The results presented in this paper allow observables, such as the location and extent of a spatially resolved HXR source, to be simply related to moments of the original distribution. The zeroth order HXR moment is in fact the thick target bremsstrahlung integral. In this sense, the theory presented here is a generalisation of the results of Brown (1971). The key result of this paper can be summarised as follows: A potentially observable moment , where corresponds to either HXRs (X) or heating (H), can be expressed as simple function of the moments of the injected particle distribution
An important issue when using moments is the effect of any lower or upper cutoff (or breaks) in the electron spectrum. For harder spectra (smaller spectral indices), moments can be very sensitive to them, and can even become infinite, even when the HXR emission is finite. These problems are mathematical, and are due to infinite energy limits in the integrals. In reality, there must be an upper cutoff, but in practice effective cut-offs will be introduced according to how the moment is measured in the observations. It would be interesting to exploit this cut-off sensitivity to devise a method of identifying spectral features at unobservably high energies. For example, if the depth range of particle beam heating can be estimated in a flare, and the spectral index of the particle distribution is also known, it would be possible to estimate the upper cutoff .
The spatial HXR moments are probably of most immediate interest. If the location and size of a HXR source can be measured in a particular flare, this can be compared with and . Such a comparison requires an assumption about the density structure. For spatially resolved HXR emission in the corona, such as the HXR above the loop top sources (Masuda 1994, Masuda et al. 1995), a density structure along the loop must be assumed. If a constant density is supposed, then the conversion of column depth to length is straightforward. However, potential complications arise if high density pockets exist as was proposed as an explanation for the HXR coronal source by Wheatland & Melrose (1995). In such a case an average density might be assumed. If a model is assumed for how the density varies through the transition region and into the chromosphere, then statements can be made about the expected footpoint brightness. For example, if it is supposed that the energy is released at the loop-top, then one would expect relatively bright footpoints if , where is the column depth from the loop-top to the transition region.
Without further knowledge of the distribution, it is not possible to make precise theoretical statements such as: "68% of the emission lies within one standard deviation of the centroid". However, the reality is that making such definite statements from current observations is not possible, for several reasons. Even the best observed example of a coronal HXR source to date was subject to observational limitations leading to debate about its detailed characteristics (Alexander & Metcalf 1997). This is partly due to the fact that HXR coronal sources are weak, and partly due to the characteristics of the instrument and how the image is constructed. Instruments such as the Yohkoh Hard X-ray Telescope (HXT), and the High Energy Solar Spectroscopic Imager (HESSI) rely on images being reconstructed by the Maximum Entropy Method (or some similar method) to yield an image that both matches the observed data and satisfies some further regularisation constraint. This latter constraint is required to make up for "missing information", and is usually taken to be that the image is the smoothest one that fits the observations. Clearly, the detailed structure of HXR sources will be affected by this smoothing constraint in a way that is difficult to quantify. This means that it is not possible at present to make a detailed comparison between theory and observation. For example, it is not realistic make deduction concerning the electron distribution from the shape of a HXR source. Estimating the observational effects on the broad properties of the distribution (i.e. moments) is therefore a more realistic proposition.
Given the observed location and extent of the HXR source, it is possible to estimate moments of the injected distribution using the expressions derived in Sect. 4.2. Specifically, the observed location and extent of a HXR source are simply related to the location and extent of the injection region, and the average injected pitch angle and the spread in injected pitch angle. That is, the first two moments of the HXR spatial distribution give information on the first two spatial and pitch angle moments of the injected electron distribution. Clearly, four quantities of interest cannot be extracted from two observables. Either assumptions must be made, or, preferably, other observations must be used. One possibility is to try and estimate the pitch angle moments using the expressions given for or . These are averages weighted in terms of the number of electrons that are emitting HXRs at a given photon energy . In particular, is very simply related to the average pitch angle of the injected distribution, and for larger values becomes nearly equal to it. Information on these can be obtained from directional or polarimetric properties of the HXRs. At present, methods for observing these properties in a single flare do not exist. The possibility of gaining such information about the injected distribution provides motivation for making stereoscopic and polarimetric HXR observations in the future. Another possibility might be to deduce pitch angle moments from observations of H impact polarisation (Henoux & Vogt 1998).
For coronal sources, under the assumption of a constant density coronal loop, temporal characteristics of the acceleration process can be deduced using the results of Sect. 4.4. The interesting aspect of this result is that the (square of the) width of the time profile of the HXRs is composed of two terms: the (square of the) width of the time profile of the injection plus a term proportional to (squared). is the collisional stopping time for an electron of energy equal to the photon energy of interest, , and is proportional to . The peak of the HXR burst is also delayed with respect to the peak of injection by a time proportional to . These results can either be applied to the whole observed time profile of a Hard X-ray burst, or to the individual "spikes" of which it is composed. They should be useful in the time of flight analyses performed by Aschwanden and co-authors (Aschwanden et al. 1997 and references therein).
© European Southern Observatory (ESO) 2000
Online publication: October 30, 19100