Power-law Mass Function Research Articles

A Volume-limited Sample of Ultracool Dwarfs. II. The Substellar Age and Mass Functions in the Solar Neighborhood

We present the most precise constraints to date for the mass and age distributions of single ultracool dwarfs in the solar neighborhood, based on an updated volume-limited sample of 504 L, T, and Y dwarfs within 25 pc. We develop a Monte Carlo approach using the 〈V/Vmax〉 statistic to correct for incompleteness and obtain a space density of (1.83−0.15+0.16)×10−2 pc−3 for spectral types L0–Y2. We calculate bolometric luminosities for our sample, using an updated “super-magnitude” method for the faintest objects. We use our resulting luminosity function and a likelihood-based population synthesis approach to simultaneously constrain the mass and age distributions. We employ the fraction of young L0–L7 dwarfs as a novel input for this analysis that is crucial for constraining the age distribution. For a power-law mass function dNdM∝M−α , we find α=0.58−0.20+0.16 , indicating an increase in numbers toward lower masses, consistent with measurements in nearby star-forming regions. For an exponential age distribution b(t) ∝ e −β t we find β = −0.44 ± 0.14, i.e., a population with fewer old objects than often assumed, which may reflect dynamical heating of the Galactic plane as much as the historical brown dwarf birthrate. We compare our analysis to that of Kirkpatrick et al., who used a similar volume-limited sample. Although our mass function measurements are numerically consistent, their assumption of a flat age distribution is disfavored by our analysis, and we identify several important methodological differences between our two studies. Our calculation of the age distribution of solar neighborhood brown dwarfs is the first based on a volume-limited sample.

Testing Primordial Black Hole Dark Matter with Atacama Large Millimeter Array Observations of the Gravitational Lens B1422+231

We examine the flux density ratio anomaly in the quadruply imaged strong gravitational lens, B1422+231, and consider the contribution of 10–103M⊙ primordial black holes (PBHs) as a potential dark matter constituent. We describe the first flux density ratio measurement of B1422+231 in the millimeter-wave band using the Atacama Large Millimeter Array (ALMA). The flux density of the quasar at 233 GHz is dominated by synchrotron emission and the source size is estimated to be less than 66.9 pc. The observed flux density ratios at 233 GHz are similar to those measured in other wave bands, which cannot be explained by a simple smooth mass model of the lens galaxy. We examine the probability of the flux density ratio anomaly arising from PBH microlensing using ray tracing simulations. The simulations consider the cases where 10% and 50% of dark matter are 10–103M⊙ PBHs with a power law mass function. The simulated scenarios are consistent with the ALMA observations, so PBH dark matter cannot be ruled out as a cause of flux density ratio anomalies. Our analysis shows that the anomalous flux density ratio for B1422+231 can be explained by a lens model with a significant fraction of dark matter being PBHs. This study demonstrates the potential for new constraints on PBH dark matter using ALMA observations of multiply imaged strong gravitational lenses.

Pre-supernova feedback sets the star cluster mass function to a power law and reduces the cluster formation efficiency

Context. The star cluster initial mass function is observed to have an inverse power law exponent around 2, yet there is no consensus on what determines this distribution, and why some variation is observed in different galaxies. Furthermore, the cluster formation efficiency (CFE) covers a range of values, particularly when considering different environments. These clusters are often used to empirically constrain star formation and as fundamental units for stellar feedback models. Detailed galaxy models must therefore accurately capture the basic properties of observed clusters to be considered predictive. Aims. We study how feedback mechanisms acting on different timescales and with different energy budgets affect the star cluster mass function and CFE. Methods. We use hydrodynamical simulations of a dwarf galaxy as a laboratory to study star cluster formation. We test different combinations of stellar feedback mechanisms, including stellar winds, ionizing radiation, and supernovae (SNe). Results. Each feedback mechanism affects the CFE and cluster mass function. Increasing the feedback budget by combining the different types of feedback decreases the CFE by reducing the number of massive clusters. Ionizing radiation is found to be especially influential. This effect depends on the timing of feedback initiation, as shown by comparing early and late feedback. Early feedback occurs from ionizing radiation and stellar winds with onset immediately after a massive star is formed. Late feedback occurs when energy injection only starts after the main-sequence lifetime of the most massive SN progenitor, a timing that is further influenced by the choice of the most massive SN progenitor. Late feedback alone results in a broad, flat mass function, approaching a log-normal shape in the complete absence of feedback. Early feedback, on the other hand, produces a power-law cluster mass function with lower CFE, albeit with a steeper slope than that usually observed.

