Stability of Expanding Jets

Serguei Komissarov & Oliver Porth

University of Leeds and Purdue University

Introduction

• AGN jets are remarkably stable compared to their terrestrial counterparts. They propagate over distances up to one billion(!) of initial jet radius;• They exhibit substantial radial expansion. The radius of M87 jet increases by ~ one million times. Expansion is know to have a stabilising effect (e.g. Moll et al. 2008); • They propagate through atmospheres with rapidly decreasing pressure (and density).

We are out to check that:

These are related. The stability is due to the loss of connectivity, caused by the rapid decline of external pressure.

Stability and Causality

Only global instabilities can threaten the jet survival.

For a global instability to develop, the jet has to be causally connected in the transverse direction. The condition is

µM > µj

µM

µj = R j =r

- Mach angle (for the fastest wave),

- jet opening angle.

µMµj

R j

r

Stability and Causality

• Hot jets:

Jets confined by external pressure, pext / r¡ · :

µM =µj / r1¡ · =2

The jet expands freely – not causally connected – when · > 2

• Poynting-dominated relativistic jets:

The same conclusion! (Komissarov at al.,2009, Lyubarsky 2009)

· = 2 is a critical value.

Stability and Causality

· ' 2 is a typical value for AGN. ( Begelman et al. 1984 )

Inner Edge of Radiation-Supported Accretion Disk:

Radio Lobes:

r ' 1014cm pext ' 107dyncm¡ 2

r ' 1023cm pext ' 10¡ 11dyncm¡ 2

In coronas of ellipticals ( 100pc – 10kpc ),

Expect closer to the core -> free expansion.

· ' 1:2:

· > 2

· ' 2

Steady-State Jets via 1D simulations

One can use 1D time-dependent simulations to construct approximate 2D steady-state solutions.

Example. 2D steady-state continuity equation:

@½¡ v1

@x1 +@½¡ v2

@x2 = 0:

Suppose . Replace v1 with c and x1 with ct to obtainv1 ' c

@½¡@t

+@½¡ v2

@x2 ' 0 - 1D time-dependent continuity eq.

Boundary conditions: Replace at the 2D jet boundary with at the 1D “jet” boundary.

V = Vext(x1)V = Vext(ct)

Steady-State Jets via 1D simulations

Test model: Axisymmetric Relativistic MHD jets

The initial solution describes a 1D cylindrical jet in magnetostaticequilibrium (Komissarov 1999); purely azimuthal magnetic field.

b

p

Kink-unstable in uniform externalmedium O’Neil et al.(2012)

/ r

/ 1=r

Steady-State Jets via 1D simulations

Test model: Axisymmetric Relativistic MHD jets

R j ;0 = 1:0; ¡ j ;0 ' 3; ¾j ;0 ' 0:2; pext;½ext / t¡ · :

Expanding 3D Jets in a Periodic Box

The approach: Introduce time-dependent external pressure,

to study the role of expansion on stability of jets using the periodic box setup. To allow perturbations of the external gas we use the forcing approach:

pext / t¡ · ;

dpext

dt= ¡

(pext ¡ p̂ext(t))tr

;

p̂ext(t) / t¡ · ; tr = ¹R j et

c:

The same approach is used to control other parameters of the externalgas. The value of is such that for the results are not strongly influenced by the forcing, which inhibits wave emitted by the jet.

Perturbation (kinks):

vr (r;Á;z) = (¢ v=N ) exp(¡ r=rm)NX

n=1

cosÁsin(2¼nz=L z)

N = 4; ¢ vr = 0:01;

z - distance along the jet, Lz – box size (Mizuno et al., 2011)

Expanding 3D Jets in a Periodic Box

Constant external pressure,

The jet is destroyed over the distance ~ 100 initial radii (c=1)

Expanding jet,

The jet propagatesat least ten timesfurther (1000 initial radii; end of simulations)

It develops irregular“knotty” structure

Jet energetics; 3D versus 1D (steady-state)

At r < 500 Rj,0 , the 3D solutionfollows closely the steady-state one.

At r > 500 Rj,0 (the highly-nonlinear phase), wave losses and magnetic dissipation in the 3D model.

Preliminary conclusions

• Rapid decline of external pressure is the main factor behind the observed stability of astrophysical jets;

• Jets are expected to flare when entering flat sections of external atmospheres. Instability -> Dissipation -> Emission. Bulk acceleration is another likely outcome for highly magnetized relativistic jets.

• This is still “work in progress”;

• Rapid decline of external pressure is identified as the main factor behind the observed stability of astrophysical jets;

• Jets are expected to flare when entering flat sections of external atmospheres. Instability -> Dissipation -> Emission. Bulk acceleration is another likely outcome for highly magnetized relativistic jets.

The End

Preliminary conclusions