**Problem Set #3**

**ASTR 3830**

- A cluster of galaxies has one hundred galaxies, each like the Milky Way, within a radius of 300kpc. Over the time since the Big Bang, 10% of the stellar mass (1% of dark matter) from each galaxy has been stripped and now forms a hot intra-cluster medium. Estimate the temperature and luminosity of the cluster in the x-rays.

- How old would the cluster have to be to have a cooling flow in center? Should it have one?

- Let’s derive a formula for the growth of black holes through accretion.
- Write down the formula for the maximum dm/dt when the black hole is accreting at the Eddington limit.
- Solve the simple differential equation to get a formula for the mass of the black hole as a function of time.
- How
long will it take to grow a black hole from 100M
_{¤}to 10^{7}M_{¤}?

- Let’s take a more careful look at ripping stars apart to feed a black hole.
- Derive a formula for the tidal radius inside which a star like the sun will be ripped apart as it flies past a black hole.
- The cross section for this interaction is the area of that circle minus the area of the central black hole. Modify the formula to represent the effective cross sectional area of a black hole to tidal disruption as a function of its mass.
- At what mass does this function peak?
- Is there any observational evidence of this effect?

- Assume that the accretion disk around an Eddington-limited AGN is a blackbody. What is the temperature of the accretion disk as a function of the mass of the central object? Derive a formula to support your assertion.