# Characteristic energy

In astrodynamics the characteristic energy (${\displaystyle C_{3}\,\!}$) 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 ${\displaystyle \epsilon }$ equal to the sum of its kinetic and potential energy:

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

where ${\displaystyle \mu =GM}$ is the standard gravitational parameter of the massive body with mass ${\displaystyle M}$ and ${\displaystyle r}$ 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:

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

where ${\displaystyle v_{\infty }}$ is the asymptotic velocity at infinite distance. Note that, since the kinetic energy is ${\displaystyle {\tfrac {1}{2}}mv^{2}}$, C3 is in fact twice the specific orbital energy (${\displaystyle \epsilon }$) 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:

${\displaystyle C_{3}<0\,}$

## Parabolic trajectory

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

${\displaystyle C_{3}=0\,}$

## Hyperbolic trajectory

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

${\displaystyle C_{3}={\mu \over {a}}\,}$

where

${\displaystyle \mu \,=GM}$ is the standard gravitational parameter,
${\displaystyle a\,}$ 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.[1]