A&A 383, 309-318 (2002)
DOI: 10.1051/0004-6361:20011683
M. Pohl^{1} - I. Lerche^{2} - R. Schlickeiser^{1}
1 - Institut für Theoretische Physik,
Lehrstuhl IV: Weltraum- und Astrophysik, Ruhr-Universität Bochum,
44780 Bochum, Germany
2 -
Department of Geological Sciences, University of South Carolina,
Columbia, SC 29208, USA
Received 20 July 2001 / Accepted 22 November 2001
Abstract
To address the important issue of how kinetic energy of
collimated blast waves is converted into radiation, Pohl & Schlickeiser
(2000) have recently investigated the relativistic two-stream instability
of electromagnetic turbulence. They have shown that swept-up
matter is quickly isotropized in the blast wave, which provides
relativistic particles and, as a result, radiation.
Here we present new calculations for the electrostatic instability in such
systems. It is shown that the electrostatic instability is faster than
the electromagnetic instability for highly relativistic beams. However,
even after relaxation of the beam via the faster electrostatic turbulence,
the beam is still unstable with respect to the electromagnetic waves,
thus providing the isotropization required for efficient production
of radiation. While the emission spectra in the model of
Pohl & Schlickeiser have to be modified, the basic characteristics
persist.
Key words: instabilities - plasmas - turbulence - BL Lacertae objects: general
Here we consider the energisation of relativistic particles in jets by interactions with the surrounding medium. The rapid variability displayed in the -raylight curves of blazars requires the emission regions be less than 0.01 pc in size. Also, VLBI observations of blazars indicate that the jets are not continuously filled emission regions, but consist of individual structures which relativistically move along a common trajectory and can be followed over years. On larger scales optical data still show individual knots, but also emission in-between (e.g. Boksenberg et al. 1992). Detailed spectral studies (e.g. Meisenheimer et al. 1996) indicate that the spectrum of the synchrotron radiating electrons is fairly independent of position, suggesting that many acceleration sites with uniform characteristics are operational on scales of 100 pc.
Accounting for these findings we model a jet as a channeled outflow with relativistic bulk velocity V, consisting of isolated plasma clouds or blast waves which contain cold electrons and protons of density (see Fig. 1). These clouds may correspond to the individual components observed in VLBI images of blazars, whereas many of them and secondary electrons of escaped nucleons would contribute to the optical and X-ray jets observed on larger scales in nearby AGN such as M87. For convenience we assume that the outflow is directed parallel to the uniform background magnetic field. We also neglect a possible expansion of the jet cloud in this paper. Expansion is expected to occur on much longer timescales than the effects discussed here.
Figure 1: Sketch of the basic geometry. The thickness of the channeled blast wave d, measured in its rest frame, is much smaller than its halfdiameter. The blast wave moves with a bulk Lorentz factor through ambient matter of density . | |
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Each cloud of protons and electrons propagates into the surrounding interstellar medium that consists of cold protons and electrons of density n_{i}^{*} (quantities indexed with an asterisk are measured in the laboratory (galaxy) frame). Viewed from the coordinate system comoving with the outflow, the interstellar protons and electrons represent a proton-electron beam propagating with relativistic speed -V antiparallel to the uniform magnetic field direction. This situation is unstable and waves will be excited which backreact on the incoming beam. In our earlier analysis (Pohl & Schlickeiser 2000) the stability of this beam was examined under the assumption that the background magnetic field is uniform and directed parallel to the direction of motion. It was shown that the beam very quickly excites low-frequency electromagnetic waves, which quasi-linearly isotropize the incoming interstellar electrons and protons in the blast wave plasma, thus providing relativistic particles. These processes are essentially those thought to operate in the first half-cycle of relativistic shock acceleration, but rather than assuming the existence of a hydrodynamical shock and a particular intensity of MHD turbulence, kinetic theory was used to self-consistently calculate the build-up of turbulence and its backreaction on the incoming interstellar particles.
