Celestial mechanics in the context of "Joseph-Louis Lagrange"

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.

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).