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Closed-Form Statistical Relations Between Projected Separation, Semimajor Axis, Companion Mass, and Host Acceleration

T0 review · 0 major / 2 minor · reviewed 2026-05-16 · grok-4.3

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Pith's one-line read Analytic probability density functions relate a companion's projected separation, mass, and the acceleration it causes on its host for randomly oriented orbits.

desk verdict This paper supplies closed-form PDFs for the projected-separation-to-semimajor-axis ratio and mass-acceleration relations under isotropic orbits, which removes a repeated integration step in fitting codes. read the letter →

arxiv 2601.14688 v2 submitted 2026-01-21 astro-ph.SR astro-ph.EPastro-ph.IM

Timothy D Brandt This is my paper · ORCID
classification astro-ph.SRastro-ph.EPastro-ph.IM
keywords projectedseparationsemimajoraxiscompanionmasshostaccelerationprobabilitydensityfunctionsKeplerianorbitsradialvelocitytrendsastrometric
checked against Foundation.AbsoluteFloorClosureCost.FunctionalEquation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper derives closed-form statistical relationships between a companion's projected separation, its mass, and the acceleration it induces on the host star. These relationships are expressed as probability density functions that hold regardless of the specific orbital elements, as long as the orbits are randomly oriented in space. A separate closed-form expression gives the distribution of the ratio between projected separation and semimajor axis for fixed eccentricity, which can be integrated over any eccentricity distribution. These analytic forms simplify calculations that involve derivatives, such as in sampling methods. The results are verified against more complex expressions from Keplerian orbit equations.

What carries the argument

Analytic probability density functions for the relationships between projected separation, semimajor axis, companion mass, and host acceleration under isotropic orbital orientations.

What would settle it

A large sample of systems with measured projected separations, masses, and accelerations showing distributions that deviate significantly from the predicted analytic PDFs would indicate the isotropic orientation assumption fails.

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Extended reading notes

Core claim

I derive the statistical relationship between a radial velocity or astrometric acceleration (a trend), a companion's mass, and the projected separation of the companion. These relationships, expressed as probability density functions, are analytic and independent of all Keplerian orbital elements so long as orbits are randomly oriented in space. I also derive a closed-form expression for the probability distribution of the ratio of the projected separation to the semimajor axis at fixed eccentricity. This expression can be numerically integrated over eccentricity for an arbitrary distribution of eccentricities.

Load-bearing premise

Orbits are randomly oriented in space with an isotropic distribution of inclinations and longitudes of ascending node.

Editorial extensions

If this is right

  • These closed-form expressions are especially useful for calculations requiring derivatives, such as Hamiltonian Monte Carlo sampling.
  • The closed-form for the projected separation to semimajor axis ratio can be integrated numerically over any given eccentricity distribution.
  • Verification shows agreement with equivalent but more complex expressions in the literature based on Keplerian orbit equations.
  • The provided Jupyter notebook includes all figures and calculations for direct reproduction and extension.

Reading between the lines

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

  • These relations could enable faster statistical modeling of companion populations by avoiding full sampling over all orbital elements.
  • Applications to radial velocity or astrometric surveys might yield more efficient mass and orbit constraints from trend data alone.
  • Further tests could examine how the PDFs change if orbital orientations deviate from isotropy, such as in young clusters.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript derives closed-form analytic probability density functions relating a companion's projected separation, semimajor axis, mass, and the host's radial-velocity or astrometric acceleration. These PDFs are independent of all other Keplerian orbital elements provided the orbits are isotropically oriented. A separate closed-form expression is given for the distribution of the projected-separation-to-semimajor-axis ratio at fixed eccentricity; this expression can be integrated numerically over an arbitrary eccentricity distribution. All derivations are verified by direct numerical comparison to established Keplerian expressions in the literature, and a Jupyter notebook containing the calculations and figures is supplied.

