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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.^{ [1] }^{ [2] }

Define T as the time required for a particular body to complete one orbit. In time T, the radius vector sweeps out 2π radians, or 360°. The average rate of sweep, n, is then

which is called the * mean angular motion * of the body, with dimensions of radians per unit time or degrees per unit time.

Define τ as the time at which the body is at the pericenter. From the above definitions, a new quantity, M, the *mean anomaly* can be defined

which gives an angular distance from the pericenter at arbitrary time t.^{ [3] } with dimensions of radians or degrees.

Because the rate of increase, n, is a constant average, the mean anomaly increases uniformly (linearly) from 0 to 2π radians or 0° to 360° during each orbit. It is equal to 0 when the body is at the pericenter, π radians (180°) at the apocenter, and 2π radians (360°) after one complete revolution.^{ [4] } If the mean anomaly is known at any given instant, it can be calculated at any later (or prior) instant by simply adding (or subtracting) n⋅δt where δt represents the small time difference.

Mean anomaly does not measure an angle between any physical objects (except at pericenter or apocenter, or for a circular orbit). It is simply a convenient uniform measure of how far around its orbit a body has progressed since pericenter. The mean anomaly is one of three angular parameters (known historically as "anomalies") that define a position along an orbit, the other two being the eccentric anomaly and the true anomaly.

The mean anomaly M can be computed from the eccentric anomaly E and the eccentricity e with Kepler's Equation:

Mean anomaly is also frequently seen as

where M_{0} is the *mean anomaly at epoch* and t_{0} is the * epoch *, a reference time to which the orbital elements are referred, which may or may not coincide with τ, the time of pericenter passage. The classical method of finding the position of an object in an elliptical orbit from a set of orbital elements is to calculate the mean anomaly by this equation, and then to solve Kepler's equation for the eccentric anomaly.

Define ϖ as the * longitude of the pericenter *, the angular distance of the pericenter from a reference direction. Define ℓ as the * mean longitude *, the angular distance of the body from the same reference direction, assuming it moves with uniform angular motion as with the mean anomaly. Thus mean anomaly is also^{ [5] }

Mean angular motion can also be expressed,

where μ is a gravitational parameter which varies with the masses of the objects, and a is the semi-major axis of the orbit. Mean anomaly can then be expanded,

and here mean anomaly represents uniform angular motion on a circle of radius a .^{ [6] }

Mean anomaly can be calculated from the eccentricity and the true anomaly f by finding the eccentric anomaly and then using Kepler's equation. This gives, in radians:

where atan2(y, x) is the angle from the x-axis of the ray from (0, 0) to (x, y). (Note that the arguments are often reversed.)

For parabolic and hyperbolic trajectories the mean anomaly is not defined, because they don't have a period. But in those cases, as with elliptical orbits, the area swept out by a chord between the attractor and the object following the trajectory increases linearly with time. For the hyperbolic case, there is a formula similar to the above giving the elapsed time as a function of the angle (the true anomaly in the elliptic case), as explained in the article Kepler orbit. For the parabolic case there is a different formula, the limiting case for either the elliptic or the hyperbolic case as the distance between the foci goes to infinity – see Parabolic trajectory#Baker's equation.

Mean anomaly can also be expressed as a series expansion:^{ [7] }

- with

A similar formula gives the true anomaly directly in terms of the mean anomaly:^{ [8] }

A general formulation of the above equation can be written as the equation of the centre : ^{ [9] }

In mathematics, an **ellipse** is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the special type of ellipse in which the two focal points are the same. The elongation of an ellipse is measured by its eccentricity , a number ranging from to .

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 mechanics and physics, **simple harmonic motion** is a special type of periodic motion where the restoring force on the moving object is directly proportional to the magnitude of the object's displacement and acts towards the object's equilibrium position. It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.

**Tidal acceleration** is an effect of the tidal forces between an orbiting natural satellite and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller body first, and later the larger body. The Earth–Moon system is the best-studied case.

**Orbital elements** are the parameters required to uniquely identify a specific orbit. In celestial mechanics these elements are considered in two-body systems using a Kepler orbit. There are many different ways to mathematically describe the same orbit, but certain schemes, each consisting of a set of six parameters, are commonly used in astronomy and orbital mechanics.

**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.

The **Gaussian gravitational constant** is a parameter used in the orbital mechanics of the Solar System. It relates the orbital period to the orbit's semi-major axis and the mass of the orbiting body in Solar masses.

In orbital mechanics, the **eccentric anomaly** is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit. The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the true anomaly and the mean anomaly.

In celestial mechanics, **true anomaly** is an angular parameter that defines the position of a body moving along a Keplerian orbit. It is the angle between the direction of periapsis and the current position of the body, as seen from the main focus of the ellipse.

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 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 orbital mechanics, **mean motion** is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

**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 two-body, Keplerian orbital mechanics, the **equation of the center** is the angular difference between the actual position of a body in its elliptical orbit and the position it would occupy if its motion were uniform, in a circular orbit of the same period. It is defined as the difference true anomaly, *ν*, minus mean anomaly, *M*, and is typically expressed a function of mean anomaly, *M*, and orbital eccentricity, *e*.

A **pendulum** is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.

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 orbital mechanics, **Kepler's equation** relates various geometric properties of the orbit of a body subject to a central force.

- ↑ Montenbruck, Oliver (1989).
*Practical Ephemeris Calculations*. Springer-Verlag. p. 44. ISBN 0-387-50704-3. - ↑ Meeus, Jean (1991).
*Astronomical Algorithms*. Willmann-Bell, Inc., Richmond, VA. p. 182. ISBN 0-943396-35-2. - ↑ Smart, W. M. (1977).
*Textbook on Spherical Astronomy*(sixth ed.). Cambridge University Press, Cambridge. p. 113. ISBN 0-521-29180-1. - ↑ Meeus (1991), p. 183
- ↑ Smart (1977), p. 122
- ↑ Vallado, David A. (2001).
*Fundamentals of Astrodynamics and Applications*(2nd ed.). El Segundo, CA: Microcosm Press. pp. 53–54. ISBN 1-881883-12-4. - ↑ Smart, W. M. (1953).
*Celestial Mechanics*. London, UK: Longmans, Green, and Co. p. 38. - ↑ Roy, A.E. (1988).
*Orbital Motion*(1st ed.). Bristol, UK; Philadelphia, PA: A. Hilger. ISBN 0852743602. - ↑ Brouwer, Dirk (1961).
*Methods of celestial mechanics*. Elsevier. pp. e.g. 77.

- Glossary entry
*anomaly, mean*at the US Naval Observatory's*Astronomical Almanac Online*

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