289 - 291 ] Binary Systems 323 
For equipartition of energy in the stars to be produced by any physical 
agency whatever, pressure of radiation, high speed electrons, molecular 
bombardment, or other physical agency of any kind whatever, the agency 
in question must have been in thermodynamical equilibrium with matter at 
a temperature of T 8 x 10 62 degrees. No such physical temperature is known, 
or can be imagined, in the universe, so that we must conclude that the 
observed equipartition arises from the gravitational interaction of the stars 
themselves, and this inevitably leads to ages of the order of those just 
calculated. 
The Dynamics of Binary Systems. 
291. In terms of the analogy with the Kinetic Theory of Gases (§ 271) 
we have treated the stars as molecules of a gas, and have investigated the 
time necessary for their velocities to approximate to the distribution specified 
by Maxwell’s law; we have in fact calculated what Maxwell describes as the 
“Time of Relaxation.” 
In terms' of the same analogy, binary systems may be treated as diatomic 
molecules. We have already found the distribution of orbits in the final 
steady state (§ 273), and have examined to what extent observed binary 
systems conform to this distribution. We can form a second estimate of 
stellar ages by calculating the time necessary to establish this approximation 
to the final steady state law of distribution. 
Let us first examine the effect of the forces from passing or distant stars 
on the eccentricity and period of a single binary system. 
The gravitational forces which an outside star of mass M 0 at x 0 , y 0 , z 0 
exerts at a point x, y, z of a binary system can be derived from a potential V, 
or >yM 0 /r, which can be expanded in the form 
v= T^o 
[(x 0 - xf + (y 0 - y) 2 + (z 0 - z) 2 f 
ryM 0 7 M . x 
= R + TF ( XX ° + yy ° + ZZ ^ 
+ [3 ( xx 0 + yy 0 + zz 0 ) 2 - (x 2 + y 2 + z 2 ) {x 2 + y 2 + V)] + • • • (2911), 
where R stands for {x 2 + y 0 2 + z 2 )K The forces are of the type 
X = x 0 + 0 (2x 0 2 - y 0 2 - z 0 2 ) + 3 yx 0 y 0 + 3 zx Q z Q ] + ... (291-2), 
and the total force at x, y, z will be the sum of a number of such forces, one 
from every star whose gravitational field of force is perceptible. 
The first term £ 7 M u x 0 /R 3 in the total force is constant over the whole 
system, and so merely gives rise to an acceleration of the system as a whole, 
without affecting the orbit of its components. The remaining terms represent