Parabolic trajectory

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In astrodynamics or celestial mechanics a parabolic trajectory is an 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.

1 Velocity
2 Equation of motion
3 Energy
4 See also

[edit] Velocity

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

$v=\sqrt{2\mu\over{r}}$

where:

$r\,\!$ is radial distance of orbiting body from central body,
$\mu\,\!$ 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 ( $v\,$ ) 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:

$v=\sqrt{2}\cdot v_O$

where:

$v_O\,\!$ is orbital velocity of a body in circular orbit.

[edit] Equation of motion

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

$r={{h^2}\over{\mu}}{{1}\over{1+\cos\nu}}$

where:

$r\,$ is radial distance of orbiting body from central body,
$h\,$ is specific angular momentum of the orbiting body,
$\nu\,$ is a true anomaly of the orbiting body,
$\mu\,$ is standard gravitational parameter.

[edit] Energy

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

$\epsilon={v^2\over2}-{\mu\over{r}}=0$

where:

$v\,$ is orbital velocity of orbiting body,
$r\,$ is radial distance of orbiting body from central body,
$\mu\,$ is standard gravitational parameter.

[edit] See also

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