How to correctly determine the power in non-thermal electrons from observed X-ray spectra

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In an attempt to remedy this situation, Emslie (2003) included consideration of the finite temperature of the target in modifying the systematic energy loss rate of the accelerated electrons; such considerations come into play as the electron energies approach a few kT. The reduced energy losses (relative to the cold target results) at low energies make the energy content in the accelerated electrons less sensitive to the assumed low-energy cutoff, to the extent that it can formally be extended down to zero, thus providing an upper bound on the energy content.  However, Emslie (2003) neglected the effects of energy diffusion on the evolution of the energy spectrum, which, as emphasized by Galloway et al. (2005), is critically important at energies of a few kT and is a necessary ingredient for describing the thermalization of the fast electrons in a warm target. Jeffrey et al. (2014) showed that the effects of diffusion in both energy and space must be included in a self-consistent analysis of electron transport in a warm target.
In an attempt to remedy this situation, Emslie (2003) included consideration of the finite temperature of the target in modifying the systematic energy loss rate of the accelerated electrons; such considerations come into play as the electron energies approach a few kT. The reduced energy losses (relative to the cold target results) at low energies make the energy content in the accelerated electrons less sensitive to the assumed low-energy cutoff, to the extent that it can formally be extended down to zero, thus providing an upper bound on the energy content.  However, Emslie (2003) neglected the effects of energy diffusion on the evolution of the energy spectrum, which, as emphasized by Galloway et al. (2005), is critically important at energies of a few kT and is a necessary ingredient for describing the thermalization of the fast electrons in a warm target. Jeffrey et al. (2014) showed that the effects of diffusion in both energy and space must be included in a self-consistent analysis of electron transport in a warm target.
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[[File:warm_target_fig2.png|left|500px|alt=Figure 1: The model in pictures. Left: the cold plasma target with non-thermal electrons propagating downwards. Right: Warm-cold model with low energy electrons collisionally diffusing and high energy free-streaming.]]
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[[File:warm_target_fig2.png|left|200px|alt=Figure 2: The model in pictures. Left: the cold plasma target with non-thermal electrons propagating downwards. Right: Warm-cold model with low energy electrons collisionally diffusing and high energy free-streaming.]]
This nugget (following Kontar et al 2015) highlights the role of thermalization of fast electrons in a model that includes both for a  warm corona and a cold chromosphere (see Figure 1). The injected/or accelerated electrons first propagate in a coronal  plasma that has a temperature comparable to the electron energy, and then collisionally stop in the cold plasma below.  The effects of energy loss and diffusion are markedly different in these two regions.  Hence, contrary to the case of a purely cold target (in which the spatially-integrated hard X-ray yield is independent of the density profile of the target), in such a composite target the relationship between the hard X-ray flux and the accelerated electron energy content needs to take into account the spatial characteristics of the emitting region, in particular the extent of the warm target region compared to that of the overall flaring region. The resulting mean electron flux spectrum shows a pile-up of thermal electrons (Fig 2), driven primarily by the effects of energy diffusion in the coronal target.  This limits the maximum rate of electron injection and hence provides the much sought-after upper bound on the total injected power .
This nugget (following Kontar et al 2015) highlights the role of thermalization of fast electrons in a model that includes both for a  warm corona and a cold chromosphere (see Figure 1). The injected/or accelerated electrons first propagate in a coronal  plasma that has a temperature comparable to the electron energy, and then collisionally stop in the cold plasma below.  The effects of energy loss and diffusion are markedly different in these two regions.  Hence, contrary to the case of a purely cold target (in which the spatially-integrated hard X-ray yield is independent of the density profile of the target), in such a composite target the relationship between the hard X-ray flux and the accelerated electron energy content needs to take into account the spatial characteristics of the emitting region, in particular the extent of the warm target region compared to that of the overall flaring region. The resulting mean electron flux spectrum shows a pile-up of thermal electrons (Fig 2), driven primarily by the effects of energy diffusion in the coronal target.  This limits the maximum rate of electron injection and hence provides the much sought-after upper bound on the total injected power .
These results show that the  use of cold thick-target model can lead to erroneous estimates of the number and energy content in the accelerated electrons especially in cases of warm-dense coronal X-ray sources and/or soft X-ray sources. Instead, we advocate the use of the more physically complete target model, including the effect of electron thermalization. We have developed a formula that allows explicit determination of this relationship, given the temperature T and extent L of the hot coronal region. A model fit routine  f_warm_thick.pro has been  developed for OSPEX. This routine allows the determination of  the minimum cut-off value and hence the maximum (upper bound) power in non-thermal electrons. For some flares, the low energy cut-off is constrained with an uncertainty of only a couple of keV, providing an accurate estimate of the accelerated electron number/power.
These results show that the  use of cold thick-target model can lead to erroneous estimates of the number and energy content in the accelerated electrons especially in cases of warm-dense coronal X-ray sources and/or soft X-ray sources. Instead, we advocate the use of the more physically complete target model, including the effect of electron thermalization. We have developed a formula that allows explicit determination of this relationship, given the temperature T and extent L of the hot coronal region. A model fit routine  f_warm_thick.pro has been  developed for OSPEX. This routine allows the determination of  the minimum cut-off value and hence the maximum (upper bound) power in non-thermal electrons. For some flares, the low energy cut-off is constrained with an uncertainty of only a couple of keV, providing an accurate estimate of the accelerated electron number/power.

