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### 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? 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 ~6109 K, for protons at ~1013 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 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

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(2mekT)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’)

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

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

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

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