Blog Post 21, WS 7.1, Problem 1: White Dwarf Pressure

White Dwarfs are supported internally again the force of gravity by "electron degeneracy" pressure.  The maximum mass that can be supported by this exotic form of pressure is 1.4 M(sun) (also known as the Chandrasekhar Mass.)  The radius of our white dwarf is approximately twice the radius of the Earth, or \(12\times 10^8\) cm.
Given this mass M, and radius R, derive an algebraic expression for the eternal pressure of a white dwarf with these properties.  Start with Virial Theorem, and assume that the white dwarf is an ideal gas with uniformly distributed mass.

As the question prompted, lets start with Virial Theorem.
\[K=-\frac{1}{2}U\]
Next we should apply the kinetic energy of a particle in 3D space:
\[K_p=\frac{3}{2}kT\]
Where k is the Boltzmann Constant and T is the temperature.
I few assume that we have N particles, the the total kinetic energy is:
\[K=\frac{3}{2}NkT\]
Now let's add our usual expression for U.
\[K=-\frac{1}{2}U\]

\[\frac{3}{2}NkT=\frac{GM^2}{2R}\]

Simplifying we get:

\[3NkT=\frac{GM^2}{R}\]

We want to solve for pressure, so let's look at ideal gas law.

\[PV=NkT\]

We have some similar variables here.  This is a very good thing.

Let's solve both of our equations for NkT.

\[NkT=PV\]

\[NkT=\frac{GM^2}{3R}\]

Now we can set them equal to one another.

\[PV=\frac{GM^2}{3R}\]

Solving for P we get:

\[P=\frac{GM^2}{3RV}\]

V isn't in our given variables, but since the star is a sphere, we can express V in terms of R.

\[V=\frac{4}{3}\pi R^3\]

\[P=\frac{GM^2}{3R(\frac{4}{3}\pi R^3)}\]
\[\boxed{P=\frac{GM^2}{4\pi R^4}}\]
Qualitatively this makes sense, because as Mass increases pressure increases, and as Radius decreases, pressure increases.  This is the perfect recipe for an explosion.  Interestingly, (derived in AY16), the Radius of a White Dwarf Scales with Mass to the 1/3.  \(M \sim \frac{1}{R^3}\)
Thus:\[P \sim R^{-10}\]
Or:
\[P \sim M^{3.33}\]
This is cool because we can scale pressure to a single variable, so with a critical mass, we also can find a critical pressure and radius.

I worked with B. Brzycki, G. Grell, and N. James on this problem.