Extreme ultraviolet (EUV) context data from the Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) on-board the Solar Dynamics Observatory spacecraft (Pesnell et al., 2012, SDO) is utilized as a large FOV context to our spectroscopic observations, for co-alignment between IRIS and NuSTAR observations, and to account for NuSTAR’s inherent pointing uncertainty when performing solar observations, as discussed in Section 2.3. The AIA 94 Å channel contains two temperature response peaks. The higher of the two is centered around 7 MK (e.g., O’Dwyer et al., 2010; Boerner et al., 2012; Testa and Reale, 2012), making it sensitive to temperatures measured by NuSTAR for flaring plasma. AIA 94 Å context images were made for the 18:20 UT flare, as well as the background time utilized in NuSTAR analysis. We isolate the higher temperature response peak in the 94 Å ~channel due to Fe xviii~via a linear combination of observed flux from AIA’s 94, 171, and 211 Å channels (Del Zanna, 2013). We have also used AIA 304 Å and 1600 Å images as context to our observations and to co-align the IRIS and AIA observations. Both filters show the cooler emission from the small ribbon that is also visible in the SJI images (see Figure 2).

We also analyze photospheric vector magnetograms and line-of-sight magnetic field intensities provided by the Helioseismic and Magnetic Imager (Scherrer et al., 2012, HMI) on board SDO. We use HMI data to derive a magnetic field extrapolation and obtain information about the magnetic connectivity of the flare loops under study (Sect. 6), which is useful to compare the results of the spectroscopic observations and models (Sect. 7). Both AIA images and HMI line-of-sight magnetic field intensity maps were processed and corrected for instrumental effects by using the standard SolarSoft routine aia_prep.pro.

Since July 2013, IRIS has been providing far ultraviolet (FUV) and near ultraviolet (NUV) images and spectra of the solar atmosphere, from the photosphere to low corona, at very high spatial (0.33–0.4″), spectral (2.7 km s⁻¹ per pixel) and temporal resolution (down to 1s or less in high-cadence flare mode, De Pontieu et al., 2014). The IRIS spectrograph channel observes line and continua formed over a broad range of temperatures, from logT [K] ≈ 3.5–7. Simultaneously, the IRIS Slit-Jaw Imager (SJI) provides high-resolution (0.33″) context images in four individual filters (C ii 1330 Å, Si iv 1400 Å, Mg ii k 2796 Å and Mg ii h wing 2803 Å). Thanks to its unique instrumental capabilities, IRIS has significantly improved our understanding of the energy deposition in the lower atmosphere during flares (e.g., see De Pontieu et al., 2021, for a recent review).

The IRIS dataset under study was a part of a medium coarse 8-step raster observation, that ran between 13:59:21–22:58:59UT on 29 April 2021. A “coarse” raster means that there is a 2″ separation between consecutive IRIS slit positions. The medium raster FOV was ≈4′′× 60″, and the raster cadence was ≈75s, with an average raster step cadence of 8 s. IRIS SJI images in the 1330 Å filter were also taken with a cadence of 10 s over a ≈59′′× 60″ FOV. In this work, we use level 2 IRIS data, which have been corrected for a number of instrumental issues, including geometric, dark and flat-field calibration, and correction for the wavelength orbital variation (De Pontieu et al., 2014; Wülser et al., 2018).

The Nuclear Spectroscopic Telescope ARray (NuSTAR), is a NASA small explorer mission launched in 2012. The instrument consists of two co-aligned direct-focusing HXR telescopes, with a 12′× 12′ FOV and angular resolution FWHM of 18′′ (Harrison et al., 2013). The identical CdZnTe solid-state focal plane detectors are made up of four pixel detector arrays (Madsen et al., 2015). Both detectors are used to improve sensitivity, and are defined as focal plane module A (FPMA) and focal plane module B (FPMB). NuSTAR is primarily an astrophysical observatory, with a total observation range of 3–79 keV. When used for solar observations, NuSTAR is limited to observing between 2.5–13 keV due to high count rates at low energies dominating the livetime during solar observations (Grefenstette et al., 2016). NuSTAR also experiences an uncertainty in absolute pointing of one to two arcmin when performing solar observations, which is accounted for by co-aligning NuSTAR data to AIA 94 Å images utilizing Sunpy’s calculate_match_template_shift function.

NuSTAR’s spatial data during this orbit is complex, with the active region producing two flares in different locations during a relatively short timescale. The first flare, occurring at 18:08 UT, is located at the eastern side of the active region. After this flare, the second flare occurs at 18:20 UT in the west, as shown in Figure 3. A background emission was measured at 18:40 UT after the second flare, with a background temperature of 4.4 MK found. Due to the quick succession of the flares, plasma heated above the background temperature is still present in the eastern region during the impulsive phase of the second flare. Because of this circumstance, we have carefully chosen source and background regions for spectroscopy, which will be further discussed in Sect. 4.2.

