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.

These conditions are described below in the examples given. The main problem to which we apply the random walk procedure to the horseshoe family of orbits used in a theory to help explain the formation of the Moon. After that, another application is briefly described. 522 EDWARD BELBRUNO

3. Horseshoe orbits, breakout, and random walk. The application of random walk to the dynamics of horseshoe orbits in the three-body problem, described in this paper, is motivated by the current generally accepted theory for the formation of the Moon, called the ’giant impact hypothesis’ [10, 6]. This describes how the early Earth was hit by a Mars-sized object(P3), about .1 mass of the Earth, and from that collision, the debris coalesced to form the Moon. What was not addressed in this hypothesis was where the Mars-sized object(impactor) came from. A theory of the origin of the impactor and how it could have collided with the Earth was published in 2005 by Belbruno and Gott [2]. The general idea is that P3 initially formed by material collecting near one of the equilateral Lagrange points L4,L5. Then, as more material collects there, and grows by accretion in mass, small velocity displacements on the forming object cause it to move in horseshoe orbits [13, 12] moving back and forth in approximately Earth’s orbit. The horseshoe motion eventually bifurcates into another type of motion called breakout, where P3 cycles about the Sun. This results in collision with the Earth. Our goal in this section is to explain, more generally, how random walk can be used to model the orbital dynamics of P3 from its initial condition at L4 or L5 to breakout, and to Earth collision. The evolution of the horseshoe orbits to breakout motion can be modeled using random walk in a reasonably accurate fashion as we will see in Section 3.4. The model we’ll consider to describe the random walk of P3 is planar restricted three-body problem between P3, Earth, Sun. At the end of this section we will briefly describe how this modeling generalizes to be more physically realistic.

3.1. Restricted Three-Body problem model. The particle P3 would require a realistic mass of .1 times the Earth’s mass to be faithful to the giant impact hypothesis. However, for the purposes of this paper we assume it’s mass is zero. Given the relatively large masses of the Sun and Earth this is fine, and as discussed in [2] yields accurate results. Let P1 represent the Sun, P2 the Earth, and P3 the third mass particle. We will model the motion of P3 by the differential equations for the restricted three-body problem. It assumes: 1. P1, P2 move in mutual Keplerian circular orbits about their common center of mass which is placed at the origin of an inertial coordinate system X, Y . 2. The mass of P3 is zero. Letting mk represent the masses of Pk, k = 1, 2, 3, then m3 = 0, and we assume that m2/m1 = .000003. Let ω be the constant frequency of circular motion of m1 and m2, ω = 2π/P , where P is the period of the motion. We consider a rotating coordinate system (x, y) which rotates with the constant frequency ω as P1 and P2. In the x–y coordinate system the positions of P1 and P2 are fixed. Without loss of generality, we can set ω = 1 and place P at (µ, 0) and P at ( 1+ µ, 0). Here we normalize the mass of m to 1 2 − 1 1 µ and m2 to µ, µ = m2/(m1 + m2)= .000003 . The equations of motion for P3 are− x¨ 2y ˙ = x +Ωx − (6) y¨ + 2x ˙ = y +Ωy, d ∂Ω where ˙ dt , Ωx ∂x , ≡ ≡ 1 µ µ Ω= − + , r1 r2 RANDOM WALK IN THE THREE-BODY PROBLEM 523

1 2 2 r1 = distance of P3 to P1 = [(x µ) + y ] 2 , and r2 = distance of P3 to P2 = 1 − [(x +1 µ)2 + y2] 2 , see Figure 1. We note that the units of position, velocity and time− are dimensionless. To obtain position in kilometers, the dimensionless position (x, y) is multiplied by 149, 600, 000 which is the distance of the Earth to the Sun. To obtain the velocity in km/s, s = seconds, the velocityx, ˙ y˙ is multiplied by the circular velocity of the Earth about the Sun, 29.78 km/s. For (6), t = 2π corresponds to 1 year. We will occasionally mention the physical units to give a sense of the real parameters, although this is not necessary.

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Figure 1. Rotating coordinate system and locations of the La- grange points.

It is noted that (6) is invariant under the transformation, x x, y y, t t. This implies that solutions in the upper half-plane are sym→metric→− to solutions→ in− the lower-half plane with the direction of motion reversed. This implies, as noted in the introduction, that all the results we will obtain for L4 are automatically true for L5, and thus L4 need only be considered. System (6) of differential equations has five equilibrium points at the well known Euler-Lagrange points Lk, k =1, 2, 3, 4, 5, wherex ¨ =y ¨ = 0 andx ˙ =y ˙ = 0. Placing P3 at any of these locations implies it will remain fixed at these positions for all time. The relative positions of Lk are shown in Figure 1. Three of these points are collinear and lie on the x-axis, and the two that lie off of the x-axis are called equilateral points. The three collinear Lagrange points Lk, k = 1, 2, 3 lying on the x-axis are un- stable. This implies that a gravitational perturbation of P3 at any of the collinear Lagrange points will cause P3 to move away from these points as time progresses. We will focus our attention on the equilateral points for this paper. These points are stable, so that if P3 were place at these points and gravitationally perturbed a sufficiently small amount, it will remain in motion near these points for all time. This stability result for L4,L5 is subtle and was a motivation for the development of the so called Kolmogorov-Arnold-Moser(KAM) theorem on the stability of motion of quasi-periodic motion in general Hamiltonian systems of differential equations [1]. A variation of this theorem was applied to the stability problem of L4,L5 by 524 EDWARD BELBRUNO

