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Sphere of influence (astrodynamics)

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 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).
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 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.

Contents

Table of selected SOI radiiEdit

The table shows the values of the sphere of gravity of the bodies of the solar system in relation to the Sun.[1]:

Body SOI radius (106 km) SOI radius (body radii)
Mercury 0.112 46
Venus 0.616 102
Earth 0.924 145
Moon 0.0661 38
Mars 0.576 170
Jupiter 48.2 687
Saturn 54.6 1025
Uranus 51.8 2040
Neptune 86.8 3525

These are all taken relative to the Sun, except for the Moon, which is relative to the Earth.

Increased accuracy on the SOIEdit

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

 

Averaging over all possible directions we get

 

DerivationEdit

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

Let's 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 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    
Frame acceleration    
Secondary acceleration    
Perturbation, tidal forces    
Perturbation ratio      

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    
Frame acceleration    
Secondary acceleration    
Perturbation, tidal forces    
Perturbation ratio      

The distance to the sphere of influence must thus satisfy   and so   is the radius of the sphere of influence of body  

See alsoEdit

ReferencesEdit

  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.
  • 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.
  • Project Pluto