WHERE DID THE MOON COME FROM?

WHERE DID THE MOON COME FROM?

Edward Belbruno

Program in Applied and Computational Mathematics, Fine Hall, Princeton University, Washington Road, Princeton, NJ 08544-1000;

[email protected]

and

J. Richard Gott III

Department of Astrophysical Sciences, Peyton Hall, Princeton University, Ivy Lane, Princeton, NJ 08544-1001; [email protected]

Receivved 2004 May 26; accepted 2004 November 18

ABSTRACT

The current standard theory of the origin of the Moon is that Earth was hit by a giant impactor the size of Mars,causing ejection of iron-poor impactor mantle debris that coalesced to form the Moon. But where did this Mars-sized impactor come from? Isotopic evidence suggests that it came from 1 AU radius in the solar nebula, andcomputer simulations are consistent with its approaching Earth on a zero-energy parabolic trajectory. But howcould such a large object form in the disk of planetesimals at 1 AU without colliding with Earth early on, beforehaving a chance to grow large or before its or Earth’s iron core had formed? We propose that the giant impactorcould have formed in a stable orbit among debris at Earth’s L4 (or L5) Lagrange point. We show that such aconfiguration is stable, even for a Mars-sized impactor. It could grow gradually by accretion at L4 (or L5), buteventually gravitational interactions with other growing planetesimals could kick it out into a chaotic creepingorbit, which we show would likely cause it to hit Earth on a zero-energy parabolic trajectory. We argue that thisscenario is possible and should be further studied.

Key words: accretion, accretion disks — celestial mechanics — methods: analytical —methods: n-body simulations — Moon — planets and satellites: formation

1. INTRODUCTION

The currently favored theory for the formation of the Moonis the giant-impactor theory formulated by Hartmann & Davis(1975) and Cameron & Ward (1976). Computer simulationsshow that a Mars-sized giant impactor could have hit Earth on azero-energy parabolic trajectory, ejecting impactor mantle de-bris that coalesced to form the Moon. Further studies of thistheory include Benz et al. (1986, 1987, 1989), Cameron &Benz(1991), Canup & Asphaug (2001), Cameron (2001), Canup(2004a), and Stevenson (1987). We summarize evidence fa-voring this theory: (1) It explains the lack of a large iron core inthe Moon. By the late time that the impact had taken place, theiron in Earth and the giant impactor had already sunk into theircores. So, when the Mars-sized giant impactor hit Earth in aglancing blow, it expelled debris, poor in iron and primarilyfrom the mantle of the giant impactor, which eventually coa-lesced to form the Moon (cf. Canup 2004a, 2004b). Computersimulations (assuming a zero-energy parabolic trajectory for theimpactor) show that iron in the core of the giant impactor meltsand ends up deposited in Earth’s core. (2) It explains the low (3.3 gcm�3) density of the Moon relative to Earth (5.5 g cm�3), againdue to the lack of iron in the Moon. (3) It explains why Earth andthe Moon have the same oxygen-isotope abundances—Earth andthe giant impactor came from the same radius in the solar nebula.Meteorites originating from the parent bodies of Mars and Vesta,from different neighborhoods in the solar nebula, have differentoxygen-isotope abundances. The impactor theory is able to ex-plain the otherwise paradoxical similarity between the oxygen-isotope abundance in Earth combined with the difference in iron.This is perhaps its most persuasive point. (4) It explains, becausethe cause is a somewhat unusual event, whymost planets (such asVenus and Mars, Jupiter and Saturn) are singletons, without a

large moon like Earth’s. Competing ideas have not had compa-rable success. For example, the idea that Earth and the Moonformed together as sister planets in the same neighborhood failsbecause it does not explain the difference in iron. Similarly, theidea that the Moon formed elsewhere in the solar nebula and wascaptured into an orbit around Earth fails because its oxygen-isotope abundances would have to be different. That a rapidlyspinning Earth could have spun off the Moon (from mantle ma-terial) is not supported by energy and angular momentum con-siderations, it is argued.Still, the giant-impactor theory has some puzzling aspects.

Planets are supposed to grow from planetesimals by accretion.How did an object as large as Mars form in the solar nebula atexactly the same radial distance from the Sun without havingcollided with Earth earlier, before it could have grown so large?Indeed, such may have been the case during the formation ofVenus and Mars, for example. It is also hard to imagine anobject as large as Mars forming in an eccentric Earth-crossingorbit. One might expect large objects forming in the solar neb-ula to naturally have nearly circular orbits in the ecliptic plane,like Earth and Venus. Besides, a Mars-sized object in an ec-centric orbit would not be expected to have identical oxygenabundances relative to Earth and would collide with Earth on ahyperbolic trajectory, not the parabolic trajectory that the suc-cessful computer simulations of the great-impact theory havebeen using. (Recent collision simulations by Canup [2004a]place an upper limit of 4 km s�1 on the impactor’s velocity atinfinity while approaching Earth, setting an upper limit on itseccentricity of P0.13.) The Mars-sized object needs to form ina circular orbit of radius 1 AU in the solar nebula but curiouslymust have avoided collision with Earth for long enough for itsiron to have settled into its core. Is there such a place to form thisMars-sized object?

1724

The Astronomical Journal, 129:1724–1745, 2005 March

# 2005. The American Astronomical Society. All rights reserved. Printed in U.S.A.

Yes: Earth’s Lagrange point L4 (or L5), which is at a radius of1 AU from the Sun, with a circular orbit 60

�behind Earth (or

60�ahead for L5). After the epoch of gaseous dissipation in the

inner solar nebula has passed, we are left with a thin disk ofplanetesimals interacting under gravity. The three-body prob-lem shows us that the Lagrange point L4 (or, equivalently, L5)for Earth is stable for a body of negligible mass even though itrepresents a maximum in the effective potential. Thus, plane-tesimals can be trapped near L4, and as they are perturbed theywill move in orbits that can remain near this location. Thisremains true as Earth grows by the accretion of small plane-tesimals. Therefore, over time it might not be surprising to see agiant impactor growing up at L4 (or L5). In x 4, we argue thatthere are difficulties in having the giant impactor come from alocation other than L4 (or L5).

Examples of planetesimals remaining at the Lagrange pointsof other bodies include the well-known Trojan asteroids atJupiter’s L4 and L5 points. As another example, asteroid 5261Eureka has been discovered atMars’s L5 point. (There are five ad-ditional asteroids also thought to be Mars Trojans: 1998 VF31,1999 UJ7, 2001 DH47, 2001 FG24, and 2001 FR127.) The Saturnsystem has several examples of bodies existing at the equilateralLagrange points of several moons, which we discuss further in anote after x 4.

We propose that the Mars-sized giant impactor could formas part of debris at Earth’s L4 Lagrange point. (It could equallywell form at L5, but as the situation is symmetric, we willsimply refer in the rest of the paper, unless otherwise indicated,to the object forming at L4, the argument being the same inboth cases.) As the object forms and gains mass at L4, wecan demonstrate that its orbit about the Sun remains stable.Thus, it has a stable orbit about the Sun, and remaining atL4 keeps it from collision with Earth as it grows. Furthermore,this orbit is at exactly the same radius in the solar nebula as thatof Earth, so the oxygen-isotope abundances should be identi-cal. The object is allowed to gradually grow and there is timefor its iron to settle into its core, and the same also happenswith Earth. The configuration is stable provided the mass ofEarth and the mass of the giant impactor are both below 0.0385times the mass of the Sun, which is the case. But we numeri-cally demonstrate that gravitational perturbations from othergrowing planetesimals can eventually kick the giant impactorinto a horseshoe orbit and finally into an orbit that is cha-otically unstable in nature, allowing escape from L4. The giantimpactor can then enter an orbit about the Sun that is at anapproximate radial distance of 1 AU and which will graduallycreep toward Earth—leading, with high probability, to a nearlyzero-energy parabolic collision with Earth. Once it has en-tered the chaotically unstable region about L4, a collision withEarth is likely. We discuss this phenomenon in detail in x 4.For references on the formation of planetesimals and relatedissues, see Goldreich & Ward (1973), Goldreich & Tremaine(1980), Ida & Makino (1993), Rafikov (2003), and Wetherill &Stewart (1989). We are considering instability of motion nearL4 due to encounters by planetesimals. (The instability ofJupiter’s outer Trojan asteroids due to the gravitational effects ofJupiter over time, studied by Levison et al. [1997], is a differentprocess.)

Horseshoe orbits connected with Earth exist. In fact, an as-teroid with a 0.1 km diameter, 2002 AA29, has recently beendiscovered in just such a horseshoe-type orbit, which currentlyapproaches Earth to within a distance of only 3.6 million kilo-meters (Connors et al. 2002). Horseshoe orbits about the Sun

of this type are also called Earth co-orbiting trajectories, whichare in 1:1 mean motion resonance. An interesting pair of ob-jects in horseshoe orbits about Saturn are discussed in thenote after x 4. A theoretical study of the distribution of ob-jects in co-orbital motion is given by Morais & Morbidelli(2002).

In this paper, we describe a special set of collision orbits withEarth that exist as a result of escape from L4 due to planetesi-mal perturbations. The perturbations cause a gradual peculiar-velocity increase of the mass forming at L4, so that it eventuallyachieves a critical escape velocity to send it toward a parabolicEarth collision approximately in the plane of motion of Earthabout the Sun. The region in velocity space where escape fromL4 occurs in this fashion is relatively narrow. This mechanismtherefore involves a special set of L4 ejection trajectories thatcreep toward collision with Earth. At the end of x 3, we present afull simulation in three dimensions of the collision of a Mars-sized impactor with Earth, assuming a thin planetesimal diskand using the general three-body problem, where planetesimalencounters with both the impactor and Earth are done in arandom fashion. The Appendix discusses the dynamics of therandom planetesimal encounters.

The paper presents several main results: We show that astable orbit at L4 exists in which a Mars-sized giant impactorcould grow by accretion without colliding with Earth. We showthat perturbations by other planetesimals eventually can causethe giant impactor to escape from L4 and send it onto a horse-shoe orbit and then onto a creeping chaotic trajectory with an ap-preciable probability of having a near-parabolic collision withEarth. In x 4, we argue that this scenario fits in extremely wellwith giant-impactor theory and explains the identical oxygen-isotope abundances of Earth and the Moon. The solar systemitself provides a testing ground for our model. As we have men-tioned, the Trojan asteroids show that planetesimals can remaintrapped at Lagrange points, and in the added note we point outthat the system of Saturn’s moons provides examples wherethe phenomenon we are discussing can be observed, supportingour model. Finally, both in the added note and in the Appendixwe discuss prospects for future work.

The spirit of this paper is to suggest the intriguing possibilitythat the hypothesized Mars-sized impactor could have origi-nated at L4 (or L5). We hope that this will lay the groundworkfor more detailed simulations and work in the future.

2. MODELS AND A STABILITY THEOREM

Let P1 represent the Sun, P2 represent Earth, and P3 representa third mass particle. We will model the motion of P3 withsystems of differential equations for the restricted and generalthree-body problems.

The first preliminary model, and key to this paper, is theplanar circular restricted three-body problem, which makes thefollowing two assumptions: (1) P1 and P2 move in mutualKeplerian 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. Thus, P3 is gravitationallyperturbed by P1 and P2, but not conversely. Lettingmk representthe masses of Pk (k ¼ 1, 2, 3), then m3 ¼ 0, and we assume thatm2=m1 ¼ 0.000003. Let ! be the constant frequency of circularmotion of m1 and m2, ! ¼ 2�=P, where P is the period of themotion. We consider a coordinate system (x, y) that rotates withthe same constant frequency ! as P1 and P2. In the x-y coordi-nate system, the positions of P1 and P2 are fixed.Without loss ofgenerality, we can set ! ¼ 1 and place P1 at (�, 0) and P2 at

ORIGIN OF THE MOON 1725

(�1þ �, 0). Here we normalize the mass ofm1 to 1� � andm2

to �, where � ¼ m2=(m1 þ m2) ¼ 0.000003. The equations ofmotion for P3 are

x� 2y ¼ xþ �x; ð1aÞyþ 2x ¼ yþ �y; ð1bÞ

where overdots indicate total differentiation with respect totime, �x � @�=@x, and �y � @�=@y, with

� ¼ 1� �

r1þ �

r2;

r1 is the distance from P3 to P1 (¼½(x� �)2 þ y2�1=2), and r2is the distance from P3 to P2 (¼½(xþ 1� �)2 þ y2�1=2) (seeFig. 1). The right-hand side of equations (1a)–(1b) representsthe sum of the radially directed centrifugal force FC ¼ (x; y)and the sum FG ¼ (�x;�y) of the gravitational forces due toP1 and P2. The units of position, velocity, and time are dimen-sionless. To obtain a position in kilometers, the dimensionlessposition (x, y) is multiplied by 149,600,000 km, which is thedistance between Earth and the Sun. To obtain the velocity inkilometers per second, the velocity (x, y) is multiplied by thecircular velocity of Earth about the Sun, 29.78 km s�1. Forequation (1), t ¼ 2� corresponds to 1 year.

We note that equation (1) is invariant under the trans-formation x ! x, y ! �y, t ! �t. This implies that solu-tions in the upper half-plane are symmetric to solutions inthe lower half-plane with the direction of motion reversed.This implies, as noted in x 1, that all the results we obtain for L4will automatically be true for L5, and thus only L4 need beconsidered.

The system of differential equations in equation (1) hasfive equilibrium points at the well-known Lagrange points Lk,k ¼ 1, 2, 3, 4, 5, where x ¼ y ¼ 0 and x ¼ y ¼ 0. (This impliesthat FC þ FG ¼ 0.) If P3 is placed at any of these locations itwill remain fixed at that position for all time. The relative po-sitions of the Lk are shown in Figure 1. The locations of theLagrange points for arbitrary � 2 (0, 1) are a function of �.Three of these points are collinear and lie on the x-axis, and thetwo that lie off of the x-axis are called equilateral points. Wenote that the labeling of the locations of the Lagrange pointsvaries throughout the literature. We are using labeling consis-tent with that of Szebehely (1967), where, in Figure 1, L2 isinterior to P2 and P1, and L4 lies above the x-axis. (Note thatthis means L4 is 60� behind Earth.)

