THE MASS-LUMINOSITY RELATION
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Still higher mass has with some plausibility been assigned to v Sagittarii
(M x sin 3 i = 260, M 2 sin 3 i = 54), but the determination seems uncertain*.
(4) The Sun.
Mass 1-985.10 33 gm.; radius 6-951.10 10 cm.
Although we are no longer dealing with material of low density we
suppose tentatively that the theory of a perfect gas applies. The methods
of calculation have been shown by the previous examples. The results are—
1 - j8 = -0499,
Pm ~ 1*411,
Pc = 76-5,
T c = 3-95.10 7 ,
k c = 177-0,
L = 5-62.10 33 .
The value of L found by direct measurement of the solar radiation is
3-78.10 33 . The difference of calculated and observed values is equivalent
to 0 m -43, the sun being fainter than predicted from its mass and radius.
Although this is in the direction corresponding to the deviation of terrestrial
gases from the perfect gas laws there is no reason to attribute it to such a
cause, since it is within the margin of error. For example, it might be due
to a slightly lower molecular weight in the sun than in Capella due to the
very much higher temperature. Many other sources of small deviations
can be suggested.
It may be remarked that by using in our calculations the radius of the
sun and the effective temperature of Capella we have magnified any
discrepancy. All our results are differential with respect to Capella which
was used to determine k x . A more direct method of comparison, using
the spectral type instead of the radius of the sun, is as follows. Since
Capella and the sun are of the same spectral type we neglect at first any
difference of effective temperature. Then, using Lee M k (1 — /?)-' we find
that the ratio of the U s is exactly 100 or 5-00 magnitudes. Hence the
sun’s bolometric magnitude is — 0-40 + 5-00 = + 4-60. To allow for the
more diffuse condition of Capella we have assigned it an effective tempera
ture 9 per cent, lowerf. The change in the factor gives a correction
0 m -08, raising the sun’s bolometric brightness to + 4 m -52 as compared
with the observed value + 4 m -85. The discrepancy is 0 m -33.
* H. Ludendorff, Berlin Sitzungsberichte, 1924, p. 67. [According to later in
formation the determination must be rejected altogether. Another massive system
(Boss 46) with masses 35 and 32 has been found by J. A. Pearce; this is the second
largest mass known.]
f It is better not to refer to the actual effective temperatures here—to avoid
suspicion of a vicious circle. The effective temperature of the sum is derived from the
observed L which we are holding in reserve for the final test.