Arenstorf Orbit

class otp.arenstorf.ArenstorfProblem

Bases: otp.Problem

ARENSTORFPROBLEM A simplified 3-body problem This problem models the trajectory of a satellite of negligible mass under the gravitational pull of two orbiting bodies. These can be considered the Moon and Earth with masses MU and MU’=1-MU, respectively. The reference frame is chosen such that the Earth is at position (-MU, 0) on the plane, and the moon is at position (MU’, 0). The satellite’s motion is described by the second-order ODEs

y1’’ = y1 + 2 y2’ - MU’ (y1 + MU) / D1 - MU (y1 - MU’) / D2, y2’’ = y2 - 2 y1’ - MU’ y2 / D1 - MU y2 / D2,

where

D1 = ((y1 + MU)^2 + y2^2)^(3/2), D2 = ((y1 - MU’)^2 + y2^2)^(3/2).

This is converted into a system of first order ODEs in which the state contains two position variables and two velocity variables:

y = [y1; y2; y1’; y2’].

Sources:

Arenstorf, Richard F. “Periodic Solutions of the Restricted Three Body Problem Representing Analytic Continuations of Keplerian Elliptic Motions.” American Journal of Mathematics, vol. 85, no. 1, 1963, p. 27.

Hairer, Ernst, et al. Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd ed., vol. 8, Springer-Verlag, 1993, pp. 129-130.

Parameters

class otp.arenstorf.ArenstorfParameters

Bases: otp.Parameters

Parameters for the Arenstorf problem.

Constructor Summary

ArenstorfParameters(varargin)

Create an Arenstorf parameters object.

Parameters:: varargin – A variable number of name-value pairs. A name can be any property of this class, and the subsequent value initializes that property.

Property Summary

Mu: The mass of one body, while the other body has mass \(\mu' = 1 - \mu\).

Presets

class otp.arenstorf.presets.Canonical

Bases: otp.arenstorf.ArenstorfProblem

CANONICAL One period of a satellite moving in an Earth-Moon system Source:

Hairer, Ernst, et al. Solving Ordinary Differential Equations I: Nonstiff Problems. 2nd ed., vol. 8, Springer-Verlag, 1993, pp. 129-130.

Constructor Summary

Canonical(varargin)

CANONICAL Construct a canonical Arenstorf problem OBJ = CANONICAL(MU) Uses a Moon of mass MU. The time span and initial conditions use the same data type as MU. Single precision, vpa, and other types are supported.

OBJ = CANONICAL() Uses a MU corresponding to the Moon