The three collinear Lagrange points Lk (k ¼ 1, 2, 3) that lieon the x-axis are unstable. This implies that a gravitationalperturbation of P3 at any of the collinear Lagrange points willcause P3 to move away from these points as time progresses,since their solutions near any of these points are dominated byexponential termswith positive real eigenvalues (Conley 1969).The two equilateral Lagrange points are stable, so that if P3 wereplaced at one of these points and gravitationally perturbed by asmall amount, it would remain in motion near that point for alltime. This stability result for L4 and L5 is subtle and was amotivation for the development of the so-called Kolmogorov-Arnold-Moser (KAM) theorem on the stability of quasi-periodicmotion in general Hamiltonian systems of differential equations(Arnold 1961, 1989; Siegel & Moser 1971). A variation of thistheorem was applied to the stability problem of L4 and L5 byDeprit & Deprit-Bartolome (1967). Their result can be sum-

marized as follows and represents a major application of KAMtheory:

L4 and L5 are locally stable if 0 < � < �1, where�1 ¼ 1

2(1� 1

9

ffiffiffiffiffi69

p) � 0.0385, so long as � 6¼ �k for k ¼ 2, 3, 4,

where �2 ¼ 12(1� 1

45

ffiffiffiffiffiffiffiffiffiffi1833

p) � 0.0243, �3 ¼ 1

2(1� 1

15

ffiffiffiffiffiffiffiffi213

p) �

0.0135, and �4 � 0.0109.

(For further details connected to this result, see Belbruno &Gott 2004.1) In our case, Earth has� = 0.000003, which is substan-tially less than �1 and the exceptional values �k (k ¼ 2, 3, 4), sothat L4 is clearly stable for the case of the Earth-Sun system.An integral of motion for equation (1) is the Jacobi energy,

given by

J ¼ �(x2 þ y2)þ (x2 þ y2)þ �(1� �)þ 2�: ð2Þ

Thus �(C ) ¼ {(x, y, x, y) 2 R4 | J ¼ C, C 2R} is a three-dimensional surface in the four-dimensional phase space (x, y,x, y) such that the solutions of equation (1) that start on �(C )remain on it for all time. The quantityC is called the Jacobi con-stant. The manifold �(C ) exists in the four-dimensional phasespace. Its topology changes as a function of the energy value C.This can be seen if one projects � into the two-dimensionalposition space (x, y). This yields the Hill regionsH(C ), in whichP3 is constrained to move. The qualitative appearance of theHill regions for different values of C is described in Belbruno(2004) and Szebehely (1967). As C decreases in value, P3 has ahigher 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, CP3, in which case the Hill region is the entireplane. Thus, P3 is free to move throughout the entire plane.In the next section we will initially use the planar circular

restricted three-body problem to obtain insight into the motionof P3 near L4. Ultimately, we are interested in the general three-dimensional three-body problem for the mass points Pk , k ¼ 1,2, 3, of respective masses mk . Unlike the restricted problem, m3

need not be zero and P1 and P2 are not defined by constantcircular Keplerian motion. Instead, P1 and P2 will be giveninitial conditions for uniform circular Keplerian motion be-tween Earth (P2) and the Sun (P1) assuming that Earth is 1 AUdistant from the Sun. However, for m3 6¼ 0 this circular motionwill not be constant. For small values ofm3, the deviation of the

P2

L4

L5

L1 L2 L3

µ

P1

x2

x1–1 + µ

Fig. 1.—Rotating coordinate system and locations of the Lagrange points.

1 Draft of May 19, which is a longer, more detailed version of the presentpaper.

BELBRUNO & GOTT1726 Vol. 129

motion of P1 and P2 from circular will in general be very small.Later in this paper we will set

m3 ¼ 0:1m2; ð3Þ

where m2 is the mass of Earth and so � ¼ 0.000003. Thus, m3

is a Mars-sized impactor.The differential equations for the general three-dimensional

three-body problem in inertial coordinates (X1, X2, X3) are de-fined by the motion of the three mass particles Pk of massmk > 0 (k ¼ 1, 2, 3), moving in three-dimensional (X1, X2, X3)-space under the classical Newtonian inverse square gravita-tional force law. We denote the Cartesian coordinates of thekth particle by the vector Xk ¼ (Xk1, Xk2, Xk3)2R3. The differ-ential equations defining the motion of the particles are thengiven by

mk Xk ¼X3j¼1j 6¼k

Gmjmk

r2jk

Xj � Xk

rjk(k ¼ 1; 2; 3); ð4Þ

where rjk ¼ jXj � Xk j ¼ ½P3

i¼1 (Xji � Xki)2�1=2 is the Euclidean

distance between the k th and jth particles, G is the universalgravitational constant, and overdots again represent differenti-ation with respect to time. Equation (4) expresses the fact thatthe acceleration of the k th particle, Pk , is due to the sum ofthe forces of the other two particles ({Pi | i ¼ 1, 2, 3, i 6¼ k}). Thetime variable t 2 R1. Without loss of generality, we place thecenter of mass of the three particles at the origin of the coor-dinate system.

We note that the stability result of Deprit &Deprit-Bartolome(1967) provides conditions for the stability of P3 with respectto L4; however, it does not necessarily provide conclusionsabout instability. It has been proved more generally by Siegel &Moser (1971) that in the three-body problem, if m3 satisfies

27(m1m2 þ m2m3 þ m3m1) > (m1 þ m2 þ m3)2;

then the motion is unstable. This is not satisfied in our case,since m1 ¼ 1� �, m2 ¼ �, and m3 ¼ 0.1� with � ¼ 0.000003.However, failure to satisfy this condition does not guaranteestability, and a deeper analysis is required, such as KAM theory.In this more general case, a result like that of Deprit & Deprit-Bartolome (1967) is not available.

We have verified in the general three-dimensional three-body problem defined by equation (4) with m1 ¼ 1� �, m2 ¼�, m3 ¼ 0.1�, and � ¼ 0.000003, and more generally in thethree-dimensional model for the solar system, that L4 is stablefor a numerical integration time span of 10 Myr. The modelof the solar system we used includes the nine planets and ismodeled as an N-body problem with circular coplanar initialconditions using the current masses of the planets and radii ofthe planetary orbits. The integration time span of 10 Myr issuitable for the purposes of our analysis.

3. CHAOTIC CREEPING ORBITS LEADINGTO PARABOLIC EARTH COLLISION

While we have shown through full solar system modelingthat L4 is stable, it might be argued that this stability could beperturbed by other planetesimals, and in fact it is exactly thisprocess that we are investigating (see Appendix). We expectthat gravitational perturbations from other planetesimals will,by means of a random-walk process in peculiar velocity, cause

the Mars-sized impactor to eventually escape from L4. Wenumerically demonstrate in this section that there exists afamily of trajectories leading from L4 to parabolic Earth col-lision. Producing these trajectories shows that Earth collision islikely when P3 escapes L4. P3 escapes L4 once it achieves acritical peculiar velocity—in the rotating frame.

To describe the construction of the parabolic Earth-collidingtrajectories, we will begin first with the planar restricted prob-lem. Then we will show that the results hold up as we make themodel more realistic.

Assumingm3 ¼ 0we consider the system of equation (1) andplace P3 precisely at L4. As long as the velocity of P3 relativeto L4 is zero, then P3 will remain at L4 for all time.

The velocity vector at L4 for P3 is given by v ¼ (x, y). Let�2 [0, 2�] be the angle that vmakes with the local axis throughL4 that is parallel to the x-axis. Thus, v ¼ (V cos � , V sin � ),where V � jvj ¼ (x2 þ y2)1=2.

When V 6¼ 0, and if t ¼ 0 is the initial time for P3 at L4, thenfor t > 0, P3 need not remain stationary at L4. If V is sufficientlysmall, then from the result of Deprit &Deprit-Bartolome (1967)the velocity of P3 should remain small for all t > 0, and P3should remain within a small bounded neighborhood of L4.This follows by continuity with respect to initial conditions.However, as V (0) � V (t)jt¼0 increases, the resulting motion ofP3 need not stay close to L4 for t > 0. This is investigated next.

We fix � and, fixing P3 at L4 at t ¼ 0, we gradually increaseV(0) and observe the motion of the solution curve �(t) ¼ (x(t),y(t)) for t > 0 for each choice of V(0). This is done by numericalintegration of equation (1). (All the numerical integrations inthis paper are done using the numerical integrator NDSolve inMathematica 4.2 until further notice.) The following generalresults are obtained, which we first state and then illustrate witha number of plots. [ In all the plots of orbits of the restrictedproblem (eq. [1]) in the (x, y)-plane that are labeled ‘‘Sun-centered,’’ the translation x ! xþ �, y ! y has been applied,which puts the Sun at the origin and Earth at the point (�1, 0)].

R1.—For each choice of �2 [0, 2�], as V(0) is graduallyincreased from V (0) ¼ 0, and where �(0) ¼ (x(0), y(0)) is atL4, the trajectory �(t) for t > 0 remains in small arclike regionsabout L4, which, as V(0) increases, evolve into thin horseshoeregions containing L4 and lying very near to Earth’s orbit aboutthe Sun. As V(0) increases further, the horseshoe region beginsto close in on itself, nearly forming a continuous annular ringabout the Sun, coming close to connecting at Earth. It is foundthat there exists a well-defined critical value of V(0) ¼ V*(0)where the ring closes at Earth, and then the motion of P3 bi-furcates from amotion constrained to the horseshoe-like region,where it never makes a full cycle about the Sun, to a motionwhere it continuously cycles about the Sun, repeatedly passingclose to Earth, and no longer in the horseshoe-like motion. Formost values of � , V*(0) has an approximate value between0.200 and 0.600 km s�1. We refer to this continuously cyclingmotion for V(0) ¼ V*(0) as breakout. Breakout continues tooccur for V(0) > V*(0).

R2.—In breakout motion for V(0)kV*(0) or V(0) ¼ V*(0),the trajectory �(t) for t > 0 traces out a dense set of orbits in athin annular region, repeatedly passing near Earth, where theflybys at Earth periapsis all appear to be approximately para-bolic. The breakout orbit is chaotic in nature, so that smallchanges in V(0) result in breakout trajectories that are in generalsignificantly different in appearance and still restricted to a thinannular region about P1. The breakout trajectories as they cycleabout the Sun have a high likelihood of colliding with Earth.Moreover, for each � a near-parabolic collision trajectory is

ORIGIN OF THE MOON 1727No. 3, 2005

readily found for V(0)kV*(0). The collision orbits move nearEarth’s orbit and gradually approach Earth for collision. (Thisgradual motion approximately along Earth’s orbit we refer to ascreeping.) The collision orbits can creep to collision along thedirect or retrograde direction with respect to P1.

3.1. Demonstration of R1

We choose an arbitrary velocity direction v for P3 at L4at t ¼ 0, where v points in the vertical positive y-direction. Inour simulations, the location of L4 is (�0.5,

ffiffiffi3

p=2), and the

y-coordinate is input with the value 0.866025404. Beginningwith � ¼ �=2, we choose a magnitude V¼ V(0)¼ 0.001 and nu-merically integrate the system of differential equations (eq. [1])forward for t 2 [0, 1000]. This velocity magnitude is small,and since L4 is stable, P3 remains in a thin arclike region ap-proximately of radius 1, shown in Figure 2. P3 starts at thelocation x(0) ¼ (0.5,

ffiffiffi3

p=2) and moves down in the prograde

direction with respect to the Sun. As it moves, it performs manysmall loops as shown in Figure 2. These loops occur because thesemimajor axis of the orbit of P3 has changed slightly from 1and the orbit of P3 has a slight nonzero ellipticity, both due to theaddition of V(0). So, as it moves in its approximately ellipticalmotion over the course of 1 yr, it falls slightly behind andforward with respect to Earth when it is at its apoapsis andperiapsis, respectively. Each loop forms in 1 yr. Thus, for t ¼1000 there are 1000=(2�) loops. P3 moves down to a minimallocation where y is approximately 0.7, and then it turns aroundand moves in the upward direction where the small loops pointin the opposite direction to when it was moving downward. Thesuperposition of the loops makes a braided pattern, as can beseen in the lower half of Figure 2. P3 stays in this boundedarclike region since L4 is stable, and the velocity V(0) is rela-tively small. [ If V (0) ¼ 0, then P3 stays fixed at L4 for all time.]Because the velocity magnitude is small, P4 has a Kepler energynearly that of L4, and so its semimajor axis with respect to theSun deviates from 1 by a negligible amount. Thus, as it moves,it stays nearly on a circle of radius 1. That is, in an inertialcoordinate system it stays approximately on Earth’s orbit aboutthe Sun. As long as V(0) is small, which it is throughout thispaper, the trajectories of P3 remain close to Earth’s orbit andmove with small loops, in the rotating coordinate system. Theparticle P3 creeps slowly along Earth’s orbit, initially in aprograde fashion and then in a retrograde fashion away fromEarth.

We repeat the above procedure, slightly increasing the valueof V(0) to 0.004 at L4 at t ¼ 0. Since V(0) has increased, as canbe seen in Figure 3, where t 2 [0, 1000], P3 creeps further in itsEarth-like orbit about the Sun. Since V(0) is small, the trajectory

of P3 deviates slightly from a circle of radius 1. This deviationslightly increases as V(0) increases. The addition of V(0) at L4causes P3 to have a slightly smaller value of the Jacobi integral,slightly less than 3 (CP 3). This means that P3 becomes moreenergetic and thus can creep further along Earth’s orbit. In-creasing V(0) by 0.001 to 0.005 causes the increased creepingshown in Figure 4, where t 2 [0, 1000]. In Figure 5, V(0) is in-creased to 0.009. P3 leaves L4, moves downward in a progradefashion to slightly behind Earth, and then turns around andmovesin a retrograde fashion on its Earth-like orbit about the Sun untilit approaches Earth from the front, turning around and then mov-ing in a prograde fashion.A braided pattern results because the Earth-like orbit is tra-

versed twice, with loops pointing in the inner and outer direc-tions. The resulting complicated-looking trajectory is symmetric

-0.6 -0.5 -0.4 -0.3 -0.2

0.7

0.75

0.8

0.85

0.9

0.95

Fig. 2.—Trajectory �(t), V(0) ¼ 0.001, t 2 [0, 1000], x vs. y (i.e., x-axis ishorizontal, y-axis is vertical), Sun-centered.

-1 -0.5 0.5 1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

Fig. 3.—Same as Fig. 2, but for V(0) = 0.004.

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Fig. 4.—Same as Fig. 2, but for V(0) ¼ 0.005.


with respect to the x-axis as a consequence of the symmetrymentioned earlier for the restricted problem. The width of theregion near Earth’s orbit in which P3 moves has slightly in-creased because of the increase in V(0).

