534 SCIENCE PROGRESS 



will be found possible to measure diameters even smaller than 

 this. Several other stars have been examined, which have 

 been found to have diameters less than this amount. 



Betelgeuse is a giant red star, and the above angular diameter 

 corresponds (on the best information available as to the 

 parallax of the star) to an actual diameter of about 300 times 

 the sun's diameter. Since the masses of the stars do not vary- 

 very greatly, its density must be extremely small. It is, in 

 fact, amongst the giant red stars of low density and large actual 

 diameter that the stars of measurable angular diameter may 

 be expected to be found. The probable diameters of the stars 

 have been theoretically discussed recently by H. N. Russell 

 {Pub. Ast. Soc. Pacific, 32, 307, 1920). Russell points out that 

 the angular diameter of a star can readily be determined if its 

 surface brightness is known. In fact, if d is the apparent 

 diameter of a star, m its visual apparent magnitude, and / its 

 surface brightness, it can easily be shown that — 



d oc io-°'^V~*- 



The constant of the proportionality can be found from the 

 data for the sun, if the sun's surface brightness is taken as 

 unity. Then, putting y= — 2-5 log/, so that j represents the 

 change in stellar magnitude corresponding to a change in 

 surface brghtness in the ratio / to i, we obtain — 



m-] 



d — o"'Oo87 (0-631) 



The angular diameter can be determined if / can be found. 

 This can only be done indirectly, the basic assumption involved 

 being that the star radiates as a black body. The difference 

 of surface brightness of two stars (when expressed in stellar 

 magnitudes) is proportional to the difference of colour indices : 

 if i is the colour index, 



k — h = k (4 — h) 



k is the same for all stars. If the colour indices are standardised 

 to a system in which the index for type ^o is zero and for type Ko 

 is + I 'GO, Russell shows, from various lines of argument, that 

 the value of k can be approximately taken as 4-0 ; j can then 

 be taken for any star as equal to four times its colour index 

 on the standard scale, and the angular diameters can then be 

 determined from the formula given above. If the parallax of 

 the star is known, its linear diameter can be calculated. 



The results obtained are at first sight rather surprising : 

 thus Sirius, the brightest star in the sky, whose angular 

 diameter Stephan attempted to measure, presumably in the 

 belief that its diameter would be relatively large, is found to 



