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One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of Rényi divergences - are strictly ordered between the two tuples, then for sufficiently large n , there exists a stochastic matrix taking the n -fold Kronecker power of each input distribution to the n -fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

We present the lifetime star formation histories (SFHs) for six ultra-faint dwarf (UFD; $M_V>-7.0$, $ 4.9<\log_{10}({M_*(z=0)}/{M_{\odot}})<5.5$) satellite galaxies of M31 based on deep color-magnitude diagrams constructed from \textit{Hubble Space Telescope} imaging. These are the first SFHs obtained from the oldest main sequence turn-off of UFDs outside the halo of the Milky Way (MW). We find that five UFDs formed at least 50\% of their stellar mass by $z=5$ (12.6~Gyr ago), similar to known UFDs around the MW, but that 10-40\% of their stellar mass formed at later times. We uncover one remarkable UFD, \A{XIII}, which formed only 10\% of its stellar mass by $z=5$, and 75\% in a rapid burst at $z\sim2-3$, a result that is robust to choices of underlying stellar model and is consistent with its predominantly red horizontal branch. This "young" UFD is the first of its kind and indicates that not all UFDs are necessarily quenched by reionization, which is consistent with predictions from several cosmological simulations of faint dwarf galaxies. SFHs of the combined MW and M31 samples suggest reionization did not homogeneously quench UFDs. We find that the least massive MW UFDs ($M_*(z=5) \lesssim 5\times10^4 M_{\odot}$) are likely quenched by reionization, whereas more massive M31 UFDs ($M_*(z=5) \gtrsim 10^5 M_{\odot}$) may only have their star formation suppressed by reionization and quench at a later time. We discuss these findings in the context of the evolution and quenching of UFDs., Comment: 18 pages, 14 figures, 5 appendices, accepted for publication in ApJ

We present the color-magnitude diagrams and star formation histories (SFHs) of seven ultra-faint dwarf galaxies: Horologium 1, Hydra 2, Phoenix 2, Reticulum 2, Sagittarius 2, Triangulum 2, and Tucana 2, derived from high-precision Hubble Space Telescope photometry. We find that the SFH of each galaxy is consistent with them having created at least 80% of the stellar mass by z ~ 6. For all galaxies, we find quenching times older than 11.5 Gyr ago, compatible with the scenario in which reionization suppresses the star formation of small dark matter halos. However, our analysis also reveals some differences in the SFHs of candidate Magellanic Cloud satellites, i.e., galaxies that are likely satellites of the Large Magellanic Cloud and that entered the Milky Way potential only recently. Indeed, Magellanic satellites show quenching times about 600 Myr more recent with respect to those of other Milky Way satellites, on average, even though the respective timings are still compatible within the errors. This finding is consistent with theoretical models that suggest that satellites' SFHs may depend on their host environment at early times, although we caution that within the error bars all galaxies in our sample are consistent with being quenched at a single epoch.

We perform a comprehensive determination of the systemic proper motions of 74 dwarf galaxies and dwarf galaxy candidates in the Local Group based on Gaia early data release 3. The outputs of the analysis for each galaxy, including probabilities of membership, will be made publicly available. The analysis is augmented by a determination of the orbital properties of galaxies within 500 kpc. We adopt the flexible Bayesian methodology presented by McConnachie \& Venn (2020), which takes into account the location of the stars on the sky, on the colour-magnitude diagram and on the proper motion plane. We apply some modifications, in particular to the way the colour-magnitude diagram and spectroscopic information are factored in, e.g. by including stars in several evolution phases. The bulk motions are integrated in three gravitational potentials: two where the Milky Way is treated in isolation and has a mass 0.9 \& 1.6 $\times 10^{12}$M$_{\odot}$ and the time-varying potential by Vasiliev et al. (2021), which includes the infall of a massive Large Magellanic Cloud (LMC). We are able to determine bulk proper motions for 73 systems, and we consider reliable 66 of these measurements. For the first time, systemic motions are presented for galaxies out to a distance of 1.4 Mpc, in the NGC~3109 association. The inclusion of the infall of a massive LMC significantly modifies the orbital trajectories of the objects, with respect to orbit integration in static Milky Way-only potentials, and leads to 6 galaxies being likely associated to the LMC, 3 possibly associated and 1 recently captured object. We discuss the results of the orbit integration in the context of the relation of the galaxies to the system of Milky Way satellites, implications for the too-big-to-fail problem, impact on star formation histories, and tidal disruption., Comment: Submitted to A&A. 47 pages

