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arxiv: 1907.02048 · v1 · pith:5O56DJRLnew · submitted 2019-07-03 · 🌌 astro-ph.EP · astro-ph.IM· stat.AP

Bias and robustness of eccentricity estimates from radial velocity data

N. C. Hara , G. Bou\'e , J. Laskar , J.B. Delisle , N. Unger This is my paper

Pith reviewed 2026-05-25 09:25 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IMstat.AP
keywords eccentricityradial velocityposterior distributionsmodel misspecificationplanetary systemsBayes factorsnoise spectrumorbital parameters
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The pith

Posterior sampling with a free error term yields reliable eccentricity estimates from radial velocity data if numerical convergence is confirmed and noise lacks a strong peak at half the planetary period.

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

The paper examines whether eccentricity values fitted to radial velocity time series represent genuine orbital properties or arise as artifacts from incomplete models. It first characterizes standard estimators under the assumption that the model is correct, then tests performance when the data include unmodeled effects such as additional planets, non-Gaussian noise, or correlated noise whose covariance is unknown. The central result is that sampling the posterior distribution of an eccentricity parameter, while allowing an extra free error term, produces trustworthy values once numerical convergence is verified and the noise power spectrum shows no strong feature exactly at half the candidate period. This matters because eccentricity encodes the dynamical history of a system, so systematic bias would distort conclusions about migration, scattering, or resonance. The authors further show that Bayes factors can help separate true eccentricity from an inner companion in 2:1 resonance and outline residual tests that flag model inadequacy.

Core claim

Data analysis via posterior distributions, with a model including a free error term gives reliable results provided two conditions. First, convergence of the numerical methods needs to be ascertained. Secondly, the noise power spectrum should not have a particularly strong peak at the semi period of the planet of interest.

What carries the argument

Posterior sampling of orbital eccentricity in a radial-velocity model augmented by one free error term, validated on simulated data that include missed companions and colored noise.

If this is right

  • Eccentricities reported for Proxima b and 55 Cnc f can be accepted once the two conditions are verified.
  • Bayes factors supply a quantitative test for distinguishing an eccentric single planet from a 2:1 resonant pair.
  • Simple residual diagnostics can reveal when the assumed noise model is inadequate.
  • It remains difficult to decide whether an apparently eccentric orbit is produced by an undetected inner companion locked in 2:1 resonance.

Where Pith is reading between the lines

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

  • If the two conditions are routinely satisfied in published analyses, many existing eccentricity catalogs may not require wholesale revision.
  • The same posterior-plus-free-error approach could be applied to other orbital elements whose estimates are sensitive to noise modeling.
  • Extending the noise-spectrum test to unevenly sampled or multi-instrument data sets would make the reliability criterion more practical.

Load-bearing premise

The range of simulated model errors adequately covers the misspecifications that actually occur in real radial-velocity series.

What would settle it

A real radial-velocity data set in which the noise power spectrum shows a strong peak at half the reported period and the eccentricity posterior shifts substantially once that peak is removed or modeled explicitly.

read the original abstract

Eccentricity is a parameter of particular interest as it is an informative indicator of the past of planetary systems. It is however not always clear whether the eccentricity fitted on radial velocity data is real or if it is an artefact of an inappropriate modelling. In this work, we address this question in two steps: we first assume that the model used for inference is correct and present interesting features of classical estimators. Secondly, we study whether the eccentricity estimates are to be trusted when the data contain incorrectly modelled signals, such as missed planetary companions, non Gaussian noises, correlated noises with unknown covariance, etc. Our main conclusion is that data analysis via posterior distributions, with a model including a free error term gives reliable results provided two conditions. First, convergence of the numerical methods needs to be ascertained. Secondly, the noise power spectrum should not have a particularly strong peak at the semi period of the planet of interest. As a consequence, it is difficult to determine if the signal of an apparently eccentric planet might be due to another inner companion in 2:1 mean motion resonance. We study the use of Bayes factors to disentangle these cases. Finally, we suggest methods to check if there are hints of an incorrect model in the residuals. We show on simulated data the performance of our methods and comment on the eccentricities of Proxima b and 55 Cnc f.

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

2 major / 3 minor

Summary. The manuscript investigates bias in eccentricity estimates from radial velocity data. It first examines classical estimators assuming a correct model, then tests the robustness of posterior sampling (with a free error term) under misspecifications including missed companions, non-Gaussian noise, and correlated noise with unknown covariance. The central claim is that such posterior distributions yield reliable eccentricities provided MCMC convergence is verified and the noise power spectrum lacks a strong peak at the planet's semi-period. The authors further evaluate Bayes factors for distinguishing true eccentricity from 2:1 mean-motion resonance signals and propose residual-based checks for model inadequacy, demonstrating performance on simulated data and commenting on Proxima b and 55 Cnc f.

