CAPTURE MODEL IN THE RESTRICTED THREE-BODY PROBLEM
Szenkovits Ferenc1, Makó Zoltán2, Csillik Iharka3, Bálint Attila4
PhD, lecturer1, PhD student, assistant2, PhD student3, Master student4
Babes-Bolyai University Cluj-Napoca
1. INTRODUCTION
The phenomenon of capture is very important with applications in study of comets and asteroids. This phenomenon can be studied using the model of the restricted three-body problem.
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.
Belbruno and Marsden (1997) discussed the motion of ten short-period comets strongly perturbed by Jupiter. In their views the motion of comet about Jupiter is stable if, when starting with elliptical initial condition with respect to Jupiter and returns to a reference plan through its initial condition without first having moved around the Sun (Fig. 1).
It is a difficult, unsolved problem to give the necessary conditions of this stability. In this paper we give a sufficient condition to decide if an orbit with given initial condition is stable.
Our “capture domain” of capture effect 2 is a subset of that initial conditions, from which arise stable motions.
The capture domain of a capture effect is determined using numerical methods.
2. THE CAPTURE DOMAIN
We characterise the phenomena of capture using the model of the circular restricted three-body problem in polar coordinates.
The circular restricted three-body problem is the following:
i.)The two “primaries” P1 and P2 have masses m1 and m2, they move under their mutual attraction and their motion is circular;
ii.)The third body has an infinitesimal mass m3.
It is natural to consider this problem in a rotating frame of reference in which the primaries have fixed positions (Fig. 2). The origin is usually at the center of mass with the primaries along the x-axis.
This leads to the two following equations of motion of m3 (see e.g. Szebehely, 1967):
(1)
where
For example, in case of the Sun-Jupiter system = 9.538810-4.
The period of P2 around of the center of mass is 2 and the mean-motion is equal to 1.
This problem has a well known integral of motion, the “Jacobi integral”:
The system (1) is equivalent with (Marchal, 1990)
(2)
In polar coordinates (r, ), r = r2, (Fig. 2) we have:
(3)
and equations (2) are transforming in:
(4)
where
With initial conditions where
Definition 1. The capture effect of the P2 to initial condition is if
Definition 2. The capture domain of effect is
.
Observation 1. It is easy to see:
.
Observation 2. It is of interest to see if
is true?
Theorem 1. The initial condition with is in S if and only if
(5)
for any .
3. NUMERICAL RESULTS
The structure of capture domain S for different values of is studied using the method of sections. Fixing two components of initial conditions , the two-dimensional section of S is approximated numerically using criteria (5).
Example: For we give a numerical approximation of the capture domains, for different values of , when r0 0.35. In Fig. 3. to each nuance of grey correspond different capture domains. The grade of darkness increase with the capture effect .. The black zone is the 2-effect capture domain, subset of that initial conditions, from which arise stable motions. It is interesting the structure of the 2-effect capture domain.
The orbits corresponding to different initial conditions pointed in Fig 3. (A1, …, A8) are plotted in Fig. 4-11. We can see that orbits arising from A1, A3, A4 return to their reference directions without first having moved around the P1, so these represent stable motions. Points from the other zones (A2, A5, A6, A7, A8) determine motions with capture effect less than 2. We chose points A1, A2, A3 to illustrate that the stable zone is not simple connex.
4. REFERENCES
[1]Belbruno, E.: Hopping in the Kuiper Belt and significance of the 2:3 resonance, The Dynamics of Small Bodies in the Solar System, 1999, pp. 37-49.
[2]Belbruno, E., Marsden, B. G.: Resonance hopping in comets, The Astronomical Journal, Vol. 113 No. 4, 1997, pp. 1433-1444.
[3]Brunini, A.: On the satellite capture problem, Celestial Mechanics, Vol. 64, 1996, pp. 79-92.
[4]Duncan, M. J., Levison, H. F., Budd, S. M.: The dynamical structure of the Kuiper Belt, The Astronomical Journal, Vol. 110, 1995, pp. 3073.
[5]Marchal, C.: The three-body problem, Elsevier, Studies in Astronautics, 1990.
[6]Morbidelli, A.: Chaotic diffusion and the origin of comets from 2:3 resonance in the Kuiper Belt, Icarus, 127, 1997, pp. 1.
[7]Murison, M. A.: The fractal dynamics of satellite capture in the circular restricted three-body problem, The Astronomical Journal, Vol. 98 No. 6, 1989, pp. 2346-2386.
[8]Roy, A.E., Orbital motion, Adam Hilger, Bristol and Philadelphia, 1988.
[9]Szebehely, V.: Theory of orbits, Academic Press, New-York, 1967.
[10]Yu, Q., Tremaine, S.: Resonant capture by inward-migrating planets, The Astronomical Journal, Vol. 121 Issue 3., 2001, pp. 1736-1740.