page_title: Sphere of influence (astrodynamics)
section_title: Derivation
hierarchical_section_title: Sphere of influence (astrodynamics) / Derivation
caption_attribution_description: English: Hill sphere and Sphere of influence for planets in Solar System and Pluto
context_page_description: A sphere of influence 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 orbit around the larger body.
  and are the masses of the smaller and the larger object, respectively.
  In the patched conic approximation, once an object leaves the planet's SOI, the primary/only gravitational influence is the Sun.
context_section_description: 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 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 .
  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.
  The distance to the sphere of influence must thus satisfy  and so  is the radius of the sphere of influence of body
caption: Hill sphere and Sphere Of Influence for _Solar system bodies.
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