180 Statistics under extreme conditions
problem nowhinges on the strengthofthegravitationalattraction, afactor which
becomes of importance well before any corrections for the relativistic Fermi velocity
ofthe neutrons. Itisalltodowiththehorizon thresholdofablackhole. As soon as
the escape velocityfor a particlefrom thegravitationalfieldofa starbecomes equal
to the speed of light, then the star must collapse to a black hole. The limiting radius
for this to occuristhe‘Schwarzschildradius’R 0 givenbyc=( 2 GM/R 0 )^1 /^2 ,i.e.
R 0 = 2 GM/c^2. Combiningthis idea with (15.5) we see that the maximum mass ofa
neutron star is then given by
M^4 /^3 =Km
− 5 / 3
n c^2 /^2 G (15.7)Substitutiongives the mass limit from (1 5 .7) to be 2.8 solar masses. Again this is
probably an over-estimate of the true value, thought to be somewhat less than 2 solar
masses.
Asafootnote, itisinterestingtoaskwhetherthereisanyfirmobservationalevidence
for neutron stars. The answer seems to lie in the discovery (by Anthony Hewish and
Jocelyn Bell in the late 19 6 0s) of pulsars. These are objects which emit radio waves
(or other electromagnetic radiation) in regular bursts, i.e. pulses. The pulses havea
periodinthe range milliseconds to secondswhichischaracteristictothe particular
pulsar. Theoriginofpulsars was notimmediatelyobvious andtheywere nicknamed
at first ‘little green men’. However, it was soon realized that highly dense stars, white
dwarfs or neutron stars, mustbeinvolved.When the numbers were putinto possible
theories, theobservationaldetailsoftheemittedradiationgave convincingevidence
that the likeliest candidate for a pulsar is a rapidly rotating neutron star. Such an
object will have aroundit a remnant ofelectrons andanintense magneticfield(about
108 T, compared to our terrestrial field of 1 mT) which confines these electrons. The
accelerating electrons are responsible for the emission of the radiation. Since the
rotationalspeedofthe starisrelativistic, theradiationisemittedpredominantly in
the forward motion direction; this effect is called ‘synchrotron radiation’, the basis of
many modern research machines designed to provide a well-collimated and intense
source ofradiation. Thepulsar effect can nowbeunderstoodas a sort of lighthouse
effect as the star rotates, so longas the magnetic field axis is different from the axis
of rotation.
In summary,wehave seen that the application ofalittle statisticalphysics can
throw much light on a number of interestingquestions about stars. There remain
many other questions of importance, such as whether superfluidity (Chapter 14) of
thedensegas playsaroleinthese systems, a realpossibilityin neutron stars. But
physics would not be an interesting subject if all the answers were easily available.