524 



SCIENCE 



[N. a Vol. XXXIV. No. 877 



Then s = rir sin i, w = vir sin i,, and by gravi- 

 tational theory 



--KHI 



(-0. 



where the constant K- is 39.7 of the sun's mass; 

 the astronomical unit and year are taken as units. 

 Hence 



:39.7J^/s^n^sin^ 



(-a- 



The last three factors are unknown, but their 

 average values can be predicted on principles of 

 probability. This gives the formula 



M = 



IStt^ 



(1) 



The average of the values given by (1) for a large 

 number of stars will be practically correct. The 

 percentage of individual cases in "which the error 

 of this formula may be expected to lie between 

 given limits has been calculated, showing that the 

 mean derived from ten cases will probably be 

 •within 10 per cent, of the truth. 



4. This method has been applied to all available 

 physical systems brighter than the sixth magni- 

 tude, 276 in all, raising the whole number of stars 

 discussed to 349. All the principal spectral types 

 are "well represented. 



For about half these stars, the relations between 

 mass and surface brightness are very similar to 

 those already found among stars for which orbits 

 have been computed. The other half (including 

 all the stars of type B and some of every other 

 type) are very much trighter in proportion to 

 their mass than those previously studied. These 

 stars are probably similar to the stars of great 

 luminosity to which Hertzsprung has called atten- 

 tion under the name of ' ' giant stars. ' ' The 

 others may be called ' ' dwarf stars. ' ' In type A 

 the two kinds run together, but among the redder 

 stars they are more and more widely separated, 

 though a few intermediate cases exist. 



Among the giant stars the relation of mass and 

 brightness is much the same for all spectral types, 

 the values of pJ-^/^ ranging from 0.004 to 0.002 

 (the sun being the standard). Among the dwarf 

 stars the brightness falls off very rapidly with 

 increasing redness. p/~V- being 0.03 for type A, 

 0.30 for F, 1.2 for G, 4.8 for K, and rising to 

 over 2,000 for certain faint stars of large parallax 

 whose spectra appear to be of types Ki or M 

 (with the exception of one aberrant star of type 

 A). 



5. The average masses of the giant and dwarf 



stars of each spectral type have been determined 

 from the parallactic motions (assuming constant 

 luminosity for the stars of each group 25 to 60 

 in number). 



All the giant stars appear to be similar in mass 

 — a system of this sort having about 10 times the 

 mass of the sun. The average masses of the 

 dwarf stars decrease with increasing redness 

 — that of a system of type F being some three 

 times the sun 's mass, and of one of type K rather 

 less than the sun's. 



The average light emission of a pair of giant 

 stars is from 150 to 250 times that of the sun 

 (the latter for type B). That of the dwarf stars 

 diminishes from 30 for type A to 4.5 for F, 1.3 

 for G, 0.3 for K and 0.01 for the faint stars 

 already spoken of. It appears therefore that the 

 more massive stars are by far the brightest. 



6. The average densities of stars of types B, A 

 and F can be found with the aid of Algol-variables 

 of these types, and the surface brightness may 

 then be deduced from the values of pJ'^/- already 

 found. 



Allowing for the fact that in the average Algol- 

 variable the brighter star is decidedly the smaller 

 of the two, the following values are found — those 

 for type F being uncertain, and the sun being 

 throughout the standard. 



Density 0.04! 0.09 0.13 



Surf ace brightness 4.5? 7.2 15.0 



Dwarf Stars 

 A F 



0.45 0.6? 

 6.1 1.7? 



The very faint stars of spectra K5 and M, even 

 if of ten times the sun 's density, can not exceed 

 1/30 of its surface brightness. 



These values agree closely with those derived 

 from the work of Wilsing and Scheiner on stellar 

 temperatures (based on the distribution of energy 

 in the spectrum). 



They afford an independent confirmation of the 

 hypothesis that the effective surface temperature 

 of a star is the principal factor which determines 

 its spectral type. 



7. Assuming that the surface brightness of giant 

 and dwarf stars of the same spectral type is the 

 same (and interpolating values for spectrum K 

 with the aid of Wilsing and Scheiner 's results) 

 it is found that the mean density of the giant 

 stars increases steadily with decreasing redness 

 from less than 1/10,000 that of the sun for type 

 M, and 1/1,000 for type K, to 1/8 for type B. 

 That of the dwarf stars increases with increasing 



