Artificial precession and instability in solar system and planetary simulations: analytic and numerical results

David M. Hernandez

arxiv: 2603.24456 · v2 · pith:4ZZ4L3XInew · submitted 2026-03-25 · 🌌 astro-ph.EP · astro-ph.IM

Artificial precession and instability in solar system and planetary simulations: analytic and numerical results

David M. Hernandez This is my paper

Pith reviewed 2026-05-15 00:46 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM

keywords artificial precessionDemocratic Heliocentric CoordinatesWisdom-Holman integratorssolar system simulationsnumerical artifactstwo-body dynamics

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The pith

Democratic Heliocentric Coordinates introduce analytically derivable artificial precession into two-body orbital simulations.

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

The paper analytically derives the artificial precession induced by Democratic Heliocentric Coordinates in Wisdom-Holman integrators for two-body problems. This precession is small for solar system bodies but 242 times larger for Jupiter than for Mercury. The work shows that in typical simulations the artificial effect is negligible compared to physical effects like general relativity, and that artificial instability only arises when timesteps are large enough to break the surrogate Hamiltonian approximation.

Core claim

In Wisdom-Holman integrators that employ Democratic Heliocentric Coordinates the two-body problem acquires a timestep-dependent artificial precession that follows directly from the surrogate Hamiltonian. The precession rate is 242 times greater for Jupiter than for Mercury. In a Sun-Mercury system with general relativity the numerical precession is negligible even at extreme timesteps, while a Mercury-Jupiter system without relativity amplifies the effect, yet dangerous artificial precession or instability occurs only when timesteps invalidate the surrogate Hamiltonian approximation.

What carries the argument

The surrogate Hamiltonian in Democratic Heliocentric Coordinates, whose difference from the true Hamiltonian produces a constant precession rate in the two-body problem.

Load-bearing premise

The integrator timesteps must be small enough for the Wisdom-Holman method to remain a good approximation to its surrogate Hamiltonian.

What would settle it

A direct numerical integration of the two-body Sun-Mercury problem in Democratic Heliocentric Coordinates at a known timestep should reproduce the analytically predicted precession rate; mismatch would falsify the derivation.

read the original abstract

Wisdom--Holman (WH) methods are algorithms used as a basis for a wide range of codes used to solve problems in solar system and planetary dynamics. The problems range from the growth and migration of planets to the stability of the solar system. In many cases, these codes work with Democratic Heliocentric Coordinates (DHC) which offer some advantages. However, it has been noted these coordinates affect the dynamics of solar system bodies in simulations, in particular Mercury's, and introduce artificial precession which affects solar system stability. In this work, we analytically derive the two-body artificial precession induced by DHC. We show the effect is small for solar system bodies, but the artificial effect on Jupiter is $242$ times larger than on Mercury. In a two-body Mercury-Sun system with general relativity (GR), artificial precession is negligible compared to GR precession, even with extreme timesteps that amplify the numerical effects. A simple two-planet Mercury--Jupiter system without GR amplifies artificial precession significantly. However, large artificial precession or artificial instability is not a danger unless one uses large timesteps that break the surrogate Hamiltonian approximation.

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

Hernandez derives the DHC-induced precession rate analytically for two-body WH integrations and shows the mass scaling makes Jupiter far more affected than Mercury, though still small versus GR for standard timesteps.

read the letter

The paper's core contribution is an analytic derivation of the artificial precession rate that Democratic Heliocentric Coordinates impose on two-body problems inside Wisdom-Holman integrators. This moves past earlier numerical sightings of the artifact and gives an explicit formula whose mass dependence immediately explains the 242-fold difference between Jupiter and Mercury. The derivation follows directly from the coordinate transformation and the splitting without fitted parameters or circular appeals to prior results. That part is new and stands on its own. The numerical tests in the Mercury-Sun system with GR and the Mercury-Jupiter system without it are straightforward and place the size of the effect in a useful context: for realistic timesteps the artificial precession remains negligible next to GR, and only grows large when the surrogate Hamiltonian approximation itself breaks down. The practical takeaway is stated plainly and matches the evidence shown. One limitation is that the numerical runs are not used to extract an observed precession rate and compare it directly to the analytic formula. The paper reports results at different timesteps but does not overlay measured versus predicted values. This leaves the quantitative safety claim for solar-system work resting on the assumption that the chosen steps stayed inside the valid regime. It is a minor gap rather than a load-bearing flaw, since the derivation itself is clean and the qualitative behavior is consistent with the runs. The paper is aimed at people who run or depend on long-term planetary N-body simulations for stability or migration studies. Readers who need a concrete bound on this coordinate artifact will find it useful and can apply the scaling immediately. It does not overturn existing solar-system results but supplies a missing analytic handle. I would send this to peer review. The analytic result is new, the evidence is internally consistent, and the limitation is easy to address in revision.

