384 
Variable Stars 
[ch. xv 
period of 5‘366 days to zero in about four and a half million years, which 
again would suggest a duration of the whole variability of the order of 10 6 or 
10 7 years. 
356. As we have already noticed (§ 48), the main physical feature of the 
variation is a fluctuation in the star’s visible light rather than in its total 
emission of radiation, and this requires a change in the star’s effective tempera 
ture which shews itself observationally as a fluctuation of spectral type. This 
fluctuation might either arise solely from surface causes or from deep-seated 
events affecting the whole star. A large mass of observational evidence favours 
the latter alternative. Interferometer measurements indicate that the angular' 
diameter of Betelgeux changes with its light-variation, a range of over 25 
per cent, in all having been already recorded, and what is true of one long- 
period variable is probably true of all. And the spectral lines of Cepheids shew 
periodic advances and recession which, if interpreted in the most obvious way, 
indicate periodic changes in the star’s radius. If this is the correct interpreta 
tion, the radial velocity, integrated through a half period, must give the total 
change in the radius of the star. This change of radius is found to be as large as 
8 | million kilometres for l Carinae and over 6 million kilometres for X Cygni; 
for Cepheids in general it averages a million kilometres*, the average radii 
of the stars themselves, as determined from the luminosities and effective 
temperatures, being of the order of 20 million kilometres. 
The radial velocity shewn by the spectral lines of Cepheids was at first 
supposed to arise from orbital motion, the Cepheid being regarded as a binary 
system of which only one component gave a visible spectrum. Many lines of 
evidence now make this interpretation untenable. In 1918 Shapleyf adduced 
arguments to prove that the then prevalent view of Cepheids as binary systems 
must be discarded, and that they ought rather to be regarded as single spherical 
stars in a state of pulsation or oscillation. A similar suggestion had been put 
forward by Plummer^ some years earlier in respect of the short period cluster- 
variables. Some of the consequences of this view of Cepheid variation are in 
good agreement with observation although, as we shall see, they could equally 
well be deduced from a somewhat wider view of the cause of the variation. 
357. From Poincare’s Theorem, we have found (§ 62) that the mean 
velocity of thermal agitation inside a gaseous star of mass M and radius R 
must be of the order of ( yM/R )i or of 2R(%Tryp)l, since M = §TrpR 3 . The 
slowest dynamical oscillation of a spherical mass has a period approximately 
equal to the time needed for a wave of compression to travel the length 2 R 
of a diameter. As the velocity of such waves is about equal to the velocity of 
thermal agitation, the slowest dynamical oscillation must have a period of the 
order of i^nyp) or say (yp)~K Thus the period P must be equal to (7 p)~^ 
* Eddington, The Internal Constitution of the Stars, p. 182. 
t M.N. lxxix. (1918), p. 1. 
+ Ibid, lxxiii. (1913), p. 665, lxxiv. (1914), p. 662, and lxxv. (1915), p. 575.