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Physics & Motion Model

This section describes the physical motion model used in the Moonlander simulation. It covers gravitational modeling, thrust-induced acceleration, numerical integration of motion, and the computation of experienced g-loads.

Lunar Gravity Model

The gravitational acceleration is modeled as a radial force pointing toward the center of the Moon. Its magnitude decreases with the square of the distance from the lunar center.
From orbital mechanics follows:

a⃗grav=−μ∣∣r⃗∣∣3r⃗\vec{a}_{grav} = - \frac{\mu}{||\vec{r}||³} \vec{r}
Or rather:
a⃗grav=−μ∣∣r⃗∣∣2\vec{a}_{grav} = - \frac{\mu}{||\vec{r}||2}

Where:
• r⃗\vec{r} is the spacecraft position vector
• ∣∣r⃗∣∣||\vec{r}||distance from the lunar center
• μ\mu is the lunar gravitational constant

Thrust Acceleration

The acceleration produced by the spacecraft's engines is calculated by dividing the thrust force by the spacecraft mass and applying it along the thrust direction:

a⃗thrust=d⃗⋅FTm\vec{a}_{thrust} = \vec{d} \cdot \frac{F_{T}}{m}

Where:
• a⃗thrust\vec{a}_{thrust} is the thrust acceleration vector [m/s²]
• d⃗\vec{d} is the unit vector in the direction of the thrust force
• FTF_{T} is the thrust magnitude produced by the engine [N]
• mm is the current spacecraft mass [kg]

Total Acceleration

The total acceleration acting on the spacecraft is the superposition of thrust-induced acceleration and gravitational acceleration.

a⃗=a⃗grav+a⃗thrust\vec{a} = \vec{a}_{grav} + \vec{a}_{thrust}
a⃗=μ∣∣r⃗∣∣3r⃗+d⃗⋅FTm\vec{a} = \frac{\mu}{||\vec{r}||³} \vec{r} + \vec{d} \cdot \frac{F_{T}}{m}

Motion Integration

Position and velocity are updated using constant-acceleration kinematic equations over a discrete time step Δt\Delta t.

v⃗(t+Δt)=v⃗(t)+a⃗⋅Δt\vec{v}(t + \Delta t) = \vec{v}(t) + \vec{a} \cdot \Delta t
Or rather:
v⃗(t+Δt)=v⃗(t)+(μ∣∣r⃗∣∣3r⃗+d⃗⋅FTm)⋅Δt\vec{v}(t + \Delta t) = \vec{v}(t) + (\frac{\mu}{||\vec{r}||³} \vec{r} + \vec{d} \cdot \frac{F_{T}}{m}) \cdot \Delta t
As well as position:
p⃗(t+Δt)=p⃗(t)+v⃗(t)⋅Δt+12a⃗⋅Δt2\vec{p}(t + \Delta t) = \vec{p}(t) + \vec{v}(t) \cdot \Delta t + \frac{1}{2} \vec{a} \cdot \Delta t^2
Or rather:
p⃗(t+Δt)=p⃗(t)+v⃗(t)⋅Δt+12(μ∣∣r⃗∣∣3r⃗+d⃗⋅FTm)⋅Δt2\vec{p}(t + \Delta t) = \vec{p}(t) + \vec{v}(t) \cdot \Delta t + \frac{1}{2} (\frac{\mu}{||\vec{r}||³} \vec{r} + \vec{d} \cdot \frac{F_{T}}{m}) \cdot \Delta t^2

This approach provides a stable and computationally efficient integration scheme suitable for real-time simulation.

G-Load (Proper Acceleration)

The experienced g-load is computed from the proper acceleration, which excludes gravitational acceleration.

aproper=atotal−g⃗a_{proper} = a_{total} - \vec{g}
gload=∥aproper∥g0g_{load} = \frac{\|a_{proper}\|}{g_0}

Where g_0 = 9.80665 m/s^2 is standard Earth gravity. This value represents the acceleration actually felt by the spacecraft structure and crew.

Summary

  • Radial inverse-square gravity model for the Moon
  • Vector-based thrust and acceleration computation
  • Discrete-time kinematic integration
  • Physically correct g-load estimation