Deprit & Deprit-Bartolom´ein 1967 [14]. Their result is summarized in the following result and represents an important application of KAM theory,

Theorem 3.1. L ,L are locally stable if 0 <µ<µ , µ = 1 (1 1 √69) .0385, 4 5 1 1 2 − 9 ≈ 1 1 √ 1 1 √ and µ = µk, k = 2, 3, 4 µ2 = 2 (1 45 1833) .0243, µ3 = 2 (1 15 213) .0135,6 µ .0109. − ≈ − ≈ 4 ≈ In our case, the Earth has µ = .000003 which is substantially less than µ1 and the exceptional values µk, k =2, 3, 4 so that L4 is clearly stable for the case of the Earth, Sun system. An integral of motion for (6) is the Jacobi energy given by J = (x ˙ 2 +y ˙2) + (x2 + y2)+ µ(1 µ)+2Ω. (7) − − Thus Σ(C) = (x, y, x,˙ y˙) R4 J = C, C R is a three-dimensional surface in the four-dimensional{ phase∈ space| (x, y, x,˙ y˙),∈ such} that the solutions of (2) which start on Σ(C) remain on it for all time. C is called the Jacobi constant. The manifold Σ(C) exists in the four-dimensional phase space. It’s topology changes as a function of the energy value C. This can be seen if we project Σ into the two-dimensional position space (x, y). This yields the Hill’s regions (C) where P3 is constrained to move. The qualitative appearance of the Hill regionsH (C) for H different values of C are described in [3]. As C decreases in value, P3 has a higher velocity magnitude at a given point in the (x, y)-plane. In this paper we will be considering cases where C is slightly less than 3, C < 3, ∼ where the Hill’s region is then the entire plane. Thus, in this case P3 is free to move throughout the entire plane.

3.2. Transition from horseshoe orbits to breakout motion. In order to rigor- ously define our concepts and motions, we describe how to construct a special family of horseshoe orbits which originate at L4 and evolve towards breakout in a tradi- tional deterministic manner as the velocity of the orbits increases in a predictable fashion. We then mention how Earth collision can result with high probability. After this is done, we reproduce the evolution towards breakout using the non- deterministic process of random walk and show that the velocity of the horseshoe orbits increases as predicted by (2).

We consider System (6) and place P3 precisely at L4. As long as the velocity of P3 relative to L4 is zero, then P3 will remain at L4 for all time. The velocity vector at L for P is given by v = (x, ˙ y˙). Let α [0, 2π] be the 4 3 ∈ angle that v makes with the local axis through L4 that is parallel to the x-axis. Thus, v = V (cos α, sin α), V v = x˙ 2 +y ˙2. When V =∗ 0 and if t = 0 is≡ the | | initialp time for P at L , then for t > 0, P 6 3 4 3 need not remain stationary at L4. If V is sufficiently small, then by Theorem 3.1 the velocity of P3 should remain small for all t > 0, and P3 should remain within a small bounded neighborhood of L4. This follows by continuity with respect to initial conditions. However, as V (0) V (t) increases, then the resulting motion ≡ |t=0 of P3 need not stay close to L4 for t> 0. This is investigated next.

We fix α and fixing P3 at L4 at t = 0, we gradually increase V (0) and observe the motion of the solution curve γ(t) = (x(t),y(t)) for t> 0 for each choice of V (0). This is done by numerical integration of System (6). (All the numerical integrations RANDOM WALK IN THE THREE-BODY PROBLEM 525 in this paper are done using the numerical integrator NDSolve of Mathematica 4.2.) The following general results are obtained which we first state, and then illustrate with a number of plots (In all of the plots of orbits for the restricted problem, in the x, y plane which are labeled ’Sun centered’ the translation x x+µ, y y has been applied which puts the Sun at the origin, and the Earth at→the point (→1, 0)). − Summary A: Transition of Horseshoe Orbits to Breakout Motion For each choice of α [0, 2π], as V (0) is gradually increased from V (0) = 0, ∈ and where γ(0) = (x(0),y(0)) is at L4, the trajectory γ(t) for t > 0, remains in small arc-like regions about L4, which as V (0) increases, evolve into thin horseshoe regions containing L4 and lying very near to the Earth’s orbit about the Sun. As V (0) increases further, the horseshoe region begins to close on itself, approaching forming a continuous annular ring about the Sun, coming close to connecting at the Earth. It is found that there exists a well defined critical value of V (0) = V ∗(0) where the ring closes at the Earth, and then the motion of P3 bifurcates from a motion constrained to the horseshoe-like region where it never makes a full cycle about the Sun, to motion where it continuously cycles about the Sun, repeatedly passing close to the Earth, and no longer in the horseshoe motion. We refer to this continuously cycling motion for V (0) = V ∗(0) as breakout. Breakout continues to > occur for V (0) V ∗(0) ∼ Demonstration of Summary A