We note that the general appearance of the orbit of the as-teroid 2002 AA29, mentioned in x 1 (Connors et al. 2002), isremarkably very similar in appearance to Figure 5. Unlike theplanar orbit considered here, 2002 AA29 has a inclination of10

�with respect to the plane of Earth’s orbit. Its oscillation pe-

riod is approximately 95 yr. The approximate period of the or-bit in Figure 5 is about 159 yr, which is not too dissimilar. Orbitsof this type are called horseshoe orbits. The horseshoe orbitsare constrained to a region we refer to as a horseshoe region,so P3 cannot move past Earth. Such a region is discussed byMurray & Dermott (1999). Let � be the polar angle measuredfrom the positive x-axis for the position of P3. The horseshoeorbits have the property that � 6¼ �. This means that P3 willnot fly by Earth. For other papers on this motion, see Christou(2000), Hollabaugh & Everhart (1973), Mikkola & Innanen(1990), Namouni (1999), and Weissman & Wetherill (1974).

When V(0) reaches 0.011, P3 is able to escape from the thinhorseshoe-like region and fly by Earth, as seen in Figure 6. Thisachieves breakout motion, where P3 then cycles about the Sunin only one direction. In Figure 6, this cycling is in the retro-grade direction. This actual cycling is not shown in this figure,since for the time range given, breakout into cycling motionoccurs when t ¼ 988, on the outer retrograde trajectory. P3 firstleaves L4, moves near Earth, and then moves back up in aretrograde fashion, going all the way around the Sun to near andin front of Earth, and then moving around the Sun again in aprograde fashion to its location behind Earth; it finally moveson the outer trajectory in a retrograde fashion back to just aheadof Earth, when it crosses by Earth at t ¼ 988 (crossing the x-axisnear Earth), performing the cycling breakout motion after thattime. This transition from creeping horseshoe motion to creep-ing breakout motion is what is desired for this paper. The tran-sition from horseshoe motion to breakout motion represents abifurcation from one type of motion to a different type. We areinterested in the likelihood of Earth collision while in breakout

motion just after the bifurcation. This represents breakoutmotion with minimal energy.

This transitional breakoutmotion has two important properties:1. P3 moves in a thin annular region about the Sun; and2. P3 repeatedly flies by Earth.

These properties imply the following: Since the annular regionis thin, the Earth flybys are, in general, close. The close Earthflybys are approximately parabolic in nature, as we will dem-onstrate, and as P3 flies by Earth its actual velocity vector isapproximately tangent to Earth’s orbit. This implies that P3gains a negligible velocity increase due to gravity assist as itflies by Earth, as we will show. This guarantees that P3 will con-tinue to move in an Earth-like orbit about the Sun and continueto cycle. This implies that as P3 moves around the Sun, it willdensely fill the thin annular region it moves in. This means thatit has a high likelihood of colliding with Earth. We will dem-onstrate that collision readily occurs in these creeping breakoutorbits.

In fact, the previous case where V(0) ¼ 0.011, which is thefirst breakout motion we computed, leads immediately to col-lision at t ¼ 1384:7176 (or 220.3847 yr). In our expositionbelow, we will use a slightly different value ofV(0) that happensto achieve collision at an even earlier time.

Breakout motion is seen in Figure 7. Observe that a shift fromV(0) ¼ 0.009 to V(0) ¼ 0.012 causes a qualitatively different-looking picture, in which the bifurcation between horseshoeand breakout motion can clearly be seen. The case just con-sidered is for the direction � ¼ �=2. The same procedure pro-duces critical values of V(0) ¼ V*(0) leading to breakoutmotion, from horseshoe motion, for any value of � 2 [0, 2�].A set of these for �-increments of �=8 are listed in Table 1 ofBelbruno & Gott (2004). This is graphically shown in Figure 8.In this figure, the length of each line is equal to the value ofV*(0) in that direction. In this way, a smooth variation of V*(0)as a function of � is numerically obtained. There is a sharpspike in the value of V*(0), which has a maximum at 5.102�=8of 0.22. There is also a similar maximum near the value13.5�=8. These are not typical: almost all the values of V*(0)are in the range of values illustrated. The minimum value of

-1 -0.5 0.5 1

-1

-0.5

0.5

1


-1 -0.5 0.5 1

-1

-0.5

0.5

1

Fig. 6.—Same as Fig. 2, but for V(0) = 0.011.


V*(0) ¼ 0.0057 is for � ¼ (9�=8) � 0.01. Multiplying thevalues of V*(0) by 29.78 yields a range of velocity valuesgenerally between 180 m s�1 and 1.2 km s�1. Note that the twodirections corresponding to the maximal spikes in velocity seenin Figure 8 approximately lie near the Sun and anti-Sun di-rections. In this figure the Sun is toward the lower right.

Note that the method, or algorithm, used above to estimatethe critical velocities V*(0) at L4 leading to breakout motion issimilar in nature to the method, described in Belbruno (2004),of estimating transitional stability regions, called weak stabilityboundaries, between capture and escape about the Moon. Thiscapture region has important applications. It was used by E. B.to find a new type of low-energy route to the Moon in 1990in which lunar capture is automatic (Belbruno & Miller 1990).This special lunar transfer was designed in order to resurrecta Japanese lunar mission and enable the spacecraft Hiten tosuccessfully reach the Moon in 1991 October with almost nofuel (Belbruno 1992; Frank 1994). More general references onthis are Adler (2000), Belbruno (2004), and Belbruno & Miller(1993).

Other methods could be used to study the bifurcation fromhorseshoe to breakout motion, such as computing suitable sur-

faces of section to the trajectories in phase space and thenmonitoring the iterates of intersecting trajectories on the sec-tion. This would give a more complete picture of the phase spacenear breakout motion, but this approach is not necessary for ourpurposes. The algorithm we have described accurately deter-mines when bifurcation occurs.We have also performed an analysis to understand the rela-

tionship of the critical breakout velocities as a function of themass of P2. It is found that, roughly,

V �(0) / m1=32 :

As an example, we consider the two cases m2 ¼ 0.1mE andm2 ¼ 0.01mE, where mE ¼ 0.000003. When m2 ¼ 0.1mE, thenfor � ¼ 0, �=2, �, and 3�=2, we obtain V*(0) ¼ 0.004, 0.007,0.004, and 0.007, respectively, and for m2 ¼ 0.01mE we obtainV*(0) ¼ 0.002, 0.003, 0.002, and 0.003, respectively. These re-sults imply that V*(0) � 0.6m

1=32 for� ¼ 0 or �, and when � ¼

�=2 or 3�=2, V *(0) � m1=32

. (Thus, a giant impactor trappedin a stable orbit about L4 and unperturbed will remain trappedthere as the proto-Earth grows by accretion. Breakout velocityincreases as the proto-Earth grows, thus postponing breakout,but at late times after the proto-Earth has reached essentially itsfull mass, according to our scenario, perturbations can drive thegiant impactor to breakout.)This concludes the demonstration of R1.

3.2. Demonstration of R2

We first show how to readily find trajectories from L4 thatcollide with Earth. The value of � ¼ �=2 is again considered,and we consider the case V(0) ¼ 0.012 k V*(0) shown inFigure 7. Plotting the distance r2 between P3 and Earth for t 2[0,1000] reveals the times of the various Earth flybys. It wasfound that the case of V(0) ¼ 0.012, for the given range of t,had very close Earth flybys but no actual collision. Randomlyaltering this value of V(0) yielded a collision on our secondrandom choice of values of V(0) ¼ 0.0119981. This is seen byplotting r2 as a function of time, shown in Figure 9.By magnifying the regions near minima of r2, it can be seen

which ones may yield collision. In this case, counting from leftto right, we determined that the first, fourth, and fifth minimayield distant flybys at over 1 million kilometers. Thus, in thesecases the Earth flybys are not of interest. The third flyby missesEarth by about 18,000 km. However, it is found that the second

-1 -0.5 0.5 1

-1

-0.5

0.5

1


-0.015-0.01-0.005 0.005 0.01 0.015

-0.015

-0.01

-0.005

0.005

0.01

0.015

Fig. 8.—Initial velocity directions v(0) at L4 whose magnitudes correspondto the associated critical breakout velocity V *(0).

200 400 600 800 1000

0.5

1

1.5

2

Fig. 9.—Variation of distance r2 from P3 to P2 (Earth) as a function oft 2 [0, 1000] in dimensionless units, for V(0) ¼ 0.0119981.


flyby in fact collides with Earth. The time of collision withthe surface of Earth is at t ¼ 360:181558; corresponding to57.3247 yr. This time is calculated as when the center of theimpactor, viewed as a circle of radius rI ¼ 3397 km, intersectsthe surface boundary of Earth, which is at a radial distancerE ¼ 6378:14 km from Earth’s center. This can be seen inFigure 10.

We are assuming that each point on the trajectory of theimpactor at any given time is at the center of the impactor. Moreaccurately, however, collision actually occurs a few momentsearlier, when the surface boundary (circle) of the impactortouches the surface boundary (circle) of Earth, that is, whenr2 ¼ rI þ rE ¼ 9775:14 km, where for rI we take the radiusof Mars, since this is a Mars-sized object, or, in dimension-less units, when r2 ¼ 0.00065342. In general, we assume colli-sions mathematically occur in our numerical simulations whenr2� 0.00065342.

We now show what the collision trajectory looks like anddiscuss its properties. The collision trajectory, ‘‘Cl,’’ is shown

in Figure 11. It starts at L4, moves in a prograde fashion towardEarth, turns around, and in a retrograde motion moves aroundthe Sun to collide with Earth. A view of this orbit in its final9.57 yr is shown in Figure 12.

Collision with Earth itself and the final 9.14 hr of the tra-jectory are shown in Figure 13. In this figure we stopped thetrajectory of the impactor before its Earth periapsis. However,if it were continued beyond collision it would reach its peri-apsis point approximately on the x-axis inside Earth’s radius att ¼ 360:1817, where r2 ¼ 0.00003 (see Fig. 14). If it were ex-tended so that t 2 [0, 1000], the trajectory would be as shownin Figure 15. This figure for V(0) ¼ 0.0119981 is shown to com-pare with Figure 7 for V(0) ¼ 0.012, indicating the sensitive, orchaotic, nature of the breakout motion, where a difference inV(0) of 0.000002 yields a qualitatively different-appearing tra-jectory. The chaotic nature of the motion near breakout is alsoseen in Figure 16, which is a plot of r2 for V(0) ¼ 0.0119986,when compared with Figure 9, where V(0) ¼ 0.0119981.

It can be seen that although the difference in V(0) is0.0000005 at L4, there is a significant difference in the quali-tative appearance of the two plots. This is caused by the fact thatinfinitesimally small changes in V(0) at L4 can cause slightlydifferent Earth flyby conditions, which, over long time spans,t 2 [0, 1000], can cause the trajectory to change noticeably ifany of the flybys are close. However, as we will see in thefollowing, close flybys will only yield negligible Kepler en-ergy increases with respect to the Sun. So although the tra-jectory may have a qualitatively different appearance, it willstill have approximately the same Kepler energy before andafter close flybys. The change in the trajectories for tinychanges in V(0) observed is typical of chaotic motion in gen-eral, and it is a sign that a hyperbolic invariant set likely existsin the phase space for the breakout motion of P3. A hyperbolic

360.175 360.182

0.00005

0.0001

0.00015

0.0002

0.00025

Fig. 10.—Illustration of the distance r2 from P3 to P2 (Earth) going belowEarth’s radius, 0.0000426346 (i.e., 6378.14 km), proving collision has oc-curred. Again, V(0) ¼ 0.0119981. The plot is t vs. r2 in dimensionless units.

-1 -0.5 0.5 1

-1

-0.5

0.5

1

Fig. 11.—Entire collision trajectory (x vs. y), originating at L4 at t ¼ 0and colliding with Earth when t ¼ 360:18, or equivalently 57.32 yr. V(0) ¼0.0119981, Sun-centered.

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

-1

-0.8

-0.6

-0.4

-0.2

Fig. 12.—Orbit approaching collision with Earth (x vs. y). A time durationof 9.57 yr is shown, t 2 [300, 360.181558], axes Earth-centered.


invariant set in general is a Cantor set all of whose points havea saddle-like structure produced by a transverse homoclinicorbit, whose existence is given by the Smale-Birkhoff theorem(Belbruno 2004). The use of the term chaotic in a strict math-ematical sense denotes the existence of a hyperbolic invariantset.

We remark that Cl represents a physical collision with thesurface of Earth, where, at periapsis below Earth’s surface, r2 ¼0.00003. It turns out that actual pure collisions where r2 � 0to high precision are readily found as well. For example, V(0) ¼0.011998 leads to a pure collision at t ¼ 360:1898. In this type

of collision, P3 asymptotically approaches the collision mani-fold, which is a set of measure zero.Since the motion of P3 repeatedly passes near Earth in break-

out motion, Earth tends to readily pull P3 toward pure andphysical collisions. The set, or manifold, of pure-collision tra-jectories is a subset of physical-collision trajectories and, infact, is a set of measure zero in the four-dimensional phasespace of position and velocity (Belbruno 2004). Since they area set of measure zero, their near-occurrence reflects the factthat the flybys of Earth are close and that Earth has a consid-erable gravitational focusing effect when the trajectory is nearlyparabolic.It turns out that Cl is approximately parabolic at collision.

This is seen by plotting the Kepler energy E2 of Cl with respectto Earth. In inertial Earth-centered coordinates X ¼ (X1, X2),

-1.0008 -1.0006 -1.0004 -1.0002

-0.0004

-0.0003

-0.0002

-0.0001

0.0001

0.0002

Fig. 14.—Continuation of collision trajectory across the x-axis when t ¼360:1816; x vs. y, Earth-centered.

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-1

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0.5

1

Fig. 15.—Extended collision trajectory, through collision, t 2 [0, 1000](i.e., t 2 [0, 159.15] yr); x vs. y, Sun-centered.

200 400 600 800 1000

0.5

1

1.5

2

Fig. 16.—Variation of r2, t 2 [0, 1000], for V(0) ¼ 0.0119986. Comparewith Fig. 9, where V(0) differs by 0.0000005.

-1.00015 -1.0001 -1.00005

0.00005

0.0001

0.00015

Fig. 13.—Collision of the impactor with Earth, x vs. y. The final 9.14 hr oftrajectory are shown, Earth-centered.


E2 ¼ 12jX j2 � �=jXj. In barycentric rotating coordinates x ¼

(x1, x2) � (x, y), E2 is transformed into

E2 ¼1

2jxj2 � �

r2þ 1

2r22 � L; ð5Þ

where r2 ¼ ½(x1 þ 1� �)2 þ x22 �1=2

and L ¼ x1x2 � x2(x1 þ 1��) (Belbruno 2002, 2004). The quantity L is the angular mo-mentum of P3. In Figure 17, we have evaluated E2 along Cl andplotted it as a function of t 2 [0, 360.181558]. From Figure 18,E2 = 0.000054 at collision, which is nearly parabolic. (If thecollision were purely parabolic then we would have E2 ¼ 0.)This yields a very slight hyperbolicity whose hyperbolic excessvelocity with respect to Earth, V1 ¼ (2E2)1=2, has the value0.0104. This is equal to V*(0) for � ¼ �=2 to within 0.0014. Inscaled coordinates, V1 ¼ 310 m s�1 and V*(0) ¼ 328 m s�1.V1 is close to V*(0) because at Earth periapsis V*(0) is like thevelocity at infinity. This velocity is approximately maintainedalong the orbit as it approaches collision. During actual Earthflyby, at periapsis the velocity with respect to Earth increases asa result of the attraction of Earth, and for this collision orbit,V ¼ 0.44 at Earth periapsis.

