Spectral and Imaging Diagnostics of Spatially-Extended Turbulent Electron Acceleration and Transport in Solar Flares

The main diagnostic tool of past (e.g., RHESSI) and current (e.g., SolO/STIX) flare X-ray instrumentation is high resolution spectroscopy ($\sim$few keV, space-integrated and space-resolved) constraining the bulk of the accelerated electron properties to-date. Similar to Stackhouse & Kontar (2018), the top panel of Figure 2 shows how the flare spectra change due to varying $\uptau_{acc}$ with the shape of the spectrum changing significantly from $\uptau_{acc}=800\uptau_{c}$ to $\uptau_{acc}=6000\uptau_{c}$. When the acceleration timescale is high, i.e $\uptau_{acc}=6000\uptau_{c}$, $nVF$ remains mostly thermal. However, as $\uptau_{acc}$ decreases, $nVF$ increases and a non-thermal tail forms (harder spectrum). As expected, the spectral index $\delta_{nVF}$ increases as acceleration timescale increases ranging from $\delta_{nVF}=0.5$ at $\uptau_{acc}=800\uptau_{c}$ to $\delta_{nVF}=7.6$ at $\uptau_{acc}=6000\uptau_{c}$⁵⁵5We note that in a traditional thick-target analysis this leads to an injected $\delta\approx 0.5+2=2.5$ and $\delta\approx 7.6+2=9.6$ respectively.. In Figure 2 (top), a single power law is fitted to the data over an appropriate energy range to find the spectral index (shown by the solid lines).

Figure 3 shows the full-flare $\delta_{nVF}$ versus $\uptau_{acc}$, for all spatial extents (3a), spatial functions (3b), and velocity dependencies (3c). Simulations ‘with’ and ‘without’ short timescale turbulent scattering, case i) and ii), are shown with crosses and circles respectively. Shaded rectangles indicate the addition of a background thermal component, $nVF_{th}$ (with a maximum ${\rm EM}=10^{49}$ cm^-3), that may change the inferred value of $\delta_{nVF}$.

For each spectra, the spectral index $\delta_{nVF}$ is determined from $E=20$ keV or greater (to avoid the influence of the thermal part at lower energies) until the highest suitable energy, with a maximum value of $E=100$ keV. We choose this maximum value for two reasons: (1) most simulation outputs are noisy after this energy and (2) most solar flare spectra (e.g. RHESSI, STIX) are fitted between this range above the background. If the highest suitable energy was $<30$ keV $\delta_{nVF}$ was fitted from a minimum of $15$ keV and the fits were studied carefully by eye. Several simulations have an increasing energy spectrum as a result of very high acceleration (i.e $\sigma_{3},\rm{g},\alpha_{4}$ at $\uptau_{acc}=800\uptau_{c}$), and the spectral index of these simulations has been removed from Figure 3 since we do not see increasing spectra in solar flare data. Similarly, several simulations have only thermal spectra, where $\delta_{nVF}$ could not be determined.

Considering Figure 3, as $\delta_{nVF}$ increases, $\uptau_{acc}$ increases for all simulations. For the simulations studied, $\sigma_{3},\rm{g},\alpha_{4}$ provides a lower boundary on $\delta_{nVF}$ at a given $\uptau_{acc}$, producing the hardest spectra (Figure 3c). Increasing the velocity dependency increases the spectral index such that $\delta_{nVF}[\alpha_{3}>\alpha_{4}]$ for a given acceleration timescale. No values for $\sigma_{3},\rm{g},\alpha_{2}$ are included, as the energy spectra was thermal for all acceleration timescales.

Figure 3a shows that as the spatial extent increases, $\delta_{nVF}$ decreases, such that $\delta_{nVF}$[$\sigma_{1}>\sigma_{3}>\sigma_{7}$], as expected, since electrons experience more acceleration in an acceleration region of greater spatial extent, leading to harder spectra. In contrast, changing $H(z)$ (Figure 3b) does not noticeably change $\delta_{nVF}$ with $\uptau_{acc}$. In general, both l and r with extent $3\arcsec$ ($\sigma_{3},\textit{l},\alpha_{3}$ and $\sigma_{3},\textit{r},\alpha_{3}$, respectively) produce larger $\delta_{nVF}$ than $\rm{g}$ of the same spatial extent ($\sigma_{3},\rm{g},\alpha_{3}$) and a more apt comparison may be with a $\rm{g}$ with a smaller spatial extent, i.e $\sigma_{1},\rm{g},\alpha_{3}$.

