On the Capture of Interstellar Objects by our Solar System
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On the Capture of Interstellar Objects by our Solar System
Kevin J. Napier , Fred C. Adams ,
1, 2 and Konstantin Batygin Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA
ABSTRACTMotivated by recent visits from interstellar comets, along with continuing discoveries of minor bodiesin orbit of the Sun, this paper studies the capture of objects on initially hyperbolic orbits by our solarsystem. Using an ensemble of ∼
500 million numerical experiments, this work generalizes previoustreatments by calculating the capture cross section as a function of asymptotic speed. The resultingvelocity-dependent cross section can then be convolved with any distribution of relative speeds todetermine the capture rate for incoming bodies. This convolution is carried out for the usual Maxwelliandistribution, as well as the velocity distribution expected for rocky debris ejected from planetarysystems. We also construct an analytic description of the capture process that provides an explanationfor the functional form of the capture cross section in both the high velocity and low velocity limits.
Keywords:
Solar system (1528), Dynamical evolution (421), Small solar system bodies (1469), Kuiperbelt (893), Oort cloud (1157) INTRODUCTIONThe past few years have witnessed the detection of two interstellar bodies passing through the solar system onhyperbolic orbits. The discoveries of the irregular body ‘Oumuamua (Meech et al. 2017) and the comet Borisov(Jewitt & Luu 2019) sparked immediate interest in characterization of these objects and facilitated wide-rangingspeculation regarding the possibility that our solar system is more broadly contaminated by minor bodies of extra-solar origin (e.g., Siraj & Loeb 2019; Namouni & Morais 2020). Although no current evidence indicates that anyspecific objects in the solar system are of extrinsic origin (Morbidelli et al. 2020), the question of whether or not anysuch objects reside in interplanetary or trans-Neptunian space is of considerable interest. Motivated by these issues,this paper reconsiders the capture of external bodies by our solar system. The calculation of the capture cross sectionsis the first step in assessing whether or not the solar system presently contains quasi-permanently trapped interstellarbodies. This treatment also provides constraints on the expected orbits of any such material.The dynamics of the outer solar system represents one of the oldest problems in theoretical astrophysics. Startingmore than two centuries ago, classic studies include the long-term stability of the solar system (Lagrange 1776; Laplace1799-1825), the origin of comets (Laplace 1846), and orbital anomalies that led to the discovery of Neptune (Le Verrier1846; Adams 1846). Over recent decades, the outer solar system has revealed itself to be increasingly complicated,with the discovery of the Kuiper Belt (Luu & Jewitt 2002), dozens of dwarf planets (starting with Brown et al. 2004, orperhaps Tombaugh 1946), high-inclination objects (Becker et al. 2018), and aligned extreme trans-Neptunian objects(Sheppard & Trujillo 2016) that led to the hypothesis of a possible ninth planet (Batygin & Brown 2016; Batyginet al. 2019). The more recent discovery of interstellar objects (Meech et al. 2017; Jewitt & Luu 2019) adds to theintrigue. Both the complex orbital architecture of the solar system and the presence of interloping objects motivatesthis present study. The goal is to determine cross sections for the capture of foreign bodies by the solar system, andto obtain a deeper understanding of the capture process.
Corresponding author: Kevin J. [email protected] a r X i v : . [ a s t r o - ph . E P ] F e b Napier et al.
The possible capture of interstellar bodies by the solar system also has a long history. The general problem ofinteracting binaries was considered by Heggie (1975), where the subset of ‘resonant’ encounters lead to capture.Subsequent studies have carried out numerical explorations of the capture process specifically for our solar system,often considering only the Sun-Jupiter system (see, e.g., Valtonen & Innanen 1982; Valtonen 1983; Siraj & Loeb 2019).Additional studies consider capture for specific scenarios, including capture by compact objects (Pineault & Duquet1993), capture of interstellar objects from the field (Lingam & Loeb 2018; Hands & Dehnen 2020), the formation ofwide binaries (Kouwenhoven et al. 2010), and the possible capture of Planet Nine (Li & Adams 2016; Mustill et al.2016). Most of these previous studies calculate the capture rate by sampling a given distribution of encounter speedsbetween the incoming body and the solar system. These studies generally use the field star velocity distribution, withdispersion ∼
40 km/s (Binney & Tremaine 2008) or that appropriate for the solar birth cluster (Portegies Zwart 2009;Adams 2010; Pfalzner 2013; Parker 2020), where the velocity dispersion is expected to be ∼ The objective of this paper is to extend the aforementioned previous work concerning the capture of interstellarbodies by the solar system. Whereas most studies determine capture rates and cross sections for a given distributionof velocities, this work finds the cross section σ ( v ∞ ) as a function of relative velocity. The results can then be integrated(after the fact) for any distribution of velocities of interest. This approach is much more computationally expensive thanprevious treatments, but is made possible with current computational capabilities. Specifically, this paper reports theresults from ∼ × fly-by simulations. In addition, we carry out the simulations for solar systems models includingall four giant planets. Although earlier work (Heggie 1975; Pineault & Duquet 1993) provides analytic estimates forthe cross sections, exact forms are not available (primarily due to the lack of an analytic solution to the gravitationalthree-body problem). We revisit this issue using a different (but equivalent) set of approximations. We then comparethe numerical and analytic results for the cross section as a function of velocity, and find good agreement. DYNAMICS OF THE ROCK CAPTURE PROCESSThis section presents an analytic description of the rock capture process. The capture of an incoming body occursthrough the time dependence of the gravitational potential of the solar system. In this treatment, we consider theincoming orbit in two regimes. In the outer regime, at large distances, the rock executes a hyperbolic orbit aboutthe center of mass of the solar system. In the inner regime, at closer distances, the rock can enter into the sphere ofinfluence of individual solar system members (e.g., the Sun or Jupiter), and then be described by a hyperbolic orbitaround that body. Under favorable conditions, the deflection by the solar system body during the close encounter canlead to energy loss and capture in the center of mass frame. This effect is essentially the inverse of the gravitationalslingshot mechanism by which satellites are boosted through planetary encounters. Note that by dividing the orbitinto two regimes, we are implicitly assuming that 3-body effects are not important.For the sake of definiteness, we consider only one planet at a time, and work in the limit where the masses of therock µ , the planet m , and the star M obey the ordering µ (cid:28) m (cid:28) M . (1)The incoming orbit of the rock is characterized by its asymptotic speed v ∞ and impact parameter b . For given inputvariables ( v ∞ , b ), we can define the orbital elements and related physical quantities, including the specific energy andangular momentum, E = 12 v ∞ and J = bv ∞ , (2)the semi-major axis and eccentricity, | a | = − a = GMv ∞ and e = 1 + b /a , (3)and the perihelion distance r p = p = a (1 − e ) = | a | ( e − . (4) As one example, the distribution of speeds for planets ejected from crowded solar systems has the approximate form dP/dv = 4 v/ (1 + v ) (e.g., Moorhead & Adams 2005). apture of Interstellar Objects a <
0. To fully characterize the orbit, one must also specify theinclination angle of the incoming trajectory.It is useful to define the effective cross section for hyperbolic orbits to enter the giant planet region of the solarsystem. In order for the incoming rock to experience the time-dependence of the gravitational potential, the perihelion r p must be smaller than the semi-major axis a p of the planet of interest. This condition implies that the impactparameter b is bounded from above by b ≤ a p + 2 a p | a | , where a is the semi-major axis of the incoming orbit. Thenominal cross section σ for orbit crossing is thus given by σ = π (cid:2) a p + 2 a p | a | (cid:3) ≈ π GMv ∞ a p , (5)where the final equality holds for essentially all incoming speeds of interest ( v ∞ < GM/a p ). The capture cross sectionwill be some fraction of the fiducial cross section (5). Gravitational Slingshot Mechanism for Close Encounters
For the inner regime defined above, we consider close encounters of incoming rocks on initially hyperbolic trajectorieswith much larger target bodies (either the Sun or one of the giant planets). We can define the coordinate system sothat the rock approaches the target body from the +ˆ x direction and from the +ˆ y direction, where the angle of theincoming trajectory is θ in the center of mass frame (see the first panel in Figure 1). The rock initially has speed v ∞ in the inertial reference frame and the target body has speed U . In the frame of the target, the incoming rock hasvelocity v tar = ( − [ v x − U ] , v y ) (6)and the outgoing velocity has the form v tar = (+[ v x − U ] , v y ) . (7)This second equation assumes that the encounter is symmetric, i.e., the outgoing trajectory of the rock is the mirrorimage of the incoming trajectory. This approximation thus assumes that the larger body does not change its velocity(consistent with the ordering of equation [1]) and that the encounter time is short compared to the orbital period. Inthe center of mass reference frame, the incoming velocity has the form v cm = ( − v x , v y ) = ( − v cos θ, v sin θ ) , (8)whereas the outgoing velocity becomes v cm = (+[ v x − U ] , v y ) = (+[ v cos θ − U ] , v sin θ ) . (9)The final speed is then given by the expression v = [ v cos θ − U ] + v sin θ = v − U v cos θ + 4 U . (10)Note that this discussion assumes that the encounter is symmetric in the reference frame of the capturing body. Thisapproximation is expected to be valid because only close encounters with the body result in capture events, and suchclose encounters will be symmetric to leading order.The discussion thus far has implicitly assumed that the target is moving in the − ˆ x direction. In general, however,the target can also have a ˆ y component to its velocity. Because of the geometry of the encounter, however, only theˆ x component of the rock velocity changes (in this approximation). We can thus incorporate the more general case byinterpreting the velocity U as the component of the target velocity in the − ˆ x direction. With this definition of U , thefinal speed still obeys equation (10). Note that the interpretation of this fiducial cross section would be more complicated if the planetary orbit had significant eccentricity.Nonetheless, one can always scale the results to the expression of equation (5).
Napier et al.
PLANETARY CAPTURE P e r t u r b e d O r b i t Jupiter Orbit
Barycentric Frame U n p e r t u r b e d O r b i t v q b a s y m p t o t e Planet-centric Frame
Jupiter's Sphere of Influence
SOLAR CAPTURE Barycentric Frame Heliocentric Frame
Figure 1.
