Basins of attraction in the restricted four-body problem of
Marañhao.
T. Kalvouridis and M. Croustalloudi
National Technical University of Athens,
Faculty of Applied Sciences, Department of Mechanics,
5 Heroes of Polytechnion Ave., 157 73, Zografou, Athens, Greece
[email protected], e-mail: [email protected]
Abstract: In this presentation we are concerned with some new aspects of the
dynamics of the restricted four-body problem studied by Marañhao. The problem
describes the motion of a body of negligible mass under the Newtonian gravitational
attraction of three much bigger bodies (called the primaries) that form a collinear
configuration. More precisely, we study the attracting regions and we investigate their
dependence on the parameters of the system.
Keywords: restricted four-body problem, ring problem, regular polygon
configurations, attracting regions, equilibrium points.
1. Introduction
The restricted four-body problem describes the motion of a body of negligible mass
under the Newtonian gravitational attraction of three much bigger bodies (called the
primaries) moving in circular periodic orbits around their center of mass fixed at the
origin of the coordinate system (Marañhao, 1995; Marañhao and Llibre, 1999). At any
moment, the primaries form a collinear equilibrium configuration. Two of these
primaries have equal masses and are located symmetrically with respect to the third
primary that has different mass and rests motionless at the center of mass of the
system (Figure 1).
The above configuration can be considered as an extension of the Copenhagen case of
the restricted three-body problem and could be used to approximate the motion of a
small body in systems consisting of a planet and two co-orbital satellites being in
syzygy with the planet. It can also be considered as a special case of the ring problem
studied by Kalvouridis (1999a,b,c, 2001, 2003, 2004), Croustalloudi and Kalvouridis
(2003), and Arribas and Elipe (2004). The ring or regular polygon configuration of N
bodies was based on an idea of Maxwell (1890) who originally investigated it. After
him many investigators based their work on this model. We mention here the works
of Goudas (1991), Sheeres (1992), Sheeres and Vinh (1993).
The basic N-body ring formation consists of ν =N-1 bodies that have equal masses
and are located at the vertices of a regular polygon, and of a Nth primary that has a
different mass and is located at the center of the mass of the system (Figure 2). A
small body S of negligible mass compared to the masses of the primaries moves in the
combined gravitational field produced by the primaries. The problem is characterized
by two parameters, namely the number of the peripheral primaries ν, and the ratio
β = m0 / m , where m0 is the mass of the central primary and m the mass of a
peripheral body. For ν=2 and β ≠ 0, the general ring configuration reduces to the
above four-body problem, while for ν=2 and β=0 we take the Copenhagen case of the
restricted three-body problem.
In this presentation we study the attracting regions and we investigate their
dependence on the parameters of the system. As it is known, there are six equilibrium
positions of the particle S if β ≠ 0. These points are found with the application of an
iterator that is activated when an initial value is given and stops when the
aforementioned positions are reached with some predetermined accuracy. The set of
the initial points, which lead to a particular equilibrium point, constitutes a “basin of
attraction” or an attracting region. In this sense, every equilibrium position is an
“attractor” regardless of its real state of stability, since we can reach it even if we start
from different initial values and follow different paths of successive approximating
steps. The results we get, show in many cases a fractal-type structure of these regions
and reveal the chaotic character of the system.
z
P(x,y)
r2
r1
P2
P0
y
O
x
P1
Figure 1. The restricted four-body problem of Marañhao.
2. Equations of motion and fundamental geometric relations
Looking at the general ring configuration of the primaries in Figure 2 we have the
following geometric relations
ψ =
(ν − 2 ) π
2π
, θ = ψ / 2 , and ϕ =
ν
ν
In the synodic coordinate system P0xy, the particle motion on the plane of the
primaries’ revolution is described by the following dimensionless differential
equations
∂U
&x& − 2 y& =
∂x
,
∂U
&y& + 2 x& =
∂y
Figure 2. The configuration of the primaries in the general ring problem and the
synodic coordinate system P0xy
where
1 2
1 β ν 1
(
x + y2 )+  + ∑ 
2
∆  r0 i =1 ri 
is the potential function, r0 and ri , i = 1,2..,ν are the distances of the particle from
each individual primary. Furthermore,
∆ = M ( Λ + βM 2 ) ,
U (x , y ) =
ν
sin 2 θ cos(ν + 1 − i )θ
sin 2 θ , Μ = [2(1 − cosψ )]1 2 = 2 sin θ ,
2
Λ=
=
sin 2 (ν + 1 − i )θ
i =2
i =2 sin( i − 1)θ
ν
∑
∑
For our case ν=2 and β ≠ 0.