We choose an arbitrary velocity direction v for P3 at L4 at t = 0, where v points in the vertical positive y direction. In our simulations, the location of L4 is √3 ( .5, 2 ), and the y coordinate is input with the value .866025404. Beginning with α−= π/2, we choose− a magnitude V = V (0) = .001, and numerically integrate the system of differential equations (2) forward for t [0, 1000]. This velocity magnitude ∈ is small, and since L4 is stable P3 remains in a thin arc-like region approximately of radius 1 shown in Figure 2. P3 starts at the location x(0) = (.5, .5√3), and

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Figure 2. γ(t), V (0) = .001,t [0, 1000], x vs y (i.e. x-axis is horizontal, y-axis is vertical), Sun∈ centered. moves down in the posigrade direction with respect to the Sun. As it moves, it performs many small loops as are shown in Figure 2. These loops occur since 526 EDWARD BELBRUNO the semi-major axis of the orbit of P3 has changed slightly from 1 and the orbit of P3 has a slight nonzero ellipticity, both due to the addition of V (0). So, as it moves in its approximate elliptical motion over the course of one year it falls slightly behind and forward with respect to the Earth when it is at its apoapsis and periapsis, respectively. Each loop forms in one year. Thus, for t = 1000, there are 1000/(2π) loops. We choose t = 1000 to obtain a clear pattern to the motion of the particle. P3 moves down to a minimal location where y is approximately .7, and then it turns around and moves in the upward direction where the small loops point in the opposite direction when it was moving in the downward direction. The superposition of the loops makes a braided pattern as seen in the lower half of Figure 2. P3 stays in this bounded arc-like region since L4 is stable, and the velocity V (0) is relatively small. Because the velocity magnitude is small, P4 has a Kepler energy nearly that of L4, and so its semi-major axis with respect to the Sun deviates from 1 by a negligible amount. Thus, as it moves, it stays nearly on a circle of radius 1. That is, in an inertial coordinate system it stays approximately on Earth’s orbit about the Sun. As long as V (0) is small, which it is throughout this paper, the trajectories of P3 remain close to the Earth’s orbit and move with small loops, in the rotating coordinate system. The particle P3 moves slowly along the Earth’s orbit initially in a posigrade fashion, and then in a retrograde fashion away from the Earth. The above procedure is repeated, where we slightly increase the value of V (0) to .004 at L4 at t = 0. Since V (0) has increased, then as is seen in Figure 3, where t [0, 1000], P moves further in its Earth-like orbit about the Sun. Since V (0) is ∈ 3

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Figure 3. γ(t), V (0) = .004,t [0, 1000], x vs y, Sun centered. ∈ small, the trajectory of P3 deviates slightly from a circle of radius 1. This deviation slightly increases as V (0) increases. The addition of V (0) at L4 causes P3 to have a slightly smaller value of the Jacobi integral, to be slightly less than 3 (C < 3). ∼ This means that P3 becomes more energetic, and thus can move further along the RANDOM WALK IN THE THREE-BODY PROBLEM 527

Earth’s orbit. Increasing V (0) by .001 to .005 causes the increased motion shown in Figure 4, where t [0, 1000]. ∈ 1

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Figure 4. γ(t), V (0) = .005,t [0, 1000], x vs y, Sun centered. ∈

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Figure 5. γ(t), V (0) = .009,t [0, 1000], x vs y, Sun centered. ∈

In Figure 5, V (0) is increased to .009. P3 leaves L4, moves downward in a posigrade fashion to slightly behind the Earth, then turns around and moves in a retrograde fashion on its Earth-like orbit about the Sun, until it approaches the Earth from the front turning around and then moving in a posigrade fashion. A braided pattern results due to the fact the Earth-like orbit is traversed twice, with loops pointing in the inner and outer directions. The resulting complicated 528 EDWARD BELBRUNO looking trajectory is symmetric with respect to the x-axis due the symmetry men- tioned earlier for the restricted problem. The width of the region near the Earth’s orbit in which P3 moves has slightly increased due to the increase in V (0). Let θ be the polar angle measured from the positive x-axis for the position of P3. The horseshoe orbits have the the property that θ = π. This means that P3 will not fly by the Earth. 6 When V (0) reaches .011, P3 is able to escape from the thin horseshoe-like region and fly by the Earth as is seen in Figure 6.

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Figure 6. γ(t), V (0) = .011,t [0, 1000], x vs y, Sun centered. ∈

This achieves breakout motion where P3 then cycles about the Sun only in one direction. In Figure 6 the cycling is in the retrograde direction. This actual cycling is not shown in this figure since for the time range given, breakout into cycling motion occurs when t = 988, on the outer retrograde trajectory. P3 first leaves L4 moves near the Earth, then back up in a retrograde fashion going all the way around the Sun to near and in front of the Earth, then moving around the Sun again in a posigrade fashion to its location behind the Earth, then finally it moves on the outer trajectory in a retrograde fashion back to just ahead of the Earth when it crosses by the Earth at t = 988 (crossing the x-axis near the Earth), then performing the cycling breakout motion after that time. This transitional breakout motion has two important properties: 1. P3 moves in a thin annular region about the Sun, 2. P3 repeatedly flys by the Earth.

