arXiv subjects

Laurent Chemin

Publications and source records attributed to Laurent Chemin.

At least 19 recordsLinked to original sources

arXiv2026

MIGHTEE-HI / LADUMA: Investigating the link between baryons and dynamics with 130 resolved HI-selected galaxies

The baryonic Tully-Fisher relation (bTFR) and the radial acceleration relation (RAR) link the observed dynamics in galaxies to that expected from their baryonic mass distributions. The relations' small intrinsic scatters place strong constraints on galaxy formation models, dark matter properties and theories of modified dynamics, yet detailed measurements beyond the very local Universe remain limited. We use 130 purely HI-selected galaxies with resolved HI kinematics and baryonic mass profiles to measure the bTFR and RAR up to $z\approx0.09$. We measure a tight RAR with an acceleration scale $a_0=(1.50\pm0.05)\times10^{-10},{\rm m,s^{-2}}$ and an intrinsic scatter of $0.096\pm0.006$ dex, consistent with local results. We fit the bTFR in the `inverse' direction, conditioning on $M_{\rm bar}$ to mitigate HI flux-related selection effects, measuring a logarithmic slope of $0.27\pm0.01$ (corresponding to a forward slope of $3.72\pm0.16$), with vertical intrinsic scatter $\sigma_\perp\approx0.05$ dex. Fitting the general $\delta$-family of MOND interpolating functions to the RAR, we infer $\delta=4.10^{+1.4}_{-0.68}$, consistent with the value required by Solar System gravitational constraints and a null Wide Binary Test. We find no significant redshift evolution in the RAR acceleration scale for our pure HI-selected sample. However, the bTFR zero-point shows an apparent evolutionary trend that is strongly dependent on the fit direction: the traditional forward fit yields an $8.7\sigma$ preference for $z$ evolution, while for our fiducial inverse fit, this reduces to $3.4\sigma$, within $\approx2\sigma$ of the RAR evolution constraint. This suggests selection effects bias the forward fit; a careful consideration of such effects will be required in future endeavours to robustly measure the redshift evolution of dynamical scaling relations.

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arXiv2026

WLM: Dynamics of an isolated Dwarf Irregular Galaxy Under Ram Pressure in the Local Group

WLM is an archetypal dwarf irregular galaxy that has not experienced interactions with major Local Group galaxies within the past 8 Gyr. It has recently been shown that WLM is losing its gas due to ram pressure forces exerted by the surrounding intergalactic medium (IGM). In this work, we explore how ram pressure may also affect the WLM gas kinematics, and we show that its dynamics is especially perturbed at its outskirts, explaining the asymmetric rotation between the approaching and receding sides. Moreover, we have been able to decompose WLM in two main components, a compact one with a solid-body rotation that resembles a bar-like structure, and a more extended one with a characteristic double-horn profile suggesting an edge-on disk. The former is relatively unaffected by ram pressure while the latter has its dynamics considerably affected by ram pressure. This study shows that mass estimates of a dwarf galaxy like WLM should account for a full modeling of its dynamical components, especially accounting for its asymmetric rotation curve.

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arXiv2026

MUSE-ALMA Haloes XIII. Molecular gas in $z \sim 0.5$ HI-selected galaxies

We present new results from the MUSE-ALMA Haloes survey, covering 79 galaxies associated with strong HI absorption at redshift about 0.5. Our ALMA Cycle 10 observations add 39 systems to the initial 21, bringing the total to 60 galaxies. CO emission is detected in 9 new galaxies, and in 12 of 60 total, doubling the number of CO-emitting HI-selected galaxies and probing 1.2 dex deeper in molecular gas mass than previous studies. These galaxies span a wide range of stellar masses and metallicities. By comparing CO(2-1) and CO(3-2) properties with star formation rates and gas-phase metallicities from VLT/MUSE and HST, we find a dual behaviour in star formation efficiency: low-mass systems follow main-sequence scaling relations, while high-mass systems show suppressed star formation. This diversity indicates that HI absorbers trace both evolved and younger galaxies, providing a key step toward completing the baryon census at redshift about 0.5.

