328 
The Ages of the Stars 
[ch. XII 
m = 10 84 , m 0 /m 0 ' — 0’70 and m/m — 0 90, formula (294’2) gives 2 at = 1'7 x 10 -88 , 
whence, inserting the value a = 52 x 10 -88 , we find 
t— 1'7 x 10 20 secs. = 5 4 x 10 12 years. 
This gives us an entirely independent estimate of the time interval 
between types B and F or 0. It is of the same order of magnitude as the 
previous estimates, and to this extent is entirely satisfactory, but it must be 
admitted that it is based on rather slender evidence, and for this reason can 
not claim great precision. 
More detailed calculations have been given by Vogt*, Shajnf and SmartJ, 
who have discussed the question from different aspects. Vogt and Smart have 
reached conclusions agreeing in the main, although different in detail, from 
those we have just reached. 
The Birth of the Stars. 
295. If .its exact age could be assigned to every star, the present distri 
bution of stars would enable us to study the rates at which stars had come 
into being at different epochs in the past. We have nothing like sufficient 
information for a detailed study of this question to be profitable, but we may 
examine the broader question of whether the galactic system of stars was 
bom in a continuous steady stream or by an approximately instantaneous 
creation. 
Let f(m) denote the luminosity-function in any system of stars, so that 
f(m)dm is the number of stars of absolute magnitudes between m and 
m + dm. Consider the hypothetical case in which exactly similar stars are 
born in a continuous steady stream so that the distribution of stars by 
luminosity does not change, at least in respect of its brighter members. The 
analytical condition that the distribution by luminosity shall not change is 
readily found to be 
/(to) ^ = constant (295T), 
where dm/dt denotes the rate at which m changes for a star of absolute 
magnitude m. 
Let a star’s luminosity L be supposed connected with its mass M by the 
relation L = M 3 of § 118 (p. 130). Combining this with the fundamental 
equation for the loss of mass by radiation 
L = — C 
dM 
dt ’ 
dm 
we readily find that varies as M 2 , and so as 10 _0,267m . From equation 
(295‘1) it follows that the luminosity-function in a steady distribution must 
be of the form 
/(m) = G 10°- 267w 
* Zeitschrift fiir Physik, 2 (1924), p. 142. 
t M.N. lxxxv. (1925), p. 245. 
..(295-2), 
X Ibid. p. 423.