Microflare Heating of a Solar Active Region Observed with NuSTAR, Hinode/XRT, and SDO/AIA

For each of the NuSTAR pointings, we chose a region at the same location, and of the same area, as those used in the SDO/AIA and Hinode/XRT analysis to produce spectra of the microflare heating. These are circular as the NuSTAR software can only calculate the response files for such regions, but do cover the flaring loop region (rectangular box, Figure 2), and are shown in Figure 4. The spectra and NuSTAR response files were obtained using NuSTAR Data Analysis software v1.6.0. These were then fitted using the XSPEC (Arnaud 1996) software¹⁰ , which simultaneously fits the spectra from each telescope module (FPMA and FPMB) instead of just adding the data sets. We also use XSPEC as it allows us to find the best-fit solution using Cash statistics (Cash 1979), which helps with the non-Gaussian uncertainties we have for the few counts at higher temperatures.

We fitted the spectra with a single thermal model, using the APEC model with solar coronal abundances (Feldman et al. 1992), and the fit results are shown in Figure 6. For the first and fourth NuSTAR pointings, before and after the microflares, the spectra are well fitted by this single thermal model showing similar temperatures and emission measures (3.3 MK and $6.3\times {10}^{46}$ cm⁻³, then 3.2 MK and $7.0\times {10}^{46}$ cm⁻³). Above 5 keV, there are very few counts, and this is due to a combination of the low livetime of the observations (164 and 152 s dwell time with about 2% livetime fraction resulting in effective exposures of around 3.5 s) and the high likelihood that the emission from this region peaked at this temperature before falling off very sharply at higher temperatures. These temperatures are similar to the quiescent ARs previously studied by NuSTAR (Hannah et al. 2016), although those regions were brighter and more numerous in the field of view, resulting in an order-of-magnitude worse livetime. The low livetime has the effect of limiting the spectral dynamic range, putting most of the detected counts at the lower energy range and no background or source counts at higher energies (Grefenstette et al. 2016; Hannah et al. 2016).

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The two NuSTAR spectra from during the microflare, the second (impulsive phase) and third (decay phase, weaker peak), both show counts above 5 keV and produce higher temperature fits (5.1 MK and 3.5 MK). This is expected as there should be heating during the microflare, but neither fit matches the observed spectrum well, particularly during the impulsive phase. This shows that there is additional hot material during these times that a single-component thermal model cannot accurately characterize. For the spectrum during the impulsive phase, the second NuSTAR pointing, we tried adding additional thermal components to the fit, as shown in Figure 7. We started by adding in a second thermal component fixed with the parameters from the pre-microflare spectrum, found from the first NuSTAR pointing (left spectrum in Figure 6), to represent the background emission. We did this as NuSTAR's pointing changed during these two times (changing the part of the detector observing the region, and hence the instrumental response) so we could not simply subtract the data from this pre-flare background time. The other thermal model component was allowed to vary and produced a slightly better fit to the higher energies and a higher temperature (5.6 MK). However, this model still misses counts at higher energies.

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So, we tried another fit where the two thermal models were both allowed to vary and this is shown in the right panel of Figure 7. Here, there is a substantially better fit to the data over the whole energy range, fitting a model of 4.1 MK and 10.0 MK. The hotter model does seem to match the bump in emission between 6 and 7 keV, which at these temperatures would be due to line emission from the Fe K-shell transition (Phillips 2004). Although this model better matches the data, it produces substantial uncertainties, particularly in the emission measure. This is because it is fitting the few counts at higher energies which have a poor signal-to-noise ratio. It should be noted that for the thermal model, the temperature and emission measure are correlated and so the upper uncertainty on the temperature relates to the lower uncertainty on the emission measure, and vice versa. Therefore, this uncertainty range covers a narrow diagonal region of parameter space, which we include later in Figure 11. These fits do however seem to indicate that emission from material up to 10 MK is present in this microflare and that the NuSTAR spectrum in this case is observing purely thermal emission. A non-thermal component could still be present, but the likely weak emission, combined with NuSTAR's low livetime (limiting the spectral dynamic range), leaves this component hidden. Upper limits to this possible non-thermal emission are calculated in Section 5.2.

