Wit 
THE OUTSIDE OF A STAR 
333 
Note that the ratio T e /T 0 depends solely on the law of darkening, so 
that for the sun it could be deduced at once from the observed law of 
darkening without reference to the theory of the temperature distribution 
in the interior. 
By the same method we can find J (9) at any level r x . Making the 
appropriate modifications of (227-15) we have 
J (6) = H sec 6 [ (2 + 3r) e -(T-Ti > secô dr 
J ti v 2/ 
J (9) = H sec 9' I (2 + 3r) e~< Ti ~ T > 8ecff ' dr 
Jo 
Hence performing the integration 
J (9) = H (2 + 3 cos 9 + 3r 4 ) 
(■ e '=*-0<!) 
(0<2) (230-31), 
J (9) = H{(2- 3cos0') (1 - e - T * aece ') + 3r x } (#'<|) (230-32). 
Substitute these values in (225-5) so as to obtain the values of J, H and 
K. We give separately the parts corresponding to the outward radiation 
(J+) and inward radiation («/_). The results are— 
J+ = H (4 + f T )> J- = H (|-+ fr — U 2 (r) + I U 3 (t)), 
J = H (2 + 3t - U 2 (r) + f U 3 (r)), 
= H (1 + fr), H_ = H (- fr + ( 6 r 2 (t) - u s (t))), 
> (230-4), 
where 
dy (230-5). 
H = H (1 + It (U t (t) - U 3 (t))), 
K+ = H (H + |t), = H (- + $T - C/ 4 (t) + f C7 5 (t)), 
K (2 + 3t - 3C/ 4 (r) + |C/ 5 (r)) 
rr» 
,(t)= 1i r 
231. Consider now how these results are to be used for a second 
approximation to the temperature distribution. Equation (225-42) is 
rigorous and H is constant, so that 
K == tH + const (231-1). 
Formerly we set K = \J the factor ^ coming from the mean value of cos 2 9 
over a sphere. We can now replace this by the ratio of K to J given in 
(230-4), which is the mean value of cos 2 9 properly weighted according to 
the distribution of the radiation in regard to direction as determined in 
(230-31) and (230-32). Hence 
J = 3/ (tH + const.), 
2 + 3 t- U 2 (t) + |27 3 (r) 
where 
/ = 
.(231-2). 
2 + 3t — 3£7 4 (t) + |H 5 (t) 
The constant has already been determined since J = f H at the boundary, 
so that acT 4 = J=fH (Jjf + 3r) (231-3),