Chaotic Exchange of Solid Material Between Planetary Systems: Implications for Lithopanspermia

2.1. General planetary system model

Consider a general planetary system (S) that consists of a central star (P ₁) and a system of N planets ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1 , i = 2 , \ldots , N$$\end{document} , with N≥3) on co-planar orbits that are approximately circular, where the labeling is not reflective of the relative distances from P ₁. We assume that the mass of the star (m ₁) is much larger than the masses of any of the planets (m_i , i.e., m ₁>> m_i , for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$i = 2 , \ldots , N$$\end{document} ) and that the mass of one of the planets (P ₂) is much larger than the masses of all the other planets (i.e., m ₂>> m_i , for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$i = 3 , \ldots , N$$\end{document} —this condition is fulfilled in the case of our solar system, where P ₂ is Jupiter). A meteoroid (P ₀) is considered to have a negligible mass (m ₀=0) with respect to the mass of any of the planets and therefore does not gravitationally perturb the circular orbit of P ₂. Without loss of generality, we view S to consist of P ₁ and the planet P ₂, moving around P ₁ in a circular orbit of radius Δ<500 AU. The motion of P ₀ within this system is the classical three-dimensional restricted three-body problem, hereafter RP3D. If P ₀ is constrained to move in the plane of motion of P ₂ and P ₁, we have the planar circular restricted three-body problem, hereafter RP2D. The differential equations for RP3D are well known (see Belbruno, 2004).

Because we are interested in the possibility of lithopanspermia to and from the Solar System, we have assumed in the present study that P ₂ is a Jupiter-mass planet. However, recent observational results by the Kepler mission indicate that planetary systems with several Neptune-mass planets may be more common than jovian systems (Batalha et al., personal communication). The weak transfer mechanism described in this paper could also be applied in the low-mass planet systems and in some regards enhanced. We leave the study of these cases for future work.

2.2. Weak capture and weak escape

A convenient way to define the capture of P ₀ with respect to P ₁ or P ₂ in RP3D is by using the concept of “ballistic capture.” The two-body Kepler energy (E_k ) of P ₀ with respect to one of the bodies P_k (k=1, 2) is \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}\begin{align*}E_k = \frac { 1 } { 2 } v_k^2 - - \frac { m_k} { r_k } \tag { 1 } \end{align*}\end{document}

where v_k is the velocity magnitude of P ₀ relative to P_k , and r_k is the distance of P ₀ from P_k . The Kepler energy is a function of the trajectory, E_k (φ(t)), where φ(t)=(r(t), v(t)) is the solution of RP3D for the trajectory of P ₀ in inertial coordinates, t is time, r=(r ₁, r ₂, r ₃) is the position vector from the center of the inertial coordinate system to P ₀ and v=(v ₁, v ₂, v ₃) is the velocity vector of P ₀. Ballistic capture takes place when E_k <0, while ballistic escape occurs at the transition from E_k =0 and E_k >0. We are interested when P ₀ changes from hyperbolic motion with respect to P ₂ to ballistic capture. These trajectories are referred to as “weak capture,” and Belbruno (2004; Belbruno et al., 2010) showed that they are generally unstable and chaotic. The region around P ₂ where weak capture occurs in position-velocity space is called a weak stability boundary (WSBa). This region around P ₂ (WSB(P ₂)) results from the gravitational perturbation of P ₁. It can be viewed as a location where a particle is tenuously and temporarily captured by P ₂; the particle will first move around P ₂ for a short time with negative Kepler energy that approaches zero, and then, increasing above zero, it will yield to hyperbolic escape. Ballistic escape from P ₂ is referred to as “weak escape” because it occurs near WSB(P ₂).

As is evident by numerical integration of P ₀ around P ₂, weak capture generally occurs for relatively short time spans. For example, in the case of the Earth-Moon system [where μ=m ₂/(m ₁+m ₂)=0.012], weak capture around the Moon occurs for time spans of days or weeks; in the case of the Sun-Jupiter system, the timescale would be months to a few years (Belbruno and Marsden, 1997; Belbruno, 2007).

In our solar system, the existence of the weak stability boundary and the viability of weak capture was demonstrated by the Japanese spacecraft Hiten; by using a trajectory designed by Belbruno (2007; Belbruno and Miller, 1993), Hiten was captured into an orbit around the Moon in 1991 without the use of rockets to slow down. Weak capture at the Moon was also achieved in 2004 by the ESA spacecraft SMART1 (Racca, 2003; Belbruno, 2007). In another application, weak escape from Earth's L ₄ (or L ₅) Lagrange point was invoked to suggest a low-energy transfer to Earth for the hypothetical Mars-sized impactor that is thought to have triggered the “giant impact” origin of the Moon (Belbruno and Gott, 2005).

