arxiv: 2605.00986 · v1 · submitted 2026-05-01 · 🌀 gr-qc · astro-ph.HE· astro-ph.SR

Recognition: unknown

Series solutions to the TOV equations

Paulo Luz

Authors on Pith no claims yet

Pith reviewed 2026-05-09 18:36 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEastro-ph.SR

keywords Tolman-Oppenheimer-Volkoff equationsseries solutionsPadé approximantscompact starsequation of statestellar massstellar radiusanalytic approximations

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The pith

The Tolman-Oppenheimer-Volkoff equations admit general power-series solutions whose coefficients are fixed by the equation of state and its derivatives, yielding closed-form approximations to stellar mass and radius.

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

This paper constructs general power-series solutions to the Tolman-Oppenheimer-Volkoff equations that describe the interior of compact stellar objects. An explicit algorithm generates the series coefficients from any chosen equation of state together with its thermodynamic derivatives. When the equation of state is analytic or meromorphic, the solutions inherit corresponding regularity properties, and Padé approximants extend the usable domain across isolated poles. The resulting analytic forms supply direct closed-form expressions for the total mass and radius that are tested on affine and polytropic models and then matched across discontinuities for piecewise equations of state.

Core claim

What carries the argument

The power-series expansion of the metric and density functions around the stellar center, with coefficients computed recursively from the equation of state and its derivatives, extended by Padé approximants for meromorphic cases.

If this is right

Closed-form expressions for stellar radius and mass become available once the equation of state is specified.
Series solutions for different pieces of a piecewise equation of state can be matched at transition surfaces to produce a global stellar model.
The analyticity of the stellar solution is directly tied to the regularity properties of the equation of state.
Padé approximants recover accurate global properties even when the raw power series has a limited radius of convergence.

Where Pith is reading between the lines

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

The same coefficient algorithm could be differentiated to produce analytic expressions for additional interior profiles such as pressure or sound speed.
Rapid evaluation of mass-radius pairs for families of equations of state would become feasible, supporting systematic surveys of possible compact-object properties.
The matching procedure for piecewise equations of state supplies a template that could be applied to models containing first-order phase transitions.

Load-bearing premise

The equation of state must be regular enough that the power-series coefficients exist and the Padé approximants converge to the true interior solution.

What would settle it

A side-by-side comparison in which the mass and radius obtained from the series-plus-Padé expressions for a polytropic fluid differ by more than a few percent from a high-precision numerical integration of the same TOV system.

Figures

Figures reproduced from arXiv: 2605.00986 by Paulo Luz.

**Figure 1.** Figure 1: Pressure profile of the Heintzmann solution (26) assuming the parameters view at source ↗

**Figure 2.** Figure 2: Radius of Strange stars as a function of the bag constant, assuming central energy density view at source ↗

**Figure 3.** Figure 3: Compactness parameter of Strange stars as a function of the bag constant, assuming central view at source ↗

read the original abstract

We present general series solutions to the Tolman-Oppenheimer-Volkoff equations for compact stellar objects. We develop an algorithm to compute the coefficients of the power series in terms of the equation of state and its derivatives with respect to the thermodynamic variables. Using these results, we establish general properties of analytic solutions and their relation to the regularity of the equation of state. Applying the theory of Pad\'e approximants, we derive series representations for meromorphic functions whose domains of convergence may include isolated poles. These analytic solutions are then used to obtain closed-form expressions to approximate the radius and mass of stellar objects. We apply the formalism to specific models, namely fluids with affine equations of state and polytropic fluids, and compare the results with those obtained from numerical integration. Lastly, we extend the formalism to piecewise equations of state, deriving series solutions that can be matched across transition hypersurfaces.

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.

Desk Editor's Note private letter to a colleague

The paper gives a general recursive algorithm for power series solutions to the TOV equations from an arbitrary regular EOS, then uses Padé approximants to approximate mass and radius with direct numerical checks.

read the letter

This paper gives a general recursive algorithm for computing the coefficients of power series solutions to the TOV equations, where each coefficient depends on the equation of state and its derivatives evaluated at the central values. They follow this with Padé approximants to handle larger radii and produce approximate closed forms for the total mass and radius of the star. The approach is new in its generality for arbitrary analytic or meromorphic equations of state. It establishes some properties of the solutions tied to EOS regularity. The applications to affine and polytropic fluids include side-by-side comparisons with numerical integrations, which appear to match well. The extension to piecewise equations of state, with series matched at the transition points, is a nice practical step. On the soft side, the method requires the EOS to be regular enough for the series to exist, which the paper states clearly. Real equations of state for neutron stars are often not analytic everywhere, so this is best for theoretical models. The paper does not seem to include detailed convergence proofs or a priori error bounds, relying instead on the numerical checks. That is a minor gap rather than a flaw. This kind of work is for researchers in general relativistic astrophysics who want analytic tools for quick calculations or to gain insight into how the EOS affects stellar properties. It is the sort of paper that deserves a serious referee because the core idea is well-defined, the examples are concrete, and the results can be checked independently. I would recommend sending it out for peer review. The technical content is sound enough to warrant feedback from experts in the area.

