# On the Capture of Interstellar Objects by our Solar System

Kevin J. Napier Department of Physics, University of Michigan, Ann Arbor, MI
48109, USA  Fred C. Adams Department of Physics, University of Michigan, Ann
Arbor, MI 48109, USA  Department of Astronomy, University of Michigan, Ann
Arbor, MI 48109, USA  Konstantin Batygin Geological and Planetary Sciences,
California Institute of Technology, Pasadena, CA 91125, USA

###### Abstract

Motivated by recent visits from interstellar comets, along with continuing
discoveries of minor bodies in orbit of the Sun, this paper studies the
capture of objects on initially hyperbolic orbits by our solar system. Using
an ensemble of $\sim 500$ million numerical experiments, this work generalizes
previous treatments by calculating the capture cross section as a function of
asymptotic speed. The resulting velocity-dependent cross section can then be
convolved with any distribution of relative speeds to determine the capture
rate for incoming bodies. This convolution is carried out for the usual
Maxwellian distribution, as well as the velocity distribution expected for
rocky debris ejected from planetary systems. We also construct an analytic
description of the capture process that provides an explanation for the
functional form of the capture cross section in both the high velocity and low
velocity limits.

Solar system (1528), Dynamical evolution (421), Small solar system bodies
(1469), Kuiper belt (893), Oort cloud (1157)

##  1 Introduction

The past few years have witnessed the detection of two interstellar bodies
passing through the solar system on hyperbolic 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-ranging speculation 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 any specific objects in the solar system are of extrinsic
origin (Morbidelli et al., 2020), the question of whether or not any such
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
sections is the first step in assessing whether or not the solar system
presently contains quasi-permanently trapped interstellar bodies. 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. Starting more than two centuries ago, classic
studies include the long-term stability of the solar system (Lagrange, 1776;
Laplace, 1799-1825), the origin of comets (Laplace, 1846), and orbital
anomalies that led to the discovery of Neptune (Le Verrier, 1846; 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, or perhaps
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; Batygin et al.,
2019). The more recent discovery of interstellar objects (Meech et al., 2017;
Jewitt & Luu, 2019) adds to the intrigue. Both the complex orbital
architecture of the solar system and the presence of interloping objects
motivates this present study. The goal is to determine cross sections for the
capture of foreign bodies by the solar system, and to obtain a deeper
understanding of the capture process.

The possible capture of interstellar bodies by the solar system also has a
long history. The general problem of interacting 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 & Duquet, 1993), capture of
interstellar objects from the field (Lingam & Loeb, 2018; Hands & Dehnen,
2020), the formation of wide 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 speeds between the incoming body and the solar
system. These studies generally use the field star velocity distribution, with
dispersion $\sim 40$ 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 $\sim 1$ km/s
(Lada & Lada, 2003). Notice, however, that the velocity distribution for rocks
(or planets) ejected by solar systems will not generally have a simple
Maxwellian form.111As one example, the distribution of speeds for planets
ejected from crowded solar systems has the approximate form
$dP/dv=4v/(1+v^{2})^{3}$ (e.g., Moorhead & Adams 2005).

The objective of this paper is to extend the aforementioned previous work
concerning the capture of interstellar bodies by the solar system. Whereas
most studies determine capture rates and cross sections for a given
distribution of velocities, this work finds the cross section
$\sigma(v_{\infty})$ 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 than previous treatments,
but is made possible with current computational capabilities. Specifically,
this paper reports the results from $\sim 5\times 10^{8}$ fly-by simulations.
In addition, we carry out the simulations for solar systems models including
all four giant planets. Although earlier work (Heggie, 1975; Pineault &
Duquet, 1993) provides analytic estimates for the cross sections, exact forms
are not available (primarily due to the lack of an analytic solution to the
gravitational three-body problem). We revisit this issue using a different
(but equivalent) set of approximations. We then compare the numerical and
analytic results for the cross section as a function of velocity, and find
good agreement.

