98
Gaseous Stars
[ch. Ill
Transforming to absolute bolometric magnitude m, by using the relation
m = — 2*5 log E + a constant, this becomes
m = cons — \ log log (\ + 1) — 2 log p. — 2 log T e .. .(90'5).
appeared completely. The luminosity of a star of given mass M can be
anything from zero to infinity, and a star can adjust its configuration to any
surface temperature accompanies low luminosity and vice versa.
The constant on the right of equation (90’5) admits of evaluation in terms
of constants of nature and the coefficients in Kramers’ opacity-law. Eddington*
average error proved to be about magnitudes, the stars having only about
an eighth of the luminosity that Kramers’ law would require if they were
given mass and luminosity, the average error in logT e is of the order of 1 £,
so that the stars have some 13 times the effective temperatures and so only
about a two-hundredth part of the radius that Eddington’s discussion would
assign to them ; on his model, a star of the mass and luminosity of the sun
would be as big as Betelgeux.
Eddington accordingly treated the constant on the right of equation (90‘5)
as adjustable f, selecting its value so that the formula gave the right absolute
magnitude for Capella. Using this value for the constant, and taking p = 2T1,
the formula was found to shew a fair agreement with observation which we
shall discuss later (§ 118).
Russell first drew attention to a general objection affecting not only
Eddington’s model, for which l = $, but all stellar models for which l has a
positive value.
If l = \, as in Eddington’s model, then kG is constant, whence we readily
Inserting Kramers’ value for k, we obtain, since p/T s is constant on this
model,
so that G assumes a large negative value. Thus the model demands a very
The supposed “mass-luminosity” law mentioned in § 76 has now dis-
given emission of radiation by selecting a suitable surface temperature. Low
found that when the constant was evaluated in this way, the absolute magni
tudes given by the formula did not agree with those of observed stars. The
purely gaseous stars built on the model 1= | or kG = constant. For a star of
find that
In the outer regions of a star p is small and T^ decreases rapidly with r ,
M.N. lxxxiv. (1924), p. 104.
f l.c. p. 308.