THE COEFFICIENT OF OPACITY 
221 
for testing a variation of k with T. There remain Capella and the Cepheids 
which are not on the main series. Capella has an internal temperature 
| of that of the main series. If the exponent is 3 instead of 3|- we have 
assigned to the main series half its proper opacity compared with Capella 
and have therefore predicted a magnitude 0 m -75 too bright. The sun, 
Sirius and a Centauri are actually 0 m -3 fainter than the original prediction, 
so that they lie about half-way between the results for the two assumptions. 
It seems then scarcely possible to decide from the observations whether 
pI/jlT- or p/fiT 3 is nearer to the true law; but p/pT^ seems to be definitely 
ruled out, and there is considerable probability that the exponent is above 
rather than below 3. 
The Target for Electron-Capture. 
151. We first make some numerical calculations as to the number of 
captures and expulsions concerned in the stellar absorption and emission. 
Consider the conditions at the centre of Capella as given in § 13. 
T = 7*2.10 6 , p = 0-0547, p = 2-1. 
The value of ak c found from the mass and absolute magnitude is 133, 
which for a = 2-5 gives P = ^ 
This is strictly the opacity coefficient and not the absorption coefficient, 
but we shall not here trouble about the difference. The emission per gm. 
Per second is (74-5) fac y 4 _ 3 . 26 1025 ergs (15M). 
By (40-93) the average energy of a quantum of radiation at temperature 
T is 2-1ORT, where R is Boltzmann’s constant. The average quantum 
is thus 2-67.10 -9 ergs (151-2). 
Hence by division the number of quanta emitted per gm. per sec. is 
1-22.10 34 (151-3). 
A correcting factor may be necessary since the average quantum concerned 
in k may not be equal to the average quantum present in the radiation, but 
this will not affect the order of magnitude. 
The number of hydrogen atoms in a gram is 6- 02.10 23 ; hence the number 
of particles of average weight 2-1 h is 
2-87.10 23 . 
Allowing about 1 ion to every 20 free electrons, this makes the number 
of free electrons per gm. 2-74.10 23 (151-4). 
By (151-3) and (151-4) each free electron is responsible for the emission of 
4-45.10 10 quanta per second (151-5).