. This would, in principle, be preferable, due to the fewer assumptions needed and our lack of photometric coverage of the infrared for our LAEs. However, samples of LAEs with Balmer line detections are rare at the redshifts concerned by this study, and sample sizes are often small. Using photometry, we could gain an appreciable sample size using just one cluster, although more would be preferable (and soon possible with the current and upcoming JWST observations). 

Studies of emission lines with slit spectroscopy for these galaxies (especially lensed LAEs and therefore potentially extended sources, both by lensing and due to the extended nature of haloes) are not without difficulties themselves. Parts of flux can be missed in slits, especially slits as small as JWST/NIRSpec's MSA (02 to 045; <cit.> provide a good discussion of these difficulties, see also <cit.>), and sometimes, flux can be missed altogether, such as in <cit.>.
 
Section <ref> outlines the determination of for individual galaxies, describing the data and process used. In Section <ref>, we describe the global evolution of , the two LFs used for this calculation, and the results thereof. A comparison of the two methods to each other and to studies in the literature is presented in Section <ref> along with consequences for reionisation and a discussion of the validity of comparing the UV and LAE LFs. Finally, Section <ref> offers conclusions and an outlook for determination in the near future.

Throughout this paper, we adopt a value for the Hubble constant of H_0=70 km s^-1Mpc^-1, and the cosmology used is Ω_Λ=0.7 and  Ω_m=0.3. The adopted IMF is that of <cit.>. All values of luminosity and absolute UV magnitude are given corrected for magnification. 






§ ESCAPE FRACTION OF INDIVIDUAL LAES





 §.§ Data: Combining spectroscopy and photometry


In order to perform analysis of this genre, photometry and spectroscopy of sufficient quality and depth is essential. For our purposes we combined public MUSE integral field unit (IFU) spectroscopic data <cit.> (094.A-0115, 095.A-0181, 096.A-0496) with the latest, deepest combination of Hubble Space Telescope (HST) and James Webb Space Telescope (JWST) photometry from the UNCOVER survey (PIs Labbé and
Bezanson, JWST-GO-2561, <cit.>) of the lensing cluster A2744. This cluster is extremely well studied and benefits from a great number of multiple images to constrain its lens models. Below, we detail the two sets of observations and how they are used for this study.




  §.§.§ MUSE IFU spectroscopy


The use of an IFU allowed us to blindly select LAEs, rather than performing spectroscopic follow-up on UV-selected targets. This ensures our sample has a simple  reproducible selection function, namely luminosity limited. MUSE has a field of view of 1×1 arcmin^2 and a spectral resolving power of R ∼ 3000 at λ=808 nm <cit.>. Wavelength coverage ranges between 4750 Å and 9350 Å, meaning the line can be detected between redshifts of 2.9 and 6.7.

The A2744 lensing cluster used in this work has integration times varying between 3.5 and 7 hours. It forms part of a data release by <cit.>[<https://cral-perso.univ-lyon1.fr/labo/perso/johan.richard/MUSE_data_release/>] and the LAEs used in this work are comprehensively described as part of the Lensed MUSE Arcs (LLAMA) Sample, in <cit.>.The full process of LAE detection in MUSE data is detailed in <cit.> and <cit.>. We give an abridged version here. 

Emission line sources are identified in MUSE narrow-band datacubes using the MUSELET software <cit.>[<https://mpdaf.readthedocs.io/en/latest/muselet.html>]. Thereafter the Source Inspection package <cit.> is used to identify line emitters as LAEs and assign redshifts. During this process, the redshift confidence of each LAE is determined, based on the emission lines seen, the shape of the line, ancillary HST data and lensing considerations. The confidence scale ranges from 1 for a tentative redshift attribution to 3 for a very secure redshift attribution. In this work, only sources that have been identified with redshift confidence levels of two and three are used, as well as a S/N greater than three. 

The flux of these LAEs is derived using SExtractor <cit.> on a continuum-subtracted Narrow-Band sub-cube with a size of 15"× 15"× 20Å. The full process is described in <cit.>.



  §.§.§ HST + JWST photometry and galaxy properties


The photometric catalogues of the UNCOVER project <cit.> comprise photometry in 15 bands ranging from the F435W HST filter to the F444W JWST/NIRCam filter. This gives excellent, deep coverage up to 4.4, which corresponds to ∼580 nm in the rest frame at z=6.7, the highest redshift of the MUSE LAEs. We used the catalogue optimised for faint and compact sources (0.32 apertures). 5σ depths in the 15 filters range between 27.16 and 29.56 in AB
magnitudes. Of the 154 total LAE images, 121 were detected, which corresponds to 99 individual LAEs. We rejected two of these LAEs as being clearly contaminated by bright cluster galaxies and a further LAE, as it lies on a critical line and has a magnification factor of 137±1500. As such, its properties would likely not be well constrained (see, for example, ). Having thus blindly matched these -selected galaxies to their UV counterparts, their SEDs were fitted with the SED-fitting code CIGALE <cit.>. The details of the fitting procedure are outlined in <cit.>; here we give a brief outline.

Taking advantage of the secure spectroscopic redshifts of these LAEs, we fit their SEDs using two different SFHs: a single exponentially decaying burst with a range of timescales and a double burst model with a second, delayed burst at a time τ after the initial burst, to be fit <cit.>. Ages for the bursts range from 10 to 700Myr and mass fractions in the second burst range from 0.001 to 0.65. The stellar template library of <cit.> is employed with metallicities ranging from 0.001 to 0.02, together with the nebular emission models from <cit.>. Gas metallicities range between 0.0004 and 0.02. Dust attenuation is taken into account by means of the Calzetti attenuation law <cit.>. 

The best-fit SFH and model was carried forward in this analysis (see  for comparisons between the two SFHs). One of the parameters in the CIGALE output is the intrinsic dust-corrected star formation, which is used in the computation of . 



  §.§.§ Lensing models


For studies involving lensing clusters, the lens models used are pivotal to the reliability of the results, as these govern the lensing magnifications that are assigned to each galaxy and therefore the correction made to their observed luminosity. The lens models used for this work for the A2744 lensing cluster are described in <cit.>. As the Hubble Frontier Fields (HFF) and A2744 in particular have been so well studied over the years, these clusters have very well constrained mass models, drawing on many multiple images identified behind each cluster. While systematic uncertainties are still associated to the particular lens model used <cit.>, the quantity of multiple images for these particular clusters render the statistical error on the mass distribution as low as 1% <cit.>. Typical, accepted magnifications in the sample range from 1.5 to 25, with the maximum magnification being 137. We note that escape fraction measurements for individual galaxies (as in the following Section), are not affected by magnification. 



 §.§ Escape fraction




To calculate the escape fraction on an object-per-object basis for the 96 individual LAEs behind the A2744 lensing cluster, from the data described in Section <ref>, we compare the SFR inferred from the flux, to the dust-corrected SFR from the CIGALE SED fitting process. The SFR inferred from the flux is calculated using the prescription in <cit.> and the conversion factor of 8.7 between and Hα luminosities, using a Case B recombination scenario <cit.> (T=10^4 K). 

The ratio of these two SFRs gives the escape fraction of photons, . The redshift evolution of this quantity is shown in Fig. <ref>. The sample median is shown as the red dashed line and the sample median in  is shown as the cyan dashed line. They are consistent to within each other's errors, even though  use a different prescription for finding . The median escape fraction uncertainty of our sample (18%) is shown by the black error bar. Each LAE is colour-coded by its absolute UV magnitude, calculated from the filter that sees the rest frame 1500 Å emission. One can clearly see that photons escape more easily from UV-fainter objects, a result already remarked upon by <cit.>, <cit.>. To clarify this further, we can fit a relation between the absolute UV magnitude and for each galaxy. This is shown in Fig. <ref>, where the galaxies are now colour-coded by redshift-bin and median error bars for each bin are shown in the corresponding colour. The best-fit to this data is

    log()=(0.27±0.02) M_UV+(4.2±0.5).

This is clearly a tighter relation than the one shown by the redshift versus , with an uncertainty on the slope and normalisation of about 10 %. These quantities, M_UV and are not independent as M_UV is related to the intrinsic SFR of a galaxy, so this relation is to be expected, but it is nevertheless noticeable that below M_UV=-18 there are few galaxies with values less than 10%. However, care must be taken when interpreting this result, as incompleteness can play a role as we start to probe fainter galaxies (towards the right of Fig. <ref>). To better assess this, in Fig. <ref> we plot three regions where we expect to be incomplete as a function of UV magnitude and lensing magnification. This is based on the limiting flux for a MUSE detection (of an LAE). The three magnification regions plotted are μ=1.5, the minimum magnification included in the sample (the area covered by MUSE observations do not reach the areas with less magnification than this, most of the area has μ>2), μ=9, the maximum magnification for an image included in the sample (we note that there are LAEs not included and images not included with μ>9) and μ=137, the maximum magnification of any LAE image in the A2744 lensing cluster. These three regions delineate the area of this plot where we expect to become incomplete in terms of detecting a faint galaxy with a small value with MUSE. We can see that this effect likely becomes important towards the faint end of this plot, at M_UV≳-17. However, at the brighter end of this graph, we do not expect the incompleteness to shape the relation we observe.
 
We did see LAEs undetected in the continuum that might be expected to fill this region of the graph. However, they are uniquely objects of high escape fraction (based on upper limits of the UV continuum, and no dust correction). These objects are denoted by the black triangles in Fig. <ref>: lower limits on and upper limits on M_UV.

We note also that around this M_UV value of -18 that we start to see objects with >100%. >100% means that we are seeing more emission than we should, given the standard conversions between and UV luminosities and SFR detailed above. Reasons for this may include our assumptions in making those conversions, such as constant SFH, ages greater than ∼ 50Myr and the case B recombination scenario <cit.>, becoming less valid (see, for example, ). This may point concurrently to our models failing to correctly estimate the dust content of these galaxies and thereby underestimating the intrinsic SFR. However, this would mean that some property of these galaxies is heavily attenuating the UV emission, but not attenuating the emission accordingly. <cit.> postulate that geometry or the presence of large outflows may play a role in disconnecting emission and dust such that one may see a high from a heavily dust-attenuated system.  

We also observed LAEs for which the continuum is completely undetected in the deepest photometry to date <cit.>, and for these galaxies is, naturally, very high, often greater than 100% (as calculated from 2σ upper limits on the UV continuum). This suggests that there is more than simply model inadequacy at play here. We discuss these objects further in Section <ref>.  

Unlike M_UV, we see no significant evolution of with redshift, although we have relatively few objects at redshifts higher than 6. It is clear from Fig. <ref> that we are affected by missed objects above z=6, both in the high-regime (as these objects have low M_UV) and the low-escape fraction regime, as objects with low are harder to detect at higher redshifts. The greater presence of skylines and decreased MUSE sensitivity towards longer wavelengths are additional challenges to detect LAEs towards the high end of our redshift range. 

The comparison to the results in  suggest that does not evolve significantly over the redshift range 1.8<z<6. This is in tension with certain literature results such as <cit.>, who find that increases with redshift (0.6<z<6). However, this study was performed in a similar way to the global analysis of that we present in the following sections. Hence, we return to this point after having taken both methods into consideration.

The scatter in the individual results shown in Fig. <ref> is to be expected, given the wide variety of conditions within galaxies, even LAEs, expected to be mostly young and highly star forming. Dust content of these galaxies likely varies, which plays a significant role in the escape of photons <cit.>. Additionally and as mentioned, geometry and line-of-sight effects are also expected to have a hand in the amount of photons that reach the observer <cit.>.


While we see consistency between the sample median of  and ours, the offset, as well as the smaller scatter of our results, is to be expected based on the differing methods for finding .  assume an intrinsic UV slope of -2.23 in order to calculate the dust reddening E(B-V). As this UV slope likely depends on factors other than dust, such as metallicity, age and mass of the stellar population <cit.>, and as there is evidence for slopes bluer than this value <cit.>, we prefer not to use this assumption. If there are bluer intrinsic slopes than this, we would expect the corresponding E(B-V) values from  to be higher, and therefore, the intrinsic SFR would be higher, and the calculated escape fractions would be lower. This is what we observed in our results. We also saw less scatter in , as we do not depend on estimations of the UV slope, which are known to have significant scatter (as well as uncertainty and possibly bias, e.g. ).

Our sample median, 22.5 %, agrees with determinations at slightly lower redshift from <cit.>, who both quote escape fractions of ∼30 % for LAEs. Most recently, also using JWST, <cit.> have found a sample median of 28 % for 10 LAEs at 5<z<8.

However, we note that the scatter on individual determinations of remains a very significant consideration when assessing samples of the kind in Fig. <ref>. It is therefore unlikely that we will be able to specify a precise fraction for escape at any given redshift. It is clear that the trend of with M_UV is more useful. 

We note also that while individual values are better determined using Hα fluxes and the Balmer decrement, the intrinsic scatter seen remains similar, with values ranging from <1% escape to 100% (see, for example, ). For our sample, we can take additional encouragement from the fact that the dust content of these low-mass LAEs, at these redshifts, is both expected to be, and observed to be, very low. We therefore do not expect many cases where our SED fitting significantly underestimates the dust attenuation of a galaxy. Both methods of determining are valuable then, and a more detailed comparison between them for the same sample of galaxies is desirable. 





§ EVOLUTION OF THE GLOBAL ESCAPE FRACTION





One can derive estimates of in a global manner by comparing the LF of LAEs and that of the `total' galaxy population. Integrating both of these LFs gives the luminosity density and the total luminosity density . From this one can derive the respective SFRDs. Then taking the ratio of these SFRDs gives <cit.>. In order to perform this calculation, we start from the two state-of-the-art LFs in lensing fields, the LAE LF from <cit.> and the UV LF from <cit.>, assumed here to be the `total' galaxy population. This assumption and its consequences for the results of this study are discussed in detail in Sect. <ref>.

Before presenting the derivation of , we briefly detail both LFs used, starting with the LAE LF from <cit.>.



 §.§ luminosity function


The LAE LF is computed by using the non-parametric V_max method to estimate the volume of the survey in which we can detect sources individually. The inverse of this parameter is the contribution of the source to the numerical density of the galaxies. The procedure to compute this parameter using data collected by lensing observations by MUSE/VLT was developed by <cit.> and presented in <cit.> with some improvements related to the magnification and completeness values of the sources. These completeness values reflect the expected number of galaxies actually in the field based on the observed galaxies and the noise statistics of the field. The sample in <cit.> includes 600 LAEs behind 17 lensing clusters observed by MUSE/VLT, identified with highly secure redshifts, z_conf = 2,3 (high line S/N, clearly asymmetric line profile, etc) as defined in <cit.>. These data are within a redshift interval 2.9<z<6.7 and covering four orders of magnitude in luminosity 39< log  L [erg s^-1]<43. These luminosity values were corrected for the magnification of the lensing effect using models from the work of <cit.>. 

The evolution of the LF with redshift was investigated in four redshift intervals 2.9<z<6.7, 2.9<z<4.0, 4.0<z<5.0 and 5.0<z<6.7 (labelled z_all, z_35, z_45 and z_56) and the Schechter function used to fit the LF points. Using the data sample collected with the help of gravitational lensing is efficient to study the LF at faint luminosities (log L [erg s^-1] < 41). However, it becomes less sensitive when examining the function surrounding the characteristic luminosity value L* of the Schechter function. To tackle this problem, <cit.> used the average LF values obtained from the literature in the same luminosity and redshift bins as a constraint at the bright end of the LF. A turnover in the shape of the LF was observed at luminosities log L [erg s^-1] < 41 when studying the evolution in the z_45 and z_56 redshift intervals. This was explained by inefficient cooling of gas in small dark matter haloes <cit.>. To accommodate the potential turnover in these two redshift bins, a modified Schechter function was introduced by multiplying an exponential term exp -(L_T/L)^m to the traditional Schechter function. The best-fit results suggest that m is about unity and log L_T [erg s^-1] is ∼ 40 and ∼ 40.7 for the redshift intervals z_45 and z_56 respectively.  Accounting for the effects of source selection, completeness cut, and flux measurement, the final slope values in the four redshift intervals are -2.00±0.50, -1.97±0.50, -2.28±0.50 and -2.06±0.60. These values are in good agreement with results from other literature studies within a 1σ deviation.  The best-fit value of the characteristic luminosity values are log L* [erg s^-1] = 42.85^+0.10_-0.10, 42.87^+0.11_-0.1, 42.97^+0.13_-0.11, 43.09^+0.10_-0.08 and the normalisation factors are ϕ*[10^-4Mpc^-3] = 7.41^+2.70_2.70, 6.56^+3.20_-2.40, 4.06^+2.70_-1.70 and 3.49^+2.11_+1.50 in redshift bins z_all, z_35, z_45 and z_60, respectively.  

In our analysis we also include results for the LAE LF at different redshift intervals from literature studies to build a general picture, and provide brief descriptions of these works here.  <cit.> studied the LF evolution of ∼ 120 LAEs behind four lensing clusters observed by MUSE, with the same redshift range as <cit.> and a luminosity range of 39≲log L_Lyα[erg/s]≲43. In the blank field observations, <cit.> presented the LF evolution with redshift of LAEs with luminosities greater than log L  [erg/s] = 41. Due to lacking data at faint luminosities, <cit.> fixed the slope value of the Schechter function at α=-1.5 (z=3.1, 3.7, 5.7).  fixed this parameter at α = -1.7 (z=1.9-3.8), while <cit.> found α=-1.6^+0.12_-0.12, -1.78^+0.10_-0.12, -1.69 (fixed) at z=1.95-3.0, 3.0-4.55, 4.55-6.6, respectively. However, at a higher redshift and a brighter luminosity range, this value has been found to be much steeper, varying from -2.26 to -2.56 at redshift z=5.7, and from -1.86 to -2.5 at redshift z=6.6 <cit.>; or ∼ -1.98 at redshift z=5.4±0.4 <cit.>, depending on the luminosity limitations used for fitting.



 §.§ Ultraviolet luminosity function



<cit.> presented a new derivation of the UV LF in the redshift range 2.0< z< 9.0 including over 2500 lensed galaxies behind the HFF clusters using HST observations from the HFF <cit.>, Grism Lens-Amplified Survey from Space (GLASS) <cit.> and Cluster Lensing And Supernova survey with Hubble (CLASH) <cit.> programmes. This LF probes down to M_UV<-16 over the full redshift range we are considering in this study. A smooth flattening of α was found, from α=-2.28±0.10 at redshift z=9 to α=-1.53±0.03 at redshift z≈2. Over our redshift range, 3≲ z≲7, the faint-end slope is found to evolve from α∼-1.60 to α∼-2.05. The shape of the LF and its evolution agrees well with other results in the literature, including determinations from blank fields (see , as well as other references in ).


We treat this LF as the LF of the total “parent” population in order to calculate the total SFRD. In this way, inclusive of completeness corrections, we assume that both LFs represent the “total” of their respective populations within the quoted luminosity and absolute UV magnitude limits. This is an assumption that we return to and discuss in Section <ref>.

All LFs, UV and LAE, are shown on a common scale (SFR) in Fig. <ref>. The LFs of <cit.> and <cit.> are highlighted in thicker, solid lines. 





 §.§ Global evolution results


The results from the LF in each redshift interval are integrated with luminosity to map out the evolution of luminosity density with redshift:

    ρ_Lyα= ∫ L_Lyα ϕ_Lyα  dL_Lyα.

This is then converted to the SFRD based on the calibration of <cit.>, assuming the factor of 8.7 between intrinsic and Hα flux and a case B recombination <cit.>:

    SFRD_Lyα [M_⊙ yr^-1 Mpc^-3] = 7.9 × 10^-42×ρ_Lyα / 8.7.


The UV luminosity density is converted into the equivalent SFRD using the form SFRD = κ×ρ_UV <cit.>, in which κ is the conversion factor, sensitive to the recent SFH, metal-enrichment history and the IMF. The term ρ_UV is the UV luminosity density, which is expressed in units of erg s^-1 Hz^-1, estimated by integration the UV LF with M_UV. The widely used value of the conversion factor from <cit.> is κ = 1.4 × 10^-28 , where a Salpeter IMF <cit.> is assumed. <cit.> used a Chabrier IMF <cit.> for this conversion, so we correct for this, using a factor of 1.96. Going forward, the SFRD obtained from UV LF is treated as the total SFRD of the entire galaxy population. We estimated the total UV SFRD down to M_UV=-13 at redshifts z = 2, 3, 4, 5, 6, 7  (see Table <ref> for the values used, after correction to the Salpeter IMF). Subsequently we interpolated the SFRD to redshifts z=2.5, 3.1, 3.5, 3.7, 4.5, 4.7, 5.4, 5.7 to adapt to redshift bins used in the literature studies.



The ratio between the LAE SFRD and the total, dust-corrected SFRD in a given redshift interval gives the global escape fraction. This definition was applied in the works of <cit.>, <cit.>, , <cit.> and <cit.>. The results for the global escape fraction derived in this work are shown in Figs <ref> and <ref>. We compared our results to the results derived using several LAE LFs in the literature, for which we adopted an identical approach to that described above <cit.>. We used the same UV LF for each comparison. 

The lower limits of the respective LF integrations are worth noting. We used two different limits, the first was 0.04L*  and the second was 0.0006L* (where L* is the characteristic galaxy luminosity of the Schechter function). The first limit, as suggested in <cit.> is designed to compare our study to previous results using brighter galaxies. It corresponds to limits of log L_Lyα [erg s^-1]∼41 and M_UV∼-17 respectively. The results for calculated with this limit are shown in Fig. <ref>. The second limit, 0.0006L*, corresponds to the faint regimes that we reach with the LAE and UV LFs (log L_Lyα[erg s^-1 ]∼39.5 and M_UV∼-13) and is shown in Fig. <ref>. The vertical error bars are estimated from published uncertainty values in ϕ*, α, L* (LAE LFs) and ϕ*, α, M* (UV LF). The horizontal error bars display the redshift intervals of each survey.  As <cit.> and <cit.> probed the LAE LF at the same redshift intervals, the data points derived from the LF of <cit.> have been shifted by 0.1 in redshift. Table <ref> presents the results for each study using this integration limit, as well as the SFRD value derived. By comparing the results obtained with the different limits we can assess the impact of the faint galaxies that we reach with the latest LAE and UV LFs.   

