In astrodynamics or celestial mechanics, a hyperbolic trajectory or hyperbolic orbit (from Newtonian theory: hyperbola shape) is the trajectory of any object around a central body with enough velocity to escape the central object's gravitational field; expressed as orbital eccentricity designated by any number more than 1. Under simplistic assumptions a body traveling along this trajectory will coast towards infinity, settling to a final excess velocity relative to the central body. As with parabolic trajectories, all hyperbolic trajectories are also escape trajectories. The specific energy of a hyperbolic trajectory orbit is positive. Planetary flybys, used for gravitational slingshots, can be described within the planet's sphere of influence using hyperbolic trajectories.
Parameters describing a hyperbolic trajectory Like an elliptical orbit, a hyperbolic trajectory for a given system can be defined (ignoring orientation) by its semi major axis and the eccentricity. However, with a hyperbolic orbit other parameters may be more useful in understanding a body's motion. The following table lists the main parameters describing the path of body following a hyperbolic trajectory around another under standard assumptions and the formula connecting them.
Semi-major axis, energy and hyperbolic excess velocity
The semi major axis ( a {\displaystyle a\,\!} ) is not immediately visible with a hyperbolic trajectory but can be constructed as it is the distance from periapsis to the point where the two asymptotes cross. Usually, by convention, it is negative, to keep various equations consistent with elliptical orbits. The semi major axis is directly linked to the specific orbital energy ( ϵ {\displaystyle \epsilon \,} ) or characteristic energy C 3 {\displaystyle C_{3}} of the orbit, and to the velocity the body attains at as the distance tends to infinity, the hyperbolic excess velocity ( v ∞ {\displaystyle v_{\infty }\,\!} ).
v ∞ 2 = 2 ϵ = C 3 = − μ / a {\displaystyle v_{\infty }^{2}=2\epsilon =C_{3}=-\mu /a} or a = − μ / v ∞ 2 {\displaystyle a=-{\mu /{v_{\infty }^{2}}}}
where: μ = G m {\displaystyle \mu =Gm\,\!} is the standard gravitational parameter and C 3 {\displaystyle C_{3}} is characteristic energy, commonly used in planning interplanetary missions Note that the total energy is positive in the case of a hyperbolic trajectory (whereas it is negative for an elliptical orbit).
Eccentricity and angle between approach and departure With a hyperbolic trajectory the orbital eccentricity is greater than 1. The eccentricity is directly related to the angle between the asymptotes. With eccentricity just over 1 the hyperbola is a sharp "v" shape. At e = 2 {\displaystyle e={\sqrt {2}}} the asymptotes are at right angles. With e > 2 {\displaystyle e>2} the asymptotes are more than 120° apart, and the periapsis distance is greater than the semi major axis. As eccentricity increases further the motion approaches a straight line. The angle between the direction of periapsis and an asymptote from the central body is the true anomaly as distance tends to infinity ( θ ∞ {\displaystyle \theta _{\infty }\,} ), so 2 θ ∞ {\displaystyle 2\theta _{\infty }\,} is the external angle between approach and departure directions (between asymptotes). Then
θ ∞ = cos − 1 ( − 1 / e ) {\displaystyle \theta {_{\infty }}=\cos ^{-1}(-1/e)\,} or e = − 1 / cos θ ∞ {\displaystyle e=-1/\cos \theta {_{\infty }}\,}
Impact parameter and the distance of closest approach
The impact parameter is the distance by which a body, if it continued on an unperturbed path, would miss the central body at its closest approach. With bodies experiencing gravitational forces and following hyperbolic trajectories it is equal to the semi-minor axis of the hyperbola. In the situation of a spacecraft or comet approaching a planet, the impact parameter and excess velocity will be known accurately. If the central body is known the trajectory can now be found, including how close the approaching body will be at periapsis. If this is less than the planet's radius an impact should be expected. The distance of closest approach, or periapsis distance, is given by:
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