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:
$\epsilon = \frac v^2 - \frac = \text = \frac C_3,$
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 ''C''₃ 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
$C_3 = -\frac < 0$
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
$C_3 = -\frac$

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:
$C_3 = \frac > 0$
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 $\sqrt\text = 3.5\text$. 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.

C₃ (km²/s²) to get from Earth to various planets: Mars 12, Jupiter 80, Saturn or Uranus 147. To Pluto (with its orbital inclination) needs about 160–164 km²/s².''New Horizons Mission Design''
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See also

*Specific orbital energy
*Orbit
*Parabolic trajectory
*Hyperbolic trajectory

References

*

Footnotes

{{Orbits
Astrodynamics
Orbits
Energy (physics)