To investigate the global evolution of escape fraction with redshift, we recall the power-law form that was proposed by  and <cit.>: f_escp^Lyα = C ×   (1+z)^ξ. The coefficient C is constrained strongly by the evolution of escape fraction at low redshift while the exponent ξ is defined by the evolution at high redshift. In Fig. <ref>, the best fits calculated for this work are shown as red and blue dash-dotted lines, fitting all the points in the graph and only those of <cit.>, respectively. In Fig. <ref>, the red fit is to all the data. The green dashed line shows the fit from <cit.>, which used the UV LF obtained from <cit.> and an LF integration down to 0.04L* (fitted to data over the range 2<z<6). We leave this fit in both plots to guide the eye as to the difference when integrating to a fainter limit.  use the results of <cit.> and an integration limit of 0. The fit of  is shown in Fig. <ref> to provide the comparison to earlier results which have been integrated to 0.

We find C=14.3±14.0 × 10^-4 and ξ=2.3±0.6 when fitting to all the literature and C=12±43 × 10^-4 and ξ=2.8±2.1 when only fitting the results from <cit.> with the integration limit at 0.04L*. With the lower integration limit of 0.0006L*, we find C=4.1± 7.7 × 10^-4 and ξ=3.6± 1.2 when fitting our results together with all the ancillary data in Fig. <ref>. Because the other studies included in this graph do not probe the faint regime that our LFs do (equivalent to the integration limit of 0.0006L*), and they therefore require an extrapolation outside the limits of the datasets used. During this fitting process, we set the weight of each point equal to the weight of the equivalent point from our study (i.e. the cyan point in the same redshift bin). This relation is qualitatively consistent with the equivalent relation derived in <cit.> (green dashed line, noting that this relation is derived using a different UV LF and a brighter integration limit) and suggests that will reach 100% around z=8. 

 








§ DISCUSSION



With the individual and global results now in hand, we can compare them and consider what these can tell us about and its evolution.



 §.§ Global evolution



Fig. <ref> shows the determination of using the LF integration limit 0.04L*. We plotted this graph in order to compare our results to previous literature determinations of the escape fraction, which used samples of galaxies brighter than this work, and to assess the impact of the bright (M_UV<-17) and faint populations (M_UV<-13) on the evolution of . We plotted results derived using both the modified (cyan) and the general Schechter (pink) functions in order to show the difference in these results created by the presence or lack of a turnover in the LAE LF. However, at this integration limit, the behaviour of the modified Schechter function is close to that of the traditional Schechter function, so we did not see a significant difference in SFRD_Lyα. As the current best derivations of the LF come from the modified Schechter functions, going forth we discuss the results related to these (cyan points).

We observed a good agreement at all redshifts between the results derived in this work and the results from the literature. We saw little evolution between z∼3.5 and z∼4.5, followed by a jump in at z∼6, but at each redshift, our results agree well with the established literature. When fitting our data (blue line) compared to all the results (red line), we observed an enhancement of , likely due to the constraints placed by the literature in the redshift range 2<z<4. Our points do lie systematically above the points from the literature, indicating a greater SFRD based on the latest LAE LF of <cit.>. We note, however, that the LFs of <cit.> and <cit.> have not been included in Figs <ref> and <ref> due to large uncertainties, especially at the highest redshifts and faint luminosities. These uncertainties contribute to unrealistically high (and poorly constrained) escape fractions based on these LFs.

Additionally, the choice of completeness cut has an impact. In <cit.>, a completeness cut of 1% was chosen (i.e. LAEs with completeness values <1% are not taken into account in the LF determination), in order to maximise the sample size, while still negating the effect of the most incomplete and therefore uncertain sources. Studies have typically chosen different completeness cuts, or none at all. For example, <cit.> chose 10%, rejecting a larger number of sources. In order to check the effect this has, we recomputed using the LF cut at 10% completeness. This yielded values similar to those seen in the literature: 10-15% escape. Therefore, we consider our results completely consistent with what has been found previously when considering a brighter integration limit (0.04L*).




The green dashed fit of <cit.> lies slightly higher than the literature data. We attribute this to the significant update in the UV LF since the publishing of that study (the UV LFs of <cit.> and <cit.> were used for this fit). This emphasises the importance of the UV LF in these calculations and the value of the latest determination from <cit.>. The values used in this work, inclusive of dust and IMF correction are shown in Table <ref>. 

The values we derive for are considerably higher when we extend the LF integrations down to 0.0006L*, particularly for the highest redshift bin, z∼6, where we see an increase of 0.4 dex with respect to the 0.04L* integration. In fact, this point is consistent with the fit to the data in , which is integrated to zero, that is, taking into account the full population by assuming the validity of the two LFs down to zero. This large escape fraction value is unsurprising, as we see a high number density of faint galaxies in the LAE LF, and these populations typically have higher (see Fig. <ref>, as well as ). In the z∼6 bin, is consistent with 100%, i.e. a similar SFRD derived from LAEs as from the UV-selected population. While still consistent at 1σ with the fit derived by  (using an LF integration to 0), our point lies above points from the literature. We note that no other studies have had access to the faint regime of LAEs used in this study and as such, the literature LFs we compare to are integrated beyond the limits of their samples to create this graph. Reasons for this enhanced value for faint LAEs at z∼6 include the effect of the LAE incompleteness treatment, which acts most strongly for faint objects at the highest redshifts. Effectively, this would mean that we are counting in the LAE LF a significant number of faint LAEs not seen in the continuum.

This result may also point to a general scenario where faint galaxies (i.e. M_UV∼-13) generally see an extremely high production and escape fraction of photons (see also ). This appears to be a clear effect when integrating down to the L* limit our sample reaches: faint galaxies experience a higher escape of photons, especially at higher redshifts. This result is also in agreement with the results presented in Fig. <ref> as in this plot, between M_UV∼-17 and M_UV∼-13, we see LAEs with exclusively high escape fractions (almost all above 10% and many around 100%). This result implies that at z∼6 and beyond, faint galaxies tend to be highly star forming and chemically unevolved, which should allow high escape fractions. On the other hand, seen in the uncertainties in the faint end of the LAE LF, it is clear that there is still a large intrinsic scatter in the populations (once more in agreement with the results in Section <ref>). 

It is worth noting, however, that if the IMF of these faint, reionisation-era LAEs is top-heavy with respect to the assumed Salpeter IMF, this may also explain an elevated . Such an IMF would increase the number of ionising photons, from massive stars, which would in turn increase the emission of the galaxy. A higher ionising photon production efficiency, ξ_ion has been observed for some LAEs (, see also ) although there is no firm consensus on how much higher. If ξ_ion were indeed to be 25.5-25.6, this would mean a factor of 2-3 more ionising photons produced. This would in turn mean more photons produced and an overestimation of in our results by a similar factor (we note that this would apply to all the results in Figs. <ref> and <ref>). This is an important dependance on the high-mass slope of the IMF and the production of ionising photons, and could be particularly relevant to the faintest galaxies towards the epoch of reionisation, which are more likely to host exotic stellar populations. 
 
In summary, we find that the LAE SFRD matches the UV SFRD during the final stages of reionisation, and is ∼20% of it at redshifts between 3 and 5. For both bright and faint integration limits, we see this jump in the highest redshift bin (z∼6) compared to little evolution between redshifts of 3 and 5. This reflects a significant evolution of the LAE LF in this redshift bin, therefore suggesting that the LAE population evolves significantly between the end of reionisation (z∼6) and later times (z∼4.5). Although it is worth noting the uncertainty on our z∼6 result that we attribute to a combination of several factors: a lower dust content in galaxies at this redshift and above combined with higher specific SFRs in our sample of faint galaxies. As mentioned, the completeness correction for the LAE LF acts most strongly on the faint, high-redshift LAEs, making this a possible additional contributor. 



 §.§ Comparison of the global and individual evolutions



It is immediately clear that the global evolution of shows a more significant trend than the individual evolution, despite significant uncertainties, especially in the highest redshift bin. This becomes additionally clear when incorporating the agreement in the literature for the global evolution.

For our datasets, this is likely due at least in part to the far greater sample size used in the global evolution: 2500 UV-selected galaxies and 600 LAEs, compared to ∼100 LAEs for which is calculated (Fig. <ref>). For a smaller sample such as the one used for the individual tracking of , particularly considering the lack of sufficient galaxies at redshifts greater than 6, any redshift evolution is washed out by the scatter in individual results. The results of  display a similar lack of evolution over 1.9<z<3.8 when taking into account the significant scatter of individual galaxies. The same is even true for the more recent studies of <cit.> and <cit.>, using Balmer decrements to calculate , albeit with smaller sample sizes. The intrinsic variety of escape scenarios for photons from different galaxies (even bearing in mind that all these systems are selected as LAEs) is evident. 

The trend seen in the global evolution of (integration limit 0.0006L*), towards ∼80% around z=6, is not seen in the individual results. Likely this is at least in part due to the incompleteness of the sample used to calculate for individual galaxies at this redshift, as both LAEs with very small escape fractions and LAEs with very large escape fractions are harder to observe. For the LAEs with large escape fractions this difficulty comes from the faintness of the associated continuum: the galaxies with the highest values tend to be faint in terms of UV magnitude (Fig. <ref>). This incompleteness should be accounted for in the two LFs used, so we do not expect this to play a significant role for the global evolution.

Contrasting these two results, global and individual, it is clear to see that a large sample size, as well as considered completeness corrections, are necessary to observe any significant redshift evolution of . Indeed, further exploration of the faintest regimes of LAEs (and UV-selected galaxies) would be desirable in order to better constrain the faint end of the LFs and reduce the uncertainties on the highest-redshift points. For example, it is still very challenging to observe more than a few galaxies with log L_Lyα<40 at z>5.5 (see, for example, Fig. 3 of ).

We note also that the global comparison is effectively the escape fraction of a given volume of the Universe, bounded by the luminosity and redshift limits of the surveys used. This calculation takes into account all galaxies, not just those actually selected as LAEs, as is the case for the individual results. We might therefore reasonably expect to see a smaller escape fraction due to the inclusion of galaxies not selected as LAEs. In the lower and intermediate redshift bins, the results are consistent with the sample median shown by the red dashed line in Fig. <ref>; however in the highest redshift bin, the global calculation of appears to yield a larger value. Taking the median of the galaxies from Fig. <ref> above z=5 would yield a higher result (36% escape), and it is likely that we simply do not detect enough LAEs of high escape fraction at higher redshifts in our limited sample behind A2744. As this is just one cluster, extending these measurements to other clusters efficient in lensing galaxies above z=5 is very desirable. 

The galaxies detected by their emission but not in the continuum (the black triangles in Fig. <ref>) naturally all have high escape fractions and are counted in the LAE LF, so this may go some way towards explaining the discrepancy we observe between our different methods of calculating . Additionally, and as mentioned, the completeness correction added to the LAE LF acts strongly in the faint, high-redshift regime so we also expect this to play a role. The results in Fig. <ref> suggest that this may be a stronger role than the equivalent completeness correction added to the UV LF. 

Contrary to the lack of similarity in redshift evolution in the global and individual results, there is a similar trend in terms of an evolution with the UV magnitude of the host galaxy. In Sect.<ref> this is established as a much more significant trend and the results when comparing LFs support this. When integrating to fainter limits we see significantly higher values of across the redshift range probed. 



 §.§ The overlap of the LAE and ultraviolet luminosity functions





Here we revisit the assumption made in order to compare the two LFs using single integration limits in terms of L* and treat them as representative of their respective “total” populations. This, in principle, is troublesome as there is no direct mapping between luminosity and UV magnitude; for every given luminosity bin, there are a range of UV magnitudes that galaxies display. In fact, this encodes exactly what we want to measure, the escape of photons. It is clearly invalid to compare a LF based on just the brightest LAEs to the complete UV LF, including the faintest sources. Therefore, in order to make the comparison discussed above and shown in Figs. <ref> and <ref>, we have to assume that we adequately represent the “total” populations with both LFs, within the UV and luminosity integration limits used. This approach is discussed at length in <cit.> and  and we revisit this in the context of our dataset.

That we adequately represent both and UV populations is reasonably evident for the brightest objects, they are certain to be included in their respective surveys. This is less evident for the fainter regimes explored. For example, it is useful to ask whether the faintest LAEs included in the LAE LF are adequately represented by the faint end of the UV luminosity function. In order to assess this, we can look at the faintest LAEs and compare them to the UV magnitude regimes included in the UV LF of <cit.>.

It is by now well established that certain LAEs remain undetected in even the deepest photometric searches to date (see, for example, ). Additionally, some LAEs display faint continuum that would not pass the criteria for many UV-based selections, be they based on colour-colour cuts <cit.> or photometric redshifts <cit.>. We note that preliminary searches conducted by JWST/NIRCam in lensing clusters yields almost exactly the same number of undetected LAEs <cit.>. On the other hand, nebular lines are detected in these systems, so they are likely to be real detections <cit.>. With future, deep JWST (and HST) surveys, we will be able to shed more light on this issue.

This raises concerns as to whether the UV LF truly represents the entirety of the galaxy population, or whether there are a significant number of `UV-dark' LAEs that are counted in the LAE LF but not in the UV LF. This is important to consider given the high values of we derive when integrating to 0.0006L* (Fig. <ref>).

A way that we can assess which of these two cases is closer to reality is to compare the UV magnitude regime where these HST-dark LAEs are expected to lie to the regime covered by the UV LF in <cit.> and the integration limits we use. Upper limits on the UV magnitude of these LAEs can be calculated by manually extracting the continuum level from the HST images, as in <cit.>. These objects, denoted by upward-pointing triangles to indicate upper estimates on M_UV, are displayed in Fig. <ref>. This shows that for the clusters considered by the UV LF in question, and for the photometry in question, the UV-dark LAEs would be expected to appear around M_UV=-16, with many residing between this and M_UV=-14, and some down to M_UV∼-12. 

The UV LF in <cit.>, inclusive of the correction for selection incompleteness, extends past M_UV=-16 at all redshifts relevant for this work, and extends past M_UV=-14 at z∼3 and z∼6, and close to it at z∼4 and z∼5. At z∼3, the coverage extends all the way to M_UV=-12. The integration limit (0.0006L*) used for the calculation of from the UV LF corresponds to M_UV=-13, as denoted by the horizontal dashed line in Fig. <ref>.

There are four LAEs estimated to have upper UV magnitude limits fainter than this in the sample of four lensing clusters in <cit.>. As the non-detection of these LAEs likely depends at least in part on the relative depths of the spectroscopy and photometry, it is difficult to estimate what effect this has across our sample, but we can reasonably expect this to have a minor effect, considering the small number of LAEs in this position in comparison to the sample size used for the two LFs. 

We can additionally reassure ourselves by comparing what the UV luminosities should be from the faintest LAEs assuming different levels of escape. The faintest LAEs included in the LAE LF are no fainter than L_Lyα=10^39 erg s^-1, which equates to an absolute UV magnitude of M_UV∼-10.4 assuming 100% escape (meaning this value is a lower (faint) limit on the absolute UV magnitude). If we assume a more likely value of our sample median from Section <ref> of 22.5%, this UV magnitude becomes M_UV=-12.2. This is very faint compared to the reach of the UV LF we use, below the integration limit of M_UV=-13. However, using this integration limit and an escape fraction of 22.5%, we recover a value L_Lyα=2.4×10^39 erg s^-1. There is only one LAE in our sample close to this luminosity (see Figure 3 of ). On the other hand, following this analysis shows that faint LAEs with L_Lyα<10^40 erg s^-1 are not accounted for in the UV LF with its integration limit set at 0.0006L* as in Fig. <ref> if they have >95%. Across the board, this effect will bias our results for low.

We can perform the same exercise in an inverse manner, to assess whether we are likely to see the emission from the faintest UV objects included in the UV LF. The LAE luminosity expected from a galaxy of M_UV=-12 is 10^39.6 erg s^-1, assuming 100% escape. Assuming 22.5% escape, this value will be 10^39.0 erg s^-1. So for galaxies of this absolute magnitude, we can expect to account for all the LAEs in the population (when taking into account the completeness correction). However the escape fraction can be lower (down to 0%), meaning that we will miss the objects of very low UV magnitude and very low escape. Unlike missing objects in the UV selection, this biases our results towards high escape values. This counter-bias can alleviate the bias introduced by missing UV-faint objects as described in the previous paragraph, and indeed this is the motivation behind choosing one value of L_⋆ down to which the two LF integrations are calculated.

In summary, and in light of the high values for found in Fig. <ref>, for these results to be overestimated, a large population of continuum-undetected LAEs must be playing a role (together with the effects introduced by their incompleteness) in the LAE LF that are not accounted for in the UV LF, even with the completeness treatment described in <cit.>. This potential effect is important for studies looking to evaluate the total SFRD of the Universe. 

Alternatively, if we assume that this effect is negligible and the high value derived at z∼6 when including our whole faint sample is reflecting a real similarity between SFRD_Lyα and SFRD_UV, this indicates that galaxies at z≳6 are emitting significantly higher amounts of photons than later in the Universe. This supports a picture of galaxies towards reionisation as chemically unevolved and highly star forming, creating a lot of emission and allowing it to escape. 




 §.§ Revisiting the contribution of LAEs to cosmic reionisation





Seeing as the escape of photons is a crucial quantity for the assessment of the LAE contribution to cosmic reionisation, we can now revisit this in light of our results. There is significant debate in the literature about the relative contribution of the LAE population to reionisation, (see, for example ). All of these results are, naturally, dependent on the values assumed in the various studies. The value used to calculate this contribution is often pivotal in the classification of this contribution. We can now recalculate the LAE contribution taking into account our results.  

We started by considering the ionising emissivity from SFGs, using the following formalism <cit.>: 

    ṅ_ion(z) = ρ_UV(z)  ξ_ion ,

where ṅ_ion is expressed as the product of the UV luminosity density ρ_UV(z) <cit.>, global UV luminosity to ionising photon conversion factor ξ_ion <cit.>, and the ionising photon escape fraction . 
We reformulated Eq. <ref>, as we are interested in the quantity ṅ_ion(z) for the LAE population only: 

    ṅ_ion, Lyα(z) = ρ_Lyα(z)   ξ_ion^Lyα ,

where ρ_Lyα is the luminosity density, ξ_ion^Lyα is the conversion factor of the observed luminosity to the number of ionising photons, and is the Lyman continuum escape fraction of the LAEs. Ignoring collisionally excited emission and using Case B recombination (; see also ) as well as the intrinsic flux ratio between and Hα of 8.7, the ionising emissivity of LAEs, corrected for , is written as:

    ṅ_ion, Lyα(z) = ρ_Lyα (z)  ×/8.7   c_Hα×  (1-),

where c_Hα is the coefficient factor of the Hα emission line, ∼ 1.25-1.35 × 10^-12 for Case B recombination. It remains to estimate the individual parameters appearing in the above function. 

- The LAE luminosity density (ρ_Lyα): this parameter was presented in Sect. <ref>. 

- The escape fraction of photons, . We cannot correct ρ_Lyα by the escape fraction derived in Sect. <ref> as this, by construction, corrects ρ_Lyα to the total value of the parent population (ρ_total) that we take from <cit.>. We can nonetheless use the value we have derived in Sect. <ref>. We discuss both our sample median, =22.5% and the sample median for the LAEs at z>5, =36%. 

- LyC escape fraction, : This parameter is poorly understood, in particular at high redshift, due to the increasing IGM neutrality, which absorbs Lyman continuum photons along the light of sight. However, recent progress has been made in local galaxies (high-redshift analogues) <cit.> and in LAEs up to z∼3 <cit.>. However, results from these studies, as well as simulations <cit.>, suggest that there is a significant scatter in the across the LAE population. Therefore, placing one value on , such as , is a simplification. Many observational studies point to values between 10% and 15%: <cit.> derive an underlying population value of 12%, <cit.> obtain values between 3% and 15%, <cit.> give an upper limit of 16% and <cit.> find values between 1% and 34%, with most of their galaxies having <15%. Based on the SPHINX simulations <cit.> and the model for Lyman continuum escape developed by <cit.>, is 1.17%, 0.4% at M_UV = -17, -13 (we note that this is for all galaxies, not just those selected as LAEs).

The ability to turn Eq. <ref> into a summation, with each LAE contributing its own ρ_Lyα, and with consequently following a relation such as those presented in <cit.>, <cit.> or <cit.> would be an improvement. Increased statistics, especially for faint LAEs and at redshifts greater than 5 would be necessary to make this approach feasible. This is within reach for JWST in the coming years, as observed samples increase in size and more faint galaxies are uncovered.

A very recent approach by <cit.> attempts this individual approach for a large sample of JWST/NIRCam-observed galaxies (not just LAEs) (observed by the JADES programme, ). The authors conclude that faint galaxies (M_UV≥-18.5) are potentially responsible for ∼80% of the reionisation budget. The majority of our sample of LAEs fall within this bracket (see, e.g. Fig. <ref>).

With ṅ_ion, Lyα, at z=6, from Eq. <ref>, we can compare this value to the ionising emissivity required for reionisation as derived by <cit.>, ṅ_ion,crit=50.92 (we note that this value uses a fiducial clumping factor C_HII=3). In Table <ref>, we show the range of results, applying different values for ρ_Lyα, and . For ρ_Lyα derived from our results, (the LF of ) we find LAE contributions ranging from minor (5-10%) for a low (0.4%) to dominant >100% for =10%. Changing between the sample median and z>5 median does not qualitatively change the outcome. As changes more significantly depending on the chosen value, this represents the main effect on the outcome. In several cases we see unrealistic contributions (>100%) for LAEs based on =10%, suggesting that this value may be too high and not representative of our whole sample. Crucial work remains to correlate with observable parameters (such as parameters) at higher redshifts and constrain the escape of ionising photons for the LAE population at z∼6 and above. 

We investigate the effect on the LAE reionisation contribution in the case of a completeness cut in the calculation of the LAE LF at 10%. We take the results of <cit.> as emblematic of this scenario (as mentioned, the LF of <cit.> is consistent with that of <cit.> if this cut is applied). In this case, the contribution to reionisation derived for the LAE population is significantly less (1-2% for =0.4% and 30-50% for =10%). This emphasises the sensitivity of these results to the completeness treatment applied for the LAE population.
 
We additionally show the difference between the modified Schechter function, which we have chosen to adopt throughout this study, and the general Schechter function (noted GSF in Table <ref>. Using the general Schechter function roughly doubles the LAE contribution to reionisation. 

