322 The Ages of the Stars [ch. xii 
The time necessary to produce any specified change S'F 2 in T 2 is 
accordingly 
i = 2Sr* (1 (289-2). 
No matter how great S'k 2 may be, this time can never exceed 1/2 aM 2 , 
which, as formula (2891) shews, is the time since the typical star was 
of infinite mass. 
If we take M = 2 x 10 83 , the value of a already given makes this time' 
equal to 8 x 10 12 years, while the value of C is found to be 2 x 10 13 years, 
from formula (287'4). With these numerical values, formula (289’2) be 
comes 
t = (1 — e- 0 ’ 45 * 2 ) x 8 x 10 12 years, 
from which we may calculate the following values for t (in years): 
q/ = 30° 60° 90° 180°, 
i=2 x 10 12 3 x 10 12 5 x 10 12 8 x 10 12 . 
290. It is by no means clear what value of T - will best represent the 
degree of approach to the final steady state which is shewn by the velocities 
of the stars, the more so as the actual velocities seem to conform much 
better to the steady state law than the distribution of their directions. The 
final steady state is of course only given by 'F = oo , but T' = 180° ought to 
give a very good approximation to it, and possibly something of the order of 
M5 r = 90° would represent the observed degree of approach. Without specifying 
the actual angle, we may say that the calculations just given indicate that 
a time of the general order of 5 x 10 12 years would suffice to bring about the 
observed degree of approach. This time must probably be extended sub 
stantially to allow for the fact that the stars spend part of their lives in 
regions in which the star-density is far less than that we have assumed. 
Such calculations as the foregoing can lay but little claim to accuracy, 
but are important as providing positive information as to the actual ages of 
the stars. In § 118 we calculated the time needed for a star to radiate away 
a specified amount of its mass, and this gave the age of the star if we 
assumed that it had originally been far more massive than now. But we are 
now in possession of a means of estimating the time, at least as regards 
order of magnitude, throughout which the stars have actually existed. 
For, unless the observed approach to equipartition of energy is a pure 
accident, which is almost incredible, it can only have been produced by 
gravitational encounters between the stars themselves. If the kinetic energy 
of the stars is interpreted as a physical temperature, the value of H already 
calculated shews that this temperature must be of the order of 1’8 x 10 62 
degrees centigrade, and this figure amply rules out all possibility of the 
approach to equipartition having resulted from the action of physical agencies.