Parabolic trajectory

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 ${\displaystyle C_{3}=0}${\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.

In 1609, Galileo wrote in his 102nd folio ^{[1] [2]} (MS. Gal 72^[3] ) about parabolic trajectory calculations,^[1] later found in Discorsi e dimostrazioni matematiche intorno a due nuove scienze as projectiles impetus.^[4]

The orbital velocity (${\displaystyle v}${\displaystyle v}) of a body travelling along a parabolic trajectory can be computed as:

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

where:

• ${\displaystyle r}${\displaystyle r} is the radial distance of the orbiting body from the central body,
• ${\displaystyle \mu }${\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 (${\displaystyle v}${\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:

${\displaystyle v={\sqrt {2}},円v_{o}}${\displaystyle v={\sqrt {2}},円v_{o}}

where:

• ${\displaystyle v_{o}}${\displaystyle v_{o}} is orbital velocity of a body in circular orbit.

For a body moving along this kind of trajectory the orbital equation is:

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

where:

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

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

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

where:

• ${\displaystyle v,円}${\displaystyle v,円} is the orbital velocity of the orbiting body,
• ${\displaystyle r,円}${\displaystyle r,円} is the radial distance of the orbiting body from the central body,
• ${\displaystyle \mu ,円}${\displaystyle \mu ,円} is the standard gravitational parameter.

This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0:

${\displaystyle C_{3}=0}${\displaystyle C_{3}=0}

Barker's equation relates the time of flight ${\displaystyle t}${\displaystyle t} to the true anomaly ${\displaystyle \nu }${\displaystyle \nu } of a parabolic trajectory:^[5]

${\displaystyle t-T={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}${\displaystyle t-T={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}

where:

• ${\displaystyle D=\tan {\frac {\nu }{2}}}${\displaystyle D=\tan {\frac {\nu }{2}}} is an auxiliary variable
• ${\displaystyle T}${\displaystyle T} is the time of periapsis passage
• ${\displaystyle \mu }${\displaystyle \mu } is the standard gravitational parameter
• ${\displaystyle p}${\displaystyle p} is the semi-latus rectum of the trajectory, given by ${\displaystyle p=h^{2}/\mu }${\displaystyle p=h^{2}/\mu }

More generally, the time (epoch) between any two points on an orbit is

${\displaystyle t_{f}-t_{0}={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D_{f}+{\frac {1}{3}}D_{f}^{3}-D_{0}-{\frac {1}{3}}D_{0}^{3}\right)}${\displaystyle t_{f}-t_{0}={\frac {1}{2}}{\sqrt {\frac {p^{3}}{\mu }}}\left(D_{f}+{\frac {1}{3}}D_{f}^{3}-D_{0}-{\frac {1}{3}}D_{0}^{3}\right)}

Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit ${\displaystyle r_{p}=p/2}${\displaystyle r_{p}=p/2}:

${\displaystyle t-T={\sqrt {\frac {2r_{p}^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}${\displaystyle t-T={\sqrt {\frac {2r_{p}^{3}}{\mu }}}\left(D+{\frac {1}{3}}D^{3}\right)}

Unlike Kepler's equation, which is used to solve for true anomalies in elliptical and hyperbolic trajectories, the true anomaly in Barker's equation can be solved directly for ${\displaystyle t}${\displaystyle t}. If the following substitutions are made

${\displaystyle {\begin{aligned}A&={\frac {3}{2}}{\sqrt {\frac {\mu }{2r_{p}^{3}}}}(t-T)\\[3pt]B&={\sqrt[{3}]{A+{\sqrt {A^{2}+1}}}}\end{aligned}}}${\displaystyle {\begin{aligned}A&={\frac {3}{2}}{\sqrt {\frac {\mu }{2r_{p}^{3}}}}(t-T)\\[3pt]B&={\sqrt[{3}]{A+{\sqrt {A^{2}+1}}}}\end{aligned}}}

then

${\displaystyle \nu =2\arctan \left(B-{\frac {1}{B}}\right)}${\displaystyle \nu =2\arctan \left(B-{\frac {1}{B}}\right)}

With hyperbolic functions the solution can be also expressed as:^[6]

${\displaystyle \nu =2\arctan \left(2\sinh {\frac {\mathrm {arcsinh} {\frac {3M}{2}}}{3}}\right)}${\displaystyle \nu =2\arctan \left(2\sinh {\frac {\mathrm {arcsinh} {\frac {3M}{2}}}{3}}\right)}

where

${\displaystyle M={\sqrt {\frac {\mu }{2r_{p}^{3}}}}(t-T)}${\displaystyle M={\sqrt {\frac {\mu }{2r_{p}^{3}}}}(t-T)}

A radial parabolic trajectory is a non-periodic trajectory on a straight line where the relative velocity of the two objects is always the escape velocity. There are two cases: the bodies move away from each other or towards each other.

There is a rather simple expression for the position as function of time:

${\displaystyle r={\sqrt[{3}]{{\frac {9}{2}}\mu t^{2}}}}${\displaystyle r={\sqrt[{3}]{{\frac {9}{2}}\mu t^{2}}}}

where

• ${\displaystyle \mu }${\displaystyle \mu } is the standard gravitational parameter
• ${\displaystyle t=0\!,円}${\displaystyle t=0\!,円} corresponds to the extrapolated time of the fictitious starting or ending at the center of the central body.

At any time the average speed from ${\displaystyle t=0\!,円}${\displaystyle t=0\!,円} is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.

To have ${\displaystyle t=0\!,円}${\displaystyle t=0\!,円} 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 later the height above the surface is three times the radius, etc.

• Kepler orbit
• Parabola

• Orbits

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