Highlighting the effects of both the function chosen to fit the LAE LF and the completeness cut applied serves to reinforce the need for a larger sample at z>5 and for faint galaxies. Better statistics in this parameter space will constrain the faint end of the LF more strongly, discriminating between the modified and general Schechter functions. With deeper observations of faint galaxies, the completeness for the faint LAEs will also improve (i.e. the sample will be more complete at a given luminosity) and the chosen cut will become a less important factor.
 
The chosen value of is clearly determinant in this calculation with LAEs contributing very little for small values of and providing the dominant contribution for reasonable values of observed in many lower-redshift LAEs and low-mass galaxies acting as high-redshift analogs. Using the ρ_Lyα from <cit.> and =36%, a sample average of just 6.5% would be enough for LAEs to provide all the necessary ionising photons at z=6. If the values of observed in these low-redshift LAEs are maintained across the LAE population into reionisation, it is certainly conceivable that these galaxies could have been responsible for reionisation in its entirety. We note that the results in this study do not constrain the reionisation scenario beyond z=6.7 and that LAE LFs at z≥7 are needed to track the contribution of LAEs further into the EoR. 


 




§ CONCLUSIONS


We have derived the escape fraction of photons across the redshift range 2.9<z<6.7 using two methods, namely, comparing dust-corrected star formation to star formation derived from emission and comparing the SFRD from the UV `parent' population LF and the SFRD derived from the LAE LF. Here we summarise the main conclusions of the work.

– We determined the values of for 96 individual galaxies behind the A2744 lensing cluster, finding a significant scatter and a sample median of 22.5 %. 

– We observed a significant evolution of with absolute UV magnitude, following the relation log()=(0.28±0.03) M_UV+(4.3±0.6). Notably, we saw very few LAEs with <10 % among objects fainter than M_UV=-18.

– We expect the line-incompleteness of MUSE detections to play a role here, but only for the faintest objects, M_UV≳-14.

– For the global redshift evolution of , we derived results in agreement with previous determinations in the literature when using a bright LF integration limit (0.04L*). appears to reach 100% around z=11-12.

– When we assessed with an integration limit that reflects the faint limits of our sample (0.0006L*), we found that becomes consistent with 100% escape by z∼6. This result supports galaxies being chemically unevolved and highly star forming at these redshifts, producing a high quantity of photons and allowing them to escape. 

– With both bright and faint integration limits, we observed little evolution between z=3-5 and a jump at z∼6, suggesting a rapid evolution of the LAE population towards reionisation, although this effect is at <1σ for our dataset. 

– We saw more significant redshift evolution in our global assessment than for individual galaxies due to the large intrinsic scatter in the LAE population combined with the far greater statistics in the respective LFs used for the global comparison.

– We revisited the LAE contribution to the reionisation process, taking into account our new values for and a range of motivated values for . The results depend heavily on the assumed values for . Taking the latest values observed in LAEs at z∼3, we find that LAEs can provide all the ionising emissivity needed for reionisation. 

– We demonstrated the additional strong dependence of the LAE contribution on the completeness cuts and fitting form of the LAE LF. This stresses the urgent need for better statistics at z>5 and log L_Lyα [erg/s]<40.

At the moment, it is challenging to improve the two luminosity functions used in this work, as they already use the full extent of deep observations in lensing clusters from HFF and MUSE. Further clusters to be included in this sample are desirable, particularly to better constrain the high-redshift (z∼6) faint end of the LAE LF. This will be made possible through further MUSE-observed lensing clusters, such as those presented in <cit.>. This will also help bin the results into smaller redshift bins to better constrain the precise evolution in the range 3<z<6. Observations of LAEs at higher redshift from JWST, such as <cit.> are starting to build larger sample sizes, but they are not yet at the quantity of LAEs and UV-selected galaxies used for the results in this work. In particular, intrinsically faint galaxies have poor statistics, although this can be improved with the continuing observations of lensing clusters by JWST/NIRCam and JWST/NIRSpec.

On the other hand, it is possible, with further JWST observations, photometry, and spectroscopy in order to improve the determinations for individual galaxies. The Balmer decrement provides accurate estimations of dust attenuation along with SED fitting utilising a greater number of photometric bands and a more extended coverage. Studying these systems in more detail will reveal the range of different escape scenarios leading to the varied values that we have observed. This can, in turn, lead to better understanding of galaxy evolution around the epoch of reionisation and how LAEs affect this process. 




This work is done based on observations made with ESO Telescopes
at the La Silla Paranal Observatory under programme IDs 060.A-9345, 092.A-0472,
094.A-0115, 095.A-0181, 096.A-0710, 097.A0269, 100.A-0249, and
294.A-5032. Also based on observations obtained with the NASA/ESA Hubble
Space Telescope, retrieved from the Mikulski Archive for Space Telescopes
(MAST) at the Space Telescope Science Institute (STScI). STScI is operated by
the Association of Universities for Research in Astronomy, Inc. under NASA
contract NAS 5-26555. 
All plots in this paper were created using Matplotlib (Hunter 2007). 
Part of this work was supported by the French CNRS, the Aix-Marseille University, the French Programme National de Cosmologie et Galaxies (PNCG) of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES.
This work also received support from the French government under the France 2030 investment plan, as part of the Excellence Initiative of Aix-Marseille University - A*MIDEX (AMX-19-IET-008 - IPhU).
Financial support from the World Laboratory, the Odon Vallet Foundation and VNSC is gratefully acknowledged. Tran Thi Thai was funded by Vingroup JSC and supported by the Master, PhD Scholarship Programme of Vingroup Innovation Foundation (VINIF), Institute of Big Data, code VINIF.2023.TS.108. This research was funded by Vingroup Innovation Foundation under project code VINIF.2023.DA.057.        


aa 
 

 


§ LAE LUMINOSITY DENSITY AND ITS STAR-FORMING RATE DENSITY




 





 Paper Title: MHD lensing in inhomogeneous ISM for qualitative understanding of the morphology of supernova remnants.



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V_LSR V_LSR V_rot   V_rotV_expaM_⊙ 
^∘ T_bI_CO I_HI I_HI 
    X_ HIX_ COX_ HIX_ COkm s^-1 km s^-1  pc K km s^-1Alfvén Alfvén VV velocityH cm^-3V_ unitV_ unitB_ unitρ_ unitt_ unitL_ unitp_r∂̣p_θp_ϕ cot sin sin cos costantan /ß
    
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    high-low-MHD lensing in ISM and SNR morphology]MHD lensing in inhomogeneous ISM for qualitative understanding of  the morphology of supernova remnantsY. Sofue]Yoshiaki Sofue 
     
    Institute of Astronomy, The University of Tokyo, Mitaka, Tokyo 186-0015, Japan2024[
        [
        Accepted for publication in Astrophysics & Space Sciences (ApSS)
    ====================================================================
    
          § ABSTRACT 
    Morphological evolution of expanding shells of fast-mode magnetohydrodynamic (MHD) waves through an inhomogeneous ISM is investigated in order to qualitatively understand the complicated morphology of shell-type supernova remnants (SNR).  
    Interstellar clouds with high  velocity act as concave lenses to diverge the MHD waves, while those with slow  velocity act as convex lenses to converge the waves to the focal points. 
    By combination of various types of clouds and fluctuations with different  velocities, sizes, or wavelengths, the MHD-wave shells attain various morphological structures, exhibiting filaments, arcs, loops, holes, and focal strings, mimicking old and deformed SNRs.  
       ISM: magnetic fields —  ISM: clouds —  ISM: supernova remnant — magnetohydrodynamics (MHD)
    
    
    § INTRODUCTION
    
     
    The interaction of expanding blast waves of supernova remnants (SNR) with the interstellar medium is one of the most fundamental subjects of interstellar physics 
    <cit.>.
    The shock compression of gas clouds and magnetic fields by SNRs is also an important issue in the high-energy astrophysics for investigating the acceleration mechanisms of cosmic-rays. 
    Extensive observations of the interacting clouds near SNRs have been obtained from γ-ray to radio wavelengths, on which several models for cosmic-ray accelerations have been proposed
    <cit.>. 
    
    Multi-wavelengths observations from γ to radio have been obtained also for investigating the morphological evolution of SNRs interacting with the ISM, which are used at the same time to diagnose the physical conditions and magnetic fields of the ISM
    <cit.>. 
    In order to explain the observed properties and morphologies of SNRs, a number of numerical simulations of hydro- and magneto-hydrdoyanamics have been obtained about the interaction of shock waves with the inhomogeneous/turbulent ISM and stellar winds
    <cit.>.
    
    
    The current studies of the SNR-cloud interaction have been devoted mainly to study the effects of SNRs as the active source on the shocked medium to change the physical conditions interior to the SNR shell. 
    The interaction of an SNR with the ISM takes place most directly at the contact surface of the outermost blast wave, which is observed as a thin shell with optical emission due to the recombination process between the blast and cold ISM as well as the non-thermal radio emission due to the Fermi acceleration of cosmic-rays at the highest compression surface.
    
    
    At the same time, the morphology of SNRs also manifests the past paths of propagation of the shell front through the interstellar medium, because the blast wave propagation is strongly influenced by the density and magnetic fields in the ISM.
    This means that the morphology of SNRs may be used to diagnose the gas density and magnetic field distributions in the ISM before the passage of SNR shells <cit.>.
    Therefore, morphological study of images of aged SNRs in optical  <cit.>
    and radio-continuum emissions
    <cit.>, 
    would provide with a tool to diagnose the interstellar medium by analysing the shapes that have been deformed during the propagation. 
    
    In this paper we aim at understanding the morphology of old shell-type SNRs by the deformation of expanding shells in the inhomogeneous ISM.
    We show that various types of morphology of old SNRs are qualitatively understood by the lensing effect due to the variation of propagation speed of the waves in the ISM.
    Since the treatment of the wave in this paper is linear, we discuss only the geometrical shapes of the front, or the morphology, of old SNRs.
    Non-linear physical quantities such as the density, magnetic strength, cosmic-ray acceleration, emission rates, etc.. cannot be calculated in the present linear approximation.  
    However, the relative distribution of the brightness of wave front, which defines the morphology, can be mapped by the distribution of column density of wave packets projected on the sky. 
    Thereby, the brightness of a sheet of MHD wave is highest, when it is seen edge on and the tangential sheet shines brightest, whereas a face-on sheet is almost invisible. 
    
    In section <ref> we formulate the method using fast-mode MHD waves. In section <ref> we describe result of simulation for  large-scale deformation of shell fronts. Section <ref> deals with more complicated morphology, and section <ref> compares the model with the well-known SNRs having unique morphology.
    In section <ref> we present some speculative considerations towards understanding the morphology of SNRs in more general cases based on the MHD model, and discuss the difficulty and limitation of the model. Section <ref> summarizes the results. 
    
    
    
    
    § MHD LENSING
    ßFast-mode MHD wave and  velocity in the ISM
    
    The  velocity in a medium with magnetic-field strength B and gas density ρ is given by
    
    V=B/√(4 πρ).
    
    Its variation with the cloud density depends on the coupling property of the magnetic field and gas.
    There are three types of coupling:
    
      *  (a) Frozen-in field depends on the gas density as B∝ρ^2/3, which leads to  velocity varying as ∝ρ^1/6. This happens when the magnetized gas cloud is highly turbulent.
        
      *  (b) Cylindrical magnetic field uniformly filled with the gas varies as B∝ρ, so that we have V∝ρ^1/2. This occurs in stretched clouds and filaments extending along the magnetic lines of force.
        
      *  (c) Gas flowing along magnetic lines of force with the field strength kept consant yields variation of the  velocity as V∝ρ^-1/2. 
        Such a flow is found in a magnetic tube inflating from the Galactic plane by the Parker instability with a condensed cloud at the bottom of the tube.
    
    Interstellar clouds in cases (a) and (b) yield higher  than the ambient medium, so that they act as concave lenses to diverge the MHD waves.
    Clouds in case (c) act as convex lenses to converge the waves.
    In this paper we represent the property of a cloud by its  variation relative to the mean value, and represent the variation by equation <ref>.
    
     
    MHD disturbances excited by explosive events such as due to SNe in the interstellar space propagate as spherical shock waves in the initial expansion phase. 
    In the fully expanded phase when the expanding velocity becomes comparable to the local  velocity or sound velocity, they propagate as the fast-mode MHD compression wave (hereafter, MHD wave), Alfvén wave, and the sound wave.  
    
    The typical  velocity is ∼ 10 with magnetic strength B∼ 5 μG and gas density ρ∼ 1 H cm^-3, whereas the sound velocity is c_ s∼ 0.1 and 1.3  for molecular and HI gases with temperatures T_ mol∼ 20 K and T_ HI∼ 100 K, respectively.
    Therefore, among the three modes, the fast MHD wave is most efficient to convey the energy into the ISM, which propagates at the group velocity approximately equal to the Alfvén velocity, v_ g≃, where the  velocity is higher than the sound velocity, or the magnetic pressure is higher than the thermal pressure <cit.>. 
    
       ß vs turbulent velocities
        
    One concern that disturbs the above assumption is that the  velocity is comparable to the turbulent velocity as the consequence of the energy balance (equilibrium) between the magnetic and thermal energy densities in the Galactic disc.
    Because the Eikonal equations hold locally, the wave propagation is superposed by the local turbulent motion.
    This effect is significant on the ray paths slowly bent (refracted) by low- clouds.
    On the other hand, it does not affect so much the ray paths against strongly disturbed (reflected) ray paths by high- lensing.
    In this paper, we assume that the turbulent motion is negligible, which, even if existed, disturbs the result on the same order as that due to low- lensing.
    The amount of ray deflection is much smaller than that by high- lensing, and the entire morphological shape is more effectively determined by the high- lensing.  
    
    
    ßPropagation of MHD waves by Eikonal equations
    
    The propagation of MHD waves is traced by solving the Eikonal equations developed for the Solar coronal Moreton waves  <cit.>.
    The method has been applied to the SNRs  <cit.> as well as to the Galactic Centre explosions <cit.>.
    Gas clouds with higher density with constant and weaker magnetic strength act as a convex lens (low- lens) gathering the waves.
    On the other hand, gas clouds with frozen-in magnetic field and gaseous cavity (hole) with constant and higher magnetic strength act as a concave lens (high- lens) reflecting the waves.         
    
    The Eikonal equations describing the propagation of an MHD wave packet are given as follows <cit.>.
    drdt=V p, dθdt=V rp, dϕdt=V rp sinθ, ddt=-pṾṛ +Vrp(^2+^2), ddt=-prṾθ̣ 
    - Vrp(-^2   θ), ddt=-psinθṾϕ̣ - Vrp( +   θ),  
    where
    V=V(r,θ,ϕ)=V(x,y,z) is the  velocity, the vector p=(, , )= grad Φ is defined by the gradient of the eikonal Φ, p=(^2+^2+^2)^1/2, (r,θ, ϕ) and (x,y,z) are the polar and Cartesian coordinates.   
    
    An interstellar cloud with lower  velocity deflects the path toward its center, acting as a convex lens, and the converging waves are focused around its focal point to produce amplified wave front.
    On the other hand, a cloud with higher  velocity acts as a concave lens, and diverges the waves to produce a hole with the deflected waves making a ring.
    
    Figure <ref> shows the propagation of MHD wave packets produced at the center intervened by clouds with higher and lower  velocities than the ambient velocity, where the  variation is expressed by Gaussian function with an e-folding scale radius of 0.5.
    
    In the following simulations, we solve the Eikonal equations assuming the functional form of the  variation in the space, which is expressed in the following form:
    
    V=V_0[1+ϵ f(x,y,z)],
    
    where V_0 is the mean  in the ISM, f(x,y,z) is a function expressing the shape of the turbulence, and ϵ represents the relative amplitude of variation.
    Here, ϵ≥ -1, and low- and high- clouds have negative and positive ϵ, respectively. 
    
    
    
    
    § LARGE-SCALE DEFORMATION
    ßLopsidedness by  gradient 
    
    The gradient of  velocity produces a lopsided elongation of the front shape, so that the lower- side is stopped, and the shell expands faster in the higher- side.
    Figure <ref> (A) shows a case when the wave expansion takes place in an ISM with one-directional  variation expressed by 
    V =V_0  [1+exp (z/h)],
    where V_0 and h=0.5 are constants. 
    Hereafter, the coordinates are (x,y,z) measured in arbitrary unit.
    The time is measured in unit of unity scale divided by , =1/V_0.
    In this diagram, the wave front at t=0.25, 0.5, 0.75 and 1  are plotted.
    The SN explosion takes place at (x,y,z)=(0,0,0) spherically, and the wave front is spherical in the initial stage.
    As it expands, the front is elongated in the z direction of increasing , and attains an open cone structure.
    The front finally forms a one-sided curved sheet open toward positive higher- direction.
    
    cosh cosh
    
    If the wave expansion takes place in a disc with increasing  gradient toward both sides expressed by a cosh function,
    V=V_0 [1+0.5 cosh(z/0.5)],
    
    the front shape becomes bi-conical, as shown in figure <ref> (B) for z_0=0. 
    If the disc is thin enough compared with the expanding shell radius, the front becomes bipolar, exhibiting a dumbbell-shaped bubbles for negative-Gaussian disc (figure <ref> (C)). 
    If the disc has higher- than the surrounding ISM having positive-Gaussian distribution, the front attains bipolar,or bilateral caps (figure <ref> (D)), somehow mimicking such an axisymmetric bilateral SNR as G296.5+10.0 <cit.>.
     
    ßHigh- and low- walls           
     
    If the MHD-wave front encounters a wall with sufficiently high  with finite thickness, the front is reflected and the shell is deformed to become a half-spherical shape.
    Figure <ref> (A) shows an example of such an encounter, where a horizontal wall is located at z_0=0.5 expressed by
    
    V=V_0 [1+exp ((z-z_0)/h)^2]
    
    with z=0.5.
    
    If the wall has lower , the front is strongly refracted toward the wall and absorbed there, resulting in a half shell, and dense, flat front along the wall is formed, as shown in panel (B) of figure <ref>.
    
     
    
    
    ßClouds with low- vs high-: Convex vs concave lensesexp expA low- cloud acts as a convex lens for the ray paths of the MHD wave packets.
    Figure <ref> shows front propagation of a spherical MHD wave front encountering a low- cloud, where the  velocity is expressed by a negative Gaussian function as
    
    V=V_0[1+ϵ exp(-(r/r_0)^2)].
    
    Here, r=[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^1/2 is the distance from the cloud center at (x,y,z)=(x_0,y_0,z_0)∼ (0.5,0,5,0,5), r_0∼ 1 is the radius of the cloud, and ϵ∼ -0.5 is the negative amplitude of .
     
    The front is reflected by the lens and bent toward the focal point of the cloud.
    Accordingly, the wave forms a concave surface shape and the amplitude increases. 
    
    If  of the cloud is small enough, the front evolves into an elephant-trunk shaped elongated lobe. 
    However, the focusing effect to the trunk's head amplifies the magnetic field together with wave density, which works to increase the  velocity and suppresses the growth of the trunk.
     
    	        
      
    On the other hand, when the  velocity of the cloud is higher than the ambient medium, it acts as a concave lens.
    The ray paths are diverged by the lens, and the front expands locally, making a vacant hole.
    Thereby, reflected rays are locally converged on the edge of the hole, and form a focal ring (loop) of amplified waves.
    Figure <ref> shows an example of evolution of the front through such a high- cloud with ϵ∼ +0.5. 
     
    
    ßLarge-scale deformation by large cloudsIC443 and Sh 2
    Large-scale deformation such as observed for the SNRs IC 443  (figure <ref>) can be 
    explained by interaction of a larger-sized clouds and density fluctuations than the size of the expanding front.
    When a spherical front encounters a wall or a sheet with higher-, the front suddenly expands faster and the shape becomes lopsided, so that one side of the shell is more extended.
    
    Our MHD-wave simulation of IC 443 is shown in figure <ref>, where the large-scale deformation of the shell is reasonably mimicked in coincidence with that obtained by the 3D numerical simulations <cit.>.
    ßManatee (W50) SNR
    
    When the SN explosion takes place in a high- tube, the front expands in the direction of the axis of the tube, attaining a spindle shape, as shown in  figure <ref>.
    Here, the magnetic tube is further superposed with oblique sheets of variable , which produce the oblique stripes and wavy deformation of the front.
    The simulated shape mimics the Manatee nebula (W50), whose large-scale morphology  <cit.> can thus be possibly explained by the interaction of the SNR exploded in a magnetic tube,
    
    although more sophisticated model has been required to explain its extremely peculiar properties related to the central active source 
    <cit.>. 
    
    The off-plane location and large angle of the major axis to the Galactic plane of the Manatee nebula <cit.> suggests that the SNR exploded in a vertical magnetic tube inflating from the Galactic disc due to the Parker instability. 
    The overlaid oblique stripes are also explained by corrugation of the front by sheer inhomogeneity of the medium, where the entire region is embedded in sheets obliquely tilted by 45 from the x axis having sinusoidal variation of  with δ∼ 0.01 and λ∼ 0.2.
    
    ßCorrugated filaments by sheet fluctuationFilamentary loop
    
    If ISM has sheet-like fluctuations in one direction, the MHD front is disturbed to form wavy fluctuations near the equator parallel to the sheets' plane.
    Figure <ref> shows a case when the  variation is sinusoidal in the y direction by ∼ 1 percent of the ambient value,
    
    V=V_0[1+Σ_j ϵ_j cos (2π y/λ_i)],
    
    where ϵ_i ∼ 0.01 and λ_i ∼ 0.1.
    
    The perturbation influences the rays that propagate near the (x,z) plane (y∼ 0) across the explosion center, yielding wavy front around the equator.
    When the shell is seen from the pole direction (y axis) axis, the projected shell exhibits wavy loop and arcs due to the tangential view of the corrugated surface of figure <ref>) (A).
    When the shell is viewed from the x direction, filament structure is observed as equatorial filaments across the center.
     
    
    Thin filaments in Cygnus Loop
    
    Filamentary structure in the tangential projection of the outer shell is more ubiquitous in older SNRs.
    The very thin filaments in the Cygnus Loop as shown in the right panel of figure <ref> (B) <cit.> can be naturally reproduced by a tangential view of corrugated front, as simulated by the MHD-wave model shown in figure <ref> (B).
    
