Sphere of Influence
The sphere of influence of a celestial body is the region in space where the gravitational attraction of the body predominates over the attraction of all other celestial bodies. The concept can be defined more precisely according to the particular problem being considered. In studying the motion of comets outside the solar system, for example, the sun’s sphere of influence is taken as the region in which the forces of attraction of the stars are so small in comparison with the attractive force of the sun that they can be neglected. In investigating the motion of comets, other small bodies, and space probes inside the solar system, the spheres of influence of the planets are considered.
If a small body is located within the sphere of influence of a planet, it is useful to study the body’s motion in a coordinate system associated with the planet; the effect of the sun’s attraction is treated as a perturbation (seeCELESTIAL MECHANICS). The sphere of influence of a planet is found by disregarding the attraction of the other planets and is the region in which F1/R1 < F/R. Here, R is the acceleration imparted by the sun to a body in heliocentric motion (motion referred to the center of the sun), F is the perturbing acceleration introduced into the body’s motion by the attraction of the planet, R1 is the acceleration imparted by the planet to a body in planetocentric motion, and F1 is the perturbing acceleration introduced into this motion by the attraction of the sun. Outside the planet’s sphere of influence, it is more useful to regard heliocentric motion as primary.
The surface bounding the sphere of influence of a planet is close in shape to a spheroid. The center of the spheroid coincides with the center of the planet, and the spheroid’s polar axis is directed toward the sun. The polar radius ρp and the equatorial radius ρe of the spheroid are given by the equations
where r is the radius vector of the planet and m is the planet’s mass divided by the mass of the sun. Since pe = 1.15 pp and r varies only slightly, the sphere of influence of the planet is in practice regarded as a planetocentric sphere with the radius
where a is the semimajor axis of the planet’s orbit.
| Table 1. Spheres of influence of the planets | |
|---|---|
| Planet | ρ(AU) |
| Mercury ............... | 0.001 |
| Venus ............... | 0.004 |
| Earth ............... | 0.006 |
| Mars ............... | 0.004 |
| Jupiter ............... | 0.322 |
| Saturn ............... | 0.364 |
| Uranus ............... | 0.346 |
| Neptune ............... | 0.580 |
Table 1 gives the value of ρ in astronomical units (AU) for all the planets except Pluto. For Pluto, ρ = 0.22. Because, however, of the considerable variation in the radius vector r of Pluto, the radius of the sphere of influence varies from 0.18 to 0.30 AU.
In analyzing the motion of space vehicles heading toward the moon, the concept of the sphere of influence of the moon is used. The sphere of influence of the moon is defined in much the same way as that of a planet. In this case, the influences of the earth and the moon on the space vehicle are compared. The value of ρ for the sphere of influence of the moon is approximately 66,000 km.