2.3. Parabolic motion and low velocity escape from S

In the absence of P ₂ in the RP2D, a parabolic orbit Q(t) for P ₀ around P ₁ is a planar two-body Keplerian parabolic trajectory. When P ₂ is considered, there is a “chaotic layer” in Q-space near the parabolic trajectory that consists of infinitely many parabolic and near-parabolic trajectories (Xia, 1992; Belbruno, 2004). This layer has a positive measure and a two-dimensional transversal slice in four-dimensional position, and velocity space yields a fractal structure. This layer is obtained when a parabolic trajectory is approximately at its periapsis with respect to P ₁ and where it also has an approximate periapsis passage with respect to P ₂, slightly beyond P ₂ (i.e., at a radial distance from P ₁ slightly larger than Δ). The periapsis passage near P ₂ is done in such a way so that P ₀ is slightly hyperbolic with respect to P ₂ [approximately in WSB(P ₂)]. This imparts a small gravity assist to P ₀. In this case, there are infinitely many possible trajectories near the original parabolic trajectory, some of which do not move out infinitely far from P ₁ and eventually fall back toward P ₁ for another possible flyby of P ₂. Other near-parabolic trajectories will escape P ₁ on hyperbolic trajectories with respect to P ₁ with a positive escape velocity, σ. Because these hyperbolic trajectories lie near parabolic trajectories, σ will be small. We refer to these as “low velocity escape trajectories” or “minimal energy escape trajectories.” Within the chaotic layer mentioned above exist infinitely many parabolic trajectories that hyperbolically escape P ₁ with small escape velocity σ. The distance R=R _esc(m ₁) from P ₁ at which P ₀ achieves this escape velocity can be obtained by the formula σ=(2Gm ₁/R)^1/2, where G is the gravitational constant. The fact that these escape trajectories are nearly parabolic and have very small escape velocities implies that their Kepler energy with respect to P ₁ is nearly zero, for sufficiently large R.

In summary, in the circular restricted three-body problem (P ₀, P ₁, P ₂), there is a chaotic layer that replaces the regular parabolic trajectories of the two-body problem (P ₀, P ₁); weak transfer can take place in this chaotic layer because trajectories in this set have low escape velocities.

2.4. Model for the minimal energy transfer of solid material within a cluster

To achieve a minimal energy transfer of the meteoroid P ₀ from S to another system S*, consider that the system S is embedded in a star cluster; this will introduce additional gravitational perturbations on P ₀ which will interact with the gravitational field of P ₁ and form a weak stability boundary around P ₁ where the motion of P ₀ will be chaotic. Because the transfer mechanism requires low relative velocities, we consider an open star cluster with a low-velocity dispersion, with relative stellar velocities U≈1 km/s [for comparison, older globular clusters have stellar velocity dispersions of several tens of kilometers per second—Binney and Tremaine (1988), Meylan and Heggie (1997)]. After the cluster disperses, the relative distances and relative velocities between a star and its closet neighbors increase. For example, the Sun's current closest neighbor α-Centauri (not necessarily a cluster sibling) is now 2.6·10⁵ AU away (4.28 pc) and moving with a relative velocity of 6 km/s. The latter is significantly higher than the ∼1 km/s required for the weak transfer mechanism. In this paper, we consider the transfer of solid material between stars in a cluster before the cluster starts to disperse.

Imagine that P ₀ makes a minimal energy escape from P ₁ in the P ₁, P ₂ orbital plane by making a slightly hyperbolic flyby of P ₂ at WSB(P ₂) (where the latter is the weak stability boundary around P ₂ created by the gravitational perturbation of P ₁); P ₀ moves away from P ₁ with a escape velocity σ near zero. P ₀ then gets to WSB(P ₁), the weak stability boundary of P ₁ caused by the gravitational perturbation of the other N−1 stars in the cluster (where CS_{N −1} represents this set of N−1 stars, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\{ P_2 , P_3 , \ldots , P_N \} $$\end{document} ); this boundary is located at a distance R _esc(m ₁) from P ₁, at which the combined gravitational force of the stars in the cluster is comparable to the gravitational force of P ₁. As described previously, the motion in this region is chaotic and lies in the transition between capture and escape from P ₁ b. Although WSB(P ₁) is a complicated fractal set, nonspherical in shape, the sphere of radius R _esc(m ₁) around the central star lies approximately in the part of WSB(P ₁) where the motion is least stable and the trajectories are slightly hyperbolic with respect to P ₁ (García and Gómez, 2007; Belbruno et al., 2010). We refer to trajectories that escape P ₁ from WSB(P ₁) as “weak escape” trajectories (note that low-velocity escape does not imply weak escape unless it occurs at a weak stability boundary). The structure of the set of near-parabolic trajectories that escape P ₁ by flyby of P ₂ yields infinitely many such trajectories. In particular, this implies that, for any point p on the circle of radius R _esc(m ₁) around P ₁, there is a nearly parabolic trajectory that will pass by p with velocity σ, weakly escaping P ₁. This is illustrated in Fig. 1 (see also Fig. 1 in Moro-Martín and Malhotra, 2005).

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FIG. 1.

Schematic representation of the weak transfer process. It consists of a meteoroid weakly escaping from a planetary system and its subsequent weak capture by a neighboring planetary system in the stellar cluster. The meteoroid P ₀ flies by planet P ₂ and weakly escapes the central star P ₁ at a distance R _esc(m ₁). R _esc(m ₁) is approximately where the weak stability boundary of P ₁ is located, caused by the gravitational perturbation of the other N−1 stars in the cluster. The motion in this region is chaotic and lies in the transition between capture and escape from P ₁. The meteoroid P ₀ is then weakly captured by the neighboring cluster star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} at a distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} , moving to periapsis \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$r_{ \rm p}^*$$\end{document} with respect to \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} (motion projected onto a plane). The numerical computation pieces together trajectories from flyby of P ₂ to the distance R _esc(m ₁) from P ₁ using RP3D and from R _esc(m ₁) to the distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} . It has been demonstrated that the piecing together of solutions of two different three-body problems at WSB(P ₁) and WSB( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} ) can be done in a well-defined and smooth manner (see Belbruno, 2004; Marsden and Ross, 2006).