Free-floating Planet Mass Function from MOA-II 9 yr Survey toward the Galactic Bulge

We present the first measurement of the mass function of free-floating planets (FFPs), or very wide orbit planets down to an Earth mass, from the MOA-II microlensing survey in 2006–2014. Six events are likely to be due to planets with Einstein radius crossing times t E < 0.5 days, and the shortest has t E = 0.057 ± 0.016 days and an angular Einstein radius of θ E = 0.90 ± 0.14 μas. We measure the detection efficiency depending on both t E and θ E with image-level simulations for the first time. These short events are well modeled by a power-law mass function, dex−1 star−1 with for M/M ⊙ < 0.02. This implies a total of FFPs or very wide orbit planets of mass 0.33 < M/M ⊕ < 6660 per star, with a total mass of star−1. The number of FFPs is times the number of planets in wide orbits (beyond the snow line), while the total masses are of the same order. This suggests that the FFPs have been ejected from bound planetary systems that may have had an initial mass function with a power-law index of α ∼ 0.9, which would imply a total mass of star−1. This model predicts that Roman Space Telescope will detect FFPs with masses down to that of Mars (including with 0.1 ≤ M/M ⊕ ≤ 1). The Sumi et al. large Jupiter-mass FFP population is excluded.

Constraints on primordial-black-hole population and cosmic expansion history from GWTC-3

Gravitational waves (GWs) from compact binary coalescences provide an independent probe of the cosmic expansion history other than electromagnetic waves. In this work, we assume the binary black holes (BBHs) detected by LIGO-Virgo-KAGRA (LVK) collaborations are of primordial origin and constrain the population parameters of primordial black holes (PBHs) and Hubble parameter H(z) using 42 BBHs from third LVK GW transient catalog (GWTC-3). Three PBH mass models are considered: lognormal, power-law, and critical collapse PBH mass functions. By performing a hierarchical Bayesian population analysis, the Bayes factor strongly disfavors the power-law PBH mass function against the other two in GWTC-3. The constraints on standard Λ CDM cosmological parameters are rather weak and in agreement with current results. When combining the multi-messenger standard siren measurement from GW170817, the Hubble constant H 0 is constrained to be 69-8 +19 km s-1 Mpc-1 and 70-8 +26 km s-1 Mpc-1 at 68% confidence for the lognormal and critical collapse mass models, respectively. Furthermore, we consider a mixed ABH+PBH model, in which we assume LVK BBHs can come from both the astrophysical black hole (ABH) and PBH channels. We find that the ABH+PBH model can better describe the mass distribution in GWTC-3 than any single ABH or PBH model, thus improving the precision to constrain the Hubble constant. With the increased BBH events, the mixed ABH+PBH model provides a robust statistical inference for both the population and cosmological parameters.

Evidence for a bottom-light initial mass function in massive star clusters

ABSTRACT We have determined stellar mass functions of 120 Milky Way globular clusters and massive Large Magellanic Cloud/Small Magellanic Cloud star clusters based on a comparison of archival Hubble Space Telescope photometry with a large grid of direct N-body simulations. We find a strong correlation of the global mass function slopes of star clusters with both their internal relaxation times and their lifetimes. Once dynamical effects are being accounted for, the mass functions of most star clusters are compatible with an initial mass function described by a broken power-law distribution N(m) ∼ mα with break masses at 0.4 and 1.0 M⊙ and mass function slopes of αLow = −0.3 for stars with masses m &amp;lt; 0.4 M⊙, αHigh = −2.30 for stars with m &amp;gt; 1.0 M⊙, and αMed = −1.65 for intermediate-mass stars. Alternatively, a lognormal mass function with a characteristic mass log MC = −0.36 and width σC = 0.28 for low-mass stars and a power-law mass function for stars with m &amp;gt; 1 M⊙ also fit our data. We do not find a significant environmental dependence of the initial mass function on cluster mass, density, global velocity dispersion, or metallicity. Our results lead to a larger fraction of high-mass stars in globular clusters compared to canonical Kroupa/Chabrier mass functions, increasing the efficiency of self-enrichment in clusters and helping to alleviate the mass budget problem of multiple stellar populations in globular clusters. By comparing our results with direct N-body simulations, we finally find that only simulations in which most black holes are ejected by natal birth kicks correctly reproduce the observed correlations.

