Gravitational Capture by the Major Primary in the Restricted Four-Body Problem

Paper number

IAC-06-C1.P.5.02

Author

Dr. Antonio Prado, National Institute for Space Research, Brazil

Year

2006

Abstract

The ballistic gravitational capture is a characteristic of some dynamical systems in celestial mechanics, as in the restricted four-body problem that is considered in this paper. The basic idea is that a spacecraft (or any particle with negligible mass) can change from a hyperbolic orbit with a small positive energy around a celestial body into an elliptic orbit with a small negative energy without the use of any propulsive system. The forces responsible for this modification in the orbit of the spacecraft are the gravitational forces of other bodies involved in the dynamics. In this way, those forces are used as a zero cost control, equivalent to a continuous thrust applied in the spacecraft. 
The application of this phenomenon in spacecraft trajectories is recent in the literature. The first demonstration of this was made by Belbruno in 1987. The first real application of a ballistic capture transfer was made during an emergency in a Japanese spacecraft, in 1990. 
Examining the literature related to the weak stability boundaries, one finds several definitions of ballistic gravitational capture, depending on the dynamical system considered. Those differences exist to account for the different behavior of the systems. In the restricted four-body problem, the system considered in the present paper, ballistic gravitational capture is assumed to occur when the massless particle stays close to one of the two primaries of the system for some time. A permanent capture is not required, because in this model it does not exist and the phenomenon is always temporary, which means that after some time of the approximation the massless particle escapes from the neighborhood of the primary. 
For the practical purposes of studying spacecraft trajectories, the majority of the papers available in the literature study this problem looking at the behavior of the two-body energy of the spacecraft with respect to the primary that performs the gravitational capture. A quantity called C3  , where V is the velocity of the spacecraft relative to the primary considered, r is the distance of the spacecraft from this primary and m is the dimensionless gravitational parameter of that primary, that is twice the total energy of a two-body system is defined.
Based upon this definition, it is possible to see that the value of C3 is related to the velocity variation (DV) needed to insert the spacecraft in its final orbit around the primary. In the case of a spacecraft approaching the Earth, it is possible to use the gravitational force of the Moon and the Sun to lower the value of C3 with respect to the Earth, so the fuel consumption required to complete this maneuver is reduced. 
The objective of the present paper is to study in some detail the ballistic gravitational capture performed by the first primary in a four body system. Analytical equations for the forces involved in this maneuver are derived to estimate their magnitude and to show the best directions of approach for the maneuver. The Earth-MoonSun system is used for the numerical evaluation.

Abstract document

IAC-06-C1.P.5.02.pdf

Manuscript document

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