Three-dimensional non-LTE Stokes inversion of the solar Mg II h & k lines with partial frequency redistribution and J-state interference for unambiguous chromospheric vector magnetometry

The polarization of the Mg II h & k resonance doublet (279.6/280.3 nm), which forms in the upper solar chromosphere, encodes the magnetic field through the joint action of scattering polarization, the Hanle, Zeeman, and magneto-optical effects. The only existing full-Stokes inversions of CLASP2 observations of these lines (HanleRT-TIC) are one-dimensional and must introduce ad-hoc radiation-field tensor components ($J^{\dagger}$) that mimic unmodeled three-dimensional radiative transfer; these components are degenerate with the magnetic field vector, so the inferred field azimuth, and sometimes inclination, is not uniquely determined. The open problem is to develop and validate a three-dimensional non-LTE Stokes inversion of Mg II h & k that includes partial frequency redistribution, J-state interference between the h & k upper levels, and horizontal radiative transfer, thereby eliminating the ad-hoc parametrization; to quantify and mitigate the remaining degeneracies between symmetry breaking and the magnetic field orientation; and to make such full-physics inversions computationally feasible for realistic datasets, within the intrinsic Hanle sensitivity window (field strengths of roughly 2–100 G for the k-line core).

Background

The solar chromosphere is the weakly ionized atmospheric layer between the photosphere and the corona whose magnetic field governs the energy transport and heating of the outer solar atmosphere, yet its vector magnetic field is still essentially unmeasured, especially outside active regions. A promising diagnostic is the Mg II h & k resonance doublet (k at 279.6 nm, h at 280.3 nm), which forms over a wide height range up to almost the base of the corona. The four Stokes parameters ($I$ intensity, $Q$ and $U$ linear polarization, $V$ circular polarization) of these lines are shaped by several coupled physical mechanisms. Anisotropic irradiation of the atoms produces population imbalances and quantum coherences among magnetic sublevels (atomic level polarization), so that scattered light is linearly polarized even without a magnetic field. A weak magnetic field modifies this scattering polarization through the Hanle effect, which is sensitive to fields between roughly $1/10$ of and about ten times the critical Hanle field — for the Mg II k core, approximately $5$–$100$ G, with an empirical sensitivity floor near $2$ G and saturation of the transverse-field response near $100$ G. The Zeeman effect contributes circular polarization and, through the magneto-optical terms of the polarized transfer equation, couples $Q$ and $U$. Because h and k share a common lower level and have upper levels with total angular momenta $J=1/2$ and $J=3/2$, quantum interference between these two upper $J$-levels (J-state interference) produces sizable linear polarization in the near and far line wings, sensitive to the longitudinal field via magneto-optical effects. Modeling these signals requires solving the radiative transfer problem out of local thermodynamic equilibrium (non-LTE) with partial frequency redistribution (PRD), the effect by which scattered photons partly remember their absorption frequency, which is essential in these strong resonance lines. Belluzzi & Trujillo Bueno (2012) predicted the doublet's scattering polarization and Alsina Ballester et al. (2016) established its magnetic sensitivity; the CLASP2 (2019) and CLASP2.1 (2021) sounding-rocket experiments measured all four Stokes parameters across the doublet in a plage and in quiet regions near the limb (Ishikawa et al. 2021). The HanleRT-TIC inversion code (radiative transfer by HanleRT; inversion by TIC, the code of Alsina Ballester et al. 2022, which handles PRD, J-state interference, and arbitrary magnetic fields) is 1D plane-parallel: it neglects horizontal radiative transfer. In the solar atmosphere, horizontal temperature and density inhomogeneities and gradients of horizontal macroscopic velocities break the axial symmetry of the pumping radiation field non-magnetically, producing Stokes $U$ signals that a 1D model cannot explain. Li et al. (2024) therefore added ad-hoc complex radiation-field tensor components $J^{\dagger\,2}_{1}$ and $J^{\dagger\,2}_{2}$ (quantifying the lack of axial symmetry of the pumping field) as free parameters, which allowed successful fits of CLASP2 profiles and field strengths of 1–20 G in the quiet upper chromosphere. However, different combinations of $J^{\dagger}$ components and magnetic-field orientations reproduce the same observed $Q$ and $U$: Euler-angle rotation of the radiation tensor into the magnetic reference frame makes distinct parameter sets equivalent, and Hanle ambiguities (distinct inclination/azimuth pairs giving identical $Q,U$) leave the azimuth, and in the absence of detectable Stokes $V$ also the inclination, undetermined, with field-strength scatter of order $\pm 10$ G. A self-consistent treatment of these symmetry-breaking effects requires solving the 3D non-LTE polarized transfer problem with PRD and J-state interference — a long-standing computational barrier. Recent progress on 3D forward solvers (Benedusi et al. 2023; the TRIP code of Benedusi et al. 2026, which reached 3D non-LTE polarized transfer with PRD for the two-level Ca I 4227 Å line) has not yet delivered the two-term-atom, Mg II-specific solver, and 3D Stokes inversion frameworks exist only in the complete-redistribution (CRD) limit (Štěpán et al. 2022; Štěpán et al. 2024, for prominences).

