Editing Characteristic energy

In [[astrodynamics]], the '''characteristic energy''' (<math>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 [[length]]<sup>2</sup> [[time]]<sup>−2</sup>, i.e. [[velocity]] squared, or [[energy]] per [[mass]].

Every object in a [[two-body problem|2-body]] [[ballistics|ballistic]] trajectory has a constant [[specific orbital energy]] <math>\epsilon</math> equal to the sum of its specific kinetic and specific potential energy:
<math display="block">\epsilon = \frac{1}{2} v^2 - \frac{\mu}{r} = \text{constant} = \frac{1}{2} C_3,</math>
where <math>\mu = GM</math> is the [[standard gravitational parameter]] of the massive body with mass <math>M</math>, and <math>r</math> is the [[Polar coordinate system|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''<sub>3</sub> is ''twice'' the [[specific orbital energy]] <math>\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 display="block">C_3 = -\frac{\mu}{a} < 0</math>
where
*<math>\mu = GM</math> is the [[standard gravitational parameter]],
*<math>a</math> is the [[semi-major axis]] of the orbit's [[ellipse]].

If the orbit is circular, of radius ''r'', then
<math display="block">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 display="block">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 display="block">C_3 = \frac{\mu}{|a|} > 0</math>
where
*<math>\mu = GM</math> is the [[standard gravitational parameter]],
*<math>a</math> is the [[semi-major axis]] of the orbit's [[hyperbola]] (which may be negative in some convention).

Also,
<math display="block">C_3 = v_\infty^2</math>
where <math>v_\infty</math> is the [[asymptotic]] velocity at infinite distance.  Spacecraft's velocity approaches <math>v_\infty</math> 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 ''C<sub>3</sub>'' is [[Forest Ray Moulton]]'s textbook ''An Introduction to Celestial Mechanics''.  In the second edition (1914) 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]] ''c<sub>3</sub>'' by the equation 
::''ẋ<sup>2</sup> + ẏ<sup>2</sup> = 2k<sup>2</sup> M/r + c<sub>3</sub>'', 
where ''M'' is the total mass of the two bodies and ''k<sup>2</sup>'' is Moulton's notation for the [[gravitational constant]].  He defines ''c<sub>1</sub>'', ''c<sub>2</sub>'', and ''c<sub>4</sub>'' to be other constants of the motion.  The notation ''C<sub>3</sub>'' probably became popularized via the [[Jet Propulsion Laboratory|JPL]] technical report TR-32-30 ("Design of Lunar and Interplanetary Ascent Trajectories", Victor C. Clarke, Jr., March 15, 1962), which used Moulton's terminology.<ref>[https://cbboff.org/UCBoulderCourse/documents/history_c3.pdf "The History of the Term C3"], Chauncey Uphoff, Fortune Eight Aerospace Industries, Inc., December 19, 2001.  Accessed December 21, 2024.  [https://web.archive.org/web/20241220054625id_/https://cbboff.org/UCBoulderCourse/documents/history_c3.pdf Archived on December 20, 2024] by the [[Wayback Machine]].</ref><ref>{{Internet Archive|introcelestial00moulrich|''An Introduction to Celestial Mechanics''|page=162|at=no}}, Forest Ray Moulton, New York: The Macmillan Company, 2nd revised edition, 1914, Chapter 5, §83-88.</ref><ref>"Design of Lunar and Interplanetary Ascent Trajectories", Victor C. Clarke Jr., Technical Report 32-30, JPL, March 15, 1962.</ref>

==Examples==
[[MAVEN]], a [[Mars]]-bound spacecraft, was launched into a trajectory with a characteristic energy of 12.2&nbsp;km<sup>2</sup>/s<sup>2</sup> with respect to the Earth.<ref>[http://www.nasaspaceflight.com/2013/11/atlasv-launch-maven-mars-mission Atlas V set to launch MAVEN on Mars mission], nasaspaceflight.com, 17 November 2013.</ref> When simplified to a [[two-body problem]], this would mean the MAVEN escaped Earth on a hyperbolic trajectory slowly decreasing its speed towards <math>\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 [[heliocentric orbit|elliptical orbit around the Sun]]. But the maximal velocity on the new orbit could be approximated to 33.5&nbsp;km/s by assuming that it reached practical "infinity" at 3.5&nbsp;km/s and that such Earth-bound "infinity" also moves with Earth's orbital velocity of about 30&nbsp;km/s.

The [[InSight]] mission to Mars launched with a C<sub>3</sub> of 8.19&nbsp;km<sup>2</sup>/s<sup>2</sup>.<ref>{{Cite web |url=https://www.ulalaunch.com/docs/default-source/launch-booklets/mob_insightfinal.pdf |title=InSight Launch Booklet |last=ULA |date=2018 }}</ref> The [[Parker Solar Probe]] (via Venus) plans a maximum C<sub>3</sub> of 154&nbsp;km<sup>2</sup>/s<sup>2</sup>.<ref>{{Cite web|url=http://parkersolarprobe.jhuapl.edu/The-Mission/index.php#Launch |title=Parker Solar Probe: The Mission |author=JHUAPL |website=parkersolarprobe.jhuapl.edu |language=en |access-date=2018-07-22}}</ref>

Typical ballistic C<sub>3</sub> (km<sup>2</sup>/s<sup>2</sup>) to get from Earth to various planets: Mars 8-16,<ref>[https://digitalcommons.usu.edu/smallsat/2022/all2022/257/ ''Delta-Vs and Design Reference Mission Scenarios for Mars Missions'']</ref> Jupiter 80, Saturn or Uranus 147.<ref>NASA studies for Europa Clipper mission</ref> To Pluto (with its orbital inclination) needs about 160–164&nbsp;km<sup>2</sup>/s<sup>2</sup>.<ref>[http://www.boulder.swri.edu/pkb/ssr/ssr-mission-design.pdf ''New Horizons Mission Design'']</ref>

==See also==
*[[Specific orbital energy]]
*[[Orbit]]
*[[Parabolic trajectory]]
*[[Hyperbolic trajectory]]

==References==
*{{cite book | last=Wie | first=Bong | title=Space Vehicle Dynamics and Control | url=https://archive.org/details/spacevehicledyna00wieb_0 | url-access=registration | publisher=[[American Institute of Aeronautics and Astronautics]] | location=[[Reston, Virginia]] | date=1998 | series=AIAA Education Series | chapter=Orbital Dynamics | isbn=1-56347-261-9 }}<!--| accessdate=2009-07-05-->

==Footnotes==
{{Reflist}}
{{Orbits}}
[[Category:Astrodynamics]]
[[Category:Orbits]]
[[Category:Energy (physics)]]