Joel E. Tohline
tohline@rouge.phys.lsu.edu
The following text was drawn from the "Polytropes.html" page of Tohline's on-line textbook.
From TABLE 4 of Ch67, Chapter IV, § 5
|
|||
Radius x1 |
Mass [ - x2 dQ/dx ] x = x1 |
rcentral/rmean | |
---|---|---|---|
n = 0 | Ö6 | 2Ö6 | 1 |
n = 0.5 | 2.7528 | 3.7871 | 1.8361 |
n = 1 | p | p | p2/3 |
n = 1.5 | 3.65375 | 2.71406 | 5.99071 |
n = 3 | 6.89685 | 2.01824 | 54.1825 |
n = 5 | ¥ | Ö3 | ¥ |
Once x1 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 Kn 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 x1 = p. Therefore, when he sets R = 1, we know that the Lane-Emden scale factor an = (R/x1) = 1/p. If he sets R = p, then an = 1. |
Mass:
Central Pressure:
The total mass of a spherical polytrope is,
In Charles Bradley's numerical model for an n = 1 polytrope, we know that
[ - x2 dQ/dx
] x = x1 =
p. Furthermore, he is setting
rcentral = 1. Hence,
From eq. [III.A.32], above, we see that,