When Radio Waves Heat the Sky: Electrons Defy the Rules of Thermal Physics

High above the Arctic, powerful radio transmitters can pump energy directly into the thin air of the upper atmosphere, heating the electrons that drift there in a process scientists call artificial ionospheric heating. For decades, researchers studying this phenomenon have relied on a convenient simplification: that the heated electrons behave like a gas in thermal equilibrium, described by a single temperature. A new study by Margaretha Myrvang and Björn Gustavsson of UiT The Arctic University of Norway, published in Annales Geophysicae, shows just how wrong that assumption can be. By solving the fundamental equations of electron transport with unprecedented care, the pair demonstrates that electrons in the D region of the ionosphere, when heated by high-frequency radio waves, arrange themselves into distributions that look nothing like the familiar bell-shaped Maxwellian curve that has underpinned ionospheric physics for generations.

The D region, spanning roughly 60 to 100 kilometers in altitude, is a peculiar place. It is a weakly ionized plasma, meaning that for every free electron there are between a hundred million and a trillion neutral molecules. In such an environment, electrons collide far more often with neutral particles than with each other, and the electron-neutral collision frequency exceeds even the frequency at which electrons spiral around Earth’s magnetic field lines. This makes the D region a realm where individual collisions, not collective plasma behavior, dictate what happens to each electron. When a high-power high-frequency radio wave sweeps through this region, electrons absorb energy from the wave’s oscillating electric field through a process known as Ohmic heating. Elastic collisions scatter the electrons in random directions, converting the ordered energy of the wave into thermal motion, while less frequent inelastic collisions drain energy away by exciting the neutral molecules around them.

It is those inelastic collisions that give the electron distribution its strange and beautiful shape. Myrvang and Gustavsson’s numerical model, a modern re-implementation of a kinetic solver first derived by physicist Peter Stubbe in 1981, tracks how electrons gain energy from the radio wave and then lose it in discrete chunks as they excite vibrational, rotational, and electronic states in molecular nitrogen, molecular oxygen, and atomic oxygen. The most dramatic effect comes from molecular nitrogen, whose vibrational states have large cross sections for electron impact in the energy range between roughly 2 and 3.5 electron volts. When enough electrons reach those energies, they slam into nitrogen molecules and lose about 0.29 electron volts per collision, exciting the molecule’s vibrational modes. The result, visible in the model output, is a sharp cut-off in the electron distribution at approximately 2 electron volts, an effective cliff that truncates the population of energetic electrons far more abruptly than any Maxwellian would allow.

The technical achievement of the new work lies in how meticulously it handles this energy degradation. Where earlier solvers treated electrons as jumping between discrete velocity points, losing exactly the excitation energy in a single step, the Norwegian team used continuous, energy-dependent cross sections for every inelastic process and carefully partitioned degrading electrons across the finite velocity bins of their numerical grid. They also corrected subtle errors in the original formulation, noting that some of the factors in Stubbe’s published equations carried incorrect units. Starting from a Maxwellian guess, their iterative scheme converges, with a damping factor applied for stability, within about sixty iterations under typical ionospheric conditions. Neutral densities come from standard atmospheric models, the neutral temperature is set to 200 kelvin, and the electron density is assumed to be ten billion per cubic meter, values representative of the quiet D region.

To make sure the kinetic solver was trustworthy, the authors built an entirely independent check: a Monte Carlo simulation that follows one million individual electrons as they are accelerated by the oscillating electric field of the heating wave between collisions. Collision times are drawn from exponential random distributions, scattering is assumed isotropic, a good approximation for electrons below about 5 electron volts, and each collision is classified as elastic or inelastic according to the best available laboratory cross sections. The simulation captures the full time evolution of the electron population, from a cold 300-kelvin Maxwellian start to a steady state in which the energy pumped in by the radio wave exactly balances the energy lost to the neutral gas. That steady state arrives remarkably quickly at lower altitudes, in about three milliseconds at 80 kilometers, but takes as long as ten seconds at 120 kilometers, where collisions are rarer and the electrons take longer to settle.

