Homework Set D

Physics 310 – Cosmology
Homework Set D
1. Suppose (unrealistically) that a globular cluster of mass M and radius a has uniform
density, and it satisfies the Virial theorem.
(a) What is the potential energy of the globular cluster (given in class)? What is the
kinetic energy, according to the Virial theorem? What is the total energy?
(b) A passing black hole “stirs things up” and increases the kinetic energy of every
star in this galaxy by a factor of 32 , without affecting the potential energy. What
is the total energy now?
(c) The globular cluster, over time, becomes Virialized again; i.e., the Virial theorem
is again satisfied, though the total energy and mass remain unchanged. What
must be the potential and kinetic energy in this case?
(d) Assuming the globular cluster is still a uniform sphere, what must its new radius
2. Near the surface of the Sun, the temperature is approximately T  5777 K .
(a) Using Wien’s law, find the peak wavelength for the Sun in nm.
(b) Find the total energy density at this temperature, in J/m3.
3. In class we found formulas for the total energy density u, the energy per unit wave
number du dk , and the energy per unit wavelength du d  . In this problem, I’d like
you go find the energy per unit frequency du df
(a) Write an expression for k in terms of f, and find dk df .
(b) Find an expression for du df strictly in terms of f.
(c) [extra credit] Find a formula for the frequency where du df is maximum as a
function of T. You will have to solve an equation numerically.