- Source: Characteristic energy
In astrodynamics, the characteristic energy (
C
3
{\displaystyle 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
ϵ
{\displaystyle \epsilon }
equal to the sum of its specific kinetic and specific potential energy:
ϵ
=
1
2
v
2
−
μ
r
=
constant
=
1
2
C
3
,
{\displaystyle \epsilon ={\frac {1}{2}}v^{2}-{\frac {\mu }{r}}={\text{constant}}={\frac {1}{2}}C_{3},}
where
μ
=
G
M
{\displaystyle \mu =GM}
is the standard gravitational parameter of the massive body with mass
M
{\displaystyle M}
, and
r
{\displaystyle 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
ϵ
{\displaystyle \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
=
−
μ
a
<
0
{\displaystyle C_{3}=-{\frac {\mu }{a}}<0}
where
μ
=
G
M
{\displaystyle \mu =GM}
is the standard gravitational parameter,
a
{\displaystyle a}
is the semi-major axis of the orbit's ellipse.
If the orbit is circular, of radius r, then
C
3
=
−
μ
r
{\displaystyle 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
{\displaystyle 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
=
μ
|
a
|
>
0
{\displaystyle C_{3}={\frac {\mu }{|a|}}>0}
where
μ
=
G
M
{\displaystyle \mu =GM}
is the standard gravitational parameter,
a
{\displaystyle a}
is the semi-major axis of the orbit's hyperbola (which may be negative in some convention).
Also,
C
3
=
v
∞
2
{\displaystyle C_{3}=v_{\infty }^{2}}
where
v
∞
{\displaystyle v_{\infty }}
is the asymptotic velocity at infinite distance. Spacecraft's velocity approaches
v
∞
{\displaystyle 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
12.2
km/s
=
3.5
km/s
{\displaystyle {\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.
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.
Footnotes
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- Jarak (tumbuhan)
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- Nitrogen
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- Characteristic energy
- Thermal energy
- Boltzmann constant
- Characteristic energy length scale
- Parabolic trajectory
- Specific orbital energy
- Characteristic polynomial
- Characteristic X-ray
- Characteristic velocity
- Electron capture
