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arxiv: 2607.00459 · v1 · pith:KIH2K42Jnew · submitted 2026-07-01 · 🌌 astro-ph.SR · math.OC· physics.space-ph

Optimization Algorithm for Determining the Source Surface Radius Based on Parker Solar Probe in situ Measurements from Encounters 1 to 19

Shiouhe Wang , Fang Shen , Yi Yang , Xueshang Feng , Jiansen He This is my paper

Pith reviewed 2026-07-02 06:27 UTC · model grok-4.3

classification 🌌 astro-ph.SR math.OCphysics.space-ph
keywords PFSS modelsource surface radiusParker Solar Probeopen magnetic fluxsolar cycleoptimization algorithmmagnetic polaritycoronal magnetic field
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The pith

The optimal source surface radius for PFSS models increases from solar minimum into the ascending phase when tuned to Parker Solar Probe data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an optimization algorithm to locate the best source surface radius in the Potential Field Source Surface model. It minimizes the mean squared error between the model's extrapolated radial field and Parker Solar Probe measurements after Parker spiral backmapping and radial scaling across encounters 1-19. The work proves the forward problem is well-posed and that a unique optimal radius exists. Results show this optimal radius grows as solar cycle 25 moves from minimum into its ascending phase. The tuned radius improves agreement on open flux while preserving or improving magnetic polarity accuracy relative to the fixed value of 2.5 solar radii.

Core claim

The paper proves the well-posedness of the PFSS forward problem and establishes existence and uniqueness of the optimal source surface radius. It defines the objective functional as the mean squared error between PFSS extrapolation and Parker Solar Probe radial magnetic field measurements after Parker spiral backmapping and radial scaling for Encounters 1-19. The optimization algorithm is validated with an analytical solution and cross-checked with ACE data. The optimal R_ss values generally increase from solar minimum into the ascending phase of solar cycle 25, and the resulting models improve open flux agreement while preserving or improving polarity prediction accuracy relative to 2.5 R_s

What carries the argument

The mean squared error objective functional between PFSS extrapolation and backmapped, radially scaled Parker Solar Probe radial field measurements, used to optimize the source surface radius.

If this is right

  • The optimal source surface radius increases from solar minimum into the ascending phase of solar cycle 25.
  • The optimized PFSS solution improves agreement with observed open flux relative to the standard 2.5 solar radii.
  • Polarity prediction accuracy is preserved or improved compared with the fixed 2.5 solar radii choice.
  • Pareto frontiers show open flux as the dominant metric during solar minimum and polarity accuracy as dominant during the ascending phase.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A cycle-dependent source surface radius could reduce systematic mismatches in models that currently assume a constant value.
  • The same optimization procedure applied to other in-situ datasets could test whether the radius trend holds at different heliocentric distances.
  • If backmapping assumptions introduce bias, the derived radii may partly reflect those procedural choices rather than purely coronal properties.

Load-bearing premise

The mean squared error between PFSS extrapolation and backmapped PSP radial field measurements after radial scaling accurately captures the true coronal magnetic topology and open flux without significant contamination from local structures or model assumptions in the backmapping procedure.

What would settle it

If the optimal source surface radius fails to increase when the algorithm is rerun on Parker Solar Probe encounters from later in the ascending phase of solar cycle 25, the reported trend would be contradicted.

Figures

Figures reproduced from arXiv: 2607.00459 by Fang Shen, Jiansen He, Shiouhe Wang, Xueshang Feng, Yi Yang.

