174 
[CH. VI 
The Evolution of the Stars 
The curves along which T e has constant values are found to lie somewhat 
as shewn in fig. 15, No two of these curves can ever intersect since the value 
of T e is uniquely determined by equation (158'3). It is readily verified that 
as M increases, f(M) increases more rapidly than M, so that the curves are 
convex to the axis of M at every point. 
A general idea of the shape of the curve is obtained by returning to the 
rough approximation E <x M 3 , which we found (§ 118) to represent the relation 
between E and M with tolerable accuracy for actual stars. This gives a 
moderately good approximation to the form of f (M) but gives a better ap 
proximation still to the luminosities of actual stars, so that it already takes 
partial account of deviations from the gas-laws. The law E oc M 3 was tested 
only for stars on or near the main sequence, and so contains no temperature 
factor. To extend it to stars of all effective temperatures, we write it in the 
form 
/ T \ 0,8 
E=ccM 3 ( 7 ^ ) (158-5). 
Here T em is the effective temperature of the main-sequence stars of 
mass M, and a is a constant. The dependence of E on temperature is that 
given by equation (158"3) while the dependence of E on mass is that given 
by our empirical law E oc M 3 , so long as we limit ourselves to main-sequence 
stars. The law is neither very definite or very accurate, but it serves for a 
general discussion of the kind in which we are now engaged, in which 
simplicity is more important than precision. 
When the emission is given by equation (158-5), the curves of constant 
effective temperature approximate closely to the cubical parabolas 
E 
= constant, 
since the changes in ( T em ) 0,8 are slight in comparison with those in M 3 . In 
fig. 15 the curves actually drawn are cubical parabolas. 
159. We now attempt to trace the evolutionary path followed by a star 
in this diagram as the emission of radiation causes its mass to diminish. 
Consider first the ideal case in which the star consists entirely of one 
single type of material w r hich liberates energy at a given constant rate per 
gramme. In this case the star’s emission E is always exactly proportional to 
its mass M. Thus throughout its evolution 
~ = cons (159-1). 
The star’s path in fig. 15 is determined by the condition that E/M shall 
remain constant, and so is a straight line through the origin such as AO. A 
glance at the diagram shews that the effective temperature of such a star 
would continually increase as the star aged, its spectrum passing through