Breakout motion is seen in Figure 7. It is observed that a shift from V (0) = .009 to V (0) = .012 causes a qualitatively different looking picture, where the bifurcation between horseshoe and breakout motion is clearly seen.

These properties imply that there is a high probability of collision of P3 with the Earth after breakout occurs, as we’ll show in the next section. This is the case since P3 obtains a negligible gravity assist to increase the velocity of P3 after each RANDOM WALK IN THE THREE-BODY PROBLEM 529

Earth fly-by, and therefore the cycling will in general occur over and over, staying within a thin annular region containing the Earth. This yields a high likelihood of collision. In fact, the previous case where V (0) = .011, which is the first breakout motion we computed, leads immediately to collision at t = 1384.7176 (or 220.3847 years). In our exposition below, we will use a slightly different value of V (0) which happens to achieve collision at an even earlier time. (It is remarked that by collision, what we mean in this paper is physical collision with the Earth, where P3 moves to within the combined radii of P3, rI , and the radius of the Earth, rE. More precisely, r r + r . (In physical units, r =6378 km, r =3397 km, the radius of Mars.) 2 ≤ I E E I A pure collision in the mathematical sense where r2 = 0 is not considered in this paper. However, the numerical investigations showed that this also occurs, but it is not as likely has a physical collision.)

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Figure 7. γ(t), V (0) = .012,t [0, 1000], x vs y, Sun centered. ∈ The case just considered is for α = π/2. The same procedure produces critical values of V (0) = V (0)∗ leading to breakout motion, from horseshoe motion, for any value of α [0, 2π]. A set critical values for α increments of π/8 are graphically ∈ shown in Figure 8. In this figure, the length of each line is equal to the value of V ∗(0) in that direction. There is a sharp spike in the value of V ∗(0) which has a maximum at5.102π/8 of .22. There is also a similar maximum near the value of 13.5π/8. These are not listed since they are not typical: almost all the values of V ∗(0) are in the range of values illustrated. The minimum value of V ∗(0) = Vmin∗ (0) = .0057 is for α = (9π/8) .01. (Multiplying the values of V ∗(0) by 29.78 yields a range of velocity values generally− between .18 km/s and 1.2 km/s.) Note that the two directions corresponding to the maximal spikes in velocity seen in Figure 8 approximately lie near the Sun and anti-Sun directions. In this figure the Sun is toward the lower right. This variation can be made much smoother by choosing more of a refinement of α which is not necessary for the purposes of this paper. Note that the method, or algorithm, used above to estimate the critical velocities V ∗(0) at L4 leading to breakout motion, is similar in nature to the method of 530 EDWARD BELBRUNO

0.015 0.01 0.005

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Figure 8. Initial velocity directions v(0) at L4 whose magnitude corresponds to the associated critical breakout velocity V ∗(0) (di- mensionless velocity units). estimating transitional stability regions, called weak stability boundaries, between capture and escape about the Moon described in [3]. The recent work by Garcia and Gomez [9] shows the complex nature of this region. 1 Other methods could be used to study the bifurcation from horseshoe to breakout motion such as the computation of suitable surfaces of section to the trajectories in phase space, and then monitoring the iterates of intersecting trajectories on the section. This would give a more complete knowledge of the phase space near breakout motion, but this approach is not necessary for our purposes. The algorithm we have described accurately determines when bifurcation occurs. This concludes the demonstration of Summary A.

3.3. Creeping collision orbits. We briefly describe how collision trajectories can be readily found once breakout has been achieved. An estimate is given for the probability of collision showing that it is likely.

It is first shown how to readily find trajectories from L4 which collide with the Earth. The value of α = π/2 is again considered, and we consider the case V (0) = > .012 V ∗(0) shown in Figure 7. The fly-by periapsis distances of r2 between P3 and the∼ Earth for t [0, 1000] reveals the times of the various Earth fly-bys. It was found that the case∈ of V (0) = .012, for the given range of t, had very close Earth fly-bys, but no collision. Randomly altering this value of V (0) yielded a collision on the second random choice of values of V (0) = .0119981. This is seen by plotting r2 as a function of time shown in Figure 9 Only the second fly-by in Figure 9 collides with the Earth. The time of collision is t = 360.181558, corresponding to 57.3247 years.

The collision orbit, , is shown in Figure 10. It starts at L4, moves in a posigrade fashion toward the Earth,C turns around and in a retrograde motion moves around the Sun to collide with the Earth. A view of this orbit in its final 9.57 years is shown in Figure 11. The collision orbit is seen to move relatively slowly near the

1This capture region has important applications that have been verified in space. It was used to find a new type of low energy route to the Moon in 1990 where lunar capture is automatic [4]. This special lunar transfer was designed in order to resurrect a Japanese lunar mission and enable the spacecraft Hiten to successfully reach the Moon in October 1991 with almost no fuel. RANDOM WALK IN THE THREE-BODY PROBLEM 531

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Figure 9. Variation of distance r2 of P3 to P2 as a function of t [0, 1000] in dimensionless units, V (0) = .0119981. ∈

Earth’s orbit which we refer to as a creeping motion. So, we refer to a creeping collision orbit.

