A loose group of about fifteen galaxies has a measured line-of-sight velocity…

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Astrophysics & Cosmology · Gravitation and central-force motion

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

Problem

A loose group of about fifteen galaxies has a measured line-of-sight velocity dispersion

v_{\rm disp} = 350\,\mathrm{km\,s^{-1}}

and a projected radius

R_{\rm cl} = 0.40\,\mathrm{Mpc}

(

1\,\mathrm{Mpc}=3.086\times10^{22}\,\mathrm{m}

). Model the group as a self-gravitating, isotropic, uniform-density sphere in a steady state. ((a)) Estimate the group's virial 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)) With only fifteen galaxies observed, and no way to be certain each one is gravitationally bound to the group rather than a foreground or background galaxy seen in projection along the same line of sight, state which assumption behind the estimate is weakest, and in which direction a few such interloper galaxies would bias the measured

v_{\rm disp}

and hence the estimated mass.

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