Examples of capture events. In each frame, the dotted black line denotes the rock’s initial (unbound) orbit; thesolid black line denotes the rock’s trajectory after a perturbation; and the arrows specify the direction of the orbit. The yellowcircle represents the Sun, the filled red circle represents Jupiter’s sphere of influence drawn at the epoch of the rock’s closestapproach, and the red line marks Jupiter’s orbit. The top row shows a capture by Jupiter: the left panel is in the frame of thesolar system’s barycenter, and the right panel is in Jupiter’s rest frame. Note that the rock gets well inside of Jupiter’s sphereof influence, but does not actually collide with the planet. The bottom row shows a capture by the Sun: the left panel is inthe frame of the solar system’s barycenter, and the right panel is in the Sun’s rest frame. If the target (Sun or planet) has acomponent U of its velocity moving away from the incoming rock as it approaches periapsis, then the encounter causes the rockto lose energy in the inertial reference frame, thereby allowing the rock to potentially enter into a bound orbit. Solar Close Encounters apture of Interstellar Objects v = v ∞ + 2 GMr , (11)where r < a J is the location of the rock. It will then execute a (hyperbolic) orbit around the Sun. Due to the motionof the Sun about the center of mass of the solar system, the post-encounter velocity will be given by v = v − U v cos θ + 4 U , (12)where U is the component of the solar velocity in the direction of the perihelion of the orbit and θ defines the shapeof the hyperbola. Capture of the rock requires that v < v − v ∞ , so that we obtain the constraint v ∞ < U v cos θ − U . (13)Here, the angle θ is determined by the parameters of the original hyperbolic orbit about the Sun, so thatcos θ = 1 e = | a | ( a + b ) / , (14)where a is the semi-major axis and b is the impact parameter. The capture constraint thus becomes v ∞ < α mM (cid:18) GMa J (cid:19) / (cid:20) v ∞ + 2 GMr p (cid:21) / | a | ( a + b ) / + O (cid:18) m M (cid:19) , (15)where we have assumed that the speed U is some fraction of the speed of the Sun in its orbit about the center of mass.Specifically, we define the parameter α such that U ≡ α mM (cid:18) GMa J (cid:19) / , (16)where m is the mass of Jupiter. Since the speed of the incoming rock v (cid:29) U for the close encounters of interest,we ignore the U term. Finally, we evaluate the rock velocity at the perihelion position r p (see equation [4]), as thislocation corresponds to where the close encounter takes place. Working to consistent order, the expression for thecapture constraint can be written in the form v ∞ < α mM (cid:18) GMa J (cid:19) / (cid:20) GM | a | b (cid:21) / = 8 α mM (cid:18) GMa J (cid:19) / GMbv ∞ . (17)The constraint can be written as a limit on the impact parameter, i.e., b < α mM (cid:18) GMa J (cid:19) / v − ∞ a J ≈
93 au α (cid:18) v ∞ / s (cid:19) − . (18)If one requires that the rocky body is not only captured, but is captured into an orbit with semi-major axis lessthan some maximum value a max , then the left-hand-side of equation (17) can be replaced with v ∞ + v , where v ≡ GM/a max . Finally, note that this treatment implicitly assumes that
U >
0. If the Sun is moving in the oppositedirection, the encounter would cause the incoming rocky body to gain energy, and capture does not take place.Given the approximations presented above, the resulting cross section for capture can be written in the from σ = 64 πα a J (cid:16) mM (cid:17) (cid:18) GMa J (cid:19) v ∞ ( v ∞ + v ) . (19)This cross section is specified up to the constant α , which is expected to be of order (but less than) unity. This formis consistent with those derived earlier by other means (Heggie 1975; Pineault & Duquet 1993; Valtonen 1983). Noticethat this derivation breaks down for sufficiently high incoming speeds, v ∞ > ∼ Napier et al.
Planetary Close Encounters
Another channel for capture occurs through close encounters with the giant planets, most often Jupiter, which willbe considered in this discussion. Equation (5) represents the cross section for an incoming rock to enter the sphere ofradius a J . Only a fraction of the incoming trajectories f = R SoI / a J will enter the sphere of influence of Jupiter, delineated by R SoI ≈ a J ( m/M ) / (Bate et al. 1971). However, not all of the orbits that enter the sphere of influencewill pass close enough to the planet to experience significant deflection. As a result, we must estimate the smallerfraction f of trajectories that allow for capture.As a rough approximation, significant deflection requires cos θ to be of order (but still less than) unity, which inturn implies b hp ∼ | a | hp (equation [14]), where ( a hp , b hp ) correspond to the elements of the hyperbolic orbit around theplanet. When the rock encounters the planet, its speed in the solar reference frame is given by equation (11) evaluatedat r ≈ a J . The asymptotic speed ( v ∞ ) hp for the hyperbolic orbit about the planet depends on the planetary motion,but will typically be of the same order. We can thus write( v ∞ ) hp = v ∞ + β GMa J ≡ v ∞ + v , (20)where β is a dimensionless factor of order unity and where the second equality defines the velocity scale v z . Thesemi-major axis of the hyperbolic planetary encounter is given by( | a | ) hp = Gm ( v ∞ ) hp = Gmv ∞ + v ∼ mM a J . (21)Since we require b hp < ∼ | a | hp and | a | hp (cid:28) R SoI , the fraction f = a hp / a J . The resulting cross section for capture dueto planetary encounters has the form σ = γπ GMv ∞ a J (cid:18) Gma J ( v ∞ + v ) (cid:19) = γπa J (cid:16) mM (cid:17) (cid:18) GMa J (cid:19) v ∞ ( v ∞ + v ) , (22)where we have introduced a dimensionless factor γ that is expected to be of order unity. Note that this expression hasa form similar to that of equation (19), which corresponds to the capture cross section for solar encounters. Keep inmind, however, that the velocity scales are different and are expected to obey the ordering v x < v z .2.4. Energy Distribution of Newly Bound Orbits
Using the results from the previous section, we can write the post-encounter speed of the rock in the form v ≈ v − U v cos θ . (23)The semi-major axis a b of the bound orbit is defined so that GMa b = 2 GMr − v = 4 U v cos θ − v ∞ . (24)Let us now define a scale length b according to b ≡ mM (cid:18) GMa J (cid:19) / a J v ∞ α ∼
100 au (cid:18) v ∞ / s (cid:19) − . (25)With this construction, the semi-major axis of the bound orbit is given by GMa b v ∞ = b b − ⇒ a b = a bb − b = a χ − χ , (26)where we let a = | a | denote the (magnitude of) the semi-major axis of the initial hyperbolic orbit, and where thefinal equality defines χ ≡ b/b . The criterion for obtaining a bound orbit (from the previous section) is equivalent Note that the sphere of influence, as defined here, corresponds to the location where the incoming trajectory switches from a two-bodyproblem with central mass M to a two-body problem with central mass m in the matched conics approximation. The boundary R SoI iscomparable to, but not equivalent to, the Hill radius R H = a ( m/ M ) / . apture of Interstellar Objects b < b ( χ < b , the distribution of impact parameters willbe weighted towards larger values. This finding, in turn, implies that typical bound orbits will have final semi-majoraxes comparable to the starting (negative, hyperbolic) semi-major axis of the incoming orbit. For v ∞ = 1 km/s, forexample, bound orbits will typically have a b ∼ a b < ∼
100 au, which in turn implies that b < ∼ b / ∼
10 au.If we assume that the impact parameters b are uniformly distributed over an area, with a maximum value b , thenthe probability distribution for the dimensionless quantity χ has the simple form dP = 2 χdχ . Using equation (26), wecan determine the probability distribution for the semi-major axes of the bound orbits, i.e., dPda b = 2 a b a ( a b + a ) . (27)As written, this distribution is normalized over the interval 0 < a b < ∞ .Note that the distribution of equation (27) corresponds to the semi-major axes of the bodies when they are captured.The orbital elements of the captured objects will continue to evolve (e.g., through continued close encounters with theplanets), so that quasi-stable orbits will display a different distribution (which should be explored further in futurework). NUMERICAL RESULTSThe cross sections derived in the previous section made use of a number of approximations. In this section we usea suite of more than 500 million simulations to numerically compute the capture cross section.3.1.
