arXiv subjects

Martha P. Haynes

Publications and source records attributed to Martha P. Haynes.

At least 19 recordsLinked to original sources

arXiv2026

H-alpha Star Formation Rates of the SHIELD Dwarf Galaxies

The Survey of HI in Extremely Low-mass Dwarfs (SHIELD) galaxy sample is an HI-selected catalog of galaxies that occupy the extreme low-mass end of the HI mass function. Our team is carrying out a comprehensive multi-wavelength observational program targeting these gas-rich, star-forming dwarf galaxies. The current paper presents H-alpha narrowband observations of the full SHIELD sample. We detect 55 of the 82 galaxies (67%) with nebular H-alpha emission, and infer sensitive upper limits to the H-alpha fluxes for the 27 non-detections. We derive H-alpha luminosities and star formation rates (SFRs), or upper limits, for all 82 SHIELD galaxies. We also present, for the first time, deep broadband and continuum-subtracted H-alpha images plus a summary of the properties for the entire sample. Our analysis of the H-alpha data re-affirms previous results which show that at low levels of star formation (log(SFR_{H-alpha}) < -2.5), H-alpha becomes an increasingly poor tracer of the recent average SFR as measured by far-UV (FUV) luminosities. However, we show that this effect is not strictly correlated with the mass of the galaxy. Some very low-mass SHIELD galaxies with higher H-alpha-based SFRs show good agreement with SFR_{FUV}. The SHIELD galaxies show no correlation between H-alpha SFR and gas mass fraction, but reveal a modest correlation between H-alpha specific SFR and gas mass fraction. Overall, the star formation efficiency exhibited by the SHIELD dwarfs is rather low, despite the gas-rich nature of the sample (68% have M_{gas} > M_{stellar}).

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arXiv2026

UV Star-Formation Rates of the SHIELD Dwarf Galaxies

The Survey of HI in Extremely Low-mass Dwarfs (SHIELD) is a multi-wavelength observational project targeting gas-rich, star-forming dwarf galaxies at the faint end of the HI mass function. We present near-ultraviolet (NUV) and far-ultraviolet (FUV) flux measurements obtained from GALEX survey images and use these fluxes to derive FUV star-formation rates (SFRs) for all 75 SHIELD galaxies with GALEX data. This paper represents the first published analysis that makes use of the full SHIELD sample. We compare the FUV SFRs to a variety of physical quantities to better understand the nature of SHIELD galaxies. When comparing the H$α$ SFRs to the FUV SFRs for SHIELD and other local galaxy surveys, we confirm and solidify previous results that show that H$α$-based SFRs for dwarf galaxies can grossly underestimate the true rate of star formation, emphasizing the highly stochastic nature of star formation in extremely low-mass galaxies. We further show that FUV SFRs appear to be robust tracers of star formation down to the very lowest galaxy masses included in this study. We show that baryonic mass is the best mass-related tracer for the prediction of the FUV SFR of the galaxies in our sample. Not surprisingly, the SHIELD dwarf galaxies exhibit long gas-depletion timescales and large gas mass fractions: fully 68% of the SHIELD galaxies have gas masses that are larger than their stellar masses.

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arXiv2023

A Search for Gas-Rich Dwarf Galaxies in the Local Universe with ALFALFA and the WIYN One Degree Imager

We present results from an optical search for Local Group dwarf galaxy candidates associated with the Ultra-Compact High Velocity Clouds (UCHVCs) discovered by the ALFALFA neutral hydrogen survey. The ALFALFA UCHVCs are isolated, compact HI clouds with projected sizes, velocities, and estimated HI masses that suggest they may be nearby dwarf galaxies, but that have no clear counterpart in existing optical survey data. We observed 26 UCHVCs with the WIYN 3.5-m telescope and One Degree Imager (ODI) in two broadband filters and searched the images for resolved stars with properties that match those of stars in typical dwarf galaxies at distances <2.5 Mpc. We identify one promising dwarf galaxy candidate at a distance of ~570 kpc associated with the UCHVC AGC 268071, and five other candidates that may deserve additional follow-up. We carry out a detailed analysis of ODI imaging of a UCHVC that is close in both projected distance and radial velocity to the outer-halo Milky Way globular cluster Pal 3. We also use our improved detection methods to reanalyze images of five UCHVCs that were found to have possible optical counterparts during the first phase of the project, and confirm the detection of a possible stellar counterpart to the UCHVC AGC 249525 at an estimated distance of ~2 Mpc. We compare the optical and HI properties of the dwarf galaxy candidates to the results from recent theoretical simulations that model satellite galaxy populations in group environments, as well as to the observed properties of galaxies in and around the Local Group.