Detecting Isolated Stellar-mass Black Holes with the Roman Telescope

Isolated stellar-mass black holes (ISMBHs) are potentially discernible through microlensing observations, because they are expected to be long-duration microlensing events. In this work, we study the detection and characterization of ISMBHs using Roman observations. We simulate a large ensemble of such events, as they will be seen by Roman, and estimate the errors in the physical parameters of the lens objects, including their masses, distances, and proper motions, by calculating Fisher and covariance matrices. Since the ∼2.3 yr time gap between Roman’s first three observing seasons and its latter three seasons may lower the efficiency of its realization of microlensing events and characterization of ISMBHs, we additionally consider a scenario where we add a small number of additional observations—1 hr of observations, every 10 days, when the Bulge is observable during the large time gap—which is equivalent to a total of an additional day of observations with the Roman telescope. These extra observations increase Roman’s efficiency in terms of characterizing ISMBHs by ∼1%–2%, and, more importantly, they improve the robustness of the results, by avoiding possible degenerate solutions. By considering uniform and power-law mass functions (, α = 2, 1, 0.5) for ISMBHs in the range of [2, 50]M ⊙, we conclude that the Roman telescope will determine the physical parameters of the lenses (the mass, distance, and relative lens–source angular velocity) within <5% uncertainty, with efficiencies of 21% and 16%–18%, respectively. By considering these mass functions, we expect that during its mission the Roman telescope will detect and characterize 3–4, 15–17, and 22–24 ISMBHs through astrometric microlensing, with the relative errors for all physical parameters being less than 1%, 5%, and 10%, respectively. Microlensing events due to ISMBHs with masses ≃10–25M ⊙ that are located close to the observer, with D l ≲ 0.5D s, while the source is inside the Galactic disk, can be characterized with the fewest errors.

Beyond the Local Volume. II. Population Scaleheights and Ages of Ultracool Dwarfs in Deep HST/WFC3 Parallel Fields

Ultracool dwarfs (UCDs) represent a significant proportion of stars in the Milky Way, and deep samples of these sources have the potential to constrain the formation history and evolution of low-mass objects in the Galaxy. Until recently, spectral samples have been limited to the local volume (d < 100 pc). Here, we analyze a sample of 164 spectroscopically characterized UCDs identified by Aganze et al. in the Hubble Space Telescope (HST) WFC3 Infrared Spectroscopic Parallel Survey (WISPS) and 3D-HST. We model the observed luminosity function using population simulations to place constraints on scaleheights, vertical velocity dispersions, and population ages as a function of spectral type. Our star counts are consistent with a power-law mass function and constant star formation history for UCDs, with vertical scaleheights of 249 pc for late-M dwarfs, 153 pc for L dwarfs, and 175 pc for T dwarfs. Using spatial and velocity dispersion relations, these scaleheights correspond to disk population ages of 3.6 Gyr for late-M dwarfs, 2.1 Gyr for L dwarfs, and 2.4 Gyr for T dwarfs, which are consistent with prior simulations that predict that L-type dwarfs are on average a younger and less dispersed population. There is an additional 1–2 Gyr systematic uncertainty on these ages due to variances in age-velocity relations. We use our population simulations to predict the UCD yield in the James Webb Space Telescope PASSAGES survey, a similar and deeper survey to WISPS and 3D-HST, and find that it will produce a comparably sized UCD sample, albeit dominated by thick disk and halo sources.

