Astron. Astrophys. 353, 473-478 (2000)

4. Estimate of the maximum Lorentz factor

Consider now the acceleration of electrons via rotating magnetospheres in AGN. Imagine an electron which moves along a rotating magnetic field line towards the light cylinder. Generally, one expects that there are two processes which could limit the energy gain of a particle:

First, there are inverse-Compton energy losses due to interaction with accretion disk photons: low energy accretion disk photons are scattered to higher energies by the accelerated electrons so that the photons gain energy while the electrons lose energy. Near the disk the electrons might encounter a very strong disk radiation field, which substantially limits the maximum attainable energy (this need not be the case if electrons are accelerated far away from the disk, e.g. Bednarek et al. 1996). The maximum energy, which an electron is able to reach under the influence of inverse-Compton scattering is given at the point where the acceleration time scale equals the cooling time scale. In the case, where the energy of the photon in the electron rest frame is small compared to the energy of the electron (Thomson scattering), the cooling time scale for inverse-Compton losses can be approximated by (e.g. Rybicki & Lightman 1979)

where is the energy density of the disk radiation field and .

If one uses Eq. (10), the acceleration time scale may be written as:

By equating this two time scales we obtain an estimate for the maximum electron Lorentz factor .

A second, general constraint, which was neither considered by Machabeli & Rogava (1997) nor used in the calculation by Gangadhara & Lesch (1997), is given by the breakdown of the bead-on-the-wire approximation which occurs in the vicinity of the light cylinder. Beyond this point, where the Coriolis force exceeds the Lorentz force [see condition Eq. (5)], the particle leaves the magnetic field line so that the rotational energy gain ceases. Hence the acceleration mechanism becomes ineffective. In the case of AGN, where the magnetic field strength is much smaller than in pulsars, this constraint may be quite important.

For illustration, we apply our calculations in the following to a typical AGN using a central black hole mass and a light cylinder radius  cm, where denotes the solar mass. The Eddington luminosity, i.e. the maximum luminosity of a source of mass which is powered by spherical accretion, is given by . Typically, we may express the disk luminosity as , with . Thus, the equipartition magnetic field strength at the radius r is given by . Electrons are assumed to be injected at an initial position with a characteristic escape velocity from the last marginally stable orbit around a black hole of c. By applying the two constraints above, we get three generic regimes for the acceleration of electrons by rotating magnetospheres:

1. the region, in which inverse-Compton losses dominate entirely over the energy gains, leading to an inefficient acceleration (generally in the case of Eddington accretion, i.e. ).

2. the region, in which inverse-Compton losses are important but not dominant (generally the sub-Eddington range: ). In this case the acceleration mechanism works, but there exists a maximum Lorentz factor given at the position where the energy gain is exactly balanced by losses. This is illustrated in Fig. 4, where we calculate the cooling and the acceleration time scale as a function of the Lorentz factor for . For this value, the maximum Lorentz factor is roughly . Typically, the maximum Lorentz factors in this range are of the order of 100 to 1000 (see Fig. 5).

3. the region, in which the inverse-Compton losses are rather unimportant (generally ). In this case, the maximum Lorentz factor is determined by the breakdown of the bead-on-the-wire approximation [see Eq. (5)], which yields a general upper limit for the Lorentz factor of the order of 1000. This limit is found if one approximates by the light velocity which amounts to the highest value for the Lorentz forces. The results are shown in Fig. 6, where we also allow the injection position to vary. We wish to note, that the results, presented in Fig. 6, depend essentially on the assumed intrinsic magnetic field strength and the size of the light cylinder radius (i.e. the angular velocity). Generally, for a sufficient approximation, the maximum Lorentz factor is given by:

   

   Thus, even if one uses a magnetic field strength of G, which is roughly three times the corresponding equipartition field, the maximum Lorentz factor does not exceed .

Fig. 4. Cooling times scale for inverse-Compton scattering, Eq. (17), and acceleration time scale , Eq. (18), as a function of the Lorentz factor using and . The maximum electron Lorentz factor, given at the position where the cooling time scale equals the accelerations time scale, is approximately 150.

Fig. 5. Maximum electron Lorentz factor attainable under the influence of inverse-Compton losses as a function of the disk luminosity for (dotted) and (solid), where and being the energy density of the disk radiation field.

Fig. 6. Maximum electron Lorentz factor as a function of the initial injection position for c and G (i.e. a disk luminosity ). The dotted line shows the decrease in efficiency of energy gain , while the dashed line indicate the relativistic limit for injection given by the condition .

© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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