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arXiv2023

Parametrized Asymmetric Neutral hydrogen Disk Integrated Spectrum Characterization (PANDISC) I: Introduction to A Physically Motivated H I Model

Modelling the integrated H I spectra of galaxies has been a difficult task due to their diverse shapes, but more dynamical information is waiting to be explored in Hi line profiles. Based on simple assumptions, we construct a physically motivated model for the integrated Hi spectra: Parametrized Asymmetric Neutral hydrogen Disk Integrated Spectrum Characterization (PANDISC). The model shows great flexibility in reproducing the diverse Hi profiles. We use Monte-Carlo Markov Chain (MCMC) for fitting the model to global H I profiles, producing statistically robust quantitative results. Comparing with several samples of H I data available in the literature , we find the model-fitted results agree with catalogued velocity widths (e.g., W50) down to the lowest S/N. The model is also shown to be useful for applications like the baryonic Tully-Fisher relation (BTFR) and profile-based sample control. By comparing v_r to v_flat , we uncover how the H I width is affected by the structure of the rotation curve, following a trend consistent with the difference in the BTFR slope. We also select a sample of spectra with broad wing-like features suggestive of a population of galaxies with unusual gas dynamics. The PANDISC model bears both promise and limitations for potential use beyond H I lines. Further application on the whole ALFALFA sample will enable us to perform large scale ensemble studies of the H I properties and dynamics in nearby galaxies.

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arXiv2023

Catching Tidal Dwarf Galaxies at a Later Evolutionary Stage with ALFALFA

We present deep optical imaging and photometry of four objects classified as "Almost-Dark" galaxies in the ALFALFA survey because of their gas-rich nature and extremely faint or missing optical emission in existing catalogs. They have HI masses of $10^7$-$10^9$ $M_{\odot}$ and distances of $\sim$9-100 Mpc. Observations with the WIYN 3.5m telescope and One Degree Imager reveal faint stellar components with central surface brightnesses of $\sim$24-25 $\mathrm{mag}\,\mathrm{arcsec}^{-2}$ in the g-band. We also present the results of HI synthesis observations with the Westerbork Synthesis Radio Telescope. These Almost-Dark galaxies have been identified as possible tidal dwarf galaxies (TDGs) based on their proximity to one or more massive galaxies. We demonstrate that AGC 229398 and AGC 333576 likely have the low dark matter content and large effective radii representative of TDGs. They are located much farther from their progenitors than previously studied TDGs, suggesting they are older and more evolved. AGC 219369 is likely dark matter dominated, while AGC 123216 has a dark matter content that is unusually high for a TDG, but low for a normal dwarf galaxy. We consider possible mechanisms for the formation of the TDG candidates such as a traditional major merger scenario and gas ejection from a high velocity fly-by. Blind HI surveys like ALFALFA enable the detection of gas-rich, optically faint TDGs that can be overlooked in other surveys, thereby providing a more complete census of the low-mass galaxy population and an opportunity to study TDGs at a more advanced stage of their life cycle.

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arXiv2022

A Generalist, Automated ALFALFA Baryonic Tully-Fisher Relation

The Baryonic Tully-Fisher Relation (BTFR) has applications in galaxy evolution as a testbed for the galaxy-halo connection and in observational cosmology as a redshift-independent secondary distance indicator. We use the 31,000+ galaxy ALFALFA sample -- which provides redshifts, velocity widths, and HI content for a large number of gas-bearing galaxies in the local universe -- to fit and test an extensive local universe BTFR. This BTFR is designed to be as inclusive of ALFALFA and comparable samples as possible. Velocity widths measured via an automated method and $M_{b}$ proxies extracted from survey data can be uniformly and efficiently measured for other samples, giving this analysis broad applicability. We also investigate the role of sample demographics in determining the best-fit relation. We find that the best-fit relations are changed significantly by changes to the sample mass range and to second order, mass sampling, gas fraction, different stellar mass and velocity width measurements. We use a subset of ALFALFA with demographics that reflect the full sample to measure a robust BTFR slope of $3.30\pm0.06$. We apply this relation and estimate source distances, finding general agreement with flow-model distances as well as average distance uncertainties of $\sim0.17$ dex for the full ALFALFA sample. We demonstrate the utility of these distance estimates by applying them to a sample of sources in the Virgo vicinity, recovering signatures of infall consistent with previous work.

