Joel E. Tohline
tohline@rouge.phys.lsu.edu
The following text was drawn from the "Polytropes.html" page of Tohline's online textbook.
From TABLE 4 of Ch67, Chapter IV, § 5


Radius x_{1} 
Mass [  x^{2} dQ/dx ] _{x = x1} 
r_{central}/r_{mean}  

n = 0  Ö6  2Ö6  1 
n = 0.5  2.7528  3.7871  1.8361 
n = 1  p  p  p^{2}/3 
n = 1.5  3.65375  2.71406  5.99071 
n = 3  6.89685  2.01824  54.1825 
n = 5  ¥  Ö3  ¥ 
Once x_{1} has been determined for a given polytropic index, it is clear from the definition of x [III.A.19 & III.A.20], that the total radius of a spherical polytrope is, 
[Equation III.A.31]
=
Ch67, Chapter IV, Eq. (62)
or, expressing K_{n} in terms of the model's central (maximum) density and pressure [II.C.2], 
Now, in Charles Bradley's numerical model for an n = 1 polytrope, we know that x_{1} = p. Therefore, when he sets R = 1, we know that the LaneEmden scale factor a_{n} = (R/x_{1}) = 1/p. If he sets R = p, then a_{n} = 1. 
Mass:
The total mass of a spherical polytrope is,
In Charles Bradley's numerical model for an n = 1 polytrope, we know that
[  x^{2} dQ/dx
] _{x = x1} =
p. Furthermore, he is setting
r_{central} = 1. Hence,

Central Pressure:
From eq. [III.A.32], above, we see that,