Physics & Motion Model

This section describes the translational motion model used in the Moonlander simulation framework. The implemented model combines Newtonian gravitation, thrust-induced acceleration, and discrete-time numerical integration to propagate spacecraft motion.

The current implementation focuses on computational efficiency, numerical robustness, and deterministic reproducibility for real-time simulation and autonomous landing research.

Origin of the Lunar Gravity Model

The gravitational acceleration model is derived from Newton's law of universal gravitation combined with Newton's second law of motion.

Newton's law of gravitation states that two masses attract each other with the force:

Where:
• $F_G$ is the gravitational force [N]
• $G$ is the universal gravitational constant
• $m_1,m_2$ are the interacting masses [kg]
• $r$ is the distance between both masses [m]

Using Newton's second law:

and solving for acceleration yields:

Since celestial mechanics frequently uses the combined quantity:

the gravitational acceleration becomes:

This expression describes only the scalar magnitude of the gravitational acceleration.

To obtain the full vector acceleration directed toward the lunar center, the normalized position vector is introduced:

The resulting vector formulation becomes:

The negative sign indicates that the gravitational acceleration always points toward the center of the Moon.

Lunar Gravity Model

The Moonlander simulation currently uses a central-body gravity model based on Newtonian point-mass gravitation.

The gravitational acceleration acting on the spacecraft is computed as:

Where:
• $\mathbf{a}_{grav}$ is the gravitational acceleration vector [m/s²]
• $\mu$ is the lunar gravitational parameter [m³/s²]
• $\mathbf{r}$ is the spacecraft position vector [m]
• $||\mathbf{r}||$ is the spacecraft distance from the lunar center [m]

This formulation naturally produces the inverse-square dependence of gravity while preserving the correct radial acceleration direction.

Thrust Acceleration

The spacecraft propulsion system generates acceleration by applying a thrust force vector along the engine thrust direction.

The generated thrust force vector is:

Where:
• $F_T$ is the scalar thrust force magnitude [N]
• $\hat{\mathbf{d}}$ is the normalized thrust direction vector [-]

Applying Newton's second law yields the thrust-induced acceleration:

Where:
• $\mathbf{a}_{thrust}$ is the thrust acceleration vector [m/s²]
• $m$ is the current spacecraft mass [kg]

Total Translational Acceleration

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

Substituting both acceleration models gives:

Discrete-Time Motion Integration

Spacecraft motion is propagated in discrete simulation steps using constant-acceleration kinematic relations over one timestep$\Delta t$.

Velocity propagation:

Position propagation:

Substituting the complete acceleration model yields:

This integration scheme provides a numerically stable and computationally efficient propagation method suitable for deterministic real-time simulation.

Proper Acceleration and G-Load

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

The corresponding g-load becomes:

Where:
• $g_0$ is standard Earth gravity [m/s²]

This quantity represents the acceleration experienced by the spacecraft structure and potential crew.

Key Characteristics

• Newtonian central-body gravity model
• Vector-based thrust force representation
• Physically consistent translational acceleration model
• Discrete-time rigid-body motion propagation
• Proper acceleration and g-load estimation
• Deterministic and real-time capable implementation