The addition of short timescale turbulent scattering and/or a background thermal component increases the spectral index. Generally, as the acceleration timescale increases, the range of the spectral index increases. Whilst it is still possible to distinguish between different velocity dependencies at larger $\uptau_{acc}$, it is difficult to distinguish between other parameters such as spatial extent.

Imaging spectroscopy (spatially resolved spectra) is an essential tool for constraining the properties of electron acceleration and transport in flares. In Figures 4a-c we plot examples of spatially resolved energy spectra showing separate spectra for the coronal looptop (4b) and chromospheric footpoints (4c) alongside the full flare integrated spectrum (4a). A comparison of the spectral indices for the full flare $\delta_{nVF}$, looptop $\delta_{nVF}^{LT}$ and footpoint $\delta_{nVF}^{FP}$ sources are shown in the legends. Similar to observational imaging spectroscopy, we define the looptop region at $-5\arcsec<z<5\arcsec$, and the footpoints sources at $20\arcsec<|z|<30\arcsec$  ⁶⁶6Here, the shown footpoint spectra is the mean spectra of both footpoint spectra.. As in Figure 2, the example spectra in Figure 4a-4c shows the control simulation for different $\uptau_{acc}$. A power law fit (thick solid line) is used to find the spectral index.

In general, for each simulation $\delta_{nVF}$ is harder than the looptop spectral index $\delta_{nVF}^{LT}$. Similarly, the footpoint spectral index $\delta_{nVF}^{FP}$ is harder than $\delta_{nVF}$, such that, $\delta^{FP}_{nVF}>\delta_{nVF}>\delta^{LT}_{nVF}$, as expected. For example, the control simulation with $\uptau_{\rm acc}=1000\uptau_{c}$ gives $\delta^{FP}_{nVF}=0.5>\delta_{nVF}=0.7>\delta^{LT}_{nVF}=1.9$.

Figure 4d, 4e, and 4f plot the difference $\delta^{LT}_{nVF}-\delta^{FP}_{nVF}$ for different spatial extents (4d), spatial functions (4e), and velocity dependencies (4f). The larger the value of $\alpha$ (velocity dependence), the greater the spectral difference $\delta^{LT}_{nVF}-\delta^{FP}_{nVF}$ with $\sigma_{3},\rm{g},\alpha_{4}$ showing $\delta^{LT}_{nVF}-\delta^{FP}_{nVF}\approx 2$ for all $\uptau_{acc}$. In this case, electrons are accelerated rapidly and behave similar to an injected distribution in the corona that streams to the chromosphere, approximately following the simple thick-target expectation. The smaller the spatial extent, the larger the spectral difference $\delta^{LT}_{nVF}-\delta^{FP}_{nVF}$ since electrons escape the acceleration region faster. This may also explain why the l and r spatial functions generally have larger spectral differences compared to $\rm{g}$, since the spatial extent is $3\arcsec$ (see Figure 1b).

Previous work (e.g, Chen & Petrosian, 2012) shows that a spectral index difference less than 2 is often consistent with the confinement of electrons in the corona. Firstly, this is mirrored in our results when the acceleration becomes less efficient. Also, we do not need to necessarily invoke different turbulent spectra or exotic plasma waves that preferentially accelerate electrons at $90^{\circ}$ (Chen & Petrosian, 2012), for such confinement. For example, as $\uptau_{acc}$ increases, the spectral index difference is smaller for all simulation runs apart from the extreme $\sigma_{3},\rm{g},\alpha_{4}$ case that has efficient acceleration across all $\uptau_{acc}$. Our different scattering cases i) and ii) also change the spectral index difference. When turbulent scattering acts on longer timescales (circles in Figure 4) closer to the collisional time, the spectral difference is $\lesssim 2$ for all cases. When the turbulent scattering acts on shorter timescales (crosses), the spectral index difference tends to increase slightly at low $\uptau_{acc}$, which is mainly due to the formation of steeper spectra in these cases. Further, the spectral index difference increases when a larger background thermal component is present since the non-thermal component in the corona is hidden by the large (steep) thermal background.

