212
VARIABLE STARS
in the ratio L 2 + \L x R<?\a‘ l : L 2 . This, however, is not true. A globe
illuminated from without does not present a uniformly bright disc, and
the formula (144-1) tells us nothing about the brightness as seen in a
specified direction. Allowing for this, the simple geometrical theory gives
the ratio L 2 + f L^ja* : L 2 (144-2).
In addition, there is a small effect due to “darkening at the limb”
which affects both the ordinary emission and the re-emission (§ 227).
At full phase the reflected radiation has an advantage, since it comes
mainly from the centre of the disc and avoids the darkening. This is found
to increase the albedo in the ratio -j-|, the increased brightness for the
observer who sees the full phase being at the expense of reduced brightness
in other directions.
Since R 2 /a is often rather large it is well to retain higher powers. A
more accurate expression for the increase is then
^L x {sin 2 <f> + (2 + cos 3 <f> — 3 cos (/>)/sin <p} (144-3),
where sin </> = R 2 /a.
It may be added that the variation of this added brightness with phase
is proportional to - (sin i p — ip cos ip), the phase-angle ip being reckoned
77
from zero at “new.” The observed reflection coefficients have been cal
culated on the assumption that the variation is proportional to |(1 — cos ip).
It so happens that this introduces no error in the reflection coefficient, but
it makes an appreciable difference in the calculation of the ellipsoidal
elongation of the stars. Determinations of the elongation of the stars
under each other’s attraction must ultimately yield important information
for the development of our theory; but at present the treatment is too
crude for our purposes. 145
145. Results for those variables in which the reflection effect has been
thoroughly studied are collected in Table 28 a.
The type given is that of the brighter component; the type of the
fainter can be estimated from the ratio of the surface luminosities
by reference to Table 16 (where J is measured in magnitudes). The unit
of heat intensity used for L x and L r is the maximum for the system,
i.e. L x + L 2 + L r = 1. The calculated value of L r is found from (144-3),
and the observed value is taken directly from the published discussions of
the photometric data; the probable error assigned by the investigator is
given in the last column. (The V s in the table refer to the light or heat
in the direction towards the observer, and seen by him except in so far as
the eclipse interferes.) No. 9 depends on selenium photometry; the others
depend on visual observations.