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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. It is not to be confused with the sphere of activity which extends well beyond the sphere of influence.^{[1]}
The most common base models to calculate the sphere of influence is the Hill sphere and the Laplace sphere, but updated and particularly more dynamic ones have been described.^{[2]}^{[3]} The general equation describing the radius of the sphere of a planet:^{[4]}
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_{SOI} 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.
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):^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}
Body | SOI | Body Diameter | Body Mass (10^{24} kg) | Distance from Sun | |||||
---|---|---|---|---|---|---|---|---|---|
(10^{6} km) | (mi) | (radii) | (km) | (mi) | (AU) | (10^{6} mi) | (10^{6} km) | ||
Mercury | 0.117 | 72,700 | 46 | 4,878 | 3,031 | 0.33 | 0.39 | 36 | 57.9 |
Venus | 0.616 | 382,765 | 102 | 12,104 | 7,521 | 4.867 | 0.723 | 67.2 | 108.2 |
Earth + Moon | 0.929 | 577,254 | 145 | 12,742 (Earth) | 7,918 (Earth) | 5.972 (Earth) |
1 | 93 | 149.6 |
Moon | 0.0643 | 39,993 | 37 | 3,476 | 2,160 | 0.07346 | See Earth + Moon | ||
Mars | 0.578 | 359,153 | 170 | 6,780 | 4,212 | 0.65 | 1.524 | 141.6 | 227.9 |
Jupiter | 48.2 | 29,950,092 | 687 | 139,822 | 86,881 | 1900 | 5.203 | 483.6 | 778.3 |
Saturn | 54.5 | 38,864,730 | 1025 | 116,464 | 72,367 | 570 | 9.539 | 886.7 | 1,427.0 |
Uranus | 51.9 | 32,249,165 | 2040 | 50,724 | 31,518 | 87 | 19.18 | 1,784.0 | 2,871.0 |
Neptune | 86.2 | 53,562,197 | 3525 | 49,248 | 30,601 | 100 | 30.06 | 2,794.4 | 4,497.1 |
An important understanding to be drawn from the above table is that "Sphere of Influence" here is "Primary". For example, though Jupiter is much larger in mass than say, Neptune, its Primary SOI is much smaller due to Jupiter's much closer proximity to the Sun.
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^{[4]}
Averaging over all possible directions we get:
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 .
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 to 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