Characteristic energy explained

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.\epsilon equal to the sum of its specific kinetic and specific potential energy:$$\backslash epsilon\; =\; \backslash frac\; v^2\; -\; \backslash frac\; =\; \backslash text\; =\; \backslash frac\; C\_3,$$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.\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), withwhere

\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$$C\_3\; =\; -\backslash frac$$

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:$$C\_3\; =\; \backslash frac$$

> 0where\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,$$C\_3\; =\; v\_\backslash infty^2$$where

v _{infty} is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches v _{infty} 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 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

\sqrt{ 12 . 2 } km/s = 3 . 5 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]

Typical ballistic C₃ (km²/s²) to get from Earth to various planets: Mars 8-16,^[4] Jupiter 80, Saturn or Uranus 147.^[5] To Pluto (with its orbital inclination) needs about 160–164 km²/s².^[6]

See also

• Specific orbital energy
• Orbit
• Parabolic trajectory
• Hyperbolic trajectory

References

• Book: Wie, Bong . Space Vehicle Dynamics and Control . registration . . . 1998 . AIAA Education Series . Orbital Dynamics . 1-56347-261-9 .

Notes and References

1. http://www.nasaspaceflight.com/2013/11/atlasv-launch-maven-mars-mission Atlas V set to launch MAVEN on Mars mission
2. Web site: InSight Launch Booklet . ULA . 2018 .
3. Web site: Parker Solar Probe: The Mission . JHUAPL . parkersolarprobe.jhuapl.edu . en . 2018-07-22.
4. https://digitalcommons.usu.edu/smallsat/2022/all2022/257/ Delta-Vs and Design Reference Mission Scenarios for Mars Missions
5. NASA studies for Europa Clipper mission
6. http://www.boulder.swri.edu/pkb/ssr/ssr-mission-design.pdf New Horizons Mission Design