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Register a new userMeasuring the mass distribution in stellar systems
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Scott Tremaine
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One of the fundamental tasks of dynamical astronomy is to infer the distribution of mass in a stellar system from a snapshot of the positions and velocities of its stars. The usual approach to this task (e.g., Schwarzschilds method) involves fitting parametrized forms of the gravitational potential and the phase-space distribution to the data. We review the practical and conceptual difficulties with this approach and describe a novel statistical method for determining the mass distribution that does not require determining the phase-space distribution of the stars. We show that this new estimator out-performs other distribution-free estimators for the harmonic and Kepler potentials.
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Cluster mass profiles are tests of models of structure formation. Only two current observational methods of determining the mass profile, gravitational lensing and the caustic technique, are independent of the assumption of dynamical equilibrium. Both techniques enable determination of the extended mass profile at radii beyond the virial radius. For 19 clusters, we compare the mass profile based on the caustic technique with weak lensing measurements taken from the literature. This comparison offers a test of systematic issues in both techniques. Around the virial radius, the two methods of mass estimation agree to within about 30%, consistent with the expected errors in the individual techniques. At small radii, the caustic technique overestimates the mass as expected from numerical simulations. The ratio between the lensing profile and the caustic mass profile at these radii suggests that the weak lensing profiles are a good representation of the true mass profile. At radii larger than the virial radius, the lensing mass profile exceeds the caustic mass profile possibly as a result of contamination of the lensing profile by large-scale structures within the lensing kernel. We highlight the case of the closely neighboring clusters MS0906+11 and A750 to illustrate the potential seriousness of contamination of the the weak lensing signal by unrelated structures.
In this chapter, we review some historical understanding and recent advances on the Initial Mass Function (IMF) and the Core Mass Function (CMF), both in terms of observations and theories. We focus mostly on star formation in clustered environment since this is suggested by observations to be the dominant mode of star formation. The statistical properties and the fragmentation behaviour of turbulent gas is discussed, and we also discuss the formation of binaries and small multiple systems.
Mass-to-light versus colour relations (MLCRs), derived from stellar population synthesis models, are widely used to estimate galaxy stellar masses (M$_*$) yet a detailed investigation of their inherent biases and limitations is still lacking. We quantify several potential sources of uncertainty, using optical and near-infrared (NIR) photometry for a representative sample of nearby galaxies from the Virgo cluster. Our method for combining multi-band photometry with MLCRs yields robust stellar masses, while errors in M$_*$ decrease as more bands are simultaneously considered. The prior assumptions in ones stellar population modelling dominate the error budget, creating a colour-dependent bias of up to 0.6 dex if NIR fluxes are used (0.3 dex otherwise). This matches the systematic errors associated with the method of spectral energy distribution (SED) fitting, indicating that MLCRs do not suffer from much additional bias. Moreover, MLCRs and SED fitting yield similar degrees of random error ($sim$0.1-0.14 dex) when applied to mock galaxies and, on average, equivalent masses for real galaxies with M$_* sim$ 10$^{8-11}$ M$_{odot}$. The use of integrated photometry introduces additional uncertainty in M$_*$ measurements, at the level of 0.05-0.07 dex. We argue that using MLCRs, instead of time-consuming SED fits, is justified in cases with complex model parameter spaces (involving, for instance, multi-parameter star formation histories) and/or for large datasets. Spatially-resolved methods for measuring M$_*$ should be applied for small sample sizes and/or when accuracies less than 0.1 dex are required. An Appendix provides our MLCR transformations for ten colour permutations of the $grizH$ filter set.
Halo stars orbit within the potential of the Milky Way and hence their kinematics can be used to understand the underlying mass distribution. However, the inferred mass distribution depends sensitively upon assumptions made on the density and the velocity anisotropy profiles of the tracers. Also, there is a degeneracy between the parameters of the halo and that of the disk or bulge. Here, we decompose the Galaxy into bulge, disk and dark matter halo and then model the kinematic data of the halo BHB and K-giants from the SEGUE. Additionally, we use the gas terminal velocity curve and the Sgr A$^*$ proper motion. With $R_odot = 8.5$kpc, our study reveals that the density of the stellar halo has a break at $17.2^{+1.1}_{-1.0}$ kpc, and an exponential cut-off in the outer parts starting at $97.7^{+15.6}_{-15.8}$kpc. Also, we find the velocity anisotropy is radially biased with $beta_s= 0.4pm{0.2}$ in the outer halo. We measure halo virial mass $M_{text{vir}} = 0.80^{+0.31}_{-0.16} times 10^{12} M_{odot}$, concentration $c=21.1^{+14.8}_{-8.3}$, disk mass of $0.95^{+0.24}_{-0.30}times10^{11} M_{odot}$, disk scale length of $4.9^{+0.4}_{-0.4}$ kpc and bulge mass of $0.91^{+0.31}_{-0.38} times 10^{10} M_{odot}$. The mass of halo is found to be small and this has important consequences. The giant stars reveal that the outermost halo stars have low velocity dispersion interestingly suggesting a truncation of the stellar halo density rather than a small overall mass of the Galaxy. Our estimates of local escape velocity $v_{rm esc} = 550.9^{+32.4}_{-22.1}$ kms$^{-1}$ and dark matter density $rho^{rm DM}_{odot} = 0.0088^{+0.0024}_{-0.0018} M_{odot} {rm pc^{-3}} $ ($0.35^{+0.08}_{-0.07}$ GeV cm$^{-3}$) are in good agreement with recent estimates. Some of the above estimates are depended on the adopted value of $R_odot$ and of outer power-law index of the tracer number density.
Most globular clusters have half-mass radii of a few pc with no apparent correlation with their masses. This is different from elliptical galaxies, for which the Faber-Jackson relation suggests a strong positive correlation between mass and radius. Objects that are somewhat in between globular clusters and low-mass galaxies, such as ultra-compact dwarf galaxies, have a mass-radius relation consistent with the extension of the relation for bright ellipticals. Here we show that at an age of 10 Gyr a break in the mass-radius relation at ~10^6 Msun is established because objects below this mass, i.e. globular clusters, have undergone expansion driven by stellar evolution and hard binaries. From numerical simulations we find that the combined energy production of these two effects in the core comes into balance with the flux of energy that is conducted across the half-mass radius by relaxation. An important property of this `balanced evolution is that the cluster half-mass radius is independent of its initial value and is a function of the number of bound stars and the age only. It is therefore not possible to infer the initial mass-radius relation of globular clusters and we can only conclude that the present day properties are consistent with the hypothesis that all hot stellar systems formed with the same mass-radius relation and that globular clusters have moved away from this relation because of a Hubble time of stellar and dynamical evolution.
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