Sphere of influence (astrodynamics)

For the term relating to black holes, see sphere of influence (black hole).

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.

The general equation describing the radius of the sphere ${\displaystyle r_{\text{SOI}}}$ of a planet: ^[1]

$${\displaystyle r_{\text{SOI}}\approx a\left({\frac {m}{M}}\right)^{2/5}}$$

where

• ${\displaystyle a}$ is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
• ${\displaystyle m}$ and ${\displaystyle M}$ 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.

Table of selected SOI radii

Dependence of Sphere of influence r_SOI/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): ^{[1] [2] [3] [4] [5] [6] [7]}

Body	SOI			Body Diameter		Body Mass (1024 kg)	Distance from Sun
	(106 km)	(mi)	(radii)	(km)	(mi)		(AU)	(106 mi)	(106 km)
Mercury	0.117	72,700	46	4,878	3,031	0.33	0.39	36	57.9
Venus	0.616	382,765	102	12,104	7,521	4.867	0.723	67.2	108.2
Earth + Moon	0.929	577,254	145	12,742 (Earth)	7,918 (Earth)	5.972 (Earth)	1	93	149.6
Moon	0.0643	39,993	37	3,476	2,160	0.07346	See Earth + Moon
Mars	0.578	359,153	170	6,780	4,212	0.65	1.524	141.6	227.9
Jupiter	48.2	29,950,092	687	139,822	86,881	1900	5.203	483.6	778.3
Saturn	54.5	38,864,730	1025	116,464	72,367	570	9.539	886.7	1,427.0
Uranus	51.9	32,249,165	2040	50,724	31,518	87	19.18	1,784.0	2,871.0
Neptune	86.2	53,562,197	3525	49,248	30,601	100	30.06	2,794.4	4,497.1

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 ${\displaystyle \theta }$ from the massive body. A more accurate formula is given by ^[1]

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

Averaging over all possible directions we get:

$${\displaystyle {\overline {r_{\text{SOI}}}}=0.9431a\left({\frac {m}{M}}\right)^{2/5}}$$

Derivation

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 ${\displaystyle C}$.

Geometry and dynamics to derive the sphere of influence

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 ${\displaystyle a_{A}={\frac {Gm_{B}}{R^{3}}}(r_{B}-r_{A})}$, 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. ${\displaystyle \chi _{A}={\frac {|g_{B}-a_{A}|}{|g_{A}|}}}$. 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 ${\displaystyle A}$, ${\displaystyle \chi _{A}\rightarrow 0}$ and ${\displaystyle \chi _{B}\rightarrow \infty }$, 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	${\displaystyle g_{A}={\frac {Gm_{A}}{r^{2}}}}$	${\displaystyle g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r\approx {\frac {Gm_{B}}{R^{2}}}}$
Frame acceleration	${\displaystyle a_{A}={\frac {Gm_{B}}{R^{2}}}}$	${\displaystyle a_{B}={\frac {Gm_{A}}{R^{2}}}\approx 0}$
Secondary acceleration	${\displaystyle g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r}$	${\displaystyle g_{A}={\frac {Gm_{A}}{r^{2}}}}$
Perturbation, tidal forces	${\displaystyle g_{B}-a_{A}\approx {\frac {Gm_{B}}{R^{3}}}r}$	${\displaystyle g_{A}-a_{B}\approx {\frac {Gm_{A}}{r^{2}}}}$
Perturbation ratio ${\displaystyle \chi }$	${\displaystyle \chi _{A}\approx {\frac {m_{B}}{m_{A}}}{\frac {r^{3}}{R^{3}}}}$	${\displaystyle \chi _{B}\approx {\frac {m_{A}}{m_{B}}}{\frac {R^{2}}{r^{2}}}}$

Hill sphere and Sphere Of Influence for Solar System bodies.

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 ${\displaystyle A}$

See also

• Hill sphere
• Sphere of influence (black hole)

References

1. 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.
2. Understanding Space: Artemis I Flight Day Eight: Orion Exits The Lunar Sphere Of Influence.
3. The Size of Planets
4. How Big Is the Moon?
5. The Mass of Planets
6. Moon Fact Sheet
7. Planet Distance to Sun, How Far Are The Planets From The Sun?

General references

• 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.

External links

• Project Pluto