Non-magnetic radiation-MHD simulations of the solar chromosphere with non-equilibrium hydrogen ionization and H2 molecule formation (Leenaarts et al. 2011) predict that expanding post-shock gas cools into pockets of 1660 K or below between roughly 0.8 and 3.4 Mm above the photosphere, with the simulated value being an upper bound set by an artificial heating floor. Whether the real quiet chromosphere ever gets this cold is unknown: the quiet Sun is threaded by turbulent small-scale-dynamo fields and network-internetwork 'magnetic carpet' connections whose Joule and ambipolar-diffusion heating could keep temperatures far higher, and it is not known whether regions effectively uninfluenced by magnetic fields exist on the spatial and temporal scales of cool-pocket formation. Resolving the problem requires quantitative determination of the quiet-chromospheric magnetic-heating contribution (3D radiation magnetohydrodynamic (RMHD) simulations combining roughly 10 km resolution, network-cell-scale horizontal extent, small-scale-dynamo and magnetic-carpet fields, and non-equilibrium H/H2/He equations of state) together with observational tests at ALMA millimeter wavelengths and in off-limb molecular lines, accounting for non-equilibrium chemistry that can suppress molecular signatures of cold gas.

Background

The solar chromosphere is the highly dynamic layer, a few thousand kilometers thick, that connects the 5800 K photosphere to the million-degree corona. Classic one-dimensional semi-empirical models of the quiet Sun place its lower boundary, the 'temperature minimum', near 4300 K. Infrared observations of the fundamental and overtone bands of carbon monoxide (CO), however, have long implied gas at or below 3000-3500 K at chromospheric heights, leading to the 'thermal bifurcation' picture in which CO radiative cooling creates locally cold pockets inside an otherwise warm chromosphere (Ayres, Plymate and Keller 2006, ApJS 165, 618). Whether a non-magnetic chromosphere could stay as warm as the semi-empirical models was already questioned by Carlsson and Stein (1995, ApJ Letters), whose dynamic simulations showed that acoustic shocks alone do not produce a static temperature rise.

Leenaarts, Carlsson, Hansteen and Gudiksen (2011, A&A 530, A124) revisited this question with a two-dimensional radiation-MHD simulation (magnetohydrodynamics coupled to radiative transfer) of a completely non-magnetic solar atmosphere extending from the upper convection zone to the lower corona. The simulation treated hydrogen ionization and H2 molecule formation in non-equilibrium — meaning the level populations and molecular abundances follow time-dependent rate equations rather than instantly relaxing to local thermodynamic equilibrium (LTE) — and used non-LTE radiative transfer for the strongest transitions. Two thermostat mechanisms emerged: in the dense low chromosphere, exothermic H2 formation (4.48 eV per molecule) reheats gas that cools below about 2000-3000 K; but higher up the H2 formation rate becomes too slow to act, so expanding post-shock gas cools almost adiabatically. The run contained pockets of gas at 1660 K between about 0.8 and 3.4 Mm above the photosphere, with such low temperatures common between 0.7 and 2.5 Mm. Because the 1660 K floor was produced by an artificial (ad-hoc) heating term, and the radiative heating was overestimated, the simulation gives an upper bound on the minimum temperature of a non-magnetic chromosphere: the authors concluded it is impossible to avoid such low temperatures with radiation-hydrodynamical processes alone.

The catch is that the real quiet Sun is magnetic. The quiet photosphere carries a turbulent field of order 130 G (spectropolarimetric estimates summarized in Leenaarts et al. 2011), plausibly maintained by a small-scale (local) dynamo — turbulent convection amplifying magnetic field without any large-scale imposed flux — and network fields may connect back into the internetwork as a 'multiscale magnetic carpet' (the network is the lattice of concentrated magnetic flux outlining supergranular convection cells; the internetwork is the weak-field cell interior), strengthening the chromospheric field. In the weakly ionized chromosphere, neutral-ion interactions (ambipolar diffusion, the drift of neutrals through the magnetized plasma) can additionally generate currents and dissipate magnetic energy as heat. If quiet-chromospheric regions effectively uninfluenced by magnetic fields do not exist on the spatial and temporal scales of cool-pocket formation, this magnetic heating could prevent real temperatures from ever reaching the simulated low values, and the minimum temperature of the quiet chromosphere would be substantially higher.