Figure 1
Figure 1. Figure 1: (a) Illustration of the solution domain for PFSS extrapolation with a spherical [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Geometric setting of the free boundary problem for PFSS extrapolation. The [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (a) PSP/FIELDS radial magnetic field Br measurements in hourly time bins for CR2210, shown as boxplots used in the IQR outlier detection, with outliers marked by open circles. (b) PSP/FIELDS Br after the IQR cleaning and 20 minutes averaging, with red and blue denoting positive and negative polarity, respectively. (c) PSP/SWEAP radial solar wind velocity vr used for Parker spiral backmapping. The orange po… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Iteration process for the analytical validation. The blue gray curve shows the [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between BPSP,k r and Bk r for Encounters 1-19. Purple curves show the reference solution with Rss = 2.50Rs, black curves show the optimized solution from Algo￾rithm 1, and red and blue points denote positive and negative BPSP,k r , respectively. –24– [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Comparison between BPSP,k r and Bk r for PSP Encounter 1 during solar min￾imum. Red and blue points denote positive and negative BPSP,k r , respectively. The purple dashed curve denotes the reference Bk r with Rss = 2.5Rs, and the black solid curve denotes the optimized solution. (b) Comparison for PSP Encounter 17 during the ascending phase. ascending phase of solar cycle 25 [PITH_FULL_IMAGE:figures/… view at source ↗
Figure 7
Figure 7. Figure 7: (a) The optimal Rss for PSP Encounters 1-19. The red polyline with square mark￾ers shows the encounter by encounter optimization results, the black curve shows the spline used only to visualize the trend, and the blue curve shows the sunspot number over the same Carrington rotation range on the secondary vertical axis. (b) The optimal Rss for ACE datasets. The orange polyline with square markers shows the … view at source ↗
Figure 8
Figure 8. Figure 8: (a) Iteration process for PSP Encounter 9. The blue gray curve shows the evolving [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Open flux and polarity prediction accuracy for the PSP based evaluation. The [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Pareto analysis for PSP Encounter 17 in parameter space. The blue curve [PITH_FULL_IMAGE:figures/full_fig_p030_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Pareto frontiers for PSP Encounters 1-19 calculated with Algorithm 2. In each [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
read the original abstract

The Potential Field Source Surface (PFSS) extrapolation is a method for estimating the large scale coronal magnetic field from photospheric magnetograms. The source surface serves as the outer boundary of its solution domain, and is typically a spherical surface. An appropriate source surface radius ($R_{ss}$) enables more accurate identification of the coronal magnetic field topology and estimation of the open flux, thereby potentially enhancing the accuracy of space weather modeling. We prove the well-posedness of the PFSS forward problem and establish the existence and uniqueness of the optimal source surface by combining compactness of the admissible set with continuity of the objective functional. The objective functional is the mean squared error (MSE) between PFSS extrapolation and Parker Solar Probe (PSP) radial magnetic field measurements after Parker spiral backmapping and radial scaling for Encounters 1-19. The optimization algorithm is validated with an analytical solution, and Advanced Composition Explorer (ACE) in situ measurements are used as an independent cross-validation dataset. Additional evaluation metrics and Pareto analysis are used to identify the dominant metrics between open flux and polarity prediction accuracy. Our results show that the optimal $R_{ss}$ derived from the algorithm generally increase from solar minimum into the ascending phase of solar cycle 25. The optimized solution improves open flux agreement while preserving or improving polarity prediction accuracy relative to $2.5R_{s}$. The Pareto frontiers show a transition for dominant metrics from open flux during solar minimum to polarity prediction accuracy during the ascending phase.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript develops an optimization algorithm to determine the optimal source surface radius R_ss for PFSS extrapolations by minimizing the mean squared error between the model radial field at the source surface and Parker Solar Probe in-situ measurements (after Parker spiral backmapping and radial scaling) across encounters 1–19. It claims a proof of well-posedness of the PFSS forward problem together with existence and uniqueness of the optimum via compactness and continuity arguments, reports that the resulting optimal R_ss values increase from solar minimum into the ascending phase of cycle 25, and states that the optimized solutions improve open-flux agreement while preserving or improving polarity accuracy relative to the conventional 2.5 R_s choice. Independent validation with ACE data and Pareto analysis of open-flux versus polarity metrics are also presented.

Significance. If the backmapping and scaling assumptions hold, the work supplies a reproducible, data-driven procedure for selecting a solar-cycle-dependent R_ss that could improve open-flux estimates used in space-weather modeling. The explicit proof of well-posedness and the use of an independent ACE cross-validation set are positive features that strengthen the methodological contribution.

major comments (3)
  1. [abstract (objective functional)] Abstract, paragraph on objective functional: the MSE is formed after Parker spiral backmapping and 1/r² radial scaling; no sensitivity tests to the assumed solar-wind speed profile or to possible non-radial flow distortions are reported. Because the headline trend (increasing optimal R_ss with cycle phase) is produced by this minimization, the absence of such tests leaves open the possibility that the reported variation is an artifact of the mapping procedure rather than a signature of changing coronal topology.
  2. [abstract (cross-validation)] Abstract and results on cross-validation: while ACE data are described as an independent validation set, the manuscript does not state whether the R_ss values optimized on PSP encounters are applied unchanged to ACE or whether a separate optimization is performed; without this clarification the claimed independence of the trend confirmation cannot be assessed.
  3. [abstract (well-posedness)] Abstract, well-posedness claim: the compactness-plus-continuity argument establishes existence and uniqueness for the mathematical optimization problem as formulated, yet supplies no argument that the chosen objective functional remains a faithful proxy once local coronal structures or transients (unrepresented by PFSS) contaminate the PSP samples; this gap directly affects the physical interpretation of the optimized R_ss sequence.
minor comments (2)
  1. The admissible set for R_ss and the precise form of the radial-scaling operator should be stated explicitly (including any bounds or discretization) to allow readers to reproduce the compactness argument.
  2. Figure captions and table headings should indicate the exact number of PSP samples retained after quality filtering for each encounter.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed report. We address each major comment below, clarifying the manuscript's approach and indicating where revisions will be made.

