Physics 410/510 First Homework Assignment

Due: Feb 7 at the end of Class

1. Derive the virial theorem (2T + W = 0) using some argument. Explain when this theorem can be applied to a gravitating system

2. Use either the library (Astrophysical Journal) or the Web to find out more about the thick disk component of our Galaxy. Give a brief description of it and explain how it helps to resolve the Oort limit problem discussed in class.

3. For a cluster of galaxies of velocity dispersion 1500 km/s and radius 1 Mega Parsec, apply the virial theorem to determine the total mass of the cluster. If the cluster contains 200 galaxies of luminosity 10¹¹ solar luminosities, what does the M/L value for each galaxy have to be in order to account for the mass. Can you think of any possible flaw in this analysis?

4. For a point mass object, show that circular velocity declines as 1/R (like it does in our solar system).

5. Determine, by any means necessary, the functional form that a gravitational potential must have in order to produce rotation velocities that are flat (that is, they are independent of radius after some radius). In may be simpler to think about this in terms of a density distirbution. What must the density distribution within a halo be, as a function of R, to produce constant circular velocities after some radius.