Stability of the relativistic rotating electron-positron jet and superluminal motion of knots

V.I. Pariev
e-mail: pariev@rasfian.serpukhov.su

Lebedev Physical Institute, Leninsky Prospect 53, Moscow B-333, 117924, RUSSIA

Abstract:

Contents

• 1. Introduction
• 2. Description of the jet configuration
• 3. The results of linear stability analysis
• 4. The phenomenon of standing wave
• Acknowledgments
• References

1. Introduction

Possibly the most intriguing feature of numerous extragalactic radio sources is the existence of a narrow, well collimated radio jets. It is these jets that are believed to be responsible for the transportation of a great amount of energy from the central compact areas of the galaxies to their distant radio emitting parts (called radiolobes). From the observations of the superluminal motions of bright knots along the jets one must conclude for the velocity of the flow to be relativistic with the Lorentz factor being of the order of 5 to 10. One of the important problem in the physics of extragalactic jets is their stability over the large distances. Up to now there have been many works in which the stability of the jets is investigated under different assumptions for the velocity of the matter in a jet and the influence of the magnetic field on the flow dynamics. The perturbed modes are considered both for cylindrical geometry of a jet and for the plane boundary between the jet and the outside media if the wavelength is small enough. However, the effect of a strong electric field on the stability of a relativistic flow of magnetized plasma is not yet clear. Indeed, at the speed of the hydrodynamic flow the electric field in plasma with high conductivity is of the order of the magnetic one . The charge density is equal to and the current density in a stationary flow or for small time dependence is . We see that the ratio of the electric force, , acting on the unit volume, to the magnetic force, , is of the order of for the relativistic case. Thus, when considering relativistic models with a significant magnetic field it is necessary to involve the electric force and for the time-dependent case to add the displacement current. As far as we know, the stability of a cylindrical sheared relativistic flow in the presence of the magnetic and electric fields has not been investigated previously.

We started this investigation with the simplest case where the energy density of the electromagnetic fields is much greater than the energy density of the matter (including the rest mass energy). This is the so-called force-free approximation. This case is reversal to the pure hydrodynamics one when the electromagnetic field is absent. Using the force-free approximation one can hope to take into account the influence of the electrodynamic effects on the stability and simultaneously obtain substantially simplified problem which admits an analytic solution for axisymmetric perturbations.

The terms in the momentum equation which are proportional to the mass and the fluid pressure, are therefore small compared to the electromagnetic force , so it is possible to set (we use units where ). The force-free approximation together with the ideal hydrodynamics approximation (which means an infinite conductivity of plasma and consequently the absence of the electric field in the frame moving with the element of the medium) can be applied to the neighbourhood of the massive black hole, which is thought of as the central engine of active galactic nuclei. Such an approach was developed by Blandford & Znajek (1977) and Macdonald (1984) (see also chapter IV of the book ``Black Hole: The Membrane Paradigm'' by Thorne et al. (1986) and chapter VII of the book ``Physics of Black Hole'' by Novikov & Frolo (1986) and references therein). Here the strong magnetic field is expected to be of the order of . We assume the black hole to have typical values of its mass and rotation parameter: , . Then the electron density of is enough to screen the longitudinal (along the magnetic field) electric field component so that the MHD approximation becomes possible. In the inner part of the flow connected with the black hole by magnetic field lines the particle density cannot exceed the value significantly, because the particles constrained to move along the magnetic surfaces do not escape the black hole and the pairs production is only made possible in the presence of the longitudinal electric field which vanishes for . The necessity of particle creation in the magnetosphere of supermassive black hole is pointed out by Blandford & Znajek (1977). The realistic process of pair production in a thin gap near the event horizon was elaborated by Beskin et al. (1992). In this case the energy density of the fields is times greater than the rest energy density of the pairs. That makes the force-free approximation adequate.

