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 gt Fermi-Dirac statistics.
  Pauli exclusion principle gt 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, pdp)
• 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,pdp), 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
p0 p
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 ? 8) 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

8
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

9
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 nekT gt nikT
• In the degenerate case, the electron pressure is
  much greater than nekT, 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)