    	
    
    § TURBULENT AND PECULIAR MORPHOLOGY
    ßRandomly distributed high- and low- clouds
    
    When the MHD wave front expands into a turbulent medium, the front is deformed to produce clumps, filaments, arcs, loops and holes.
    We simulate three cases with randomly distributed 100 Gaussian clouds having  random sizes, amplitudes, and positions in the plotted frame. 
    
    Figure <ref> compares the time evolution of MHD fronts at t=1 and 2t_ unit, expanding into turbulent medium filled with randomly distributed high-V (left) and low-V (right) Gaussian clouds.
    High-V clouds deform the shell more strongly than low-V clouds. 
    
     
    If the medium is filled with high- clouds, the front is largely deformed to exhibit filaments, arcs, and holes as shown in figure <ref> (A).
    When the interstellar medium is composed of low- random clouds, 
    the front is deformed to make randomly focusing, concave, and locally amplified clumps, as shown in  figure <ref> (B).
    If the medium is mixed with low- and high-V clouds, the front results in a complicated deformed shell with the mixture of concave clumps, holes, and filaments as shown in figure <ref> (D).    
    Here, the  distribution in the ISM is given superposition of random Gaussian  clouds expressed by
     V=1+Σ_i δ V_i 
    with
    δ V_i=v_i exp(-(x-x_i)^2+(y-y_i)-2+(z-z_i)^2/w_i^2),
    
    where 
          w_i=0.05+0.1×ℛ(4,i),       x_i=2(ℛ(1,i)-0.5),      y_i=2(ℛ(2,i)-0.5),      z_i=2(ℛ(3,i)-0.5),      v_i=2(ℛ(5,i)-0.5), 
    and ℛ(j,i) represents random number between 0 and 1, as shown in figure <ref> (C).
    
    
               
     
    Expansion into a turbulent ISM filled with high-V clumps results in a front filled with cell-like clumps of arc-shaped filaments.  
    As the wave front expands further, the turbulent ISM with high-V clumps causes more disturbed structure.
    The Vela nebula would be an example of such deformed front, where a number of filamentary arcs and loops are observed as shown in figure <ref> (B).
    
    ßExpansion into sheets and tubes: oblique stripes 
    
    Plane parallel sheets with sinusoidal  variation creates longitudinal stripes.  
    If it is superposed by sinusoidal variation in the y and z directions, mimicking high- and low- tubes, the longitudinal stripes appear to cover the surface as shown in figure <ref>, where the wavelengths are taken as λ_y=λ_z=1.   
     
    MHD wave front expanding into ordered ISM with directional filaments with high- results in a shell covered by filaments and stripes parallel to the perturbing ISM. 
    Low- filaments result in less deformed filamentary surface.
    Figure <ref> (A) shows the simulated results.
    
    
    
    However, if the deformed filaments are viewed from the direction parallel to the filament axes, the deformed front resembles the result for random high- clouds.
    Figure <ref> (B) shows the same result for filamentary ISM turbulence viewed from two different directions: 
    The left panel  is a view perpendicular to the filaments, and the right is parallel.
    The projection parallel to the filaments resembles to random-cloud cases.
    
    
    ßMorphological evolution
     
    We have so far presented snap shots of the simulated result.
    Besides snap shots, it is also interesting to see how the morphology evolves during the expansion. 
    In figure <ref> and <ref> we show the time variation of an expanding MHD wave front in a turbulent medium filled with sinusoidal and randomly distributed high- and low- clouds. 
     
    The shell expands spherically in the initial stage, until the size becomes comparable to the scale size of the turbulence.
    After a while, the front is deformed to exhibit asymmetric and elongated shell due to the spatial gradients in the  distribution.
    As it further expands, the deformation is amplified to attain a turbulent morphology covered by randomly distributed filaments, arcs, holes, and concave focal clumps.  
    
    
    
    § QUALITATIVE UNDERSTANDING OF PECULIAR MORPHOLOGY OF SNRS
    ßCygnus Loop: Encounter with low- bars 
    
    The Cygnus Loop is known as the most typical shell-type SNR, but it also shows peculiarly open and complicated morphology divided by a bunch of long straight filaments crossing the center <cit.>.
    It consists of largely deformed arcs and filaments:
    The Veil Nebulae (NGC 6992) composes the eastern limb of a bunch of bright arcs and filaments, and the western limb also composes a bunch of veil nebulae and extends to the south.
    The entire SNR is largely deformed and is opened toward the south, making a long protrusion.
    The most spectacular structure is the straightly extending bunch of vertical (north-south) filaments, which separate the whole shell into eastern and western halves.
    
    These striking features have been well simulated by a lopsided outburst of the progenitor star in a uniform ISM
    <cit.>.
    We here try to simulate such peculiar morphology qualitatively by the lensing effect, where an initially spherical shell expands into an inhomogeneous medium.
    
    We assume that the front encounters a vertical (NS-oriented) long bar-shaped cloud with low- and a high- cloud near the southern edge.
    Figure <ref> shows the results compared with a photograph of the SNR, where blue and red dots represent the low and high  clouds, respectively.
    In the top panel we show a stereoscopic view of the simulated results, one panel rotated 10 from the other perpendicular to the line of sight to explore the 3D structure of the SNR.
     
    
    ßThe τ nebula: two orthogonal low- bars
    
    The τ nebula (N49/DEM-L-90) in the LMC is an SNR with the most strange morphology, exhibiting two orthogonal bunches of straight filaments.
    In our simulation, such morphology is understood as a result of an encounter of the MHD front with two orthogonal bar-shaped clouds with low-.
    Figure <ref> shows the simulation result and assumed low- clouds, as compared with the HST optical image.
    
    
    ßSNR S147: loops, arcs, and converging filaments 
    
    The old shell-type SNR S147 has been simulated in our early paper <cit.> based on the optical image <cit.>.
    In figure <ref> we reproduce our early work by the present simulation.
    We further performed higher-resolution simulation, and show the result in the lower panels of the figure in comparison with an optical image of S147 in the Hα line.
    
    Low- clouds 1 and 2 causes the concave filaments due to the focusing effect. 
    High- clouds 3 and 4 leads to the convex deformation in the eastern and western edges.
    
         
    
    Figure <ref> enlarges the NE filamentary region compared with the simulation.
    Due to the focusing effect by the low- cloud, wave front is strongly deformed and many converging waves produce a condensation of overlapped filaments, leading to a concave bright clump.
    
    
    
    § DISCUSSION
    ßLimitation of the method and speculative application of the model to SNRs
    
    By definition, the present MHD wave simulation cannot be applied to the early phase of SNR expansion, when the mass of plowed gas is less than the mass of the progenitor. 
    Also, the method is not applicable to SNRs with expansion velocity much faster than the  velocity with V_ expa>> 10 as well.
    However, we argue that the result would be applied to shell-type SNRs in order to qualitatively understand their morphology 
    
    Strictly speaking, the Eikonal equation with linear wave treatment is not applicable to waves of finite amplitude, but such waves do not exist in nature, and this is true for any linear theory of astrophysics.
    Nevertheless, linear theories are widely used to understand essential aspects of phenomena.
    Here we discuss the possibility of applying the MHD wave method to more general SNR cases.
    We consider whether shock waves are subject to lensing effects, or reflection and refraction by an inhomogeneous ISM.   
    
    There have been extensive studies of the interaction of expanding shells of SNRs with interstellar clouds <cit.> 
    in the premise that the interaction may trigger star formation and acceleration of cosmic rays by the compression of the magnetized molecular gas. 
    There have been also a number of hydrodynamical and MHD numerical simulations of the interaction  with the clouds 
    <cit.>. 
    
    Commonly seen in these simulations is the implosion of the shock front converging toward the cloud's center axis by refraction of propagation direction according to the deceleration of shock velocity due to momentum conservation.
    As a result, the shock wave suffers from a similar lensing effect to the MHD-wave focusing around the low- cloud's center, as shown in figures <ref> (A) and <ref>. 
    On the other hand, no numerical simulation has been obtained yet about shock waves interacting with a gaseous cavity or a high- cloud containing compressed frozen-in magnetic fields. 
    However, we may speculate that a shock wave encountering such a cavity would behave similarly to MHD-waves that expand inside the cavity and make a hole in the front surrounded by a focal ring, as shown in figure <ref> (B).
    Also, a shock wave inside high- cloud would become sub-Alfvénic due to the compression of frozen-in fields, which will propagate as an MHD-wave, so that the behavior would become more similar to that of MHD waves. 
    
    In summary, shock waves in SNRs behave similarly to MHD waves and result in similar morphological evolution.
    The difference is that shock waves disturb/destroy the clouds they pass through, while MHD waves are passive and do not disturb the environment.
    The deformation of the shock front may be weaker than that of the MHD wave front, but the sense of deformation is similar to each other.
    We therefore believe that the morphological peculiarities observed in shell-type SNRs can be qualitatively explained by the lensing effect of the inhomogeneous ISM.
    Figure <ref> shows some examples of such SNRs and the corresponding MHD wave models. 
    
    
    	        
      
    
    
    Through the simulation, we learnt that the shape of MHD wave front suffers from significant deformation due to the lensing effect by the inhomogeneous medium.
    This means in turn that a simplified analysis to find association of shell-shaped
    molecular gas around a round SNR edge, as often obtained about molecular-gas association <cit.>, may not be sufficient to study the past interaction with clouds. 
    
    ßComment on the 'mixed morphology'
    
    The term, 'mixed morphology' SNR, is often used to represent an SNR with different morphologies observed at different wavelengths such as in radio and X rays, which originate from the outer shell and interior hot gas, respectively <cit.>.
    Since the physical properties of the two components are quite different, the difference in their distributions is trivial.
    That means, the term may better be read as 'mixed physics', because the existence of such multiple conditions with different spatial distribution is trivial as the general property of a shock-wave driven shell by point explosion.
    
    On the other hand, we have shown that even the radio (or optical) shell, which is the outermost surface of the SNR in contact with the ISM, shows partially different morphologies within the same SNR.
    Such "mixed morphologies" with the same physical condition within a single SNR shell are explained by interactions with interstellar clouds with different  velocities.
    
    ßDifferent kinds of 'filament'
    
    We have thus far shown that the filaments in SNRs are classified into two categories, either strings or sheets.
    
      *  (a) Sheet filaments:  
        Many of the observed 'filaments' are tangential view of corrugated (curved) sheets projected on the sky, but not real strings. 
        The corrugation occurs by the reflection and refraction of the wave front by the lens effect of high- and low- interstellar fluctuations acting as concave and convex lenses, respectively.    
        Traditionally, most of the shell-type SNRs have been recognized by their loop shapes of tangential-sheet filaments near their edges.    
    
      *  (b) String filaments
          A real string (line)-structured filament is formed in such a case that a low- tube or an elongated low- cloud acts as a cylindrical lens and the waves are converged onto a focal string (line).
          The straight filaments crossing the Cygnus Loop and the τ nebula are shown to be filaments of this type.
          It must be noted that the string filament exhibits higher amplification of the wave due to the focusing effect compared to the corrugated-sheet filaments.
          An extreme case of wave amplification of this type is the focusing by a spherical low- cloud to the focal point, where the wave exhibits a small and concave spherical front with strongly amplified wave. 
    
      *  (c) Ring around hole
      The third type of filament is a focusing ring/loop around a hole on the shell surface produced by the diverging reflection of the wave front by a high- cloud.
      The tangled rings and arcs of filaments in the Vela nebula and S147 are of this type, which were produced by the diverging-lens effect through many high- clouds.
      
      *  (d) Focus of converging front:
      Another type of filament is a dense cluster of converging waves due to focusing effect by a low- cloud. The north-eastern clump of high density condensation of filaments in S147 is a typical case of such focusing filaments.  
    
    
    These types of filaments have different formation mechanisms, and hence, different physical conditions.
    
    Sheet filaments are the tangential view of  wave fronts whose large line-of-sight depth makes the apparent brightness much higher than the surroundings.
    
    On the other hand, the string filaments, as well as the converging clumps by convex lenses, are formed by focusing of the waves onto a line or a point, so that the wave amplitude is increased by a factor proportional to the surface-area ratio between before and after the focusing, often amounting to orders of magnitudes.
    Such difference of real amplitudes would result in different emission mechanisms and physical conditions in the filaments. However, discussing such difference quantitatively is beyond the present simulation on the linear wave approximation.
    It would be a subject for the future work and observations.
    
    ßSNRs with unique morphology
    
    The 'Manatee' nebula (W50) is an SNR having the most strange morphology.
    While efforts have been made to explain its elongated shape by interaction of a round shell and a precessing jet from the central compact object, SS433, <cit.>, the oblique stripes superposed on the surface have been difficult to be understood.
    In the present MHD model, however, such unique morphology has been relatively easily reproduced by assuming interaction with the ISM with cigar-shaped high- clouds superposed by sinusoidal variation of . 
    
    The τ nebula (N49 in LMC) is also unique for its orthogonal bunches of numerous long and straight filaments <cit.>.
    This morphology is relatively simply explained in the present MHD model as due to interaction of the SNR surface with two orthogonal bar-shaped clouds having low-.
    The central long filaments across the Cygnus Loop is classified into the same category due to interaction with an elongated cloud having low-.
    
    
    
    
    
    § SUMMARY
    
    
    By the simulation we showed qualitatively that most of the morphologies observed in shell-type SNRs can be reasonably explained by the lensing effect due to high- and low- interstellar clouds.
    Most of the "filaments" in SNRs are shown to be tangential views of corrugated sheets of the wave fronts.
    Real string-shaped filaments are formed by cylindrical focusing by cigar-like low  clouds, and arcs and rings around holes are formed by reflection around high- clouds.
    The various types of "filaments" on SNR surfaces are, thus, understood as the manifestation of ISM fluctuations, through which the fronts have propagated, suffering from the lensing effect.
    However, since the front is treated as a linear MHD wave, such non-linear quantities as the gas density, temperature, magnetic strength, cosmic-ray acceleration, and electromagnetic emissions cannot be calculated in the present method.
    
    The SNR morphology can be used to constrain the ISM conditions such as the wavelength of fluctuations, cloud sizes, direction of elongated clouds, and variation of relative amplitudes of the  velocity.
    However, the observed shape is a result of combination of the reflection and refraction paths of the waves, the filaments are significantly displaced from the original position of the lenses (clouds), which makes it not easy to quantitatively determine the parameters of the clouds.
    Deconvolution of the observed SNR filaments back into the real ray paths as well as the distribution of  velocities in the ISM would be a subject for the future.
    
    
    
    
    
    § ACKNOWLEDGEMENTS
     
    The computations were carried out at the Astronomy Data Center of the National Astronomical Observatory of Japan. 
    
    
    
    § DATA AVAILABILITY
     
    The images of the supernova remnants were downloaded mainly from the home page of APOD (Astronomy Picture of the Day) of NASA at
    https://apod.nasa.gov/apod/, 
    and from the Hubble site at
    https://hubblesite.org/images.
    Some were taken from the web sites described in the figure captions for the individual images.
    
    
    
    § CONFLICT OF INTEREST
    
    The author has no conflict of interest. 
     
    
    
    
    
    
    
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      Ref: Submission ID 54c32948-3562-46e5-a46a-dda03b437fca
    
    Decision: minor revision
    
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    REVIEWER REPORTS
    
    Reviewer Comments:
    
    Reviewer 1
    The author presents a fast-mode MHD wave simulation of  old and slow SNR shells expanding through inhomogeneous ISM. The model qualitatively shows how these inhomogeneities, i.e. dense inert clouds of magnetized material, can act like convex and concave lenses diverging the MHD waves. Combinations of clouds with different density and magnetization (i.e. Alfven velocities) could lead to various complex morphological structures of these SNR shells. The paper is convincing, well-presented and I recommend it for publication modulo the minor following requests:
    
    Page 1.
    There are a few groups of colleagues involved into the SNR morphology subject. To put it in a broader context of MHD simulations, some more references can be added in the introduction. This might help the paper be more visible.
    
    Some people to consider: Salvatore Orlando's group (+Bocchino/Petruk/Miceli), Pablo Velazquez's group (+Shneiter/Toledo-Roy), Li Zhang's group (+Jun Fang/Huan Yu), and Vikram Dwarkadas.
    
    [A] Thank you very much fot this comments. I include a more number of references.
    
    Page 2.
    At the beginning of paragraph 2.3 it is not clear why the cloud should have "constant OR weaker/higher" B-fields. Was it supposed to be written "constant AND weaker/higher"?
    
    [A] I have rewritten the paragraph.
    
    Page 4.
    1) Figure 2 has no (C) panel.
    2) "as shown in panel (C) of figure 3" – The author probably means here panel (B).
    3) The last sentence in 3.2 subsection refers to panel (C) of Fig 3, but Fig 3 has only panels (A) and (B).
    
    [A] Thank you for this notice. I have rewritten the caption and the text accordingly.
    
    4) "A low-V cloud acts as a concave lens for the ray paths of the MHD wave packets." – It was probably meant to be "convex lens".
    
    [A] Thank you.
    
    Page 8.
    "Concave clumps, holes, and filaments as shown in figure 8 (E)." – I guess, there should be figure 8 (D).
    
    [A] Thank you.
    
    Page 12.
    1) Figure 14 (A) has two nearly identical 3D renders of the wave front. The author could leave just one of them (or change the view angle more than 10 degrees).
    
    [A] Fig. 14 (A) is a "streogram" of the wave front, so that the reader may watch the left and right panels with the left and right eyes, respectively. I added remarks.
    
    2) Caption of Fig 14: "Cygnus loop in panel (E)" – The picture of Cygnus loop has no letter, and it should be (D) probably.
    
    [A] Thank you.
    
    And finally, the summary could be slightly more elaborated (e. g. the SNR limitations from subsection 6.1 could be added).
    
    [A] I added some more lines, thank you.


 Paper Title: Simulating intermediate black hole mass measurements for a sample of galaxies with nuclear star clusters using ELT/HARMONI high spatial resolution integral-field stellar kinematics.



firstpage–lastpage
Multiple Greedy Quasi-Newton Methods for Saddle Point Problems
    1st Minheng Xiao*
dept. Integrated System Engineering 

Ohio State University

Columbus, USA 

minhengxiao@gmail.com*
2nd Shi Bo
dept. Mathematics and Statistics 

Boston University

Boston, USA 

shibo@bu.edu
3rd Zhizhong Wu
College of Engineering 

UC Berkeley

Berkeley, USA 

ecthelion.w@gmail.com
    August 5, 2024
=================================================================================================================================================================================================================================================================================================================




§ ABSTRACT 
The fraction of low-mass galaxies hosting an intermediate-mass black hole (IMBH, with masses M_ BH≈ 10^2-10^5 M_⊙), is sensitive to how black hole seeds formed in the early Universe but is observationally still unconstrained.   In this paper, we assemble a sample of dwarf galaxies within 10 Mpc hosting bright nuclear star clusters (NSCs) that could host IMBHs.  For a subset of them, we use their observed surface brightness from Hubble Space Telescope () images, an assumed synthetic spectrum of their stellar population, Jeans Anisotropic Model (JAM) of the stellar dynamics, and the HSIM simulator software to create mock observations with the High Angular Resolution Monolithic Optical and Near-infrared Integral (HARMONI) field spectrograph for the Extremely Large Telescope (ELT).   We analyze the simulated data cube like real data, using JAM to infer the IMBH mass and its error in a Bayesian framework. Our simulations show that the ELT/HARMONI instrument can clearly detect the existence of IMBH demographics in NSCs down to a mass of about 0.5% of the NSC. 






galaxies: general – galaxies: supermassive black holes – galaxies: nuclei – galaxies: kinematics and dynamics – galaxies: evolution – galaxies: formation








§ INTRODUCTION


Both theoretical <cit.> and observational <cit.> studies have firmly established a correlation between the mass of central supermassive black holes (SMBHs, M_ BH>10^6 ) and the large-scale properties of nearby massive galaxies <cit.>. These properties encompass the stellar mass <cit.> and luminosity <cit.> of the central spherical bulge of stars, as well as its statistical dispersion of stellar velocities <cit.>. These robust black hole–galaxy scaling correlations strongly suggest a close connection between the growth of SMBHs and the formation and evolution of their host galaxies <cit.>.

Additionally, the evolutionary trajectories of SMBHs over cosmic time provide invaluable constraints that are vital for a comprehensive understanding of the assembly and development of both galaxies and their central black holes <cit.>. These connections offer profound insights into the intertwined cosmic story of black holes and galaxies, shedding light on the underlying physical processes that have shaped the universe we observe today.

In the cosmos surrounding us, central black holes are commonly classified into two primary categories: stellar-mass black holes and SMBHs. The stellar-mass black holes possess masses ranging from a few to several tens of solar masses, and their origins are well-established as the remnants of massive stars that conclude their lifecycles by undergoing supernova explosions. In contrast, the formation mechanisms of SMBHs remain enigmatic <cit.>. It is conceivable that SMBHs could grow from smaller stellar-mass “seeds” <cit.>. Alternatively, these enigmatic objects may have originated as massive seeds, formed through the direct collapse of pristine gas <cit.> in extreme conditions shortly after the Big Bang.

Presently, a significant disparity exists in the limited resolving powers of both space-based and ground-based telescopes, resulting in a substantial gap <cit.> that separates these two distinct populations of black holes. The nature of this gap, situated between the most massive stellar-mass black holes detected by LIGO  <cit.> and the lowest mass dynamically confirmed central black hole  <cit.>, remains a subject of uncertainty. This gap may arise from the absence of black hole populations within this mass range, termed “intermediate-mass” black holes (IMBH, M_ BH∼10^2-5 ), or it could be attributed to our current limitations in detecting them.




 §.§ Previous IMBH determinations

Over the past decades, extensive studies have provided dynamical M_ BH measurements for approximately 120 galaxies <cit.>, in which there are 23 IMBHs. These measurements determine black hole masses by analyzing stellar motions <cit.>, ionised gas dynamics <cit.>, CO-molecular gas dynamics <cit.>, [CI]-atomic gas dynamics <cit.>, or accretion disks <cit.>. Remarkably, these investigations have unveiled the ubiquity of black holes in high-mass galaxies <cit.>. However, the situation is notably less clear in lower-mass galaxies, where central black holes are anticipated to be IMBHs <cit.> or, in some cases, absent altogether <cit.>.