The lightcurves for this NuSTAR observation are shown in Figure 8. The middle two panels are livetime-corrected count rates for NuSTAR, separated into low and high energies. The top panel shows the total GOES short- and long-wavelength X-ray Sensor (XRS) SXR flux. The bottom panel includes both the high and low energy NuSTAR lightcurves, normalized to arbitrary units to best compare their time profiles. The derivatives of the NuSTAR 3–7 keV count rate and the long-wavelength GOES XRS flux, with a 2-min boxcar average to account for noise, are also included.

The NuSTAR lightcurve displays noticeable similarities to those of larger flares and the standard flare model. The higher energy NuSTAR lightcurve shows an earlier peak time and greater impulsivity than the lower energy range. The change in lightcurve profile is at ∼7keV, which matches well to the found cutoff energies for accelerated electrons in spectral fitting (discussed in Section 4.2). Notably, the NuSTAR low energy and GOES SXR derivatives match the profile of the higher energy HXR lightcurve, exhibiting the Neupert effect (Neupert, 1968). The Neupert effect is often used to address the potential for non-thermal signatures. The derivative of SXR or EUV emission matching that of the higher energy HXRs suggests that thermal emission results from heating by accelerated electron beams (Neupert, 1968; Veronig et al., 2002; Dennis et al., 2003).

Spectroscopy of the 18:20 UT flare was performed using the X-ray spectral fitting package OSPEX and was independently confirmed using the package XSPEC. A 2-min window during the impulsive phase of the flare, highlighted in red in Figure 8, was used for spectral fitting. This early-flare phase was used in order to best search for signatures of accelerated electrons.

Solar observations result in significantly higher count rates than those experienced by NuSTAR when observing its astrophysical targets. Therefore, the detectors must be checked for pileup (discussed in Grefenstette et al., 2016; Appendix C). Detector pileup may occur when multiple photons enter a single pixel within a single measurement time, but are only recorded as a single event Harrison et al. (2013). NuSTAR events track the pattern of pixels in which charge was measured, or “grade.” Pulse pileup conditions may be checked by looking at the event rate of “unphysical grades”—combinations that are impossible to achieve from a single photon. The number of events per grade was calculated for the unphysical grades, and found to be a negligible fraction of the total events. Therefore, pileup effects are considered to be negligible in this analysis. Due to the significantly low detector livetimes ( < 1%), a gain calibration correction must be performed when doing spectroscopy on NuSTAR solar events. This gain correction is performed by allowing the gain slope of FPMA and FPMB to be independent free parameters during a spectral fit, and is discussed in detail in the Appendix of Duncan et al. (2021). For this event, the gain correction was performed using the XSPEC spectroscopy package.

In this microflare, a thermal background of temperature 4.4 MK was measured at 18:40–18:42 UT and utilized in spectral fitting. The 4.4 MK background temperature is similar to thermal temperatures found when performing spectral fitting on the same region during other non-flaring intervals on the same day. No gain correction was performed for the background interval, as it was not found to be necessary based on the criteria described in Duncan et al. (2021).

Counts used for spectral fitting of both the flare and the background were limited to the western side of the active region, in order to eliminate emission from the previous eastern 18:08 UT flare. The eastern NuSTAR source lies ∼100 arcsec from the source of interest. Given that NuSTAR’s PSF drops by at least an order of magnitude at a distance of ∼100 arcsec (Koglin et al., 2011; Madsen et al., 2015), and given that the eastern source is much fainter than our source of interest, the contamination from the eastern source at our flare site is negligible. Spectral fitting over the 2-min window was performed for this eastern post-flare region, in order to assess the physical implications of excluding emission from that side. A double thermal model, including a temperature very close to the 4.4 MK background and a higher temperature component, was found to be the best fit for the eastern area. This indicates that this additional thermal component is leftover cooling plasma from the 18:08 UT flare and is unrelated to the western flare. Since the eastern region showed no non-thermal signatures and is outside of IRIS’s FOV, the area was excluded from NuSTAR spectral fitting in order to best compare the results with IRIS observations and RADYN simulations.

Spectral fitting in XSPEC (Arnaud, 1996) was used to ascertain the gain correction and for an initial assessment of whether non-thermal electrons were present in this microflare. Finding that particle acceleration was indeed present, we then used the OSPEX fitting tool for the rest of the analysis. OSPEX allows to directly fit a thick-target non-thermal electron distribution to the X-ray data. NuSTAR non-thermal flares are assumed to be thick-target sources in that all electrons presumably lose their suprathermal energy to collisions with the ambient plasma within the observation region.

The best fit statistic is achieved by the model that includes a non-thermal component, indicating the presence of accelerated electrons. A double thermal fit was also considered, but the double thermal model requires the higher temperature plasma component to have a superhot temperature of ∼ 50 MK, a temperature that has only been observed in large M or X class flares (Caspi et al., 2014).

The OSPEX thermal plus non-thermal model results are shown in Figure 9, and their parameter results are listed in Table 1. Both a cold thick target model (background + vth + thick2) and a warm thick target model (background + thick warm) were fit. The warm thick target fit did not require an additional thermal component, and the plasma temperature and density were set as free parameters. The fit parameters obtained by the warm thick target model closely resemble those achieved using the cold thick target model, affirming the use of the cold thick target model as an accurate approximation of this flare.

Thermal and non-thermal energies are calculated for the cold thick target model. The thermal energy may be calculated by assuming an isothermal plasma for each vth component. The thermal energy may be up to a few times greater than this isothermal approximation due to cooler components NuSTAR is not sensitive to (Aschwanden et al., 2015). The thermal energy is given by. U T = 3 k B T EM f V [ erg ] (1) U T ≈ 1 × 1 0 28 [ erg ] (2) with the emission measure EM and temperature T taken from the vth fit. The filling factor f is assumed to be unity. The thermal energy is an order of magnitude estimate. For the thick warm calculation, EM f V is replaced with nV, with n representing the emission measure. The volume V was calculated from the AIA 94 Å images, solving for the area and converting to volume via A ^3/2 = V. The AIA 94 Å data was cropped to the region used for spectroscopy, and thresholded to only the flaring region by subtracting the 18:40 UT background.

The non-thermal energy is found by multiplying the non-thermal power by the spectral observation window, in this case 2 minutes. If the electron spectrum F(ϵ) above the cutoff energy is assumed to be a power law of index δ, the non-thermal energy for the vth + thick2 model may be calculated as. U N E > E C = P E > E C Δ t = ∫ E C ∞ F ϵ E d E Δ t ≈ 1.6 × 1 0 − 9 δ − 1 δ − 2 N E C Δ t [ erg ] (3) U N E > E C = 2 . 3 − 0.9 + 0.9 × 1 0 28 [ erg ] (4) with N representing the number of electrons per second, δ the electron spectral index, and E _C the cutoff energy—all parameters fit by OSPEX.

The IRIS observations provide crucial information about the response of the plasma to the energy release in the lower atmosphere. At the same time, NuSTAR observations provide direct constraints on the energy distributions of the accelerated electrons in the corona. In order to find evidence for the physical mechanisms driving the flare, we run hydrodynamic simulations using the RADYN code (e.g., Carlsson and Stein, 1992; Allred et al., 2005; 2015; Carlsson et al., 2015), which solves the equation of radiative hydrodynamics on a 1-dimensional (1D) adaptive grid (Dorfi and Drury, 1987). A key property of RADYN is the ability to perform non-local thermodynamic equilibrium (non-LTE) radiative transfer for species which are important for chromospheric energy balance (e.g., H, He and Ca). Other atomic species are included as background continuum opacity sources (in LTE) using the Uppsala opacity package (Gustafsson, 1973). The radiative losses for optically thin lines are calculated using the CHIANTI 7.1 (Dere et al., 1997; Landi et al., 2013) database assuming ionization and thermal equilibrium.

RADYN simulates flare heating assuming different possible physical mechanisms: accelerated non-thermal electrons streaming from the corona to the chromosphere, whose propagation is treated using the Fokker-Planck equation (Allred et al., 2015); in-situ heating in the corona and consequent energy transport to the lower atmosphere via thermal conduction; or dissipation of Alfvén waves (Kerr et al., 2016). Since we have direct observations of accelerated electrons from NuSTAR, here we focus on the first mechanism.

We run simulations which covered a range of electron beam parameters and initial conditions of the flare loops, as summarized below.