The protons carry the bulk of the power, because velocities are conserved in the isotropization process and hence the protons have a factor of 2000 more kinetic energy than the electrons. The protons produce high energy emission via the production of neutral pions in inelastic pp collisions. In parallel charged pions would provide secondary electrons which emit synchrotron radiation as well as bremsstrahlung and inverse Compton scattering. As discussed in Pohl & Schlickeiser (2000), the high energy emission produced by these particles has characteristics typical of BL Lacertae objects. The observed secular variability of the -rayemission of AGN would be related to the existence (or non-existence) of a relativistic blast wave in the sources, and thus to the availability of free energy in the system. The observed fast variability, on the other hand, would be caused by density inhomogeneities in the interstellar medium through which the blast wave propagates.
In parallel to the -rays, neutrinos are emitted whose spectrum and flux would be closely correlated with those of the -rays, which permits one to use the -ray light curves of blazars to very efficiently search for neutrino emission as a diagnostic for an hadronic origin of the high energy radiation (Schuster et al. 2002).
In the initial stage virtually no radio emission would be produced, for the plasma frequency and the synchrotron self-absorption frequency would be too high. After the initial -rayemission phase, when the blast wave has decelerated and expanded, a mm-radio flare would build up, which behaviour is preferentially observed in radio to -raycorrelation studies of EGRET sources (Mücke et al. 1996).
Here we expand on the previous treatment by calculating the two-stream instability for longitudinal, electrostatic waves. In contrast to electromagnetic waves, which scatter the particles in pitch angle but preserve their kinetic energy until the distribution is isotropized, the electrostatic waves change the particles' energy until a plateau distribution is established.
For charge e, mass m particles under the action of an
electrical field
,
in the direction parallel to an ambient magnetic field the perturbations,
,
to the original distribution function satisfy
A plot of k^{2} versus a is given in Fig. 2. To be noted from the figure is that there are always two real values of a, occurring in a> 0and a< -V, respectively. In the region
the dispersion relation returns two real positive values of a, provided k^{2} exceeds a minimum value,
,
given by
Figure 2: The dispersion relation for longitudinal waves. Note that there is always a real solution in the range , and another real solution in . In the remaining range the dispersion relation returns two real values of a only if k^{2} exceeds . | |
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Figure 3: An enlargement of the dispersion relation for longitudinal waves in the phase velocity range where instability occurs. For small perturbation parameters, , is essentially flat, unless is very close to . | |
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The square of the growth rate is
Figure 4: The growth rate as a function of the wave number k for three values of the perturbation parameter . Most of the instability occurs in a small wavenumber range close to . | |
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Figure 5: The growth rate as a function of the phase velocity for three values of the perturbation parameter . The perturbation parameter controls only the speed of the instability, not its spectral form. | |
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The time-dependent behavior of the intensities I(k,t) of the
excited waves is given by (Lerche 1967; Lee & Ip 1987)
To describe the long-term influence of the excited waves on the beam particles,
which are the particles that can resonate with the waves, one uses the
quasi-linear Fokker-Planck equation.
Because the longitudinal waves act with an electric vector only, and because
that vector parallels the magnetic field, the phase space density for the
resonant particles then has only its momentum parallel to the ambient field
influenced by the longitudinal turbulence. Hence, the corresponding
Fokker-Planck equation reads
Note that some care has to be exercised in treating the -function in Eq. (27) for two reasons. First, slight variations in ( ) correspond to large variations in for relativistic particles. Second, during the evolution of the system the initial beam may disperse, for example through interactions with electromagnetic turbulence, so that a range of would correspond to a given . We consider here two extreme cases:
(1) when the variation in phase speed is much larger than the variation in transverse speed;
(2) when the variation in transverse speed is much larger than the variation in phase speed.
Consider each case in turn.
In this situation one can effectively set the resonant particles to a cold gas approximation and deal only with the bulk flow.
One conventionally argues that the wave generation is sufficiently rapid that the local (in time) wave intensity has reached a quasi-steady state in the sense that it alters quasi-adiabatically as the particles change their bulk streaming parameters to feed the waves.