##  2 Dynamics of the Rock Capture Process

This section presents an analytic description of the rock capture process. The
capture of an incoming body occurs through the time dependence of the
gravitational potential of the solar system. In this treatment, we consider
the incoming orbit in two regimes. In the outer regime, at large distances,
the rock executes a hyperbolic orbit about the center of mass of the solar
system. In the inner regime, at closer distances, the rock can enter into the
sphere of influence of individual solar system members (e.g., the Sun or
Jupiter), and then be described by a hyperbolic orbit around that body. Under
favorable conditions, the deflection by the solar system body during the close
encounter can lead to energy loss and capture in the center of mass frame.
This effect is essentially the inverse of the gravitational slingshot
mechanism by which satellites are boosted through planetary encounters. Note
that by dividing the orbit into 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 the rock $\mu$, the planet $m$, and the star
$M$ obey the ordering

$\mu\ll m\ll M\,.$ (1)

The incoming orbit of the rock is characterized by its asymptotic speed
$v_{\infty}$ and impact parameter $b$. For given input variables
$(v_{\infty},b)$, we can define the orbital elements and related physical
quantities, including the specific energy and angular momentum,

$E={1\over 2}v_{\infty}^{2}\qquad{\rm and}\qquad J=bv_{\infty}\,,$ (2)

the semi-major axis and eccentricity,

$|a|=-a={GM\over v_{\infty}^{2}}\qquad{\rm and}\qquad e^{2}=1+b^{2}/a^{2}\,,$
(3)

and the perihelion distance

$r_{p}=p=a(1-e)=|a|(e-1)\,.$ (4)

Note that, by convention, the semi-major axis $a<0$. To fully characterize the
orbit, one must also specify the inclination 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 solar system. 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 impact parameter $b$ is
bounded from above by $b^{2}\leq a_{p}^{2}+2a_{p}|a|$, where $a$ is the semi-
major axis of the incoming orbit. The nominal cross section $\sigma_{0}$ for
orbit crossing is thus given by

$\sigma_{0}=\pi\left[a_{p}^{2}+2a_{p}|a|\right]\approx 2\pi{GM\over
v_{\infty}^{2}}a_{p}\,,$ (5)

where the final equality holds for essentially all incoming speeds of interest
($v_{\infty}^{2}<GM/a_{p}$). The capture cross section will be some fraction
of the fiducial cross section (5).222Note 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).

###  2.1 Gravitational Slingshot Mechanism for Close Encounters

Figure 1: Examples of capture events. In each frame, the dotted black line
denotes the rock’s initial (unbound) orbit; the solid black line denotes the
rock’s trajectory after a perturbation; and the arrows specify the direction
of the orbit. The yellow circle represents the Sun, the filled red circle
represents Jupiter’s sphere of influence drawn at the epoch of the rock’s
closest approach, and the red line marks Jupiter’s orbit. The top row shows a
capture by Jupiter: the left panel is in the frame of the solar system’s
barycenter, and the right panel is in Jupiter’s rest frame. Note that the rock
gets well inside of Jupiter’s sphere of influence, but does not actually
collide with the planet. The bottom row shows a capture by the Sun: the left
panel is in the 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 a component $U$ of
its velocity moving away from the incoming rock as it approaches periapsis,
then the encounter causes the rock to lose energy in the inertial reference
frame, thereby allowing the rock to potentially enter into a bound orbit.

For the inner regime defined above, we consider close encounters of incoming
rocks on initially hyperbolic trajectories with much larger target bodies
(either the Sun or one of the giant planets). We can define the coordinate
system so that the rock approaches the target body from the $+{\hat{x}}$
direction and from the $+{\hat{y}}$ direction, where the angle of the incoming
trajectory is $\theta$ in the center of mass frame (see the first panel in
Figure 1). The rock initially has speed $v_{\infty}$ in the inertial reference
frame and the target body has speed $U$. In the frame of the target, the
incoming rock has velocity

${\bf v}_{1tar}=\left(-[v_{x}-U],v_{y}\right)$ (6)

and the outgoing velocity has the form

${\bf v}_{2tar}=\left(+[v_{x}-U],v_{y}\right)\,.$ (7)

This second equation assumes that the encounter is symmetric, i.e., the
outgoing trajectory of the rock is the mirror image 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. In the center of
mass reference frame, the incoming velocity has the form

${\bf
v}_{1cm}=\left(-v_{x},v_{y}\right)=\left(-v\cos\theta,v\sin\theta\right)\,,$
(8)

whereas the outgoing velocity becomes

${\bf v}_{2cm}=\left(+[v_{\rm
x}-2U],v_{y}\right)=\left(+[v\cos\theta-2U],v\sin\theta\right)\,.$ (9)

The final speed is then given by the expression

$v_{2}^{2}=[v\cos\theta-2U]^{2}+v^{2}\sin^{2}\theta=v^{2}-4Uv\cos\theta+4U^{2}\,.$
(10)

Note that this discussion assumes that the encounter is symmetric in the
reference frame of the capturing body. This approximation is expected to be
valid because only close encounters with the body result in capture events,
and such close encounters will be symmetric to leading order.

The discussion thus far has implicitly assumed that the target is moving in
the $-{\hat{x}}$ direction. In general, however, the target can also have a
${\hat{y}}$ component to its velocity. Because of the geometry of the
encounter, however, only the ${\hat{x}}$ component of the rock velocity
changes (in this approximation). We can thus incorporate the more general case
by interpreting the velocity $U$ as the component of the target velocity in
the $-{\hat{x}}$ direction. With this definition of $U$, the final speed still
obeys equation (10).

###  2.2 Solar Close Encounters

When the rock is far away from the Sun, it effectively orbits the center of
mass of the system. When the radial distance of the rock becomes sufficiently
small, however, its orbit is determined by the location of the Sun. We can
delineate the boundary between these two regimes by considering the
acceleration (and hence forces) in an accelerating reference frame. As
expected, the orbit of Jupiter provides an approximate boundary between the
outer problem (hyperbolic orbit about the center of mass of the solar system)
and the inner problem (close encounter with the Sun).

When the rock enters the sphere of influence of the Sun, its speed is given by

$v^{2}=v_{\infty}^{2}+{2GM\over r}\,,$ (11)

where $r<a_{J}$ is the location of the rock. It will then execute a
(hyperbolic) orbit around the Sun. Due to the motion of the Sun about the
center of mass of the solar system, the post-encounter velocity will be given
by

$v_{2}^{2}=v^{2}-4Uv\cos\theta+4U^{2}\,,$ (12)

where $U$ is the component of the solar velocity in the direction of the
perihelion of the orbit and $\theta$ defines the shape of the hyperbola.
Capture of the rock requires that $v_{2}^{2}<v^{2}-v_{\infty}^{2}$, so that we
obtain the constraint

$v_{\infty}^{2}<4Uv\cos\theta-4U^{2}\,.$ (13)

Here, the angle $\theta$ is determined by the parameters of the original
hyperbolic orbit about the Sun, so that

$\cos\theta={1\over e}={|a|\over(a^{2}+b^{2})^{1/2}}\,,$ (14)

where $a$ is the semi-major axis and $b$ is the impact parameter. The capture
constraint thus becomes

$v_{\infty}^{2}<4\alpha{m\over M}\left({GM\over
a_{J}}\right)^{1/2}\left[v_{\infty}^{2}+{2GM\over
r_{p}}\right]^{1/2}{|a|\over(a^{2}+b^{2})^{1/2}}+{\cal O}\left({m^{2}\over
M^{2}}\right)\,,$ (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 $\alpha$ such that

$U\equiv\alpha{m\over M}\left({GM\over a_{J}}\right)^{1/2}\,,$ (16)

where $m$ is the mass of Jupiter. Since the speed of the incoming rock
$v\gg{U}$ for the close encounters of interest, we ignore the $U^{2}$ term.
Finally, we evaluate the rock velocity at the perihelion position $r_{p}$ (see
equation [4]), as this location corresponds to where the close encounter takes
place. Working to consistent order, the expression for the capture constraint
can be written in the form

$v_{\infty}^{2}<4\alpha{m\over M}\left({GM\over
a_{J}}\right)^{1/2}\left[{4GM|a|\over b^{2}}\right]^{1/2}=8\alpha{m\over
M}\left({GM\over a_{J}}\right)^{1/2}{GM\over bv_{\infty}}\,.$ (17)

The constraint can be written as a limit on the impact parameter, i.e.,

$b<8\alpha{m\over M}\left({GM\over
a_{J}}\right)^{3/2}\,v_{\infty}^{-3}a_{J}\approx 93\,\,{\rm
au}\,\,\alpha\left({v_{\infty}\over 1\,\,{\rm km/s}}\right)^{-3}\,.$ (18)