Significance. If the derivations hold, the analytic PDFs remove the need to marginalize over unknown orbital elements in many statistical applications and are especially convenient for gradient-based methods such as Hamiltonian Monte Carlo. The provision of machine-verifiable code and the explicit statement of the isotropic-orientation assumption constitute clear strengths that increase the utility and reproducibility of the results for exoplanet and binary-star population studies.

minor comments (2)
  1. The abstract states that the results are 'independent of all Keplerian orbital elements'; it would be clearer to add the qualifying clause 'under the isotropic-orientation assumption' already present in the body of the text.
  2. In the verification section, the manuscript compares the new expressions to 'equivalent but more complex expressions in the literature'; citing the specific references (e.g., the exact equations being reproduced) would allow readers to trace the numerical checks more easily.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review and recommendation to accept the manuscript. We appreciate the recognition of the utility of the derived closed-form PDFs for applications in exoplanet and binary-star population studies, as well as the value placed on the provided Jupyter notebook for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

Derivation self-contained from isotropic orientations and Keplerian geometry

full rationale

The paper derives closed-form PDFs for relations among projected separation, semimajor axis, companion mass, and acceleration directly from the explicit assumption of random orbital orientations in space combined with standard Keplerian orbit equations. These expressions are independent of other orbital elements by construction under that assumption, are verified by direct comparison to prior literature expressions, and include reproducible code. No load-bearing step reduces to a fitted parameter renamed as prediction, self-citation chain, or self-definitional loop. The central results follow from the stated geometric prior without circular reduction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The derivations rest on the single domain assumption of isotropic orbital orientations; no free parameters are introduced and no new physical entities are postulated.

assumptions (1)
  • domain assumption Orbits are randomly oriented in space (uniform distribution of inclinations and nodes)
    This isotropy assumption is invoked to obtain expressions independent of all Keplerian elements.

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Cite this review

Pith. "Pith review of Closed-Form Statistical Relations Between Projected Separation, Semimajor Axis, Companion Mass, and Host Acceleration." pith.science (2026). https://pith.science/paper/2601.14688

@misc{pith2026260114688,
  author       = {Pith},
  title        = {Pith review of: Closed-Form Statistical Relations Between Projected Separation, Semimajor Axis, Companion Mass, and Host Acceleration},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2601.14688}},
  note         = {Machine review of arXiv:2601.14688}
}
read the original abstract

I derive the statistical relationship between a radial velocity or astrometric acceleration (a trend), a companion's mass, and the projected separation of the companion. These relationships, expressed as probability density functions, are analytic and independent of all Keplerian orbital elements so long as orbits are randomly oriented in space. I also derive a closed-form expression for the probability distribution of the ratio of the projected separation to the semimajor axis at fixed eccentricity. This expression can be numerically integrated over eccentricity for an arbitrary distribution of eccentricities. I verify my results with empirical comparisons to equivalent but more complex expressions in the literature based on the equations of Keplerian orbits. The closed-formed expressions derived here would be especially useful for any calculation that requires derivatives, e.g., Hamiltonian Monte Carlo. I also provide a Jupyter notebook including all figures and calculations.

Figures

Figures reproduced from arXiv: 2601.14688 by the authors.

Figure 1
Figure 1. — Empirical comparison of the probability density func￾tions for ΦRV and Φast (Equations (17) and (8)) to the equivalent formulae from T99 that are written as functions of the Keplerian orbital elements. The empirical functions are histograms using 107 points generated assuming random phases and orientations and a uniform distribution of eccentricities up to 0.8; they are indistin￾guishable from the analytic results… view at source ↗
Figure 2
Figure 2. — Two-dimensional probability density d 2p/(dψ de) as a function of eccentricity e and ψ, the ratio of projected separation to semimajor axis. The probability densities are calculated using Equations (33) and (34), and are shown using linear (left) and logarithmic (right) color scales. The region ψ > 1 + e is forbidden on physical grounds, while there is an integrable singularity along the line ψ = 1 − e. 0.0 0.5 1.… view at source ↗
Figure 3
Figure 3. — One-dimensional distributions of ψ, the ratio of projected separation to semimajor axis, for several distributions of eccentricity. The left panel includes shaded histograms that have been computed from the equations of a Keplerian orbit using Monte Carlo. Solid lines are computed using Equations (33), (34), and (37), with the latter using numerical quadrature in one dimension. notebook can serve as a template to … view at source ↗

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