7.1. ASTROPHYSICAL FLUID DYNAMICS 403
In this case, we have
M(r) =4 π
3ρ 0 r^3 for 0≤r≤R,ρ 0 =3
4 πm
R^3.
Thus we obtain the following solution of (7.1.36)-(7.1.38) with (7.1.33):
p(r) =ρ 0
(
1 −^2 Gmr2
c^2 R^3) 1 / 2
−
(
1 −^2 cGm (^2) R
) 1 / 2
3
(
1 −^2 cGm (^2) R
) 1 / 2
−
(
1 −^2 Gmr
2
c^2 R^3) 1 / 2
(7.1.39) ,
eu=[
3
2
(
1 −
2 Gm
c^2 R) 1 / 2
−
1
2
(
1 −
2 Gmr^2
c^2 R^3) 1 / 2 ]^2
(7.1.40) ,
ev=[
1 −
2 Gmr^2
c^2 R^3]− 1
(7.1.41).
The functions (7.1.39)-(7.1.41) are the TOV solution. By (7.1.12), the solution (7.1.40) and
(7.1.41) yields the metric
ds^2 =−[
3
2
(
1 −
2 Gm
c^2 R) 1 / 2
−
1
2
(
1 −
2 Gmr^2
c^2 R^3) 1 / 2 ]^2
(7.1.42) c^2 dt^2
+
[
1 −
2 Gmr^2
c^2 R^3]− 1
dr^2 +r^2 (dθ^2 +sin^2 θdφ^2 ),which is called the TOV metric.
7.1.3 Differential operators in spherical coordinates
In Subsection7.1.1, we gave the Navier-Stokes equations on general Riemannianmanifolds.
For astrophysical fluid dynamics, we mainly concern the equations on 3Dspheres. Hence in
this subsection we discuss the basic differential operators (7.1.2)-(7.1.8) under the spherical
coordinate systems(θ,φ,r).
For a 3DsphereM, the Riemannian metric is given by
(7.1.43) ds^2 =α(r)dr^2 +r^2 dθ^2 +r^2 sin^2 θdφ^2
whereα(r)>0 represents the relativistic effects:
(7.1.44) α=
1 no relativistic effect,
(
1 −2 Gm
c^2 r)− 1
for the Schwarzschild metric (7.1.26),
(
1 −
2 Gmr^2
c^2 R^3)− 1
for the TOV metric (7.1.42).