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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 4pi r3, and integrate from zero to infinity.

b) Show that the first-term becomes -3N<sigmar2>

Show that the second term becomes +2N[<sigmar2> - <sigmat2>]

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

c) Show that: <sigmar2 + 2sigmat2> = <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 rho0 out to some radius rmax.

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.

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