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The science
Units. Lengths are in GM/c² (1.477 km per solar mass) and times in GM/c³ (4.93 µs per solar mass), so G = c = M = 1. The event horizon is at r = 2.
Probe orbit. A freely falling mass in the Schwarzschild metric conserves energy E and angular momentum L per unit mass. In proper time τ: (dr/dτ)² = E² - V(r), V = (1 - 2/r)(1 + L²/r²), dφ/dτ = L/r². The simulator integrates the second-order form r'' = -1/r² + L²/r³ - 3L²/r&sup4; with RK4 and adaptive steps. The last term is purely relativistic: it turns the closed Kepler ellipse into a rosette.
Orbit shape. With semi-latus rectum p and eccentricity e (r = p/(1 + e cosχ)) the constants are E² = ((p-2)² - 4e²) / (p(p - 3 - e²)), L² = p² / (p - 3 - e²). Stable bound orbits need p > 6 + 2e. For e = 0 the limit p = 6 is the innermost stable circular orbit (ISCO) with E = 2√2/3 (5.7 % of the rest energy is radiated away by matter spiralling in to it) and L = 2√3.
Periapsis shift. Per orbit the angle swept is 4√(p/(p-6+2e)) K(k), k² = 4e/(p-6+2e); the excess over 2π is the precession, ≈ 6π/p for large p (Mercury's 43 arcsec per century has the same origin). The sim measures it by integrating and compares with this closed form.
Clock rate. A static clock far away ticks at 1. The probe clock runs at dτ/dt = (1 - 2/r)/E (gravity and motion combined). The local speed seen by a static observer is v = √(1 - (1 - 2/r)/E²).
Light. For photons with impact parameter b: d²u/dφ² = -u + 3u², u = 1/r. Rays with b < 3√3 = 5.196 are captured; far away the deflection is 4/b + 15π/(4b²) (1.75 arcsec for a ray grazing the Sun). The unstable photon sphere is at r = 3.
Honest simplifications. Non-rotating, uncharged hole; the probe is a test mass (no gravitational radiation, tides or back-reaction); the orbit lies in the equatorial plane. The accretion disc is decoration (not simulated) and the picture is not ray-traced: it shows the geometry in Schwarzschild coordinates, not the lensed image a camera would record. Real discs also contain gas and magnetic fields, and most real holes spin.