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Specific orbital energy
In astrodynamics the specific orbital energy (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy () and kinetic energy () per unit mass. According to the orbital energy conservation equation (also referred to as vis-viva equation) it is the same at all points of the trajectory:
- is the orbital speed of the orbiting body
- is the orbital position of the orbiting body
- is the standard gravitational parameter
- is the specific relative angular momentum of the orbiting body
- is the orbit eccentricity
Equation forms for different orbits
For an elliptical orbit specific orbital energy equation simplifies to:
For a parabolic orbit this equation simplifies to:
For a hyperbolic trajectory this specific orbital energy equation takes form:
It is related to the hyperbolic excess velocity (the orbital velocity at infinity) by
It is relevant for interplanetary missions.
Rate of change
For an elliptical orbit the rate of change of the specific orbital energy with respect to a change in the semi-major axis is:
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.
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 this is (2a - R)g ; 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.
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 delta-v to reach this orbit is 8.1 km/s (the actual delta-v 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) these figures are:
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.
Taking into account the rotation of the Earth, the delta-v is up to 0.46 km/s less (starting at the equator and going east) or more (if going west).
- a is the acceleration due to thrust (the time-rate at which delta-v is spent)
- g is the gravitational field strength
- v is the velocity of the rocket
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.
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 (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 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:
See also Specific energy change of rockets.
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