We also study the usefulness of the ratio diagnostic $\delta^{LT}_{nVF}/\delta^{FP}_{nVF}$ with $\uptau_{acc}$, seen in Figures 4g, 4h, and 4i. For all acceleration regions, $\delta^{LT}_{nVF}/\delta^{FP}_{nVF}$ decreases as $\uptau_{acc}$ increases. When $\uptau_{acc}\leq 1000\uptau_{c}$ the ratio $\delta^{LT}_{nVF}/\delta^{FP}_{nVF}>1$. At high acceleration timescales ($\uptau_{acc}\geq 2000\uptau_{c}$), the spectrum is mainly thermal, and the ratio begins to flatten remaining at $\sim 1$.

Shorter timescale turbulent scattering generally leads to smaller spectral index ratios (again due to the steeper footpoint spectra), as does the addition of a background thermal component with increasing EM. Furthermore, the greater the velocity dependence or spatial extent, the greater the ratio. Once again, the spatial functions are harder to separate. However, $\rm{g}$ often has a greater ratio than l and r. Thus in general, the more efficient the acceleration in the loop, the larger the ratio $\delta^{LT}_{nVF}/\delta^{FP}_{nVF}$.

Although differences in spectral index between looptop and footpoint sources are often discussed in observational work (e.g., Battaglia & Benz, 2006; Chen & Petrosian, 2012; Simões & Kontar, 2013), here we find that the ratio of spectral indices $\delta^{LT}_{nVF}/\delta^{FP}_{nVF}$ is the more reliable diagnostic for indicating the acceleration timescale when there is a small background thermal component. Whereas, the spectral difference is a stronger diagnostic for the velocity dependence.

Instruments such as RHESSI and STIX provide us with spatially resolved data of solar flares in several energy bins. Figures 5d-5i plot energy-dependent $\varphi$ versus $E$. For realistic comparison with imaging data, we use the following energy bins: $E=3-6~\text{keV},6-12~\text{keV},12-25~\text{keV},25-50~\text{keV},50-100~\text{keV}$. Figure 5 shows simulations without turbulent scattering (5d-5f) and with short-timescale turbulent scattering (5g-5i).

As energy increases, $\varphi$ decreases. Thus, as $E$ increases the footpoint emission becomes more dominant, as expected. The examples shown in Figures 5d-5i compare different spatial extents, $\sigma_{1},\rm{g},\alpha_{3}$ (5d and 5g), $\sigma_{3},\rm{g},\alpha_{3}$ (5e and 5h), and $\sigma_{7},\rm{g},\alpha_{3}$ (5f and 5i) for all $\uptau_{acc}$⁷⁷7At higher energies, data may be missing at larger acceleration timescales.. We choose to compare spatial extent $\sigma$, since this property produces a larger variation in $\varphi$ with $\uptau_{acc}$ as shown in Figures 5a - 5c. For a given $\sigma$, we see differences in $\varphi$ with energy for different $\uptau_{acc}$. For example, at lower $\uptau_{acc}$ and for a smaller spatial extent, we see larger $\varphi$, as electrons experience less acceleration. For the other studied variables (not shown), such as velocity dependence, as $\alpha$ increases we see a larger spread in $\varphi$ with $\uptau_{acc}$ for a given energy. Acceleration regions with r show slightly less variation in $\varphi$ with $\tau_{acc}$, similar to simulations with $\alpha_{2}$. Whereas, the l and $\rm{g}$ distributions show a larger range of $\varphi$ versus $\uptau_{acc}$ for each energy bin.

Shorter timescale turbulent scattering decreases energy dependent $\varphi$ by up to an order of magnitude, with this difference increasing with energy $E$, due to the decrease in higher energy footpoint emission.

Adding a background thermal component increases $\varphi$ by several orders of magnitude (not shown). However, electrons with energy $\geq 25$ keV are not dominated by the thermal component at acceleration timescales $\leq 2000\uptau_{c}$. Overall, if a flare has a small background thermal component, $\varphi$ may help to constrain all properties of the acceleration region. However, if a flare has a large background thermal component, energy-independent $\varphi$ becomes obsolete and energy-dependent $\varphi$ is not useful for $E\leq 25$ keV.