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arXiv2022

Searches for Extremely Metal Poor Galaxies using ALFALFA-selected Dwarf Galaxies

We present a study of nearby dwarf galaxies selected from the ALFALFA blind HI survey. A primary goal of the project was to utilize a non-standard selection method with the hope of detecting previously unrecognized extremely metal-poor (XMP) galaxies. The study was motivated by the recent discovery of two XMP galaxies $-$ Leo P and Leoncino $-$ which were both originally found via the ALFALFA survey. We have obtained narrowband H$α$ images for 42 dwarf systems, many of which are located in the local void in front of the Pisces-Perseus Supercluster. Spectra for eleven of the best candidates resulted in the determination of metal abundances for ten of the systems. None were found to be extremely metal poor, although one system (AGC 123350) was found to have an oxygen abundance of log(O/H)+12 = 7.46, or $\sim$6\% solar. One of the galaxies in our sample exhibits a high oxygen abundance for its luminosity, suggesting the possibility that it may have a tidal origin.

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arXiv2022

Dark matter halos and scaling relations of extremely massive spiral galaxies from extended HI rotation curves

We present new and archival atomic hydrogen (\hi) observations of \galnum\ of the most massive spiral galaxies in the local Universe ($M_\star>10^{11} \, \mathrm{M}_\odot$). From 3D kinematic modeling of the datacubes, we derive extended \hi\ rotation curves, and from these, we estimate masses of the dark matter halos and specific angular momenta of the discs. We confirm that massive spiral galaxies lie at the upper ends of the Tully-Fisher relation (mass vs velocity, $M \propto V^{4}$) and Fall relation (specific angular momentum vs mass, $j \propto M^{0.6}$), in both stellar and baryonic forms, with no significant deviations from single power laws. We study the connections between baryons and dark matter through the stellar (and baryon)-to-halo ratios of mass $f_\mathrm{M} \equiv M_\star/M_\mathrm{h}$ and specific angular momentum $f_\mathrm{j} \equiv j_\star/j_\mathrm{h}$ and $f_\mathrm{j,bar} \equiv j_\mathrm{bar}/j_\mathrm{h}$. Combining our sample with others from the literature for less massive disc-dominated galaxies, we find that $f_\mathrm{M}$ rises monotonically with $M_\star$ and $M_\mathrm{h}$ (instead of the inverted-U shaped $f_\mathrm{M}$ for spheroid-dominated galaxies), while $f_\mathrm{j}$ and $f_\mathrm{j,bar}$ are essentially constant near unity over four decades in mass. Our results indicate that disc galaxies constitute a self-similar population of objects closely linked to the self-similarity of their dark halos. This picture is reminiscent of early analytical models of galaxy formation wherein discs grow by relatively smooth and gradual inflow, isolated from disruptive events such as major mergers and strong AGN feedback, in contrast to the more chaotic growth of spheroids.

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arXiv2022

Young, blue, and isolated stellar systems in the Virgo Cluster. II. A new class of stellar system

We discuss five blue stellar systems in the direction of the Virgo cluster, analogous to the enigmatic object SECCO 1 (AGC 226067). These objects were identified based on their optical and UV morphology and followed up with HI observations with the VLA (and GBT), MUSE/VLT optical spectroscopy, and HST imaging. These new data indicate that one system is a distant group of galaxies. The remaining four are extremely low mass ($M_\ast \sim 10^5 \; \mathrm{M_\odot}$), are dominated by young, blue stars, have highly irregular and clumpy morphologies, are only a few kpc across, yet host an abundance of metal-rich, $12 + \log (\mathrm{O/H}) > 8.2$, HII regions. These high metallicities indicate that these stellar systems formed from gas stripped from much more massive galaxies. Despite the young age of their stellar populations, only one system is detected in HI, while the remaining three have minimal (if any) gas reservoirs. Furthermore, two systems are surprisingly isolated and have no plausible parent galaxy within $\sim$30' ($\sim$140 kpc). Although tidal stripping cannot be conclusively excluded as the formation mechanism of these objects, ram pressure stripping more naturally explains their properties, in particular their isolation, owing to the higher velocities, relative to the parent system, that can be achieved. Therefore, we posit that most of these systems formed from ram pressure stripped gas removed from new infalling cluster members, and survived in the intracluster medium long enough to become separated from their parent galaxies by hundreds of kiloparsecs, and that they thus represent a new type of stellar system.