In Belbruno & Gott (2004) it is analytically shown that incritical or near-critical breakout motion, all close Earth flybys,including collision trajectories, are approximately parabolic atperiapsis. For critical breakout trajectories, which start at L4 attime t ¼ 0, V(0) ¼ V*(0). For near-critical breakout motion weassume that V(0) k V*(0). Notationally, V(0)

� V*(0) includesboth these cases.

We next define the terms ‘‘close Earth flyby’’ and ‘‘approx-imately parabolic.’’ Let �(t) be a trajectory that performs a flybyof Earth, with a periapsis distance r2 at some time t. We say thatthis is a close Earth flyby if r2 � 100,000 km or, in dimen-sionless coordinates, r2 � 0.000668. The figure of 100,000 kmis arbitrarily chosen since for weakly hyperbolic flybys of Earthbeyond this distance, the effect of an Earth gravity assist isnegligible. Physical collisions are included as close flybys.

We use E2 to determine the type of collision, which is com-puted at Earth periapsis. So, in the case of physical collision at

Earth’s surface, we propagate the trajectory to Earth periapsiswithin Earth. This point occurs a very short time after physicalcollision, which for the case of Cl is only 12 minutes. At theperiapsis point, E2 is evaluated at the trajectory state of positionand velocity. If jE2jk 0, the collision or collision trajectory iscalled approximately parabolic. It could be slightly elliptical,slightly hyperbolic, or purely parabolic. It turns out, as we willsee, that for breakout or near-breakout motion, the flybys are allapproximately parabolic. The following result is obtained (fordetails, see Belbruno & Gott 2004):

For the set of critical or near-critical breakout velocities atL4, the value of E2 at the close Earth flybys at periapsis is

E2 � 12½V (0)�2

�12½V �(0)�2k 0; ð6Þ

that is, the close Earth flybys are approximately parabolic.This is true for all the values of � except those values in smallneighborhoods of 5.102�=8 and 13.5�=8 (see comment below).This implies that a trajectory �(t) starting near critical break-out velocity at L4 for t ¼ t0 will satisfy equation (6) for anyfuture time t > t0 corresponding to any close Earth flyby atperiapsis.

More precisely, as shown in Belbruno & Gott (2004), for atrajectory starting at L4, at Earth periapsis on a close Earth flybyat a distance r2 ¼ �2,

E2 ¼ 12½V (0)�2 þO0(�)þO1(�1)þO2(�2); ð7Þ

where O0 ¼ 3�, O1 ¼ 2�1, O2 ¼ 2�2 cos �2, r2 ¼ �2 <0.000668T1, r1 ¼ 1þ �1, |�1|< 0.000668, j�1j 0, and �2 0. This relation yields equation (6) for small values of �, �1,and �2.

Let P3 be at L4 (or L5) at t ¼ 0, and let v(0) be the initialvelocity, with magnitude V (0) ¼ jv(0)j. Then, the Jacobi inte-gral J has the value J ¼ C0 ¼ 3� ½V (0)�2 � �(1� �). This im-plies that for the set of critical breakout velocities V*(0) at L4for � 2 [0, 2�] (see Fig. 8), C0P 3.

As mentioned earlier, there are two sharp spikes in thebreakout velocities shown in Figure 8 of values 0.22 and 0.25that occur for � of 5.102�=8 and 13.5�=8, respectively. How-ever, most values of V*(0) vary between approximately 0.006

50 100 150 200 250 300 350

0.5

1

1.5

2

Fig. 17.—Kepler energy E2 with respect to P2 as a function of t along theentire collision trajectory. It can be seen that E2 ! 0.

360.178 360.182

0.0000543

0.0000544

0.0000545

0.0000546

0.0000547

Fig. 18.—Energy E2 as a function of t at the end of the collision trajectory,approaching the value of 0.0000541 at periapsis below Earth’s surface, wherer2 ¼ 0.00003 (i.e., 4488 km), for t ¼ 0.00014 (i.e., 12 minutes) beyond Earthcollision, t 2 [360.175, 360.1817].


and 0.05 if two intervals in � of total width approximately0.157 radians are deleted near where the spikes occur. Thisimplies that for nearly all of the values of � , equation (6) im-plies that approximately

E22½0:000018; 0:0012�: ð8Þ

Thus, P3 is approximately parabolic at collision. The rangegiven by equation (8) is a crude estimate of E2. The observedvalue for Cl of E2¼ 0.000054 is contained within this interval.A sharper estimate can be made using equation (7). For Cl,V(0) ¼ 0.0119981, and at flyby periapsis, below the surfaceof Earth, r2 ¼ �2 = 0.00003. This occurs approximately on thex1-axis, implying r1 ¼ ��1 ¼ �0.00003 and �2 ¼ 0. Substi-tution of V(0), r1, r2, and �2 into equation (7) yields the valueE2 � 0.00007. Noting that the numerically observed value ofE2 at periapsis for Cl is E2 = 0.000054, the predicted value isin error by only 0.000016, demonstrating the accuracy of thepredicted values.

We remark that in all the cases numerically observed, E2 wasnot negative (i.e., the orbit was not elliptical) during the flyby.Some such flybys are likely to exist because of the chaotic na-ture of breakout motion; however, the probability of findingtrajectories with elliptical flyby states is apparently small. Theirchaotic structure and low probability of occurrence is studiedin Belbruno (2004). When E2 is near zero, this defines weakcapture as studied in Belbruno (2004), where P3 will, in general,move about Earth in a chaotic fashion generally leading toescape or collision. However, as remarked, the case of interesthere is that in which E2 is very slightly hyperbolic.

The retrograde collision trajectory Cl emanating from L4 ispaired with another symmetric collision trajectory Cl* ema-nating from L5, which is symmetric to Cl and moves in aprograde fashion about the Sun. This follows from the sym-metry of solutions mentioned in x 2. It will collide with Earth inthe fourth quadrant, as shown in Figure 13.

Probability of collision at breakout for the restricted prob-lem.—A measure of the likelihood of finding collision tra-jectories is now described. This is done for the four basic initialvelocity directions at L4:� ¼ 0, �=2, �, and 3�=2. For an initialvelocity of P3 at L4 for a given � , we assume the correspondingbreakout velocity V*(0) shown in Figure 8. The orbit of P3 ispropagated from L4 for t 0, and since it is in breakout motion,we know that it will not be in horseshoe motion but will cycleabout the Sun and repeatedly fly past Earth. We can numericallydemonstrate that collision with Earth is likely. This intuitivelymakes sense, since the flybys will be close and the annularregion supporting the breakout motion is narrow. Now, for agiven initial velocity at L4 for t ¼ 0 we see from Figure 8 thatV*(0) is given up to three digits. For a given value of V*(0),depending on � we propagate the trajectory for up to t ¼ 4000,which corresponds to 637 yr, and see whether collision hasoccurred. The value t ¼ 4000 is chosen arbitrarily for conve-nience and is fairly small in astronomical terms. If no collisionoccurred in that time, then we give V*(0) a random perturbationby adding to it a random number 0.000mn, where m and n arepositive random integers ranging from 0 to 9. For a choice of mand n the trajectory is propagated again. If collision does notoccur, we repeat the process again for a different choice of mand n, continuing trials until success is achieved.

For � ¼ 0 and � ¼ �=2, we required two random trials forsuccess, where success means we achieve collision within t ¼4000 ¼ 637 yr. For � ¼ �, three random trials were requireduntil we had success, and for � ¼ 3�=2 six random trials were

required. Therefore, we have achieved success in four randomtrials out of 13. This gives our best estimate of the probabilityPof success for 0 � t � 4000 as

P � 4=13: ð9Þ

If we had not limited ourselves to t ¼ 4000, the probabil-ity would have been larger. This probability is discussed in fur-ther detail in the Appendix. We have run a sufficient numberof trials to produce a rough order-of-magnitude estimate ofthis probability, which is sufficient for our purposes, but a largenumber of additional trials could establish this number to higheraccuracy.Note that the gravitational focusing on P3 to cause a collision

is substantial. This is related to the fact that the breakout motionis occurring at a fixed energy for the planar restricted problem.The fixed energy yields a three-dimensional energy surface ob-tained from the Jacobi integral. As is proved in Belbruno (2004),the manifolds leading to collision at P2 are two-dimensional, andalthough they are a set of measure zero, the particle P3 is readilyable to move asymptotically close to these surfaces and tocollision after the gravitational focusing. The collision mani-folds on the Jacobi integral surface separate the phase space, soit is fairly easy for P3 to get near the collision manifold. Inhigher dimensions this separation of the phase space on theJacobi surface does not occur, and the collision manifold ismore elusive.This concludes the demonstration of R2.It is interesting that these creeping chaotic orbits seem to lead

naturally to collision with Earth (as proposed by the giant-impactor theory) rather than to capture into a bound orbit (as inthe sister-planet theory). If one wanted to have a sister-planettheory, of course, L4 would be a promising place for the Moonto start out. So it is significant that our chaotic creeping orbits(slightly hyperbolic/nearly parabolic) lead naturally to collisionrather than to capture. This favors the giant-impactor theory.The sister-planet theory would, of course, also have a problemwith the difference in iron between Earth and the Moon.

3.3. Random Walk, �V Accumulation,and Relevvant V(0) Rangge

In determining V*(0) at L4 above, we kept P3 fixed at L4 andgradually increased V(0) for a given velocity direction. Thisyields a well-defined set of V*(0)(� ) for � 2 [0, 2�].We now consider a more realistic way that P3 could increase

its velocity in a gradual fashion. The mechanism for this is toassume that P3 is randomly being perturbed by encounteringother planetesimals (whether by gravitational encounter or di-rect collision) and, in each encounter, it acquires an instanta-neous kick�V. So, it is not kept fixed at L4. To make this morerealistic, we assume that the times of encounter are random,within a large range, and the directions � of the kicks are ran-dom. The only thing we normalize is the magnitude of the�V ’s, which for convenience is held fixed.Thus, P3 starts at L4 with a zero velocity, and at time

t ¼ t1 ¼ 0 a velocity V (0) ¼ �V is applied in a random di-rection. This yields a vector v1 with magnitude �V. P3 moveson a trajectory �(t) in a neighborhood of L4, assuming that thevalue of �V is small. At a random time t2 > 0 another velocityvector v2 of random direction and magnitude �V is vectoriallyadded to P3’s velocity at t ¼ t2. Then the trajectory is propa-gated for t > t2 until at another random time t3 > t2 a randomvector v2 of magnitude �V is vectorially added to P3’s velocity


vector at t ¼ t3, and this process continues, creating a sequencetk of times, tkþ1 > tk , and velocities vk , k ¼ 1, 2, 3, . . . .

While the �V ’s are being applied, the trajectory �(t) isgradually moving farther from L4, but since the velocity di-rections vk are applied randomly, the path of the trajectory �(t)will move farther away from L4 for some time spans and thenmove toward L4 for others. However, as k increases, one wouldexpect, by the principle of random walks, for P3 to eventuallyescape L4 and creep toward Earth for k sufficiently large whenthe velocities vk (k ¼ 1, 2, 3, . . .) applied on the trajectory � (tk)gradually accumulate to a sufficiently large magnitude forbreakout to occur. If the vk were all applied in the direction ofmotion of P3 at tk , then the magnitudes �V would add, pro-ducing a cumulative velocity addition of k �V at the k th step.However, the directions of vk are random, and by the principleof a random walk, the number of encounters k before ejectionoccurs should be expected to instead satisfyffiffiffi

kp

�V � 0:006 ð10Þ

for k sufficiently large (and�V sufficiently small), where 0.006is approximately the minimum value 0.0057 of {V*(0)}. Thismakes dynamical sense, since as the �V ’s are applied, the tra-jectory �(t) would seek to minimize the Jacobi energy, and hencethe velocity, along its path. We found in all our numerical sim-ulations the following result:

For a given value of �V, the number k of random vk appli-cations required for breakout to occur approximately satisfiesequation (10).

We now describe this process of random-walk �V accumu-lation, and its verification.

For convenience we choose�V = 0.001 and start at L4 withzero initial velocity. Equation (10) implies that k should satisfyffiffiffik

p�V � 0.006, which yields k ’ 36. It was found that break-

out occurred when k ¼ 36, as predicted. Random kicks ofvelocity �V were applied in random directions � k and afterrandom time intervals tk . (As in all future runs, the times be-tween kicks are just chosen to be large enough to randomize theposition. The real time for random-walk–out is expected to bemuch longer in years—perhaps 30 Myr, as considered in theAppendix.) From the foregoing, we have the following result:

(Random-walk�V accumulation) Under a realistic assump-tion of random walk, the peculiar velocity for P3 accumulates inproportion to the square root of the number of encounters until itreaches a breakout state. Since a random walk is isotropic, thepeculiar velocity is likely to encounter the breakout state firstat a point near the minimum value of 0.006 of the set {V*(0)},thus giving equation (10).

Therefore, substituting V*(0) = 0.006 into equation (6) im-plies that for close Earth flybys resulting from the random-walkprocess at or near breakout, E2 � 0.000018. This implies, atclose Earth flyby resulting from the random-walk process, anominal value of V1 = 0.006, which is 179m s�1.When P3 per-forms a close flyby of Earth, after passage through periapsis itwill receive a gravity assist and increase, or decrease, its ve-locity with respect to the Sun. Ameasure of this velocity changeis observed because of the bending of the trajectory of P3 as itpasses through periapsis. For example, this bending is clearlyseen in Figure 14. The more distant the flyby, then in generalthe less the bending. The maximum bending is obtained frompure-collision trajectories, where the bending angle is �. Itwas determined by Broucke (1994) that the resulting change

in magnitude of the velocity, �v, with respect to the Sun due togravity assist is maximally 2V1. Thus, for each close Earthflyby, the expected maximum gain in velocity magnitude is ap-proximately 358 m s�1. In general, it will be less.

The maximum velocity of 358 m s�1 is a relatively smallnumber and will have little effect on a breakout trajectory whenit has a close Earth flyby. This velocity is less than 1.2% of theorbital velocity, inducing eccentricities into the trajectory ofP3 after flybys of at most this order. It is found in general thatwithin time spans on the order of 2000 time units, there aregenerally only one or two close Earth flybys. This implies thatP3 will remain in breakout motion about the Sun in a relativelythin annular region for very long periods of time, generally tensof thousands of time units, and repeatedly pass by Earth withoutbeing ejected.

