The news hook. Artemis II is the first crewed flight around the Moon since 1972, a roughly ten-day mission of four astronauts in the Orion capsule (2026). The mission is reported to use a free-return trajectory: after the burn that leaves Earth orbit, the capsule swings round the far side of the Moon and gravity brings it back to Earth with no further engine burn needed. This page models that class of trajectory with a generic capsule and service module; it is not a model of the real mission design, its dates or its numbers.
The restricted three-body problem. In a frame rotating with the Moon's circular orbit (angular rate ω = 2.665×10−6 rad/s, Earth-Moon distance 384 400 km), with μ = MMoon/(MEarth+MMoon) = 0.01215, a massless craft obeys x'' = 2y' + x − (1−μ)(x+μ)/r13 − μ(x−1+μ)/r23, y'' = −2x' + y − (1−μ)y/r13 − μy/r23 (lengths in Earth-Moon distances, time in 1/ω). It has one exact invariant, the Jacobi constant C = x² + y² + 2(1−μ)/r1 + 2μ/r2 − v², which the page monitors as a check on the integrator. The motion has no closed-form solution; it is integrated with the adaptive Dormand-Prince 5(4) method (relative tolerance 10−10), with Hermite interpolation for playback and event detection (closest approach, entry interface).
Trans-lunar injection. From a 185 km circular orbit (v = 7.793 km/s) an impulsive prograde burn raises the apogee to the Moon's distance. For an apogee of exactly 384 400 km the vis-viva equation v² = μE(2/r − 1/a) gives Δv ≈ 3.135 km/s, the Hohmann-type lower bound; the free-return paths here need 3.14 to 3.19 km/s because the Moon's pull and its motion matter. A real burn takes several minutes (finite-burn and gravity losses add roughly 0.1 km/s), and the real mission first spends about a day in Earth orbit; the model uses one instant burn, planar geometry and the Moon on a perfectly circular orbit, and ignores the Sun, the Earth's oblateness and the Moon's inclination.
The free-return property. A craft that arrives at the Moon's orbit ahead of the Moon is pulled backwards by it; passing round the far side turns the velocity through about 100° and sends it back toward Earth. In the rotating frame the path is a loop around the Moon joined to the loop of the parking orbit, the familiar "figure eight" outline of free-return diagrams. For each burn size the timing of the burn (where in the parking orbit it happens) is solved by a scan plus bisection so that the return perigee is 36 km above the surface, which corresponds to an entry flight-path angle of −6.5° at 120 km altitude.
Patched conics. The textbook shortcut treats the trip as three two-body problems: an Earth-centred conic until the craft enters the Moon's sphere of influence (radius a(MMoon/MEarth)2/5 = 66 000 km), a Moon-centred hyperbola through the encounter, and a new Earth-centred conic after exit. It uses universal-variable Kepler propagation. The perilune it predicts is good to a few hundred km, but the return perigee it predicts is wrong by thousands of km, because the Moon's tidal pull on the long Earth-centred legs is neglected. Experiment 2 measures this.
Sensitivity. Near the Moon a path bends by an angle that depends strongly on how close it passes; a 1 km change in perilune moves the return point by tens of km at Earth, and a 1 m/s change in the main burn moves the perilune by hundreds of km. A few centimetres per second at the burn change the entry angle by degrees, so every real lunar mission plans correction burns (here one prograde impulse at a chosen time; the earlier it is made, the smaller it needs to be). The Auto-correct button finds that burn with a bracketing scan and bisection on the return perigee.
Re-entry corridor. The capsule meets the atmosphere at about 11 km/s (more than the 7.8 km/s of a return from low Earth orbit). If the flight-path angle at the 120 km entry interface is too shallow the capsule skips out and may not come back; if it is too steep the deceleration exceeds what people or the structure can take. The planar entry integrator uses an exponential atmosphere (ρ = 1.225 e−h/7.25 km kg/m³), a ballistic coefficient of 390 kg/m², a gravity term and the centrifugal term (v²/r − g), and for the lifting capsule a lift-to-drag ratio of 0.3 with a simple feedback bank law that holds the load factor near 3 g. For a purely ballistic entry the peak deceleration of the flat-Earth Allen-Eggers solution is amax = vE² sin|γE| / (2eH) (the Checks button confirms the integrator reproduces it); with Earth curvature at 11 km/s the real peak is far lower, because the trajectory flattens. The shaded corridor is the one of this simplified model; the outlined bar is a representative published range of roughly −5.5° to −7.5° for Apollo-class capsules, which depends on the vehicle and its guidance. Heating, an Earth-rotating atmosphere, banking to steer downrange and the parachutes are not modelled.
Sources. Szebehely, Theory of Orbits (1967); Bate, Mueller and White, Fundamentals of Astrodynamics (1971); Curtis, Orbital Mechanics for Engineering Students; Allen and Eggers, NACA Report 1381 (1958); the NASA Apollo and Artemis mission documentation for the mission profile. Numbers shown are for a generic spacecraft and are illustrative.
Civil, educational space-flight dynamics only. The capsule and service module are a generic design; no real vehicle livery, logo or mission patch is depicted.