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.