The restricted 2+2 body problem was stated by Whipple (1984) as a particular case of the general
After many years of thorough study of the restricted three-body problem and the new issues imposed by the flight missions in the 60s, 70s, and 80s, many investigators have focused their scientific interest to new N-body models (
The original configuration consists of two big spherical, homogeneous bodies P1 and P2, called hereafter the primaries, with masses
The configuration of the 2+2 body problem.
Among the works treating this problem, we can mention the papers of Whipple and White [
In what follows, we shall numerically investigate some new aspects of the 2+2 body problem including the parametric variation of the equilibrium states (Section
As we have mentioned before, the system consists of two primaries
By considering a synodic coordinate system O
Here we note that for the normalization process we have used the total mass of the primaries
There is a Jacobian-type integral of motion
Whipple has proved that all equilibrium positions
We have made all the computations from scratch by considering the four coordinates of the minor bodies as independent variables, and we have adjusted the numerical methods used to a new and more general treatment of the problem. Obviously, this results in more complex and extended expressions as well. Further details will be given in a later section of this paper. Figure
Sun-Jupiter-binary asteroids (case A).
Collinear equilibrium solutions
|
|
|
|
Jacobian constant |
Stability | |
---|---|---|---|---|---|---|
|
|
0 |
|
0 | 3.037482 | U |
|
|
0 |
|
0 | 3.037482 | U |
|
|
0 |
|
0 | 3.038764 | U |
|
|
0 |
|
0 | 3.038764 | U |
|
|
0 |
|
0 | 3.000966 | U |
|
|
0 |
|
0 | 3.000966 | U |
Inline (
|
|
|
|
Jacobian constant |
Stability | |
---|---|---|---|---|---|---|
|
|
|
|
|
2.999844 | U |
|
|
|
|
|
2.999844 | U |
|
|
|
|
|
2.999844 | U |
|
|
|
|
|
2.999844 | U |
| ||||||
|
|
|
|
Jacobian constant |
Stability | |
| ||||||
|
|
|
|
|
2.999844 | S |
|
|
|
|
|
2.999844 | S |
|
|
|
|
|
2.999844 | S |
|
|
|
|
|
2.999844 | S |
Earth-Moon-dual satellite system (case B).
Collinear equilibrium solutions
|
|
|
|
Jacobian constant |
Stability | |
---|---|---|---|---|---|---|
|
|
0 |
|
0 | 3.772852 | U |
|
|
0 |
|
0 | 3.772852 | U |
|
|
0 |
|
0 | 3.792084 | U |
|
|
0 |
|
0 | 3.792084 | U |
|
|
0 |
|
0 | 3.582608 | U |
|
|
0 |
|
0 | 3.582608 | U |
Inline (
|
|
|
|
Jacobian constant |
Stability | |
---|---|---|---|---|---|---|
|
|
|
|
|
3.553904 | U |
|
|
|
|
|
3.553904 | U |
|
|
|
|
|
3.553904 | U |
|
|
|
|
|
3.553904 | U |
| ||||||
|
|
|
|
Jacobian constant |
Stability | |
| ||||||
|
|
|
|
|
3.553904 | S |
|
|
|
|
|
3.553904 | S |
|
|
|
|
|
3.553904 | S |
|
|
|
|
|
3.553904 | S |
Lagrangian points of the system Sun-Jupiter-point-like mass
|
|
|
|
---|---|---|---|
|
|
0 |
|
|
|
|
|
|
|
0 |
|
|
|
|
|
|
|
|
|
Lagrangian points of the system Earth-Moon-point-like mass.
|
|
|
|
---|---|---|---|
|
|
0 |
|
|
|
0 |
|
|
|
0 |
|
|
|
|
|
|
|
|
|
Distances between the minor bodies in a pair of equilibria in the Sun-Jupiter-binary asteroids and the Earth-Moon-dual satellites systems.
