Reformulating polarized radiative transfer for astrophysical applications (I). A formalism allowing non-local Magnus solutions

TL;DR

This work reframes polarized radiative transfer in Stokes space as rotations in a Lorentz/Poincaré group, enabling a non-local, non-constant integration of the RTE through the Magnus expansion. It derives a compact, exact-like homogeneous evolution operator expressed in terms of integrated Lorentz matrices, preserving Lie-group structure even when propagation matrices do not commute. To address the inhomogeneous term, it introduces a 5×5 homogeneous reformulation that yields an inhomogeneous evolution operator via a novel function $\boldsymbol{\phi}$, enabling efficient, analytic treatment of emissivity contributions. The framework is validated analytically and numerically, implemented in the HAZEL-Exp code, and promises accelerated NLTE computations and new avenues for cross-domain problems sharing the Lorentz/Poincaré algebra.

Abstract

The solar atmosphere is diagnosed by solving the polarized radiative transfer problem for plasmas in Non-Local Thermodynamic Equilibrium (NLTE). A key challenge in multidimensional NLTE diagnosis is to integrate efficiently the radiative transfer equation (RTE), but current methods are local, i.e. limited to constant propagation matrices. This paper introduces a formalism for non-local integration of the RTE using the Magnus expansion. We begin by framing the problem in terms of rotations within the Lorentz / Poincare group (Stokes formalism), motivating the use of the Magnus expansion. By combining the latter with a highly detailed algebraic characterization of the propagation matrix, we derive a compact analytical evolution operator that supports arbitrary variations of the propagation matrix and allows to increasingly consider any order in the Magnus expansion. Additionally, we also reformulate the inhomogeneous part of the RTE, again using the Magnus expansion, and introducing the new concept of inhomogeneous evolution operator. This provides the first consistent, general, and non-local formal solution to the RTE that is furthermore efficient and separates integration from the algebraic formal solution. Our framework is verified analytically and computationally, leading to a new family of numerical radiative transfer methods and potential applications such as accelerating NLTE calculations. With minor adjustments, our results apply to other universal physical problems sharing the Lorentz / Poincare algebra in special relativity and electromagnetism.

Figures (6)

• Figure 1: Vector fields (arrows) and solutions (red lines) in different wavelengths for the homogeneous RTE. Left: rotational flow due to magneto-optical effects for $\mathbf{K}$ constant in plane Q-U. The trajectory deviates from circularity to describe a spiral due to the accumulated error in a numerical method breaking Lie structure. Middle: idem for hyperbolic dichroic pseudo-rotation in I-V plane. Right: idem in QUV space for variable $\mathbf{K}$.
• Figure 2: Local homogeneous solutions (white circles) to the RTE evolving in the ordinary 3D space of the ray (orange) and in the 4D space of the Poincaré group (red). They result of locally applying the exponential map to linear vector fields (blue) in the Lie algebra.
• Figure 3: Geometry associated to $a_{\pm}$ and $a^2_{\pm}$. Left panel: steps deducing triangle r-q-h with a corner at $\lambda_1$. (1) As $|a_+|=\sqrt{h}$ is the side the romboid, $\vec{a}_{+}$ can be set from $\eta_0$ to the point $\lambda_1 +i\lambda_3$ (as if its origin were at $\eta_0$); (2) Obtain $\alpha$ through the area of the romboid $A_{\diamondsuit}=2|\hat{a}||\tilde{a}|=|q|=h\sin(2\alpha)$; (3) As $h^2=q^2+r^2$ and $\sin(2\alpha)=q/h$, then h, q, and r can be associated with sides of a rectangular triangle with a corner at $\lambda_1$; (4) Identify auxiliary triangles with sides $\hat{d},\tilde{d}$ and $\hat{D},\tilde{D}$, finding: $\hat{d}+i\tilde{d}=(\sqrt{h}-1)\cdot(|\hat{a}|+i|\tilde{a}|)$ and $\hat{D}+i\tilde{D} =\sqrt{h}\cdot(|\hat{a}|+i|\tilde{a}|)$. 5.- Finally, we identify a vector $a^2_{\pm}=r+iq$ enclosed in the diagonal of the rectangle of sides r-q-h with origin in $O'=\hat{O}'+i\tilde{O}'=\lambda_1-re^{i\alpha}$. The distance $C=OO'$ is then: $C = [(\hat{O}')^2+(\tilde{O}')^2]^ {1/2}=[\lambda^2_1-2r\cos(\alpha)\lambda_1+r^2]^{1/2}$. Right panel: $a^2_+$ and $a^2_-$ when their origin $O'$ is chosen at $\lambda_2$.
• Figure 4: Mandelbrot fractal arising when iterating the square of the propagation vector $C+a^2_{\pm}$ for different values of C. The iteration only converges in the white region. Two vectors for $C=\eta_0$ are represented.
• Figure 5: Stokes profiles in the Na${\rm I}$ D1 line for an arbitrary parametric solar atmosphere (Appendix \ref{['sec:atm_hazel_exp']}) at solar disk center ($\mu=1$) with $7$ points (bottom panels) and $97$ points (top panels). The methods are (see Sec. \ref{['sec:intro']}): Trapezoidal, DELO order 1, DELOpar order 1.5, DELOparabolic order 2, Evolution Operator, Magnus degraded to piecewise constant (M0), Magnus-order 1 with order-3 integrals (M1), Magnus-order 1 with trapezoidal integrals (Mtrap), Magnus-order 2 with order-3 integrals (M2).