sphere of influence (astrodynamics)

The most common base models to calculate the sphere of influence is the Hill sphere and the Laplace sphere, but updated and particularly more dynamic ones have been described.{{Cite journal |last=Cavallari |first=Irene |last2=Grassi |first2=Clara |last3=Gronchi |first3=Giovanni F. |last4=Baù |first4=Giulio |last5=Valsecchi |first5=Giovanni B. |date=May 2023 |title=A dynamical definition of the sphere of influence of the Earth |journal=Communications in Nonlinear Science and Numerical Simulation |publisher=Elsevier BV |volume=119 |pages=107091 |arxiv=2205.09340 |bibcode=2023CNSNS.11907091C |doi=10.1016/j.cnsns.2023.107091 |issn=1007-5704 |s2cid=248887659}}{{Cite journal |last=Araujo |first=R. A. N. |last2=Winter |first2=O. C. |last3=Prado |first3=A. F. B. A. |last4=Vieira Martins |first4=R. |date=December 2008 |title=Sphere of influence and gravitational capture radius: a dynamical approach |journal=Monthly Notices of the Royal Astronomical Society |publisher=Oxford University Press (OUP) |volume=391 |issue=2 |pages=675–684 |bibcode=2008MNRAS.391..675A |doi=10.1111/j.1365-2966.2008.13833.x |issn=0035-8711 |doi-access=free |hdl-access=free |hdl=11449/42361}}

The general equation describing the radius of the sphere $r\_\backslash text\{SOI\}$ of a planet:{{Cite book |last=Seefelder |first=Wolfgang |url=https://books.google.com/books?id=NVg_vYHePt0C&pg=PA76 |title=Lunar Transfer Orbits Utilizing Solar Perturbations and Ballistic Capture |publisher=Herbert Utz Verlag |year=2002 |isbn=978-3-8316-0155-4 |location=Munich |page=76 |access-date=July 3, 2018}}

$$r\_\backslash text\{SOI\}\; \backslash approx\; a\backslash left(\backslash frac\{m\}\{M\}\backslash right)^\{2/5\}$$

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 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.

File:Sphere of influence on ratio m M.png

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):{{Cite web |last=Vereen |first=Shaneequa |date=23 November 2022 |title=Artemis I – Flight Day Eight: Orion Exits the Lunar Sphere Of Influence |url=https://blogs.nasa.gov/artemis/2022/11/23/artemis-i-flight-day-eight-orion-exits-the-lunar-sphere-of-influence |publisher=NASA Blogs}}{{Cite web |date=23 May 2013 |title=The Size of Planets |url=https://planetfacts.org/size-of-planets-in-order |website=Planet Facts}}{{Cite web |date=4 June 2012 |title=How Big Is the Moon? |url=https://planetfacts.org/how-big-is-the-moon |website=Planet Facts}}{{Cite web |date=9 May 2012 |title=The Mass of Planets |url=https://www.outerspaceuniverse.org/the-mass-of-planets-how-much-do-the-planets-in-our-solar-system-weigh.html |website=Outer Space Universe}}{{Cite web |title=Moon Fact Sheet |url=https://nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html |website=NASA Space Science Data Coordinated Archive}}{{Cite web |date=5 March 2021 |title=Planet Distance to Sun, How Far Are The Planets From The Sun? |url=https://www.cleverlysmart.com/planet-distance-to-sun-how-far-are-the-planets-from-sun/ |website=CleverlySmart}}

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 $\backslash theta$ from the massive body. A more accurate formula is given by

$$r\_\backslash text\{SOI\}(\backslash theta)\; \backslash approx\; a\backslash left(\backslash frac\{m\}\{M\}\backslash right)^\{2/5\}\backslash frac\{1\}\{\backslash sqrt[10]\{1+3\backslash cos^2(\backslash theta)\}\}$$

Averaging over all possible directions we get:

$$\backslash overline\{r\_\backslash text\{SOI\}\}\; =\; 0.9431\; a\backslash left(\backslash frac\{m\}\{M\}\backslash right)^\{2/5\}$$

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 $C$.

