 Specific orbital energy

In the gravitational twobody problem, the specific orbital energy (or visviva energy) of two orbiting bodies is the constant sum of their mutual potential energy () and their total kinetic energy (), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as visviva equation), it does not vary with time:
where
 is the relative orbital speed;
 is the orbital distance between the bodies;
 is the sum of the standard gravitational parameters of the bodies;
 is the specific relative angular momentum in the sense of relative angular momentum divided by the reduced mass;
 is the orbital eccentricity;
 is the semimajor axis.
It is expressed in J/kg = m^{2}s^{−2} or MJ/kg = km^{2}s^{−2}. For an elliptical orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity (parabolic orbit). For a hyperbolic orbit, it is equal to the excess energy compared to that of a parabolic orbit. In this case the specific orbital energy is also referred to as characteristic energy.
Contents
Equation forms for different orbits
For an elliptical orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to:^{[1]}
where
 is the standard gravitational parameter;
 is semimajor axis of the orbit.
Proof:

 For an elliptical orbit with specific angular momentum h given by

 we use the general form of the specific orbital energy equation,

 with the relation that the relative velocity at periapsis is

 Thus our specific orbital energy equation becomes

 and finally with the last simplification we obtain:
For a parabolic orbit this equation simplifies to
For a hyperbolic trajectory this specific orbital energy is either given by
or the same as for an ellipse, depending on the convention for the sign of a.
In this case the specific orbital energy is also referred to as characteristic energy (or ) and is equal to the excess specific energy compared to that for an parabolic orbit.
It is related to the hyperbolic excess velocity (the orbital velocity at infinity) by
It is relevant for interplanetary missions.
Thus, if orbital position vector () and orbital velocity vector () are known at one position, and is known, then the energy can be computed and from that, for any other position, the orbital speed.
Rate of change
For an elliptical orbit the rate of change of the specific orbital energy with respect to a change in the semimajor axis is
where
 is the standard gravitational parameter;
 is semimajor axis of the orbit.
In the case of circular orbits, this rate is one half of the gravity at the orbit. This corresponds to the fact that for such orbits the total energy is one half of the potential energy, because the kinetic energy is minus one half of the potential energy.
Additional energy
If the central body has radius R, then the additional energy of an elliptic orbit compared to being stationary at the surface is
 For the Earth and a just little more than R / 2 this is (2a − R)g ; the quantity 2a − R is the height the ellipse extends above the surface, plus the periapsis distance (the distance the ellipse extends beyond the center of the Earth); the latter times g is the kinetic energy of the horizontal component of the velocity.
Examples
ISS
The International Space Station has an orbital period of 91.74 minutes, hence the semimajor axis is 6,738 km.
The energy is −29.6 MJ/kg: the potential energy is −59.2 MJ/kg, and the kinetic energy 29.6 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 3.4 MJ/kg, the total extra energy is 33.0 MJ/kg. The average speed is 7.7 km/s, the net deltav to reach this orbit is 8.1 km/s (the actual deltav is typically 1.5–2 km/s more for atmospheric drag and gravity drag).
The increase per meter would be 4.4 J/kg; this rate corresponds to one half of the local gravity of 8.8 m/s².
For an altitude of 100 km (radius is 6471 km):
The energy is −30.8 MJ/kg: the potential energy is −61.6 MJ/kg, and the kinetic energy 30.8 MJ/kg. Compare with the potential energy at the surface, which is −62.6 MJ/kg. The extra potential energy is 1.0 MJ/kg, the total extra energy is 31.8 MJ/kg.
The increase per meter would be 4.8 J/kg; this rate corresponds to one half of the local gravity of 9.5 m/s². The speed is 7.8 km/s, the net deltav to reach this orbit is 8.0 km/s.
Taking into account the rotation of the Earth, the deltav is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).
Voyager 1
For Voyager 1, with respect to the Sun:
 = 132,712,440,018 km^{3}s^{−2} is the standard gravitational parameter of the Sun
 r = 17 billion kilometers
 v = 17.1 km/s
Hence:
 146 km^{2}s^{−2}  8 km^{2}s^{−2} = 138 km^{2}s^{−2}
Thus the hyperbolic excess velocity (the theoretical orbital velocity at infinity) is given by
 16.6 km/s
However, Voyager 1 does not have enough velocity to leave the Milky Way. The computed speed applies far away from the Sun, but at such a position that the potential energy with respect to the Milky Way as a whole has changed negligibly, and only if there is no strong interaction with celestial bodies other than the Sun.
Applying thrust
Assume:
 a is the acceleration due to thrust (the timerate at which deltav is spent)
 g is the gravitational field strength
 v is the velocity of the rocket
Then the timerate of change of the specific energy of the rocket is : an amount for the kinetic energy and an amount for the potential energy.
The change of the specific energy of the rocket per unit change of deltav is
which is v times the cosine of the angle between v and a.
Thus, when applying deltav to increase specific orbital energy, this is done most efficiently if a is applied in the direction of v, and when v is large. If the angle between v and g is obtuse, for example in a launch and in a transfer to a higher orbit, this means applying the deltav as early as possible and at full capacity. See also gravity drag. When passing by a celestial body it means applying thrust when nearest to the body. When gradually making an elliptic orbit larger, it means applying thrust each time when near the periapsis.
When applying deltav to decrease specific orbital energy, this is done most efficiently if a is applied in the direction opposite to that of v, and again when v is large. If the angle between v and g is acute, for example in a landing (on a celestial body without atmosphere) and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the deltav as late as possible. When passing by a planet it means applying thrust when nearest to the planet. When gradually making an elliptic orbit smaller, it means applying thrust each time when near the periapsis.
If a is in the direction of v:
Earth orbits
orbit centertocenter
distancealtitude above
the Earth's surfacespeed period/time in space specific orbital energy minimum suborbital spaceflight (vertical) 6,500 km 100 km 0.0 km/s just reaching space 61.3 MJ/kg ICBM up to 7,600 km up to 1,200 km 6 to 7 km/s time in space: 25 min 27.9 MJ/kg Low Earth orbit 6,600 to 8,400 km 200 to 2,000 km circular orbit: 6.9 to 7.8 km/s
elliptic orbit: 6.5 to 8.2 km/s89 to 128 min 17.0 MJ/kg Molniya orbit 6,900 to 46,300 km 500 to 39,900 km 1.5 to 10.0 km/s 11 h 58 min 4.7 MJ/kg GEO 42,000 km 35,786 km 3.1 km/s 23 h 56 min 4.6 MJ/kg Orbit of the Moon 363,000 to 406,000 km 357,000 to 399,000 km 0.97 to 1.08 km/s 27.3 days 0.5 MJ/kg See also
References
 ^ Wie, Bong (1998). "Orbital Dynamics". Space Vehicle Dynamics and Control. AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. p. 220. ISBN 1563472619.
Voyager program Concepts Voyager 1 Work Pale Blue Dot
 Family Portrait (Voyager)
 Voyager Golden Record
 Contents of the Voyager Golden Record
 Voyager: Sounds of the Cosmos
Voyager 2 Work Voyager Golden Record
 Contents of the Voyager Golden Record
 Voyager: Sounds of the Cosmos
Voyager team  Stamatios Krimigis
 Carolyn Porco
 Raymond Heacock
 Jim Blinn
 Edward C. Stone
 Timothy Ferris
Observations Categories: Celestial mechanics
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 Astrodynamics
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