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arXiv2025

The Gaia RVS Benchmark Stars. III . Confirmed and New High Radial Velocity Stars From Gaia DR3

High-velocity stars are interesting targets to unveil the formation of the Milky Way. In fact they can be recently accreted from an infalling dwarf galaxies or they can be the result of a turbulent merging of galaxies. Gaia is providing the community a way to select stars for their kinematics and the radial velocity, one of the speed components, is derived from the Gaia RVS spectrum. High absolute radial velocity values are sensitive to be false positive. They are rare and as such they are more easily impacted than lower velocity stars, by contamination by very low SNR spurious measures. We here investigate a sample of 26 stars with Gaia absolute radial velocity in excess of 500___km/s with spectroscopic follow-up observations with UVES. For all but one star the extreme radial velocity is confirmed and these stars are all metal-poor and, as expected, enhanced in the elements. The complete chemical inventory confirm the large star-to-star scatter for the heavy elements and their on average larger ratio over iron with respect to the Sun, in agreement with the literature.

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arXiv2025

Investigating the HI distribution and kinematics of ESO444-G084 and [KKS2000]23: New insights from the MHONGOOSE survey

We present the HI distribution, kinematics, mass modeling, and disk stability of the dwarf irregular galaxies ESO444-G084 and [KKS2000]23 using high-resolution, high-sensitivity MHONGOOSE survey data from MeerKAT. ESO444-G084 shows centrally concentrated HI emission, while [KKS2000]23 exhibits irregular high-density clumps. Total HI fluxes measured down to 10^19 and 10^18 cm^-2 are nearly identical, indicating that the increased HI diameter at lower column densities results mainly from the larger beam, with no significant extra emission detected. We derive total HI masses of (1.1 +/- 0.1) x 10^8 and (6.1 +/- 0.3) x 10^8 solar masses for ESO444-G084 and [KKS2000]23, respectively. Using PyFAT and TiRiFiC, we extract 3D rotation curves that reveal disk-like kinematics in both galaxies. ESO444-G084 shows a warp beyond ~1.8 kpc and a fast-rising curve consistent with a centrally concentrated dark matter distribution, while [KKS2000]23's more gradual rise suggests a more extended halo. Mass modeling with an isothermal halo and stellar mass-to-light ratios of 0.20 for ESO444-G084 and 0.18 for [KKS2000]23 yields consistent results. We analyze disk stability using spatially resolved Toomre Q and gas-to-critical surface density ratios, linking these with H-alpha and FUV-based star formation. ESO444-G084 supports localized star formation despite global stability, while [KKS2000]23 appears gravitationally unstable yet lacks H-alpha, suggesting that turbulence, gas depletion, or past feedback suppresses star formation. No inflows or outflows are detected, indicating internal processes regulate star formation. This study highlights the interplay between HI morphology, kinematics, dark matter distribution, and disk stability, showing how internal processes shape dwarf galaxy evolution.

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arXiv2025

Uncovering Extraplanar Gas in UGCA 250 with the Ultra-deep MHONGOOSE Survey

We use the neutral atomic hydrogen (HI) observations of the edge-on galaxy UGCA 250, taken as part of the MeerKAT HI Observations of Nearby Galactic Objects - Observing Southern Emitters (MHONGOOSE) survey to investigate the amount, morphology, and kinematics of extraplanar gas. The combination of high column density sensitivity and high spatial resolution of the survey over a large field of view is ideal for studying the underlying physics governing the extraplanar gas. These data reveal 9 additional detections within the field of view along with UGCA 250, with 8 of them being within $\sim$ 200 km s$^{-1}$ of the galaxy's systemic velocity. The galaxy seems to have a tail-like feature extending away from it in the southern direction up to $\sim$ 41 kpc (in projection). We also detect a cloud at anomalous velocities, but we did not find any optical counterpart. We construct a detailed tilted ring model for this edge-on galaxy to gain a deeper understanding of the vertical structure of its neutral hydrogen. The model that best matches the data features a thick disc with a scale height of $\sim$ 3$\pm$1 kpc and an HI mass of about 15$\%$ of the total HI mass. This extraplanar gas is detected for the first time in UGCA 250. Our analysis favours a mixed origin for the extraplanar gas in UGCA 250, likely arising from a combination of internal stellar feedback and external tidal interactions.

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arXiv2024

Looking At the Distant Universe with the MeerKAT Array: the HI Mass Function in the Local Universe

We present measurements of the neutral atomic hydrogen (HI) mass function (HIMF) and cosmic HI density ($\Omega_{\rm HI}$) at $0 \leq z \leq 0.088$ from the Looking at the Distant Universe with MeerKAT Array (LADUMA) survey. Using LADUMA Data Release 1 (DR1), we analyze the HIMF via a new "recovery matrix" (RM) method that we benchmark against a more traditional Modified Maximum Likelihood (MML) method. Our analysis, which implements a forward modeling approach, corrects for survey incompleteness and uses extensive synthetic source injections to ensure robust estimates of the HIMF parameters and their associated uncertainties. This new method tracks the recovery of sources in mass bins different from those in which they were injected and incorporates a Poisson likelihood in the forward modeling process, allowing it to correctly handle uncertainties in bins with few or no detections. The application of our analysis to a high-purity subsample of the LADUMA DR1 spectral line catalog in turn mitigates any possible biases that could result from the inconsistent treatment of synthetic and real sources. For the surveyed redshift range, the recovered Schechter function normalization, low-mass slope, and "knee" mass are $\phi_\ast = 3.56_{-1.92}^{+0.97} \times 10^{-3}$ Mpc$^{-3}$ dex$^{-1}$, $\alpha = -1.18_{-0.19}^{+0.08}$, and $\log(M_\ast/M_\odot) = 10.01_{-0.12}^{+0.31}$, respectively, which together imply a comoving cosmic HI density of $\Omega_{\rm HI}=3.09_{-0.47}^{+0.65}\times 10^{-4}$. Our results show consistency between RM and MML methods and with previous low-redshift studies, giving confidence that the cosmic volume probed by LADUMA, even at low redshifts, is not an outlier in terms of its HI content.

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arXiv2023

Detection of the Keplerian decline in the Milky Way rotation curve

Our position inside the Galactic disc had prevented us from establishing an accurate rotation curve, until the advent of Gaia, whose third data release (Gaia DR3) made it possible to specify it up to twice the optical radius. We aim to establish a new rotation curve of the Galaxy from the Gaia DR3, by drastically reducing uncertainties and systematics, and with the goal to provide a new estimate of the mass of the Galaxy. We have compared different estimates, established a robust assessment of the systematic uncertainties, and addressed differences in methodologies, particularly regarding distance estimates. This results in a sharply decreasing rotation curve for the Milky Way, the decrease in velocity between 19.5 and 26.5 kpc is approximately 30 km s$^{-1}$. We have identified, for the first time, a Keplerian decline of the rotation curve, starting at $\sim$ 19 kpc and up to $\sim$ 26.5 kpc from the Galaxy center, while a flat rotation curve is rejected with a significance of 3$σ$. The total mass is revised downwards to $2.06^{+0.24}_{-0.13}\times 10^{11}\ M_{\odot}$, in agreement with an absence of significant mass increase at radii larger than 19 kpc. The upper limit of the total mass was evaluated by considering the upper values of velocity measurements, which leads to a strict, unsurpassable, limit of $5.4\times 10^{11}\ M_{\odot}$.

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arXiv2023

A Machine Learning Approach to Galactic Emission-Line Region Classification

Diagnostic diagrams of emission-line ratios have been used extensively to categorize extragalactic emission regions; however, these diagnostics are occasionally at odds with each other due to differing definitions. In this work, we study the applicability of supervised machine-learning techniques to systematically classify emission-line regions from the ratios of certain emission lines. Using the Million Mexican Model database, which contains information from grids of photoionization models using \texttt{cloudy}, and from shock models, we develop training and test sets of emission line fluxes for three key diagnostic ratios. The sets are created for three classifications: classic \hii{} regions, planetary nebulae, and supernova remnants. We train a neural network to classify a region as one of the three classes defined above given three key line ratios that are present both in the SITELLE and MUSE instruments' band-passes: [{\sc O\,iii}]$\lambda5007$/H$β$, [{\sc N\,ii}]$\lambda6583$/H$α$, ([{\sc S\,ii}]$\lambda6717$+[{\sc S\,ii}]$\lambda6731$)/H$α$. We also tested the impact of the addition of the [{\sc O\,ii}]$\lambda3726,3729$/[{\sc O\,iii}]$\lambda5007$ line ratio when available for the classification. A maximum luminosity limit is introduced to improve the classification of the planetary nebulae. Furthermore, the network is applied to SITELLE observations of a prominent field of M33. We discuss where the network succeeds and why it fails in certain cases. Our results provide a framework for the use of machine learning as a tool for the classification of extragalactic emission regions. Further work is needed to build more comprehensive training sets and adapt the method to additional observational constraints.