A possible mass distribution of primordial black holes implied by LIGO-Virgo

The LIGO-Virgo Collaboration has so far detected around 90 black holes, some of which have masses larger than what were expected from the collapse of stars. The mass distribution of LIGO-Virgo black holes appears to have a peak at ∼ 30M ☉ and two tails on the ends. By assuming that they all have a primordial origin, we analyze the GWTC-1 (O1&O2) and GWTC-2 (O3a) datasets by performing maximum likelihood estimation on a broken power law mass function f(m), with the result f ∝ m 1.2 for m < 35 M ☉ and f ∝ m -4 for m > 35 M ☉. This appears to behave better than the popular log-normal mass function. Surprisingly, such a simple and unique distribution can be realized in our previously proposed mechanism of PBH formation, where the black holes are formed by vacuum bubbles that nucleate during inflation via quantum tunneling. Moreover, this mass distribution can also provide an explanation to supermassive black holes formed at high redshifts.

Towards a fully consistent Milky Way disk model

Aims.We present an updated version of the semi-analytic Just-Jahrei (JJ) model of the Galactic disk and constrain its parameters in the Solar neighbourhood.Methods.The new features of the JJ model include a simple two-component gaseous disk, a star-formation rate (SFR) function of the thick disk that has been extended in time, and a correlation between the kinematics of molecular gas and thin-disk populations. Here, we study the vertical number density profiles andW-velocity distributions determined from ∼2 × 106local stars of the secondGaiadata release (DR2). We also investigate an apparent Hess diagram of theGaiaDR2 stars selected in a conic volume towards the Galactic poles. Using a stellar evolution library, we synthesise stellar populations with a four-slope broken power-law initial mass function, the SFR, and an age-metallicity relation. The latter is consistently derived with the observed metallicity distribution of the local Red Clump giants from the Apache Point Observatory Galactic Evolution Experiment (APOGEE). Working within a Bayesian approach, we sample the posterior probability distribution in a multidimensional parameter space using the Markov chain Monte Carlo method.Results.We find that the spatial distribution and motion of theGaiaDR2 stars imply two recent SF bursts centered at ages of ∼0.5 Gyr and ∼3 Gyr and characterised by a ∼30% and ∼55% SF enhancement, respectively, relative to a monotonously declining SFR continuum. The stellar populations associated with this SF excess are found to be dynamically hot for their age: they haveW-velocity dispersions of ∼12.5 km s−1and ∼26 km s−1. The new JJ model is able to reproduce the local star counts with an accuracy of ∼5%.Conclusions.UsingGaiaDR2 data, we self-consistently constrained 22 parameters of the updated JJ model. Our optimised model predicts two SF bursts within the last ∼4 Gyr, which may point to recent episodes of gas infall.

Two populations of LIGO-Virgo black holes

We analyse the LIGO-Virgo data, including the recently released GWTC-2dataset, to test a hypothesis that the data contains more than one population of black holes. We perform a maximum likelihood analysis including a population of astrophysical black holes with a truncated power-law mass function whose merger rate follows from star formation rate, and a population of primordial black holes for which we consider log-normal and critical collapse mass functions. We find that primordial black holes alone are strongly disfavoured by the data, while the best fit is obtained for the template combining astrophysical and primordial merger rates. Alternatively, the data may hint towards two different astrophysical black hole populations. We also update the constraints on primordial black hole abundance from LIGO-Virgo observations finding that in the 2–400 mass range they must comprise less than 0.2% of dark matter.

Mass–radius relation of intermediate-age disc super star clusters of M82

ABSTRACT We present a complete set of structural parameters for a sample of 99 intermediate-age super star cluster (SSCs) in the disc of M82, and carry out a survival analysis using the semi-analytical cluster evolution code emacss. The parameters are based on the profile-fitting analysis carried out in previous work, with the mass-related quantities derived using a mass-to-light ratio for a constant age of 100 Myr. The SSCs follow a power-law mass function with an index α = 1.5, and a lognormal size function with a typical half-light radius, Rh = 4.3 pc, which is both comparable with the values for clusters in the Magellanic Clouds, rather than in giant spirals. The majority of the SSCs follow a power-law mass−radius relation with an index of b = 0.29 ± 0.05. A dynamical analysis of M82 SSCs using emacss suggests that 23 per cent of the clusters are tidally limited, with the rest undergoing expansion at present. Forward evolution of these clusters suggests that the majority would dissolve in ∼2 Gyr. However, a group of four massive compact clusters, and another group of five SSCs at relatively large galactocentric distances, are found to survive for a Hubble time. The model-predicted mass, Rh, μV, and core radius of these surviving SSCs at 12 Gyr are comparable with the corresponding values for the sample of Galactic globular clusters.

