## 1. IntroductionHard X-Ray (HXR) and radio emission from the Sun tells us that non-thermal distributions of electrons are present during the impulsive stage of a solar flare (Brown & Smith 1980). These electrons have energies (tens to hundreds of keV) much greater than the ambient particles of the solar atmosphere ( keV), and there is good reason to believe their particle distributions are anisotropic and, in some cases, beam-like (Karlicky 1997). Careful interpretation of the HXR and radio observations of the impulsive stage of flares can lead us to a better understanding of how the energy is first released, then given to non-thermal particles, and then lost causing subsequent, more gradual solar flare phenomena. Observations of gamma-rays also tell us that protons and ions are accelerated during a flare, and that their total energy content is arguably similar to that of the electrons (Trottet et al. 1998; Ramaty et al. 1995). The work presented in this paper is primarily aimed at electrons, because they can produce HXRs which can be imaged, though many of the results presented here can be easily extended to describing protons. The observation of coronal, impulsive HXR sources (Masuda et al.
1995) has been interpreted as being the most direct evidence yet of
the location of a flare energy release site. It is plausible to make
such an interpretation because the short collisional life-times of HXR
producing electrons (relative to protons/ions that produce gamma-ray
lines) mean that the HXRs they emit give us the earliest, and
spatially nearest, information on where energy is being released. It
is interesting to examine the assumptions implicit in the last
sentence, they are that: 1) the acceleration of particles is localised
in space and time (in the Masuda case to the loop-top), and 2) the
electrons are not a secondary product of energy release, e.g.
accelerated by ions or protons which were the first to receive energy.
Possible scenarios exist that explain the observations with
alternative assumptions that are equally plausible. For example, one
possibility is that the energy is first released into waves, which
then accelerate particles, e.g. Miller et al. (1997). Models invoking
the existence of density pockets (Wheatland & Melrose, 1995) can
then be used to explain the loop-top source, without energy release
taking place there. The fact that the HXR source was clearly above the
soft X-ray loop top is significant as it raises the problematic issue
of energy transfer across magnetic field lines, as tentatively
suggested by Conway & MacKinnon (1998b). (Note: A prior paper
Conway & MacKinnon (1998a) proposes a model based on the erroneous
result that electron-cyclotron maser emission can travel There are of course a great many effects to be considered when
modelling particle transport. These range from straightforward
propagation along the magnetic field, to complicated wave-particle
interactions and instabilities. Indeed, the term "transport" could
even include the particle acceleration itself. The formalism presented
at the start of this paper provides a frame-work to address any of
these effects in terms of moments of the distribution. Whether it is
practical to do so is another matter, and depends on the availability
of solution to stochastic differential equations. Such solutions are
available for the case of an arbitrary density and constant magnetic
field strength (Conway et al. 1998), and these are used to yield
useful results in this paper. The use of the results and methods are
two-fold. Firstly, it provides mathematical results that can be used
to link observations and theory without recourse to detailed numerical
simulations. Secondly, in more complicated cases (i.e. beyond effects
of propagation and collisions) it can provide specific results that
can be used to verify that the distributions from numerical
simulations have the correct moments. We chose to concentrate our
attention on propagation and collisional effects on this paper for two
reasons. Firstly, the availability of simple mathematical results for
this case, and also because these effects Two physical properties of the solar atmosphere play a key rôle in particle transport. The first is the magnetic field which guides charged particles so that they `spiral' along the field lines. The magnetic field does not directly affect the energy of a charged particle; it only alters the pitch angle in regions of changing field strength. The second key property is the density of the `cold' (i.e. thermal energy energetic particle energy) background particles. As energetic particles move through a background media, they interact with its particles via the Coulomb force. At any given moment an energetic particle will experience collisions with a very large number of background particles. This has two implications. Firstly, it is a scattering process and must be treated statistically. This means that a particle with given velocity at some time, cannot have its velocity calculated uniquely for some later time. Secondly, because there are so many collisions per second, Coulomb collisions will change the velocity of the particle in a continuous way on observational timescales. That is, the change in a particle's velocity in, say, a second, is not due to any single collision, but due to the effects of a great many collisions. Coulomb collisions are regarded as distant, in the sense that the
vast number of distant collisions, involving only small changes to
All previous descriptions of energetic particle distributions accounting for stochastic effects in the solar atmosphere have been numerical in nature (Leach & Petrosian 1981; Kovalev & Korolev 1981; Bai 1982; Hamilton et al. 1990; MacKinnon & Craig 1991). Analytic results that add valuable insight have also been developed over the last three decades, but, until now, only for the deterministic (i.e. mean) aspects of particle transport. The mean scattering approach, as it has been called, was first used by Brown (1971) and developed further by Emslie (1978), and later formulated and solved in a more general form, in terms of the continuity equation, by Vilmer et al. (1986) and Craig et al. (1985). In this paper variations across the magnetic field are ignored, e.g. the background density and particle distribution are only considered as varying along the magnetic field lines. Basic physics and recent observations strongly suggest that such quantities can vary across field lines. Also, the time profile of hard X-rays may be thought to be composed of many individual spikes, with the time density of spikes being greatest at the HXR peak. Present HXR observations do not have sufficient spatial or temporal resolution to investigate these possibilities directly and so they are not considered in this paper. However, the theory presented here is not inconsistent with them, if quantities such as the density and particle distribution are thought of as averages across the field line, and over a length of time greater than the width of an elementary spike. This paper's main purpose is to provide mathematical expressions relating (directly and indirectly) observable moments, to the moments of the injected distribution. Sect. 2 sets out the general mathematical formalism. Sect. 3 considers results specific to HXR emission and energy deposition. Sect. 4 shows how to derive expressions for time-independent moments for the case of a non-uniform background density and constant magnetic field, and to derive temporal moments for the uniform density case. Sect. 5 discusses the implications and uses of the results. © European Southern Observatory (ESO) 2000 Online publication: October 30, 19100 |