We present the color-magnitude diagrams and star formation histories (SFHs) of seven ultra-faint dwarf galaxies: Horologium 1, Hydra 2, Phoenix 2, Reticulum 2, Sagittarius 2, Triangulum 2, and Tucana 2, derived from high-precision Hubble Space Telescope photometry. We find that the SFH of each galaxy is consistent with them having created at least 80% of the stellar mass by z ~ 6. For all galaxies, we find quenching times older than 11.5 Gyr ago, compatible with the scenario in which reionization suppresses the star formation of small dark matter halos. However, our analysis also reveals some differences in the SFHs of candidate Magellanic Cloud satellites, i.e., galaxies that are likely satellites of the Large Magellanic Cloud and that entered the Milky Way potential only recently. Indeed, Magellanic satellites show quenching times about 600 Myr more recent with respect to those of other Milky Way satellites, on average, even though the respective timings are still compatible within the errors. This finding is consistent with theoretical models that suggest that satellites' SFHs may depend on their host environment at early times, although we caution that within the error bars all galaxies in our sample are consistent with being quenched at a single epoch.

We perform a comprehensive determination of the systemic proper motions of 74 dwarf galaxies and dwarf galaxy candidates in the Local Group based on Gaia early data release 3. The outputs of the analysis for each galaxy, including probabilities of membership, will be made publicly available. The analysis is augmented by a determination of the orbital properties of galaxies within 500 kpc. We adopt the flexible Bayesian methodology presented by McConnachie \& Venn (2020), which takes into account the location of the stars on the sky, on the colour-magnitude diagram and on the proper motion plane. We apply some modifications, in particular to the way the colour-magnitude diagram and spectroscopic information are factored in, e.g. by including stars in several evolution phases. The bulk motions are integrated in three gravitational potentials: two where the Milky Way is treated in isolation and has a mass 0.9 \& 1.6 $\times 10^{12}$M$_{\odot}$ and the time-varying potential by Vasiliev et al. (2021), which includes the infall of a massive Large Magellanic Cloud (LMC). We are able to determine bulk proper motions for 73 systems, and we consider reliable 66 of these measurements. For the first time, systemic motions are presented for galaxies out to a distance of 1.4 Mpc, in the NGC~3109 association. The inclusion of the infall of a massive LMC significantly modifies the orbital trajectories of the objects, with respect to orbit integration in static Milky Way-only potentials, and leads to 6 galaxies being likely associated to the LMC, 3 possibly associated and 1 recently captured object. We discuss the results of the orbit integration in the context of the relation of the galaxies to the system of Milky Way satellites, implications for the too-big-to-fail problem, impact on star formation histories, and tidal disruption., Comment: Submitted to A&A. 47 pages

The nuclear stellar disc (NSD) is, together with the nuclear star cluster (NSC) and the central massive black hole, one of the main components in the central parts of our Milky Way. However, until recently, only few studies of the stellar content of the NSD have been obtained due to extreme extinction and stellar crowding. With a dedicated KMOS (VLT, ESO) spectroscopic survey, we study the kinematics and global metallicities of the NSD based on the observations of K/M giant stars. We trace radial velocities and metallicities which were derived based on spectral indices (Na I and CO) along the NSD and compare those with a Galactic Bulge sample of APOGEE (DR16) and data from the NSC. We find that the metallicity distribution function and the fraction of metal-rich and metal-poor stars in the NSD are different from the corresponding distributions and ratios of the NSC and the Galactic Bulge. By tracing the velocity dispersion as a function of metallicity, we clearly see that the NSD is kinematically cool and that the velocity dispersion decreases with increasing metallicity contrary to the inner Bulge sample of APOGEE ($\rm |b| < 4^{o}$). Using molecular gas tracers ($\rm H_{2}CO$, CO(4-3)) of the Central Molecular Zone (CMZ) we find an astonishing agreement between the gas rotation and the rotation of the metal-rich population indicating that the metal-rich stars could have formed from gas in the CMZ. On the other hand, the metal-poor stars show a much slower rotation profile with signs of counter-rotation indicating a different origin of these stars. Coupling kinematics with global metallicities, our results demonstrate that the NSD is chemically and kinematically distinct with respect to the inner Bulge indicating a different formation scenario., Comment: accepted for publication in A\&A