3. Equilibrium points and their parametric variation
In the restricted four-body problem and for every value of parameter β ( β ≠ 0 ), there
are 6 equilibrium points that correspond to the three equilibrium zones A1, C2, and C1
of the ring problem. The two points of zone A1 are disposed along the syzygies’ axis
of the primaries in symmetric positions with respect to the origin O and between the
central primary and the peripheral ones. The two points of zone C1 lie on the same
axis but beyond the peripheral primaries. Finally the two points of zone C2 lie on the
y-axis (Figure 3). As the mass parameter decreases the two points of A1 approach
each other by moving towards the origin from opposite directions, where they
coincide when β=0. Therefore, for the limit value β=0, we take the five Lagrange
equilibrium positions of the Copenhagen case of the restricted three-body problem.
1
0.8
0.6
0.4
0.2
P0
y
P2
P1
0
-1.5
-1
-0.5
0
0.5
1
1.5
A1
C1
C2
A1
C1
C2
-0.2
-0.4
-0.6
-0.8
-1
x
Figure 3. The equilibrium positions in the restricted four body problem for β=2 (blue)
and β=0.2 (red)
4. Determination of the Attracting Regions
The equilibrium positions of the particle are found by solving the system of algebraic
equations [Szebehely, 1967],
Ux=0
Uy=0
This system is solved with the application of a numerical iterator, provided that an
initial approximation is given. This approximation represents a point on plane ( x, y ) .
1.5
y
1.25
1
0.75
0.5
P2
0.25
Po
-0.5
P1
0.5
1.5
x
-0.25
-0.5
1
Starting
point
Figure 5. The consecutive steps followed by the numerical iterator and the crooked
path-line leading to an equilibrium position
The iterator stops when some predetermined accuracy is reached. The consecutive
steps of the determination of successive approximating points form a crooked pathline leading to an equilibrium position (Figure 5). From this point of view, the
equilibrium position (whatever its real state of stability is) can be considered as an
“attractor”. We call the set of the initial points that lead to the points of a particular
equilibrium zone, an “attracting region”.
In order to achieve our purposes we have selected and used the Newton-Raphson
method that still remains a fast, simple and accurate computational tool. For our
problem this algorithm takes the form,
x ( n ) = x ( n −1 ) −
y
(n)
= y
U xU yy −U yU xy
( n −1 )
2
U yyU xx −U xy
+
x ( n −1 ) , y ( n −1 )
U xU yx −U yU xx
2
U yyU xx −U xy
x ( n −1) , y ( n −1 )
x ( n ) , y ( n ) , are the values of x and y at the n -th step of the iteration process.
The determination of the attracting regions has been accomplished by applying a
double scanning of plane ( x, y ) within intervals xo ∈ [ −1.0, 0) ∪ (0,1.0] ,
y0 ∈ [−1.0, 0) ∪ ( 0,1.0] with a step equal to 0.005. In all the considered cases we used
10-8 as the accuracy criterion to terminate the iterative process. We have recorded all
the initial points that lead to the equilibrium points of a particular zone. In Figs. 6, 7
and 8, we present the results obtained. We would also like to emphasize that we
thought it would be more convenient to consider a common basin of attraction for all
the equilibrium points which belong to a particular equilibrium zone, instead of
dealing with separate basins of attraction. Therefore, the points of plane ( x, y ) were
separated in three “groups”. We symbolize with R(A1), R(C2) and R(C1) the basins of
attraction of the equilibrium zones respectively. The attracting regions of each zone
present, as it is expected, all the symmetry elements of the primaries’ arrangement.
