293, 294] Mass-ratio in Binary Systems 327 
TABLE XXYI. Mass-ratios in Spectroscopic Binaries (Aitken). 
Number 
of Stars 
Spectral 
Class 
Average 
Period (days) 
Average 
mlm' 
Range of 
m/m' 
16 
B 
5-16 
073 
0-39 to 1-00 
29 
A 
9-58 
0-69 
0-17 to 0-99 
11 
F 
8-73 
0-93 
0-51 to 1-00 
10 
O 
9*06 
0-89 
0'71 to 1-00 
1 
Gb 
5-4 
0-88 
0-88 
Owing to the emission of radiation, the masses m, m' of the components 
of a binary system will change in accordance with the equation 
dm 
dt 
= - C*E. 
Simple differentiation gives 
d . fm\ _ 1 dm _ 1 dm _ _ ~ 2 (E _ E' \ 
dt °g {m ) m dt m dt \m my 
C*(G-G')... (2941). 
Here G and (?' are the rates of generation of energy, in ergs per gramme, 
of the two components. In spectroscopic binaries the more massive com 
ponent has almost invariably the larger value of G, so that equation (294’1) 
shews that, as a spectroscopic binary ages, the ratio m/m of its masses ought 
continually to approximate to unity. Table XXYI reveals a definite, although 
not very marked, tendency in this direction in actual binaries. 
If we like to introduce definite assumptions, we can calculate the ages of 
stars of different spectral classes from the extent to which their two com 
ponents have approached equality of mass. Let us, for instance, assume that 
the emission of radiation is given by our approximate law L= M 3 , so that, 
as in § 118, the mass M at any time t is given in terms of the mass M 0 at 
time t = 0, by the relation 
M = M 0 (1 + 2atM 0 2 )~K 
The ratio of the masses of the two constituents of a binary is now 
given by 
(m \ 2 _ /m 8 \ 2 1 + 2 atm 0 ' 2 
\m) Uo/ 1 + 2a£m 0 2 ’ 
Solving for t we find that the time-interval between mass-ratios m 0 /m 0 ' 
and m/m is given by 
2 at = 
(294-2). 
As an illustration let us calculate the time necessary for an A or B type 
binary of mass ratio 0-70, whose smaller component has five times the mass of 
the sun, to change into an F or G type binary with a mass ratio 0‘90. Putting