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arxiv: 2605.12982 · v1 · pith:4556DV4Onew · submitted 2026-05-13 · 🌌 astro-ph.EP · astro-ph.IM

On the Reparameterization Between Cartesian Position-Velocity Vectors and Orbital Elements in the Kepler Problem

Kento Masuda , Kansuke Nunota This is my paper

Pith reviewed 2026-06-30 21:45 UTC · model grok-4.3

classification 🌌 astro-ph.EP astro-ph.IM
keywords Jacobian determinantorbital elementsCartesian state vectorsKepler problemmicrolensingastrometric orbit fittingreparameterizationMCMC sampling
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The pith

A corrected analytic Jacobian for orbital-element to Cartesian transformation removes a singularity in prior microlensing models and improves MCMC sampling.

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

The paper derives compact closed-form expressions for the Jacobian determinants of the transformation between orbital elements and Cartesian position-velocity vectors in the Kepler problem. These expressions allow consistent transformation of probability densities when reparameterizing priors for Bayesian inference. It identifies that the Skowron et al. (2011) formulation for binary microlensing adopts an incorrect definition of the longitude of the ascending node, rendering the Jacobian singular, and supplies a corrected derivation showing the final formula is essentially unchanged under a proper node definition. An explicit comparison of gradient-based MCMC in astrometric orbit fitting demonstrates that sampling from Cartesian state vectors yields higher efficiency and robustness than sampling from orbital elements.

Core claim

The central claim is that the Jacobian determinant of the map from orbital elements to Cartesian state vectors admits a compact analytic expression; when this expression is used to revisit the Skowron et al. (2011) parameterization, the reported singularity is traced to an incorrect choice of reference axis for the longitude of the ascending node, and the corrected Jacobian remains effectively identical once the node is defined with respect to an orbit-independent axis.

What carries the argument

The Jacobian determinant of the analytic transformation between the six orbital elements and the six Cartesian position-velocity components in the two-body Kepler problem.

If this is right

  • Bayesian priors on orbital elements can be transformed to Cartesian coordinates without numerical differentiation or approximation.
  • The corrected Jacobian permits consistent density reparameterization in microlensing light-curve modeling.
  • Cartesian-state sampling in gradient-based MCMC for astrometric orbits produces higher acceptance rates and shorter autocorrelation times than element-based sampling.
  • The same closed-form expressions apply unchanged once the ascending node is measured from an axis independent of the binary orbit.

Where Pith is reading between the lines

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

  • The same Jacobian expressions could be reused in other two-body fitting problems where orbital elements are poorly constrained, such as visual binaries or radial-velocity planets.
  • Adopting Cartesian state vectors as the sampling space may simplify the incorporation of additional constraints like sky-plane accelerations or proper-motion measurements.
  • The approach suggests testing whether similar reparameterizations reduce mixing times in joint fits that combine microlensing, astrometry, and radial velocity.

Load-bearing premise

The Skowron et al. (2011) parameterization inadvertently adopts an incorrect definition of the longitude of the ascending node with respect to the sky-projected binary axis at a reference epoch, rendering the intermediate Jacobian formally singular.

What would settle it

Direct numerical evaluation of the Jacobian matrix entries under the original Skowron et al. definition at a reference epoch where the node aligns with the binary axis, checking whether any determinant vanishes.

Figures

Figures reproduced from arXiv: 2605.12982 by Kansuke Nunota, Kento Masuda.

Figure 1
Figure 1. Figure 1: Our simulated data (circles) along with 20 models drawn from the posterior (gray lines). We choose the +x-axis in the positive Dec direction and +y-axis in the positive RA direction (left of this diagram). Thus the +z-axis points toward the observer. to isolate the impact of the reparameterization. Here we employ a Markov Chain Monte Carlo (MCMC) algorithm that makes explicit use of the gradient of the pos… view at source ↗
Figure 2
Figure 2. Figure 2: Corner plot comparing posterior samples obtained by directly sampling the orbital elements (blue) and by sampling the Cartesian state vectors at the reference epoch (orange). The same physical prior is imposed in both parameterizations through the Jacobian transformation, so the two posteriors overlap as expected. Samples are shown in both orbital-element coordinates (first six entries) and Cartesian phase… view at source ↗
Figure 3
Figure 3. Figure 3: The effective sample size (ESS) per core-minute for the selected parameters sampled with (orange) and without (blue) reparameterizing to Cartesian state vectors. find qualitatively consistent improvements across a range of configurations when adopting the Cartesian state-vector parameterization. We emphasize that this benchmark is intended as a relative comparison between the two parameterizations, rather … view at source ↗
read the original abstract

