THE OUTSIDE OF A STAR 335 
80 per cent, of the heat is emitted, 10 per cent, being above r x and 10 per 
cent, below r 2 , and regard these conventionally as the upper and lower 
boundaries of the photosphere. 
To find r x and r 2 we must insert these as limits in the integral in 
(227-15) 
sec 6 j (2 + 3 t) e~ TSece dr = j (2 + 3z cos 6) e~ z dz 
= — e~ z (2 + 3 cos 6 + 3z cos 6) 
(232-1). 
It is then easy to determine the values of z = r- sec 6 between which 
(232-1) increases by any given fraction of its total increment from 2 = 0 
to 00 . 
For the centre of the disc — 
Setting cos 6 = 1 we find 
71 = 0-25, t 2 = 3-4 (232-2). 
By (226-5) the temperatures at these levels are 
T x = 0-911 T e , T 2 = 1-322 T e (232-3), 
where T e is the effective temperature of the star; or if T e ' is the effective 
temperature of the centre of the disc 
T x = 0-8677, T 2 = 1-2577. 
For the integrated disc, or for a region of the disc where cos 6 = §— 
We find 
r x = 0-134, T 2 = 2-21 (232-4), 
and the temperatures are 
T x = 0-8807 7 ,, T 2 = 1-212 T e (232-5). 
It will be seen that the range of temperature in the photosphere is 
determined without any knowledge of the law of variation of the absorp 
tion coefficient 1c. 
233. The hydrostatic equation dP — — gpdx continues to be valid at 
the outside of a star so that 
d (pa + Pb) = - gpdx. 
But since the radiation pressure is not strictly isotropic p R is here the 
vertical component of the pressure. The pressure of radiation in a vertical 
direction is K/c (cf. the definition of p R ' in (74-2)). Hence by (225-42) 
dp R = dK/c = Hdr/c = — kpHdx/c, 
lcH 
so that dpn = — (dpa + dp R ) (233-1), 
c 9 
which is the same as (81-4) for the stellar interior. 
Conformably with the first approximation we set p R = ^aT 4 . The 
exact value is ^aT^/f, where / is given in Table 45; but it would not be 
proper to introduce / without attending to the other modifications in 
volved in a second approximation.