NATURE 



[September 12, 1901 



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Density and Figure of Close Binary Stars. 



When one star revolves round another in an orbit of which 

 the plane is coincident, or nearly so, with the line of sight, an 

 eclipse of one star by the other will take place every revolution. 

 We have, in such circumstances, the well-known phenomenon 

 of Algol variation. 



It is evident that the duration, extent, and nature of the 

 eclipse will depend upon the size and brightness of the com- 

 ponent stars, and on the magnitude, inclination, and eccentricity 

 of the orbit in which they move. 



In practice, of course, the problem is the indirect one of de- 

 termining the elements of orbital movement of the system from 

 the observed variation in the star's brightness. That is to say, 

 having ascertained, either photometrically or photographically, 

 the manner in which the light of the star varies, we determine 

 the physical conditions which have produced this variation. 



There is no field of astronomical research in the present day 

 so interesting, or so rich in future possibilities, as that which 

 deals with close binary stars. We have only to instance the 

 recent discovery, by Prof. W. W. Campbell, of the Lick Ob- 

 servatory, that at least one star in every five or six is a spectro- 

 scopic binary {Astrophysical Journal, vol. xiii. p. 89), as an 

 indication of the vastness of the field. 



If Prof. Campbell's estimate of the number of binary systems 

 be correct, then there ought, on the consideration that such 

 systems may revolve in any plane, to be at least 800 Algol 

 variables brighter than the ninth magnitude. At present only 

 twenty-two such systems are known to astronomers. 



Then, again, the certainty that in variable stars of the Algol 

 type — that is, binary stars revolving round one another almost 

 or actually in contact— we have the first stage in the evolution 

 of a stellar system, gives a unique interest to any investigation, 

 whether spectroscopic or photometric, which has as its purpose 

 the delineation of the conditions of magnitude and movement of 

 such systems. 



Of the many problems intimately related to a determination 

 of the elements of orbital movement of any close binary system, 

 two are, I think, of peculiar interest, as bearing directly on the 

 evolution of such systems. 



(i) When we have ascertained the size and brightness of the 

 component stars of any system, and also the form, position, and 

 magnitude of the orbit in which they revolve, we can directly 

 deduce the mean density of the system. A full investigation of 

 this and allied problems is given by Merian in Comptes rendiis 

 (122, 1254). 



A nomenclature adopted by the present writer, in the Astro- 

 physical Journal (vol. X. p. 308), meets, I think, more directly 

 the simpler conditions of the problem of the density of a close 

 binary star. 



Putting / = time, in days, of revolution ; 



r = semi-axis major of the orbit of the system ; 

 / = ratio of the radius of companion (i) to semi- 

 axis major ; 

 q = ratio of the radius of companion (2) ; 

 Wj = mass of companion (l) ; 

 «o = mass of companion (2) ; 

 A, = density of companion (l) ; 

 A., = density of companion (2) ; 



'(the sun's radius, mass and density are taken as unity) then 

 the simple relation 



O'oi^i; m. 



Ai = — -^ . ! — (I) 



p-'l- m^ + m.i 



_ 0-OI35 m„ 



q'f nil + w/2 



(2) 



will determine the values of Aj and Ao when the relative masses 

 of the two component stars are known. 



If p=q, 



then A, + A., = °'°'3i. . . . 



- ft'' 



NO. 1663, VOL. 64] 



It will be at once evident that, since 

 and 



(3) 



must always be less than unity, 



A, < °:?'35. 



A, ^ o:°L35 



These two relations express the limit in one direction of the 

 density of any binary system when the size of the component 

 stars, and the period of variation, has been ascertained from 

 an examination of the light-curve of the variable. In the 

 Astrophysical Journal (\o\. x. p. 315), Prof H. N. Russell, of 

 the Princeton University, from considerations similar to the 

 foregoing, deals with the light-variation of seventeen out of the 

 twenty-two Algol variables, deducing from their variation their 

 densities. 



He finds the mean density of the seventeen stars considered 

 to be 



019, 



the density o. water being unity. 



In the same _/<>«r«(j/ (vol. x. p. 314), the writer discussed the 

 light-changes of four southern Algol variables which had been 

 under observation for some years at Lovedale, South Africa. 



The me^n density of these four stars was found to be 0'I3 

 that of the sun. If the sun's density be taken as i -44 times the 

 density of water, then this result would yield as the mean 

 density of the stars considered, 



0-187. 



Since this article was written, two new southern Algol vari- 

 ables have been discovered — one at the Cape Royal Observatory 

 and the other at Lovedale. Further, a new photometric equa- 

 torial, specially constructed by Messrs. Cooke for variable star 

 work, has made it possible to secure observations, at Lovedale, 

 of all the eight southern Algol variables of considerable accuracy. 



A reduction of these observations has just been completed, 

 and an examination of the results gives, as the mean density of 

 all the southern Algol variables at present known, viz. eight, 

 the value 



0-176. 



Of the three investigations just given in brief, the first two 

 were independently conceived, carried out, and completed. Yet 

 the results are practically accordant. 



There is just the barest possibility that this agreement may 

 be fortuitous ; such a remote possibility exists in all investi- 

 gations. 



It is much more probable, however, that the agreement 

 between the results indicates the truth of the conclusion. And 

 this conclusion is that the average density of close binary stars 

 — that is, of bodies just forming, by the compulsion of their 

 inherent forces, into a dual existence — is one-sixth that of water 

 or one-eighth that of the sun. 



It is not the purport of the present paper to follow the 

 investigation to its legitimate termination — that is, to discover 

 in what agreement is the result just obtained with the theoretical 

 conditions of density consonant with a rotating ellipsoid on the 

 limit of bipartition. .\ broad general agreement, however, is 

 evident even on an elementary judgment, for if the result had 

 been that the average density of close binary systems, or of the 

 actual density of any one system, was, say, much greater than 

 that of the earth, then it would be difficult to understand how 

 separation could take place under these conditions. On the 

 other hand, no violence is done to our appreciation of what is 

 reasonable when we find that all close binary systems have a 

 density much less than water ; in some single cases, indeed, we 

 meet with densities approaching that of a gaseous nebula. 

 Such a condition of tenuity is evidently favourable to disruption. 



Any investigation of the light-changes of an Algol variable, 

 having extreme accuracy in view, must of necessity consider the 

 form of the stars alternately eclipsing one another. 



It is evident that the rate of decrease or increase of eclipse 

 will be more rapid the more ellipsoidal in figure the component 

 stars are. 



In the case of two spheres eclipsing one another, the amount 

 of obscuration, or the total amount of light emitted by the 



