Characteristic energy

In astrodynamics, the characteristic energy ( $C_{3}\,\!$ ) is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length²time⁻², i.e. energy per mass.

Every object in a 2-body ballistic trajectory has a constant specific orbital energy $\epsilon$ equal to the sum of its kinetic and potential energy:

\epsilon ={\tfrac {1}{2}}v^{2}-\mu /r=constant={\tfrac {1}{2}}C_{3}

where $\mu =GM$ is the standard gravitational parameter of the massive body with mass $M$ and $r$ 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.

Note that C₃ is twice the specific orbital energy $\epsilon$ 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:

C_{3}<0\,

Parabolic trajectory

A spacecraft leaving the central body on a parabolic trajectory has exactly the energy needed to escape and no more:

C_{3}=0\,

Hyperbolic trajectory

A spacecraft that is leaving the central body on a hyperbolic trajectory has more than enough energy to escape:

C_{3}=-{\mu  \over {a}}\,

where

\mu \,=GM

is the standard gravitational parameter,

a\,

is the semi-major axis of the orbit's hyperbola (which is negative by convention).

Also:

C_{3}=v_{{\infty }}^{2}\,\!

where $v_{{\infty }}$ is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches $v_{{\infty }}$ as it is further away from the central object's gravity.

Examples

MAVEN, a Mars-bound spacecraft, was launched into a trajectory with a characteristic energy of 12.2 km²sec⁻²with respect to the Earth.^[1] When simplified to a two-body problem, this would mean the MAVEN escaped Earth on a hyperbolic trajectory slowly decreasing its speed towards ${\sqrt {1}}2.2km/s=3.5km/s$ But since the Sun's gravitational field is much stronger than Earth's, the two-body solution is insufficient. The characteristic energy with respect to Sun was negative, and MAVEN – instead of heading to infinity – entered an elliptical orbit around the Sun. But the maximum velocity on the new orbit could be approximated to 33.5 km/s by assuming that it reached practical "infinity" at 3.5 km/s and that such Earth-bound "infinity" also moves with Earth's orbital velocity of about 30 km/s.

References

Wie, Bong (1998). "Orbital Dynamics". Space Vehicle Dynamics and Control. AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. ISBN 1-56347-261-9.

Footnotes

↑ Atlas V set to launch MAVEN on Mars mission, nasaspaceflight.com, 17 November 2013.

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