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Figure 10. Entire collision orbit (x vs y)- Originating at L4 at t = 0 and colliding with the earth when t = 360.18, V (0) = .0119981, Sun centered.

Collision with the Earth itself and the final 9.14 hours of the trajectory are shown in Figure 12. This figure is shown since the location of where P3 collides with the Earth is consistent with the giant impact hypothesis. 532 EDWARD BELBRUNO

-0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3

-0.2

-0.4

-0.6

-0.8

-1

Figure 11. Orbit approaching collision with the Earth (x vs y). Time duration of 9.57 years shown, t [300, 360.181558], axis earth centered. ∈

It is observed that since the motion of P3 repeatedly passes near to the Earth in breakout motion, the Earth tends to readily pull P3 toward pure and physical collisions. The set, or manifold, of pure collision trajectories are a subset of physical collision trajectories, and, in fact, are a set of measure zero in the four-dimensional phase space of position and velocity (Belbruno 2004). Since they are a set of measure zero, their near occurrence is reflective of the fact the fly-bys of the Earth are close and that the Earth has a considerable gravitational focusing effect when the trajectory is near parabolic. As mentioned previously, pure collision is observed to occur. An important characteristic of the collision trajectory is its relatively low Kepler > energy, E2 it has with respect to the Earth when near to collision, E2 0, which we refer to as near parabolic. This is also consistent with the giant impact∼ hypothesis. The fact that the collision orbits are near parabolic is briefly summarized:

It can be analytically shown that in the critical or near critical breakout motion, all close Earth fly-bys, including collision trajectories, are near parabolic at periap- sis. For critical breakout trajectories, which start at L4 at time t = 0, V (0) = V ∗(0). > For near critical breakout motion we assume that V (0) V ∗(0). Notationally, ∼ V (0) ≥ V ∗(0) includes both of these cases. (We say that this is a close Earth fly-by if r ∼ 100, 000 km, or in dimensionless coordinates, r .000668. The figure of 2 ≤ 2 ≤ RANDOM WALK IN THE THREE-BODY PROBLEM 533

0.00015

0.0001

0.00005

-1.00015 -1.0001 -1.00005

Figure 12. Collision of the impactor with the Earth, x vs y. Final 9.14 hours of trajectory shown, earth centered.

100,000 km is arbitrarily chosen since for weakly hyperbolic fly-bys of the Earth beyond this distance, the effect of an Earth gravity assist is negligible.) It can be shown that for the set of critical or near critical breakout velocities at L4, the value of E2 at the close Earth fly-bys at periapsis has the value

1 2 1 2 > E V (0) ≥ V ∗(0) 0; (8) 2 ≈ 2 ∼ 2 ∼ That is, the close Earth fly-bys are near parabolic. This is true for all the values of α except those values in small neighborhoods of 5.102π/8, 13.5π/8 (see comment below).

This implies that a trajectory γ(t) starting near critical breakout velocity at L4 for t = t0 will satisfy (8) for any future time t>t0 corresponding to any close Earth fly-by at periapsis. As mentioned earlier, there are two sharp spikes in the breakout velocities shown in Figure 8 of values .22, .25 which occur for α = 5.102π/8, 13.5π/8, respectively. However, most values of V ∗(0) vary between approximately .006 and .05 if two intervals in α of total width approximately .157 radians are deleted near where the spikes occur. This implies that for nearly all of the values of α, (8) yields, E [.000018, .0012]. (9) 2 ∈ Thus, P3 is near parabolic at collision. The range given by (9) is a rough estimate of E2 that can be made more precise. The observed value for of E2 = .000054 is contained within this interval. C