Simulation Details
We sample rocks of mass 10 − M (cid:12) isotropically on the sphere at a barycentric distance of 10 au. Each rock’s velocityunit vector is uniquely defined by its position on the sphere, pointing directly toward the solar system barycenter. Wethen assign each rock an impact parameter at some random angle in its plane tangent to the sphere. We randomlysample the impact parameter uniformly given the condition that the maximum pericenter distance q max ≤
12 au—comfortably above the largest pericenter distance for capture not attributable to chance close encounters with a giantplanet. Finally we scale the rock’s velocity unit vector by a factor v = (cid:114) v ∞ + 2 µr (28)where µ = G (cid:80) i m i and i ∈ { Sun, Jupiter, Saturn, Uranus, Neptune } . In Equation 28, v ∞ is the rock’s field (orcluster) velocity at infinity, and the second term accounts for the kinetic energy that the rock gains by falling frominfinity to a barycentric distance r .The above procedure gives us a state vector, from which we compute a body’s Keplerian orbital elements. To savecomputation time, we use these elements to propagate each rock along its unperturbed hyperbolic orbit to a barycentricdistance of 1,000 au. This approximation (that the solar system is a point mass with all of its mass at the barycenter)should be accurate to about one part in 10 , since the solar system’s quadrupole term goes like r − . Once we haveperformed the analytic propagation of the rock, we use NASA’s development ephemerides to initialize the solar systemat a random date in a 200–year range around the arbitrarily chosen Julian Date 2459010 .
5. This ensures that ourresults are not affected by some exceptional coincidence in the initial phases of the giant planets’ orbits.When we have initialized our rock and the solar system, we use
Rebound’s IAS15 integrator (Rein & Spiegel 2015)to evolve the system numerically. For each simulation, we conserve the system’s total energy to better than one partin 10 —much smaller than the fraction of the system’s energy attributable to the rock. Therefore we are confidentin the accuracy of our integrations.For each integration, there are three possible outcomes: the rock may be captured; undergo a collision with anotherbody; or be ejected from the system. If at any point during the simulation the rock’s energy drops below zero, weconsider it to be captured and end the simulation. If the rock undergoes a collision or if the rock is unbound andexiting the solar system with a barycentric distance greater than 40 au, we end the simulation and determine that therock was not captured. We then follow up on our captured objects, integrating for 51% of an orbital period to ensurethat each object is truly bound (as opposed to having a transient bound osculating semi-major axis due to the phases Napier et al. of the giant planets). If during our followup the object’s apocenter distance exceeds 1 parsec, we consider it to be lostto cluster or galactic tides.Current models of solar system formation predict that the giant planets formed in a more compact arrangement, andthen migrated to their current orbits. To account for this we ran a set of simulations with the compact solar systemmodel presented in Tsiganis et al. (2005). The cross section we calculate with this model differs from that calculatedusing the solar system at the current epoch by less than 1 percent, so our calculations should be equally applicable tothe pre-and-post-instability architectures of the solar system.3.2.
Capture Cross Section
Since we sampled events uniformly in impact parameter, we can calculate the capture cross section as σ = 2 πb max N (cid:88) i b i δ i (29)where b max is the maximum impact parameter sampled, N is the number of events, and δ i is a Kronecker delta that is1 if the event resulted in capture, and 0 otherwise. We display our results in Figure 2. As we expect, σ ( v ∞ ) goes like v − ∞ in the low-speed limit, and like v − ∞ in the high-speed limit. To facilitate the use of the cross section in analyticcalculations we fit σ ( v ∞ ) with the simple function σ ( v ∞ ) = σ u ( u + 1) (30)where u ≡ v ∞ /v σ and v σ is a velocity scale determined by the properties of the planet ejecting the rock. We find thedata are best fit by parameter values σ = 232 ,
250 au and v σ = 0 . σ for the cross section obtained from fitting our numerical results can be compared to the analyticestimates of the previous section. If we evaluate equation (19) in the high speed limit, then agreement between theanalytic and numerical estimates implies that σ = 64 πα a J ( m/M ) ( v σ /v J ) , where v J is the orbital speed of Jupiter.The expressions are equal if the dimensionless parameter α ≈ .