We now compute the trajectory of P ₀ from the distance WSB(P ₁) until it encounters another planetary system S* centered on star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} of mass \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} . This is done by using a different set of differential equations; during this motion the trajectory of P ₀ is relatively undisturbed. The meteoroid is then weakly captured at WSB( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} ) at the distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R^* = R_{ \rm cap} ( m_1^* )$$\end{document} from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} . Figure 1 gives a schematic representation of this process. After P ₀ is weakly captured at the distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , say at time t=T, it moves toward \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} for t>T. Analogous to system S, we assume that system S* has a dominant planet \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_2^*$$\end{document} orbiting \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} in a circular orbit at a radial distance Δ*. As t increases from T, P ₀ will move to a periapsis distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$r_p^*$$\end{document} from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$r_p^*$$\end{document} can range from 0 (collision with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} ) to \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} . However, unlike the escape of P ₀ from P ₁, where P ₀ had a periapsis near the location of P ₂ and in the same orbital plane of P ₂, the approach of P ₀ into the S* system is three-dimensional and need not be restricted to lie in the \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_2^*$$\end{document} orbital plane nor have a periapsis of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} near \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_2^*$$\end{document} . The resulting motion of P ₀ for t>T will, in general, be complicated and a priori not known. There is no way to predict whether P ₀ will pass near or collide with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_2^*$$\end{document} without doing numerical simulations that are outside the scope of this paper. For this reason, we conclude our search at t=T, when P ₀ is weakly captured at \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , and calculate the probability of this weak capture to take place. An order-of-magnitude estimate of the weak capture probability is derived in Section 3 based on geometrical considerations, while in Section 4 we calculate the probability numerically, using Monte Carlo simulations.

3.1. Approximate location of the weak stability boundary

We calculate R _esc(m ₁) and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} as a function of the stellar mass. For weak escape to take place, the velocity σ of the meteoroid at the distance R _esc(m ₁) from the star P ₁ must be sufficiently small. Because we are considering slow transfer within an open cluster with a characteristic dispersion velocity U≈1 km/s, we require that σ is significantly smaller than U, that is, of the order of 0.1 km/s. This is much smaller than the nominal values of several kilometers per second used by the Monte Carlo methods of Melosh (2003) and Adams and Spergel (2005).

To place the above choice for σ in context, we considered the velocity distribution of weakly escaping test particles from the Solar System, using a three-body problem between the Sun (P ₁), Jupiter (P ₂), and a massless particle (P ₀). To be consistent with our framework, we modeled this, using RP2D where the test particle moves in the same plane of motion as Jupiter, which we assumed to be in a circular orbit at 5 AU. The trajectory of the test particle is numerically integrated by a standard Runge-Kutta scheme of order six and numerical accuracy of 10⁻⁸ in the scaled coordinates. The initial condition of the test particle is an elliptic trajectory very close to parabolic with periapsis distance r _p=5 AU and apoapsis distance r _a=40,000 AU. (Note that such orbits are not dissimilar to those of known long-period comets in the Solar System.) For each numerical integration, we assume that Jupiter is at a random point in its orbit when the test particle starts from apoapsis at 40,000 AU and falls toward P ₁. We record the velocity of the test particle at the time it escapes with respect to the Sun (i.e., when the Kepler energy with respect to the Sun is positive) after performing a sufficient number of Jupiter flybys. Figure 2 shows the distribution of v _∞: out of 670 cases, 58% have v _∞≤0.1 km/s, and 79% have v _∞≤0.3 km/s. Based on these results, we assume the velocity σ of the meteoroid at the distance R _esc(m ₁) from the star to be in the range 0.1–0.3 km/s.

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FIG. 2.

Velocity distribution of 670 low-velocity test particles escaping from the Solar System. The numerical model is the planar circular restricted three-body problem of the Sun, Jupiter, and a massless particle.

For a given σ, the location of the part of the weak stability boundary that gives the most unstable motion is approximately given by the sphere of radius R _esc(m ₁)=2Gm ₁/σ ², where m ₁ is the mass of the star and σ is in the range 0.1–0.3 km/s (see Fig. 3). Beyond this boundary, when P ₀ weakly escapes P ₁, we assume that the meteoroid will move at a nearly constant velocity σ with respect to the star.

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FIG. 3.

The diagonal solid lines plot the spatial scale of the weak stability boundary for escape, R _esc(m ₁)=2Gm ₁/σ ², as a function of stellar mass, m ₁, for two values of the escape velocity, σ=0.1 and 0.3 km/s. The diagonal dashed line plots the weak stability boundary for capture, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* ) = 2Gm_1^* / U^2$$\end{document} , for the average relative velocity of stars in the cluster, U=1 km/s. The horizontal dashed-dotted lines indicate the spatial scale of star clusters, R _cluster, while the horizontal dotted lines indicate the mean interstellar distance, D, for clusters consisting of N=100, N=1000, and N=4300 members.

