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  "slug": "wide-binary-angle-inversion",
  "title": "An exact projected-angle likelihood and population inversion for wide-binary eccentricities",
  "shortTitle": "One angle, an exact likelihood: reading wide-binary eccentricities",
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  "oneLine": "For wide binary stars seen at a random moment from a random direction, the angle between separation and motion has a closed-form likelihood; its distribution determines the population's eccentricity distribution exactly, and simple averages of the angle estimate eccentricity summaries directly.",
  "abstract": "A wide binary star takes thousands of years to complete one orbit, so its eccentricity usually cannot be found by following the orbit. Astronomers instead use the angle Γ between the pair's projected separation and projected relative velocity, whose distribution over many binaries depends on the population's eccentricity law. In the ideal model, orbital phase is a random time in the orbit and orientation is uniformly random in three dimensions. For this model the candidate derives the likelihood in closed form, p(γ|e)=[acosh(1/e)−1{|cos γ|>e} acosh(|cos γ|/e)]/(π sin²γ), and shows that its mixtures are classical Abel transforms. This gives an explicit inverse for the population law, including atoms, and answers the uniqueness question raised in Appendix B of Hwang, Ting and Zakamska (2022): different eccentricity laws never give the same angle law. The inverse is continuous in the weak topology but discontinuous in total variation, so fine structure can be nearly invisible. Moments need no inversion: for example E√(1−e²)=(π/2)E sin³Γ and E e²=1/2+E cos 2Γ−(1/2)E cos 4Γ. These are the only symmetric unbiased single-angle observables for their targets, and they give near-efficient estimators of power-law indices. Two exact selection results bear on Gaia samples: thermal populations keep a uniform angle law under selections that ignore sky position angles, and binning by projected separation preserves the model under a scale-free log-uniform semimajor-axis law. Simulations of Kepler orbits check representative formulas. No observational catalogue is analysed. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.",
  "datePublished": "2026-10-31",
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  "keywords": [
    "wide binaries",
    "orbital eccentricity",
    "Gaia",
    "projected angle",
    "Abel transform",
    "identifiability",
    "inverse problems",
    "Kepler orbits",
    "selection effects",
    "method of moments"
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  "keyResults": [
    "Exact likelihood (Theorem 3.1, proof): for eccentricity 0<e<1, p(γ|e)=[acosh(1/e)−1{|cos γ|>e} acosh(|cos γ|/e)]/(π sin²γ); circular orbits give −log|cos γ|/(π sin²γ).",
    "Population inversion (Theorem 4.1, proof): mixtures over any eccentricity law are classical Abel transforms, so the angle law determines the eccentricity law, atoms included, by an explicit inverse. This answers the uniqueness question in Appendix B of Hwang, Ting and Zakamska (2022).",
    "Power laws (Section 5, proof): f(e)=(α+1)e^α gives p_α(γ)=B((α+1)/2,1/2)(1−|cos γ|^(α+1))/(2π sin²γ); a flat angle law forces the thermal law f(e)=2e.",
    "Moments without inversion (Theorems 6.1–6.2, Corollary 7.1, proof): E√(1−e²)=(π/2)E sin³Γ and E e²=1/2+E cos 2Γ−(1/2)E cos 4Γ; each target has at most one symmetric unbiased single-angle observable. Power-law index estimators reach at least 96% efficiency on the computed grid (Section 7.2).",
    "Selection (Propositions 8.1–8.2, proof): thermal populations keep flat angles under selections that ignore sky position angles; binning by projected separation is exact under a log-uniform semimajor-axis law and otherwise reweights by an explicit factor.",
    "Stability limit (Propositions 9.1–9.2, proof): the inverse is continuous in the weak topology but discontinuous in total variation, so fine structure in the eccentricity law can be nearly invisible in the angles."
