POLYTROPIC GAS SPHERES 
91 
of dTjdr at any point in a star would require a sink of energy within the 
star, since heat must flow in the direction of the temperature gradient. 
Problem I. To find the minimum value of the central pressure in a star 
of mass M and radius R subject only to the condition that the density does 
not decrease inwards. 
First consider an arbitrary distribution of density satisfying the con 
dition dp ¡dr < 0. Remove a small part Sp 1 of the density in the shell r 1 to 
r i + dr x , and add the matter to the shell between r 2 and r 2 + d,r 2 , thereby 
increasing the density by 8 p 2 . If 8 M is the mass transferred 
SM = 4:vr 1 2 dr 1 Sp 1 = 4:7rr 2 2 dr 2 8p 2 (64-1). 
The temperature distribution must, of course, be altered in the way 
necessary to maintain the new distribution of density in equilibrium. 
We take r 2 < r x . The pressure at any point within r 2 will be increased 
for two reasons. Firstly, the matter between r x and r 2 is now under the 
attraction of the mass 8M, whereas formerly 8 M exerted no resultant 
force on it. Its weight is thus increased and contributes more pressure. 
Secondly, $M in its new position contributes more pressure, viz. g 2 Sp 2 dr 2 
instead of gi8p l dr 1 . By (64T) these are in the ratio 
QiK ' 9il r i 2 
which is greater than 
l/r 2 : 1/g 
because g cannot increase faster than r when the density does not increase 
outwards. 
Thus any transfer of matter inwards increases the central pressure. 
The minimum pressure occurs when the matter is as far from the centre 
as is consistent with the limitations of the problem. We can go on reducing 
the central pressure by transferring matter outwards until we arrive at 
uniform density. The central pressure has then the value for a uniform 
sphere 
3 GM 2 
^ ~ ‘877 R l ’ 
which is the required minimum value. 
65. Problem II. To find the minimum value of the central temperature 
in a star of mass M and radius R composed of perfect gas of constant molecular 
weight p, subject only to the condition that density and temperature do not 
decrease inwards. (Radiation pressure is neglected.) 
We first show that in the distribution giving the minimum central 
temperature either the density gradient or the temperature gradient 
vanishes at every point. 
For, if not, consider three consecutive spherical shells with densities 
Pl > p 2 > Ps and temperatures T x > T 2 > T 3 . Remove a small mass from