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Random Walk in the Three-Body Problem and Applications

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Random Walk in the Three-Body Problem and Applications DISCRETE AND CONTINUOUS Website: http://aimSciences.org DYNAMICAL SYSTEMS SERIES S Volume 1, Number 4, December 2008 pp. 519–540 RANDOM WALK IN THE THREE-BODY PROBLEM AND APPLICATIONS Edward Belbruno Department of Astrophysical Sciences, Princeton University Princeton, New Jersey 08544, USA Abstract. The process of random walk is described, in general, and how it can be applied in the three-body problem in a systematic manner. Several applications are considered. The main one which is a focus of this paper is on the evolution of horseshoe orbits and their transition to breakout motion in the restricted three-body problem. This connection is related to their use for an Earth-impactor in a theory on the formation of the Moon. We briefly discuss another application on the instability of asteroid orbits. 1. Introduction. Typically, the three-body problem, which describes the motions of three particles, of given masses, under the Newtonian gravitational force law, is studied in a deterministic fashion, where the motion of the particles is explicitly determined by a system of differential equations. In this way, the trajectory of any of the particles, P , at a given point in phase space is locally well defined, and unique, provided the differential equations are sufficiently differentiable near the point. Then, in forward, or backward, time the trajectory of the particle, for a given initial condition, is defined from moment to moment as time progresses. Both the position and the velocity of the particle vary smoothly and in a predictable way, given by the differential equations. In this paper, we consider a different situation, where the motion of a given particle, P , is not determined only by the differential equations. Rather, the velocity direction at a discrete set of times for P is randomly chosen. The methodology we will choose for the choice of the randomly chosen velocity direction is based on the principle of random walk. Random walk is a discrete process in time whereas the motion of particles described by the three-body problem is a continuous one. The approach we take in this paper is to combine the two processes, described in Section 3.4. We assume that at a discrete set of times along a trajectory of the restricted three-body problem, a tiny velocity is applied to P with a random direction, but with a constant magnitude. For the purposes of this paper and our modeling, it is reasonable to use a constant velocity magnitude, which we discuss in Section 3.4. It is interesting that under certain conditions this methodology can yield accurate results in the three-body problem, which we will describe in Section 3.4. The accuracy is measured by the degree to which the resultant velocity of P increases as a result of the small velocities being given at random times along its trajectory, and 2000 Mathematics Subject Classification. Primary: 37N05, 60G35, 70F05, Secondary: 37M05, 37M20, 37J20, 37J25. Key words and phrases. Three-body problem, random walk, horseshoe orbit, bifurcation, sta- bility, stochastic process, hyperbolicity, collision. 519 520 EDWARD BELBRUNO it is demonstrated that it increases according to the random walk methodology. This is applied in the restricted three-body problem to the family of horseshoe orbits, used in a theory for the formation of the Moon [2]. Another application we briefly describe is applied to the problem of ejection from the Sun-Jupiter system [11]. We will discuss the application of random walk to other situations that may arise especially when considering sensitive dynamics. The process of random walk has been extensively applied to many different types of problems in physics and there are a number of references (See [5, 8]). In this paper, an attempt is to formulate its application to the restricted three-body problem in a precise manner. 2. Applying random walk in the Three-Body problem. Random walk in general can be viewed as a discrete process where a particle undergoes a series of displacements. It is a stochastic process where the rule for the change in the displacements is not based on any previous information. The simplest is random walk in one dimension which assumes the particle moves on a line L. It is assumed that the particle can either move forward or backward in steps of equal length, with equal probability of 1/2. After N 1, steps, the particle could be at any of the following positions: N, N +1, ≥N +2, ..., 1, 0, 1, ..., N 1,N. If you ask of the probability of the− particle− being− at a point−n after N steps,− then one obtains N N a Bernoulli distribution for the probability, C(N+n)/2(1/2) , which yields a root mean displacement of √N [5]. This idea can be generalized to higher dimensions and in more general situations. Let’s assume we have a particle moving in three-dimensions. We assume it under- goes a random vector displacement b1 to a new location, of magnitude b = b1 . | | It then receives another vector displacement b2 also of magnitude b. This process continues so that the particle receives N vector displacements bk, k =1, ..., N, all of magnitude b = bk . If N is sufficiently large, then, setting B = b1 +b2 +...+bN and B = B , it is| found| that | | B b√N. (1) ≈ B is called the mean free path. The path of the particle appears jagged. In order to apply random walk to the three-body problem, we would also like to make use of the continuous flow given by the differential equations. To do this we consider velocity space instead of position space since that will allow us to apply random walk. As in the above situation, we assume that we have a particle at a given location in three-dimensional space and it undergoes a series of random velocity displacements vk all of equal magnitude v. Setting V = v1 + v2 + ... + vN and V = V , similar to (1) it is found that | | V v√N. (2) ≈ It is assumed that when the velocity change is given, the particle’s resulting velocity is changing but the actual trajectory in physical space is not considered. It would result by following the path in a linear manner for a short random time before the next velocity is prescribed, resulting in a jagged path. A way to derive (2) is the following: Consider a particle at a location in three- dimensional space. When it undergoes a velocity displacement w followed by an- other z, then V 2 = w2 + z2 2wz cos θ, (3) − RANDOM WALK IN THE THREE-BODY PROBLEM 521 where V = w + z, V = V , w = w , v = z . θ is the angle between w, z. From the principle of random walk,| | θ |[0,|2π] is| randomly| chosen and cos θ takes on its average value for θ [0, 2π]. Under∈ this hypothesis, we can replace (3) by ∈ V 2 = w2 + z2 2wz < cos θ >, (4) − where 1 2π < cos θ>= cos(θ)dθ. (5) 2π Z0 Since < cos θ >= 0. Assuming w = z , and applying this N times, for N sufficiently large, yields (2). It is important to note that the derivation of (2) assumes that the probabil- ity distribution of our random walk has a zero mean. This assumption, although restrictive, works well for our modeling described in Section 3.4. Also, we are assum- ing in the derivation of (2) that the velocity increments vk have equal magnitude. This restriction is discussed in Section 3.4. It is noted that the value if N used in the derivation of (2) needs to be ’sufficiently large’. The values of N that are determined to provide a validation of (2) are also discussed in Section 3.4. This is accomplished by checking the validity of Equation 11. We can apply (2) to the three-body problem by filling in the motion in space between velocity perturbation changes at random time intervals. To do this, we will consider the motion of one of the three mass point particles, P . We will assume that this particle moves in a manner we describe below relative to the two other particles, for a random time span t1, assuming it starts with zero velocity. At the end of this time, the particle undergoes a small random velocity perturbation, v1 of magnitude v, due to a perturbation we’ll describe below, added to the velocity the particle already has at the given time. P then moves according to the differential equations for a random time t2. At the end of this time another random velocity perturbation v2, also of magnitude v, is added to the velocity of the trajectory and the motion of P continues for another random time t3, and the process continues. If we now let V be the actual velocity magnitude of P , then in examples we will describe below, it turns out that (2) is satisfied. This gives the resultant velocity magnitude of the moving particle due to small velocity perturbations given at random times. This relation gives an estimate of how the resultant velocity on P increases. In order to apply the random walk procedure as just described to the three-body problem, the motion of the particle P being considered needs to have three special properties: a.) There needs to be a well defined way to prescribe the application of small ve- locity perturbations at different random times. b.) The magnitudes of the velocity perturbations need to be equal. c.) The motion of interest that the particle is performing remains stable enough so that a sufficiently high number N of velocity perturbations can take place where (2) is satisfied.
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