Figures 6a-6f plot angle-integrated $nVF$ versus $z$ for the control simulation using $\uptau_{acc}=800\uptau_{c}$ (6a and 6d), $2000\uptau_{c}$ (6b and 6e) and $6000\uptau_{c}$ (6c and 6f), with (6d – 6f) and without (6a – 6c) the inclusion of shorter timescale turbulent scattering, using a $1\arcsec$ spatial binning. In this figure, $z=0\arcsec$ indicates the loop apex with the chromospheric boundary at $\pm 20\arcsec$.

The spatial distribution can change drastically for different energy bins. For all spatial distributions and acceleration timescales, electrons with energy $E=3-6$ keV have the highest electron flux in the coronal looptop, but $nVF$ for this energy range is significantly smaller in the footpoints. As electron energy increases, $nVF$ in the looptop decreases. Regardless of acceleration timescale, the two highest energy bins ($E=25-50$ keV, $50-100$ keV) do not show a peak in the coronal looptop. This is consistent with observations where coronal X-ray sources are usually observed at $\approx 10-30$ keV.

At high acceleration timescales, nVF at both the chromospheric boundary ($|z|=20^{\prime\prime}$) and the loop apex ($z=0^{\prime\prime}$) is dominated by low energy electrons. At lower acceleration timescales (e.g. $\uptau_{acc}=800\uptau_{c}$, Figure 6a), the footpoint emission generally increases. However the energy of the electrons at the footpoints appears to depend on the spatial function used. For $\rm{g},\alpha_{3}$ simulations (i.e, $\sigma_{1},\rm{g},\alpha_{3}$, $\,\sigma_{3},\rm{g},\alpha_{3}$, and $\sigma_{7},\rm{g},\alpha_{3}$) the footpoint emission is comprised of all energy bins, and as the spatial extent increases, higher energy electrons become more prominent. In comparison, simulations with l and r have significantly less high energy particles in the footpoints. Instead, even at a low acceleration timescale, the $6-12$ keV emission is still the most prominent.

The acceleration timescale at which high energy emission ($E=50-100$ keV) dominates over low energy emission ($E=15-20$ keV) can be seen in the Figures 6g-6i which shows $\eta$ for different acceleration timescales, where

The dashed line is $\eta=1$. Simulations with and without turbulent scattering are shown by crosses and circles, respectively. The addition of a background thermal component is represented by the shaded rectangles, creating a range of values for $\eta$. Several data points are missing, particularly at high acceleration timescales, as there are no electrons with energy $50-100$ keV for that simulation.

Let us now consider simulations without turbulent scattering and no background thermal component. For all acceleration regions, $\eta$ increases over several orders of magnitude as acceleration timescale increases from $\uptau_{acc}=800\uptau_{c}$ to $\uptau_{acc}=6000\uptau_{c}$. When $\uptau_{acc}\geq 1000\uptau_{c}$, low energy electrons dominate for all simulated regions (except for $\alpha_{4}$). Whereas when $\uptau_{acc}=800\uptau_{c}$ acceleration regions which have large footpoint emission (i.e, $\sigma_{3},\rm{g},\alpha_{3}$, $\,\sigma_{7},\rm{g},\alpha_{3}$, and $\sigma_{3},\rm{g},\alpha_{4}$) also have values of $\eta<1$ , as high energy electrons dominate.

The greater the spatial extent or the velocity dependence the smaller $\eta$ for a given acceleration timescale, such that $\eta[\sigma_{1}>\sigma_{3}>\sigma_{7}]$ and $\eta[\alpha_{2}>\alpha_{3}>\alpha_{4}]$. However, the separation of the velocity dependencies is much clearer than the spatial extents. Once again, $\alpha_{2}$ and $\alpha_{4}$ show two extremes of $\eta$, where $\alpha_{4}$ is the lower boundary for a given acceleration timescale. This is clearly shown in Figure 6j which shows how $\eta$ changes for all spatial functions and acceleration timescales. Unfortunately, simulations using $\alpha_{2}$ only have electrons with energy greater than 50 keV for $\uptau_{acc}=800\uptau_{c}$. For this timescale $\alpha_{2}$ creates an upper boundary for $\eta$. Using this diagnostic, it is difficult to separate the spatial functions l, r, and $\rm{g}$.

The addition of short timescale turbulent scattering generally increases $\eta$ by up to an order of magnitude. The greater the acceleration timescale, the smaller the change in $\eta$ due to shorter timescale turbulent scattering. The addition of a background thermal component increases $\eta$ by several orders of magnitude, removing most trends in the graphs. The only property of the acceleration region that could still be easily determined is the velocity dependence.

