5. Effect of plasma electrons
Central to the above mechanism for HXR production is the assumption that the neutral beam survives as such along the stopping length of the protons without the beam electrons being stripped from them. As discussed by Brown et al. (1998a) and by Simnett & Haines (1990) the presence in the background gas of free electrons, which we have ignored here so far, modifies the electrodynamics and can result in electron stripping. Since there will always be some free background electrons present in the real solar atmosphere, even in the low chromosphere, we reconsider here whether a neutral beam can in fact survive the propagation process. In doing so we correct an error in the discussion of this problem by Brown et al. (1998a) which may change their conclusions under some conditions.
When a neutral beam enters a region with free background electrons these respond to the p-e charge separation electric field and so contribute to the return current. Denoting by , and the current densities of the beam electrons, beam protons, and ambient electrons respectively, steady current neutralisation then requires . Background electron motion results in so that and since in the beam implies so that the electrons fall progressively further behind the protons. Whether this effect is large enough to affect our conclusions regarding HXR production depends on how large a current the background can carry in response to the electric field and this will depend on the free electron density and on the effective collisionality (resistivity) of the background plasma. The dynamics of this process are described by Eqs. (16)-(18) of Brown et al. (1998a) which, in the above notation (dropping subscript e for simplicity and denoting the electric field by ) are
and is the collisonal drag force. If the beam persists long enough, the solution of these equations approaches the steady state (Brown et al. 1998a)
with determined by
The error in Brown et al. (1998a) was the argument that (35) implies (meaning that the beam electrons are stripped from the protons and join the plasma electron drift) "since is monotonic". In Brown et al. the treatment was solely 1-D and deterministic and the beam and plasma were cold so no spread in were included. In real plasmas is, however, only monotonic () for drift speeds well above the thermal speed but turns over at around the thermal speed and tends to for slow drifts. Unless the plasma is very highly conductive (i.e. hot) the plasma drift speed , and hence the plasma drift current , at which (33) reaches a steady state are smaller than () found by Brown et al. In short, the plasma electrons are more collisional than the beam electrons and their contribution to neutralising will be less likely to result in beam stripping than suggested by Brown et al. To quantify this for general drift speed we adopt (with at temperature T)
which for is
where is the background plasma temperature. (Here we have essentially taken the Spitzer conductivity limit for a fully ionised gas and ignored any anomalous effects - these classical are pessimistic in minimising the collisionality and hence maximising ). For the beam electrons () it is
where we have added the factor f from Eq. (1) to allow as before for the fact that, as they propagate, beam electrons become `warm' relative to the bombarding background.
Inserting (37) and (38) in (35) to find we get
and inserting this in (34) gives
Once the beam electron heating, which we have described in previous Sections, is well developed we approach giving
with . Thus and stripping is negligible even for quite high beam densities - even moreso in the chromosphere than in the corona since falls across the transition zone
At the start of beam propagation, however, we have and so
This suggests that if the above steady state is approached and the beam starts and in the corona then very substantial electron stripping may occur () even for an of 10 MeV unless: , which is unlikely; and/or the beam density is very high ( - i.e. essentially all of the plasma is accelerated; and/or substantial anomalous resistivity (collisionality) exists. If on the other hand the beam started in the chromosphere the facts that falls by a factor across the transition zone, that the number density of free electrons drops with depth, and that the beam becomes denser as the protons slow down, all make survival of the neutral beam more likely. This analysis rests on consideration of the steady state. This is approached on a timescale . At the start of the beam the relevant numbers yield
which is short compared to the time for the beam to reach the chromosphere or to stop unless and .
© European Southern Observatory (ESO) 2000
Online publication: December 17, 1999