5. Application to HXR loop top sources
As mentioned earlier, the distribution function itself is generally not useful because of its complexity. However, the low order moments can be used to investigate problems mathematically, without recourse to numerical simulations. In particular, previously, any problem where the spatial spreading of electrons was of interest could not be treated by mean scattering and therefore required a detailed numerical simulation. Here, we present such a problem and show how it can be addressed using the second order moment of the distribution.
Masuda et al. (1994) reported and analysed observations of an "above the loop top" HXR source during the limb flare that occurred on 13 January 1992, which have been re-analysed recently by Alexander and Metcalf (1997). In this event, three distinct patches of HXRs were observed, two corresponding to the footpoints, the third apparently being situated above the soft X-Ray loop. One of several models for this was suggested by Fletcher (1995) who pointed out that the loop top source could be explained using electron transport effects, without any need for assuming a magnetic trap or plasma density enhancement at the loop top (eg Wheatland and Melrose 1995). In this model, any electrons injected into the loop top with large pitch angles () remain there until they are scattered. The resultant concentration of electrons at the loop top causes an increase in the thin target HXR bremsstrahlung emission there. We shall refer to this as "µ-trapping" for obvious reasons. Intuitively, it might be thought that this cannot yield a bright enough source, and that the situation is not improved by increasing n, since both the rate of electrons being scattered out of the loop top and the thin target HXR flux are proportional to the density n - hence the emission is independent of density. We show here that this assertion is not correct and that increasing the density can increase the source brightness. We shall consider how electrons spread away from the loop top, where they are injected. We shall also assume that , a reasonable assumption for electrons with keV and if the density is less than cm-3, since the loop top's dimensions are cm. We will also assume that the density n is constant across the loop top.
Starting with (15) for :
With x assumed to be small, expanding to the the lowest non-zero power of x gives . Since the density is uniform, we can return from our de-dimensionalised column depth variables x and y to t (time after injection) and z (distance from loop apex). This gives
This means that we would expect an electron to leave the loop top region in time given by:
where we take the loop top to be a region that extends from to . In a steady state situation, ie where the number of electrons in the loop top region is not changing, and there is a constant rate of injection R, we have
This means that the thin target HXR emission from these electrons will be proportional to , which is proportional to . That is a denser background plasma does cause the source to be brighter.
The treatment presented here is only supposed to be illustrative and a more complete development of these ideas will be presented in a future paper.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998