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, to ...
a parabolic trajectory is a
Kepler orbit
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
with the
eccentricity
Eccentricity or eccentric may refer to:
* Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal"
Mathematics, science and technology Mathematics
* Off-center, in geometry
* Eccentricity (graph theory) of a v ...
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 In astrodynamics, the characteristic energy (C_3) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass.
Every object in ...
).
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
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
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 heat a ...
hyperbolic trajectories from negative-energy
elliptic orbit
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it ...
s.
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
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
,
*
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 when ...
.
At any position the orbiting body has the
escape velocity
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
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
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
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
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is ...
.
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 traj ...
an
orbital equation
In astrodynamics, an orbit equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a for ...
becomes:
:
where:
*
is radial distance of orbiting body from
central body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
,
*
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 positi ...
of the
orbiting body
In astrodynamics, an orbiting body is any physical body that orbits a more massive one, called the primary body. The orbiting body is properly referred to as the secondary body (m_2), which is less massive than the primary body (m_1).
Thus, m_2 ...
,
*
is a
true anomaly
In celestial mechanics, true anomaly is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus ...
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 when ...
.
Energy
Under standard assumptions, the
specific orbital energy
In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy (\varepsilon_p) and their total kinetic energy (\varepsilon_k), divided ...
(
) of a parabolic trajectory is zero, so the
orbital energy conservation equation
In astrodynamics, the ''vis-viva'' equation, also referred to as orbital-energy-invariance law, is one of the equations that model the motion of orbiting bodies. It is the direct result of the principle of conservation of mechanical energy which ...
for this trajectory takes the form:
:
where:
*
is
orbital velocity of orbiting body,
*
is radial distance of orbiting body from
central body
A primary (also called a gravitational primary, primary body, or central body) is the main physical body of a gravitationally bound, multi-object system. This object constitutes most of that system's mass and will generally be located near the syst ...
,
*
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 when ...
.
This is entirely equivalent to the
characteristic energy In astrodynamics, the characteristic energy (C_3) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length2 time−2, i.e. velocity squared, or energy per mass.
Every object in ...
(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 speci ...
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
In orbital mechanics, Kepler's equation relates various geometric properties of the orbit of a body subject to a central force.
It was first derived by Johannes Kepler in 1609 in Chapter 60 of his ''Astronomia nova'', and in book V of his '' Epi ...
, 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
In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
. 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:
: