The Impact of Errors in the Shape Function of Rotating Neutron Stars in the Oblate Schwarzschild Approximation

Charlee Amason; John Ming Ngo; Sharon M. Morsink

REVIEW

The Impact of Errors in the Shape Function of Rotating Neutron Stars in the Oblate Schwarzschild Approximation

John Ming Ngo, Charlee Amason, Sharon M. Morsink

High Energy Astrophysical Phenomena · Submitted 27 Aug 2026

Pith's verdict: Solid error analysis of shape functions, with a clear caveat about the OS approximation for dOmega.

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arXiv ID2608.27744 (version 2)

Pith IDS73L2LBR, Pith's permanent record of this paper and its review

CitationCharlee Amason, John Ming Ngo, Sharon M. Morsink. "The Impact of Errors in the Shape Function of Rotating Neutron Stars in the Oblate Schwarzschild Approximation." arXiv:2608.27744 (2026). https://pith.science/paper/S73L2LBR

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Editor's note

read the letter

The paper does something genuinely useful: it compares five shape functions against a common reference (RNS) over a broad parameter range, quantifies both radius and derivative errors, and propagates them into the differential solid angle dOmega under the OS approximation. The finding that derivative errors dominate near the limb and that polar-radius modeling is the main bottleneck for the PVLS shape function is actionable. The cross-validation test for universality is well done. Credit where it’s due: the methodology is sound, the public codes are used properly, and the limitations are stated honestly.

The soft spot is the one the stress-test flags. The headline numbers—Figure 8’s ~10% local dOmega error for a NICER-like pulsar—rest entirely on the oblate Schwarzschild approximation. The paper acknowledges that OS neglects frame dragging and the rotational quadrupole, and defers a check to future work. That’s a real hole: if the exact metric amplifies or shifts the error patterns, the quantitative conclusions could change. The authors don’t verify that OS faithfully propagates shape-function errors for even one representative model. That said, the paper’s main recommendations (model the derivative, improve the polar radius, stay within training range) are robust regardless of the metric choice, because they follow from the radius and derivative errors alone. The dOmega numbers are suggestive, not definitive.

The paper is worth engaging. It’s the first joint error analysis of these shape functions, and it gives practical guidance for building better ones. I’d accept it for peer review, with a request that the authors add a cross-check using a full numerical metric (even for a few cases) or at least strengthen the caveat that the dOmega error maps are OS-specific.

Referee report

Significance

If the results hold, this paper makes a practically important contribution for neutron star radius inference from X-ray flux modelling. It is the first systematic, EOS-spanning comparison of all existing shape functions, revealing not just error magnitudes but the distinct morphologies and the critical role of the derivative. The demonstration that simple corrections can reduce errors to ~0.5% supports the feasibility of improved empirical shape functions. The physics insight that derivative errors dominate near the limb, and that integrated solid angle can hide cancellation, is valuable for the community developing pulse-profile modelling codes. The work is thorough, the numerical methodology (RNS at high resolution, NS-SWORD) is sound, and the code is publicly available.

Major objections (2)

[Section IV and Figure 8] The quantitative dΩ error maps (e.g., the ~10% near-limb error for the NICER-like model) rely entirely on the oblate Schwarzschild (OS) approximation, which neglects frame dragging and the rotational quadrupole of the exterior metric. The paper acknowledges this limitation (Section V B) and defers a full-metric calculation to future work. However, because these numbers are the headline quantitative result and the basis for the statement that shape-function errors can produce substantial local flux errors, the authors should provide at least a brief estimate of how much the OS approximation itself might alter the error propagation. For a few representative models (e.g., the NICER-like case and the 600 Hz case), comparing dΩ computed with OS against dΩ computed with a full numerical metric (from RNS or Lorene) would bound the uncertainty. If such a comparison is impractical, a detailed, quantitative discussion of the expected systematic bias from the OS approximation should be added to Section IV or V, including an argument for why the general error pattern (area vs. projection, derivative dominance) is robust even if the exact percentages change.
[Section III B, Figure 3] The analysis of the PVLS shape function correctly identifies that the polynomial fit for the polar radius limits the overall accuracy, raising the maximum error from 0.25% to 2.77%. This is a key finding. However, the paper does not show whether the remaining errors after applying the correction fit (Section III E) are dominated by the polar radius error or by latitudinal shape errors. The authors should quantify, for PVLS, how much of the residual error after correction (panels b–e in Figure 6) stems from the polar radius fit versus the ANN surface reconstruction. This would sharpen the recommendation that “modeling the polar radius … may be one of the key problems.”