For instance, initial measurements conducted on galaxies within the Local Group (D < 3.5 Mpc) boasting stellar masses exceeding 10^9  indicated a notable absence of central IMBHs <cit.>. However, recent observations have hinted at the detections of a few IMBHs residing at the cores of dwarf galaxies <cit.> and massive globular clusters (GCs) like <cit.> and <cit.>. These IMBHs were identified through the disturbances observed in the motions of stars and gas within these objects. Additionally, the presence of IMBHs has been inferred from X-ray and radio emissions, believed to originate from matter falling onto a black hole, with the most compelling detection being the off-nuclear X-ray source HLX-1 <cit.>. Furthermore, considering some M_ BH–galaxy’s macroscopic properties scaling relations <cit.>, IMBH populations are expected to inhabit the centers of ∼10^8-9  galaxies (as referenced therein), ∼10^6-7  ultra-compact dwarf galaxies <cit.>, and ∼10^5-6  GCs. <cit.>, with decreasing mass from 10^5  to 10^2 .   

However, if IMBHs do indeed exist in low-mass galaxies, their demographics bear significant relevance to several intriguing astrophysical inquiries. The predictions stemming from stellar-mass seeding black holes suggest a high “occupation fraction” – i.e. a high likelihood that IMBHs are hosted by 10^9  galaxies, estimated to be around 80–100%  <cit.>. Conversely, the massive seeding scenario proposes a lower fraction of IMBHs in such galaxies <cit.>. The occupation fraction serves as a critical metric for quantifying black hole number density, thereby influencing our expectations regarding the rate of stellar tidal disruptions <cit.>. It also plays a pivotal role in studies aimed at determining the number of black holes likely to be discovered in the nuclei of stripped galaxies <cit.>. Moreover, obtaining compelling evidence of an IMBH census holds vital importance in constraining their formation mechanisms and their relationship with SMBHs and stellar-mass black hole populations <cit.>. Such findings will further refine predictions for gravitational-wave experiments <cit.>, enhancing our understanding of the broader astrophysical implications associated with IMBHs.

Furthermore, an imperative aspect of the quest for IMBH demographics lies in its profound implications for our understanding of galaxy formation, co-evolution across cosmic epochs, and the broader formation of cosmic structures <cit.>. The core strategy in the search for these IMBHs involves scrutinizing the dynamic signatures exhibited by stars and gas at the centers of nuclear star clusters <cit.> in dwarf galaxies, as well as within Galactic and intergalactic globular clusters <cit.>. These signatures arise due to the presence of a massive, compact, dark, and centrally concentrated mass that exceeds what would be anticipated from dynamical mass-segregation processes (discussed in Section <ref>). In very nearby stellar systems, it is plausible to expect roughly one star per central spaxel, facilitating the measurement of radial velocities for individual stars. However, in more distant systems, we continue to rely on the integrated light distributions of the bulk of stars within a single spaxel, provided that the spaxel falls within the black hole’s sphere of influence (SOI[The spherical space-time region surrounding the black hole where its gravitational force dominates other matter’s gravity, which is also defined as the black hole vicinity whose enclosed mass equals twice the black hole mass (or enclosed galaxy-mass equals the black hole mass).]) radius (R_ SOI= GM_ BH/σ^2_⋆, where G represents the gravitational constant).

To precisely measure the motions induced by IMBHs within their SOI, both spectroscopic and photometric observations must marginally resolve the R_ SOI. For IMBHs within a distance of ≈10 Mpc, this R_ SOI spans from a few thousandths of a parsec to a few parsecs <cit.>. These observational scales plunge below the diffraction limits of instruments like Gemini or the Very Large Telescope (VLT) equipped with adaptive optics (AO) techniques. These instruments provide a point spread function (PSF) with a full width at half maximum FWHM_ PSF≈008-01, equivalent to a range from a few parsecs to a few tens of parsecs for the same sample. The current Enhanced Resolution Imager and Spectrograph (ERIS) on VLT promises a spatial resolution of about ≈50 milli-arcseconds (mas). However, even this level of resolution falls short in probing the SOI of a typical 10^4  IMBH at a distance D≈10 Mpc because such a small black hole exerts minimal gravitational influence on its surroundings. Furthermore, while the James Webb Space Telescope  () enhances the flux sensitivity of ongoing black hole surveys, it does not substantially improve spatial resolution. Consequently, compelling evidence for the existence of IMBH populations remains elusive beyond the Local Group, and current observational facilities do not allow unambiguous confirmation.




 §.§ The role of ELT/HARMONI

The unanswered question regarding the existence of IMBH populations awaits the arrival of the Extremely Large Telescope (ELT) and its High Angular Resolution Monolithic Optical and Near-infrared Integral (HARMONI) field spectrograph <cit.> along with the Multi-AO Imaging Camera for Deep Observations (MICADO) imager <cit.>. These instruments are equipped with AO and offer unprecedented levels of sensitivity and spectral/angular resolution, boasting R = λ/Δλ∼7,100 and ∼17,400 for HARMONI. The HARMONI system, as reported in <cit.> (hereafter N23), provides an FWHM_ PSF≈12 milli-acrseconds (mas), while MICADO reaches an FWHM_ PSF≈10 mas <cit.>. The combination of HARMONI integral field spectroscopy (IFS) and MICADO imaging on the ELT platform will have the capacity to detect IMBHs at distances up to 10 Mpc (as detailed in this work). These IMBHs represent a crucial missing link between stellar-mass black holes and SMBHs and may have served as seeds for the formation of present-day SMBHs in the early Universe <cit.>.

The utilization of HARMONI will enable us to conduct indepth and precise three-dimensional (3D) kinematic studies of stellar populations within NSCs – locations where IMBHs are believed to be situated (as expounded in N23 for much higher mass black holes). Additionally, with MICADO, we can achieve substantially higher spatial resolution (a few mas compared to a hundred of mas with ). Consequently, the observable volume will expand by over tenfold compared to current capabilities for the search for nearby IMBHs. This extended reach will enable a significant increase in the number of direct measurements of black hole masses that can be conducted, ranging from a few to several tens. Ultimately, the ELT will facilitate dynamic measurements of prominent young star clusters. The unequivocal detection of IMBHs within such clusters carries profound implications for understanding the formation of black hole seeds during the early Universe <cit.>. 

We outline the specific selection criteria we have devised for the identification of our HARMONI IMBH survey sample in Section <ref>, while presenting the key properties of this sample in Section <ref>. Subsequently, in Sections <ref> and <ref>, we elucidate the dynamical and photometric models we employ in our simulations, respectively. Moving on to Section <ref>, we undertake near-infrared (NIR) IFS simulations utilizing the HARMONI SIMulator <cit.> software designed for ELT observations with the HARMONI instruments. Within this section, we present the simulated datacubes and the methodologies employed for extracting stellar kinematics. In Section <ref>, we engage in a comprehensive discussion of the potential applications of dynamical modeling to measure the masses of central black holes using the observations, accompanied by a consideration of their limitations. Finally, our findings are summarized and concluded in Section <ref>.

In this paper, we adopt a flat Universe model with a Hubble constant of H_0≈70 km s^-1 Mpc^-1, a matter density of Ω_ m,0≈0.3, and dark energy density Ω_Λ,0≈0.7, which are in accordance with constraints derived from both the Planck collaboration <cit.> and WMAP <cit.>. We ensure uniformity by applying a foreground extinction correction <cit.> and utilizing the interstellar extinction law proposed by <cit.> for all quantities. Furthermore, to maintain consistency throughout our work, we employ the AB-photometric magnitude system <cit.> and orient all kinematic maps, aligning the horizontal and vertical directions with the galaxy’s major and minor axes, respectively.







§ SAMPLE SELECTION


We first describe the selection process to find the best dwarf galaxies hosting bright NSCs, where IMBHs probably reside and are observable with ELT in Section <ref>, and then we summarize the selection criteria in Section <ref>.




 §.§ Sample selection process


Our target selection process involves two primary steps. Initially, we employed the same methodology as outlined in N23, utilizing homogeneously obtained NIR Ks-band (≈2.2 μm) photometric data sourced from the Two Micron All Sky Survey <cit.> and the 2MASS redshift survey <cit.>. This initial step was aimed at identifying dwarf galaxies within proximity of ≲10 Mpc. These selected targets exhibit a stellar mass range falling within 2×10^6≲ M_⋆≲5×10^10 , measured using either the simplification of a uniform mass-follow-light hypothesis with the mass-to-light ratio of _K=1 (/) <cit.> or the equation (2) of <cit.> (i.e. log(M_⋆) = 10.58 - 0.44 ×(M_K + 23) and -23.3≤ M_K≤-13.3 mag), which is 1.8 mag overlapping the low-M_K boundary of the ATLAS^ 3D survey <cit.>. It is important to note that <cit.> and <cit.> have demonstrated that the Ks-band is less susceptible to dust absorption compared to optical wavelengths, by a factor of 5–10, and that _K exhibits less variability, by a factor of 2–3. Subsequently, in the second step of our selection process, we conducted a thorough cross-referencing of these preselected targets with all available surveys focusing on nearby low-mass galaxies known to host luminous NSCs, where IMBHs are likely to reside.

The necessity for the second step arises due to certain limitations associated with the 2MASS data. Specifically, 2MASS employs the de Vaucouleurs “effective” aperture for measuring a galaxy’s half-light radius, denoted as R_e  (the radius encompassing half of the galaxy’s total light), and extrapolates the surface brightness profile of the galaxy to derive its total flux, a technique described in detail by <cit.>. In practice, this involves integrating the flux in incremental annular regions, commencing from the center (r>5), and moving along the elliptical contours of the galaxy until the half-light point is reached. Subsequently, interpolation is used to construct a precise value for R_e. While this approach is robust for larger galaxies and remains largely unaffected by the PSF circularizing effects, it becomes increasingly susceptible to inaccuracies when estimating R_e for dwarf galaxies.

Furthermore, it is important to note that 2MASS observations tend to possess relatively shallow photometric depths, a characteristic highlighted by <cit.>. This shallow depth may potentially result in an underestimation of the galaxy’s Ks-band luminosity, thus complicating the accurate determination of Sérsic indices for light profiles in sources with relatively compact dimensions. As a result of these limitations, relying solely on 2MASS (and 2MRS) data for the identification of dwarf galaxies becomes inherently unreliable. Nevertheless, it is worth emphasizing that these datasets serve as valuable starting points for preselecting candidate galaxies that are characterized by less dust extinction, proximity, and relatively faint luminosity, while also providing robust estimates of their stellar masses.






To determine the luminosities and stellar masses of galaxies based on their observed apparent magnitudes, we followed a methodology akin to the approach employed in the Most Massive Black Holes (MMBH) IFS survey, which targeted ultra-massive galaxies with ELT/HARMONI (N23). A crucial prerequisite for this process is knowledge of the distances to the target galaxies. Among the extensive 1.65 million galaxies catalogued in the 2MASS extended source catalog <cit.>, approximately 100,000 of them possess redshift-independent distances available from the NASA/IPAC Extragalactic Database (NED[<https://ned.ipac.caltech.edu/>]), denoted as NED-D[<http://ned.ipac.caltech.edu/Library/Distances/>] <cit.>. For these galaxies, we conducted matching with the 2MRS galaxies and adopted their NED-D distances, which are estimated with a typical accuracy of approximately (10–20)%. In cases where the 2MRS targets lacked NED-D distances (but less likely for the nearby targets), we derived distances based on their redshifts, incorporating corrections for peculiar motions arising from the Virgo cluster listed in NED <cit.>. These corrections were crucial to ensure the accuracy of redshift distances, which were then employed in our preliminary survey with a distance criterion of 2≲ D≲10 Mpc. More specifically, the redshift distances were calculated using the formula D=V_ cosmic/H_0, where V_ cosmic represents the cosmic velocity. This cosmic velocity, in turn, was calculated as the sum of the observed heliocentric velocity (V_ hel), a correction for the Local Group centroid motion <cit.>, and the correction for the Virgo cluster infall (V_ in, Virgo), which amounted to 200  at the Local Group.








Using the obtained distances, we proceeded to transform the observed magnitudes into absolute magnitudes, denoted as M_K=K_T - 5log D - 25 - 0.11× A_V <cit.>. In this equation, K_T , signifying k_m_ext, represents the apparent magnitude in the Ks-band, measured within an isophotal aperture derived from a single Sérsic surface-brightness profile, extrapolated to the inner-unresolved profile <cit.>. Additionally, A_V corresponds to the Galactic extinction in the Landolt V-band, sourced from <cit.>, and incorporates the reddening relation proposed by <cit.>, featuring R_V = A_V/E(B-V) = 3.1. Subsequently, we computed the Ks-band luminosities of the galaxies based on their M_K values and estimated the stellar mass of each galaxy under the assumption of a uniform mass-follow-light characterized by the mass-to-light ratio of _K=1 (/) <cit.>.

With a focus on the lower stellar-mass range of 2×10^6≲ M_⋆≲5×10^10 , which aligns with the capabilities of ELT for observation (as outlined in Table <ref>), we initially identified 570 galaxies from the 2MRS survey. To ensure the accuracy of our selections, we conducted a meticulous verification process using the Update Nearby Galaxies Catalogue <cit.>. This process involved confirming critical details such as apparent Ks-band magnitudes (or luminosities) and stellar masses. It is important to highlight that, during this step, we excluded sources with 2MASS luminosities and mass that did not align with their values from <cit.>. Due to the unreliable R_e of 2MASS dwarf galaxies, which are likely included in our initial screening for a sample, their apparent magnitudes (K_T), absolute magnitudes (M_K), and consequently, their estimated stellar masses (M_⋆) are also considered unreliable. As a result of this thorough examination, we excluded a vast majority of targets that are not present in the catalog or targets with Ks-band estimated luminosities and masses that do not match their values reported in the catalog, resulting in a final selection of approximately 110 dwarf galaxies that met our criteria.

To ensure the efficiency of our quest to detect and accurately measure IMBH masses that might be nestled within low-mass galaxies, we embarked on a search for galaxies housing NSCs exhibiting central stellar-velocity dispersions of σ_⋆, c<75 . The high stellar density within NSCs promises robust signal-to-noise ratios (S/N) during relatively brief exposure times, facilitating unambiguous and minimally error-prone dynamical measurements. Our choice of a velocity dispersion threshold aligns with our objective to identify IMBHs, based on the M_ BH–σ_⋆ scaling relation derived by <cit.>, extrapolated to the mass range relevant to IMBHs. Notably, the velocity dispersion criterion significantly impacts the required spatial resolution for HARMONI’s imaging capabilities, which must resolve the SOI of IMBHs. To address this demand, we established a canonical lower limit of angular resolution at R_ SOI≈10 mas for the entire sample, factoring in the trade-off between sensitivity and instrumental efficiency, as per the methodology described in equation (1) and section 2.1 from N23. 

Next, we meticulously scoured the literature for essential information pertaining to NSCs within the selected dwarf galaxies. This entailed gathering data on NSC properties such as mass, effective radius, metallicity, stellar populations, ages, and . Additionally, we sought details regarding the stellar velocity dispersions of these NSCs, as well as the half-light radii of the 110 dwarf galaxies that had successfully passed our initial inspection. Our comprehensive search encompassed both photometric sources <cit.> and spectroscopic surveys <cit.>. We also incorporated insights from detailed case studies <cit.>. Through this thorough investigation, we identified 40  dwarf galaxies hosting NSCs within the pool of 110 low-mass galaxies that met our stringent criteria.



This final sample shown in Fig. <ref> encompasses a diverse array of galaxy types, including 4 dwarf ellipticals, 6 lenticulars, 25 spirals, and 5 irregulars. These galaxies' properties, their NSCs' properties, their possible IMBHs, and related references are recorded in Table <ref> of the supplementary material. Furthermore, it encompasses a wide spectrum of large-scale structures such as bulge, bulgeless, bar, and disc configurations. Notably, 17 of these NSCs benefit from high-resolution Hubble Space Telescope () images, while the remaining NSCs have archival imagery from the Digitized Sky Survey (DSS[The dataset Digital Object Identifier (DOI): https://www.ipac.caltech.edu/doi/irsa/10.26131/IRSA44110.26131/IRSA441]), and Sloan Digital Sky Survey (SDSS).  This meticulously curated selection of galaxies has been aptly named the “HARMONI IMBH sample”. 








 §.§ Selection criteria


We formed the selected criteria listed as the following selection steps:

(i) The explored volume is opened out to the distant range of 2≲ D≲10 Mpc, where hosts -23.3≤ M_K≤-13.3 mag dwarf galaxies. These galaxies contain the well-known NSCs with central velocity dispersion of σ_⋆, c<75 , enforcing the possible IMBHs.

(ii) Observable with ELT that satisfies the limit on the zenith distance for a good AO correction: |δ + 24^∘|< 45^∘, where δ is the sky declination (N23).

(iii) The sample must exclude all targets within the Galactic equatorial plane and Galactic bulge regions where highly contaminated by dust: |b|≤8^∘. Here, b is the Galactic latitude.

Table <ref> provides a concise summary of the selection criteria applied in our process, while Table <ref> presents general properties of the low-mass galaxies and NSCs featured in the final HARMONI IMBH sample. It is important to note that in the finalization of our selected targets, an additional prerequisite emerges – namely, the availability of a tip-tilt star in close proximity to the science target (or a natural-guide star, NGS). This star serves as a reference point in the sky for the correction of atmospheric turbulence effects during ground-based IFS observations. To meet this requirement, the NGS stars should possess a magnitude of m_H < 19 in Vega mag ( or <20.4 AB mag) and be situated at distances ranging from 12 to 60 from the center of the science target <cit.>.

We diligently searched for such essential NGS stars following the guidance outlined in N23, utilizing Gaia and SDSS archival databases, and showing them as the white crosses along with their H-band photometric AB magnitudes in Fig. <ref> when available for each individuals galaxy. For Sloan NGS, we converted their SDSS magnitude to AB magnitude following N23 section 2.1. However, for Gaia NGS, which uses the Vega magnitude system, we then converted it to the AB magnitude system by applying the Gaia data release 3 documentation, specifically section 5.5.1 titled “Photometric relationships with other photometric systems,” <cit.>.

Roughly 60% of galaxies within our sample comfortably satisfy the tip-tilt star requirement for Laser Tomography Adaptive Optics (LTAO[LTAO mode on the ELT/HARMONI instrument needs one NGS at least to work simultaneously with six other artificial off-axis Laser Guide Stars (LGS). The system brings the required NGS to be more than 10,000 times fainter than that from the classical AO used on Gemini and VLT.]) correction. In cases where galaxies lack NGS meeting the tip-tilt star criteria for LTAO correction, we suggest employing the NSCs themselves for open-loop AO correction <cit.>. 







§ PROPERTIES OF THE HARMONI IMBH SAMPLE






 §.§ Galaxy stellar mass and size
 

The left panel of Fig. <ref> illustrates the distribution of our HARMONI IMBH sample (depicted as red dots) within the phase space of Ks-band absolute magnitude versus distance. This analysis confines the explored volume to a range of 2≲ D≲10 Mpc and encompasses a wide span of Ks-band absolute magnitudes, ranging from -13.3 to -23.3 mag. Consequently, the sample comprises a diverse set of nearby dwarf stellar systems, with stellar masses ranging from slightly above 2×10^6  to just under 5×10^10 . The plot also includes data from larger survey volumes, such as ATLAS^ 3D <cit.>, MASSIVE <cit.> and MMBH (N23), for reference. This overlay emphasizes the distinct position of the HARMONI IMBH survey sample within the hierarchy of galaxy structures, as it occupies the lowest permissible mass range for a stellar system. 

In Section <ref>, we determined the galaxy photometric stellar masses based on the K-band luminosities, assuming a constant M/L_K=1 (/) <cit.>. We also cross-verified the stellar masses for the HARMONI IMBH sample using an alternative formula from equation (2) of <cit.>. This formula, calibrated from 260 ATLAS^ 3D early-type galaxies, is expressed as log(M_⋆) = 10.58 - 0.44 ×(M_K + 23). The differences between the two methods did not exceed 10%. In addition, the sizes of galaxies in the sample were estimated using R_e = 1.61× j_r_eff, where j_r_eff represents the 2MASS XSC keyword semimajor axis of the isophote enclosing half of the total galaxy light in the J-band <cit.>. The preference for using j_r_eff over k_r_eff was primarily due to the superior signal-to-noise ratio (S/N) of the J-band data in comparison to the equivalent K-band data <cit.>. However, we adopted j_r_eff ≡ R_e as the size estimate only when it exceeded 1.5 kpc in cases where this information was not available in the literature. Otherwise, we relied on the values provided in the literature. The sample thus boats dwarf galaxies with an effective radius range of 0.3 < R_e≲ 5 kpc.








 §.§ Our IMBH sample on the mass–size diagram


We constructed a mass–size diagram (M_⋆–R_e) for the HARMONI IMBH sample based on the collected stellar masses and sizes. This diagram was then overlaid with data from the 2MRS sources before the application of selection criteria. Here, we removed all 2MRS galaxies those have R_e<3 because these R_e are unreliable (Section <ref>). The resulting plot, presented in the right panel of Fig. <ref>, reveals interesting trends. At lower masses (M_⋆ < M_ crit≈ 2 × 10^11 ), the lines representing constant σ_⋆ (also indicating constant M_ BH) trace the mass concentration and mass density (or bulge mass fraction) of galaxies. This suggests that the central M_ BH of these galaxies behaves similarly to other galaxy properties. The evolution of SMBHs is thus linked to various factors, including optical color, molecular-gas fraction, dynamical–, initial mass function (IMF) normalization, age, metallicity, and α-element abundance <cit.>. Notably, their M_ BH growth follows a similar trend to that of galaxy properties influenced by star formation within the host galaxies <cit.>. We also added the ATLAS^ 3D <cit.>, MASSIVE <cit.>, and MMBH (N23) galaxies on this M_⋆–R_e diagram to highlight the distinct distribution of this lowest mass and compact galaxies sample.

Our HARMONI IMBH sample spans an extensive range of galaxy masses, covering more than four orders of magnitude. It is worth noting that our sample distributes right below and fully consistent with the upper limit for dwarfs and spirals, the dashed blue line expressed as (R_e/ kpc)=8×[M_⋆/(10^10  M_⊙)]^0.24. The sample distributes in a similar manner of the dwarfs in fig. 9 of <cit.>, which is typically followed by 99% of early-type galaxies. 