We present a new derivation of the star formation history (SFH) of the dSph galaxy Fornax in two central regions, characterized by unprecedented precision and age resolution. It reveals that star formation has proceeded in sharp bursts separated by periods of low level or quiescent activity. The SFH was derived through colour-magnitude diagram (CMD) fitting of two extremely deep Hubble Space Telescope CMDs, sampling the centre and one core radius. The attained age resolution allowed us to single out a major star formation episode at early times, a second strong burst 4.6 +/- 0.4 Gyr ago and recent intermittent episodes similar to 2-0.2 Gyr ago. Detailed testing with mock stellar populations was used to estimate the duration of the main bursts and study the occurrence of low-level star formation between them. The SFHs in both regions show common features, with activity at the same epochs and similar age-metallicity relationship. However, clear indications of a spatial gradient were also found, with mean age increasing with radius and star formation episodes being more prolonged in the centre. While some galaxy evolution models predict bursty SFHs in dwarf galaxies and thus a secular origin of the observed SFH cannot be excluded in Fornax, other evidence points to possible mergers or interactions as the cause of its bursty SFH. In particular, we calculated the Fornax orbit relative to the closest dwarfs and the Milky Way and observed a correspondence between the main intermediate-age and young events and peri-passages of Fornax around the Milky Way, possibly indicating tidally induced star formation.

Context. In the central few degrees of the bulge of the Milky Way there is a flattened structure of gas, dust, and stars, known as the central molecular zone, that is similar to nuclear disks in other galaxies. As a result of extreme foreground extinction, we possess only sparse information about the (mostly old) stellar population of the nuclear disk. Aims. In this work we present our KMOS spectroscopic survey of the stars in the nuclear disk reaching the old populations. To obtain an unbiased data set, we sampled stars in the full extinction range along each line of sight. Methods. We also observed reference fields in neighboring regions of the Galactic bulge. We describe the design and execution of the survey and present first results. Results. We obtain spectra and five spectral indices of 3113 stars with a median S/N of 67 and measure radial velocities for 3051 stars. Of those, 2735 sources have sufficient S/N to estimate temperatures and metallicities from indices. Conclusions. We derive metallicities using the CO 2-0 and Na I K-band spectral features, where we derive our own empirical calibration using metallicities obtained with higher-resolution observations. We use 183 giant stars for calibration spanning in metallicity from -2.5 to 0.6 dex and covering temperatures of up to 5500 K. The derived index based metallicities deviate from the calibration values with a scatter of 0.32 dex. The internal uncertainty of our metallicities is likely smaller. We use these metallicity measurements, together with the CO index, to derive effective temperatures using literature relations. We publish the catalog in this paper. Our data set complements Galactic surveys such as Gaia and APOGEE for the inner 200 pc radius of the Milky Way, which is not readily accessible by those surveys owing to extinction. We will use the derived properties in future papers for further analysis of the nuclear disk. © ESO 2021.

