Inferring spherical mass distributions using the projected mass estimator

Various workers have applied the projected mass estimator (PME) to infer the radial mass distribution M(r) in stellar systems ranging in scale from the Galactic center to dwarf spheroidals galaxies and globular cluster systems of external galaxies. The PME was originally used to infer the total mass of stellar systems according to the relation M = f〈ν_z,i²r_⊥,i〉/G, where the factor f depends on the stellar orbit distribution (e.g., isotropic) and how coextensive the mass distribution is compared to the tracer population. We examine the general expression of the PME for a spherically symmetric mass distribution and an arbitrary sampling volume. For a cylinder centered on the distribution, which corresponds to the observational case of evaluating the PME within apertures of increasing radius R, boundary terms that arise from the finite sampling volume make appreciable contributions when R → 0. Numerical calculations and Monte Carlo simulations demonstrate that the PME can overestimate M(r) by factors of at least 3-4 inside the core of a self-gravitating distribution. More importantly, the functional form of the PME as R → 0 is attributable to the normalization f = f(R), not the function M(r). Analytic "η-models" show that the PME can infer the presence of a compact object at the center of a stellar distribution only when its mass greatly exceeds the mass of the cluster. Attempts to overcome volume incompleteness by computing the PME within a series of concentric annuli are also examined. The precise term that gives rise to M(r) in the case of spherical sampling cancels out when the tracer is sampled in concentric annuli. What is measured is an average of M(r) along the line of sight, weighted by the local tracer density and velocity dispersion gradients. The PME often yields results that resemble M(r), but this is not precisely what is being measured. The generalized estimator can be compared with observations if one has some model of the mass distribution and the tracer population.