322 
THE OUTSIDE OF A STAR 
Multiply this first by day and integrate over a sphere, then by da> cos 6 
and integrate over a sphere; we obtain 
(225-41), 
dr Ic 
dK 
dr 
= H (225-42), 
where 
J 
1 
47T 
J (6) dco, H 
1 
4-77 
J (6) cos ddcu, 
K = 
4c7T 
J ( 6) cos 2 9da) 
(225-5). 
We are here considering how the net flow per sq. cm. H, established by 
the liberation of energy through the whole interior, makes its way out 
through the last few thousand kilometres; hence H is to be taken as 
constant. Then by (225-41) 
j = JcJ (225-6), 
and by (225-42) K = Hr + const (225-7). 
The foregoing analysis is essentially the same as in § 74. 
First Approximation. 
226. Approximate treatments for the outer layers of a star have been 
developed by Schwarzschild, Jeans, Lindblad, Milne and others*. The 
following approximation appears to be most natural from our point of 
view. We set 
J (6) = a constant = J 1 for 6 < ^ 
J (6) = another constant = J 2 for 6 > ^ 
(226-1). 
(The constancy is as regards 6, J x and J 2 being functions of t.) The 
approximation consists in ignoring direction except for the broad dis 
tinction of inwards and outwards. 
Substituting for J (6) in (225-5) we havef 
J — \ (J x + J 2 ), 
Hence by (225-42) 
H — \ (J 1 — J 2 ), 
dJ 
dr 
= m, 
so that 
J = 3 Ht + const. 
K = 1J...(226-2). 
The constant is determined from the condition that at the boundary 
* Our approximations and formulae are usually equivalent to those given by 
Milne, except that the second approximation in § 230 is on different lines, 
f The average value of cos 0 over a hemisphere is ± \, and of cos 2 6 is ^.