PY 228: STELLAR ASTROPHYSICS

HOMEWORK #4: DUE MARCH 28


Let's build a white dwarf!

If a white dwarf is supported entirely by degenerate electron pressure, then the structure of the star is independent of temperature. In this case the relevant equations reduce to the equation of hydrostatic equilibruium, Poisson's equation for the gravitational potential of the star, and the equation of state for a degenerate electron gas.

Your mission, should you decide to accept it, is to compute the size and density of a solar mass white dwarf. The first step in your mission is to combine the three relevant equations into a single, second-order differential equation, known as the Lane-Emden Equation. The second step is to solve this equation numerically for the radius of the white dwarf, given a mass (assume one solar mass). After that, finding the density should be a piece of cake. The hand-out from class should help you through this homework with a minimum of pain. (No pain - no gain.)

The Lane-Emden Equation can be solved numerically for arbitrary adiabatic index, gamma. It can be solved analytically for gamma = 2 and gamma = 6/5. I will give you lots of extra-credit if you can derive either of these analytic solutions.

The numerical solution of the Lane-Emden Equation can presumably be found with the aid of Maple. However, I was not able to get a correct solution (w should cross zero for y between 3 and 4). If you can find out what is wrong with my Maple script, please let me know. Instead of Maple, I used a simple fortran program to compute the solution using a Runge-Kutta algorithm. If you are getting frustrated (as I was with Maple), you are welcome to borrow this code (also available in /ncsu/py228_info/polytrope.f).