They generally consist of some “compact” parts, that is areas all the points of which
lead to the equilibrium positions of this particular zone. S (A1), S (C2) and S (C1)
symbolize these areas respectively. Furthermore, we have found dispersed points that
lie on the boundaries of the “compact” regions of this or other zones. D(A1), D(C2)
and D(C1) symbolize these subsets of points respectively. We must stress that these
boundaries are not clearly defined. This is the reason why the word “compact” is
placed in inverted commas. Therefore, each basin of attraction is the join of the
compact areas and of the dispersed points. Figures 6 to 8 show the attracting regions
of the three equilibrium zones for β=2. We have marked with black arrows some
“compact” areas and regions of dispersed points.
A compact region with a fractal structure surrounds each equilibrium position of zone
A1. From its boundaries leap tentacles formed by dispersed points that terminate to the
boundaries of the compact region of the symmetric equilibrium position of that zone
(Figure 6). Several other dispersed points also accumulate rather a long way from the
x-axis in strip-like areas. Between these two concentrations no other points of this
zone exist. When the mass parameter augments, then the compact regions shrink,
while other compact areas are formed in a distance from the first ones. In the latter
case, the dispersed points diffuse, thus occupying a larger part of the xy plane.
Regarding the attracting region of C2, it is the biggest one in comparison to the
respective regions of the other two zones, as is evident in figure 7. It covers most of
the surface of the xy plane but presents considerable gaps near the x-axis. As mass
parameter β increases, the attracting region extends and comes closer to the x-axis.
The attracting region of zone C1 consists of four compact regions that are
symmetrically disposed with respect to the origin. These regions are also symmetric
with respect to the x-axis. Two of them, those that are closer to the origin, are wider
than the other two. The dispersed points either form tentacles that come out from the
boundaries of the compact regions, or they form shapes that surround the origin and
resemble the wings of a windmill (Figure8). The equilibrium points of C2 lie
eccentrically inside these areas. As parameter β increases the compact regions shrink,
while the dispersed points diffuse covering a more extended area of the xy plane.
Yo
1
0.8
0.6
S(A1)
0.4
0.2
P2
-1
-0.8
-0.6
-0.4
-0.2
P1
P0
0
0
0.2
0.4
Xo
0.6
0.8
1
-0.2
-0.4
D(A1)
-0.6
-0.8
-1
The attracting region of equilibrium zone A1, R(A1), for ν=2, β=2.0
Figure 6. The attracting region of equilibrium zone A1 for β=2.0
primary
A1
C2
C1
Figure 7. The attracting region of equilibrium zone C2 for β=2.0
Yo
1
0.8
0.6
S(C1)
0.4
0.2
P2
P1
P0
Xo
0
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-0.2
-0.4
D(C1)
-0.6
-0.8
primary
A1
C2
C1
-1
The attracting region of equilibrium zone C 1, R(C1), for ν=2, β=2.0
Figure 8. The attracting region of equilibrium zone C1 for β=2.0
When the initial approximation, lies in the central part of the “compact” area of an
equilibrium zone, then the process ends in an equilibrium position of that zone.
However, if it lies on its boundaries or on the boundaries of the “compact” areas of
other zones, then the prediction of their final destination becomes extremely difficult.
If, for example, we select the following points lying very close to each other in one
“unsafe” region, (0.5299, 0.08310999), (0.5299, 0.083109999), (0.5299, 0.0831099),
(0.52990005, 0.08310999), then the process will lead us to different equilibrium
points belonging either to the same (Figures 9b and 9d), or to different equilibrium
zones (Figures 9a and 9c).
0.4
0.2
0.2
-2
-1.5
-1
-0.5
0.5
-0.2
1
-1.5
-1
-0.5
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1.5
-1
1
-1
(a) Initial point (0.5299,0.08310999).
Final target zone A1
-2
0.5
-0.2
(b) Initial point (0.5299,0.083109999).
