342 
THE OUTSIDE OF A STAR 
Secondly, k'/k > 133. Very high absorption coefficients are required. 
The element producing the line can scarcely be expeoted to constitute 
more than 1 or 2 per cent, of the whole material ; h is probably of the order 
100 to 1000; so that the absorption coefficient of the pure element for the 
monochromatic radiation must be 10 6 or 10 7 . 
If the material does not extend to the surface but stops say at r x = 0-2 
we find from (236-2) that for jp — 0 
so that however intense the absorption below r x = 0-2 the blackening is 
not more than 1:3. The filling up of the line is caused by the photospheric 
emission above t x = 0-2, but this emission is about double the emission 
from the same stratum in neighbouring parts of the spectrum. This is 
because the material of high opacity which backs it acts very much like 
a mirror. It stops radiation of the frequency of the absorption line from 
going deeper into the star, so that it all has to come outwards. 
Finally, consider a thin layer of material of great opacity near the 
boundary. From (237-3) we find that when pr x ' -> 0 
This holds only when jpr x is small and r x then is much smaller. As r x 
increases, the value must tend to the limit (238-1). 
The main points which emerge are that very high absorption coefficients 
are required to give strong blackening of the lines and that the blackness 
increases as the square root of k'. Also greater contrast than 1 : 3 cannot 
be produced by absorption below r = 0-2 however strong. For this reason 
we think that the reversing layer, i.e. the region which is most effective 
in determining the darkness of the absorption lines, should not be placed 
much lower than r = 0-2. 
Estimates of the blackness of observed absorption lines are at present 
rather contradictory. Kohlschiitter and Shapley obtain a contrast of 
1 : 3 or 4 in the strongest lines ; Schwarzschild about 1 : 10 ; H. H. Plaskett 
1 : 10 for faint lines*. 
It is clear that we must be able to detect lines with a contrast-ratio of 
less than 1:2; otherwise the double spectra in spectroscopic binaries 
could never be observed. The usual estimate is that a line just becomes 
observable when the intensity in it is of the intensity of the surrounding 
spectrum. 
The theoretical difficulties increase if very high contrast is insisted on, 
since that may involve impossibly high absorption coefficients. The difficulty 
= 0-32 (238-3), 
(238-4). 
* C. H. Payne, Stellar Atmospheres, p. 51.