THE OUTSIDE OF A STAR 
323 
(t = 0) there is no inflowing radiation, and hence J 2 = 0; then by (226-2) 
J = 2H ■ A ° COrdingly J = H {2 + 3r) (226-3). 
The energy-density is J/c and the effective temperature T of the 
radiation is therefore given by 
The effective température T 0 of the radiation at the boundary is then 
The effective temperature of the star T e is by (31-1) 
Other approximations by Jeans and Milne give this factor as 1-278 
and 1-232 respectively. Our second approximation in § 230 gives 1-230. 
is at a temperature not greatly less than the photospheric temperature, 
so that the material is not exposed to radiation differing widely in quality 
from that which would be present in thermodynamical equilibrium. For 
this reason a judicious application of the results of thermodynamical 
equilibrium is often permissible in dealing with the outside of a star. 
In particular, the material will take up approximately the same 
temperature T as the radiation, and the results (226-4)-(226-7) will be 
taken to refer to the temperature of the material* and the effective 
temperature of the radiation indiscriminately. 
227. Accepting the first approximation (226-8) as giving with sufficient 
accuracy the emission j at different depths we can calculate how the 
intensity of the emergent radiation varies with direction. 
A ray travelling at an angle d to the vertical has to traverse an optical 
thickness r sec 6 before emergence and is reduced by absorption to the 
fraction e~ T8eo0 of its original intensity. Considering an oblique cylinder 
of unit cross-section, a length ds(=dx sec 6) gives an emission within 
solid angle da> equal to pdx sec d .jdcoj^v which is reduced before emergence 
Jjc — aT 4 
by the definition of effective temperature in § 29. Hence 
acT 4 = H (2 + 3t) 
(226-5). 
(226-4), 
Hence 
acT* = 4:H .... 
T e = 2* T 0 = 1-189 T 0 
(226-62). 
..(226-7). 
It is important to have found that the whole outer atmosphere of a star 
By (225-6) and (226-3) 
j = JcH (2 -f 3 t) 
(226-8). 
j — pdx sec 6 e~ T8ec0 = — H -—sec 6 (2 + 3r) e TSecô dr (227T), 
J 4tt r 47 T