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Wikipedia / Sphere of influence (astrodynamics)

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. As such it is the dominating part of a gravitational field. It is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moonss, 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.

Models

The most common base models to calculate the sphere of influence (sometimes also called gravisphere or gravity sphere) 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 a planet:

where

  • is the semimajor axis of the smaller object's (usually a planet's) orbit around the larger body (usually the Sun).

  • 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 rSOI 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

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):

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

from the massive body. A more accurate formula is given by

Averaging over all possible directions yields:

Derivation

.

.

to the separating surface.

Gravity well

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.

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.

See also

References

General references