THE OUTSIDE OF A STAR 
329 
/ 
we introduce variations of the absorption coefficient in order to account 
for the difference between the broken curve and dotted curve in Fig. 5, 
we shall necessarily alter the centre-limb contrast and presumably make 
it less accordant with observation. 
The predicted energy-curve allowing for the spread of temperature 
but not allowing for any variation of k v with v is shown by the dotted curve 
in Fig. 5. It is based on Milne’s calculations. 
( b) Variation of k v . 
By (228-5) if k v is constant j v varies with v, and vice versa. There is 
no known physical hypothesis that suggests constancy of k v , but the 
hypothesis that j v is independent of v is plausible. 
On Kramers’ theory of absorption as developed for X rays in the 
interior of a star we found that j v was independent of v up to the guillotine 
limit (§157). It is a wide extrapolation to extend this to optical frequencies ; 
but since the general principles of optical and X ray absorption are the 
same, we shall adopt the hypothesis tentatively. We may say at once that 
it will not bring about the desired agreement with the observed energy 
curve of the sun ; but it is a proper first step to exhibit divergences from 
a curve which has a physical meaning rather than from a curve which 
corresponds to a purely mathematical abstraction with no physical 
interpretation. 
From the analysis in § 157 it follows that if j v is independent of v 
k„ 105 ft e” - 1 
k 128 a x 3 
(229-2), 
where x = hv/RT, ft = 1-151, a — 1-0823. The mean coefficient k in the 
present discussion is evidently the coefficient of opacity k 2 of § 157. From 
(226-5) and (226-61) 8 THT 8 Ax 
dr = 3TT = 3 X ° Â’ 
where x 0 = hv/RT 0 . Hence 
r„ = ijdr = ™Px 0 *l e ^dx (229-3). 
16 a 
•! X„ 
Having tabulated the integral of (e x — l)/x 8 we can calculate the values 
of x (and therefore T) for any optical depth t v and then proceed as in 
Table 43. 
The calculation would be long and it is doubtful if the labour of an 
accurate computation would be justified. A short method will give 
sufficient accuracy for our purpose. Although x varies as we descend in the 
star we shall be content to calculate kjk for a mean value x e = }ivjRT e 
and treat it as constant. By (228-2) and (226-5) the effective temperature 
is the temperature at an optical depth given by r sec 6=1. The effective 
temperature for a particular frequency v will also correspond very nearly