Similar to the integrated spatial distribution, in the chromosphere ($|z|\geq 20\arcsec$) we study $\eta_{\rm FP}$ where

Figures 6j and 6k show $\eta$ and $\eta_{\rm FP}$ for each simulation, respectively. Acceleration timescales are shown in different colors, indicated in the legend. Compared to $\eta$ for the entire spatial distribution, for $\eta_{\rm FP}$ the lower energy bin has been increased to better suit the chromosphere. When a background thermal component is not included in the corona, both $\eta$ and $\eta_{\rm FP}$ are almost identical. The differences between $\eta$ and $\eta_{\rm FP}$ are centred around $\eta=1$. When $\eta>1$, $\eta_{\rm FP}$ may be slightly smaller than $\eta$. Whereas when $\eta<1$, $\eta_{\rm FP}\gtrsim\eta$. However these differences are always less than an order of magnitude.

As with $\eta$, from $\eta_{\rm FP}$ the velocity dependence and spatial extents may be identified. However unlike $\eta$, $\eta_{\rm FP}$ does not suffer from additional background thermal effects and thus, is a useful diagnostic for studying acceleration properties.

Figure 7 shows $nVF(E=6-12$ keV) against $nVF(E=50-100$ keV) for each acceleration timescale, indicated by a different color, and each simulation, shown by a different symbol. Simulations ‘without’ turbulent scattering are shown on the top and simulations with turbulent scattering are shown on the bottom. The black line is the identity line, when $nVF(E=6-12$ keV) $=nVF(E=50-100$ keV).

To the left of the identity line, for all simulations without turbulent scattering (except $\sigma_{3},\rm{g},\alpha_{4}$), as acceleration timescale increases, $nVF(E=50-100$ keV) increases over several orders of magnitude. In comparison, $nVF(E=6-12$ keV) remains fairly consistent, increasing by just one order of magnitude across all simulations. There is a slight increase in $nVF(E=6-12$ keV) which peaks to the left of the identity line. Unlike the other simulations, for simulations with $\alpha_{4}$, $nVF(E=50-100$ keV) remains fairly constant, and $nVF(E=6-12$ keV) decreases over an order of magnitude as acceleration timescale increases.

Similar trends can be seen in simulations with turbulent scattering, except the range of values of $nVF(E=50-100$ keV) is smaller, increasing over three orders of magnitude instead of six. Generally, simulations which include shorter timescale turbulent scattering have a lower $nVF(E=50-100$ keV).

For simulations without turbulent scattering, the only data to the right of the identity lines comes from simulations which use $\alpha_{4}$ for $800\uptau_{c}<\uptau_{acc}<2000\uptau_{c}$. These data points represent the simulations in which high energy emission ($E=50-100$  keV) dominates over low energy emission in the footpoints ($E=6-12$ keV).

Overall, $\eta$ can be used to determine the velocity dependence. If the background thermal component is small the spatial extent and spatial function may also be determined. Furthermore, plotting $nVF(E=6-12$ keV) against $nVF(E=50-100$ keV), again for a small thermal component, may indicate if short timescale turbulent scattering is present.

In a collisional thick-target model, the depth electrons travel into the chromosphere changes with electron energy, with higher energy electrons moving deeper into the chromosphere.

Figure 8 shows emission in the chromosphere. Figures 8a and 8d plot $nVF$ versus $|z|$ in the chromosphere using a $0.1\arcsec$ spatial resolution with no turbulent scattering, whilst Figures 8b and 8e show the spatial distribution with short timescale turbulent scattering. An acceleration timescale of $800\uptau_{c}$ (Figures 8a-8c) and an acceleration timescale of $2000\uptau_{c}$ (Figures 8d-8f) are both shown. The colors show different electron energy bins, given in the legend.

Consider the simulations without short timescale (‘no’) turbulent scattering (Figures 8a and 8d). The spatial distribution in the chromosphere changes drastically depending on the amount of acceleration in the loop. When there is little acceleration (e.g., $\uptau_{acc}=6000\uptau_{c}$) all energy bins show a peak in $nVF$ at the chromospheric boundary and $nVF$ decreases with chromospheric depth. Alternatively, when there is efficient acceleration ($\uptau_{acc}=800\uptau_{c}$) there is an initial peak in $nVF$ at the chromospheric boundary, where lower energy electrons are dominant. Then, we also see a secondary peak at $\approx 2\arcsec$ into the chromosphere, where higher energy electrons are dominant.