Problem Statement

Develop and validate a three-dimensional non-LTE Stokes inversion of the Mg II h & k doublet that includes the full physics of its polarization — atomic level polarization, partial frequency redistribution, J-state interference between the h & k upper levels, and the Hanle, Zeeman, and magneto-optical effects, together with horizontal radiative transfer in realistic (magneto-hydrodynamic or semi-empirical 3D) chromospheric models — so that the magnetic field vector (strength, inclination, and azimuth) in the upper chromosphere can be inferred unambiguously from full-Stokes observations, without the ad-hoc $J^{\dagger}$ radiation-tensor parametrization of Li et al. (2024). Concretely, the problem comprises jointly supporting deliverables toward this single objective: (i) a forward solver for the Mg II h & k Stokes profiles in 3D models with PRD and J-state interference in the presence of magnetic fields, verified against established 1D codes in controlled limits; (ii) inversion machinery built on it whose recovery of field strength, inclination, and azimuth is quantified on synthetic data from known 3D models, replacing the ad-hoc symmetry-breaking parameters with self-consistently computed radiation-field tensors; (iii) a characterization of the remaining degeneracies and Hanle ambiguities — the combinations of radiation-field anisotropy and field orientation that yield indistinguishable Stokes $Q$ and $U$ — including how far complementary constraints (additional spectral lines, e.g. in the CLASP2/2.1 window, or higher-sensitivity Stokes $V$) restore uniqueness in azimuth and inclination; and (iv) a demonstration of computational feasibility at dataset scale, e.g. via validated surrogate or machine-learning acceleration of the full polarized non-LTE problem. The intrinsic Hanle sensitivity window of the k-line core (roughly $2$–$100$ G, with saturation of the transverse-field response above about $100$ G) is a physical property of the diagnostic that delimits, but does not define, the problem: within that window, the task is to make the vector-field inference unique and computable.

Completion boundary. Parts (i)–(iv) are jointly required, and no proper subset of them resolves the task. Concretely: (i) the forward solver must be verified against established 1D codes in controlled limits, with the level of agreement stated quantitatively; (ii) the inversion's recovery of field strength, inclination, and azimuth on synthetic data from known 3D models must be quantified by explicit retrieval errors and by the parameter ranges, within the $2$–$100$ G Hanle window, over which the recovery is unique; (iii) the surviving degeneracies between symmetry-breaking radiation-field structure and field orientation must be enumerated, together with the conditions under which each vanishes and the effect of the stated complementary constraints; and (iv) dataset-scale feasibility must be demonstrated by a reported computational cost (time and memory) and by the accuracy preserved at that cost. The uniqueness clause is to be read as requiring the field vector to be either uniquely inferred or its residual ambiguity explicitly delimited: a quantified negative result — a demonstrated irreducible ambiguity that survives the self-consistent 3D treatment and the stated complementary constraints — satisfies (iii) and does not by itself fail the task.

References

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