# Characteristic energy explained

In astrodynamics, the **characteristic energy** (

) is a measure of the excess

specific energy over that required to just barely escape from a massive body. The units are

length^{2} time^{−2}, i.e.

velocity squared, or

energy per

mass.

equal to the sum of its specific kinetic and specific potential energy:

*\epsilon*=

*v*^{2}-

=constant=

*C*_{3,}

where

is the

standard gravitational parameter of the massive body with mass

, and

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.

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

where

is the

standard gravitational parameter,

is the

semi-major axis of the orbit's

ellipse.

If the orbit is circular, of radius *r*, then

## Parabolic trajectory

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

## Hyperbolic trajectory

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

where

is the

standard gravitational parameter,

is the

semi-major axis of the orbit's

hyperbola (which may be negative in some convention).

Also,

where

is the

asymptotic velocity at infinite distance. Spacecraft's velocity approaches

as it is further away from the central object's gravity.

## Examples

MAVEN, a Mars-bound spacecraft, was launched into a trajectory with a characteristic energy of 12.2 km^{2}/s^{2} 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

. 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 C_{3} of 8.19 km^{2}/s^{2}.^{[2]} The Parker Solar Probe (via Venus) plans a maximum C_{3} of 154 km^{2}/s^{2}.^{[3]}

C3 (km^{2}/s^{2}) from Earth to get to various planets : Mars 12, Jupiter 80, Saturn or Uranus 147.^{[4]} To Pluto (with its orbital inclination) needs about 160–164 km^{2}/s^{2}.^{[5]}

## See also

## References

## Notes and References

- http://www.nasaspaceflight.com/2013/11/atlasv-launch-maven-mars-mission Atlas V set to launch MAVEN on Mars mission
- Web site: InSight Launch Booklet . ULA . 2018 .
- Web site: Parker Solar Probe: The Mission . JHUAPL . parkersolarprobe.jhuapl.edu . en . 2018-07-22.
- NASA studies for Europa Clipper mission
- http://www.boulder.swri.edu/pkb/ssr/ssr-mission-design.pdf
*New Horizons Mission Design*