Two-body (patched-conic) transfer from a low Earth parking orbit to a lunar orbit, drawn in the non-rotating Earth frame with the Moon moving on its orbit.
The science
Parking orbit and burn. A circular orbit of radius r1 = RE + h has speed vc = √(μ/r1), μ = 398 600.44 km³/s². A prograde burn Δv1 at perigee gives vp = vc + Δv1 and an ellipse with semi-major axis from the vis-viva equation v² = μ(2/r − 1/a), i.e. 1/a = 2/r1 − vp²/μ. The minimum-energy (Hohmann) transfer has apogee exactly at the target radius r2.
Time. The Hohmann coast is half an ellipse: T = π√(a³/μ). For a faster (higher-energy) orbit the ship crosses r2 before apogee; the time follows from Kepler's equation M = E − e sin E = n t, n = √(μ/a³), with r = a(1 − e cos E).
Rendezvous. The Moon moves at ωm = √(μ/D³) (circular, Earth-only gravity, D = 384 400 km). To meet it, the Moon must lead the ship's launch point by νarr − ωmT. A phase error φ leaves a miss of about 2 r2 sin(φ/2).
Capture. Relative to the Moon the arrival speed is v∞ = |vmoon − varr| (taken collinear), the periapsis speed is vp = √(v∞² + 2μm/rp) and the capture burn into a circular lunar orbit is vp − √(μm/rp), μm = 4902.8 km³/s².
Numerical checks. (1) The Kepler-equation time to reach r2 is compared with an RK4 integration of the two-body equations (step 4 s). (2) The RK4 speed at r2 is compared with vis-viva. (3) The specific energy at perigee and at arrival from the integration are compared.
Honest simplifications. Patched conics: the Moon's gravity acts only after arrival, the Earth and Moon orbits are circular and coplanar, the burn is impulsive, and the arrival geometry is idealised (collinear velocities). Real lunar missions use a free-return or low-energy design with 3-body dynamics; this page shows the textbook picture. The 3D page's earlier “energy” slider was a visual time scale; here the extra Δv is physical.