Inferring Spherical Mass Distributions Using the Projected Mass Estimator: Possibility or Pitfall?

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Session 22 -- Galactic Structure; Galactic Center
Display presentation, Monday, 9, 1995, 9:20am - 6:30pm

[22.07] Inferring Spherical Mass Distributions Using the Projected Mass Estimator: Possibility or Pitfall?

J.W. Haller, F. Melia (Dept. of Physics and SO, U. of Arizona)

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, dwarf spheroidals, and globular cluster systems of external galaxies. The PME was originally used to infer the total mass of a stellar system according to the relation $M=f\ \times/G$ where the factor $f$ depends on the stellar orbit distribution (e.g., isotropic) and how co-extensive the mass distribution is compared to the tracer population. We here 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, corresponding to the observational case of evaluating the PME within apertures of increasing radius $R$, boundary terms arising from the finite sampling volume make appreciable contributions when the aperture radius $R\rightarrow0$. Numerical calculation and Monte Carlo simulations demonstrate that the PME overestimates $M(r)$ by factors of at least 3 - 4 inside the core. More importantly, the functional form of the PME as $R\rightarrow0$ is attributable to the factor $f=f(R)$, not the function $M(r)$. Analytical ``$\eta$-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. Previous attempts to overcome volume incompleteness have computed the PME within a series of concentric annuli. Here the term giving rise to the total mass in the case of complete volume sampling, and through which one intends to infer $M(r)$, actually cancels out. Thus, using the PME often gives 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.

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