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Encyclopedia > Specific orbital energy

In astrodynamics the specific orbital energy $epsilon,!$ (or vis-viva energy) of an orbiting body traveling through space under standard assumptions is the sum of its potential energy ($epsilon_p,!$) and kinetic energy ($epsilon_k,!$) 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: Astrodynamics is the study of the motion of rockets, missiles, and space vehicles, as determined from Sir Isaac Newtons laws of motion and his law of universal gravitation. ... In physics, an orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity. ... In astrodynamics, an orbiting body () is a body that orbits central body (). Under standard assumptions in astrodynamics: it is orders of magnitude lighter than central body (i. ... Space is a general or specialized concept of a local, relative, containing, or otherwise relevant area â€”where all objects within have a relationship with (the) space which follows various (theoretically) defineable rules. ... For most of the problems in astrodynamics involving two bodies and standard assumptions are usually the following: A1: and are the only objects in the universe and thus influence of other objects is disregarded, A2: The orbiting body () is far smaller than central body (), i. ... Potential energy is stored energy. ... Kinetic energy is energy that a body possesses as a result of its motion. ... In astrodynamics vis-viva equation (also referred to as orbital energy conservation equation) is one of the fundamental and useful equations that govern the motion of orbiting bodies. ... A trajectory is an imagined trace of positions followed by an object moving through space. ...

$epsilon=epsilon_k+epsilon_p={v^2over{2}}-{muover{r}} =-{1over{2}}{mu^2over{h^2}}left(1-e^2right)$

where:

It is expressed in J/kg = m2s-2 or MJ/kg = km2s-2. The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ... In astrodynamics or celestial dynamics orbital state vectors (sometimes State Vectors) are vectors of position () and velocity () that together with their time () ( epoch) uniquely determine the state of an orbiting body. ... In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting... In astrodynamics specific relative angular momentum () of orbiting body () relative to central body () is the relative angular momentum of per unit mass. ... In astrodynamics, under standard assumptions any orbit must be of conic section shape. ... Look up second in Wiktionary, the free dictionary. ... A kilometre (American spelling: kilometer), symbol: km is a unit of length in the metric system equal to 1000 metres (from the Greek words Ï‡Î¯Î»Î¹Î± (khilia) = thousand and Î¼Î­Ï„ÏÎ¿ (metro) = count/measure). ... Look up second in Wiktionary, the free dictionary. ...

## Equation forms for different orbits GA_googleFillSlot("encyclopedia_square");

For an elliptical orbit specific orbital energy equation simplifies to: In astrodynamics or celestial mechanics a elliptic orbit is an orbit with the eccentricity greater than 0 and less than 1. ...

$epsilon = -{mu over{2a}},!$

where:

• $mu,!$ is the standard gravitational parameter
• $a,!$ is semi-major axis of the orbiting body

For a parabolic orbit this equation simplifies to: In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ... In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. ...

$epsilon=0,!$

For a hyperbolic trajectory this specific orbital energy equation takes form: In astrodynamics or celestial mechanics a hyperbolic trajectory is an orbit with the eccentricity greater than 1. ...

$epsilon = {mu over{2a}},!$

In this case the specific orbital energy is also referred to as characteristic energy (or $C_3,!$) and is equal to the excess specific energy compared to that for an escape orbit (parabolic orbit). In astrodynamics a characteristic energy () is a measure of the energy required for an interplanetary mission that requires attaining an excess orbital velocity over an escape velocity required for additional orbital maneuvers. ... An escape orbit (also known as C3 = 0 orbit) is the high-energy parabolic orbit around the central body. ... In astrodynamics or celestial mechanics a parabolic trajectory is an orbit with the eccentricity equal to 1. ...

It is related to the hyperbolic excess velocity $v_{infty} ,!$ (the orbital velocity at infinity) by The orbital speed of a body, generally a planet, a natural satellite, an artificial satellite, or a multiple star, is the speed at which it orbits around the barycenter of a system, usually around a more massive body. ...

