Characteristic energy

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.

Every object in a 2-body ballistic trajectory has a constant specific orbital energy $$\epsilon$$ equal to the sum of its specific kinetic and specific potential energy: $$\epsilon = \frac{1}{2} v^2 - \frac{\mu}{r} = \text{constant} = \frac{1}{2} C_3,$$ 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.

Note that C3 is twice the specific orbital energy $$\epsilon$$ 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 $$C_3 = -\frac{\mu}{a} < 0$$ where
 * $$\mu = GM$$ 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 $$C_3 = -\frac{\mu}{r}$$

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: $$C_3 = \frac{\mu}{|a|} > 0$$ where
 * $$\mu = GM$$ is the standard gravitational parameter,
 * $$a$$ is the semi-major axis of the orbit's hyperbola (which may be negative in some convention).

Also, $$C_3 = v_\infty^2$$ 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. When simplified to a two-body problem, this would mean the MAVEN escaped Earth on a hyperbolic trajectory slowly decreasing its speed towards $$\sqrt{12.2}\text{ km/s} = 3.5\text{ km/s}$$. 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. The Parker Solar Probe (via Venus) plans a maximum C3 of 154 km2/s2.

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