Mathematical Principles of Theoretical Physics

448 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

whereλ= (δ,Re), and theδ-factor and the Rayleigh number are defined by

(7.4.12) δ=

2 M 0 G

c^2 r 0

, Re=

M 0 Gr 0 r 1 β

κ ν

T 0 −T 1

r 1 −r 0

.

TheLλis the derivative operator (i.e. the linearized operator) ofF(u,p)atu 0 :

Lλ=DF(u 0 ,ρ),

andGis the higher order operator.
Then, we consider the dynamic transition of (7.4.11). Let ̃vλbe a stable transition solution
of (7.4.11). Then the function

(7.4.13) ̃u=u 0 + ̃vλ

provides the physical information of the galaxy.

7.4.3 Spiral galaxies

Spiral galaxies are divided into two types: normal spirals(S-type) and barred spirals (SB-
type). We are now ready to discuss these two sequences of galaxies by using the spiral
galaxy model (7.4.4)-(7.4.5).
Let the stationary solutions of (7.4.4)-(7.4.5) be independent ofφ, given by

Pr= 0 , Pφ=P ̃φ(r), T=T ̃(r), p=p ̃(r).

The heat source is approximatively taken asQ=0. Then the stationary equations of (7.4.4)-
(7.4.5) are

(7.4.14)

rP ̃φ′′+ 2 P ̃φ′−

1

r

P ̃φ−δr^0

r

(

rP ̃φ′′+

3

2

P ̃φ′−

P ̃φ

2 r

)

= 0 ,

∂p ̃

∂r

=

1

rρ

P ̃φ^2 −^1

r^2

ρ( 1 −β ̃T)MrG,

d

dr

(r^2

dT ̃

dr

) = 0 ,

P ̃φ(r 0 ) =ζ 0 , P ̃φ(r 1 ) =ζ 1 , T ̃(r 0 ) =T 0 , T ̃(r 1 ) =T 1.

The first equation of (7.4.14) is an elliptic boundary value problem, which has a unique
solutionPφ. Sinceδr 0 /ris small in the domain (7.4.1), the first equation of (7.4.14) can be
approximated by

P ̃φ′′+^2

r

Pφ′−

1

r^2

P ̃φ= 0 ,

which has an analytic solution as

(7.4.15) P ̃φ=β 1 rk^1 +β 2 rk^2 , k 1 =

√

5 − 1

2

, k 2 =−

√

5 + 1

2

.