Lecture 15: Electron Degeneracy Pressure

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Lecture 15: Electron Degeneracy Pressure As the core contracts during shell H-burning we reach densities where the equation of state becomes significantly modified from that of the ideal gas law.

1 Lecture 15: Electron Degeneracy Pressure As the core contracts during shell H-burning we reach densities where the equation of state becomes significantly modified from that of the ideal gas law. The reason is that quantum effects begin to restrict the ability of the electrons to assume the momentum states that they would otherwise wish to. In particular, the Pauli Exclusion Principle limits the number of electrons that can be put in any cell of 6-dimensional phase space to 2, of opposite spin. N 2 V dxdydzdpx dp y dp z all momenta Number of available cells in phase space When the pressure begins to deviate from what you would have calculated from the ideal gas, (i.e. becomes higher) we refer to the gas as becoming degenerate. Note that it is the electrons that become degenerate first, way before the atoms, so it is their pressure that supports the star. Hence, this is referred to as electron degeneracy pressure. Why the electons? Basically, particles cannot be packed together any more tightly than their DeBroglie wavelengths, h λ DeBroglie = h p

2 In a gas, the available kinetic energy will be equally distributed among particles of all mass, so that classically, E = constant = 1 and 2 mv2 = p2 p = 2mE 2m Hence, higher mass particles (atoms) have higher momenta and smaller DeBroglie wavelengths. You can, therefore, pack many more atoms into the same volume of space before they start showing effects of degeneracy. Since electrons are at least 18 times less in mass than any atoms, atomic degeneracy is a negligible effect. Generally, the atoms remain non-degenerate while the star (or core) is supported by the degenerate electrons. In a non-degenerate gas, electrons distribute themselves over momentum space in accordance with the Maxwell-Boltzmann law for velocities. When densities are high, however, the lower velocity states are all filled and electrons are forced to higher states than they would otherwise desire. Higher momenta mean higher transport of momentum across any surface and, hence, higher pressure. The star can support itself partially or entirely by this degeneracy pressure.

3 Complete Degeneracy Fermi Momentum Number of particles Partial Degeneracy Non-degenerate Maxwell-Boltzmann One component of momentum vector

4 Complete Degeneracy occurs when all momentum states below some Fermi momentum are filled, while all those above it are empty. If the Fermi momentum is in the non-relativistic regime (p_f << m_e c), then we refer to this as non-relativistic degeneracy. If the Fermi momentum is in the relativistic regime (p_f >> m_e c), then this is ultra-relativistic degeneracy. Obviously, the lower momenta states will fill first, which pushes any further electrons to higher states. When the higher states are not completely filled we have Partial degeneracy. Note that increasing the temperature of the gas can lift degeneracy by spreading the Maxwell Boltzmann distribution over more momenta states. Similarly, lowering the temperature of a gas can induce degeneracy. Relationship between electron density and Fermi momentum in complete degeneracy. The inequality of the equation above is replaced with an equality, in this case, and we have: n e = N V = 2 pf h p 2 dp dω n e = 8π p f h π

5 To calculate the pressure associated with electron degeneracy, we return to the fundamental expression for pressure derived in the book, namely: In the non-relativistic case, we have that the magnitude of the momentum vector is just mv, so p = m e v which leads to: or, in terms of electron density: P e = 1 Electron density can be replaced with mass density (rho) as follows: n e = P e = 1 8π h Pf n e (p)pvdp p m e dp = 8π h P e = Kn 5 e where K = h2 5m e 2m h (2X + Y + Z) = n e = X m H + 2Y m H + Z p 5 f 5m e [ ] 2 8π 2m H (X + X + Y + Z) = 2m h 2m h (1 + X)

6 It is often the case that X = in a completely degenerate zone (for example in white dwarfs). In that case, the electron degeneracy pressure can be written as: [ ] 2 [ P e = h2 1 5m e 8π 2m H which is the expression on your problem set. ] 5 5 In the ultra-relativistic case, the velocity of the electrons approximates the speed of light, c, so the pressure integral becomes: P e = 1 8π Pf h c p dp = 8π h cp f or, replacing the Fermi momentum with the electron density P e = Kn e where K = hc [ ] 1 8π If X =, then P e = hc [ ] 1 1 8π 2m H which is expression on your (revised) problem set. ]

7 9 8 7 Log T 6 5 Radiation Pressure NONDEGENERATE Sun Gas Pressure DEGENERATE Non-relativistic Relativistic Log Density (gm/cm^)

AST1100 Lecture Notes

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