82 
Gaseous Stars 
[ch. Ill 
Eddingtons Model. 
76. The first attack on these equations was made by Eddington*, who 
has discussed their solution subject to the special assumption that k(x has 
a constant value throughout the star. This special assumption facilitates the 
mathematical treatment of the problem enormously, for when it is made, 
equation (75'6) admits of immediate integration in the form 
^^Pr=Pg+Pr ( 761 )- 
This shews that when k(7 is assumed to be constant throughout a star, the 
ratio of p Q to p R is also constant. Inserting their values for p Q and p R the 
equation becomes 
n 
P = 
3 R V kG 
Thus p varies as T 3 , so that p varies as p 7 throughout the star, and this 
is the relation of pressure to density which prevails in a sphere of gas 
arranged in adiabatic equilibrium with /e = |. Thus Eddington’s assumption 
restricts us to stars arranged in the same way as a sphere of gas in adiabatic 
equilibrium with k = f or n = 3. But, as we shall now see, such a sphere of 
gas has very weird and wonderful properties, which make it unsuitable to be 
used as a model of actual stars. 
In genera], the total mass of a sphere of gas is 
M = 47r 
J pr 2 dr, 
and, on inserting the appropriate values for p and r from equations (65’4) 
and (65‘6), this becomes 
< 76 ' 3 >' 
This shews that in general M depends on p e and on ©. For a mass of 
matter of given heat-energy ® is given. As a star formed of such matter 
expands or contracts adiabatically, © retains its value unchanged through the 
expansion or contraction, but p c , the central density, changes. Equation (76'3) 
shews that a given mass of matter for which the value of © is assigned can 
always find a position of equilibrium by changing its value of p c —i.e. by 
expanding or contracting. If n is greater than 3, (3 — n)/2n is negative, so 
that large masses are diffuse and small ones are compact. When n is less 
than 3 , the reverse is the case ; large masses are compact and small ones are 
diffuse. The closer the value of n is to 3, the more rapidly the density varies 
with the mass. Finally as n touches the value 3, the addition or subtraction 
of the slightest amount of mass causes the star to rush through the whole 
range of values from p c = 0 to p c = oo . 
* M.N. lxxvii. (1917), pp. 16 and 596, and lxxxiv. (1927), pp. 104 and 308; see also The 
Internal Constitution of the Stars, Chap. VI.