For SIRs, CIRs, and Beyond: Polarization Ratio to Feature Location

Christopher J. Scott; Craig DeForest; Curt A. de Koning; Dusan Odstrcil; Sarah Gibson; Vic Pizzo

REVIEW 1 major objections 5 minor 81 references

For SIRs, CIRs, and Beyond: Polarization Ratio to Feature Location

T0 review · 1 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper claims that, in the small-Sun and small-feature limit, the polarization ratio of Thomson-scattered white light reduces to cos² of the scattering angle, so a single measured ratio gives a feature's line-of-sight position up to a f

desk verdict Solid analytic framework for PUNCH polarization-ratio images, with honest limits; the single-feature assumption is the main practical caveat, but the paper already admits it. read the letter →

arxiv 2607.23043 v1 pith:NF562NMW submitted 2026-07-25 physics.space-ph astro-ph.IMastro-ph.SR

Curt A. de Koning , Dusan Odstrcil , Sarah Gibson , Craig DeForest , Vic Pizzo , Christopher J. Scott This is my paper · ORCID

classification physics.space-phastro-ph.IMastro-ph.SR

keywords polarizationratioThomsonscatteringstreaminteractionregioncorotatingheliosphericwhite-lightimagingsuperparticleconstructionspherePUNCHmission

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that polarization-ratio images from the PUNCH heliospheric imager can place small, isolated solar-wind features (SIRs/CIRs) along the line of sight. For a narrow feature, the ratio of radially- to tangentially-polarized Thomson-scattered radiance depends only on location, not on the density profile: PR = cos²χc, yielding two candidate scattering angles symmetric about the Thomson sphere. For finite-width features, the paper derives exact closed forms for two toy densities (a radially expanding slab and a compression pulse), each showing that one PR measurement constrains a curve in (location, width) space rather than a unique pair. Under the single-feature assumption, the systematic location error from the point-particle approximation is generally below 10% at the tangent 'bean' positions for features narrower than about 15° half-width. The paper also warns that the single-feature assumption is fragile: with multiple features along one line of sight, PR becomes a brightness-weighted average over unknown positions, widths, and densities—an unresolved tension.

What carries the argument

The central object is the polarization ratio PR = BR/BT, the ratio of radially-polarized to tangentially-polarized Thomson-scattered radiance in the small-Sun limit. Its load-bearing property is that in the point-particle (superparticle) limit it collapses to cos²χc, where χc is the scattering angle at the feature's location; this is established by taking the line-of-sight density to a delta function (Eq. 28) or by taking the width Δχ→0 in either finite-width model (Eqs. 21, 24). The Thomson sphere—the sphere with the Sun and observer as antipodal points—marks χc = 90° and separates the two candidate locations. The two toy densities, a radially expanding slab (boxcar with r⁻² falloff) and a

What would settle it

A direct test is to compute synthetic polarization-ratio images from a time-dependent MHD simulation of the solar wind at solar maximum (with fragmented density islands), apply the single-feature inversion to each pixel, and compare the recovered χ_spc to the true brightness-weighted centroid of the line-of-sight density. If the inversion systematically outputs a location that matches no actual density feature—or if the two candidate solutions both miss the true position by more than the quoted 10%—then the central claim fails for realistic conditions.

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Extended reading notes

Core claim

The central claim is that for any line-of-sight electron density that is narrow compared with the Thomson-scattering geometry, the polarization ratio PR becomes cos²χc, independent of the density profile. This is shown by the superparticle construction—collapsing all scatterers on a line of sight into one point particle—which gives PR = cos²χspc, so that χspc = cos⁻¹(±√PR), with the plus sign placing the feature inside the Thomson sphere and the minus sign placing it outside. For extended features, the closed-form expressions (Eq. 21 for the radially expanding slab, Eq. 24 for the compression pulse) show that PR depends on both the central scattering angle χc and the half-width Δχ, making th

Load-bearing premise

The load-bearing premise is that each line of sight contains exactly one isolated, background-subtracted, symmetric finite feature whose density matches one of the two toy models; if multiple features are present, the measured polarization ratio becomes a brightness-weighted average over unknown positions, widths, and densities, and the recovered 'location' has no clear physical meaning.

