Elliptical orbit in the context of "Kepler's third law"

In astronomy, Kepler's laws of planetary motion, published by Johannes Kepler in 1609 (except the third law, which was fully published in 1619), describe the orbits of planets around the Sun. These laws replaced the circular orbits and epicycles of Copernicus's heliostatic model of the planets with a genuinely heliocentric theory that described how planetary velocities vary following elliptical orbits. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the longer its orbital period.

The orbital plane of a revolving body is the geometric plane in which its orbit lies. Three non-collinear points in space suffice to determine an orbital plane. A common example would be the positions of the centers of a massive body (host) and of an orbiting celestial body at two different times/points of its orbit.

The orbital plane is defined in relation to a reference plane by two parameters: inclination (i) and longitude of the ascending node (Ω).

A Klemperer rosette is a gravitational system of (optionally) alternating heavier and lighter bodies orbiting in a symmetrical pattern around a common barycenter. It was first described by W.B. Klemperer in 1962, and is a special case of a central configuration.

Klemperer described rosette systems as follows:

Hiʻiaka, formal designation (136108) Haumea I, is the larger, outer moon of the trans-Neptunian dwarf planet Haumea. Discovered by Michael E. Brown and the Keck Observatory adaptive optics team on 26 January 2005, it is named after Hiʻiaka, the patron goddess of the Big Island of Hawaii and one of the daughters of Haumea. The moon follows a slightly elliptical orbit around Haumea every 49.5 days, at a distance of 49,400 km (30,700 mi).

Hiʻiaka is an elongated and irregularly shaped body with a mean diameter of 369 km (229 mi), making it the sixth-largest known moon of a trans-Neptunian object. It has a very low bulk density between 0.46 g/cm and 0.69 g/cm, which indicates it is mostly made of loosely-packed water ice and rock. Telescope observations have shown that Hiʻiaka has a highly reflective surface made of crystalline water ice, much like Haumea itself. Hiʻiaka rotates about its axis every 9.68 hours, and appears to rotate sideways with respect to its orbit around Haumea. Like its smaller sibling moon Namaka, Hiʻiaka is believed to be a fragment of Haumea that was ejected in the aftermath of a giant impact 4.4 billion years ago.

Namaka (full designation (136108) Haumea II) is the smaller, inner moon of the trans-Neptunian dwarf planet Haumea. Discovered by Michael E. Brown and the Keck Observatory adaptive optics team in the fall of 2005, it is named after Nāmaka, a water spirit and one of the daughters of Haumea in Hawaiian mythology. Namaka follows a highly elliptical orbit that is highly tilted by roughly 13 degrees with respect to Haumea's equator. Namaka is heavily perturbed by both the gravitational influence of Haumea's larger, outer moon Hiʻiaka and the variable gravitational field of Haumea's elongated shape.

With a diameter of around 150 km (93 mi), Namaka is predicted to have an irregular shape and a chaotic rotation. It has a reflective surface made of fresh water ice, similar to that of Haumea and Hiʻiaka. Like Hiʻiaka, Namaka is believed to be a fragment of Haumea that was ejected in the aftermath of a giant impact 4.4 billion years ago.

2018 AG₃₇ is a distant trans-Neptunian object and centaur that was discovered 132.2 ± 1.5 AU (19.78 ± 0.22 billion km) from the Sun, farther than any other currently observable known object in the Solar System. Imaged in January 2018 during a search for the hypothetical Planet Nine, the confirmation of this object was announced in a press release in February 2021 by astronomers Scott Sheppard, David Tholen, and Chad Trujillo. The object was nicknamed "FarFarOut" to emphasize its distance from the Sun.2018 AG₃₇ was discovered when it was near aphelion, the farthest point from the Sun in its elliptical orbit. The object is estimated to be at least 400 km (250 mi) in diameter. Because of its extreme distance, 2018 AG₃₇ appears extremely faint with an apparent magnitude of 25—only visible to the largest telescopes in the world.