Minor objections (8)

[Abstract] The abstract states “errors in dΩ may be understood as coming from an area contribution … and a projection contribution.” It would be helpful to explicitly mention that the projection contribution grows near the limb, as this is a central result.
[Section I, page 3] The sentence “We will not (in this paper) provide a new, improved shape function” is clear, but the paper does provide correction surfaces (Section III E). The term “correction surface” could be misleading if a reader thinks it is a full shape function. Consider a brief clarification that the correction surface is a proof-of-principle demonstration, not a final product.
[Section II A, equations (7)–(8)] The notation uses the same symbol for the impact parameter b and the metric function B in equation (14). While the context is different, this could cause confusion. Consider a footnote or different symbol for the metric potential.
[Section III D, end of page 12] The sentence “The same extremal radius error is shown for three of the shape functions in panel (a) of Figure 6” references panel (a), but Figure 6 has five panels (a)–(e). In the online version, panel (a) is identified, but ensure the caption and text are explicit for each panel.
[Section IV B, page 18] The description of Figure 7 for the i=0° panel states “the error in the differential solid angle is approximately twice the error in the radius at this latitude.” This is correct from equation (43). However, the reference to the “lower panels” is ambiguous; there are two rows (AM and PVLS). Specify which row.
[Section IV B, page 20] In the paragraph starting “The median locations and angular widths …”, the green circle and lines in Figure 8 are described. The color scale for the error maps in Figure 8 uses a different range than previous figures. The caption should include the percentile-based color scaling, as done for Figures 7 and 9.
[Table I] The table includes spin frequencies for several stars but no corresponding compactness or spin parameter. Since these parameters are central to the paper, consider adding columns with typical ζ and ϵ values (or referencing Table II) to help the reader connect observed stars to the parameter space studied.
[Section V C, item 3] The recommendation to model the polar radius accurately is well supported, but the text “improving the modeling of Rpole is a method to reduce the relative error to as low as 0.25%” implicitly assumes the polar radius is the only source of error. Qualify this statement to note that the ANN itself contributes ~0.25% error in the best case.

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The paper

The paper claims

Five shape functions for rotating neutron stars produce systematic radius errors that propagate into large local errors in the observable solid angle, with corrections reducing radius errors to ~0.5%.

The reading

The paper examines five shape functions used to model the oblate surface of rotating neutron stars in the oblate Schwarzschild (OS) approximation. It finds that while radius errors are typically below 1% for moderate spins, errors in the derivative of the shape function cause substantially larger local errors in the differential solid angle (up to ~10% for a NICER-like pulsar) because the projection contribution dominates near the limb. Simple polynomial corrections reduce the radius errors of several shape functions to within approximately 0.5%. The paper provides guidelines for constructing more accurate shape functions, emphasizing that both the shape and its derivative must be modeled accurately and that the polar radius must be predicted precisely.

What carries the argument

The central objects are the shape function R(θ) and its derivative R'(θ) for the oblate surface of a rotating neutron star. The error propagation into the differential solid angle dΩ (a key factor in the observed X-ray flux) is analyzed within the oblate Schwarzschild (OS) approximation. An analytic approximation derived from the Beloborodov formula separates the dΩ error into an area contribution (≈2E_R) and a projection contribution dominated by E_{R'}, explaining why small radius errors can produce large dΩ errors near the limb where cosσ → 0.

Extended reading notes

Core claim

The central claim is that the five existing shape functions—MLCB, BBPO, AM, SPYY, and PVLS—all introduce systematic errors in the surface radius and its derivative, and these errors propagate into the differential solid angle dΩ, a geometric factor in the observed flux. Although radius errors are often <1% at moderate spins, the derivative error generates a projection term that amplifies dΩ errors near the limb. For a NICER-like pulsar (PSR J0740+6620), local dΩ errors reach ~10% over about 20% of the rotational phase. Simple polynomial corrections in colatitude, compactness, and spin reduce the bulk radius errors to within ±0.5%, but residual EOS-dependent offsets hint at a possible limit to quasi-universality. The analysis also reveals that the PVLS shape function's accuracy is dominated by errors in its analytical fit for the polar radius rather than the neural network's surface interpolation.