Slow transfer to a neighboring planetary system enables the meteoroid to arrive at the distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} with a relative velocity with respect to the target star ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} ) that is similar or smaller than its parabolic escape velocity at that distance, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\sigma^* = ( 2Gm_1^* / R_{ \rm cap} ( m_1^* ) ) ^{1 / 2}$$\end{document} , where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} is the mass of the target star; if its velocity is higher than σ*, it will not be captured and will fly by. Since the relative velocity between stars in the cluster is U≈1 km/s, the meteoroid that weakly escaped from star P ₁ moves toward the target star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} with velocity U±σ (see Fig. 1). Because σ is small relative to U and can be neglected, the relative velocity of the meteoroid with respect to the target star is≈U. Therefore, weak capture can occur at the distance at which σ*≈U≈1 km/s, i.e., \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* ) = 2Gm_1^* / U^2$$\end{document} . Figure 3 shows R _esc(m ₁) and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} as a function of the stellar mass. The horizontal lines indicate the range of cluster sizes (dashed-dotted lines) and mean interstellar distances (dotted lines) for clusters consisting of N=100, 1000, and 4300 members, respectively; clusters with this range of sizes are the birthplaces of a large fraction of stars in the Galaxy (Lada and Lada, 2003). The radius of the cluster depends on the number of stars and is given by R _cluster=1pc(N/300)^1/2 (Adams, 2010). For N=100, 1000, and 4300, R _cluster is approximately 2.1·10⁵ AU, 6.5·10⁵ AU, and 1.3·10⁶ AU, respectively. We can estimate the average interstellar distance within a cluster, D=n ^−1/3, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$n = 3N / 4 \pi R_{ \rm cluster}^3$$\end{document} is the average number density of stars in the cluster. D is approximately 7·10⁴ AU, 10⁵ AU, and 1.2·10⁵ AU for a cluster with N=100, 1000, and 4300 members, respectively. The choice of N=4300 is explained in Section 4.2.

It is of interest to compare the results in Fig. 3 to those of Melosh (2003). The latter considers the exchange of meteoroids between field stars by using hyperbolic trajectories; it estimates a cross section of 0.025 AU² for meteoroid capture by a planetary system with a Jupiter-mass planet located at 5 AU. Under the scenario considered in the present work, the order-of-magnitude estimate for the weak capture cross section would be \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\pi \cdot R_{ \rm cap}^2$$\end{document} ; for a solar-mass star, R _cap ∼ 2·10³ AU (Fig. 3) so that the weak capture cross section is ∼10⁷ AU² [i.e., many orders of magnitude larger than the Melosh (2003) estimate]. This indicates that weak escape and capture within an open cluster can enhance drastically the probability of transfer. A recent study in which enhanced capture probability within a stellar cluster has been invoked is that of Levison et al. (2010). They modeled the exchange of comets within a stellar cluster, assuming that each star is surrounded by a large disk of comets with q=30 AU and a=1000–5000 AU, and with a population similar to that of the Sun; they concluded that up to 90% of comets in the Oort Cloud may have an extrasolar origin.

3.2. Constraints on stellar masses for weak transfer

From Fig. 3, we can set constraints on the stellar mass m ₁ that could allow weak escape from P ₁ to take place. The idea is simple: if for a given σ (within the range 0.1–0.3 km/s, see Section 3.1), R _esc(m ₁)<D, that is, the weak escape boundary is located within the distance of the next neighboring star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , then weak transfer is possible because at the time the meteoroid passes near the star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} its velocity is similar to the mean stellar velocity dispersion (U ∼ 1 km/s), and there is a significant probability of capture (which we quantify in Section 3.3). Conversely, weak transfer by the process schematically represented in Fig. 1 is much less likely to take place if R _esc(m ₁)>D. This is because, at the time the meteoroid passes near a neighboring star, its velocity is too high, and as a consequence it will likely fly by. Figure 3 shows that, for σ=0.1 km/s, the condition R _esc(m ₁)<D for weak escape is satisfied for m ₁<0.4 M _Sun, m ₁<0.57 M _Sun, and m ₁<0.68 M _Sun for clusters with N=100, 1000, and 4300 members, respectively. If we consider the higher, but still acceptable, value σ=0.3 km/s, Fig. 3 shows that the stellar mass limits for weak escape are m ₁<3.5 M _Sun, m ₁<5.1 M _Sun, and m ₁<6.1 M _Sun for clusters with N=100, 1000, and 4300 members, respectively. A meteoroid escaping nearly parabolically from any star with a mass larger than these limits will achieve a velocity of σ<0.1–0.3 km/s only at a distance larger than the mean interstellar distance in the cluster; weak transfer is not likely to take place under such conditions.

Of particular interest is the case of the Solar System as the source of the meteoroids. It has been estimated that the Sun's birth cluster consisted of N=4300±2800 members (Adams, 2010). For a 1 M _Sun star in such a cluster, we find that the parabolic escape velocity at the mean interstellar distance of D=1.2·10⁵ AU is σ=(2Gm ₁/D)^1/2=0.12 km/s. Because this value of σ lies within the range of values of interest for weak escape, we conclude that the meteoroids originating in the early Solar System could have met the conditions for weak escape in the Sun's birth cluster.

