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.