145 
132-134] 
Stability of Liquid Stars 
the rate of generation of energy per unit mass of the stellar matter. This 
requires us to give a small positive value to a and a small negative value to /3. 
134. The second condition of stability, the dynamical condition, is that D 
shall be positive in equation (108*2) or that 
Here a and /3 have the meanings just explained, n is defined through the 
opacity formula, the coefficient of opacity being supposed proportional (§107) 
to fipT ~ {3+n) ; X is the ratio of material pressure to gas-pressure in the central 
regions of the star, and i is a quantity which exists only when there are 
deviations from the gas-laws, the material pressure p G being supposed pro 
portional to p l+s T. 
To our first approximation 3a 4- /3 is zero; calculation shews that even to 
the second approximation, in which changes of ionisation are taken into 
account, 3a + /3 is, generally speaking, very small and negative—i.e. the effect 
of changes of temperature outweighs that of changes of density, so that 
3a + yS takes the sign of /3. With Kramers’ formula for the opacity n = §, and 
in general n may be assumed positive. 
Thus the first term on the right of the above inequality is negative, and 
since 7 + n — /3 is in any case positive, the inequality can only be satisfied by 
assigning a positive value to s. This merely reiterates our former conclusion 
that stability requires deviations from the gas-laws, but we now see that 
such deviations are necessary whatever the mass of the star, and relation 
(134*1) enables us to estimate the amount of the necessary deviations. 
To a sufficiently good approximation for our present purpose, we may 
neglect a and /3 and put n = \. Relation (134*1) now assumes the form 
and we find as condition for stability 
For stars of moderate mass, X is large, and the condition for stability is 
that s must be greater than ^; for stars of large mass X is comparatively 
small, and s must have a value substantially higher than 
Values of s in the neighbourhood of s = ^ look small until we make 
numerical calculations of what they involve. The calculations of Chapter hi 
have shewn that the density in the central regions of a gaseous star may well 
be about 100 times the mean density of the star, and so more than 100 times 
that in the outer regions of the star in which Boyle’s law is obeyed. Thus 
with s = ^ the additional factor p s in the pressure requires a pressure in the 
centre of the star of at least (100)*" or 1*107 times that given by Boyle’s law. 
(3a + /3 — n)+ x+ - 4 (7+?i —/3)>0 
,(134*1). 
(134*2).