3.4. Collisions in Three Dimensions andthe Mars-sized Impactor

Thus far we have constrained P3 to lie in the plane of motionof Earth and the Sun in the planar restricted three-body prob-lem. It that situation, it was seen that collision trajectories arereadily found. When the motion of P3 has an out-of-planecomponent, z, added to it, it is more complicated. In this casewe have the three-dimensional circular restricted three-bodyproblem, defined in exactly the sameway as the planar problem,except that P3 can move in the z-direction (see Szebehely 1967;Belbruno 2004). By continuity with respect to initial condi-tions, it must be the case that collisions in the planar case willpersist in the three-dimensional case if |z| is sufficiently small. Away to obtain an approximate measure of the maximum allowedz-motion is to consider a collision trajectory from L4 to Earthgenerated at or near critical breakout and then see how muchz can be added at L4 and still maintain collision with Earth,which is now a three-dimensional sphere. This is a relativelystraightforward calculation.

In the Appendix we show that Earth collisions should per-sist provided that |z|P 0.0034. This implies a thin disk of plan-etesimals. In the Appendix we also show that a disk of thickness28rE easily satisfies this requirement. This has an angular widthof 4A1 as seen from the Sun. Although this seems thin, it turnsout that the inner B and A rings of Saturn, extending from92,000 to 140,210 km, have a thickness of 0B18–1B7 as seenfrom the center of Saturn, which is even thinner than required inour situation.

The modeling of most interest in this paper is for the generalthree-dimensional three-body problem defined by equation (4),wherem3 ¼ 0.1m2 6¼ 0, so aMars-sized Earth impactor is mod-eled. This is used by Canup & Asphaug (2001). Although themotion of Earth (P2) is given initial conditions for uniform cir-cular motion about the Sun (P1), it need not remain circular astime progresses, as a consequence of the gravitational pertur-bations of P3. This property makes the problem more interest-ing. To better understand this and to see its effect on collisiontrajectories, the general planar three-body problem is first con-sidered. It is defined from equation (4) by setting Xk3 ¼ 0, wherek ¼ 1, 2, 3.

As with the restricted problem, a rotating coordinate system(x, y) is chosen, this time rotating with the meanmotion of Earthabout the Sun. In this system Earth is not fixed on the�x axis asin the restricted problem, since it is perturbed by P3, especiallyduring flybys.

Just as in the restricted three-body problem, breakout fromL4 can be defined for the general planar three-body problemwith exactly the samemethodology as for the restricted problem,


by gradually increasing the velocity of P3 at L4 until breakoutis achieved, for any given velocity direction. Nearly identicalresults are obtained as in R1 (Fig. 8), and the analysis is notduplicated for this situation. The desired random-walk �Vaccumulation process is defined exactly as before for this cur-rent problem, and we have verified that the same results areobtained.

We determine the breakout of P3 to Earth collision as wedid in the restricted problem, by gradually increasing the veloc-ity at L4 until a critical velocity is reached. That is, we are notmodeling the random-walk process and are copying the pro-cedure we performed for the restricted problem. This is done tokeep the modeling as close as possible to the restricted problemin order to better understand how P3 perturbs P2 and the effectthis has on obtaining collision trajectories.

An important difference in using the planar three-bodyproblem instead of the restricted problem is observed when thebreakout velocity is determined for P3 at L4. As the velocitymagnitude V(0) for t ¼ 0 is gradually increased at L4, and P3moves in the horseshoe regions about the Sun, the location ofEarth moves in its orbit, approximately maintaining its 1 AUdistance from the Sun but shifting its angular position withrespect to the Sun. This is because as P3 creeps farther andfarther away from L4 approximately on Earth’s orbit, it cangravitationally perturb Earth when it moves relatively near toEarth, since its mass is now 1

10that of Earth. Analogous to the

restricted problem, when V(0) gets close to the breakout value,the horseshoe region begins to close on itself in a symmetricway with respect to Earth. However, since Earth has shifted itslocation, the symmetric closing point is not on the x-axis as inFigure 5 for the restricted problem but at another location, thelocation of Earth, approximately 1 AU from the Sun. This isillustrated in Figure 19, where V(0) � |(x, y)| ¼ 0.160 km s�1

and � ¼ 0 is assumed. This is fairly close to breakout, for thegiven velocity direction, which occurs for V(0) ¼ 0.205 km s�1.The final location of Earth when breakout is achieved is in thethird quadrant about 20� from the negative y-axis, which meansthat the x-axis is not fixed to Earth but to the mean motion.

The cases we have examined indicate that the likelihood forcollision to occur in this problem is similar to that of the re-stricted problem. However, the dynamics of collision is morecomplicated.

We now examine a collision trajectory, Cl2 and the associ-ated dynamics. Cl2 occurs in the breakout state for V (0) ¼0:205 km s�1. This is only discussed briefly here, as the detailscan be found in Belbruno & Gott (2004). It starts at L4 for t ¼ 0with a velocity of 0.205 km s�1 in the positive x-direction. Earthis located initially on the negative x-axis at 1 AU distance fromthe Sun, at the origin. Cl2 is plotted in Figure 20.In this figure, in the rotating coordinate system P3 escapes

L4, moves down toward Earth, then and turns around and movesin a retrograde motion (clockwise) about the Sun, continuing tothe third quadrant; then, in a prograde motion (counterclock-wise) with respect to the Sun, it circles the Sun and moves pastthe negative x-axis, to Earth collision. Unlike Cl � Cl1, Cl2 isa prograde collision orbit.Now, as P3 has moved in this collision orbit, Earth has moved

also. It has moved in the prograde direction and then, towardthe end, in a complicated motion in the retrograde direction.This is seen in Figure 21.The motion of Earth about 30 years prior to collision with

P3 is complicated. The final phase of Earth’s motion, much en-larged, is shown in Figure 22. It consists of many small loops,one for each year, caused by the perturbation of P3 as it nearscollision. In Figure 22 this looping motion is shown in largescale about 4 years prior to collision with P3. Earth is moving ina retrograde fashion in this figure, starting at the lower right andending in collision near the center of the coordinate system. Thevery end of Earth’s trajectory is actually parabolic in appear-ance (visible in Fig. 24), but this is too small to be detected inFigure 22.In the same time frame as in Figure 22, we show P3 on Cl2

moving to collision with Earth in Figure 23. The small blacksmudge at lower right is the complicated motion of P2 (shownenlarged in Fig. 22) prior to collision, which, relative to thescale of Cl2, is too small to be clearly seen.The actual Earth collision with P3 is shown in Figure 24,

which shows the relative motions of Earth and P3. We haveintegrated the motion of P3 through collision to better showthe relative motions. In this figure, Cl2 is the larger parabolic-type curve, and Earth moves in the smaller curve. Earth movesclockwise from left to the right, and P3 moves clockwise fromright to the left. The span of the vertical axis is approximately

Fig. 19.—Trajectory �(t) for V(0) ¼ 0.160 km s�1, t 2 [0, 200] yr, x vs. y,Sun-centered.

Fig. 20.—Orbit Cl2 from L4 at t ¼ 0 to Earth collision at t ¼ 108:628235 yrin the third quadrant. Sun-centered, x vs. y.


20,000 km, so that one-half of this distance represents the radiiof Earth and P3 added together. This implies that actual physicalcollision between Earth and P3 occurs in this figure when P3 isnear the start of its motion at lower right. We have verified thatthis is a near-parabolic collision as in Cl1.

3.5. Three-dimensional Simulation of Collisionin an Anisotropic Thin Planetesimal Diskvvia Random-Walk Encounter Dynamics

We now consider the main model of this paper for the nu-merical simulation of a collision trajectory by a Mars-sized im-pactor. So, the planar three-body problem just considered is nowgeneralized to three dimensions as given by equation (4). Att ¼ 0 Earth has the same initial position as in the planar case, andP3 starts at L4. We now more realistically model the breakoutof P3 from L4 by the random-walk �V accumulation process.

We assume that�V’s are imparted to P3 at random times. Wewill also assume that at these random times separate indepen-dent�V’s are imparted to Earth. As described in the Appendix,we are assuming a thin planetesimal disk, and at the randomtimes two types of velocity kicks are applied to both Earth andP3. One velocity kick is assumed to be in a random directionin the plane, labeled �Vk, and the other is perpendicular to theplane randomly either up or down and labeled �V?. The velocitykicks applied to Earth have a subscript of ‘‘E,’’ and those rel-ative to P3 have no subscript. In the Appendix we estimate themagnitudes of these velocities. The magnitudes of the velocitykicks for P3 are given by equation (A6) in the Appendix, and themagnitudes for Earth are given by equation (A8). In this case,breakout occurs when k ¼ 14. After these directions are input,the trajectory is propagated for a random time t that varies be-tween 0 and 100 yr. The process is then repeated.

Fig. 22.—Motion of Earth (note enlarged scale) about 4 years prior tocollision with P3 on Cl2 at t ¼ 108:628235 yr (x vs. y).

Fig. 23.—Orbit Cl2, 4 years prior to collision with Earth at t ¼ 108:628235 yr(x vs. y).

Fig. 24.—Relative motions of Earth and P3 near collision. Cl2 moves onthe larger parabolic curve clockwise from right to left, and Earth moves on thesmaller parabolic curve clockwise from left to right (x vs. y). The time du-ration is 1.9 hr.

Fig. 21.—Motion of Earth from t ¼ 0 to collision with P3 on Cl2 att ¼ 108:628235 yr, x vs. y.


Once the breakout state was achieved in the 14th step, and thepropagation terminated after t14 ¼ 93:1 yr, it was found that byextending it another 6.9 to 100 yr, no collision occurred. We thenwent back to the beginning of the 14th step. A different set ofrandom values were given to the velocity-kick directions forboth Earth and P3, keeping the time of integration to be from 0to 100 yr. Collision again did not occur. We then again wentback to the beginning of the 14th step and again picked randomvalues, and collision also did not occur. We then made a thirdadditional random trial for the velocity-kick directions at thebeginning of the 14th step and found collision did occur whent14 � 4:56 yr.

This yields a probability of collision P ¼ 14(because in one of

four random trials we succeeded); we discuss this further in theAppendix. (As in the discussion following eq. [9], our numberof trials is sufficient to give a rough estimate of this probability,which is all we require, and additional trials could establish thisnumber to higher accuracy.)

The collision trajectory �(t) is plotted in Figure 25 from theinitial value in the breakout state at the 14th step using therandomly chosen velocity-kick directions in the final attempt.The time from its initial state is 4.5615948 yr, and this finalportion of the trajectory �(t) is shown. In Figure 25 this is theupper curve, and the motion is in the downward direction. Thesmaller lower curve shows the motion of Earth, which movesin the upward direction. The collision is seen to take place nearthe x-axis (when the center of P3 hits Earth’s surface). Actualphysical collision occurs slightly earlier, when the surface of theimpactor hits the surface of Earth. If we continue the trajectory�(t) through Earth’s surface to Earth periapsis, the periapsisdistance is only approximately 200 km. The time of periapsis is4.5616056 yr (see Belbruno & Gott 2004 for more details).

The initial conditions for Earth and P3 at the beginning ofbreakout, 4.5615948 years prior to Earth collision, are explic-itly given in Belbruno & Gott (2004). This breakout state re-sults from the random-walk process previously described after13 velocity kicks at times ti, i ¼ 1, 2, . . . , 13, where t14 ¼

4:5615948 yr. The total time T for the motion of P3 to reach thisstate is T ¼

P14i¼1 ti ¼ 728:2 yr. The position of Earth at break-

out shows that it has migrated in its orbit a considerable distancein a prograde fashion from its initial position on the negativex-axis to the first (upper right) quadrant approximately along itsorbit. The velocity of Earth has slightly changed to 29.773 kms�1 from 29.78 km s�1 as a result of perturbations from P3.Even though collision with Earth is 4.56 yr away, the velocityof P3 is 29.698 km s�1, which differs from that of Earth by only75 m s�1.We next describe the collision trajectory of P3 with Earth.

The coordinate system is the same that we used in describingthe motion of Cl2, that is, a coordinate system rotating with themean motion of Earth about the Sun. It is convenient to useJacobi coordinates q ¼ (qx , qy, qz) andQ¼ (Qx , Qy, Qz), whereq is the relative vector of Earth with respect to the Sun and Qis the vector from the center of mass of the binary pair P1-P2 toP3 (Belbruno 2004). As with the planar three-body problem weconsidered, we use a rotating coordinate system that initiallyrotates in the plane of Earth about the Sun, and with the meanmotion of Earth about the Sun.Along the collision trajectory �(t) of P3 the z-variation be-

tween Earth and P3, given by qz � Qz, oscillates between approx-imately +2000 and �2000 km. This is shown in Figure 26.We show the relative motion of Earth and P3 near collision inFigure 27. In this figure the time duration is only 0.53 hr. Theorbits of Earth and P3 have been continued beyond collision toget a better understanding of the dynamics. The vertical axisspans approximately 10,000 km, so that actual collision of thesurface of the impactor with the surface of Earth would occurnear the beginning of the trajectory �(t) in the upper right quad-rant. This dynamics is analogous to that of Cl2 near collision,shown in Figure 24.As a final comment, it has been verified that with the inclu-

sion of full solar systemmodeling, as described at the end of x 2,the process of obtaining the collision trajectory �(t) is perturbedby a negligible amount, and a nearby collision trajectory can beconstructed. This is due to the fact that at breakout, the motionof P3 stays close to 1 AU radial distance from the Sun. We alsonote that in our analysis of the motion of P3 after breakout,it performs several close flybys of Earth for several hundredyears. During this time, collision is fairly likely to occur. Ouranalysis has shown that it is most likely to occur soon after

Fig. 25.—Collision between Earth (lower curve) and impactor (uppercurve) in the first quadrant. Earth moves in the upward (prograde) direction,and the impactor moves in the downward (retrograde) direction. Collision oc-curs slightly above the x-axis. Time duration: 4.5615948 yr. Planar projection,x vs. y.

Fig. 26.—Variation of qz � Qz. Time duration: 4.5615948 yr.


breakout, which for the trajectory �(t) was only 4.6 yr. It hasbeen found that after breakout has occurred, P3 generally makesvery close flybys of Earth for up to approximately 500 yr, andthen the flyby distance becomes steadily larger. The repetitiveflybys increase the semimajor axis and eccentricity of the tra-jectory, making collision less likely as P3 is no longer restrictedto staying as close to 1 AU radial distance from the Sun as it didimmediately after breakout. Also, P3 will acquire larger devi-ations in the out-of-plane directions. Soon after breakout, theseeffects are significantly less pronounced. In this sense, soonafter breakout P3 has a good chance to creep into an Earthcollision; however, if it does not have one relatively soon, thenthe flybys themselves will eventually make collision less likely.Overall, as we have shown, the probability of collision withEarth rather promptly after breakout is appreciable (of order 1

4).

As discussed previously, running additional cases could es-tablish this number to greater precision.

4. DISCUSSION

We have shown that a giant impactor could have formed at L4(L5) and then escaped on a creeping chaotic trajectory to impactEarth, with a near-parabolic encounter in agreement with sim-ulations. We note that there are difficulties if the giant impactorcame from a location other than L4 (or L5). To illustrate this,assume that it came from elsewhere.

Since the Earth’s orbit and Venus’s are nearly circular andcoplanar even at the current epoch, after 4.5 Gyr of perturba-tions, this suggests that the early disk of planetesimals in theneighborhood of Earth was quite thin and that the planetesimalsin the disk were in orbits that had low eccentricity (e) and in-clination (i), e � iT1. The critical impact parameter for col-lision with Earth for a small planetesimal is

bm ¼ rE½1þ (V 2es=V

2pec)�

1=2;

where Vpec is the peculiar velocity of the planetesimal, that is,Vpec � eVorb � e ; 30 km s�1, and Ves is the escape velocityfrom the surface of Earth (see Appendix). If Vpec=Vorb< 0:004,

then bm � rE(Ves=Vpec) > e ; 1 AU � i ; 1 AU � (Vpec=Vorb) ;1 AU, and the planetesimals whose semimajor axes are within adistance of bm of Earth’s 1 AU distance from the Sun will likelysuffer collision with Earth within a short number of years, sincewe expect the orbits to be chaotic and the impact parameter withEarth is less than the critical impact parameter bm for collisionwith Earth. This will clear out a region of bm around 1 AUexcept for planetesimals in stable orbits around L4 (or L5).Planetesimals at nearly 1 AU from the Sun—and not at L4(or L5)—will quickly be accreted by Earth before having thechance to grow large by accretion themselves. C. R. Cowley hasnoted this problem,2 saying, ‘‘Advocates of the Big Whackhypothesis usually say that the impactor must have been formednear the Earth. This is neither probable nor impossible. It is notprobable because the Earth could have readily swept up mate-rials that would have formed the other body. It is not impossiblebecause we do not know the precise conditions of the accu-mulation of the Earth, and cannot say how improbable assem-bly of the putative impactor near one astronomical unit reallywas.’’ The stable location at L4 (or L5) answers the probabilityquestion, offering a reasonably likely scenario for forming thegiant impactor near 1 AU without the material first being sweptup by Earth. Once this cleared-out region of bm has beenestablished, there will be no further quick accretion onto Earth,because the planetesimal’s orbits will not take them to withinan impact distance bm from Earth. Then they will have to diffusein by two-body relaxation—from perturbations by other plan-etesimals and planets. This two-body relaxation process willslowly put planetesimals into the gap region again, and therewill be quick accretion from the gap. The giant impactor isexpected to be one of the later impactors to hit Earth because thesuccessful simulations of the formation of the Moon start withEarth already at nearly its current mass, showing that its sub-sequent accretion (after the giant impactor hit) is assumed to besmall (Canup&Asphaug 2001). The giant impactor should alsobe expected to be one of the latter impacts because planetesimals,including Earth, grow by accretion with time, and that wouldhave also allowedmore time for the giant impactor to have grownby accretion itself.

If the giant impactor is one of the later impactors, as arguedby Canup & Asphaug (2001) (after most accretion for Earthhas been completed), then if it is not from L4 (or L5) it mustoriginally come from either significantly outside 1 AU or sig-nificantly inside 1 AU. But then it would violate one of the keyadvantages of the great-impactor theory, namely, point 3 in x 1,which explains why Earth and theMoon have the same oxygen-isotope abundance—because Earth and the giant impactor camefrom the same radius in the solar nebula. Meteorites from dif-ferent neighborhoods in the solar nebula (those associated withparent bodies Mars and Vesta, for example) have different oxygen-isotope abundances. The impactor theory is able to explain theotherwise paradoxical similarity between the oxygen-isotopeabundance in Earth and the Moon combined with the differencein iron.

Earth has oxygen-isotope abundances that are an averageover all the planetesimals it has accreted—some initially frominside 1 AU and some from outside. A giant impactor formingoutside 1 AU and drawn in by two-body interactions wouldhave oxygen-isotope abundances intermediate between those ofEarth and Mars and therefore not identical with Earth’s. Stan-dard giant-impact theory has the Moon formed primarily out of

2 University of Michigan lecture notes, 2002.

Fig. 27.—Relative motions of Earth (smaller parabolic curve) and theimpactor (larger parabolic curve) near collision. Earth moves counterclock-wise from left to right, and the impactor moves counterclockwise from right toleft. Time duration: 0.53 hr. Planar projection, x vs. y.


mantle material from the giant impactor (see Canup 2004a). Therest of the material in the giant impactor is absorbed by Earth,and the iron core of the giant impactor eventually finds its wayinto Earth’s core, leaving the Moon iron-depleted relative toEarth. TheMoon has been found to have a small core, and this isassumed to be from giant-impactor material. (See Wanke 1999for a discussion of how the giant-impactor theory can accom-modate this.) If the Moon derives from giant-impactor materialthen it would have, the theory proposes, isotopic abundancesidentical to Earth’s if it was formed near 1 AU, and this isobserved to be the case (Clayton & Mayeda 1996; Wiechertet al. 2001; Wanke 1999; Lodders & Fegley 1997). But if thegiant impactor came from significantly outside 1 AU, its iso-topic abundances would be significantly different from thoseof proto-Earth. Furthermore, since MI is only 10% the mass ofEarth, this would pollute the proto-Earth’s isotopic abundanceswith only a 10% contribution from the giant impactor. Thiswould give Earth and the Moon different isotopic abundances,if the giant impactor came from significantly outside 1 AU. Asimilar problem occurs if the giant impactor originated signif-icantly inside 1 AU, if the oxygen-isotope abundances inside1 AU are heterogeneous as well. (At present we have no mete-orites in our possession whose parent bodies are thought to beMercury or Venus. So we currently have no data for oxygenabundances inside 1 AU.)

On the other hand, consider what happens if the giant im-pactor originated at L4 (or L5). It is in a stable orbit, so it is notimmediately accreted onto Earth, and can grow large and hitEarth later, alleviating the problemmentioned by Cowley. It sitsnicely at 1 AU and accretes exactly the same type of materialEarth does, some diffusing from outside 1 AU, and some frominside. The integral of the oxygen-isotope abundances of theaccretion should be identical to that of Earth. Eventually, per-turbations kick the giant impactor out of its stable orbit, and itcollides quickly with Earth. When the giant impactor hits Earthand kicks out the Moon, since Earth and giant impactor haveidentical isotope ratios, Earth and the Moon should haveidentical isotopic abundances even though they are polluted todifferent extents by giant-impactor material. This yields agree-ment with the giant-impactor model. Since this is one of thelater accretion events for Earth in terms of the accumulation ofits mass, the oxygen-isotope abundances for Earth and theMoon will not be further significantly changed by post–giant-impactor accretion.

Thus, we propose the following scenario: Debris remainsat L4 (as the Trojan asteroids prove). From this debris a giantimpactor starts to grow like Earth through accretion as de-scribed above. As the forming giant impactor reaches a suffi-cient mass (�0.1mE), it gradually moves away from L4 throughgravitational encounters with other remaining planetesimalsand it randomly walks in peculiar velocity. It gradually movesfarther and farther from L4 approximately on Earth’s orbit in ahorseshoe orbit at 1 AU, until it acquires a peculiar velocity ofapproximately 180 m s�1. The giant impactor then undergoesbreakout motion in which it performs a number of cycles aboutthe Sun, repeatedly passing near Earth. In a time span roughlyon the order of 100 years, it collides with Earth on a near-parabolic orbit.

We have presented here a mechanism for the origin of aMars-sized Earth impactor and described the path it would taketo arrive at Earth collision via a special class of slowly movingchaotic collision trajectories. The analysis shows that Earthcollision along these trajectories is likely. Approaches for fur-ther work are discussed in the Appendix.

Note added in manuscript.—As we have discussed, the giantimpactor could have grown up in a stable orbit at Earth’s L4 (orL5) point, where a stable orbit is possible and an object couldremain and be able to grow by accretion without hitting Earthearly on. We expect this phenomenon could occur when therewas a thin disk of planetesimals (in nearly circular orbits). Wehave noted that Saturn’s rings are an example of such a thin diskof planetesimals (in this case, chunks of ice plus some dirt)observable today. Saturn’s regular icy moons (inside the orbit ofTitan) are all in nearly circular orbits of low eccentricity, sug-gesting that they formed out of a thin disk of planetesimals (icechunks) rather like Saturn’s rings today, only larger in extent.In such a situation we might expect our scenario to operate.Therefore it is quite interesting that we can find examples of ob-jects at L4 (or L5), or escaping from L4 (or L5), in the Saturnsystem. Saturn’s moon Helene co-orbits at the L5 point (60�

ahead) of the larger moon Dione. Helene has a largest diame-ter of 36 km and Dione has a diameter of 1120 km. Saturn’smoons Telesto (diameter 34 km) and Calypso (diameter 34 km)occupy both the L4 and L5 points relative to Saturn’s moonTethys (diameter 1060 km). We would say that Helene, Telesto,and Calypso originated in a planetesimal disk (of ice chunks) atthese stable Lagrange points and have grown in place there,surviving till the present without colliding with Dione or Te-thys. The rest of the planetesimals (ice chunks) have accretedonto the regular moons of Saturn. (Saturn’s rings themselveslie inside the Roche limit, where the formation of large objectsby accretion is forbidden.) While these Lagrange moons aresmall relative to the primary, growth of larger objects with re-spect to the primary is also possible. Saturn’s moons Epimetheus(119 kmdiameter) and Janus (179 kmdiameter) co-orbit in horse-shoe orbits just like the one we found for the giant impactor nearbreakout (Fig. 5). We would say that Epimetheus formed at aLagrange point of Janus and grew along with it by accretionfrom the planetesimal disk. Later perturbations by other plan-etesimals kicked it out into a horseshoe orbit just short ofbreakout. Thus, an object (Epimetheus) nearly as large as theprimary (in this case Janus) can form and end up in a horseshoeorbit. Just a little more perturbation and Epimetheus wouldachieve breakout and likely collide with Janus. These provideexamples of the phenomena described in this paper that can beobserved today.A similar pair of co-orbiting objects in horseshoe orbits in

another solar system could be easily detected using stellar radialvelocity data. This would appear as a planet in circular orbitabout the star whose mass was observed to mysteriously vary.The mass variation would be approximately sinusoidal in timewith a period significantly longer than the orbital period of theprimary. For example, if the secondary had a mass 0.1 times thatof the primary ( like the giant impactor), then this would showup as a nearly sinusoidal variation of 10% in the deduced massof the primary. If the twowere nearly equal in mass, there wouldbe a 100% variation in the mass. (When they were near eachother at one end of the horseshoe orbit, the effective massperturbing the star would be nearly doubled, and when theycirculated to be on opposite sides of the star their perturbationwould temporarily vanish.) We should have a look among theknown extrasolar planets for such cases. Granted, we are cur-rently able to see only gas giant planets (which may have evenmigrated inward) rather than the terrestrial ones we are con-sidering, but it would still be interesting to look. If one foundsuch a case, it would be easy to prove.Also of particular interest is the Earth co-orbiting asteroid

2002 AA29, which is in a horseshoe orbit relative to Earth. Of


course, in addition to the giant impactor there can be otherLagrange point debris, particularly at the other stable Lagrangepoint not occupied by the giant impactor. This material mayhave been kicked out early on by other planetesimals or by thegiant impactor itself as it escaped into a horseshoe orbit. Doesany of this material survive to the present day? Asteroid 2002AA29 (diameter of less than 0.1 km) is in a horseshoe orbit at1 AU, virtually identical to the horseshoe orbits found by us inFigure 5. This asteroid approaches Earth closely (5.8 millionkilometers away) once every 95 years while circling the Sun at1 AU. It was near one of these close approaches that it wasdiscovered in 2002. After a number of cycles, it is brieflycaptured for a period of 50 yr as a quasi satellite of Earth beforereturning to the 95 yr horseshoe orbit cycle. Its rather largeinclination (10N7) saves it from collision with Earth. This objectmay have originated near L4 (or L5) and have been kicked outinto a horseshoe orbit ( perhaps by the giant impactor itself ). Ifthat is so, it could be composed of the same material that alsoformed the seeds for Earth and the giant impactor. A samplereturn from this asteroid thus offers the possibility of obtainingsome primordial material from the same reservoir that producedEarth and the Moon. In this case, it should have oxygen-isotopeabundances similar to those found for Earth and the Moon andan iron abundance similar to that of Earth. The final oxygen-isotope abundances and iron abundances of Earth reflect notonly their seed material (originally from 1 AU), but also theintegral of the abundances accreted later, from material origi-nally inside and outside 1 AU. Thus, any slight differences inoxygen-isotope abundances would be helpful in illuminat-ing the accretion process. It would of course be very interest-ing to measure the age of a 2002 AA29 sample. On the otherhand, if the sample has oxygen-isotope abundances identicalwith Earth and the Moon but is poor in iron like the Moon, thatwould suggest it was part of the splash material kicked out bythe giant impactor at near escape velocity that did not coalesceonto the Moon but rather ended circling the Sun at 1 AU andthen became trapped at L4 (or L5), where it moved for per-haps a considerable time before finally being kicked out. Ifthe sample has completely different oxygen-isotope abundancesfrom those of Earth and the Moon, that would indicate an ori-gin elsewhere in the solar nebula (not at 1 AU), and we wouldthen have to explain how it somehow got perturbed into a low-eccentricity horseshoe orbit at 1 AU. (Most Earth-crossing as-teroids perturbed into their current orbits from the main beltshould have much larger eccentricities, according to Ipatov &Mather 2004.) Bottke et al. (1996) have previously suggestedthat low-eccentricity objects near 1 AU could have an origintracing back to the Earth-Moon system, and radar results sug-gest (Ostro et al. 2003) that 2002 AA29 has a high albedo,which supports this hypothesis (according to Connors et al.2002). A sample return from asteroid 2002 AA29 is thus ofparticular scientific interest and may provide important clues asto the origin of Earth and the giant impactor that formed theMoon.