Apparently, the force-free approximation is also valid for the inner parts of the jet, which are close to the axis of symmetry and are connected with the black hole Lovelace et al. (1987). To show this consider the conservation of the particle flow and the conservation of the current in the process of possible recollimation of this inner part of the jet. Continuity of the particle flow implies that , where is the radius of the jet, is the Lorentz factor of the plasma flow, is the particle concentration in the reference frame comoving with the plasma flow and the velocity of the particles is relativistic everywhere. Conservation of the current flowing inside the jet leads to that the magnetic field after recollimation will be predominantly toroidal and scale as . We see therefore that the ratio , so after recollimation to larger radii (say, parsecs) the jet remains to be force-free if the Lorentz factor of the flow will not reach an unbelievably high value of .

2. Description of the jet configuration

After this general introduction, we will formulate a particular model case which we have considered and briefly outline the results obtained.

We investigate the stability of the force-free MHD jet and the propagation of disturbances along it. The equilibrium configuration of the jet has cylindrical symmetry. This means that all quantities describing the jet depend on the distance from the jet axis and do not depend on the coordinate along the jet and rotational angle . The boundary of the jet has the shape of a cylinder. We suggest that the jet propagates in a medium which has a density greater than that of the jet but the temperature and pressure are small, so the condition of impermeability is fulfilled and the boundary is at rest. The poloidal magnetic field is assumed to be uniform and parallel to the jet axis. The fluid moves along spirals because of the radial electric field. In this case, the stationary magnetic configuration is governed by the force balance in radial direction . When all quantities are time independent and the velocity of the matter can be decomposed as follows

where are cylindrical coordinates; , and are the unit vectors in the cylindrical coordinate frame; , . The electric field is directed radially and is equal to

Here can be treated as the angular rotation velocity of magnetic field lines Thorne et al. (1986). Stationary jet structure in our model is entirely determined by the function . The toroidal magnetic field is

To avoid the problem of a closing current loop somewhere outside the jet it is natural to demand the vanishing of the total poloidal current through the jet. This will lead to , where is the jet boundary Istomin & Pariev (1994). Bearing this in mind we chose for numerical calculations , where .

Because of neglecting the inertia of the mass flow compared to the electromagnetic forces, when dealing with the force-free approximation, stationary jet configuration and the results of the investigation of small amplitude disturbances propagation do not depend on the value of the velocity component parallel to the magnetic field . This is seen from the fact that the coefficients of basic equations (3.5) and (3.6, see below) governing the stability problem do not contain the quantity at all. Our consideration is irrelevant also to the conductivity of the outside medium, because there are no perturbations penetrating in that region.

3. The results of linear stability analysis

We perform linear stability analysis using the common method of small perturbations: , . The linearised set of equations can be reduced to one second order ordinary differential equation on the radial component of the magnetic field perturbation . To fulfill boundary conditions

one has to solve the eigenvalue problem for . We use the temporal approach for investigating the stability, i.e. we seek for complex values of for real . In the case , i.e. for axisymmetric or 'pinch type' disturbances, it was proved that must be real for any real , so the jet is stable with respect to these perturbations Istomin & Pariev (1994).

The investigation in the case of is much more involved. The second-order differential equation governing the perturbation of the radial component of the magnetic field was derived in Istomin & Pariev (1994). It has the form

where the prime denotes the differentiation with respect to . The coefficients and are infinite in 3 cases:

1. or . This is the resonant surface . For nonrelativistic MHD consideration there exist local Suydam modes in the vicinity of Bondeson et al. (1987). However this is not the case for the force-free approximation (we see it below).
2. . This point can be interpreted as the resonance of the perturbation with the fast magnetosonic wave which in the force-free approximation always has the speed equal to the speed of light.
3. . This is the resonance of the perturbation with the Alfvén wave.

