View source for Sphere of influence (astrodynamics)

{{Short description|Region of space gravitationally dominated by a given body}}{{for|the term relating to black holes|sphere of influence (black hole)}}
{{No footnotes|date=March 2016}}
{{Astrodynamics}}
A '''sphere of influence''' ('''SOI''') in [[astrodynamics]] and [[astronomy]] is the [[oblate spheroid|oblate-spheroid]]-shaped region around a [[celestial body]] where the primary [[gravity|gravitational]] influence on an [[orbit]]ing 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 [[natural satellite|moon]]s, 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.

The general equation describing the [[radius]] of the sphere <math>r_{SOI}</math> of a planet:
: <math>r_{SOI} \approx a\left(\frac{m}{M}\right)^{2/5}</math>
where
: <math>a</math> is the [[semimajor axis]] of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
: <math>m</math> and <math>M</math> are the [[mass]]es 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<sub>SOI</sub> relies on the presence of the Sun and a planet, the term is only applicable in a [[n-body problem|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.

==Table of selected SOI radii==
[[File:Sphere of influence on ratio m M.png|thumb|Dependence of Sphere of influence r<sub>SOI</sub>/a on the ratio m/M]]
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).<ref>{{cite book |last= Seefelder|first= Wolfgang |year= 2002|title= Lunar Transfer Orbits Utilizing Solar Perturbations and Ballistic Capture|url= https://books.google.com/books?id=NVg_vYHePt0C&pg=PA76|location= Munich|publisher= [[Herbert Utz Verlag]]|page= 76|isbn= 3-8316-0155-0||access-date=July 3, 2018}}</ref>:

{| class="wikitable"
|-
! Body !! SOI radius (10<sup>6</sup> km) !! SOI radius (body radii)
|-
| [[Mercury (planet)|Mercury]] || align="center" | 0.112 || align="right" | 46
|-
| [[Venus]] || align="center" | 0.616 ||align="right" | 102
|-
| [[Earth]] || align="center" | 0.924 ||align="right" | 145
|-
| [[Moon]] || align="center" | 0.0661 ||align="right" | 38
|-
| [[Mars]] || align="center" | 0.576 ||align="right" | 170
|-
| [[Jupiter]] || align="center" | 48.2 ||align="right" | 687
|-
| [[Saturn]] || align="center" | 54.6 ||align="right" | 1025
|-
| [[Uranus]] || align="center" | 51.8 ||align="right" | 2040
|-
| [[Neptune]] || align="center" | 86.8 ||align="right" | 3525
|}
==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 <math>\theta</math> from the massive body. A more accurate formula is given by

<math>r_{SOI}(\theta) \approx a\left(\frac{m}{M}\right)^{2/5}\frac{1}{\sqrt[10]{1+3\cos^2(\theta)}}</math>

Averaging over all possible directions we get

<math>\overline{r_{SOI}} = 0.9431 a\left(\frac{m}{M}\right)^{2/5}</math>

==Derivation==
Consider two point masses <math> A</math> and <math> B</math> at locations <math> r_A</math> and <math> r_B</math>, with mass <math> m_A</math> and <math> m_B</math> respectively. The distance <math> R=|r_B-r_A|</math> separates the two objects. Given a massless third point <math> C </math> at location <math> r_C </math>, one can ask whether to use a frame centered on <math> A</math> or on <math> B</math> to analyse the dynamics of <math> C </math>.
[[File:Sphereofinfluence.png|thumb|
Geometry and dynamics to derive the sphere of influence
]]

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

{| class="wikitable"
|-
!  !! Frame A !! Frame B
|-
| Main acceleration || align="center" | <math> g_A </math> || align="center" | <math> g_B </math>
|-
| Frame acceleration || align="center" | <math> a_A </math> || align="center" | <math> a_B </math>
|-
| Secondary acceleration || align="center" | <math> g_B </math> || align="center" | <math> g_A </math>
|-
| Perturbation, tidal forces || align="center" | <math> g_B-a_A </math> || align="center" | <math> g_A-a_B </math>
|-
| Perturbation ratio <math> \chi </math>  || align="center" | <math> \chi_A = \frac{|g_B-a_A|}{|g_A|} </math> ||align="center" | <math> \chi_B = \frac{|g_A-a_B|}{|g_B|} </math>
|}
As <math> C </math> gets close to <math> A </math>, <math> \chi_A \rightarrow 0 </math> and <math> \chi_B \rightarrow \infty </math>, and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which <math> \chi_A = \chi_B </math> separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say <math> m_A \ll  m_B </math>, it is possible to approximate the separating surface. In such a case this surface must be close to the mass <math> A </math>, denote  <math> r </math> as the distance from <math> A </math> to the separating surface.

{| class="wikitable"
|-
!  !! Frame A !! Frame B
|-
| Main acceleration || align="center" | <math> g_A = \frac{G m_A}{r^2} </math> || align="center" | <math> g_B \approx \frac{G m_B}{R^2} + \frac{G m_B}{R^3} r \approx \frac{G m_B}{R^2} </math>
|-
| Frame acceleration || align="center" | <math> a_A = \frac{G m_B}{R^2}  </math> || align="center" | <math> a_B = \frac{G m_A}{R^2} \approx 0 </math>
|-
| Secondary acceleration || align="center" | <math> g_B \approx \frac{G m_B}{R^2} + \frac{G m_B}{R^3} r </math> || align="center" | <math> g_A = \frac{G m_A}{r^2} </math>
|-
| Perturbation, tidal forces || align="center" | <math> g_B-a_A \approx \frac{G m_B}{R^3} r </math>  || align="center" | <math> g_A-a_B \approx \frac{G m_A}{r^2} </math>
|-
| Perturbation ratio <math> \chi </math>  || align="center" | <math> \chi_A \approx \frac{m_B}{m_A} \frac{r^3}{R^3} </math> ||align="center" | <math> \chi_B \approx \frac{m_A}{m_B} \frac{R^2}{r^2}</math>
|}
[[File:Hill sphere and SOI.png|thumb|Hill sphere and Sphere Of Influence for _Solar system bodies.]]
The distance to the sphere of influence must thus satisfy <math> \frac{m_B}{m_A} \frac{r^3}{R^3} = \frac{m_A}{m_B} \frac{R^2}{r^2}</math> and so <math> r = \left(\frac{m_A}{m_B}\right)^{2/5} R</math> is the radius of the sphere of influence of body <math> A </math>

==See also==
* [[Hill sphere]]
* [[Sphere of influence (black hole)]]

==References==
{{reflist}}

== General references ==

*{{cite book |last       = Bate
 |first      = Roger R.
 |author2    = Donald D. Mueller
 |author3    = Jerry E. White
 |title      = Fundamentals of Astrodynamics
 |date       = 1971
 |publisher  = [[Dover Publications]]
 |location   = New York
 |isbn       = 0-486-60061-0
 |pages      = 333–334
 |url-access = registration
 |url        = https://archive.org/details/fundamentalsofas00bate
}}

*{{cite book 
|last=Sellers
|first=Jerry J.
|author2=Astore, William J. |author3=Giffen, Robert B. |author4=Larson, Wiley J. 
|editor=Kirkpatrick, Douglas H.
|title= Understanding Space: An Introduction to Astronautics
|edition=2nd
|date=2004
|publisher=McGraw Hill
|isbn=0-07-294364-5
|pages= 228, 738}}

*{{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 ==
*[http://www.projectpluto.com/find_orb.htm#sphere_influence Project Pluto]

[[Category:Astrodynamics]]
[[Category:Orbits]]