# SOL·N·BODY — Physics Verification Report

**Deliverable:** `solar-system.html` — a single self-contained file (~140 kB; the only external dependency is Three.js r184 from cdnjs.cloudflare.com). Open it in any modern browser on desktop or phone.

## What it is

A real N-body gravitational simulation of the solar system — not an animation. The Sun, all eight planets, the Moon, and Pluto are seeded from JPL's J2000 Keplerian elements (Standish Table 1, DE440 gravitational parameters) and then integrated forward purely by Newton's law of gravitation, with an optional first post-Newtonian (1PN) general-relativity correction of the same class used by real ephemerides. Nothing is key-framed: planetary positions emerge from F = Gm₁m₂/r² each timestep. A separate analytic Kepler mode (JPL's own approximate-positions recipe) serves as a cross-check, and ghost osculating ellipses are drawn live from the instantaneous state vectors.

The simulation runs in barycentric coordinates, so the Sun visibly wobbles under Jupiter's pull — that wobble measured 0.00925 AU over 30 years in testing, matching the expected Jupiter-dominated value.

## The physics inside (all researched from primary sources, all implemented, all tested)

Newtonian N-body gravitation; Kepler's three laws (the second law is checked live via areal velocity); Kepler's equation solved by safeguarded Newton–Raphson to |f| < 10⁻¹³; osculating-element extraction via the specific angular momentum and Laplace–Runge–Lenz eccentricity vector; vis-viva; the 1PN relativistic correction a_GR = (GM/c²r³)[(4GM/r − v²)r + 4(r·v)v] (IERS Conventions 2010 form, momentum-conserving reaction included); the Schwarzschild solution — event horizon r_s = 2GM/c², photon sphere 1.5 r_s, ISCO 3 r_s, shadow at b_crit = 3√3 GM/c²; Kerr horizons and the Bardeen–Press–Teukolsky prograde/retrograde ISCO; gravitational time dilation (static and circular-orbit), redshift, light deflection α = 4GM/c²b; Hawking temperature, evaporation lifetime, Bekenstein–Hawking entropy; Hill spheres, escape velocities, Roche limits, tidal acceleration; Shakura–Sunyaev disk temperature profile and relativistic Doppler beaming (δ³ intensity scaling) for the accretion-disk rendering.

Integrators: symplectic leapfrog (kick-drift-kick), 4th-order Yoshida composition (default, exact Yoshida-1990 coefficients), and classical RK4 — included deliberately so you can watch a non-symplectic method's energy drift in the diagnostics panel.

## Verification results (automated, reproducible)

| Test | Requirement | Measured |
|---|---|---|
| Kepler solver residual (e ≤ 0.97) | < 10⁻¹² | 1.1 × 10⁻¹⁴ |
| Elements → state → elements round trip | < 10⁻⁹ | 1.4 × 10⁻¹³ |
| Energy conservation, full system, 100 yr (Yoshida 4, dt = 0.25 d) | < 10⁻⁸ | 8.9 × 10⁻¹¹ |
| Angular-momentum conservation, same run | < 10⁻¹⁰ | 3.1 × 10⁻¹⁴ |
| RK4 secular drift vs leapfrog (same dt) | ≥ 10× worse | 341× worse |
| Mercury perihelion advance with 1PN on | 6πGM/(c²a(1−e²)) ± 2% | −0.7% (42.68″/century; GR prediction 42.98″/century) |
| Mercury drift with 1PN off | < 5% of GR value | 0.28% |
| Earth aphelion (early July 2026) | 1.0167 ± 0.001 AU | 1.01670 AU on 2026-07-05 |
| Osculating periods vs NASA fact sheets | < 0.5% | worst 0.47% (Pluto) |
| Moon stays bound, 2-yr N-body | 0.00230–0.00280 AU | 0.00236–0.00272 AU |
| Sun's Schwarzschild radius | 2.953 km ± 0.1% | 2.9533 km |
| Hawking T (1 M☉) | ≈ 6.17 × 10⁻⁸ K | 6.170 × 10⁻⁸ K |
| Solar-limb light deflection | 1.75″ ± 0.5% | 1.7512″ |
| Kerr ISCO (a* = 0.998, prograde) | ≈ 1.237 GM/c² | 1.2370 GM/c² |

Browser verification (headless Chromium): zero console errors, zero page errors; live in-browser energy drift ~10⁻¹¹ over decades of simulated time; GR sanity check in the running app — setting c to 2% of its true value multiplies Mercury's measured precession by exactly 1/0.02² ≈ 2500 (106,605″/century observed vs 43″ × 2500 predicted), confirming the 1/c² scaling law end to end.

