Characteristic energy
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 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 equal to the sum of its specific kinetic and specific potential energy:
Note that C3 is twice the specific orbital energy of the escaping object.
Non-escape trajectory[edit]
A spacecraft with insufficient energy to escape will remain in a closed orbit (unless it intersects the central body), with
- 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[edit]
A spacecraft leaving the central body on a parabolic trajectory has exactly the energy needed to escape and no more:
Hyperbolic trajectory[edit]
A spacecraft that is leaving the central body on a hyperbolic trajectory has more than enough energy to escape:
- is the standard gravitational parameter,
- is the semi-major axis of the orbit's hyperbola (which may be negative in some convention).
Also,
Examples[edit]
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 . 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[edit]
References[edit]
- 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[edit]
- ^ Atlas V set to launch MAVEN on Mars mission, nasaspaceflight.com, 17 November 2013.
- ^ ULA (2018). "InSight Launch Booklet" (PDF).
- ^ JHUAPL. "Parker Solar Probe: The Mission". parkersolarprobe.jhuapl.edu. Retrieved 2018-07-22.
- ^ Delta-Vs and Design Reference Mission Scenarios for Mars Missions
- ^ NASA studies for Europa Clipper mission
- ^ New Horizons Mission Design