Solar flares are complicated phenomena that cover a broad range of heights in the solar atmosphere and produce electromagnetic emission in a broad range of wavelengths. The flares occur basically due to the sudden magnetic reconnection processes in the solar corona, which result in plasma heating, acceleration of charged particles, etc. (e.g., Benz and Güdel, 2010; Emslie et al., 2012).

To obtain a comprehensive picture of a flare, we need observations in different spectral ranges: e.g., the hard X-rays and white-light and ultraviolet (UV) continuum emissions are produced by non-thermal electrons mainly in the chromosphere at the footpoints of the coronal flaring loops; in the corona, the same electrons produce the microwave continuum emission due to the gyrosynchrotron mechanism; the soft X-rays and extreme ultraviolet (EUV) emission reflect the dynamics of the hot thermal plasma in the corona. Both the evolution of the spatially resolved images and the delays between the emissions at different wavelengths can reflect the dynamics of acceleration and transport of the non-thermal particles (e.g., Aschwanden, 2002). The recent commissioning of such solar-oriented astronomical instruments as the Siberian Radioheliograph (SRH, Altyntsev et al., 2020) and the Hard X-Ray Imager on board the Advanced Space-based Solar Observatory (ASO-S/HXI, Su et al., 2019; Gan et al., 2023), which provide imaging spectroscopy observations in the microwave and hard X-ray ranges, respectively, offers new opportunities to study the solar flares.

In addition to the multiwavelength observations, understanding the nature of solar flares requires data-constrained modeling, which enables us to estimate the physical parameters in the flaring regions and to link the observed phenomena with the underlying processes of energy release and particle acceleration and transport. The recent advances in this field include, e.g., the case studies by Kuznetsov and Kontar (2015), Kuroda et al. (2018), Fleishman et al. (2018), Fleishman et al. (2021b), Fleishman et al. (2023), where the 3D structures of flares were reconstructed, and the spatial and energy distributions of energetic electrons and their dynamics were determined.

Here we investigate the GOES M5.8 class solar flare that occurred on 2023-03-06, at ∼02:15–03:30 UT; it was one of the first strong flares observed by the SRH and ASO-S/HXI, as well as by other instruments. We present the results of observations and 3D modeling, and analyze the factors affecting the transport of non-thermal electrons.

The microwave images of the flare were obtained using the Siberian Radioheliograph (SRH, Altyntsev et al., 2020). This instrument consists of three independent antenna arrays, two of which (for the frequency bands of 2.8–5.8 and 5.8–11.8 GHz) were operable at the considered date, thus providing imaging observations with the spatial resolutions of 15″− 30″and 12″− 24″, respectively. The observations were performed at 16 equidistant frequencies in each frequency band, i.e., at 32 frequencies in total, with the time resolution of ∼ 3 s. The flux calibration was performed using the estimated microwave flux from the quiet-Sun regions (Zirin et al., 1991). To obtain a better alignment of the microwave images with magnetograms and images in other spectral ranges, we also performed simulations of the thermal gyroresonance emission from a non-flaring active region (AR 13245) just before the considered flare (at 02:10 UT) using the GX Simulator code (Nita et al., 2018; Nita et al., 2023; see also Fleishman et al., 2021a), and determined the position deviations between the observed and synthetic microwave images; the shifts needed to remove those deviations were then applied to all observed images throughout the flare. In addition to the imaging observations, we used the spatially unresolved measurements by the Nobeyama Radiopolarimeters (NoRP, Shimojo and Iwai, 2023), Palehua station of the Radio Solar Telescope Network (RSTN ¹ ), and Chashan Broadband Solar millimeter spectrometer (CBS, Shang et al., 2022; Shang et al., 2023) in the microwave range at a number of frequencies from 1 to 40 GHz.