Load-bearing premise

The analysis assumes that the oblate Schwarzschild (OS) approximation is accurate enough to isolate the impact of shape-function errors on the flux, specifically that the error in dΩ computed within OS meaningfully represents the error in the full relativistic flux calculation; if the OS systematically amplifies or cancels errors differently from the exact rotating metric, the quantitative conclusions about the magnitude and pattern of dΩ errors would change.

Editorial extensions

If this is right

Current shape functions, particularly AM used in NICER analyses, may introduce systematic biases in inferred radii for faster-rotating pulsars; the paper recommends using SPYY or PVLS for rapidly rotating accreting millisecond X-ray pulsars.
Future shape functions must be calibrated over the compactness and spin range of the targeted observations and must model the polar radius accurately, as errors there dominate the PVLS surface error.
The error decomposition shows that shape functions should minimize errors in both R(θ) and R'(θ); a function with small R error but large derivative error will still produce large dΩ errors.
Gravitational lensing and atmospheric beaming may partly mitigate the limb errors, but the paper argues that integrated solid angle is a poor metric because cancellations mask local inaccuracies.
The universality tests suggest that a better shape function could be constructed by fitting a polynomial correction to an existing one like SPYY, using a larger and more representative EOS set.

Reading between the lines

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

The ~10% dΩ error near the limb may be partially offset by the beaming of the Hydrogen atmosphere, which suppresses limb emission; the net effect on inferred radius could be smaller than the geometric error alone.
The EOS-dependent offsets remaining after correction hint at a fundamental floor to quasi-universality; a truly universal shape function may be impossible, and separate calibrations for different EOS classes (e.g., hadronic vs. bare quark stars) may be necessary.
The error propagation framework could be extended beyond the OS approximation to full general-relativistic metrics (e.g., Kerr-like) to assess whether frame-dragging and quadrupole terms alter the shape-function error budget.
The PVLS polar radius fit error (0.25% to 2.77%) suggests that improving the analytic R_pole relation is a more impactful target than refining the neural network itself.