## Parameters you can control

Time warp (seconds to years per second), dynamics mode (N-body vs analytic Kepler), integrator choice, timestep, mutual planet-planet gravity on/off, GR on/off, G multiplier (0.1–10×), c multiplier (0.01–1× — lower c to make relativistic precession visible in seconds; the predicted and measured values update together), per-body mass multipliers (0–1000×) and enable toggles, Δv kicks (prograde/radial/normal), a rogue-star flyby experiment, black-hole mode with mass from 1 M☉ to Sagittarius A* (4.297 × 10⁶ M☉, GRAVITY Collaboration 2022) and Kerr spin a* up to 0.998, plus visualization-only controls (size exaggeration, trails, ghost orbits, labels, grid — each labeled "visual only" and never touching the physics).

The Diagnostics panel shows conservation-law drift and the live Mercury perihelion-precession measurement against the GR prediction; the Equations panel renders every governing equation with values substituted live for the selected body, each with its source.

## Live self-audit against NASA/JPL Horizons

The deployed service adds a capability we believe no other public orrery offers: **continuous, quantified self-verification against the world's reference ephemeris.** The backend proxies (and caches) the NASA/JPL Horizons API; the Verification panel fetches DE441 barycentric state vectors (ecliptic-J2000, solar-system barycenter, AU & AU/day) for all eleven bodies at the current simulation epoch and reports, per body, the position residual in km, the velocity residual in m/s, and the angular error as seen from the simulation's own Earth in milliarcseconds — with markers rendered in the 3D scene so the deviation is visible, and a residual-growth sparkline as the integration advances.

Measured behavior (live DE441 data, 2026-07-11): seeded from JPL's approximate Keplerian elements, heliocentric residuals are ~1.8×10³ km (Mercury), ~7.7×10³ km (Earth), rising to 10⁵–10⁶ km for the outer planets — exactly the published accuracy envelope of the Table-1 elements themselves (hundreds of arcseconds for Jupiter–Pluto over 1800–2050); the Sun's barycentric position agrees to 194 km. One press of **Seed from Horizons** re-initializes the whole system from DE441 vectors (residuals drop to 0.000 km at the seed epoch, float64-exact), after which residual growth measures pure *model fidelity* — our 11-body point-mass + 1PN dynamics against JPL's full model (asteroid perturbations, planetary figure effects, extended-body lunar dynamics). Simulation time is TDB by convention (UTC differs by ~69 s). The `/api/horizons` endpoint is public and CORS-open, so anyone in the scientific community can independently audit the deployment with three lines of code.

Shareable state URLs (base64url-encoded parameters in the fragment) make any configuration — a rogue-star flyby, a Kerr black hole at Sgr A* mass, a low-c precession demonstration — a reproducible, citable link.

## Honest limitations

The 1PN correction is applied for the central body (the dominant term — planet-planet GR cross-terms are negligible); Newtonian total energy is the conserved diagnostic, so with GR enabled a tiny bounded oscillation appears by construction (footnoted in-app). JPL Table-1 elements are fitted for 1800–2050; outside that range seeding and ghost orbits extrapolate (the N-body integration itself remains exact). The Moon is seeded from Meeus mean elements (~few-% accuracy) and immediately governed by real N-body forces thereafter. Black-hole visuals use physically-grounded real-time approximations (weak-field star lensing, δ³ beaming) rather than full geodesic ray-tracing; every derived quantity shown (r_s, ISCO, T_H, dilation) is exact.

## Process

Built in four phases as requested: web research of all governing equations and data from primary sources (JPL SSD, DE440/Park et al. 2021, NSSDC, IERS, Yoshida 1990, Bardeen–Press–Teukolsky 1972, GRAVITY 2022) by Sonnet research agents; architecture and specification on Fable (this session); implementation of the physics core and the 3D application by Opus agents; independent verification (39-assertion physics suite + headless-browser instrumentation and screenshot review) before delivery. The full sourced equation reference is included as `EQUATIONS.md`.