Editorial extensions

If this is right

A PUNCH WFI polarization-ratio image of an isolated, narrow SIR/CIR can be inverted to a line-of-sight position with two solutions, one inside and one outside the Thomson sphere.
For finite-width features, one PR measurement constrains only one parameter; an independent estimate of either location or width is required, with the choice depending on monotonicity regions of the closed-form expressions.
The leading edge of an SIR, where the angular width is smallest, gives the most accurate location estimate under superparticle construction.
Two line-of-sight probes through the same feature—one at the leading edge, one at a broad tangent—can, in principle, recover both location and width.
The same polarization-ratio framework extends to CMEs through a hollow-shell density variant, whose closed form has the same functional structure as the SIR compression-pulse result.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: The front/back ambiguity means polarization-ratio localization will produce paired ghost positions on opposite sides of the Thomson sphere; independent constraints (in-situ measurements, a second viewpoint, or elongation-time tracks) will be needed to select one candidate.
Inference: At solar maximum, when PUNCH's main mission occurs, many lines of sight will likely contain several density islands; the brightness-weighted mean behavior of PR suggests the recovered location will be biased toward the brightest feature, so combining PR with total-radiance and morphological information may partially lift the degeneracy.
Inference: The quoted "below 10%" errors are probably lower bounds, because the ground-truth densities used in the error budget are only toy models; a testable extension would be to synthesize polarization-ratio images from a high-fidelity, time-dependent MHD simulation with known true density and compare the inverted positions against the actual density-weighted centroids.
Inference: The mirror symmetry PR(χc) = PR(π−χc) implies that any single-view inversion inherits a fundamental ambiguity that cannot be resolved by improving measurement precision alone; the paper's Figure 21 error map should be read as a best-case, single-feature, single-model estimate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

One thing to know: this is a genuinely useful analytic methods paper for PUNCH, and its central result—in the small-Sun, small-feature, single-feature limit, PR always reduces to cos²χc—is proven cleanly and honestly. The closed-form PR expressions for a radially expanding slab (Eq. 21), a compression pulse (Eqs. 24–25), and a cavity (Eq. 61) are new, as is the first systematic error map for SuperParticle Construction (Figure 21) and the multi-feature weighted-mean result (Eqs. 51–59). I independently re-derived the main integrals; they check out, with correct limits (PR→cos²χc as Δχ→0 and PR→0.25 as ε→0). The paper is also unusually candid about its own limits. The Q.a test, the Section 7 multi-feature collapse, and the 'minimum values' language in Section 8 are prominent and not buried. That honesty is a real strength.

The soft spots are mostly the flip side of that honesty. The single-feature line-of-sight assumption is load-bearing for the practical claim that PR images can localize SIR/CIR features. The paper's own WSA-ENLIL figures show patchy, fragmented density islands, and PUNCH's prime mission is at solar maximum, so multi-feature lines of sight will be common rather than exceptional. Section 7 shows that with K≥2 features, PR becomes a brightness-weighted mean over unknown positions, widths, and densities; in that case Eq. 36's χspc may not correspond to any physical feature. The stress-test note is right about that. But the paper already says this is a 'final, unresolved tension.' So it's a scope limitation, not a hidden flaw. Similarly, the Figure 21 error budget assumes one toy density as ground truth, so the quoted uncertainties are lower bounds; the authors concede this. The ±error claims around Figure 11 are visually propagated from an assumed PR uncertainty, and no code or data are shipped. Those are minor issues.

Who should read this: anyone preparing to interpret PUNCH polarization-ratio images, and anyone working on Thomson-scattering diagnostics of SIRs or CMEs. It's a preparatory framework, not an observational result. The negative results—the location–width degeneracy, the front/back ambiguity, the multi-feature collapse—are themselves the useful output. It deserves a serious referee: the math is sound, the scope is clearly stated, and the limitations are handled honestly. I would send it to peer review.

Referee Report

1 major / 5 minor

Summary. The paper investigates the polarization ratio PR=B_R/B_T for Thomson-scattered white light in the small-Sun limit. It introduces two line-of-sight electron density models: a radially expanding slab (boxcar in angular coordinate with r^-2 falloff, Eq. 16) and a compression pulse (Eq. 23). Substituting these into the PR integral (Eq. 9) yields closed-form expressions, Eq. (21) and Eq. (24), showing PR depends on the feature's central angular position χc and half-width Δχ. In the small-feature limit both models reduce to PR→cos²χc, giving the location inversion χspc=cos⁻¹(±√PR) (Eq. 36) with a two-fold front/back ambiguity. The paper quantifies the systematic error of this superparticle approximation in Figure 21 and analyzes multi-feature lines of sight in Section 7, where PR becomes a brightness-weighted mean. The authors are explicit that the single-feature assumption is a caveat and that the multi-feature case remains an unresolved tension.

