Class Activity: Virializing the Night Away Problems 5-9

If we have a star that has ceased fusion in the core, as we add more mass to the star, the radius should increase until the star is massive enough that it begins to shrink under its own weight. Then, when the star is small enough, it becomes so dense that it cannot get any denser due to the uncertainty principle--electrons can't be in the same state as other electrons, so they must occupy other energy levels instead and this means there is an upper bound to the density of a star. Once this bound is reached, it cannot get more dense, and increasing mass must increase the radius again.


By the virial theorem (twice the kinetic energy plus the gravitational energy of a system is 0), we can determine the relationship between the kinetic energy of the electrons in a white dwarf and the gravitational energy of the dwarf.

where U is the gravitational energy, which is found by integrating the gravitational energy over the mass of the entire star, and T is the kinetic energy, which is the number of electrons n_e times the thermal energy per particle.


We know that degeneracy pressure becomes important in a white dwarf when the inter-particle spacing is of order of the de Broglie wavelength, which is given by:

where h is Planck's constant and p is the momentum of a particle. The uncertainty in the electron's position is also of order of the de Broglie wavelength, since uncertainty in position is on the order of the position. Since the inter-particle spacing is dependent on the number density of electrons, N_e, so is the uncertainty in the position of the electrons, and we have:

The inter-particle spacing cubed is essentially the volume per particle, which is the inverse of the number density, which is the particles per volume. The kinetic energy of a particle is related to the momentum of the particle by:

The uncertainty in momentum is on the order of the momentum of the particle, and by the Heisenberg uncertainty principle in 3 dimensions,

we can relate the uncertainty in position and the uncertainty in momentum.

This is the relationship between the kinetic energy per particle in the white dwarf and the dwarf's electron number density, where h is Planck's constant and m is the mass of an electron. To relate the electron number density to the total mass and radius of the white dwarf, we realize that the total mass will be the number of electrons times the mass of an electron plus the number of protons times the mass of a proton plus the number of neutrons times the mass of a neutron. We can take the mass of an electron to be negligible when compared to the mass of a proton. We can take the number of electrons to be some multiple of the number of neutrons, and let's call this factor A. The mass of a proton is approximately equal to the mass of a neutron. Then we have:

where n_e is the number of electrons, and V is the volume of the star. This is the relationship between the number density of electrons and the mass and radius of the white dwarf. We substitute this back into the virial theorem (see above), using the previously found expression for the kinetic energy per particle and for the electron number density, keeping in mind that the total kinetic energy is the number density times the volume times the kinetic energy per particle:

In the last couple steps we've dropped all constants to obtain the relationship between the radius and mass of a white dwarf.


Secondary author: Melodie Kao