A&A 381, 357-360 (2002)
DOI: 10.1051/0004-6361:20011448
Y. D. Xu
Physics Department, Shanghai Jiaotong University, 200030 Shanghai, PR China
Received 20 July 2001 / Accepted 5 October 2001
Abstract
We consider the magnetic acceleration of charged particles in
rotating magnetospheres of active galactic nuclei (AGNs).
The accelerating particle loses its kinetic energy due to the inverse Compton
scattering with the photons emitted from the disk. The disk radiation is
anisotropic, so that the inverse Compton energy loss of the particle
depends sensitively on the direction of motion of the particle. We find that
the maximum Lorentz factor the accelerating electron can attain near the
light cylinder is mainly determined by the direction of motion of the electron.
In the cases of
,
the maximum Lorentz
factor of a magnetically accelerated electron can be as high as a few
thousand, if the electron is moving close to the normal direction to the
disk. The maximum Lorentz factor becomes relatively low if
.
Key words: galaxies: active, nuclei - acceleration of particles - accretion disks
Recently, Rieger & Mannheim (2000, 2001) explored this mechanism in detail and estimated the maximum Lorentz factor of the charged particles under the assumption of an idealized rotating magnetosphere. They showed that the maximum Lorentz factor of a charged particle cannot be larger than 1000. Their main conclusions are as follows:
We consider an anisotropic radiation field from the accretion disk and assume that an electron with Lorentz factor
moves along
the magnetic field with an inclined angle
to the normal
direction to the disk (see Fig. 1). In the electron rest frame [
], the
number of photons scattered by the electron to the direction
(
,
)
is
(1) |
(2) |
Figure 1: The geometry used in the calculations. | |
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Since the scattering number is a Lorentz invariant, we can obtain the scattering
number measured in inertial frame [S]:
(3) |
The power of a scattered photon is available:
0pt
= | |
= | (4) |
(5) |
(6) |
(7) |
(8) |
(9) |
Assuming the accretion disk has an infinite parallel plane geometry, we can integrate Eq. (7) over :
(10) |
(11) |
(12) |
= | |
(13) |
(14) |
(15) |
(16) |
So the cooling time scale for the inverse Compton energy loss can be
expressed as:
(17) |
(18) |
Thus, we can obtain the maximum Lorentz factor of the electron by equating these two time scales given by Eqs. (17) and (18).
In order to show the influence of the anisotropic disk radiation field on the inverse Compton energy loss, we plot as functions of in Fig. 2. We note that value of is insensitive to the Lorentz factor , if . (For a large , in Eq. (13).) increases rapidly with , i.e., the inverse Compton energy loss is strongly reduced in small case (close to the normal direction to the disk).
Figure 2: The dependence of given by Eq. (13) on , where is adopted. | |
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Figure 3: Acceleration time scale (dotted) and inverse Compton cooling time scale (Eq. (17)) for , and , as functions of the electron Lorentz factor , where is adopted. | |
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In our calculations of the maximum electron Lorentz factor , we use the same parameters as those used by Rieger Mannheim (2000) for comparison: a light cylinder radius cm, an initial position and an initial velocity of the injected electron. We assume the mass of the central black hole in the AGN , where denotes the solar mass. We also express the disk luminosity as , and is the maximum luminosity corresponding to the mass of the black hole.
The two time scales and as functions of the Lorentz factor are shown in Fig. 3. We use and , , in the calculations. In this figure, is denoted by the dotted line, and the other three solid lines correspond to for the different incoming directions , and . It can be seen that the electron can attain a high , when the electron is accelerated close to the normal direction to the disk.
The maximum Lorentz factor as a function of for the different electron incoming directions is plotted in Fig. 4. From right to left, each line corresponds to , , and , respectively.
Figure 4: Maximum electron Lorentz factor as a function of the disk luminosity , for and , respectively. | |
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We considered the centrifugal particle acceleration mechanism for an electron moving along a rotating magnetic field line in AGNs. The electron gains energy from centrifugal acceleration, while it also loses energy due to the interaction with the photons emitted from the disk (inverse Compton scattering). The maximum Lorentz factor is determined by equating the acceleration time scale and the cooling time scale.
The radiation field of the disk is anisotropic, so the inverse Compton energy loss will depend on the direction of motion of the electron. We therefore calculate the inverse Compton energy loss in the case of an anisotropic radiation field from the accretion disk. We found that the inverse Compton energy loss sensitively depends on the angle , i.e., the angle between the axis of the accretion disk and the direction of motion of the electron (see Fig. 1). The maximum Lorentz factor of the electron is therefore significantly influenced by this angle. We can see in Fig. 2, that the maximum Lorentz factor varies with the direction of motion of the electron for fixed disk luminosity. is less than 200, when the electron is moving parallel to the disk plane. This is similar to the results given by Rieger & Mannheim (2000). The value of could be higher than 1000, if the angle . It would be very high if the electron moves close to the normal direction of the disk ( ). It indicates that the electron can be magnetically accelerated more efficiently if the direction of motion is close to the normal direction to the disk. This is consistent with the well known phenomenon that jets are always perpendicular to the disk plane.
In Fig. 4, we found that the electron can be magnetically accelerated to in sub-Eddington cases ( ), if the electron is moving close to the normal direction to the disk ( ). In the cases of , the inverse Compton energy loss dominates over the energy gain from the magnetic field line, even if the electron is moving close to the disk axis. It implies that the centrifugal acceleration of the electron is mainly determined by the disk luminosity for high Eddington cases. It is therefore worthwhile to explore the relation between the radio properties and the accretion type in AGNs, which would be a useful test on this acceleration mechanism of electrons. This is beyond the scope of the present work, but it will be considered in future work.
Besides the inverse Compton scattering with the soft photons emitted from the disk, the emission from the broad-line region (BLR) will interact with the electron. The total broad-line luminosity in the AGN is about 10 percent of its optical continuum luminosity (Cao & Jiang 1999, 2001). The radius of the BLR is usually much more larger than light cylinder (Kaspi et al. 2000). So, the inverse Compton energy loss due to the interaction with the soft photons from the BLR can be neglected compared with that caused by the disk radiation, even if the electron is moving close to the disk axis.
Acknowledgements
I thank the support from NSFC (No. 10173016).