Original abstract

Measurements of thermal X-ray flux emitted by neutron stars can constrain their radii and the equation of state of cold dense matter. Accurate flux modeling of rapidly rotating neutron stars requires an approximation for their oblate surface shape, usually given by a shape function. We examine the accuracy of five shape functions by comparing them against numerical relativistic stellar surfaces for seven representative equations of state over a broad range of compactness and spin. We characterize their errors in the surface radius and their derivative with respect to colatitude, test the extent of quasi-universality, and propagate these errors into the differential solid angle d{\Omega}, a geometric factor underlying the observed flux, under the oblate Schwarzschild approximation. We find that the shape functions produce distinct error patterns associated with their forms and parameter ranges. These errors remain structured in compactness, spin, and colatitude. Simple polynomial corrections substantially reduce the radius errors of several shape functions, with most corrected errors within approximately 0.5%. We show that errors in d{\Omega} may be understood as coming from an area contribution, which scales with twice the relative radius error, and a projection contribution controlled primarily by the error in the surface derivative. The projection contribution grows towards the limb, where small surface errors can produce much larger local errors in d{\Omega}. We provide a set of guidelines for the construction of improved quasi-universal shape functions that can be used in future radius estimation efforts. Our recommendations include that both the shape and its derivative be modeled accurately, that more accurate predictions of the polar radius be developed, and that the shape function be calibrated over the same compactness-spin range that is to be tested.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,293 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.34 · human participants 0.01 · controlled comparison 0.50 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.79 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Measurements of thermal X-ray flux emitted by neutron stars can constrain their radii and the equation of state of cold dense matter. Accurate flux modeling of rapidly rotating neutron stars requires an approximation for their oblate surface shape, usually given by a shape function. We examine the accuracy of five shape functions by comparing them against numerical relativistic stellar surfaces for seven representative equations of state over a broad range of compactness and spin. We characterize their errors in the surface radius and their derivative with respect to colatitude, test the extent of quasi-universality, and propagate these errors into the differential solid angle d{\Omega}, a geometric factor underlying the observed flux, under the oblate Schwarzschild approximation. We find that the shape functions produce distinct error patterns associated with their forms and parameter ranges. These errors remain structured in compactness, spin, and colatitude. Simple polynomial corrections substantially reduce the radius errors of several shape functions, with most corrected errors within approximately 0.5%. We show that errors in d{\Omega} may be understood as coming from an area contribution, which scales with twice the relative radius error, and a projection contribution controlled primarily by the error in the surface derivative. The projection contribution grows towards the limb, where small surface errors can produce much larger local errors in d{\Omega}. We provide a set of guidelines for the construction of improved quasi-universal shape functions that can be used in future radius estimation efforts. Our recommendations include that both the shape and its derivative be modeled accurately, that more accurate predictions of the polar radius be developed, and that the shape function be calibrated over the same compactness-spin range that is to be tested.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Measurements of thermal X-ray flux emitted by neutron stars can constrain their radii and the equation of state of cold dense matter.	background 1.00	0.06	0.02	hedged possibility 0.98
s1	Accurate flux modeling of rapidly rotating neutron stars requires an approximation for their oblate surface shape, usually given by a shape function.	background 1.00	0.05	0.02	asserted 1.00
s2	We examine the accuracy of five shape functions by comparing them against numerical relativistic stellar surfaces for seven representative equations of state over a broad range of compactness and spin.	contribution 0.61	0.95	0.04	asserted 0.95
s3	We characterize their errors in the surface radius and their derivative with respect to colatitude, test the extent of quasi-universality, and propagate these errors into the differential solid angle d{\Omega}, a geometric factor underlying the observed flux, under the oblate Schwarzschild approximation.	method 0.65	0.95	0.05	asserted 0.80
s4	We find that the shape functions produce distinct error patterns associated with their forms and parameter ranges.	result 1.00	0.96	0.21	asserted 1.00
s5	These errors remain structured in compactness, spin, and colatitude.	result 0.99	0.88	0.11	asserted 1.00
s6	Simple polynomial corrections substantially reduce the radius errors of several shape functions, with most corrected errors within approximately 0.5%.	result 1.00	0.95	0.25	asserted 1.00
s7	We show that errors in d{\Omega} may be understood as coming from an area contribution, which scales with twice the relative radius error, and a projection contribution controlled primarily by the error in the surface derivative.	interpretation 0.57	0.93	0.19	hedged possibility 0.63
s8	The projection contribution grows towards the limb, where small surface errors can produce much larger local errors in d{\Omega}.	interpretation 0.54	0.87	0.10	asserted 0.91
s9	We provide a set of guidelines for the construction of improved quasi-universal shape functions that can be used in future radius estimation efforts.	future work 0.92	0.85	0.19	asserted 0.98
s10	Our recommendations include that both the shape and its derivative be modeled accurately, that more accurate predictions of the polar radius be developed, and that the shape function be calibrated over the same compactness-spin range that is to be tested.	future work 1.00	0.50	0.23	asserted 0.92

What kind of paper the classifier reports this is: simulation study 0.72 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Diagram illustrating the oblate Schwarzschild (OS)

**Figure 2.** FIG. 2. Percent error in shape functions

**Figure 3.** FIG. 3. Histogram of the distribution of errors relative to

**Figure 4.** FIG. 4. Signed derivative error

**Figure 5.** FIG. 5. Representative examples of the dependence of the extremal radius error on compactness and spin parameter. The

**Figure 6.** FIG. 6. Empirical all-EOS correction surface fits to the radius errors of AM, SPYY(s), and PVLS. For each shape function,

Figures reproduced from arXiv:2608.27744 by the authors.

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 97 sentences without a confident label

Uses the method or tool of 4 papers

Our implementation of the PVLS shape function follows the sample Python code published with their paper [ref] .

