Mathematical Principles of Theoretical Physics

7.4. GALAXIES 455

Jet

Accretion

(a) (b)

Accretion

Jet

Jet

Figure 7.11: (a) A jet in the latitudinal circulation withk=1 cell, two jets in the latitudinal
circulation withk=2 cells.

the bifurcated solution of (7.4.26), which can be expressed as

Pr=R^2 sQr,

whereQris independent ofRs. Thus, the radial force (7.4.35) nearr=Rsis approximatively
written as

(7.4.37) fr=

ν

1 −Rs/r

QRs, r=Rs+ ̃r for 0< ̃r≪Rs.

LetfEbe the lower limit of the effective force, which is defined as that the total radial
forceFrin the third equation of (7.4.26) is positive providedfr>fE:

Fr>0 if fr>fE.

LetREbe the effective distance:

fr>fE if Rs<r<Rs+RE.

Then, it follows from (7.4.37) that

(7.4.38) RE=kRs (k=νQRs/fE).

It is clear that there is a critical distanceRcsuch that

(7.4.39)

a jet forms ifRE>Rc orRs>k−^1 Rc,

no jet forms ifRE<Rc orRs<k−^1 Rc.

The criterion (7.4.39) is the condition for jet generation.
The condition (7.4.39) can be equivalently rewritten as that there is a critical massMc
such that the galactic nucleus is active if its massMis bigger thanMc, i.e. M>Mc. By
(7.4.21), we have
105 M⊙<Mc or 106 M⊙<Mc.