110 
RADIATIVE EQUILIBRIUM 
Since (77-32) is the fundamental equation used in astronomical theory, 
the astronomical researches lead directly to a determination of the 
coefficient of opacity. Attempts to calculate the absorption by a considera 
tion of the physical processes occurring in the matter at given temperature 
and density usually depend on (77-31) and lead naturally to the mean 
coefficient of absorption. Thus the physical and the astronomical results 
are not strictly comparable until the averaging factor has been allowed for. 
We shall, of course, take account of Rosseland’s correction when practic 
able. But in some cases the theories discussed are not sufficiently detailed 
to give dj for separate frequencies and only the total emission can be found ; 
exact correction is then impossible. Averaging factors very commonly 
occur in statistical investigations and they are not usually so large as to 
disturb the rough kind of agreement which is the most we can hope for 
in these problems. Rosseland’s factor is, however, liable to be of surprising 
importance and we must discuss it in some detail. 
78. Since k 2 has the character of a harmonic mean and k x of an arith 
metic mean k 2 is likely to be smaller than k x . This prediction is not certain 
in all cases because the weighting is not the same for k x and k 2 . 
The fact that k 2 is a harmonic mean is highly important because it 
restricts the range within which we must seek for sources of absorption 
which contribute importantly to stellar opacity. Suppose, for example, 
that we have studied exhaustively a range of frequency v x to v 2 which 
contains § of the whole weight of k 2 ; that is to say, such that 
Let the weighted mean value of 1 /k v for this range be 1/k'. We can now 
set an upper limit to k 2 , because—to take the worst possible case—even 
if k v is infinite outside this range (77-4) gives 
Whatever happens beyond the limits v x and v 2 , the opacity cannot be 
increased more than 50 per cent. 
An upper limit to k 2 is especially valuable because the danger is that 
we may be unaware of some important mechanism of absorption and 
emission. Indeed at the present moment physical theory apparently does 
not indicate sufficient absorption to agree with astronomical observation 
and we should be glad to find an additional mode of absorption. The 
result just proved shows that we can narrow down our search to processes 
capable of absorbing and emitting frequencies between v x and v 2 , and 
relieves us from an exhaustive discussion of very low and very high 
so that 
1__2 1 i. 1 1 
k 2 3 k' 3 oo ’ 
k 2 — f k'.