If one requires that the rocky body is not only captured, but is captured into
an orbit with semi-major axis less than some maximum value $a_{\rm max}$, then
the left-hand-side of equation (17) can be replaced with
$v_{\infty}^{2}+v_{\rm x}^{2}$, where $v_{\rm x}^{2}\equiv$ $GM/a_{\rm max}$.
Finally, note that this treatment implicitly assumes that $U>0$. If the Sun is
moving in the opposite direction, 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

$\sigma=64\pi\alpha^{2}a_{J}^{2}\left({m\over M}\right)^{2}\left({GM\over
a_{J}}\right)^{3}{1\over v_{\infty}^{2}(v_{\infty}^{2}+v_{\rm x}^{2})^{2}}\,.$
(19)

This cross section is specified up to the constant $\alpha$, which is expected
to be of order (but less than) unity. This form is consistent with those
derived earlier by other means (Heggie, 1975; Pineault & Duquet, 1993;
Valtonen, 1983). Notice that this derivation breaks down for sufficiently high
incoming speeds, $v_{\infty}\,\raise 1.29167pt\hbox{$>$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,8$ km/s, as shown in Appendix A.

###  2.3 Planetary Close Encounters

Another channel for capture occurs through close encounters with the giant
planets, most often Jupiter, which will be considered in this discussion.
Equation (5) represents the cross section for an incoming rock to enter the
sphere of radius $a_{J}$. Only a fraction of the incoming trajectories
$f_{1}=R_{\scriptscriptstyle SoI}^{2}/4a_{J}^{2}$ will enter the sphere of
influence of Jupiter,333Note that the sphere of influence, as defined here,
corresponds to the location where the incoming trajectory switches from a two-
body problem with central mass $M$ to a two-body problem with central mass $m$
in the matched conics approximation. The boundary $R_{\scriptscriptstyle SoI}$
is comparable to, but not equivalent to, the Hill radius
$R_{H}=a(m/3M)^{1/3}$. delineated by $R_{\scriptscriptstyle
SoI}\approx{a_{J}}(m/M)^{2/5}$ (Bate et al., 1971). However, not all of the
orbits that enter the sphere of influence will pass close enough to the planet
to experience significant deflection. As a result, we must estimate the
smaller fraction $f_{2}$ of trajectories that allow for capture.

As a rough approximation, significant deflection requires $\cos\theta$ to be
of order (but still less than) unity, which in turn implies
$b_{hp}\sim|a|_{hp}$ (equation [14]), where $(a_{hp},b_{hp})$ correspond to
the elements of the hyperbolic orbit around the planet. When the rock
encounters the planet, its speed in the solar reference frame is given by
equation (11) evaluated at $r\approx a_{J}$. The asymptotic speed
$(v_{\infty})_{hp}$ for the hyperbolic orbit about the planet depends on the
planetary motion, but will typically be of the same order. We can thus write

$(v_{\infty}^{2})_{hp}=v_{\infty}^{2}+\beta{GM\over a_{J}}\equiv
v_{\infty}^{2}+v_{\rm z}^{2}\,,$ (20)

where $\beta$ is a dimensionless factor of order unity and where the second
equality defines the velocity scale $v_{\rm z}$. The semi-major axis of the
hyperbolic planetary encounter is given by

$(|a|)_{hp}={Gm\over(v_{\infty}^{2})_{hp}}={Gm\over v_{\infty}^{2}+v_{\rm
z}^{2}}\sim{m\over M}a_{J}\,.$ (21)

Since we require $b_{hp}\,\raise 1.29167pt\hbox{$<$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,|a|_{hp}$ and $|a|_{hp}\ll R_{\scriptscriptstyle
SoI}$, the fraction $f_{2}=a_{hp}^{2}/4a_{J}^{2}$. The resulting cross section
for capture due to planetary encounters has the form

$\sigma={\gamma\pi\over 2}{GM\over v_{\infty}^{2}}a_{J}\left({Gm\over
a_{J}\left(v_{\infty}^{2}+v_{\rm z}^{2}\right)}\right)^{2}={\gamma\pi
a_{J}^{2}\over 2}\left({m\over M}\right)^{2}\left({GM\over
a_{J}}\right)^{3}{1\over v_{\infty}^{2}\left(v_{\infty}^{2}+v_{\rm
z}^{2}\right)^{2}}\,,$ (22)

where we have introduced a dimensionless factor $\gamma$ that is expected to
be of order unity. Note that this expression has a form similar to that of
equation (19), which corresponds to the capture cross section for solar
encounters. Keep in mind, however, that the velocity scales are different and
are expected to obey the ordering $v_{\rm x}<v_{\rm z}$.