Reparameterization from the standard set of orbital elements to Cartesian position-velocity vectors can be computationally advantageous for orbit inference problems, particularly when orbital elements are weakly constrained. Here we present compact analytic expressions for the Jacobian determinants of this transformation and its variants, which enable consistent transformation of prior probability densities under reparameterization and are therefore useful for a Bayesian treatment of such problems. We then use these results to clarify the application of this reparameterization in microlensing and astrometric contexts. We first revisit the widely used formulation of lens orbital motion during binary microlensing events presented by Skowron et al (2011). We show that their parameterization inadvertently adopts an incorrect definition of the longitude of the ascending node with respect to the sky-projected binary axis at a reference epoch, which renders the intermediate Jacobian formally singular. Using our closed-form expression, we provide a corrected analytic derivation of the Jacobian for this transformation and show that the resulting formula remains effectively unchanged when the longitude of the ascending node is properly defined with respect to an axis independent of the binary orbit. We also perform an explicit quantitative comparison of astrometric orbit fitting using a gradient-based Markov Chain Monte Carlo algorithm under the two parameterizations, and find that reparameterizing to Cartesian state vectors improves sampling efficiency and robustness relative to orbital-element sampling.

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

1 major / 2 minor

Summary. The paper derives compact analytic expressions for the Jacobian determinants of the transformation between standard orbital elements and Cartesian position-velocity vectors in the Kepler problem. It revisits the Skowron et al. (2011) parameterization for binary microlensing lens orbital motion, claims that their definition of the longitude of the ascending node (with respect to the sky-projected binary axis at a reference epoch) inadvertently renders the intermediate Jacobian singular, supplies a corrected closed-form derivation, demonstrates that the formula is effectively unchanged under a proper independent-axis definition, and reports a quantitative MCMC comparison in astrometric orbit fitting showing improved sampling efficiency and robustness when using Cartesian state vectors.

Significance. If the identification of the singularity and the corrected Jacobian hold, the closed-form expressions enable consistent prior-density transformations under reparameterization, which is valuable for Bayesian inference in microlensing and astrometric orbit problems. The explicit MCMC comparison provides concrete evidence favoring Cartesian parameterization for gradient-based sampling when orbital elements are weakly constrained. The work supplies reproducible analytic results rather than fitted parameters.

major comments (1)
  1. [Abstract] Abstract (and the corresponding derivation section): the central claim that the Skowron et al. (2011) choice of reference axis for the longitude of the ascending node renders the intermediate Jacobian formally singular must be supported by an explicit computation of the Jacobian matrix or its determinant showing a zero eigenvalue or vanishing determinant; without this step the assertion that the definition is 'incorrect' (rather than a removable convention) and the need for the 'corrected' derivation remain unverified load-bearing points.
minor comments (2)
  1. The manuscript should specify the exact reference epoch and coordinate axes used in the Skowron et al. (2011) formulation versus the corrected version to allow direct reproduction of the Jacobian.
  2. The MCMC comparison section would benefit from reporting the specific number of chains, burn-in length, and convergence diagnostics (e.g., Gelman-Rubin statistic) alongside the efficiency metrics.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and constructive feedback on the manuscript. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the corresponding derivation section): the central claim that the Skowron et al. (2011) choice of reference axis for the longitude of the ascending node renders the intermediate Jacobian formally singular must be supported by an explicit computation of the Jacobian matrix or its determinant showing a zero eigenvalue or vanishing determinant; without this step the assertion that the definition is 'incorrect' (rather than a removable convention) and the need for the 'corrected' derivation remain unverified load-bearing points.

