Why windows. Earth (365.256 d) laps Mars (686.980 d) every synodic period, 1/Ts = |1/TE − 1/TM|, i.e. 779.9 days or about 26 months. Cheap transfers exist only when the planets are in the right relative position, so the astronomical Mars launch windows recur about every 26 months; one opens in late 2026, and Mars missions are much in the news.
Hohmann transfer. Between circular, coplanar orbits of radius 1 AU and 1.524 AU the minimum-energy ellipse has semi-major axis at = (r1+r2)/2, flight time π√(at³/μ☉) ≈ 259 days, heliocentric burns of 2.94 km/s (v∞ leaving Earth) and 2.65 km/s (v∞ at Mars). Mars must lead Earth by φ = 180° − 360° · tH/TM ≈ 44° at departure. Leaving from a 300 km parking orbit costs Δv = √(v∞² + 2μE/r) − √(μE/r) ≈ 3.6 km/s; capture into a low Mars orbit adds roughly 2 km/s, so the total is about 5.6–5.7 km/s.
Lambert's problem. Real windows are not Hohmann: orbits are eccentric (Mars e = 0.093) and inclined (1.85°). For a chosen departure date and arrival date the planets' positions r1, r2 and the flight time are fixed, and Lambert's theorem says exactly one conic (for a given direction) joins them. This lab solves it with universal variables (Stumpff functions C(z), S(z) and bisection on z, after Bate, Mueller and White; the same problem is solved by Izzo's 2015 algorithm used in modern tools). The result is the departure and arrival velocities; v∞ is their difference from the planet's velocity, C3 = v∞dep² is the launch energy. Transfers sweeping less than 180° round the Sun are Type I, more than 180° Type II; both appear in the plot.
Porkchop plot. Evaluating Lambert's solution for every (departure, arrival) pair and contouring C3, arrival v∞ and Δv gives the characteristic closed "pork-chop" islands. The lowest valley is the best launch window; its width in time is how long you can wait. A late launch costs more C3 (see story 4). Capture: a hyperbolic arrival at v∞ into an orbit with periapsis rp and apoapsis ra needs Δv = √(v∞²+2μM/rp) − √(2μMra/(rp(ra+rp))); a high elliptical capture orbit is far cheaper than a low circular one. Aerocapture uses the atmosphere instead; here it is only a rough, illustrative model (feasible if the entry speed is below about 7.5 km/s, then a 0.15 km/s trim).
Launch vehicle. The payload curve is a one-stage rocket-equation toy (dry mass 3 t, propellant 28 t, exhaust velocity 4.4 km/s, starting from the parking orbit). It shows the shape, not a real rocket's capability.
Planet positions. Keplerian elements with secular rates from the JPL approximate-positions table for 1800–2050 (E. M. Standish), solved with Kepler's equation. Good to a few hundredths of a degree for Earth and about 0.1° for Mars (tens of thousands of km) over the dates used: right for teaching and for finding the window, not for navigation.
Simplifications. Patched conics: the Sun alone governs the cruise, each planet's gravity only acts at departure and arrival. No gravity-assist, no mid-course corrections, no multi-revolution solutions, no launch-site or arrival-geometry constraints, no inclination-matching cost at Mars, Earth treated as the Earth–Moon barycentre. In the 3D view the planets and the craft are enlarged to be visible. For background see Curtis, Orbital Mechanics for Engineering Students; Vallado, Fundamentals of Astrodynamics and Applications; Izzo (2015) Revisiting Lambert's problem; the NASA Mars Climate Orbiter, MAVEN and Mars 2020 mission design papers for real window examples.
Educational orbital mechanics only: a generic robotic science spacecraft. No real vehicle, brand or logo is depicted.