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arXiv2023

A new resonance-like feature in the outer disc of the Milky Way

Modern astrometric and spectroscopic surveys have revealed a wealth of structure in the phase space of stars in the Milky Way, with evidence of resonance features and non-equilibrium processes. Using Gaia's third data release, we present evidence of a new resonance-like feature in the outer disc of the Milky Way. The feature is most evident in the angular momentum distribution of the young Classical Cepheids, a population for which we can derive accurate distances over much of the Galactic disc. We then search for similar features in the outer disc using a much larger sample of red giant stars, as well as a compiled list of over 31 million stars with spectroscopic line-of-sight velocity measurements. While much less evident in these two older samples, the distribution of stars in action-configuration space suggests that resonance features are present here as well. The position of the feature in action-configuration space suggests that the new feature may be related to the Galactic bar, but other possibilities are discussed.

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arXiv2022

The disturbed outer Milky Way disc

The outer parts of the Milky Way's disc are significantly out of equilibrium. Using only distances and proper motions of stars from Gaia's Early Data Release 3, in the range |b|<10°, 130°<l<230°, we show that for stars in the disc between around 10 and 14 kpc from the Galactic centre, vertical velocity is strongly dependent on the angular momentum, azimuth, and position above or below the Galactic plane. We further show how this behaviour translates into a bimodality in the velocity distribution of stars in the outer Milky Way disc. We use an N-body model of an impulse-like interaction of the Milky Way disc with a perturber similar to the Sagittarius dwarf to demonstrate that this mechanism can generate a similar disturbance. It has already been shown that this interaction can produce a phase spiral similar to that seen in the Solar neighbourhood. We argue that the details of this substructure in the outer galaxy will be highly sensitive to the timing of the perturbation or the gravitational potential of the Galaxy, and therefore may be key to disentangling the history and structure of the Milky Way.

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arXiv2022

WISDOM project -- XI. Star Formation Efficiency in the Bulge of the AGN-host Galaxy NGC 3169 with SITELLE and ALMA

The star formation efficiency (SFE) has been shown to vary across different environments, particularly within galactic starbursts and deep within the bulges of galaxies. Various quenching mechanisms may be responsible, ranging from galactic dynamics to feedback from active galactic nuclei (AGN). Here, we use spatially-resolved observations of warm ionised gas emission lines from the imaging Fourier transform spectrograph SITELLE at the Canada-France-Hawaii Telescope (CFHT) and cold molecular gas (CO(2-1)) from the Atacama Large Millimeter/sub-millimeter Array (ALMA) to study the SFE in the bulge of the AGN-host galaxy NGC 3169. After distinguishing star-forming regions from AGN-ionised regions using emission-line ratio diagnostics, we measure spatially-resolved molecular gas depletion times (τ_dep = 1/SFE) with a spatial resolution of \approx 100 pc within a galactocentric radius of 1.8 kpc. We identify a star-forming ring located at radii 1.25 \pm 0.6 kpc with an average τ_dep of 0.3 Gyr. At radii < 0.9 kpc, however, the molecular gas surface densities and depletion times increase with decreasing radius, the latter reaching approximately 2.3 Gyr at a radius \approx 500 pc. Based on analyses of the gas kinematics and comparisons with simulations, we identify AGN feedback, bulge morphology and dynamics as the possible causes of the radial profile of SFE observed in the central region of NGC 3169.

