Demonstration of Satellite Motion Around a Planet

This page presents a detailed mathematical demonstration of satellite motion around a planet, incorporating the 2ème loi de Newton and orbital mechanics principles.

The demonstration begins by establishing the force equation for a satellite orbiting at altitude h:

Definition: F = GMm / $R+h$², where G is the gravitational constant, M is the planet's mass, m is the satellite's mass, R is the planet's radius, and h is the orbital altitude.

Using the Frenet frame, the acceleration is expressed as:

Formula: a² = (du/dt)² T² + V² N², where u is the unit vector.

The demonstration then proceeds to determine the satellite's rotational velocity:

Highlight: By equating the centripetal force to the gravitational force, we derive V² = GM / $R+h$.

To show that the motion is uniform circular motion:

Example: The derivative of velocity with respect to time (du/dt) is zero, indicating constant velocity and thus uniform circular motion.

The orbital period is derived:

Formula: T = 2π√$(R+h)³ / GM$

Finally, the demonstration concludes by verifying Kepler's Third Law:

Highlight: The ratio T² / $R+h$³ is shown to be constant, confirming Kepler's Third Law for planetary orbits.

This comprehensive analysis provides students with a deep understanding of satellite motion, integrating key concepts from Mouvement des satellites et des planètes Terminale.