Revision as of 21:32, 3 November 2015

Solar flare hard X-rays provide us with the ability to diagnose the properties of the energetic particles responsible for the hard X-ray emission. However, an accurate determination of the total number of injected electrons (or, equivalently, their energy content) has challenged modelers for decades. Solar flare hard X-ray spectra in the nonthermal domain are typically power-law forms. Using a cold target model (that retains only the effect of energy loss in the dynamics of emitting electrons) requires that the injected electron flux spectra are also power-laws. Hence the total injected energy flux is dominated by the lower limit of the injected energy spectrum. As a result the concept of a “low-energy cutoff” has been used by many authors in order to keep the number and energy content integrals finite. In practice, while fitting with OSPEX, we take this cutoff to be the maximum value of the cut-off energy consistent with the hard X-ray data, thus establishing lower limits to the number/energy content in the accelerated electrons. This procedure results in an estimate of the minimum (lower bound) energy in the nonthermal electrons. However, a similar upper bound cannot be straightforwardly determined from this approach, so that the overall number/energy content remains highly uncertain.

Figure 1: The model in pictures. Left: the cold plasma target with non-thermal electrons propagating downwards. Right: Warm-cold model with low energy electrons collisionally diffusing and high energy free-streaming.

In an attempt to remedy this situation, Emslie (2003) included consideration of the finite temperature of the target in modifying the systematic energy loss rate of the accelerated electrons; such considerations come into play as the electron energies approach a few kT. The reduced energy losses (relative to the cold target results) at low energies make the energy content in the accelerated electrons less sensitive to the assumed low-energy cutoff, to the extent that it can formally be extended down to zero, thus providing an upper bound on the energy content. However, Emslie (2003) neglected the effects of energy diffusion on the evolution of the energy spectrum, which, as emphasized by Galloway et al. (2005), is critically important at energies of a few kT and is a necessary ingredient for describing the thermalization of the fast electrons in a warm target. Jeffrey et al. (2014) showed that the effects of diffusion in both energy and space must be included in a self-consistent analysis of electron transport in a warm target.

Figure 2: The model in pictures. Left: the cold plasma target with non-thermal electrons propagating downwards. Right: Warm-cold model with low energy electrons collisionally diffusing and high energy free-streaming.

This nugget (following Kontar et al 2015) highlights the role of thermalization of fast electrons in a model that includes both for a warm corona and a cold chromosphere (see Figure 1). The injected/or accelerated electrons first propagate in a coronal plasma that has a temperature comparable to the electron energy, and then collisionally stop in the cold plasma below. The effects of energy loss and diffusion are markedly different in these two regions. Hence, contrary to the case of a purely cold target (in which the spatially-integrated hard X-ray yield is independent of the density profile of the target), in such a composite target the relationship between the hard X-ray flux and the accelerated electron energy content needs to take into account the spatial characteristics of the emitting region, in particular the extent of the warm target region compared to that of the overall flaring region. The resulting mean electron flux spectrum shows a pile-up of thermal electrons (Fig 2), driven primarily by the effects of energy diffusion in the coronal target. This limits the maximum rate of electron injection and hence provides the much sought-after upper bound on the total injected power . These results show that the use of cold thick-target model can lead to erroneous estimates of the number and energy content in the accelerated electrons especially in cases of warm-dense coronal X-ray sources and/or soft X-ray sources. Instead, we advocate the use of the more physically complete target model, including the effect of electron thermalization. We have developed a formula that allows explicit determination of this relationship, given the temperature T and extent L of the hot coronal region. A model fit routine f_warm_thick.pro has been developed for OSPEX. This routine allows the determination of the minimum cut-off value and hence the maximum (upper bound) power in non-thermal electrons. For some flares, the low energy cut-off is constrained with an uncertainty of only a couple of keV, providing an accurate estimate of the accelerated electron number/power.

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