We have established the range of stellar masses that could in principle allow weak escape to take place. We now estimate the probability of weak capture by a neighboring planetary system. Whether or not the transfer of meteoroids from one star to any star of a given mass \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} takes place will depend on the relative capture cross section of the target star and on the number of potential targets. We discuss these factors below.

3.3. Relative capture cross section of the target star

In the case of weak transfer, assuming that the meteoroids escape isotropically from P ₁, the relative capture cross section of the target star would be given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$C_S = G_{ \rm f} ( R_{ \rm cap} ( m_1^* ) / D ) ^2$$\end{document} , where D is the distance between the two stars, R _cap is the weak capture boundary (illustrated in Fig. 1 and estimated in Fig. 3), and the factor G _f accounts for gravitational focusing. The assumption of isotropy is not justified in the framework of the restricted three-body problem, but it is expected in the presence of the cluster stars and of the galactic potential that can isotropize the escaping meteoroids' orbits, in analogy with the isotropization of the Oort Cloud of comets (Dones et al., 2004).

The gravitational focusing factor is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}\begin{align*}G_f = 1 + \Big( \frac { v_ { \rm esc } } { v_ { \infty } } \Big) ^2 \tag { 2 } \end{align*}\end{document}

where v _∞ is the velocity at infinity and v _esc is the escape velocity at the distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} . In the case of weak capture, the term G _f increases the cross section due to enhanced gravitational focusing. For example, as we saw previously in the case of the Sun and Jupiter, v _∞≈0.1–0.3 km/s, while at the distance R _cap=40,000 AU, v _esc≈0.2 km/s. In this case, G _f≈2, which doubles the capture cross section. This situation occurs in our capture methodology. In the formulation for determining \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} , the meteoroid has an approximate relative approach velocity to the target star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} of roughly U=1 km/s, which represents v _∞. However, R _cap is determined so that this same value of velocity is taken for v _esc from the target star. That is, we are assuming v _∞≈v _esc≈1 km/s. This implies also that G _f=2. This is a conservative estimate and does not make use of the nature of weak capture dynamics. In fact, the value of G _f can be substantially increased, as we demonstrate below, if at a given value of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R_{ \rm cap} ( m_1^* )$$\end{document} , v _∞ is smaller than the approximate value of U=1 km/s, while v _esc≈U; in this case, v _∞→0, which implies G _f→∞ .

3.4. Number of potential targets

We denote \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_{ \rm IMF} ( m_1^* )$$\end{document} as the probability of finding a star of mass \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} in the star cluster. This probability is referred to as the initial mass function (IMF), and it can be inferred from observations of stellar clusters. It is found that a wide range of clusters, varying from large clusters like the Trapezium to smaller clusters like Taurus, as well as older field stars, show very similar distributions of stellar masses down to the hydrogen burning limit at ∼0.1 M _Sun (Lada and Lada, 2003, and references therein). The IMF can be characterized by the broken power-law, ξ(M)=ξ₁M^−2.2 for 0.6–100 M _Sun, and ξ(M)=ξ₂ M ^−1.1 for 0.1–0.6 M _Sun, where ξ(M)dM is the number of stars with mass (M, M+dM). There is a steep decline into the substellar brown dwarf regime and a possible second peak, but we will ignore objects below the hydrogen burning limit. The resulting average stellar mass is ∼0.88 M _Sun. To calculate \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P + { \rm IMF} ( m_1^* )$$\end{document} (square symbols in Fig. 4), we use a logarithmic binning of masses with d(logM)=0.1 and normalize the distribution to unity, which gives ξ₁=0.19 and ξ₂=0.34.

OPEN IN VIEWER

FIG. 4.

The diagonal dashed lines plot the probability for weak capture of meteoroids, G _f (R _cap/D)², as a function of the target stellar mass ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} ), for three values of the number of stars in a star cluster. For G _f, we take the conservative value of 2. The solid lines plot the probability for weak capture of meteoroids by a neighbor star of stellar mass equal to that of the source star ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1 = m_1^*$$\end{document} ). The symbols indicate the stellar mass distribution.

An upper limit to the probability that a meteoroid escaping from a star of mass m ₁ (within the range described in Section 3.2) will get captured by a neighboring star of mass \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} is approximately given by the relative capture cross section of its weak stability boundary, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$G_{ \rm f} ( R_{ \rm cap} ( m_1^* ) / D_{m^*1 - m1} ) ^2$$\end{document} , where R _cap is the radius of the weak stability boundary for capture of the target star, and D_{m *1−m1} is the average distance between two stars of masses \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} and m ₁, respectively. If we assume that this distance is the average interstellar distance D estimated in Section 3.1, an upper limit to the weak capture probability for a solar-type star is approximately 1.3·10⁻³, 6.5·10^−4, and 4.5·10⁻⁴, for a cluster of N=100, 1000, and 4300 members, respectively (see dashed line of Fig. 4). However, the distance to a star of a given stellar mass will not be D necessarily, but it will depend on the distribution of stellar masses in the cluster (the IMF). The simplest case is when both stars have equal masses \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^* = m_1$$\end{document} . In this case, the average interstellar distance would be D_{m 1}∼(1/n_{m 1})^1/3, where n_{m 1} is the average number density of stars with mass m ₁, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$n_{m1} = 3N_{m1} / ( 4 \pi R_{ \rm cluster}^3 )$$\end{document} , and N_{m 1} is the total number of stars in the cluster with mass m ₁, N_{m 1}=N·P _IMF(m ₁). Because n_{m 1}=n·P _IMF(m ₁), we get that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$D_{m1} \,\sim\, D \cdot ( P_{ \rm IMF} ( m_1^* ) ) ^{ - 1 / 3}$$\end{document} . This means that the transfer probability between two stars of equal mass is given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$G_{ \rm f} ( R_{ \rm cap} ( m_1^* ) / D ) ^2 \cdot ( P_{ \rm IMF} ( m_1^* ) ) ^{2 / 3}$$\end{document} , where D is the average distance between any two stars in the cluster (regardless of their mass). As we have discussed above, the value of the focusing factor G _f can be very large. A conservative estimate of the capture probability can be done by assuming G _f=2. Figure 4 shows that the capture probability between two planetary systems with solar-type central stars \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$( m_1^* = m_1 = 1 M_{ \rm Sun} )$$\end{document} are 1.9·10⁻⁴, 8.1·10⁻⁵, and 5.6·10⁻⁵, for a cluster of N=100, 1000, and 4300 members, respectively.

