Scientists are increasingly focused on understanding the solar chromosphere, a region of the Sun’s atmosphere for which diagnostic tools remain limited compared to other layers. Dufresne from DAMTP, University of Cambridge, Osborne from SUPA School of Physics and Astronomy, University of Glasgow, and Del Zanna et al. present new research modelling the potential of neutral carbon Rydberg lines as a diagnostic tool for chromospheric conditions. Their work utilises the radiative transfer code Lightweaver to demonstrate how these ultraviolet emissions respond to changes in temperature, density and turbulence, revealing their sensitivity to atmospheric perturbations. This research is significant because it establishes a pathway for utilising a wealth of Rydberg line data, particularly that soon to be gathered by the Solar-C EUVST, to reconstruct detailed models of the solar chromosphere and improve our understanding of this crucial region.

Despite the complexity of chromospheric conditions, hundreds of Rydberg lines emitted by neutral atoms have been observed at ultraviolet wavelengths. Recent advancements in atomic data now allow researchers to investigate the diagnostic potential of these lines, specifically those emitted by carbon.

This work utilises the radiative transfer code Lightweaver to model the formation of these lines and assess their sensitivity to variations in temperature, density, and micro-turbulent velocity within the solar atmosphere. To simplify the modelling process, the study focuses on lines originating from levels with a principal quantum number of 10 or greater, where levels are expected to be in Saha-Boltzmann equilibrium with the ground state of singly-charged ions.
Optical depth effects are demonstrably present in the modelled lines, and their response to atmospheric perturbations indicates their suitability for reconstructing atmospheric properties through inversion techniques. The research establishes a foundation for utilising a multitude of such lines, emitted by various elements at different heights, promising a significant increase in diagnostic capabilities.

The modelling relies on the principle that, for higher energy levels, the atomic physics is considerably simplified when levels are in Saha-Boltzmann equilibrium, avoiding the need for complex non-local thermodynamic equilibrium calculations. This approach requires only the relative populations of long-lived levels, the energies of the Rydberg states, and the spontaneous decay rates of the lines.

The study builds upon recent data, including atomic data for carbon Rydberg lines from Storey et al. (2023) and improved atomic rates for estimating ion fractions, as presented by Dufresne et al. (2025). Analysis reveals that the Rydberg lines form at different heights in the lower chromosphere, with series limits at 1101.1 Å, 1240.3 Å, and 1445.7 Å.

This height stratification, coupled with the lines’ sensitivity to atmospheric parameters, suggests they can provide valuable insights into the complex structure of the chromosphere. The work is particularly relevant given the imminent launch of the Solar-C EUV High-throughput Spectroscopic Telescope (EUVST), which is expected to observe a large number of these lines with unprecedented spatial and spectral resolution, enabling detailed reconstructions of the solar chromosphere.

Modelling Rydberg line emission under collisional dominance and local thermodynamic equilibrium

Lightweaver, a radiative transfer code version 0.13.0, underpinned the modelling of carbon Rydberg lines emitted from the solar chromosphere. The research focused on lines originating from levels with principal quantum number, anticipating Saha-Boltzmann equilibrium with the singly-charged ion ground state.

This simplification bypassed the need for a large-scale collisional-radiative model, reducing computational demands while still capturing essential Rydberg level populations. Optical depth effects were carefully considered, and the lines’ responses to atmospheric changes indicated their potential for atmospheric reconstruction via inversions.

The study addressed the non-linear problem of population distribution by leveraging the high densities within the solar chromosphere. Collisional processes dominate, allowing Rydberg level populations to be accurately described using the Saha-Boltzmann equation, proportional to the carbon ion density.

This enabled a two-part radiative transfer calculation, initially solving for long-lived levels with a limited atomic model, then incorporating Rydberg levels with fixed Saha-Boltzmann populations. The code employed a steady-state, 1D radiative transfer scheme, including non-LTE calculations for hydrogen, helium, carbon, oxygen, magnesium, aluminium, silicon, sulphur, and iron, alongside LTE treatment for nitrogen, calcium, and nickel.

New atomic models for carbon, silicon, and sulphur were implemented, utilising the Fontenla et al. network model as the input atmosphere, chosen for its expected strong contribution to line intensity. Partial redistribution was applied for hydrogen Lyman-α and -β lines, crucial for accurately modelling the Rydberg series at 1240 Å, which forms on the wings of Lyman-α.

Data from Storey et al. regarding level energies and radiative decay rates were added to the carbon model, enabling the calculation of Rydberg level populations via the Saha-Boltzmann equation, considering electron temperature, ionisation potential, and electron number density. The emergent intensity was then calculated through integration of the source function and monochromatic optical depth, defining a contribution function and ultimately, the response function to atmospheric perturbations.

Carbon Rydberg line spectra modelling reveals wavelength-dependent agreement with chromospheric observations

Radiative transfer calculations demonstrate significant improvement over previous optically thin models when analysing carbon Rydberg lines emitted from the solar chromosphere. Synthetic spectra from these calculations closely match observed line intensities for shorter wavelengths, exhibiting good agreement in both line shapes and blends.

Specifically, the intensities of the lines, excluding the continuum, align very well with observations for the two series at shorter wavelengths. However, the series decaying to the 1S term, at wavelengths longer than 1450 Å, show discrepancies, appearing further from observational data. In the longer wavelength region, numerous molecular and unidentified lines complicate the identification of carbon lines and assessment of blending.

Candidate lines near 1456.0 Å exhibit intensities approximately two to three times lower than observed values in the synthetic models. Comparison with optically thin calculations by Storey et al. (2023) reveals that the new radiative transfer calculations, performed using Lightweaver, represent a substantial advancement.

These calculations compute lines and continua self-consistently from the same atmospheric model, eliminating the need for emissivity scaling factors applied in previous studies. The radiative transfer calculations also improve predictions for lower n states, where optically thin results previously overpredicted observed intensities.

Population levels between n=10 and n=20, sharing the same parent and total angular momentum J, are within 10% of each other. Furthermore, if these levels were not in Saha-Boltzmann equilibrium, their populations would be even lower, resulting in reduced line intensities. Discrepancies between synthetic and observed intensities for stronger lines are likely due to opacity effects, with the carbon ion populations potentially underestimated, particularly for series decaying to the 1D and 1S states.

Carbon Rydberg line modelling constrains chromospheric temperature, density and velocity structures

Modelling of carbon Rydberg lines offers a promising new avenue for diagnosing the solar chromosphere. This research demonstrates the diagnostic potential of hundreds of ultraviolet Rydberg lines emitted in this region by modelling their formation using the radiative transfer code Lightweaver. The study explores how these lines respond to variations in atmospheric temperature, density, and micro-turbulent velocity, revealing their utility in reconstructing chromospheric properties through atmospheric inversions.

Optical depth effects are evident within the lines, and their sensitivity to atmospheric perturbations suggests they can be effectively used to map the chromosphere. The authors simplified modelling by focusing on lines originating from levels expected to be in Saha-Boltzmann equilibrium with the ground state of singly-charged carbon, reducing computational demands.

While acknowledging the complexity of full non-LTE calculations, the research establishes that, under chromospheric densities, Rydberg level populations can be reasonably approximated using this simplification. Future observations from the forthcoming Solar-C EUV High-throughput Spectroscopic Telescope are expected to benefit significantly from this work, enabling the analysis of numerous lines from multiple elements across varying atmospheric heights.