Settling this requires, on the modeling side, three-dimensional RMHD simulations that combine sufficient horizontal extent (a network cell is about 30 Mm across) to capture the magnetic carpet, grid spacing of order 10 km or finer to resolve local-dynamo field generation, and the same non-equilibrium equation of state (hydrogen, H2, and helium) and radiative treatments as the non-magnetic baseline. Known modeling complications include the unphysical 10^4 K thermostat that LTE helium ionization produces (removable with non-equilibrium helium), the parameterized multi-group radiative losses that approximate the strongest chromospheric lines, and finite numerical resolution that underestimates high-frequency acoustic-wave excitation. On the observational side, the Atacama Large Millimeter/submillimeter Array (ALMA) measures chromospheric brightness temperatures in the optically thick sub-millimeter continuum where LTE applies, and off-limb spectroscopy in molecular lines (H2, CO) would directly prove cold gas at high chromospheric heights — although chemical non-equilibrium may suppress molecular abundances even where the gas is cold, so absence of molecular lines would not imply absence of cool gas.

Problem Statement

Determine the minimum gas temperature of the quiet solar chromosphere and, in particular, whether pockets of gas at temperatures of order 2000 K or below — as predicted for a non-magnetic chromosphere by radiation-MHD simulations with non-equilibrium hydrogen ionization and H2 formation — actually occur in the real Sun. This requires answering, as jointly supporting parts of one objective:

  1. Whether quiet-Sun chromospheric regions effectively uninfluenced by magnetic fields — that is, regions where magnetic heating is too weak to prevent post-shock gas from cooling into the low-temperature pockets found in the non-magnetic baseline — exist on the spatial and temporal scales relevant to the formation of the simulated cool pockets, or whether heating from the turbulent small-scale (local) dynamo field, the multiscale magnetic carpet connecting network and internetwork, and neutral-ion (ambipolar-diffusion) current generation in the weakly ionized chromosphere keeps quiet-chromospheric temperatures substantially above the non-magnetic values.
  2. A quantitative determination of the chromospheric magnetic-heating contribution, through 3D radiation-MHD simulations that combine horizontal extent sufficient to represent a network cell (of order 30 Mm) and the magnetic carpet, grid spacing of order 10 km or finer to capture local-dynamo field generation, and the same non-equilibrium equation of state (hydrogen, H2, and helium) and non-LTE radiative treatments as the non-magnetic baseline, with a term-by-term energy budget (radiative, acoustic/shock, resistive/Joule, ambipolar).
  3. An observational determination of whether such cool pockets exist, via ALMA sub-millimeter brightness-temperature measurements and off-limb molecular (H2, CO) spectroscopy, with the interpretation accounting for chemical non-equilibrium that can suppress molecular signatures even where the gas is cold.

The problem is solved when the minimum temperature of the quiet chromosphere is established with its physical cause identified — magnetic heating versus radiative-hydrodynamic cooling — and the result is consistent between the modeling and observational constraints.

