In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity 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 C3 = 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.
The orbital velocity (
v
v=\sqrt{2\mu\overr}
r
\mu
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
v=\sqrt{2}vo
vo
For a body moving along this kind of trajectory the orbital equation is:
r={h2\over\mu}{1\over{1+\cos\nu}}
r
h
\nu
\mu
Under standard assumptions, the specific orbital energy (
\epsilon
\epsilon={v2\over2}-{\mu\overr}=0
v
r
\mu
This is entirely equivalent to the characteristic energy (square of the speed at infinity) being 0:
C3=0
Barker's equation relates the time of flight
t
\nu
t-T=
| 1 | \sqrt{ | |
| 2 |
| p3 | |
| \mu |
where:
D=\tan
| \nu | |
| 2 |
T
\mu
p
p=h2/\mu
More generally, the time between any two points on an orbit is
tf-t0=
| 1 | \sqrt{ | |
| 2 |
| p3 | |
| \mu |
Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit
rp=p/2
t-T=\sqrt{
| |||||||||
| \mu |
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
t
\begin{align} A&=
| 3 | \sqrt{ | |
| 2 |
| \mu | ||||||
|
then
\nu=2\arctan\left(B-
| 1 | |
| B |
\right)
With hyperbolic functions the solution can be also expressed as:[2]
\nu=2\arctan\left(2\sinh
| ||||||
| 3 |
\right)
where
M=\sqrt{
| \mu | ||||||
|
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:
r=\sqrt[3]{
| 9 | |
| 2 |
\mut2}
where
t=0
At any time the average speed from
t=0
To have
t=0