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 ...
and
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 radial 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 zero
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. Two objects in a radial trajectory move directly towards or away from each other in a straight line.
Classification
There are three types of radial trajectories (orbits).
*
Radial elliptic trajectory
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 ...
: an orbit corresponding to the part of a degenerate ellipse from the moment the bodies touch each other and move away from each other until they touch each other again. The relative speed of the two objects is less than 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 ...
. This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit. If the
coefficient of restitution
The coefficient of restitution (COR, also denoted by ''e''), is the ratio of the final to initial relative speed between two objects after they collide. It normally ranges from 0 to 1 where 1 would be a perfectly elastic collision. A perfectl ...
of the two bodies is 1 (perfectly elastic) this orbit is periodic. If the coefficient of restitution is less than 1 (inelastic) this orbit is non-periodic.
*
Radial parabolic trajectory
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 ca ...
, a non-periodic orbit where the relative speed of the two objects is always equal to the escape velocity. There are two cases: the bodies move away from each other or towards each other.
*
Radial hyperbolic trajectory
In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fa ...
: a non-periodic orbit where the relative speed of the two objects always exceeds the escape velocity. There are two cases: the bodies move away from each other or towards each other. This is a hyperbolic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit.
Unlike standard orbits which are classified by their
orbital eccentricity
In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values betwee ...
, radial orbits are classified by their
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 ...
, the constant sum of the total kinetic and potential energy, divided by the
reduced mass
In physics, the reduced mass is the "effective" Mass#Inertial mass, inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, how ...
:
:
where ''x'' is the distance between the centers of the masses, ''v'' is the relative velocity, and
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 ...
.
Another constant is given by:
:
*For elliptic trajectories, w is positive. It is the inverse of the
apoapsis distance
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any elli ...
(maximum distance).
*For parabolic trajectories, w is zero.
*For hyperbolic trajectories, w is negative, It is
where
is the velocity at infinite distance.
Time as a function of distance
Given the separation and velocity at any time, and the total mass, it is possible to determine the position at any other time.
The first step is to determine the constant w. Use the sign of w to determine the orbit type.
:
where
and
are the separation and relative velocity at any time.
Parabolic trajectory
:
where ''t'' is the time from or until the time at which the two masses, if they were point masses, would coincide, and ''x'' is the separation.
This equation applies only to radial parabolic trajectories, for general parabolic trajectories see
Barker's equation.
Elliptic trajectory
:
where ''t'' is the time from or until the time at which the two masses, if they were point masses, would coincide, and ''x'' is the separation.
This is the
radial Kepler equation.
See also
equations for a falling body Lection 0
A set of equations describing the trajectories of objects subject to a constant gravitational force under normal Earth-bound conditions. Assuming constant acceleration ''g'' due to Earth’s gravity, Newton's law of universal gravitati ...
.
Hyperbolic trajectory
:
where ''t'' is the time from or until the time at which the two masses, if they were point masses, would coincide, and ''x'' is the separation.
Universal form (any trajectory)
The radial Kepler equation can be made "universal" (applicable to all trajectories):
:
or by expanding in a power series:
:
The radial Kepler problem (distance as function of time)
The problem of finding the separation of two bodies at a given time, given their separation and velocity at another time, is known as the
Kepler problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
. This section solves the Kepler problem for radial orbits.
The first step is to determine the constant
. Use the sign of
to determine the orbit type.
:
Where
and
are the separation and velocity at any time.
Parabolic trajectory
::
See also
position as function of time in a straight escape orbit.
Universal form (any trajectory)
Two intermediate quantities are used: w, and the separation at time t the bodies would have if they were on a parabolic trajectory, p.
:
Where t is the time,
is the initial position,
is the initial velocity, and
.
The
inverse radial Kepler equation is the solution to the radial Kepler problem:
:
Evaluating this yields:
:
Power series can be easily differentiated term by term. Repeated differentiation gives the formulas for the velocity, acceleration, jerk, snap, etc.
Orbit inside a radial shaft
The orbit inside a radial shaft in a uniform spherical body
[Strictly this is a contradiction. However, it is assumed that the shaft has a negligible influence on the gravity.] would be a
simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
, because gravity inside such a body is proportional to the distance to the center. If the small body enters and/or exits the large body at its surface the orbit changes from or to one of those discussed above. For example, if the shaft extends from surface to surface a closed orbit is possible consisting of parts of two cycles of simple harmonic motion and parts of two different (but symmetric) radial elliptic orbits.
See also
*
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 ...
*
Kepler problem
In classical mechanics, the Kepler problem is a special case of the two-body problem, in which the two bodies interact by a central force ''F'' that varies in strength as the inverse square of the distance ''r'' between them. The force may be ei ...
*
List of orbits
Summary
A simple list of just the common orbit abbreviations.
List of abbreviations of common Earth orbits
List of abbreviations of other orbits
Classifications
The following is a list of types of orbits:
Centric classifications
* Gal ...
References
* Cowell, Peter (1993), Solving Kepler's Equation Over Three Centuries, William Bell.
External links
* Kepler's Equation at Mathworl
{{DEFAULTSORT:Radial Trajectory
Orbits
Astrodynamics
Johannes Kepler