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A **sphere of influence** (**SOI**) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a celestial body where the primary gravitational influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons, despite the presence of the much more massive but distant Sun. In the patched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different masses using a two body approximation, ellipses and hyperbolae, the SOI is taken as the boundary where the trajectory switches which mass field it is influenced by.

- Table of selected SOI radii
- Increased accuracy on the SOI
- Derivation
- See also
- References
- General references
- External links

The general equation describing the radius of the sphere of a planet:

where

- is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
- and are the masses of the smaller and the larger object (usually a planet and the Sun), respectively.

In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun (until the object enters another body's SOI). Because the definition of r_{SOI} relies on the presence of the Sun and a planet, the term is only applicable in a three-body or greater system and requires the mass of the primary body to be much greater than the mass of the secondary body. This changes the three-body problem into a restricted two-body problem.

The table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun (with the exception of the Moon which is reported relative to Earth):^{ [1] }

Body | SOI radius (10^{6} km) | SOI radius (body radii) |
---|---|---|

Mercury | 0.117 | 46 |

Venus | 0.616 | 102 |

Earth + Moon | 0.929 | 145 |

Moon | 0.0661 | 38 |

Mars | 0.578 | 170 |

Jupiter | 48.2 | 687 |

Saturn | 54.5 | 1025 |

Uranus | 51.9 | 2040 |

Neptune | 86.2 | 3525 |

The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance from the massive body. A more accurate formula is given by^{[ citation needed ]}

Averaging over all possible directions we get^{[ citation needed ]}

Consider two point masses and at locations and , with mass and respectively. The distance separates the two objects. Given a massless third point at location , one can ask whether to use a frame centered on or on to analyse the dynamics of .

Consider a frame centered on . The gravity of is denoted as and will be treated as a perturbation to the dynamics of due to the gravity of body . Due their gravitational interactions, point is attracted to point with acceleration , this frame is therefore non-inertial. To quantify the effects of the perturbations in this frame, one should consider the ratio of the perturbations to the main body gravity i.e. . The perturbation is also known as the tidal forces due to body . It is possible to construct the perturbation ratio for the frame centered on by interchanging .

Frame A | Frame B | |
---|---|---|

Main acceleration | ||

Frame acceleration | ||

Secondary acceleration | ||

Perturbation, tidal forces | ||

Perturbation ratio |

As gets close to , and , and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say , it is possible to approximate the separating surface. In such a case this surface must be close to the mass , denote as the distance from to the separating surface.

Frame A | Frame B | |
---|---|---|

Main acceleration | ||

Frame acceleration | ||

Secondary acceleration | ||

Perturbation, tidal forces | ||

Perturbation ratio |

The distance to the sphere of influence must thus satisfy and so is the radius of the sphere of influence of body

**Mass** is the quantity of *matter* in a physical body. It is also a measure of the body's *inertia*, the resistance to acceleration when a net force is applied. An object's mass also determines the strength of its gravitational attraction to other bodies.

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

The **tidal force** is a gravitational effect that stretches a body along the line towards the center of mass of another body due to a gradient in gravitational field from the other body; it is responsible for diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within the Roche limit, and in extreme cases, spaghettification of objects. It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearest side is attracted more strongly than the farthest side. It is this difference that causes a body to get stretched. Thus, the tidal force is also known as the differential force, as well as a secondary effect of the gravitational field.

In celestial mechanics, **escape velocity** or **escape speed** is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account the fact that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.

In Newtonian physics, **free fall** is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

An **equatorial bulge** is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.

The **orbital period** is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.

**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.

A **trajectory** or **flight path** is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete trajectory is defined by position and momentum, simultaneously.

**Newton's law of universal gravitation** is usually stated as that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The publication of the theory has become known as the "first great unification", as it marked the unification of the previously described phenomena of gravity on Earth with known astronomical behaviors.

The **Hill sphere** of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.

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 celestial mechanics the **specific angular momentum** plays a pivotal role in the analysis of the two-body problem. One can show that it is a constant vector for a given orbit under ideal conditions. This essentially proves Kepler's second law.

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 physics, **gravitational acceleration** is the acceleration of an object in free fall within a vacuum. This is the steady gain in speed caused exclusively by the force of gravitational attraction. At a fixed point on the Earth's surface, all bodies accelerate in vacuum at the same rate, regardless of the masses or compositions of the bodies; the measurement and analysis of these rates is known as *gravimetry*.

A **photon sphere** or **photon circle** is an area or region of space where gravity is so strong that photons are forced to travel in orbits. The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole:

For most numbered asteroids, almost nothing is known apart from a few physical parameters and orbital elements and some physical characteristics are often only estimated. The physical data is determined by making certain standard assumptions.

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 geophysics and physical geodesy, a **geopotential model** is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field.

- ↑ Seefelder, Wolfgang (2002).
*Lunar Transfer Orbits Utilizing Solar Perturbations and Ballistic Capture*. Munich: Herbert Utz Verlag. p. 76. ISBN 3-8316-0155-0 . Retrieved July 3, 2018.

- Bate, Roger R.; Donald D. Mueller; Jerry E. White (1971).
*Fundamentals of Astrodynamics*. New York: Dover Publications. pp. 333–334. ISBN 0-486-60061-0.

- Sellers, Jerry J.; Astore, William J.; Giffen, Robert B.; Larson, Wiley J. (2004). Kirkpatrick, Douglas H. (ed.).
*Understanding Space: An Introduction to Astronautics*(2nd ed.). McGraw Hill. pp. 228, 738. ISBN 0-07-294364-5.

- Danby, J. M. A. (2003).
*Fundamentals of celestial mechanics*(2. ed., rev. and enlarged, 5. print. ed.). Richmond, Va., U.S.A.: Willmann-Bell. pp. 352–353. ISBN 0-943396-20-4.

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