is paired with another symmetric collision trajectory ∗ emanating from L5 whichC is symmetrical to and moves in a posigrade fashionC about the Sun. This C 534 EDWARD BELBRUNO follows by the symmetry of solutions. It will collide with the Earth in the 4th quadrant as shown in Figure 12. Probability of Collision at Breakout for the Restricted Problem A measure of the likelihood of finding collision trajectories is now described. This is done for the four basic initial velocity directions at L4: α = 0,π/2, π, 3π/2. For an initial velocity of P3 at L4 for a given α we assume the corresponding breakout velocity V ∗(0) as shown in Figure 8. The orbit of P3 is propagated from L4 for t 0 and since it is in breakout motion we know that it will not be in horseshoe motion,≥ but will cycle about the Sun and repeatedly fly past the Earth. We can numerically demonstrate that collision with the Earth is likely. This intuitively makes sense since the fly-bys will be close and the the annular region supporting the breakout motion is narrow. Now, for a given initial velocity at L4 for t =0 we see from Figure 8 that V ∗(0) is given up to three digits. For a given value of V ∗(0), depending on α, we propagate the trajectory for up to t = 4000, which corresponds to 637 years, and see if collision has occurred. t = 4000 is chosen arbitrarily, for convenience and is fairly small in astronomical terms. If no collision occurred in that time, then we give V ∗(0) a random perturbation by adding to it the random number .000mn, where m,n are positive random integers ranging from 0 to 9. For a choice of m,n the trajectory is propagated again. If collision does not occur, we repeat the process again for a different choice of m,n, continuing trials until success is achieved. For α =0,π/2, we required two random trials for success, where success means we achieve collision within t = 4000 = 637years. For α = π, three random trials were required until we have success, and for α =3π/2, six random trials were required for success. Therefore we have achieved success in four random trials out of thirteen. This gives our best estimate of the probability of success for 0 t 4000 as P ≤ ≤ 4 . (10) P ∼ 13 If we had not limited ourselves to t = 4000 the probability would have been larger. We have run a sufficient number of trials to produce a rough order of magnitude estimate of this probability which is sufficient for our purposes, but a large number of additional trials could establish this number to higher accuracy. It is noted that the gravitational focusing on P3 to cause a collision is substantial. This is related to the fact the breakout motion is occurring at a fixed energy for the planar restricted problem. The fixed energy yields a three-dimensional energy surface obtained from the Jacobi integral. The manifolds leading to collision at P2 are two-dimensional, and although they are a set of measure zero, the particle P3 is readily able to move asymptotically close to these surfaces and to collision after the gravitational focusing. The collision manifolds on the Jacobi integral surface separate the phase space, so it is fairly easy for P3 to get near to the collision manifold. In higher dimensions this separation of the phase space on the Jacobi surface does not occur, and the collision manifold is more elusive.

3.4. Random walk leading to breakout. In determining V ∗(0) at L4 above, we kept P3 fixed at L4 and gradually increased V (0) for a given velocity direction. This yields a well defined set of V ∗(0)(α) for α [0, 2π]. ∈ We now consider a more realistic way that P3 would increase its velocity in a gradual fashion. The mechanism for this is to assume that P3 gets instantaneous RANDOM WALK IN THE THREE-BODY PROBLEM 535 changes in velocity, we refer to as ∆V kicks, at random discrete times. So, in this case, once P3 receives a ∆V kick when starting at L4 with zero velocity, it immediately moves away from L4 on its trajectory defined by the restricted three- body problem. More precisely, we assume that the times of encounters are random, within a large range, and the direction α of the kicks are random. The only thing we normalize is the magnitude of the ∆V ’s which for convenience is held fixed. | | The motivation for this modeling is that P3 would be getting random velocity kicks due to having encounters with material (small planetesimals) that were presumed to be abundant at the time of the early solar system. If these planetesimals were assumed to be very small, then the velocity kicks would similarly be very small, and assuming that they would be approximately equal is reasonable. From a strict mathematical sense though, this is a restrictive assumption. It is sufficient for our toy model, but a more in depth analysis of the random walk process would ideally assume that the velocity ∆V kicks would not be constant. However, if these kicks were assumed to be sufficiently| | small, then assuming that they would be of equal magnitude may suffice in certain general situations and provide insight into the more general situation. Thus, P3 starts at L4 with a zero velocity, and at time t = t1 = 0 a velocity V (0) = ∆V , is applied in a random direction. This yields a vector v1 with magni- tude ∆V . P3 moves on a trajectory γ(t) in a neighborhood of L4, assuming that the value of ∆V is small. At a random time t2 > 0 another velocity vector v2 of random direction and magnitude ∆V is vectorially added to P3’s velocity at t = t2. Then the trajectory is propagated for t>t2 until at another random time t3 > t2 a random vector v2 of magnitude ∆V is vectorially added to P3’s velocity vector at t = t3, and this process continues creating a sequence tk of times, tk+1 >tk, and velocities vk, k =1, 2, 3,.... While the ∆V ’s are being applied, the trajectory γ(t) is gradually moving further from L4, but since the velocity directions vk are applied randomly, the path of the trajectory γ(t) will move further away from L4 for some time spans, and then move toward L4 for others. However, as k increases, one would expect, by the principle of random walk, for P3 to eventually escape L4 and creep toward the Earth for k sufficiently large when the velocities vk, k = 1, 2, 3,... applied on the trajectory γ(tk), gradually accumulate to a sufficiently large magnitude for breakout to occur. If the vk were all applied in the direction of motion of P3 at tk, then the magnitudes ∆V would add producing an cumulative velocity addition of k∆V at the kth step. However, the directions of vk are random, and by the principle of a random walk, the number of encounters k before ejection occurs should be expected to instead satisfy √k∆V .006 (11) ≈ for k sufficiently large(and ∆V sufficiently small) where .006 is approximately the minimum value .0057 of V ∗(0) . This makes dynamical sense, since as the ∆V ’s are applied, the trajectory{ γ(t)} would seek to minimize the Jacobi energy, and hence the velocity, along its path. We found in all our numerical simulations that for a given value of ∆V , the number k of random vk applications required for breakout to occur approximately satisfies (11). We refer to this as random walk ∆V accumulation. Equation 11 can be used to check the accuracy of the random walk model in the restricted three-body problem. We describe the process of random walk ∆V 536 EDWARD BELBRUNO accumulation with several examples illustrating the accuracy. For convenience we choose ∆V = .001 and start at L4 with zero initial velocity. (11) implies that k should satisfy √k∆V .006, which yields k 36. We propagate a trajectory using the random walk∼ model where a random∼ velocity direction is chosen at each random time ti,i = 1, 2, ..., k, where the velocity magnitude is fixed at .001. We monitor the magnitude of the resultant velocity of P3 at the end of the kth step until the velocity reaches the value of .006, and record the value of k. The first simulation yielded a value of k = 36 as predicted by (11). We repeated this two more times, and found the values of k = 37, 34, respectively. In these experiments, the random walk model gave reasonably accurate results. This also implied that k = 36 satisfied the criteria of being approximately large enough, at least for the few cases we ran. These simulations are lengthy with the software we are currently using, so only three cases were run with this velocity value. We also ran another simulation using the value of ∆V = .0015. Equation 11 predicted that k 9, and we obtained a value of 6. This accuracy is not as good as for the previous∼ velocity value, probably because the predicted value of k is not large enough. The accuracy of this method should be performed with more simulations using smaller values of ∆V , and larger values of k. This is a project for further work. Our results are summarized in Table 1. (The random time intervals are chosen to be large enough to randomize the position which we took to lie within [0, 4000].)

