A capture orbit is the high-energy parabolic orbit that allows the capture other than crashing directly to the central body's surface (or atmospheric re-entry).
Capture orbit is identical to escape orbit - just the direction is reverse. Both 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:
where
-
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.
See also