232 
THE COEFFICIENT OF OPACITY 
160. We shall now try to calculate the reduction of the coefficient of 
opacity when the guillotine is placed at too low a frequency to be neg 
lected. 
We have agreed that the emission consists of the classical spectrum 
Qdv extending from hv = 0 to 
hv — |raF 2 + ifj (160*1), 
where in an ordinary mixture of elements 0 may generally be taken to 
represent the average energy-level down to which the atoms are ionised*. 
We could substitute RT for |mF 2 since this is its mean value allowing for 
the greater frequency of capture of the slower electrons. This approxima 
tion gives the emission and absorption quantitatively ; qualitatively there 
is a certain amount of shifting of the frequencies, but in a mixture of 
elements this cancels out to a large extent and for most purposes it is a 
fair approximation qualitatively. But if we use this representation to 
calculate the opacity, its qualitative defect becomes conspicuous. It 
leaves a region of the spectrum perfectly transparent ; and if any region is 
transparent the mean opacity of the whole is zero. There is, of course, no 
danger in actual stars of very high transparency for any frequency; even 
if Kramers’ absorption left a window, electron-scattering would prevent 
the transparency exceeding a moderate limit. However, by treating 
Kramers’ absorption a little more carefully we can avoid introducing this 
spurious high transparency. 
If y = è m ^ 2 > ^e number of free electrons with energy between y and 
y + dx is proportional to e~ x l RT V dy. Remembering that the emission per 
electron in a range dv is proportional to 1/F, the emission from electrons 
between y and y + dx is proportional to e~ x / RT dx- This gives the relative 
intensity of the partial spectrum contributed by electrons of energy y, and 
in accordance with the previous discussion we take it to extend -with this 
uniform intensity up to frequency (y + ip)/h and there terminate. The 
total intensity at (y + ip)/h is obtained by integrating over those partial 
spectra which extend up to or beyond this frequency; the result is pro 
portional to .a, 
e ~xl RT dx 
'x 
or to e~*t RT . Hence, instead of taking the spectrum to continue with 
uniform intensity Q to RT + ifj and there terminate abruptly, we must 
take it to have uniform intensity Q up to ifj and afterwards to have 
intensity Qe~ x l RT at ifj + y. This gives the same total intensity. 
It is easily seen that all Kramers’ lines give rise to bands starting 
* This may be modified when there is ionisation of the K electrons; but in the 
chief stellar applications a low position of the guillotine accompanies low ionisation, 
so that the modification does not arise.