Capture orbit

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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.

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:

$r=(4.5\mu t^2)^{1/3}\!\,$

where

$\mu\!\,$ is the standard gravitational parameter
$t=0\!\,$ 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.

v • d • e Orbits
Orbit types	Box orbit • Circular orbit • Ecliptic orbit • Elliptic orbit • Highly Elliptical Orbit • Graveyard orbit • Hyperbolic trajectory • Inclined orbit • Osculating orbit • Parabolic trajectory • Capture orbit • Escape orbit • Semi-synchronous orbit • Subsynchronous orbit • Synchronous orbit
Earth-centered orbits	Geosynchronous orbit • Geostationary orbit • Sun-synchronous orbit • Low Earth orbit • Medium Earth Orbit • Molniya orbit • Near equatorial orbit • Orbit of the Moon • Polar orbit • Polar sun synchronous orbit • Tundra orbit
Other orbits	Areosynchronous orbit • Areostationary orbit • Lissajous orbit • Lunar orbit • Heliocentric orbit • Heliosynchronous orbit
Orbital elements	Inclination ( $i\,\!$ ) • Longitude of the ascending node ( $\Omega\,\!$ ) • Eccentricity ( $e\,\!$ ) • Argument of periapsis ( $\omega\,\!$ ) • Semi-major axis ( $a\,\!$ ) • Mean anomaly at epoch ( $M_o\,\!$ )
Other orbital parameters	True anomaly ( $v\,$ ) • Semi-minor axis ( $b\,$ ) • Linear eccentricity ( $\epsilon\,$ ) • Eccentric anomaly ( $E\,$ ) • Mean longitude ( $L\,$ ) • True longitude ( $l\,$ ) • Orbital period ( $T\,$ )
Orbital maneuvers	Bi-elliptic transfer • Geostationary transfer orbit • Gravitational slingshot • Hohmann transfer orbit • Orbital inclination change • Orbit phasing • Space rendezvous
Other orbital mechanics topics	Apsis • Celestial coordinate system • Delta-v budget • Epoch • Ephemeris • Equatorial coordinate system • Ground track • Interplanetary Transport Network • Kepler's laws of planetary motion • Lagrangian point • List of orbits • n-body problem • Orbit equation • Orbital state vectors • Retrograde and direct motion • Specific orbital energy • Specific relative angular momentum