244
THE COEFFICIENT OF OPACITY
gold is 2-46. In Capella gold would retain at least the 10 inner electrons
and perhaps some of the M electrons as well; this raises /x to at least 2-8.
Changing ¡jb from 2-1 to 2-8 increases 1 — /3 from 0-28 to 0-41 and reduces
/3/(1 — /3) 2 from 9-2 to 3-5. This precisely cancels the increase in Z 2 /A*.
169. There is one way in which k t and k a can be reconciled by an
assumed chemical composition of the star, namely, by mixing a consider
able proportion of hydrogen with a heavier element, say, iron. Hydrogen
alone would be comparatively transparent to stellar radiation, since its K
line is of comparatively low frequency. The actual k t would be much less
than the value given by (158-2), since the guillotine cuts off most of the
classical radiation capable of affecting the opacity. But a mixture of
hydrogen and iron may be more opaque than either element separately.
Suppose that we have a mixture containing 15 hydrogen atoms
(A = 1, Z = 1) to 1 iron atom {A = 56, Z = 26). The hydrogen atoms will
be ionised, and the iron atom (in Capella) will have lost 24 electrons;
hence the weight 71 corresponds to 16 ions and 39 electrons. We have
M = U= 1-29, /=¿« = 0-41.
In applying (158-2) A should be taken as 71 instead of 56, since the number
of iron ions per gram is now 1/11h , and the hydrogen ions are ineffective.
We must also recalculate T with the new value of ¡x. We find that (1 — /3)
is now 0-100 and the central temperature of Capella is reduced in the ratio
1-30.
By these changes in (158-2) the absorption of the iron is increased in
the ratio 2-38 so that 7 0
k t = 11-8.
The absorption by the hydrogen ions is negligible. The chief function of
the hydrogen is to lower the molecular weight; but in addition it nearly
pays for its own inactivity by providing more numerous electrons for the
iron to capture.
The astronomical coefficient is reduced in proportion to 1 — /3, and
now becomes 7 , 0 -
k a — 18-7
which is in good enough agreement with k t .
For any star of small mass such as the sun the above mixture would
increase k t /k a 37 times, so that the original discordance would be very
much over-corrected. About half the quantity of hydrogen (7 atoms to
1 iron atom) is needed in order to obtain the same agreement as for Capella.
It may be urged that this would be in keeping with the view that the stars
* The adopted molecular weight 2-1 was not intended to correspond strictly
to iron but to allow for some admixture of lighter elements; so that by taking
Z 2 jA for iron we have rather minimised the discrepancy.