Mathematical Principles of Theoretical Physics

446 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

galaxy nucleus

galaxy disk

Figure 7.6: A schematic diagram of spiral galaxy.

for which the spherical coordinates reduce to the polar coordinate system(φ,r):

(7.4.2) (θ,φ,r) =

(π

2

,φ,r

)

for 0≤φ≤ 2 π, r 0 <r<r 1.

The metric satisfying the gravitational field equations (7.1.62) of the galaxy nucleus is the
Schwarzschild solution:

(7.4.3)

g 00 =−

(

1 +

2

c^2

ψ

)

, ψ=−

M 0 G

r

,

g 11 =α(r) =

(

1 −

δr 0

r

)− 1

, δ=

2 M 0 G

c^2 r 0

,

wherer 0 <r<r 1 andM 0 is the mass of galactic nucleus.
With (7.4.2) and (7.4.3), the 2D fluid equations (7.1.65)-(7.1.68) are written as

(7.4.4)

∂Pφ

∂ τ

+

1

ρ

(P·∇)Pφ=ν∆Pφ−

1

r

∂p

∂ φ

,

∂Pr

∂ τ

+

1

ρ

(P·∇)Pr=ν∆Pr−

1

α

∂p

∂r

−ρ( 1 −βT)

MrG

αr^2

,

∂T

∂ τ

+

1

ρ

(P·∇)T=κ ̃∆T+Q,

∂ ρ

∂ τ

+divP= 0 ,

supplemented with boundary conditions:

(7.4.5)

Pφ(r 0 ) =ζ 0 , Pr(r 0 ) = 0 , T(r 0 ) =T 0 ,

Pφ(r 1 ) =ζ 1 , Pr(r 1 ) = 0 , T(r 1 ) =T 1.

Hereαis as in (7.4.3), andMris the total mass in the ballBr.

2.Elliptical galaxies.Elliptical galaxies are spherically-shaped, defined in a spherical-
annular domain, as shown in Figure7.7:

(7.4.6) Ω={x∈R^3 |r 0 <|x|<r 1 }