Significance. The analytical results are significant for the PUNCH mission: they provide closed-form, parameter-free predictions for the polarization ratio of idealized SIR/CIR features, and they demonstrate both the power and the fundamental degeneracies of PR-based localization. The manuscript's main strengths are its self-contained first-principles derivation (Eq. 3 → Eq. 9 → Eqs. 21/24), explicit lists of assumptions, and honest treatment of limitations, including the multi-feature degeneracy and the remark that the Figure 21 errors are lower bounds. The limiting result PR→cos²χc is robust and density-independent, making it a useful rule of thumb for PUNCH WFI data. The practical applicability is limited to isolated, small features, but that limitation is clearly stated in the text; the mathematical framework is sound.

major comments (1)

[Section 4.1, Eq. (16)] The displayed density has sin²χ in the denominator: n_e = n_⊙/(r_obs² sin²ε sin²χ). This contradicts Eq. (18), where the same density is expanded as (n_⊙/2r_obs² sin²ε)[1−cos2χc cos2Δ+...], and the text statement that Eq. (16) has a maximum at χc=90°. With n(r)=n_⊙/r² and r=r_obs sinε/sinχ, the correct χ-dependence is n_e ∝ sin²χ. As written, Eq. (16) would give a minimum at 90° and would not lead to Eq. (21). Please correct the denominator/numerator and check the derivation of Eq. (21) accordingly (the subsequent equations suggest the intended form is n_e = n_⊙ sin²χ/(r_obs² sin²ε)).

minor comments (5)

[Abstract] The abstract states that polarization ratio images 'will provide three-dimensional location information' without qualification. Given the paper's own Q.a requirement and the multi-feature analysis in Section 7, I recommend adding a condition such as 'for an isolated, single feature along the line of sight.'
[Figure 29 caption] The caption says the bottom panel is a plot of Equation 114 twice; it should be Equation 117 for the bottom panel.
[Eqs. (21), (27)] Please define sinc(x)=sin(x)/x at first use.
[Section 5] The phrase 'the polarization ratio always reduces to PR→cos²χc' is too strong without restating the small-Sun and small-feature limits in the same sentence. The surrounding text is careful, but this sentence should carry the qualifiers.
[Figure 21 and Section 8] Consider explicitly labeling the quoted errors as systematic errors due to the superparticle approximation under the two toy ground-truth densities, to avoid confusion with measurement noise. The text makes this point, but a caption note would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central PR-to-location result is derived self-contained by integration and limiting arguments; self-citations are contextual only.

full rationale

The derivation chain is self-contained. Equation 3 is the standard Thomson-scattering radiance integral; the small-Sun approximation converts it to Equations 4–9. Substituting the two stated model densities (Equations 16 and 23) and integrating gives the closed-form polarization ratios (Equations 21 and 24). The small-feature limit is taken independently both by letting Δχ→0 in those formulas (Equations 30–31) and by a delta-sequence argument (Equations 32–35), yielding PR→cos²χ_c. Equation 36 is simply the algebraic inversion of Equation 29. No parameter is fitted to any dataset, and no load-bearing result is imported from the authors' prior work: SuperParticle Construction is explicitly stated as an assumption in Section 5 and then used in this paper to derive Equation 29; the de Koning (2017) citation supplies the name and context, not the proof. The same holds for the other self-citations (de Koning 2014; Pizzo et al. 2011): they are contextual. The paper's own acknowledged limitations—Q.a, Section 7, and the 'final, unresolved tension'—describe the validity domain (single-feature lines of sight) and show that with multiple features PR becomes a brightness-weighted mean (Equations 54, 56, 59). That is an honest limitation of the practical claim, not circular reasoning. Figure 21 is a forward error analysis that assumes a toy ground-truth density and then computes the SPC mislocation; it is not a fitted prediction. Thus there is no circular step by construction.