The agreement between the two methods is convincing at most altitudes. At 80 and 100 kilometers, the Boltzmann solver and the Monte Carlo simulation produce energy distributions that match well above 0.1 electron volts, with second-moment temperatures of roughly 1750 and 1230 kelvin respectively in the kinetic calculation. Below 0.1 electron volts the two methods diverge somewhat, a discrepancy the authors attribute to the approximate analytical expressions used for rotational excitation and elastic collisions in the Boltzmann approach, approximations that matter most where the neutral density is highest. At 120 kilometers a more serious problem emerges: the electron-neutral collision frequency drops below the electron gyro-frequency, violating the solver’s assumption that electrons are unmagnetized. There the Boltzmann method overestimates the temperature significantly, predicting 350 kelvin where the Monte Carlo simulation, which fully accounts for the magnetic field, finds only 219 kelvin. The lesson is clear: the kinetic solver is a superb tool for the D region, but it cannot simply be extended upward into the magnetized E and F regions without modification.

Perhaps the most provocative result concerns temperature itself. The authors show that even when a temperature can be defined for the heated electrons, by taking the second moment of the distribution as a measure of total thermal energy, that single number conceals more than it reveals. At 90 kilometers with an electric field of 2.0 volts per meter, the second-moment temperature is a scorching 5124 kelvin, yet the actual distribution bears little resemblance to a Maxwellian at that temperature. An alternative measure, an energy-dependent effective temperature derived from the local slope of the distribution, swings wildly across the energy spectrum. It peaks near 10,000 kelvin around 1.1 electron volts, where electrons degraded from higher energies pile up in a relatively flat region of phase space, then plunges in the 1.8 to 2.7 electron volt range where nitrogen vibrational excitation drains the population, and finally settles toward the neutral gas temperature at the lowest energies. Undulations between 0.1 and 1.2 electron volts trace the resonance peaks in the vibrational cross sections of molecular oxygen.

These findings matter far beyond the niche community of ionospheric heating experiments. Maxwellian assumptions are baked into standard calculations of electron cooling rates, radio wave absorption, refractive indices, and D region chemistry. If the true distribution deviates sharply from Maxwellian whenever electrons are heated above roughly 500 to 600 kelvin, then every one of those derived quantities may need revisiting. The shape of the non-Maxwellian distribution is not fixed either: it changes with altitude and with the strength of the heating field, growing more distorted as more energy is injected. At 70 kilometers, where frequent collisions keep electrons from gaining much energy between impacts, the distribution stays relatively close to Maxwellian even at strong fields, with temperatures rising only from about 400 to 1600 kelvin as the electric field increases from 0.75 to 4.0 volts per meter. Higher up, at 90 to 100 kilometers, the same field range drives temperatures from a few hundred kelvin to between 5000 and 6000 kelvin, and the deviations become dramatic.

This paper is the first of a pair. In the companion study, the authors compute electron cooling rates for vibrational excitation of nitrogen and oxygen and for the excitation of fine structure levels in atomic oxygen, using the non-Maxwellian distributions rather than Maxwellian approximations. Together, the two papers promise to sharpen the tools scientists use to interpret experiments at facilities such as the EISCAT heating facility in Norway, where radio-induced electron heating has been measured in situ since the year 2000. They also serve as a reminder of a deeper truth in physics: temperature is a bookkeeping device, not a description. In the thin, collision-dominated air of the upper atmosphere, the electrons tell a richer story, one that can only be read by solving the kinetic equations in full, one collision at a time.

Subject of Research: Non-Maxwellian electron energy distributions in the ionospheric D region under artificial high-frequency radio wave heating

Article Title: Non-Maxwellian electron distributions in the D region during artificial heating – Part 1: Model development and electron temperature

Article References: Myrvang, M., & Gustavsson, B. (2026). Non-Maxwellian electron distributions in the D region during artificial heating – Part 1: Model development and electron temperature. Annales Geophysicae, 44(2), 921-935. https://doi.org/10.5194/angeo-44-921-2026

Image Credits: AI Generated

DOI: 10.5194/angeo-44-921-2026

Keywords: ionosphere, D region, artificial heating, electron temperature, non-Maxwellian distribution, Boltzmann equation, Monte Carlo simulation, HF radio waves, vibrational excitation, molecular nitrogen, electron cooling, space physics