Celestial mechanics in the context of Reference plane

In celestial mechanics, an orbit is the curved trajectory of an object under the influence of an attracting force. Known as an orbital revolution, examples include the trajectory of a planet around a star, a natural satellite around a planet, or 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.

For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion.

Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle' and μέτρον (métron) 'measure') is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.

Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.

Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia or Giuseppe Ludovico De la Grange Tournier; 25 January 1736 – 10 April 1813), also reported as Giuseppe Luigi Lagrange or Lagrangia, was an Italian and naturalized French mathematician, physicist and astronomer. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics.

In 1766, on the recommendation of Leonhard Euler and d'Alembert, Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, Prussia, where he stayed for over twenty years, producing many volumes of work and winning several prizes of the French Academy of Sciences. Lagrange's treatise on analytical mechanics (Mécanique analytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1788–89), which was written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Isaac Newton and formed a basis for the development of mathematical physics in the nineteenth century.

In celestial mechanics, escape velocity or escape speed is the minimum speed needed for an object to escape from contact with or orbit of a primary body, assuming:

• Ballistic trajectory – no other forces are acting on the object, such as propulsion and friction
• No other gravity-producing objects exist.

Although the term escape velocity is common, it is more accurately described as a speed than as a velocity because it is independent of direction. Because gravitational force between two objects depends on their combined mass, the escape speed also depends on mass. For artificial satellites and small natural objects, the mass of the object makes a negligible contribution to the combined mass, and so is often ignored.

Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the nature of the heavenly bodies, rather than their positions or motions in space—what they are, rather than where they are", which is studied in celestial mechanics.

Among the subjects studied are the Sun (solar physics), other stars, galaxies, extrasolar planets, the interstellar medium, and the cosmic microwave background. Emissions from these objects are examined across all parts of the electromagnetic spectrum, and the properties examined include luminosity, density, temperature, and chemical composition. Because astrophysics is a very broad subject, astrophysicists apply concepts and methods from many disciplines of physics, including classical mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.

In celestial mechanics, the Lagrange points (/ləˈɡrɑːndʒ/), also called the Lagrangian points or libration points, are points of equilibrium for small-mass objects under the gravitational influence of two massive orbiting bodies. Mathematically, this involves the solution of the restricted three-body problem.

Normally, the two massive bodies exert an unbalanced gravitational force at a point, altering the orbit of whatever is at that point. At the Lagrange points, the gravitational forces of the two large bodies and the centrifugal force balance each other. This can make Lagrange points an excellent location for satellites, as orbit corrections, and hence fuel requirements, needed to maintain the desired orbit are kept at a minimum.

In astrodynamics or celestial mechanics, an elliptical orbit or eccentric orbit is an orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. Some orbits have been referred to as "elongated orbits" if the eccentricity is "high" but that is not an explanatory term. For the simple two body problem, all orbits are ellipses.

In a gravitational two-body problem, both bodies follow similar elliptical orbits with the same orbital period around their common barycenter. The relative position of one body with respect to the other also follows an elliptic orbit.

A planetary-mass object (PMO), planemo, or planetary body (sometimes referred to as a world) is, by geophysical definition of celestial objects, any celestial object massive enough to achieve hydrostatic equilibrium and assume an ellipsoid shape, but not enough to sustain core fusion like a star.

The purpose of this term is to classify together a broader range of celestial objects than just "planet", since many objects similar in geophysical terms do not conform to conventional astrodynamic expectations for a planet. Planetary-mass objects can be quite diverse in origin and location, and include planets, dwarf planets, planetary-mass moons and free-floating planets, which may have been ejected from a system (rogue planets) or formed through cloud-collapse rather than accretion (sub-brown dwarfs).

In astronomy, an epoch or reference epoch is a moment in time used as a reference point for some time-varying astronomical quantity. It is useful for the celestial coordinates or orbital elements of a celestial body, as they are subject to perturbations and vary with time. These time-varying astronomical quantities might include, for example, the mean longitude or mean anomaly of a body, the node of its orbit relative to a reference plane, the direction of the apogee or aphelion of its orbit, or the size of the major axis of its orbit.

The main use of astronomical quantities specified in this way is to calculate other relevant parameters of motion, in order to predict future positions and velocities. The applied tools of the disciplines of celestial mechanics or its subfield orbital mechanics (for predicting orbital paths and positions for bodies in motion under the gravitational effects of other bodies) can be used to generate an ephemeris, a table of values giving the positions and velocities of astronomical objects in the sky at a given time or times.

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and are widely used for studying periodic phenomena through Fourier analysis.

The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant, the secant, and the cotangent functions, which are less used. Each of these six trigonometric functions has a corresponding inverse function and has an analog among the hyperbolic functions.

In celestial mechanics, a Kepler orbit (or Keplerian orbit, named after the German astronomer Johannes Kepler) 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 most applications, there is a large central body, the center of mass of which is assumed to be the center of mass of the entire system. By decomposition, the orbits of two objects of similar mass can be described as Kepler orbits around their common center of mass, their barycenter.

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

Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbital plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers.

A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.In this case, not only the distance, but also the speed, angular speed, potential and kinetic energy are constant. There is no periapsis or apoapsis. This orbit has no radial version.

Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the orbital plane.

In astrodynamics or celestial mechanics a parabolic trajectory is a Kepler orbit with the eccentricity (e) 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 ${\displaystyle C_{3}=0}$ orbit (see Characteristic energy).

Under standard assumptions a body traveling along an escape orbit will coast along a parabolic trajectory to infinity, with velocity relative to the central body tending to zero, and therefore will never return. Parabolic trajectories are minimum-energy escape trajectories, separating positive-energy hyperbolic trajectories from negative-energy elliptic orbits.

Immanuel Velikovsky (/ˌvɛliˈkɒfski/; Russian: Иммануи́л Велико́вский, IPA: [ɪmənʊˈil vʲɪlʲɪˈkofskʲɪj]; 10 June [O.S. 29 May] 1895 – 17 November 1979) was a Russian-American psychoanalyst, writer, and catastrophist. He is the author of several books offering pseudohistorical interpretations of ancient history, including the U.S. bestseller Worlds in Collision published in 1950. Velikovsky's work is frequently cited as a canonical example of pseudoscience and has been used as an example of the demarcation problem.

His books use comparative mythology and ancient literary sources (including the Old Testament) to argue that Earth suffered catastrophic close contacts with other planets (principally Venus and Mars) in ancient history. In positioning Velikovsky among catastrophists including Hans Bellamy, Ignatius Donnelly, and Johann Gottlieb Radlof [de], the British astronomers Victor Clube and Bill Napier noted "... Velikovsky is not so much the first of the new catastrophists ...; he is the last in a line of traditional catastrophists going back to mediaeval times and probably earlier." Velikovsky argued that electromagnetic effects play an important role in celestial mechanics. He also proposed a revised chronology for ancient Egypt, Greece, Israel, and other cultures of the ancient Near East. The revised chronology aimed at explaining the so-called "dark age" of the eastern Mediterranean (c. 1100–750 BC) and reconciling biblical accounts with widely accepted archaeology and Egyptian chronology.