APPLICATION TO BINARY STARS. 153 



VII. APPLICATION TO BINARY STARS. 



Direct telescopic observations prove the existence of very many binary 

 stars. According to Hussey and Aitken about one star in 18 of those 

 brighter than the ninth magnitude is a visual double. There are many 

 stars whose spectra periodically consist of double lines. This phenomenon 

 is taken as indicating that these stars are binary systems, made up of 

 two approximately equal components, whose orbital planes pass nearly 

 through the earth. There are many more stars whose spectral lines period- 

 ically shift. This is interpreted as meaning that these systems are binaries 

 in each of which one component is relatively non-luminous. There are many 

 variable stars whose light curves are explained by supposing they are eclips- 

 ing binary systems. Unless the interpretations of these phenomena are 

 very much in error a considerable fraction of all the stars are binaries. 

 Granting that the interpretations are correct, the evolution of binary sys- 

 tems is a standard celestial process. We are raising the question here 

 whether these binaries may have originated from parent masses by fission 

 without external disturbing factors, and if so at what stage of condensation. 



Before considering the question of the fission of single masses into binary 

 systems, we shall write down some of the implications of the interpreta- 

 tions of the spectra of spectroscopic binaries, particularly as regards limits 

 on their masses and densities. 



Consider first the case where both spectra are visible. Let i be the com- 

 plement of the inclination of the plane of the orbit to the line of sight. 

 Suppose the orbit is circular. Let v be the maximum observed radial 

 velocity, mj and m^ the two masses, P the period of revolution, and a the 

 major semi-axis of the orbit. 



Then we have 



Pv ^ r. 27ra? , Pv^ , ,, 



r = 27ra P = — , ^1 + ^2 = 77-7^ -r-. (14) 



cost fc-^m, + m, 2;rfc^cos3* ^ ^ 



Since i can not be determined in a star which is not also a visual binary we 

 have, reducing the units so that when v is expressed in kilometers per second 

 and P in mean solar days, the sum m^ -1- m^ expressed in terms of the sun's 

 mass by the relation ^^^ 



wii + m^^^Pv' (15) 



Suppose the spectrum of only one star, m^, is visible, and let v^ be its 

 maximum observed radial velocity with respect to that of the center of 

 gravity of the system. Let r^ be the radius of its orbit, assumed to be cir- 

 cular, around the center of gravity of the system. Then we have 



Pv, _ , p 2;rr^Kwi + W2) 

 -27rri m^T^ = m^r^ a^r^-^r^ P = ^^^-3 



cost 



m,= 



whence ^ \ mJ (1^) 



^ 27rfc^ cos^ i 



