324 
THE OUTSIDE OF A STAR 
by (226-8). Hence integrating, the total amount emerging is 
(227-15) 
4tt Jo 
= H — {I + *cos6) 
(227-2). 
This is called the “law of darkening” since it gives the variation of bright 
ness over the apparent disc of the star. As we approach the edge we view 
the surface by more and more oblique rays. Between the centre (0 = 0) 
and the limb (d = the brightness changes in the ratio f : 1 or very 
nearly 1 magnitude. This (approximate) theoretical formula is in close 
agreement with observations of the solar disc. 
The total amount of radiation emerging from unit area of the star 
(not of the disc ) is obtained by multiplying (227-2) by the factor cos 6 for 
foreshortening and integrating over a hemisphere. The result is H —as it 
should be*. 
The effective temperature of a particular region on the disc is given by 
This follows from (227-2) because for this point of the disc we see only the 
radiation emerging in the direction 0, and compare it with a black body 
giving the same flow in all directions. 
Hence at the centre of the disc acT e 4 = 5 H, so that the effective 
for the star as a whole, e.g. the effective temperature at the centre of the 
solar disc is 6070° against 5740° for the integrated radiation of the sun. 
The effective temperature of the integrated disc is the same as that of 
a region where cos 0 = §. 
The Spectral Energy Curve. 
228. The average depth r m from which the emergent radiation has 
come is r00 
* The early approximations of Sehwarzschild and Jeans do not satisfy this 
check. See Milne, Monthly Notices, 81, p. 364. 
acT/ = 2 H (1+1 cos 0) 
(227-3). 
temperature at the centre is (f)^ or 1-0574 times the effective temperature 
J o 
r(2 + 3r) e~ T8eoe dr (2 + 3t) e~ TBece dr 
= cos 0(1 + 3 cos 9) -r (1 + f cos 9). 
By (226-5) the temperature T m at r m is given by 
acT m 4 = 2 H {1 + f cos 0(1 + 3 cos 0)/( 1 + f cos 0)} 
_ 1 + 3 cos 0 + 1 cos 2 0 
1 + f cos~0 
(228-1).