146 Liquid Stars [oh. v
G = f 4nrpr 2 f(T, p) dr
.( 135 * 1 ),
The total pressure, it is true, is only increased by 107 per cent., but this
increase must be contributed solely by the atomic nuclei with their bound
electrons, the free electrons being so minute that their pressure will always obey
Boyle’s law. Even if we suppose that there are only 90 free electrons to each
atomic nucleus (which will soon prove to be an under-estimate), the atoms
will contribute only one-ninety-first part of the total pressure when Boyle’s
law is obeyed, so that their additional contribution of 10’7 per cent, of the
whole pressure is 91 x 107 per cent, of their own pressure as given by Boyle’s
law, or say 10 times this pressure. Thus even a value s = -fe requires the
nuclei to be so closely packed at the centre of the star that the pressure they
exert is about 11 times that given by Boyle’s law. This is what we may
describe as a semi-fluid state.
This represents the minimum deviation from Boyle’s law which is
adequate to ensure stability. Our analysis has shewn that unless the atoms
in the star’s central regions are packed so close as to provide a firm unyielding
base of the kind just described, the star will be liable to start contracting or
expanding, this contraction or expansion continuing unchecked until a firm
base is formed at its centre.
In the average stable star the deviations from Boyle’s law must naturally
be more than the minimum; the discussion of § 131 suggested that in actual
stars the pressure of the nuclei may be about 40 times that given by Boyle’s
law, and so nearly 4 times the minimum required for stability.
135. The thermodynamical stability criterion discussed in the last chapter
(cf. formula (1091)) did not involve s at all; s enters only in the second
(dynamical) criterion which has just been discussed. As, however, it is clear
that the fictitious assumed law p oc p 1+s cannot represent the actual facts of
deviations from Boyle’s law with any accuracy, it becomes important to
examine what form is assumed by the second stability criterion when this
veiy special law is no longer assumed to hold.
Let a star’s emission of radiation be plotted against its radius as in
fig. 9. We have already seen how a star of given mass can assume con
figuration of different radii, in which the star will emit radiation at different
rates. Each of these configurations will be represented by a single point in
the diagram. Let the curved line MM' be supposed to represent, in a purely
diagrammatic way, the various configurations of equilibrium which can be
assumed by a star of specified mass M as its rate of internal generation of
energy changes.
Let the internal rate of generation of energy at every point of a star
be supposed to depend quite generally on the temperature T and density p
at that point, so that the total rate of generation of energy G of the star will
be represented by an integral of the type