Stellar Structure

1. 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

2. Pressure – do we need to modify our simple expressions? Pgas • (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109 K, for protons at ~1013 K • Quantum effects important at high enough density (see next slide) • Both must be considered – but only for electrons

3. 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 h3 in phase space: • Derive approximately, using Pauli and Heisenberg (see blackboard) • Hence number of states in (p, p+dp) and volume V p x

4. 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 ne≥ 2(2mekT)3/2/h3 (5.13) • Consider extreme case, when quantum effects dominate (limit T → 0 – no thermal effects, but may have relativistic effects from ‘zero-point energy’)

5. Completely degenerate electron gas: definition and electron density • Zero temperature – all states filled up to some maximum p; all higher states empty: • p0 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, ne, in terms of p0 • What about the pressure of such a gas? N(p)/p2 p0p

6. 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 p0 • 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

7. 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 (p0→ 0), and the extreme relativistic (E.R.) limit of very large Fermi momentum (p0→ ∞); 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