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 $ϵ$ equal to the sum of its kinetic and potential energy:

$\frac{1}{2}{v}^2-\mu /r=constant=\frac{1}{2}{C}_3$

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:

$C_3=v_{\mathrm{\infty }}^2$

where $v_{\mathrm{\infty }}$ is the asymptotic velocity at infinite distance. Note that, since the kinetic energy is $\frac{1}{2}m{v}^2$, C₃ is in fact twice the specific orbital energy ($ϵ$) 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:

$C_3=\frac{\mu }{a}$

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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