Equation of state for electron degenerate matter

[aleazk]

Hi, this may be a silly question. In the study of white dwarfs, why we change from the non-relativistic equation of state, P(ρ)=K.ρ^(5/3), to the relativistic case, P(ρ)=K'.ρ^(4/3), when there is an increase in the central density of the star, i.e., why the electrons, which compose the Fermi gas, become relativistic in the high density regime?. Thanks in advance.

 

[Ken G]

That's not a silly question, it's a question that would have seemed unthinkably bizarre to anyone before the year 1920. But it all has to do with the surprising fact that to contain an electron into a small space requires that the electron have a lot of momentum. This is the heart of the concept of "degeneracy pressure", which is what you are talking about.

Now, at first glance it seems completely crazy that the electron should have a lot of momentum if it is contained in a small space (by high density)-- shouldn't being highly localized be the opposite of having lots of momentum? Not according to the Heisenberg uncertainty principle, which says that momentum and position are complementary so to know the second to high precision requires that the first might be very high, and not according to the deBroglie wavelength, which says that to confine an electron in a small space requires a short deBroglie wavelength, which in turn requires a large momentum (the deBroglie wavelength is deeply connected with the uncertainty principle).

Now, mind you the momentum can't have a particular direction, because to be tightly confined the particle can't shoot off in some direction. So we are really talking about the magnitude of the momentum, or the momentum squared, which are the two ways of talking about kinetic energy (the first for highly relativistic particles, the second for nonrelativistic particles, and that's the difference you refer to above).

So we have that, if the density is high, then indistinguishable Fermions that are not allowed to occupy the same state must be highly confined, meaning that they must have very small deBroglie wavelengths (which scale inversely to the cube root of the density, that's the interparticle spacing), meaning that they must have proportionately large expected momentum, meaning that they must have large kinetic energy, meaning that they must exert high pressure. How you make the connection is different for relativistic or non-relativistic-- if relativistic, the pressure is 1/3 the kinetic energy density, and if nonrelativistic, the pressure is 2/3 the kinetic energy density. Also, if relativistic, the kinetic energy density scales with the density times the momentum per particle, and if nonrelativistic, the kinetic energy density scales with the density times the square of the momentum per particle. That's it, that's all you need, to get the dependences you cited above.