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