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Introduction to the N-body problem

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4-body Method:

The four-body problem is especially interesting because of its application to strong interactions between two binaries, both in isolation and in star clusters.

An accurate treatment of two encountering close binaries is very difficult because of the sensitive nature of the solutions. Here we make use of the so-called chain regularization method which was developed by Seppo Mikkola to study such systems (see publications).

The main idea is to connect the four particles by a set of three chain vectors chosen to maximize the respective interparticle forces, while the contributions from the three weaker interactions are added in a more direct way. Thus the three-body regularization represents a special case with two chain distances, leaving the third and largest to be treated directly.

Each chain vector describes two-body solutions with the property of increased accuracy at decreasing separations, unlike in conventional calculations. The sequence of the existing chain is checked frequently and modified according to changes in the configuration for maximum efficiency. Going from triples to four-body systems increases the choice of initial conditions and gives rise to a greater variety of outcomes.

Now we may observe escape of one particle and be left with a triple for further study, sometimes in the form of a hierarchical (or stable) system. There is also the possibility that one or both binaries may shrink significantly and thereby acquire high escape velocity (recoil effect).

Finally, the fascinating process known as exchange of components may take place. A full exploration of the available parameter space would require a huge effort and is still waiting to be carried out.

Initial Conditions

The orbit calculation requires the mass, coordinates and velocity components of each body. Although the code is three-dimensional, it is most convenient to carry out the calculations in the same plane as the display. The choice of initial values should be reasonable, i.e. masses no different than of a factor of about one million and no two initial positions nearly identical. Also note that the accuracy of long life-times may not be reliable. The movie stops if one of the particles escapes, leaving behind a binary (for bound systems), or the specified calculation time (in units of the orbital crossing) is exceeded.

Simple examples of point-mass collision orbits can be studied by placing all three bodies on the same axis and assigning appropriate masses and coordinates, with strictly radial velocities. Note that in linear configurations it is impossible for the middle body to escape. Such deceptively simple systems can still reveal considerable complexity beyond the most powerful mathematical analysis.

Click here to try out the three-body simulations

Click here to go to the four-body introduction