Region of space gravitationally dominated by a given body
A
sphere of influence
(
SOI
) in
astrodynamics
and
astronomy
is the
oblate-spheroid
-shaped region where a particular
celestial body
exerts the main
gravitational
influence on an
orbiting
object. 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 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 the
sphere of activity
which extends well beyond the sphere of influence.
[1]
Models
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]
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.
[2]
[3]
The general equation describing the
radius
of the sphere
of a planet:
[4]
![{\displaystyle r_{\text{SOI}}\approx a\left({\frac {m}{M}}\right)^{2/5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2714e6d7f0dad2482f560e0a341da91d7506b06)
where
is the
semimajor axis
of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).
and
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
[
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]
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):
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Body
|
SOI
|
Body Diameter
|
Body Mass (10
24
kg)
|
Distance from Sun
|
(10
6
km)
|
(mi)
|
(radii)
|
(km)
|
(mi)
|
(AU)
|
(10
6
mi)
|
(10
6
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
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]
The Sphere of influence is, in fact, not quite a sphere. The distance to the SOI depends on the angular distance
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 )}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05a5c6c2fe78947b0074f5e9b7e2bffab910ec63)
Averaging over all possible directions we get:
![{\displaystyle {\overline {r_{\text{SOI}}}}=0.9431a\left({\frac {m}{M}}\right)^{2/5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb366565bfa4c096cab14fbae7d19d05950b27a7)
Derivation
[
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]
Consider two point masses
and
at locations
and
, with mass
and
respectively. The distance
separates the two objects. Given a massless third point
at location
, one can ask whether to use a frame centered on
or on
to analyse the dynamics of
.
Geometry and dynamics to derive the sphere of influence
Consider a frame centered on
. The gravity of
is denoted as
and will be treated as a perturbation to the dynamics of
due to the gravity
of body
. Due to their gravitational interactions, point
is attracted to point
with acceleration
, 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.
. The perturbation
is also known as the tidal forces due to body
. It is possible to construct the perturbation ratio
for the frame centered on
by interchanging
.
|
Frame A
|
Frame B
|
Main acceleration
|
![{\displaystyle g_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7ba6d327db9359457da13a61d21888cd7b0037) |
|
Frame acceleration
|
![{\displaystyle a_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/187a574b62266ad7c236172391d575b150e69bc9) |
|
Secondary acceleration
|
![{\displaystyle g_{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c4b43aa900ad18b2b3712865fcfcb7d24534219) |
|
Perturbation, tidal forces
|
![{\displaystyle g_{B}-a_{A}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e259c77282cd6862c5a6d50ddec38f124aa72cf7) |
|
Perturbation ratio
![{\displaystyle \chi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437) |
![{\displaystyle \chi _{A}={\frac {|g_{B}-a_{A}|}{|g_{A}|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/405de39ad1ea9628b9a639497fefb0dce7818fa5) |
|
As
gets close to
,
and
, and vice versa. The frame to choose is the one that has the smallest perturbation ratio. The surface for which
separates the two regions of influence. In general this region is rather complicated but in the case that one mass dominates the other, say
, it is possible to approximate the separating surface. In such a case this surface must be close to the mass
, denote
as the distance from
to the separating surface.
|
Frame A
|
Frame B
|
Main acceleration
|
![{\displaystyle g_{A}={\frac {Gm_{A}}{r^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ae94ab4f19cf34d88f8880a1eea5f9cd0ca868) |
|
Frame acceleration
|
![{\displaystyle a_{A}={\frac {Gm_{B}}{R^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220f3fa15340c431ffadff2cc05b10ec677708b7) |
|
Secondary acceleration
|
![{\displaystyle g_{B}\approx {\frac {Gm_{B}}{R^{2}}}+{\frac {Gm_{B}}{R^{3}}}r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab89d5f70c0208386612d9bb40031688a41f698c) |
|
Perturbation, tidal forces
|
![{\displaystyle g_{B}-a_{A}\approx {\frac {Gm_{B}}{R^{3}}}r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfd4409a2b8cab54afa7b972a43bfe7baae3bbee) |
|
Perturbation ratio
![{\displaystyle \chi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/656111758322ace96d80a9371771aa6d3de25437) |
![{\displaystyle \chi _{A}\approx {\frac {m_{B}}{m_{A}}}{\frac {r^{3}}{R^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c223d1b5c593b674c345d1f0035bd6f95fcd5e) |
|
Hill sphere and Sphere Of Influence for Solar System bodies
The distance to the sphere of influence must thus satisfy
and so
is the radius of the sphere of influence of body
Gravity well
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]
Gravity well
is a metaphorical name for the sphere of influence, highlighting the
gravitational potential
that shapes a sphere of influence, and that needs to be accounted for to escape or stay in the sphere of influence.
See also
[
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]
References
[
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]
- ^
Souami, D.; Cresson, J.; Biernacki, C.; Pierret, F. (2020). "On the local and global properties of gravitational spheres of influence".
Monthly Notices of the Royal Astronomical Society
.
496
(4): 4287?4297.
arXiv
:
2005.13059
.
doi
:
10.1093/mnras/staa1520
.
- ^
Cavallari, Irene; Grassi, Clara; Gronchi, Giovanni F.; Bau, Giulio; Valsecchi, Giovanni B. (2023). "A dynamical definition of the sphere of influence of the Earth".
Communications in Nonlinear Science and Numerical Simulation
.
119
. Elsevier BV: 107091.
arXiv
:
2205.09340
.
Bibcode
:
2023CNSNS.11907091C
.
doi
:
10.1016/j.cnsns.2023.107091
.
ISSN
1007-5704
.
S2CID
248887659
.
- ^
Araujo, R. A. N.; Winter, O. C.; Prado, A. F. B. A.; Vieira Martins, R. (2008-12-01).
"Sphere of influence and gravitational capture radius: a dynamical approach"
.
Monthly Notices of the Royal Astronomical Society
.
391
(2). Oxford University Press (OUP): 675?684.
Bibcode
:
2008MNRAS.391..675A
.
doi
:
10.1111/j.1365-2966.2008.13833.x
.
hdl
:
11449/42361
.
ISSN
0035-8711
.
- ^
a
b
c
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
.
- ^
Understanding Space: Artemis I Flight Day Eight: Orion Exits The Lunar Sphere Of Influence.
, 23 November 2022
- ^
The Size of Planets
, 23 May 2013
- ^
How Big Is the Moon?
, 4 June 2012
- ^
The Mass of Planets
, 9 May 2012
- ^
Moon Fact Sheet
- ^
Planet Distance to Sun, How Far Are The Planets From The Sun?
, 5 March 2021
General references
[
edit
]
- 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
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]