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Astron. Astrophys. 323, 271-285 (1997)

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8. Astrophysical applications

8.1. Cooling processes

In relativistic astrophysical leptonic jets, synchrotron and Inverse Compton processes are the two main sources of cooling mechanism. The total IC power emitted by a relativistic particle with a Lorentz factor [FORMULA] is given by the Eq. (3). This expression is general and does not depend on the nature of the soft photon source; an accretion disk, synchrotron radiation in synchro-Compton process, or clouds diffusing primary radiations. This power equals the opposite of the energy lost per second by an ultra-relativistic particle [FORMULA]. The corresponding IC cooling time is (with the notation associated with the Eq. (54))

[EQUATION]

The same relativistic particle with a pitch-angle [FORMULA] in a magnetic field B will emit synchrotron radiation with a total power given by

[EQUATION]

The resulting synchrotron cooling time is

[EQUATION]

For a longitudinal magnetic field ([FORMULA]) the condition the synchrotron process dominates if

[EQUATION]

where [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. At a pitch-angle [FORMULA] where the IC losses are strongly reduced (see Eq. (8)) the condition (96) simplifies as

[EQUATION]

As explained in the previous sections, the nature of the kinetic theory of the relativistic beam depends on the pair beam density. The evolution of the pair population has already been described in MHP in the framework of two flow model (Sol et al. (1989)); a pair beam confined in the magnetic structure of a MHD jet launched by a magnetized accretion disk. One of the most important result is the existence of a compact region where the pair beam density reaches a maximum value. The localization of this region above the central object (at a distance [FORMULA]) is set by a double condition on the opacity to pair production and the Thomson opacity of the pair beam

[EQUATION]

where [FORMULA] is the beam width at [FORMULA]. From the condition fulfilled by the Thomson opacity, for a jet width of [FORMULA] the maximal pair density scales as

[EQUATION]

For an Eddington accretion rate [FORMULA], the distance of the compact region derived from Eq. (78) in MHP, for a power law energy distribution with an index s is

[EQUATION]

Extragalactic sources have an index [FORMULA], and black hole mass of order [FORMULA] leading to a maximal pair density of order of [FORMULA] and a compact zone located at [FORMULA]. Galactic objects with sometimes steeper index up to 4, and black hole with solar mass have maximal pair density of order [FORMULA] and a compact zone located at [FORMULA]. The different values of [FORMULA] indicate the extension of the non-relativistic zone; the relativistic regime typically dominates for [FORMULA].

Let us return now to the comparison of the different cooling processes. The soft disk photon energy density scales as

[EQUATION]

This estimation have been made for a disk emitting at an Eddington luminosity [FORMULA], and a distance above the central object z is measured in Schwarzschild radius unit [FORMULA]. The condition (97) then transforms as

[EQUATION]

At distances [FORMULA], the condition (97) is fulfilled for magnetic fields [FORMULA] for extragalactic black holes, and for [FORMULA] for galactic black holes, but due to re-absorption, the real magnetic values are surely underestimated. From Eq. (96) if the magnetic energy density increases, the opening angle of the reduced losses cone decreases as [FORMULA]. The synchrotron losses then contribute to sharpen the cone, but do not question its existence.

In the same way, an external source of photon (apart from the disk) cannot be considered as a dominant source of cooling in the compact region. As showed by MHP, the Compton power emitted by the relativistic particles due to their interaction with soft photons diffused by surrounding clouds is not efficient before [FORMULA], far from the regions involved here.

Thereafter, we will uniquely consider the role of soft photons from the accretion disk in the cooling processes of the relativistic pair plasma.

8.2. The non relativistic regime

We now turn to the kinetic theory developed in full details in the previous sections. As described before, unstable modes arises by the streaming of the relativistic population in the ambient electron-proton plasma. In this scheme, the plasma waves are supported by the medium with the greater mass energy density. For tenuous pair population, the ambient plasma energy density overcomes the pair one. All the theory is derived in the MHD jet frame.

8.2.1. Conditions for the non relativistic regime

The non relativistic regime is achieved when the pair energy density is smaller than the MHD plasma one. This gives

[EQUATION]

Let us give an estimate of [FORMULA].

Sol et al. (1989), and Rosso & Pelletier (1994), argued that the MHD jet at VLBI scales has an Alfvén velocity in the range

[EQUATION]

So defining [FORMULA] such that [FORMULA] we obtain in the jet

[EQUATION]

The proton density [FORMULA] on the disk surface corresponding to an Alfvén speed [FORMULA] for [FORMULA] is of order of

[EQUATION]

The proton density [FORMULA] typically drops by a factor of [FORMULA] for an isothermal disk (Ferreira & Pelletier (1995)), leading to a proton density in the MHD jet of order of [FORMULA] for extragalactic sources and [FORMULA] for galactic sources. Note that because of the electro-neutrality in the ambient medium, the cold electrons have a density [FORMULA]. In fact, the Bohm diffusion of the protons from the MHD flow to the pair plasma region implies a diffusion time much longer than the transit time. The proton density in the beam region is surely lower than the previous estimates.

8.2.2. Astrophysical signatures

We now emphasize on the astrophysical consequences of the kinetic theory derived previously.