From these spectral fits, we estimated the GOES 1–8 Å flux¹¹ to be $5.3\times {10}^{-9}$ Wm⁻² for the impulsive phase and $4.0\times {10}^{-9}$ Wm⁻² for the pre-flare time. This means that the background-subtracted GOES class for the impulsive phase is equivalent to ∼A0.1 and would be slightly larger during the subsequent peak emission time.

To check the compatibility of the NuSTAR, Hinode/XRT, and SDO/AIA observations, we compared the observed fluxes from Hinode/XRT and SDO/AIA to synthetic fluxes obtained from the NuSTAR thermal fits. For the NuSTAR two-thermal fit (Figure 7, right panel), we multiplied the emission measures by the SDO/AIA and Hinode/XRT temperature response functions at the corresponding temperatures and then added the two fluxes together to get a value for each filter channel.

The Hinode/XRT temperature response functions were created using xrt_flux.pro with a CHIANTI 7.1.3 (Dere et al. 1997; Landi et al. 2013) spectrum (xrt_flux713.pro¹² ) with coronal abundances (Feldman et al. 1992) and the latest filter calibrations that account for the time-dependent contamination layer present on the CCD (Narukage et al. 2014). The SDO/AIA temperature response functions are version 6 (v6; using CHIANTI 7.1.3) and obtained using aia_get_response.pro with the "chiantifix," "eve_norm," and "timedepend_date" flags. The comparison of the observed and synthetic fluxes is shown in Figure 8.

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We found that the SDO/AIA 94 Å Fe xviii synthetic flux is near the observed value, as expected; however, there is a consistent discrepancy for Hinode/XRT. The observed fluxes should match the synthetic fluxes from the NuSTAR spectral fits as they are sensitive to the same temperature range. Other authors have found similar discrepancies (Testa et al. 2011; Cheung et al. 2015; Schmelz et al. 2015), and there is the suggestion that the Hinode/XRT temperature response functions are too small by a factor of ∼2–3 (see Schmelz et al. 2015). We have therefore multiplied the Hinode/XRT temperature response functions by a factor of two (Figure 8, top right) and find a closer match to the synthetic values derived from the NuSTAR spectral fits. The main effect of these larger temperature response functions is that it requires there to be weaker emission at higher temperatures to obtain the same Hinode/XRT flux.

Recovering the line-of-sight DEM, $\xi ({T}_{j})$, involves solving the ill-posed inverse problem, ${g}_{i}={{\boldsymbol{K}}}_{i,j}\xi (T)$, where g_i [DN s⁻¹ px⁻¹] is the observable and ${{\boldsymbol{K}}}_{i,j}$ is the the temperature response function for the ith filter channel and the jth temperature bin. Numerous algorithms have been developed for the DEM reconstruction, and we use two methods to recover the DEM: Regularized Inversion¹³ (RI; Hannah & Kontar 2012) and the xrt_dem_iterative2.pro method¹⁴ (XIT; Golub et al. 2004; Weber et al. 2004).

The regularized inversion (RI) approach recovers the DEM by limiting the amplification of uncertainties using linear constraints. Uncertainties on the DEM are also found on both the DEM and temperature resolution (horizontal uncertainties); see Hannah & Kontar (2012). XIT is a forward-fitting iterative least-squares approach, using a spline model. Uncertainties in the final DEM are calculated with Monte Carlo (MC) iterations with input data perturbed by an amount randomly drawn from a Gaussian distribution with the standard deviation equal to the uncertainty in the observation. The resulting spread of these MC iterations indicates the goodness of fit.