    Authors: We agree that an explicit computation of the Jacobian matrix (or its determinant) for the Skowron et al. (2011) parameterization is needed to verify the singularity claim. In the revised manuscript we will insert the full intermediate Jacobian matrix under their definition of the longitude of the ascending node, compute its determinant, and demonstrate that it vanishes identically. This addition will also clarify why the singularity is not merely a removable convention but arises directly from the reference-axis choice. revision: yes

Circularity Check

0 steps flagged

No circularity in Jacobian derivations or Skowron et al. critique

full rationale

The paper derives compact analytic expressions for Jacobian determinants of the orbital-element to Cartesian transformation and its variants as independent first-principles results. It references Skowron et al. (2011) only to contextualize a claimed definitional difference in the longitude of the ascending node, then supplies its own closed-form correction and shows the formula is effectively unchanged under a proper axis choice. An MCMC comparison is performed as an explicit numerical test. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the derivations stand on their own algebraic content and are externally falsifiable via direct matrix evaluation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, ad hoc axioms, or invented entities. It derives mathematical results from established orbital mechanics.

axioms (1)
  • standard math Standard definitions and properties of Keplerian orbital elements and their relation to Cartesian coordinates
    The transformation and Jacobian calculations rely on classical orbital mechanics without introducing new axioms.

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Works this paper leans on

27 extracted references · 21 canonical work pages · 2 internal anchors

  1. [1]

    Agol, E., & Fabrycky, D. C. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte, 7, doi: 10.1007/978-3-319-55333-7 7

    work page doi:10.1007/978-3-319-55333-7 2018

  2. [2]

    P., Rhie, S

    Bennett, D. P., Rhie, S. H., Nikolaev, S., et al. 2010, ApJ, 713, 837, doi: 10.1088/0004-637X/713/2/837

    work page doi:10.1088/0004-637x/713/2/837 2010

  3. [3]

    2016, A&A, 587, A89, doi: 10.1051/0004-6361/201527388

    Beust, H., Bonnefoy, M., Maire, A.-L., et al. 2016, A&A, 587, A89, doi: 10.1051/0004-6361/201527388

    work page doi:10.1051/0004-6361/201527388 2016

  4. [4]

    L., De Rosa, R

    Blunt, S., Nielsen, E. L., De Rosa, R. J., et al. 2017, AJ, 153, 229, doi: 10.3847/1538-3881/aa6930

    work page doi:10.3847/1538-3881/aa6930 2017

  5. [5]

    , keywords =

    Blunt, S., Wang, J. J., Angelo, I., et al. 2020, AJ, 159, 89, doi: 10.3847/1538-3881/ab6663

    work page doi:10.3847/1538-3881/ab6663 2020

  6. [6]

    2021, MNRAS, 505, 126, doi: 10.1093/mnras/stab1376

    Bozza, V., Khalouei, E., & Bachelet, E. 2021, MNRAS, 505, 126, doi: 10.1093/mnras/stab1376

    work page doi:10.1093/mnras/stab1376 2021

  7. [7]

    2018, JAX: composable transformations of Python+NumPy programs, 0.2.5 http://github.com/google/jax

    Bradbury, J., Frostig, R., Hawkins, P., et al. 2018, JAX: composable transformations of Python+NumPy programs, 0.2.5 http://github.com/google/jax

    2018

  8. [8]

    2012, A&A, 542, A41, doi: 10.1051/0004-6361/201118346 Ferrer-Ch´ avez, R., Blunt, S., & Wang, J

    Chauvin, G., Lagrange, A.-M., Beust, H., et al. 2012, A&A, 542, A41, doi: 10.1051/0004-6361/201118346 Ferrer-Ch´ avez, R., Blunt, S., & Wang, J. J. 2021, Research Notes of the American Astronomical Society, 5, 162, doi: 10.3847/2515-5172/ac151d

    work page doi:10.1051/0004-6361/201118346 2012

  9. [9]

    2016 , month =

    Foreman-Mackey, D. 2016, The Journal of Open Source Software, 24, doi: 10.21105/joss.00024

    work page doi:10.21105/joss.00024 2016

  10. [10]

    W., Lang, D., et al.\ 2013, , 125, 306

    Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067 Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2023, A&A, 674, A1, doi: 10.1051/0004-6361/202243940

    work page doi:10.1086/670067 2013

  11. [11]