Assumptions & free parameters 6 free parameters · 8 assumptions · 0 invented entities

Everything the central claim rests on: the small-Sun truncation of the van de Hulst coefficients, two toy density families (r⁻² boxcar slab and cos²ᶜ pulse), optically thin single Thomson scattering with perfect background subtraction, and exactly one symmetric feature per line of sight. No parameter is fitted to make PR come out as claimed: n⊙, n₀, and u cancel from the ratio; χc and Δχ are the two unknowns the method estimates, with illustrative values only in plots. The Figure 21 error analysis validates SPC against the same toy densities used in the derivation — model-consistency, not independent validation, and flagged as 'minimum values' by the authors. No invented physical entities: the superparticle is a mathematical δ-function proxy, not a new mechanism (from de Koning 2017).

free parameters (6)

χc (feature central angular position)
Geometric unknown the method aims to recover; assigned illustrative values in plots (e.g., 103° in Fig. 8, 142° in Fig. 13, 26°/157° in the PR=0.80 example). Not fitted to any data.
Δχ (feature angular half-width)
Second geometric unknown in Eqs. 21/24; varied across Figures 10–15. The paper's central degeneracy result is that one PR measurement cannot determine both χc and Δχ.
q (compression pulse exponent) = 1 (q=0 and q=2 also given)
Whole-number exponent in Eq. 23 chosen for tractability; q=1 emphasized throughout, q=0 reduces to a constant-density pulse attributed to Gibson et al. (2026).
Assumed PR measurement uncertainty = ±0.01
Illustrative input in Figure 11; propagated by contour reading to claims like χc = 26°±1° and Δχ = 5°±3°. Not derived from instrument noise or background-subtraction error.
Δ_TS = 3.5° (Thomson-sphere width in Section 6 radiance comparison) = 3.5°
Chosen scaling for comparing pB/tB at bean vs Thomson-sphere locations; authors state conclusions do not depend on the choice.
Parker spiral solar wind speeds and rotation rate = vr,BG=400 km/s; vr,CH=575 km/s; Ω=−2.66622×10⁻⁶ s⁻¹
Cartoon SIR in Figure 2 and bean tangent geometry (Eqs. 10–11, 42–47); standard ambient parameters, not fitted to the PR result.

assumptions (8)

domain assumption Optically thin corona/heliosphere, single Thomson scattering, perfect background subtraction
Standard white-light imaging premise behind Eq. 3, cited to Billings (1966), Hundhausen (1993), Howard and Tappin (2009), and Inhester (2016).
domain assumption Small-Sun limit: van de Hulst coefficients truncated at O(ω²)
Appendix A, Eqs. 70–73; paper gives ≤3% error at r>5R☉ and ≤1% at r>9R☉. Limits validity to PUNCH WFI (r≳10R☉), not NFI.
domain assumption Single isolated feature along the line of sight (paper's Q.a answered 'yes')
Load-bearing premise. Section 7 shows K features turn PR into a weighted mean (Eqs. 54/56/59) that the paper concedes is a 'final, unresolved tension' to invert. The paper's own ENLIL figures (Figs. 4–5) show fragmented densities.
domain assumption SIR density is r⁻² radially expanding slab within a boxcar (Eq. 16)
Section 4.1 justifies r⁻² via in-situ scalings: Elliott et al. (2012) n_p ∝ r⁻²°⁰, Perrone et al. (2019) r⁻¹¹⁶, Allen et al. (2021) r⁻². Also a modeling convenience: PR becomes scale-free under r⁻².
ad hoc to paper Compression pulse density n₀cos²ᶜ[π(χ−χc)/(2Δχ)] with zero ambient (Eq. 23)
Smooth bump chosen for closed-form integrability; represents a background-subtracted compression enhancement. No physical justification beyond shape; q=0 reduces to a constant-density pulse from Gibson et al. (2026).
ad hoc to paper Boxcar/pulse symmetric about χc and fully inside (ε, π)
Symmetric interval assumption (Eq. 17). Figure 9 shows the physically accessible width Δacc < Δχ and center χacc ≠ χc otherwise, so near-observer or near-180° features are excluded from the closed forms.
domain assumption Archimedean (Parker) spiral stream interface with constant radial speeds for tangent/bean geometry
Section 6, Eqs. 10–11 and 42–47; standard solar wind model, and the bean result follows Sheeley and Rouillard (2010).
standard math Standard math: Law of Sines, delta-sequence limit of bump functions, Taylor expansions
Used throughout Appendices A–D; the δ-sequence argument for PR→cos²χc (Eqs. 32–35) follows Arfken, Weber, and Harris (2013).