Context. The nuclear stellar disc (NSD) is, together with the nuclear star cluster (NSC) and the central massive black hole, one of the main components in the central parts of our Milky Way. However, until recently, only a few studies of the stellar content of the NSD have been obtained owing to extreme extinction and stellar crowding. Aims. We study the kinematics and global metallicities of the NSD based on the observations of K/M giant stars via a dedicated KMOS (VLT, ESO) spectroscopic survey. Methods. We traced radial velocities and metallicities, which were derived based on spectral indices (Na I and CO) along the NSD, and compared those with a Galactic bulge sample of APOGEE (DR16) and data from the NSC. Results. We find that the metallicity distribution function and the fraction of metal-rich and metal-poor stars in the NSD are different from the corresponding distributions and ratios of the NSC and the Galactic bulge. By tracing the velocity dispersion as a function of metallicity, we clearly see that the NSD is kinematically cool and that the velocity dispersion decreases with increasing metallicity contrary to the inner bulge sample of APOGEE (|b|< 4°). Using molecular gas tracers (H2CO, CO(4-3)) of the central molecular zone (CMZ), we find an astonishing agreement between the gas rotation and the rotation of the metal-rich population. This agreement indicates that the metal-rich stars could have formed from gas in the CMZ. On the other hand, the metal-poor stars show a much slower rotation profile with signs of counter-rotation, thereby indicating that these stars have a different origin. Conclusions. Coupling kinematics with global metallicities, our results demonstrate that the NSD is chemically and kinematically distinct with respect to the inner bulge, which indicates a different formation scenario. © M. Schultheis et al. 2021.

We present a new derivation of the star formation history (SFH) of the dSph galaxy Fornax in two central regions, characterized by unprecedented precision and age resolution. It reveals that star formation has proceeded in sharp bursts separated by periods of low level or quiescent activity. The SFH was derived through colour-magnitude diagram (CMD) fitting of two extremely deep Hubble Space Telescope CMDs, sampling the centre and one core radius. The attained age resolution allowed us to single out a major star formation episode at early times, a second strong burst 4.6 +/- 0.4 Gyr ago and recent intermittent episodes similar to 2-0.2 Gyr ago. Detailed testing with mock stellar populations was used to estimate the duration of the main bursts and study the occurrence of low-level star formation between them. The SFHs in both regions show common features, with activity at the same epochs and similar age-metallicity relationship. However, clear indications of a spatial gradient were also found, with mean age increasing with radius and star formation episodes being more prolonged in the centre. While some galaxy evolution models predict bursty SFHs in dwarf galaxies and thus a secular origin of the observed SFH cannot be excluded in Fornax, other evidence points to possible mergers or interactions as the cause of its bursty SFH. In particular, we calculated the Fornax orbit relative to the closest dwarfs and the Milky Way and observed a correspondence between the main intermediate-age and young events and peri-passages of Fornax around the Milky Way, possibly indicating tidally induced star formation.

We present a new derivation of the star formation history (SFH) of the dSph galaxy Fornax in two central regions, characterised by unprecedented precision and age resolution. It reveals that star formation has proceeded in sharp bursts separated by periods of low-level or quiescent activity. The SFH was derived through colour-magnitude diagram (CMD) fitting of two extremely deep Hubble Space Telescope CMDs, sampling the centre and one core radius. The attained age resolution allowed us to single out a major star formation episode at early times, a second strong burst $4.6\pm0.4$ Gyr ago and recent intermittent episodes $\sim2-0.2$ Gyr ago. Detailed testing with mock stellar populations was used to estimate the duration of the main bursts and study the occurrence of low-level star formation between them. The SFHs in both regions show common features, with activity at the same epochs and similar age-metallicity relationship. However, clear indications of a spatial gradient were also found, with mean age increasing with radius and star formation episodes being more prolonged in the centre. While some galaxy evolution models predict bursty SFHs in dwarf galaxies and thus a secular origin of the observed SFH cannot be excluded in Fornax, other evidence points to possible mergers or interactions as the cause of its bursty SFH. In particular, we calculated the Fornax orbit relative to the closest dwarfs and the Milky Way and observed a correspondence between the main intermediate-age and young events and peri-passages of Fornax around the Milky Way, possibly indicating tidally-induced star formation., Comment: 20 pages, 12+3 figures, accepted for publication in MNRAS; corrected error estimation (Figure 5, Sec. 3.3), added Sec. 4.3, appendices, more references, moderate revision of text and figures

