Pressure

As the central temperature of a star increases, the source of the pressure support changes.

Radiation Pressure

• Small T (small M) ====> P ~ N k T (gas pressure)
• High T (large M) ====> P ~ a T**4 / 3 (radiation pressure)

• For the Sun
  • T ~ 15,000,000 Kelvin, N ~ 2 x 10**26 particles per cubic centimeter ===> P(gas) ~ 4.2 x 10**18 ergs per cubic centimeter

    P(radiation) ~ 1.3 x 10**14 ergs per cubic centimeter (where we used a = 7.6x10**(-15) (c.g.s. units)

    P(gas)/P(radiation) >> 1 and the Sun is supported by gas pressure.

• As M increases (T increases) and the importance of radiation pressure increases. It turns out that around M ~ 50 - 100 times the mass of the Sun, radiation pressure would provide most of the pressure support for Main Sequence stars. However, what happens is that a star supported by pure radiation pressure is not stable. It has been suggested that the Upper Mass Limit for Main Sequence stars may be related to this fact.

Degenerate Electron Pressure

• At lower T (which means lower kinetic energies for the gas particles), an interesting thing can happen.

  A gas of electrons has a large number of energy states stretching from almost no energy to essentially an infinite amount of energy (Note--this is like an atom where there are essentially an infinite number of available energy states in an atom). Note that the energy states do not depend on the temperature of the gas (again, this is like an atom). Even if the gas T = 0 Kelvin, the energy structure is the same.

  However, if you look at a gas of electrons, all of the states are not going to be occupied. Depending upon how the particles are excited, the low energy states will be preferred or the higher energy states will be preferred. In a low temperature gas, particles only have enough energies to occupy the low energy states.

  • So, lower T ===> lower kinetic energy (and thus a smaller number of energy states which are easily accessible to the gas particles). Why is this a problem? It is a problem because of a Quantum Mechanical effect based on something known as the

    Pauli Exclusion Principle

  • The Pauli Exclusion Principle states that only one electron can occupy a given energy state. (This rule is actually more general in that it also applies to protons, neutrons, neutrinos -- particles known collectively as fermions). That is, there can only be 1 electron in each unique energy state. You cannot force 2 electrons into 1 energy state.

  • So imagine that you have a box of electrons at some T. Increase the density of the particles in the box by adding in some electrons. Initially, if most of the energy states are unoccupied, you can place electrons into the low energy states ===> you can add low energy electrons to the box. This does not take much of an energy investment on your part and is quite easy to do.

    Note that by adding electrons to the box, you increased the density of particles in the box. This is effectively like compressing the gas. So, initially, it is easy to compress the gas (this is the ordinary gas pressure case).

  • For high temperatures and/or low densities, the Pauli Exclusion Principle does not pose a problem, because the number of accessible low energy states is large (or there are so few particles) that there is no strong competition for low energies.

  • At low T (and/or high densities), problems arise. At low T, because electrons will have a hard time filling the high energy states, the lowest energy states will start to get filled-up. This means that if you now try to increase the density of the particles in the box by adding electrons, you must add the electrons to the highest energy states ===> you must add very high energy electrons and the investment on your part is large.

  • This means that to increase the density of the box (compress the gas) requires a large amount of exertion (energy input) on your part. That is, it is hard to compress the gas. Note that this effect arises even if the temperature of the gas is ) Kelvins. The resistance to compression depends only on the fact that you have filled up the lowest energy states (essentially meaning that you are at high density)!

  • This type of resistance to compression (a type of pressure) is know as

    Electron Degenerarcy Pressure

  Electron Degeneracay pressure becomes important for Main Sequence stars right around masses of 0.1 times the mass of the Sun. Hmmmm.