    B., Stern, H

    Gelman, A., Carlin, J. B., Stern, H. S., et al. 2014, Bayesian data analysis, 3rd edn., Texts in statistical science (CRC Press). https://ci.nii.ac.jp/ncid/BB13945229

    2014

  12. [12]

    The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo

    Hoffman, M. D., & Gelman, A. 2011, arXiv e-prints, arXiv:1111.4246. https://arxiv.org/abs/1111.4246

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  13. [13]

    M., Marois, C., Macintosh, B

    Konopacky, Q. M., Marois, C., Macintosh, B. A., et al. 2016, AJ, 152, 28, doi: 10.3847/0004-6256/152/2/28 14

    work page doi:10.3847/0004-6256/152/2/28 2016

  14. [14]

    2019 , publisher =

    Kumar, R., Carroll, C., Hartikainen, A., & Martin, O. 2019, The Journal of Open Source Software, 4, 1143, doi: 10.21105/joss.01143

    work page doi:10.21105/joss.01143 2019

  15. [15]

    Markley, F. L. 1995, Celestial Mechanics and Dynamical Astronomy, 63, 101, doi: 10.1007/BF00691917

    work page doi:10.1007/bf00691917 1995

  16. [16]

    2025, jnkepler: Differentiable N-body model for multi-planet systems,, Astrophysics Source Code Library, record ascl:2505.006 http://ascl.net/2505.006

    Masuda, K. 2025, jnkepler: Differentiable N-body model for multi-planet systems,, Astrophysics Source Code Library, record ascl:2505.006 http://ascl.net/2505.006

    2025

  17. [17]

    E., Livingston, J

    Masuda, K., Libby-Roberts, J. E., Livingston, J. H., et al. 2024, AJ, 168, 294, doi: 10.3847/1538-3881/ad83d3

    work page doi:10.3847/1538-3881/ad83d3 2024

  18. [18]

    D., & Dermott, S

    Murray, C. D., & Dermott, S. F. 1999, Solar system dynamics (Cambridge University Press)

    1999

  19. [19]

    Neal, R. M. 2012, arXiv e-prints, arXiv:1206.1901. https://arxiv.org/abs/1206.1901 O’Neil, K. K., Martinez, G. D., Hees, A., et al. 2019, AJ, 158, 4, doi: 10.3847/1538-3881/ab1d66

    work page doi:10.3847/1538-3881/ab1d66 2012

  20. [20]

    V., & Tremaine, S

    Peiris, H. V., & Tremaine, S. 2003, ApJ, 599, 237, doi: 10.1086/378638

    work page doi:10.1086/378638 2003

  21. [21]

    Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro

    Phan, D., Pradhan, N., & Jankowiak, M. 2019, arXiv preprint arXiv:1912.11554

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Poisson, E., & Will, C. M. 2014, Gravity

    2014

  23. [23]

    2011, ApJ, 738, 87, doi: 10.1088/0004-637X/738/1/87

    Skowron, J., Udalski, A., Gould, A., et al. 2011, ApJ, 738, 87, doi: 10.1088/0004-637X/738/1/87

    work page doi:10.1088/0004-637x/738/1/87 2011

  24. [24]

    2023, AJ, 166, 164, doi: 10.3847/1538-3881/acf5cc

    Thompson, W., Lawrence, J., Blakely, D., et al. 2023, AJ, 166, 164, doi: 10.3847/1538-3881/acf5cc

    work page doi:10.3847/1538-3881/acf5cc 2023

  25. [25]

    2020, MNRAS, 491, 1941, doi: 10.1093/mnras/stz3181

    Tremaine, S. 2020, MNRAS, 491, 1941, doi: 10.1093/mnras/stz3181

    work page doi:10.1093/mnras/stz3181 2020

  26. [26]

    2023, Dynamics of Planetary Systems

    Tremaine, S. 2023, Dynamics of Planetary Systems

    2023

  27. [27]

    2021, Bayesian Analysis, 16, 667, doi: 10.1214/20-BA1221

    Vehtari, A., Gelman, A., Simpson, D., Carpenter, B., & B¨ urkner, P.-C. 2021, Bayesian Analysis, 16, 667, doi: 10.1214/20-BA1221

    work page doi:10.1214/20-ba1221 2021