267-269] Motion subsequent to Fission 257 



as in 62. An elliptic orbit will be described only if f remains sensibly con- 

 stant ; in this case the " mean motion " n is given by 



M+JU' 



this reducing to our former equation (121) when the orbit is circular. 

 The orbital momentum of the system is readily found to be 



where I is the semi-latus rectum, equal to a (1 e 2 ), whence, on adding the 

 rotational momenta, the total angular momentum is found to be 



M = Mfra + MWu' + MM ' -. (1 + )* J* ......... (583). 



Let us suppose that in the earliest stage of existence the components 

 rotate fairly close to one another with a common angular velocity in an 

 approximately circular orbit of radius R. In this case G> = o>' = n, and formula 

 (583) becomes 



M = jftF + M'k'* + fijjfi ^ 2 (1 + ?) ( M + ^O ^ " - 

 an equation which has already been given in 64. 



269. Consider first the extreme case in which the masses are supposed 

 homogeneous and incompressible. To obtain some idea of the ratio of division 

 of M into its rotational and orbital parts, I have calculated the ratios of the 

 separate terms in M for Darwin's figures of closest approach, the data being 

 those already tabulated in 65. The results prove to be as follows : 



4- 0-4 0-5 I'O 



M 



Rotational mom. of M' '039 '046 '077 



M 1 '160 -135 -077 



Orbital momentum '801 '819 '846 



Total 1-000 1-000 I'OOO 1-000 



With very few exceptions all known binary stars have values of M'jM 

 lying between 0'4 and I'O. Excluding the few systems for which M'/M is less 

 than 0*4, it appears that the orbital momentum must initially be at least 

 80 per cent, of the whole if the components move in circular orbits ; it would 

 of course be still greater if they moved in stable elliptical orbits. 



Thus no matter for how long tidal friction or other similar tendencies act, 

 the orbital momentum cannot, in the whole course of a binary star's history, 

 j. c. 17 