Equil.states | Sun-Jupiter-binary asteroids | Earth-Moon-dual satellites | ||
---|---|---|---|---|
Distances between |
Distances between |
|||
|
Dimensionless | Physical (Km) | Dimensionless | Physical (m) |
|
4.94 · 10−6 |
|
2.53 · 10−8 | 9.72 |
|
4.65 · 10−6 |
|
2.19 · 10−8 | 8.41 |
|
6.93 · 10−6 |
|
3.4 · 10−8 | 13.07 |
|
6.94 · 10−6 |
|
3.42 · 10−8 | 13.15 |
|
7.75 · 10−5 |
|
1.63 · 10−7 | 62.84 |
The equilibrium solutions
Distribution of the collinear equilibrium solutions in the neighborhood of
Distribution of the pairs of the inline and perpendicular equilibria in the neighbourhood of the triangular Lagrangian point
In this section we investigate the parametric dependence of the equilibrium positions of the two minor bodies which were calculated in the previous paragraph. Obviously, when the two minor masses are equal, then
Variation of the distances (dist) of the equilibrium positions of the minor bodies from their neighboring Lagrangian points
Variation of the distances of the equilibrium positions of the minor bodies from their neighboring Lagrangian point with mass parameter
As we have already mentioned, the inline solutions are located on a straight line which forms a small angle
(a) The angles
In Figure
In Figure
Figure If If If
Whipple proved that the collinear and inline triangular solutions are unstable for every value of the mass parameters. However, for some values of these parameters, the perpendicular solutions are stable (see Tables
Regions of stability (red) and instability (green) for the perpendicular equilibrium points. Points A and B show, respectively, the Sun-Jupiter-binary asteroids system and the Earth-Moon-dual satellites one.
As we have mentioned before, the exact locations of the equilibria of the system are found numerically with the use of an iterative scheme. In everyday practice, the process starts when an initial approximation is given and stops when an equilibrium solution is found with a predetermined accuracy. We have used the well-known Newton-Raphson method since it is a fast (it converges quadratically), simple, and accurate computational tool. For comparison reasons, we also have investigated the attracting regions of both the restricted three-body problem which is characterized by two degrees of freedom and the restricted 2+2-body problem with four degrees of freedom. Here we note that we have obtained similar results by using various numerical methods and that the whole process aims to locate the initial values that lead to a particular equilibrium state in the fastest possible way, thus saving computing time.
By considering systems with two degrees of freedom (
All the initial points that lead to the same equilibrium point form a set of points which constitutes the so-called attracting region or attracting domain of this equilibrium point (Croustalloudi and Kalvouridis [
(a) Attracting domains of all Lagrangian points for the Sun-Jupiter-asteroid system:
The equations of motion which describe the two-dimensional motion of a point-like small mass on the
Our conclusions regarding this problem can be summarized as follows. The attracting region The attracting region Attracting region Attracting region
In this case there are four independent variables (
The quantities
We have applied the above technique to many different cases of the 2+2 body problem, and hereafter we have considered the results of the Sun-Jupiter-2 asteroids system, where
Here we note that, although the basic idea of the processes as described in the two considered problems is the same, the behavior of the algorithm is quite different. As we have mentioned in the previous section (Section
However, in the 2+2 BP and for a given initial value
Since the initial approximation
Attracting sets (a)
Also, it is important to emphasize that, whenever an initial value
Bar chart with the percentages of the converging and nonconverging points for the Sun-Jupiter-binary asteroids system.
In what follows we give a short description of the main characteristics of each attracting set
As we have mentioned before, the set
This set consists of two subsets, namely, the attracting subsets
It results from the union of the attracting subsets
It consists of four attracting subsets. Two of them,
Here we note that, if instead of keeping S20 constant throughout the scanning process, we shall consider the initial position S10 constant, then the patterns of the attracting sets
As an example, we present in Figures
Attracting subsets of the inline and the perpendicular equilibria around
Number of points of the attracting subsets
We have elaborated anew on the problem proposed by Whipple, by following a more general approach and by applying it to two problem cases: (i) the Sun-Jupiter-binary asteroids system and (ii) the Earth-Moon-dual satellites system. We have numerically studied the parametric variation of the equilibrium states, as well as the formation, the structure, and the parametric dependence of the attracting regions of these equilibrium states. Our comments, conclusions, and remarks can be classified into three parts. The first part includes the general remarks which concern the efficiency of the numerical algorithm when applied to both problems: the 2+2 BP (four degrees of freedom) and the RTBP (two degrees of freedom). A second part includes the conclusions concerning the location and the parametric variation of the equilibrium states of the two minor bodies
The distances of all the equilibrium positions from their neighboring Lagrangian points, except for those of the perpendicular ones, slightly change when
As
For comparison reasons we have investigated the attracting regions of both the restricted three-body problem and the restricted 2+2 BP. Comparing the results obtained by the two problems we confirm similarities and differences. Among these similarities, we refer to the ones concerning that in both problems the attracting region
Mass of the Sun: Mass of Jupiter: Mean distance of Jupiter from Sun: 5.2 AU = 777920000 km. Reduced mass of Jupiter: Mass of Reduced mass of Mass of Reduced mass of Reduced mass of
Mass of the Earth: Mass of the Moon: Mean distance of the Moon from Earth: 384400 km. Reduced mass of the Moon: Mass of Reduced mass of Mass of Reduced mass of Reduced mass of