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

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 f are needed. These have been calculated analytically by Conway et al. (1998) for the case of non-relativistic electrons in a constant magnetic field . Here we develop the use of the preceding theory by using those results.

4.1. Stochastic variables and single particle moments

A single particle distribution function, , µ, s; , , , , describes the distribution associated with a single particle, that was injected at time , with energy , pitch angle cosine at position . We will consider the case where the background density distribution is arbitrary but the magnetic field strength is constant. For this case, MacKinnon & Craig (1991) showed that the Fokker-Planck equation is equivalent to 3 coupled Stochastic Differential Equations (SDEs), also known as Îto equations:

where . These express the progress of a particle as a series of increments in each of the dependent variables: , and (or z, v and µ in MacKinnon & Craig 1991). The hat symbol is used to emphasise that these variables are stochastic variables . Stochastic variables do not necessarily have a given value at a given time; they have a distribution of possible values that is described by the single particle distribution . Each equation contains a dt term; this represents a conventional deterministic change in that variable with the independent variable t. The equation also contains a term; this term represents a stochastic change in in an infinitesimal time step dt. is known as a Wiener process, which formally is a (Dirac) delta-correlated white noise time series. It can be defined in a number of other ways. For example, it can be considered as the limit of a discrete white noise time series of time-step , as . Integration of from 0 to time t gives , which is a white noise, normally (Gaussian) distributed stochastic variable with mean 0 and variance . To clarify the meaning of , it is helpful to consider how it would be implemented in a numerical code. Let the code's time-step be , and let be the coefficient of dt in the equation. The equation is implemented by adding to the previous value, plus a zero-mean, variance 2, Gaussian distributed random number multiplied by . (Note the convention of the Wiener process having a variance of 2, rather than 1).

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 n is not spatially uniform, then the density at the particle's position will become a stochastic quantity. This means that, in general, is also a stochastic variable. If the density is uniform however, then the equation becomes deterministic and can be integrated to give an explicit function of time. This is a tremendous simplification, as it makes the system of equations (nearly) linear, and thus easily soluble. The same simplification can be made for the non-uniform density if the path depth P is used as the independent variable instead of t. The cost is of course that the time dependence is no longer explicit. Changing from t to P yields the following set of equations:

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 P. This means that these equations are essentially linear in nature (despite the appearance of in the stochastic term). The correspondence of the notation of Conway et al. (1998) to the present notation (the right hand sides) is as follows: , , where is the path depth (integrated along a particle's path) required to stop a particle of initial energy .

It is clear that is a deterministic function of P. In other words, it has a mean that is a function of P and always has zero variance. The physical reason for this deterministic relationship is that the energy of the fast particle is assumed to be much greater than the thermal energies of the background particles. As a result it is pulled inexorably back into the background distribution. Once a particle's energy nears the thermal energy, neglected terms in the fast particle Fokker-Planck equation will become important, and this treatment will no longer be appropriate. Contrast this with situation when the µ distribution is far from being isotropic, e.g. for a directed beam with (zero pitch angle). As can be seen from the equation in (11), the stochastic term vanishes for this case, and the deterministic term pulls to lower values. As this happens, the stochastic term grows, and by the stopping time, will in fact b be distributed uniformly, i.e. the particle velocity will have an isotropic distribution.

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 t after its injection. Similarly, is the particle's expected location. The spread about this position is measured by . Similar expressions can be used when working with path depth P as the independent variable. The moments denoted by represent averages over the transports effects, together with the initial distribution, weighted according to the cross-section of interest .

4.2. Time independent: arbitrary density

The 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 N is given by

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 P to energy and using the Bethe-Heitler cross-section (8) will result in having to deal with integrals of the form

where are integers. The second integral, which is more convenient to evaluate, is obtained by reversing the order of integration of E and . Explicit higher and lower energy limits, and respectively, are placed on the injected distribution function here because we will see later that the spatial HXR variance can become infinite if electrons of infinite energy are present. The practical justification for assuming upper and lower cutoffs is discussed in Sect. 4.3. To proceed from here requires assumptions to be made about the form of h. We will assume that the distribution is a power law, and that it is separable in and , i.e. it can be written as follows . Here , where is commonly used electron flux power law index. is defined so that the total number of electrons injected per unit time is given by

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 x and y:

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 are undefined. Careful consideration of the integration in these cases can yield well-behaved mathematical 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).

Fig. 1a-f. Plots of the apparent size of a HXR source in terms of: column depth, i.e. ; and , the spatial extent along a loop of constant density  cm-3. For the largest values, L is unrealistically large, which can be interpreted as being too large. Curves for three different values are shown on each plot: is the upper (solid) curve, is middle (dotted) curve and is the lowest (dashed) curve. The injected distribution is isotropic, i.e. and .

The heating moments are easier to calculate and the relevant expressions for µ N, where , are

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.

Fig. 2a-d. Plots of the mean column depth of energy deposition (solid) and the standard deviation (dashed) for a variety of parameters.

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. Cutoffs

It 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 P is a deterministic function of time t:

where is the stopping time. For the same reason, N and s are simply related to each other: . Given the extra freedom of choosing how the injection function h varies with time, specific results can only be obtained by making further assumptions. Here we are only concerned with calculating and , which tells us the time, and the time interval, during which the HXR emission at a particular photon energy will take place. These times are defined in relation to the moments and , which are obtained by integrating multiplied by and respectively, over all and also . Although it is possible to calculate the time-dependent moments of µ and s (E is deterministic in time here), as defined in (5), doing so requires more information than the first two temporal moments of h provide. Further assumptions must be made relating to specific circumstances in order to give useful results, so we will not consider this case in this paper.

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 t) of the form:

where E (and therefore ) is a function of . Changing variable to E using the relation

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 q in (17), gives the final results:

where

This result is valid for . A more general expression can be calculated using the general expression for q in (16). Note also that the form used for h above assumes that the distribution is separable in and - i.e. the injection spectrum is time-independent. The time delay between the HXR flux's peak at photon energy and the peak in the rate of electron injection is simply . This is clearly proportional to which itself is proportional to . Numerical evaluation of the factor involving show that it decreases with . This is expected as larger means relatively more low energy electrons, and so shorter electron life-times. The same conclusions can be made for the width of the HXR peak of emission, . Note that all of these conclusions can be radically changed, and become much less intuitive, when approaches 5 and the upper cut-off (or break in the power law) is not much larger than .

© European Southern Observatory (ESO) 2000

Online publication: October 30, 19100
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