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
The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value is measured in meters per second squared, then the numerical values will be in meters per second, in meters, and in radians per second.
Velocity
The speed (or the magnitude of velocity) relative to the central object is constant:[1]: 30
, is the mass of both orbiting bodies , although in common practice, if the greater mass is significantly larger, the lesser mass is often neglected, with minimal change in the result.
the kinetic energy of the system is equal to the absolute value of the total energy
the potential energy of the system is equal to twice the total energy
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.[citation needed]
Delta-v to reach a circular orbit
Maneuvering into a large circular orbit, e.g. a geostationary orbit, requires a larger delta-v than an escape orbit, although the latter implies getting arbitrarily far away and having more energy than needed for the orbital speed of the circular orbit. It is also a matter of maneuvering into the orbit. See also Hohmann transfer orbit.
Orbital velocity in general relativity
In Schwarzschild metric, the orbital velocity for a circular orbit with radius is given by the following formula:
where is the Schwarzschild radius of the central body.
Derivation
For the sake of convenience, the derivation will be written in units in which .
The four-velocity of a body on a circular orbit is given by:
( 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:
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:
^ a b cLissauer, Jack J.; de Pater, Imke (2019). Fundamental Planetary Sciences : physics, chemistry, and habitability. New York, NY, USA: Cambridge University Press. p. 604. ISBN 9781108411981.