366 
THE OUTSIDE OF A STAR 
The deficit fr'cr must be made up by gas pressure; hence 
-^ = §t'<№ (254-3), 
or = | T'og/k', 
where k' is the monochromatic coefficient of absorption. 
In integrating this equation we can neglect the slight variation of k' 
due to the inert ions present. Hence 
Vo = lag r' 2 Ik', 
3 tigg-S* 
p 4 W'T K ’ 
Then dr’ = - k'pdx = - dx. 
Since T is practically constant* we have by integration 
4 WT 
t' = - , r (254-51), 
3 fjiag (x + Xq) 
where x 0 is a constant of integration. Hence by (254-4) 
" • -V".* £ t < 254 ' S2 >- 
The value of x 0 can be found roughly. By (226-2) 
* (Ji ~ J%) = 2 H\ \ (// + J 2 ') = J' = 2H' (1 + fr'), 
so that J x ' - 4H' (1 + fr') (254-61). 
For radiation just outside the H and K lines the absorption in the chromo 
sphere is negligible, so that the boundary values apply, viz. 
J x = № (254-62). 
By the definition of r, H' = rH\ hence J x ' will be equal to J x at a depth 
V given by , (1 + f V ) = 1, 
SO that T 0 ' = | (1 — r)/r (254-63). 
Clearly the chromosphere cannot have greater depth than this, for the 
outflowing radiation in the absorption lines cannot have greater intensity 
than the surrounding spectrum. Hence r 0 ' should be an upper limit to 
the optical thickness of the chromosphere. The theory is, however, some 
what of an extrapolation since in the lower part of the chromosphere other 
ions besides Ca + will be present and the simple conditions will no longer 
* If the only exchange of energy between molecular speeds and radiation is by 
the scattering of the free electrons, the whole radiation of the sun is equally effective 
and the chromosphere takes up the uniform temperature T 0 . There may, however, 
be some conversion of the radiation v 12 into molecular speeds ; in that case the speeds 
will increase according to the intensity of v 12 and the chromosphere will be slightly 
hotter towards its base.