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