The initial stage of the considered flare (until ∼ 02:32 UT) was observed also by the Hard X-Ray Imager on board the Advanced Space-based Solar Observatory (ASO-S/HXI, Su et al., 2019; Gan et al., 2023). This instrument provides imaging spectroscopy observations of the solar X-ray emission in the energy range of ∼ 10 − 400 keV with a spatial resolution down to about 3.2″ at 30 keV. After ∼ 02:32 UT, the X-ray data from the ASO-S/HXI became unreliable due to a strong parasite signal caused by the radiation belt particles. More continuous (but spatially unresolved) hard X-ray data were provided by the Konus-Wind spectrometer on board the Wind spacecraft (KW, Lysenko et al., 2022). In the considered event, this instrument operated in the waiting mode and recorded the X-ray count rates in the energy ranges of 19–78, 78–323, and 323–1,280 keV with the time resolution of ∼ 3 s. The spatially unresolved soft X-ray data were provided by the Geostationary Operational Environmental Satellite (GOES ² ).

In addition to the above observations, we used the data from the instruments on board the Solar Dynamic Observatory: UV and EUV images from the Atmospheric Imaging Assembly (SDO/AIA, Lemen et al., 2012) and magnetograms from the Helioseismic and Magnetic Imager (SDO/HMI, Scherrer et al., 2012). All instruments used in this study are summarized in Table 1.

The considered GOES M5.8 class solar flare occurred on 2023-03-06 in the active region AR 13243 near the western solar limb, at N18W64. Figure 1 demonstrates the light curves of the flare at several selected microwave frequencies and X-ray energy ranges. The SRH light curves represent the microwave fluxes integrated over the 200″× 200″ area centered at the flare. As has been said above, the reliable ASO-S/HXI data are available only before ∼ 02:32 UT; no flare-related X-ray signal above 50 keV has been detected during that time interval. No flare-related X-ray signal in the KW 323–1,280 keV channel has been detected as well.

From the light curves, one can notice that the flare actually consisted of two separate (but closely related) flaring events, separated by the vertical thick dashed grey line in Figure 1. The first part of the flare (before ∼02:33 UT) was mostly “thermal” (cf. Fleishman et al., 2015, and references therein): the X-ray spectrum was relatively soft, with no significant flux above ∼ 50 keV, but relatively high fluxes at lower energies. The microwave emission demonstrated a good correlation with the hard X-rays above ∼ 20 keV (which indicates its non-thermal origin), but was relatively weak. The GOES soft X-ray flux was sufficiently high, too, and demonstrated a noticeable delay with respect to the non-thermal emissions.

In the second, “cooler” part of the event (after ∼ 02:33 UT), the X-ray spectrum became considerably harder, with the KW 78–323 keV flux considerably higher, but the KW 19–78 keV and GOES 1–8 Å fluxes lower than during the first part of the flare. The non-thermal microwave emission, too, reached much higher intensities than during the first part of the flare. One can distinguish the impulsive phase of the flare (∼02:34–02:43), which was characterized by a prominent hard X-ray emission with multiple local peaks, corresponding likely to separate acts of magnetic reconnection. The microwave emission demonstrated firstly a similar dynamics with multiple peaks (especially at the frequencies of ∼ 10 GHz) corresponding to the hard X-ray peaks; however, in contrast to the hard and soft X-rays, the microwave emission demonstrated also an overall increasing trend likely caused by a gradual accumulation of energetic particles in the flaring loop(s). The microwave emission reached a maximum at ∼ 02:43 UT. After that, the hard X-ray emission dropped rapidly to the background level, and the microwave and soft X-ray emissions demonstrated a gradual decay that lasted for up to ∼ 50 min.

A notable feature of the considered event was a filament eruption that occurred immediately before the flare. Figure 2 demonstrates a sequence of the SDO/AIA 335 Å EUV images of the flaring region. The eruption started at ∼ 02:11 UT, i.e., well before the brightenings in the microwave and X-ray ranges. At 02:19:01 UT, when the first microwave and hard X-ray peak was observed, the filament had already risen up to a height of about 20,000 km. We have found no correlation between the filament parameters and the microwave and X-ray emissions. Therefore, although the filament eruption could trigger the magnetic reconnection and thus initiate the flare, at later stages (after the trigger) the evolutions of the flare and the filament likely diversified and became independent of each other.

