2. The Source of Extreme Flare Energies

The accelerated electrons (and other types of particles such as protons and ions) are generally thought to be produced during the process of magnetic reconnection in the corona. Figure 1 illustrates the context (Kazachenko et al. Reference Kazachenko, Albelo-Corchado and Tamburri2022) in which magnetic reconnection occurs below an erupting filament, which can develop into a coronal mass ejection. In the wake of the eruption, oppositely directed field lines pinch together due to an imbalance of pressure and tension forces. Reconnection produces fast outflow jets directed away from and toward the solar surface. In the reconnection region (middle; Mei et al. Reference Mei, Shen and Wu2012) and throughout the outflow, the particles may gain energy through a myriad of mechanisms (right; Ji et al. Reference Ji, Daughton and Jara-Almonte2022). To name a few, there could be curvature-drift acceleration in magnetic islands (regions of pinched off field regions, which manifest as flux ropes in 3D; Dahlin et al. Reference Dahlin, Drake and Swisdak2014, Reference Dahlin, Drake and Swisdak2017), stochastic acceleration in turbulence within the fast outflow jets (Larosa & Moore Reference Larosa and Moore1993), and shock drift acceleration at a termination shock that forms as the outflow rams into the previously reconnected flare loops below (Mann et al. 2009).

Figure 1.

Composite graphic that illustrates the range of scales and processes involved in particle acceleration in solar flares. Magnetic reconnection occurs in the wake of an erupting filament that straddles a magnetic polarity inversion line (left; from Reference Kazachenko, Albelo-Corchado and TamburriKazachenko et al. (2022)). The reconnecting current sheet breaks up into a number of plasmoids (mid; from Mei et al. (Reference Mei, Shen and Wu2012)). Particle acceleration can occur in a myriad of ways around an X-point (right; from Reference Ji, Daughton and Jara-AlmonteJi et al. (2022)). Efficient acceleration may occur in a series of plasmoids, in the turbulent outflow jets, and at a termination shock lower in the corona. Figures reproduced with permission from the authors.

Figure 1 Long description

Panel A: A diagram showing magnetic reconnection in the wake of an erupting filament. Key components include overlaying field lines, an expanding flux rope, flare ribbon, and secondary dimmings. Panel B: A diagram depicting the reconnecting current sheet breaking up into a number of plasmoids. Panel C: A diagram illustrating particle acceleration around an X-point with magnetic islands/plasmoids and turbulent acceleration and shock acceleration.

The sources of particle acceleration are macroscopic electric fields. The electric field is given by the change in the magnetic field as it pinches off: $\frac{\partial B}{\partial t} = -c \nabla \times E$ . Simulations show that the out-of-plane (z-directed) reconnection electric field can be decomposed into three terms (Huang & Bhattacharjee Reference Huang and Bhattacharjee2016)

(1) \begin{equation} \bar{E}_z = -(\bar{v} \times \bar{B})_z -\overline{(\tilde{v} \times \tilde{B})}_z + \eta \bar{J}_z\end{equation}

which correspond to the convective field due to the bulk (average) flow (e.g., in a jet, in contracting ends of magnetic islands, or in a termination shock), the convective field due to turbulent motions, and the direct electric field at an X-point of reconnected field lines. The plasma heating goes into both increasing the thermal energy and producing power-law tails. The heating is related to the electric field by $Q = J \cdot E$ . In the case of the first term of Eq. 1, which could conceivably correspond to the velocity of retracting ends of a plasmoid, a curvature drift current density, $J_{\rm{curv}}$ , is generated as particles zip around the ends of a plasmoid. This is a type of first-order Fermi acceleration (Drake et al. Reference Drake, Swisdak and Che2006).

Observations have not clearly discriminated between the various proposed mechanisms of particle acceleration in solar flares. I suggest that one of the best strategies is to leverage the observational effects of the reconnection electric field on the surrounding plasma. Namely, the hydrogen lines broaden dramatically due to the Stark effect in the presence of an electric field. Casini & Foukal (Reference Casini and Foukal1996) propose novel observational targets: the $12-8$ and $15-9$ transitions of hydrogen around $\lambda = 11 \mu$ m. These transitions show strong ( $\approx$ 20%) linear polarization in the presence of electric fields of 10 V cm $^{-1}$ , which is the same order as in reconnection models (Lin & Forbes Reference Lin and Forbes2000). However, it is not clear if all of the hydrogen is completely ionized in the reconnection region, thus causing the lines to be very faint. There have been recent efforts to observe reconnection fields with optical hydrogen lines during reconnection events within the lower atmosphere (Anan et al. Reference Anan, Casini and Uitenbroek2024).

