58-6o] The Double-Star Problem 55 



DARWIN'S PROBLEM. 



60. In a paper of very great importance, Sir G. Darwin* has discussed 

 the double-star problem in the case in which both masses are supposed fluid 

 so that each is subject to distortion from the tidal forces generated by the 

 other. The discussion falls naturally into two parts the determination of 

 figures of equilibrium and the determination of the stability or instability of 

 these figures. 



In Roche's problem the secondary was assumed to be a rigid sphere, so 

 that its potential could be written down in the form of formula (88), 



In Darwin's problem, the secondary is a mass of fluid of a shape deter- 

 mined by the mutual tidal actions between the two bodies, and an expansion 

 such as the foregoing is no longer permissible. To a first approximation 

 both bodies may be regarded as ellipsoids. Darwin assumes the bodies to 

 be distorted ellipsoids and expresses the distortions in terms of ellipsoidal 

 harmonics. The amount of this distortion is found to be in every case quite 

 small, so that the supposition that the figures are actually ellipsoidal is 

 found to give a tolerably accurate solution. In illustration of this the 

 following figures may be quoted from Darwin's paper f; they express the 

 proportional increase &a/a in the semi-major axis of the primary which 

 would be produced by the removal of the ellipsoidal constraint when the 

 masses are at the closest distance consistent with stability (cf. 64 below). 



MJM'= 0-4 0-7 1*0 



Sa/a in direction towards secondary T 1 T -^ $ 



Ba/a in direction away from secondary % -^ -$. 



The amount of these corrections is shewn by the dotted lines in figures 

 11, 12 and 13 (p. 64 below). 



For configurations in which the masses are further apart than this 

 minimum distance the error in the ellipsoidal solution will of course be 

 less, so that the assumption that the figures are ellipsoidal is seen to give a 

 very fair approximation. 



* Phil. Trans. 206 A (1906), p. 161, or Coll. Works, in. p. 436. The actual paper occupies 

 88 pages in each place, so that it will be understood that only the merest outline of it is given 

 here. And, to avoid the complicated methods of ellipsoidal harmonic analysis employed by 

 Darwin, I have substituted a simpler discussion of the fundamental equations, deriving them 

 in a form analogous to the equations of Roche already discussed. 



t I.e. p. 510. 



