Characteristic energy

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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.

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−2 with 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 12.2 km/s = 3,5 km/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.

See also

References

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Footnotes