Using the Modified Lognormal Power-law Distribution to Model the Mass Function of NGC 1711

Abstract A determination of the mass function (MF) of stellar clusters can be quite dependent on the range of measured masses, the fitting technique, and the analytic function that is being fit to the data. Here, we use Hubble Space Telescope/WFPC2 data of NGC 1711, a stellar cluster in the Large Magellanic Cloud, as a test case to explore a range of possible determinations of the MF from a single data set. We employ the analytic modified lognormal power-law (MLP) distribution, a hybrid function that has a peaked lognormal-like body and a power-law tail at intermediate and high masses. A fit with the MLP has the advantage that the resulting best-fit function can be either a hybrid function, a pure lognormal, or a pure power law, in different limits of the function. The completeness limit for the observations means that the data contains masses above ∼0.90 M ⊙. In this case, the MLP fits yield essentially a pure power-law MF. We demonstrate that the nonlinear regression/least-squares approach is not justified since the underlying assumptions are not satisfied. By using maximum-likelihood estimation, which is independent of binning, we find a best-fit functional form , where α = 1.72 ± 0.05 or 1.75 ± 0.05 for two different theoretical isochrone models, respectively. Furthermore, we explore the possibility of systematic errors in the determination of the power-law index due to the depth of the observations. When we combine the observational data with artificially generated data from the lognormal Chabrier initial MF for masses below 0.90M ⊙, the best-fit MLP is a hybrid function but with a steeper asymptotic slope i.e., α = 2.04 ± 0.07. This illustrates the systematic uncertainties in commonly used MF parameters that can depend on the range of data that is fitted.

Self-consistent proto-globular cluster formation in cosmological simulations of high-redshift galaxies

ABSTRACT We report the formation of bound star clusters in a sample of high-resolution cosmological zoom-in simulations of z ≥ 5 galaxies from the Feedback In Realistic Environments project. We find that bound clusters preferentially form in high-pressure clouds with gas surface densities over $10^4\, \mathrm{ M}_{\odot }\, {\rm pc}^{-2}$, where the cloud-scale star formation efficiency is near unity and young stars born in these regions are gravitationally bound at birth. These high-pressure clouds are compressed by feedback-driven winds and/or collisions of smaller clouds/gas streams in highly gas-rich, turbulent environments. The newly formed clusters follow a power-law mass function of dN/dM ∼ M−2. The cluster formation efficiency is similar across galaxies with stellar masses of ∼107–$10^{10}\, \mathrm{ M}_{\odot }$ at z ≥ 5. The age spread of cluster stars is typically a few Myr and increases with cluster mass. The metallicity dispersion of cluster members is ∼0.08 dex in $\rm [Z/H]$ and does not depend on cluster mass significantly. Our findings support the scenario that present-day old globular clusters (GCs) were formed during relatively normal star formation in high-redshift galaxies. Simulations with a stricter/looser star formation model form a factor of a few more/fewer bound clusters per stellar mass formed, while the shape of the mass function is unchanged. Simulations with a lower local star formation efficiency form more stars in bound clusters. The simulated clusters are larger than observed GCs due to finite resolution. Our simulations are among the first cosmological simulations that form bound clusters self-consistently in a wide range of high-redshift galaxies.

Modelling the Effects of Dark Matter Substructure on Globular Cluster Evolution with the Tidal Approximation