File:Sphereofinfluence.png

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 to their gravitational interactions, point $A$ is attracted to point $B$ with acceleration $a\_A\; =\; \backslash frac\{Gm\_B\}\{R^3\}\; (r\_B-r\_A)$, 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. $\backslash chi\_A\; =\; \backslash frac$

|}

As $C$ gets close to $A$, $\backslash chi\_A\; \backslash rightarrow\; 0$ and $\backslash chi\_B\; \backslash rightarrow\; \backslash infty$, and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which $\backslash chi\_A\; =\; \backslash chi\_B$ separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say $m\_A\; \backslash ll\; m\_B$, 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.

File:Hill sphere and SOI.png

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

Gravity well (or funnel) is a metaphorical concept for a gravitational field of a mass, with the field being curved in a funnel-shaped well around the mass, illustrating the steep gravitational potential and its energy that needs to be accounted for in order to escape or enter the main part of a sphere of influence.{{cite book | last=May | first=Andrew | title=How Space Physics Really Works: Lessons from Well-Constructed Science Fiction | publisher=Springer Nature Switzerland | publication-place=Cham | date=2023 | isbn=978-3-031-33949-3 | doi=10.1007/978-3-031-33950-9 | page=}}

An example for this is the strong gravitational field of the Sun and Mercury being deep within it.{{cite journal | last=Mann | first=Adam | title=NASA mission set to orbit Mercury | journal=Nature | date=2011-03-08 | issn=0028-0836 | doi=10.1038/news.2011.142 | doi-access=free | url=https://www.nature.com/articles/news.2011.142.pdf | access-date=2025-03-03 | page=}} At perihelion Mercury goes even deeper into the Sun's gravity well, causing an anomalistic or perihelion apsidal 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 by Albert Einstein through his formulation of gravity with the speed of light, and the corresponding general relativity theory, eventually being one of the first cases proving the theory.

File:Schwarzschild effective potential.svg of schwarzschild geodesics for various angular momenta. Each point on the curves represent a radius or circular orbit and the curve represents their stability depending on the energy of their particle, with orbits therefore normally not remaining circular and migrating along the curve. At small radii, the energy drops precipitously, causing the particle to be pulled inexorably inwards to $r\; =\; 0$. However, when the normalized angular momentum $\backslash frac\{a\}\{r\_\backslash text\{s\}\}\; =\; \backslash frac\{L\}\{mcr\_\backslash text\{s\}\}$ equals the square root of three, a metastable circular orbit is possible at the radius highlighted with a green circle. At higher angular momenta, there is a significant centrifugal barrier (orange curve) or energy hill{{cite book | last=Wheeler | first=John Archibald | title=A journey into gravity and spacetime | publisher=Scientific American Library | publication-place=New York | date=1999 | isbn=978-0-7167-5016-1 | page=173ff}} and an unstable inner radius, highlighted in red.]]

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

{{reflist}}

• {{Cite book |last=Bate |first=Roger R. |url=https://archive.org/details/fundamentalsofas00bate |title=Fundamentals of astrodynamics |last2=Mueller |first2=Donald D. |last3=White |first3=Jerry E. |date=1971 |publisher=Dover Publications |isbn=978-0-486-60061-1 |series=Dover books on astronomy |location=New York |pages=[https://archive.org/details/fundamentalsofas00bate/page/333 333–334] |url-access=registration}}
• {{Cite book |last=Sellers |first=Jerry Jon |url=https://archive.org/details/understandingspa0000sell |title=Understanding space: an introduction to astronautics |last2=Astore |first2=William J. |last3=Giffen |first3=Robert B. |last4=Larson |first4=Wiley J. |date=2015 |publisher=McGraw-Hill Companies |isbn=978-0-9904299-4-4 |editor-last=Kirkpatrick |editor-first=Douglas |editor-link=McQuade |editor-mask=Marilyn |edition=4nd |publication-place=New York |pages=[https://archive.org/details/understandingspa0000sell/page/n245 228], 738 |url-access=registration}}
• {{Cite book |last=Danby |first=J. M. A. |url=https://archive.org/details/fundamentalsofce0000danb_q3w0 |title=Fundamentals of celestial mechanics |date=1992 |publisher=Willmann-Bell |isbn=978-0-943396-20-0 |edition=2nd |location=Richmond, Va., U.S.A |pages=352–353}}

• [http://www.projectpluto.com/find_orb.htm#sphere_influence Project Pluto]

Category:Astrodynamics

Category:Orbits