Case ∆V N Error 1 .0010 36 0 2 .0010 37 1 3 .0010 34 -2 4 .0015 6 -3

Table 1 Error from predicted iterations in the random walk procedure.

From the above, we have the following result, Random Walk ∆V Accumulation Under a realistic assumption of random walk, the resultant velocity for P3 accumulates proportional to the square root of the number of encounters until it reaches a breakout state. Since a random walk is isotropic the resultant velocity is likely to encounter the breakout state first at a point near the minimum value of .006 of the set V ∗(0) thus giving (11). { } Therefore, substituting V ∗(0) = .006 into (8) implies that for close Earth fly- bys resulting from the random walk process at or near breakout, E2 .000018. This implies that at close Earth fly-by resulting from the random walk≈ process, a nominal value of V = .006 is obtained which is .179 km/s. When P3 does a close fly-by of the Earth,∞ after passage through periapsis it will receive a gravity assist and increase, or decrease, its velocity with respect to the Sun. A measure of this velocity change is observed due to the bending of the trajectory of P3 as it passes through periapsis. The more distant the fly-by, then in general the less the bending. The maximum bending is obtained from pure collision trajectories, where the bending angle is π. The resulting change in magnitude of the velocity, δv with respect to the Sun due± to gravity assist is maximally, 2V . Thus, for each close Earth fly-by, the expected maximum gain in velocity magnitud∞ e is approximately .358 km/s. In general, they will be less. RANDOM WALK IN THE THREE-BODY PROBLEM 537

The maximal velocity of .358 km/s is a relatively small number and will have little effect on a breakout trajectory when it has a close Earth fly-by. It is found in general that within time spans on the order of 2000 time units, there are generally only one or two close Earth fly-bys. This implies that P3 will remain in breakout motion about the Sun in a relatively thin annular region for very long periods of time, generally tens of thousands of time units, and repeatedly pass by the Earth without being ejected. This enhances the probability that collision will occur. It is noted that the three conditions needed to apply the random walk procedure listed at the end of Section 2 have been satisfied in this situation. Velocity kicks, or perturbations, were applied to P3 in a well defined manner. Their magnitudes were all equal, and the horseshoe motion as it transitioned to breakout is stable enough to acquire the velocity changes in a gradual manner so that the ∆V velocity accumulation could occur.

Realistic Modeling We have described how to model the transition of horseshoe motion to breakout and to collision using random walk. This was done for the planar circular restricted three-body problem. In order to apply this to a more realistic setting, the modeling needs to be considerably extended, with m3 = .1m2. The motion of P3 should be in three-dimensions using the general three-dimensional three-body problem. The Earth’s orbit should not be constrained to be uniformly circular. Also, the random velocity kicks need to more accurately given. They should be given in three dimensions to both the Earth and P3, and in a way that is consistent with the solar nebula structure at the time P3 was forming and moving in its orbit from L4 (or L5). As described in [2], a thin anisotropic disc of planetesimals is assumed and the other planets of the solar system are modeled as well. It is shown that the random walk works accurately as predicted by the ∆V velocity accumulation equation. It is noted that in the three-dimensional case, the probability of collision occurring 1 from breakout is not as high, where a value of about 4 was obtained, instead of the 4 value stated previously for the restricted three-body problem of about 13 .