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arXiv2022

AGC 226178 and NGVS 3543: Two deceptive dwarfs towards Virgo

The two sources AGC 226178 and NGVS 3543, an extremely faint, clumpy, blue stellar system and a low surface brightness dwarf spheroidal, are adjacent systems in the direction of the Virgo cluster. Both have been studied in detail previously, with it being suggested that they are unrelated normal dwarf galaxies or that NGVS 3543 recently lost its gas through ram pressure stripping, and that AGC 226178 formed from this stripped gas. However, with HST ACS imaging we demonstrate that the stellar population of NGVS 3543 is inconsistent with being at the distance of the Virgo cluster, and that it is likely a foreground object at approximately 10 Mpc. Whereas the stellar population of AGC 226178 is consistent with it being a very young (10-100 Myr) object in the Virgo cluster. Through a re-analysis of the original ALFALFA HI detection we show that AGC 226178 likely formed from gas stripped from the nearby dwarf galaxy VCC 2034, a hypothesis strengthened by the high metallicity measured with MUSE VLT observations. However, it is unclear whether ram pressure or a tidal interaction is responsible for stripping the gas. AGC 226178 is one of at least five similar objects now known towards Virgo. These objects are all young and unlikely to remain visible for over ~500 Myr, suggesting that they are continually produced in the cluster.

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arXiv2021

Non-circular flows in HIghMass galaxies in a test of the late accretion hypothesis

We present H-alpha velocity maps for the HIghMass galaxies UGC 7899, UGC 8475, UGC 9037 and UGC 9334, obtained with the SITELLE Imaging Fourier Transform Spectrometer on the Canada-France-Hawaii Telescope, to search for kinematic signatures of late gas accretion to explain their large atomic gas reservoirs. The maps for UGC 7899, UGC 9037, and UGC 9334 are amenable to disk wide radial flow searches with the DiskFit algorithm, and those for UGC 7899 and UGC 9037 are also amenable to inner-disk kinematic analyses. We find no evidence for outer disk radial flows down to Vr ~ 20 km/s in UGC 9037 and UGC 9334, but hints of such flows in UGC 7899. Conversely, we find clear signatures of inner (r ~ 5 kpc) noncircularities in UGC 7899 and UGC 9037 that can be modelled as either bisymmetric (which could be produced by a bar) or radial flows. Comparing these models to the structure implied by photometric disk-bulge-bar decompositions, we favour inner radial flows in UGC 7899 and an inner bar in UGC 9037. With hints of outer disk radial flows and an outer disk warp, UGC 7899 is the best candidate for late accretion among the galaxies examined, but additional modelling is required to disentangle potential degeneracies between these signatures in H I and H-alpha velocity maps. Our search provides only weak = constraints on hot-mode accretion models that could explain the unusually high H I content of HIghMass galaxies.

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arXiv2021

The ALFALFA Almost-Dark Galaxy AGC~229101: A Two Billion Solar Mass HI Cloud with a Very Low Surface Brightness Optical Counterpart

We present results from deep HI and optical imaging of AGC 229101, an unusual HI source detected at v$_{\rm helio}$ = 7116 km/s in the ALFALFA survey. Initially classified as a candidate "dark" source because it lacks a clear optical counterpart in SDSS or DSS2 imaging, AGC 229101 has $10^{9.31\pm0.05}$ solar masses of HI, but an HI line width of only 43$\pm$9 km/s. Low resolution WSRT imaging and higher resolution VLA B-array imaging show that the source is significantly elongated, stretching over a projected length of ~80 kpc. The HI imaging resolves the source into two parts of roughly equal mass. WIYN pODI optical imaging reveals a faint, blue optical counterpart coincident with the northern portion of the HI. The peak surface brightness of the optical source is only $μ_{g}$ = 26.6 mag arcsec$^{-2}$, well below the typical cutoff that defines the isophotal edge of a galaxy, and its estimated stellar mass is only $10^{7.32\pm0.33}$ solar masses, yielding an overall neutral gas-to-stellar mass ratio of M$_{\rm HI}$/M$_*=$~98$_{+111}\atop^{-52}$. We demonstrate the extreme nature of this object by comparing its properties to those of other HI-rich sources in ALFALFA and the literature. We also explore potential scenarios that might explain the existence of AGC~229101, including a tidal encounter with neighboring objects and a merger of two dark HI clouds.