read point-by-point responses

  1. Referee: [abstract (objective functional)] Abstract, paragraph on objective functional: the MSE is formed after Parker spiral backmapping and 1/r² radial scaling; no sensitivity tests to the assumed solar-wind speed profile or to possible non-radial flow distortions are reported. Because the headline trend (increasing optimal R_ss with cycle phase) is produced by this minimization, the absence of such tests leaves open the possibility that the reported variation is an artifact of the mapping procedure rather than a signature of changing coronal topology.

    Authors: The manuscript does not report sensitivity tests to solar-wind speed profiles or non-radial flows, which is a valid observation. The backmapping follows standard Parker spiral assumptions with a representative speed. In revision we will add an appendix with sensitivity tests using a range of constant speeds (300-600 km/s) and a variable-speed model, quantifying the effect on the optimal R_ss sequence and the reported trend. This will allow readers to assess robustness directly. revision: yes

  2. Referee: [abstract (cross-validation)] Abstract and results on cross-validation: while ACE data are described as an independent validation set, the manuscript does not state whether the R_ss values optimized on PSP encounters are applied unchanged to ACE or whether a separate optimization is performed; without this clarification the claimed independence of the trend confirmation cannot be assessed.

    Authors: The R_ss values optimized on PSP encounters 1-19 were applied unchanged to the corresponding ACE intervals; no separate optimization was performed on ACE. This is the intended meaning of 'independent cross-validation.' We will revise the abstract and Section 4 to state this procedure explicitly, removing any ambiguity about the validation design. revision: yes

  3. Referee: [abstract (well-posedness)] Abstract, well-posedness claim: the compactness-plus-continuity argument establishes existence and uniqueness for the mathematical optimization problem as formulated, yet supplies no argument that the chosen objective functional remains a faithful proxy once local coronal structures or transients (unrepresented by PFSS) contaminate the PSP samples; this gap directly affects the physical interpretation of the optimized R_ss sequence.

    Authors: The compactness-continuity argument establishes existence and uniqueness strictly for the mathematical problem as posed. It does not address whether the MSE remains a faithful proxy when PFSS assumptions are violated by transients or unresolved coronal structure. The manuscript presents the optimized R_ss as an empirical, data-driven choice within the PFSS framework and reports the resulting open-flux improvement as an observed outcome. We will add a limitations paragraph in the discussion acknowledging this gap and its implications for physical interpretation. revision: partial

Circularity Check

0 steps flagged

No circularity; optimization result is direct empirical output of MSE fit with independent validation

full rationale

The paper defines an objective functional as MSE between PFSS Br (after backmapping/scaling) and PSP data for encounters 1-19, proves mathematical existence/uniqueness of the minimizer via compactness and continuity, applies the algorithm, and reports the resulting trend in optimal R_ss plus open-flux improvement. This is a standard data-driven optimization whose output trend is not forced by definition or prior self-citation; ACE serves as external cross-validation and an analytical solution validates the solver. No step reduces the central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of the PFSS potential-field assumption, the accuracy of Parker spiral backmapping and radial scaling for in-situ data, and the choice of MSE as the objective that correctly represents coronal topology.

free parameters (1)
  • R_ss
    The source surface radius is the parameter being optimized via the MSE objective on PSP data.
axioms (2)
  • domain assumption PFSS forward problem is well-posed with existence and uniqueness of optimal source surface
    Claimed in abstract but not verifiable without full text.
  • domain assumption Backmapping and radial scaling preserve the relevant magnetic field information for comparison
    Central to the objective functional definition.

pith-pipeline@v0.9.1-grok · 5820 in / 1462 out tokens · 23460 ms · 2026-07-02T06:27:54.774726+00:00 · methodology

discussion (0)

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