Simulating High-Energy Protons in the Solar Atmosphere to Explain Gamma-Ray Emission

When the Sun unleashes its most powerful flares, something remarkable happens: for minutes, and sometimes for hours after the initial explosion, our star continues to glow in gamma rays. This high-energy emission is the signature of protons accelerated to energies exceeding 300 million electron volts slamming into the solar atmosphere, and it represents one of the most direct windows scientists have into the acceleration of ions on the Sun. Now, a new study published in the journal Solar Physics has delivered a detailed numerical treatment of how these high-energy protons travel through the Sun’s dense outer layers and what kind of gamma-ray fingerprints they leave behind — work that could help resolve one of the most persistent puzzles in modern solar physics.

The research, carried out by Alexey Kuzmitskiy and Alexey Kochanov of the Institute of Solar-Terrestrial Physics in Irkutsk, Russia, tackles the problem head-on by solving the non-stationary Ginzburg–Syrovatskii particle transport equation in the well-known “leaky box” approximation. This equation, originally developed in the 1960s to describe how cosmic rays propagate through the Galaxy, describes how a population of particles distributed in energy evolves over time inside a confined region, balancing continuous injection against losses and escape. Applied to the solar atmosphere, the leaky-box framework allows the researchers to track a population of accelerated protons as they cascade downward through the chromosphere and photosphere, losing energy and interacting with ambient nuclei along the way.

The physics at stake is intricate. Fast protons passing through the Sun’s atmosphere surrender energy through ionization of the ambient gas and through inelastic nuclear collisions. Those same collisions, however, are precisely what produce the observable signal: when a proton above the kinematic threshold strikes another nucleus, it can generate neutral pions, which decay almost immediately into pairs of gamma-ray photons. The decay of these pions is the primary source of the high-energy gamma-ray emission detected by instruments such as the Large Area Telescope on NASA’s Fermi Gamma-ray Space Telescope. The collisions also regenerate protons through nuclear reactions, meaning that the proton population does not simply deplete monotonically — secondary protons can themselves go on to produce pions, subtly reshaping the resulting emission spectrum over time.

What distinguishes the new work is its comprehensive, time-dependent treatment of these coupled processes. Kuzmitskiy and Kochanov computed the full evolution of the proton energy spectrum as the particles propagate into photospheric and deeper layers, incorporating ionization and collisional energy losses, nuclear regeneration, and neutral pion production within a single numerical scheme. From the calculated secondary particle distributions they then derived the two observables that matter most to solar astronomers: the time-dependent spectra of the emitted gamma rays and the corresponding light curves — the brightness of the emission as a function of time. Nine figures in the published paper chart these quantities across the different injection scenarios the team explored.

The scenarios themselves matter enormously, because where and how accelerated protons are injected determines how long the gamma-ray emission persists. The simplest case is free precipitation: protons accelerated during a flare are injected directly downward and plunge into the dense atmosphere, where they lose energy rapidly. In this picture, the gamma-ray emission should be brief, essentially tracking the impulsive phase of the flare itself. Yet observations have repeatedly defied this expectation. Since the earliest detections of pion-decay gamma rays from solar flares in the 1980s, and especially since Fermi-LAT began routinely monitoring the Sun, a class of events known as long-duration gamma-ray flares has stubbornly refused to fit the simple picture, with high-energy emission continuing for many hours after the impulsively accelerated particles should have exhausted themselves.

The new modeling directly addresses this tension by considering an alternative injection geometry: precipitation preceded by trapping in the coronal portions of magnetic loops. The Sun’s corona above an active region is threaded with arches of magnetic field, and charged particles spiraling along these field lines can be confined, bouncing between mirror points while slowly leaking out — the “leaky” part of the leaky box. Kuzmitskiy and Kochanov considered both low magnetic loops, where the confinement region sits relatively close to the photosphere, and high loops extending far into the corona. The calculations show that trapping fundamentally changes the temporal behavior of the emission. Instead of a sharp burst, the gamma-ray light curve acquires an extended tail, sustaining high-energy emission on time scales ranging from a fraction of a minute to several tens of minutes. For the largest coronal loops, the trapping time could potentially stretch to the order of hours — exactly the range needed to account for the most extreme long-duration gamma-ray flares observed by Fermi-LAT.

