A globular cluster has a measured line-of-sight velocity dispersion and a radius ( )

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Galactic structure and stellar dynamics · Galactic structure and stellar dynamics

Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector

Problem

A globular cluster has a measured line-of-sight velocity dispersion

v_{\rm disp} = 8.0\,\mathrm{km\,s^{-1}}

and a radius

R_{\rm cl} = 5.0\,\mathrm{pc}

(

1\,\mathrm{pc} = 3.086\times10^{16}\,\mathrm{m}

). Model the cluster as a self-gravitating, isotropic, uniform-density sphere in a steady state. ((a)) Estimate the cluster's mass using

M = 5v_{\rm disp}^2R_{\rm cl}/G_{\rm N}

, in kilograms and in solar masses (

M_\odot = 1.989\times10^{30}\,\mathrm{kg}

). ((b)) Compute the crossing time

R_{\rm cl}/v_{\rm disp}

in years. Taking the cluster's age to be

1.2\times10^{10}\,\mathrm{yr}

, estimate how many crossings have occurred and use this to argue whether the steady-state assumption is safe here. ((c)) Real globular clusters are strongly centrally concentrated, unlike a uniform sphere. Which single assumption behind the mass estimate is weakened most by this, and in which direction does it bias the estimated mass?

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