A&A 421, 771-774 (2004)
DOI: 10.1051/0004-6361:20047025
A. Széll ^{1} - B. Steves ^{1} - B. Érdi ^{2}
1 - Glasgow Caledonian University,
Cowcaddens Road, G4 0BA Glasgow, UK
2 - Eötvös Loránd University,
Pázmány Péter Sétány,
Budapest, Hungary
Received 7 January 2004 / Accepted 7 April 2004
Abstract
Numerical escape criteria is presented for the
Caledonian Symmetric Four-Body problem (CSFBP). The numerical experiments show
that escapes can be detected very early with the help of the
method. Integrating a huge amount of orbits of the symmetric
four-body system we found that for the equal mass case the
double binary escape, and in the planetary case the single
bodies escape are the most likely outcome of the disintegration
of the system.
Key words: celestial mechanics
The determination of whether a body escapes a system or not is a very difficult problem. There are no known analytical methods for the general n-body problem (n>2) which detect escapes. However, it is possible to determine numerical criteria for the detection of escapes. Numerically these criteria can be verified, but as of yet the validity of such criteria has not been proven analytically.
In the paper of Shebalin & Tippens (1996) a numerical escape criterion is given for the general problem of three bodies P_{1}, P_{2}, and P_{3}. They divided the system into three binary subsystems, ie. three sets of "composite binaries", or binaries P_{1}P_{2}, P_{1}P_{3}, and P_{2}P_{3}. They defined three energy-like parameters, corresponding to the energies of the three binary subsystems. They found that these parameters correlate with an escape of the system, or in other words these parameters can predict escape. Plotting the three energy-like parameters as functions of time shows that they fluctuate below and sometimes above zero. Shebalin & Tippens (1996) found that if all the parameters become greater than zero at the same time, then the system would break up in a finite time.
In our present work we use their idea in order to find escape criteria for a symmetric four-body problem called the Caledonian Symmetric Four Body Problem (CSFBP) first defined by Steves & Roy (1998, 2000, 2001). In their papers they investigated the hierarchical stability and evolution of the CSFBP using an analytical stability criterion which they showed depends solely on a parameter they called the Szebehely constant C_{0}, where C_{0}is a function only of the total energy and angular momentum of the system.
The hierarchical stability of the CSFBP was also investigated numerically by Sz ll et al. (Sz ll 2003), in which they confirmed the analytical criteria and explored in detail the dominant hierarchical configurations for different mass ratios between the bodies. The CSFBP has two dynamical symmetric pairs of mass m and Mwith mass ratio . As part of their study, they followed the evolution of about 70 000 different CSFBP orbits having 23 different pairs of where and 0.001.
During these orbits many escapes of the system bodies occurred. It became a valuable saving of the computer time to be able to recognize those systems which would result in escapes long before they had been numerically integrated to break up point.
The aim of this paper is to derive such an early detection method for escapes in the CSFBP and to study the dominant characteristics of escapes in the symmetrical four body problem.
At first we define the four-body configuration. Then we introduce energy-like parameters that can be used to detect escapes. We apply these criteria to investigate the distribution of the different types of escape configurations in the symmetric four-body problem.
Figure 1: The symmetric four-body configuration at any time t. | |
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Let us consider four mass-points
P_{1}, P_{2}, P_{3}, P_{4} in the two dimensional
Euclidean space with position vectors
,
and velocity vectors
,
i=1,2,3,4. Let the
origin be at the centre of mass C of the whole system and let it
be at rest. Let M_{1},
M_{2}, M_{3}, M_{4} be the masses of the
mass-points. Let us introduce the following symmetry conditions
It is straightforward from Eq. (1) that
We shall use the following notations:
Figure 2: Typical time evolution of the E_{1}, E_{2}, E_{3}, and E_{4} parameters in the case of a) "12'' type of escape (two binaries escaping) and b) "13'' type of escape (two binaries escaping). | |
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Figure 3: Typical time evolution of the E_{1}, E_{2}, E_{3} and E_{4} parameters in the case of a) "14'' type of escape (two singles escaping, leaving a binary) and b) "23'' type of escape (two singles escaping, leaving a binary). | |
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In this section we define four new, energy-like parameters: E_{1}, E_{2}, E_{3}, and E_{4}, that can be used to detect escapes.
We recall, that in the Newtonian two-body problem the
energy of the system can be written as
Thus E_{1} can be interpreted as the formal energy expression of a two-body system that is formed by a point mass located at c_{12} with mass , and another point mass at c_{34} with the same mass. But in the present case E_{1} is not a constant with respect to time t. E_{1} = E_{1}(t), since Eq. (3) is not an integral of motion of the CSFBP (the variations of v_{c12} and r_{c12c34} can be determined from the equations of motion).
Still, (3) can be used to predict the disruption of the system. If the energy E_{1} at a given time was positive, then the two-body system corresponding to Eq. (3) at that time would break up. For increasing positive energies E_{1}, the points c_{12} and c_{34} would diverge from each other. According to our numerical experiments in the case of a "12'' type of escape, i.e when the system breaks into two binaries, with the P_{1} P_{2} and the P_{3} P_{4} binaries diverging from each other, the function E_{1}(t) behaves typically as shown in Fig. 2a. It can be seen that E_{1}(t) oscillates with positive minimum values that continue to increase.
