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CHAPTER XII 
THE OUTSIDE OF A STAR 
225. The fundamental equation (71-2) for the radiative flow of energy 
fails when the radiation is not enclosed by matter at approximately uniform 
temperature ; in particular, the analysis breaks down at the photosphere 
where radiation is escaping freely into outer space. 
The approximations used in the preceding Chapters relate to matter 
at a temperature of some millions of degrees. In addition to a re-examina 
tion of the fundamental equation, physical approximations of a different 
kind will be needed in the treatment of the cool outer layers. A higher 
standard of approximation is generally desirable, since observational 
comparisons are more abundant and more direct. There is so little in our 
previous work that can safely be applied to the outside of a star that the 
simplest course is to begin de novo. 
Take the axis of x upwards along the vertical, so that — x is the depth 
below the surface. We shall be concerned with depths small in comparison 
with the radius and therefore neglect the curvature of the surface. Let 
J (6) dco/4:7T be the flow per sq. cm. per sec. of radiation travelling in direc 
tions within an infinitesimal solid angle da> at an inclination 6 to the 
vertical. Consider a cylinder of unit cross-section with its axis in the 
direction d, the element of length along the cylinder being 
Then the flow J (d) dœ/47r will in the length ds lose by absorption 
and gain by emission (jdaoj^Tr) pds. Here j is the emission per gm. per sec., 
of which jdm/4:7T is in directions within dco. Hence we have 
ds = dx sec d. 
(J (d) d(X)j4:7T) kpds 
dJ (d) , T ... 
— — kpj (d) +jp, 
or cos d = — kpJ (d) +jp (225-1) 
or 
a formula equivalent to (74-1). 
Let r be the “optical depth” below the surface defined by 
r=l kpdx (225-21), 
so that 
dr — — kpdx 
(225-22). 
..(225-3). 
Then by (225-1) 
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