## 3. Particle orbits in the current sheetLet 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 , 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 : (Zhu & Parks 1993). Here the minus sign corresponds to the
electric current in the positive 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
, the motion consists of the
acceleration along the electric field
and finite oscillations along the
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 defines the acceleration length as the displacement along the electric field, which corresponds to when the magnetized electrons initially inside the sheet at leave the sheet along the field lines: The displacement perpendicular to the electric field is given by a similar 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, . Models for
fast collisionless reconnection suggest that the average transverse
field is of order (Hill 1975).
According to Hill (1975), the ratio
should be of order the reconnection Mach number 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
cm, which is much less than the
length of synchrotron losses from Eq. (2). Given the small
acceleration length , 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 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 erg s 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
, associated with a current sheet
with dimensions Such annihilation would simplify the field geometry, making the nonreconnecting axial field 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. © European Southern Observatory (ESO) 1999 Online publication: September 2, 1999 |