Statistics in astrophysical systems 179whereKisaknown constant (roughly known thatis, sinceitincludesα). Infact,
assumingα= 0 .6, simple substitution gives in SI units (i) for electrons as in a white
dwarfK= 6. 6 × 10 −^28 ,and(ii)for neutrons asin a neutron starK= 3. 6 × 10 −^31.
All thatremainsistorelate thenumberNofgaseousfermions to the totalmassMof
the star. For a neutron star, we have simplyM=Nmnwheremnis the neutron mass.
Forawhitedwarf,therelationdependsindetailon the (unknown) composition;if itis
heliumthenM= 2 Nmneffectively,since eachelectronhas two nucleons associated
with it. Any other composition gives a relation of similar magnitude. For both types
of star, (15.5) thus becomesM^1 /^3 R=constant,i.e. mass×volume = constant.
This is an intriguingresult, verifyingthe alarminginverse relation between mass and
volume. A heavy stable star of this type issmallerthan a lighter one. The density
increasesdramaticallyas the square ofthe mass. Thisideagives the correct order of
magnitude for the properties of these stars mentioned above. Usingthe values ofK
given above, it follows that, for a stellar mass equal to that of the Sun ( 2 × 1030 kg),
theradius ofawhitedwarfisabout 7000 kmandtheradius ofa neutron staris a mere
1 2 km. Hence the amazinglyhigh densities.
The upper masslimit. Itis nothardtobelieve that these resultsimplythat very
heavywhite dwarfs or neutron stars cannot be stable at all. Certainly,aswehave
seen, the heavier the star the smaller and more dense it will become. Moreover,
our treatmentisonlyapproximate, andathighenough densities otherfactors come
into play.
In the case of a white dwarf the new factor is the high Fermi velocity of the
electrongas. As relativistic speeds are approached, we needmove our calculation of
the pressure of the gas from (1 5 .2) towards (1 5 .3). It is now easy to see that a limit is
involved. If we combine( 15 .3)with( 15 .4)we obtain(instead of( 15 .5))
K 1 N^4 /^3 /R^4 =M^2 /R^4 (15.6)with the constantK 1 =1.1× 10 −^15 in SI units.
This is another intriguing result, since the radius of the star does not appear:R^4
cancelsR^4 .The onlymovingpart in (15.6) is the mass. For a white dwarf star with
M= 2 Nmnas above, the equation solves for a massM= 3. 4 × 1030 kg, i.e. about
1 .7 solar masses. The implication is that lighter white dwarfs are stable, with a radius
effectivelygiven by(15.5). As the stargets heavier, the star collapses even faster
than the non-relativistic expression would indicate, until for stars heavier than 1. 7
solar masses thereisnostablesolution; therelativistic gas simply cannot generatea
highenoughpressure to overcome thegravitationalattraction. Hence our prediction
is that the maximum size of a stable white dwarf is 1.7 solar masses. (Incidentally
the correct valuefor thislimit, when the correctdensity profilein a starisincludedis
actually1.44 solar masses, the‘Chandrasekharlimit’. Our simple treatmentis not so
bad!)
Aneutron star also getsinto problems athighenoughmass andthus at extremely
high density.Relativityagainlimits thevalidityofthesimple treatment,but the