equation of state of a degenerate gas
Thus far, we have assumed that stars are composed of ideal gases and
hence the pressure is given by the ideal gas
of state we have just derived. We might expect this assumption
to break down at sufficiently high densities, when the gas particles
in a star are packed so closely together that one can no longer
neglect the interactions between them.
One might have thought that electrostatic effects are responsible
for the first departures from an ideal gas, but this is not so. In
fact, the first such departures are due to interactions between the
free electrons in the highly ionised gas because electrons have to
obey Pauli's exclusion principle. One way of stating Pauli's
exclusion principle is that:
no more than two electrons (of opposite spin) can occupy the same
The quantum state of an electron is given by the 6 values which describe
its position and momentum, i.e. x, y, z,
px, py, pz.
It is not possible, however, to know both the position and momentum
of an electron with complete accuracy. There is an uncertainty
x in any position coordinate and
p in the corresponding momentum
coordinate, such that
h / 4.
This is known as Heisenberg's uncertainty principle and it means that,
instead of thinking of a quantum state as a point in six-dimensional
position-momentum space (also known as phase space), we can
think of a quantum state as a volume, h3, of
phase space. Pauli's exclusion principle then tells us that no more than
2 electrons are permitted to occupy each volume h3
of phase space.
Let us now consider what happens at the centre of a star as the density
of electrons is increased. The electrons will become increasingly crowded
together in position space and eventually a stage will be reached when
two electrons with almost the same momenta will occupy almost the same
point in position space. This volume of phase space will then be full
(according to Pauli's exclusion principle). It will not be possible to
squeeze in an additional electron to this region of position space unless its
momentum is significantly different (so that it occupies a different region
of phase space).
The additional electron will therefore possess a higher momentum than it
would have had at the same temperature in an ideal gas.
Higher momentum implies
that such a gas will exert a
greater pressure than predicted by the ideal gas equation of state.
The effect of the Pauli exclusion principle on the Maxwellian
distribution of electron momenta.
The way in which the Maxwellian distribution of electron momenta
is modified by the Pauli exclusion principle is shown in
figure 11. Curve A shows the Maxwellian
distribution of electron momenta in an ideal gas. At sufficiently
high densities, the Maxwellian distribution begins to violate Pauli's
exclusion principle (denoted by the horizontal dashed line) at low
momenta. These electrons must then occupy higher momentum states than
predicted by a Maxwellian distribution. This situation is shown by
curve B, where dashed portions of the distribution above the value
defined by the Pauli exclusion principle are effectively transferred
to higher momenta, as indicated by the solid curve. As the density
increases still further, more and more of the low momenta electrons
are transferred to higher momentum states, as shown by curve
A gas in which the Pauli exclusion principle is important is known
as a degenerate gas. Because the ions in such a gas have
higher momenta than the electrons, they are less likely to violate
Pauli's exclusion principle. The pressure due to the ions can then
be treated as an ideal gas, but the pressure due to the degenerate
electrons is much larger and hence the gas obeys a different
equation of state, which we shall now derive.
Consider a group of electrons occupying a volume V of
position space which have momenta lying in the range p
and p + p. The volume
of momentum space occupied by these electrons is given by the
volume of a spherical shell of radius p and thickness
The volume of phase space occupied by these electrons is given by the
volume they occupy in position space multiplied by
the volume they occupy in momentum space, i.e.
The number of quantum states in this volume of phase space is then
simply the above volume divided by the volume of a quantum state,
(4 p2V / h3)
If we now define Npp as the number of electrons in V with
momenta in the range p and p + p (so Np is the number of
electrons per unit momentum range), Pauli's exclusion principle tells
(8 p2V / h3)
We will now define a completely degenerate gas as one in which
all of the momentum states up to some critical value p0
are filled, while the states with momenta greater than
p0 are empty. Strictly speaking, this occurs
at absolute zero. Hence, for:
p2V / h3, and for
p > p0,
Np = 0.
The total number of electrons, N, in volume, V, is
then given by the integral of Npp, i.e.:
(8 V / h3)
8 p03V /
The pressure, P, of a gas is the mean rate of transport of
momentum across unit area. This can be written
as follows (see Appendix 3 of Tayler for a derivation):
P = (1/3)
(Np / V ) pvp dp,
where vp is the velocity of an electron with momentum
p. To evaluate this integral, we must use the relation between
p and vp given by the special theory
p = mevp /
(1 - vp2 / c2)
which can be rearranged to give,
vp = (p / me) (1 +
where me is the rest mass of an electron. Hence,
combining the three expressions for Np, P
and vp derived above, we obtain an expression for
the pressure of a completely degenerate gas:
3 h3 me)
p4 dp /
We will not evaluate this integral here (for the solution, see
Appendix 3 of Tayler). Instead we will consider two limiting cases:
- a non-relativistically degenerate gas (i.e.
p0 << mec)
In this case,
(1 + p2 /
3 h3 me)
p4 dp =
15 h3 me.
8 p03V /
3 h3 and defining the electron density,
ne = N / V, we obtain:
p0 = (h / 2)
(3 ne / )1/3.
Combining the last two equations gives:
(1 / 20) (3 / )2/3
(h2 / me) ne5/3.
- a relativistically degenerate gas (i.e.
p0 >> mec). This
occurs when the velocity of an electron approaches that of light,
in which case its momentum approaches infinity.
In this case, we can neglect 1 in comparison with the
me2c2 term in the
pressure equation, i.e.
(1 + p2 /
(8 c /
p3 dp =
8 c p04/
p0 = (h / 2)
(3 ne / )1/3
and substituting, we obtain:
(1 / 8) (3 / )1/3
h c ne4/3.
Our aim is to obtain an equation of state for a degenerate gas in
terms of density and chemical composition so that it is compatible with the
other equations of stellar structure. Hence we must convert the
electron density ne to mass
density . We can do this using
similar arguments to those given in the derivation of
mean molecular weight.
For each mass of
hydrogen, mH, there is one electron.
For helium and heavier elements there is (approximately) half an
electron for each mH. Thus:
(X / mH) +
((1 - X) / 2mH) =
((1 + X) / 2mH).
where X is the mass fraction of hydrogen.
We can now use this expression to finally write:
Pgas = K1 5/3
equation of state of a non-relativistically degenerate gas
Pgas = K2 4/3
equation of state of a relativistically degenerate gas
(h2 / 20 me) (3 / )2/3
[(1 + X) / 2mH]5/3, and
(h c / 8)
(3 / )1/3
[(1 + X) / 2mH]4/3.
Hence in a degenerate gas, the pressure depends only on the density and
chemical composition and is independent of temperature.
Of course, there is not a sharp transition between relativistically
degenerate and non-relativistically degenerate gases. Similarly, there
is not a sharp transition between an ideal gas and a completely
degenerate gas. There is a region of temperature and density in which
some intermediate and much more complicated equation of state must be
used, a situation known as partial degeneracy.
What types of stars are the above equations applicable to?
We will see later in the course that there
are stars in which no nuclear
fusion is occurring and in which it is the outward-acting force due to
degeneracy pressure that balances the inward-acting
gravitational force. White dwarfs, brown dwarfs and neutron stars
are examples of such stars, in which the degeneracy pressure
due to electrons (in the case of white dwarfs and brown dwarfs)
or neutrons (in the case of neutron stars)
balances the force of gravity. We will also see that many stars
temporarily develop degenerate cores as they evolve off the
©Vik Dhillon, 19th November 2013