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

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2. General formalism

Consider a cross-section [FORMULA] that describes a process involving an energetic charged particle of energy E, pitch angle cosine µ, that is at a distance s from some fixed reference point on a magnetic field line. We assume that a charged particle will follow a single field line, which is a good approximation in many astrophysical scenarios. The two specific cross-sections that we are concerned with in this paper are the Hard X-ray (HXR) bremsstrahlung cross-section and Coulomb energy loss cross-section.

Given the particle distribution function [FORMULA] (number of energetic electrons per unit E, per unit µ, per unit s at time t), the total rate of the process described by [FORMULA] at time t, [FORMULA], can be expressed:

[EQUATION]

where [FORMULA] is the density of background particles, and [FORMULA] is the speed of the energetic particle. (The semi-colon is used to emphasise that t is the independent variable, an issue that will become more important later. Hereafter, multiple integral symbols, integral limits and function arguments like those that appear in the above equation will be omitted for clarity, unless the specifically needed.) The definition of f is such that [FORMULA] if [FORMULA], where [FORMULA] is the number of particles in the whole system at time t. Note that [FORMULA] does not mean zero energy, it means that the particle has joined the thermal distribution and can no longer be described by the energetic particle Fokker-Planck equation.

Eq. (1) can be regarded as the zeroth time dependent moment of the distribution f weighted by the cross-section [FORMULA] (we choose to phrase it this way to emphasise the physics - it could equally well be regarded as the zeroth moment of the distribution [FORMULA]). More generally we can write that the time dependent moment of a function [FORMULA], weighted by cross-section [FORMULA] is

[EQUATION]

This equation also serves to define the expectation operator [FORMULA], which in this equation represents an average of q over the distribution f weighted by cross section [FORMULA]. For example, [FORMULA] tells us the mean position at which the process described by [FORMULA] is taking place. Note that [FORMULA] is the total rate of the process described by cross-section [FORMULA]. The meaning of the expectation operator is most easily understood in the context of stochastic variables, discussed later in Sect. 4.

2.1. The single particle distribution

Expressions for the moments could be obtained by solving the Fokker-Planck equation and inserting the solution into (2). Alternatively, Conway et al. (1998) showed how moments can be obtained without explicit solution of the Fokker-Planck equation. However, the moments derived in that paper were for a single particle distribution. To use these results it is first necessary to show how the full distribution function can be constructed from single particle distributions . A single particle distribution, denoted by [FORMULA] here, is the distribution function of a particle injected at a given instant in time, with given values of energy and pitch angle, at a given position (represented here by [FORMULA], [FORMULA], [FORMULA] and [FORMULA] respectively). From a frequentist point of view one can also view it as the distribution function of a number of particles injected with the same initial conditions at the same time, normalised by the number of injected particles. Physically, a single particle distribution only contains information on particle transport, and contains no features from any initial distribution of particles. Mathematically, the single particle distribution [FORMULA] is defined as the distribution evolved from initial condition [FORMULA].

To express a general distribution f in terms of single particle distributions [FORMULA] requires the introduction of an injection rate function [FORMULA], which is the number of particles injected per unit time at time [FORMULA], per unit [FORMULA], [FORMULA] and [FORMULA]. The general distribution f can then be formed by adding together single particle distributions, weighted by the injection function evaluated at their injection parameters (variables with subscript 0):

[EQUATION]

It is useful to introduce the single particle expectation operator, [FORMULA] (in Conway et al. 1998 this was written `E[FORMULA]'). The single particle expectation of q is defined as

[EQUATION]

This gives the expected value of q for a particle at time t that had energy [FORMULA], pitch angle cosine [FORMULA] at its injection location [FORMULA] at [FORMULA]. For example, [FORMULA] gives the expected position of the particle at time t, and [FORMULA] gives the expected variance about that position. It is important to remember that because we are dealing with the single particle expectation operator, this spread in position is entirely due to transport effects, and not to any initial distribution of particles.

2.2. The time dependent case

By inserting (3) into (2), changing the order of integration, and using the definition of the single particle moments (4) we obtain

[EQUATION]

This is the important general result that underlies the subsequent theory developed in this paper. Its importance arises from the fact that observables of some process described by [FORMULA] (the left hand side of (5)), such as HXR bremsstrahlung, can be directly related to moments of the injected distribution h (right hand side). The description of the particle transport is solely contained in the single particle expectation, which will be dealt with in more detail in Sect. 4.

