## 2. General formalism
Consider a cross-section that
describes a process involving an energetic charged particle of energy
Given the particle distribution function
(number of energetic electrons per
unit where is the density of background
particles, and is the speed of the
energetic particle. (The semi-colon is used to emphasise that Eq. (1) can be regarded as the zeroth time dependent moment of
the distribution This equation also serves to define the expectation operator
, which in this equation represents
an average of ## 2.1. The single particle distributionExpressions 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 To express a general distribution It is useful to introduce the single particle expectation operator,
(in Conway et al. 1998 this was
written `E'). The single particle
expectation of This gives the expected value of ## 2.2. The time dependent caseBy inserting (3) into (2), changing the order of integration, and using the definition of the single particle moments (4) we obtain 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
(the left hand side of (5)), such as
HXR bremsstrahlung, can be directly related to moments of the injected
distribution ## 2.3. The time independent caseTo remain general to the case of an arbitrary background density
distribution , we are forced to
consider the problem as time independent, i.e. that Our aim is now to re-express (5) with the assumption that With , this is simply equating
particle number (per unit It is the two integrals that are equal. In general, an interval of 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
Using (6) with , we can replace the right hand side of the above equation as follows: 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 The time independent, arbitrary background density version of (5) is therefore To be completely general to any distribution of density, the column
depth is used in the above equation
in place of . This is a
straightforward change of variable that requires a model for
if © European Southern Observatory (ESO) 2000 Online publication: October 30, 19100 |