Gravity & Orbits
Orbital mechanics, solar system, tidal forces, and N-body simulations
Schwarzschild Orbit Precession (Rosette)
Schwarzschild geodesic in the φ-form d²u/dφ² + u = 1/L² + 3u² (G = c = M = 1) integrated by RK4. The closed Newtonian ellipse is replaced by an orange precessing rosette with apsidal advance Δφ ≈ 6πM/[a(1 − e²)] per orbit — the same mechanism that produces the historic 43″/century perihelion shift of Mercury. Horizon r = 2M and ISCO r = 6M annotated.
ISCO & Photon Sphere (V_eff)
Schwarzschild effective potential V_eff(r) for massive (timelike) and photon (null) test particles in geometric units. Sliding angular momentum L collapses the stable / unstable circular pair into the innermost stable circular orbit r_ISCO = 6M (the inner edge of accretion discs); for photons the unstable photon sphere r = 3M defines the inner ring of black-hole shadow images.
Einstein Ring & Paczyński Microlensing
Point-mass thin lens (weak-field GR): lens equation β = θ − θ_E²/θ gives two images θ_± = ½(β ± √(β² + 4θ_E²)) with magnifications μ_± = ½[(u² + 2)/(u√(u² + 4)) ± 1], u = β/θ_E. Animated source transit at impact parameter u₀ over timescale t_E renders the canonical symmetric Paczyński light curve and the full Einstein ring θ_E = √(4GM·D_LS/(c² D_L D_S)) at perfect alignment.
Gravitational Wave Binary Chirp (Inspiral)
Leading-order post-Newtonian inspiral of a compact binary: f(τ) ∝ τ^(−3/8), strain h(t) ∝ M_c^(5/3) f^(2/3) / D_L. Tune component masses m₁, m₂ and luminosity distance D_L; live h(t) and f(t) traces with the orbiting bodies on the side. The chirp mass M_c = (m₁m₂)^(3/5)/(m₁+m₂)^(1/5) is the very quantity LIGO/Virgo measures from the early inspiral; the frequency freezes at the Schwarzschild ISCO.
Shapiro Time Delay (4th GR Test)
A radio signal grazing the Sun picks up an excess one-way travel time Δt ≈ (2GM/c³) ln[(r_E + r_E cos α)(r_R + r_R cos β)/b²] on top of the Newtonian light-time. Cassini, Mariner and Viking presets, with the round-trip delay readout in microseconds and an animated bent-photon path against a straight Newtonian baseline. The Cassini 2003 conjunction constrains |γ_PPN − 1| < 2 × 10⁻⁵ — the strongest weak-field GR test to date.
Three-Body Figure-Eight
Equal masses: Chenciner–Montgomery choreography in 2D (RK4, periodic orbit).
Restricted 3-Body (map)
CRTBP: escape vs collision vs chaos proxy; μ slider.
Multistage Rocket (Tsiolkovsky)
Δv per stage; sum vs single-stage with same total propellant.
Orbital Debris & Kessler (toy)
LEO shell: n = N/V, collision rate ∝ N²; optional fragment cascade.
Gravity-Assist Fly-By
Planet frame |u_out|=|u_in| rotated by δ; star frame v = V + u — Δ|v| from moving planet.
Orbit Simulator
Launch satellites. Achieve circular, elliptical orbits, or escape.
Solar System
Interactive scaled model with time controls and orbital data.
Gravity Sandbox
Place masses and watch N-body gravitational interactions unfold.
Kepler's Laws
Elliptical orbits with equal-area sweeps visualized.
Escape Velocity
Launch from different planets and see trajectory results.
Gravitational Lensing
Massive objects bending light. Visual distortion effects.
Lagrange Points L1–L5
CRTBP effective potential; L1–L5; Coriolis test particle.
Earth–Moon Tides
Equilibrium tide bulges; orbit speed; ~12.4 h spacing note.
Binary Star (circular)
COM orbits; r₁,r₂; Kepler T² ∝ a³/(M₁+M₂).
Roche Limit
Fluid d ≈ 2.456 R_p (ρ_p/ρ_s)^(1/3); vs orbit distance.
Space Elevator Tether
1D tension vs height; peak near GEO (normalized model).
Hohmann Transfer
Coplanar circles r₁,r₂; transfer ellipse; Δv₁, Δv₂ from vis-viva.
Geostationary Orbit
ω²r = GM/r² from sidereal vs solar day; Earth-fixed view with sub-satellite longitude.
Orbital Decay (Atmosphere)
Toy perigee drag: shrinking, circularizing ellipse; ISS lifetime intuition.
Mercury Perihelion Precession
GR Δω per orbit vs Newton; ~43″/century readout; amplified animation.
Earth–Moon Barycenter Wobble
Heliocentric path of Earth’s center: barycentric ellipse plus lunar epicycle (exaggerated).
Oberth Effect
Same prograde Δv at peri vs apo on one ellipse; higher ε when burning deep.