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The **argument of periapsis** (also called **argument of perifocus** or **argument of pericenter**), symbolized as *ω*, is one of the orbital elements of an orbiting body. Parametrically, *ω* is the angle from the body's ascending node to its periapsis, measured in the direction of motion.

For specific types of orbits, words such as **perihelion** (for heliocentric orbits), **perigee** (for geocentric orbits), **periastron** (for orbits around stars), and so on may replace the word **periapsis**. (See apsis for more information.)

An argument of periapsis of 0° means that the orbiting body will be at its closest approach to the central body at the same moment that it crosses the plane of reference from South to North. An argument of periapsis of 90° means that the orbiting body will reach periapsis at its northmost distance from the plane of reference.

Adding the argument of periapsis to the longitude of the ascending node gives the longitude of the periapsis. However, especially in discussions of binary stars and exoplanets, the terms "longitude of periapsis" or "longitude of periastron" are often used synonymously with "argument of periapsis".

In astrodynamics the **argument of periapsis***ω* can be calculated as follows:

- If
*e*< 0 then_{z}*ω*→ 2π −*ω*.

- If

where:

**n**is a vector pointing towards the ascending node (i.e. the*z*-component of**n**is zero),**e**is the eccentricity vector (a vector pointing towards the periapsis).

In the case of equatorial orbits (which have no ascending node), the argument is strictly undefined. However, if the convention of setting the longitude of the ascending node Ω to 0 is followed, then the value of *ω* follows from the two-dimensional case:

- If the orbit is clockwise (i.e. (
**r**×**v**)_{z}< 0) then*ω*→ 2π −*ω*.

- If the orbit is clockwise (i.e. (

where:

*e*and_{x}*e*are the_{y}*x*- and*y*-components of the eccentricity vector**e**.

In the case of circular orbits it is often assumed that the periapsis is placed at the ascending node and therefore *ω* = 0. However, in the professional exoplanet community, *ω* = 90° is more often assumed for circular orbits, which has the advantage that the time of a planet's inferior conjunction (which would be the time the planet would transit if the geometry were favorable) is equal to the time of its periastron.^{ [1] }^{ [2] }^{ [3] }

**Precession** is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In other words, if the axis of rotation of a body is itself rotating about a second axis, that body is said to be precessing about the second axis. A motion in which the second Euler angle changes is called *nutation*. In physics, there are two types of precession: torque-free and torque-induced.

**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.

The **longitude of the ascending node** is one of the orbital elements used to specify the orbit of an object in space. It is the angle from a specified reference direction, called the *origin of longitude*, to the direction of the ascending node, as measured in a specified reference plane. The ascending node is the point where the orbit of the object passes through the plane of reference, as seen in the adjacent image. Commonly used reference planes and origins of longitude include:

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 celestial mechanics, the **longitude of the periapsis**, also called **longitude of the pericenter**, of an orbiting body is the longitude at which the periapsis would occur if the body's orbit inclination were zero. It is usually denoted *ϖ*.

**Mean longitude** is the ecliptic longitude at which an orbiting body could be found if its orbit were circular and free of perturbations. While nominally a simple longitude, in practice the mean longitude does not correspond to any one physical angle.

In celestial mechanics **true longitude** is the ecliptic longitude at which an orbiting body could actually be found if its inclination were zero. Together with the inclination and the ascending node, the true longitude can tell us the precise direction from the central object at which the body would be located at a particular time.

**Orbital inclination change** is an orbital maneuver aimed at changing the inclination of an orbiting body's orbit. This maneuver is also known as an **orbital plane change** as the plane of the orbit is tipped. This maneuver requires a change in the orbital velocity vector at the orbital nodes.

**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 astronomy, a **resonant trans-Neptunian object** is a trans-Neptunian object (TNO) in mean-motion orbital resonance with Neptune. The orbital periods of the resonant objects are in a simple integer relations with the period of Neptune, e.g. 1:2, 2:3, etc. Resonant TNOs can be either part of the main Kuiper belt population, or the more distant scattered disc population.

In celestial mechanics, the **Kozai mechanism** or **Lidov–Kozai mechanism** or **Kozai–Lidov mechanism**, also known as the **Kozai**, **Lidov–Kozai** or **Kozai–Lidov****effect**, **oscillations**, **cycles** or **resonance**, is a dynamical phenomenon affecting the orbit of a binary system perturbed by a distant third body under certain conditions, causing the orbit's argument of pericenter to oscillate about a constant value, which in turn leads to a periodic exchange between its eccentricity and inclination. The process occurs on timescales much longer than the orbital periods. It can drive an initially near-circular orbit to arbitrarily high eccentricity, and *flip* an initially moderately inclined orbit between a prograde and a retrograde motion.

