86 Gaseous Stars [oh. m 
solution, passing through 0 in the figure, to which all other solutions ap 
proximate very rapidly as x increases. 
The physical interpretation of this is very simple. The surface of the star 
is determined by^ o = 0, and so by A = 0, so that the axis of x represents the 
star’s surface. The value of x, although small at the star’s surface, is not 
actually zero, and the various solutions correspond to different values of x at 
the surface, and so to different effective temperatures of the star. But we see 
that very shortly they all run together; the influence of the special conditions 
which prevail at the surface .soon disappears as we pass inwards into the star, 
and at a short distance from the surface we are on the asymptotic solution, so 
that A, depends only on x, and no longer on the effective temperature of the 
star’s surface. 
The star’s surface temperature and effective temperature T e are determined 
by the star’s emission of radiation, and so by the rate at which energy is 
being generated in the star’s interior. The bunch of solutions shewn in Fig. 7 
branch out from the asymptotic solution, and it is this branching out which 
enables us to find a solution suitable for every given rate of internal generation 
of energy by the star. 
81. Equation (79‘5) may also be solved by analytical methods. To do 
this it is convenient to regard £ (| — 0 as a small quantity and search for an 
expansion in powers of this small quantity. 
If, to a first approximation, we neglect — l) altogether, we find that A 
is a root of 
= x (\ + 1) 
■(81*1).