136 



SCIENCE 



[N. S. Vol. XLIX. No. 1258 



have just seen that it also gives with an 

 increase in diameter by a factor of 2.7 or of 

 volume by a factor of about 21. Hence it 

 follows that Cepheids of any given period 

 have about five times the mass of those of half 

 the period. This conclusion (which holds 

 good only for the Cepheids proper, and not at 

 all for the cluster variables) depends to some 

 degree upon the assumed relation betwen color 

 index and surface brightness; but it is hard 

 to see how any juggling with the data can 

 escape from the conclusion that Cepheid va- 

 riation sets in, if at all, at different stages of 

 evolution among stars of different masses — 

 the most massive stars reaching the critical 

 condition, (whatever it may be) at the lowest 

 density. 



Among the cluster variables, which are very 

 similar to one another in color and brightness, 

 . and hence probably in diameter, both density 

 and mass must be greatest for the stars of 

 shortest period. This reversal of the relation- 

 ship may be connected with the fact noted by 

 Shapley, that the absolute magnitude of these 

 stars mark an inferior limit of luminosity, 

 below which Cepheid variation appears not 

 to occur. 



To attempt a numerical estimate of the 

 densities and masses of these stars is pre- 

 carious, as we do not know the exact nature 

 of the mode of pulsation. But the assump- 

 tion seems plausible that for a star of given 

 density, the period of the Cepheid pulsation 

 is not likely to be very different from that of 

 the fundamental oscillation of a gaseous mass 

 of the same density. According to Emden, 

 this period would be two hours for a mass of 

 the density of the sun. This leads to the 

 rough -estimate that a Cepheid of 7 days 

 period is of about 1/7,000 of the sun's density. 

 With the diameter previously estimated this 

 would mean a mass 2.7 times that of the sun. 

 A Cepheid of Si days' period would have about 

 half the sun's mass, and one of 40 days' period 

 about 150 times the sun's mass. A cluster 

 variable of 12 hours period would be of 1/36 

 the density, and 2 J times the mass of the sun. 

 These numerical values are extremely un- 

 certain but it is of interest to note that they 



appear to be of quite a reasonable order of 

 magnitude. The masses calculated for the 

 Cepheids of moderate period, which are bril- 

 liant giant stars, are probably rather low — 

 which suggests that the Cepheids of long period 

 are really stars of exceptionally great mass, 

 as would follow, on Eddington's recent theory, 

 from their exceptionally great luminosity. 

 Eddington has very recently called attention 

 to another important deduction that may be 

 drawn from the study of variables of this 

 type. The period of Delta Cephei is shorten- 

 ing by about one second in twenty years. On 

 the pulsation hypothesis, this would indicate 

 a gradual increase of density, but at so slow a 

 rate that it would take the star three million 

 years to double in density, and probably ten 

 million to pass from class G (its present type) 

 to class F. 



If studies of the secular variation in period 

 of other Cepheids confirm this date of change, 

 there will be direct evidence that the rate of 

 stellar evolution is exceedingly slow and that 

 the time scale of cosmieal processes is of very 

 great length. 



But what time remains must be devoted to 

 the other classes of variable stars. The irregu- 

 lar variables need detain us but for a moment, 

 for, beyond the fact that their variations are 

 usually of small amplitude, their spectra 

 practically all of classes M or N, and their 

 luminosities probably comparable with those of 

 other giant stars of these types, we know 

 practically nothing about them. 



The variahles of long period form a very defi- 

 nite natural class, with periods ranging from 

 about SO to 800 days, but exhibiting a nearly 

 normal frequency distribution about a mean 

 value of 300 days. The range of variation is 

 much greater than for any other class of va- 

 riables except the Novae, averaging about 

 four magnitudes, and sometimes reaching 

 seven or eight — a change of a thousand-fold 

 in light. 



They are far from being regular time-keep- 

 ers. For almost all the stars that have been 

 carefully followed, the .times of maxima de- 

 viate from any uniform period by much more 



