94 
POLYTROPIC GAS SPHERES 
If T m is the mean temperature 
( 66 - 1 ). 
This is a minimum when, for each step of (M r ) 2 , r is as great as possible. 
This evidently leads to the uniform sphere. Hence by (62-2), the minimum 
value is 
67. A class of problems arises in which the polytropic condition (55-1) 
applies only to part of the star. 
If the non-poly tropic region is interior to the poly tropic region 
Tables 4-7 are not applicable. For they give a particular solution of the 
general differential equation which conforms to certain boundary con 
ditions at the centre; when the central region is excluded these conditions 
are no longer effective and another solution of the differential equation 
must be taken. The gravitational field which controls the polytropic part 
is modified by the excess or defect of mass in the central non-polytropic 
part. 
An example of this would be afforded by a star obeying (55-1) only so 
long as its material can be treated as a perfect gas. Then, if the density is 
great enough, the equation breaks down near the centre. Tables 4-7 are 
not applicable even to the outer part which satisfies (55-1); and a new 
solution of the differential equation must be found by numerical calcula 
tion de novo. There is little of a general character to be said about this 
class of problem; but progress in particular cases that arise can usually 
be made by numerical calculation. 
If the non-polytropic region is on the outside, Tables 4-7 can be used 
for the interior poly tropic part. An instance of this has been treated in 
§ 65 where Table 7 was used. A more important problem of this type 
arises in the following way. We can scarcely expect that in an actual star 
any simple physical law will hold uniformly from a central temperature 
of 10,000,000° to an outside temperature of 10,000°. Making use of the 
simplified properties of matter at temperatures of several million degrees 
( 66 - 2 ). 
It is interesting to compare this with (65-7). 
Incomplete Poly tropic Regions.