This result carries significant weight because the origin of long-duration gamma-ray emission remains actively debated. Competing explanations include continuous acceleration at shock waves driven by coronal mass ejections low in the corona, acceleration in large-scale coronal loops behind the flare site, and the return of particles that escaped into interplanetary space and back-precipitated onto the Sun. Behind-the-limb flares — events where the flare site is hidden from view but gamma rays are still detected from visibly emitted regions of the Sun — have been particularly challenging, since they suggest that the particle reservoir extends far from the electromagnetic flare itself. By quantifying how long trapped protons in loops of different sizes can keep the emission alive, the new calculations provide a quantitative benchmark against which these competing scenarios can be tested. If the observed decay time of a gamma-ray event matches the trapping time expected for a loop of plausible coronal dimensions, trapping becomes a viable explanation; if the emission outlasts any reasonable trapping time, continuous acceleration elsewhere is required.

The authors also positioned their results within the broader landscape of previous theoretical work. The problem of high-energy gamma-ray production in solar flares has a long pedigree, stretching back to analytic models of the 1970s and Monte Carlo simulations of the 1990s that followed individual particles through model atmospheres. More recently, sophisticated Monte Carlo toolkits such as FLUKA have been applied to the problem, tracking pion production and decay with great detail. A key motivation for the new study is reliability: to confidently infer the energy spectrum of the parent protons from the shape of the observed gamma-ray spectrum, theorists need robust and mutually consistent calculations of both the particle transport and the emission. The team compared their computed spectra both with calculations from previous decades and with recent Monte Carlo results, providing a cross-check between fundamentally different computational approaches — deterministic solutions of a transport equation versus stochastic particle-by-particle simulation. Agreement between such independent methods strengthens confidence in the predicted spectra and in the spectral inversions that observers perform on Fermi-LAT data.

The choice of the Ginzburg–Syrovatskii framework also gives the work an appealing economy. Rather than simulating millions of individual particle trajectories, the transport-equation approach evolves the distribution function directly, making it computationally efficient enough to explore a wide grid of injection spectra, loop sizes, and trapping parameters. The model atmosphere underlying the calculations draws on standard semi-empirical models of the solar chromosphere and photosphere, with the chemical composition of the Sun taken from modern determinations, ensuring that the nuclear interaction rates and energy-loss profiles reflect the best available knowledge of the target material the protons traverse.

For the observational community, the practical payoff is a set of theoretical light curves and spectra against which real events can be matched. The Fermi-LAT solar flare catalog, along with studies of individual spectacular events such as the behind-the-limb flares of 2014, 2017, and 2022, has accumulated a rich sample of high-energy gamma-ray light curves with decay times spanning orders of magnitude. Upcoming and proposed instruments, including the GAMMA-400 gamma-ray telescope and mission concepts devoted to solar particle acceleration, will extend these measurements to higher energies and finer time resolution. Models like the one developed by Kuzmitskiy and Kochanov are the interpretive machinery that turns such observations into statements about where protons are accelerated, how they are confined, and how much energy the Sun deposits into its atmosphere in the form of energetic ions.

The work also touches on broader questions in space weather. High-energy protons that escape the Sun rather than precipitating form solar energetic particle events, which pose radiation hazards to satellites and astronauts. Understanding the shared origins of precipitating and escaping protons — and the role of coronal trapping in sorting them — informs statistical relationships that researchers have drawn between long-duration gamma-ray emission and solar energetic particle events. The study was supported by the Ministry of Science and Higher Education of the Russian Federation, and its detailed numerical solutions, covering free precipitation and trapping in both low and high coronal loops, now stand as a reference toolkit for anyone seeking to decode the gamma-ray glow that follows the Sun’s most violent explosions.