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arXiv2022

LADUMA: Discovery of a luminous OH megamaser at $z > 0.5$

In the local Universe, OH megamasers (OHMs) are detected almost exclusively in infrared-luminous galaxies, with a prevalence that increases with IR luminosity, suggesting that they trace gas-rich galaxy mergers. Given the proximity of the rest frequencies of OH and the hyperfine transition of neutral atomic hydrogen (HI), radio surveys to probe the cosmic evolution of HI in galaxies also offer exciting prospects for exploiting OHMs to probe the cosmic history of gas-rich mergers. Using observations for the Looking At the Distant Universe with the MeerKAT Array (LADUMA) deep HI survey, we report the first untargeted detection of an OHM at $z > 0.5$, LADUMA J033046.20$-$275518.1 (nicknamed "Nkalakatha"). The host system, WISEA J033046.26$-$275518.3, is an infrared-luminous radio galaxy whose optical redshift $z \approx 0.52$ confirms the MeerKAT emission line detection as OH at a redshift $z_{\rm OH} = 0.5225 \pm 0.0001$ rather than HI at lower redshift. The detected spectral line has 18.4$σ$ peak significance, a width of $459 \pm 59\,{\rm km\,s^{-1}}$, and an integrated luminosity of $(6.31 \pm 0.18\,{\rm [statistical]}\,\pm 0.31\,{\rm [systematic]}) \times 10^3\,L_\odot$, placing it among the most luminous OHMs known. The galaxy's far-infrared luminosity $L_{\rm FIR} = (1.576 \pm 0.013) \times 10^{12}\,L_\odot$ marks it as an ultra-luminous infrared galaxy; its ratio of OH and infrared luminosities is similar to those for lower-redshift OHMs. A comparison between optical and OH redshifts offers a slight indication of an OH outflow. This detection represents the first step towards a systematic exploitation of OHMs as a tracer of galaxy growth at high redshifts.

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arXiv2021

Faint objects in motion: the new frontier of high precision astrometry

Sky survey telescopes and powerful targeted telescopes play complementary roles in astronomy. In order to investigate the nature and characteristics of the motions of very faint objects, a flexibly-pointed instrument capable of high astrometric accuracy is an ideal complement to current astrometric surveys and a unique tool for precision astrophysics. Such a space-based mission will push the frontier of precision astrometry from evidence of Earth-mass habitable worlds around the nearest stars, to distant Milky Way objects, and out to the Local Group of galaxies. As we enter the era of the James Webb Space Telescope and the new ground-based, adaptive-optics-enabled giant telescopes, by obtaining these high precision measurements on key objects that Gaia could not reach, a mission that focuses on high precision astrometry science can consolidate our theoretical understanding of the local Universe, enable extrapolation of physical processes to remote redshifts, and derive a much more consistent picture of cosmological evolution and the likely fate of our cosmos. Already several missions have been proposed to address the science case of faint objects in motion using high precision astrometry missions: NEAT proposed for the ESA M3 opportunity, micro-NEAT for the S1 opportunity, and Theia for the M4 and M5 opportunities. Additional new mission configurations adapted with technological innovations could be envisioned to pursue accurate measurements of these extremely small motions. The goal of this White Paper is to address the fundamental science questions that are at stake when we focus on the motions of faint sky objects and to briefly review instrumentation and mission profiles.

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arXiv2020

Anisotropy of random motions of gas in Messier 33

(Abridged) We study the properties of anisotropic and axisymmetric velocity ellipsoids from maps of the gas velocity dispersion in nearby galaxies. This data allow us to measure the azimuthal-to-radial axis ratio of gas velocity ellipsoids, which is a useful tool to study the structure of gaseous orbits in the disk. We also present the first estimates of perturbations in gas velocity dispersion maps by applying an alternative model that considers isotropic and asymmetric random motions. High-quality velocity dispersion maps of the atomic medium at various angular resolutions of the nearby spiral galaxy Messier 33, are used to test the anisotropic and isotropic velocity models. The velocity dispersions of hundreds of individual molecular clouds are also analyzed. The HI velocity dispersion of M33 is systematically larger along the minor axis, and lower along the major axis. Isotropy is only possible if asymmetric motions are considered. Fourier transforms of the HI velocity dispersions reveal a bisymmetric mode which is mostly stronger than other asymmetric motions and aligned with the minor axis of the galaxy. Within the anisotropic and axisymmetric velocity model, the stronger bisymmetry is explained by a radial component that is larger than the azimuthal component of the ellipsoid of random motions, thus by gaseous orbits that are dominantly radial. The azimuthal anisotropy parameter is not strongly dependent on the choice of the vertical dispersion. The velocity anisotropy parameter of the molecular clouds is observed highly scattered. Perturbations such as HI spiral-like arms could be at the origin of the gas velocity anisotropy in M33. Further work is necessary to assess whether anisotropic velocity ellispsoids can also be invoked to explain the asymmetric gas random motions of other galaxies.

