Classification of orbits

When a body moves under the gravitational influence of a central body — a satellite around the Earth, the Earth around the Sun — the shape of its trajectory is not arbitrary: it depends on a single number, the total energy $E_\text{tot} = E_\text{cin} + E_\text{pot,G}$. The sign of this sum decides whether the body will remain bound or manage to escape.

Classification of orbits

• $E_\text{tot} < 0$: elliptical orbit (bound body)
• $E_\text{tot} = 0$: parabolic trajectory (escape at the limit)
• $E_\text{tot} > 0$: hyperbolic trajectory (escape with excess energy)

The intuition is the same as for a potential well. The potential energy is negative and tends to zero at infinity. If the total energy is negative, the body does not have enough kinetic energy to reach infinity (where $E_\text{pot}\to 0$): it remains trapped in a closed orbit. If it is exactly zero, it reaches infinity with zero velocity: the parabola is the boundary case. If it is positive, it reaches infinity with residual velocity: the hyperbola.

Classification of orbits based on total energy: ellipse (bound), parabola (escape at the limit), hyperbola (escape).

Links

Topics: Gravitazione Concepts: Orbite · Conservazione dell’energia meccanica

Related exercises: True or false on energy and orbits · Speed of a comet at perihelion · Derivation of escape velocity