106 
RADIATIVE EQUILIBRIUM 
Using the exact formula (73-6) in which curvature is taken into account 
we have 
Hence equating coefficients of the corresponding harmonics on both 
sides, we obtain the following series of equations 
a typical point in the interior of a star is cut off almost completely from 
any direct radiation from matter differing in temperature by more than 
0°-001 (§71). Allowing a full difference of 0°-001 we can have the point 
illuminated from one direction by the radiation from matter at say 
4.000. 000-001 degrees and from another direction from matter at 
4.000. 000-000 degrees. It is this unequal illumination which is responsible 
for the asymmetry of E (0). With the above numbers the proportional 
difference of intensity of the radiation in the two directions is 1 part in 
10 9 . Hence none of the coefficients B, C, D, ... in (75-1) can exceed 10 -9 A. 
Choose a unit of length comparable with the radius of the star and 
consider a point not unduly near the centre so that dBjdr and B/r are of 
the same order of magnitude as B; and so on for the other coefficients. 
Since in (75-32) dC/dr and C/r are of order not greater than 10 _9 H, it 
follows that kpB is of the same order as dAjdr or A. Hence Jcp must be 
of order 10 9 (as can be verified directly from the values we have given 
for the stellar opacity). Then from (75-33) we see that JcpC is of order 
10~ 9 A, so that G is of order 10~ 18 H. It can now easily be shown that 
D, ( E ), ... are of orders 10 -27 A, 1CD 36 A, etc.* 
* Since the equations show that kpC, kpD, kp (E) are of order not greater than 
10 -9 A, G and all subsequent coefficients are not greater than 1(V 18 A. This being 
proved the equations now show that kpD, kp (E), etc. are not greater than 10 -18 A; 
hence D and all subsequent coefficients are not greater than 10~ 27 A; and so on. 
(The symbol (E) is used to avoid confusion with the energy-density E.) 
By the properties of zonal harmonics 
cos e.P n = ( nP n _ x + (n+ 1) P n+1 )/(2n + 1), 
(75-31), 
(75-34). 
We have already seen that the opacity of stellar material is such that 
(75-32), 
(75-33),