Let E_{2} be
(4) |
Note, that E_{2} is very similar to E_{1} in nature, only the binaries studied are now P_{1} P_{3} and P_{2} P_{4}, instead of the previous P_{1} P_{2} and P_{3} P_{4} cases. The arguments given in Sect. 3.1 can also be applied here. The E_{2}(t) curve can therefore be used to detect "13'' type of escapes.
In Fig. 2b the E_{2}(t) function can be seen for a "13'' type of escape. It can be seen that the minimum values of E_{2} are positive and continue to increase. When this behaviour is found the eventually system breaks up exhibiting a "13'' escape.
Let E_{3} be
Thus E_{3} can be interpreted as the formal energy of a two-body system that is formed by a point mass located at P_{1} with mass , and another point mass at P_{4} with the same mass. But again in this case E_{3} is not a constant with respect to time t. E_{3} = E_{3}(t), since v_{a} and r_{1} can be determined from the equations of motion to which Eq. (5) is not an integral.
If the bodies P_{1} and P_{4} were placed very far from each other, then E_{3} would tend to be a constant, since the system could be approximated by a two-body system formed by P_{1} and P_{4} with masses . This is the reason why E_{3} is defined as the energy-like variable belonging to the "14'' type of escape.
Now we can repeat the train of thought we used for the "12'' type hierarchy case. If the energy E_{3}(t) at a given time was positive, then the two-body system corresponding to Eq. (5) at that time would break up. For increasing positive energy P_{1} and P_{4} would diverge from each other.
According to our numerical experiments in the case of a "14'' type of escape, the E_{3}(t) function behaves typically as shown in Fig. 3a. It can be seen that the E_{3}(t) function is positive, first increasing and then almost constant indicating escape has occurred.
Let E_{4} be
(6) |
Note, that E_{4} is very similar to E_{3} in nature, only the central binary pair is now P_{1} and P_{4}, instead of the previous pair P_{2} and P_{3}. The arguments given in Sect. 3.3 can also be applied here. The E_{4}(t) curve can therefore be used to detect "23'' type of escapes.
In Fig. 3b the E_{4}(t) function can be seen for a "23'' type of escape. It can be seen that the E_{4}(t) function is positive and almost constant in time after a certain limit.
We examined in the aggregate about 15 000 orbits which ended in escapes, for four different values of , for , 0.1, 0.01, and 0.001, in order to investigate the distribution of the different kinds of escape configurations. For greater values the system can be a model for quadruple stellar systems, while for smaller values the system can be a model for planetary systems.
During the numerical studies we chose the initial conditions to satisfy the symmetry conditions of the CSFBP and initially the bodies were collinear.
We detected the escapes with the help of our escape criteria. However, the integration was not stopped when an escape was detected. It was always executed for 4 000 000 time steps. Since the step-size was set to 0.001, the result was a numerical integration time of 4 000 time units. Since the gravitational constant G and the mass of the more massive bodies were set to one, one time unit was about 368 years (Széll 2003). Thus the total integration time was about 1 500 000 years. We found that in each case, when the escape criteria observed an escape, at the end of the integration time two pairs or two of the bodies were always located considerably far from the baricentre of the system.
The results of the integrations can be seen in Table 1. Physically there is no difference between the "12'' and "13'' type of escapes. They are both double binary escapers. In the case of "14'' and "23'' type of escapes there are two single bodies escapers. In the equal masses case they are similar. For there are two possibilities: the two smaller ("14'' type) or the two greater ("23'' type) bodies can escape. The numerical integrations showed that always the two smaller bodies escape from the system. Thus in Table 1 we tabulate only the two different kind of outcomes: double binary escapes when the system falls apart into two escaping binaries, and two single escapers when the two smaller single bodies escape ("14'' type of escapes).
Table 1: Percentage of the total number of escape configuration.
The table shows that as the parameter decreases, the single body escapes become dominant. For the most likely escape is the double binary escape. Thus four-star systems break up most likely into two binary systems. In the case the escape of 2 single bodies escape is dominant, but the double binary escapes are still considerable. For small values of only the single escapers are considerable. Thus for planetary systems, the escape of the planets is the most likely outcome.
In this paper we presented numerical escape criteria for the Caledonian Symmetric Four-Body Problem. By approximating the symmetrical four body problem on the verge of break up as a two body problem, we were able to derive four energy like parameters E_{1}, E_{2}, E_{3} and E_{4} which were related to each of the four types of possible escapes. A study of the energy like parameters for 15 000 orbits integrated to escapes showed that the energy like parameters exhibit distinctive behaviour when the system is about to break up. this behaviour can be used as an escape criteria to predict escapes of the CSFBP. For example when either E_{3} or E_{4} becomes a positive value increasing to a constant, this indicates that a "14'' or "23'' type of escape respectively will occur. When either E_{1} or E_{2} has a consecutive minimum values which are positive and increasing and the energy parameters have some negative value, then this indicate that a "12'' or "13'' type of escape respectively will occur. The CSFBP symmetry conditions were employed in the derivation of energy parameters, but it is likely that the same train of thought can be applied for arbitrary systems. This will be a subject of future research.
Numerous orbits were investigated with the help of the numerical escape criteria. We found that for stellar systems the most likely escape configuration is the double binary escape, i.e. the system falls apart into two binary-star configurations. For small values of , the system can be a model for two stars two planets systems. The integrations show that the most likely escape is the escape of the two planets.
Acknowledgements
This work was partly supported by the British-Hungarian Scientific and Technological Foundation under grant number GB-53/01.