Addendum 6: Stellar Death: White Dwarfs (Chapter 18)

First define some constants and dimensional units needed below

1. Minimum temperature required for fusion. We can roughly estimate the minimum temperature required for two nuclei (e.g. hydrogen = protons) to undergo fusion. They must have enough kinetic energy to overcome the repulsive electrical force caused by their positive charges. If they can get close enough together, the strong nuclear force will attract them (this is what binds nuclei together). Unlike gravity and the Coulomb (electrical) force, the strong nuclear force has a finite range, a ~ 10^-12 m. If the protons are moving fast enough, and can collide with a distance less than a, the strong nuclear force will bind them together.

Analagous to the gravitational binding energy we derived earlier, the energy associated with two protons with charge q and separated by distance r is:

e₀ is a constant called the 'permittivity of free space'.

From a previous worksheet, we know that the mean kinetic energy of a particle in a gas of temperature T is

Equating these and solving for the temperature gives:

Example: What minimum temperature is required for hydrogen to undergo fusion to helium?

charge of electron and proton are equal and opposite

range of strong nuclear force

This is very close to the correct value determined using quantum mechanics, about 15 million K.

2. Mass-Radius relation for a white dwarf. This derivation requires several steps.

(a) Pressure of a gas as a function of speed, density. Consider gas particles in a box with number density n and with an average speed Vx along some direction x. The pressure on the sides of the box can be calculated by considering that pressure is force per unit area, and force is the rate of change (time derivative) of linear momentum (mass x velocity):

definition of pressure and force

definition of momentum

number of particles hitting area Dx² on side of box per time interval Dt

so

(b) Degenerate gases and the Heisenberg uncertainty principle. The electrons in a white dwarf star are squeezed together so tightly that they are 'degenerate': they occupy every allowed volume in phase space. This is a result of the Pauli exclusion principle, wihch state that two particles cannot have the same quantum states. One consequence is that their uncertainty in position and speed in any direction is set by the Heisenberg uncertainty principle:

so

solving for Vx and equating...

Since there is 1 particle in a box with dimension Dx on a side, we have

or

So the pressure is:

where hbar is h/2*p

(c) Gravitational pressure .In a stable white dwarf, this electron degeneracy pressure is balanced by the pressure due to gravity. We can estimate this pressure (roughly!) by considering a sphere of mass M divided into 2 hemispheres. The force between them is:

Since the pressure is force/area, the average pressure is the gravitational force divided by the cross-sectional area betweeen the hemispheres:

(d) Equate degenerate gass pressure with gravity. Finally, we equate the gravitational and degenerate gas pressures:

We can eliminate the number density:

where m_n is the mass of each particle (e.g. carbon atom for a carbon whte dwarf)

So

Cancel common factors and rearrange:

the numerical factor:

is close enough to 1 in this approximate calculation that we can drop it, so obtain the final result:

Notice that the right hand side is a constant - this means that more massive white dwarfs must be smaller!

For a carbon white dwarf m_n ~ 6m_p

Let's plot the mass vs radius of a white dwarf. We can use Earth radii for the units of radius, since white dwarfs are so small.