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Radial trajectory

Radial trajectory

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In astrodynamics
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 motion and Newton's law of universal gravitation. It...

 and celestial mechanics
Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics is a subfield which focuses on...

 a radial trajectory is a Kepler orbit
Kepler orbit
In celestial mechanics, a Kepler orbit describes the motion of an orbiting body as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space...

 with zero angular momentum
Angular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...

. 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: a nonperiodic 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
    Escape velocity
    In physics, escape velocity is the speed at which the kinetic energy plus the gravitational potential energy of an object is zero gravitational potential energy is negative since gravity is an attractive force and the potential is defined to be zero at infinity...

    . This is an elliptic orbit with semi-minor axis = 0 and eccentricity = 1. Although the eccentricity is 1 this is not a parabolic orbit.

  • Radial parabolic trajectory, a nonperiodic 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: a nonperiodic 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
Orbital eccentricity
The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit...

, radial orbits are classified by their specific orbital energy
Specific orbital energy
In the gravitational two-body problem, the specific orbital energy \epsilon\,\! of two orbiting bodies is the constant sum of their mutual potential energy and their total kinetic energy , divided by the reduced mass...

, the constant sum of the total kinetic and potential energy, divided by the reduced mass
Reduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...

:


where x is the distance between the centers of the masses, v is the relative velocity, and is the standard gravitational parameter
Standard gravitational parameter
In astrodynamics, the standard gravitational parameter μ of a celestial body is the product of the gravitational constant G and the mass M of the body.\mu=GM \ The SI units of the standard gravitational parameter are m3s−2....

.

Another constant is given by:

  • For elliptic trajectories, w is positive. It is the inverse of the apoapsis distance (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.

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
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 either attractive or repulsive...

. This section solves the Kepler problem for radial orbits.

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 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 probem:


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 would be a simple harmonic motion
Simple harmonic motion
Simple harmonic motion can serve as a mathematical model of a variety of motions, such as the oscillation of a spring. Additionally, other phenomena can be approximated by simple harmonic motion, including the motion of a simple pendulum and molecular vibration....

, 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.

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