4.1. Modeling procedure

To model the motion of P ₀ from the distance R _esc(m ₁) from P ₁ to S* and with initial velocity σ, we consider a general Newtonian three-dimensional three-body problem that is perturbed by the gravitational force of the stellar cluster. This model gives the motion of P ₀, P ₁, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} in an inertial coordinate system (r ₁, r ₂, r ₃). We assume the gravitational perturbing force of the cluster is obtained from a spherically symmetric Hernquist potential \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}\begin{align*}U_C ( r ) = \frac { GM } { r + a } \tag { 3 } \end{align*}\end{document}

(Hernquist, 1990), where M is the total cluster mass, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$r^2 = r_1^2 + r_2^2 + r_3^2$$\end{document} , and a is the cluster scale length, which is approximately the radius of the cluster, R _cluster. The center of mass of the cluster is at the origin, r =(0, 0, 0).

The system of differential equations modeling the motion of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_0 , P_1 , P_1^*$$\end{document} , of mass \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_0 , m_1 , m_1^*$$\end{document} , respectively, is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}\begin{align*}\dot { \bf r } _k = { \bf v } _k , \quad \dot { \bf v } _k = m_k^ { - 1 } \frac { \partial V } { \partial { \bf r } _k } \tag { 4 } \end{align*}\end{document}

where r_k =(r_{k 1}, r_{k 2}, r_{k 3}) and v_k =(v_{k 1}, v_{k 2}, v_{k 3}) are the position and velocity vectors, respectively, of the k ^th particle, P_k (with k=0, 1, 2; P ₂ represents \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , and m ₂ represents \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} ). The gravitational potential is V=U+m_k U_C (r_k ), where r_k is the distance of P_k to the origin, r_k =| r_k |, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}\begin{align*}U = \mathop\sum_ { j = 0 \atop j \neq k } ^2 \frac { Gm_j m_k } { r_ { j / k } } \tag { 5 } \end{align*}\end{document}

U=U( r ₁ , r ₂ , r ₃ ) is a function of nine variables r_kj , j=1, 2, 3, and \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document}\begin{align*} \frac { \partial V } { \partial { \bf r } _k }= \left( \frac { \partial V } { \partial r_ { k1 } } , \frac { \partial V } { \partial r_ { k2 } } , \frac { \partial V } { \partial r_ { k3 } } \right) \tag { 6 } \end{align*}\end{document}

where r_j/k is the distance between P_j and P_k , r_j/k =| r_j − r_k |. m ₀ is approximately 0 relative to m ₁, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^*$$\end{document} .

In the Monte Carlo model, the trajectory of motion for P ₀ is determined from (4) by providing its initial position at the distance R=R _esc(m ₁) from P ₁ at t=0. The initial velocity of P ₀ with respect to P ₁ at the distance R is chosen from a distribution of weak escape velocities (see Fig. 2). P ₀ then weakly escapes P ₁. The initial positions and velocities of P ₁ and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} are also given at t=0. Their initial separation distance depends on the cluster properties and is a function of the number of stars in the cluster N. After the initial positions and velocities of P ₀, P ₁, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} are given, the trajectory of P ₀ is propagated for t>0. We search for the condition of the first weak capture of P ₀ with respect to \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , given by a negative value of the Kepler energy of P ₀ with respect to \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^* , E_1^* < 0$$\end{document} . This implies that for P ₀, v _∞→0, and therefore G _f→∞ . This occurs at the weak stability boundary of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , WSB( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} ), at a distance \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$R^* = R_{ \rm cap} ( m_1^* )$$\end{document} from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} . This stability boundary is formed around \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} due to the resultant gravitational perturbation of the N−1 remaining stars of the cluster. We are interested in cases where the time of propagation T is smaller than the cluster dispersal timescale, which is a function of the number of stars in the cluster, N. Under the conditions described above and using a Monte Carlo approach, we sample millions of trajectories of P ₀ weakly escaping P ₁ using Runge-Kutta-Nyström 12^th/10^th order, variable step, symplectic integrator (Dormand et al., 1987). The Monte Carlo method calculates the number of these particles to be weakly captured by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} , that is, the probability of weak transfer.