The properties of galaxies with masses exceeding 10^9  tend to align with the constant σ_⋆ lines, implying that their M_ BH may conform to the predictions of the M_ BH–σ_⋆ relations <cit.>. Below this mass and due to the incompleteness at low M_⋆ and low R_e of the 2MASS survey, we have very little information of the relations among constant σ_⋆, M_⋆, and R_e (particularly those with σ_⋆<30 ). Although, compare our HARMONI IMBH sample with the dwarfs sample in fig. 9 of <cit.> – which has properly R_e – it is clear that our HARMONI IMBH sample follows the expected distribution for dwarfs. Consequently, any relation among these parameters that have found based on the primary distribution of ≳10^9  galaxies should be stopped when the galaxy mass goes below this limit. However, recent observations <cit.> suggested that black hole mass in these lower-mass galaxies does not follow to the established black hole–galaxy scaling relations, indicating the involvement of distinct mechanisms in the coevolution of these two crucial components at the low-mass regime.

The shift in the black hole–galaxy scaling relations, particularly in the low-mass regime, has prompted various explanations, as discussed in studies such as <cit.>. However, a definitive answer remains elusive due to the limitations of our current observational capabilities in detecting these elusive IMBHs. The dependencies between M_ BH and galaxy properties in this lower-mass range are challenging to discern with existing data. Our proposed sample of nearby IMBHs leverages the unique advantages of ELT/HARMONI, which offers exceptional angular and spectral resolutions as well as sensitivity. This initiative aims to overcome these challenges and potentially uncover new insights into the previously unexplored realms of IMBH and galaxy coevolution.





 §.§ Nuclear star clusters and central massive black holes


Our HARMONI IMBH sample includes galactic nuclei with NSCs. Among them, 16 NSCs have known masses and effective radii falling within the ranges of 10^5<M_ NSC≲10^8   and 3 < R_e < 27 pc, respectively. For the remaining NSCs, this information is unavailable. We have compiled data on these NSCs as well as their host galaxies’ properties in Table <ref> in the supplementary material. In Fig. <ref>, we have plotted these NSCs alongside those obtained from literature sources, including both photometric and spectroscopic surveys.

Out of the 60 selected targets, two IMBHs have had their masses estimated through stellar kinematics, measured from marginally resolved VLT/SINFONI observations: NGC 5102 with an estimated mass of ≈9.1×10^5  and NGC 5206 with an estimated mass of ≈6.3×10^5  <cit.>. <cit.> found a firm signature of an IMBH with a possible mass range of 3×10^5-4.3×10^6  in the nucleus of NGC 3593 based on ALMA observation of circumnuclear CO(2-1)-molecular gas disc. Additional dynamical models presented in <cit.> suggested that the IMBH in NGC 300, with kinematics measured from VLT/UVES spectra <cit.>, likely has a mass below 10^5 . Similarly, the IMBH in NGC 3621, with kinematics measured from the Echellette Spectrograph at the Keck Observatory <cit.>, is estimated to have a mass below 3×10^6  <cit.>.

In the regime of the dwarf galaxies with masses below 7×10^9 , the behavior of the black hole mass–galaxy scaling correlations remain largely unconstrained. It strongly depends on the mass distribution of the currently elusive IMBH population found in the nearby universe <cit.>. Recent work by <cit.> also provided evidence that the dynamical mass constraints of a sample of IMBHs within 3.5 Mpc, in a sample of five early-type galaxies, fall well below the predictions from the <cit.>, <cit.>, and <cit.> M_ BH–σ_⋆ relations.





 §.§ Simulated targets


To explore the extreme ranges of observability, we created mock IFS observations with HSIM of two low-mass galaxies with NSCs: NGC 300, the closest, and NGC 3115 dw01, farther from Earth. These galaxies represent our IMBH survey’s overall characteristics.

NGC 300 is a spiral galaxy of type SA(s)d, characterized by a lack of a bulge <cit.> and inclined at an angle of i=(42.3±3.0)^∘ <cit.> projected on the sky. Located approximately 2.2 Mpc away from us <cit.>, this galaxy exhibits a steadily increasing size of its NSC from ultraviolet (UV) to infrared (IR) wavelengths <cit.>. The NSC in NGC 300 possesses a central velocity dispersion σ_⋆=13.3±2.0 , as determined from high-resolution VLT/UVES spectroscopy <cit.>, or σ_⋆=13.3±0.3  when measured using high-resolution VLT/X-Shooter spectroscopy <cit.>. The NSC has a dynamical mass of M_ NSC≈10^6  measured from VLT/UVES spectroscopy <cit.>. The galaxy has a total stellar mass of 2.2×10^9  <cit.>. 

The integration of Jeans Anisotropic Model (JAM) and the stellar kinematics measurements from the <cit.> high-resolution VLT/UVES spectroscopy found an M/L_I≈0.6 (/) and suggested an IMBH mass in the NGC 300's NSC with an upper limit of ≈10^5  (or ≈10% of M_ NSC), while their best-fit estimate is ≈10^2  <cit.>. 

The star formation history (SFH) of NGC 300’s nucleus has been independently analyzed by <cit.> and <cit.>, utilizing different spectra, spectral ranges, and simple stellar population (SSP) models. Their findings reveal a relatively consistent picture of the stellar population. More than 50% of the stars in the nucleus formed over 5 Gyr ago, characterized by low metallicity ([Fe/H] ∼-1 dex). Subsequent star formation episodes involving young populations (around 10 Myr old) occurred until a few hundred million years ago, resulting in higher metallicity, eventually reaching values similar to solar metallicity about 1 billion years ago. Despite these later episodes, their contributions to the total light spectrum and mass remain modest, exceeding 10% and 1%, respectively. Remarkably, no evidence of emission lines in the spectra implies that the contribution of a very young population in NGC 300’s NSC is likely from an extended horizontal branch <cit.> or blue straggler stars <cit.> in the nucleus.

NGC 3115 dw01 (PGC029300) falls under the classification of a dwarf elliptical galaxy <cit.>. Positioned at a distance of roughly 9.7 Mpc from our vantage point <cit.>, this galaxy boasts a total stellar mass of 8.9×10^8   determined from its B-V colour, and houses a substantial NSC with an effective radius of 6.61±0.09 pc and a mass of 7.2×10^6 , as gauged from / WFPC2 F814W images, resulting in a photometric M/L_ F814W≈1.4 (/) <cit.>. Kinematic assessments carried out at CTIO (R≈10,000), utilizing the 4 m telescope and its echelle spectrograph, yield a central velocity dispersion of σ_⋆=32±5 . However, measurements confined within the galaxy’s effective radius (≈1.2 kpc) result in a value of around ≈45 <cit.>.

Due to the absence of information regarding stellar populations and SFH in NGC 3115 dw01, as well as the galaxy's morphology, we opted for simplicity in our HSIM IFS simulations. We employed Stellar Population Synthesis (SPS) spectra with a fixed age of 5 Gyr and Solar metallicity (z002) and inclination i=42^∘, which are similar to the nucleus of NGC 300.







§ DYNAMICAL MODEL





 §.§ Jeans Anisotropic Model (JAM)


Our HARMONI IMBH sample comprises dwarf galaxies hosting bright NSCs with significant rotational characteristics <cit.>. Notably, a considerable fraction of NSCs exhibits substantial rotation, as evidenced by the V/σ_⋆ ratio falling within the range of 0.1 to 0.6 <cit.>, indicating their tendency toward cylindrical projections <cit.>. Consequently, we generated mock kinematics using a dynamical model based on the solution of the Jeans equations. Our model assumed axisymmetry with a cylindrical alignment of the velocity ellipsoid, providing a robust approximation for a broad spectrum of dynamics, encompassing both fast and slow rotators <cit.>. This model was implemented as JAM_ cyl and was used to predict the mean velocity. In doing so, we assumed a model in which the velocity ellipsoid was axially symmetric around the vertical direction (σ_z ≠σ_r = σ_ϕ). This choice aligns with the cylindrical limit and is particularly suitable for modeling the expected rotators of NSCs. This assumption was achieved by setting align=‘cyl’  in the jam_axi_proj procedure of JamPy[jampy v7.2.0; available from < https://pypi.org/project/jampy/>] package <cit.>.





 §.§ Predicted velocity dispersion profiles






Before presenting detailed simulations of the predicted kinematics due to IMBHs of different masses in NSC, here we try to understand the quantitative behaviour we should expect. This is particularly important in this paper because we have to decide how to extrapolate the inner surface brightness profile of NSCs, at radii much smaller than resolvable with .

In their classic paper, <cit.> studied the qualitative behaviour of the projected velocity dispersion (σ_p) profiles in elliptical galaxies with central SMBHs. They described the predicted velocity-dispersion profile of isotropic galaxy models described by a double power-law density profile, with general inner density ρ(r)∝ r^-γ and fixed outer density ρ(r)∝ r^-4 <cit.>, and a smooth transition between these two power-law regimes. Here, γ is the power-law index.

Here, we illustrate the behaviour of the velocity dispersion for the more general Core-Sérsic model <cit.>, which parametrizes the projected surface brightness (not the intrinsic density) as follows

    I(R) = I' [1 + (R_b/R)^α]^γ/αexp[-b_n_ Ser(R^α + R_b^α/R_e^α)^1/(α  n_ Ser)]
 
where 

    I'=I_b 2^-γ/αexp[b_n_ Ser2^1/(α  n_ Ser)( R_bR_e)^1/n_ Ser]
 
and b_n_ Ser is a function of the various parameters (n_ Ser, α, γ, R_b, and R_e) that can be determined by solving the relation (A10) of <cit.> when the enclosed luminosity at R_e is equal to half of the total luminosity,  2L(R_e) = L_T. In numerical practice, <cit.> use the asymptotic expansion theorem for the 1/R^n law to solve for b_n_ Ser as an analytical function of the Sérsic index n_ Ser as b_n_ Ser = 2n_ Ser - 13 + 4405n_ Ser + 4625515n^2_ Ser +  O(1n^3_ Ser). Here, n_ Ser is the Sérsic index, which controls the shape of the outer Sérsic part. R_e is the effective radius of the profile. R_b is the break radius, which is the point at which the surface brightness changes from the outer Sérsic part to the inner power-law regime of the profile. I_b is the intensity at the break radius (converted to surface-brightness μ_b in Table <ref>). α is the sharpness parameter that controls the sharpness of the transition between the outer Sérsic and inner power-law regimes.

Outside the inner break at radius R=R_b this is a <cit.> profile of projected half-light radius R_e, but gradually transitions to a power law surface brightness Σ(R)∝ R^-γ at smaller radii R≪ R_b.

We adopt the Core-Sérsic profile because it can describe the profile of real NSCs. However, we allow for a cusp in the inner slope to parametrize our ignorance of the profile of NSC at radii which are inaccessible with current instruments, but which will become observable with the ELT. In our analysis, we model the NSC in isolation, even though they are embedded in the large-scale gravitational potential of the host galaxy. However, the latter has an insignificant effect on the predicted σ_p profiles at the centre of the NSC. In particular, in our case, R_e represents the size of the NSC, not the size of the host galaxy.

To compute the predictions for the σ_p profiles we first used the mge_fit_1d procedure in the MgeFit package[v5.0 of the Python package from <https://pypi.org/project/mgefit/>] <cit.> to fit the Core-Sérsic profiles in the range R_b/100<R<20R_e using 20 Gaussians. Then used the jam_sph_proj procedure in the JamPy package <cit.> to compute the isotropic velocity dispersion for different values of the Sérsic index n_ Ser and the core inner slope γ. In all our tests we adopted a fixed α=2 value, which produces a smooth transition between the outer Sérsic and inner power-law profiles. We show the results in Fig. <ref>. The qualitative behaviour of the profiles is independent of the absolute normalization of their total masses M_⋆ and R_e. It can be applied unchanged to describe the σ_p profile of NSC or giant elliptical galaxies. However, in this Fig., we adopt as mass and size reference the values (M_ NSC/) = 6.2 and R_e=3.1 pc. These are the median values for NSC of late-type galaxies with M_⋆<10^9 M_⊙ <cit.>. 

In Fig. <ref>, we adopt either no black hole, or black hole masses of 0.5% and 1% of the mass of the NSC. These values are the same characteristic fractions observed for normal galaxies <cit.>. However, nothing is currently known about the mass of IMBH in NSC. In fact, NSCs may not have IMBHs at all, or they may be dominated by the IMBH mass. But the qualitative behaviour remains unchanged.

For γ>0 and without black holes the σ_p profile decreases towards the nucleus, regardless of n_ Ser. This is a well-known general feature of realistic galaxy models. It appeared in puzzling contrast with the early observations <cit.> until it became clear that all massive galaxies contain SMBHs. With an IMBH, the σ_p profile starts increasing towards the centre, inside the IMBH's SOI, asymptotically approaching the Keplerian rise σ_p∝ R^-1/2 (for 0<γ<2), regardless of the profile slope, as shown by <cit.>.

<cit.> also pointed out that[<cit.> studied densities ρ(r)∝ r^-γ', which we convert to the corresponding surface brightness Σ(R)∝ R^-γ using γ=γ'+1.], when γ=0, namely when the inner surface brightness approaches a constant value, the asymptotic behaviour becomes qualitatively different. In that case, without an IMBH, the σ_p profile approaches a constant positive value, instead of dropping towards zero. We confirm the behaviour for the Core-Sérsic profile with a flat inner core regardless of n_ Ser. With an IMBH, the σ_p profile rises towards the centre, but it does so less steeply than the γ>0 case, as predicted for the double power-law models.

What is new in this study is the fact that we explore a range of Sérsic indices, including the extreme case n_ Ser=1/2, which corresponds to a Gaussian profile, when γ=0. We find that in this extreme situation, the σ_p profile becomes only weakly sensitive to the black hole mass. This is quite important for the present study, because we are approximating the surface brightness of the NSC using the multi-Gaussian expansion (MGE) method <cit.>. And we are fitting photometric data that have a much lower resolution than the one we will be able to obtain with the ELT. This implies that our model may end up being described by a Gaussian in the central parts, well inside the width of the PSF of our photometry. This is precisely the key region, near the black hole, where we want to produce an accurate σ_p prediction.

In practice, we should not worry about NSC having flat surface brightness profiles and isotropic orbital distributions. This is because such models are unphysical and cannot exist in real stellar systems. Instead, if an IMBH is present, isotropic models produce a density cusp that rises as ρ(r)∝ r^-1/2 near the black hole <cit.>, or Σ(R)∝ R^-3/2 in surface brightness <cit.>. One would require extreme tangential anisotropies for flat profiles γ=0 to be allowed, as indicated by the cusp-slope vs central anisotropy theorem <cit.>. We have no reason to think NSC satisfies this requirement. However, what is important is that, lacking real high-resolution photometry at the resolution of HARMONI, we extrapolate the observed surface brightness using a non-zero inner power slope, when constructing the mock kinematics of the NSCs. Failing to do so, we would generate predictions for σ_p profiles that vastly underestimate the central rise that we expect from high-resolution HARMONI observations.





 §.§ MARCS synthetic library of stellar spectra


To create mock HSIM IFS simulations, we required stellar population synthesis (SPS) spectra[Available from <https://marcs.astro.uu.se/>] that contain information about the stellar populations <cit.>, based on the Model Atmospheres with a Radiative and Convective Scheme (MARCS) synthetic library of theoretical spectra, originally developed by <cit.>. Although MARCS synthetic library spectra are not as reliable as an empirical stellar library, it offers broad wavelength coverage spanning from 1,300 Å to 20 μm, high spectral resolution with σ = 6.4  (equivalent to R = λ/Δλ∼20, 000), and fine sampling with 100,724 flux points (Δλ≈0.065 Å). We assumed a Salpeter IMF, an age of 5 Gyr, and Solar metallicity (z002) for both NGC 300’s NSC <cit.> and NGC 3115 dw01 (Section <ref>). The SPS spectra were truncated within the wavelength range of 1.0–2.5 μm to match the HARMONI/J, H, K, H-high, K-short, and K-long gratings.







§ GALAXY PHOTOMETRY MODELS















 §.§  images and their PSF models


The /Wide Field and Planetary Camera 2 (WFPC2) Planetary Camera (PC) imaging details utilized in this study are provided in Table <ref>. Specifically, the nucleus of NGC 300 was observed on May 6, 2001, using the PC1 chip with the F814W filters. These observations comprised three exposures, each lasting 213 seconds. Similarly, the nucleus of NGC 3115 dw01 was observed on November 29, 1995, employing the PC1 chip with both F547M and F814W filters. These observations consisted of one and two exposures, each lasting 160 seconds, for the F547M and F814W filters, respectively. The data were obtained under Project ID 8599 (PI: Boeker) for NGC 300 and Project ID 5999 (PI: Phillips) for NGC 3115 dw01. We accessed the reduced and drizzled images from the /Hubble Legacy Archive (HLA), with a pixel scale of 0045.

For our analysis, it was imperative to meticulously characterize the PSFs used to deconvolve the  profiles into their intrinsic forms (Section <ref>).  To achieve this, we generated simulated images of the /WFPC2 PSFs in each filter using tiny1 and tiny2  routines within the Tiny Tim software package <cit.>, which creates a model  PSF based on the instrument, detector chip, detector chip position, and filter used in the observations. To ensure that the model PSFs were processed in the same way as the  images, we produced individual versions of each PSF corresponding to their realistic exposure in each filter and each galaxy on a subsampled grid with sub-pixel offsets, using the same four-point box dither pattern as the /WFPC2 exposures. Next, in order to account for the effect of electrons leaking into neighbouring pixels on the CCD, each model PSF was convolved with the appropriate charge diffusion kernel. The same filters set of PSFs of each galaxy were then combined and resampled onto a final grid with a pixel size of 0045 using Drizzlepac/AstroDrizzle <cit.>. The FWHM information for these ultimate PSF models is provided in Table  <ref>.






 §.§ Surface brightness profiles and galaxy mass models


The  data, summarized in Table <ref>, served as the basis for our measurements of the galaxies’ surface-brightness profiles. We conducted fits to analytical functions and extrapolated these profiles using the best-fit parameters obtained from these fits, under certain assumptions adequate for presenting the simulations and dynamical models.

Our initial step involved extracting radial surface brightness profiles of the observed stars. For NGC 300, this was accomplished in the F814W filter, while for NGC 3115 dw01, data from both the F814W and F547M filters were used. We employed the Image Reduction and Analysis Facility (IRAF) ellipse  task <cit.> to conduct this task. The ellipse routine systematically integrated the flux of stars within concentric annuli, allowing for variations in position angles and ellipticities along the semi-major axis of the galaxy. Notably, fixing both position angles and ellipticities did not yield any significant changes in our results <cit.>. To obtain the inherently radial profiles in IRAF, we initiated spatial imaging deconvolution using simulated images of the /WFPC2 Tiny Tim PSF generated in Section <ref>. Subsequently, we converted the average flux within each annulus, measured in counts/s, into surface brightness expressed in mag arcsec^-2. This conversion was carried out using the photometric information derived from the broad-band filters and camera specifications, as provided in Table <ref>.  The large-scale profile of NGC 300 (μ_0(r>2)) was taken from its Wide Field Imager (WFI) observed in R band with the MPG/ESO 2.2-metre telescope, which presented in <cit.> and is available at <https://www.eso.org/public/images/eso1037a/>. Here, we also calibrated our inset /WFPC2 F814W profile (μ_0(r≤2)) with the same  profile from <cit.> using the AB magnitude system.

Next, we applied analytical functions to fit these inherently radial surface-brightness profiles of NGC 300 and NGC 3115 dw01. These functions involved combinations of a Core-Sérsic model <cit.> and a pure-Sérsic model <cit.>. Here, while the Core-Sérsic model is used to fit the NSC component, the pure-Sérsic model was applied to fit the extended component of the galaxy. This approach was adopted due to the morphological complexity revealed by the high-resolution  and wide-field MPG/ESO observations of galaxies, which often exhibit multiple components like NSCs, bulges, disks, bars, and a constant sky/galaxy background. This detailed morphological inspection is also useful to tell us what Sérsic index (n_ Ser) best fits these cores and predict the kinematics trend (see Section <ref>). Furthermore, the pixel size of the /WFPC2 is 45 mas. This means that a single pixel contains over 20 spaxels of the proposed 10 mas mode adopted for this HARMONI IMBH survey. In fact, it is more than 11 times larger than the pixel size for ELT/MICADO images (i.e. 4 mas). We thus have no clue of the stellar information at this scale, and this region is indeed embedded deeply in the core of an NSC, which is controlled by the power-law behaviour of the Core-Sérsic model. In this work, we adopted the power-law index γ=0.1 as in N23 for both galaxies, although our tests with varying γ provide γ=0.08 for NGC 300 and γ=0.13 for NGC 3115 dw01. Here, the <cit.> pure-Sérsic model has a form:  


    I(R) = I_eexp{-b_n[(R/r_e)^1/n-1]}
 
In which, we denoted n is the Sérsic index of the pure-Sérsic profile to distinguish with that n_ Ser of the Core-Sérsic profile in equation (1), which controls the degree of curvature of the profile. An additional benefit of employing the pure-Sérsic model for the extended component of a galaxy is that, when n=0.5, the model simplifies to a Gaussian profile. Similarly, for n=1, the model represents an exponential profile, and for n=4, it corresponds to the de Vaucouleurs profile. And again, the coefficient b_n is approximately as b_n=2n-1/3. r_e is the effective radius of the profile. I_e is the intensity at r_e, which is converted to surface-brightness density μ_e shown in the legends of Fig. <ref> and Table <ref>.  

We applied a non-linear least-squares algorithm using the Python MPFIT[Available from <http://purl.com/net/mpfit.>] function <cit.> to iteratively fit the combined function of a Core-Sérsic and a pure-Sérsic above for both for NGC 300 and NGC 3115 dw01 to their corresponding spatially deconvolved profiles, thereby obtaining the best-fit parameters homogeneously.   After the fits, we extrapolated the best-fit surface brightness profiles towards the galaxies' centres to a scale of 4 mas required for this IMBH survey’s imaging with ELT/MICADO, as demonstrated in Fig. <ref>. The best-fit parameters and their associated uncertainties derived from these surface brightness profile fits are meticulously documented in Table <ref>.  

