272 



KNOWLEDGE. 



[Decembeb 1, 1894. 



than it should be if placed at the distance indicated by the 

 hypothetical parallax. I have computed the hypo- 

 thetical parallax and the corresponding magnitude of the 

 sun for all the binary stars for which an orbit has hitherto 

 been computed, and I find that in the great majority of 

 cases the star is brighter than the sun would be if placed 

 at the distance indicated by the hypothetical parallax. 

 This fact would imply tliat most of the binary star systems 

 — at least those with spectra of the solar type — have a 

 smaller mass than that of the sun, and are at a less 

 distance than that indicated by the hypothetical 

 parallax. This will appear from the following con- 

 sideration. If p be the parallax of a binary star, a the 

 semi-axis major of the orbit in seconds of arc, P the period 

 in years, m and ?»' the masses of the components, and M 



the mass of the sun, we have the formula m + m^ = -rp.. M. 



whence p= 



M. Now if III + i)i' = M, or the 



mass of the system is equal to the sun's mass, we have 

 /)= p^ , which is the well-known expression for the hypo- 

 thetical parallax. But if )« + ?«' = » M, we have, taking 

 From this it follows that the larger the 



M=l, n= 



p^ P"-- 



parallax^ is, the smaller is the mass of the system, and 

 conversely, the smaller the parallax the greater the mass 

 will be. Now, as in most cases the binary is brighter 

 than the sun would be if placed at the distance corre- 

 sponding to the hypothetical parallax, it follows that to 

 make the sun of equal brightness with the star it should 

 be placed at a less distance than that indicated by the 

 hypothetical parallax — that is, the parallax must be 

 increased, and consequently the mass of the binary star 

 diminished. This reasoning, of course, only applies to 

 those binaries which have spectra of the solar type, for 

 stars of the first or Sirian type are, as is well known, 

 much brighter than the sun is in proportion to their mass. 

 Let us first consider the binaries having spectra of the 

 solar type, in which I find some remarkable differences 

 between the star's apparent brightness and the sun's cor- 

 responding magnitude, if we assume that they have the 

 same mass as the sun. I have omitted those in which 

 this difl'erence does not much exceed two magnitudes. 



y Leonis. — This is the most remarkable of all the 

 binaries, having spectra of the second or solar type. Its 

 " relative brightness," calculated by a well-known formula, 

 is very high, being about 93 times brighter than ^ Ursae 

 Majoris taken as a standard, and which has also a 

 spectrum of the solar type, although slightly different 

 according to the Draper Catalogue, the spectrum of 

 y Leonis being noted as of class K and that of Ursse as 

 class G. According to an orbit computed by Dr. Doberck, 

 the hypothetical parallax of y Leonis is only 0036". 

 Placed at this distance, the sun would shine as a star of 

 only 8-29 magnitude. The magnitude of y Leonis, as 

 measured with the photometer at Harvard Oljservatory, is 

 2-24. There is, therefore, a difference of G-05 magnitudes, 

 which denotes that y Leonis is about 2G3 times brighter 

 than the sun would be if placed at the distance indicated 

 by the hypothetical parallax. The sun placed at a distance 

 corresponding to a parallax of 0'58 ' would appear of the 

 same brightness as y Leonis. But if the star had so large 

 a parallax it would probably have been detected ere this. 

 With a parallax of 0-58 ', I find that the mass of y Leonis 

 would be less than ^jrijoth of the sun's mass, which also 

 seems improbable. It is, consequently, very difficult to 

 give any satisfactory explanation of the great brilliancy of 

 this binary star. The accuracy of the orbit computed for 

 this pair (period 407 years) is of course very doubtful, as 



the arc described since its discovery has been small. Its 

 slow motion, however, indicates that the period must be 

 long. The spectrum is not exactly the same as that of the 

 sun, and the star may be hotter and brighter. 



Tt Cephei. — The spectrum of this star is, according to 

 the Draper Catalogue, of the second type, but doubtful 

 (H '?). According to an orbit computed by Prof. Glasenapp, 

 its hypothetical parallax is 0-032", which would reduce the 

 sun to a star of magnitude 8-G. The star's magnitude 

 measured at Harvard is 4-48, so that there is a difference 

 of 4-12 magnitudes, denoting that the star — if of the same 

 mass as the sun — is 44^ times brighter. To make the sun 

 equal in brightness to the star, the parallax should be 

 increased to 0-208", and the mass of the binary system 

 diminishad to . J^th of the sun's mass. But as the spectra 

 are not identical, perhaps the sun and tt Cephei are not 

 exactly comparable. 



[u Leonis. — In this case the hypothetical parallax is, 

 from Dr. Doberck's orbit, 022 , a parallax which would 

 reduce the sun to a magnitude of only 9-36. The star's 

 measured magnitude is 5-55, which gives a difference of 

 3-81 miignitudes. This would imply that the star is about 

 33^ times brighter than the sun, if both bodies were of 

 the same mass. To make the sun equal in brightness to 

 the star, the parallax should be increased to 0-127", and 

 the mass of the binary system diminished to yiT^th of the 

 sun's mass. The spectrum being, however, of class G of 

 the Draper Catalogue, the sun being of class F, the two 

 bodies are not perhaps strictly comparable. 



/? Delphini. — This is a binary star with a spectrum of 

 class F, or that of the sun, so that the two bodies should 

 be comparable in intrinsic brilliancy. From my orbit for 

 this pair — which cannot be far from the truth — the 

 hypothetical parallax is 0-052 ', which would reduce 

 the sun to a star of 7-49 magnitude. As the star was 

 measured 3-74 at Harvard, we have a difference of 3-75 

 magnitudes, denoting that the binary — if of the same 

 mass as the sun — must be nearly fifty times brighter. As 

 the spectrum is of the same type this seems improbable, 

 and we must conclude that the star's parallax is more than 

 0-052". The sun, placed at a distance corresponding to a 

 parallax of 0-292", would shine as bright as the star appears 

 to us. With this parallax, the mass of the binary would 

 be reduced to -f i-th of the mass of the sun. It seems 

 improbable, however, that so comparatively large a parallax 

 as 0-292" should remain undetected. 



r Ophiuclii. — From Dr. Doberck's orbit of this pair, the 

 hypothetical parallax is 0-033". At the distance thug 

 indicated the sun would be reduced to a star of 8-48 

 magnitude. The star's photometric magnitude is 4-93, 

 which gives a difference of 3-55 magnitudes, denoting that 

 the star would be — if of the same mass — about twenty-six 

 times brighter than the sun. A parallax of about 0-17" 

 would make the sun equal to the star in brightness, and 

 this parallax would reduce the mass of the binary to 

 -J J„th of the sun's mass. 



Z, Aquarii. — This binary has the longest period yet 

 determined, namely, 1578 years, according to Doberck. 

 Of course, this great length of period renders the accuracy 

 of the orbit very uncertain, but its slow motion and 

 comparatively great distance between the components 

 (a = 7-64") makes it an interesting object in the present 

 discussion, particularly as its spectrum is that of the sun 

 (F). With the above period and semi- axis major the 

 hypothetical parallax is 0050'', which would reduce the 

 sun to a star of 7-33 magnitude. As the photometric 

 magnitude of the star is 3-81, we have a difference of 3-52 

 magnitudes, or nearly the same as in the case of t Ophiuchi. 

 To reduce the sun to the brightness of the star it should 



