Solar Wind Helios 2 Analysis — Bi-Kappa VDF Fitting & Radial Evolution

Analysis of Helios 2 solar wind proton velocity distribution functions (VDFs) using bi-kappa distribution fitting, and a comprehensive power-law study of the radial evolution of 50+ plasma and field parameters between 0.29 and 1 AU.

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Overview

The solar wind is a weakly collisional plasma, meaning its particle velocity distributions routinely deviate from a Maxwellian — developing temperature anisotropies, field-aligned beams, and enhanced suprathermal tails. The bi-kappa distribution captures these non-Maxwellian features through a single spectral index κ, which converges to a Maxwellian as κ → ∞ and produces progressively heavier power-law tails as κ decreases.

This project fits bi-kappa distributions to every Helios 2 proton VDF and extracts the proton core number density n_p, bulk velocity v, parallel temperature T_∥, perpendicular temperature T_⊥, and spectral index κ. These fitted parameters, together with the concurrent magnetic field, are used to compute a comprehensive set of derived plasma quantities and characterise their radial dependence via power-law fits f ∝ r^α.

Key results

Quantity	Power-law index α	CGL / Parker prediction
Proton density n_p	−2.075	−2 (mass flux)
Total magnetic field B	−1.530	— (mixed)
B_r (radial component)	−1.873	−2 (Parker)
B_t (tangential component)	−1.214	−1 (Parker)
Perp. temperature T_⊥	−1.315	−2 (CGL)
Par. temperature T_∥	−0.825	0 (CGL)
Bulk speed V	≈ 0	0 (terminal velocity)
Perp. entropy S_⊥	+2.290	0 (CGL)
Par. entropy S_∥	+0.559	0 (CGL)
Proton magnetic moment μ_p	+0.215	0 (CGL invariant)

The κ index decreases systematically from ~4–5 at 1 AU to ~2.5–3 at 0.3 AU, indicating progressively stronger non-Maxwellian character closer to the Sun — consistent with greater wave activity and velocity filtration effects in the inner heliosphere.

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Notebook

SWHA_Helios2_FINAL.ipynb

The single self-contained analysis notebook. It runs end-to-end from raw Helios 2 data to all figures and the power-law index table. The notebook is structured as follows:

Section	Description
1. Data loading	Load Helios 2 proton and magnetic field data via heliopy
2. VDF inspection	Plot raw 3D VDFs, identify population boundaries (maxima/minima)
3. Population cut	Isolate proton core from beam and alpha particles in E/q space
4. Bi-kappa fitting	Fit drifting bi-kappa distributions to each proton VDF
5. Parameter extraction	Extract n_p, v, T_∥, T_⊥, κ; rotate to RTN frame
6. Derived quantities	Compute pressures, betas, Mach numbers, entropy proxies, etc.
7. Radial fits	Power-law OLS fits f ∝ r^α for all quantities
8. Figures	Generate all publication figures

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Bi-Kappa Distribution

The proton core is modelled as a drifting, gyrotropic bi-kappa distribution in the magnetic-field-aligned frame:

$$f_\kappa(v_\parallel, v_{\perp 1}, v_{\perp 2}) = \frac{n_p}{\pi^{3/2} \kappa^{3/2} w_\parallel w_\perp^2} \frac{\Gamma(\kappa+1)}{\Gamma(\kappa - 1/2)} \left[ 1 + \frac{1}{\kappa} \left( \frac{(v_\parallel - u_\parallel)^2}{w_\parallel^2} + \frac{v_{\perp 1}^2 + v_{\perp 2}^2}{w_\perp^2} \right) \right]^{-(\kappa+1)}$$

with κ > 3/2 and physical temperatures

$$T_{\parallel/\perp} = \frac{m_p w_{\parallel/\perp}^2}{2k_B} \cdot \frac{\kappa}{\kappa - 3/2}$$

Fitting is done by minimising squared residuals in linear (not log) phase-space density using a bounded Levenberg–Marquardt algorithm, with the fitting window restricted to the proton core region below the beam saddle point.

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Installation

git clone https://github.com/salastro/swha.git
cd swha
pip install -r requirements.txt
jupyter notebook notebooks/SWHA_Helios2.ipynb

Requirements

numpy
scipy
sympy
matplotlib
pandas
seaborn
mpl-scatter-density

A full requirements.txt is provided. Python 3.9+ is recommended. The notebook was developed and tested on Python 3.11.

Data

The Helios 2 proton dataset used here is the reprocessed dataset from Stansby et al. (2018), downloaded automatically via heliopy. Magnetic field data are from the Helios E2/E3 fluxgate magnetometers, also available through heliopy. No manual data download is required.

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References

• Parker (1958) — Dynamics of the interplanetary gas and magnetic fields. ApJ, 128, 664.
• Vasyliunas (1968) — A survey of low-energy electrons in the evening sector of the magnetosphere. JGR, 73, 2839.
• Marsch et al. (1982) — Solar wind protons: three-dimensional velocity distributions and derived plasma parameters measured between 0.3 and 1 AU. JGR, 87, 52.
• Stansby et al. (2018) — A new inner heliosphere proton parameter dataset from the Helios mission. Sol. Phys., 293, 155. [DOI]
• Stansby et al. (2019) — Alpha particle thermodynamics in the inner heliosphere fast solar wind. A&A, 623, L2. [DOI]
• Perrone et al. (2019) — Radial evolution of the solar wind in pure high-speed streams: HELIOS revised observations. MNRAS, 483, 3730. [DOI]
• Livadiotis (2015) — Kappa distribution in the solar wind. JGR, 120, 1607.

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License

MIT — see LICENSE.