A capture orbit is a reverse escape orbit. It is a parabolic orbit with as special case a straight line in the direction of the center of the central body. If it intersects the central body or its atmosphere the object will crash into the central body or there will be atmospheric re-entry. Otherwise the orbit continues as escape orbit.
Both the capture orbit and the escape orbit are also known as "C3 = 0 orbit".
For capture velocities see the table in escape velocity.
Position as function of time
Finding the position as function of time corresponds to solving a differential equation. In the theoretical case of falling on a central body in a straight trajectory there is a rather simple expression for the solution:
, is the standard gravitational parameter
corresponds to the extrapolated time of the fictitious arrival at the center of the central body; either t is negative, or t is the time until this arrival, counting down.
To count down to impact at the surface, apply a time shift; for the Earth (and any other spherically symmetric body with the same average density) as central body this time shift is 6 minutes and 20 seconds; seven of these periods earlier the height above the surface is three times the radius, etc.
Compare free-fall time.
* List of orbits
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