###  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

$v_{2}^{2}\approx v^{2}-4Uv\cos\theta\,.$ (23)

The semi-major axis $a_{b}$ of the bound orbit is defined so that

${GM\over a_{b}}=2{GM\over r}-v_{2}^{2}=4Uv\cos\theta-v_{\infty}^{2}\,.$ (24)

Let us now define a scale length $b_{0}$ according to

$b_{0}\equiv 8{m\over M}\left({GM\over a_{J}}\right)^{3/2}{a_{J}\over
v_{\infty}^{3}}\alpha\,\sim 100\,\,{\rm au}\left({v_{\infty}\over 1\,{\rm
km/s}}\right)^{-3}\,.$ (25)

With this construction, the semi-major axis of the bound orbit is given by

${GM\over a_{b}v_{\infty}^{2}}={b_{0}\over b}-1\qquad\Rightarrow\qquad
a_{b}={a_{0}b\over b_{0}-b}={a_{0}\chi\over 1-\chi}\,,$ (26)

where we let $a_{0}=|a|$ denote the (magnitude of) the semi-major axis of the
initial hyperbolic orbit, and where the final equality defines
$\chi\equiv{b}/b_{0}$. The criterion for obtaining a bound orbit (from the
previous section) is equivalent to the requirement $b<b_{0}$ $(\chi<1)$. Since
the cross section depends on $b^{2}$, the distribution of impact parameters
will be weighted towards larger values. This finding, in turn, implies that
typical bound orbits will have final semi-major axes comparable to the
starting (negative, hyperbolic) semi-major axis of the incoming orbit. For
$v_{\infty}=1$ km/s, for example, bound orbits will typically have $a_{b}\sim
1000$ au. In order to obtain tighter orbits comparable to the size of the
solar system (or even the Kuiper belt), we need $a_{b}\,\raise
1.29167pt\hbox{$<$}\kern-7.5pt\lower 3.01385pt\hbox{$\sim$}\,100$ au, which in
turn implies that $b\,\raise 1.29167pt\hbox{$<$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,b_{0}/10\sim 10$ au.

If we assume that the impact parameters $b$ are uniformly distributed over an
area, with a maximum value $b_{0}$, then the probability distribution for the
dimensionless quantity $\chi$ has the simple form $dP=2\chi d\chi$. Using
equation (26), we can determine the probability distribution for the semi-
major axes of the bound orbits, i.e.,

${dP\over da_{b}}={2a_{b}a_{0}\over(a_{b}+a_{0})^{3}}\,.$ (27)

As written, this distribution is normalized over the interval
$0<a_{b}<\infty$.

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
the planets), so that quasi-stable orbits will display a different
distribution (which should be explored further in future work).

##  3 Numerical Results

The cross sections derived in the previous section made use of a number of
approximations. In this section we use a suite of more than 500 million
simulations to numerically compute the capture cross section.

###  3.1 Simulation Details

We sample rocks of mass $10^{-9}$ $M_{\odot}$ isotropically on the sphere at a
barycentric distance of $10^{9}$ au. Each rock’s velocity unit vector is
uniquely defined by its position on the sphere, pointing directly toward the
solar system barycenter. We then assign each rock an impact parameter at some
random angle in its plane tangent to the sphere. We randomly sample the impact
parameter uniformly given the condition that the maximum pericenter distance
$q_{\text{max}}\leq 12$ au—comfortably above the largest pericenter distance
for capture not attributable to chance close encounters with a giant planet.
Finally we scale the rock’s velocity unit vector by a factor

$v=\sqrt{v_{\infty}^{2}+\frac{2\mu}{r}}$ (28)

where $\mu=G\sum_{i}m_{i}$ and
$i\in\\{\text{Sun,Jupiter,Saturn,Uranus,Neptune}\\}$. In Equation 28,
$v_{\infty}$ is the rock’s field (or cluster) velocity at infinity, and the
second term accounts for the kinetic energy that the rock gains by falling
from infinity to a barycentric distance $r$.

The above procedure gives us a state vector, from which we compute a body’s
Keplerian orbital elements. To save computation time, we use these elements to
propagate each rock along its unperturbed hyperbolic orbit to a barycentric
distance 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 $10^{9}$, since the solar system’s quadrupole term goes like $r^{-3}$.
Once we have performed the analytic propagation of the rock, we use NASA’s
development ephemerides to initialize the solar system at a random date in a
200–year range around the arbitrarily chosen Julian Date $2459010.5$. This
ensures that our results 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 part in
$10^{14}$—much smaller than the fraction of the system’s energy attributable
to the rock. Therefore we are confident in the accuracy of our integrations.

For each integration, there are three possible outcomes: the rock may be
captured; undergo a collision with another body; or be ejected from the
system. If at any point during the simulation the rock’s energy drops below
zero, we consider it to be captured and end the simulation. If the rock
undergoes a collision or if the rock is unbound and exiting the solar system
with a barycentric distance greater than 40 au, we end the simulation and
determine that the rock was not captured. We then follow up on our captured
objects, integrating for $51\%$ of an orbital period to ensure that each
object is truly bound (as opposed to having a transient bound osculating semi-
major axis due to the phases of the giant planets). If during our followup the
object’s apocenter distance exceeds 1 parsec, we consider it to be lost to
cluster or galactic tides.

Current models of solar system formation predict that the giant planets formed
in a more compact arrangement, and then migrated to their current orbits. To
account for this we ran a set of simulations with the compact solar system
model presented in Tsiganis et al. (2005). The cross section we calculate with
this model differs from that calculated using the solar system at the current
epoch by less than 1 percent, so our calculations should be equally applicable
to the 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

$\sigma=\frac{2\pi b_{\text{max}}}{N}\sum_{i}b_{i}\delta_{i}$ (29)

where $b_{\text{max}}$ is the maximum impact parameter sampled, $N$ is the
number of events, and $\delta_{i}$ is a Kronecker delta that is 1 if the event
resulted in capture, and 0 otherwise. We display our results in Figure 2. As
we expect, $\sigma(v_{\infty})$ goes like $v_{\infty}^{-2}$ in the low-speed
limit, and like $v_{\infty}^{-6}$ in the high-speed limit. To facilitate the
use of the cross section in analytic calculations we fit $\sigma(v_{\infty})$
with the simple function

$\sigma(v_{\infty})=\frac{\sigma_{0}}{u^{2}(u^{2}+1)^{2}}$ (30)

where $u\equiv v_{\infty}/v_{\sigma}$ and $v_{\sigma}$ is a velocity scale
determined by the properties of the planet ejecting the rock. We find the data
are best fit by parameter values $\sigma_{0}=232,250$ au2 and
$v_{\sigma}=0.4179$ km/s. Keep in mind that these cross sections apply for
capture into any bound orbit.

The scale $\sigma_{0}$ for the cross section obtained from fitting our
numerical results can be compared to the analytic estimates of the previous
section. If we evaluate equation (19) in the high speed limit, then agreement
between the analytic and numerical estimates implies that $\sigma_{0}$ =
$64\pi\alpha^{2}a_{J}^{2}(m/M)^{2}(v_{\sigma}/v_{J})^{6}$, where $v_{J}$ is
the orbital speed of Jupiter. The expressions are equal if the dimensionless
parameter $\alpha\approx 0.21$. The analytic and numerical results are in
agreement for all incoming speeds if we identify the scales $v_{x}$ and
$v_{\sigma}$, which is equivalent to considering captures with a maximum
(post-encounter) semimajor axis $a_{\rm max}\approx 5090$ au. Notice also that
$v_{\sigma}\sim{v_{x}}\sim(Gm/a_{J})^{1/2}$ (see also Appendix C). Similarly,
equation (22) agrees with the numerical result in the high speed limit if the
dimensionless parameter $\gamma\approx 5.8$.

Figure 2: Capture cross section (in au2) as a function of the asymptotic speed
$v_{\infty}$ (in km/s). The black points represent the numerically calculated
cross sections, and the corresponding error bars represent one standard
deviation. The red curve represents the best fit of equation (30) to the data.
The blue dashed line shows a power-law of the form $\sigma\sim
v_{\infty}^{-6}$, as expected in the limit of high velocity. The green dashed
line shows a power-law of the form $\sigma\sim v_{\infty}^{-2}$, as expected
in the low velocity limit. The shaded regions indicate the parameter space
where captures due the Sun (green) and planets (blue) dominate, although the
boundaries are not sharp.

