Parabolic trajectory
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In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity equal to 1. When moving away from the source it is called an escape orbit, otherwise a capture orbit.

Under standard assumptions a body traveling along an escape orbit will coast to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectory is a minimum-energy escape trajectory.

Velocity

Under standard assumptions the orbital velocity () of a body traveling along parabolic trajectory can be computed as:

where:

• is radial distance of orbiting body from central body,
• is standard gravitational parameter.

At any position the orbiting body has the escape velocity for that position.

If the body has the escape velocity with respect to the Earth, this is not enough to escape the Solar System, so near the Earth the orbit resembles a parabola, but further away it bends into an elliptical orbit around the Sun.

This velocity () is closely related to the orbital velocity of a body in a circular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory:

where:

• is orbital velocity of a body in circular orbit.

Equation of motion

Under standard assumptions, for a body moving along this kind of trajectory an orbital equation becomes:

where:

• is radial distance of orbiting body from central body,
• is specific angular momentum of the orbiting body,
• is a true anomaly of the orbiting body,
• is standard gravitational parameter.

Energy

Under standard assumptions, specific orbital energy () of parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes form:

where:

• is orbital velocity of orbiting body,
• is radial distance of orbiting body from central body,
• is standard gravitational parameter.

See also

• Hyperbolic trajectory
• Orbit
• Circular orbit
• Elliptic orbit
• Orbital equation
• List of orbits

v • d • e Orbits   Types General Box · Capture · Circular · Elliptical / Highly elliptical · Escape · Graveyard · Hyperbolic trajectory · Inclined / Non-inclined · Osculating · Parabolic trajectory · Parking · Synchronous (semi · sub) Geocentric Geosynchronous · Geostationary · Sun-synchronous · Low Earth · Medium Earth · Molniya · Near-equatorial · Orbit of the Moon · Polar · Tundra Other Areosynchronous · Areostationary · Halo · Lissajous · Lunar · Heliocentric · Heliosynchronous   Parameters Classical  Inclination ·  Longitude of the ascending node ·  Eccentricity ·  Argument of periapsis ·  Semi-major axis ·  Mean anomaly at epoch Other  True anomaly ·  Semi-minor axis ·  Linear eccentricity ·  Eccentric anomaly ·  Mean longitude ·  True longitude ·  Orbital period   Maneuvers Bi-elliptic transfer · Geostationary transfer · Gravity assist · Hohmann transfer · Inclination change · Phasing · Rendezvous · Transposition, docking, and extraction   Other orbital mechanics topics Apsis · Celestial coordinate system · Delta-v budget · Epoch · Ephemeris · Equatorial coordinate system · Gravity turn · Ground track · Interplanetary Transport Network · Kepler's laws of planetary motion · Lagrangian point · Low energy transfers · n-body problem · Oberth effect · Orbit equation · Orbital speed · Orbital state vectors · Perturbation · Retrograde and direct motion · Specific orbital energy · Specific relative angular momentum List of orbits

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