VARIABLE STARS
205
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the mass. The observed progression of spectral type with mass is in the
opposite direction.
Our conclusion is that the suggestions in § 136 and § 137 both lead to
serious difficulties. On the whole, the difficulties of the former seem to be
of the more fundamental kind; whereas the difficulties of the latter may
perhaps be set down as numerical misfits natural to an early stage in the
development of a complex theory.
Miscellaneous Problems.
139. In the investigation of § 127 the square of the amplitude has
been neglected. In typical Cepheids & may amount to T ’ ¥ and P x to nearly
so that the second order terms are quite considerable; these will give
rise to terms containing cos 2 nt.
For a treatment of the theory with retention of terms of the second
order, reference may be made to Monthly Notices, 79, p. 183. The com
putations are there carried far enough to show that the complete formula
for will be of the form
g x = a x cos nt — a 2 cos
where a x and a 2 are both positive. Hence the velocity of recession has the
^ orm V = b x sin nt — b 2 sin 2 nt
with b x and b 2 positive. This represents a velocity-curve having the general
characteristics of the observed velocity-curves of Cepheids, viz. a sharp
decrease from maximum to minimum receding velocity and a slower
return to maximum with indications of a hump in the curve. The equivalent
elliptic orbit has its periastron at a> = 90°.
The close similarity of the light-curve and velocity-curve and the
relation of phase between them has not as yet received adequate theoretical
explanation. It is not that any opposition of theory and observation has
been found; but the difficulty of the mathematics has hitherto proved too
great an obstacle. We have found that in the adiabatic region of the star
the outward flow of radiation is greatest at the time of maximum com
pression; this is true for the whole region (§ 133) or, at any rate, for the
outer part of it (§ 137). But the greatest outward flow from the surface
is observed to occur simultaneously with maximum velocity of approach,
that is to say, a quarter-period later. Presumably the retardation occurs
in the non-adiabatic region near the surface. The leakage wave discussed
in § 133 is 90° behind the adiabatic wave in phase, and it grows in import
ance as we approach the surface. It would, however, be too crude a de
duction to attribute the 90° retardation of flux to the leakage wave
supplanting the adiabatic wave. Undoubtedly there will be some re
tardation in the non-adiabatic region, but no definite prediction of the
amount of retardation can yet be made.