Recovering Ion Distribution Functions: II. Gyrotropic Slepian Reconstruction of Solar Wind Electrostatic Analyzer Measurements

TL;DR

This work extends the Slepian Basis Reconstruction (SBR) framework to gyrotropic ion distribution functions (GDF) for solar wind measurements with limited field-of-view. By exploiting gyrotropy, the g-SBR method reconstructs continuous VDFs from partial ESA data using three frameworks—polar-cap Slepians, 2D Cartesian Slepians, and a hybrid combination—anchored in field-aligned coordinates and a gyroaxis determined via MCMC/L-BFGS optimization. The approach preserves essential kinetic structures and plasma moments, enabling accurate density, velocity, and temperature estimates even when only a fraction of the VDF is observed (e.g., ≥90% recovery with as little as 20% of data). The accompanying gdf Python package provides open-source tools to implement these reconstructions, bridging particle observations with kinetic theory and simulations and supporting future heliophysics missions like Helioswarm.

Abstract

Velocity distribution functions (VDF) are an essential observable for studying kinetic and wave-particle processes in solar wind plasmas. To experimentally distinguish modes of heating, acceleration, and turbulence in the solar wind, precise representations of particle phase space VDFs are needed. In the first paper of this series, we developed the Slepian Basis Reconstruction (SBR) method to approximate fully agyrotropic continuous distributions from discrete measurements of electrostatic analyzers (ESAs). The method enables accurate determination of plasma moments, preserves kinetic features, and prescribes smooth gradients in phase space. In this paper, we extend the SBR method by imposing gyrotropic symmetry (g-SBR). Incorporating this symmetry enables high-fidelity reconstruction of VDFs that are partially measured, as from an ESA with a limited field-of-view (FOV). We introduce three frameworks for g-SBR, the gyrotropic Slepian Basis Reconstruction: (A) 1D angular Slepian functions on a polar-cap, (B) 2D Slepian functions in a Cartesian plane, and (C) a hybrid method. We employ model distributions representing multiple anisotropic ion populations in the solar wind to benchmark these methods, and we show that the g-SBR method produces a reconstruction that preserves kinetic structures and plasma moments, even with a strongly limited FOV. For our choice of model distribution, g-SBR can recover $\geq90\%$ of the density when only $20\%$ is measured. We provide the package \texttt{gdf} for open-source use and contribution by the heliophysics community. This work establishes direct pathways to bridge particle observations with kinetic theory and simulations, facilitating the investigation of gyrotropic plasma heating phenomena across the heliosphere.

Figures (10)

• Figure 1: Flow diagram illustrating the steps starting from the ESA measured 3D VDF leading to our GDF reconstruction and final moment calculation.
• Figure 2: Schematic figure showing the pre-processing steps before fitting the VDF using the gyrotropic model in Eqn. (\ref{['eqn:f_gyro_1DSlep']}). Panel (A) represents the VDF in the instrument frame before any pre-processing, the example magnetic field vector is entirely contained in the $v_x-v_y$ plane. For simplicity, we only show the FOV restricted ESA grids (using dots) in a single plane of elevation. Panel (B) demonstrates the shift of the VDF to the plasma frame for a certain bulk velocity $\boldsymbol{U}$. Panel (C) shows the final magnetic field aligned VDF, resulting in the $(v_{\parallel}, v_{\perp})$ axes. The bulk velocity vector in the instrument frame is shown with the blue arrow and the magnetic field direction is shown using the green arrow.
• Figure 3: Fitting architecture when using 1D Slepians on a polar cap. Panel (A) shows the distribution of grid points in black after rotating to FAC followed by boosting the frame to induce a maximum angle of $\Theta$ to the grid points about the origin. The series of operations of obtain the FAC is demonstrated in Fig. \ref{['fig:FAC']} and the frame boosting is performed according to Eqn. (\ref{['eqn:frame_boost']}). The set of B-splines corresponding to the synthetic ESA grids projected into FAC is shown in panel (B). The local support in radius for the red B-spline highlighted is shown as the shaded red arc in panel (A). All grid points encompassed by this B-spline are marked with a larger size. Panel (C) shows the Slepian functions optimally concentrated within angle $\Theta$ about the gyroaxis. The distribution of the grid points within the red arc of panel (A) is overplotted on each of the Slepian functions $S_{\alpha}(\theta)$. The vertical red dashed line marks $\Theta$ used for the polar cap shown in panel (A).
• Figure 4: MCMC corner plots corner of the Bayesian gyroaxis optimization using 8 walkers and 2000 steps each. The final solution from the scipy minimization $\widetilde{\boldsymbol{U}}$ was used as the initial guess about which walkers were randomly initialized. The $0.5$ quantile is indicated along with the 0.14 ($\approx -1\sigma$) and 0.86 ($\approx +1\sigma$) quantile error bars.
• Figure 5: Super-resolution of a synthetic VDF using for $\boldsymbol{\hat{b}} = 0.987 \, \boldsymbol{\hat{x}} -0.154 \, \boldsymbol{\hat{y}} -0.029 \, \boldsymbol{\hat{z}}$ and $\mathbf{u}_{\mathrm{bulk}}\mathrm{[km/s]} = -433.505 \, \boldsymbol{\hat{x}} + 107.615 \, \boldsymbol{\hat{y}} + 32.554 \, \boldsymbol{\hat{z}}$ for our synthetic VDF described in Sec. \ref{['sec:synthetic_demo']}. The left panel shows the super-resolved GDF as the background colormap. The synthetic data is overplotted as scattered points on an irregular grid as obtained from the boosted FAC. According to the calculations in Eqns. (\ref{['eqn:frame_boost']})-(\ref{['eqn:Theta_calc']}), $\Delta v_{\parallel} = 463.37$ km/s, $\Theta = 57.03^{\circ}$ and $N^{\mathrm{2D}}_{\mathrm{polcap}} = 4$ with 21 knots placed logarithmically according to Eqn. (\ref{['eqn:dlnv']}) across all the FAC grids. The white-dashed line marks the convex hull encompassing all the grids with significant counts. The right panel shows the trade-off plot between data-misfit and model-misfit. The knee of this L-curve, indicated by the black 'x', is used to find the optimal regularization constant $\mu$ to be used in Eqn. (\ref{['eqn:polcap_inv']}) to obtain the model coefficients $\mathbf{M_A}$. In this case, the B-spline regularization is found to be $\mu = 2.64 \times 10^{-7}$ from the knee of the trade-off curve.