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[[File:OrbitalEccentricityDemo.svg|thumb|right|The blue path in this image is an example of a hyperbolic trajectory]]
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[[File:Gravity Wells Potential Plus Kinetic Energy - Circle-Ellipse-Parabola-Hyperbola.png|thumb|250px|A hyperbolic trajectory is depicted in the bottom-right quadrant of this diagram, where the [[gravity well|gravitational potential well]] of the central mass shows potential energy, and the kinetic energy of the hyperbolic trajectory is shown in red. The height of the kinetic energy decreases as the speed decreases and distance increases according to Kepler's laws. The part of the kinetic energy that remains above zero total energy is that associated with the hyperbolic excess velocity.]]


In [[astrodynamics]] or [[celestial mechanics]] a '''hyperbolic trajectory''' is the trajectory of any object around a [[central body]] with more than enough velocity to escape the central object's gravitational pull. The name derives from the fact that according [[Newton's law of universal gravitation|Newtonian theory]] such an orbit has the shape of a [[hyperbola]]. In more technical terms this can be expressed by the condition that the [[orbital eccentricity]] is greater than one.
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Under standard assumptions a body traveling along this trajectory will coast to infinity, arriving there with hyperbolic excess velocity relative to the central body. Similarly to [[parabolic trajectory]] all hyperbolic trajectories are also [[escape trajectory|escape trajectories]]. The [[specific orbital energy|specific energy]] of a hyperbolic trajectory orbit is positive.
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Planetary flybys, used for [[Gravity assist|gravitational slingshots]], can be described within the planet's [[Sphere of influence (astrodynamics)|sphere of influence]] using hyperbolic trajectories.
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==Hyperbolic excess velocity==
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{{see also|Characteristic energy}}


Under standard assumptions the body traveling along hyperbolic trajectory will attain in infinity an [[Kinetic energy|orbital velocity]] called hyperbolic excess velocity (<math>v_\infty\,\!</math>) that can be computed as:
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:<math>v_\infty=\sqrt{\mu\over{-a}}\,\!</math>
 
where:
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*<math>\mu=Gm\,\!</math> is [[standard gravitational parameter]],
 
*<math>a\,\!</math> is the negative [[semi-major axis]] of [[orbit]]'s [[hyperbola]].
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The hyperbolic excess velocity is related to the [[specific orbital energy]] or characteristic energy by
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:<math>2\epsilon=C_3=v_{\infty}^2\,\!</math>
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==Velocity==
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Under standard assumptions the [[Kinetic energy|orbital velocity]] (<math>v\,</math>) of a body traveling along a hyperbolic trajectory can be computed as (Vallado):
 
:<math>v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}</math>
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where:
 
*<math>\mu\,</math> is [[standard gravitational parameter]],
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*<math>r\,</math> is radial distance of orbiting body from [[central body]],
 
*<math>a\,\!</math> is the negative [[semi-major axis]].
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Under standard assumptions, at any position in the orbit the following relation holds for [[Kinetic energy|orbital velocity]] (<math>v\,</math>), local [[escape velocity]](<math>{v_{esc}}\,</math>) and hyperbolic excess velocity (<math>v_\infty\,\!</math>):
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:<math>v^2={v_{esc}}^2+{v_\infty}^2</math>
 
 
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Note that this means that a relatively small extra [[delta-v]] above that needed to accelerate to the escape speed, results in a relatively large speed at infinity.
 
 
</ul>
==Angle between approach and departure==
Let the angle between approach and departure (between asymptotes) be <math>2\theta\,</math>.
 
:<math>\theta=\cos^{-1}(1/e)\,</math> and <math>e=1/\cos\theta\,</math>
 
where:
*<math>e\,</math> is the orbital [[eccentricity (orbit)|eccentricity]], which is greater than 1 for hyperbolic trajectories.
 
==Distance of closest approach==
The distance of closest approach, also called the [[periapse]] distance and the focal distance, is given by <math>-a(e-1)\,</math>
 
==Energy==
Under standard assumptions, [[specific orbital energy]] (<math>\epsilon\,</math>) of a hyperbolic trajectory is greater than zero and the [[orbital energy conservation equation]] for this kind of trajectory takes form:
:<math>\epsilon={v^2\over2}-{\mu\over{r}}={\mu\over{-2a}}</math>
where:
*<math>v\,</math> is [[Kinetic energy|orbital velocity]] of orbiting body,
*<math>r\,</math> is radial distance of orbiting body from [[central body]],
*<math>a\,</math> is the negative [[semi-major axis]],
*<math>\mu\,</math> is [[standard gravitational parameter]].
 
==Radial hyperbolic trajectory==
A radial hyperbolic trajectory is a non-periodic [[Radial trajectory|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. Although the eccentricity is 1 this is not a parabolic orbit.
 
==Relativistic two-body problem==
In context of the [[two-body problem in general relativity]], trajectories of objects with enough energy to escape the gravitational pull of the other, no longer are shape like an hyperbola. Nonetheless, the term "hyperbolic trajectory" is still used to describe orbits of this type.
 
 
==See also==
*[[Orbit]]
*[[Orbital equation]]
*[[Kepler orbit]]
*[[List of orbits]]
*[[Planetary flyby]]
 
==References==
{{refbegin}}
* {{cite book |last=Vallado |first=David A. |title=Fundamentals of Astrodynamics and Applications, Third Edition |year=2007 |isbn=978-1-881883-14-2 |publisher=Hawthorne Press |location=Hawthorne, CA.}}
{{refend}}
{{reflist}}
 
==External links==
* http://homepage.mac.com/sjbradshaw/msc/traject.html
* http://www.go.ednet.ns.ca/~larry/orbits/ellipse.html
* http://www.braeunig.us/space/orbmech.htm#hyperbolic
 
{{orbits}}
 
[[Category:Orbits]]

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