We present a detailed study of the composition of 20 M giants in the Galactic center with 15 of them confirmed to be in the Nuclear Star Cluster. As a control sample we have also observed 7 M giants in the Milky Way Disk with similar stellar parameters. All 27 stars are observed using the NIRSPEC spectograph on the KECK II telescope in the K-band at a resolving power of R=23,000. We report the first silicon abundance trends versus [Fe/H] for stars in the Galactic center. While finding a disk/bulge like trend at subsolar metallicities, we find that [Si/Fe] is enhanced at supersolar metallicities. We speculate on possible enrichment scenarios to explain such a trend. However, the sample size is modest and the result needs to be confirmed by additional measurements of silicon and other \textalpha-elements. We also derive a new distribution of [Fe/H] and find the most metal rich stars at [Fe/H]=+0.5 dex, confirming our earlier conclusions that the Galactic center hosts no stars with extreme chemical composition., Comment: 13 pages, 10 figures, accepted for publication in ApJ, Figure 7 corrected

The Milky Way nuclear star cluster (MWNSC) is a crucial laboratory for studying the galactic nuclei of other galaxies, but its properties have not been determined unambiguously until now. Aims. We aim to study the size and spatial structure of the MWNSC. Methods. This study uses data and methods that address potential shortcomings of previous studies on the topic. We use $0.2''$ angular resolution $K_{s}$ data to create a stellar density map in the central $86.4$pc x $21$pc at the Galactic center. We include data from selected adaptive-optics-assisted images obtained for the inner parsecs. In addition, we use IRAC/Spitzer mid-infrared (MIR) images. We model the Galactic bulge and the nuclear stellar disk in order to subtract them from the MWNSC. Finally, we fit a S\'ersic model to the MWNSC and investigate its symmetry. Results. Our results are consistent with previous work. The MWNSC is flattened with an axis ratio of $q=0.71 \pm0.10$, an effective radius of $R_{e}=(5.1 \pm1.0)$ pc, and a S\'ersic index of $n=2.2 \pm0.7$. Its major axis may be tilted out of the Galactic plane by up to $-10$ degrees. The distribution of the giants brighter than the Red Clump (RC) is found to be significantly flatter than the distribution of the faint stars. We investigate the 3D structure of the central stellar cusp using our results on the MWNSC structure on large scales to constrain the deprojection of the measured stellar surface number density, obtaining a value of the 3D inner power law of $\gamma=1.38\pm0.06_{sys}\pm0.01_{stat}$. Conclusions. The MWNSC shares its main properties with other extragalactic NSCs found in spiral galaxies. The differences in the structure between bright giants and RC stars might be related to the existence of not completely mixed populations of different ages. This may hint at recent growth of the MWNSC through star formation or cluster accretion., Comment: 20 pages

We use Gaia DR2 systemic proper motions of 45 satellite galaxies to constrain the mass of the Milky Way using the scale free mass estimator of Watkins et al. (2010). We first determine the anisotropy parameter $\beta$, and the tracer satellites' radial density index $\gamma$ to be $\beta$=$-0.67^{+0.45}_{-0.62}$ and $\gamma=2.11\pm0.23$. When we exclude possible former satellites of the Large Magellanic Cloud, the anisotropy changes to $\beta$=$-0.21^{+0.37}_{-0.51}$. We find that the index of the Milky Way's gravitational potential $\alpha$, which is dependent on the mass itself, is the parameter with the largest impact on the mass determination. Via comparison with cosmological simulations of Milky Way-like galaxies, we carried out a detailed analysis of the estimation of the observational uncertainties and their impact on the mass estimator. We found that the mass estimator is biased when applied naively to the satellites of simulated Milky Way halos. Correcting for this bias, we obtain for our Galaxy a mass of $0.58^{+0.15}_{-0.14}\times10^{12}$M$_\odot$ within 64 kpc, as computed from the inner half of our observational sample, and $1.43^{+0.35}_{-0.32}\times10^{12}$M$_\odot$ within 273 kpc, from the full sample; this latter value extrapolates to a virial mass of $M_\mathrm{vir\,\Delta=97}$=$1.51^{+0.45}_{-0.40} \times 10^{12}M_{\odot}$ corresponding to a virial radius of R$_\mathrm{vir}$=$308\pm29$ kpc. This value of the Milky Way mass lies in-between other mass estimates reported in the literature, from various different methods., Comment: 17 pages, 10 figures, 2 tables. Accepted for publication in MNRAS