Universal description of the Neutron Star's surface and its key global properties: A Machine Learning Approach for nonrotating and rapidly rotating stellar models · PVLS: Papigkiotis, Vardakas, Likas, Stergioulas (2025) CITEMARK103Z · 1.00
The shape function is computed using the Hartle-Thorne [ref] slow-rotation approximation, and then fit to the function [equation] where $R_0$ is the equatorial circumferential radius in the non-spinning limit and $ξ_2$ is a coefficient computed in the slow-rotation formalism [ref] and depends on the dimensionless parameters $ζ$ and $ϵ$.

Relations Between Neutron-Star Parameters in the Hartle-Thorne Approximation · BBPO: Bauböck, Berti, Psaltis, & Özel (2013) CITEMARK69Z · 0.86
We can gain a more intuitive understanding of this dependence by employing the Beloborodov approximation [ref] , which relates $ψ$ and $α$ through the expansion [equation] Since the coefficient of the term of order $(1-cosα)^3$ is very small [ref] , this approximation is surprisingly good even for photons emitted in directions far from the radial direction.

Gravitational Bending of Light Near Compact Objects · Analytic Approximation to the Differential Solid Angle Error · 0.98

Uses data or values from 8 papers

The EOS library included the EOS catalog (A, B, C, F, G, L, N, O) of [ref] , APR [ref] , a suite of hybrid quark stars with mixed quark-hadron phases, [ref] , and hyperon-rich models [ref] .

Hybrid stars that masquerade as neutron stars · MLCB: Morsink, Leahy, Cadeau, & Braga (2007) CITEMARK58Z · 0.90
Four of the EOSs are taken from the CompOSE database (CompOSE labels are given below with our abbreviations in square brackets) [ref] : a hadronic EOS using a chiral SU(3) model "DS(CMF-2) with crust" [CMF2] [ref] ; two EOSs using the Thomas-Fermi approximation and two different density functionals, "XMLSLZ(NL3)" [NL3], and "XMLSLZ(PKDD)" [PKDD] [ref] ; and a hybrid hadronic-quark EOS "KBH(QHC21A)" [QHC21A] [ref] .

Equations of state for supernovae and compact stars · EOSs · 0.94
Four of the EOSs are taken from the CompOSE database (CompOSE labels are given below with our abbreviations in square brackets) [ref] : a hadronic EOS using a chiral SU(3) model "DS(CMF-2) with crust" [CMF2] [ref] ; two EOSs using the Thomas-Fermi approximation and two different density functionals, "XMLSLZ(NL3)" [NL3], and "XMLSLZ(PKDD)" [PKDD] [ref] ; and a hybrid hadronic-quark EOS "KBH(QHC21A)" [QHC21A] [ref] .

Implications of NICER for neutron star matter: the QHC21 equation of state · EOSs · 0.94

Compares with 2 papers

This should be contrasted with the errors computed in the original paper by [ref] , which are generally much smaller (see, for example, their Figure 21).

Universal description of the Neutron Star's surface and its key global properties: A Machine Learning Approach for nonrotating and rapidly rotating stellar models · Polar Radius in the PVLS Shape Function · 0.98
Earlier work [ref] compared the errors in equatorial radius resulting if a perfectly spherical stellar surface was assumed in the analysis of data for a star spinning at 600 Hz, with an extended hotspot, viewed at an inclination angle of $90^∘$, a case that would have much larger errors than what we show here.

Determining Neutron Star Properties by Fitting Oblate-Star Waveforms To X-ray Burst Oscillations · Errors in a NICER-like Rotation Powered Pulsar · 0.80

Cites as background 12 papers

In practice, several shape functions have been introduced [ref] , and they make different assumptions about the colatitude dependence, the dimensionless-parameter dependence, and the EOS set used for calibration.

The Oblate Schwarzschild Approximation for Light Curves of Rapidly Rotating Neutron Stars · Introduction · 0.98
In practice, several shape functions have been introduced [ref] , and they make different assumptions about the colatitude dependence, the dimensionless-parameter dependence, and the EOS set used for calibration.

Relations Between Neutron-Star Parameters in the Hartle-Thorne Approximation · Introduction · 0.98
In practice, several shape functions have been introduced [ref] , and they make different assumptions about the colatitude dependence, the dimensionless-parameter dependence, and the EOS set used for calibration.

Universality of the Acceleration Due to Gravity on the Surface of a Rapidly Rotating Neutron Star · Introduction · 0.98

Discussion

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