A major milestone in 2D simulations of the reconnecting region has been the production of a power-law tail over several orders of magnitude (Drake et al. Reference Drake, Arnold and Swisdak2019; Arnold et al. Reference Arnold, Drake and Swisdak2021). In these simulations, $\approx 2-30$ % of the background plasma electron population is accelerated into power-law tails. Going forward, a closer look at how the predictions for the extreme tails compare to observations of hard X-rays could facilitate further advancements. In the simulations, the highest energy electrons form within $5-10$ Alfven times, where the Alfven time is $\approx 3$ s for a length scale of $L=10^9$ cm, a magnetic field of $B=100$ G, and an ambient density of $n_e = 5\times 10^9$ cm $^{-3}$ . These representative values give timescales of $15-30$ s to accelerate the highest energy electrons. This is longer than the shortest timescales of tens to hundreds of milliseconds in hard X-ray ( $E=29-278$ keV) light curves seen in about 10% of large events (Kiplinger et al. Reference Kiplinger, Dennis and Frost1983; Aschwanden et al. Reference Aschwanden, Kliem and Schwarz1998). Further, energy-dependent hard X-ray time delays place constraints on the acceleration time to be much shorter than the propagation time for the highest energy electrons (Aschwanden Reference Aschwanden1996), which must arrive at the chromosphere first.

3. Chromospheric Footpoints: The Fate of Nonthermal Particle Kinetic Energy

The kinetic energy of the accelerated electrons is transferred to the chromospheric plasma through Coulomb collisions, thus causing an explosive temperature increase, mass motions, and density changes throughout many atmospheric layers. Recent high-resolution images in the optical, combined with hard X-ray data from RHESSI, have led to inferences of enormous flux densities of electron beams of $10^{12}-10^{13}$ erg cm $^{-2}$ s $^{-1}$ (Krucker et al. Reference Krucker, Hudson and Jeffrey2011). These extreme values are surprising because particle-in-cell simulations suggest that instabilities should affect beam propagation through the corona (Lee & Büchner Reference Lee and Büchner2011). The beams should decelerate close to the acceleration site rather than propagate to the chromosphere. Nevertheless, we observe strong nonthermal hard X-ray chromospheric sources.

There have been efforts to measure the beam densities in the corona, but the results systematically vary by nearly a factor of 100. Fleishman et al. (Reference Fleishman, Nita and Chen2022) infer the beam density to be $10^{10}$ cm $^{-3}$ from radio observations, but Kontar et al. (Reference Kontar, Emslie and Motorina2023) leverage HXR observations to argue for values that are two orders of magnitude smaller in the same event. Krucker & Battaglia (Reference Krucker and Battaglia2014) suggest large beam densities, while Oka et al. (Reference Oka, Krucker and Hudson2015) apply a different model to arrive at about 5x smaller density for the same flare. Volpara et al. (Reference Volpara, Massa and Krucker2024) trace the systematic changes in the particle distribution along the coronal loops for the first time, suggesting that beams do indeed propagate, but the beam densities they infer are also relatively low. There is clearly a lot to understand about the conditions when energy propagation becomes problematic for the standard “dilute beam” paradigm. New 3D MHD models of flares and eruptions have arrived, but these also show that thermal flux densities reach levels around $10^{12}$ erg cm $^{-2}$ s $^{-1}$ or larger (Rempel et al. Reference Rempel, Chintzoglou and Cheung2023). The electrodynamics of particle beams is certainly a problem that needs to be addressed in more detail in the future.

3.1. The Thermal Response of the Chromosphere

Although we cannot measure the electric fields that are responsible for particle acceleration, we can observe the chromospheric thermal response in optical and ultraviolet spectra. In stellar flares, the continuum response is sensitive to the beam densities and energies that impact and heat the chromosphere. Figure 2 shows several radiative-hydrodynamic flare models (Kowalski et al. Reference Kowalski, Allred and Carlsson2024) of the flaring chromosphere. High energy electron beams that heat the lower chromosphere to $T \approx 10^4$ K generate the dark blue spectrum, while lower energy beams heat the mid to upper chromosphere to $T \approx 10^4$ K produce the light blue spectrum. These model spectra are remarkably different: the dark blue spectrum rises in the optical toward shorter wavelengths where there is a small Balmer jump, while the light blue spectrum is flat in the optical and has a large Balmer jump (the peaks are also notably displaced within the near-UV).

Figure 2.

RHD model spectra calculated with the RADYN code (Carlsson & Stein Reference Carlsson and Stein1995), illustrating the dramatic differences between heating to $T \approx 10^4$ K in the low and upper chromosphere. Figure reproduced from Kowalski et al. (Reference Kowalski, Allred and Carlsson2024) with permission from the authors.

Many spectral observations in the impulsive phase of flares from active M dwarfs (e.g., Kowalski et al. Reference Kowalski, Hawley and Wisniewski2013) are much more consistent with the dark blue model spectrum in Figure 2. They show a rise in the blue-optical and a smaller Balmer jump than predicted by heating the upper chromosphere with fluxes of electron beams that one would expect in solar flares. Previous observations from broadband photometry have constrained the peak to the near-UV and U-band (Hawley & Fisher Reference Hawley and Fisher1992; Hawley et al. Reference Hawley, Allred and Johns-Krull2003). However, detailed spectra in the NUV have rarely been obtained (Kowalski et al. Reference Kowalski, Wisniewski and Hawley2019) to show the expected peak and turnover of the continuum radiation toward shorter wavelengths.

