# 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:

$\tfrac{1}{2} v^2 - \mu/r = constant = \tfrac{1}{2} C_3$

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.[1]