Conservation of Energy.

a) The potential energy in the sun is the energy that would be liberated if it collasped. For a sphere of uniform density this is given by:

Where G is the universal constant, M is the mass of the sun, and R is the radius of the sun. Solve this set of integrals, find values for the constant and properties, and calculate the potential energy of the sun (astronomers usually do this calculation in ergs). This is actually an approximation. To see a more accurate derivation check out this page.

b) The thermal energy of the sun can be described by:

Where N is the number of particles, k is the Boltzmann factor, and T is the average temperature in the sun. This equation comes from assuming that hydrogen is an ideal gas and calculating its kinetic energy. Using 8E6 K for the average internal temperature of the sun and 2E57 for the number of particles, calculate the energy. Compare your answer with part A. Because of the approximation in the previous part, your answer may be off by a factor of 2.

Pressure. This problem will find the pressure profile of the sun. To begin, we will assume that the sun is a spherically symmetrical object. It's mass then, is simply the sum of its density at each radius times the surface area at that radius. In calculus, it is writen as follows:

or in differential form:

We are assumping hydrostatic equilibrium which means that the pressure in terms of radius can be writen as follows:

Solve the differential form of the mass equation for the density. Then substitute it in for the density in this pressure equation. Multiply each side of the pressure equation by dr and integrate from the center of the sun to the surface (from the r=0 to R). Use known quantities to then calculate the pressure at the center of the sun.

 

A bit of help: When you're doing the last integral. The pressure at the surface is zero. How much mass is there at the center? How much mass is enclosed at the surface?