For the DEM analysis, we calculated the uncertainties on the Hinode/XRT and SDO/AIA data. The non-statistical photometric uncertainties for Hinode/XRT were calculated from xrt_prep.pro (Kobelski et al. 2014), and photon statistics were calculated from xrt_cvfact.pro¹⁵ (Narukage et al. 2011, 2014). The uncertainties on the SDO/AIA data were computed with aia_bp_estimate_error.pro (Boerner et al. 2012), and an additional 5% systematic uncertainty was added in quadrature to both the Hinode/XRT and SDO/AIA data to account for uncertainties in the temperature response functions. The Hinode/XRT and SDO/AIA data and uncertainties have been interpolated to a common time step and averaged over the NuSTAR observational duration. The uncertainty for the NuSTAR values in specific energy bands was determined as a combination of the photon shot noise and a systematic factor (of 5%) to account for the cross-calibration between NuSTAR's two telescope modules (FPMA and FPMB). The NuSTAR temperature response functions for each energy range and telescope module (shown in Figure 8) were calculated using the instrumental response matrix for the regions shown in Figure 4.

The resulting DEMs obtained for the impulsive phase are shown in Figure 9 (left) with the quality of the recovered DEM solution shown as residuals between the input and recovered fluxes (right). XIT is used with the addition of 300 MC iterations where outlier XIT MC solutions have been omitted. We have used all available filters with the exception of Hinode/XRT Be-Thick due to large uncertainties that are the result of a lack of counts (Figure 3) and SDO/AIA 335 Å due to the observed long-term drop in sensitivity (see Figure 1 in Boerner et al. 2014). The standard Hinode/XRT responses (Figure 9, top) lead to disagreement between the two methods for DEM recovery, notably at the peak and at higher temperatures (${\chi }_{\mathrm{XIT}}^{2}=2.77$, ${\chi }_{\mathrm{RI}}^{2}=1.01$). Using the Hinode/XRT responses multiplied by a factor of two results in the methods having much better agreement (${\chi }_{\mathrm{XIT}}^{2}=1.02$, ${\chi }_{\mathrm{RI}}^{2}=1.00$), and the DEM solutions result in smaller residuals, specifically in the Hinode/XRT filters. These final DEMs (Figure 9, bottom) show a peak at ∼3 MK and little material above 10 MK.

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To understand how much of this material has been heated out of the background during the microflare, we performed DEM analysis for the pre-flare NuSTAR time (∼11:10 UT). There is no Hinode/XRT data for this time so we determined the DEM using NuSTAR and SDO/AIA data. The DEMs for the pre-flare observations are shown in Figure 10. These DEMs for each method peak at a similar temperature (∼3 MK) and fall off very sharply to ∼5 MK. During the microflare, there is a clear addition of material up to 10 MK (Figure 10, bottom).

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We also represent the DEMs as the emission measure distributions (EMDs; $\xi (T){dT}$), which allows us to compare the DEM results to the NuSTAR spectral fits, shown in Figure 11. Here we have also overplotted the EM loci curves, ${\mathrm{EM}}_{i}={g}_{i}/{K}_{i}$, which are the upper limits of the emission based on an isothermal model, with the true solution lying below all of the EM loci curves. The NuSTAR thermal model fits are the isothermal (in the pre-flare phase) or two-thermal (impulsive phase) fits to the multi-thermal plasma distribution, and so represent an approximation of the temperature distribution and emission measure. These models produce the expected higher emission measure values compared to the EMD and are consistent with the EM loci curves.

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For an isothermal plasma at a temperature T and emission measure EM, the thermal energy is calculated as

$$\begin{eqnarray}&&{U}_{{T}_{I}}=3{k}_{{\rm{B}}}T\sqrt{\mathrm{EM}{fV}}\quad [\mathrm{erg}],\end{eqnarray} \tag{ 2 }$$