To model the microwave emission of the considered flare, we used the GX Simulator code (Nita et al., 2015; 2023). This code allows one to create a 3D magnetic field model of the flaring region using the nonlinear force-free field extrapolation, to select a flaring loop (or loops), to fill the flaring loop(s) with thermal and non-thermal electrons, and to compute the corresponding gyrosynchrotron and free-free microwave emission using the “fast gyrosynchrotron codes” by Fleishman and Kuznetsov (2010); Kuznetsov and Fleishman (2021). For comparison with the observations, the computed microwave emission maps were then convolved with the SRH beam.

As has been said above, during the first part of the flare, the emission was likely produced in an arcade consisting of multiple flaring loops. A model of such a structure would have too many free parameters. In addition, the lack of microwave data in the optically thin frequency range does not allow us to constrain reliably the spectrum of the emitting electrons; therefore, we do not consider that time interval here. In contrast, during the second part of the flare, the observed structure of the emission sources could be described reasonably well by a single-loop model. Based on the available images in the microwave, UV, and EUV ranges, we selected the flaring loop shown in Figure 8 that provided the best agreement with the observations. The loop had the length of 53,000 km and rose up to the height of 17,000 km; the magnetic field strength (at the loop axis) varied from 170 G at the loop top up to 1,600 and 1,380 G in the north-eastern and south-western footpoints, respectively. By analogy with a number of previous simulations (e.g., Kuznetsov and Kontar, 2015; Kuroda et al., 2018; Fleishman et al., 2021b; 2023; Wu et al., 2024), in order to reduce the number of free parameters, the magnetic structure of the loop (determined by the selected axial magnetic field line) was assumed to be the same at all times throughout the impulsive phase of the flare, and only the parameters of the energetic electrons varied.

For the energetic electrons, we adopted a single power-law energy distribution function in the form of f(E) ∝ E ^−δ , with the electron energy E in the range from 0.01 to 10 MeV, and the electron number density equal to n _b; the pitch-angle distribution was assumed to be isotropic. The spatial distribution of the energetic electrons within the flaring region was described by the model function in the form of n b r , s = n b 0 ⁡ exp − r 2 2 σ r 2 s exp − s − s 0 2 2 σ s 2 , where s and r are the coordinates along and across the selected flaring loop, respectively, with the coordinate s measured relative to the loop top and positive in the direction towards the north-eastern footpoint, and the coordinate r measured relative to the loop axis; σ _s and σ _r are the characteristic scales of the distribution in the respective directions, and n _b0 is the peak electron number density. Following the magnetic flux conservation, the transverse scale σ _r0 varied along the loop as σ r ( s ) / σ r 0 = B 0 / B ( s ) , where B ₀ and σ _r0 are the magnetic field strength and the transverse scale σ _r at the loop top, and B(s) is a local magnetic field strength.

We note that the above model is oversimplified and accounts for only the basic characteristics of the energetic electrons in the flaring region. Therefore, our aim was to reproduce: a) the total (spatially integrated) microwave emission spectra of the flare, primarily in the optically thin frequency range (above ∼ 10 GHz), and b) the 2D locations of the microwave source peaks, as well as the microwave brightness distributions along the flaring loop at high frequencies (namely, at 11.80 GHz). The model parameters that provided the best agreement with the observations at four different times (corresponding to the microwave emission peaks) are presented in Table 2, while the corresponding synthetic and observed images and spectra are shown in Figure 9.

From Table 2 and Figure 9, one can notice that the energetic electrons in the considered event (during the second part of the flare) were likely concentrated near the top of the flaring loop. Even at 02:34:56 UT, the estimated displacement of the electron distribution from the loop top s ₀ was relatively small, while the observed emission was concentrated near a footpoint due to a stronger magnetic field there. With time, the electron distribution peak approached gradually the loop top (s ₀ decreased). The energetic electron distribution along the flaring loop broadened gradually with time, and the effective thickness of the loop increased as well (both σ _s and σ _r increased). The spectral index of the energetic electrons δ varied with time, in agreement with the estimations based on the observed microwave spectral index δ _MW (see Figure 7) and the empirical formula by Dulk and Marsh (1982). The parameter n _b0 is not representative, because it depends strongly on the low-energy cutoff of the electron distribution, which was not reliably known and was chosen arbitrarily. A more reliable characteristic is the total number of energetic electrons at high energies, say, above 1 MeV. According to Table 2, this number (N _{1–10 MeV}) increased gradually with time during the impulsive phase of the flare, which reflected the process of accumulation of the energetic electrons in the flaring loop.