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Pith. "Pith review of For SIRs, CIRs, and Beyond: Polarization Ratio to Feature Location." pith.science (2026). https://pith.science/paper/NF562NMW

@misc{pith2026260723043,
  author       = {Pith},
  title        = {Pith review of: For SIRs, CIRs, and Beyond: Polarization Ratio to Feature Location},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NF562NMW}},
  note         = {Machine review of arXiv:2607.23043}
}

read the original abstract

The Polarimeter to UNify the Corona and Heliosphere (PUNCH) mission will remotely observe solar wind transients with high signal-to-noise ratio, high-cadence, high-resolution polarized white-light images. Using different polarization states, an important PUNCH data product will be polarization ratio images. In the small-Sun limit, and using two simple line-of-sight density distributions with a finite angular width that can approximate a stream or corotating interaction region (SIR/CIR), we analytically investigate how the polarization ratio will provide three-dimensional location information and what the uncertainty in this estimated location is.

Figures

Figures reproduced from arXiv: 2607.23043 by the authors.

**Figure 1.** The Thomson-scattering triangle illustrates the simple geometrical relationships that exist between the linear and angular quantities that recur frequently in applied white-light polarimetry. See text for details. possible. The quantities defined in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** A stationary, pure Parker spiral cartoon of an SIR. The black spirals represent slow solar wind and the red spirals represent fast solar wind originating from a coronal hole. The interaction between the fast and slow solar wind creates a leading compression region, indicated by the blue spirals. Separating the fast solar wind and the compression region is the stream interface, indicated by a single purple curve. The… view at source ↗

**Figure 3.** A WSA-ENLIL simulation of a single fast solar wind stream interacting with a featureless slow solar wind. This simulation depicts an SIR that is clearly distinguishable from the background solar wind. Unlike the stationary, pure Parker spiral model in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (27 more)

**Figure 4.** Figure 4: A steady-state, or forecast-style, WSA-ENLIL simulation of an SIR. The simulation depicts an SIR that has recently swept over Earth. Another SIR is forming east of the L5 point and will impact Earth on 2013-05-31T16:00. These SIRs show significant substructure compared…

**Figure 5.** Figure 5: Time-dependent WSA-ENLIL simulations of the solar wind using WSA maps that update hourly. Both panels depict multiple islands of enhanced solar wind density. An arbitrary line of sight probing the heliosphere may pass through zero, one, or more disconnected islands tha…

**Figure 6.** Figure 6: Heliocentric radius, r, as a function of viewing direction, ε, for a plasma parcel moving radially outward from the Sun. Radial outward motion from the Sun is equivalent to fixed-angle-ψ motion, where ψ is measured with respect to the Sun-observer line. As indicated in…

**Figure 7.** Figure 7: The intermittent release of plasma parcels moving radially outward from the Sun at a constant speed of 400 km s−1 . The rotation of the Sun causes an apparent convergence of plasma parcels when viewing east of the observer-Sun line and an apparent divergence when viewi…

**Figure 8.** Figure 8: Variation of the radially expanding solar wind density as a function of angular position along a line of sight at angle ε. The green curve shows the variation in electron density along the entire line of sight, from the observer out to infinity in linear position, or f…

**Figure 9.** Figure 9: For a given central feature location, χc, the left and right panels demonstrate two outcomes when ∆χ overflows the observer’s sunward line of sight. Both examples assume that the LOS density is symmetric about χc. In the left panel, the central angular position of the …

**Figure 10.** Figure 10: The polarization ratio, Equation 21, calculated from the radially expanding slab density. The image shows a colored contour plot of P R as a function of χc and ∆χ for a line of sight at ε = 15◦. See text for details. although we may talk about the polarization ratio o…

**Figure 11.** Figure 11: The polarization ratio, Equation 21, calculated from the radially expanding slab density for a line of sight at ε = 15◦. The image highlights three values of P R: in light blue, P R = 0.10 ± 0.01; in green, P R = 0.20 ± 0.01; and in dark red, P R = 0.80 ± 0.01. See te…

**Figure 12.** Figure 12: The polarization ratio, P R, for a radially expanding slab, calculated from Equation 21. The left panel plots P R as a function of ∆χ for different values of χc. The right panel plots P R as a function of χc for different values of ∆χ. These plots assume that the obs…

**Figure 13.** Figure 13: Density variation through the compression region of an SIR as a function of angular position along a line of sight at angle ε. For this plot, we have chosen ε = 15◦, χc = 142◦, and ∆χ = 33◦. when q = 1 and χc = ∆χ = 90◦ , then Equation 23 simplifies to Equation 16, th…

**Figure 14.** Figure 14: The polarization ratio, Equation 24, calculated from the compression pulse density for q = 1. The image shows a colored contour plot of P R as a function of χc and ∆χ for a line of sight at ε = 15◦. See text for details. both the density profile and the geometry of th…

