Sphere of influence (astrodynamics)
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A sphere of influence (SOI) in astrodynamics and astronomy is the oblate-spheroid-shaped region around a
celestial body An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical object, physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''bod ...
where the primary
gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...
influence on an orbiting object is that body. This is usually used to describe the areas in the Solar System where
planets A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a young ...
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 In astrodynamics, the patched conic approximation or patched two-body approximation is a method to simplify trajectory calculations for spacecraft in a multiple-body environment. Method The simplification is achieved by dividing space into vario ...
, 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. The general equation describing the radius of the sphere r_ of a planet: : r_ \approx a\left(\frac\right)^ where : a is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun). : m and M 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 rSOI relies on the presence of the Sun and a planet, the term is only applicable in a
three-body Three body may refer to: ;Science *Three-body problem, a problem in physics and classical mechanics *Euler's three-body problem, a problem in physics and astronomy *Three-body force, a force appearing in a three-body system ;Science fiction * ''Th ...
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.


Table of selected SOI radii

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): An important understanding to be drawn from the above table is that "Sphere of Influence" here is "Primary". For example, though Jupiter is much larger in mass than say, Neptune, its Primary SOI is much smaller due to Jupiter's much closer proximity to the Sun.


Increased accuracy on the SOI

The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance \theta from the massive body. A more accurate formula is given by r_(\theta) \approx a\left(\frac\right)^\frac Averaging over all possible directions we get: \overline = 0.9431 a\left(\frac\right)^


Derivation

Consider two point masses A and B at locations r_A and r_B, with mass m_A and m_B respectively. The distance R=, r_B-r_A, separates the two objects. Given a massless third point C at location r_C , one can ask whether to use a frame centered on A or on B to analyse the dynamics of C . Consider a frame centered on A . The gravity of B is denoted as g_B and will be treated as a perturbation to the dynamics of C due to the gravity g_A of body A . Due their gravitational interactions, point A is attracted to point B with acceleration a_A = \frac (r_B-r_A) , 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. \chi_A = \frac . The perturbation g_B-a_A is also known as the tidal forces due to body B . It is possible to construct the perturbation ratio \chi_B for the frame centered on B by interchanging A \leftrightarrow B . As C gets close to A , \chi_A \rightarrow 0 and \chi_B \rightarrow \infty , and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which \chi_A = \chi_B separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say m_A \ll m_B , it is possible to approximate the separating surface. In such a case this surface must be close to the mass A , denote r as the distance from A to the separating surface. The distance to the sphere of influence must thus satisfy \frac \frac = \frac \frac and so r = R\left(\frac\right)^ is the radius of the sphere of influence of body A


See also

* Hill sphere * Sphere of influence (black hole)


References


General references

* * *{{cite book, last1=Danby, first1=J. M. A., title=Fundamentals of celestial mechanics, date=2003, publisher=Willmann-Bell, location=Richmond, Va., U.S.A., isbn=0-943396-20-4, pages=352–353, edition=2. ed., rev. and enlarged, 5. print.


External links


Project Pluto
Astrodynamics Orbits