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Problems on Rotational Motions in Gravitating Systems

An interesting set of questions in statics concerns the properties of moments of forces exerted on each other by randomly distributed points forming a system S. In dynamics the questions concern angular momenta of subsystems a contained in S; as a function of the time.

Let us imagine the following situation: E consists of a number of mass points mi, m2, ... mn located at t = 0 at positions rl,... rn given at random, say, in a unit sphere (let us assume, for example, a uniform probability distribution for the position of each point). We assume further Newtonian attractive forces Fij acting between any two points mi, mj. Denote by Gi the sum over all j of forces acting upon the point mi and let us imagine the vector Gi applied at the point ri. It is clear that the sum over all i of Gi is equivalent to zero. What we propose to study is, at first, the statistical behavior of the forces Gi if summed over subsystems a of the whole set with the following questions to be investigated: let p be any number and let us consider subsystems located in a circle with radius p and an arbitrary center ro. Let us form the sum of all Gi located in such and we obtain, in general, a single force $ and a couple P referred to ro with magnitude which we shall call 0. Both ( and 0 thus computed are functions of p and ro but we can integrate these quantities over all initial positions ro and will obtain )p and Op, a single force and moment, which will be now functions of p alone. It is our aim to obtain these functions for a random dynamical system, that is to say, the expected values in a random distribution. This can be obtained in practice by computing, on the machine, these quantities for a large number of systems each chosen by a random process. Our statistical computations will be confined at first to plane

cases, the three dimensional systems requiring too great a memory at present.

The next, more interesting, thing to study is the following. In a situation as described above at time t = 0 let t increase. Motions of our

points will ensue and we intend to investigate the angular momenta of subsystems o as functions of the size of a and of $q in time.

Such a situation is perhaps exemplified by star clusters. What we want to study are dynamical systems with many particles, but not gases; that is to say, by mean free path for "collision" we mean an appreciable change in the velocities due to gravitational forces acting between just two of our mass points. It is known that clusters or galaxies possess a rotational motion as a whole. These could, perhaps, originate as follows: the original distribution of matter now separated in galaxies was more or less uniform and random-like. Our system E can be imagined infinite. Finite subsystems have angular momenta as a result of fluctuations in the distribution and then, if fluctuations of density occurred also, some subsystems would isolate themselves, stay together due to gravitation (cf. the work of Jeans) and if the whole space expanded with time these condensations would have kept all or most of their angular momenta due to the original fluctuations in the system of vector forces and as the condensations receded from each other, their non-zero angular momenta would have stayed constant in time. Another way to look upon the problem is to study the distribution of vorticity of finite subsystem a in a very large or infinite system E whose points exert forces on each other.

Our proposed calculations consist then of producing a sizeable number of randomly chosen systems E and following the behavior of subsystems in time, i.e., using a discrete series of time intervals or "cycles" on the machine and computing the following set of averages: L(ro, p), the angular momentum as a function of the radius of the subsystem p and position ro. For a system of 100 points four positions along a radius with four values of p at each point are adequate. The running time per cycle for 100 mass points is less than five minutes with this machine. The running time increases as the square of the number of mass points, but statistics can be improved by running many problems with few mass points, in preference to the less economical method of increasing the number of mass points.

The total kinetic and potential energy is calculated. The system should stabilize at some given size. In this steady state the number of double and triple "stars" is of interest. The total energy serves as a check on the problem.

The moment of inertia of the complete system is calculated, and finally the number of particles in each of the sixteen subsystems mentioned above. The latter numbers permit an approximate plot of the density distribution of the system. The largest value of p is chosen to be of the order of the dimension of the system.

Several hundred cycles will be necessary for results on this problem.

Appended are graphs (Figs. 2 to 5) showing the results of 100 cycles run so far.