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arXiv2021

Galaxy Properties at the Faint End of the HI Mass Function

The Survey of HI in Extremely Low-mass Dwarfs (SHIELD) includes a volumetrically complete sample of 82 gas-rich dwarfs with M_HI~<10^7.2 Msun selected from the ALFALFA survey. We are obtaining extensive follow-up observations of the SHIELD galaxies to study their gas, stellar, and chemical content, and to better understand galaxy evolution at the faint end of the HI mass function. Here, we investigate the properties of 30 SHIELD galaxies using Hubble Space Telescope imaging of their resolved stars and Westerbork Synthesis Radio Telescope observations of their neutral hydrogen. We measure tip of the red giant branch (TRGB) distances, star formation activity, and gas properties. The TRGB distances are up to 4x greater than estimates from flow models, highlighting the importance of velocity-independent distance indicators in the nearby universe. The SHIELD galaxies are in under-dense regions, with 23% located in voids; one galaxy appears paired with a more massive dwarf. We quantify galaxy properties at low masses including stellar and HI masses, SFRs, sSFRs, SFEs, birthrate parameters, and gas fractions. The lowest mass systems lie below the mass thresholds where stellar mass assembly is predicted to be impacted by reionization. Even so, we find the star formation properties follow the same trends as higher mass gas-rich systems, albeit with a different normalization. The HI disks are small ( <0.7 kpc) making it difficult to measure the HI rotation using standard techniques; we develop a new methodology and report the velocity extent, and its associated spatial extent, with robust uncertainties.

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arXiv2021

What determines the HI gas content in galaxies?: morphological dependence of the HI gas fraction across M*-SFR plane

We perform a stacking analysis of the HI spectra from the Arecibo Legacy Fast ALFA (ALFALFA) survey for optically-selected local galaxies from the Sloan Digital Sky Survey (SDSS) to study the average gas fraction of galaxies at fixed stellar mass ($M_*$) and star formation rate (SFR). We first confirm that the average gas fraction strongly depends on the stellar mass and SFR of host galaxies; massive galaxies tend to have a lower gas fraction, and actively star-forming galaxies show higher gas fraction, which is consistent with many previous studies. Then we investigate the morphological dependence of the HI gas mass fraction at fixed $M_*$ and SFR to minimize the effects of these parameters. We use three morphological classifications based on parametric indicator (Sérsic index), non-parametric indicator (C-index), and visual inspection (smoothness from the Galaxy Zoo 2 project) on the optical image. We find that there is no significant morphological dependence of the HI gas mass fraction at fixed $M_*$ and SFR when we use C-index. In comparison, there exists a hint of diminishment in the HI gas mass fraction for "smooth" galaxies compared with "non-smooth" galaxies. We find that the visual smoothness is sensitive to the existence of small-scale structures in a galaxy. Our result suggests that even at fixed $M_*$ and SFR, the presence of such small-scale structures (seen in the optical image) is linked to their total HI gas content.

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arXiv2021

VLA Imaging of HI-bearing Ultra-Diffuse Galaxies from the ALFALFA Survey

Ultra-diffuse galaxies have generated significant interest due to their large optical extents and low optical surface brightnesses, which challenge galaxy formation models. Here we present resolved synthesis observations of 12 HI-bearing ultra-diffuse galaxies (HUDs) from the Karl G. Jansky Very Large Array (VLA), as well as deep optical imaging from the WIYN 3.5-meter telescope at Kitt Peak National Observatory. We present the data processing and images, including total intensity HI maps and HI velocity fields. The HUDs show ordered gas distributions and evidence of rotation, important prerequisites for the detailed kinematic models in Mancera Piña et al. (2019b). We compare the HI and stellar alignment and extent, and find the HI extends beyond the already extended stellar component and that the HI disk is often misaligned with respect to the stellar one, emphasizing the importance of caution when approaching inclination measurements for these extreme sources. We explore the HI mass-diameter scaling relation, and find that although the HUDs have diffuse stellar populations, they fall along the relation, with typical global HI surface densities. This resolved sample forms an important basis for more detailed study of the HI distribution in this extreme extragalactic population.