where k_B is the Boltzmann constant, f the filling factor, and V the plasma volume (e.g., Hannah et al. 2008). Using the two-thermal fit (Figure 7, right), we calculated the thermal energy during the impulsive phase, finding ${U}_{{T}_{I}}=0.9\times {10}^{28}$ erg (t_I = 116 s). Here, the equivalent loop volume, ${V}_{E}={fV}$, was calculated as a volume of a cylinder enclosing only the flaring loop with length L ∼ 50'' and diameter d ∼ 6''. This thermal energy includes both the microflare and background emission. We found the pre-flare energy (using fit parameters; Figure 6, left) as ${U}_{{T}_{{I}_{0}}}=0.9\times {10}^{28}$ erg (and ${t}_{{I}_{0}}=164$ s). The resulting heating power during the microflare from the thermal fits to the NuSTAR spectra is then ${P}_{{T}_{{I}_{F}}}={U}_{{T}_{I}}/{t}_{I}-{U}_{{T}_{{I}_{0}}}/{t}_{{I}_{0}}\,=2.5\,\times \,{10}^{25}$ erg s⁻¹.

The thermal energy can also be estimated for a multi-thermal plasma using

$$\begin{eqnarray}&&{U}_{T}=3{k}_{{\rm{B}}}{V}_{E}^{1/2}\displaystyle \frac{{\int }_{T}T{\xi }_{V}(T)\,{dT}}{\sqrt{{\int }_{T}{\xi }_{V}(T)\,{dT}}}\quad [\mathrm{erg}]\end{eqnarray} \tag{ 3 }$$

as described in Inglis & Christe (2014), with the filling factor, f = 1, and ${\xi }_{V}(T)={n}^{2}{dV}/{dT}$ in units of cm⁻³ K⁻¹. For the RI and XIT DEM solutions, we find values of ${U}_{{T}_{\mathrm{RI}}}=1.1\,\times {10}^{28}$ erg and ${U}_{{T}_{\mathrm{XIT}}}=1.2\times {10}^{28}$ erg during the impulsive phase of the microflare. For the pre-flare thermal energies, we find ${U}_{{T}_{{\mathrm{RI}}_{0}}}=1.2\times {10}^{28}$ erg, and ${U}_{{T}_{{\mathrm{XIT}}_{0}}}=1.2\times {10}^{28}$ erg, and this then gives values of the heating power during the impulsive phase of the microflare as ${P}_{{T}_{{\mathrm{RI}}_{F}}}=2.3\times {10}^{25}$ erg s⁻¹ and ${P}_{{T}_{{\mathrm{XIT}}_{F}}}\,=3.0\times {10}^{25}$ erg s⁻¹. All of these approaches give a similar value for the heating, about $2.5\times {10}^{25}$ erg s⁻¹, over the microflare's impulsive period, and a summary of these values with uncertainties are given in Table 1. It should be noted that these values are lower limits as the estimates ignore losses during heating.

Table 1.  Summary of Thermal Energies of AR 12333

Method	${U}_{{T}_{0}}$ a	${U}_{T}$ b	${P}_{{T}_{F}}$
	[×1028 erg]	[×1028 erg]	[×1025 erg s−1]
NuSTAR fit	${0.9}_{-0.1}^{+0.1}$	${0.9}_{-0.2}^{+0.6}$	${2.5}_{-1.6}^{+5.4}$
RI	${1.2}_{-0.1}^{+0.1}$	${1.1}_{-0.1}^{+0.1}$	${2.3}_{-1.0}^{+0.9}$
XIT	${1.2}_{-0.1}^{+0.1}$	${1.2}_{-0.1}^{+0.1}$	${3.0}_{-0.7}^{+0.6}$

Notes. The uncertainties on the energies and power derived from the NuSTAR fit are $2.7\sigma$ (90% confidence), and those from RI/XIT are $1\sigma$.

^a164 s observation. ^b116 s observation.

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From the analysis of 25,705 RHESSI events (Table 1 in Hannah et al. 2008), microflare thermal energies were found to range from ${U}_{T}={10}^{26\mbox{--}30}$ erg (5%–95% range; from a 16 s observation). This is equivalent to ${P}_{T}=6.3\times {10}^{24\mbox{--}28}$ erg s⁻¹, and therefore the thermal power from our NuSTAR microflare is in the lower range of RHESSI observations. This is as expected as NuSTAR should be able to observe well beyond RHESSI's sensitivity limit to small microflares.

