Mathematical Principles of Theoretical Physics

7.4. GALAXIES 447

galaxy nucleus

Figure 7.7: A schematic diagram of elliptical galaxy.

The metric is as in (7.4.3), and the corresponding fluid equations (7.1.65)-(7.1.68) are in
the form:

(7.4.7)

∂P

∂ τ

+

1

ρ

(P·∇)P=ν∆P−∇p−ρ( 1 −βT)

M 0 G

αr^2

~k,

∂T

∂ τ

+

1

ρ

(P·∇)T=κ∆ ̃T+Q,

∂ ρ

∂ τ

+divP= 0 ,

supplemented with the physically sound conditions:

(7.4.8)

Pr= 0 ,

∂Pθ

∂r

= 0 ,

∂Pφ

∂r

=0 atr=r 0 ,r 1 ,

T(r 0 ) =T 0 , T(r 1 ) =T 1.

3.Galaxy dynamics.Based on both models (7.4.4)-(7.4.5) and (7.4.7)-(7.4.8), we outline
below the large scale dynamics of both spiral and ellipticalgalaxies.
Let the models be abstractly written in the following form

(7.4.9)

du

dt

=F(u,ρ),

whereu= (P,T,p)is the unknown function, andρis the initial density distribution, which is
used as a control parameter representing different physical conditions.
First, we consider the stationary equation of (7.4.9) given by

(7.4.10) F(u,ρ) = 0.

Letu 0 be a solution of (7.4.10), and consider the deviation fromu 0 as

u=v+u 0.

Thus, (7.4.9) becomes the following form

(7.4.11)

dv

dt

=Lλv+G(v,λ,ρ),