Editing Orbit (section)

==Newton's laws of motion==
===Newton's law of gravitation and laws of motion for two-body problems===
In most situations, relativistic effects can be neglected, and [[Newton's laws]] give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on a body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two-point masses or spherical bodies, only influenced by their mutual gravitation (called a [[two-body problem]]), their trajectories can be exactly calculated. If the heavier body is much more massive than the smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a [[coordinate system]] that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing the coordinate system at the center of the mass of the system.

===Defining gravitational potential energy===
Energy is associated with [[gravitational field]]s. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational ''[[potential energy]]''. Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses, the gravitational energy decreases to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances.

===Orbital energies and orbit shapes===
When only two gravitational bodies interact, their orbits follow a [[conic section]]. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total [[energy]] ([[kinetic energy|kinetic]] + [[potential energy]]) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the [[escape velocity]] for that position, in the case of a closed orbit, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy.

An open orbit will have a parabolic shape if it has the velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a [[hyperbola]] when its velocity is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever.

All closed orbits have the shape of an [[ellipse]]. A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is called the [[perigee]], and when orbiting a body other than earth it is called the periapsis (less properly, "perifocus" or "pericentron"). The point where the satellite is farthest from Earth is called the [[apogee]], apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the ''[[line of apsides|line-of-apsides]]''. This is the major axis of the ellipse, the line through its longest part.

===Kepler's laws===
Bodies following closed orbits repeat their paths with a certain time called the period. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be
formulated as follows:

# The orbit of a planet around the [[Sun]] is an ellipse, with the Sun in one of the focal points of that ellipse. [This focal point is actually the [[barycenter]] of the [[Solar System|Sun-planet system]]; for simplicity, this explanation assumes the Sun's mass is infinitely larger than that planet's.] The planet's orbit lies in a plane, called the '''[[Orbital plane (astronomy)|orbital plane]]'''. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits about particular bodies; things orbiting the Sun have a [[perihelion]] and [[aphelion]], things orbiting the Earth have a [[perigee]] and [[apogee]], and things orbiting the [[Moon]] have a [[perilune]] and [[apolune]] (or [[periselene]] and [[aposelene]] respectively). An orbit around any [[star]], not just the Sun, has a [[periastron]] and an [[apastron]].
# As the planet moves in its orbit, the line from the Sun to the planet sweeps a constant area of the [[Orbital plane (astronomy)|orbital plane]] for a given period of time, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its [[perihelion]] than near its [[aphelion]], because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time."
# For a given orbit, the ratio of the cube of its [[semi-major axis]] to the square of its period is constant.

===Limitations of Newton's law of gravitation===
Note that while bound orbits of a point mass or a spherical body with a [[Newtonian gravitational field]] are closed [[ellipse]]s, which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by the slight oblateness of the [[Earth]], or by [[Theory of relativity|relativistic effects]], thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed [[ellipse]]s characteristic of Newtonian [[two-body motion]]. The two-body solutions were published by Newton in [[Philosophiae Naturalis Principia Mathematica|Principia]] in 1687. In 1912, [[Karl Fritiof Sundman]] developed a converging infinite series that solves the [[three-body problem]]; however, it converges too slowly to be of much use. Except for special cases like the [[Lagrangian point]]s, no method is known to solve the equations of motion for a system with four or more bodies.

===Approaches to many-body problems===
Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms:
:One form takes the pure elliptic motion as a basis and adds [[perturbation (astronomy)|perturbation]] terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets, and other bodies are known with great accuracy, and are used to generate [[ephemeris|tables]] for [[celestial navigation]]. Still, there are [[secular phenomena]] that have to be dealt with by [[Parameterized post-Newtonian formalism|post-Newtonian]] methods.
:The [[differential equation]] form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (''F = ma''). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an [[initial value problem]]. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated.<ref>{{Cite journal |last1=Carleton |first1=Timothy |last2=Guo |first2=Yicheng |last3=Munshi |first3=Ferah |last4=Tremmel |first4=Michael |last5=Wright |first5=Anna |title=An excess of globular clusters in Ultra-Diffuse Galaxies formed through tidal heating |journal=Monthly Notices of the Royal Astronomical Society |year=2021 |volume=502 |pages=398–406 |doi=10.1093/mnras/stab031 |doi-access=free |arxiv=2008.11205 }}</ref>