In
astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of ...
or
celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
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 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
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
hyperbolic trajectories from negative-energy
elliptic orbits.
Velocity
The
orbital velocity (
) of a body travelling along parabolic trajectory can be computed as:
:
where:
*
is the radial distance of orbiting body from
central body,
*
is the
standard gravitational parameter
In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
.
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 (
) 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:
:
where:
*
is
orbital velocity of a body in
circular orbit.
Equation of motion
For a body moving along this kind of
trajectory
A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete tr ...
an
orbital equation becomes:
:
where:
*
is radial distance of orbiting body from
central body,
*
is
specific angular momentum
In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative posit ...
of the
orbiting body,
*
is a
true anomaly of the orbiting body,
*
is the
standard gravitational parameter
In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
.
Energy
Under standard assumptions, the
specific orbital energy (
) of a parabolic trajectory is zero, so the
orbital energy conservation equation for this trajectory takes the form:
:
where:
*
is
orbital velocity of orbiting body,
*
is radial distance of orbiting body from
central body,
*
is the
standard gravitational parameter
In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
.
This is entirely equivalent to the
characteristic energy (square of the speed at infinity) being 0:
:
Barker's equation
Barker's equation relates the time of flight
to the true anomaly
of a parabolic trajectory:
:
where:
*
is an auxiliary variable
*
is the time of periapsis passage
*
is the standard gravitational parameter
*
is the
semi-latus rectum
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a sp ...
of the trajectory (
)
More generally, the time between any two points on an orbit is
:
Alternately, the equation can be expressed in terms of periapsis distance, in a parabolic orbit
:
:
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
. If the following substitutions are made
:
then
:
With hyperbolic functions the solution can be also expressed as:
[ Eq.(40) and Appendix C.]
:
where
:
Radial parabolic trajectory
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:
: