A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
A circular orbit is depicted in the top-left quadrant of this diagram, where the gravitational potential well of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy remains constant throughout the constant speed circular orbit.
At the top of the diagram, a satellite in a clockwise circular orbit (yellow spot) launches objects of negligible mass: (1 - blue) towards Earth, (2 - red) away from Earth, (3 - grey) in the direction of travel, and (4 - black) backwards of the direction of travel.
Dashed ellipses are orbits relative to Earth. Solid curves are perturbations relative to the satellite: in one orbit, (1) and (2) return to the satellite having made a clockwise loop on either side of the satellite. Unintuitively, (3) spirals farther and farther behind whereas (4) spirals ahead.
The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of is measured in meters per second per second, then the numerical values for will be in meters per second, in meters, and in radians per second.
( is constant on a circular orbit, and the coordinates can be chosen so that ). The dot above a variable denotes derivation with respect to proper time .
For a massive particle, the components of the four-velocity satisfy the following equation:
We use the geodesic equation:
The only nontrivial equation is the one for . It gives:
From this, we get:
Substituting this into the equation for a massive particle gives:
Assume we have an observer at radius , who is not moving with respect to the central body, that is, their four-velocity is proportional to the vector . The normalization condition implies that it is equal to:
The dot product of the four-velocities of the observer and the orbiting body equals the gamma factor for the orbiting body relative to the observer, hence: