Mathematical Principles of Theoretical Physics

7.1. ASTROPHYSICAL FLUID DYNAMICS 411

Then, in view of (7.1.28)-(7.1.31), the equation (7.1.76) can be expressed by

(7.1.79)

1

α

(

1

r^2

−

α′

rα

)

−

1

r^2

=−

8 πG

c^2

ρ 0 +

1

2 α ψ

ψ′φ′,

1

α

(

1

r^2

+

ψ′

rψ

)

−

1

r^2

=

8 πG

c^2

p+

1

α

φ′′−

1

2 α^2

α′φ′,

1

α

[

ψ′′

ψ

−

1

2

(

ψ′

ψ

) 2

−

α′ψ′

2 α ψ

+

1

r

(

ψ′

ψ

−

α′

α

)]

=

16 πG

c^2

p+

2

rα

φ′,

where the pressurepsatisfies the stationary equations of (7.1.78) withP=0 as follows

(7.1.80)

p′=−

c^2

2

ψ′ρ[ 1 −β(T−T 0 )],

κ

αr^2

d

dr

(

r^2

dT

dr

)

=−Q(r).

The functionsψandαsatisfy the boundary conditions (7.1.33), i.e.

(7.1.81) ψ(r 0 ) = 1 −
2 Gm
c^2 r 0

, α(r 0 ) =

(

1 −

2 Gm

c^2 r 0

)− 1

.

In addition, for the ordinary differential equations (7.1.79)-(7.1.81), we also need the bound-
ary conditions forψ′,φ′andT. Since−^12 c^2 ψ′represents the gravitational force, the condition
ofψ′atr=r 0 is given by

(7.1.82) ψ′(r 0 ) =

2 mG

c^2 r^20

.

Based on the Newton gravitational law,φ′is very small in the external sphere; also see
(Ma and Wang,2014e). Hence we can approximatively take that

(7.1.83) φ′(r 0 ) = 0.

Finally, it is rational to take the temperature gradient in the boundary condition as follows

(7.1.84)

∂T

∂r

(r 0 ) =−A (A> 0 ).

Let the stationary solution of the problem (7.1.79)-(7.1.81) be given by ̃p,T ̃,ψ,α,φ′.
Make the translation transformation

P→P, p→p+ ̃p, T→T+T ̃.

Then equations (7.1.78) are rewritten in the form

(7.1.85)

∂P

∂ τ

+

1

ρ

(P·∇)P=ν∆P−∇p+

c^2 ρ

2 α

dψ

dr

β~kT,

∂T

∂ τ

+

1

ρ

(P·∇)T=κ ̃∆T−

1

ρ

dT ̃

dr

Pr,

divP= 0 ,