Mathematical Principles of Theoretical Physics

450 CHAPTER 7. ASTROPHYSICS AND COSMOLOGY

dictate the dynamic behaviors of spiral galaxies, which aredetermined by the physical pa-
rameters:

(7.4.20) ζ 0 ,ζ 1 ,r 0 ,r 1 ,κ,ν,β,γ=

T 0 −T 1

r 1 −r 0

,δ=

2 M 0 G

c^2 r 0

,Mr=M 0 + 4 π

∫r 1

r 0

r^2 ρdr.

Based on the dynamic transition theory in (Ma and Wang,2013b), we have the following
physical conclusions:

• If the parameters in (7.4.20) make the first eigenvalueλ 1 <0, then the spiral galaxy is
  of S0-type.


• Ifλ 1 >0, then the galaxy is one of the typesSa,Sb,Sc,SBa,SBb,SBc, depending on the
  structure of(Pφ^0 ,Pr^0 )in (7.4.19).


• Letλ 1 >0 and the first eigenvector(Pφ^0 ,Pr^0 )of (7.4.19) have the vortex structure as
  shown in Figure7.8. Then the number of spiral arms of the galaxy isk/2, wherekis
  the vortex number of(Pφ^0 ,Pr^0 ). Hence, ifk=2, the galaxy is of theSBc-type.

Figure 7.8: The vortex structure of the first eigenvector(Pφ^0 ,Pr^0 ).

The reason behind the number of spiral arms beingk/2 is as follows. First the number of
vortices in Figure7.8is even, and each pair of vortices have reversed orientations. Second, if
the orientation of a vortex matches that of the stationary solutionPφ(r)of (7.4.14), then the
superposition ofPφ(r)andPφ^0 of (7.4.19) gives rise to an arm; otherwise, the counteraction

ofPφ(r)andPφ^0 with reversed orientations reduces the energy momentum density, and the
region becomes nearly void.

Remark 7.17.There are three terms in (7.4.18), which may generate the transition of (7.4.17):

F 1 =

(

0 ,−

k 1 T

r^2

,−

k 2 Pr

r^2

)

, k 2 =

ρ βMrG

α ν

, k 2 =

r 0 r 1 γ

κ ρ

,

F 2 =

(

1

ν

(

P ̃φ

r

+

dP ̃φ

dr

)

Pr,−

2 P ̃φ

α νr

Pφ, 0

)

,

F 3 =

(

−

1

2 α^2 r

dα

dr

∂

∂r

(rPφ),

1

2 α

∂

∂r

(

1

α

dα

dr

Pr), 0

)

.