104
RADIATIVE EQUILIBRIUM
The amount absorbed in the cylinder is
E (9) - cdS . kpds
4:77
(73-3).
Finally, a certain amount will be emitted by the material in the cylinder.
This will be emitted indiscriminately in all directions so that the amount
within doi is the fraction dayf^v of the whole. So far as ordinary thermal
emission is concerned the amount will depend on the temperature; but
we must here include also any sub-atomic energy liberated in the form of
aether waves. If j is the total radiation emitted per gram per second, the
amount from the mass pdsdS in the cylinder emitted within da> is
Balancing the gains (73T) and (73-4) and the losses (73-2) and (73-3)
by the cylinder in a steady state, we have
along ds the axis of the cylinder meets successive radii at a diminishing
angle. Hence
74. First consider the case when r is large and the curvature of the
stratification in the star can be neglected. Then (73-6) reduces to
Let E be the total energy-density of the radiation, H the net outward
flow per second across unit surface perpendicular to r, and p R ' the actual
pressure of the radiation in the radial direction (allowing for the imperfect
isotropy). Then
(73-4).
(73-5).
When E (9) is a function of r and 9 only, as in a star, we have
d d. sin 9 d
The second term on the right takes account of the fact that proceeding
cos e | E (9) (6) J±- k P E (6)
(73-6).
(74-1).
\
(74-2).
The proof of the last two equations follows at once from the discussions
in §§ 31 and 22 respectively.