## 4. Derivation of moment expressions
Application of the above theory requires information on the single
particle distribution function , that
describes the propagation of a single injected particle. More
precisely, moments of ## 4.1. Stochastic variables and single particle momentsA single particle distribution function,
, where . These express the progress
of a particle as a series of increments in each of the dependent
variables: ,
and
(or The presence of the Wiener process is only explicit in the
equation, though
appears in the
equation, making
a stochastic variable also. If the
density where and
. The middle equation can be
integrated straightforwardly, allowing the appearances of
in the first and second equations to
be substituted with an explicit function of It is clear that is a
deterministic function of Now that stochastic variables have been introduced, it is much
easier to interpret the meaning of the expectation operator. For
example, is the expected pitch angle
cosine for a particle with given injection parameters
() at a time ## 4.2. Time independent: arbitrary densityThe results below apply to the case of continuous steady injection of a distribution of electrons. So as to remain general to any density distribution, we will work with column depth (i.e. density integrated along the magnetic field). We wish to answer the following questions: 1) Where is the centroid of the HXR emission? 2) What is the apparent size of the HXR source? 3) Where is the electron distribution's energy mainly deposited? 4) What is the extent of this energy deposition region? We will answer the first two questions in detail, but just state the results for the last two, as the calculations involved are very similar. Before proceeding, we quote the main results of Conway et al. (1998) in the notation of this paper, adding terms that generalise those results, which assumed injection at , to have injection at : where and the stopping path
depth is . It is interesting to note
that the final () spread of the
electrons' positions in terms of column depth So for a completely directed, zero pitch angle () monoenergetic beam, this spread will be , which is about a quarter of the stopping depth in terms of . For a distribution with only pitch angles initially (), the final spread is . Note that in using such expressions to make rough estimates of HXR source sizes or positions at photon energy , should be used instead of . This leads to smaller values for such quantities. Inserting the above expressions into (7), changing variable from
path depth where are integers. The second
integral, which is more convenient to evaluate, is obtained by
reversing the order of integration of Evaluating the integral of (15) yields where . By splitting the integral at , it can be evaluated to where and
. The function
is the same as the standard Beta
function, except that its limits are replaced by At this point we can make two simplifying assumptions, which are commonly made elsewhere in HXR bremsstrahlung calculations: and the photon energy of interest is higher than the lower cutoff of the injected distribution, this means we can replace all occurrences of with . This results in a much simpler expression: Notice that is simply the HXR thick target expression. We are now in a position to calculate the desired moments. Firstly, from (11) we can write that, for : Therefore, the average pitch angle of the electrons emitting HXRs is simply: which is actually independent of the photon energy . is the average pitch angle of the injected distribution. To calculate the spread of pitch angles, we need to calculate , which for is given by In a similar fashion, the following two column depth moments can be shown to be: where Note that remains finite only
for , and
for
. The appearance of a factor
in a denominator does not
necessarily mean that a moment is undefined at
, though the above expressions
Assuming an upper cut-off, for whatever reason, can lead to counter-intuitive results, especially in light of the usual situation where it is assumed that only a lower cutoff is important. One surprise is that the size of a HXR source does not simply scale as . This is illustrated in Fig. 1 for an isotropic source, for a range of photon energies, and for three upper cut-off values, keV, keV and keV.. The departure from scaling is most pronounced for low values of . The tangent of these curves only accords with a line of gradient for large and . When becomes a sizeable fraction of , the source size can even decrease with photon energy. This behaviour was observed for the 13th Jan. 1992 (Masuda) flare (Masuda et al. 1995), which had a hard (low ) spectrum. There are two explanations for this counter-intuitive effect. One is simply that for close to , only the highest energy electrons contribute, and do so only for a short time, during which they cannot move too far. Secondly, the relatively long collisional time for these higher energy electron means that those with large pitch angle electrons will take longer to diffuse away from their starting position than electrons with lower energy. This can be seen from the expression derived for `' in Conway et al. (1998).
The It is interesting to note that the pitch angle moments are independent of the electron energy spectrum. In contrast, the column depth moments can be very sensitive to assumptions made about the electron spectrum, especially the upper cutoff . In general, this will correspond to a real cutoff in the electron distribution. However, in certain applications an effective upper cutoff may be appropriate, as previously discussed in relation to HXR moments. For example, in computing the H impact polarisation due to a particle beam, knowledge of pitch angle moments is needed in a restricted range of depths for particular electron energies (Henoux & Vogt 1998). Plots of the average column depth of heat deposition and the spread of depths are shown for selected values of , , and , in Figs. 2a, 2b and 2c.
The sensitivity of spatial moments to high energy features in the spectrum, and also their insensitivity to the lower cut-off, could lead to new diagnostics of otherwise unobservably high energy electrons. To achieve this, calibration of the relationships is needed from observations where both the spatial and spectral information is available. Also, any effective upper cut-off, as discussed above, must be carefully considered in applying the results in this way. ## 4.3. CutoffsIt is clear that cutoffs in the electron spectrum have to be assumed to avoid unphysically infinite moments. We now discuss what determines these cutoffs in practice, giving particular attention to the upper cutoff because it will often be the most problematic in applying the present theory. It is important to realise that the value of the upper cutoff, , need not arise from basic physics alone, but can often depend on how moments are deduced from the observations. As such, the relevance of the considerations discussed below can vary from application to application. Another point to note is that a literal cut-off is not necessarily required, a more gradual roll-over, or a break to a different spectral index can serve the same purpose. The most obvious justification for assuming a cutoff in the electron spectrum is if there is evidence for one in the observed HXR photon spectrum. However, lower cutoffs are usually masked by thermal emission and upper cutoffs are only rarely observed. An upper cutoff at a few tens of MeV was reported by Trottet et al. (1998) for the flare of 11 June 1990. In general, will probably be less than 20 keV and will be greater 1 MeV; though more restrictive limits will be possible for some large flares. Depending on the value of , and the moment in question, one or other of these cutoffs may dominate the moment expressions, and the other may be set to its extreme value (i.e. zero or infinity). In many cases of interest, an effective cutoff can be more important than any real cutoff in the electron spectrum. For example, an upper cutoff of 10 MeV in the electron spectrum cannot possibly be relevant in determining the apparent size of a coronal HXR (say 10-100 keV) source. This is because electrons at energies of a few MeV will contribute much less to this emission than electrons at energies of a few tens of keV. The effective upper cutoff in this case would depend on both the observed photon energy range and the energy corresponding to the column depth of the coronal material. Similarly in applying the results to H impact polarisation, only electrons in certain ranges of energy and depth are relevant, thus introducing effective lower and upper cutoffs. ## 4.4. Time dependent: constant density
The time dependent case can only be considered if the density is
uniform in space, because then the path depth where is the stopping time. For
the same reason, Firstly, the temporal moments of the HXR intensity distribution, are defined: remembering that is the HXR intensity as discussed in Sect. 2. It is therefore clear that the normalised moments in time are given by: and . Using (5) to expand (25) with the Bethe-Heitler cross-section given
by (8) will result in time integrals (in where gives the integral in the following form: This integral must also be integrated over and . The integration over can be written in terms of introduced in (15). Doing so gives: The integral over simply means
replacing instances of with
. Doing this assuming that
and using the expression for
where This result is valid for . A more
general expression can be calculated using the general expression for
© European Southern Observatory (ESO) 2000 Online publication: October 30, 19100 |