386
Variable Stars
[CH. XV
* Harvard, Circular , No. 314 (1927).
accordingly seen to represent the relation P oc p~\ the relation which we
have just seen (§ 357) must connect the period P and the density p of a
pulsating mass of gas.
359. Let us examine the form assumed by the law when differences in
mass and effective temperature are taken into account.
A star’s bolometric luminosity is proportional to 4mR 3 T e *, so that, if m
denotes the star’s absolute bolometric magnitude,
— 0*4 m = 4 log T e + 2 log R + constant (3591),
and from the approximate relation (§ 118) that the bolometric luminosity is
proportional to M 3 or to (|7 rpR 3 ) 3 ,
— 0*4m = 3 log p + 9 log R + constant (359*2).
Eliminating R between these two equations and replacing p from the
relation that P oc p~^, we obtain
log P + 0*23m + 3 log T e = constant (359*3).
The agreement of this formula with observation is almost uncanny. In
the following table the first four columns express observational data collected
and averaged by Shapley*, and the fifth column gives the absolute bolometric
magnitude obtained by applying the bolometric correction from § 48. The
sixth column gives the quantity which ought to be constant according to
formula (358*2), while finally the last column gives the value of the left-hand
member of equation (359*3).
TABLE XXXII. Observed and Calculated Data for Cepheid Variables.
Spectral
Type
Effect.
Temp.
log P
m Y u
Wlbol
log P
+ 0-3m vU
log P
+ 0‘23 m
+ 3 log T t
A 0
10000
-0*56
-0*3
-0-6
-0*6
11*3
Ab
8500
-0*31
-0*3
-0-4
-0*4
11*4
F 0
7400
-0*06
-0*6
-0-6
-0*2
11*4
Fb
6500
+ 0-23
-1*0
-1*0
-0*1
11-4
Fib
6000
+ 0-40
-1-4
-1*4
0*0
11-4
GO
5500
+ 0 59
-1-8
-1*9
0*0
11-4
G 2'b
5050
+ 0*85
-2-4
-2*6
+0*1
11-4
Gb
4600
+ 1*22
-3*9
-4-2
0*0
11-3
Gib
4300
+ 1-62
-5-4
-5*9
0*0
11*2
360. The success of the law Pp^ = constant as shewn in this table is so
striking that one is tempted at first sight to suppose that the Cepheid
variables, at any rate within the range covered by the table, might unhesi
tatingly be treated as pulsating spheres, in accordance with Shapley’s sugges
tion. When, however, the absolute values of the quantities are evaluated