2.3. The time independent case

To remain general to the case of an arbitrary background density distribution [FORMULA], we are forced to consider the problem as time independent, i.e. that h is independent of [FORMULA]. The fundamental reason for this will be made clear below. In order to do this a new independent variable needs to be introduced - this is path depth P. It is defined so that a particle moving at speed v, through a background of density n will encounter a path depth [FORMULA] during an infinitesimal time dt. We call P "path depth" because it measures the amount of background material the particle has encountered along its path. In general, the particle will follow a helical path along a magnetic field line. Column depth, which we denote by its conventional symbol N, differs in that it is a measure of background material along a given magnetic field line (or sometimes along the vertical direction). The column depth traversed by a particle in time dt is defined as [FORMULA]. A fundamental difference between the two is that path depth can only be discussed with reference to an energetic particle, whereas column depth can be used independently much like a distance measure, e.g. "the column depth to the transition region is [FORMULA]". We are careful to distinguish between path depth and column depth because they are easily confused, especially in 1D treatments - see MacKinnon & Brown (1989) for a discussion on this subject. The physical significance of path depth is that it measures the total amount of background material experienced by a particle up to a given point in its life. In the case of collisional interactions with background particles, the particle's energy E is a deterministic function of P, i.e. it will have a particular value for a given P, unlike pitch angle, say, which will have a distribution of values. In contrast, E and P are in general stochastic functions of t. The reasons behind this will be made clear later in Sect. 4. For now it is enough to note that because the relation between E and t is stochastic, the stopping time of a particle, which is needed as an upper limit in the time integrals, does not in general have a deterministic value. For this reason we use P rather than t as the independent variable.

Our aim is now to re-express (5) with the assumption that h is independent of [FORMULA] and express [FORMULA] as a function of P rather than t. To achieve this, we must redefine the single particle moment to be a function of P: the function [FORMULA] is the distribution of particles with initial parameters [FORMULA] at [FORMULA], once they have traversed a path depth P. Unless the density is spatially uniform, particles injected at the same time will not all reach the same path depth P at some later time t. However, the advantage in using path depth is that in dealing with collisions with background particles, the energy distribution of [FORMULA] is a Dirac delta function in E ([FORMULA]). This is a mathematical statement of the fact that the energy of the particle is only dependent on the amount of background material encountered - which is exactly what is quantified by P. We can therefore write that t and P forms of [FORMULA] are related by the following relation, for any function [FORMULA]:

[EQUATION]

With [FORMULA], this is simply equating particle number (per unit µ and s). Two cautionary notes are needed here. The above relation does not represent a simple change of variable, i.e.

[EQUATION]

It is the two integrals that are equal. In general, an interval of [FORMULA] corresponds to many different time intervals because different particles with the same initial conditions will take different paths through the background media, sampling different densities as they do so. The situation is analogous to the use of a differential emission measure where a volume integral, perhaps performed over a set of concentric spherical surfaces, is replaced by an integral over surfaces of constant temperature (Craig & Brown, 1976).

To re-express (5) under the time independent assumption that h is independent of [FORMULA], consider the time integral over the single particle expectation

[EQUATION]

Using (6) with [FORMULA], we can replace the right hand side of the above equation as follows:

[EQUATION]

It is now clear, at least mathematically, why an arbitrary density distribution necessitates a time-independent treatment. The entire mathematical argument, from (6) onwards, is predicated on the assumption that h can be removed from the [FORMULA] integral. If this were not the case, then the presence of [FORMULA] in h presents an insurmountable problem because we are not simply making a change of variables from [FORMULA] to P. Physically, it is not meaningful to write the injection rate h as a function of P, because P is defined in relation to the evolution of a particle, and cannot be related directly to an external independent variable such as time [FORMULA]. The fundamental reason for this problem is that we made a prior assumption that P and E are deterministically related. This assumption was made because all currently known solutions to the single particle moments have this property. In fact the deterministic relation of E and P is guaranteed for any process affecting particle energy, that has a cross-section proportional to nv.

The time independent, arbitrary background density version of (5) is therefore

[EQUATION]

To be completely general to any distribution of density, the column depth [FORMULA] is used in the above equation in place of [FORMULA]. This is a straightforward change of variable that requires a model for [FORMULA] if N and s are to be related.

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Online publication: October 30, 19100
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