• Energy flux (F) = 3 ⋅ 10⁸–5 ⋅ 10⁹ ergs s⁻¹ cm⁻² (3F8–5F9)

We use values of E _C and δ which are close to those provided by the NuSTAR spectral analysis, within uncertainties. We simulate a broader range of values for the energy flux, since this parameter is not completely constrained by NuSTAR, as it depends also on the area over which the electrons deposit their energy (as discussed in Sect. 6). We use the same “plage-like” atmospheres presented in Polito et al. (2018), and we also assume half-loop lengths of 15 Mm. Our choice of loop length and initial temperatures is motivated by the larger parameter studies presented in Polito et al. (2018) and Testa et al. (2020). In particular, these studies have demonstrated that simulations with different loop lengths and the same initial temperature provide very similar trends (e.g., Figure A1 of Polito et al., 2018). Testa et al. (2014) and Polito et al. (2018) also showed that using hotter and denser initial loop atmospheres (e.g., 5 MK loop with apex density of ≈ 10¹⁰ cm³) results in less heating of the lower atmosphere for smaller events such as this B1 flare under study. Finally, given the uncertainty in the duration of the blue shifts in individual IRIS pixels due to the relatively long raster cadence of this observation, we assume here a heating duration of 10 s for simplicity, and we refer to future work for a more extended investigation of the effects of longer duration heating on the models (e.g., Testa et al., 2020; Cho et al., 2023).

We synthesize the emission of the IRIS Si iv spectral line using the values of density, temperature, and bulk velocity at each grid point and timestep from the RADYN simulations and atomic data from CHIANTI v.10 (Dere et al., 1997; Del Zanna et al., 2021), assuming photospheric abundances (Asplund et al., 2009) and equilibrium ionization. We follow Eq 1 of Polito et al. (2018) to convert the synthetic spectra to units of DN s⁻¹ pixel⁻¹. The time-velocity spectra described in Sect. 5.3 are then obtained by integrating the synthetic emission in each RADYN grid point along the loop as a function of time, assuming an exposure time of 1s and taking into account the instrumental broadening of 0.026 Å.

To synthesize the optically thick Mg ii and C ii spectra, we provide the input from our RADYN flare atmospheres (temperature, electron density, bulk velocity, hydrogen atomic level populations) as a function of time to the radiation transport code RH15D (Pereira and Uitenbroek, 2015). RH15D solves the equation of non-LTE radiation transport and atomic level populations and allows us to take into account the effects of partial redistribution (PRD), which can be important for the Mg ii lines (Leenaarts et al., 2013a). The NLTE radiation transport equations were solved for H, Mg ii and C ii, with additional species solved in LTE as sources of background opacity. We add a microturbulence of 7 km s⁻¹ as an additional source of line broadening mechanism in the chromosphere, consistent with the values reported in Carlsson et al. (2015), as well as a recent study by Sainz Dalda and De Pontieu (2022) based on inversions of IRIS Mg ii profiles during flares. The synthetic Mg ii and C ii NUV spectra were converted to IRIS count rates using the same procedure described in Polito et al. (2018); Testa et al. (2020); Polito et al. (2023).

Figure 10 (top panel) shows Si iv synthetic spectra as a function of velocity (x-axis) and time (y-axis) for different RADYN flare simulations, assuming an initial apex temperature of 1 MK, E _C = 8keV, δ = 10, and different values of energy flux, according to the legend. The vertical dotted white lines indicate the position of the line at rest (assuming the reference wavelength available in CHIANTI v.10), and negative values here indicate blueshifts. The time-velocity plots show that when the energy flux is below 5 ⋅ 10⁸ ergs s⁻¹ cm⁻² (5F8), the line is mostly blueshifted over time. On the other hand, as the flux energy increases, the line becomes more at-rest or red shifted. As described in detail in Polito et al. (2018), this different behavior is due to the fact that, in case of more gentle fluxes, the electrons can deposit their energy below the height formation of Si iv–and thus driving the Si iv evaporation–for a longer time. In these simulations it takes longer for the loop density to rise, meaning that the electrons can deposit energy in the lower atmosphere for a longer time. In fact, when the density is high enough, the electrons get stopped at higher heights and eventually drive a downflow of Si iv plasma. This shift from upflow to downflow happens much quicker in simulations with stronger energy flux (see Polito et al., 2018, for more details).

Figure 10 (bottom panels) shows Si iv synthetic spectra for models with an initial apex temperature of 3 MK. As also discussed in Polito et al. (2018), this hotter initial atmosphere is also denser (with apex density 10^9.6 cm⁻³ compared to 10^8.7 cm⁻³ for the 1MK loop), meaning that the electrons need comparatively higher energy to be able to heat the transition region and therefore drive an increase of intensity and brightenings in the Si iv line. This phenomenon is illustrated in Figure 10A), which shows that an energy flux of at least 1.2 ⋅ 10⁹ ergs s⁻¹ cm⁻² (1.2F9) is needed to drive a response in the Si iv line. These results demonstrate that in order to reproduce the Si iv blueshift, one needs a certain combination of parameters for the electron beam distribution, which will also depend on the initial conditions.

