Lunar Descent

What is modelled

A lander falls toward the Moon with gravity of 1.62 m/s². The engine pushes it up and burns propellant at a rate fixed by the thrust and the exhaust speed. To land softly the engine must remove all the downward speed before the ground arrives. Starting the burn as late as possible with full thrust, the suicide burn, saves propellant because gravity loss is smallest, but it leaves no margin for error.

Equations

• Motion m dv/dt = T − m g (upward), mass flow dm/dt = − T / (Isp g0).
• Rocket equation Δv = Isp g0 ln(m0 / m1).
• Hover needs T = m g, burning m g / (Isp g0) kg per second.
• Free fall from height h with speed v₀ takes t = (−v0 + √(v0² + 2 g h)) / g.
• Constant-thrust burn (exhaust speed v_e, mass flow rate q): speed v(t) = v0 + g t − ve ln(m0/m(t)) and distance s(t) = v0 t + ½ g t² − ve [t + (m(t)/q) ln(m(t)/m0)].
• Burn start height for constant mass h = v² / (2 (T/m − g)).

Checks run in the page

• A full burn in empty space against the rocket equation.
• The hover fuel flow against m g / (Isp g0).
• The free-fall time from the integrator against the closed form.
• A constant-thrust burn from the integrator against the closed-form speed and distance.

Honest simplifications

Purely vertical motion, flat surface, constant gravity, ideal engine with instant throttle response and constant specific impulse, and no sloshing, no pitch control and no sensor noise. The real Moon landings needed attitude control, terrain avoidance and a rotating target.

• Press Autopilot landing and watch the engine stay off until the braking curve is reached.
• Press Late burn to see a negative margin turn the same descent into a crash.
• Press Manual hover and fly the throttle slider yourself to match the gravity.