Astronomy:Characteristic energy

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

• [math]\displaystyle{ \mu = GM }[/math] is the standard gravitational parameter,
• [math]\displaystyle{ a }[/math] is the semi-major axis of the orbit's ellipse.

If the orbit is circular, of radius r, then [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ C_3 = \frac{\mu}{|a|} \gt 0 }[/math] where

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

Also, [math]\displaystyle{ C_3 = v_\infty^2 }[/math] where [math]\displaystyle{ v_\infty }[/math] is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches [math]\displaystyle{ 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. When simplified to a two-body problem, this would mean the MAVEN escaped Earth on a hyperbolic trajectory slowly decreasing its speed towards [math]\displaystyle{ \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. 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.

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. https://archive.org/details/spacevehicledyna00wieb_0.