Abstract We present direct N-body simulations of tidally filling 30,000 M⊙ star clusters orbiting between 10 kpc and 100 kpc in galaxies with a range of dark matter substructure properties. The time-dependent tidal force is determined based on the combined tidal tensor of the galaxy’s smooth and clumpy dark matter components, the latter of which causes fluctuations in the tidal field that can heat clusters. The strength and duration of these fluctuations are sensitive to the local dark matter density, substructure fraction, sub-halo mass function, and the sub-halo mass-size relation. Based on the cold dark matter framework, we initially assume sub-halos are Hernquist spheres following a power-law mass function between 105 and 1011M⊙ and find that tidal fluctuations are too weak and too short to affect star cluster evolution. Treating sub-halos as point masses, to explore how denser sub-halos affect clusters, we find that only sub-halos with masses greater than 106M⊙ will cause cluster dissolution times to decrease. These interactions can also decrease the size of a cluster while increasing the velocity dispersion and tangential anisotropy in the outer regions via tidal heating. Hence increased fluctuations in the tidal tensor, especially fluctuations that are due to low-mass halos, do not necessarily translate into mass loss. We further conclude that the tidal approximation can be used to model cluster evolution in the tidal fields of cosmological simulations with a minimum cold dark matter sub-halo mass of 106M⊙, as the effect of lower-mass sub-halos on star clusters is negligible.

Simulating star clusters across cosmic time – I. Initial mass function, star formation rates, and efficiencies

ABSTRACT We present radiation-magneto-hydrodynamic simulations of star formation in self-gravitating, turbulent molecular clouds, modelling the formation of individual massive stars, including their UV radiation feedback. The set of simulations have cloud masses between mgas = 103 M⊙ and 3 × 105 M⊙ and gas densities typical of clouds in the local Universe ($\overline{n}_{\rm gas} \sim 1.8\times 10^2$ cm−3) and 10× and 100× denser, expected to exist in high-redshift galaxies. The main results are as follows. (i) The observed Salpeter power-law slope and normalization of the stellar initial mass function at the high-mass end can be reproduced if we assume that each star-forming gas clump (sink particle) fragments into stars producing on average a maximum stellar mass about $40{{\ \rm per\ cent}}$ of the mass of the sink particle, while the remaining $60{{\ \rm per\ cent}}$ is distributed into smaller mass stars. Assuming that the sinks fragment according to a power-law mass function flatter than Salpeter, with log-slope 0.8, satisfy this empirical prescription. (ii) The star formation law that best describes our set of simulation is ${\rm d}\rho _*/{\rm d}t \propto \rho _{\rm gas}^{1.5}$ if $\overline{n}_{\rm gas}\lt n_{\rm cri}\approx 10^3$ cm−3, and ${\rm d}\rho _*/{\rm d}t \propto \rho _{\rm gas}^{2.5}$ otherwise. The duration of the star formation episode is roughly six cloud’s sound crossing times (with cs = 10 km s−1). (iii) The total star formation efficiency in the cloud is $f_*=2{{\ \rm per\ cent}} (m_{\rm gas}/10^4~\mathrm{M}_\odot)^{0.4}(1+\overline{n}_{\rm gas}/n_{\rm cri})^{0.91}$, for gas at solar metallicity, while for metallicity Z &amp;lt; 0.1 Z⊙, based on our limited sample, f* is reduced by a factor of ∼5. (iv) The most compact and massive clouds appear to form globular cluster progenitors, in the sense that star clusters remain gravitationally bound after the gas has been expelled.

Constraints on Upper Cutoffs in the Mass Functions of Young Star Clusters

Abstract We test claims that the power-law mass functions of young star clusters (ages ≲ few × 108 yr) have physical upper cutoffs at . Specifically, we perform maximum likelihood fits of the Schechter function, , to the observed cluster masses in eight well-studied galaxies (LMC, SMC, NGC 4214, NGC 4449, M83, M51, Antennae, and NGC 3256). In most cases, we find that a wide range of cutoff masses is permitted ( ). We find a weak detection at in one case (M51) and strong evidence against this value in two cases. However, when we include realistic errors in cluster masses in our analysis, the constraints on M * become weaker and there are no significant detections (even for M51). Our data are generally consistent with much larger cutoffs, at . This is the predicted cutoff from dynamical models in which old globular clusters and young clusters observed today formed by similar physical processes with similar initial mass functions.

