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A radial hyperbolic trajectory is a non-periodic trajectory on a straight line 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.
It is called hyperbolic motion because the equation describing the path of the object through spacetime is a hyperbola, as can be seen when graphed on a Minkowski diagram whose coordinates represent a suitable inertial (non-accelerated) frame.
For a hyperbolic trajectory this specific orbital energy is either given by =. or the same as for an ellipse, depending on the convention for the sign of a . In this case the specific orbital energy is also referred to as characteristic energy (or C 3 {\displaystyle C_{3}} ) and is equal to the excess specific energy compared to that for a ...
Furthermore, the equation was derived on the assumption of an elliptical orbit, and so it does not hold for parabolic or hyperbolic orbits. These difficulties are what led to the development of the universal variable formulation , described below.
A hyperbola is an open curve with two branches, ... or the scattering trajectory of a ... can be described by several parametric equations: Through hyperbolic ...
The velocity equation for a hyperbolic trajectory has either ... It is an open orbit corresponding to the part of the degenerate ellipse from the moment the bodies ...
Here, the total turn is analogous to turning number, but for open curves (an angle covered by velocity vector). The limit case between an ellipse and a hyperbola, when e equals 1, is parabola. Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity.
Every object in a 2-body ballistic trajectory has a constant specific orbital energy equal to the sum of its specific kinetic and specific potential energy: = = =, where = is the standard gravitational parameter of the massive body with mass , and is the radial distance from its center. As an object in an escape trajectory moves outward, its ...