When we do account for shorter timescale turbulent scattering (Figures 8b and 8e) the chromospheric emission is reduced across all energy bins. Furthermore, the secondary peak in $nVF$ seen at small $\uptau_{\rm acc}$ almost fully disappears in the two simulations shown in Figure 8.

Figures 8c and 8f show the average electron depth in the chromosphere versus electron energy for $\uptau_{\rm acc}=800\uptau_{c}$ and $\uptau_{\rm acc}=2000\uptau_{c}$, respectively. For a comparison with data, here the average depth is calculated using the first moment of $nVF$ versus $z$ for each energy bin⁸⁸8We use a mean value of the chromospheric depth since it is a better comparison with similar published observational results i.e., Kontar et al. (2010) where the centroid positions of HXR footpoint sources are determined to sub-arcsecond accuracy.. Here, a depth of $0\arcsec$ corresponds to the top of the chromosphere. All simulated acceleration regions without shorter timescale turbulent scattering are shown in different colors, given in the legend.

Changing the velocity dependence creates upper and lower boundaries of electron depth. For any energy, electrons in $\alpha_{2}$ simulations appear at locations closer to the chromospheric boundary ($0\arcsec$) and for $\alpha_{4}$, electrons are located deeper in the chromosphere. Between these two boundaries are the other simulations (using $\alpha_{3}$). As $\uptau_{acc}$ decreases, electrons of a give energy are located deeper in the chromosphere. Thus, the initial depth of the chromospheric emission (given by low energy electrons) may help indicate the acceleration timescale and velocity dependence. At low acceleration timescales of $\uptau_{acc}=800\uptau_{c}$, the electron depth into the chromosphere increases linearly with energy. For $\sigma_{3},\rm{g},\alpha_{4}$, we see this trend for both acceleration timescales shown. However, the gradient of depth with energy increases as $\uptau_{acc}$ increases. This comparison cannot be made for other simulated acceleration regions, as at high acceleration timescales the distributions are more scattered; this may be due to a lack of electrons at higher energies. The scattered distribution at higher acceleration timescales makes it difficult to separate the different spatial extents, which is possible at $\uptau_{acc}=800\uptau_{c}$. Such that, electron depth into the chromosphere for $\sigma_{1}<\sigma_{3}<\sigma_{7}$, for a given energy. It is difficult to distinguish between the l and r spatial functions. Once again, these functions may be better compared to a $\rm{g}$ distribution of a smaller spatial extent, i.e., $\sigma_{1},\rm{g},\alpha_{3}$.

Figure 9 shows the average depth for electrons of energy $15$ keV $\leq E<100$ keV, versus acceleration timescale. As acceleration timescale increases, depth decreases, as expected. Although chromospheric $nVF$ is lower for simulations with shorter timescale turbulent scattering, the average depth for all electron energies between $15$ keV $\leq E<100$ keV, is approximately the same, as seen in Figure 9.

The velocity dependence changes depth versus acceleration timescale such that we see greater chromospheric depths for larger $\alpha$. The greater the spatial extent, the greater the depth. However, it is difficult to distinguish between $\sigma_{1}$ and $\sigma_{3}$. The different spatial functions cannot be identified easily either.

Overall, studying the emission in the chromosphere may help to constrain the acceleration timescale by studying (1) the gradient of depth versus energy and (2) the depth of low energy electrons ($E\lesssim 15$ keV) in the chromosphere. This diagnostic is useful for limb flares, such as that studied in Kontar et al. (2010). If the acceleration timescale is low, depth vs energy may also indicate the spatial extent of the acceleration region. Alternatively, if the acceleration timescale is high, the average depth versus acceleration timescale may constrain the velocity dependence.

The spatial distribution in the coronal loop, $-20\arcsec<z<20\arcsec$ (see the middle panel of Figure 2) can be approximated with a Gaussian distribution in the majority of cases. For each acceleration region, the full width at half maximum (FWHM) [arcseconds] was determined for electrons with energy $10$ keV $<E<15$ keV⁹⁹9The energy range is chosen to allow comparison with observational results, e.g., Jeffrey et al. (2015), where the FWHM of a coronal looptop source is determined from X-ray data. using: