Characteristic energy

In astrodynamics, the characteristic energy (<math>C_3</math>) 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 <math>\epsilon</math> equal to the sum of its specific kinetic and specific potential energy: <math display="block">\epsilon = \frac{1}{2} v^2 - \frac{\mu}{r} = \text{constant} = \frac{1}{2} C_3,</math> where <math>\mu = GM</math> is the standard gravitational parameter of the massive body with mass <math>M</math>, and <math>r</math> 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 <math>\epsilon</math> 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 <math display="block">C_3 = -\frac{\mu}{a} < 0</math> where

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

If the orbit is circular, of radius r, then <math display="block">C_3 = -\frac{\mu}{r}</math>

Parabolic trajectory

A spacecraft leaving the central body on a parabolic trajectory has exactly the energy needed to escape and no more: <math display="block">C_3 = 0</math>

Hyperbolic trajectory

A spacecraft that is leaving the central body on a hyperbolic trajectory has more than enough energy to escape: <math display="block">C_3 = \frac{\mu}{|a|} > 0</math> where

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

Also, <math display="block">C_3 = v_\infty^2</math> where <math>v_\infty</math> is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches <math>v_\infty</math> 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.<ref>Atlas V set to launch MAVEN on Mars mission, nasaspaceflight.com, 17 November 2013.</ref> When simplified to a two-body problem, this would mean the MAVEN escaped Earth on a hyperbolic trajectory slowly decreasing its speed towards <math>\sqrt{12.2}\text{ km/s} = 3.5\text{ km/s}</math>. 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.<ref>ULA (2018). "InSight Launch Booklet" (PDF).</ref> The Parker Solar Probe (via Venus) plans a maximum C₃ of 154 km²/s².<ref>JHUAPL. "Parker Solar Probe: The Mission". parkersolarprobe.jhuapl.edu. Retrieved 2018-07-22.</ref>

Typical ballistic C₃ (km²/s²) to get from Earth to various planets: Mars 8-16,<ref>Delta-Vs and Design Reference Mission Scenarios for Mars Missions</ref> Jupiter 80, Saturn or Uranus 147.<ref>NASA studies for Europa Clipper mission</ref> To Pluto (with its orbital inclination) needs about 160–164 km²/s².<ref>New Horizons Mission Design</ref>

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.