21. The analytic and numerical results are in agreementfor all incoming speeds if we identify the scales v x and v σ , which is equivalent to considering captures with a maximum(post-encounter) semimajor axis a max ≈ v σ ∼ v x ∼ ( Gm/a J ) / (see also Appendix C).Similarly, equation (22) agrees with the numerical result in the high speed limit if the dimensionless parameter γ ≈ . v ∞ ≤
15 km/s. This upper limit is invoked fora number of reasons. Due to the steep power-law fall-off of the capture cross section, relatively few capture events takeplace at higher speeds, so additional computation leads to diminishing returns. In addition to the steep dependencewith v ∞ , the numerical data indicate that the power-law begins to break at a comparable speed. Some type of break isexpected: For v ∞ greater than ∼
10 km/s, capture by close encounters with the Sun becomes ineffective (see AppendixA). It is noteworthy that the capture cross section at v ∞ ≈
13 km/s is comparable to the geometrical area of the Sun( ∼ × − au ). For larger encounter speeds, incoming rocky bodies are thus more likely to collide with the Sunthan be captured into a bound orbit. ANALYSIS OF CAPTURED OBJECTSIn this section we examine the orbital elements of our captured objects to gain insight into the mechanics of thecapture process. In Figure 3, we show the impact parameter (and pericenter distance) distribution of the unperturbedorbits of our captured objects for asymptotic speeds v ∞ of 1 and 2 km/s. Each histogram displays a clear relativepeak at the pericenter distances corresponding to the orbit of Jupiter, along with a much smaller peak for the orbitof Saturn. Comparison of the two histograms indicates that somewhere between 1 and 2 km/s, the dominant captureprocess switches from that due to the motion of the solar system barycenter to close encounters with a giant planet(especially Jupiter).In Figure 4, we show the post-capture eccentricity as a function of semi-major axis for a subset of the capturedobjects with v ∞ of 1 km/s (top panel) and 2 km/s (bottom panel). The figure also includes equi-pericenter curves For completeness, we note that due to gravitational focusing, the collision cross section with the Sun is larger than the capture cross sectionfor speeds v ∞ > − apture of Interstellar Objects v (km/s) ( a u ) v v = u (1 + u ) , u v v Solar Captures Planetary Captures
Figure 2.
Capture cross section (in au ) as a function of the asymptotic speed v ∞ (in km/s). The black points representthe numerically calculated cross sections, and the corresponding error bars represent one standard deviation. The red curverepresents the best fit of equation (30) to the data. The blue dashed line shows a power-law of the form σ ∼ v − ∞ , as expected inthe limit of high velocity. The green dashed line shows a power-law of the form σ ∼ v − ∞ , as expected in the low velocity limit.The shaded regions indicate the parameter space where captures due the Sun (green) and planets (blue) dominate, althoughthe boundaries are not sharp. corresponding to integer multiples of the spheres of influence of Jupiter and Saturn. The numerical results for capturesdisplay a relative overdensity of points with pericenter distances at Jupiter and Saturn, indicating that these capturesare (likely) attributable to close encounters.0 Napier et al. b (au) R e l a t i v e F r a c t i o n q (au) b (au) R e l a t i v e F r a c t i o n q (au) Figure 3.
Histograms of the (unperturbed) impact parameter distribution of captured objects for asymptotic speeds v ∞ of 1km/s (top) and 2 km/s (bottom). For convenience, we also indicate the pericenter distance of the unperturbed orbit. Both plotsshow relative peaks at pericenter distances corresponding to the orbits of Jupiter and Saturn. As v ∞ increases, close encounterswith the giant planets become more important for capture. e a (au) e Jupiter InfluenceSaturn Influence
Figure 4.
Post-capture eccentricity versus semi-major axis of captured objects for incoming speeds of 1 km/s (top) and at2 km/s (bottom). The orange and red regions correspond to integer multiples of the radius of influence centered at the thesemimajor axis of the orbits of Jupiter and Saturn, respectively.
In Figure 5, we show the kernel density representations for the post-capture inclination semi-major axis, and eccen-tricity for v ∞ = 0.5, 1, and 2 km/s. While captures become increasingly rare for higher-velocity events, the resultingsemi-major axes of the captured objects are typically smaller than those for objects captured in low-velocity events.This trend is important for assessing object retention, as captured bodies with semi-major axes a (cid:38) apture of Interstellar Objects v ∞ increases, the low-eccentricity tail of thedistribution becomes fatter. i K e r n e l D e n s i t y a (au) K e r n e l D e n s i t y e L o g a r i t h m i c D e n s i t y Figure 5. (Top) Gaussian kernel density estimate of the post-capture inclination distribution (with i measured from the eclipticplane) of captured objects at speeds v ∞ = 0.5, 1, and 2 km/s. (Center) Gaussian kernel density estimate of the post-capturesemi-major axis distribution of captured objects for v ∞ = 0.5, 1, and 2 km/s. (Bottom) Relative fraction (in logarithmic scale)of the post-capture eccentricity distribution of captured objects for v ∞ = 0.5, 1, and 2 km/s. Note that these curves representprobability distribution functions; there will be fewer captures in the high-speed case than in the low-speed case.5. APPLICATIONS5.1.
Velocity Averaged Cross Sections
This paper determines the velocity dependent cross section σ ( v ∞ ), which can be fit with a function of the form σ ( v ∞ ) = σ u (1 + u ) where u ≡ v ∞ v σ . (31)2 Napier et al.
