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v t e

Asphere of influence (SOI) inastrodynamics andastronomy is theoblate spheroid-shaped region where a particularcelestial body exerts the maingravitational influence on anorbiting object. This is usually used to describe the areas in theSolar System whereplanets dominate the orbits of surrounding objects such asmoons, despite the presence of the much more massive but distantSun.

In thepatched conic approximation, used in estimating the trajectories of bodies moving between the neighbourhoods of different bodies 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. It is not to be confused with thesphere of activity which extends well beyond the sphere of influence.^[1]

The most common base models to calculate the sphere of influence is theHill sphere and theLaplace sphere, but updated and particularly more dynamic ones have been described.^[2][3]The general equation describing theradius of the sphere${\displaystyle r_{\text{SOI}}}$ of a planet:^[4]$${\displaystyle r_{\text{SOI}}\approx a{\left(\frac{m}{M}\right)}^{2/5}}$$where

• ${\displaystyle a}$ is thesemimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
• ${\displaystyle m}$ and${\displaystyle M}$ are themasses 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 ofr_SOI relies on the presence of the Sun and a planet, the term is only applicable in athree-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):^{[4][5][6][7][8][9][10]}

An important understanding to be drawn from this 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.

The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance${\displaystyle \theta }$ from the massive body. A more accurate formula is given by^[4]$${\displaystyle r_{\text{SOI}}(\theta )\approx a{\left(\frac{m}{M}\right)}^{2/5}\frac{1}{\sqrt[10]{1+3{\cos }^2(\theta )}}}$$

Averaging over all possible directions we get:$${\displaystyle \overline{r_{\text{SOI}}}=0.9431a{\left(\frac{m}{M}\right)}^{2/5}}$$

Consider two point masses${\displaystyle A}$ and${\displaystyle B}$ at locations${\displaystyle r_A}$ and${\displaystyle r_B}$, with mass${\displaystyle m_A}$ and${\displaystyle m_B}$ respectively. The distance${\displaystyle R=|r_B-r_A|}$ separates the two objects. Given a massless third point${\displaystyle C}$ at location${\displaystyle r_C}$, one can ask whether to use a frame centered on${\displaystyle A}$ or on${\displaystyle B}$ to analyse the dynamics of${\displaystyle C}$.

Consider a frame centered on${\displaystyle A}$. The gravity of${\displaystyle B}$ is denoted as${\displaystyle g_B}$ and will be treated as a perturbation to the dynamics of${\displaystyle C}$ due to the gravity${\displaystyle g_A}$ of body${\displaystyle A}$. Due to their gravitational interactions, point${\displaystyle A}$ is attracted to point${\displaystyle B}$ with acceleration${\displaystyle a_A=\frac{G{m}_B}{R^3}(r_B-r_A)}$, 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.${\displaystyle {\chi }_A=\frac{|g_B-a_A|}{|g_A|}}$. The perturbation${\displaystyle g_B-a_A}$ is also known as the tidal forces due to body${\displaystyle B}$. It is possible to construct the perturbation ratio${\displaystyle {\chi }_B}$ for the frame centered on${\displaystyle B}$ by interchanging${\displaystyle A\leftrightarrow B}$.

	Frame A	Frame B
Main acceleration	${\displaystyle g_A}$	${\displaystyle g_B}$
Frame acceleration	${\displaystyle a_A}$	${\displaystyle a_B}$
Secondary acceleration	${\displaystyle g_B}$	${\displaystyle g_A}$
Perturbation, tidal forces	${\displaystyle g_B-a_A}$	${\displaystyle g_A-a_B}$
Perturbation ratio${\displaystyle \chi }$	${\displaystyle {\chi }_A=\frac{|g_B-a_A|}{|g_A|}}$	${\displaystyle {\chi }_B=\frac{|g_A-a_B|}{|g_B|}}$

As${\displaystyle C}$ gets close to${\displaystyle A}$,${\displaystyle {\chi }_A\to 0}$ and${\displaystyle {\chi }_B\to \mathrm{\infty }}$, and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which${\displaystyle {\chi }_A={\chi }_B}$ separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say${\displaystyle m_A\ll m_B}$, it is possible to approximate the separating surface. In such a case this surface must be close to the mass${\displaystyle A}$, denote${\displaystyle r}$ as the distance from${\displaystyle A}$ to the separating surface.

	Frame A	Frame B
Main acceleration	${\displaystyle g_A=\frac{G{m}_A}{r^2}}$	${\displaystyle g_B\approx \frac{G{m}_B}{R^2}+\frac{G{m}_B}{R^3}r\approx \frac{G{m}_B}{R^2}}$
Frame acceleration	${\displaystyle a_A=\frac{G{m}_B}{R^2}}$	${\displaystyle a_B=\frac{G{m}_A}{R^2}\approx 0}$
Secondary acceleration	${\displaystyle g_B\approx \frac{G{m}_B}{R^2}+\frac{G{m}_B}{R^3}r}$	${\displaystyle g_A=\frac{G{m}_A}{r^2}}$
Perturbation, tidal forces	${\displaystyle g_B-a_A\approx \frac{G{m}_B}{R^3}r}$	${\displaystyle g_A-a_B\approx \frac{G{m}_A}{r^2}}$
Perturbation ratio${\displaystyle \chi }$	${\displaystyle {\chi }_A\approx \frac{m_B}{m_A}\frac{r^3}{R^3}}$	${\displaystyle {\chi }_B\approx \frac{m_A}{m_B}\frac{R^2}{r^2}}$

The distance to the sphere of influence must thus satisfy${\displaystyle \frac{m_B}{m_A}\frac{r^3}{R^3}=\frac{m_A}{m_B}\frac{R^2}{r^2}}$ and so${\displaystyle r=R{\left(\frac{m_A}{m_B}\right)}^{2/5}}$ is the radius of the sphere of influence of body${\displaystyle A}$

Gravity well (or funnel) is a metaphorical concept for agravitational field of a mass, with the field being curved in a funnel-shaped well around the mass, illustrating the steepgravitational potential andits energy that needs to be accounted for in order to escape or enter the main part of a sphere of influence.^[11]

An example for this is the strong gravitational field of theSun andMercury being deep within it.^[12] Atperihelion Mercury goes even deeper into the Sun's gravity well, causing an anomalistic or perihelionapsidal precession which is more recognizable than with other planets due to Mercury being deep in the gravity well. This characteristic of Mercury's orbit was famously calculated byAlbert Einstein through his formulation of gravity with thespeed of light, and the correspondinggeneral relativity theory, eventually being one of the firstcases proving the theory.

• Hill sphere
• Sphere of influence (black hole)
• Clearing the neighbourhood

• Bate, Roger R.; Mueller, Donald D.; White, Jerry E. (1971).Fundamentals of astrodynamics. Dover books on astronomy. New York:Dover Publications. pp. 333–334.ISBN 978-0-486-60061-1.
• Sellers, Jerry Jon; Astore, William J.; Giffen, Robert B.; Larson, Wiley J. (2015).Marilyn (ed.).Understanding space: an introduction to astronautics (4nd ed.). New York:McGraw-Hill Companies. pp. 228, 738.ISBN 978-0-9904299-4-4.
• Danby, J. M. A. (1992).Fundamentals of celestial mechanics (2nd ed.). Richmond, Va., U.S.A: Willmann-Bell. pp. 352–353.ISBN 978-0-943396-20-0.

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• Astrodynamics
• Orbits

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