Eddington Inversion Techniques

• Eddington inversion is a method that reconstructs stellar and dark matter distribution functions from observed density and potential profiles using Abel-type integral equations.
• It underpins the Milne–Eddington approach in solar magnetometry by simplifying radiative transfer with a linear source function to infer average atmospheric parameters.
• Advanced techniques—including regularization, machine learning, and Bayesian frameworks—enhance spatial resolution and quantify uncertainties in astrophysical models.

Eddington inversion encompasses a family of analytical and numerical techniques that exploit the relationship between observable distributions (e.g., polarized light spectra, star counts, surface brightness) and the underlying physical state of an astrophysical system (e.g., vector magnetic field distributions, dark matter phase-space densities), under the simplifying assumption of a (quasi-)constant parameter set. Originally formalized for the derivation of stellar and dark matter distribution functions from observed density and potential profiles, and independently for the extraction of average atmospheric quantities in magnetic field diagnostics via the Milne–Eddington (ME) approximation, Eddington inversion is foundational in astrophysical inference.

1. Mathematical Foundations and Classic Formulations

At its core, the Eddington inversion method is an application of Abel-type integral equations that connect integrated or projected observables (such as the spatial mass density ρ(r), surface brightness, or Stokes profiles) to an underlying distribution function f (e.g., the phase-space DF in galaxies or the constant physical atmosphere parameters in solar physics). For a non-rotating, spherically symmetric, isotropic system, the classical Eddington inversion for a phase-space DF reads: $$\rho(r) = 4\pi\sqrt{2} \int_0^{\Psi(r)} f(\mathcal{E}) \sqrt{\Psi(r) - \mathcal{E}}\, d\mathcal{E}$$ where $\rho(r)$ is the mass density, $\Psi(r) = \Phi_0 - \Phi(r)$ is the relative potential, and $f(\mathcal{E})$ is the DF as a function of relative energy. The inversion yields: $$f(\mathcal{E}) = \frac{1}{\sqrt{2}\pi^2} \frac{d}{d\mathcal{E}} \int_0^{\mathcal{E}} \frac{d\rho}{d\Psi}\frac{d\Psi}{\sqrt{\mathcal{E} - \Psi}}$$ For the Milne–Eddington formalism applied to radiative transfer in stellar atmospheres, the defining assumption is that all relevant atmospheric parameters (magnetic field vector, bulk velocity, opacity, Doppler width, etc.) are constant with optical depth, and the source function is linear: $S(\tau) = S_0 + S_1 \tau$ This assumption enables an analytical solution to the polarized radiative transfer equation and allows for direct inversion algorithms to retrieve “average” physical quantities from observed Stokes profiles.

2. Eddington Inversion in Solar Magnetometry and the Milne–Eddington Paradigm

Eddington inversion is central in modern solar spectropolarimetric diagnostics, most notably through the Milne–Eddington approximation. Here, inversion codes (e.g., MILOS, VFISV, HeLIx⁺, ASP/HAO) infer the magnetic vector and thermodynamic parameters from observed Stokes profiles by fitting analytical solutions to the radiative transfer equation under the ME model. These solutions assume a linear source function and fixed atmospheric parameters, yielding nine free parameters per pixel (see (Suárez et al., 2010, Borrero et al., 2014)):

• Source function (S₀, S₁)
• Line-to-continuum opacity ratio (η₀)
• Doppler width (Δλ_D)
• Damping parameter (a)
• Magnetic field strength (B)
• Field inclination (γ)
• Field azimuth (χ)
• Line-of-sight velocity (v_LOS)

The practical ME inversion workflow involves generating synthetic profiles from these parameters via the Unno–Rachkovsky solution and minimizing a merit function (typically χ²) comparing synthetic to observed profiles. For widely used lines (Fe I 630.1/630.2 nm, Fe I 1.56 μm), simultaneous dual-line ME inversion improves parameter constraints through the increased number of observables (Suárez et al., 2010).

3. Physical Interpretation and Systematic Limitations

The output of an ME Eddington inversion is a spatial map of “average” atmospheric parameters that are, in effect, weighted means of the true, height-dependent atmospheric stratification over the line formation region (Suárez et al., 2010, Borrero et al., 2014). Empirically, different parameters correspond to different effective formation heights, which may vary with local structure (granule vs. intergranular lane) and physical conditions. Uncertainties arising from the ME assumption of vertical homogeneity dominate over photon noise for high-quality data, with rms errors of order 30 G (B), 13–20° (field angles), and ≈500 m s⁻¹ (v_LOS).

ME inversions do not assign a parameter to a unique physical layer; e.g., the formation depth for B is typically log τ ∼ −1.1, while for v_LOS it is log τ ∼ −1.0—with these depths shifting to log τ ≈ −0.3 to −0.7 in granules and log τ ≈ −1.3 to −1.7 in intergranular lanes (Suárez et al., 2010). This spatial variability must be considered in interpreting inversion maps.

A further limitation is the inability of the ME model to fit asymmetric and/or multi-lobed Stokes profiles arising from steep gradients or discontinuities in atmospheric parameters. In such cases, ME inversion returns an “average” value that may not correspond well to any physical layer, but remains useful for statistical analysis.

4. Algorithmic Advances, Regularization, and Machine Learning Approaches

Conventional ME inversion methods are computationally demanding—each pixel’s inversion is non-linear and high-dimensional. Recent work introduces extensive optimizations:

• Weight-rescaling to equalize Stokes I, Q, U, V impact (Centeno et al., 2014),
• Parameter reduction and regularization to mitigate degeneracies (e.g., fixing damping, penalizing large η₀, variable transformations for faster convergence),
• Hybrid explicit/implicit spectral evaluation to accelerate the forward model,
• Parallelization (MPI, GPU) for scalability (Harker et al., 2012, Harker, 2017).

