80 POLYTROPIC GAS SPHERES 
relation is necessary. The search for this will lead us far into the study of 
the thermodynamics of the star. In this Chapter we content ourselves 
with laying down an arbitrary connection between P and p and tracing 
the consequences. In general, whether the gas is perfect or imperfect, 
any value of the pressure can be made to correspond to given density 
by assigning an appropriate temperature; our procedure thus amounts 
to imposing a particular temperature distribution on the star. This will 
onlv correspond to possible actual conditions if the temperature distribu 
tion is such that it can maintain itself automatically. 
The third relation is taken to be of the form 
P = Kpi (55-1), 
where k and y are disposable constants. By taking different values of y 
a variety of temperature distributions are brought under survey, and 
among these the actual distribution or one closely approximating to it 
will be included. 
When the equation (55-1) is obeyed the distribution is called polytropic. 
By differentiation, we have 
dP = y/cp y_1 dp. 
Hence by (54-4) 
yKp y ~ 2 dp = dcf>, 
and on integration 
— Kpy- 1 = <f> + const (55-2). 
y - 1 
The zero from which the gravitation potential <f> is measured, is arbitrary. 
The usual convention is to make <f> vanish at an infinite distance from all 
matter. But in this subject it is more convenient to take the zero of <f> at 
the boundary of the star; the additive constant in (55-2) is then zero. 
Write 
y=l + l/?i (55-3). 
Then by (55-2) 
f c) « 
p = {(» + 1)4 
and P = - P (55-42). 
n + 1 v ' 
Equation (54-6) accordingly becomes 
V + lfr + «V*-° (55-S). 
where ' ¡(re '"I')«:" (55 ' 6 >- 
The procedure will consist in solving this differential equation so as to 
determine ^ as a function of r. Then p and P are found by (55-41) and 
(55- 42). In order to standardise the solution we introduce two new variables 
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