Lecture #6a-- The cause of supernovae



1) Unique characteristics of degeneracy pressure

Degeneracy pressure has two unique aspects that lead to violent behavior:
i) It does not rely on temperature, so does not increase much with temperature, so does not act like a thermostat to nuclear fusion-- excess temperature leads to excess fusion which leads to even more excess temperature which leads to runaway fusion. This does not cause stars to explode, because eventually it gets hot enough where gas pressure takes over and regulates the fusion, but this does lead to the "helium flash" at the start of the helium burning main sequence (also called the "horizontal branch").
ii) It is less sensitive to addition of mass than gravity is, so adding mass gives gravity an advantage. If the degeneracy pressure is for nonrelativistic particles, then this merely means that adding mass causes the equilibrium volume to shrink. But when the degeneracy pressure is for highly relativistic particles, gravity wins catastrophically, leading to a supernova.

2) Nonrelativistic degeneracy pressure

The thermodynamic way to determine the pressure of a ball of gas is to adiabatically squeeze its volume a little and ask how much work that required, and then form the ratio of the work required to the volume reduction. So if we knew the kinetic energy of the electrons in a stellar volume V, as a function of that V, we could find the pressure. We will not do this in detail as a function of location, just the overall characteristic scaling. Using the Heisenberg uncertainty principle, we can find how the minimum kinetic energy scales with volume (you should do this). We simply take the HUP, cube both sides, and find that the minimum momentum cubed is inversely proportional to the volume per particle. For nonrelativistic particles, this means the minimum kinetic energy (of each particle) to the 3/2 power is inversely proportional to volume per particle, which in turn means that the minimum kinetic energy per particle is proportional to the mass of the star over the volume, all raised to the 2/3 power. This in turn means that the total minimum kinetic energy is the mass to the 5/3 power times the volume to the -2/3 power. The degeneracy pressure scales with the derivative of the energy with respect to volume, so scales like the mass to the 5/3 power times the volume to the -5/3 power, which is also the density to the 5/3 power. So the nonrelativistic degeneracy pressure depends only on density, and scales like density to the 5/3 power.
3) Competition with gravity
If the volume of a star shrinks adiabatically, the gravitational energy that is released must go into kinetic energy. If that amount of energy is more than the work required to squeeze the volume against the pressure, then gravity is able to crush the star. If, on the other hand, the gravitational energy released is lee than the work done against pressure, it means that if instead the star expands, the pressure will do more work than is needed to go into the gravitational potential energy, and the pressure explodes the star. Thus to have force balance, a change in volume must be accompanied with a work done by pressure that is the same as the change in the gravitational potential energy. Again this is a global scaling argument that is independent of the detailed radial structure inside the star, which is assumed to remain self-similar.
The kinetic energy due to nonrelativistic degeneracy scales with mass to the 5/3 and volume to the -2/3, so its gradient with volume scales with mass to the 5/3 and volume to the -5/3, i.e., like degeneracy pressure. The gravitational potential energy scales with mass squared times volume to the -1/3, so its gradient with volume scales with mass squared times volume to the -4/3. Equating these to find the equi