Method and assumptions
Planet positions come from JPL’s approximate Keplerian elements with secular rates (Standish), fit for 1800–2050. Errors are a few arcminutes for the inner planets and up to roughly a degree for the outer ones; outside that span accuracy falls off. Earth means the Earth–Moon barycenter. All vectors are heliocentric ecliptic J2000; the Sun’s gravitational parameter is 1.32712440018 × 10¹¹ km³/s².
Transfers are patched conics. The heliocentric arc is a Lambert solution in universal variables (bisection on z, with golden-section search for the minimum flight time on each multi-revolution interval; multi-revolution answers are checked by propagation and discarded when the geometry is degenerate). Planet spheres of influence are treated as points.
- C3 is the square of the departure hyperbolic excess speed, the figure launch vehicle performance charts use.
- Departure Δv is from a circular parking orbit: √(v∞² + 2μ/r) − √(μ/r).
- Capture Δv lowers the arrival hyperbola into an orbit with the chosen periapsis altitude and eccentricity, burning at periapsis.
- DLA and RLA (Earth departures) are the declination and right ascension of the outgoing asymptote in the equatorial frame. A launch site can reach the asymptote directly only if |DLA| is below its latitude-dependent limit.
- Powered flyby: the periapsis radius is solved so that the incoming and outgoing hyperbolas together give the turn angle the two Lambert legs require; the Δv is the speed mismatch at that periapsis. If the turn would need a periapsis below the minimum altitude, the flyby is marked and the missing rotation is charged as an extra burn.
- Deep-space maneuver model (MGA-1DSM): the decision vector is the launch date, launch v∞ (magnitude and two direction angles), and for each leg its duration and the fraction of it before the maneuver; each flyby adds a periapsis radius and B-plane angle. The spacecraft coasts from the planet, burns once, then follows a Lambert arc to the next planet. Flybys are exactly unpowered. Optimization is differential evolution (rand/1 and best/1, dithered F, CR 0.9) followed by a Nelder–Mead polish, seeded from the powered-flyby solution.
Not modeled: launch vehicle performance, third-body perturbations, finite burns, and planetary oblateness. In the powered-flyby model, missions that relied on a deep-space maneuver (Cassini between its two Venus flybys, Juno) show a larger flyby Δv than they flew, and exactly resonant same-planet legs (Galileo’s two-year Earth loop) are degenerate. Switch to the deep-space maneuver model for those.
The mission list covers 38 flights: Pioneer, Voyager, Galileo, Ulysses, Cassini, New Horizons, Juno, JUICE and Europa Clipper to the outer planets; nineteen Mars missions from Mariner 4 to Tianwen-1; Mariner 2 and 5, Venera 7, Pioneer Venus and Magellan to Venus; Mariner 10 and MESSENGER to Mercury. Dates are launch and encounter days and capture orbits are approximate, so the figures are reconstructions rather than flown values. Landers use the atmospheric entry arrival, which reports the entry speed at 125 km.
Deep-space maneuver presets were solved offline with dates free to move a few days. MESSENGER comes out at C3 14.9 km²/s² (flown 16.4) and orbit insertion 0.85 km/s (flown 0.86), with Mercury flybys at 200 to 2,000 km. JUICE’s August 2024 lunar–Earth double flyby is modeled as an Earth flyby only, so its first flyby altitude differs from the flown one. Europa Clipper’s Mars flyby comes out near 530 km and its Earth flyby near 3,350 km.
Checks against flown missions with the DSM model: Cassini C3 16.1 km²/s² (flown 16.6), Venus–Venus maneuver 0.43 km/s (flown about 0.45), Jupiter flyby 10.3 million km (flown 9.7). Galileo C3 13.4 (flown about 13). Juno C3 28.9 (flown 31.1), maneuvers 0.68 km/s (flown about 0.8), orbit insertion 0.53 km/s (flown 0.54).
Moon missions
The Earth–Moon module does not use patched conics. The spacecraft is integrated with an adaptive Dormand–Prince 5(4) method under the gravity of Earth and the Moon, including the Moon’s pull on Earth, with the Moon placed by the Astronomical Almanac’s low-precision lunar series (about 0.3° in longitude). The Sun, Earth’s oblateness and the Moon’s mascons are left out. All burns are impulsive and tangential.
- Outbound: the translunar injection is placed on a circular parking orbit in the Moon’s orbital plane; its size and position are solved by Newton iteration so the spacecraft passes behind the Moon at the chosen altitude and time. Lunar orbit insertion circularizes at pericynthion in one burn (Apollo used two).
- Return: the trans-Earth injection burn and its timing within one lunar orbit are solved so the osculating Earth orbit reaches the entry interface (122 km) at the chosen flight-path angle at the chosen time.
- Free return: the burn is solved so the Moon’s gravity alone swings the spacecraft round the far side and back to the entry corridor, the figure-eight path Apollo 8, 10, 11 and 13 started on.
Apollo event times in the presets are approximate. Apollo 11 comes out at TLI 3.16 km/s, LOI 0.91 km/s, TEI 0.98 km/s and entry at 11.03 km/s after 30 lunar revolutions, against roughly 3.18, 0.94, 1.0 and 11.0 flown. Apollo 13’s unaided free return passes the Moon three minutes before the flown time at the flown 254 km. Artemis II, solved from its TLI time with the flown 6,550 km far-side pass as the only target, reaches the Moon five minutes after the flown time and the entry interface about 15 minutes before the flown splashdown.
When several free-return paths fit the targets, the one whose Moon passage falls nearest the time you enter is kept.
Using the view
- Drag to rotate, shift-drag or right-drag to pan, scroll or pinch to zoom, double-click to fit.
- Space plays or pauses; F fits the view to the trajectory.
- Settings are remembered in this browser. Export writes JSON or CSV files with state vectors in km and km/s.
References
- E. M. Standish, Keplerian Elements for Approximate Positions of the Major Planets, JPL Solar System Dynamics.
- H. D. Curtis, Orbital Mechanics for Engineering Students, ch. 3, 5 and 8.
- D. A. Vallado, Fundamentals of Astrodynamics and Applications.
- R. R. Bate, D. D. Mueller, J. E. White, Fundamentals of Astrodynamics.