Variable Stars 
383 
353-355] 
are probably some 100 million stars. Shapley * finds that only about 50 known 
Cepheids lie within this distance of the sun. Thus probably only about one 
star in a million is a Cepheid variable. 
The proportion of long-period variables is necessarily even smaller, since 
less than one star in a million has the luminosity of the long-period variables 
and of the stars of the requisite luminosity only a fraction are variables. At 
a guess perhaps about one star in ten millions is a long-period variable. 
355. In searching for a clue to the physical interpretation of variability 
we naturally consider first whether it is the effect of some peculiar accident 
which happens only to a few stars, or whether it is a normal condition which 
affects all or many stars in the course of their natural development. 
The only accidents which we can imagine are of the nature of collisions or 
of close approaches by other stars. Calculations have shewn (§ 286) that actual 
collisions must be excessively rare, while encounters at distances close enough 
to produce physical effects in the stars can be only one degree less so. It is 
difficult to estimate for how long the effects of a collision or encounter would 
continue to produce a variation in a star’s light, but unless we allow a very 
long period indeed the number of variable stars in the sky would appear to be 
too great for such an origin to be assigned to their variability. Moreover, stars 
of all masses and of all luminosities would be equally liable to accidents of 
this type, so that, on this hypothesis, it would be hard to explain why, as a 
rule, only stars of high mass and high luminosity shew variability. On the 
other hand, it should not be overlooked that the cluster-variables are especially 
frequent in globular star-clusters (from which they take their name), and that 
in these the stars are packed so closely that near encounters must be far 
more frequent than in the galactic system as a whole. 
Nevertheless, surveying the question as a whole, it seems very improbable 
that variability can be attributed to the occurrence of an accident, and we seem' 
forced to conclude that it is more of the nature of a passing phase in the 
normal development either of every star or at least of a considerable proportion 
of stars. In favour of this view is the circumstance that most, and possibly 
all, of the if-type giants of high luminosity are found to be long-period 
variables. This of itself would almost suffice to rule out the accident theory, 
at least in its application to long-period variables. 
If variability is a phase in the development of the normal star, the figures 
given above shew that it must be of short duration. If a star’s average life is 
10 13 years, and only one star in a million is variable at any given instant, then 
variability can only last for about 10 7 years. Hertzsprungf finds that the 
period of the typical Cepheid variable 8 Cephei is decreasing at the rate of a 
tenth of a second per annum. This rate of decrease would reduce the present 
* Astrophys. Journ. xLvm. (1928), p. 279, or Mt Wilson Contribution, No. 151. 
f Observatory, xlti. (1919), p. 338.