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CHAPTER VII 
THE MASS-LUMINOSITY RELATION 
105. We give some examples showing how the formulae of Chapter vi 
are employed in actual calculations*. 
(1) Capella (brighter component). 
Mass, 4-18 x O; absolute visual magnitude, — 0 m *26; type G 0, assumed 
to indicate effective temperature 5200°. 
The way in which the above data were obtained has been explained 
in § 13. Applying the correction — Am from Table 16, the absolute bolo- 
metric magnitude is — 0 m *40. The sun’s absolute visual magnitude is 
taken as + 4 m *9 which at 5740° corresponds to bolometric magnitude 
+ 4 m *85. The difference of 5-25 bolometric magnitudes indicates a rate 
of radiation 126 times faster than the sun (log 10 126 = 5-25 x 0-4). Hence 
L = 126 x 3-78.10 33 = 4-8.10 35 ergs per second. 
From the relation L = vacR^Tf (equation (87*2)) 
R = 9-55.10 11 cm. 
We have also M = 4-18 x 1*985.10 33 = 8*30.10 33 gm. 
Hence p m = M/±7rR 3 = *00227 gm./cm. 3 
By the formulae for the polytrope n = 3 we have (as already calculated 
in § 59) 
p c = *1234 gm./cm. 3 , 
P c = 6*11.10 13 dynes per sq. cm. 
We assume that the average molecular weight can be taken as p, = 2*11. 
By interpolation in Table 14, or by direct solution of the fundamental 
quartic equation 
1 - 0 = *00309 (4-18) 2 (2-11 ) 4 
we find 1 — /? = 0*283. 
The central temperature can now be calculated from 
(1 ~ P) P c = &T C *, 
or (without troubling to calculate P c first) from (87*1) 
T*/p c - 3 * (1 - fifanP. 
This gives T c = 9*08.10 6 degrees. 
The differences from the figures given in Chapter i are due to our 
neglect here of variation of p, with temperature. 
* The physical and astronomical constants required are given in Appendix I.