Sparsity-driven and spatially-coupled inversion codes implement global regularization by minimizing the χ² function in a transformed (e.g., wavelet) domain, enforcing that only a sparse subset of the transform coefficients are nonzero. This approach exploits the spatial correlation in parameter maps and yields reconstructions with higher noise resilience and better physical plausibility, while straightforwardly accounting for the telescope PSF (Ramos et al., 2015).

Physics-informed neural networks (PINNs) further generalize Eddington inversion, embedding the ME analytic model within the loss functional of a deep network that maps (x, y, t) coordinates to ME parameter sets (Jarolim et al., 19 Feb 2025, Li et al., 12 Jul 2025). This generates intrinsically regularized, spatio-temporally coupled magnetograms and allows direct inclusion of PSF effects. The framework achieves rapid, robust inversion (sub-minute full-disc maps), resolves fine-scale structure in both strong and weak-field regimes, and exhibits robust recovery of the magnetic field vector (correlation coefficients ≈0.9 with established pipelines).

Recent Bayesian approaches based on variational inference and normalizing flows enable rapid, multi-parameter posterior sampling, quantifying ambiguities and uncertainties in ME (and general radiative transfer) inversions at negligible online cost after training (Baso et al., 2021).

Technique	Benefit	Limitation
ME pixelwise inversion	Analytical, fast, statistical overview	Lacks gradient/asymmetry modeling
Sparse/global inversion	Spatial regularization, PSF deconvolution	Assumes smooth parameter fields
PINN (Physics-Informed NN)	Fast, robust, spatio-temporal coherence	Still ME approximation
Normalizing flows Bayesian	Fast, full posterior, uncertainty quantified	Needs comprehensive training set

5. Extensions and Modifications: Non-LTE Effects and Atmospheric Structure

The classical ME approach is tailored for LTE (photospheric) lines. For chromospheric lines formed in non-LTE regimes (e.g., Mg I b₂, Ca II), the traditional linear source function is insufficient to reproduce observed asymmetric or emission-core line shapes. A modified ME source function with additional depth-dependent exponential terms: $$S(\tau) = S_0 + S_1 \tau + A_1 e^{-\alpha_1\tau} - A_2 e^{-\alpha_2\tau}$$ enables fits that capture core emission and broad wings, with negligible new trade-offs or degeneracies between the added parameters and the magnetic/kinematic ones (Dorantes-Monteagudo et al., 2021). This generalized approach remains robust against moderate noise and provides improved inference for chromospheric diagnostics.

6. Eddington Inversion in Galactic Dynamics and Dark Matter Astrophysics

Beyond radiative transfer, Eddington inversion is pivotal in dynamical astronomy for reconstructing the phase-space DF of a system from observed densities and potentials. This is vital for dark matter haloes: $$\rho(r) = 4\pi \sqrt{2} \int_0^{\Psi(r)} f(\mathcal{E}) \sqrt{\Psi(r) - \mathcal{E}} d\mathcal{E}$$

$$f(\mathcal{E}) = \frac{1}{\sqrt{2}\pi^2} \int_0^{\mathcal{E}} \frac{d^3\rho}{d\Psi^3} \sqrt{\mathcal{E} - \Psi} d\Psi$$

Applied to observed surface brightness of galaxies, Eddington inversion allows testing the physical consistency of a proposed stellar profile and a dark matter potential: if the computed f(ε) < 0 for any ε, that model is ruled out. For low-mass galaxies, profiles with a cored stellar density (dρ/dr → 0 at r→0) are incompatible with NFW (cuspy) potentials, a result robust even when allowing for relaxed axisymmetry or variable velocity anisotropy (Almeida et al., 23 Jul 2024). This provides a photometry-only constraint on DM halo structure.

Boundary effects are nontrivial: neglecting the finite halo radius introduces pathological divergences in the DF near escape energy (e.g., a 1/√𝒜 divergence), which must be regularized through artificial profile smoothing or King-like DF truncation (Lacroix et al., 2018).

7. Astrophysical Consequences, Validation, and Outlook

ME Eddington inversion remains the standard in high-resolution solar magnetic field inference as evidenced by modern pipeline codes for Hinode/SP, SDO/HMI, and BBSO/NIRIS (Centeno et al., 2014, Harker, 2017, Jarolim et al., 19 Feb 2025, Li et al., 12 Jul 2025). Comparisons between independent ME codes on synthetic (MHD) and observed data yield agreement within ΔB ≲ 35–130 G and Δγ ≲ 1–5°, depending on the comparison metric and stratification (Borrero et al., 2014). However, attempts to recover intrinsic field components (e.g., via variable magnetic filling factor) are susceptible to substantial systematic biases from foreshortening, PSF mixing, noise, and the ME model’s neglect of gradients (Centeno et al., 2023).

For galactic and DM astrophysics, Eddington inversion constrained by photometry now provides a method to rule out incompatible halo potentials in low-mass systems; in particular, NFW-like DM halos are inconsistent with strictly cored stellar light profiles when spherical symmetry and velocity isotropy hold (Almeida et al., 23 Jul 2024).

The evolution of Eddington inversion methods—through spatial and temporal regularization, Bayesian sampling, machine-learning acceleration, and analytic generalizations—continues to advance the field across both stellar and galactic scales, providing physically grounded, interpretable, and computationally efficient inference frameworks in the analysis of large, complex astrophysical datasets.