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arXiv2020

GaiaNIR: Combining optical and Near-Infra-Red (NIR) capabilities with Time-Delay-Integration (TDI) sensors for a future Gaia-like mission

ESA recently called for new "Science Ideas" to be investigated in terms of feasibility and technological developments -- for technologies not yet sufficiently mature. These ideas may in the future become candidates for M or L class missions within the ESA Science Program. With the launch of Gaia in December 2013, Europe entered a new era of space astrometry following in the footsteps of the very successful Hipparcos mission from the early 1990s. Gaia is the successor to Hipparcos, both of which operated in optical wavelengths, and Gaia is two orders of magnitude more accurate in the five astrometric parameters and is surveying four orders of magnitude more stars in a vast volume of the Milky Way. The combination of the Hipparcos/Tycho-2 catalogues with the first early Gaia data release will give improved proper motions over a long ~25 year baseline. The final Gaia solution will also establish a new optical reference frame by means of quasars, by linking the optical counterparts of radio (VLBI) sources defining the orientation of the reference frame, and by using the zero proper motion of quasars to determine a non-rotating frame. A weakness of Gaia is that it only operates at optical wavelengths. However, much of the Galactic centre and the spiral arm regions, important for certain studies, are obscured by interstellar extinction and this makes it difficult for Gaia to deeply probe. Traditionally, this problem is overcome by switching to the infra-red but this was not possible with Gaia's CCDs. Additionally, to scan the entire sky and make global absolute parallax measurements the spacecraft must have a constant rotation and this requires that the CCDs operate in TDI mode, increasing their complexity.

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arXiv2018

Asymmetric Drift in the Andromeda Galaxy (M31) as a Function of Stellar Age

We analyze the kinematics of Andromeda's disk as a function of stellar age by using photometry from the Panchromatic Hubble Andromeda Treasury (PHAT) survey and spectroscopy from the Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo (SPLASH) survey. We use HI 21-cm and CO ($\rm J=1 \rightarrow 0$) data to examine the difference between the deprojected rotation velocity of the gas and that of the stars. We divide the stars into four stellar age bins, from shortest lived to longest lived: massive main sequence stars (0.03 Gyr), more luminous intermediate mass asymptotic giant branch (AGB) stars (0.4 Gyr), less luminous intermediate mass AGB stars (2 Gyr), and low mass red giant branch stars (4 Gyr). There is a clear correlation between the offset of the stellar and the gas rotation velocity, or the asymmetric drift: the longer lived populations lag farther behind the gas than short lived populations. We also examine possible causes of the substructure in the rotation curves and find that the most significant cause of scatter in the rotation curves comes from the tilted ring model being an imperfect way to account for the multiple warps in Andromeda's disk.

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arXiv2018

A mass-velocity anisotropy relation in galactic stellar disks

The ellipsoid of stellar random motions is a fundamental ingredient of galaxy dynamics. Yet it has long been difficult to constrain this component in disks others than the Milky Way. This article presents the modeling of the azimuthal-to-radial axis ratio of the velocity ellipsoid of galactic disks from stellar dispersion maps using integral field spectroscopy data of the CALIFA survey. The measured azimuthal anisotropy is shown to be not strongly dependent on the assumed vertical-to-radial dispersion ratio of the ellipsoid. The anisotropy distribution shows a large diversity in the orbital structure of disk galaxies from tangential to radial stellar orbits. Globally, the orbits are isotropic in inner disk regions and become more radial as a function of radius, although this picture tends to depend on galaxy morphology and luminosity. The Milky Way orbital anisotropy profile measured from the Second Gaia Data Release is consistent with those of CALIFA galaxies. A new correlation is evidenced, linking the absolute magnitude or stellar mass of the disks to the azimuthal anisotropy. More luminous disks have more radial orbits and less luminous disks have isotropic and somewhat tangential orbits. This correlation is consistent with the picture in galaxy evolution in which orbits become more radial as the mass grows and is redistributed as a function of time. With the help of circular velocity curves, it is also shown that the epicycle theory fails to reproduce the diversity of the azimuthal anisotropy of stellar random motions, as it predicts only nearly radial orbits in the presence of flat curves. The origin of this conflict is yet to be identified. It also questions the validity of the vertical-to-radial axis ratio of the velocity ellipsoid derived by many studies in the framework of the epicyclic approximation.

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