The capture rate for rocky bodies by our solar system is given byΓ = n R (cid:104) σv ∞ (cid:105) , (32)where n R is the number density of rocks that the solar system encounters. The capture rate depends on the velocity-averaged cross section, which is given by the integral (cid:104) σv ∞ (cid:105) = (cid:90) ∞ v ∞ f ( v ∞ ) σ ( v ∞ ) dv ∞ , (33)where f ( v ∞ ) is the distribution of encounter velocities of the rocky bodies.The distribution of relative speeds f ( v ∞ ) depends on the environment. In the solar birth cluster, f ( v ∞ ) is determinedby the processes that eject the rocky bodies from their original planetary systems. In general, the clusters are notsufficiently long-lived for the rocks to attain a thermal distribution of speeds. Instead, they are expected to retain thevelocity distribution resulting from the ejection process. If ejection occurs through scattering interactions with giantplanets, then f ( v ∞ ) takes the approximate form f ( v ∞ ) = 4 v ∞ /v p (1 + v ∞ /v p ) , (34)where the velocity scale v p ≈ GM ∗ /a p , where a p is the semi-major axis of the planet that scatters the rocks (e.g., seeMoorhead & Adams 2005 for a derivation).Note that the distribution (34) is normalized over the entire interval 0 ≤ v ∞ ≤ ∞ . In practice, the distribution willhave a maximum value determined by the escape speed from the planets that scatter the rocky bodies. Notice alsothat the full distribution will be a convolution of the distribution of ejection speeds from each planet that scattersrocky material. As an approximation, we consider only a single distribution and interpret the velocity scale v p as atypical value. As a result, v p is expected to be comparable to the orbit speed of outer planets, i.e., v p ∼
10 km/s.Finally, we are assuming that equation (34) corresponds to the distribution of relative speeds between the rocks andthe solar system (e.g., see the discussion of Binney & Tremaine 2008).Putting the above considerations together, we can write the velocity averaged cross section in the form (cid:104) σv ∞ (cid:105) = σ v σ v p (cid:90) ∞ du (1 + η u ) (1 + u ) = σ v σ v p I ( η ) , (35)where u = v ∞ /v σ (as before), we have defined η ≡ v σ /v p , and where the second equality defines the integral function I ( η ). The dimensionless function I ( η ) can be evaluated to obtain I ( η ) = π (1 + η )(1 + 3 η ) + 3 η / η ) . (36)Note that I → π in the limit η →
0, and in practice η ∼ /
10. As a result, a good approximation for the capture crosssection takes the form (cid:104) σv ∞ (cid:105) ≈ πσ v σ v p . (37)For comparison, we can determine the velocity averaged cross section for the scenario where the cluster rocks arevirialized and have the same (Maxwellian) velocity distribution as the stars. In this limit, (cid:104) σv ∞ (cid:105) can be written inthe form (cid:104) σv ∞ (cid:105) = σ v σ s (cid:114) π (cid:90) ∞ v ∞ dv ∞ s exp[ − v ∞ / s ](1 + v ∞ /v σ ) , (38)where s is the velocity dispersion of the distribution. Note that the value of s for the distribution of relative speedsis larger than the value s for the velocity distribution of the stars in the clusters ( s = √ s ). Here we define thevariable w ≡ v ∞ /s and the parameter ξ = s/v σ , so that (cid:104) σv ∞ (cid:105) = σ v σ s (cid:114) π (cid:90) ∞ wdw exp[ − w / ξ w ) ≡ σ v σ s (cid:114) π J ( ξ ) , (39) In general, we expect the stars to reach virial equilibrium much faster than the rocky ejecta. The stars start their cluster trajectories withsub-virial speeds, but then fall toward the cluster core where interactions take place, and equilibrium is rapidly realized (in a few Myr, e.g.,Adams et al. 2006). In contrast, the rocks are ejected with speeds much larger than the virial speed and have little chance for interactions toslow them down. Moreover, the stellar virialization starts as soon as stars form, whereas the planet formation and the subsequent ejectionof rocks occurs many Myr later. apture of Interstellar Objects J ( ξ ). The exact form for J ( ξ ) can be found. If we define µ = 1 / (2 ξ ) = v σ / (2 s ), then J ( ξ ) = J ( µ ) = µ [1 − µ e µ E ( µ )] , (40)where E ( x ) is the exponential integral (Abramowitz & Stegun 1972). In the limit ξ → µ → ∞ ), the function J = 1; in the opposite limit ξ (cid:29) µ → J = µ = 1 / (2 ξ ). A good working approximation for the cross section ofequation (39) thus has the form (cid:104) σv ∞ (cid:105) ≈ σ v σ s (cid:112) /πv σ + 2 s , (41)which is exact in the limits and has a relative error less than ∼
20% over the entire range 0 ≤ ξ ≤ ∞ . One can useequations (39) and (40) if higher accuracy is required.5.2. Rock Capture in the Birth Cluster
Using the results derived above, we can estimate the total mass in rocky bodies that were captured while the Sunremained in its birth cluster. The capture rate is given by equation (32), with velocity-averaged cross section specifiedthrough equations (39) and (40). The rocky bodies will have a distribution of sizes g ( R ), which is defined here suchthat n R = (cid:90) ∞ g ( R ) dR and ρ R = (cid:90) ∞ g ( R ) m ( R ) dR , (42)where m ( R ) is the mass of the rock as a function of its size. With these definitions, the capture rate Γ can be convertedinto a mass accretion rate given by ˙ M = ρ R (cid:104) σv ∞ (cid:105) , (43)where ρ R is the mass density of the cluster in the form of rocks. Given that each planetary system in the cluster isexpected to eject a few Earth masses of rocky material (e.g., Rice & Laughlin 2019), the density ρ R is given by ρ R = αM ⊕ N ∗ V = αM ⊕ n ∗ , (44)where α is a dimensionless factor of order unity and n ∗ is the number density of stars. For completeness, note that theinclusion of icy planetesimals will increase this density estimate. In any case, the total mass in rocky bodies capturedby the solar system during its cluster phase can be written in the form(∆ M ) R = αM ⊕ (cid:20)(cid:90) ∞ n ∗ dt (cid:21) (cid:104) σv ∞ (cid:105) ≡ αM ⊕ (cid:104) n ∗ (cid:105) τ (cid:104) σv ∞ (cid:105) . (45)The final equality defines the mean density of the cluster, where τ is its effective lifetime. A number of studieshave found upper bounds on the product (cid:104) n ∗ (cid:105) τ by requiring that the solar system is not overly disrupted, includingconsiderations of the planetary orbits (Adams & Laughlin 2001; Adams 2010; Li & Adams 2015), the Kuiper Belt(Moore et al. 2020), and the orientation of the plane of the cold classicals (Batygin et al. 2020). This work indicatesthat the product is bounded by (cid:104) n ∗ (cid:105) τ < ∼ × pc − Myr.If we take v σ = 0.5 km/s, v p = 10 km/s, and σ = 2 × au , then the velocity averaged capture cross sectionbecomes (cid:104) σv ∞ (cid:105) ≈
800 au km/s. With this cross section, the total mass in captured rocky material from equation(45) is about (∆ M ) R ∼ − M ⊕ . Of course, most of this material will be ejected back into the cluster or the field.The retention rate of material in the inner Oort cloud is ∼
1% (Brasser et al. 2006), so we would expect ∼ − M ⊕ to be captured in the inner Oort cloud.Note that if rocks ejected from planetary systems in clusters follow the velocity distribution of equation (34), thensome fraction of the material will leave the cluster during its first crossing. The high speed tail of the velocitydistribution will thus be de-populated. In practice, however, most of the capture events arise from the low-speedportion of the distribution, so that the correction for the loss of high speed material is modest.We can also estimate the mass of rocks captured while the solar system is in the field. In this case, we expect therocky material to encounter the solar system with a velocity distribution comparable to that of the field stars, i.e.,a Gaussian distribution with s ∼
40 km/s. In this case, the velocity averaged cross section (cid:104) σv ∞ (cid:105) ≈ .
