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Astron. Astrophys. 353, 729-740 (2000)

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3. Analytic treatment

For the moment, as in the numerical simulations, we consider for simplicity only the case of an injected beam which is mono-energetic; with speed [FORMULA], energy [FORMULA], and of density [FORMULA] at injection. (Aspects of the case with an energy distribution will be discussed in Sect. 4.) As the beam protons decelerate by colliding with the background, these values evolve to [FORMULA], [FORMULA] and [FORMULA] at time t, distance x and atmospheric column density N along the path. The particle flux [FORMULA] is constant in a steady state (for this mono-energetic case) and corresponds, over injection area A, to a total injected beam power [FORMULA].

According to the 1D analysis of Brown et al. (1998a) (end of their Sect. 2), the accompanying electrons on average follow the protons at a tiny distance (due to the differential collisional drag) with the same mean speed, though undergoing small amplitude, high frequency, electrostatic oscillations about their mean position. The spread in the electron speed was not considered in the 1D approach, as collisions were assumed only to cause a deterministic deceleration of the electrons. In Sect. 2 above, however, we have seen in a 2D treatment where (transverse) scattering, and not just energy loss, is allowed that both the mean value and the variance of the transverse velocity increase as the beam propagates. This is because changes in transverse speeds are not inhibited by the charge separation electric field as is the longitudinal speed. This growth and spread in transverse electron energy can be visualised in the local beam frame as essentially a heating of the beam (target) electrons under the collisional effects of the bombarding atmosphere. The flow of energy here is not immediately obvious. In the beam frame, the initially cold beam electrons are apparently being heated by these collisions. This heating can be seen in the plasma background frame as the beam electrons developing a distribution in velocity whilst following the longitudinal motion of the protons. In the plasma background frame, the electrons and protons are both losing energy to the background particles. However, the electrons are also gaining energy from the protons because of the dragging and, according to the results of Sect. 2, must be doing so faster than they are losing energy to the background plasma. If left alone, these beam electrons would quickly stop, and only heat up very slightly. However, the fact that they are driven, by the electrostatic dragging action of the protons, means that they continue to heat up. We want to estimate the mean random electron kinetic energy achievable by this process during the stopping lifetime of the protons, and also to evaluate the efficiency of the resulting bremsstrahlung radiation to see if it can contribute to the flare HXR production. Though we recognise that the spread in electron energy need not be Maxwellian, it is evident from the numerical results of Sect. 2 that there is a considerable spread skewness about the mean and we will sometimes deal with this in terms of a `beam electron temperature'.

The beam protons lose energy via beam collisions in two ways. The first is via direct collisions with the background particles. This loss is dominantly to background electrons, and occurs at a rate [FORMULA] where [FORMULA], where [FORMULA] is the Coulomb logarithm, [FORMULA] and [FORMULA] is the local background density (cf Emslie 1978). Here we have assumed the background to remain cold. The second is via the electrostatic drag exerted on the beam protons by the beam electrons. Energy transfer from the beam protons to the beam electrons in their collisions with the background can be written for a mean beam electron of energy [FORMULA] as [FORMULA] where the factor [FORMULA] allows for the decline in collisional energy transfer to the beam electrons, from the background electrons and ions, as the beam electron mean random speed increases and exceeds the speed ([FORMULA]) of the `bombarding' background particles. If the beam electrons are cold at injection (rms speed [FORMULA]) then there the cold target limit [FORMULA] applies and we see that rapid heating should occur at the rate [FORMULA] which means (just as found in our simulations - Sect. 2) that the random [FORMULA] will increase on the time-scale of an electron collision time. This means, however, that in a very small fraction ([FORMULA]) of the proton stopping time the electron random speed will exceed the longitudinal velocity of the beam, i.e. of the bombarding background (or [FORMULA]), and the warm target situation (Trubnikov 1965) of decreasing f will apply for the remainder of the beam propagation. Since this will in fact apply over almost all the beam length, and since the electron energies in the very initial phase are not relevant to HXR production, we will simplify our analysis by adopting a small beam electron random energy [FORMULA] at injection and using the warm target f throughout. To do so we adopt a rough analytic fit to the graph of f in Emslie et al. (1997), viz.

[EQUATION]

where we have equated [FORMULA] here with temperature (energy units) in that graph. In this regime the evolution of [FORMULA] and [FORMULA] are then well described by

[EQUATION]

[EQUATION]

(It is important to recall here that [FORMULA] since the random electron speed is not equal to the beam speed [FORMULA]). In (2) and (3) we have retained only the lowest order terms in f: in this domain (covering almost the whole trajectory), the energy equation of the protons is dominated by their direct energy losses to the background i.e. f is assumed small enough to justify [FORMULA] but still substantial enough to result in significant energy transfer from the protons. These equations can be solved to give [FORMULA] and also [FORMULA] where [FORMULA] is the ambient column density measured from injection. (Note that we neglect a minor correction due to proton pitch angle scattering on the background protons). The solutions are

[EQUATION]

[EQUATION]

where

[EQUATION]

is the column density required to stop the protons and the maximum mean electron energy [FORMULA], attained by the electrons as the protons stop, is given by

[EQUATION]

It follows that collisional effects in the propagation of a neutral beam of particle energy say 10MeV at injection give rise to random transverse motions of the beam electrons with mean energies up to around 180 keV. (We observe, however, that in (7) the factor [FORMULA] cannot really be distinguished from unity, given the approximations made.) In Fig. 5a we show [FORMULA] (solid) and [FORMULA] (dashed) as functions of depth [FORMULA] in the target, and in Fig. 5b [FORMULA] is plotted againsts [FORMULA]. We can see that the electrons closely approach the maximum [FORMULA] rather rapidly so that the `hard' electrons capable of emitting HXRs emit throughout most of [FORMULA], the moreso since the beam emissivity is concentrated toward [FORMULA] as the beam density increases. The mechanism we have discovered is thus capable of producing, from a 10MeV neutral beam, electrons of sufficient individual energies for flare HXRB production, but we still have to determine whether the process is efficient enough to produce the observed HXRB bremsstrahlung intensities and, if so, to explore its observational properties.

[FIGURE] Fig. 5. a  The ratio of proton energy to initial proton energy [FORMULA] (solid line) and electron energy to maximum (final) electron energy [FORMULA] (dashed line) vs. the ratio of traversed column depth to beam stopping column depth [FORMULA]. b  The ratio of electron energy to maximum (final) electron energy [FORMULA] vs. the ratio of proton energy to initial proton energy [FORMULA].

We have also checked this by more exact treatment of the above problem with [FORMULA] as described in Emslie et al. (1997). Namely, the total electron heating was computed by a numerical integration of the following equation:

[EQUATION]

where [FORMULA] is the proton stopping time, C=[FORMULA], and G = [FORMULA] is the exact warm target expression. (Our [FORMULA] corresponds to µ in Emslie's notation.) The energy gain of beam electrons was computed for different proton beam energies in the range of 1-10 MeV. Results are presented in Fig. 6, where the initial electron energies (dashed line) and the resulting mean electron energies after the heating process (solid line) are shown. These results are in very good agreement with the results obtained using empirical fit used above.

[FIGURE] Fig. 6. The total energy of the beam electrons after the heating process (solid line) and their initial energies (dashed line) vs the initial proton beam energy.

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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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