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arXiv2020

The ALFALFA-SDSS Galaxy Catalog

We present an HI-optical catalog of ~ 30,000 galaxies based on the 100% complete Arecibo Legacy Fast Arecibo L-band Feed Array (ALFALFA) survey combined with data from the Sloan Digital Sky Survey (SDSS). Our goal is to facilitate public use of the completed ALFALFA catalog by providing carefully determined matches to SDSS counterparts, including matches for ~ 12,000 galaxies that do not have SDSS spectra. These identifications can provide a basis for further cross-matching with other surveys using SDSS photometric IDs as a reference point. We derive absolute magnitudes and stellar masses for each galaxy using optical colors combined with an internal reddening correction designed for small- and intermediate-mass galaxies with active star formation. We also provide measures of stellar masses and star formation rates based on infrared and/or ultraviolet photometry for galaxies that are detected by the Wide-field Infrared Survey Explorer (WISE) and/or the Galaxy Evolution Explorer (GALEX). Finally, we compare the galaxy population in the ALFALFA-SDSS sample with the populations in several other publicly-available galaxy catalogs, and confirm that ALFALFA galaxies typically have lower masses and bluer colors.

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arXiv2020

Robust HI kinematics of gas-rich ultra-diffuse galaxies: hints of a weak-feedback formation scenario

We study the gas kinematics of a sample of six isolated gas-rich low surface brightness galaxies, of the class called ultra-diffuse galaxies (UDGs). These galaxies have recently been shown to be outliers from the baryonic Tully-Fisher relation (BTFR), as they rotate much slower than expected given their baryonic mass, and to have baryon fractions similar to the cosmological mean. By means of a 3D kinematic modelling fitting technique, we show that the HI in our UDGs is distributed in "thin" regularly rotating discs and we determine their rotation velocity and gas velocity dispersion. We revisit the BTFR adding galaxies from other studies. We find a previously unknown trend between the deviation from the BTFR and the disc scale length valid for dwarf galaxies with circular speeds < 45 km/s, with our UDGs being at the extreme end. Based on our findings, we suggest that the high baryon fractions of our UDGs may originate due to the fact that they have experienced weak stellar feedback, likely due to their low star formation rate surface densities, and as a result they did not eject significant amounts of gas out of their discs. At the same time, we find indications that our UDGs may have higher-than-average stellar specific angular momentum, which can explain their large optical scale lengths.

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arXiv2020

Direct Measurement of the HI-Halo Mass Relation through Stacking

We present accurate measurements of the total HI mass in dark matter halos of different masses at z ~ 0, by stacking the HI spectra of entire groups from the Arecibo Fast Legacy ALFA Survey. The halos are selected from the optical galaxy group catalog constructed from the Sloan Digital Sky Survey DR7 Main Galaxy sample, with reliable measurements of halo mass and halo membership. We find that the HI-halo mass relation is not a simple monotonic function, as assumed in several theoretical models. In addition to the dependence of halo mass, the total HI gas mass shows strong dependence on the halo richness, with larger HI masses in groups with more members at fixed halo masses. Moreover, halos with at least three member galaxies in the group catalog have a sharp decrease of the HI mass, potentially caused by the virial halo shock-heating and the AGN feedback. The dominant contribution of the HI gas comes from the central galaxies for halos of $M<10^{12.5}h^{-1}M_{\odot}$, while the satellite galaxies dominate over more massive halos. Our measurements are consistent with a three-phase formation scenario of the HI-rich galaxies. The smooth cold gas accretion is driving the HI mass growth in halos of $M<10^{11.8}h^{-1}M_{\odot}$, with late-forming halos having more HI accreted. The virial halo shock-heating and AGN feedback will take effect to reduce the HI supply in halos of $10^{11.8}h^{-1}M_{\odot}<M<10^{13}h^{-1}M_{\odot}$. The HI mass in halos more massive than $10^{13}h^{-1}M_{\odot}$ generally grows by mergers, with the dependence on halo richness becoming much weaker.

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