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