Standard gravitational parameter

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:

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

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.

Characteristic energy can be computed as:

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

where $v_{\infty }$ is the asymptotic velocity at infinite distance. Note that, since the kinetic energy is ${\tfrac {1}{2}}mv^{2}$, C₃ is in fact 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.

Examples

MAVEN, a Mars-bound spacecraft, was launched into a heliocentric orbit with a characteristic energy of 12.2 km²sec^-2 with respect to the Earth.

See also

• Specific orbital energy
• Orbit
• Parabolic trajectory
• Hyperbolic trajectory

References

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