338 
THE OUTSIDE OF A STAR 
is a not uncommon idea that the Fraunhofer spectrum is caused by a 
cloud of cooler matter which cuts out the photospheric radiation at 
frequencies for which it is opaque and substitutes the less intense radiation 
proper to its own temperature; this is argument (6). It seems best to 
examine the whole question by an analytical investigation, tracing the 
formation of absorption lines—under idealised conditions it is true, but 
not too unlike the actual conditions. 
We assume the radiation to have the equilibrium constitution with the 
necessary exception at the absorption line which is being studied*. We 
set J' for the flow of radiation of frequency v to v + dv in the absorption 
* There is no self-contradiction in assuming equilibrium constitution of the radia 
tion right up to the boundary of a star; by making the emission coefficient,/ a suitable 
function of v and T we can construct an ideal star conforming to this condition. 
In actual stars, however, the region above the photosphere is traversed by radiation 
which is beginning to deviate appreciably from equilibrium constitution, and it 
may fairly be asked whether our assumption does not differ so much from the actual 
conditions as to make the results misleading. I think it can be shown that our 
discussion represents fairly well the typical conditions of formation of actual absorp 
tion lines. We may divide the assumption into two parts in so far as it relates to 
(a) radiation contiguous to the absorption line, ( b ) more remote parts of the spectrum. 
The latter part seems to be a fair approximation, no more harmful in this connection 
than in §§ 241-243; it is employed in most standard investigations of the outer 
layers of a star. But the part (a) is more risky; in particular an incorrect assumption 
as to the density of the contiguous spectrum directly affects calculations of the 
contrast ratio or blackness of absorption lines. But this error is not systematic; 
the actual radiation is richer in high frequencies and poorer in low frequencies than 
the equilibrium radiation substituted for it, so that lines in the blue and in the red 
are affected in opposite ways. There is in fact an intermediate region of the spectrum 
where the assumption is substantially correct; its location can be found as follows. 
According to observation the radiation at the boundary is approximately equilibrium 
radiation for temperature T e except that its density is reduced to \ by lack of in 
flowing components; thus the density is 
^Cv 3 l(e hv l RTe - 1) instead of Cv s /(e flv l RT ° - 1), 
where T 0 = T e */£. The constitution below the surface cannot be determined with 
out a detailed knowledge of the emission laws, but presumably it follows the same 
kind of relation, i.e. approximately 
dCv 3 /{e h, '/ RTe - 1) instead of Cv 3 /(e h: i RT - 1), 
where T — T e £J9. The condition that the two expressions become equal is found to 
be hv/RT e = 3-6 to 3-9 for 0*= \ to 1. For the sun the corresponding wave-lengths 
are 6900 to 6400 A. Our results should be fairly correct for absorption lines near 
this part of the spectrum. This region can be shifted to any part of the observed 
spectrum by choosing a star of appropriate temperature. 
Part (a) of the assumption is employed when we set the ordinary continuous 
emission jpds equal to kJpds (below). By (225-41) the more general expression is 
j = k (J - dH/dr). Although dHjdr vanishes for the whole radiation it does not in 
general vanish when (as here) the symbols refer to radiation of a particular frequency. 
There would thus be an additional term - kdH/dr on the right of (234-1). Unless 
this is got rid of either by assuming equilibrium constitution or by choosing the 
region of the spectrum discreetly, the analysis becomes intractable. The assumption 
is also invoked in calculating the emission due to excitation by inelastic collisions.