THE SOURCE OF STELLAR ENERGY 
303 
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stimulated by a field of y radiation of the same frequency as the y rays 
emitted in the disintegration; the amount of this stimulation can be 
calculated from Einstein’s equation. By (38-5) the emission at temperature 
T is to the emission at temperature zero in the ratio (1 - e~ hv l RT )~ 1 , where 
v is the frequency of the y rays. At stellar temperatures the increase is 
quite insignificant. 
This argument does not settle the question because it is limited to one 
mode of release of energy—and that not the mode which an astronomical 
theory is likely to propose. We shall first explain why the astronomer feels 
bound to insist on a variation with temperature and density. 
If the rate of liberation of energy were independent of p and T the 
stars would be unstable. For then the energy generated E would be 
incapable of alteration by any expansion or contraction of the radius. 
The energy radiated L is determined by the mass and (to a comparatively 
small extent) by the radius. Suppose that by exhaustion of the source or 
by slight disturbance E becomes less than L. Then the energy of the star 
diminishes at the rate L — E, so that it contracts. By hypothesis this 
does not affect E, but L increases according to the law L oc R~K Thus the 
deficit becomes worse and the star contracts indefinitely*. 
We here assume that L oc iT* or at least as a negative power of R. 
It will be remembered that this depends on the exponent n in the law 
k oc p/T n being greater than 3 — a condition which although probable 
both from theory and observation is not established as certainly as we 
could wish (§ 150). But if n were less than 3 it would scarcely overthrow 
the argument. The quantities E and L are governed by entirely different 
laws; each has an observed range of 1,000,000 : 1 in the stars; and evidently 
they would not be equal in a star unless it had some means of adjusting 
them to agreement. Taking n — 3| we have shown above that the adjust 
ment is not made by L changing towards E because it would actually 
change away from E. If n = 2\ , L changes towards E but it cannot change 
by a factor greater than 4 without going beyond known stellar conditions. 
Thus in either case the main adjustment must be made by E changing 
towards L. 
In order to give the star stability E must increase as the star contracts 
so as to oppose the contraction, i.e. it must increase with p or T or with 
both. This condition was first pointed out by H. N. Russell. The threatened 
instability is with respect to a rather long time-scale and is not catastrophic. 
Unless the star keeps E and L closely balanced it will change density at 
* This argument has been criticised by J. H. Jeans ( Monthly Notices, 85, p. 792 ). 
He objects that since the star with E < L is changing its energy it is not legitimate 
to apply equilibrium equations. He further states that when E < L the star 
expands. It seems to be sufficient to point out in reply that E = 0 corresponds to 
the Kelvin contraction hypothesis.