# What is a degenerate gas?

by Baggio
Tags: degenerate
 P: 212 hi, I'm studying stellar structures and there's always mention of a degenerate gas (I think it goes by other names such as degenerate electron gas or cold gas). The only time I've heard the word degenerate is in the context of quantum mechanics, but I think this is completely different. Very confused.. Thanks
 Sci Advisor HW Helper P: 11,948 Yes,it's actually quantum statistics...Degenerate quantum gases:Fermi & Bose.I think a fair treatment is given in every serious book on statistical mechanics.K.Huang,B.Diu,F.Schwabl,... Daniel.
 P: 212 ok thanks I'll check my book
 Emeritus Sci Advisor PF Gold P: 2,977 Degeneracy is cool stuff, if you ask me. To really appreciate it, one should step back and review the mechanisms by which large objects support themselves. To just see my response to the issue of degeneracy, check out the last few paragraphs. When I find a topic interesting, I tend to digress a lot. First of all, what do we mean by "support"? Well, the dilemma that stars face is that they're really massive and really dense (on cosmological scales), so gravity wants them to contract. Fortunately for us, however, there are several things which can prevent this collapse and hold the star up from the forces of gravity. The most common one is gas pressure. All this really means is that the particles of which the star is composed are bouncing around so fast that their momentum is holding the star up. As gravity tries to bring the particles closer to the center, they run into other particles that push them back out. Even just with this simple picture, one can sort of see where the ideal gas law (the equation generally used to calculate gas pressure) comes from. It's given by $$P=nkT$$ where P is the pressure, n is the number density, k is Boltzmann's constant, and T is the temperature. It's easy to imagine that if there are more particles per cubic centimeter, then there will be more pressure (more particles to run into). Thus the proportionality with density. Also, you might imagine that higher particle velocities would mean more pressure (each particle would pack more of a punch), and this is the origin of the temperature dependence (higher temperature means higher average particle velocity). In order to have pressure support, however, you need a pressure gradient (that is, the pressure must increase towards the center). Without this, the particles on the outside will have just as much gas pressure as those on the inside, but they will also be pushing inward due to the forces of gravity, so the net effect would be collapse. To prevent this, the gas particles on the inside must have a larger pressure than those on the outside to counteract the extra pressure from gravity. This the origin of the equation of hydrostatic equilibrium: $$\frac{dP}{dr}=-\rho g$$ where $$\rho$$ is the mass density and g is the gravitational acceleration at the radius, r. This is just saying that, in order to counteract gravity, the pressure must increase as the radius gets smaller (both sides of the equation must be negative). Anyway, the above mechanism is what the sun and most stars use to support themselves. In really massive stars, radiation pressure can actually play a role in keeping a star up. Since light carries momentum, then this can be said to work in a very similar way to gas pressure, the difference being that light is doing the pushing instead of matter. One of the problems with gas pressure, however, is that it requires continuous energy input. If not steadily supplied with energy, particles have a tendency to radiate and slow down (or "cool") as time goes on. This is why nuclear fusion is required to keep stars alive for long periods of time. The energy liberated in fusing atoms can be donated to the gas or radiation holding up the star. There are only so many atoms to fuse, however, so as the "fuel" is used up, the matter cools and there arises the need for another form of support from the push of gravity. This job is usually filled by degeneracy pressure. What happens is that, as gravity pushes on a star unable to support itself with gas pressure, the star tends to contract. As the star contracts, the ions and electrons get closer and closer together. Quantum mechanics states, however, that there are only so many states available for certain kinds of particles (specifically, fermions) in a given volume, depending on how much energy they have. As the star collapses further and further, the free electrons tend to fill the available states. At some point, the star is prevented from collapsing any further because there are no more states available for the electrons. This quantum mechanical effect is known as electron degeneracy pressure and occurs in white dwarfs, brown dwarfs, and, to some extent, the cores of active stars. Are there limits to the strength of electron degeneracy pressure? Yes, it turns out that if you push them hard enough (i.e. if the star is massive enough), the electrons will react with the protons to form neutrons (inverse beta decay). This will cause the star to collapse further. It turns out, however, that there are also limits on the number of neutrons you can have in a volume. This is because neutrons, like electrons, are fermions (have half-integer spin) and obey the Pauli exclusion principle. Neutrons can be packed much tighter, however, because their effective quantum mechanical wavelength is much smaller: $$\lambda\simeq\frac{h}{p}$$ where h is Planck's constant and p is the momentum of the particle. The momentum is proportional to the particle mass, so this is why a neutron's wavelength is generally much smaller and also why neutrons can be packed much tighter. This is also why the ratio of the radius of a typical white dwarf (held up by electron degeneracy) to a typical neutron star (held up by neutron degeneracy) is given approximately by the ratio of the neutron and electron masses. That is: $$\frac{R_{WD}}{R_{NS}}\simeq\frac{m_n}{m_e}\simeq 2000$$ Degeneracy can also play a role in more everyday circumstances, though it is generally not as straightforward. Although there is degeneracy pressure exerting forces on many solid objects, the dominant effects are generally electrostatic forces. That is, the main reason you can't squish your coffee mug is that the charges are in a stable lattice and perturbations to it will result in an electrostatic force that brings it back to the equilibrium position.