Next, we employed these derived parameters to reconstruct the two-dimensional (2D) luminosity density models extrapolated towards the central 4 mas for each galaxy. During which, we assumed  a small flattening based on the fact that not all NSCs are likely perfectly spherical and non-rotating, characterized by the ratio between the semiminor and semimajor axes q'=b/a≈0.9. This assumption also works for the extended bulge or disk of NGC 3115 dw01 as the galaxy morphology looks round through its  observations. However, the extended scale of NGC 300 revealed through the MPG/ESO 2.2-m WFI R band image suggests the galaxy's outer part (r>2) can be approximately assumed as q'=b/a≈0.75. Subsequently, we transformed these 2D luminosity density models into multiple Gaussian functions using the multi-Gaussian expansion (MGE) method <cit.>. This involved using the mge_fit_sectors routine from the MgeFit package <cit.> and encompassed 15 Gaussians spanning radii from 4 mas to approximately 1000 for NGC 300 and 15 for NGC 3115 dw01, respectively.  Detailed information regarding these MGEs can be found in Table <ref> in the supplementary material.
 
Finally, we converted the luminosity surface density into mass surface density by assuming a constant mass-to-light ratio, M/L_ F814W, dyn≈ M/L_I≈0.6 (/) for NGC 300 <cit.> and M/L_ F814W≈ M/L_I≈1.4 (/) for NGC 3115 dw01 derived in AB mag <cit.>. This stellar mass component was subsequently combined with an IMBH represented as a point mass. It is pertinent to mention that throughout our work, we disregarded (1) potential variations in M/L_I due to differences in stellar populations <cit.> and (2) the distribution of dark matter halos <cit.>. Our focus remained exclusively on the stellar kinematics within the central part of 04×04 of HARMONI field-of-view (FOV), where the central black hole’s and NSC's potential is the dominant factor.





 §.§ Discrete stars


Given our HARMONI IMBH survey’s proposed 10-mas spaxel angular resolution, it’s crucial to ensure that we have a sufficient number of stars within a single spaxel for reliable statistical estimates. This issue becomes more significant for the closest targets. To address this, we carefully investigated the number of stars within a 10-mas spaxel based on the surface brightness profile presented in Fig. <ref>. First, we converted the surface brightness density (in mag arcsec^-2) into surface luminosity density (in  arcsec^-2). Then, we accounted for the size of the 10-mas spaxel to calculate the average number of stars.

Fig. <ref> illustrates our estimated number of stars per 10-mas spaxel as a function of radius from the centre for NGC 300 and NGC 3115 dw01, one of the closest and furthest target galaxies in our HARMONI IMBH sample, respectively. For NGC 300, the number of stars per 10-mas spaxel decreases from approximately more than 100 stars at the centre to around one star at radii larger than 200 spaxels (equivalent to 215, the FOV of our HARMONI IMBH survey with 10-mas spaxel). Similarly, for NGC 3115 dw01, the number of stars per 10-mas spaxel decreases from approximately 5000 stars at the centre to around 700 stars within the FOV of our HARMONI IMBH survey. In the extreme case of the closest and easiest to be resolved and within our measuring 04×04 central region, NGC 300 maintains from approximately 100 stars per 10 mas spaxel, validating our methodology. We should also note that this matter of number of stars per 10 mas spaxel is only critical for a few central spaxels because we will Voronoi bin (see footnote 13, Section <ref>) several spaxels together at larger radii to have enough numbers of stars within spatial bins for integrated light method of the stellar orbits superposition.








§ HARMONI IFS SIMULATIONS


First, we describe the HARMONI instrument on ELT and HARMONI Simulator (HSIM) in Section <ref>. Next, we combine the mass models of NGC 300 and NGC 3115 dw01 constructed in Section 5 with HSIM to simulate their J (1.046  –  1.324 μm), H (1.435  –  1.815 μm), K (1.951  –  2.469 μm), H-high (1.538 – 1.678 μm), K-short (2.017  –  2.201 μm), and K-long (2.199  –  2.400 μm) mock datacubes in Section <ref>. Finally, we present their extracted kinematics in Section <ref>.




 §.§ HARMONI instrument and HSIM simulator


HARMONI, an instrument poised to be one of the pioneers of ELT, will usher in a new era of optical and NIR wavelength IFS ranging from 0.458 to 2.469 μm. Distinguished by its versatile capabilities, HARMONI offers IFS at four distinct spatial scales (i.e. 4×4, 10×10, 20×20, and 30×60 mas^2) and three spectral resolving powers (i.e. λ/Δλ≈3,300, ≈7,100 and ≈17,400) through an array of 13 gratings. Assisted by the ingenious design incorporates 798 hexagonal segments, each approximately 1.4 meters across, spanning a field of 39 meters of the ELT, HARMONI possesses the remarkable capability to gather spectral data from a vast field of 152×214 spaxels (equivalent to ≈32,530 spectra) equipped with laser guide star AO.

This extraordinary technical prowess positions HARMONI as an ideal instrument for unveiling the elusive population of IMBHs residing in nearby dwarf galaxies, stellar clusters such as NSCs, and extragalactic GC <cit.>. Notably, a previous study (N23) demonstrated that HARMONI could robustly measure the masses of SMBHs at distances of up to 1 Gpc (z≲0.3) within a mere hour of exposure time. 

HARMONI’s versatility extends to a wide range of scientific endeavours, from diffraction-limited imaging to ultrasensitive observations. Its capabilities encompass the study of morphology, spatially resolved populations and kinematics, abundances, and line ratios in distant sources, even in regions of faint surface brightness <cit.>. For an in-depth understanding of the instrument, readers can refer to the comprehensive description provided by <cit.> and explore further details on the HARMONI webpage[<https://harmoni-elt.physics.ox.ac.uk/>]. HARMONI’s arrival heralds a new chapter in astronomical observation, promising groundbreaking insights across various fields of astrophysics and significantly advancing our understanding of the cosmos.

 <cit.> introduced HSIM, a sophisticated pipeline designed for simulating observations with the HARMONI instrument on ELT. This software takes as input high spectral and spatial resolution IFS cubes devoid of random noise, generated as described in Section <ref>, and incorporates the physical properties of celestial targets to generate simulated data cubes. Throughout the simulation process, HSIM meticulously factors in the complex atmospheric conditions at the ELT’s observation site and realistic detector statistics/readout, thereby producing data that faithfully mimics actual observations. This paper is primarily focused on utilizing HARMONI’s LTAO IFS simulations to probe the nuclear stellar kinematics within the cores of nearby dwarf galaxies. The objective is to discern the telltale kinematic signatures associated with IMBHs and estimate their masses. These simulations also delineate the boundaries within which HARMONI can deliver feasible observables. Furthermore, they provide invaluable scientific insights, offering guidance for future research endeavours that will leverage actual data obtained with ELT/HARMONI.










 §.§ Simulations of the mock IFS datacubes



We used the dedicated HSIM pipeline to simulate the HARMONI observations with three medium (J, H, K: λ/Δλ≈7,100, σ_ instr≈18 ) and three high (H-high, K-short, K-long: λ/Δλ≈17,400, σ_ instr≈6 ) spectral resolution gratings to access the effect of kinematics errors and test the feasibility of different stellar features on M_ BH determination.

Following N23 and <cit.>, the spectrum in the K-band (comprising K-short and K-long) offers several molecular absorption lines, including commonly used CO absorption bandheads within the range of 2.29–2.47 μm (e.g. ^12CO(2–0) λ2.293 μm and ^12CO(3-1) λ2.312 μm bands), which are instrumental in measuring stellar kinematics. Additional atomic lines such as Nai λ2.207 μm, Cai λ2.263 μm, and Mgi λ2.282 μm are also essential for this purpose. In the H-band (or H-high), a plethora of atomic absorption lines (e.g. Mgi λ1.487, 1.503, 1.575, 1.711 μm, Fei λ1.583 μm, and Sii λ1.589 μm) and strong CO absorption features (e.g. ^12CO(3–0) λ1.540 μm, ^12CO(4–1) λ1.561 μm, ^12CO(5–2) λ1.577 μm, ^12CO(6–3) λ1.602 μm, ^12CO(7–4) λ1.622 μm, and ^12CO(8–5) λ1.641 μm) are available. Additionally, promising atomic and molecule absorption lines in the J-band IFS have been identified for potential use in future works. These lines are primarily related to metallicity and include Fei, Mgi, Sii, and Tii (e.g. Tii λ1.1896, 1.1953 μm; Sii λ1.1988, 1.1995, 1.2035, 1.2107 μm; Mgi  λ1.1831, 1.2087 μm; Fei λ1.1597, 1.1611, 1.1641, 1.1693, 1.1783, 1.1801, 1.1887, 1.1976 μm) <cit.>.     A subset of these potential atomic and molecular absorption lines is indicated on the six HSIM mock IFS spectra presented in Fig. <ref> in the supplementary material. These lines were identified to guide the selection of the optimal wavelength range for extracting stellar kinematics, aiming to avoid contamination from sky emission (e.g. OH lines in J band), as discussed in Section <ref>. For a comprehensive list and illustrations of these stellar absorption tracers, readers can refer to <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>. 

 It is worth noting that the atomic absorption species and molecular absorption lines in the J, H, H-high, K-short, and the blue part of the K-band IFS exhibit relatively narrow profiles and low intensity, rendering them being saturated at very high velocity dispersion in massive galaxies (as in N23) but not for NSCs, which have much lower σ_⋆. This limitation arises from the fact that SMBHs generate high-velocity dispersions (σ_⋆≳75 ) in their vicinity, causing line broadening and blending with spectral noise. However, these lines may serve as effective indicators for measuring stellar kinematics around IMBHs with lower velocity dispersions. In this simulated work focusing on IMBHs, we will evaluate the feasibility of using these lines in the J band, K-short, and the blue part of the K-band IFS to measure stellar kinematics (Section <ref>).

All essential properties of NGC 300 and NGC 3115 dw01 needed for the modelings are presented in Section <ref>, while the chosen grating IFS and HSIM simulations are shown in Table <ref>. Regarding the AO performance during HSIM simulations, we applied the LTAO mode with an NGS of 17.5 mag in H-band within the radial distance of 30, the optical (500 nm) zenith seeing of FWHM = 064, and airmass of 1.3. These parameters are defaulted in HSIM to perform median observational conditions for the Armazones site.

We conducted IFS simulations within a central measuring 04×04 with a pixel size of 10×10 mas^2. This choice ensured that the stellar kinematics, primarily influenced by the central black holes in our HARMONI IMBH sample, could be robustly detected. The exposure time for each simulation was carefully adjusted based on the gratings to ensure a S/N of at least 3 in each spaxel at the measured stellar features. Subsequently, pixel binning was employed to increase the S/N further. To replicate actual ELT observations, we incorporated multi-exposure frames and dithering by setting a Detector Integration Time (DIT) of 900 s (15 min.) for each exposure. The total exposure time was determined by the number of exposures (NDIT), which is an integer specified in the HSIM pipeline.

For this study, we adopted a Gaussian assumption for the line-of-sight velocity distribution (LOSVD). To do this, we first computed both the 2D intrinsic first (V) and second (V_ rms) moment of stellar integrated velocity using the jam_axi_proj routine from the JamPy package <cit.>, then calculated the velocity dispersion σ_⋆^2=V_ rms^2-V^2. Finally we assumed the LOSVD is a Gaussian with mean velocity dispersion equal to the JAM moments described in Section <ref>.

In the JAM_ cyl modelings, we chose to simulate three sets of kinematics corresponding to three different black hole masses  those are fractions of their NSC masses in both cases, such as 0%, 0.5%, 1% for both NGC 300 and NGC 3115 dw01. These adoptions are equivalent to M_ BH = (0, 5×10^3, and 10^4)  for NGC 300 and M_ BH = (0,  3.5×10^4, and 7×10^4)   for NGC 3115 dw01.   Here, we calculated the NSC mass for NGC 300 (M_ NSC = 1.05×10^6 ) and NGC 3115 dw01 (M_ NSC = 7.01×10^6 ) using their MGE models of Core-Sérsic profile listed in Table <ref> (in the supplementary material) and their M/L_ I≈0.6 (/) <cit.> and M/L_ F814W≈1.4 (/) <cit.>, which are consistent with their JAM model masses within 3% uncertainty. The resulting kinematic maps were computed using JAM_ cyl on a regular grid within the central regions of 04×04 and a pixel size of 5×5 mas^2. These maps were subsequently convolved with the HARMONI PSF, rebinned, and interpolated to the specific pixel-sampling scale of 10×10 mas^2 by the  HSIM software. Additionally, for the sake of simplicity, we assumed isotropic JAM_ cyl models (β_z=0) and inclination angles i=42^∘ in our simulations (discussed in Section <ref>).  

Because the HARMONI IMBH sample is restricted within ≲10 Mpc, we ignored the spectrally stellar features’ shifted towards the longer wavelength part due to redshift. Based on all these assumptions, we thus created an input noiseless-IFS cube for HSIM by employing the following steps:

(i) Given the synthetic-stellar spectrum chosen for population (SPS; Section <ref>), we logarithmically did rebin it to a scale at which the velocity scale is set as velscale = 0.5  to obtain a constant Δlogλ intervals in the spectrum.

(ii) We constructed kinematics Gaussian kernel sampled with a step of Δ V = 0.5  for each spatial position in the IFS cube at which the mean velocity and velocity dispersion (V^2 and σ_⋆) is computed by the JAM_ cyl model for that position.

(iii) We logarithmically convolved the rebinned spectrum created from step (i) with the Gaussian kernel generated from step (ii), which will be linearly interpolated to the constant wavelength step Δλ≈0.01 Å. This newly constant-wavelength step is small enough to conserve information during the interpolation.

(iv) We rebinned the spectrum by co-adding an integer number of adjacent spectral pixels to reach a step in wavelength at a minimum two times smaller than the smallest HARMONI instrumental resolution in terms of FWHM (e.g. the J, H, K gratings have Δλ≈2 Å and the H-high, K-short, K-long have Δλ≈0.5 Å), making a rigorous integral over the pixels, and no information is lost.

(v) We stored the resulting linearly-sampled, LOSVD-convolved, and noiseless spectrum in a cube. 

(vi) We estimated the surface brightness of NGC 300 and NGC 3115 dw01 (i.e. integrated intensities) using their models' best-fit parameters of a Core-Sérsic + a pure-Sérsic function inferred from Table <ref>, then assigned every spaxel’s intensity to its corresponding linearly-sampled, LOSVD-convolved, and noiseless spectrum in the cube. Since these multi-Sérsic profiles only describe the galaxies’ surface brightnesses along their major axes, these two galaxies are not all spherical. We thus computed the elliptical radius of every pixel using the relation: r^2_ ellipse = x^2 + (y/q)^2, to account for galaxy shapes, where x, y, and q are the positions of pixels (x, y) and the axis ratio from the stellar-light MGE model (Table <ref> in the supplementary material), respectively, then assigned the flux by their corresponding multi-Sérsic profile I(r_ ellipse) in equation (1).

(vii) We scaled the template spectrum in each 5×5 mas^2 spaxels in the manner so that the surface brightness in a single pixel of the /F814W-band image (in mag arcsec^-2, pixel scale of 0045) represented in the mean flux (in erg s^-1 cm^-2 Å^-1 arcsec^-2) distributed into the total number of 80 (5×5 mas^2) spaxels area. The surface brightness is necessary for estimating the required sensitivity (and thus S/N) observed with HARMONI, which should be high enough for measuring the stellar (or ionized gas emission, if detected) kinematics precisely within a reasonable exposure time (N23). 











 §.§ HSIM mock datacubes and extracted kinematics


In accordance with N23, we have identified several prominent stellar absorption features associated with molecules and atomic species commonly observed in realistic IFS datacubes. 

To begin, we have established the optimal spectral wavelength ranges for fitting stellar templates to measure stellar kinematics across the six grating bands. These ranges include λ2.28–2.40 μm for both the K- and K-long bands, λ1.45–1.74 μm for both the H- and H-high bands, and λ1.16–1.218 μm for the J-band and λ2.05–2.20 μm for the K-short band within the HSIM IFS mock datacubes. These wavelength ranges are depicted in Fig. <ref> in the supplementary material, along with all the previously identified absorption features. Subsequently, to enhance the reliability of our stellar kinematic measurements and the applicability of the identified absorption features across the six HARMONI IFS gratings, we conducted experiments by fitting the entire spectral ranges of the gratings (excluding the J-band IFS due to contaminated by the OH lines from sky emision). We then compared the results with those obtained by fitting the fixed best spectral wavelength ranges. The experiments demonstrated that the differences in kinematic measurements between these two fitting approaches are minimal, typically less than 5%. These findings validate the robustness and feasibility of employing the atomic and molecular absorption features in stellar spectra of the J, H, H-high, K-short, and the blue part of the K-band IFS, both from N23 and the present work, to measure stellar kinematics for IMBH mass estimation with minimal uncertainty. We will also discussed these statistical errors in Section <ref>; and see Figs <ref>, <ref> and <ref> (in the supplementary material) for NGC 300; also see Figs <ref>, <ref> and <ref> (in the supplementary material) for NGC 3115 dw01).

Prior to extracting stellar kinematic data from the simulated IFS data cubes in all six simulated HARMONI IFS bands, we employed a spatial binning technique to enhance the S/N to values exceeding 20 per bin over the listed above optimal spectral wavelength ranges specified for each grating band,  as determined by adopting the adaptive Voronoi Binning method <cit.>. This technique involves grouping the signals of multiple spaxels within each bin while maintaining the noise at its median level, ultimately reducing the uncertainty in the kinematic measurements for that bin. 

Subsequently, to derive the stellar kinematic measurements, we first carried out the binned spectra log-rebinning along the spectral dimension, then employed the Penalized PiXel-Fitting algorithm <cit.> to fit the binned spectra of each bin to a stellar spectral template. During the pPXF fitting process, we utilized the MARCS <cit.> version of the <cit.>  SPS models. We corrected the template continuum shape using default Legendre polynomials (specifically,  mdegree  = 0 and degree = 4) and performed the fit for the kinematic parameters V (velocity) and σ_⋆ (velocity dispersion) by specifying  moments = 2. Additionally, we accounted for the instrumental broadening inherent to the HARMONI IFS by convolving the stellar templates with the constant instrumental dispersion adopted by HSIM. 

To enhance the fidelity of our fitting procedure, we included 13 templates with ages ranging from 3 to 15 Gyr and Solar metallicities (z002). The best-fit SPS template is superimposed on sections of the simulated spectra from the central bin of all six gratings (J, H, K, H-high, K-short, and K-long), as illustrated in Fig. <ref> in the supplementary material. For reference, the residuals (data-model) are also depicted as green dots in the same panel, providing a visual representation of the fit quality.

Figs <ref> and <ref> (in the supplementary material) present the kinematic maps of the NGC 300 nucleus, while Figs <ref> and <ref> (in the supplementary material) display those of the NGC 3115 dw01 nucleus. These maps have been extracted from the HSIM mock data cubes encompassing the K-long, K-short, H-high, K, H, and J bands and modeled using the JAM_ cyl approach. Each figure comprises multiple red-bordered rectangular panels, with each panel containing kinematic measurements for three different black hole masses located at the centres of these NSCs as mentioned in Section <ref>.  We have achieved a small root-mean-squared velocity scatter, typically with values of Δ V_ rms≲1%. In each row-panel of the red-bordered rectangular plot, the kinematic maps are systematically displayed. These include maps showing the rotation (V) with the systemic velocity (V_ sys) subtracted, velocity dispersion (σ_⋆), and root-mean-squared velocity (V_ rms), which is defined as V^2_ rms=V^2 +σ_⋆^2.








The left panels of each red-bordered rectangular panel of these stellar kinematic maps of both NGC 300 and NGC 3115 dw01 displayed crucial signatures of the presence or absence of an IMBH, exhibiting a distinctive central drops in the velocity dispersion (σ_⋆) and root-mean-squared velocity (V_ rms) maps when either no black hole (M_ BH=0 ) is present or when the black hole masses are insufficient (e.g. M_ BH=5×10^3  for NGC 300 and M_ BH=3.5×10^4  for NGC 3115 dw01). This central drop in velocity dispersion is a common feature in the predicted stellar kinematics of galaxies when no or very small IMBHs are assumed, and it holds true for a range of assumed density and anisotropy profiles <cit.>. Conversely, models featuring relatively large IMBHs generate centrally rising peaks in both the velocity dispersion and root-mean-squared velocity maps at several central spaxels, which correspond to the black hole’s SOI, showing a clearly upturned trending as the IMBH mass increases.   This observation aligns with the general expectation that central velocity dispersion should increase in a Keplerian manner within a tiny region near the central IMBH, where its gravitational potential dominates over all others.

The disparity between these kinematic maps at the galaxy centrers is notably clear for both NGC 300 and NGC 3115 dw01, which represent the closest and farthest targets in our HARMONI IMBH sample, respectively. This emphasizes the exceptional spectral and spatial resolving powers of ELT/HARMONI in detecting the stellar-kinematic signatures of central IMBHs in the local Universe and accurately measuring their masses through dynamic means.

Crucially, it is evident that HARMONI has the capability to distinguish between the kinematic effects of a scenario with no IMBH and one with a 3.5×10^4  IMBH (V_ rms, M_ BH=3.5×10^4  M_⊙ - V_ rms, M_ BH=0  M_⊙≈1.0 ). This distinction remains discernible even at the distance of NGC 3115 dw01, which is ≈10 Mpc away. 

Nonetheless, both the NSCs of NGC 300 and NGC 3115 dw01 exhibit minimal rotation at their centers, with velocities of V= 0.7  for NGC 300 and V=5.0  for NGC 3115 dw01. In both cases, the dominant factor is velocity dispersion.

The kinematic maps also reveal the presence of edge effects, particularly evident in the higher velocity dispersions (and consequently higher root-mean-squared velocities) observed in the top and bottom bins. This effect causes these bins to exhibit higher values compared to the predictions from the purely JAM_ cyl model, primarily because these spaxel bins are located farther from the center. The consequence of this effect is the squared shape of the outermost surface brightness contour observed in all maps. To ensure the accuracy of our dynamical models and avoid uncertainties stemming from this instrumental issue, we have excluded all high V_ rms bins from our model for M_ BH (as described in Section <ref>).








