THE CHAOTIC VARIATION OF THE CAPTURE EFFECT

IN THE THREE BODY PROBLEM

Edit-Mária Garda-Mátyás1, Zoltán Makó,1,2, Ferenc Szenkovits2, Iharka Csillik3

1 Sapientia University, Department of Mathematics and Informatics,

530104 – Miercurea-Ciuc, Romania

2 Babeş – Bolyai University, Faculty of Mathematics and Computer Science,

400084 – Cluj-Napoca, Romania

3 Astronomical Institute of Romanian Academy, 400487 – Cluj-Napoca, Romania

INTRODUCTION

The gravitational capture is a phenomenon, when a massless particle changes its Kepler-energy around one of the primaries from positive to negative. This capture is always temporary and, after some time, the Kepler-energy changes back to positive and the massless spacecraft leaves the neighborhood of the primary. The temporary capture is when we can be used to decrease the fuel expenditure for a mission going from one of the primaries to the other primaries, like an Earth-Moon mission.

Many authors studied this problem, introducing different types of capture, like weakly capture (Belbruno, 1999), temporary capture (Brunini, 1996), longest capture (Winter, Vieira, 2000), resonant capture (Yu, Tremaine, 2001), etc.

In all these studies the time is used as measure of the capture. In this paper we try to study the phenomenon of capture using the variation of the angle of the small body around the capturing planet. We introduce the capture effect of the planet to the captured body, as the variation of the angle Dj during the capture, as long as the Kepler-energy of the small body relative to the central planet is negative (see Fig. 1). The beginning moment the capture is in the moment when the Kepler-energy of the captured body, relative to the capturing body, becomes negative. The end of capture is in the moment when the Kepler-energy becomes positive.

Definition 1. Let and the moments of the beginning and the end of capture. The variation of ’s angle around to depends continue on the time. We assume that there is a partition , such as in each interval the variation of angle is constant. The capture effect of to initial condition of is

, (1)

where (see Fig. 1).

Definition 2. The capture domain of effect is

. (2)

Fig. 1.

The capture effect

DETERMINATION OF CAPTURE EFFECT IN ER3BP

The simplex model which allow use to study the real capture phenomenon is the elliptic restricted three-body problem (ER3BP). If we study the capture phenomenon using circular restricted three-body problem (CR3BP), the Jacobi constant is greater than C1 critical value of Hill’s regions, therefore the asteroid does not enter in the Hill's region of primaries, therefore the primaries can’t capture the asteroid. In reality, Jacobi "constant" fluctuate approximately periodic, and this fluctuation is rapidly near the primaries, so the capture phenomenon is possible [10]. The elliptical orbit of the primaries perturb “Jacobi” constant in the highest degree.

We characterize the phenomenon of capture using the model of the elliptic restricted three-body problem.

The ER3BP describes the motion of three bodies under they mutual gravitational attraction, if

i) two bodies, named "primaries" and with masses and move under their mutual attraction and their motion is elliptical;

ii) the third body of this system has an infinitesimal mass and is subjected to the attraction of the two primaries.

The differential equations of motion of the ER3BP are presented using a nonuniformly rotating and pulsating coordinate system. From that follows dimensionless variables, which are introduced by using the distance between the primaries,

, (3)

as the quantity which distances and dimensionless coordinates are divided. Here a and e are the semi major axis and the eccentricity of the elliptic orbit of either primary around the other and u is the true anomaly of P2 (see Fig. 2).

Fig. 2.

ER3BP

The equations of motion of the elliptic restricted three-body problem using the true anomaly as independent variable may be written as [9]:

(4)

where

, (5)

, (6)

(7)

. (8)

The capture effect can be determined by using numerical methods. Our algorithm has the following steps:

•  The orbital elements of are determined when is in the perihelion, before the capture begins.

•  The position of the capturing planet is determined at the time γ, where γ is the time of ’s transition to the perihelion.

•  Initial conditions are determined.

•  Equations of motion is numerically integrated. At each step, the Kepler-energy

. (9)

respect to the capturing planet is evaluated. The variation of the angle is summed, from the beginning to the end of the capture.

THE VARIATION OF CAPTURE EFFECT AROUND TO EARTH

The variation of capture effect around to Earth is studied using sections. In the ER3BP model we put the following initial conditions: z=0, ż=0, the initial velocity is fixed and perpendicular to the initial position vector. Fig. 3 presents the variation of the capture effect on initial position. Gradations of gray varies from black to white when the capture effect increases from 0° to 40000°. The complicated structure of the Fig. 3 shows the complexity of the phenomenon of capture.

We are determined five zones, in which, for a given velocity, the capture is probably (see Fig. 4). These regions are chaotic, since, if we vary insignificantly the initial conditions, than the capture effect can change greatly.

Fig. 3.

The variation of capture effect around to Earth

Fig. 4.

The five positive effect zone around to Earth

ACKNOWLEDGEMENTS

This work was supported by the Research Programs Institute of Foundation Sapientia under grand E/CS/432/25.03.2003.

REFERENCES

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