The presented observations offer some clues into the non-thermal particle transport processes in the considered flare. During the first part of the flare, we have found no significant delays between the microwave and hard X-ray emissions, which indicates that the energetic electrons injected into the flaring loops then precipitated into the chromosphere nearly immediately, without experiencing a noticeable trapping. Such a behaviour is quite consistent with the above-described scenario suggesting that during the first part of the flare, the energy release processes occurred in an arcade of relatively short flaring loops with low mirror ratios.

On the other hand, during the second part of the flare, the microwave emission peaks were delayed with respect to the X-ray ones by up to ∼ 10 − 30 s. These delays were much longer than any expected time-of-flight delays, and could be naturally attributed to the particle trapping and accumulation processes, given that the energy release likely occurred in a relatively long flaring loop with the mirror ratio of up to ∼ 10 . The evolution of the trapped energetic particles in coronal magnetic tubes is governed primarily by their pitch-angle scattering due to Coulomb collisions or/and interaction with a magnetohydrodynamic turbulence, when the particles scattered into the loss cone escape from the trap (are precipitated in the chromosphere). Thus the observed hardening of the microwave emission spectrum (and hence of the energy spectrum of the trapped energetic electrons in the flaring loop) with time in the absence of the particle injection, see Figure 7, indicates that the lower-energy electrons were scattered more efficiently in the considered event, which favours the Coulomb collisions as the dominant scattering factor. In turn, the intermittent acts of energy release (highlighted by the hard X-ray emission pulses) resulted in injection of additional portions of electrons with a softer energy spectrum into the flaring loop, and this interplay between the particle injection and escape processes formed the observed hard-soft-hard pattern in the optically thin spectral index of the microwave emission. At the same time, the total number of the high-energy electrons increased more-or-less steadily with time until the end of the impulsive phase of the flare.

The observed delays between the hard X-ray emission peaks at higher and lower energies (see Figure 7) are consistent with the above scenario, because the hard X-ray emission is produced by the precipitating electrons that escaped from the magnetic trap. As has been said above, in the considered event, the lower-energy trapped electrons were scattered into the loss cone more efficiently (due to the Coulomb collisions), and therefore escaped from the trap and reached the loop footpoints first, then followed by the higher-energy electrons. This explanation implies that the energetic electrons were injected somewhere in the coronal part of the flaring loop, with an isotropic pitch-angle distribution or preferably in the direction across the local magnetic field, so that a significant fraction of the particles became trapped immediately after the injection. The suitable particle acceleration mechanisms include, e.g., stochastic acceleration or acceleration in a collapsing magnetic trap (e.g., Zharkova et al., 2011).

The delay between the hard X-ray emission pulses at the energies of E ₂ and E ₁ can be estimated as Δt ≃ τ(E ₂) − τ(E ₁), where τ is the characteristic electron scattering time due to the Coulomb collisions, given, e.g., by Eq. 12.5.11 in the monograph of Aschwanden (2005). For the energies of E ₁ = 20 keV and E ₂ = 80 keV, the observed delays of Δt ≃ 5–7 s would occur due to scattering in a thermal plasma with the density of about 3 × 10¹⁰ cm⁻³ and the temperature of a few MK. The obtained plasma density value seems to be typical of the coronal flaring loops.

We presented the results of observations and simulations of a M5.8 class solar flare that occurred on 2023-03-06 near the north-eastern solar limb. The flare was observed by a number of instruments, including the Siberian Radioheliograph, Nobeyama Radiopolarimeters, and Radio Solar Telescope Network in the microwave range, Hard X-ray Imager on board the Advanced Space-based Solar Observatory and Konus-Wind on board the Wind spacecraft in the hard X-ray range, and the Solar Dynamic Observatory in the optical, ultraviolet, and extreme ultraviolet ranges. The main results can be summarized as follows:

• The flare consisted of two separate flaring events. The first part of the flare was mostly “thermal,” with a relatively soft spectrum of energetic particles and weak microwave emission. During the second part of the flare, the spectrum of energetic particles was much harder and the microwave emission was much stronger than during the first part.