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.