Transfer orbits to/from the Lagrangian points in the restricted four-body problem
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 ( and ), called primaries, are orbiting their center of mass in circular orbits and a third body with negligible mass (not influencing the motion of the and ) 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 , so together with the initial prescribed position the complete initial state is known.
- (ii)
Integrate the equations of motion from until .
- (iii)
Check
Numerical results
The results are organized in graphs of the energy and the velocity increment 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)
Traveling between the Lagrangian Points and the Earth
Acta Astronautica
(1996)
Cited by (16)
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2013, Advances in Space ResearchCitation Excerpt :Hiday-Johnston and Howell (1994) formulated a strategy to design optimal time-fixed impulsive transfers between three-dimensional libration-point orbits based on the elliptic restricted three-body model. The four-body dynamic model was developed to consider the effects of the Sun or the Moon in the design of transfer orbits (Salmani and Büskens, 2011; Cabette and Prado, 2008; Assadian and Pourtakdoust, 2010). The Earth–Moon–Sun–Satellite system is used to study the transit trajectory properties in restricted three- and four-body problems (Circi, 2012).
Real-Time control of optimal low-Thrust transfer to the Sun-Earth halo orbit in the bicircular four-body problem
2011, Acta AstronauticaCitation Excerpt :Therefore, in spite of the fact that these orbits are the periodic solutions of the three-body problem, this model is not considered as the underlying model in this paper, since the gravitational effects of the Sun in the Earth–Moon three-body model is neglected, either way the force of the Moon in Sun–Earth/Moon model. To bridge the gap between these two three-body models, authors used four-body problem [7–11]. The special model so called bicircular four-body model places the Sun and Earth in circular orbits about their barycenter and places the Moon in circular orbits about the barycenter of the whole system.