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Astrophysical Applications of Relativity

1 Examples of relativistic fluidsThe powerful radiosource Cygnus A,(z=0.056075)

(Carilli, Perley, Bartheland Dreher).

Red represents radioemission; blue X-ray.

Astrophysical Gas Dynamics: Relativistic Applications 1/20

43 GHz VLBI image of the core of Cygnus A (Krichbaum et al., 329, 873) This shows an asymmetric two-sided jet on the milliarcsecond scale. (1 mas = 1.2 pc)


Features of jets from radio galaxies and quasars

• We believe that the jets such as those from Cygnus A are ejected at relativistic speeds from a black hole in the nucleus.

• The knots in the jets are shock waves• Two possibilities for creation of the shocks include variation in the flow velocity and the creation of disturbances in the supersonic flow via the Kelvin-Helmholtz instability

• In some quasars, in which we see one-sided jets, only, the knots move at apparent superluminal speeds.


Superluminal motion in 3C273

Two images of the quasar 3C273 taken 9 days apart. Imageslike this are used to infer apparent superluminal motions ofabout . These 43 GHz images are from Mantovani et al.A&A, 354, 497. Note that the jet is one-sided as a result of rel-ativistic beaming.

5c


2 Superluminal motion

2.1 Apparent transverse velocity

Consider an object which moves from to in a time in

the observer’s frame. Let photons be emitted from and at

times and , respectively, with

(1)

l Vt=

d Vt cos=

To observer

P1 P2

D

l Vt sin=

V

P1 P2 t

P1 P2

t1 t2

t t2 t1–=


The times at which these photons are received by the observerare:

(2)

The apparent distance moved by the object is

(3)

Hence, the apparent velocity of the object is:

t1rec t1

D Vt cos+ c

-------------------------------------+=

t2rec t2

Dc----+=

t2rec t1

rec– t Vt cosc

----------------------– t 1Vc--- cos–

= =

l Vt sin=


(4)

In terms of

(5)

vappVt sin

t 1Vc--- cos–

------------------------------------

V sin

1Vc--- cos–

------------------------------= =

vappc

----------

Vc--- sin

1Vc--- cos–

------------------------------=

app

vappc

----------= Vc---=

app sin

1 cos–------------------------=


The non-relativistic limit is just , as we would ex-

pect. However, note that this result is not a consequence of theLorentz transformation, but a consequence of light travel timeeffects resulting from the finite speed of light.

2.2 Superluminal motionFor angles close to the line of sight, the effect of this equationcan be dramatic. First, determine the angle for which the appar-ent velocity is a maximum:

(6)

vapp V sin=

dappd

--------------1 cos– cos sinsin–

1 cos– 2---------------------------------------------------------------------------------=

2–cos

1 cos– 2--------------------------------=


This derivative is zero when

(7)

and this implies that

(8)

If then and the apparent velocity of an object can belarger than the speed of light.

cos =

app sin

1 cos–------------------------

1 2–

1 2–-----------------------

1 2–------------------- = = = =

1» 1


3 Speed of advance of hotspot

This picture repre-sents the generic situ-ation of jet forcing itway through the inter-stellar medium. Thespeed of advance ofthe hotspot at the endof the jet is governedby the momentumbalance at the end of

the jet.

Je

Cocoon

Bow shock

Terminal jetshock

ISM

Contact discontinu-ity

Shocked ISM


The speed of the hotspot is effectively governed by the thrust(force) of the jet against the interstellar medium through whichit is travelling. The interaction region between jet and ISM isquite complex. However, we can analyse it approximately as fol-lows:

Bow shock

Jet

Terminal shock

Hotspot

Contact discontinuity


Let be the speed of the

hotspot in the lab frame.

We examine the balance be-tween the force per unit area ofthe jet in this frame and the forceper unit area of the incoming gasfrom the interstellar medium.

extvh2Txx

Contact discontinuity Bow shock

The force balance in the

x

Jet

vh

vh


flux of the component of momentum in the direc-tion.

(9)

For the jet,

(10)

For the external medium,

(11)

Txx = x x

Tij e p+ 2vivj

c2--------- pij+=

vivj pij for non-relativistic motion+

Txx 2 e p+ vx

2

c2----- p+=

Txx extvh2=


alancing these two fluxes of momentum:

(12)

Non-relativistic jet

It helps to gain some understanding of the physics if we first con-sider a non-relativistic, but supersonic jet. Then,

(13)

extvh2 2 e p+

vx2

c2----- p+=

Txx jetvx2 extvh

2=

vh

jetext---------- 1 2/

vjet


It is obvious from this expression, that when the jet is light com-pared to the external medium, as we believe all extragalactic jetsto be, then the speed of advance of the hotspot is much less thanthe speed of the jet.

Relativistic jet

Let us assume that the jet consists of highly relativistic particles

(for which there is good evidence) and that . Then, and

(14)

2 1»p e 3=

Txx 2 4p 2 4p2 extvh2=


The velocity of advance of the hotspot is then given by:

(15)

If the jet pressure is anything like the ambient ISM pressure (ar-guable but not definitely required) then

(16)

vh 2pjetext---------- 1 2/

2pjet

nextm------------------ 1 2/

=

vh 2pismext----------- 1 2/

2kTismmp

-------------- 1 2/

2as= =


where is the isothermal speed of sound in the ISM. Since we

have assumed that , the motion of the hotspot in the ISM issupersonic, with a Mach number of approximately , and thatis why there is a bow shock surrounding the waste material fromthe jet.

The hotspot pressure

Recall the relationship between pressure and pre-shock velocityfor a shock

(17)

as 1»

2

113---

3p2 p1+

3p1 p2+---------------------

=


It is also easy to derive

(18)

Since the hotspot region is moving slowly at a subrelativisticspeed, then we can consider the terminal jet shock to be at rest.Hence, the shock relations derived for the frame of the shock ap-ply and

(19)

2 1

1 12–

---------------98---

3p28p1---------+= =

p2p1----- 8

3---2 3–=

Hotspot pressureJet pressure

----------------------------------------phspjet--------

83---jet

2 3 1»–= =


3.1 Application to Cygnus AEstimates of the jet and hotspot pressures in Cygnus A give

(20)

This gives:

(21)

The velocity of the hotspot is:

(22)

pjet 310–10 dynes cm 2– 3

11–10 N m 2–=

phs 39–10 dynes cm 2– 3

10–10 N m 2–=

83---jet

2 3– 10= jet 2.2

vh 2pjetext---------- 1 2/

2pjet

nextmp--------------------- 1 2/

=


X-ray observations inform us that the medium surrounding Cyg-

nus A has a temperature of about and a density~

( ). Hence the velocity of advance into thesurrounding medium is

(23)

That is, the speed of advance of the hotspot is about 1/40 th ofthe velocity of the jet.

107K

10 2– cm 3– 104 m 3–

vh 0.025c


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