Local disc model in view of Gaia DR1 and RAVE data

Aims. We test the performance of the semi-analytic self-consistent Just-Jahreiß disc model (JJ model) with the astrometric data from the Tycho-Gaia Astrometric Solution (TGAS) sub-catalogue of the first Gaia data release (Gaia DR1), as well as the radial velocities from the fifth data release of the Radial Velocity Experiment survey (RAVE DR5). Methods. We used a sample of 19 746 thin-disc stars from the TGAS×RAVE cross-match selected in a local solar cylinder of 300 pc radius and 1 kpc height below the Galactic plane. Based on the JJ model, we simulated this sample via the forward modelling technique. First, we converted the predicted vertical density laws of the thin-disc populations into a mock sample. For this we used the Modules and Experiments in Stellar Astrophysics (MESA) Isochrones and Stellar Tracks (MIST), a star formation rate (SFR) that decreased after a peak at 10 Gyr ago, and a three-slope broken power-law initial mass function (IMF). Then the obtained mock populations were reddened with a 3D dust map and were subjected to the selection criteria corresponding to the RAVE and TGAS observational limitations as well as to additional cuts applied to the data sample. We calculated the quantities of interest separately at different heights above the Galactic plane, taking the distance error effects in horizontal and vertical directions into account separately. Results. The simulated vertical number density profile agrees well with the data. An underestimation of the stellar numbers begins at ∼800 pc from the Galactic plane, which is expected as the possible influence of populations from |z| &gt; 1 kpc is ignored during the modelling. The lower main sequence (LMS) is found to be thinner and under-populated by 3.6% relative to the observations. The corresponding deficits for the upper main sequence (UMS) and red giant branch (RGB) are 6% and 34.7%, respectively. However, the intrinsic uncertainty related to the choice of stellar isochrones is ∼10% in the total stellar number. The vertical velocity distribution function f(|W|) simulated for the whole cylinder agrees to within 1σ with the data. This marginal agreement arises because the dynamically cold populations at heights &lt; 200 pc from the Galactic plane are underestimated. We also find that the model gives a fully realistic representation of the vertical gradient in stellar populations when studying the Hess diagrams for different horizontal slices. We also checked and confirm the consistency of our results with the newly available second Gaia data release (DR2). Conclusions. Based on these results and considering the uncertainties in the data selection as well as the sensitivity of the simulations to the sample selection function, we conclude that the fiducial JJ model confidently reproduces the vertical trends in the thin-disc stellar population properties. Thus, it can serve as a starting point for the future extension of the JJ model to other Galactocentric distances.

The Role of Gravity in Producing Power-law Mass Functions

Abstract Numerical simulations of star formation have found that a power-law mass function can develop at high masses. In a previous paper, we employed isothermal simulations that created large numbers of sinks over a large range in masses to show that the power-law exponent of the mass function, , asymptotically and accurately approaches Γ = −1. Simple analytic models show that such a power law can develop if the mass accretion rate , as in Bondi–Hoyle accretion; however, the sink mass accretion rates in the simulations show significant departures from this relation. In this paper, we show that the expected accretion rate dependence is more closely realized provided the gravitating mass is taken to be the sum of the sink mass and the mass in the near environment. This reconciles the observed mass functions with the accretion rate dependencies, and demonstrates that power-law upper mass functions are essentially the result of gravitational focusing, a mechanism present in, for example, the competitive accretion model.

Mean proper motions, space orbits, and velocity dispersion profiles of Galactic globular clusters derived fromGaiaDR2 data

We have derived the mean proper motions and space velocities of 154 Galactic globular clusters and the velocity dispersion profiles of 141 globular clusters based on a combination of Gaia DR2 proper motions with ground-based line-of-sight velocities. Combining the velocity dispersion profiles derived here with new measurements of the internal mass functions allows us to model the internal kinematics of 144 clusters, more than 90% of the currently known Galactic globular cluster population. We also derive the initial cluster masses by calculating the cluster orbits backwards in time applying suitable recipes to account for mass loss and dynamical friction. We find a correlation between the stellar mass function of a globular cluster and the amount of mass lost from the cluster, pointing to dynamical evolution as one of the mechanisms shaping the mass function of stars in clusters. The mass functions also show strong evidence that globular clusters started with a bottom-light initial mass function. Our simulations show that the currently surviving globular cluster population has lost about 80% of its mass since the time of formation. If globular clusters started from a log-normal mass function, we estimate that the Milky Way contained about 500 globular clusters initially, with a combined mass of about $2.5 \cdot 10^8$ $M_\odot$. For a power-law initial mass function, the initial mass in globular clusters could have been a factor of three higher.