APPLICATION TO BINARY STARS. 155 



If they are separated so that their surfaces are at a distance from each 

 other equal to one-half the sum of their radii, i.e., if « = ^, equation (23) 



gives 7.4^(7^1.8 



According to Darwin's results, loc. cit., p. 232, this is about the minimum 

 distance at which homogeneous masses could revolve in stable equilibrium. 

 If they were separated farther the already high limits on the density would 

 be still greater. It follows that we must conclude either that these very 

 short period binaries and eclipsing variables are very dense, or that hetero- 

 geneous masses are more stable than homogeneous ones. 



Now let us return to the question of fission of binary stars. Denoting 

 the periods of rotation of m^ and Wj by Dj and D^ respectively, we have for 

 the total moment of momentum of the system ^ 



(2;r)Hwii + W2)^ i>i I>2 



The signs of the second and third terms in this equation are determined 

 under the hypothesis that both bodies rotate in the direction in which they 

 revolve. But this is a necessary consequence of the fission theory, and 

 therefore an allowable assumption in testing it. 



If P > Di and if P > D2 then the mutual tides of the two bodies tend to 

 bring P, Dj,, and Dj eventually to the same value. In the case of widely 

 separated visual binaries the fission theory implies that the tidal evolution 

 has proceeded far, and that Dj and Dj are closely approaching an equality 

 with P. If these conditions are satisfied and if Wj and m^ are approximately 

 equal, we see from (24) that the inequality 



M> ,,yi^',. (25) 



is nearly an equality. For example, in the case of a Centauri, assuming that 

 each component has the dimensions of the sun and that D^^D^^P approx- 

 imately, the ratio of the first term to the sum of the other two is roughly 

 10,000 to 1. In general, the greater P and the more nearly equal m^ and 

 m^, the more nearly the inequality (25) approaches an equality, and the 

 opposite. 



Let us suppose the two stars were originally in one spheroidal mass 

 w = w?i + Wj. Then equation (6) gives the relation between its density and 

 oblateness, which reduces by means of (25) to 



,f,W^»P^ ,<(l+ff!|(3+J2te,-U-3} (26) 



6(37r)HWj + W2) X- [ X j 



Now let — -=!i', then equation (26) gives 



Wj 



Letting /(//) = ^^ V^' we find for n = \ that -|^ = 0, -|^>0 and that -j- 



*See equation (6), p. 85. 



