Statistics in astrophysical systems 177inasingle stagetoablackhole. Bothofthefirst two oftheseinvolvehighly dense
matter and extremely degenerate Fermi–Dirac gases.
First, consider whitedwarfstars. They arefaint objects,because theprincipal
energysourceissimply from agradualgravitationalcollapse. Nevertheless,by
becoming dense they remain hot, with a typical core temperature of 1 07 K,simi-
lar to that ofthe core ofthe Sun. Hence thewhite colour. They are madeupofhelium
(and/or of heavier nuclei – this is not known, and it has little effect on the follow-
ing). They are extremely dense, about 1 07 to10^11 kgm−^3 ,i.e. a million times more
dense than the Earth(imagineyourbodyweightinacubicmillimetre!). At these
temperatures, the electrons are all free from the nuclei, so that there is a dense FD
gas of electrons. Coulomb forces ensure that the material remains overall neutral,
theelectronsfollow the nucleiandvice versa. The stabilityofthewhitedwarfarises
from a balance between the attractivegravitational potential energyof the nuclei and
the high kinetic energy (and hence pressure) of the FD electron gas. It turns out that
aheavywhitedwarfis never stable, thelargest possible massbeingabout 1.4 solar
masses for it not to suffer furthergravitational collapse.
In a neutron star, further collapse of the material leaves our neutral star all in the
form ofneutrons, the protons andelectronshavingcombinedbyinversebeta-decay.
The density is even higher than for a white dwarf star, around 10^13 – 1017 kgm−^3 .This
means that the neutrons are onlyafew neutron radiiapart, notfarfrom pure nuclear
matter. Again stabilityofthe star occurs when thereisabalancebetweengravitational
attraction and the outward pressure of the highly degenerate FD neutron gas. Once
again it turns out that there is an upper mass limit for stability, namely 1.5–2 solar
masses althoughan exact calculationis not easy.Alarger star must,it seems, collapse
to become a black hole in a single step.
Let usbriefly examine the stability ofthese systemsin moredetail.
Pressure ofadegenerate FD gas. In section 8.1, we workedout thepressure ofa
uniform, non-relativistic, spin-^12 ,ideal FD gas. There are three steps:
(a) The Fermi wavevector is readily shown (see (8.6)) to be equal to kkkF =
( 3 π^2 N/V)^1 /^3 .Thisisjust abitof‘waves-into-boxes’geometry.
(b) The internal energy of the gas is worked out using the (non-relativistic) dispersion
relationε=^2 k^2 / 2 m. (Note: we use the symbolmfor the particle massinthis
section,inorder to reserveMfor the mass ofa star). The answer atT=0(see
(8.8))isU=^35 Nμ,whereμis the Fermi energy (=^2 kkkF^2 / 2 m).
(c) Finally, the pressureis workedout (section 8.1.3)fromP=^23 U/V,where the^23
factor also arises from the dispersion relation (sinceε∝V−^2 /^3 ). The pressure is
thusgivenby
P=
2
5
^2
2 m( 3 π^2 )^2 /^3 (N/V)^5 /^3 ( 15 .2)In the case ofawhitedwarfstar, theelectron gas canberelativistic. Itisinstructive
tofollow the same steps asbefore, andto realize that thecalculationgeneralizes to