The highly anisotropic Compton cooling (Eq. (3)) determines a cone of half angle [FORMULA] of strongly reduced IC losses. Inside the cone, acceleration by Langmuir waves prevails at low energy, but for Lorentz factors [FORMULA] the Alfvén turbulence dominates. This factor [FORMULA] is given by the Eq. (58) with [FORMULA]

[EQUATION]

For extragalactic sources the cold electron Compton temperature [FORMULA], leading to thermal speeds [FORMULA]. In galactic objects the Compton temperature is higher; [FORMULA] and [FORMULA]. The ratio of the cyclotron to the plasma frequency is [FORMULA], leading to Lorentz factors [FORMULA].

The low energy pairs are then accelerated to Lorentz factors of order of few and feed the high energetic component of the distribution function. The balance between acceleration by Alfvén waves and the IC cooling process leads to the formation of a power-law distribution with an index [FORMULA] and with [FORMULA].

Let us have an estimate of [FORMULA].

The Inverse Compton cooling time of a relativistic particle (with a Lorentz factor [FORMULA]) is given by

[EQUATION]

where

[EQUATION]

we recall that [FORMULA] and [FORMULA], such that [FORMULA] (see Sect. 2).

Inside the cone the IC cooling frequency is

[EQUATION]

As specified in Sect. (6) the maximum Lorentz factor of the pair population is obtained for balanced acceleration and cooling times leading to a spectral index [FORMULA] (Eq. (61)). The energy distribution function inside the cone is flat and [FORMULA]. For [FORMULA], the pitch-angle scattering is

[EQUATION]

In the case of a Schwarzshild black hole with a mass M and a standard accretion disk emitting at the Eddington limit [FORMULA] (with a compactness [FORMULA]) the mean Lorentz factor of the particle distribution takes the form

[EQUATION]

The relativistic pressure is dominated by the particles inside the cone since outside the pair distribution drops as a power law and displays an accumulation of low energy particles. The relativistic pressure is reduced by a factor [FORMULA] compared to the isotropic case

[EQUATION]

We assume the equipartition with the magnetic pressure; then

[EQUATION]

Thus,

[EQUATION]

In term of the black hole mass we obtain

[EQUATION]

The compact region (of maximum of pair creation) is of order of [FORMULA] in MHP, leading to an extension of the non relativistic regime between 10 and [FORMULA]. Moreover, the magnetic field typically drops by a factor of [FORMULA] between the disk surface and the compact zone. We then estimate the maximum Lorentz factor of the particle distribution in the non relativistic regime to be of order of [FORMULA] for extragalactic black holes and [FORMULA] for galactic black holes. The resulting bulk Lorentz factors, for a soft photon source of compactness [FORMULA], cannot be greater than [FORMULA]. For [FORMULA], this gives for extragalactic sources [FORMULA] and for galactic sources [FORMULA].

8.3. The relativistic regime

As stated before, this second case may occur in regions where the pair density [FORMULA] approaches its maximum values and supports the instabilities. We can then examine the conditions for resonance between back-stream particles and pair plasma modes in this regime.

We consider the synchrotron resonance with electro-magnetic waves [FORMULA], where [FORMULA] is the cyclotron frequency of the cold electrons ([FORMULA]) or protons ([FORMULA]).

The electron back-stream triggers a synchrotron maser instability of the right handed circularly polarized resonant mode. This would contribute to an interesting coherent emission in the radio band probably with a fast variability. We report the astrophysical investigation of this phenomenon to future works.

The proton back-stream is responsible for the destabilization of left handed circularly polarized Alfvén waves. The growth rate is slower than the electronic one, but has a much stronger dynamical effect on the flow.

However, this Alfvén wave is unstable only if its wavelength is larger than Larmor radius of the energetic pairs. In other words, we can define a saturation energy level for those pairs corresponding to the equality of the wavelength with the Larmor radius, namely

[EQUATION]

such that the instability grows for [FORMULA].

The theory of the non-linear evolution remains to be done for this unusual scheme. Nevertheless we can state that the subsequent heating of the relativistic plasma leads to a saturation value of [FORMULA] given by Eq. (117).

Inversely, if the pair plasma would have a [FORMULA] larger than [FORMULA], because of radiation cooling, [FORMULA] would decrease to [FORMULA] as long as no other turbulent heating is at work. We then expect that the relativistic regime built up a distribution function of pairs such that [FORMULA]. Bearing in mind that, even coupled, the proton component is less massive, the dynamics of pairs in the Compton radiation field is not significantly modified and therefore we still have:

[EQUATION]

Combining the Eqs. (12) and (13)

[EQUATION]

and

[EQUATION]

For a soft photon source compactness of radius [FORMULA] ([FORMULA]) the Eqs. (119) and (120) give typical upper values of the bulk Lorentz factor and the internal energy of the pair plasma of order of

[EQUATION]

These results are interesting in the sense that they do not depend on all the details of the instability and the turbulence. They are directly expressed in term of an observable quantity (the compactness of the source [FORMULA]). It is worth mentioning that if we consider a power supply by the turbulence carried by the MHD jet then the internal energy of the relativistic pairs can exceed the saturation value [FORMULA]. In this case, the protons are no more coupled with the pair plasma through this process. Only rare Coulomb collisions can do this work.

The non-linear theory is supposed to provide the spectrum of the Alfvén waves that must heat the pairs. Except some peculiar spectrum, the distribution function is unlikely a power law, but more likely a quasi-monoenergetic distribution characterized by the internal energy derived above. A power law in [FORMULA] could be obtain by taking into account the inhomogeneity effect together with the Compton cooling. The index of the distribution must be steeper if the pair creation process is considered.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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