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 (such as gravity), has an orbit that is a conic section (i.e. circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory, or radial trajectory) with the central body located at one of the two foci, or *the* focus (Kepler's first law).

- Central, inverse-square law force
- Low-energy trajectories
- Categorization of orbits
- See also
- Notes
- References

If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness).

Consider a two-body system consisting of a central body of mass *M* and a much smaller, orbiting body of mass , and suppose the two bodies interact via a central, inverse-square law force (such as gravitation). In polar coordinates, the orbit equation can be written as^{ [1] }

where

- is the separation distance between the two bodies and
- is the angle that makes with the axis of periapsis (also called the
*true anomaly*). - The parameter is the angular momentum of the orbiting body about the central body, and is equal to , or the mass multiplied by the magnitude of the cross product of the relative position and velocity vectors of the two bodies.
^{ [note 1] } - The parameter is the constant for which equals the acceleration of the smaller body (for gravitation, is the standard gravitational parameter, ). For a given orbit, the larger , the faster the orbiting body moves in it: twice as fast if the attraction is four times as strong.
- The parameter is the eccentricity of the orbit, and is given by
^{ [1] } - where is the energy of the orbit.

The above relation between and describes a conic section.^{ [1] } The value of controls what kind of conic section the orbit is:

- when , the orbit is elliptic (circles are ellipses with );
- when , the orbit is parabolic;
- when , the orbit is hyperbolic.

The minimum value of in the equation is :

while, if , the maximum value is :

If the maximum is less than the radius of the central body, then the conic section is an ellipse which is fully inside the central body and no part of it is a possible trajectory. If the maximum is more, but the minimum is less than the radius, part of the trajectory is possible:

- if the energy is non-negative (parabolic or hyperbolic orbit): the motion is either away from the central body, or towards it.
- if the energy is negative: the motion can be first away from the central body, up to

- after which the object falls back.

If becomes such that the orbiting body enters an atmosphere, then the standard assumptions no longer apply, as in atmospheric reentry.

If the central body is the Earth, and the energy is only slightly larger than the potential energy at the surface of the Earth, then the orbit is elliptic with eccentricity close to 1 and one end of the ellipse just beyond the center of the Earth, and the other end just above the surface. Only a small part of the ellipse is applicable.

If the horizontal speed is , then the periapsis distance is . The energy at the surface of the Earth corresponds to that of an elliptic orbit with (with the radius of the Earth), which can not actually exist because it is an ellipse fully below the surface. The energy increase with increase of is at a rate . The maximum height above the surface of the orbit is the length of the ellipse, minus , minus the part "below" the center of the Earth, hence twice the increase of minus the periapsis distance. At the top^{[ of what? ]} the potential energy is times this height, and the kinetic energy is . This adds up to the energy increase just mentioned. The width of the ellipse is 19 minutes^{[ why? ]} times .

The part of the ellipse above the surface can be approximated by a part of a parabola, which is obtained in a model where gravity is assumed constant. This should be distinguished from the parabolic orbit in the sense of astrodynamics, where the velocity is the escape velocity.

See also trajectory.

Consider orbits which are at one point horizontal, near the surface of the Earth. For increasing speeds at this point the orbits are subsequently:

- part of an ellipse with vertical major axis, with the center of the Earth as the far focus (throwing a stone, sub-orbital spaceflight, ballistic missile)
- a circle just above the surface of the Earth (Low Earth orbit)
- an ellipse with vertical major axis, with the center of the Earth as the near focus
- a parabola
- a hyperbola

Note that in the sequence above^{[ where? ]}, , and increase monotonically, but first decreases from 1 to 0, then increases from 0 to infinity. The reversal is when the center of the Earth changes from being the far focus to being the near focus (the other focus starts near the surface and passes the center of the Earth). We have

Extending this to orbits which are horizontal at another height, and orbits of which the extrapolation is horizontal below the surface of the Earth, we get a categorization of all orbits, except the radial trajectories, for which, by the way, the orbit equation can not be used. In this categorization ellipses are considered twice, so for ellipses with both sides above the surface one can restrict oneself to taking the side which is lower as the reference side, while for ellipses of which only one side is above the surface, taking that side.

- ↑ There is a related parameter, known as the specific relative angular momentum, . It is related to by .

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 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 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.

The **orbital 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 planets, exoplanets orbiting other stars, or binary stars.

In astronautics, the **Hohmann transfer orbit** is an orbital maneuver used to transfer a spacecraft between two orbits of different altitudes around a central body. Examples would be used for travel between low Earth orbit and the Moon, or another solar planet or asteroid. In the idealized case, the initial and target orbits are both circular and coplanar. The maneuver is accomplished by placing the craft into an elliptical transfer orbit that is tangential to both the initial and target orbits. The maneuver uses two impulsive engine burns: the first establishes the transfer orbit, and the second adjusts the orbit to match the target.

**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 the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

In gravitationally bound systems, the **orbital speed** of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one body is much more massive than the other bodies of the system combined, its speed relative to the center of mass of the most massive body.

In celestial mechanics, the **mean anomaly** is the fraction of an elliptical orbit's period that has elapsed since the orbiting body passed periapsis, expressed as an angle which can be used in calculating the position of that body in the classical two-body problem. It is the angular distance from the pericenter which a fictitious body would have if it moved in a circular orbit, with constant speed, in the same orbital period as the actual body in its elliptical orbit.

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 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(m_{1}+m_{2}), or as GM when one body is much larger than the other.

In astrodynamics or celestial mechanics, a **hyperbolic trajectory** or **hyperbolic orbit** 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.

In the gravitational two-body problem, the **specific orbital 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, it does not vary with time:

In astrodynamics, the ** vis-viva equation**, also referred to as

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 between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a rosette orbit through the Galaxy.

**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 astronautics and aerospace engineering, the **bi-elliptic transfer** is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver.

In celestial mechanics, a **Kepler orbit** is the motion of one body relative to another, as an ellipse, parabola, or hyperbola, which forms a two-dimensional orbital plane in three-dimensional space. A Kepler orbit can also form a straight line. It considers only the point-like gravitational attraction of two bodies, neglecting perturbations due to gravitational interactions with other objects, atmospheric drag, solar radiation pressure, a non-spherical central body, and so on. It is thus said to be a solution of a special case of the two-body problem, known as the Kepler problem. As a theory in classical mechanics, it also does not take into account the effects of general relativity. Keplerian orbits can be parametrized into six orbital elements in various ways.

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 astrodynamics and celestial mechanics a **radial trajectory ** is a Kepler orbit with zero angular momentum. Two objects in a radial trajectory move directly towards or away from each other in a straight line.

- 1 2 3 Fetter, Alexander; Walecka, John (2003).
*Theoretical Mechanics of Particles and Continua*. Dover Publications. pp. 13–22.

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