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Astron. Astrophys. 349, 685-690 (1999)

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3. Particle orbits in the current sheet

Let us see now what happens when the assumption of one-dimensionality of the particle motion inside the current sheet is relaxed by considering the magnetic field inside the sheet. A large body of research has been devoted to this question in the context of particle acceleration on the Sun and in the geotail (e.g., Speiser 1965; Martens 1988; Zhu & Parks 1993; Litvinenko 1996). The point is that although the magnetic field itself cannot change the particle energy, it can change the electron orbit, decreasing the displacement along the electric field and hence the energy gain. The sooner the particles are ejected from the sheet, the smaller the energy gain will be.

In general, not only the reconnecting component of the field [FORMULA], but also the longitudinal (along the electric field) and the transverse (perpendicular to the plane of the sheet) magnetic field components are present in the sheet (Fig. 1). The usual approach to studying the charged particle trajectories in the sheet is to approximate the field by the first nonzero terms in the Taylor expansion inside the sheet located at [FORMULA]:

[EQUATION]

(Zhu & Parks 1993). Here the minus sign corresponds to the electric current in the positive z-direction, and a is the current sheet half-thickness. The reconnection electric field is [FORMULA]. The nonreconnecting component [FORMULA] may be assumed constant. The transverse field [FORMULA] changes sign at the center of the sheet and reaches a maximal value at its edges [FORMULA]. For [FORMULA], this component is a very slowly varying function of x. Hence [FORMULA] is often also assumed constant on a given particle trajectory (see next section, however).

The character of the charged particle motion for various relative values of the magnetic field components in the current sheet has been summarized by Litvinenko (1996). In the limit [FORMULA], the motion consists of the acceleration along the electric field [FORMULA] and finite oscillations along the y-axis caused by the Lorentz force [FORMULA] (Speiser 1965, Zhu & Parks 1993). This idealized, highly symmetric situation is unlikely to occur. In fact any sheet model requires a nonzero [FORMULA] as a result of the reconnection itself. Particle orbits in current sheets with [FORMULA] are very complex in general, but the situation is simpler in two limiting cases. If the longitudinal field is weak, usually [FORMULA], the maximum displacement along the electric field is determined by the particle gyroradius in the transverse field [FORMULA] (Speiser 1965). Since the magnetic field in extragalactic jets is known to have a significant axial component, the other limit of a strong longitudinal field, [FORMULA], should be appropriate for the jets. The strong longitudinal field [FORMULA] magnetizes the electrons and makes them follow the field lines (Litvinenko 1996). Such magnetization is much less efficient for much heavier protons that will still follow the Speiser-type orbits. In addition to the reconnection electric field, therefore, the charge separation electric field arises in the sheet because the electrons and protons follow different orbits. Particle simulations of collisionless reconnection (Horiuchi & Sato 1997), however, appear to be in a surprisingly good agreement with the results of the test particle approach (Litvinenko 1997) that is assumed in this paper.

Thus the magnetized electrons will mainly move along the magnetic field lines in the sheet. Acceleration will cease when the particles leave the sheet. Integrating the magnetic field line equations

[EQUATION]

defines the acceleration length [FORMULA] as the displacement [FORMULA] along the electric field, which corresponds to [FORMULA] when the magnetized electrons initially inside the sheet at [FORMULA] leave the sheet along the field lines:

[EQUATION]

The displacement perpendicular to the electric field is given by a similar formula [FORMULA].

As far as the numerical values are concerned, in what follows the reconnecting and nonreconnecting components of the field are assumed to be of the same order, [FORMULA]. Models for fast collisionless reconnection suggest that the average transverse field is of order [FORMULA] (Hill 1975). According to Hill (1975), the ratio [FORMULA] should be of order the reconnection Mach number M. This will be confirmed below. The current sheet half-thickness a is not an independent parameter either. As analysis of balance equations following from the conservation laws and the Maxwell equations shows (Cowley 1986), it is of order the ion skin length [FORMULA], which is [FORMULA] cm in the jet. The numerical values of the reconnecting field component [FORMULA] G and the particle density [FORMULA] cm[FORMULA] are taken as previously.

Allowing for a nonzero magnetic field inside the sheet has important consequences for electron orbits. Eq. (8) shows that the particle escape is indeed much more efficient across the sheet than along it. The average acceleration length is estimated to be [FORMULA] cm, which is much less than the length of synchrotron losses from Eq. (2). Given the small acceleration length [FORMULA], it is clear that the strong DC electric field acceleration is a local reacceleration mechanism that can occur throughout the reconnection region, which may explain constant luminosity all along the jet. Using this value of l in Eq. (5) leads to the self-consistent electric field [FORMULA] statvolt/cm, which is almost six orders of magnitude larger than the value required in the weak field model from Eq. (3). As assumed earlier, this value of E implies fast magnetic reconnection with [FORMULA] cm s-1 and [FORMULA], where [FORMULA] is the inflow speed to the sheet as determined by the electric drift.

A principal benefit of the fast reconnection model is that it provides a significant energy release rate in a single current sheet. Using the above value of the electric field in Eq. (4) gives the luminosity [FORMULA] erg s-1 for a current sheet with dimensions [FORMULA] kpc. Even larger luminosities can be achieved in longer and wider sheets. Hence just a few reconnection regions may be enough to provide the energy release even in most powerful sources. This conclusion is consistent with an estimate for the power output [FORMULA] erg/(cm[FORMULA]s) due to magnetic reconnection in a jet with somewhat different parameters (Vekstein et al. 1994).

The high rate of the magnetic energy release by virtue of fast reconnection would require the corresponding continuous supply of magnetic energy into the jet. It is easy to see that otherwise the reconnecting field component in the volume of order [FORMULA], associated with a current sheet with dimensions w and [FORMULA], would be essentially annihilated on the reconnection time scale

[EQUATION]

Such annihilation would simplify the field geometry, making the nonreconnecting axial field [FORMULA] the only nonzero component in the jet. This problem can be avoided if the jet magnetic field is continuously being dragged out from the central rotating source by the jet flow (Romanova & Lovelace 1992). It appears possible in this model that the subsequent shearing of the field by plasma flows regenerates the reconnecting component of the field thus supplying the required magnetic energy in the jet. Evidently a steady state can be reached provided the time scales for release and supply of the magnetic energy are comparable.

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© European Southern Observatory (ESO) 1999

Online publication: September 2, 1999
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