White Dwarfs: Properties and Degeneracy

Sirius B: First white dwarf discovered in 1862. Observed properties

M = 1 M_sun
L = 0.03 L_sun
T = 27,000 K

From this, we can use Stefan-Boltzmann law to derive radius:

R=0.008 R_sun ~ R_earth
density ~ 3x10⁶ gm/cm³

Not a normal star! We are seeing the central C/O core of a dead low mass (M < 8 M_{s un}) star.

Let's estimate the central pressure inside a white dwarf:

Start with hydrostatic equilibrium:

and integrate (assuming density is constant):

So we have an expression for the central pressure:

If we put in numbers for Sirius B, we get P_c ~ 4x10²³ dynes/cm²> 10⁶ P_c,sun. At these densities this high pressure is provided by electron degeneracy.

Degeneracy

Remember two things from quantum mechanics:

The Pauli Exclusion Principle from quantum mechanics: no two electrons can have the same quantum state (i.e., energy level).
The Heisenberg Uncertainty Principle:

So how much momentum does each electron have? Each electron must at least have a momentum equal to:

and how far apart are they? Higher density means the electrons are more tightly packed. If the number density of electrons in the WD is n_e, then we have a average separation of

so then each particle's momentum is

Now, what is the pressure? Think of particles bouncing around in a box -- the pressure they exert on the walls is given by

So, plugging in our expression for momentum we get

So for a degenerate gas, we have

If the density is high enough, electrons keep getting pushed to higher momentum, or higher velocities. But electrons can't move faster than the speed of light -- their speed approaches, but does not exceed, the speed of light. In this case, the pressure becomes

Or, for a relativistic degenrate gas, we have

So what the heck does all this mean?