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Astron. Astrophys. 362, 383-394 (2000)

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1. Introduction

Hard 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 ([FORMULA] 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 along magnetic field lines.) Regarding assumption 2 above, inferences made about the energy release site could well be wrong if electrons are not a primary product. Electron acceleration by protons or ions was found to be a natural consequence of a neutral beam (protons and electrons travelling together) by Karlicky et al. (2000), though this mechanism by itself cannot account for HXRs in larger flares due to energy efficiency constraints (Brown et al. 2000). It is clear that existing observations, and the observations that will be made by HESSI, need careful interpretation to yield information about how and where the particles were accelerated. Much of this interpretation is model dependent, but whatever the model, it must, at the very least, account for particle transport.

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 must be accounted for in nearly every interpretation of HXR observations. Other effects, such as stochastic particle acceleration, the electron-cyclotron maser and magnetic field convergence, are certainly of great interest, but there is no reason to suppose that they will be important in every flare. Further discussion of motivation of this work is discussed by Conway (2000).

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 [FORMULA] 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 E and µ, dominate over the few close encounters that produce large changes in these quantities (Spitzer 1962). However, the closer encounters are the ones that are important in producing the bremsstrahlung Hard X-rays, for the simple reason that an energy exchange of at least several keV is needed to produce a photon of several keV. The rate of HXR production and the rate of energy loss of an energetic electron are both proportional to a single physical quantity: the density of the background medium. This link between Coulomb collisions and HXR production is the reason why the concept of column depth is so useful. It is also the reason why it is actually more convenient to deal with moments of the HXR intensity distribution than it is to deal directly with the moments of the electron distribution. This is discussed in more detail below. The implication of this is that quantities of greatest observational interest, e.g. the moments describing the spatial location and extent of a HXR source, can be related directly to moments of the injected distribution without any need to solve the Fokker-Planck equation directly.

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.

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Online publication: October 30, 19100