Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions.
Here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the orbital plane.
Transverse acceleration (perpendicular to velocity) causes a change in direction.
If it is constant in magnitude and changing in direction with the velocity, circular motion ensues.
Taking two derivatives of the particle's coordinates concerning time gives the centripetal acceleration where: The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula.
The speed (or the magnitude of velocity) relative to the centre of mass is constant:[1]: 30 where: The orbit equation in polar coordinates, which in general gives r in terms of θ, reduces to:[clarification needed][citation needed] where: This is because
) can be computed as:[1]: 28 Compare two proportional quantities, the free-fall time (time to fall to a point mass from rest) and the time to fall to a point mass in a radial parabolic orbit The fact that the formulas only differ by a constant factor is a priori clear from dimensional analysis.
[citation needed] The specific orbital energy (
) is negative, and Thus the virial theorem[1]: 72 applies even without taking a time-average:[citation needed] The escape velocity from any distance is √2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.
For the sake of convenience, the derivation will be written in units in which
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
It gives: From this, we get: Substituting this into the equation for a massive particle gives: Hence: 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: This gives the velocity: Or, in SI units: