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,

Models

The most common base models to calculate the sphere of influence are the Hill sphere and the Laplace sphere, but updated and other models, as by Gleb Chebotaryov or particularly more dynamic ones, like the patched conic approximation, have been described.

The general equation describing the radius of the sphere <math>r_\text{SOI}</math> of a planet:

<math display="block">r_\text{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 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<sub>SOI</sub> 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

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):

<div class="noresize" style="clear: both;">

{| class="wikitable" style="text-align: center;"

|-

! rowspan=2 | Body

! colspan=3 | SOI

! colspan=2 | Body Diameter

! rowspan=2 | Body Mass (10<sup>24</sup>&nbsp;kg)

! colspan=3 | Distance from Sun

|-

! (10<sup>6</sup>&nbsp;km) !! (mi) !! (radii)

! (km) !! (mi)

! (AU) !! (10<sup>6</sup>&nbsp;mi) !! (10<sup>6</sup>&nbsp;km)

|-

| align="left" | Mercury

| 0.117 || 72,700 || 46 || 4,878 || 3,031 || 0.33 || 0.39 || 36 || 57.9

|-

| align="left" | Venus

| 0.616 || 382,765 || 102 || 12,104 || 7,521 ||4.867 || 0.723 || 67.2 || 108.2

|-

| align="left" | Earth + Moon

| 0.929 || 577,254 || 145 || 12,742 (Earth) || 7,918 (Earth) || 5.972<br/>(Earth) || 1 || 93 || 149.6

|-

| align="left" | Moon

| 0.0643 || 39,993 || 37 || 3,476 || 2,160 || 0.07346 || colspan="3" | See Earth + Moon

|-

| align="left" | Mars

| 0.578 || 359,153 || 170 || 6,780 || 4,212 || 0.65 || 1.524 || 141.6 || 227.9

|-

| align="left" | Jupiter

| 48.2 || 29,950,092 || 687 || 139,822 || 86,881 || 1900 || 5.203 || 483.6 || 778.3

|-

| align="left" | Saturn

| 54.5 || 38,864,730 || 1025 || 116,464 || 72,367 || 570 || 9.539 || 886.7 || 1,427.0

|-

| align="left" | Uranus

| 51.9 || 32,249,165 || 2040 || 50,724 || 31,518 || 87 || 19.18 || 1,784.0 || 2,871.0

|-

| align="left" | Neptune || 86.2 || 53,562,197 || 3525 || 49,248 || 30,601 || 100 || 30.06 || 2,794.4 || 4,497.1

|}

</div>

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 Neptune, its Primary SOI is much smaller due to Jupiter's 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 <math>\theta</math> from the massive body. A more accurate formula is given by

An example for this is the strong gravitational field of the Sun and Mercury being deep within it. 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.

thumb|350px|right|Gravity well illustrated with the [[Effective potential#Gravitational potential|effective radial potentials 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 <math display="inline">r = 0</math>. However, when the normalized angular momentum <math display="inline">\frac{a}{r_\text{s = \frac{L}{mcr_\text{s</math> 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 and an unstable inner radius, highlighted in red.]]

See also

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

References

General references

  • Project Pluto