08 au km/s.If we also assume that each planetary system ejected the same mass in rocks during its formative phases, then the4 Napier et al. density of rocky material will be proportional to the stellar density (we are thus assuming negligible losses). As aresult, the product (cid:104) n ∗ (cid:105) τ ∼
460 pc − Myr, and the expected mass in captured rocks is about (∆ M ) R ≈ × − M ⊕ .Using the approximate retention rate of 1%, we would expect only ∼ × − M ⊕ of these rocks to remain in theinner Oort cloud. This inventory of captured alien material from the field is exceedingly small, roughly the equivalentof one 5 km body. Rock capture during the birth cluster phase is thus expected to produce the dominant contribution(by roughly a factor of one million). These latter objects are expected to have radiogenic ages comparable to ordinarysolar system bodies, but might be identified by different (unusual) chemical composition.Note that the values presented here are highly approximate. Not all of the rocks will be captured in the inner Oortcloud, so that the retention fraction could be smaller than assumed here (most of the captured interstellar bodiesinitially have Jupiter-crossing orbits, whereas the planetesimals in the Oort cloud could have different origins). Inany case, most of the captured objects will be ejected, and some will eventually collide with the Sun. Although thesecalculations provide working order-of-magnitude estimates, in forthcoming work we will refine these projections bynumerically investigating the long-term behavior of the captured bodies from this work. CONCLUSIONSThis paper has revisited the problem of capturing interstellar objects on initially hyperbolic trajectories into boundstates. Using an ensemble of 500 million numerical fly-by simulations, the main result of this study is the determinationof our solar system’s capture cross section as a function of encounter speed (see Figure 2). The resulting capture crosssection shows the power-law velocity dependence σ ∼ v − ∞ in the limit of low speeds and the dependence σ ∼ v − ∞ inthe limit of high speeds. The capture cross section σ ( v ∞ ) over the entire range of asymptotic speeds can be fit withthe function given in equation (31).This paper also presents an analytic treatment of the capture problem using the approximation of matched conics andthe (inverse) gravitational slingshot effect (Section 2). These arguments show that capture by both close encounterswith the Sun (Section 2.2), and by close encounters with a giant planet (Section 2.3), have the same nearly velocitydependence as that seen in the numerical simulations (namely σ ∼ v − ∞ at low speeds and σ ∼ v − ∞ at high speeds).The capture events can be classified as either close encounters with the Sun or close encounters with giant planets.At low speeds, encounters with the Sun dominate the capture cross section. At higher speeds, close encounters withJupiter dominate. Close encounters with the other giant planets contribute to the cross section, but do not dominatethe dynamics. More specifically, for the particular case of v ∞ = 1 km/s, capture events due to close encounters withJupiter are ∼
100 times more likely than captures due to Saturn. The frequency of close encounters with Uranus andNeptune are smaller (than for Saturn) by an additional factor of ∼ (cid:104) vσ (cid:105) / (cid:104) v (cid:105) canbe determined for any distribution of encounter speeds. For the case of a Maxwellian distribution and a power-lawdistribution motivated by rock ejection, the mean cross section can be evaluated analytically (see Section 5.1).Finally, as an application of the capture cross section, we estimate the total mass (∆ M ) R in the Oort cloud thatoriginates from other planetary solar systems (Section 5.2). The mass accreted while the Sun lived within its birthcluster is of order (∆ M ) R ∼ − M ⊕ , about a million time larger than the mass subsequently accreted from the field.Although the capture cross section for the solar system is now well-characterized, many avenues for future researchremain. The simulations of this paper consider the capture of interstellar objects and the resulting cross sectionsinclude all capture events, independent of their residence time in the solar system as bound objects. Future workshould determine how long captured bodies can remain bound to the Sun, since many such objects are expected tobe ejected from the system or to collide with other solar system members. The residence time (ejection time) shouldthus be determined for each type of orbit displayed by the captured objects. With these results in place, one can makea refined estimate of the current population of alien objects in the solar system, along with their expected orbitalproperties. ACKNOWLEDGMENTSWe would like to thank David Gerdes, Hsing-Wen Lin, and Larissa Markwardt for helpful discussions during thepreparation of this manuscript. apture of Interstellar Objects A. UPPER BOUND ON INCOMING SPEED FOR CAPTUREIn this Appendix, we find upper limits on the asymptotic speed, i.e., the largest speed for which a rock can becaptured by the solar system. We consider both close encounters with the Sun and with one of the giant planets.We start with the capture criterion of equation (13), which we reproduce here, v ∞ < U v cos θ − U < U v , (A1)where the second inequality follows because cos θ < U >
