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

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4. Strong electric field acceleration

Turning to the question of high-energy electron acceleration, we recall that according to Eq. (5) the average electron energy gain is determined by [FORMULA], which is rather modest. As an interesting aside, we note that lower but still acceptable particle densities of order [FORMULA] cm[FORMULA] would make it possible to energize the bulk of electrons to [FORMULA] 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 [FORMULA] is much less than its average value.

Because the transverse field [FORMULA] slowly varies along the current sheet, going through zero at its center, the magnetic field projection onto the xy-plane has the geometry of a standard magnetic X-point (Fig. 1). In other words, this point is a projection of the singular magnetic field line with [FORMULA] onto the xy-plane. This is where the field lines are "cut" and "reconnected." More complicated geometries with multiple singular lines are also possible due to the tearing instability in the sheet, for instance. Since the acceleration length scale is typically much less than the length scale of [FORMULA] that can be of order w, Eq. (8) derived for [FORMULA] remains valid for [FORMULA], unless [FORMULA]. Now, however, the energy gain depends on the location in the sheet as a parameter. The electron Lorentz factor is found from Eqs. (1) and (8):


with [FORMULA].

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 [FORMULA] and out of the sheet (along the x-axis in Fig. 1) with the Alfvén speed [FORMULA] and carry the magnetized particles with them. This familiar "sling-shot effect" causes the reconnected field lines to straighten out so that [FORMULA] increases from zero at [FORMULA] to the maximum value [FORMULA] at [FORMULA] for each reconnected field line, leading to a dependence [FORMULA] in a steady state. Determining the full reconnection dynamics of the field is beyond the scope of this paper. Useful exact solutions for steady state reconnection have been given by Craig & Henton (1995). It is sufficient for our purposes to consider a simple approximation of a constant speed of the reconnected field lines along the x-axis:


Here [FORMULA], and the scale of the field variation from zero to [FORMULA] is half the current sheet width, [FORMULA]. The approximation of a constant speed [FORMULA] corresponds to a linear dependence [FORMULA] in the steady state when [FORMULA].

It is clear from Eq. (10) that higher particle energies can be reached close to the singular line where [FORMULA]. 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 [FORMULA] increases together with the energy gain. When [FORMULA] becomes large enough, it is no longer possible to assume that acceleration occurs at a fixed x and to ignore the temporal variation of the transverse field [FORMULA] on the particle orbit. What happens instead is that the magnetized electrons are carried with the reconnected field lines from the center of the sheet to its edges where [FORMULA] is larger and the energy gain [FORMULA] is smaller. This effect ultimately limits the electron energy.

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 [FORMULA] from Eq. (10). The change in [FORMULA] during the acceleration time is determined from Eq. (11) as [FORMULA]. The acceleration time [FORMULA] is evaluated for the acceleration length [FORMULA] given by Eq. (8). This leads to the estimate


This is very small in most of the current sheet. For instance, [FORMULA] for [FORMULA], so it is indeed possible to treat [FORMULA] as a parameter in most of the sheet. Nevertheless the approximation obviously breakes down at the singular line where [FORMULA] and we will use this fact later to determine the maximum energy of the accelerated electrons [FORMULA].

For energies lower than [FORMULA], 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: [FORMULA]. The energy spectrum [FORMULA] below [FORMULA] follows from the continuity equation [FORMULA] with [FORMULA] from Eq. (10). Assuming a spatially uniform inflowing distribution [FORMULA] leads to


It is interesting to point out that the power-law electron distribution [FORMULA] 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 [FORMULA] 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 [FORMULA] 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 [FORMULA] 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 [FORMULA] 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: [FORMULA] cm [FORMULA]. 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 [FORMULA] is obtained from Eqs. (10) and (12) under condition that [FORMULA]. Hence the derived [FORMULA] is actually the energy when the effects of the reconnected field line motion become noticeable. The resulting [FORMULA] corresponds to the energy of about 3 TeV. This explains the observed optical synchrotron jet emission that appears to require [FORMULA] (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 [FORMULA] s, where [FORMULA] 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 [FORMULA] in Eq. (15) does not depend on the reconnection electric field E. We repeat for clarity that for time intervals and energy gains smaller than those given by Eqs. (15) and (16) the time dependence of the magnetic field lines can be ignored, so that the particle acceleration far from the magnetic singular line can be studied assuming a constant instantaneous value of [FORMULA] that depends on [FORMULA] as a parameter. This justifies the use of Eq. (10) to derive the energy distribution in Eq. (13) for [FORMULA].

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Online publication: September 2, 1999