4. Applicability of random walk to other motions and hyperbolicity. From the previous problem we studied, we can deduce the type of motions that lend them- selves to the applicability of the random walk procedure, in the restricted three- problem problem and more general settings. The particle P3 started its motion at L4 with a zero velocity, and small velocity displacements ∆V applied at random times in random directions were sufficient to cause significant, but gradual, change in the motion of P3. For reference, we call this regular motion. This regular motion occurs in a well defined bounded narrow annular region approximately centered at a unit distance from P1. This motion is that of horseshoe orbits in this case. As more and more velocity displacements are applied at random times in accordance with the random walk procedure, the particle gradually approaches breakout. While this is occurring the horseshoes gradually increase in extent until breakout is achieved. Meanwhile, the resultant velocity accumulates in accordance with the random walk equation V √N∆V , where N is the resulting number of velocity displacement ≈ and where breakout occurs when V V ∗(0)min. Breakout is observed to occur after a short nonlinear transition near≈ the Earth. This is seen in Figure 6 as the trajectory arc that is seen passing by the Earth to start the breakout process. This 538 EDWARD BELBRUNO transition itself we refer to as unstable transition motion. After this transition mo- tion, a bifurcation to breakout occurs and it can be viewed as a motion analogous to escape from L4. So, in general, we have the evolution of regular random walk motion to an un- stable transition motion to escape. Within this very general setting, it is easier to see how to apply the random walk procedure in other types of situations. One such situation that we can consider, for example, is that of P3 assumed to move about the Sun in a near circular motion, with a very small eccentricity. In this case, we assume that P2 is Jupiter, and that P3 moves interior to Jupiter’s orbit. For such motions in the planar restricted restricted three-body, we know from the KAM theorem, that when the eccentricity of the orbit of P3 is assumed to be zero, or is sufficiently small, then the motion will be constrained to lie either on or be- tween two-dimensional invariant tori in the three-dimensional phase space, when restricted to the Jacobi energy surface, and therefore is orbitally stable. However, in the planar general three-body problem, the motion of P3 need not be stable. In this more general problem, Jupiter’s orbit is not constrained to be uniformly circular and the mass of P3 is assumed to be nonzero. Let’s assume that P3 moves in the vicinity of the outer most part of the inner asteroid belt, at a radial distance greater than .5 units from the Sun, and where Jupiter is assumed to be moving at about 1 unit, in our dimensionless coordinates. As P3 moves about the Sun in its near circular orbit, it will repeatedly make fly-by passes with respect to Jupiter, when it is at periapsis with respect to Jupiter. At these times, it will lie on the radial line between the Sun and Jupiter, which we refer to as a ’periapsis passage’, where the distance to Jupiter, up to first order, is at a minimal value, labeled d. As P3 cycles about the Sun, its orbit is gravitationally perturbed by Jupiter. This will gradually cause the orbit of P3 to change. This change can be modeled by the random walk procedure. A way to model the gravitational perturbative effect of Jupiter on the orbit of P3 is to do this only at the periapsis passages when P3 is d units from Jupiter, d< 0.5. This is done by placing the resulting cumulative gravitational perturbative effect of Jupiter between consecutive periapsis passage times at the current periapsis passage. This gives rise to a small ∆V in the orbit of P3, thereby slightly changing the near circular orbit. This is carried out by Lecar et. al [11]in a more general solar system modeling, but the procedure is applicable to our model also. As described in [11], this ∆V yields a change in the eccentricity, e, ∆e. This can be approximated as ∆V ∆e , (12) ≈ VJ where VJ is the velocity of Jupiter, modeled as circular at distance R = 1, which we normalize to 1 in dimensionless coordinates. As P3 cycles around the Sun, it receives the ∆e increments at a discrete set of random times, tk, k =1, ..., N. These values of ∆e need not be constant in general. Each time P3 receives a change in its orbit at a time tk, we measure as ∆e, this will, in general, change as P3 cycles around the Sun. However, if the orbit changes in a small amount from its near circular one on a given cycle, the value of ∆e should also change by a small amount. To first order, we assume that ∆e is constant. This makes sense if the orbit of P3 about the Sun very gradually changes its orbital parameters from one cycle about the Sun to another. Under our assumptions, as P3 cycles about the Sun in its near circular orbits, as long as P3 doesn’t pass too near RANDOM WALK IN THE THREE-BODY PROBLEM 539 to Jupiter, the variation of ∆e should be gradual. This leads to an approximation that will enable us to apply the random walk process to model an increase in e, resulting in e √N∆e. (13) ≈ It turns out that N is the number of cycles of Jupiter about the Sun. This random walk process is regular. Now, an unstable transition occurs for a short time when the eccentricity of P3 increases the orbit size so that it gets too close to Jupiter. This occurs when the orbit of P3 crosses the orbit of Jupiter occurring for, e d/R. When this occurs, the orbit of P3 changes abruptly to an escape trajectory.≈ The time to achieve the transition can be estimated as T 2 5 ≈ (1/(8µ )(d/R) )TJ , where TJ =2πR/VJ . Unlike the description of the main example of the random walk of the horseshoe orbits, this one we outlined here is heuristic in nature and many of the estimates need to be made more rigorous. However, it is found numerically that it gives fairly accurate results [11]. It follows the same evolution as the horseshoe orbit example, going from regularity to transition to escape. It is promising that this is the case and may imply other motions to consider. As a concluding comment, if ones considers the transition motion in the random walk example, or in the latter example, it is likely that a complicated hyperbolic network exists near the unstable transition motion. This would suggest a connection between motion in a hyperbolic network with random walk, which is worth studying further. Acknowledgments. I would like to thank Marian Gidea’s encouragement for me to write this material up and to Richard Gott for introducing me to the topic of random walk. I am also grateful to the comments of the reviewers which were valuable in enhancing this paper. This work is partially supported by grants from NASA and the AISR program.

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