Lunar Free-Return Trajectory (2D)

What the 2D view shows

The Earth-Moon plane of a crewed free-return trajectory: one trans-lunar-injection burn from a 185 km parking orbit sends the capsule round the far side of the Moon and back to a 36 km perigee over the Earth with no second burn. Switch between the rotating frame (Earth-Moon line fixed, the figure-eight-like loop) and the inertial frame (the Moon moves). The graphs show distance and speed, how the return perigee depends on the burn error, and the entry deceleration.

Equations (same module as the 3D page)

• Planar circular restricted three-body problem in the rotating frame, nondimensional: x″ - 2y′ = ∂Ω/∂x, y″ + 2x′ = ∂Ω/∂y, with Ω = (x² + y²)/2 + (1-μ)/r1 + μ/r2, μ = 0.01215.
• Jacobi constant C = 2Ω - v² is conserved between burns; integration uses an adaptive Dormand-Prince 5(4) scheme.
• Burn: circular speed at 185 km altitude plus dv; the burn phase angle is solved so the return perigee is 36 km (entry angle about -6.5 deg).
• Entry: planar integration with exponential atmosphere, ballistic coefficient and optional lift; skip-out, safe entry and over-g classes.

Honest simplifications

The Moon is on a circular orbit in the Earth-Moon plane; no Sun, no Earth oblateness, impulsive burns, a point-mass capsule, a generic ballistic entry model and a generic g limit. It illustrates why precision matters; it is not a mission-design tool and is not affiliated with any space agency.

What to try

• Burn 1.3 cm/s too fast: the capsule skips out. 1 cm/s too slow: too steep.
• Burn 1 m/s too fast, then Auto-correct on day 1 and compare with the ghost line.
• Lower dv to 3.14 km/s for a slow, distant flyby.
• Start the burn 5 s late.