## 4. Strong electric field accelerationTurning to the question of high-energy electron acceleration, we recall that according to Eq. (5) the average electron energy gain is determined by , which is rather modest. As an interesting aside, we note that lower but still acceptable particle densities of order cm would make it possible to energize the bulk of electrons to thus removing the injection problem for electron acceleration by shock waves or MHD turbulence (cf. Lesch & Birk 1997). It should be emphasized that the DC electric field acceleration itself will lead to much higher energies in those parts of the current sheet where is much less than its average value. Because the transverse field
slowly varies along the current sheet, going through zero at its
center, the magnetic field projection onto the with . It should be realized that the spatial variation of the field is
related to its temporal evolution. The magnetic field in the sheet is
not static. The reconnecting field lines move into the sheet with
speed and out of the sheet (along
the Here , and the scale of the field variation from zero to is half the current sheet width, . The approximation of a constant speed corresponds to a linear dependence in the steady state when . It is clear from Eq. (10) that higher particle energies can be
reached close to the singular line where
. One might think that arbitrarily
large energy gains (or at least large enough to make synchrotron
losses important) could be possible near the singular line. This is
not the case though because the acceleration time
increases together with the energy
gain. When becomes large enough, it
is no longer possible to assume that acceleration occurs at a fixed
It is straightforward to estimate when the temporal variation can be ignored. The relative error in the energy gain due to ignoring the temporal variation of the field is from Eq. (10). The change in during the acceleration time is determined from Eq. (11) as . The acceleration time is evaluated for the acceleration length given by Eq. (8). This leads to the estimate This is very small in most of the current sheet. For instance, for , so it is indeed possible to treat as a parameter in most of the sheet. Nevertheless the approximation obviously breakes down at the singular line where and we will use this fact later to determine the maximum energy of the accelerated electrons . For energies lower than , the dependence of the energy gain on the particle location as a parameter leads to a continuous electron spectrum extending to high energies. Calculation of the detailed spectrum and the total number of accelerated electrons is a complicated problem that requires numerical simulations including the effects of nonuniform magnetic fields, particle escape from the sheet, and the charge separation electric fields. Nevertheless, it can be demonstrated that a power-law spectrum may result (Litvinenko 1996). Consider acceleration in the case of a linear magnetic X-point: . The energy spectrum below follows from the continuity equation with from Eq. (10). Assuming a spatially uniform inflowing distribution leads to It is interesting to point out that the power-law electron distribution has been obtained in numerical simulations of driven collisionless reconnection in the context of extragalactic radio sources (Romanova & Lovelace 1992). It appears, however, that a somewhat steeper spectrum would be necessary to interpret observations of powerful radio galaxies (Meisenheimer et al. 1997). This is not suprising because of the simplifying assumptions used in the estimate above. An additional process of the particle loss from the sheet, caused by more complicated geometry for example, would make the actual spectrum steeper. Mori et al. (1998) considered a similar problem of charged particle acceleration in the vicinity of a singular line in the solar corona and demonstrated numerically the formation of a power-law spectrum with the index of about for a wide range of parameters. The question remains whether the maximum energy of the accelerated electrons in the current sheet is compatible with the observations that imply the presence of TeV electrons in extragalactic radio sources. Recall that it was possible to ignore the time dependence of in Eqs. (8) and (10) because the acceleration time is much less than the time scale of the field variation. Ultimately, though, the time dependence limits the energy of the accelerated electrons. Physically, the energy of a magnetized electron increases with time but the maximum possible energy decreases as the particles are carried out of the sheet and the transverse magnetic field "felt" by the particles becomes larger, which makes it easier for them to escape the current sheet. The maximum electron energy can be estimated by noting that the magnetized electrons move almost along inside the sheet, so their relativistic Lorentz factor increases with time as where a small initial energy is ignored. The maximum energy is determined by Eq. (10) with the time-dependent for a reconnected field line given by Eq. (11) and the time-dependent Lorentz factor given by Eq. (14). Making the substitutions leads to an equation for the maximum acceleration time, which is solved to give Substituting this into Eq. (14) gives the sought-after maximum electron Lorentz factor The maximum acceleration length is still quite small: cm . Thus even for the highest energies, the strong DC electric field acceleration remains a local acceleration mechanism. The same numerical values as before have been employed in these estimates. It is reasonable that the same value for is obtained from Eqs. (10) and (12) under condition that . Hence the derived is actually the energy when the effects of the reconnected field line motion become noticeable. The resulting corresponds to the energy of about 3 TeV. This explains the observed optical synchrotron jet emission that appears to require (Lesch & Birk 1998). It is possible of course that a finite efficiency of the acceleration process will somewhat decrease the estimate in Eq. (16). Clearly the maximum acceleration time above is still much less than
the time of synchrotron losses s,
where is defined by Eq. (2). In
other words, the maximum acceleration length is still much less than
the synchrotron loss length. This is in agreement with our suggestion
that the energy losses are not important for acceleration in strong
electric fields. Note that the loss length is calculated based on
Eq. (2) that holds for the electron motion perpendicular to the field
lines and gives the maximum synchrotron energy loss rate. Since this
paper argues that electrons are likely to be accelerated along the
magnetic field lines inside the current sheet, the role of the losses
should be even less noticeable. It is also interesting to note that
in Eq. (15) does not depend on the
reconnection electric field © European Southern Observatory (ESO) 1999 Online publication: September 2, 1999 |