Blog #21, Worksheet #7.1, #1, Electron Degeneracy Pressure

1. Given this mass, M and radius, R, derive an algebraic expression for for the internal pressure of a white dwarf with these properties. Start with the Virial theorem, recall that the internal kinetic energy per particle is \(\frac{3}{2}kT\), where \(k = 1.4 x 10^{-16} \, erg \, K^{-1} \) is the Boltzmann constant. You can also assume the interior of the white dwarf is an ideal gas, and its mass is uniformly distributed. 

We begin by using the fact that the internal structure of the white dwarf acts as an ideal gas. We can use the standard formula for pressure \[P = \frac{NkT}{V}\] Where N is the number of molecules, k is the Boltzmann constant \((k = 1.4 x 10^{-16} \, erg \, K^{-1})\), T is temperature, and V is the volume. In our case, \[V = \frac{4}{3} \pi R^3\]Combining these equations we get an equation for P. \[P = \frac{3NkT}{4 \pi R^3} \]Next we use the Virial theorem to try and get rid of the N in our Pressure equation. We know that \[K = - \frac{1}{2} U = \frac {3}{2} NkT\]Where \[U = \frac{GM^2}{R}\]Setting the equations equal to one another and simplifying to solve for N, we get \[N = \frac{GM^2}{3RkT}\]We then substitute this variable into our equation for P and get a final answer of \[P = \frac{3kT}{4 \pi R^3} \cdot \frac{GM^2}{3RkT} = \frac{GM^2}{4 \pi R^4}\]