Stellar encounters — one engine, four dynamical levels

What the catalogue is. A small curated list — not a complete survey. The Gaia-based catalogues (Dybczyński, Berski, Tokarek, Podlewska-Gaca, Langner & Bartczak 2022, A&A 664, A123 — 155 perturbers at a 2 pc threshold with Monte-Carlo uncertainties; Bailer-Jones, Rybizki, Andrae & Fouesneau 2018, A&A 616, A37) are the place to look for completeness. This page shows what our engine gets from the published 6D states of a representative set — and keeps the disagreements visible.

Provenance. Every astrometric value is the one SIMBAD carries for that object — parallax and proper motion from Gaia EDR3 where available, otherwise Hipparcos (Algol, α Cen and Sirius are too bright for Gaia); the radial-velocity source is printed per row. Masses marked “assumed from spectral type” are not measurements and enter the impulse metric linearly.

What is not modelled. Binaries are single point masses (Algol, α Cen, Sirius and Scholz's star are all multiple); the Sun's reflex motion is ignored; the catalogue does not search for new perturbers; and the published encounter values are quoted for comparison only.

Where the Galactic term matters — a Level-3 finding. The catalogue exposes the contrast that motivated Engine C: for GJ 710 the Milky-Way term is +122.6 AU (+1.16 %), i.e. 3.8× below its astrometric error bar (σ_b = 460.6 AU) — for this star the model band is the larger term (9,455 … 11,226 AU, 3.85 σ_b). For HD 7977 — a *nearly radial* passage — the same treatment moves the closest approach from 30,571 AU to 2,200 AU (−92.8 %) with the linearised tide, and to 2,312.5 AU (−92.4 %) in the full two-orbit integration; on top of that the six-potential band spans 2,313 … 10,663 AU. The reason is amplification, not magic: in a near-radial encounter the closest approach is a *tiny transverse velocity* divided into the radial speed (b ≈ d·v_t/|v_r| with v_t ≈ 0.05 km/s here), so any perturbation — a Galactic tide, or one σ of proper motion — is amplified by d/|v_r|. GJ 710's encounter is short and its leverage arm small; HD 7977's is 2.8 Myr long with a 76 pc arm. That is why the page quotes both the number *and* its model sensitivity.

How to read a verdict (A · B · C). A published value is not ground truth: it is "published input → published encounter". Each row therefore answers three separate questions. A — input agreement: is our 6D state the one the published analysis used? (We also run a DR2-era proxy state through the same engine, so the input revision can be isolated.) B — dynamical-model agreement: this row's S, plus the linearised-tide vs full-orbit difference Δb = b_tide − b_full. C — published-uncertainty containment: is our distance inside their 90 % interval, and if not, is that a consequence of A (the inputs), of B (the model), or of neither (their Monte-Carlo median)? A row can be "outside now, inside on the published inputs" — that is an input-revision verdict, not an engine failure. One star can also carry two rows (an ingested catalogue copy plus the curated one: HD 168442 ≡ GJ 710, HD 7977 twice): they are marked twin, and every aggregate on this page counts unique stars. Populations are stated, not implied: the catalogue is the published DR2 candidate list, so every aggregate counts only rows whose current parallax has S/N ≥ 3 (a distance exists only then), and the encounter-class subset (Galactic closest approach inside 1 pc) is reported separately. Every difference from a published value is decomposed into three named terms — input revision, our Galactic correction, their model — with the identity checked numerically on all decomposable rows: cargo run --release -p tks-ephemeris --example encounter_forensics 8.

Product A — model ensemble envelope (six named potentials). Running the same state through an ensemble of six potentials — two calibrations (MWPotential2014 and a McMillan-2017-style model) plus a Ferrers bar (a = 3.1, b = 1.0, c = 0.4 kpc, M = 1.06×10¹⁰ M☉, Ω_b = 39 km/s/kpc, exact quadrature force) and spiral arms calibrated to the observed 5 % peak radial-force contrast at R₀ — gives HD 7977: 2,313 … 10,663 AU (spread 8,350 AU = 380 % of b, larger than its σ_b = 5,348 AU) and GJ 710: 9,455 … 11,226 AU (spread 1,771 AU = 3.85 σ). So HD 7977 is published as a range, and for GJ 710 the model — not the astrometry — is the leading term once bar and arms are included: 10.4 kAU ± 0.5 (astrometric) ± 0.8 (model). Both perturbations are validated against exact limits (sphere limit, boundary continuity, curl-free, far-field monopole) in the test suite.

The band as a distribution (grid + priors). Weighting the grid the way the literature constrains it — potential (equal weights), bar pattern speed Ω_b = 39 ± 3, bar angle 25° ± 5°, arm contrast 5 ± 1.5 %, pitch 12.5° ± 2°, arm pattern speed Ω_p = 20 ± 4 km/s/kpc; 128 combinations per star — gives HD 7977: weighted 5–95 % = 1,617 … 15,452 AU (median 7,032 AU, weighted P(b < 3 kAU) = 11.8 %, P(b < 10 kAU) = 64.6 %) and GJ 710: 8,846 … 11,412 AU (median 9,655 AU). The index itself is uncertain: S = 2.83–5.41 (HD 7977) and 0.65–3.66 (GJ 710). The dominant lever is the arm contrast (dln b/dln c = +1.13); pattern speeds barely matter over ±3 Myr flight times — they rotate the pattern by only 1–3°. Every grid node and its weight: data/encounters/tks_ensemble.csv (example: encounter_ensemble).