84 
POLYTROPIC GAS SPHERES 
A more convenient way of obtaining p 0 is to express it in terms of the 
mean density p m . We have 
m if i d4>\ 
pm |7rJ? 3 G\ i-nr 3 ' dr) u= q ' 
a relation which is also evident by comparing (54-6) and (55-5). Hence 
The ratio p 0 lp m will be found at the foot of the sixth column in the tables. 
Other entries in the same column give the ratio of the central density to 
the mean density interior to the point considered. 
The last column of the tables gives numbers proportional to the mass 
M r within a sphere of radius r. The unit is found at once since the con 
cluding entry (M') corresponds to the whole mass of the star. Similarly 
the first column gives the distance from the centre in terms of a unit 
which is ascertained from the condition that the concluding entry R' 
corresponds to the radius of the star. 
58. We shall have to consider particularly stars composed of perfect 
gas. The temperature is then determined from P and p by the gas equation 
where 51 is the universal gas constant 8-26 . 10 7 , and p is the molecular 
weight in terms of the hydrogen atom*. 
But before using this equation we must notice that in it P represents 
the gas pressure only, whereas in our analysis P has been used to denote 
the whole pressure of every description acting across a surface drawn in 
the star (cf. § 54, where P is first introduced). The pressure of radiation 
is therefore to be included in P. If j8 is the ratio of the gas pressure p G to 
the whole pressure P the corrected equation runs 
* Unless otherwise stated the molecular weight will be measured in terms of 
the hydrogen atom in this book. If, however, ¡x is measured in grams the constant 
in the numerator is Boltzmann’s constant R = 1-372.10 -16 . The relation is 9i = R/H, 
where H is the mass of a hydrogen atom in grams. 
But by (55-7) 
r dr ' zdz‘ 
Hence 
(57-4). 
(57-5), 
But by (55-41) and (55-6) a 2 </> 0 n — 4:vGp 0 
Pm 
Po 
(57-6). 
51 
P = - P T 
P 
(58-1) 
Then by (55*42) 
Hence 
(58*2). 
(58*3).