Referee Report

2 major / 2 minor

Summary. The manuscript presents general power series solutions to the Tolman-Oppenheimer-Volkoff (TOV) equations for compact stellar objects. It develops an algorithm to compute the coefficients of these series in terms of the equation of state (EOS) and its derivatives with respect to thermodynamic variables. The authors establish properties of analytic solutions linked to EOS regularity, apply Padé approximants to extend convergence for meromorphic cases, and derive approximate closed-form expressions for stellar radius and mass. The formalism is demonstrated on affine and polytropic EOS with direct comparisons to numerical integrations, and extended to piecewise EOS via series matching at transition hypersurfaces.

Significance. If the derivations and comparisons hold, the work supplies a systematic analytic framework for the TOV system that complements numerical methods, particularly for exploring EOS dependence and for piecewise models relevant to phase transitions in compact objects. The recursive coefficient algorithm and Padé extension offer potential for controlled approximations without full numerical integration.

major comments (2)

[Applications to specific models] In the section on applications to polytropic fluids, the reported comparisons to numerical integration omit the truncation order of the series, the Padé order employed, and quantitative error measures (e.g., relative differences in total mass and radius as functions of central density), which are required to substantiate the accuracy of the closed-form approximations.
[Piecewise EOS extension] In the extension to piecewise equations of state, the matching of series solutions across transition hypersurfaces is outlined, but the explicit verification that the metric functions, pressure, and their first derivatives remain continuous (as required by the TOV junction conditions) is not provided, which is load-bearing for the validity of the global solution.

minor comments (2)

The notation for thermodynamic variables and their derivatives in the coefficient recursion could be clarified by an explicit table or list of definitions to aid reproducibility.
A brief discussion of the radius of convergence of the bare power series (prior to Padé) for the chosen EOS examples would strengthen the presentation of the method's domain of applicability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. We address the two major comments below.

read point-by-point responses

Referee: In the section on applications to polytropic fluids, the reported comparisons to numerical integration omit the truncation order of the series, the Padé order employed, and quantitative error measures (e.g., relative differences in total mass and radius as functions of central density), which are required to substantiate the accuracy of the closed-form approximations.

Authors: We agree that these details are necessary to fully substantiate the comparisons. In the revised manuscript we will explicitly state the truncation order of the power series (typically up to order 10), the specific Padé approximant orders employed (e.g., [5/5]), and include quantitative error measures such as tables or figures showing relative differences in total mass and radius as functions of central density. revision: yes
Referee: In the extension to piecewise equations of state, the matching of series solutions across transition hypersurfaces is outlined, but the explicit verification that the metric functions, pressure, and their first derivatives remain continuous (as required by the TOV junction conditions) is not provided, which is load-bearing for the validity of the global solution.

Authors: The matching is constructed by equating the series coefficients at the transition radius so that the metric functions, pressure, and their first derivatives are continuous by design, consistent with the TOV junction conditions. We acknowledge that an explicit verification step was not shown in the original text. In the revision we will add the explicit matching conditions together with a short demonstration that the required continuities are satisfied. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives recursive power-series solutions to the standard TOV system by substituting a power-series ansatz into the differential equations and solving for coefficients in terms of the EOS and its derivatives at the center. It then applies the classical theory of Padé approximants to extend the radius of convergence and obtains explicit approximations for stellar radius and mass by truncating the series at the surface. These steps are direct consequences of the TOV equations plus standard analytic techniques; no parameter is fitted to data and then re-labeled as a prediction, no result is defined in terms of itself, and no load-bearing step reduces to a self-citation. Direct numerical comparisons for affine and polytropic EOS further confirm that the construction is independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard Tolman-Oppenheimer-Volkoff equations of general relativity and the mathematical theory of power series and Padé approximants; no new physical entities or free parameters are introduced.

axioms (2)

standard math The Tolman-Oppenheimer-Volkoff equations correctly describe hydrostatic equilibrium for static, spherically symmetric stars in general relativity.
Invoked as the starting point for all series expansions.
domain assumption The equation of state is sufficiently differentiable or meromorphic to permit power-series expansion and Padé continuation.
Stated in the abstract as the condition for analytic solutions and their relation to EOS regularity.

pith-pipeline@v0.9.0 · 5441 in / 1334 out tokens · 44602 ms · 2026-05-09T18:36:25.380262+00:00 · methodology

discussion (0)

Reference graph

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In the second column it is indicated the value of the compactness parameter of the star; in the third column, the value ofrb found from the numerical integration; in the fourth column it is presented the value ofrb found from the analytic approximation (28); in the last column is the percent error,δ, for the analytic approximation ofrb relative to the num...

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