208 
VARIABLE STARS 
2200° the mass by Table 14 is 14 Q. But the conditions in such a star are 
so far from typical that there is risk in applying our theory; at any rate, 
the molecular weight in the interior may be greater than 2T owing to 
the low temperature, and 1 — ¡3 will be increased. Probably 10 O is a fair 
estimate of the mass of a typical long-period variable. 
Let us now consider what must be the radius of the star for a pulsation 
period of 300 days. We may take ( ya = 0-4; then by (130-4) the mean 
density is 1-1.10~ 7 . From the mass and mean density the required radius is 
350,000,000 km. = 2-3 astronomical units. 
Direct measurements of the angular diameter of o Ceti made by 
F. G. Pease with the interferometer give a diameter 0"-06, or somewhat 
larger if allowance is made for darkening at the limb. The probable 
parallax according to the foregoing discussion of absolute magnitude is 
0"-020. This gives a radius of at least 1-5 astronomical units. Alternatively 
from the bolometric magnitude at maximum combined with an effective 
temperature 2200° (estimated from the heat-index) we obtain a radius of 
1-3 astronomical units. For such a rough calculation the agreement is as 
close as could be expected; and the dimensions are accordingly consistent 
with the hypothesis that the period of about 300 days is that of the natural 
pulsations of the star. 
As regards spectral type the long-period variables fit on to the Cepheids 
in natural sequence. In the Cepheids the type reddens as the period 
increases, so that a very red type is anticipated for a 300-day period. 
Eclipsing Variables 
142. The adaptation of our theory of the mass-luminosity relation to 
calculations for eclipsing variables has been illustrated by the example of 
V Puppis, and eclipsing variables which have known orbits for both com 
ponents have been used to check the theory. Some further applications 
may be noted here. 
Theoretically it is possible to check the theory by stars with only one 
component observed spectroscopically; but it is difficult to find an example 
with data accurate enough to hold out any hope of success. 
The light-curve furnishes amongst other data the ratio e/ 2 Mi of the 
surface luminosities of the components. Hence if the spectral type or 
effective temperature of the primary is observed the effective temperature 
of the secondary can be found from Table 16. The light-curve gives also 
the ratio of the luminosities and this will suggest an approximation to the 
mass ratio M 2 /M 1 . 
Assume for trial an arbitrary value of M 2 \M X ; then we can work out 
L for each component in the manner explained for V Puppis (§ 105), 
firstly, from the calculated radius of each star, and secondly from the