Specific orbital energy


In the gravitational two-body problem, the specific orbital energy of two orbiting bodies is the constant quotient of their mechanical energy to their reduced mass.
According to the orbital energy conservation equation, it does not vary with time:
where
It is a kind of specific energy, typically expressed in units of or . For an elliptic orbit the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to escape velocity. 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.

Equation forms for different orbits

For an elliptic orbit, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's apsides, simplifies to:
where
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 and is equal to the excess specific energy compared to that for a parabolic orbit.
It is related to the hyperbolic excess velocity 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 elliptic orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is
where
In the case of circular orbits, this rate is one half of the gravitation 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 specific energy of an elliptic orbit compared to being stationary at the surface is
The quantity is the height the ellipse extends above the surface, plus the periapsis distance. For the Earth and just little more than the additional specific energy is ; which is the kinetic energy of the horizontal component of the velocity, i.e., .

Examples

ISS

The International Space Station has an orbital period of 91.74 minutes, hence by Kepler's Third Law the semi-major axis of its orbit is 6,738km.
The specific orbital energy associated with this orbit is −29.6MJ/kg: the potential energy is −59.2MJ/kg, and the kinetic energy 29.6MJ/kg. Compared with the potential energy at the surface, which is −62.6MJ/kg., the extra potential energy is 3.4MJ/kg, and the total extra energy is 33.0MJ/kg. The average speed is 7.7km/s, the net delta-v to reach this orbit is 8.1km/s.
The increase per meter would be 4.4J/kg; this rate corresponds to one half of the local gravity of 8.8m/s2.
For an altitude of 100km :
The energy is −30.8MJ/kg: the potential energy is −61.6MJ/kg, and the kinetic energy 30.8MJ/kg. Compare with the potential energy at the surface, which is −62.6MJ/kg. The extra potential energy is 1.0MJ/kg, the total extra energy is 31.8MJ/kg.
The increase per meter would be 4.8J/kg; this rate corresponds to one half of the local gravity of 9.5m/s2. The speed is 7.8km/s, the net delta-v to reach this orbit is 8.0km/s.
Taking into account the rotation of the Earth, the delta-v is up to 0.46km/s less or more.

''Voyager 1''

For Voyager 1, with respect to the Sun:
Hence:
Thus the hyperbolic excess velocity is given by
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:
Then the time-rate 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 delta-v is
which is |v| times the cosine of the angle between v and a.
Thus, when applying delta-v 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 delta-v 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. Such maneuver is called an Oberth maneuver or powered flyby.
When applying delta-v 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 and in a transfer to a circular orbit around a celestial body when arriving from outside, this means applying the delta-v 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: