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In [[astrodynamics]] the '''characteristic energy''' (<math>C_3\,\!</math>) is a measure of the excess [[specific energy]] over that required to just barely escape from a massive body. The units are [[length]]<sup>2</sup>[[time]]<sup>-2</sup>, i.e., [[energy]] per [[mass]].
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Every object in a [[two-body problem|2-body]] [[ballistics|ballistic]] trajectory has a constant [[specific orbital energy]] <math>\epsilon</math> equal to the sum of its kinetic and potential energy:
 
:<math>\tfrac{1}{2} v^2 - \mu/r = constant = \tfrac{1}{2} C_3</math>
 
where <math>\mu = GM</math> is the [[standard gravitational parameter]] of the massive body with mass <math>M</math> and <math>r</math> is the radial distance from its center. As an object in an escape trajectory moves outward, its kinetic energy decreases as its potential energy (which is always negative) increases, maintaining a constant sum.
 
Characteristic energy can be computed as:
:<math>C_3=v_{\infty}^2\,\!</math>
where <math>v_{\infty}</math> is the [[asymptotic]] [[Kinetic energy|velocity]] at infinite distance. Note that, since the kinetic energy is <math>\tfrac{1}{2} mv^2</math>, C<sub>3</sub> is in fact '''twice''' the [[specific orbital energy]] (<math>\epsilon</math>) of the escaping object.
 
==Non-escape trajectory==
A spacecraft with insufficient energy to escape will remain in a closed orbit (unless it intersects the [[central body]]) with:
:<math>C_3<0\,</math>
 
==Parabolic trajectory==
A spacecraft leaving the [[central body]] on a [[parabolic trajectory]] has exactly the energy needed to escape and no more:
:<math>C_3=0\,</math>
 
==Hyperbolic trajectory==
A spacecraft that is leaving the [[central body]] on a [[hyperbolic trajectory]] has more than enough energy to escape:
:<math>C_3={\mu\over{a}}\,</math>
where
:<math>\mu\,=GM</math> is the [[standard gravitational parameter]],
:<math>a\,</math> is the [[semi-major axis]] of the [[orbit]]'s [[hyperbola]].
 
==Examples==
[[MAVEN]], a [[Mars]]-bound spacecraft, was launched into a [[heliocentric orbit]] with a characteristic energy of 12.2 km<sup>2</sup>sec<sup>-2 </sup>with respect to the Earth.<ref>[http://www.nasaspaceflight.com/2013/11/atlasv-launch-maven-mars-mission Atlas V set to launch MAVEN on Mars mission, nasaspaceflight.com, 17 November 2013.]</ref>
 
==See also==
*[[Specific orbital energy]]
*[[Orbit]]
*[[Parabolic trajectory]]
*[[Hyperbolic trajectory]]
 
==References==
*{{cite book | last=Wie | first=Bong | title=Space Vehicle Dynamics and Control | publisher=[[American Institute of Aeronautics and Astronautics]] | location=[[Reston, Virginia]] | date=1998 | series=AIAA Education Series | chapter=Orbital Dynamics | isbn=1-56347-261-9 | accessdate=2009-07-05}}
 
==Footnotes==
{{Reflist}}
 
[[Category:Astrodynamics]]
[[Category:Orbits]]
[[Category:Energy (physics)]]

Revision as of 21:29, 15 February 2014

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