/ 
ure; 
any 
to 
for 
2 or 
ella 
lius 
ally 
t in 
lieh 
thin 
con- 
uired 
ition. 
nd it 
slab, 
may 
ralise 
The 
iding 
that 
n kw 
pdx, 
Hing 
will 
than 
slab, 
RADIATIVE EQUILIBRIUM 
101 
the sign is reversed. Note that H is reckoned per sq. cm. of the slab, not 
per sq. cm. of cross-section of the beam. 
In general radiation will be flowing both ways through the slab. 
Corresponding to the flow H + through a square centimetre from the 
negative to the positive side of the slab there will be an absorption of 
H + kpdx/c units of positive momentum, and corresponding to the flow 
H_ from the positive to the negative side there will be an absorption of 
H_ kp dx/c units of negative momentum. Hence for a net positive flow 
H = H+ — H_, there will be a net gain of Hkpdx/c units of positive 
momentum by the matter in the slab. 
We have seen that the ic-momentum — dp R acquired in the region 
per second must all be transferred to the matter by this process. Hence 
— dp R = Hkpdx/c, 
or 
H = - 
kp dx 
(71-1). 
a ac dT i 
3 kp dx 
(71-2). 
Writing for p R its value \aT i , 
Equation (71-1) shows that the net flow of radiation is, as we should 
expect, proportional to its internal pressure gradient and inversely pro 
portional to a factor kp measuring the obstructive power of the material 
screen through which it is being forced. The equation is analogous to that 
governing the flow of a material fluid through a channel or sieve. 
The equation breaks down under the same circumstances as the corre 
sponding equation for a material fluid, viz. when the flow is so rapid that 
the pressure gradient can no longer be calculated hydrostatically. This 
happens near the surface of a star. The argument cannot apply to any part 
of the star which we can see; for the fact that we see it shows that its 
radiation is not “enclosed.” But at a small depth below the photosphere 
the equation becomes a tolerable approximation; and throughout the 
main interior its accuracy is so far beyond all requirements that it may be 
used without hesitation*. 
72. A certain amount of controversy has occurred with regard to the 
derivation of this equation which reflects the time-long difference of view 
between the physicist and the mathematician. Perhaps a short digression 
on this antagonism may be permitted, for it is likely to give rise to many 
misunderstandings in problems of the kind we have to consider. I con 
ceive that the chief aim of the physicist in discussing a theoretical problem 
* Rather unexpectedly the equation remains a good approximation even in the 
extreme outer layers. The “first approximation,” described in § 226 and used generally 
throughout Chapter xii, gives (71-2) immediately. Comparison with the “second 
approximation ” developed in § 230 shows that the inaccuracy is not large.