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  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 t a and t b the moments of the beginning and the end of capture.
The variation of P3 ’s angle around to P2 depends continue on the time. We assume that
there is a partition t a  t 0  t1  t 2  ...  t n  t b , such as in each interval [t i , t i 1 ] the
variation of angle is constant. The capture effect of P2 to initial condition
( x0 , y0 , z 0 , x 0 , y 0 , z0 ) of P3 is
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n 1
 ( x0 , y 0 , z 0 , x 0 , y 0 , z 0 )    i 1   i ,
(1)
i 0
where  i   (t i ) (see Fig. 1).
Definition 2. The capture domain of effect  is
S  {( x0 , y0 , z 0 , x 0 , y 0 , z0 ) /  ( x0 , y0 , z 0 , x 0 , y 0 , z0 )   } .
(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 P2 perturb “Jacobi” constant in the highest degree.
We characterize the phenomenon of capture using the model of the elliptic
restricted three-body problem.
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The ER3BP describes the motion of three bodies under they mutual gravitational
attraction, if
i) two bodies, named "primaries" P1 and P2 with masses m1 and m2 move under
their mutual attraction and their motion is elliptical;
ii) the third body of this system has an infinitesimal mass m3 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,
l
a(1  e 2 )
,
1  e cos u
(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]:
d 2 x
dy 

,
 2 2
du

x
du

d 2 y
dx 

,
 2 2
du y
 du
 d 2 z 
,
 2 
z
 du
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(4)
where

,
1  e cos u
(5)
1 2
1   1
(x  y 2  z 2 ) 
   (1   ) ,
2
r1
r2 2
(6)


r1  ( x   ) 2  y 2 , r2  ( x  1   ) 2  y 2 ,

m2
.
m1  m2
(7)
(8)
The capture effect can be determined by using numerical methods. Our algorithm
has the following steps:
• The orbital elements of P3 are determined when P3 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 P3 ’s transition to the perihelion.
• Initial conditions are determined.
• Equations of motion is numerically integrated. At each step, the Kepler-energy
EK 
1 2

( x  y 2  z 2 )  .
2
r2
(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.
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Fig. 3.
The variation of capture effect around to Earth
Fig. 4.
The five positive effect zone around to Earth
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ACKNOWLEDGEMENTS
This work was supported by the Research Programs Institute of Foundation
Sapientia under grand E/CS/432/25.03.2003.
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