291—293] 
Binary Systems 
325 
Differentiating with respect to the time, keeping m and m! constant, 
we obtain 
Thus to obtain the rates at which the period and eccentricity change 
We may calculate dE/dt as the rate at which the forces already evaluated 
do work on the two components of the binary system. Thus 
where X, Y, Z are the components of the force already evaluated, and the 
summation is with respect to the two components of the binary system. 
Inserting the value of X from equation (291'2) and integrating (by parts) 
throughout a complete encounter, we find that the increment of energy dE 
resulting from the encounter is 
of the passing star; the orbit of the binary enters only through the quadratic 
terms in 3 yz, etc. Thus if all other factors were the same, the change of 
energy dE produced by an encounter would be proportional to the square of 
same is true as regards dh, and so also as regards dP and de. These various 
293. The orbits of the spectroscopic binaries have dimensions of the 
order of IO 11 or 10 12 cms., whereas the analysis of § 286 shews that during 
a star’s whole life it is unlikely that there will be a single approach of 
another star to within a distance of 10 14 cms. Thus formula (292‘1) is entirely 
appropriate to encounters with spectroscopic binaries. 
On the other hand the orbit of the average visual binary has linear 
dimensions of the order of 10 1S cms. (representing a period of 390 years 
when each component is of mass equal to the sun). Thus within a lifetime 
of the order of 5 x 10 12 years, a number of stars are likely to pass so near to 
either component of a binary as to be momentarily nearer than the other 
component. In such a case formula (292T) does not apply, but the two 
1 ^ _ A dE 
P ~dt ~ 2E Ht ’ 
1 dh _ e de 1 dE 
h ~dt = ~ (•! - e 2 ) di + 2Ë ~dt ' 
during an encounter with another star we must calculate dE/dt and dh/dt. 
<292 ' 1) - 
In this formula, all terms of the type ^ {...} depend solely on the motion 
the linear dimensions of the orbit of the binary. It is readily seen that the 
formulae are only valid if the passing star keeps outside the orbit of the binary.