Elsevier

Acta Astronautica

Volume 63, Issues 11–12, December 2008, Pages 1221-1232
Acta Astronautica

Transfer orbits to/from the Lagrangian points in the restricted four-body problem

https://doi.org/10.1016/j.actaastro.2008.05.005Get rights and content

Abstract

The well-known Lagrangian points that appear in the planar restricted three-body problem are very important for astronautical applications. They are five points of equilibrium in the equations of motion, what means that a particle located at one of those points with zero velocity will remain there indefinitely. The collinear points (L1, L2 and L3) are always unstable and the triangular points (L4 and L5) are stable in the present case studied (Earth–Sun system). They are all very good points to locate a space-station, since they require a small amount of ΔV (and fuel), the control to be used, for station-keeping. The triangular points are especially good for this purpose, since they are stable equilibrium points.

In this paper, the planar restricted four-body problem applied to the Sun–Earth–Moon–Spacecraft is combined with numerical integration and gradient methods to solve the two-point boundary value problem. This combination is applied to the search of families of transfer orbits between the Lagrangian points and the Earth, in the Earth–Sun system, with the minimum possible cost of the control used. So, the final goal of this paper is to find the magnitude of the two impulses to be applied in the spacecraft to complete the transfer: the first one when leaving/arriving at the Lagrangian point and the second one when arriving/living at the Earth.

The dynamics given by the restricted four-body problem is used to obtain the trajectory of the spacecraft, but not the position of the equilibrium points. Their position is taken from the restricted three-body model. The goal to use this model is to evaluate the perturbation of the Sun in those important trajectories, in terms of fuel consumption and time of flight. The solutions will also show how to apply the impulses to accomplish the transfers under this force model.

The results showed a large collection of transfers, and that there are initial conditions (position of the Sun with respect to the other bodies) where the force of the Sun can be used to reduce the cost of the transfers.

Introduction

This paper has the goal of studying models and methods used for the calculation of optimal orbital trajectories, in the sense of using a small amount of fuel [3]. It considers the transfer of the space vehicle between the Earth and the Lagrangian Points in the Sun–Earth–Moon system. The Lagrangian Points are determined using the model of the circular restricted three-body problem, but the trajectories are studied with the bi-circular restricted problem of four bodies.

So, the problem is to transfer a space vehicle between two given points with the minimal possible amount of fuel. There are several important factors in a transfer of this type, like the time used for the transfer, the state of the vehicle, etc. Therefore, in this work, the amount of fuel is the critical element of the maneuvers, although the time required by the maneuver it is also verified.

Section snippets

The planar circular restricted three-body problem

The circular planar restricted problem of three-bodies is defined as follows: two bodies revolve around their center of mass under the influence of their mutual gravitational attraction and a third body moves in the plane defined by the two revolving bodies. Therefore, the model assumes that two point masses (M1 and M2), called primaries, are orbiting their center of mass in circular orbits and a third body with negligible mass M3 (not influencing the motion of the M1 and M2) is orbiting the

The two point boundary value problem

The problem that is considered in the present paper can be formulated as:

“Find an orbit (in the restricted four-body problem context) that makes a spacecraft to leave a given point A and goes to another given point B, arriving there after a specified time of flight”. To solve this problem the following steps are used:

  • (i)

    Guess an initial velocity Vi, so together with the initial prescribed position ri the complete initial state is known.

  • (ii)

    Integrate the equations of motion from t0=0 until tf.

  • (iii)

    Check

Numerical results

The results are organized in graphs of the energy and the velocity increment (ΔV) in the rotating system against the time of flight.

Conclusions

In this paper, the bi-circular restricted four-body problem is used to find families of transfer orbits between the Earth and all the five Lagrangian points that exist in the Earth–Sun system.

Two models were used, considering or not the fourth body. From the results, it is clear that the inclusion of the fourth body can help mission designers to save fuel. The differences are large enough to be used in favor of the mission. It is also clear that the time of flight can be reduced by using the

References (3)

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