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.