0. The largest speed U possible for the Sun, relativeto the center of mass, is the orbit speed due to the giant planets, where U = mM (cid:18) G ( M + m ) a p (cid:19) / ≈ mM (cid:18) GMa p (cid:19) / , (A2)where the second expression is consistent with the ordering approximation and where we have included the reflex speeddue to only one planet. The maximum possible speed is given by the sum of the contributions of all of the planets.We can account for this complication by writing the limit in the form U < m J M (cid:18) GMa J (cid:19) / . (A3)Thus far, our upper bound has the form v ∞ < m J M (cid:18) GMa J (cid:19) / v = 8 m J M (cid:18) GMa J (cid:19) / (cid:20) v ∞ + 2 GMR (cid:12) (cid:21) / , (A4)where we have taken the minimum distance (and maximum speed) of the orbit to be given by the radius of the Sun.If we drop the first term in square brackets, the expression simplifies to the form v ∞ < m J M (cid:18) GMa J (cid:19) / (cid:18) GMR (cid:12) (cid:19) / = 8 √ Gm J ( a J R (cid:12) ) / ≈ (8 km / s) . (A5)To derive a rigorous version of the upper bound, we define two velocity components u ≡ m J M (cid:18) GMa J (cid:19) / and w ≡ (cid:18) GMR (cid:12) (cid:19) / , (A6)so that the limit takes the form v ∞ < uw (cid:20) u w (cid:21) / + u . (A7)Since u is of order the orbital speed of the Sun (specifically, a small fraction of 1 km/s) and w is the escape speed ofthe Sun ( ∼
620 km/s), we find u (cid:28) w . In this limit, the full expression of equation (A7) reduces to the form v ∞ < uw ,which corresponds to the approximation of equation (A5).We can also consider the case where the incoming rock enters into the sphere of influence of a planet and loses energythrough an inverse gravitational assist from the encounter. In order for the rock to be captured by the solar system,the final speed must be sufficiently small, i.e., v < v , (A8)6 Napier et al. where the orbital speed v orb of the planet is a measure of the depth of the gravitational potential well at the locationof the planet (the location of the encounter). We also assume that the incoming rock obeys conservation of energy sothat 12 v ∞ = E = 12 v − GMr ≈ v − v ⇒ v = v ∞ + 2 v . (A9)Using this expression in the result for the post-encounter speed from before, we obtain v ∞ + 4 U < U cos θ [ v ∞ + 2 v ] / . (A10)If we take U = v orb cos φ , this expression can be rewritten in the form v ∞ < v v ∞ cos φ (2 cos θ −
1) + 16 v cos φ (2 cos θ − cos φ ) . (A11)We can thus obtain an upper limit on the asymptotic speed for an object to be captured by taking cos θ = 1 = cos φ ,i.e., v ∞ < v v ∞ + 16 v , (A12)which leads to the bound v ∞ < v orb (1 + √ / . (A13)The fastest orbit is that of Jupiter, where v orb ≈
13 km/s, so we have the limit v ∞ < ∼
40 km/s.We thus find that the maximum possible speed (for capture) is larger for the channel involving close approachesto planets (in particular Jupiter) than for close approaches to the Sun. On the other hand, planet encounters areexpected to occur much less frequently. Note that these limits do not account for effects such as radiation pressureor atmospheric drag. It is therefore possible in principle to capture rocks with larger v ∞ , but such events would beexceptionally rare. B. ROCK CAPTURE BY CIRCUMSTELLAR DISKSRock capture could also take place due to gas drag if interstellar bodies enter the solar nebula while it retains itsgaseous component. This Appendix explores the efficacy of this process.If a rock has speed v when it enters the gaseous region of the disk, it will have final speed v f given by v f = v exp[ − ρA(cid:96)/m ] , (B14)where ρ is the gas density, A is the rock area, m is the rock mass, and (cid:96) is the length traveled. If a rock comes frominfinity with speed v ∞ , then it will have a larger velocity when it hits the disk, so that v is given by v = v ∞ + 2 GMr . (B15)In order for the rock to lose enough energy to enter into a bound orbit, the final speed must be less than the limit v f = (cid:20) v ∞ + 2 GMr (cid:21) exp (cid:20) − ρA(cid:96)m (cid:21) < GMr . (B16)The capture criterion thus becomes 2 ρA(cid:96)m > ln (cid:20) v ∞ r GM (cid:21) . (B17)If the rock passes through the disk vertically, then ρ(cid:96) = Σ, where the surface density for the solar nebula can bewritten in the form Σ( r ) = Σ (cid:18) r (cid:19) / , (B18)where Σ ≈ . If the rock passes through the nebula at an arbitrary angle θ , then ρ(cid:96) = Σ / cos θ . We canalso write A = πR and m = ρ R (4 π/ R where R is the size of the rock. If we work in the low speed limit with v ∞ = 1 km/s, and write R in units km and r in units of au, then the criterion for capture reduces to the simple form36cos θ > Rr / . (B19) apture of Interstellar Objects θ = 1, the solar nebula can thus capture incoming rocks with radii R < ∼ r = 1 au. Somewhat larger rocks can be captured for typical inclined trajectories (e.g., the limit becomes R < ∼ θ = 1 / θ → C. DIMENSIONAL ANALYSISThe capture problem has more than one dimensionless field, so that the cross section of interest cannot be directlydetermined from dimensional analysis. Nonetheless, such an analysis is useful for understanding the result.Consider the solar system to consist of only the Sun-Jupiter binary. The physical variables required to characterizethe systems are thus the masses (