Unit 6.6: Motion of Orbiting Satellites

Introduction

The motion of orbiting satellites is governed by Newton’s laws of motion and universal gravitation. Satellites remain in orbit because of the balance between their tangential velocity and the gravitational pull of the central mass (e.g., Earth). Understanding this motion is key to analyzing planetary motion, artificial satellites, and space exploration.

Key Concepts

Orbital Motion: A satellite moves in a circular or elliptical path due to gravitational attraction. The centripetal force keeping it in orbit is provided by gravity.
Centripetal Force: The force required to maintain circular motion: \[ F_c = \frac{m v^2}{r} \] where m is the satellite’s mass, v is its velocity, and r is the orbital radius.
Gravitational Force: The force exerted by a planet on a satellite follows Newton’s law of universal gravitation: \[ F_g = \frac{G M m}{r^2} \] where G is the gravitational constant, M is the planet’s mass, and m is the satellite’s mass.
Orbital Speed: For a circular orbit, equating centripetal and gravitational forces gives: \[ v = \sqrt{\frac{G M}{r}} \]
Orbital Period (T): The time required for a satellite to complete one orbit is given by: \[ T = 2\pi \sqrt{\frac{r^3}{G M}} \] (Kepler’s third law for circular orbits).

Mathematical Routines

Use v = \sqrt{GM/r} to calculate the speed of a satellite in a circular orbit.
Apply Kepler’s third law to find the period of motion.
Determine the gravitational force acting on a satellite using Newton’s law of gravitation.

Tip: The higher the orbit, the longer the period! Satellites in geostationary orbit match Earth's rotation period (24 hours).

Creating Representations

Diagrams: Draw free-body diagrams showing gravitational force as the only force acting on the satellite.
Graphs: Plot orbital radius vs. period squared to verify Kepler’s third law.

Scientific Questioning & Argumentation

Why do astronauts in the ISS appear weightless even though gravity still acts on them?
How does increasing the orbital radius affect the orbital speed and period?

Use Kepler’s laws and Newton’s laws to support your reasoning.

Exam Tip: For conceptual questions, always explain orbital motion using **centripetal force, gravitational force, and Newton’s laws**.

Practice Activities

Activity 1: Orbital Speed Calculation

A satellite orbits Earth at an altitude of 500 km. Given that Earth’s mass is **5.97 × 10¹⁴ kg**, calculate the satellite’s orbital speed.

Activity 2: Geostationary Orbits

A geostationary satellite remains above the same point on Earth. Use Kepler’s third law to find the required orbital radius for a 24-hour period.

Summary & Exam Preparation Tips

Orbital motion is governed by **gravity acting as the centripetal force**.
Use **v = √(GM/r)** for orbital speed calculations.
Kepler’s third law relates **orbital radius and period**.
Objects in orbit are in **free fall**, which explains weightlessness.