The properties of NUV flare radiation are also important beyond flare physics. They are critical in assessing the habitability and biosignatures of exoplanets around cool stars (Segura Reference Segura2018). Kowalski et al. (Reference Kowalski, Osten and Notsu2025) presents the first results from 81 orbits of NUV stellar flare monitoring with the Cosmic Origins Spectrograph on the Hubble Space Telescope. The observations cover the short wavelength NUV at $\lambda \approx 1680 - 2080$ Å and the long-wavelength NUV at $\lambda \approx 2775-3170$ Å. The NUV flare spectra are intended as a fulcrum that provides essential links between the optical and far-ultraviolet wavelengths, and between astrobiology and stellar flare physics. This is a common dataset that different communities can work with toward their respective goals.

The first results from the “Fulcrum” project feature spectra during the two largest flares on the M dwarf binary CR Dra, including the most luminous NUV luminosity that has ever been observed in detail from a stellar flare. The flares were observed simultaneously in the optical with TESS, and the largest event produced an energy in the TESS band of $\approx 10^{35}$ erg. The Hubble Space Telescope only observed the first half of the rise phase of this flare because it became too bright to observe within its conservatively set brightness limits. The spectra during the large flare event are shown in Figure 3. The spectra show that the continuum in the NUV surprisingly does not turn over, as expected from previous spectra and photometry in the U-band and optical. The short wavelength NUV continuum flux is larger than the longer wavelength NUV continuum flux, and the spectral energy distribution continues to rise toward FUV wavelengths. Kowalski et al. (Reference Kowalski, Osten and Notsu2025) discuss that only the most extreme electron beam heating scenarios can reproduce a rise into the short wavelength NUV. These fully relativistic electron beams severely challenge the standard paradigm for solar flare particle acceleration (Figure 1).

Figure 3.

A NUV stellar megaflare spectrum showing that the continuum does not turn over at $\lambda \approx 3000$ Å. According to the only explanation thus far, high energy particle beams heat the lower atmosphere and produce rising continua (dashed and solid red lines with connecting circles) into the FUV. Figure reproduced from Kowalski et al. (Reference Kowalski, Osten and Notsu2025) with permission from the authors.

Figure 3 Long description

Panel A: A line graph shows the spectral flux density (fλ) in units of erg cm^-2 s^-1 Å^-1 against the wavelength in angstroms (Å) ranging from 1700 to 2100. Multiple datasets are overlaid, including m5F11-25-4, 10^4 K optically thin slab, cF13-500-3, cF13-500-3 + X_rel m2F12-37-2.5, and AD Leo Great Flare. The graph shows fluctuations in spectral flux density with notable peaks and troughs. Panel B: Another line graph displays spectral flux density against wavelength in angstroms (Å) ranging from 2800 to 3100. Similar datasets are overlaid, including m5F11-25-4, 10^4 K optically thin slab, cF13-500-3, cF13-500-3 + X_rel m2F12-37-2.5, AD Leo Great Flare, and CR Dra Flare Event 2 – Preflare. This graph also shows variations in spectral flux density with distinct peaks and a preflare event indicated.

Broad-wavelength spectra (as in Fig 3) of solar flares have not been readily available since the 1980s. Since then, solar instrumentation has evolved toward high spatial and spectral resolution at the expense of wavelength breadth within any given spectral region. The Interface Region Imaging Spectrograph (IRIS; De Pontieu et al. Reference De Pontieu, Title and Lemen2014) has observed many solar flares at high resolution with moderate spectral coverage in the long wavelength NUV around the Mg II h and k lines. One of the most pressing goals has been to understand the extreme redshifts, broadening enhancements, and continuum increases in IRIS observations of the brightest chromospheric sources. These features are indicated in IRIS data in Fig 4 from an X-class flare. The redshifts are also prominent in many Fe II emission lines (Graham et al. Reference Graham, Cauzzi and Zangrilli2020) and in Sun-as-a-star spectra of the Balmer H $\alpha$ line (Namekata et al. Reference Namekata, Ichimoto and Ishii2022).

Figure 4.

IRIS spectra of an X-class solar flare in the long wavelength NUV. The flare spectra are seen toward the bottom where there are transient redshifts superimposed on a marked continuum increase.

It is thought that the brightest locations in solar flares (e.g., Fig 4) originate in beam-generated chromospheric condensations from high-flux density electron beam heating. One-dimensional radiative hydrodynamic models that employ electron beam heating within the standard paradigm of acceleration and propagation can reproduce some salient properties in the observed spectra. However, the models also have some striking shortcomings in their predictions for the magnitude, broadening amounts, and brightnesses of the redshifted components of emission lines. Future observations with the NSF’s Daniel K. Inouye Solar Telescope will reveal whether the same discrepancies exist for other optical lines in the blue/optical, such as the Ca II H. This emission line can be observed with the nearby Balmer H $\epsilon$ line to place new constraints on the chromospheric condensation densities. We have also started a modeling project to understand the three-dimensional radiative-hydrodynamic evolution of chromospheric condensations.