# Sphere of Influence Module¶

Sphere of Influence

Contains methods to compute radius of sphere of influence.

Laplace Radius: A laplace 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 laplace radius is taken as the boundary where the trajectory switches which mass field it is influenced by. The result is:

$a\left(\frac{m}{M}\right)^{\frac{2}{5}}$

Hill Radius: In the three body problem, if that third object stays within an extremely complex boundary called the Roche lobe, the orbit of that third object about the smaller body will be stable for at least some amount of time. The Roche lobe just touches the L1 and L2 points and fans out from there. George Hill used the L1 point to define a sphere that approximated the Roche lobe. This is still intractable; the L1 point is defined by a fifth order polynomial that cannot be solved in terms of the elementary functions. Hill further simplified things by realizing that a simple cubic equation yields a very good approximation of that intractable fifth order equation. The result is:

$a\left(\frac{m}{3M}\right)^{\frac{1}{3}}$
poliastro.threebody.soi.laplace_radius(body, a=None)

Approximated radius of the Laplace Sphere of Influence (SOI) for a body.

Parameters: body (~poliastro.bodies.Body) – Astronomical body which the SOI’s radius is computed for. a (float, optional) – Semimajor axis of the body’s orbit, default to None (will be computed from ephemerides). Approximated radius of the Sphere of Influence (SOI) [m] astropy.units.quantity.Quantity
poliastro.threebody.soi.hill_radius(body, a=None, e=None)

Approximated radius of the Hill Sphere of Influence (SOI) for a body.

Parameters: body (~poliastro.bodies.Body) – Astronomical body which the SOI’s radius is computed for. a (float, optional) – Semimajor axis of the body’s orbit, default to None (will be computed from ephemerides). e (float, optional) – Eccentricity of the body’s orbit, default to 0 (will be computed from ephemerides). Approximated radius of the Sphere of Influence (SOI) [m] astropy.units.quantity.Quantity