References

  1. J. Leenaarts, M. Carlsson, V. Hansteen, B. V. Gudiksen, 'On the minimum temperature of the quiet solar chromosphere', Astronomy & Astrophysics 530, A124 (2011). DOI: 10.1051/0004-6361/201016392, arXiv:1104.5081, https://arxiv.org/abs/1104.5081
  2. M. Carlsson, R. F. Stein, 'Does a nonmagnetic solar chromosphere exist?', The Astrophysical Journal Letters 440, L29 (1995). DOI: 10.1086/187753, https://doi.org/10.1086/187753
  3. T. Ayres, C. Plymate, C. U. Keller, 'Solar Carbon Monoxide, Thermal Profiling, and the Abundances of C, O, and Their Isotopes', The Astrophysical Journal Supplement Series 165, 618 (2006). DOI: 10.1086/504847, https://doi.org/10.1086/504847
  4. T. P. Golding, J. Leenaarts, M. Carlsson, 'Non-equilibrium helium ionization in an MHD simulation of the solar atmosphere', The Astrophysical Journal 817, 125 (2016). DOI: 10.3847/0004-637X/817/2/125, arXiv:1512.04738, https://arxiv.org/abs/1512.04738
  5. M. Carlsson, V. H. Hansteen, B. V. Gudiksen, J. Leenaarts, B. De Pontieu, 'A publicly available simulation of an enhanced network region of the Sun', Astronomy & Astrophysics 585, A4 (2016). DOI: 10.1051/0004-6361/201527226, https://doi.org/10.1051/0004-6361/201527226
  6. J. Martínez-Sykora, J. de la Cruz Rodríguez, M. Gošić, A. Sainz Dalda, V. H. Hansteen, B. De Pontieu, 'Chromospheric Heating from Local Magnetic Growth and Ambipolar Diffusion under Nonequilibrium Conditions', The Astrophysical Journal Letters 943, L14 (2023). DOI: 10.3847/2041-8213/acafe9, arXiv:2211.08472, https://arxiv.org/abs/2211.08472
  7. D. Przybylski, R. Cameron, S. K. Solanki, M. Rempel, S. Danilovic, J. Leenaarts, 'Structure and dynamics of the internetwork solar chromosphere: results of a small-scale dynamo simulation', Astronomy & Astrophysics 703, A148 (2025). DOI: 10.1051/0004-6361/202554656, arXiv:2508.19809, https://arxiv.org/abs/2508.19809
  8. Q. Noraz, M. Carlsson, G. Aulanier, 'Chromosphere of the quiet Sun - I. Shock and current-sheet dynamics and heating', Astronomy & Astrophysics 705, A86 (2026). DOI: 10.1051/0004-6361/202556375, arXiv:2511.01858, https://arxiv.org/abs/2511.01858
  9. M. Gošić, V. H. Hansteen, A. Sainz Dalda, B. De Pontieu, L. H. M. Rouppe van der Voort, 'Bifrost Models of the Quiet Sun. I. Comparison with Solar Observations', The Astrophysical Journal 990, 195 (2025). DOI: 10.3847/1538-4357/adf4db, arXiv:2508.19355, https://arxiv.org/abs/2508.19355
  10. A. Nindos, C. E. Alissandrakis, T. S. Bastian, S. Patsourakos, B. De Pontieu, H. Warren, T. Ayres, H. S. Hudson, T. Shimizu, J.-C. Vial, S. Wedemeyer, V. Yurchyshyn, 'First high-resolution look at the quiet Sun with ALMA at 3mm', Astronomy & Astrophysics 619, L6 (2018). DOI: 10.1051/0004-6361/201834113, https://doi.org/10.1051/0004-6361/201834113
  11. M. A. Loukitcheva, S. M. White, S. K. Solanki, 'ALMA Detection of Dark Chromospheric Holes in the Quiet Sun', The Astrophysical Journal Letters 877, L26 (2019). DOI: 10.3847/2041-8213/ab2191, arXiv:1905.06763, https://arxiv.org/abs/1905.06763
  12. J. M. da Silva Santos, J. de la Cruz Rodríguez, J. Leenaarts, G. Chintzoglou, B. De Pontieu, S. Wedemeyer, M. Szydlarski, 'The multi-thermal chromosphere: inversions of ALMA and IRIS data', Astronomy & Astrophysics 634, A56 (2020). DOI: 10.1051/0004-6361/201937117, arXiv:1912.09886, https://arxiv.org/abs/1912.09886
  13. C. Alissandrakis, T. S. Bastian, R. Brajša, 'The quiet Sun at mm wavelengths as seen by ALMA', Frontiers in Astronomy and Space Sciences 9, 981320 (2022). DOI: 10.3389/fspas.2022.981320, arXiv:2209.02569, https://arxiv.org/abs/2209.02569
  14. S. A. Jaeggli, P. G. Judge, A. N. Daw, 'Formation of the UV Spectrum of Molecular Hydrogen in the Sun', The Astrophysical Journal 855, 134 (2018). DOI: 10.3847/1538-4357/aaafd5, arXiv:1802.03779, https://arxiv.org/abs/1802.03779
  15. L. S. Anusha, M. van Noort, R. H. Cameron, 'Nonequilibrium Equation of State in Stellar Atmospheres', The Astrophysical Journal 911, 71 (2021). DOI: 10.3847/1538-4357/abe45d, https://doi.org/10.3847/1538-4357/abe45d
  16. J. R. Stauffer, S. Wedemeyer, K. P. Reardon, 'Nonequilibrium CO Formation in 3D Simulations of the Solar Atmosphere', The Astrophysical Journal 1004, 190 (2026). DOI: 10.3847/1538-4357/ae69c8, https://doi.org/10.3847/1538-4357/ae69c8