The **perifocal coordinate** (**PQW**) **system** is a frame of reference for an orbit. The frame is centered at the focus of the orbit, i.e. the celestial body about which the orbit is centered. The unit vectors and lie in the plane of the orbit. is directed towards the periapsis of the orbit and has a true anomaly of 90 degrees past the periapsis. The third unit vector is the angular momentum vector and is directed orthogonal to the orbital plane such that:

In astronomy, a **co-orbital configuration** is a configuration of two or more astronomical objects orbiting at the same, or very similar, distance from their primary, i.e. they are in a 1:1 mean-motion resonance..

In mathematics, the **axis–angle representation** of a rotation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector **e** indicating the direction of an axis of rotation, and an angle *θ* describing the magnitude of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector **e** rooted at the origin because the magnitude of **e** is constrained. For example, the elevation and azimuth angles of **e** suffice to locate it in any particular Cartesian coordinate frame.

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 celestial mechanics, **apsidal precession** is the precession of the line connecting the apsides of an astronomical body's orbit. The apsides are the orbital points closest (periapsis) and farthest (apoapsis) from its primary body. The apsidal precession is the first time derivative of the argument of periapsis, one of the six main orbital elements of an orbit. Apsidal precession is considered positive when the orbit's axis rotates in the same direction as the orbital motion. An **apsidal period** is the time interval required for an orbit to precess through 360°.

**Nodal precession** is the precession of the orbital plane of a satellite around the rotational axis of an astronomical body such as Earth. This precession is due to the non-spherical nature of a rotating body, which creates a non-uniform gravitational field. The following discussion relates to low Earth orbit of artificial satellites, which have no measurable effect on the motion of Earth. The nodal precession of more massive, natural satellites like the Moon is more complex.

In orbital mechanics, a **frozen orbit** is an orbit for an artificial satellite in which natural drifting due to the central body's shape has been minimized by careful selection of the orbital parameters. Typically, this is an orbit in which, over a long period of time, the satellite's altitude remains constant at the same point in each orbit. Changes in the inclination, position of the lowest point of the orbit, and eccentricity have been minimized by choosing initial values so that their perturbations cancel out. This results in a long-term stable orbit that minimizes the use of station-keeping propellant.

The hypothetical Planet Nine would modify the orbits of extreme trans-Neptunian objects via a combination of effects. On very long timescales exchanges of angular momentum with Planet Nine cause the perihelia of anti-aligned objects to rise until their precession reverses direction, maintaining their anti-alignment, and later fall, returning them to their original orbits. On shorter timescales mean-motion resonances with Planet Nine provides phase protection, which stabilizes their orbits by slightly altering the objects' semi-major axes, keeping their orbits synchronized with Planet Nine's and preventing close approaches. The inclination of Planet Nine's orbit weakens this protection, resulting in a chaotic variation of semi-major axes as objects hop between resonances. The orbital poles of the objects circle that of the Solar System's Laplace plane, which at large semi-major axes is warped toward the plane of Planet Nine's orbit, causing their poles to be clustered toward one side.

- ↑ Iglesias-Marzoa, Ramón; López-Morales, Mercedes; Jesús Arévalo Morales, María (2015). "Thervfit
*Code*: A Detailed Adaptive Simulated Annealing Code for Fitting Binaries and Exoplanets Radial Velocities".*Publications of the Astronomical Society of the Pacific*.**127**(952): 567–582. doi: 10.1086/682056 . - ↑ Kreidberg, Laura (2015). "Batman: BAsic Transit Model cAlculatioN in Python".
*Publications of the Astronomical Society of the Pacific*.**127**(957): 1161–1165. arXiv: 1507.08285 . Bibcode:2015PASP..127.1161K. doi:10.1086/683602. S2CID 7954832. - ↑ Eastman, Jason; Gaudi, B. Scott; Agol, Eric (2013). "EXOFAST: A Fast Exoplanetary Fitting Suite in IDL".
*Publications of the Astronomical Society of the Pacific*.**125**(923): 83. arXiv: 1206.5798 . Bibcode:2013PASP..125...83E. doi:10.1086/669497. S2CID 118627052.

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