VARIABLE STARS
193
/
relation ascribes too high a luminosity to the long-period Cepheids*.
Moreover, we have no reason to suppose that T is exactly the same
throughout the Table.
The value of y or T has been calculated from physical data by Fowler
and Guggenheim for several representative Cepheids (§189). Unfortu
nately, it depends considerably on the assumed chemical composition.
It is not so much a question whether the elements are mainly light or
heavy, but whether a particular group of, say, 10 consecutive elements
for which the conditions of the star are critical, is abundant. We infer
from these calculations that unless there is some peculiar accident of
composition the value of T will not be greatly less than its value for a
monatomic gas, and we may perhaps take 1*55 as probable for an average
star.
For 8 Cephei 1 — /3 = 0-45, so that with T = If
( ya )I = -493.
Since \/ p c = -165 (Table 25) we have by (130-4)
II = 3-57 days
compared with the observed period 5-37 days. Considering the uncertainties
both of the absolute magnitude and the effective temperature the agree
ment is very satisfactory.
Moreover, without using any actual estimate of the value of T we can
predict the period as accurately as the other data warrant. For the values
T = If, If, If, if, we have (ya)$ = -558, -493, -382, -000; whence
II = 3-15, 3-57, 4-60, oo days
respectively. The value 3-15 days is definitely a lower limit; there is no
upper limit but a period substantially greater than 4-6 days could only
occur as the result of an improbably close coincidence of T with the value
f. There is no special likelihood that the ratio of specific heats of a sample
of stellar matter will be in the neighbourhood of §; in fact, we infer from
Fowler’s calculations that for any likely mixture it is well above that
value. If it is considered that there is a one-tenth chance of T — f lying
between 0 and ^ (the whole possible range being 0 to f), then it results
that the chances are 9 to 1 that the period of 8 Cephei will be between
3-15 and 7-78 days.
To find the value of T which best fits the observations, the mean of
stars Nos. 5-9 gives 1 — ¡3 = -49, nvp c = -85. Hence (ya)^ = -342,
r = 1-43. The mean of Nos. 10-14 gives 1 — ¡3 = -41, n V p c = '96. Hence
(ya)% = -302, T = 1-39. We may accordingly adopt T = 1-40. According
to the theory of § 28 this value signifies that the internal energy of the
matter (energy of ionisation) is § of the translatory energy (chiefly energy
* Monthly Notices, 79, pp. 21-22.
e . 13