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 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 ${\displaystyle \epsilon }$ equal to the sum of its specific kinetic and specific potential energy:

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.

Note that C₃ is 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

where

• ${\displaystyle \mu =GM}$ is the standard gravitational parameter,
• ${\displaystyle a}$ 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

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

Also,

where ${\displaystyle v_{\infty }}$ is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches ${\displaystyle v_{\infty }}$ 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 ${\displaystyle {\sqrt {12.2}}{\text{ km/s}}=3.5{\text{ km/s}}}$. 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]

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

• Wie, Bong (1998). "Orbital Dynamics". Space Vehicle Dynamics and Control. AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. ISBN 1-56347-261-9.

Footnotes