Because of the unknown nature of the initial wave spectrum, and because other processes ignored in the development will also influence the generation of waves, one of the standard devices is to ignore the wave intensity spectrum generation and to use models of how one believes the waves spectrum has evolved to its current state. This particular decoupling of the wave and particle behaviours has been an enormous success in cosmic ray astrophysics in general and in heliospheric physics (a thorough review can be found in Schlickeiser 2001). Under this scenario one writes Eq. (26) now with I(k,t)a wave intensity spectrum to be prescribed by the user.
In addition, the parameters related to the relaxation of the plasma,V, , etc., are considered to be independent of time in evaluating the diffusion coefficient (26), but are then allowed to evolve with time once the diffusion coefficient is evaluated in order that the long term evolution of the plasma distribution function can be tracked.
The particle velocities,
,
for which resonance can be
established, are those which satisfy
When we consider the growth rate (Eq. (23)) as a function of
phase velocity, we see in Fig. 5 that
the growth rate is essentially independent of .
Neglecting
the possible effects of damping and cascading, we can therefore expect
that the intensity spectrum in phase velocity,
is flat between
and
and zero outside this range (Akhiezer et al. 1975). Upon
interaction with this wave spectrum the beam will relax to a plateau
distribution between the velocities -V and
.
The energy available for the build-up of the waves comes from the beam.
Because the wave growth is much faster than the relaxation of the beam,
we may estimate the final wave intensity spectrum as the ratio of the
energy lost by the beam during relaxation to a plateau
between the velocities -V and
and the
phase velocity range in which effective growth occurs, namely
.
Therefore we may write
Suppose the initial beam is slightly dispersed, such that all particles
have the same total momentum, P, but instead of flying exactly backwards
()
the particles homogeneously occupy a fixed solid angle element
defined by
.
This initial condition approximates the
situation after the onset of pitch angle scattering but long before the isotropization is completed. The initial distribution function in the
blast wave frame is then
The term
is identical to the right-hand-side of the
dispersion relation for an ideal beam (14), and hence equal to
.
The remaining binomial expression in x_{1} is solved by
For the expansion we require
Figure 6: The normalized phase velocity modification, , for different beam openings, , and different bulk Lorentz factors, . The perturbation parameter is held constant. The phase velocity modification scales approximately linearly with , unless is close to . | |
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Figure 7: The growth rate modification, , as a function of the unmodified phase velocity for different beam openings, , and different bulk Lorentz factors, . The perturbation parameter is held constant. The growth rate modification has a similar dependence on the parameters as has the phase velocity modification. | |
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Figure 8: The modified growth rate, , as a function of the modified phase velocity, , for different beam openings, . The perturbation parameter and the bulk Lorentz factor are held constant. Note that this plot has a linear scaling in units of 10^{-4}. The solid line applies for all , for which the modification has no visible effect. The dotted line represents the enhanced growth rate for . | |
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In Fig. 6 we show the phase velocity modification, , for different parameters. The phase velocity modification scales approximately linearly with , unless is close to , which reflects the dominance of the first order term in x_{1} over the second order term in Eq. (40) for all not too close to . Figure 7 shows the growth rate modification, , for different parameters.
In Fig. 8 we show the modified growth rate, , as a function of the modified phase velocity, . To be noted from the figure are
As one example of such models, two of us have recently published a calculation of an electromagnetic two-stream instability in a channeled relativistic outflow, a jet, and shown that the isotropization of the incoming interstellar particles would provide relativistic particles in the AGN jets, whose high energy emission has characteristics typical of BL Lacertae objects (Pohl & Schlickeiser 2000). The calculations presented here expand on these results. We have shown that for parameters suitable to explain the high energy emission spectra of AGN, the electrostatic instability is much faster than the electromagnetic instability, even in case of a weak dispersion of the incoming beam. The beam will therefore relax to a plateau-distribution in , which is, however, still unstable with respect to electromagnetic waves. The spectrum of electromagnetic waves excited by a plateau-distribution in will not be a k^{-2} spectrum as in the case of a cold beam, for it can be understood as a superposition of the wave spectra produced by cold beams with different energies, each of which would be a k^{-2} spectrum with energy-dependent upper and lower limits in k. Nevertheless, isotropization in the collimated blast wave would still occur on roughly the same time scale as calculated by Pohl & Schlickeiser (2000).