Equation (3.5) can be rewritten as a system of the two first order differential equations in terms of the radial displacement and the disturbance of the total pressure instead of the radial component of the magnetic field perturbation

This system has the same form as derived by Appert et al. (1974) for nonrelativistic case of MHD stability investigation of the plasma cylinder. The only difference is in the coefficients , , and . Particularly, . It is readily seen from equation (3.6) that the 1-st and 2-nd type singularities in equation (3.5) are only apparent as they are not the singularities of the system (equation 3.6). The only real singularity is one for which (or 3-d type). To bypass the singular point (or points) arising from it is necessary to use Landau's rule, i.e. contour of integration in the complex plane must bypass every singular point from that side as if the frequency have had small positive imaginary part. The solutions for from both sides of the 1-st and 2-nd type singular points do not depend on how they will be bypassed when doing the integration procedure. This is just what is predicted from looking at system (equation 3.6).

The eigenvalue problem is able to be solved only by means of numerical calculations. It occurred that for all involved in calculations, so the jet seems to be stable with respect to helicoidal perturbations too. This is not the proof of the stability in the strict sense because it is impossible to cover by numerical calculation the infinite range of the values of and . We are only sure that our model jet is stable with respect to perturbations having wavelength long enough, i.e. for perturbations with limited values of , and radial wavenumber. Actually, computations were performed for ,, and for the first 3 radial modes for each and . Constant in the expression for ranged in the interval from 0.1 to 20.

  
Figure 1: The dependence of the real (left plot) and imaginary parts (right plot) of . , . The first branch of the dispersion curve is indicated by solid line, the second is indicated by dashed line, and the third by dashed-dotted line.

  
Figure 2: The dependence of the real part of . , . The first branch of the dispersion curve is indicated by solid line, the second is indicated by dashed line, and the third - by dashed-dotted line. Straight lines are and .

Following the procedure outlined we have calculated the dispersion curves . The first three branches of them for and are shown in Figure 1. In Figure 2 we show the dependence of the real part of for . In this case, because of the absence of the Alfvén resonance surface inside the jet, the imaginary part of is always equal to 0. If 3 values of ,, and are the solution of the eigenvalue problem, then the values ,, and will be the solution too but for the complex conjugated function , so we depicted only the branches of having . Those having can be obtained by the reflection of Figure 1 and Figure 2 with respect to the coordinates origin.

4. The phenomenon of standing wave

The remarkable feature of the dependencies is that they have a minimum at some and . At the same time waves damping is small (it never exceeded 0.1 in our computations). Because of these, the perturbation with do not propagate, since the group velocity vanishes for . In contrast to the waves having this wave packet undergoes only diffuse broadening due to the finite value of for . It means that such oscillations form the ``standing wave'' with the wave vector . The amplitude of the ``standing wave'' will be larger than the amplitudes of other waves because it experiences a dispersion spreading only. This phenomenon is caused by the fact that the oscillations with wave vectors less and greater than propagate in the opposite directions. The phenomenon of ``standing wave'' takes place for axisymmetric perturbations too Istomin & Pariev (1994).

  
Figure 3: The dependence of the observable velocity of the standing wave pattern on the angle between the jet axis and the line of sight of the observer. Phase velocity of the perturbation is chosen equal to . If the jet is pointed directly to the observer than , if it is pointed directly from the observer than .

After a long time after initial excitation the pattern of disturbance is formed with the wave crests moving with the velocity which always exceeds the light speed. If one has relativistic electrons emitting synchrotron radiation inside the magnetic configuration we are dealing with (which is the case for extragalactic jet), then this pattern will be visible. This provides us with a new type of superluminal source. Now according to a well known formula one can calculate the observable velocity of such superluminal source in the projection onto the plane of the sky , where is the angle between the jet axis and the line of sight of the observer, is the velocity of the superluminal source. In Figure 3 the dependence of from is depicted. If then , i.e. the apparent motion of knots will be reverse, toward the core. The observer will see superluminal motion () if

It is the task for radioastronomers to detect such ``natural'' (not due to the effect of projection) superluminal motions. Bååth (1992) reported about three epoch observation of one component in 3C345 moving inward to the core, but he writes that the significance of this observation still remains to be verified. Hardee (1990) proposed another scenario which can lead to observation of backward motions of the intersection points of the shocks in nonmagnetized jets. In the frame of our model periodical structures moving backward to the core may be observable while Hardee's model predicts isolated knots.

Acknowledgments

References

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