Gravitationally bound system I have a question concerning an $N$-body gravitationally bound system. It is quite simple and straight forward but I have not found an answer to it yet. I'm not even sure if the question is actually scientifically meaningful:
Does every "orbit" of a planet in this system orbit around the gravitational center aka the center of mass of the system in any way or are there exceptions in certain cases.
 A: Gravitational N-body problem is in no way a simple problem and to answer your  question it is necessary to use some interesting and absolutely non-trivial results of the mathematical physics research of the last decades of the past century.
Indeed, the geometric definition of the center of mass as weighted average of the N-body positions implies that it will remain inside the maximum and minimum value of the cartesian components of the positions.
However, it is well known since the pioneering analysis done by Poincaré, that for N$\gt2$ a gravitational system may develop a set of chaotic, highly irregular trajectories in the phase space, corresponding to very complex behaviors of the individual bodies trajectories.  Just a few years after Poincaré's memoir, von Zeipel was able to show the possibility that in an N-body problem Newton's laws of motion allow bodies to separate infinitely far a part in a finite time. It took until 1988 to understand through Xia's work how such a mechanism can work for systems of at least five bodies. A nice summary of the main steps in this story can be read in a paper by Saari and Xia  on Notices of AMS. The main mechanism is schematically depicted in their Fig.3 that I have redrawn here:

The body on the straight line moves forwards and backwards between the two pairs of bodies orbiting in increasingly  eccentric orbits. The non lower-boundedness of the gravitational potential energy allows a runaway increase of the kinetic energy of the two pairs  which eventually results in sending them at infinite distance in a finite time (details in the cited paper).
From the point of view of the present question, it is clear that the trajectories of those five bodies could hardly be described as "orbiting" around the common center of mass in the usual meaning of such a sentence.
A: Yes, all orbits will be around the center of mass of the entire system. This is because the center of mass will give the position vector of the resultant of all the other gravitational forces on the body. Hence the center of mass will always lie inside all the orbits.