We would like to thank Scott Tremaine and Peter Goldreichfor helpful comments, and also Robert Vanderbei for use of hissolar system simulator. Partial support for this work for J. R. G.is from NSF grant AST 04-06713, and for E. B. from grants bythe NASA Office of Space Science and Goddard Space FlightCenter.

APPENDIX

PLANETESIMAL DYNAMICS IN A THIN DISKAND RANDOM ENCOUNTERS WITH THE GREAT

IMPACTOR AND EARTH

We estimate the magnitudes of the velocity perturbations onthe giant impactor and Earth due to encounters with other smallplanetesimals with a back-of-the-envelope calculation. Suchvelocity perturbations drive the impactor into breakout.

Let � be the typical mass of the remaining planetesimals (ormore precisely, the rms mass observed in the distribution; notethat this definition of � is different from the definition of � in thediscussion of the three-body problem, eq. [1]). Consider thedisk of planetesimals to be of thickness

�2(V�=Vorb)(1 AU) � 2rEV�=(4 ; 10�5Vorb);

where V� is the typical peculiar velocity of the planetesimalsand includes all radii nearer to 1 AU than either Venus orMars. The volume of the disk is then 2(V�=Vorb)�(0.85)(1 AU)

3.Let the planetesimals have peculiar velocities on the order ofV�, or equivalently orbital eccentricities e (and inclinations i)of order V�=Vorb � 0.0006, where Vorb ¼ 29:86 km s�1. Thethickness of the disk is then on the order of 30rE. For distantencounters, with impact parameter b, the deflection of the plan-etesimal by the impactor is

�V�=V� �2GMI=V

2�b; if b > bc;

1; if b � bc;

�

where bc ¼ 2GMI=V2� and MI is the mass of the giant

impactor.These are hyperbolic encounters, and the planetesimal exits

the encounter with the same magnitude of peculiar velocity V� ,but changed in vector direction. Thus, for nearby encounters�V�=V� can be at most 2. Momentum is conserved, so the kickin velocity �V received by the giant impactor is given by

�V � �V��=MI ;

and for a single collision

(�V )2 �4G2�2=V 2

� b2; if b > bc;

V 2��

2=M 2I ; if b � bc:

(

The collisions are independent, so the velocity kicks add inquadrature, giving a random walk in velocity space with time.Since the disk is thin with V�=Vorb � 0.0006, the half-thicknessof the disk is 0.0006 AU � 14rE, most of the time b314rE,and the encounters are mostly in the plane (i.e., 0 < b? < 14rE,while 0 < bk < 1 AU) and

(�V?)2=(�Vk)

2 � b2?=b2k;

implying that the major component of the �V is parallel to theplane. The number of planetesimals is n ¼ Mdisk=�, so con-sidering the geometry of the disk [seen edge-on it is a horizontalstrip of thickness 2(V�=Vorb)(1 AU)], the number of collisions


with impact parameter b (>14rE) from that strip within a time tis

�N � n4(V�=Vorb)(1 AU)

2(V�=Vorb)�(0:85)(1 AU)3db V�t:

The relevant area is 4v (1 AU)db, where v � V�=Vorb, becausethe disk has thickness of 2v (1 AU) and there are two verticalstrips of width db and height 2v(1 AU) at a distance b from thegiant impactor (one to the left and one to the right). The rangeof impact parameters we should integrate over is approx-imately 0 to 1 AU.

The impact parameter for physical impact is calculated fromthe parameters of the hyperbolic encounter. When the smallplanetesimal hits MI it will have a velocity VS , where 0.5V

2S ¼

0.5V 2� þ GMI=rI . At the maximum impact parameter for phys-

ical collision, the planetesimal will just graze MI at the perigeeof its orbit, so that VS will be tangential at that point and itsangular momentum per unit mass will be L ¼ VSrI ¼ V�bm (orequal to what it had initially). Thus, bm ¼ rIVS=V� and

bm � rI

�1þ 2GMI

rIV 2�

�1=2

� rI

�1þ bc

rI

�1=2

� bc

�r2Ib2c

þ rI

bc

�1=2

;

and since the escape velocity squared V 2es from the surface

of the giant impactor is much greater than V 2� , then rI=bc ¼

V 2�=V

2esT1 and bm � (bcrI )

1=2. [We are considering here onlygravitational encounters, not physical collisions, so bm will bethe minimum impact parameter for our integration. Of course,there will be some contribution to (�Vk)

2 from direct collisionsand accretion, but this is difficult to calculate and we will ignorethis contribution in this simple treatment. Since (bcrI)

1=2 is smallrelative to 1AU, this contribution to (�V )2 will be negligible in anycase. The effects of direct collisions on (�V?)

2 may be significant,but they are again difficult to calculate, so for simplicity we areignoring them and only considering gravitational encounters.]

Thus, we will integrate from (bcrI)1=2 to 1 AU, and in the

regime we are interested in v � 0.0006, (bc rI )1=2 < 1 AU, and

bc > 1 AU, so we are in the regime where (bc rI )1=2 < b < 1 AU

and

(�Vk)2 � Mdisk

�

4v (1 AU)V�t

2v� (0:85)(1 AU)3

Z 1 AUffiffiffiffiffiffiffibc rI

pV 2��

2

M 2I

db: ðA1Þ

This yields

(�Vk)2=V 2

orb�Mdisk

0:085�4v3

�t

1 yr

��

MI

�2

:

Breakout is achieved when (�Vk)2 � (0.006)2V 2

orb, or after atime

tbreak � 7:65 ; 10�6v�3(�=Mdisk)(MI=�)2 yr:

Since v ¼ 0:0006,

tbreak � 35;400(�=Mdisk)(MI=�)2 yr: ðA2Þ

At breakout we might expect Mdisk � 0:3MI , since we wantsubsequent accretion onto Earth to be inconsequential relativetoMI . Canup&Asphaug’s (2001) successful simulation has thegiant impactor hit a nearly formed Earth. For example, with� � MI=300 we expect n � 100 other large planetesimals of

mass � � MI=300 and tbreak � 35 Myr, long enough for ironcores to form in Earth and the giant impactor as required and inagreement with radioactive halfnium-tungsten chronometerresults (see Canup 2004a).The same calculation as done leading to equation (A1) above

could be repeated withME replacingMI, and one would see that

(�Vk)2E=(�Vk)

2I � M 2

I =M2E: ðA3Þ

Thus the rms peculiar velocity acquired by Earth due to ve-locity kicks in time t is 1

10as large as that acquired by the giant

impactor.Returning to the giant impactor, we see that for individual

collisions

(�V?)2 � (�Vk)

2(b2?=b2k);

where

hb2?i �Z V�(1 AU)=Vorb

0

x2 dx

�Z V�(1 AU)=Vorb

0

dx

� 13v2(1 AU)2:

Similar to the estimation of (�Vk)2 in equation (A1), it is found

that

(�V?)2 � Mdisk

0:85� (1 AU)22V�t

v 2

3(1 AU)2

Z 1 AUffiffiffiffiffiffiffibc rI

pV 2��

2

b2M 2I

db;

which simplifies to

(�V?)2 � Mdisk

0:85�2V�t

v 2

3

V 2�

(bc rI )1=2M 2

I

: ðA4Þ

Equations (A1) and (A4) imply

(�V?)2=(�Vk)

2 � 13v2½1 AU=(bc rI )

1=2�: ðA5Þ

Now, (bc rI )1=2 � rI (Ves=V�), and it can be shown that equa-

tion (A5) reduces to

(�V?)2=(�Vk)

2 � 13(V�v

2=Ves)(1 AU=rI ):

If V� � 0.0006Vorb ,

(�V?)2=(�Vk)

2 � 1:9 ; 10�5:

So when the giant impactor achieves breakout, �Vk � 0.006Vorband �V? � 0.000026Vorb , so if we simulate the random walk byapplying random kicks

�Vk=Vorb � 0:001; �V?=Vorb � 0:0000044 ðA6Þ

to the giant impactor at random times, after approximately 36kicks breakout should be achieved. If we followed this to itsconclusion, that would mean an inclination at breakout for thegiant impactor of i � �V?=Vorb � 0.000026 radians � 500 �0.6rE=(1 AU), which would keep it on an easy collision coursewith Earth. This also guarantees that the collision will be nearlyin the plane of the ecliptic (i.e., b?=bk � 0.6rE=0.006 AU �4 ; 10�3, or within 0N25 of the ecliptic). This should, by con-sideration of the total angular momentum, produce an orbit for


the Moon approximately in the plane of the ecliptic, as is ob-served. Some debris is ejected, so the alignment should just beapproximate—which it is. Interaction of the debris disk withEarth as the Moon forms can also naturally lead to a moderatetilt of Earth relative to the Moon’s orbit, as explained by Canup& Asphaug (2001).

The calculation for (�V?)2 can be repeated for Earth, yielding

(�V?)2E

(�Vk)2E

� (0:0006)3

3

�Vorb

VE;es

��1 AU

rE

�� 4:5 ; 10�6;

where bc ¼ 2GME=V2� ; since rE=bc¼V 2

�=V2E;es¼ (0.018 km s�1=

11.19 km s�1)2 ¼ 2:6 ; 10�6, bc ¼ 16:4 AU.Thus, by the time the giant impactor has achieved breakout

the values for the peculiar velocity of Earth will be

(�Vk)E=Vorb �0:0006; (�V?)E=Vorb � 0:0000013: ðA7Þ

Therefore, we will apply velocity kicks of

(�Vk)E=Vorb � 0:0001; (�V?)E=Vorb � 0:0000002 ðA8Þ

at random times to Earth, and after 36 kicks they should achieveby random walk the values given by equation (A7). As can beseen, the movement of Earth is less than that of the giant im-pactor, so one may say that although the motion of Earth com-plicates the situation, it is still basically the giant impactor thatis achieving breakout. Earth is basically a spectator while thegiant impactor breaks out of its stable equilibrium at L4.

Let us calculate the minimum impact parameter bm for col-lision betweenMI andME. This is a two-body problem. Impactoccurs when the separation of the centers of MI and ME is� ¼ rI þ rE. MI has a peculiar velocity relative to ME on theorder of Vbreak. Imagine shooting MI at Earth with relative ve-locity Vbreak and a minimum impact parameter bm so that it justhas a grazing collision with Earth. At that point of collision, therelative velocity is VS and the angular momentum per unit massis VS� ¼ Vbreakbm, which is the initial angular momentum perunit mass. Now, conservation of energy gives

12V 2S ¼ 1

2V 2break þ ½G(MI þME)=��;

and this yields

bm ¼ �

�1þ (V 2

es=V2break)(MI þME)rE

ME�

�1=2; ðA9Þ

whereVbreak¼ 0.006Vorb¼ 0.18 km s�1 andVes¼ 11.19 km s�1¼62.1Vbreak is the escape velocity from the surface of Earth. Now,(MI þME)=ME ¼ 1:1 and �=rE ¼ 1:53rE. Thus, equation (A9)yields

bm ¼ 1:53rE½1þ (3856:4=1:39)�1=2 ¼ 80:6rE ¼ 0:0034 AU:

If the vertical peculiar velocity obtained by the giant impactorupon breakout is less than 0.0034Vorb, as it is in the thin-diskcase we are considering, then it will not miss Earth in the ver-tical direction. It will have a typical eccentricity of 0.006 andwill therefore have a typical impact parameter with respect toEarth in the plane on the order of 0.006 AU. So, the chance ofimpact in the thin-disk case on the first pass by Earth is of order

P � bm=(0:006 AU) � 0:57;

or a substantial probability. In fact we observe P � 0.25, asdiscussed in step 6 of x 3. In other words, we had one success onhitting Earth on the first pass in four attempts. Our earlier testswith the restricted three-body problem in the plane had foursuccesses out of 13 attempts, forP � 0.3, as we discussed at theend of step 2 in x 3 and as given by equation (9). Interestingly,there we picked four directions +x, +y,�x,�y, where the valuesof Vbreak=Vorb were respectively 0.007, 0.011, 0.007, and 0.012.The above calculation indicates that, roughly, P / 1=Vbreak ,so given the values of Vbreak we would expect P � 0.49 for the x directions, where Vbreak = 0.007, and in fact, we had fiveruns with two successes at hitting Earth within 647 yr, so weobserve P = 0.4. By comparison, we would predict P � 0.30for the y cases, where Vbreak = 0.0115. In fact, in those caseswe had two successes in hitting Earth in 647 yr out of eight runs,giving P = 0.25. In both the x and y cases, the results arecomparable to our estimates. As discussed previously, these are allrough numbers that could be improved by doing additional trials.

[ In the case of a thick disk, the perturbations on the giantimpactor would be of order i � 0.006 ¼ Vbreak=Vorb, and there-fore it would have a probability of missing in the vertical di-rection of order bm=(0.006 AU ), as well as a similar probabilityof missing in the plane, so the chance of impacting Earth on thefirst pass would be of order P � (bm=[0.006 AU ])2 � 0.32,which is smaller but still appreciable.]

As we derived above in equation (A2),

tbreak � 35;400(MI=Mdisk)(MI=�) yr:

For Mdisk � 0:3MI and � � MI=300, this gives a breakout timeof �35 Myr.

Since the planetesimals are in orbits with eccentricities e �V�=Vorb and the accretion impact parameterba> (V�=Vorb)(1AU),the only planetesimals that can hit the great impactor or Earthare those at a radius from the Sun of 1 AU� ba, I < r< 1 AUþba, I and 1 AU � ba,E < r < 1 AU þ ba,E , respectively. (Recallthat the impact parameter for accretion of a planetesimal ontothe giant impactor is ba; I � rI ½1þ (V 2

es=V2�)�1=2 � 270rI � 9:1 ;

105 km, and onto Earth is ba;E � rE½1þ (V 2es=V

2� )�1=2 � 622rE �

4:0 ; 106 km>(V�=Vorb)(1AU)¼ 8:9 ; 104 km.) But once a bandof width 2ba,E is cleared out, there will be no further promptphysical collisions with Earth or the giant impactor.

So, Earth and the giant impactor will quickly clear out thatarea, but further accretion will await the scattering of plane-tesimals into that region on a timescale dictated by two-bodyrelaxation among the planetesimals. The planetesimal diskcloser to 1 AU than to Venus or Mars has limits from 0.86 to1.26 AU. Thus, to accrete the entire remaining disk, a timescaleis required similar to that for a planetesimal to random-walkup to an eccentricity of order 0.2, or equivalently to acquire apeculiar velocity in the disk on the order of �Vk � 0.2Vorb. Letus estimate this accretion timescale.

