Gravity creates the orbit
An orbit is not a force that keeps an object moving in a circle. It is the result of gravity continually changing the direction of an object's velocity.
A planet travelling around a star is constantly falling toward it, but because the planet also has sideways velocity, it continually misses the star. The result is an orbit.
Here, G is the gravitational constant, M is the mass of the central body, m is the orbiting mass and r is the distance between them.
Interactive orbit
Change the eccentricity and mass below and watch the orbit respond. The simulation uses Newtonian two-body orbital mechanics.
Kepler's laws of planetary motion
1. The law of ellipses
Planets travel in elliptical orbits with the Sun at one focus of the ellipse.
The parameter e is eccentricity and a is the semi-major axis.
2. The law of equal areas
A line joining a planet to the Sun sweeps out equal areas in equal times.
This means a planet moves faster when it is closer to the Sun and slower when it is farther away.
3. The harmonic law
The square of a planet's orbital period is proportional to the cube of its semi-major axis.
For objects orbiting the Sun, when distance is measured in astronomical units and time in years:
Orbital velocity
For a circular orbit, the required orbital velocity is:
Notice the inverse square-root relationship. An object farther from the central body requires a lower orbital velocity.
For an elliptical orbit, the velocity changes continuously. The general vis-viva equation is:
This is one of the most useful equations in orbital mechanics. It connects an object's speed, its current distance and the size of its orbit.
Escape velocity
Escape velocity is the minimum speed required for an object to escape the gravitational influence of a body without any further propulsion.
Escape velocity is exactly the square root of 2 times the circular orbital velocity at the same distance.
Why does the planet speed up?
Near the star
At perihelion, the planet is closer to the star. Gravitational potential energy is lower and kinetic energy is higher.
The planet therefore travels faster.
Far from the star
At aphelion, the planet has moved farther away. Some kinetic energy has been converted into gravitational potential energy.
The planet therefore travels more slowly.
From planets to spacecraft
These equations are not merely descriptions of planetary motion. They form the foundation of modern orbital mechanics.
The same mathematics can be used to calculate the trajectories of satellites, probes and spacecraft travelling between planets.
More advanced models introduce additional effects such as atmospheric drag, gravitational perturbations, planetary oblateness and relativistic corrections.
The simple two-body problem is therefore the starting point for an enormous amount of real-world space engineering.