In astronomy, Kepler's laws of planetary motion give a good approximations for the orbits of planets around the Sun. They were published by Johannes Kepler from 1608-1621 in three works Astronomia nova, Harmonice Mundi and Epitome Astronomiae Copernicanae. The laws were based Kepler's concept of solar fibrils adapted to the accurate astronomical data of Tycho Brahe. These laws replaced the circular orbits and epicycles of Copernicus's heliostatic model of the planets with a heliocentric model that described elliptical orbits with planetary velocities that vary accordingly. The three laws state that:

1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

The elliptical orbits of planets were indicated by calculations of the orbit of Mars. From this, Kepler inferred that other bodies in the Solar System, including those farther away from the Sun, also have elliptical orbits. The second law establishes that when a planet is closer to the Sun, it travels faster. The third law expresses that the farther a planet is from the Sun, the longer its orbital period.

Tidal heating (also known as tidal dissipation or tidal damping) occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either (or both) the surface ocean or interior of a planet or satellite. When an object is in an elliptical orbit, the tidal forces acting on it are stronger near periapsis than near apoapsis. Thus the deformation of the body due to tidal forces (i.e. the tidal bulge) varies over the course of its orbit, generating internal friction which heats its interior. This energy gained by the object comes from its orbital energy and/or rotational energy, so over time in a two-body system, the initial elliptical orbit decays into a circular orbit (tidal circularization) and the rotational periods of the two bodies adjust towards matching the orbital period (tidal locking). Sustained tidal heating occurs when the elliptical orbit is prevented from circularizing due to additional gravitational forces from other bodies that keep tugging the object back into an elliptical orbit. In this more complex system, orbital and rotational energy still is being converted to thermal energy; however, now the orbit's semimajor axis would shrink rather than its eccentricity.

A trans-lunar injection (TLI) is a propulsive maneuver, which is used to send a spacecraft to the Moon. Typical lunar transfer trajectories approximate Hohmann transfers, although low-energy transfers have also been used in some cases, as with the Hiten probe. For short duration missions without significant perturbations from sources outside the Earth-Moon system, a fast Hohmann transfer is typically more practical.

A spacecraft performs TLI to begin a lunar transfer from a low circular parking orbit around Earth. The large TLI burn, usually performed by a chemical rocket engine, increases the spacecraft's velocity, changing its orbit from a circular low Earth orbit to a highly eccentric orbit. The mission phase following TLI – while the spacecraft is flying passively towards the moon under its own momentum and influenced by terrestrial and lunar gravity – is called translunar coast. As the spacecraft begins coasting on the lunar transfer arc, its trajectory approximates an elliptical orbit about the Earth with an apogee near to the radius of the Moon's orbit. The TLI burn is sized and timed to precisely target the Moon as it revolves around the Earth. The burn is timed so that the spacecraft nears apogee as the Moon approaches. Finally, the spacecraft enters the Moon's sphere of influence, making a hyperbolic lunar swingby.

2012 VP₁₁₃ is a trans-Neptunian object (TNO) orbiting the Sun on an extremely wide elliptical orbit. It is classified as a sednoid because its orbit never comes closer than 80.5 AU (12.04 billion km; 7.48 billion mi) from the Sun, which is far enough away from the giant planets that their gravitational influence cannot affect the object's orbit noticeably. It was discovered on 5 November 2012 at Cerro Tololo Inter-American Observatory in Chile, by American astronomers Scott Sheppard and Chad Trujillo, who nicknamed the object "Biden" because of its "VP" abbreviation. The discovery was announced on 26 March 2014. The object's diameter has not been measured, but its brightness suggests it is around 450 km (280 mi) in diameter. 2012 VP₁₁₃ has a reddish color similar to many other TNOs.

2012 VP₁₁₃ has not yet been imaged by high-resolution telescopes, so it has no known moons. The Hubble Space Telescope is planned to image 2012 VP₁₁₃ in 2026, which should determine if it has significantly sized moons.

2023 KQ₁₄, informally nicknamed Ammonite, is a trans-Neptunian object (TNO) orbiting the Sun on an extremely wide elliptical orbit. It was discovered by the Subaru Telescope atop Mauna Kea on 16 May 2023, as part of an internationally led astronomical survey known as the "Formation of the Outer Solar System: an Icy Legacy" (FOSSIL) survey. 2023 KQ₁₄ is unusual because the direction of its orbital apsides is not aligned with those of previously known TNOs with high-perihelion elliptical orbits (sometimes known as sednoids), which challenges the hypothesis that an unseen distant planet ("Planet Nine") could be aligning their orbits. 2023 KQ₁₄ likely has a diameter between 220 and 380 km (140 and 240 mi).