Let � be the typical mass of the remaining planetesimals (ormore precisely, the rms mass observed in the distribution). Letthe peculiar velocities of the planetesimals be V� � 0:0006Vorb.The disk of planetesimals is to be of thickness (2 ; 0.0006)(1 AU)and includes all radii nearer to 1 AU than either Venus or Mars.The volume of the disk is then 2(V�=Vorb)�(0.85)(1AU)3. For dis-tant encounters, with impact parameter b, the deflection of a plan-etesimal upon passing another planetesimal is

�V�=V� �ffiffiffi2

pG�=V 2

�b; if b > bc ¼ffiffiffi2

pG�=V 2

� ;

1; if b � bc;

(


where we take into account the fact that the total mass of thetwo-particle system is 2�, the rms relative velocity between thetwo particles is

ffiffiffi2

pV�, and �V� is half the total change in relative

velocity.So, for a single collision

(�V�)2 �

2G2�2=V 2�b

2; if b > bc;

V 2� ; if b < bc:

(

The collisions are independent, so the velocity kicks add inquadrature, giving a random walk in velocity space and time.Since the disk is thin, bc 3 (V�=Vorb)(1 AU), and 0< b? <(V�=Vorb)(1 AU) while 0<bk< 1 AU, so the encounters aremostly in the plane and the major component of �V� is parallelto the plane. The number of planetesimals is Mdisk=�; consid-ering the geometry of the disk [seen edge-on it is a horizontalstrip of thickness 2(V�=Vorb)(1 AU)], the number of collisionswith impact parameter b [where b> (V�=Vorb)(1 AU)] from thatstrip within a time t is

�N � Mdisk

�

�2V�

Vorb

�(0:85)(1 AU)3��1

4V�

Vorb

(1 AU)ffiffiffi2

pV�t db:

The range of impact parameters b we should integrate over isapproximately bm to 1 AU. The rms impact velocity is

ffiffiffi2

pV� .

Considering the relative velocities of the two planetesimals andthat physical impact occurs when their center-to-center separa-tion is 2r� , the minimum impact parameter for physical impact is

bm � 2r�

�1þ G�

r�V 2�

�1=2

� 2r�

�1þ bcffiffiffi

2p r�

�1=2

:

Now, r� � (�=MI)1=3(3380 km) and bc�

ffiffiffi2

pG�=V

2� � (1:75 ;

108 km)(�=MI)(0.0006Vorb=V�)2. Sobc=r� � 5:18 ; 104(�=MI)

2=3

(0.0006Vorb=V�)2, and if the remaining planetesimals are large

but still smaller than the giant impactor (say, �=MI � 1=300),then bc=r� 31, bm � 23=4(bc r�)1=2, and 1 AU 3 bc 3 bm >(V�=Vorb)(1 AU) 3 r�.

[We are considering here only gravitational encounters, so bmwill be the minimum impact parameter for our integration. Ofcourse, there will be some contribution to (�Vk)

2 from directcollisions, which is difficult to calculate and will be ignored inthis simple treatment. Since bm is small relative to bc , this con-tribution to (�V�)

2 will be negligible in any case.]Thus, we will integrate from bm ¼ 23=4(bc r�)

1=2 to 1 AU:

(�Vk)2 � Mdisk

�

4ffiffiffi2

p

2�(0:85)(1 AU)2V�t

;

�Z bc

bm

V 2� dbþ

Z 1 AU

bc

2G2�2

V 2�b

2db

�:

This yields

(�Vk)2 � Mdisk

�

4ffiffiffi2

p

0:85(1 AU)

V�

Vorb

�t

1 yr

��V 2�bc þ

2G2�2

V 2�bc

�;

which reduces to

(�Vk)2 � Mdisk

�

16

0:85(1 AU)

V�

Vorb

�t

1 yr

�G�;

yielding

(�Vk)2=V 2

orb �16Mdisk

0:85MSun

V�

Vorb

�t

1 yr

�:

Thus, the accretion timescale is on the order of

ta � (0:2)2�0:85MSun

16Mdisk

��Vorb

V�

�yr;

which yields ta � 11:8(MI=Mdisk) Myr.Recall that the breakout timescale is tbreak � 35,400(MI=

Mdisk)(MI=�) yr. If we want ta > tbreak, then

�=MI k1=300:

Thus, we expect that the remaining planetesimals would havehad time to grow large but would still be smaller than the giantimpactor. If Mdisk � 0:3MI and � � MI=300, there would be onthe order of 100 large (�500 km in radius) planetesimals, and thebreakout timescale for the giant impactor would be on the orderof 35 Myr, while the timescale for accretion of the remaininglarge planetesimals would be on the order of 40Myr. This leavesenough time for Earth and the giant impactor to form and for theiriron cores to sink into their centers (in agreement with the esti-mate of 10–30 Myr for core formation from the radioactivehalfnium-tungsten chronometer; see discussion in Canup 2004a).So if we want the breakout to occur on a timescale shorter

than the remaining accretion timescale, we would want �=MI �1=300 or greater. Thus, we expect that by the time the giantimpactor is achieving breakout, there would be just a few(<100) large planetesimals left dominating the mass distribu-tion—reasonable, since the largest of the remaining plane-tesimals left at the time of breakout would have had time to growlarge. Still, we would expect the giant impactor to be the largestof the remaining planetesimals.After the giant impactor hits Earth and the Moon is formed

from the splash debris, Earth will continue to accrete the remain-ing planetesimals. The accretion cross section for Earth is on theorder of �E � �(bcRE)� �r2E(VE;es=V�)

2 � �(62rE)2. In Canup

& Asphaugh’s (2001) model, the Moon is expected to form at1.2aRoche = 3.5rE (much later it drifts out to its current location bytidal interaction). The cross section for bringing an object insidethe Moon’s orbit radius is �EM � �(3:5rE)2(V

2E;es=3:5V

2�) �

3:5�E. Of those crossing the Moon’s orbit, only two out ofseven will hit Earth on that pass. Of those crossing the Moon’sorbit, at the time they cross they will have a velocity of V 2 �V 2� þ V 2

E;es=3:5 � V 2E;es=3:5, and at this velocity the Moon’s ac-

cretion cross section is �m � �r2M½1 þ (3:5V 2M;es=V

2E;es)�

1=2 �1:16�r2M. The fraction of those that cross the Moon’s orbit thatimpact the Moon will therefore be f �1:16�r2M=4�(3:5rE)

2 �1:7 ; 10�3. So, the number of objects hitting Earth is larger thanthe number hitting the Moon by a factor of (1=3:5)=(1:7 ;10�3) � 160. If fewer than 160 large objects are eventually ac-creted, then the average number expected to hit the Moon is lessthan 1. So if the number of large objects accreting is less than100, there is an appreciable chance that all those large objectswill hit Earth and none will hit the Moon. This an importantadvantage for our model, since it does not pollute the Moon withany additional iron—leaving it iron-poor. Indeed, this reason iscited by Canup & Asphaugh (2001) in arguing that accretion onEarth and the Moon after the great impact should be small. TheMoon may be expected not to gain appreciable additional ma-terial. Still smaller planetesimals, which make an insignificant


contribution to the total remaining disk mass, may fall on Earthand the Moon—creating impact sites such as the Mare Impriumwithout adding significantly to the mass. Since our scenario de-pends on the fact that there will be some debris left in Earth’sneighborhood at the time of the formation of the Moon by thegiant impactor, it is a plus that theMoon shows some signs of lateimpacts itself.

The parameters here can be considered as a toy model at best.We have ignored dynamical friction, which could slow break-out by slowing the accumulation of peculiar velocity of a mas-sive body, but since L4 is at a peak of the effective potential,dynamical friction might even speed breakout. We have alsoignored the effects of momentum transfer perpendicular to theplane in planetesimals that actually hit the giant impactor(considering only the momentum transfer in the more frequentdistant encounters), because the latter is difficult to calculate.We have considered peculiar and parallel velocities as if theyoccurred in a slab, ignoring the Keplerian motion around theSun. We need not be married to these particular parameters, asthey just form a jumping-off point. They give us a guess as to theratio of perpendicular velocities to velocities in the plane thatmight be acquired by gravitational perturbations in a random-walk scenario. The random-walk scenario is one that should oc-cur under very general circumstances. Other model parametersand assumptions might lead to varying scenarios, but those wehave shown are a starting point for discussion. A natural con-tinuation would beN-body experiments. More complicatedmassdistributions and various eccentricity models for the planetesi-mals could be considered. There is a great deal of parameter spaceto be explored. More elaborate simulations with millions of par-ticles, including treatment of physical collisions, could simulatethe formation of Earth and the giant impactor and their growthin a cold-disk scenario. After gaseous dissipation was finishedand only planetesimals were left, we expect some debris to have

remained at L4 and L5 ( like the Trojan asteroids) because plan-etesimals in perturbed stable orbits about L4 and L5 would staythere. A large object such as a giant impactor can grow by ac-cretion at L4 (or L5). It is a matter of survival: an object in astable orbit at L4 (or L5) will survive—not hitting Earth—andby surviving can have more time to accrete other planetesimalsand grow large itself. One could see how often the second-largest object growing near 1 AU in fact started in the Lagrangepoint debris of Earth. Finally, in the cases where a giant impactorof the type required to form the Moon did indeed hit Earth, caus-ing formation of a moon like ours, one could see how often thatimpactor did in fact originate in Lagrange point debris. In otherwords, what is the probability that the giant impactor originated atL4 or L5 given that a moon like ours (with material of identicaloxygen abundance from 1 AU) is formed by collision?

There are several races going on. Earth starts forming by ac-cretion, and as it grows in mass, a stable Lagrange point at L4(and one at L5) forms. As we have shown, as Earth grows theregion of stable orbits around L4 grows in size, so a planetes-imal trapped there in a stable orbit would stay there as a proto-Earth grew. It would grow larger itself by surviving and accretingother planetesimals. It must grow to a mass on the order of0.1ME by the time it is perturbed out of its stable orbit andachieves breakout. Breakout must be achieved before all theremaining planetesimals have accreted onto Earth. After break-out, a collision with Earth on a near-parabolic trajectory islikely.

We point out the possibility that the giant impactor thatformed the Moon could have originated at L4, survived therelong enough to grow large by accretion, and eventually beenperturbed by other planetesimals onto a collision course withEarth. Awaiting numerical simulations capable of showing thisoccurring in detail, we have used examples within our own solarsystem to support our model (see note added in manuscript).

REFERENCES

Adler, R. 2000, Nature, 408, 510Arnold, V. I. 1961, Soviet Math., 2, 247———. 1989, Mathematical Methods of Classical Mechanics (2nd ed.; Berlin:Springer)

Belbruno, E. 1992, Planet. Rep., 7(3), 8———. 2002, in Proc. 53rd Inter. Astronautical Congress: Astrodynamics(Paris: Intl. Astronaut. Federation), No. IAC-02-A.6.03

———. 2004, Capture Dynamics and Chaotic Motions in Celestial Mechanics(Princeton: Princeton Univ. Press)

Belbruno, E., & Gott, J. R., III. 2004, preprint (astro-ph /0405372)Belbruno, E. A. & Miller, J. 1990, A Ballistic Lunar Capture Trajectory for theJapanese Spacecraft Hiten (Interoffice Memo. 312/90.4-1317) (Pasadena: JPL)

———. 1993, J. Guidance Control Dyn., 16, 770Benz, W., Cameron, A. G. W., & Melosh, H. J. 1989, Icarus, 81, 113Benz, W., Slattery, W. L., & Cameron, A. G. W. 1986, Icarus, 66, 515———. 1987, Icarus, 71, 30Bottke, W. F., Jr., Nolan, M. C., Melosh, H. J., Vickery, A. M., & Greenberg, R.1996, Icarus, 122, 406

Broucke, R. 1994, in Proc. Advances in Nonlinear Astrodynamics, ed. E.Belbruno (Minneapolis: Geometry Cent., Univ. Minnesota), No. 7

Cameron A. G. W. 2001, Meteoritics Planet. Sci., 36, 9Cameron A. G. W., & Benz, W. 1991, Icarus, 92, 204Cameron A. G. W., & Ward, W. R. 1976, in Lunar and Planetary Science VII(Houston: Lunar Planet. Inst.), 120

Canup, R. M. 2004a, Phys. Today, 57(4), 56———. 2004b, Icarus, 168, 433Canup, R. M., & Asphaug, E. 2001, Nature, 412, 708Christou, A. A. 2000, Icarus, 144, 1Clayton, R. N., & Mayeda, T. K. 1996, Geochim. Cosmochim. Acta, 60, 1999Conley, C. C. 1969, J. Differential Equations, 5, 136

Connors, M., Chodas, P., Mikkola, S., Weigert, P., Veillet, C., & Innanen, K.2002, Meteoritics Planet. Sci., 37, 1435

Deprit, A., & Deprit-Bartolome, A. 1967, AJ, 72, 173Frank, A. 1994, Discover, 15(9), 74Goldreich, P., & Tremaine, S. 1980, ApJ, 241, 425Goldreich, P., & Ward, W. R. 1973, ApJ, 183, 1051Hartmann, W. K., & Davis, D. R. 1975, Icarus, 24, 504Hollabaugh, W., & Everhart, E. 1973, Astrophys. Lett., 15, 1Ida, S., & Makino, J. 1993, Icarus, 106, 210Ipatov, S. I., &Mather, J. C. 2004, in Ann. NYAcad. Sci., 1017, Astrodynamics,Space Missions, and Chaos, ed. E. Belbruno, D. Folta, & P. Gurfil, 46

Levison, H. F., Shoemaker, E. M., & Shoemaker, C. S. 1997, Nature, 385, 42Lodders, K., & Fegley, B., Jr. 1997, Icarus, 126, 373Mikkola, S., & Innanen, K. A. 1990, AJ, 100, 290Morais, M. H. M., & Morbidelli, A. 2002, Icarus, 160, 1Murray, C. D., & Dermott, S. F. 1999, Solar System Dynamics (Cambridge:Cambridge Univ. Press)

Namouni, F. 1999, Icarus, 137, 293Ostro, S. J., Giorgini, J. D., Benner, L. A. M., Hine, A. A., Nolan, M. C.,Margot, J.-L., Chodas, P. W., & Veillet, C. 2003, Icarus, 166, 271

Rafikov, R. R. 2003, AJ, 126, 2529Siegel, C. L., & Moser, J. K. 1971, Lectures on Celestial Mechanics (Berlin:Springer)

Stevenson, D. J. 1987, Annu. Rev. Earth Planet. Sci., 15, 271Szebehely, V. G. 1967, Theory of Orbits (New York: Academic)Wanke, H. 1999, Earth Moon Planets, 85–86, 445Weissman, P. R., & Wetherill, G. W. 1974, AJ, 79, 404Wetherill, G. W., & Stewart, G. R. 1989, Icarus, 77, 330Wiechert, U. W., Halliday, A. N., Lee, D.-C., Snyder, G. A., Taylor, L. A., &Rumble, D. 2001, Science, 294, 345


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