SOLUTION OF THE EQUATIONS 
117 
Eliminating T we obtain 
where 
P = Kpî 
39? 4 (1 - 
/ 3 ) 
a/ji^i3 4 
.(84-2), 
.(84-3). 
The distribution is thus one of the polytropic forms discussed in 
Chapter iv, viz. that given by y = £, n = 3 (Table 6). Setting n = 3 in 
the second equation of (57-3) 
( GM\ 2 _ (4*) 3 
\M') ~ 4t tG 
4 3 39Î 4 (1 - 3) 
47 tG tt/X 4 .< 
Hence 1 - p = CMytp* (84-4), 
where C = 3^W^ = 7 ' 83 - 10 ~ , ° <84 ' S) - 
Here the mass of the star is expressed in grams and the molecular weight 
in terms of the hydrogen atom. It is more convenient to express the mass 
of the star in terms of the sun’s mass (O)- Since O = 1-985.10 33 gm., 
(84-4) becomes 
1 - /3 - -00309 (M/ O) 2 /r 4 ^ 4 (84-6). 
The value of /3 is found by solving this quartic equation; it can then 
be substituted in (83-4). We notice that 3 depends only on the mass and 
mean molecular weight of the star and is independent of its radius and 
opacity. 
Estimates of ¡i depend on a discussion of the ionisation in the interior 
(Chapter x). For most stars the value is probably about 2-2; it may be 
slightly modified according to the temperature and density of the particular 
star considered. 
Table 9, containing the values of 1 — 3 f° r various masses and assumed 
molecular weights, is given for the purpose of forming a general idea of 
the conditions. A more extensive table intended for use in practical 
calculations will be found in § 100. 
Table 9. 
Values of 1 — /3. 
Mass 
(Sun = 1) 
H = 2-2 
¡x — 3'5 
fi = 30 
I 
•004 
•026 
•738 
I 
•017 
•082 
•810 
1 
•057 
•195 
•864 
2 
•151 
•344 
•903 
4 
•292 
•492 
•931 
8 
•444 
•620 
•951 
50 
•747 
■836 
•980