320 
The Ages of the Stars 
[ch. XII 
to neutralise one another. To obtain correct results we must stop off the 
integration before it brings us to values of a as large as this. 
We must accordingly choose the lower limit a so as to correspond to 
adjacent stars, and so to v By equation (285-4), this value of a is given by 
Assigning this value to a and putting ¡3 = ^rr, equation (287 - 3) becomes 
or, inserting the numerical values already mentioned, 
The time necessary for deflections less than a right angle to produce a 
resultant deflection equal to a right angle is obtained by putting T = r, and 
is found to be 
Comparing with formula (286-3) we see that this time is only about one- 
fortieth of that needed for a single encounter to deflect the path by a right 
angle. With the values already used it is equal to about 5 x 10 13 years. 
Estimate of Stellar Ages. 
288. It is now clear that changes in the direction of a star’s path, and 
hence also exchanges of energy and momentum, are produced far more through 
the cumulative effect of a large number of small encounters, than by the 
occurrence of single big effects. With conditions such as now prevail in the 
neighbourhood of the sun, it appears that after 5 x 10 13 years only about one 
star in eighty will have had its course turned through a right angle by a single 
close encounter with another star, but one star in every two will have had its 
course turned through a right angle by an accumulation of small effects. 
We are not free to imagine a period of 5 x 10 13 years with present condi 
tions prevailing uniformly throughout, for we have already seen that in a 
period of this length, the stars would have lost the greater part of their 
masses by radiation. If the observed approach to equipartition of energy is to 
be explained as the effect of stellar encounters in past ages, we have to go 
back to times when stellar masses were greater than now and other conditions 
were consequently different from now. 
The whole galactic system must be experiencing a decrease of mass at a 
rate comparable with that of the average star. If, as was at one time sug 
gested by Poincaré, the Milky Way may be treated as an encircling band of 
stars held in relative equilibrium by slow rotation, then the radius of this belt 
a distance of closest approach which is about equal to the distance between 
(m + m'f V 0 2 ' * 
t = 0026 
(m 4- m'Y V 0 l 
7 rvfm 6 
seconds 
(287-5).