Significance. If the simulation results generalize, the work supplies concrete, actionable criteria for trusting eccentricity inferences in RV analyses, which directly affects interpretations of planetary system formation and evolution. The forward-simulation framework with known inputs is a clear strength, enabling direct quantification of bias and the identification of the power-spectrum condition as a diagnostic. This could improve standard practice in the field provided the tested misspecifications adequately represent real RV covariance structures.

major comments (2)
  1. [Simulations of misspecifications (results section)] Simulations of misspecifications (results section): The central reliability claim depends on the tested noise models spanning realistic cases; however, the correlated-noise experiments with unknown covariance appear limited to stationary or simple forms and do not explicitly include quasi-periodic red noise with power near the semi-period that commonly arises from stellar activity or instrumental effects. This leaves open whether the stated power-spectrum condition is sufficient for actual time series.
  2. [Bayes-factor analysis (section on 2:1 resonance cases)] Bayes-factor analysis (section on 2:1 resonance cases): The performance of Bayes factors in separating eccentricity from resonant companions is demonstrated only on the simulated cases; without reported sensitivity tests to prior choices on the additional companion or quantitative decision thresholds calibrated to false-positive rates, it is unclear how decisive the factors are when applied to real data such as 55 Cnc f.
minor comments (3)
  1. [Figure captions] Figure captions for the simulated-data results should explicitly list the exact functional forms and parameter ranges of the correlated-noise models employed.
  2. [Application to real systems] The discussion of Proxima b would benefit from a direct numerical comparison of the posterior eccentricity obtained here versus previously published values, including the impact of the free error term.
  3. [Notation] Notation for the additional white-noise term (denoted variously as an extra jitter or free error) should be unified across equations and text for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the scope and limitations of our results. We respond point-by-point to the major comments below.

read point-by-point responses

  1. Referee: Simulations of misspecifications (results section): The central reliability claim depends on the tested noise models spanning realistic cases; however, the correlated-noise experiments with unknown covariance appear limited to stationary or simple forms and do not explicitly include quasi-periodic red noise with power near the semi-period that commonly arises from stellar activity or instrumental effects. This leaves open whether the stated power-spectrum condition is sufficient for actual time series.

    Authors: The power-spectrum diagnostic is presented as a general criterion that follows from the frequency content of the noise rather than from any particular covariance model. Our correlated-noise experiments were chosen to illustrate cases in which the free-error-term model either succeeds or fails according to that criterion. While quasi-periodic red noise was not simulated explicitly, the underlying argument—that excess power at the semi-period can produce spurious eccentricity—remains applicable to any noise process whose spectrum exhibits that feature. We will revise the text to state this generality explicitly and to note that the diagnostic can be checked on real data by inspecting the periodogram of the residuals after subtracting the Keplerian model. revision: partial

  2. Referee: Bayes-factor analysis (section on 2:1 resonance cases): The performance of Bayes factors in separating eccentricity from resonant companions is demonstrated only on the simulated cases; without reported sensitivity tests to prior choices on the additional companion or quantitative decision thresholds calibrated to false-positive rates, it is unclear how decisive the factors are when applied to real data such as 55 Cnc f.

    Authors: The Bayes-factor calculations are offered as an exploratory tool rather than a calibrated classifier. The simulations demonstrate that the factors can be informative but are sensitive to the model assumptions, including the prior on the putative resonant companion. For 55 Cnc f we report the value obtained under our chosen priors and note the ambiguity. We agree that additional sensitivity tests would be useful; we will add a short paragraph acknowledging the dependence on prior choices and recommending that users repeat the calculation under plausible alternative priors when applying the method to real systems. revision: partial

Circularity Check

0 steps flagged

No circularity; conclusions drawn from forward simulations with known inputs

full rationale

The paper derives its main conclusions by generating simulated radial-velocity time series with prescribed eccentricities, companions, and noise properties, then recovering posteriors and comparing them to the known inputs. No equation or estimator is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise reduces to a self-citation. The two stated conditions (MCMC convergence and absence of a strong noise peak at the semi-period) are validated directly against the simulation ensemble rather than by construction. This is the normal, non-circular case of an empirical robustness study.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard assumptions of Bayesian inference for radial-velocity modeling and on the representativeness of the chosen simulation suite; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • domain assumption The radial-velocity likelihood can be written with an additive white-noise term whose amplitude is a free parameter.
    Invoked when the authors state that including a free error term yields reliable posteriors.
  • domain assumption Numerical samplers (MCMC or equivalent) can be run to convergence for the models considered.
    Explicitly listed as the first required condition for trustworthy results.

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