Orbits and Satellites

Orbital mechanics, circular orbital velocity, escape speed, and satellite trajectories.

General idea of the topic

A satellite in orbit is in free fall around the body: gravity acts as the centripetal force and continuously bends its path without it ever falling. In a circular orbit, the equality of gravitational attraction and centripetal force fixes the speed for each radius.

The topic derives the orbital speed, computes the energy of an orbit, which is negative for a bound system and equal to half the potential energy, and states Kepler's laws as a result of Newtonian gravitation.

What you will truly understand

By the end you will be able to compute orbital speed and period, the energy needed to change orbit or escape, and apply Kepler's third law to compare bodies orbiting the same center.

You will also know why a higher satellite moves more slowly.

Recommended learning route

1. Orbital Velocity: v = √(GM/r), period and relation to radius.
2. Orbital Energy: E = −GMm/(2r), binding energy and escape velocity.
3. Kepler's Laws: elliptical orbits, equal areas in equal times and T² proportional to a³.

Physical criterion you must acquire

The essential criterion is not to confuse the radius of the orbit with the height above the surface: r = R + h. A negative total energy indicates a closed orbit, zero a parabolic one and positive a hyperbolic one. In a circular orbit the kinetic energy is half the potential energy in absolute value.

Connection with real problems

• geostationary satellites and GPS.
• space stations.
• interplanetary probes.
• motion of the Moon and planets.

Final goal of the topic

The final goal is to compute and compare circular and elliptical orbits with Newtonian gravitation and Kepler's laws.