As the NuSTAR spectrum in Figure 7 is well fitted by a purely thermal model, we can therefore find the upper limits of the possible non-thermal emission. This approach allows us to determine whether the accelerated electrons could power the observed heating in this microflare. We used the thick-target model of a power-law electron distribution above a low-energy cutoff E_c [keV] given by

$$\begin{eqnarray}&&F(E\gt {E}_{c})\propto {E}^{-\delta },\end{eqnarray} \tag{ 4 }$$

where δ is the power-law index, and the power in this non-thermal distribution is given by

$$\begin{eqnarray}&&{P}_{N}(\gt {E}_{c})=1.6\times {10}^{-9}\displaystyle \frac{\delta -1}{\delta -2}{N}_{N}{E}_{c}\quad [\mathrm{erg}\ {{\rm{s}}}^{-1}],\end{eqnarray} \tag{ 5 }$$

where N_N is the non-thermal electron flux [electrons s⁻¹].

We determined the upper limits on N_N (and P_N) for a set of δ ($\delta =5,7,9$) and E_c consistent with a null detection in the NuSTAR spectrum. We performed this by iteratively reducing the model electron flux N_N until there were fewer than four counts $\gt 7\,\mathrm{keV}$, consistent with a null detection to $2\sigma$ (Gehrels 1986). We also ensured that the number of counts $\leqslant 7\,\mathrm{keV}$ are within the counting statistics of the observed counts. For each iteration, we generated the X-ray spectrum for the two-component fitted thermal model (Figure 7, right) and added to this the non-thermal X-ray spectrum for our chosen δ, E_c, and N_N, calculated using f_thick2.pro¹⁶ (see Holman et al. 2011). This was then folded through the NuSTAR response to generate a synthetic spectrum (as discussed in Hannah et al. 2016). The upper limits are shown in Figure 12 along with the three estimates of the thermal power for the background-subtracted flare, ${P}_{{T}_{{I}_{F}}}$ ("NuSTAR Fit," black), ${P}_{{T}_{{\mathrm{RI}}_{F}}}$ (pink), and ${P}_{{T}_{{\mathrm{XIT}}_{F}}}$ (blue). For a flatter spectrum of $\delta =5$, barely any of the upper limits are consistent with the required heating power. With a steeper spectrum, $\delta \geqslant 7$, a cutoff of ${E}_{c}\lesssim 7\,\mathrm{keV}$ is consistent with the heating requirement. These steep spectra indicate that the bulk of the non-thermal emission would need to be at energies close to the low-energy cutoff to be consistent with the observed NuSTAR spectrum. If we instead consider some of the counts in the 6–7 keV range to be non-thermal (e.g., the excess above the thermal model in the left panel in Figure 7), then we would obtain a higher non-thermal power, about a factor of 0.5 larger. However, this would only substantially affect the steep non-thermal spectra ($\delta \geqslant 7$) as flatter models would be inconsistent with the data below 7 keV.

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We can again compare the microflare studied here to the non-thermal energetics derived from RHESSI microflare statistics. Hannah et al. (2008) report non-thermal parameters of $\delta =4\mbox{--}10$ and ${E}_{c}=9\mbox{--}16\,\mathrm{keV}$, and non-thermal power ranges from ${P}_{N}(\geqslant {E}_{c})={10}^{25\mbox{--}28}$ erg s⁻¹. The largest upper limits that NuSTAR produces for this microflare are again at the edge of RHESSI's sensitivity. In a previous study of nanoflare heating, Testa et al. (2014) investigated the evolution of chromospheric and transition region plasma from IRIS observations using RADYN nanoflare simulations. This is one of the few non-thermal nanoflare studies, and they reported that heating occurred on timescales of $\lesssim 30\,{\rm{s}}$, characterized by a total energy ≲10²⁵ erg and ${E}_{c}\sim 10\,\mathrm{keV}$. The simulated electron beam parameters in this IRIS event are consistent with the NuSTAR-derived parameters, but in a range insufficient to power the heating in our microflare.