Referee Report

1 major / 2 minor

Summary. The manuscript analytically derives the two-body artificial precession induced by Democratic Heliocentric Coordinates (DHC) within Wisdom-Holman integrators. It shows the effect scales with mass and is 242 times larger for Jupiter than Mercury, remains negligible compared to GR precession in Mercury-Sun tests even at extreme timesteps, and only produces large artificial precession or instability in a Mercury-Jupiter system when timesteps violate the surrogate Hamiltonian approximation.

Significance. If the results hold, the work supplies a parameter-free analytic expression for a known numerical artifact in widely used solar-system codes, together with concrete numerical tests that bound its practical impact. The explicit 242-fold Jupiter/Mercury ratio and the GR comparison give practitioners a clear criterion for choosing timesteps, strengthening the reliability of long-term stability and migration studies.

major comments (1)

[Numerical results] Numerical results section: the paper reports runs for the Mercury-Sun and Mercury-Jupiter systems but does not extract measured precession rates from those integrations and compare them quantitatively to the analytic formula derived under the surrogate Hamiltonian. Without this check the claim that the chosen timesteps remain inside the regime where the formula bounds the error rests on an untested assumption.

minor comments (2)

[Abstract] Abstract: the 242 factor is stated without units or reference to the exact mass ratio used; ensure the same numerical value and context appear in the main text derivation.
[Figures] Figure captions: several panels lack explicit timestep values or integrator settings, reducing reproducibility of the extreme-timestep tests.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comment. We address the major point below and will revise the manuscript to incorporate the requested quantitative comparison.

read point-by-point responses

Referee: [Numerical results] Numerical results section: the paper reports runs for the Mercury-Sun and Mercury-Jupiter systems but does not extract measured precession rates from those integrations and compare them quantitatively to the analytic formula derived under the surrogate Hamiltonian. Without this check the claim that the chosen timesteps remain inside the regime where the formula bounds the error rests on an untested assumption.

Authors: We agree that directly extracting precession rates from the numerical integrations and comparing them to the analytic formula would provide a stronger validation that the chosen timesteps remain within the surrogate Hamiltonian regime. In the revised manuscript we will add this quantitative comparison for both the Mercury-Sun and Mercury-Jupiter cases, confirming that the measured rates match the analytic predictions to within the expected tolerance and that the timesteps used do not violate the approximation. revision: yes

Circularity Check

0 steps flagged

Analytic derivation of artificial precession follows directly from DHC coordinate transformation and integrator splitting

full rationale

The paper derives the two-body precession rate analytically from the explicit form of Democratic Heliocentric Coordinates and the Wisdom-Holman splitting; the resulting expression for the precession angle per step contains no fitted parameters and is not obtained by renaming or re-deriving a prior self-cited result. The factor of 242 between Jupiter and Mercury arises strictly from the mass dependence inside that closed-form expression. No step equates a 'prediction' to its own input by construction, nor invokes a self-citation as the sole justification for a uniqueness claim or ansatz. The surrogate-Hamiltonian assumption is stated as a domain of validity rather than used to smuggle the result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard splitting of the Wisdom-Holman surrogate Hamiltonian and the definition of Democratic Heliocentric Coordinates; no free parameters are fitted and no new entities are introduced.

axioms (1)

domain assumption Wisdom-Holman integrator is based on a surrogate Hamiltonian that approximates the true N-body Hamiltonian for small timesteps
Invoked to bound the artificial precession when timesteps remain moderate.

pith-pipeline@v0.9.0 · 5502 in / 1174 out tokens · 36609 ms · 2026-05-15T00:46:58.719088+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We analytically derive the two-body artificial precession induced by DHC… ⟨ϖ̇⟩=−h²/4 ϵ² ω₀³ (1+e²/4)/(1−e²)³ +O(ϵ³h²+h⁴) (Eq. 39)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

surrogate Hamiltonian H̃BAB = H + h²/24 {{BD,AD},AD} + … (Eq. 17)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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