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In the gravitational two-body problem, the specific orbital energy \varepsilon (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
(\varepsilon_p) and their total
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
(\varepsilon_k), divided by the
reduced mass In physics, the reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. It is a quantity which allows the two-body problem to be solved as if it were a one-body problem. Note, however, that the mass ...
. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time: \begin \varepsilon &= \varepsilon_k + \varepsilon_p \\ &= \frac - \frac = -\frac \frac \left(1 - e^2\right) = -\frac \end where *v is the relative
orbital speed In gravitationally bound systems, the orbital speed of an astronomical body or object (e.g. planet, moon, artificial satellite, spacecraft, or star) is the speed at which it orbits around either the barycenter or, if one body is much more mas ...
; *r is the orbital distance between the bodies; *\mu = (m_1 + m_2) is the sum of the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
s of the bodies; *h is the
specific relative angular momentum In celestial mechanics, the specific relative angular momentum (often denoted \vec or \mathbf) of a body is the angular momentum of that body divided by its mass. In the case of two orbiting bodies it is the vector product of their relative positi ...
in the sense of relative angular momentum divided by the reduced mass; *e is the
orbital eccentricity In astrodynamics, the orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values bet ...
; *a is the
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
. It is expressed in MJ/kg or \frac. For an
elliptic orbit In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
the specific orbital energy is the negative of the additional energy required to accelerate a mass of one kilogram to
escape velocity In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non- propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically ...
( 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.


Equation forms for different orbits

For an
elliptic orbit In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...
, the specific orbital energy equation, when combined with conservation of specific angular momentum at one of the orbit's
apsides An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion. General description There are two apsides in any ellip ...
, simplifies to: \varepsilon = -\frac where *\mu = G\left(m_1 + m_2\right) is the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
; *a is
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
of the orbit. For a parabolic orbit this equation simplifies to \varepsilon = 0. For a hyperbolic trajectory this specific orbital energy is either given by \varepsilon = . 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 C_3) and is equal to the excess specific energy compared to that for a parabolic orbit. It is related to the hyperbolic excess velocity v_\infty (the orbital velocity at infinity) by 2\varepsilon = C_3 = v_\infty^2. It is relevant for interplanetary missions. Thus, if orbital position vector (\mathbf) and orbital velocity vector (\mathbf) 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.


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 \frac where * \mu=(m_1 + m_2) is the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
; *a\,\! is
semi-major axis In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the lon ...
of the orbit. 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 -\frac+\frac = \frac 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). For the Earth and a just little more than R the additional specific energy is (gR/2); which is the kinetic energy of the horizontal component of the velocity, i.e. \fracV^2 = \fracgR, V=\sqrt.


Examples


ISS

The
International Space Station The International Space Station (ISS) is the largest Modular design, modular space station currently in low Earth orbit. It is a multinational collaborative project involving five participating space agencies: NASA (United States), Roscosmos ( ...
has an
orbital period The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting pla ...
of 91.74 minutes (5504s), hence by
Kepler's Third Law In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler between 1609 and 1619, describe the orbits of planets around the Sun. The laws modified the heliocentric theory of Nicolaus Copernicus, replacing its circular orbi ...
the semi-major axis of its orbit is 6,738km. The energy is −29.6MJ/kg: the potential energy is −59.2MJ/kg, and the kinetic energy 29.6MJ/kg. Compare with the potential energy at the surface, which is −62.6MJ/kg. The extra potential energy is 3.4MJ/kg, the total extra energy is 33.0MJ/kg. The average speed is 7.7km/s, the net
delta-v Delta-''v'' (more known as " change in velocity"), symbolized as ∆''v'' and pronounced ''delta-vee'', as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such a ...
to reach this orbit is 8.1km/s (the actual delta-v is typically 1.5–2.0km/s more for
atmospheric drag In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
and gravity drag). 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 (radius is 6471km): 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 (starting at the equator and going east) or more (if going west).


''Voyager 1''

For ''
Voyager 1 ''Voyager 1'' is a space probe launched by NASA on September 5, 1977, as part of the Voyager program to study the outer Solar System and interstellar space beyond the Sun's heliosphere. Launched 16 days after its twin '' Voyager 2'', ''V ...
'', with respect to the Sun: *\mu = GM = 132,712,440,018 km3⋅s−2 is the
standard gravitational parameter In celestial mechanics, the standard gravitational parameter ''μ'' of a celestial body is the product of the gravitational constant ''G'' and the mass ''M'' of the bodies. For two bodies the parameter may be expressed as G(m1+m2), or as GM whe ...
of the Sun *''r'' = 17
billion Billion is a word for a large number, and it has two distinct definitions: *1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English. * 1,000,000,000,000, i. ...
kilometers *''v'' = 17.1 km/s Hence: \varepsilon = \varepsilon_k + \varepsilon_p = \frac - \frac = \mathrm - \mathrm = \mathrm Thus the hyperbolic excess velocity (the theoretical orbital velocity at infinity) is given by v_\infty = \mathrm However, ''Voyager 1'' does not have enough velocity to leave the
Milky Way The Milky Way is the galaxy that includes our Solar System, with the name describing the galaxy's appearance from Earth: a hazy band of light seen in the night sky formed from stars that cannot be individually distinguished by the naked eye. ...
. 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 Thrust is a reaction force described quantitatively by Newton's third law. When a system expels or accelerates mass in one direction, the accelerated mass will cause a force of equal magnitude but opposite direction to be applied to that ...
(the time-rate at which
delta-v Delta-''v'' (more known as " change in velocity"), symbolized as ∆''v'' and pronounced ''delta-vee'', as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such a ...
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 \cdot \mathbf: an amount \mathbf \cdot (\mathbf-\mathbf) for the kinetic energy and an amount \mathbf \cdot \mathbf for the potential energy. The change of the specific energy of the rocket per unit change of delta-v is \frac 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: \Delta \varepsilon = \int v\, d (\Delta v) = \int v\, a dt


See also

* Specific energy change of rockets * Characteristic energy C3 (Double the specific orbital energy)


References

{{Voyager program Astrodynamics Orbits