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Astron. Astrophys. 331, 1103-1107 (1998)

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1. Introduction

Non-thermal particles play a central role in solar flares (eg Brown and Smith 1980) and as such knowledge of how they are distributed along the magnetic fields in flare loops is of great interest. Emission of electro-magnetic radiation, eg Hard X-ray bremsstrahlung, is direct evidence for the presence of a non-thermal particle distribution and heating of the plasma as a result of its presence can cause a variety of observable phenomena, eg chromospheric evaporation. In both cases, to interpret and model the observations, an understanding is needed of how the electrons propagate and consequently how they are distributed spatially. For understanding the directivity and polarisation of emitted radiation, the pitch angle distribution is certainly very important. More fundamentally, understanding of particle transport must be improved before observations can be interpreted meaningfully in relation to the primary energy release and particle acceleration in flares.

As high energy (energy much greater than the thermal energy of particles in the background plasma) particles travel along a magnetic field their pitch angle, ie the angle between their velocity vector and magnetic field, can be affected by: the particles in the "cold" background plasma; waves in the plasma, which they may also drive; and any convergence in the magnetic field. Here we shall concentrate on the first of these processes in the non-relativistic limit. To order [FORMULA] (the Coulomb logarithm) in the Fokker-Planck equation, scattering reduces the energy of the particles deterministically but affects the direction of the particle's motion in a stochastic way, due to the effect of very many long distance Coulomb collisions with the particles in the background plasma. In the current work, we consider only charged particles moving in the presence of a constant magnetic field, so that only the pitch angle of the electrons, [FORMULA] (or [FORMULA]) need be considered. This means, for example, that the propagation of the electrons along a field line is a stochastic process determined by both the slowing of the electrons and by the change in their pitch angle distribution.

Without resorting to the Fokker-Planck equation it is possible to determine where, on average, the electrons will be, in terms of column depth measured along the field line, N, and what their average pitch angle, µ, will be, once they have been degraded to energy E. If the injection energy and pitch angle are [FORMULA] and [FORMULA] resp. then, as was first shown in Brown (1972),




e being the charge of an electron in e.s.u. and [FORMULA] being the Coulomb logarithm, which we take to be constant. [FORMULA] can be interpreted as the column depth measured along the electron's path that is required to reduce its energy to zero (in reality the electron will join the thermal distribution of the plasma, the typical thermal energy being much less than [FORMULA]). The [FORMULA] accounts for the average reduction in pitch angle, effectively reducing the "average" electron's component of velocity along the field. These are the results of "mean scattering", for which further details can be found in Brown (1972), Emslie (1978), Tandberg-Hanssen and Emslie (1984), Craig et al. (1985) and Vilmer et al. (1986). The treatment in this paper shows how these results can be derived directly from the Fokker-Planck equation.

Mean scattering deals only with first order statistical properties of the electron distribution function. To find the distribution function itself, the Fokker-Planck equation must be solved. To date, only approximate analytic solutions for some special cases have been found eg Leach and Petrosian (1981), McTiernan and Petrosian (1990a). Also, an exact expression for the spatially integrated distribution function in terms of Legendre polynomials can be derived, see Kel'ner and Skrynnikov (1985) and Lu and Petrosian (1988). Numerical solution of the Fokker-Planck equation is required to yield further information. Several Monte Carlo type methods have been described, eg Bai (1982), Hamilton et al. (1990) and MacKinnon and Craig (1991). The last of these, referred to as MC from here on, shows how the numerical method can be formally and simply related to the Fokker-Planck equation. These numerical methods have now been exploited to model various observations: stereoscopic observations of flares including treatment of directivity, McTiernan and Petrosian (1990b); above the loop top HXR sources, Fletcher (1995); and height distribution of HXR sources, Fletcher (1996).

There is no doubt that numerical methods provide a very powerful tool, even though they can involve lengthy processing times, but an improved analytic approach is still desirable for gaining a clearer insight and intuitive understanding of the problem. This paper describes a method that can provide exact and analytic expressions for the 2nd and higher order moments of the distribution, which in turn can provide an expression for the distribution function itself. We concentrate mainly on the second order properties which are of particular importance since they describe the spread of electrons about the expected mean position. This method takes us beyond the first order properties of the mean scattering approach and allows discussion of situations previously accessible only with numerical simulations.

The problem is stated in Sect. 2and the method of solving the equations is described in Sect. 3. Sect. 4compares the first and second order moments with results from a code based on the method of MC. In Sect. 5we illustrate the usefulness of the results obtained in looking at how a population of large pitch angle electrons injected at the loop top disperse spatially. The results are then summarised and discussed in Sect. 6.

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© European Southern Observatory (ESO) 1998

Online publication: March 3, 1998