87 
[. in 
80-82] Configurations of Equilibrium 
i ap- 
! star 
s the 
3 not 
x at 
e see 
tions 
star, 
>n, so 
f the 
nined 
rgy is 
Fig. 7 
which 
ration 
To do 
for an 
that X 
This approximate solution fails near the star’s surface, because dX/dx 
may become infinite here. This circumstance, however, only represents the 
branching out which has already been noticed, and equation (8T1) will give 
one solution, which we may regard as the asymptotic solution, right up to 
the boundary of the star. 
As far as first powers of the small quantity — l), it is readily shewn* 
that the asymptotic solution is 
x L 4^+1 
[(*•+»*-« (81-2). 
The second factor alone would of course give the solution already obtained, 
the root of equation (81 ‘ 1 ). This further approximation shews that the 
amount of error involved in the earlier approximation is quite small. The 
maximum error occurs when x = oo, in which case it is of amount AX. such 
that 
X = i-A(è- 0 - 
When l — 0, this is of the order of 3 per cent., and for all small values of 
l the error is so small that we may properly neglect it. 
To a similar degree of error we might have determined A, directly from 
equation (78'2) and should have obtained X as a root of the equation 
X(X + 1) = 
UnCRyT* 
cFOam 
(81-3). 
To within the degree of error just mentioned, this gives the value of X 
throughout a star’s interior, no matter in what way c, F and G vary, provided 
only that their changes are not too abrupt. 
82. We now turn to the dynamical equations. 
The total pressure p is given by 
If T' is defined by 
where p 0 is any constant, 
(821). 
r=fl+i^T 
\ X/ p 0 
(82-2), 
P=—pT' 
mpo 
so that the pressure is identical with that in a fictitious mass of gas in 
which radiation pressure is non-existent, the density p is the same as in the 
actual star, the molecular weight has a uniform value p 0 throughout, and the 
temperature is T'. Emden’s numerical solutions give the equilibrium of just 
such masses of gas. 
L-l). 
M.N. lxxxv. (1925), p. 203.