Figure 11 shows a summary of Si iv Doppler shift velocities based on moment calculation for different models. Panel a) summarizes the Si iv Doppler shift velocity at the maximum intensity during the simulation, while panel b) shows the maximum blue shift velocity during the simulation. For both panels, we only consider spectra where the Si iv line is above a detection threshold of 10 DN for the total intensity of the line. For some simulations there is no associated data point in Figure 11B), which means that the Si iv line does not exhibit blue shifts in the detectable spectra. Figure 11 also shows that, in order to reproduce a significant (of 10 km s⁻¹ or larger) Si iv blueshift for the range of E _C obtained from the NuSTAR observations, including uncertainties (i.e. 7–9 keV), we need an energy below ≈8 ⋅ 10⁸ ergs s⁻¹ cm⁻² (8F8) for the 1MK loop or ≈5 ⋅ 10⁹ ergs s⁻¹ cm⁻² (5F9) for the 3MK loop. Further, we note that for the 5F9 model, the maximum blueshift does not occur in the brightest spectrum of the simulation. Finally, Figure 11B) shows the impact of the initial temperature (and density) of the loop atmosphere on the magnitude of the Doppler shifts, with stronger Si iv blueshift values being observed in simulations with an initial lower temperature loop. This behavior was also discussed in Polito et al. (2018) and suggests that, although the observed values of blueshifts are slightly larger (≈-25–30 km s⁻¹) than the maximum blueshifts observed in the simulations with the 3MK loop, this might be due to some extent to the details of the initial physical conditions of the loops.

Previous work has shown that studying the response of the atmosphere to the flare heating by combining IRIS spectral lines formed at different heights from the chromosphere to the transition region, we can obtain even stricter constraints on the models (Polito et al., 2018; Testa et al., 2020). With this motivation in mind, we investigate in more detail the behavior of the chromospheric lines for those models where we succeed in reproducing the Si iv blueshifts. Figure 12 (top panels) shows synthetic spectra for the Si iv (Panel a), C ii (Panel b), Mg ii (Panel c) and Mg ii triplet (Panel d) lines as a function of velocity and time for the 5F8 model with E _C = 8 keV, δ = 10 and initial loop apex temperature of 3MK. The bottom panels show the same quantities for the 2.5F9 flare model with E _C = 8 keV, δ = 10 and initial loop apex temperature of 3MK. Figure A2 in the Appendix also show the synthetic spectra of the IRIS lines for the 1.2F9 model and 3 MK loop. While these three models can to some extent explain the observed Si iv blueshifts, they are characterized by different behaviors in the synthetic spectra of the chromospheric lines. In particular, the Mg ii k and triplet chromospheric lines in the simulations with the 1 MK loop are characterized by a deeper central reversal (Figure 12, top panels c) and d)). A less pronounced center reversal (which is still deeper than that in the observed spectra) is also seen in the 1.2F9 model and 3 MK loop (Figure A2). On the other hand, the chromospheric spectra in the 2.5F9 flare model with the 3MK loop (Figure 12, bottom panels c) and d)) more closely resemble the key characteristics of the observed spectra. One main difference that remains is that, in all of the three models, the C ii line appears to be red shifted or stationary, while in the observations the line is blueshifted. Another difference between models and observations is in the values of line width, which are smaller in the models than the observations. Such discrepancies are well-known and were discussed in detail in e.g. Testa et al. (2020), Cho et al. (2023). In particular, these previous studies pointed out the difficulty associated with comparing the results of a single loop model with observed spectral lines in one IRIS pixel, where heating episodes from many individual loop strands (possibly also with different physical conditions) might overlap. In addition, physical mechanisms such as turbulence and waves, which are missing from our models, may contribute to the discrepancy between the observed and simulated line widths.

In summary, despite their limitations, the models with an initial temperature of 3MK and energy flux in the range of 1.2–2.5F9 are capable of reproducing qualitative observational characteristics of the flare spectra (e.g., Si iv blueshift, lack of Mg ii k deep central reversal and presence of increased Mg ii triplet emission). We note that the initial higher temperature for the flare loops is more consistent also with the background temperature observed by NuSTAR of ≈ 4MK.

The comparison discussed here demonstrates how the models can provide a useful tool to constrain the properties of the heating. In order to compare the simulations with the results of the NuSTAR X-ray spectral analysis, we need to estimate the area over which the electrons might deposit their energy in the lower atmosphere. In Sect.6 we discuss how to derive this value using magnetic field extrapolations, while in Sect. 7 we compare both UV and X-ray observations with the predictions of the models and summarize our conclusions.