4.2. Cluster properties

We adopt those cluster properties thought to be representative of the Sun's birth cluster. These properties are inferred from a wide range of physical considerations, including the effect of supernova explosions on the enrichment of short-lived radioactive isotopes in the solar nebula, protoplanetary disk truncation due to photoevaporation from nearby hot stars, and the dynamical disruption of planetary orbits due to close encounters with cluster stars. The observed Solar System properties that depend on the solar birth environment (such as the evidence of short-lived radionuclides in meteorites and the dynamical properties of the outer Solar System planets and Kuiper Belt) led Adams (2010) to conclude that the Sun was born in a moderately large cluster with N=1,000–10,000 and <N>=4300±2800 members, similar to the Trapezium cluster in Orion. For reference, Adams (2010) noted that approximately 50% of stars are born in systems with N≥1000, but for ∼80% of these stars the clusters dissolve quickly after 10 Myr; only ∼10% of the total number of stars would be born in open clusters with lifetimes of the order of 100–500 Myr.

Using the IMF in Section 3.4, we estimate that the total cluster mass is M _cluster=(3784±2500) M _Sun. Following Adams (2010), the properties of such a cluster would be as follows. The scale length of the cluster is approximately its size, given by a=R _cluster=1pc(N/300)^1/2=3.78±1.5 pc. With the stellar number density, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$n = 3N / ( 4 \pi R_{ \rm cluster}^3 ) = 19 \ { \rm pc}^{ - 3}$$\end{document} , the average interstellar distance is D=n ^−1/3=0.375 pc. The typical expected distance of the Sun to the center of mass of the cluster would be d _cm=2R _cluster/3=2.52±1 pc (using a radial profile for stellar density consistent with a Hernquist potential). The cluster lifetime is approximately T=2.3(M _cluster/ M _Sun)^0.6 Myr=(135–535) Myr, for N=1,000–10,000.

4.3. Modeling assumptions

For the Monte Carlo simulations, we adopted the average values of the Solar System birth cluster: N=4300, D=0.375 pc, a=3.78 pc, d _cm=2.52 pc, and a cluster lifetime T=322.5 Myr. We also made the following assumptions:

• For the duration of the simulation, the cluster size and therefore the average distance between the stars, D, are kept constant (i.e., we make the simplifying assumption that the cluster disperses instantly at the end of the simulation, at a time equal to the average cluster age).

In summary, the Monte Carlo simulation is done by randomly choosing values of the parameters v _0/1, r _0/1, φ ₀, θ ₀, θ, |v₁ | and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\mid { \bf v}_{ \bf 1}^* \mid$$\end{document} at t=0 that satisfy \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$v_{0 / 1} \,\in \,Dv_{ \infty} , r_{0 / 1}\, \in \,[ 0.02 , 0.2 ] \ { \rm pc} , \varphi_0 \in [ 0 , 2 \pi ]$$\end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\theta_0 \in [ 0 , \pi ] , \theta \in [ 0 , 2 \pi ] , \mid { \bf v}_{ \bf 1} \mid \leq 2 \ { \rm km / s} , \mid { \bf v}_{ \bf 1}^* \mid \leq 2 \ { \rm km / s}$$\end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\mid { \bf v}_{ \bf 1} - { \bf v}_{ \bf 1}^* \mid = 1 \ { \rm km / s}$$\end{document} , respectively. The Monte Carlo algorithm searches for the condition \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$E_1^* < 0$$\end{document} for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$t \in [ 0 , T_{ \max} ]$$\end{document} Myr.

4.4. Results from the Monte Carlo simulations

The Monte Carlo simulations study the capture probability within the cluster described in Section 4.2 for the following three different combinations of stellar masses: m ₁=1 M _Sun, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^* = 1 \ M_{ \rm Sun}$$\end{document} (Case 1); m ₁=1 M _Sun, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^* = 0.5 \ M_{ \rm Sun}$$\end{document} (Case 2); and m ₁=0.5 M _Sun, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$M_{ \rm Sun} , \,m_1^* = 1 \ M_{ \rm Sun}$$\end{document} (Case 3). These simulations explore the weak transfer of material between two solar-type stars and between a solar-type star and a star half its mass, the interest of the latter being that low-mass stars are the most abundant in the Galaxy, with ∼64% and 72% having mass<0.4 M _Sun and 0.6 M _Sun, respectively. For each case, the Monte Carlo simulation ran 5 million trajectories, a sufficient number so that the randomization of the initial values produces distributions of the parameters that span their respective ranges. The resulting distributions are discussed in Appendix B.

For Case 1 (weak transfer of meteoroids between two solar-type stars), the Monte Carlo simulations give a revised estimate of the weak capture probability of 1.5·10⁻³ (see Table 1). Note that the Monte Carlo simulations assume that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} with \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^* = 1 \ M_{ \rm Sun}$$\end{document} is at the average interstellar distance D=n ^−1/3 (i.e., without regard to the distribution of stellar masses in the cluster). The Monte Carlo–derived weak capture probability is approximately a factor of 3 larger than the order-of-magnitude estimate in Section 3.4, which was approximated by the relative capture cross section of the weak stability boundary of the target star, given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$G_{ \rm f} ( R_{ \rm cap} ( m_1^* ) / D ) ^2 = 4.5 \cdot 10^{ - 4}$$\end{document} , for \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1^* = 1$$\end{document} , N=4300 and using a conservative gravitational focusing factor of G _f=2. This increase in the weak capture probability estimated by the Monte Carlo simulations is likely due to the larger focusing factors that result from the nature of weak capture. Note that both estimates assume that the meteoroids are initially on weakly escaping trajectories from P ₁. For Cases 2 and 3, the Monte Carlo simulations give a weak capture probability of 0.5·10⁻³ and 1.2·10⁻³, respectively (see Table 1).

