EXERCISE

1) Derive the virial theorem from the Jeans (or "stellar hydrodynamic") equation, as follows:

For a spherical nonrotating system (e.g., Coma cluster), the Jeans equation is

a) Multiply both sides by 4 r³, and integrate from zero to infinity.

b) Show that the first-term becomes -3N<_r²>

Show that the second term becomes +2N[<_r²> - <_t²>]

Show that the third term becomes - N<rd / dr>

c) Show that: <_r² + 2_t²> = <v²> and

2) Use this equation to derive estimates for the total mass of a system, based on a sample of objects (stars, galaxies) with known <v²> and known spatial distribution n(r). Assume:

a) The mass is all in a central object, around which the "tracer" objects orbit; or

b) The mass is distributed with a constant density ₀ out to some radius r_max.

3) Compute the ratio of these two estimated masses, assuming that the "tracer" population has density law

Comment on the usefulness of the virial theorem when nothing is known a priori about the form of the matter distribution.