**Tidal circularization** is an effect of the tidal forces between a body in orbit around a central celestial object, whereby the eccentricity of the orbit is reduced over time so that it becomes less and less elliptical.

In figure 1 let's start by assuming body 1 is a star and body 2 is another star or maybe a Jupiter like planet. Initially think of body2 as a point mass. The gravity from Body 2 applied to Body 1 produces tidal bulges (see Tidal Force). Let's assume the orbital period is slower than the rotation of Body 1 (ω<Ω) as shown in figure 1. One might expect a lag angle as shown. If Body1 is 100% elastic (e.g. gas bodies are usually very elastic but a bag of sand is not very elastic) then the bulge wouldn't have a lag angle. The more inelastic, the larger the lag angle. The larger the difference in angular velocities (ω/Ω), the larger the lag angle. If ω>Ω, the lag angle will be in the other direction.

For a star we can think of inelasticity as viscosity. The main cause of inelasticity in a star seems to be convection forces inside the star.^{ [1] } When the lag angle is non zero as in figure 1 you can see that the forces F1 and F2 combine to torque body 1 clockwise because F1 is stronger. At the same time they torque the orbital motion counter clockwise: if you ignore the portion of F1 and F2 that lie along the line connecting the two bodies the remaining combined force on the entirety of body 1 is F3. Similarly F1’ and F2’ combine to produce F3’. F3 and F3’ torque the orbit counter clockwise. Side note: rotational momentum of the combined rotations is preserved.

We now have a rule of thumb: Whenever angular velocity at a given moment of the orbit is less than the angular velocity of either body (ω<Ω) then the orbital torque tries to speed up the orbit. And vice versa.^{ [1] }

Now imagine two stars orbiting each other in elliptical orbits with the special case where both are tidally locked such that over the course of an orbit the same sides face each other (ω=Ω on average). Although Ω is constant for one orbit, ω varies throughout the orbit. Figure 2 shows the path of one of the stars where G is the center of gravity of the system. When the objects are near apoapsis (red region of figure 2), ω<Ω which tries to speed up the orbit. The result of this torque makes the far side of the orbit (periapsis) farther out making the orbit more circular. This follows from the rule of thumb "if thrust is applied briefly to speed up an orbit (i.e. applied along the direction of travel), then when the object orbits half way around, that part of the orbit will be higher" and vice versa: "retrograde thrust lowers the far side of an orbit" (see orbital rules of thumb).

More importantly when Body 1 is in the green region of figure 2 and especially when it is closest to the center of gravity and therefore the tidal bulge is largest and ω/Ω is at maximum, the torque slows down the orbit (F3 in figure 1 is now negative because the lag angle is reversed) which lowers the far side of the orbit (lowers apoapsis). Lowering Apsis or raising Periapsis is basically the definition of circularizing an orbit.

The objects in orbit don't have to be two stars. One can be a planet or one can be a planet and one can be a moon. Circularization can also occur in clusters of stars orbiting an imaginary point in space at the center of gravity.^{ [2] }

Orbital circularization can be caused by either or both of the two objects in an orbit if either or both are inelastic. Cooler stars tend to be more viscous and circularize objects orbiting them faster than hot stars.^{ [3] }

If Ω/ω > 18/11 (~1.64) circularization won't occur and actually the eccentricity will increase.^{ [4] } So the bodies need to reach tidal lock first where at least one object has the same side facing the other object during the course of an orbit.^{ [1] } Tidal locking is another behavior caused by tidal forces.

In physics, **angular momentum** is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.

**Nutation** is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called *free nutation* or *Euler nutation*. A *pure nutation* is a movement of a rotational axis such that the first Euler angle is constant. Therefore it can be seen that the circular red arrow in the diagram indicates the combined effects of precession and nutation, while nutation in the absence of precession would only change the tilt from vertical. However, in spacecraft dynamics, precession is sometimes referred to as nutation.

In celestial mechanics, an **orbit** is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of 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.

**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.

**Rotation** or **rotational motion** is the circular movement of an object around a central line, known as * axis of rotation*. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a

**Tidal locking** between a pair of co-orbiting astronomical bodies occurs when one of the objects reaches a state where there is no longer any net change in its rotation rate over the course of a complete orbit. In the case where a tidally locked body possesses synchronous rotation, the object takes just as long to rotate around its own axis as it does to revolve around its partner. For example, the same side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body. However, if both the difference in mass between the two bodies and the distance between them are relatively small, each may be tidally locked to the other; this is the case for Pluto and Charon, as well as for Eris and Dysnomia. Alternative names for the tidal locking process are **gravitational locking**, **captured rotation**, and **spin–orbit locking**.

In physics, **angular acceleration** refers to the time rate of change of angular velocity. Following the two types of angular velocity, *spin angular velocity* and *orbital angular velocity*, the respective types of angular acceleration are: **spin angular acceleration** involving a rigid body about its centre of rotation, and **orbital angular acceleration** involving a point particle about a fixed origin.

**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.

**Aerobraking** is a spaceflight maneuver that reduces the high point of an elliptical orbit (apoapsis) by flying the vehicle through the atmosphere at the low point of the orbit (periapsis). The resulting drag slows the spacecraft. Aerobraking is used when a spacecraft requires a low orbit after arriving at a body with an atmosphere, as it requires less fuel than using propulsion to slow down.

**Rotational energy** or **angular kinetic energy** is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed:

**Orbital decay** is a gradual decrease of the distance between two orbiting bodies at their closest approach over many orbital periods. These orbiting bodies can be a planet and its satellite, a star and any object orbiting it, or components of any binary system. If left unchecked, the decay eventually results in termination of the orbit when the smaller object strikes the surface of the primary; or for objects where the primary has an atmosphere, the smaller object burns, explodes, or otherwise breaks up in the larger object's atmosphere; or for objects where the primary is a star, ends with incineration by the star's radiation. Collisions of stellar-mass objects are usually accompanied by effects such as gamma-ray bursts and detectable gravitational waves.

The **argument of periapsis**, 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.

In astrodynamics, the **orbital eccentricity** of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a rosette orbit through the Galaxy.

**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.

**Rotation around a fixed axis** is a special case of rotational motion around a *axis of rotation* fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result.

**Tidal heating** occurs through the tidal friction processes: orbital and rotational energy is dissipated as heat in either 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 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 and the rotational periods of the two bodies adjust towards matching the orbital period. 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.

**Stellar rotation** is the angular motion of a star about its axis. The rate of rotation can be measured from the spectrum of the star, or by timing the movements of active features on the surface.

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 farthest (apoapsis) and closest (periapsis) 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°, which takes Earth's orbit about 112,000 years, completing a cycle and returning to the same orientation.

**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.

This **glossary of astronomy** is a list of definitions of terms and concepts relevant to astronomy and cosmology, their sub-disciplines, and related fields. Astronomy is concerned with the study of celestial objects and phenomena that originate outside the atmosphere of Earth. The field of astronomy features an extensive vocabulary and a significant amount of jargon.

- 1 2 3 Zahn, Jean-Paul (30 July 2008). "Tidal dissipation in binary systems".
*Eas Publications Series*.**29**: 67–90. arXiv: 0807.4870 . Bibcode:2008EAS....29...67Z. doi:10.1051/eas:0829002. S2CID 118685663. - ↑ Mathieu, Robert D; Meibom, Søren; Dolan, Christopher J (27 January 2004). "WIYN Open Cluster Study. XVIII. The Tidal Circularization Cutoff Period of the Old Open Cluster NGC 188".
*The Astrophysical Journal*.**602**(2): L121–L123. arXiv: astro-ph/0401582 . Bibcode:2004ApJ...602L.121M. doi:10.1086/382686. S2CID 204935755. - ↑ Winn, Joshua N; Fabrycky, Daniel; Albrecht, Simon; Johnson, John Asher (12 July 2010). "Hot Stars with Hot Jupiters Have High Obliquities".
*The Astrophysical Journal Letters*.**718**(2): L145–L149. arXiv: 1006.4161 . Bibcode:2010ApJ...718L.145W. doi:10.1088/2041-8205/718/2/L145. S2CID 13032700. - ↑ Darwin, George H (1880). "On the Secular Changes in the Elements of the Orbit of a Satellite revolving about a Tidally distorted Planet".
*Philosophical Transactions of the Royal Society*.**171**(2): 889.

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