6.1. Summary of the weak transfer mechanism

We have explored a mechanism that allows the transfer of solid material between two planetary systems embedded in a cluster. This mechanism is based on the chaotic dynamics of the restricted three-body model of the meteoroid, the star, and the most massive planet in the planetary system (P ₀, P ₁, P ₂), in which a chaotic layer replaces the regular parabolic trajectories of the two-body problem of (P ₀, P ₁). Similarly, there is a chaotic layer around the target star (assumed to harbor a planetary system). Weak transfer takes place within these chaotic layers because the trajectories have low escape velocities and therefore the capture probability is enhanced. We have applied this mechanism to the problem of planetesimal transfer between planetary systems in an open star cluster, where the relative stellar velocities are sufficiently low (∼1 km/s) to allow weak escape and capture. We found that weak escape and capture within an open cluster can enhance drastically the probability of transfer compared to the scenario described by Melosh (2003) and corresponding to the exchange of meteoroids between field stars using hyperbolic trajectories: while Melosh (2003) estimated a cross section of 0.025 AU² (for capture by a planetary system with a Jupiter-mass planet at 5 AU), an order-of-magnitude estimate for weak capture cross section would be \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$\pi \cdot R_{ \rm cap}^2 \sim 10^7 { \rm AU}^2$$\end{document} (i.e., many orders of magnitude larger).

To obtain quantitative estimates, we adopted the average cluster properties inferred for the solar birth cluster (Adams, 2010), with N=4300 members, a total mass of M _cluster=3784 M _Sun (using an average stellar mass of 0.88 M _Sun resulting from the IMF), and a cluster scale length of a=1pc(N/300)^1/2=3.78 pc. Such a cluster is expected to have a lifetime of 2.3 (M _cluster/M _Sun)^0.6 Myr=322.5 Myr (ranging from 135 to 535 Myr, for N=1,000–10,000).

With the aid of Monte Carlo simulations, we estimated the probability of weak capture for meteoroids that have weakly escaped their original planetary system and are weakly captured at a neighboring system. We consider three cases where the source P ₁ and the target star \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$P_1^*$$\end{document} have masses of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1 = 1 , m_1^* = 1$$\end{document} (Case 1), of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1 = 1 , m_1^* = 0.5$$\end{document} (Case 2), and of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland, xspace} \usepackage{amsmath, amsxtra} \pagestyle{empty} \DeclareMathSizes {10} {9} {7} {6} \begin{document}$$m_1 = 0.5 , m_1^* = 1$$\end{document} (Case 3). The resulting weak capture probabilities are 0.15%, 0.05%, and 0.12%, respectively. This capture probability is much larger than the capture probabilities obtained in previous studies; for example, Adams and Spergel (2005) found capture probabilities of ∼10⁻⁶ for mean ejection speeds of ∼5 km/s, typical of hyperbolic ejecta of the Solar System.

Adopting parameters from the minimum mass solar nebula (Weidenschilling, 1977; Hayashi, 1981), considering a range of planetesimal size distributions derived from observations of asteroids and Kuiper Belt objects and theoretical coagulation models (Bernstein et al., 2004; Bottke et al., 2005; Kenyon et al., 2008), and taking into account the results from Oort Cloud formation models (Brasser et al., 2012) for the fraction of planetesimals that are subject to weak escape from the early Solar System, we estimated the number of meteoroids that may have been delivered to the weak stability boundary of the Solar System over the lifetime of the Sun's birth cluster. Using this number and the probability of the weak capture of meteoroids on weakly escaping orbits, we calculated the number of weak transfer events from the early Solar System to the nearest star in the cluster, assuming it is a solar type and harbors a planetary system. This number depends strongly on the adopted planetesimal size distribution. We found that, for the cases where the power-law size distribution at the small sizes (dN/dD∝D^{−q 2} for D<D ⁰) has index q ₂=3.5, the expected number of weak transfer events between two solar-type stars is of the order of 10¹⁴ to 3·10¹⁶; for a shallow size distribution (q ²=1.1), the number is of the order of 10⁴.

We conclude that solid material could have been transferred in significant quantity from the Solar System to other solar-type stars in its birth cluster via the weak transfer mechanism described here.

6.2. Implications for lithopanspermia

Section 6.1 indicates that the transfer of planetesimals via the weak transfer mechanism described in this paper is likely to be the dominant process for the exchange of solid material among planetary systems in a star cluster. This is of interest for lithopanspermia because if life arose in any of these systems before the cluster dispersed, this mechanism may have allowed the exchange of life-bearing rocks among the planetary systems in the cluster. Within the context of the Solar System's formation and dynamical history, in this section we discuss how much material originating from Earth's crust may have been available for weak transfer (Section 6.2.1) and, given the time constraints imposed by the weak transfer mechanism, whether there was a “window of opportunity” for lithopanspermia to take place (Section 6.2.2).