On the Numerical Solution of Boundary Value Problems of the Restricted Three-Body Problem

This document translates the original German-language article (“Zur numerischen Lösung von Randwertaufgaben des eingeschränkten Dreikörperproblems”) by Kurt Riedorff and Helmut Altmann, published as a Sonderdruck (offprint) in Luftfahrttechnik Raumfahrttechnik, Vol. 14 (1968), pp. 220–221. Published by AEG-Telefunken, Anlagen Informationstechnik.

Authors: Kurt Riedorff, Helmut Altmann

Source: Luftfahrttechnik Raumfahrttechnik 14 (1968), pp. 220–221. Sonderdruck (Offprint).

Abstract

A boundary value problem of the restricted three-body problem is examined. With a given analog computer (Rechenanlage), it is shown that, given initial conditions for the third (infinitesimal) body at two freely selectable endpoints, and the desired final boundary conditions, the program solves the trajectory of a spacecraft at the lowest possible flight time, with the condition that the trajectory at the endpoints corresponds to a given circular orbit.

1. The Restricted Three-Body Problem

The restricted three-body problem is a classical problem of mechanics. While this problem has been thoroughly studied from the standpoint of analytical mechanics, the aim here is to find a practically useful, numerically stable solution. Practical and numerically relevant solutions exist with comparatively small effort.

Following well-known investigations by E. Strömgren and collaborators, the restricted three-body problem has recently attracted attention in connection with orbital flight trajectories.

The setup is as follows: Two finite bodies (A and B) orbit their common center of mass. A third, infinitesimally small body C is set into motion in the vicinity of these two bodies. The body C is described in a rotating coordinate system (x, y) which rotates with the angular velocity of the finite bodies A and B around their common center of mass. The body A is placed at the positive x-axis and the body B at the negative x-axis. A and B are assigned the normalized masses of 1 − μ and μ respectively (where μ = mass of B / total mass). The movement of the infinitesimal body C is described by the equations of motion:

ẍ − 2ẏ − x = −(1−μ)(x−μ)/r₁³ − μ(x+1−μ)/r₂³

ÿ + 2ẋ − y = −(1−μ)y/r₁³ − μy/r₂³

where r₁ and r₂ are the distances from C to A and B respectively.

Boundary conditions: Initial values for position and velocity at both endpoints. The body C (spacecraft) moves initially at time t = 0 from body A (Moon), and at the end moves to body B (Earth) at time t = t₁.

The masses of the two finite bodies are chosen as: Earth (body B) and Moon (body A), with μ = 0.01215.

2. The Boundary Value Problem

The equations of motion yield (for normalized units) a system of four first-order differential equations. The boundary conditions are formulated as:

At time t = 0, body C departs from point A with given coordinates x₀, y₀ and velocities ẋ₀, ẏ₀.
At time t = t₁, body C arrives at point B with given coordinates x₁, y₁ and velocities ẋ₁, ẏ₁.

The condition for a circular flight path at the endpoints is:

β = √(a₀ · e⁻²) · t₁ + a · e⁻² … (6)

The minimum flight time t₁ is determined such that the boundary conditions at the endpoints correspond to a given circular orbit.

3. Results

The results show that the numerical procedure L solves the restricted three-body problem successfully. The Rechenanlage BA 741 from AEG-Telefunken was used as the computing platform. By varying the starting conditions, two solutions were found with different flight paths. These are shown in Figures 1 and 2:

Figure 1: Trajectory of the infinitesimal spacecraft for given boundary conditions at the endpoints, with v₀ = 0.5745 km/s and y₀ = 1.536 km in the y-direction.
Figure 2: Trajectory with boundary conditions v₀ = 1.629 km/s and y₀ = 0.4 km in the y-direction.

The computation system (Rechenanlage BA 741) was found to be well-suited to this class of problems with respect to the capabilities of the computing system and the convergence of the procedure. The paper concludes that the general problem has no unique solution and two solutions were identified.

[Translation covers the first 2 pages (the complete document); no additional pages remain.]