§ DISCUSSION





 §.§ Black hole mass recovering


Given the simulated IFS datacubes from six HSIM gratings and the resulting kinematic maps (as detailed in Section <ref>), we performed fitting of these kinematic maps (specifically V_ rms) to the JAM_ cyl models, aimed at determining the black hole mass and constraining the model’s dynamical parameters. This involved making certain assumptions about a central compact dark mass object (referred to as IMBH), including its mass (M_ BH), as well as considering dynamical parameters such as stellar orbitals (represented by the global anisotropy parameter β_z, equivalent to γ), stellar mass (parameterized by  relative to the best-fit stellar population estimate from the /F814W broad-band image, _ F814W, and the inclination angle (i).  

In this process, we adopted a logarithmic scale for the black hole mass to ensure a more Gaussian sampling across multiple orders of magnitude, while the other three parameters were sampled in linear scales. During the fitting process, JAM_ cyl generated kinematic models that could be compared to their corresponding simulated values (V_ rms) while accounting for the HARMONI LTAO PSF, characterized by σ_ PSF≈5 mas (or FWHM_ PSF≈12 mas; N23). 

To obtain the best-fit JAM_ cyl model, we utilized a Markov Chain Monte Carlo (MCMC) simulation to comprehensively explore the parameter space encompassing i, _ F814W, M_ BH, and β_z. The model was employed to fit the simulated kinematic data and determine the best-fit values, along with statistical uncertainties, using the adaptive Metropolis algorithm <cit.> within the Bayesian framework <cit.>. Our MCMC chains comprised a total of 3×10^4 iterations, with the initial 20% of iterations excluded as a burn-in phase. The full probability distribution function (PDF) was constructed from the remaining 80% of calculations after the burn-in phase. The best-fit parameters corresponded to the highest likelihood within the PDF, while the statistical uncertainties for all four parameters were determined at 1σ and 3σ confidence levels, representing the ranges (16–84)% and (0.14–99.86)% of the 80% remaining calculations after the burn-in phase, respectively.

During the fitting process, fixed search ranges were defined for all four model parameters (M_ BH, _ F814W, β_z, i) as follows:

• M_ BH: from 0 to 10^6  (or log_10(M_ BH/): from 0 to 6) 

• _ F814W: from 0.1 to 3 (/)

• β_z: from -1.0 to 0.99

• i: from 5^∘ to 90^∘

Additionally, initial guesses for the JAM_ cyl model parameters were fixed for _ F814W, β_z, and i. For example, for NGC 300, the initial guesses were set to (_ F814W, β_z, i) = (0.6, 0, 42), while for NGC 3115 dw01, they were set to (_ F814W, β_z, i) = (1.4, 0, 42). We also initially assumed the same values for the black hole masses as those used in the simulated IFS data, as described in Section <ref>. Specifically, we considered M_ BH=0, 5×10^3, 10^4  for NGC 300 and M_ BH=0, 3.5×10^4, 7×10^4  for NGC 3115 dw01, irrespective of the specific IFS gratings used in the simulations. All of these initial guesses remained the same across all simulated IFS gratings.

In Figs <ref>, <ref>, <ref> (in the supplementary material) for NGC 300, and Figs <ref>, <ref>, <ref> (in the supplementary material) for NGC 3115 dw01, the best-fit JAM_ cyl parameters and their associated statistical uncertainties are presented. These parameters describe the derived stellar kinematics for three different M_ BH values and across six simulated HSIM IFS gratings.

The figures depict 2D scatter plots for each parameter, with colored points representing their likelihood. White corresponds to the maximum likelihood, while black indicates a confidence level smaller than 3σ. Histograms are also included, showing the 1D distributions for each parameter. The 1D distributions were used to calculate the best-fit values and their corresponding uncertainties, which are listed in Table <ref> for NGC 300 and Table <ref> for NGC 3115 dw01 in the supplementary material.

The insert plots at the top-right corner of each red-squared panel in the figures highlight the best-fit models for each mock IFS by directly comparing the simulated V_ rms to the corresponding JAM_ cyl model, all on the same velocity scale. To provide a clearer view of the differences between the model and the data, the figures also include V_ rms residual maps ((data-model)/data ) at the middle top of each red-squared panel. These residual maps show the relative agreements or disagreements between the data and the model on a bin-by-bin basis in the V_ rms maps. In general, our constrained models exhibit excellent agreement with the mock kinematics across the 04×04 central regions of HARMONI's FOV, irrespective of the specific gratings used.

The recovered values for M_ BH and _ F814W closely match the input values used during the creation of the noiseless input cubes for the HSIM simulations. The differences are typically less than 5% for both M_ BH and _ F814W. However, it’s essential to note that these uncertainties, obtained from the MCMC routines, are formal and relatively small (e.g. the 3σ uncertainty is ≈5%). This is due to the high quality of the simulated kinematics and our choice of the lowest limits for the simulated black hole’s SOI used (i.e. M_ BH=5×10^3  for NGC 300 and  M_ BH=3.5×10^4  for NGC 3115 dw01; as discussed in Section <ref>). These choices ensure that the SOIs are well-resolved with the 10×10 mas^2  angular scale of the HARMONI simulations (mentioned in Section <ref> and for instance, these black holes have R_ SOI=13 mas at D=2.2 Mpc of NGC 300 and R_ SOI=35 mas at D=9.7 Mpc of NGC 3115 dw01). We expect that these errors produced from the realistic HARMONI observations will be larger, at least a factor of 2 or 3.

Remarkably, our analysis reveals consistent 2D parameter distributions across different black hole masses and mock IFS gratings, as evident in Figs <ref>, <ref>, and <ref>, <ref>, <ref>, <ref> (in the supplementary material). Specifically, when black hole masses exceed zero, we observe an expected negative covariance between M_ BH and _ F814W, resulting in a characteristic “banana shape” in the 3σ confidence level regions within the 2D PDF. This covariance arises due to the interplay between the potentials of central black holes and the host galaxies, where larger black holes correspond to smaller _ F814W values and vice versa.

This behaviour is attributed to the high spatial resolution of our simulations, enabled by the 10×10 mas^2 observational scale, which allows us to resolve the SOI of even the smallest black holes in NGC 300 and NGC 3115 dw01 at their respective distances. In scenarios without black holes, the 2D PDF yields upper limits on the constrained black hole masses. Notably, these upper limits are consistently roughly 10^2  for NGC 300 and 10^3  for NGC 3115 dw01. This implies that HARMONI can effectively differentiate between black holes with masses above 10^2  and 10^3  for NGC 300 at the distance of 2.2 Mpc and NGC 3115 dw01 at the distance of 9.7 Mpc away from us, respectively; but see a discussion of dynamical mass segregation in Section <ref>.

The recoveries of β_z are also notably well-constrained by the models, yielding tiny uncertainties with preferred values of β_z≈0 for both NGC 300 (slight dominance with tangential stellar orbits, β_z≲0) and NGC 3115 dw01 (slight dominance with radial stellar orbits, β_z≳0), which are totally consistent to the input β_z=0 (isotropic; see Section <ref>). In addition, it is worth noting that the inclination angle (i) is not as tightly constrained by the simulated data, spanning a wide range from 40^∘ to 90^∘ for NGC 300 but strickly remains at ≳5^∘ for NGC 3115 dw01 (except for the constrained model with the J band's kinematics, which behaves in a similar manner of the constrained models of NGC 300). This can be attributed to the nature of the kinematics observed in NGC 300 and NGC 3115 dw01, where velocity dispersion dominates, and there is little (V≈5  for NGC 3115 dw01) to almost no rotation (V≈0.7  for NGC 300). In cases where the NSCs exhibit minimal rotation, their kinematics behave almost like spherically axisymmetric systems, and in the limit of sphericity, galaxies appear similar over a wide range of inclinations. This characteristic results in less constrained inclination angles in our models. 


 


Graphically, a concise summary of the comparisons between the input black hole masses for HSIM  and their corresponding recovered values using the adamet MCMC algorithm and JAM_ cyl modelings is presented in Fig. <ref>. Additionally, the recovered black hole masses for both galaxies listed in Tables <ref> and <ref> of the supplementary material are included in this figure. In this figure, we utilize the median black hole masses recovered from all six gratings (e.g. H-high, K-short, K-long, J, H, K) IFS mock kinematics as the representative values for our simulations. The error bars, adopted within 5% (3σ) of the statistic uncertainties, are also displayed accordingly.




 §.§ Caveat of dynamical mass-segregation processes


According to a review by <cit.>, stringent dynamical constraints on a black hole of M_ BH≈3×10^3  in Galactic GCs could be measurable with ELT. Because of mass segregation, only the most massive stars (i.e. M_ star≈20  remain near the black hole. For such a black hole, their SOI  contains (by definition, see footnote 1) ≈150 such massive stars, enough for dynamical and statistical measurements from integrated light.  In this work, we adopt the lower limits for unambiguously detecting and measuring black holes in these two galaxies (i.e. also representing for our HARMONI IMBH sample entirely) equal to 0.5% of their NSCs' masses. These numbers are all above the limit of M_ BH≈3×10^3 .
 
Nevertheless, <cit.> proposed that stellar-mass black holes may undergo mass segregation on timescales considerably shorter than the relaxation time (t_ relax) of GCs, causing them to migrate toward the center. This migration enhances the central mass-to-light ratio () of GCs. Consequently, there is a potential for a notable accumulation of stellar-mass black holes at the centers of bound stellar systems, including GCs and NSCs. This concentration of mass can lead to an increase in , creating a mimicry of the stellar kinematics in the central regions of these systems and potentially resulting in a false detection of an IMBH. Observations of the GC M15 indeed show an enhancement in  <cit.>. It is important to highlight that the current simulations primarily focus on low-mass systems, which constitute a significant portion of our HARMONI IMBH sample. However, uncertainties persist regarding the extent to which stellar-mass black holes segregate and are retained during this process.
 
To avoid the false detections of IMBHs due to mass segregation, it is necessary to have accurate estimates of the stellar system's relaxation timescale as a function of radius using the following equation <cit.>:  


    t_ relax(r) ≈ (1.4×10^8 yr) σ^3_20(r) ρ_5^-1(r)(lnΛ_10)^-1.
 
where σ_20=σ_⋆/20 ,  ρ_5=ρ/10^5   pc^-3, lnΛ_10=ln(Λ)/10 (i.e. Λ is denoted as the "Coulomb logarithm" and we assumed it is lnΛ=10). Here, σ_⋆ denotes the stellar velocity dispersion, which is assumed as a constant within the central measuring 04×04 in our simulations. ρ_ dyn(r) is the intrinsic mass densities of the stellar systems. If the estimated relaxation time is approximately equal to or longer than the age of the Universe, it guarantees that there is no accumulation of stellar-mass black holes at the nuclei's centers. Therefore, the observed enhancement in velocity dispersion in the central region is a genuine signature of IMBHs. 

In order to compute t_ relax for the NSCs of NGC 300 and NGC 3115 dw01 as examples, we used σ_⋆=13.3  for NGC 300 <cit.> and σ_⋆=32  for NGC 3115 dw01 <cit.> (Section <ref>). We also used the MGE models listed in Table <ref> (in the supplementary material) to estimate ρ_ dyn(R_e) for NGC 300 (13 first Gaussians) and NGC 3115 dw01 (3 first Gaussians), after an analytic deprojection of the MGEs, using their corresponding photometric M/L_ I≈0.6 (/) <cit.> and M/L_ F814W≈1.4 (/) <cit.> and assuming the spherical mass distribution for both NSCs. These NSCs' R_e are listed in Table <ref>. Our calculations, as shown in Fig. <ref>, yield t_ relax(R_e≈1.9  pc)≈3 Myr for NGC 300 and t_ relax(R_e≈3.2  pc)≈10 Gyr for NGC 3115 dw01, suggesting mass segregation cannot be exclusively excluded from our consideration for central IMBH detections via stellar kinematics in NSCs, especially for the case of NGC 300. Nevertheless, more realistic simulations are necessary to assess whether substantial fractions of the NSC black holes can indeed undergo mass segregation and remain retained within the clusters <cit.>. 

In the case of NGC 300, which is particularly prone to mass segregation, the situation is compounded by the fact that its NSC has a size with R_e≈017, just slightly smaller than our HARMONI's central measuring region with a radius of 02. The exceptionally well-resolved kinematics and close proximity of NGC 300 exacerbate the challenges involved in accurately measuring the mass of its IMBH. This is primarily because any variation or enhancement in  due to mass segregation can reproduce the kinematics exclusively attributed to the black hole. While the simulated work of mass segregation and its effect on  variation lies beyond the scope of this paper, we intend to present this analysis in a subsequent article.


 


If t_ relax is relatively short, some stellar-mass black holes may migrate toward the center, potentially complicating the detection of IMBHs <cit.>. Recent studies have proposed that these migrating stellar-mass black holes could be identified through gravitational microlensing events, observed over extended periods in time-domain observations <cit.>. This is particularly relevant because massive star clusters are considered primary candidates for hosting binary black hole mergers detected through gravitational waves <cit.>. The theoretical expectation is that such massive clusters can retain a large population of stellar-mass black holes, which subsequently migrate to the cluster's center, providing a steady source of binary systems <cit.>. Nevertheless, the complexity of gravitational microlensing caused by stellar-mass objects fall well below the resolvable power of ELT for extragalactic stellar systems. This topic is beyond the scope of discussion in this paper.

For typical GCs, there is limited observational <cit.> and theoretical <cit.> supports for the idea that IMBH formation is favored in these environments. In constrast, many NSCs exhibit high velocity dispersions and a range of relaxation times (typically half-mass relaxation times are many gigayears), making them plausible sites for the formation of IMBHs <cit.>.





 §.§ Sensitivity limits


We have also examined the ultra-sensitivity in terms of exposure time for the instruments, conducting similar simulations for both galaxies, NGC 300 and NGC 3115 dw01. In these simulations, we progressively reduced the exposure times, consequently lowering S/N, until the simulated HARMONI IFS cubes still yielded marginally meaningful kinematic maps after spaxel binning via Vorbin, targeting a specific bin-S/N of ≈20. The resulting required exposure times for each galaxy are detailed in Column 4 of Table <ref>. It is important to note that the galaxy’s surface brightness was derived from  images (for NGC 300 we added the large-scale MPG/ESO observations for its extended disk) and extrapolated to the 4 mas regime using an analytical function discussed in Section <ref>. Additionally, we considered the flux spread across the grating’s wavelengths, as explained in Section <ref>. These required exposure times are relatively short, less than three hours, making them well-suited for ELT observations.

However, it is essential to exercise caution as our sensitivity estimates in terms of exposure time offer approximate values for the HARMONI IMBH survey. The actual sensitivity measurements may be somewhat lower, necessitating longer exposure times. This is due to the potential impact of higher spatial-resolution imaging (e.g. ≲005 from /WFPC2 and /NIRCam or a few mas from ELT/MICADO), which could resolve out some of the flux. Therefore, we are demonstrating the feasibility of our proposed science with ELT/HARMONI while acknowledging that real sensitivity measurements may require adjustments.





§ CONCLUSIONS


Guided by our mission to conclusively establish the presence of IMBHs within the Local Universe (up to a distance of ≈10 Mpc), bridging the current gap between well-understood stellar-mass black holes and SMBHs, we have utilised the potentially unparalleled capabilities of ELT and its cutting-edge instrument, HARMONI, thanks to its remarkable advancements in spatial and spectral resolution, coupled with its unprecedented sensitivity across the optical and near-infrared spectrum. Our objective is to understand the stellar dynamics of the nuclei of nearby dwarf galaxies. Through this process, we aim to uncover integrated kinematic signatures that may signify the existence of IMBHs nestled at the cores of these galaxies. Subsequently, we endeavor to precisely determine the masses of these IMBHs via rigorous dynamical modelling.

In the near future, when the ELT becomes operational, it promises to offer a comprehensive insight into the realm of black hole physics. This endeavor will also shed light on the underlying physical processes that govern the origin and co-evolution of central black holes and their host galaxies. This profound exploration will span the entire spectrum of black hole masses and galaxy masses and sizes, encompassing IMBHs <cit.> in dwarf galaxies to the colossal SMBHs residing in the most massive galactic systems (N23).

Drawing upon the extensive body of literature encompassing both photometric and spectroscopic surveys, as well as insightful case studies, we embarked on a quest to assemble a cohort of nearby nucleated dwarf galaxies. Our selection criteria, designed to ensure observability with ELT, were stringent: galaxies falling within the range of -20.3<M_K<-13.3 mag, situated in regions with |δ + 24^∘|< 45^∘ and |b|≤8^∘. This meticulous pursuit yielded a treasure trove of 45 nucleated dwarf galaxies, characterized by a stellar mass spanning 2×10^6 <M_⋆<5×10^10 . Remarkably, this selection comprises a diverse array of galaxy types, with ellipticals constituting 13%, lenticulars representing 16%, spirals accounting for 60%, and irregulars taking 11%. These galaxies play host to luminous NSCs, with masses and effective radii falling within the ranges of 10^5 <M_ NSC<6×10^7  and 3 < R_e < 27 pc, respectively. Crucially, they exhibit low central velocity dispersion, σ_⋆,  c<75 , a vital criterion ensuring the potential presence of IMBHs.

Subsequently, our sample selected strategy resulted in what we refer to as the “HARMONI IMBH sample.” By extending our attention to the galaxies with lowest stellar masses, we have pushed the limit goes below the conventional stellar and gas dynamics of current SMBH mass measurements. Additionally, the HARMONI IMBH sample is ideal to address the existing bias toward dwarf early-type galaxies, as demonstrated by previous research  <cit.>. It includes a wide range of galaxy types and spectrum of extragalactic environments, ranging from isolated galaxies to those dwelling within satellite galaxy fields.

Our planning IFS and imaging observations of the HARMONI IMBH sample will serve as Rosetta Stone for testing diverse black hole seeding formation scenarios, dating back to high-redshift epochs (approximately z∼30-12). Furthermore, these observations will be instrumental in refining models that operate at the lower end of the black hole mass function. This refinement holds significant importance, particularly in predicting the detection rate of gravitational waves with the Laser Interferometer Space Antenna (LISA). Additionally, it will afford deeper insights into the intrinsic behavior of black hole–galaxy scaling relations, particularly within the lower-mass regime <cit.>.

Our analysis of the kinematic maps extracted from the previously mentioned absorption features yielded consistent results within the margins of systemic errors. Specifically, the medium spectral resolution (J, H, K) exhibited systemic errors of Δ V_ rms, sys≲0.3 , while the high spectral resolution (H-high, K-short, K-long) demonstrated even smaller systemic errors of Δ V_ rms, sys≲0.1 .

Our comprehensive testing revealed that the medium spectral gratings (J, H, K) provided kinematic measurements consistent with those obtained from the high spectral resolution gratings (H-high, K-short, K-long) for a stellar system down to a factor of 2 with respect to their instrument errors (σ_ instr). In terms of instrument errors, the medium spectral gratings (J, H, K) exhibited an instrument error of σ_ instr≈18 , while the high spectral resolution (H-high, K-short, K-long) displayed a lower instrument error of σ_ instr≈6 . This implies that, despite NGC 300’s velocity dispersion of σ_⋆≈13  , the medium spectral gratings still delivered reliable measurements in agreement with those derived from the high spectral resolution gratings. Consequently, we can establish a velocity dispersion threshold for accurately obtaining stellar kinematics using the spectral gratings: the medium spectral gratings (J, H, K) remain effective in nuclei with σ_⋆≈10 , while the high spectral resolution gratings (H-high, K-short, K-long) can be employed down to a limit of σ_⋆≈3 .

Subsequently, we harnessed these simulated kinematics to estimate the masses of IMBHs. This was achieved by combining the kinematic data with the interpolated stellar-mass model derived from the  image (and combined with the extended MPG/ESO 2.2-m R band image for NGC 300) and employing JAM_ cyl modeling techniques. Our findings indicated that the retrieved M_ BH and _ F814W values from the simulated data were in complete agreement with our input values, demonstrating an impressive level of consistency with uncertainties of less than 5%. 

Our simulations served as a powerful demonstration of HARMONI’s capabilities, highlighting its potential as a unique facility for accurately measuring IMBH masses and delving into the black hole mass–galaxy scaling relations, particularly within the lowest-mass regimes for both host galaxies and black holes.

Our proposed survey, featuring angular resolutions of R_ SOI = 10×10 mas^2 and simulated pixel scales of 10×10 mas^2, proved to be highly effective in resolving the stellar kinematics within the central black hole’s SOI. This was achieved down the IMBHs with masses that are as small as 0.5% of their NSCs mass (i. e. 5×10^3   for NGC 300 at a distance of 2.2 Mpc and 3.5×10^4   for NGC 3115 dw01 at a distance of 9.7 Mpc. Consequently, the anti-covariance between M_ BH _ F814W, stemming from the degeneracy between the potentials of the central black holes and the host galaxies, was prominently evident in the 3σ confidence levels within the 2D PDF.

Our simulations forecasted that ELT/HARMONI could acquire high-quality IFS and stellar kinematic data within a relatively short observation time, specifically in three hours or less. This emphasizes HARMONI’s potential as a state-of-the-art instrument for advancing our understanding of the aforementioned scientific objectives.





§ ACKNOWLEDGEMENTS


DDN is grateful to the LABEX Lyon Institute of Origins (ANR- 10-LABX-0066) Lyon for its financial support within the program “Investissements d’Avenir” of the French government operated by the National Research Agency (ANR). MPS acknowledges support from grant RYC2021-033094-I funded by MCIN/AEI/10.13039/501100011033 and the European Union NextGenerationEU/PRTR.












 
Facilities: /WFPC2, 2MASS, SDSS DR12, Gaia, DSS.

Software: python 3.12   (<https://www.python.org/>), Matplotlib 3.6.0    (<https://matplotlib.org/>), numpy 1.22    (<https://www.scipy.org/install.html>), scipy 1.3.1   (<https://www.scipy.org/install.html>), photutils 0.7   (<https://photutils.readthedocs.io/en/stable//>), MPFIT (<http://purl.com/net/mpfit>), plotbin 3.1.3    (<https://pypi.org/project/plotbin/>), astropy 5.1    <cit.>, adamet 2.0.9  <cit.>, jampy 7.2.0 <cit.>, pPXF 8.2.1 <cit.>, vorbin 3.1.5   <cit.>, MgeFit 5.0.14 <cit.> and HSIM 3.11 <cit.>.





§ DATA AVAIBILITY

All data and software used in this paper are public. We provided their links in the text when discussed. 




mnras


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§ SUPPLEMENTARY FIGURES AND TABLES




 
































































































































































