In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity (e) equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an escape orbit, otherwise a capture orbit. It is also sometimes referred to as a C 3 = 0 {\displaystyle C_{3}=0} orbit (see characteristic energy). Under standard assumptions a body traveling along an escape orbit will coast along a parabolic trajectory to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectories are minimum-energy escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits.
History In 1609, Galileo wrote in his 102nd folio (MS. Gal 72) about parabolic trajectory calculations, later found in Discorsi e dimostrazioni matematiche intorno a due nuove scienze as projectiles impetus.
Velocity The orbital velocity ( v {\displaystyle v} ) of a body travelling along a parabolic trajectory can be computed as:
v = 2 μ r {\displaystyle v={\sqrt {2\mu \over r}}}
where:
r {\displaystyle r} is the radial distance of the orbiting body from the central body,
μ {\displaystyle \mu } is the standard gravitational parameter. At any position the orbiting body has the escape velocity for that position. If a body has an 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 {\displaystyle 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 = 2 v o {\displaystyle v={\sqrt {2}}\,v_{o}}
where:
v o {\displaystyle v_{o}} is orbital velocity of a body in circular orbit.
Equation of motion For a body moving along this kind of trajectory the orbital equation is:
r = h 2 μ 1 1 + cos ν {\displaystyle r={h^{2} \over \mu }{1 \over {1+\cos \nu }}}
where:
r {\displaystyle r\,} is the radial distance of the orbiting body from the central body,
h {\displaystyle h\,} is the specific angular momentum of the orbiting body,
ν {\displaystyle \nu \,} is the true anomaly of the orbiting body,
μ {\displaystyle \mu \,} is the standard gravitational parameter.
Energy Under standard assumptions, the specific orbital energy ( ϵ {\displaystyle \epsilon } ) of a parabolic trajectory is zero, so the orbital energy conservation equation for this trajectory takes the form:
ϵ = v 2 2 − μ r = 0 {\displaystyle \epsilon ={v^{2} \over 2}-{\mu \over r}=0}
where:
v {\displaystyle v\,} is the orbital velocity of the orbiting body,
r {\displaystyle r\,} is the radial distance of the orbiting body from the central body,
μ {\displaystyle \mu \,} is the standard gravitational parameter. This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0:
C 3 = 0 {\displaystyle C_{3}=0}
Barker's equation Barker's equation relates the time of flight t {\displaystyle t} to the true anomaly ν {\displaystyle \nu } of a parabolic trajectory:
t − T = 1 2 p 3 μ ( D + 1 3 D 3 ) {\displaystyle t-T={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}
where:
D = tan ν 2 {\displaystyle D=\tan {\frac {\nu }{2}}} is an auxiliary variable
T {\displaystyle T} is the time of periapsis passage
μ {\displaystyle \mu } is the standard gravitational parameter
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