Sphere of Influence

 

Sphere of Influence

 

Of the various possible perturbations to basic two-body motion, the most obvious are those due to the presence of additional bodies. Such bodies are always present and cannot be easily included in an analysis, particularly at elementary levels. It is then necessary to determine criteria for the validity of Keplerian approximations to real orbits when more than two bodies are present.
If we consider a spacecraft in transit between two planets, it is clear that when close to the departure planet, its orbit is primarily subject to the influence of that
planet. Far away from any planet, the trajectory is essentially a heliocentric orbit, whereas near the arrival planet, the new body will dominate the motion. There
will clearly be transition regions where two bodies will both have significant influence on the spacecraft motion. The location of these transition regions is
determined by the so-called sphere of influence of each body relative to the other, a concept originated by Laplace. For any two bodies, such as the sun and a planet, the sphere of influence is
defined by the locus of points at which the sun’s and the planet’s gravitational fields have equal influence on the spacecraft. The term “sphere of influence” is
somewhat misleading; every body’s gravitational field extends to infinity, and in any case, the appropriate equal-influence boundary is not exactly spherical. Nonetheless, the concept is a useful one in preliminary design. Although the relative regions of primary influence can be readily calculated for any two bodies, the concept is most useful when, as mentioned earlier, one of
the masses is much greater than the other. In such a case, the so-called classical sphere of influence about the small body has the approximate radius

 

where
r = sphere of influence radius
R = distance between primary bodies m and M
m = mass of small body
M = mass of large body