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 length² time⁻², i.e. velocity squared, or energy per mass.

Every object in a 2-body ballistic trajectory has a constant specific orbital energy $\epsilon$ equal to the sum of its specific kinetic and specific potential energy:

$${\displaystyle \epsilon ={\frac {1}{2))v^{2}-{\frac {\mu }{r))={\text{constant))={\frac {1}{2))C_{3},}$$

$\mu =GM$

$M$

$r$

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

$${\displaystyle C_{3}=-{\frac {\mu }{a))<0}$$

• $\mu =GM$ is the standard gravitational parameter,
• $a$ is the semi-major axis of the orbit's ellipse.

If the orbit is circular, of radius r, then

$${\displaystyle C_{3}=-{\frac {\mu }{r))}$$

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:

$${\displaystyle C_{3}={\frac {\mu }{|a|))>0}$$

• $\mu =GM$ is the standard gravitational parameter,
• $a$ is the semi-major axis of the orbit's hyperbola (which may be negative in some convention).

Also,

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

$v_{\infty }$

$v_{\infty }$

Examples

MAVEN, a Mars-bound spacecraft, was launched into a trajectory with a characteristic energy of 12.2 km²/s² 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 ${\displaystyle {\sqrt {12.2)){\text{ km/s))=3.5{\text{ km/s))}$. However, 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 maximal 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.

The InSight mission to Mars launched with a C₃ of 8.19 km²/s².^[2] The Parker Solar Probe (via Venus) plans a maximum C₃ of 154 km²/s².^[3]

C₃ (km²/s²) to get from Earth to various planets: Mars 12, Jupiter 80, Saturn or Uranus 147.^[4] To Pluto (with its orbital inclination) needs about 160–164 km²/s².^[5]