**Figure 15.** Figure 15: The polarization ratio, P R, calculated from the compression pulse density, Equation 24. The left panel plots P R as a function of ∆χ for different values of χc. The right panel plots P R as a function of χc for different values of ∆χ. These plots assume that the obs…

**Figure 16.** Figure 16: The Thomson-scattering triangle superimposed on an SIR. These triangles have the same constituent elements used in [PITH_FULL_IMAGE:figures/full_fig_p035_16.png]

**Figure 17.** Figure 17: The bean of locations within the equatorial plane at which a line of sight is tangent to its unique Archimedean spiral. The observer, Sun, and Thomson sphere are shown using the same colors and styles as previous figures. The observer-Sun line separates viewing direct…

**Figure 18.** Figure 18: The solution space for normalized heliocentric radius, R, and scattering angle, χc. The left panel is a plot of R vs. ε from Equation 47. The right panel is a plot of χc vs. ε derived from Equation 43. See text for details. Given R, χ ′ , and ε, we can calculate the h…

**Figure 19.** Figure 19: The left (right) panel shows the Thomson-scattered polarized (total) radiance calculated from Equation 39 (40). The Thomson-scattered radiance is normalized such that B(ε)N (ε) = sin−3 ε. The black curve is the radiance at the Thomson sphere. The colored curves are th…

**Figure 20.** Figure 20: A schematic demonstrating how the angular location and width of an SIR could be measured simultaneously. See text for details [PITH_FULL_IMAGE:figures/full_fig_p043_20.png]

**Figure 21.** Figure 21: Color contour plots of the relative error introduced through the use of SuperParticle Construction to estimate feature location for a line of sight at ε = 15◦. The top panel shows the error in χspc assuming that the true electron number-density is the radially expand…

**Figure 22.** Figure 22: The percent error in the Thomson-scattering geometry functions, comparing the geometry functions in the small-Sun limit, Equations 79–87, against the exact geometry functions, Equations 62–65. Starting with the upper left plot and moving left to right is GT in brown …

**Figure 23.** Figure 23: The Thomson-scattering geometry functions in the small-Sun limit: In the left plot Gtot is green and Gpol is pink. Both functions are centered on the Thomson sphere at χ = 90◦. The dashed line at G = 0.9 is used as a metric to describe the width of each geometry funct…

**Figure 24.** Figure 24: A variation of the Thomson-scattering triangle. In the line-of-sight coordinate system centered on the Thomson sphere, we can write r = p s 2 + d 2 (89) and sin χ ′ = sin χ = d r = d √ s 2 + d 2 . (90) where d = robs sin ε. Once again, for a fixed line of sight at ang…

**Figure 25.** Figure 25: The Thomson-scattering geometry functions in the small-Sun limit as a function of the normalized linear position, S = s/d: In the left plot Gtot is green and Gpol is pink. Both functions are centered on the Thomson sphere at S = 0. The dashed line at G = 0.9 is used a…

**Figure 26.** Figure 26: Looking along a line of sight in linear position or angular position. The left panel shows ℓ/d vs. χ for a line-of-sight direction ε = 15◦; the right panel shows ℓ/d vs. χ for a LOS direction ε = 75◦. Integrating the numerator in Equation 95 for constant ne = n0, resu…

**Figure 27.** Figure 27: Feature width in linear space vs angular space. These plots assume that the observer is viewing along a line of sight at ε = 15◦. One obvious difference is that in linear coordinates the line of sight extends from the observer at 0 out to +∞; however, in angular coord…

**Figure 28.** Figure 28: Partial derivative of the radially expanding slab polarization ratio. The panels shows a colored contour plot of the partial derivatives as a function of χc and ∆χ for a line of sight at ε = 15◦. The top panel is a plot of Equation 105 and the bottom panel is a plot o…

**Figure 29.** Figure 29: Partial derivative of the compression pulse polarization ratio. The panels shows a colored contour plot of the partial derivatives as a function of χc and ∆χ for a line of sight at ε = 15◦. The top panel is a plot of Equation 114 and the bottom panel is a plot of Equa…

**Figure 30.** Figure 30: An orthogonal, left-handed, observer-centric coordinate system. The green + is the observer at the origin of the coordinate system. The light-blue line is the observer’s line of sight. This figure represents two sets of angles that can be used to describe the orientat…

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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