$2epsilon=C_3=v_{infty}^2,!$

It is relevant for interplanetary missions.

Thus, if orbital position vector ($mathbf{r},!$) and orbital velocity vector ($mathbf{v},!$) are known at one position, and $mu,!$ is known, then the energy can be computed and from that, for any other position, the orbital speed. In astrodynamics or celestial dynamics orbital state vectors (sometimes State Vectors) are vectors of position () and velocity () that together with their time () ( epoch) uniquely determine the state of an orbiting body. ... In astrodynamics or celestial dynamics orbital state vectors (sometimes State Vectors) are vectors of position () and velocity () that together with their time () ( epoch) uniquely determine the state of an orbiting body. ...

## 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:

$frac{mu}{2a^2},!$

where:

• $mu,!$ is the standard gravitational parameter
• $a,!$ is semi-major axis of the orbiting body

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. In astrodynamics, the standard gravitational parameter () of a celestial body is the product of the gravitational constant () and the mass : The units of the standard gravitational parameter are km3s-2 Small body orbiting a central body Under standard assumptions in astrodynamics we have: where: is the mass of the orbiting... The semi-major axis of an ellipse In geometry, the term semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolas. ...

If the central body has radius R, then the additional energy of an elliptic orbit compared to being stationary at the surface is

$-frac{mu}{2a}+frac{mu}{R} = frac{mu (2a-R)}{2aR}$

• For the Earth and a just little more than R / 2 this is (2aR)g ; 2aR 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

The International Space Station has an orbital period of 91.74 minutes, hence the semi-major axis is 6738 km [1]. ISS Statistics Crew: 2 As of August 21, 2005 Perigee: 352. ...

The energy is −29.6 MJ/kg [2]: 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). General In general physics delta-v is simply the change in velocity. ... Atmospheric drag is a form of drag, which is the force that opposes an object moving through a liquid or gas. ... In astrodynamics, gravity drag is inefficiency encountered by a spacecraft thrusting while moving against a gravitational field. ...

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 [3].

For an altitude of 100 km (radius is 6471 km) these figures are:

The energy is −30.8 MJ/kg [4]: 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 [5], the net delta-v to reach this orbit is 8.0 km/s [6].

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).

## Applying thrust

Assume:

• 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 $mathbf{v} cdot mathbf{a}$: an amount $mathbf{v} cdot (mathbf{a}-mathbf{g})$ for the kinetic energy and an amount $mathbf{v} cdot mathbf{g}$ for the potential energy. Thrust is a reaction force described quantitatively by Newtons Second and Third Law. ... General In general physics delta-v is simply the change in velocity. ...

The change of the specific energy of the rocket per unit change of delta-v is

$frac{mathbf{v cdot a}}{|mathbf{a}|}$

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. In astrodynamics, gravity drag is inefficiency encountered by a spacecraft thrusting while moving against a gravitational field. ...

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:

$Delta epsilon = int v, d (Delta v) = int v, a dt$

See also Specific energy change of rockets. Tsiolkovskys rocket equation, named after Konstantin Tsiolkovsky who independently derived it, considers the principle of a rocket: a device that can apply an acceleration to itself (a thrust) by expelling part of its mass with high speed in the opposite direction, due to the conservation of momentum. ...

Results from FactBites:

 Energy (2555 words) Energy can be in several forms: mechanical potential—due to possible physical interactions with other objects (for example, gravitational potential energy); kinetic—contained in macroscopic motion; chemical—potential stored in chemical bonds between atoms; electrical—potential due to possible charge interactions; thermal—contained in the kinetic energy of individual molecules; nuclear—potential stored between constituents of nuclei. Similarly, gravitational potential energy is converted into the kinetic energy of moving water (and a turbine) in a dam, which in turn is transformed into electric energy by a generator. This mathematical relationship demonstrates the direct connection between force and potential energy: the force between two objects is in the direction of decreasing potential energy, and the magnitude of the force is proportional to the extent to which potential energy decreases.
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