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A **circular orbit** is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.

- Circular acceleration
- Velocity
- Equation of motion
- Angular speed and orbital period
- Energy
- Delta-v to reach a circular orbit
- Orbital velocity in general relativity
- Derivation
- See also
- References

Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion.

In this case, not only the distance, but also the speed, angular speed, potential and kinetic energy are constant. There is no periapsis or apoapsis. This orbit has no radial version.

Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, circular motion ensues. Taking two derivatives of the particle's coordinates with respect to time gives the centripetal acceleration

where:

- is orbital velocity of orbiting body,
- is radius of the circle
- is angular speed, measured in radians per unit time.

The formula is dimensionless, describing a ratio true for all units of measure applied uniformly across the formula. If the numerical value of is measured in meters per second per second, then the numerical values for will be in meters per second, in meters, and in radians per second.

The speed (or the magnitude of velocity) relative to the central object is constant:^{ [1] }^{:30}

where:

- , is the gravitational constant
- , is the mass of both orbiting bodies , although in common practice, if the greater mass is significantly larger, the lesser mass is often neglected, with minimal change in the result.
- , is the standard gravitational parameter.

The orbit equation in polar coordinates, which in general gives *r* in terms of *θ*, reduces to:^{[ clarification needed ]}^{[ citation needed ]}

where:

- is specific angular momentum of the orbiting body.

This is because

Hence the orbital period () can be computed as:^{ [1] }^{:28}

Compare two proportional quantities, the free-fall time (time to fall to a point mass from rest)

- (17.7% of the orbital period in a circular orbit)

and the time to fall to a point mass in a radial parabolic orbit

- (7.5% of the orbital period in a circular orbit)

The fact that the formulas only differ by a constant factor is a priori clear from dimensional analysis.^{[ citation needed ]}

The specific orbital energy () is negative, and

Thus the virial theorem ^{ [1] }^{:72} applies even without taking a time-average:^{[ citation needed ]}

- the kinetic energy of the system is equal to the absolute value of the total energy
- the potential energy of the system is equal to twice the total energy

The escape velocity from any distance is √2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.^{[ citation needed ]}

Maneuvering into a large circular orbit, e.g. a geostationary orbit, requires a larger delta-v than an escape orbit, although the latter implies getting arbitrarily far away and having more energy than needed for the orbital speed of the circular orbit. It is also a matter of maneuvering into the orbit. See also Hohmann transfer orbit.

In Schwarzschild metric, the orbital velocity for a circular orbit with radius is given by the following formula:

where is the Schwarzschild radius of the central body.

For the sake of convenience, the derivation will be written in units in which .

The four-velocity of a body on a circular orbit is given by:

( is constant on a circular orbit, and the coordinates can be chosen so that ). The dot above a variable denotes derivation with respect to proper time .

For a massive particle, the components of the four-velocity satisfy the following equation:

We use the geodesic equation:

The only nontrivial equation is the one for . It gives:

From this, we get:

Substituting this into the equation for a massive particle gives:

Hence:

Assume we have an observer at radius , who is not moving with respect to the central body, that is, their four-velocity is proportional to the vector . The normalization condition implies that it is equal to:

The dot product of the four-velocities of the observer and the orbiting body equals the gamma factor for the orbiting body relative to the observer, hence:

This gives the velocity:

Or, in SI units:

**Orbital mechanics** or **astrodynamics** is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

The **Klein–Gordon equation** is a relativistic wave equation, related to the Schrödinger equation. It is second-order in space and time and manifestly Lorentz-covariant. It is a quantized version of the relativistic energy–momentum relation. Its solutions include a quantum scalar or pseudoscalar field, a field whose quanta are spinless particles. Its theoretical relevance is similar to that of the Dirac equation. Electromagnetic interactions can be incorporated, forming the topic of scalar electrodynamics, but because common spinless particles like the pions are unstable and also experience the strong interaction the practical utility is limited.