Final target zone C2
0.4
0.4
0.2
0.2
-0.5
-0.2
0.5
1
-1.5
-1
-0.5
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
(c) Initial point (0.5299,0.0831099)
Final target zone C1
0.5
1
1.5
-1
(d) Initial point (0.52990005,0.08310999)
Final target zone C2
Figure 9. Due to the high sensitivity of the initial approximations on the boundaries
of the compact regions, the process, for very small changes of the initial values, leads
to equilibrium points belonging either to the same (cases b and d), or to different
equilibrium zones (cases a and c)
In general, the attracting regions are organized in formations that sometimes present
the self-similarity of fractal structures. These formations cannot be described with
equations. Nevertheless, we can elicit some qualitative conclusions if we study their
shape and parametric evolution. A major conclusion is that the attracting regions in
zone C2 are more expanded than those in the rest of the zones, for all the values of β .
6. Conclusions and remarks
In the above paper, we have studied the creation, evolution and parametric
dependence of the attracting regions in the restricted four-body problem by simulating
it with a special case of the ring-type N-body model. As we have seen, each
equilibrium zone has its own attracting region that consists of a “compact” part and of
dispersed points. These points are distributed on the boundaries of the compact parts
of the same or other equilibrium zones. Mass parameter β plays an important role on
the formation of both the equilibrium zones and the attracting regions. There are three
equilibrium zones with two points each but when the mass parameter approaches the
critical value 0, then the two points of equilibrium zone Α2 coincide at the origin and
we obtain the special Copenhagen case of the restricted three-body problem where
only five equilibrium points (the Lagrangian points) exist. As β increases, the
compact parts of A2 and C1 shrink, while the compact areas of C2 begin to decompose
in smaller compact parts leaving empty space between them.
References
Arribas, M. , Elipe, A.: 2004, Bifurcations and equilibria in the extended N-body ring
problem, Mechan.Res. Comm., 31, 1-8.
Croustalloudi, M., Kalvouridis, T.J.: 2003, A study of the equilibrium zones in
ring-type N-body systems, in Recent Advances in Mechanics and the Related
Fields (eds. Markellos V., Katsiaris G., Hadjidemetriou J.) in honor of
Professor Constantine Goudas, Patras, pp.89-96.
Goudas, C.,: 1991, The N-dipole problem and the rings of Saturn, in Predictability,
Stability and Chaos in N-body Dynamical Systems, NATO ASI series B: Physics,
vol. 272, pp. 371- 385.
Kalvouridis, T., J.:1999a, A planar case of the n+1 body problem: The ring problem,
Astrophys.Sp.Sci, 260 (3), 309-325.
Kalvouridis, T., J.:1999b, Periodic solutions in the ring problem, Astrophys.Sp.Sci.,
266 (4), 467-494.
Kalvouridis, T., J.:1999c, Motion of a small satellite in planar multi-body
surroundings, Mechan.Res.Com., 26, No 4, 489-497.
Kalvouridis, T., J.: 2001a, Multiple periodic orbits in the ring problem : families of
triple periodic orbits, Astrophys.Sp.Sci., 277 (4), 579-614.
Kalvouridis, T., J.: 2003, Retrograde orbits in ring configurations of N bodies,
Astrophys.Sp.Sci., 284, 1013-1033.
Kalvouridis, T., J.: 2004, On a property of zero-velocity curves in N-body ring-type
systems, Planetary and Space Science, (in press).
Marañhao, D.,L.: 1995, Estudi del flux d’un problema restringit de quatre
cossos”,Ph.D. Thesis, Universitat Autonoma de Barcelona.
Marañhao, D., Llibre, J.: 1999, Ejection- collision orbits and invariant punctured tori
in a restricted four-body problem, Cel.Mech. and Dynam. Astron., 71, 1-14.
Maxwell, C.J., 1890, On the stability of the motion of Saturn’s Rings, Scientific
Papers of James Clerk Maxwell, Cambridge University Press, vol. 1, 228.
Sheeres, D.J.: 1992, On symmetric central configurations with application to satellite
motion about rings, PhD thesis, University of Michigan.
Sheeres, D.J. Vinh, N.X.: 1993, Acta Astronaut. 29, 237-248.