IONISATION, DIFFUSION, ROTATION 287 
The material of the rotating star does not find its state of equilibrium 
by the methods which the mathematician might employ. Its motto is 
solvitur ambulando. At present we have not got so far as the solvitur, but 
we can speak confidently as to the ambulando. 
200. The general problem of radiative equilibrium of a rotating star 
has been treated by E. A. Milne* and H. von Zeipelf. The former adopted 
the approximation e = const, and the latter adopted the condition (198-1) 
which we have been discussing. Although (198-1) cannot be accepted with 
its original interpretation as a law of distribution of the subatomic source 
of energy in actual stars, we may regard either Milne’s or von Zeipel’s law 
of e as a sufficient first approximation on the same footing as our approxi 
mation Jcr) = const, for non-rotating stars. Presumably the error arising 
from the inaccuracy of the approximation will be limited as in § 91. 
Milne finds that the effect of rotation on the apparent brightness is very 
small. Imagine a typical star to be set rotating so fast that its equatorial 
radius is elongated 10 per cent.; then the luminosity will decrease 2-5 per 
cent. J No great stress is to be laid on the precise value, since no allowance 
was made for the change of k due to the alterations of density and tempera 
ture. The significance of the result is that deviations of individual stars 
from the mass-luminosity curve due to their different speeds of rotation 
will be very small. 
Milne finds further that in the star considered the effective temperature 
at the poles is 6-4 per cent, above the mean and at the equator 3-2 per cent, 
below the mean, so that the poles are brighter than the equator. This 
variation of surface brightness is found more simply in von Zeipel’s papers 
where it is deduced as a simple consequence of (198-8). Since/ (</>) = G, we 
have by (198-42) *, 
H = — C-¥ = — Cq (200-1). 
dn 
The analysis breaks down near the actual surface of the star, but we note 
that, as usual, the surface value of H must be continuous with its value 
a few thousand kilometres below the surface. Hence 
Hoc g (200-2), 
where g (= df/dn) is the value of gravity including centrifugal force §. 
The variation of brightness over the surface of a rotating star corresponds 
exactly to the variation of gravity. 
* Monthly Notices, 83, p. 118 (1923). 
| Monthly Notices, 84, p. 665 (1924). J Loc. cit. p. 139. 
§ The result H <x g for a rotating star was first given in the case of stars of very 
large mass by Jeans ( Monthly Notices, 79, p. 330). He, however, there insisted that 
the deduction was only applicable if the star had no source of energy other than 
contraction; and he has abandoned the result (for reasons not stated) in a later 
paper {ibid. 85, p. 935).