In physics and astronomy, the **Reissner–Nordström metric** is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass *M*. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric.

In astrodynamics or celestial mechanics a **parabolic trajectory** is a Kepler orbit with the eccentricity equal to 1 and is an unbound orbit that is exactly on the border between elliptical and hyperbolic. When moving away from the source it is called an **escape orbit**, otherwise a **capture orbit**. It is also sometimes referred to as a **C _{3} = 0 orbit** (see Characteristic energy).

In astrodynamics or celestial mechanics, a **hyperbolic trajectory** is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

In astrodynamics or celestial mechanics, an **elliptic orbit** or **elliptical orbit** is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In astrodynamics an **orbit equation** defines the path of orbiting body around central body relative to , without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance, has an orbit that is a conic section with the central body located at one of the two foci, or *the* focus.

In general relativity, **Schwarzschild geodesics** describe the motion of test particles in the gravitational field of a central fixed mass that is, motion in the Schwarzschild metric. Schwarzschild geodesics have been pivotal in the validation of Einstein's theory of general relativity. For example, they provide accurate predictions of the anomalous precession of the planets in the Solar System and of the deflection of light by gravity.

The **Kerr–Newman metric** is the most general asymptotically flat, stationary solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass. It generalizes the Kerr metric by taking into account the field energy of an electromagnetic field, in addition to describing rotation. It is one of a large number of various different electrovacuum solutions, that is, of solutions to the Einstein–Maxwell equations which account for the field energy of an electromagnetic field. Such solutions do not include any electric charges other than that associated with the gravitational field, and are thus termed vacuum solutions.

**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 general relativity, the **metric tensor** is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.

A **theoretical motivation for general relativity**, including the motivation for the geodesic equation and the Einstein field equation, can be obtained from special relativity by examining the dynamics of particles in circular orbits about the earth. A key advantage in examining circular orbits is that it is possible to know the solution of the Einstein Field Equation *a priori*. This provides a means to inform and verify the formalism.

The **effective potential** combines multiple, perhaps opposing, effects into a single potential. In its basic form, it is the sum of the 'opposing' centrifugal potential energy with the potential energy of a dynamical system. It may be used to determine the orbits of planets and to perform semi-classical atomic calculations, and often allows problems to be reduced to fewer dimensions.

**Alternatives to general relativity** are physical theories that attempt to describe the phenomenon of gravitation in competition to Einstein's theory of general relativity. There have been many different attempts at constructing an ideal theory of gravity.

In classical mechanics, a **Liouville dynamical system** is an exactly soluble dynamical system in which the kinetic energy *T* and potential energy *V* can be expressed in terms of the *s* generalized coordinates *q* as follows:

In general relativity, **Lense–Thirring precession** or the **Lense–Thirring effect** is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum .

** f(R)** is a type of modified gravity theory which generalizes Einstein's general relativity.

In classical mechanics, the **central-force problem** is to determine the motion of a particle in a single central potential field. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

The **pressuron** is a hypothetical scalar particle which couples to both gravity and matter theorised in 2013. Although originally postulated without self-interaction potential, the pressuron is also a dark energy candidate when it has such a potential. The pressuron takes its name from the fact that it decouples from matter in pressure-less regimes, allowing the scalar-tensor theory of gravity involving it to pass solar system tests, as well as tests on the equivalence principle, even though it is fundamentally coupled to matter. Such a decoupling mechanism could explain why gravitation seems to be well described by general relativity at present epoch, while it could actually be more complex than that. Because of the way it couples to matter, the pressuron is a special case of the hypothetical string dilaton. Therefore, it is one of the possible solutions to the present non-observation of various signals coming from massless or light scalar fields that are generically predicted in string theory.

- 1 2 3 Lissauer, Jack J.; de Pater, Imke (2019).
*Fundamental Planetary Sciences : physics, chemistry, and habitability*. New York, NY, USA: Cambridge University Press. p. 604. ISBN 9781108411981.

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