- 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
*r*^{3}, and integrate from zero to infinity. - b) Show that the first-term becomes
-3
*N*<_{r}^{2}>Show that the second term becomes +2

*N*[<_{r}^{2}> - <_{t}^{2}>]Show that the third term becomes -

*N*<*rd*/*dr*> - c) Show that: <
_{r}^{2}+ 2_{t}^{2}> = <*v*^{2}> 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 <*v*^{2}> 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.