POLYTROPIC GAS SPHERES 
81 
u and z proportional respectively to </> and r. Let <^> 0 be the value of </> at 
the centre, and let 
The change of variables from r to u, z is merely a change of units 
introduced in order to bring the differential equation and its limiting con 
ditions to a standard form. The condition du/dz = 0 at the centre follows 
from the vanishing there of g = — cty/dr. 
56. The equation (55-8) can be solved without quadratures when 
n = 0, 1 or 5 (§ 61). For other values of n we obtain a start by assuming 
that for small values of z, u can be expanded in an infinite series which 
will run 
determining the coefficients so as to satisfy (55-8). This will not carry the 
solution very far, since it will presently diverge. But beginning with the 
values of u and du/dz at a point conveniently reached by the series we can 
now carry the solution step by step outwards through the star by quad 
ratures. When u becomes zero the density vanishes, so that the boundary 
of the star is indicated. 
Extensive tables of the solution for a number of values of n have been 
calculated by R. Emden*. We select here three of the tables which will 
ultimately concern us most. The values of z, u, du/dz are the direct result 
of quadratures; the remaining columns are calculated from these and are 
required for various applications. 
The successive columns give the following physical quantities, ex 
pressed in each case in terms of a unit which will depend on the star 
considered— 
1. Distance from centre. 
2. Gravitation potential. Temperature (for a perfect gas of constant 
molecular weight). 
3. Density. 
4. Pressure. 
5. Acceleration of gravity. 
6. Reciprocal of mean density interior to the point considered. 
7. Mass interior to the point considered. 
</>=</>oU, r = z/acf>Q (w 1} 
On substituting in (55-5) we obtain 
(55-7). 
(55-8), 
with the central conditions 
u = 1, du/dz = 0, when 2 = 0. 
u = 1 — | 2 2 + a 3 z 3 + a 4 2 4 + ... 
Gaskugeln (Teubner, 1907), Chapter v.