Stellar Structure Section 5: The Physics of Stellar Interiors Lecture 10 – Relativistic and quantum effects for electrons Completely degenerate electron gas Electron density, pressure, thermal energy … as functions of Fermi momentum … relativistic effects Asymptotic forms Pressure-density relations

Pressure – do we need to modify our simple expressions? P gas (b) Gas pressure  ion-electron electrostatic interactions: small effect except at very high densities (e.g. in white dwarf stars)  relativistic effects  quantum effects (Fermi-Dirac statistics) Relativistic effects important when thermal energy of a particle exceeds its rest mass energy (see blackboard) – occurs for electrons at ~6  10 9 K, for protons at ~10 13 K Quantum effects important at high enough density (see next slide) Both must be considered – but only for electrons

Quantum and relativistic effects on electron pressure - 1 For protons, relativistic and quantum effects become important only at temperatures and densities not found in normal stars Electrons: fermions => Fermi-Dirac statistics. Pauli exclusion principle => ≤ 2 electrons/state What is a ‘state’ for a free electron? Schrödinger: 1 state/volume h 3 in phase space: Derive approximately, using Pauli and Heisenberg (see blackboard) Hence number of states in (p, p+dp) and volume V p x

Quantum and relativistic effects on electron pressure - 2 From density of states, find (see blackboard) maximum number of electrons, N(p)dp, in phase space element (p,p+dp), V Compare with N(p)dp from classical Maxwell-Boltzmann statistics Hence find (see blackboard): Quantum effects important when n e ≥ 2(2  m e kT) 3/2 /h 3 (5.13) Consider extreme case, when quantum effects dominate (limit T → 0 – no thermal effects, but may have relativistic effects from ‘zero-point energy’)

Completely degenerate electron gas: definition and electron density Zero temperature – all states filled up to some maximum p; all higher states empty: p 0 is the Fermi momentum This gives a definite expression for N(p) Hence (see blackboard), by integrating over all momenta, we can find the electron density in real space, n e, in terms of p 0 What about the pressure of such a gas? N(p)/p 2 p 0 p

Completely degenerate electron gas: pressure The general definition of pressure is: the mean rate of transfer of (normal component of) momentum across a surface of unit area This can be used, along with the explicit expression for N(p)dp, to find (see blackboard) an integral expression for the pressure, in terms of p 0 The integral takes simple forms in the two limits of non- relativistic and extremely relativistic electrons It can still be integrated in the general case, but the result is no longer simple – see blackboard for all these results

Thermal energy and asymptotic expressions (see blackboard) The total thermal energy U can also be evaluated – and is not zero, even at zero temperature: the exclusion principle gives the electrons non-zero kinetic energy The pressure and thermal energy take simple forms in two limiting cases: the classical (non-relativistic: N.R.) limit of very small Fermi momentum (p 0 → 0), and the extreme relativistic (E.R.) limit of very large Fermi momentum (p 0 → ∞); in these limits there are explicit P(  ) and U(P) relations If the gas density is simply proportional to the electron density: P   5/3 (N.R.), P   4/3 (E.R.)(5.29), (5.30) – polytropes with n = 3/2 and n = 3 respectively

Other effects Relativistic effects in non-degenerate gases (see blackboard):  the pressure behaves like an ideal gas at all temperatures  the thermal energy depends on the kinetic energy of the particles (but is the same function of P in the NR and ER limits as for degenerate gases) Thermal effects: produce a Maxwell-Boltzmann tail at high p. Total pressure does have temperature terms (see blackboard), but the thermal corrections to the degenerate pressure formula are small

Total pressure For (most) ionized gases, the electron density is larger than the ion density, so even the non-degenerate electron pressure is larger than the ion pressure: n e kT > n i kT In the degenerate case, the electron pressure is much greater than n e kT, so the ion pressure is negligible The radiation pressure is generally smaller than the ion pressure, especially at high densities Thus, to a good approximation, when electrons are degenerate, we have: (5.34)