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  "relatedWorks": [
    {
      "citation": "Hwang, Ting and Zakamska (2022), The eccentricity distribution of wide binaries and their individual measurements. Monthly Notices of the Royal Astronomical Society 512, 3383–3399. Source of the uniqueness question (Appendix B) and of the numerical likelihood that the closed form replaces.",
      "url": "https://doi.org/10.1093/mnras/stac675",
      "doi": "10.1093/mnras/stac675"
    },
    {
      "citation": "Tokovinin (1998), On the distribution of orbital eccentricities for wide visual binary stars. Astronomy Letters 24, 178–179. Proposed inferring eccentricities from projected position–velocity angles.",
      "url": "https://articles.adsabs.harvard.edu/pdf/1998AstL...24..178T",
      "doi": null
    },
    {
      "citation": "Tokovinin and Kiyaeva (2016), Eccentricity distribution of wide binaries. Monthly Notices of the Royal Astronomical Society 456, 2070–2079. Joint angle-and-speed inference.",
      "url": "https://doi.org/10.1093/mnras/stv2825",
      "doi": "10.1093/mnras/stv2825"
    },
    {
      "citation": "Tokovinin (2020), Eccentricity distribution of wide low-mass binaries. Monthly Notices of the Royal Astronomical Society 496, 987–993.",
      "url": "https://doi.org/10.1093/mnras/staa1639",
      "doi": "10.1093/mnras/staa1639"
    },
    {
      "citation": "Benisty, Evans and Davis (2022), The eccentricity distribution of wide binaries. Monthly Notices of the Royal Astronomical Society: Letters 518, L51–L56. Analytic speed and acceleration transforms of the eccentricity law: a complementary route through a different observable.",
      "url": "https://doi.org/10.1093/mnrasl/slac134",
      "doi": "10.1093/mnrasl/slac134"
    },
    {
      "citation": "Makarov (2025a), Distributions of wide binary stars in theory and in Gaia data. I. Generalized Ambartsumian (1937) approach and the family of power-law distributions of eccentricity. The Astronomical Journal 169, 113. Forward-simulation motion-angle inference with astrometric errors.",
      "url": "https://doi.org/10.3847/1538-3881/ada5fa",
      "doi": "10.3847/1538-3881/ada5fa"
    },
    {
      "citation": "Makarov (2025b), Distributions of wide binary stars in theory and in Gaia data. II. Reconstruction of sample probability density of true orbit sizes. The Astronomical Journal 170, 138. A different inverse problem (orbit sizes).",
      "url": "https://doi.org/10.3847/1538-3881/adec79",
      "doi": "10.3847/1538-3881/adec79"
    },
    {
      "citation": "Makarov (2026), Distributions of wide binary stars in theory and in Gaia data. III. Orbital momenta, masses, and manifestations of MOND. The Astronomical Journal 171, 79.",
      "url": "https://doi.org/10.3847/1538-3881/ae2757",
      "doi": "10.3847/1538-3881/ae2757"
    },
    {
      "citation": "Brandt (2026), Closed-form statistical relations between projected separation, semimajor axis, companion mass, and host acceleration. arXiv:2601.14688v2. Analytic orbital statistics for other observables.",
      "url": "https://arxiv.org/abs/2601.14688",
      "doi": "10.48550/arXiv.2601.14688"
    },
    {
      "citation": "Hamilton (2022), On the phase-mixed eccentricity and inclination distributions of wide binaries in the Galaxy. The Astrophysical Journal Letters 929, L29. Galactic tides and anisotropic orientations.",
      "url": "https://doi.org/10.3847/2041-8213/ac6600",
      "doi": "10.3847/2041-8213/ac6600"
    },
    {
      "citation": "Hwang, El-Badry, Rix, Hamilton, Ting and Zakamska (2022b), Wide twin binaries are extremely eccentric: evidence of twin binary formation in circumbinary disks. The Astrophysical Journal Letters 933, L32.",
      "url": "https://doi.org/10.3847/2041-8213/ac7c70",
      "doi": "10.3847/2041-8213/ac7c70"
    },
    {
      "citation": "El-Badry, Rix and Heintz (2021), A million binaries from Gaia eDR3: sample selection and validation of Gaia parallax uncertainties. Monthly Notices of the Royal Astronomical Society 506, 2269–2295.",
      "url": "https://doi.org/10.1093/mnras/stab323",
      "doi": "10.1093/mnras/stab323"
    },
    {
      "citation": "Jeans (1919), The origin of binary systems. Monthly Notices of the Royal Astronomical Society 79, 408–416. The thermal eccentricity law.",
      "url": "https://doi.org/10.1093/mnras/79.6.408",
      "doi": "10.1093/mnras/79.6.408"
    },
    {
      "citation": "Shatsky (2001), The long-period companions of multiple stars tend to have moderate eccentricities. Astronomy & Astrophysics 380, 238–244.",
      "url": "https://doi.org/10.1051/0004-6361:20011401",
      "doi": "10.1051/0004-6361:20011401"
    },
    {
      "citation": "Kloza and Elliott (2023), Inverse projection of axisymmetric orientation distributions. EPL 143, 22002. A fixed-axis orientation inverse with different sampling geometry.",
      "url": "https://doi.org/10.1209/0295-5075/ace4ee",
      "doi": "10.1209/0295-5075/ace4ee"
    },
    {
      "citation": "Wicksell (1925), The corpuscle problem. A mathematical study of a biometric problem. Biometrika 17, 84–99. The classical Abel-type inverse from random sections.",
      "url": "https://doi.org/10.1093/biomet/17.1-2.84",
      "doi": "10.1093/biomet/17.1-2.84"
    },
    {
      "citation": "Gorenflo and Vessella (1991), Abel Integral Equations: Analysis and Applications. Lecture Notes in Mathematics, Springer. Abel inversion and its ill-posedness.",
      "url": "https://doi.org/10.1007/BFb0084665",
      "doi": "10.1007/BFb0084665"
    },
    {
      "citation": "Groeneboom and Jongbloed (1995), Isotonic estimation and rates of convergence in Wicksell's problem. The Annals of Statistics 23, 1518–1542. Estimation theory for the stereological analogue.",
      "url": "https://doi.org/10.1214/aos/1176324310",
      "doi": "10.1214/aos/1176324310"
    },
    {
      "citation": "Hwang (n.d.), Eccentricity-of-wide-binaries: public code and simulation grids. GitHub repository. The originating public code, whose ideal likelihood grids the closed form replaces.",
      "url": "https://github.com/HC-Hwang/Eccentricity-of-wide-binaries",
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