124
SOLUTION OF THE EQUATIONS
Hence i]T ^ = 7]k/k c = k 0 /k c ,
where k 0 is the constant introduced in § 83.
If rjT~- is not quite constant the closest approximation to k 0 /k c will
be given by using the mean value of rjT ~* given at the foot of the table.
Hence (83-4) may be written
(90-1),
where a = 1-32 1-74 2-12 2-75
for e oc const. T T 2 T 4
respectively. The brightness diminishes as the concentration of the source
to the centre increases, as already foreseen. When numerical results are
required we shall generally adopt a = 2-5 representing a fairly strong
concentration to the centre.
Point-source of Energy.
91. It will appear in later Chapters that the observed brightness of
the stars is less than that predicted by present physical theories and that
it is very difficult to find a plausible explanation of the discordance. The
predicted brightness is decreased by concentrating the source of energy
towards the centre and the discordance thereby lessened. We inquire
what is the maximum possible change that can be made in this way on
the most extreme suppositions. We therefore consider now the limit when
the concentration is complete and the source of stellar energy is a point-
source at the centre*.
This problem can only be solved by very laborious numerical calcula
tions; but it seems worth while to carry out an accurate calculation for
this limiting case in order that we may know the extreme margin of error
entailed by our present ignorance of the laws of liberation of subatomic
energy.
The differential equations of the theory are
dpu = — kpHdr/c (91T),
dP = — gpdr (91*2).
For a point-source at the centre emitting the whole of the energy L
ultimately radiated by the star
H = L/4:Trr 2 }
and the adopted law of absorption is
k = k lP /T\
Monthly Notices, 85, p. 408.