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V.I. Pariev
e-mail:
pariev@rasfian.serpukhov.su
Lebedev Physical Institute, Leninsky Prospect 53, Moscow B-333, 117924, RUSSIA
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
.
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.
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:
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.
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.