Transcript Powerpoint of lecture 10

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 ~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
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(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’)
Completely degenerate electron gas:
definition and electron density
• Zero temperature – all states filled
up to some maximum p; all higher
N(p)/p2
states empty:
p0
p
• 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?
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 nonrelativistic 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.)
– polytropes with n = 3/2 and n = 3 respectively
(5.29), (5.30)