Both the light rays and the probe follow the real Schwarzschild equatorial-plane geodesic equation written for u = 1/r as a function of the swept angle φ (G = c = 1, M = rs/2):
Photon: d²u/dφ² = -u + 3Mu²
Massive: d²u/dφ² = -u + M/L² + 3Mu²
Each ray is seeded with the flat-space asymptote u(φ)=sinφ/b (a straight line at perpendicular distance b) and then integrated with 4th-order Runge–Kutta; gravity bends the path away from that line. Below the critical impact parameter bcrit = 3√3 M the ray spirals past the photon sphere (1.5 rs) and is captured — above it, it escapes with a real deflection angle Δφ.
The probe's angular momentum L is set from the classic circular-orbit relation L² = M r₀² / (r₀ − 3M), then scaled by the velocity factor; away from 1.00× the same ODE produces the perihelion precession General Relativity predicts (the rosette orbit), or — inside the innermost stable circular orbit (ISCO = 3 rs) — a plunge into the horizon. Time is advanced with the exact relation dφ/dτ = L·u², so the on-screen sweep rate is the probe's own proper-time clock, not a cosmetic animation.