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:
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 increases, maintaining a constant sum.
Note that C3 is twice the specific orbital energy of the escaping object.

Non-escape trajectory

A spacecraft with insufficient energy to escape will remain in a closed orbit, with
where
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
Also,
where is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches as it is further away from the central object's gravity.

History of the notation

According to Chauncey Uphoff, the ultimate source of the notation C3 is Forest Ray Moulton's textbook An Introduction to Celestial Mechanics. In the second edition of this book, Moulton solves the problem of the motion of two bodies under an attractive gravitational force in chapter 5. After reducing the problem to the relative motion of the bodies in the plane, he defines the constant of the motion c3 by the equation
where M is the total mass of the two bodies and k2 is Moulton's notation for the gravitational constant. He defines c1, c2, and c4 to be other constants of the motion. The notation C3 probably became popularized via the JPL technical report TR-32-30, which used Moulton's terminology.

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. 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. The Parker Solar Probe plans a maximum C3 of 154 km2/s2.
Typical ballistic C3 to get from Earth to various planets: Mars 8-16, Jupiter 80, Saturn or Uranus 147. To Pluto needs about 160–164 km2/s2.