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In the gravitational two-body problem, the **specific orbital energy** (or **vis-viva 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 vis-viva equation), it does not vary with time:

- Equation forms for different orbits
- Rate of change
- Additional energy
- Examples
- ISS
- Voyager 1
- Applying thrust
- Tangential velocities at altitude
- See also
- References

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 semi-major axis.

It is expressed in J/kg = m^{2}⋅s^{−2} or MJ/kg = km^{2}⋅s^{−2}. 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 (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.

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:^{ [1] }

where

- is the standard gravitational parameter;
- is semi-major axis of the orbit.

For an elliptic 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 a 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.

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

- is the standard gravitational parameter;
- is semi-major axis 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.

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 (the distance the ellipse extends beyond the center of the Earth). 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. , .

The International Space Station has an orbital period of 91.74 minutes (5504 s), hence by Kepler's Third Law the semi-major axis of its orbit is 6,738 km.^{[ citation needed ]}

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 1.5–2.0 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/s^{2}.

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/s^{2}. The speed is 7.8 km/s, the net delta-v to reach this orbit is 8.0 km/s.

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).

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:

Thus the hyperbolic excess velocity (the theoretical orbital velocity at infinity) 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.

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 : 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**:

Orbit | Center-to-center distance | Altitude above the Earth's surface | Speed | Orbital period | Specific orbital energy |
---|---|---|---|---|---|

Earth's own rotation at surface (for comparison— not an orbit) | 6,378 km | 0 km | 465.1 m/s (1,674 km/h or 1,040 mph) | 23 h 56 min 4.09 sec | −62.6 MJ/kg |

Orbiting at Earth's surface (equator) theoretical | 6,378 km | 0 km | 7.9 km/s (28,440 km/h or 17,672 mph) | 1 h 24 min 18 sec | −31.2 MJ/kg |

Low Earth orbit | 6,600–8,400 km | 200–2,000 km | - Circular orbit: 6.9–7.8 km/s (24,840–28,080 km/h or 14,430–17,450 mph) respectively
- Elliptic orbit: 6.5–8.2 km/s respectively
| 1 h 29 min – 2 h 8 min | −29.8 MJ/kg |

Molniya orbit | 6,900–46,300 km | 500–39,900 km | 1.5–10.0 km/s (5,400–36,000 km/h or 3,335–22,370 mph) respectively | 11 h 58 min | −4.7 MJ/kg |

Geostationary | 42,000 km | 35,786 km | 3.1 km/s (11,600 km/h or 6,935 mph) | 23 h 56 min 4.09 sec | −4.6 MJ/kg |

Orbit of the Moon | 363,000–406,000 km | 357,000–399,000 km | 0.97–1.08 km/s (3,492–3,888 km/h or 2,170–2,416 mph) respectively | 27.27 days | −0.5 MJ/kg |

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

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 orbits and epicycles with elliptical trajectories, and explaining how planetary velocities vary. The three laws state that:

- The orbit of a planet is an ellipse with the Sun at one of the two foci.
- A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

In celestial mechanics, an **orbit** is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

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.

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 stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account the fact that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

An **electric field** is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field for a system of charged particles. Electric fields originate from electric charges, or from time-varying magnetic fields. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

**Noether's theorem** or **Noether's first theorem** states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

In mathematics, the **Laplace operator** or **Laplacian** is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols , , or . In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian Δ*f* (*p*) of a function *f* at a point *p* measures by how much the average value of *f* over small spheres or balls centered at *p* deviates from *f* (*p*).

**Orbital mechanics** or **astrodynamics** is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

In the general theory of relativity, the **Einstein field equations** relate the geometry of spacetime to the distribution of matter within it.

In astrodynamics or celestial mechanics a **parabolic trajectory** is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an **escape orbit**, otherwise a **capture orbit**. It is also sometimes referred to as a **C _{3} = 0 orbit** (see Characteristic energy).

In astrodynamics or celestial mechanics, a **hyperbolic trajectory** is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics, the **characteristic energy** is a measure of the excess specific energy over that required to just barely escape from a massive body. The units are length^{2} time^{−2}, i.e. velocity squared, or energy per mass.

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, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

A **circular orbit** is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.

In astrodynamics an **orbit equation** defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or *the* focus.

A **classical field theory** is a physical theory that predicts how one or more physical fields interact with matter through **field equations**, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories. In most contexts, 'classical field theory' is specifically meant to describe electromagnetism and gravitation, two of the fundamental forces of nature.

**Spacecraft flight dynamics** is the application of mechanical dynamics to model how the external forces acting on a space vehicle or spacecraft determine its flight path. These forces are primarily of three types: propulsive force provided by the vehicle's engines; gravitational force exerted by the Earth and other celestial bodies; and aerodynamic lift and drag.

In electrodynamics, the **Larmor formula** is used to calculate the total power radiated by a nonrelativistic point charge as it accelerates. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.

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** is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The **semi-minor axis** of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle.

In theoretical physics, **relativistic Lagrangian mechanics** is Lagrangian mechanics applied in the context of special relativity and general relativity.

- ↑ Wie, Bong (1998). "Orbital Dynamics" .
*Space Vehicle Dynamics and Control*. AIAA Education Series. Reston, Virginia: American Institute of Aeronautics and Astronautics. p. 220. ISBN 1-56347-261-9.

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