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Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types oftwo-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
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v t e

Inastrodynamics orcelestial mechanics aparabolic trajectory is aKepler orbit with theeccentricity 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 anescape orbit, otherwise acapture orbit. It is also sometimes referred to as aC₃ = 0 orbit (seeCharacteristic energy).

Under standard assumptions a body traveling along an escape orbit will coast along aparabolic trajectory to infinity, with velocity relative to thecentral body tending to zero, and therefore will never return. Parabolic trajectories are minimum-energy escape trajectories, separating positive-energyhyperbolic trajectories from negative-energyelliptic orbits.

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

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

where:

• ${\displaystyle r}$ is the radial distance of the orbiting body from thecentral body,
• ${\displaystyle \mu }$ is thestandard gravitational parameter.

At any position the orbiting body has theescape 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 (${\displaystyle v}$) is closely related to theorbital velocity of a body in acircular orbit of the radius equal to the radial position of orbiting body on the parabolic trajectory:

${\displaystyle v=\sqrt{2}{v}_o}$

where:

• ${\displaystyle v_o}$ isorbital velocity of a body incircular orbit.

For a body moving along this kind oftrajectory theorbital equation is:

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

where:

• ${\displaystyle r}$ is the radial distance of the orbiting body from thecentral body,
• ${\displaystyle h}$ is thespecific angular momentum of theorbiting body,
• ${\displaystyle \nu }$ is thetrue anomaly of the orbiting body,
• ${\displaystyle \mu }$ is thestandard gravitational parameter.

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

${\displaystyle ϵ=\frac{v^2}{2}-\frac{\mu }{r}=0}$

where:

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

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

${\displaystyle C_3=0}$

Barker's equation relates the time of flight${\displaystyle t}$ to the true anomaly${\displaystyle \nu }$ of a parabolic trajectory:^[1]

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

where:

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

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

${\displaystyle t_f-t_0=\frac{1}{2}\sqrt{\frac{p^3}{\mu }}\left(D_f+\frac{1}{3}{D}_f^3-D_0-\frac{1}{3}{D}_0^3\right)}$

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

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

UnlikeKepler'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${\displaystyle t}$. If the following substitutions are made

then

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

With hyperbolic functions the solution can be also expressed as:^[2]

${\displaystyle \nu =2\arctan \left(2\sinh \frac{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\frac{3M}{2}}{3}\right)}$

where

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

A radial parabolic trajectory is a non-periodictrajectory on a straight line where the relative velocity of the two objects is always theescape 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:

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

where

• μ is thestandard gravitational parameter
• ${\displaystyle t=0}$ 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${\displaystyle t=0}$ is 1.5 times the current speed, i.e. 1.5 times the local escape velocity.

To have${\displaystyle t=0}$ 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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