Figure 9: -decay spectra calculated with and without the electrostatic instability. The spectra refer to a observed time of 0.125 hours after the blast wave ran over an isolated cloud. At this time the spectrum of swept-up particles has evolved very little. At the peak, about 50% of the beam particle energy is lost due to the effect of the electrostatic instability. There is little effect at smaller photon energies. | |
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Figure 10: The group velocity of electrostatic waves as a function of phase velocity for the initial ideal beam. The group velocity is always between about 50% and 100% of the beam velocity, which result is fairly independent of the weighted beam intensity, . | |
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Therefore, the main effect of the electrostatic instability is a
change in the spectrum of swept-up particles.
Instead of the prior result for the differential sweep-up rate
in the absence of the electrostatic instability
One important effect is nonlinear Landau damping (Sagdeev & Galeev 1969). This is the interaction between thermal particles and the beat of two electrostatic waves: if the phase velocity of the beat wave is almost equal to the thermal velocity of charged particles in the background plasma, the particles would Landau-damp the beat wave and hence be heated. If the remaining wave has a superluminal phase velocity, it will not resonate with the beam particles anymore. As a consequence, the number of scattering processes between the beam particles and the waves, which are responsible for the production of a plateau in the beam distribution function, would be drastically reduced. This process reduces the wave intensity in the resonant phase velocity range as long as the intensity of waves with superluminal phase velocity is not so strong that nonlinear interactions in this regime cause wave energy to be retransferred to resonant subluminal phase velocities.
The nonlinear Landau damping is a process resulting from weak turbulence theory: although waves interact, the wave energy is small enough that individual waves are identifiable. If the wave energy is too high, strong turbulence theory applies (Zakharov 1972). The charge distribution in the background plasma, which results from the electric field of the waves (essentially the RHS of Eq. (4)), becomes so strong that it affects the dispersion relation. As a result, the wave field will collapse into cavitons (Goldman 1984) in which the plasma has a low density. This effect will depend on the temperature of the background plasma, and on whether or not the diffusive particle motion can effectively smear out the depleted cavitons, and thus impede their build-up.
Without having performed a detailed study of these effects in a background medium with possibly time-variable temperature caused by heating processes, we can only speculate on the impact of the nonlinear effects, which would increase the time scale for the plateau production of the beam (Eq. (35)). However, because that time scale is more than two orders of magnitude shorter than the time scale for the electromagnetic instability, even if important such nonlinear effects would likely still permit the production of a plateau on a timescale much faster than the isotropization.
We do not know the fraction of the wave energy density that is channeled into heating processes of the blast wave plasma. Here we concentrate on a qualitative discussion of the possible effects.
The wave energy is transported quickly toward the backside of the collimated blast wave. At the transition between the blast wave and its wake, the dispersive properties of the blast wave change. We can, therefore, expect that a part of the wave field is reflected at the backside, and possibly again reflected at the front side and so forth. If the reflection coefficient is not small, the wave energy density inside the blast wave would increased, until nonlinear effects at the front and backside cause the waves to escape. The heating rate would be similarly enhanced.
What would be the effect of strong heating in the blast wave? As an
extreme example consider that the full wave energy flux is available
for homogeneously heating the blast wave.
The heating rate then is
The existence of radio galaxies implies that the collimation of AGN jets is well maintained on linear scales of hundreds of kiloparsecs. A semi-relativistic plasma jet could hardly remain collimated on these scales, because the expansion by internal pressure alone would cause a substantial opening of the jet. We may take the observed collimation of jets in radio galaxies as empirical evidence against excessive heating. For the fate of the electrostatic wave field, this observation indicates that the reflection efficiency of waves at the backside of the blast wave is small, whatever the cause.
Since the waves carry momentum, even a small reflection would transfer momentum to the background plasma at the backside of the blast wave, which would then "boil off'' the blast wave and fill its wake. The waves that have escaped the blast wave would fill the wake and the regions around it, and be finally damped by the interstellar plasma in the host galaxy of the AGN, thus giving rise to additional heating of the interstellar medium around the AGN jet.
Acknowledgements
Partial support by the Verbundforschung, grant DESY-05AG9PCA, is gratefully acknowledged.