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

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2. Weak electric field acceleration

The electron energy gain in the current sheet is defined by the potential drop experienced by a particle:

[EQUATION]

Here [FORMULA] is the electron Lorentz factor, [FORMULA] and e are the electron mass and charge, and E is the electric field in the sheet. Here and in what follows the relativistic case is studied, [FORMULA], so the particle speed is set to the speed of light c where possible. In the one-dimensional model, the acceleration length [FORMULA], which is the particle displacement along the electric field inside the sheet, could be as large as the current sheet length [FORMULA]. It should be expected though that the synchrotron energy losses will limit the acceleration length. The synchrotron loss length scale for an electron moving perpendicular to the magnetic field is equal to

[EQUATION]

where [FORMULA] is the reconnecting magnetic field.

It is likely that the electric field acceleration will produce beams of electrons, possibly moving with an angle [FORMULA] with respect to the magnetic field lines. In this case the energy loss rate will decrease by a factor of [FORMULA], and the loss length will increase by a factor of [FORMULA]. Since the minimum length given by Eq. (2) is already on the order of the jet length, smaller synchrotron losses would be too weak to determine the acceleration length. The reconnection geometry in Fig. 1 suggests that the electrons are accelerated along the electric field [FORMULA] and perpendicular to the reconnecting magnetic field [FORMULA] outside the current sheet, so that indeed [FORMULA] and the loss length is determined by Eq. (2). The influence of the magnetic field inside the sheet on the particle motion is ignored in the one-dimensional model.

[FIGURE] Fig. 1. Reconnecting current sheet dimensions (length [FORMULA], width w, thickness [FORMULA]) and the magnetic field projection on the xy plane. The reconnection electric field and the longitudinal magnetic field are along the z-axis. The acceleration length [FORMULA] and the synchrotron length [FORMULA] are also measured along the z-axis.

Postulating that [FORMULA] and taking [FORMULA] G and [FORMULA] according to observations (Meisenheimer et al. 1997), one obtains the electric field required in the model

[EQUATION]

The large [FORMULA] cm from Eq. (2) requires the electric field to be very small. A conventional measure of the field strength is the reconnection Alfvén Mach number [FORMULA], where [FORMULA] is the inflow speed as measured by the drift speed into the sheet, [FORMULA] cm s-1 is the Alfvén speed, [FORMULA] is the proton mass, and n is the plasma density. Assuming [FORMULA] cm[FORMULA] gives [FORMULA], so reconnection is indeed very slow.

The weak electric field model is an attractively simple approach to the problem of electron acceleration in the jets by the DC electric field. More importantly, it can be used to explicitly relate the properties of the accelerated particles to the characteristics of the energy release by virtue of magnetic reconnection (Lesch & Birk 1998). In spite of these advantages, the model has several unappealing features. First, the energy release rate in a single current sheet is very small. The Poynting flux into the current sheet determines its luminosity

[EQUATION]

which is only [FORMULA] erg s-1 if both the current sheet width w and its length [FORMULA] are assumed to be of order [FORMULA] kpc (cf. Lesch & Birk 1998). Thus [FORMULA] reconnection regions are required to explain the highest observed power of about [FORMULA] erg s-1. An unknown physical mechanism would have to produce multiple, densely packed current sheets in the jet and to ensure the stability of such highly filamented structure. It is not clear what could prevent violent interaction of the multiple electric currents with each other. Second, the acceleration length of order the total length of the current sheet is hard to achieve since it would require a highly symmetric configuration of the fields, which is problematic given the postulated highly filamentary magnetic field. Moreover, assuming that the average [FORMULA] leads to a contradiction.

Allowing for the possibility that the acceleration length is different for different particles, let us introduce the average value [FORMULA]. The next section will show that this possibility is realized in reconnecting current sheets with a nonzero magnetic field. In any case, since the magnetic field energy is converted into the particle kinetic energy in the sheet, the Poynting flux in Eq. (4) should be equal to the product of the particle influx to the sheet, [FORMULA], and the average particle energy gain, [FORMULA]. Equating the product of these two quantities to the Poynting flux gives the following self-consistency relation:

[EQUATION]

(Alfvén 1968; Vasyliunas 1980). Using [FORMULA] statvolt/cm together with the previous values for [FORMULA] and n gives [FORMULA]. Self-consistency dictates therefore that the average acceleration length is much less than [FORMULA], which is in evident conflict with the weak field model. The conclusion is that there has to exist a mechanism that limits the electron acceleration length, at least for some particles, much more efficiently than the synchrotron losses alone. One such mechanism is discussed in the remainder of this paper.

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© European Southern Observatory (ESO) 1999

Online publication: September 2, 1999
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