Statistical Physics, Second Revised and Enlarged Edition

176 Statistics under extreme conditions

wheremisthe massdifferencebetween neutron andproton. Usingtheknown mass

values,we find thatmc^2 is about 1.3 MeV (the proton mass is 938 MeV). Thus

mc^2 =kkkBTat about 1. 5 × 1010 K. Thus (15.1) suggests that at 1 011 K,the numbers

ofneutrons andprotons are substantiallyequal; we areinahigh-temperaturelimit

for this equilibrium. At 10^10 K,the ratio should be about 0.22,and at later times

(lower temperatures) it should rapidly drop to zero according to (15.1). However,

here we can use our knowledge (Chapter 13) of reactions to realize that the thermal

equilibrium situation will not last for ever. There is an activation energy for the beta-

decayreactions whichmaintain equilibriumbetween neutrons andprotons. What

happens is that at around 1 010 K(t∼ 1 second), the reactions become sufficiently

slow that the approximately 20% of neutrons are effectively frozen out in a metastable

state on the current time scale.

In the next period, nucleosynthesis is still delayed bythe high energytail of the

gamma photons. However by about 100 seconds (1 09 K) these photons are sufficiently

few to allowdeuterons andhence other nucleito start toform. Manyofthe neutrons

thus end up as helium, the alpha particle beinga verystable and thus favoured product.

Modelling suggests a few other nuclei, including a very little^7 Li. By 10 000 seconds

(10^7 K,10keV) there are nofree neutronsleft, since thosefew which didnotform

nuclei have in any case beta-decayed spontaneously into protons (half-life = 1000

seconds). Thus nofurther nucleiareformed,since two nucleons must overcome a

Coulombbarrier tofuse, andthethermalenergy is notlarge enoughto allow this.

Thus no heavy nuclei are formed at this stage, and their formation must await the

nuclear reactionsinthehot centres ofstars millions ofyearslater. We may note that

the suggestion of essentially25% helium, 75% hydrogen for the nuclei at this stage

implies a neutron/proton ratio of 1/7, entirely consistent with the above scenario.

1 5.2.2 The stability ofstars

Astrophysicistshave alwaysbeengoodat makingup names. Themajorityofobserved
stars (of which our Sun is one) show a correlation between brightness and colour.
Theseform the ‘main sequence’ stars,inwhichfaint stars are redandbright stars
are white. But there are exceptions, with the picturesque names of redgiants (and
supergiants) and white dwarfs.
Inthemain sequence, the energysourceisprimarily fromhydrogenfusion reactions
to produce helium,generatingand sustaininghigh temperatures. The correlation
relates to the black-body radiation spectrum from a hot body. The hotter it becomes,
thebrighteritis(U∝T^4 )andthewhiteritis(νmax∝T)as shownin section 9.3.1.
Thus the radiation from our Sun is essentially that of a black body with temperature
equal to that of the surface Sun temperature (about 6000 K).
Aninterestingquestioniswhathappens to a star when thehydrogenisfinally burnt
out. How does it die? Certainly initial cooling and gravitational collapse can then
occur. But withinthis context there are (atleast) three possible next stages. Oneis
formation ofawhitedwarf, oneisformation ofa neutron star andthethirdiscollapse