Mathematical Principles of Theoretical Physics

7.4. GALAXIES 451

The termF 1 corresponds to the Rayleigh-B ́enard convection with the Rayleigh number

R=k 1 k 2 =

βMrGr 0 r 1 γ

α ν κ

,

the termF 2 corresponds to the Taylor rotation which causes the instability of the basic flow
(Pφ,Pr) = (P ̃φ, 0 ), andF 3 is the relativistic effect which only plays a role in the casewhere
δ≃1.

7.4.4 Active galactic nuclei (AGN) and jets

The black hole core of a galaxy attracts a large amounts of gases around it, forming a galactic
nucleus. The mass of a galactic nucleus is usually in the range

(7.4.21) 105 M⊙∼ 109 M⊙.

Galactic nuclei are divided into two types: normal and active. In particular, an active galactic
nucleus emits huge quantities of energy, called jets. We focus in this section the mechanism
of AGN jets.

1.Model for AGN.The domain of an galactic nucleus is a spherical annulus:

(7.4.22) B=

{

x∈R^3 |Rs<|x|<R 1

}

,

whereRsis the Schwarzschild radius of the black hole core, andR 1 is the radius of the galaxy
nucleus.
The model governing the galaxy nucleus is given by (7.4.7)-(7.4.8), defined in the domain
(7.4.22) with boundary conditions:

(7.4.23)

Pr= 0 ,

∂Pθ

∂r

= 0 ,Pφ=P 0 ,T=T 0 forr=Rs,

Pr= 0 ,

∂Pθ

∂r

= 0 ,Pφ=P 1 ,T=T 1 forr=R 1.

Let the stationary solution of the model be as

Pθ= 0 , Pr= 0 , Pφ=Pφ(r,θ),

andp,ρ,Tbe independent ofφ. Then the stationary equations for the four unknown func-
tionsPφ,T,p,ρare in the form

(7.4.24)

∂p

∂ θ

=−

1

ρ

cosθ

sinθ

Pφ^2 ,

∂p

∂r

=

1

ρr

Pφ^2 −ρ( 1 −βT)

MbG

r^2

,

−ν∆ ̃Pφ+

Pφ

r^2 sinθ

+

1

2 α^2 r

dα

dr

∂

∂r

(rPφ) = 0 ,

−

κ

αr^2

∂

∂r

(r^2

∂

∂r

)T=Q(r),