Note that the ensemble of numerical simulations is confined to speeds
$v_{\infty}\leq 15$ km/s. This upper limit is invoked for a number of reasons.
Due to the steep power-law fall-off of the capture cross section, relatively
few capture events take place at higher speeds, so additional computation
leads to diminishing returns. In addition to the steep dependence with
$v_{\infty}$, the numerical data indicate that the power-law begins to break
at a comparable speed. Some type of break is expected: For $v_{\infty}$
greater than $\sim 10$ km/s, capture by close encounters with the Sun becomes
ineffective (see Appendix A). It is noteworthy that the capture cross section
at $v_{\infty}\approx 13$ km/s is comparable to the geometrical area of the
Sun ($\sim 7\times 10^{-5}$ au2). For larger encounter speeds, incoming rocky
bodies are thus more likely to collide with the Sun than be captured into a
bound orbit.444For completeness, we note that due to gravitational focusing,
the collision cross section with the Sun is larger than the capture cross
section for speeds $v_{\infty}>2-3$ km/s.

##  4 Analysis of Captured Objects

In this section we examine the orbital elements of our captured objects to
gain insight into the mechanics of the capture process. In Figure 3, we show
the impact parameter (and pericenter distance) distribution of the unperturbed
orbits of our captured objects for asymptotic speeds $v_{\infty}$ of 1 and 2
km/s. Each histogram displays a clear relative peak at the pericenter
distances corresponding to the orbit of Jupiter, along with a much smaller
peak for the orbit of Saturn. Comparison of the two histograms indicates that
somewhere between 1 and 2 km/s, the dominant capture process switches from
that due to the motion of the solar system barycenter to close encounters with
a giant planet (especially Jupiter).

Figure 3: Histograms of the (unperturbed) impact parameter distribution of
captured objects for asymptotic speeds $v_{\infty}$ of 1 km/s (top) and 2 km/s
(bottom). For convenience, we also indicate the pericenter distance of the
unperturbed orbit. Both plots show relative peaks at pericenter distances
corresponding to the orbits of Jupiter and Saturn. As $v_{\infty}$ increases,
close encounters with the giant planets become more important for capture.

In Figure 4, we show the post-capture eccentricity as a function of semi-major
axis for a subset of the captured objects with $v_{\infty}$ of 1 km/s (top
panel) and 2 km/s (bottom panel). The figure also includes equi-pericenter
curves corresponding to integer multiples of the spheres of influence of
Jupiter and Saturn. The numerical results for captures display a relative
overdensity of points with pericenter distances at Jupiter and Saturn,
indicating that these captures are (likely) attributable to close encounters.

Figure 4: Post-capture eccentricity versus semi-major axis of captured objects
for incoming speeds of 1 km/s (top) and at 2 km/s (bottom). The orange and red
regions correspond to integer multiples of the radius of influence centered at
the the semimajor 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 eccentricity for $v_{\infty}$ = 0.5, 1, and 2
km/s. While captures become increasingly rare for higher-velocity events, the
resulting semi-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 $a\gtrsim
1000$ au are more likely to be stripped from the solar system by interacting
with passing stars (in the solar birth cluster) or the galactic tides (in the
field).

It is noteworthy that capture events readily produce highly-inclined and even
retrograde objects. This finding indicates that capture is yet another
potential channel for the production of the observed populations of highly-
inclined and retrograde centaurs, which are currently best explained by the
putative Planet Nine (Batygin et al., 2019). However, it is important to note
that the orbits of the captured objects will evolve over time. As a result,
the captured objects do not represent a long-term stable population. As the
orbits of the captured objects evolve, some will become more eccentric until
they collide with the Sun; some will undergo scattering events or interactions
with the Galactic tides and be ejected from the solar system; some will be
frozen into the inner Oort cloud by passing stars; and some will continue to
evolve on stable or quasi-stable orbits (either by becoming caught in
resonances with the giant planets, or by achieving orbits that otherwise avoid
close encounters).

Because most capture events resulted in high-eccentricity orbits, we have
rather low statistics for small values of eccentricity. Despite the data
limitations, though, it is clear that as $v_{\infty}$ increases, the low-
eccentricity tail of the distribution becomes fatter.

Figure 5: (Top) Gaussian kernel density estimate of the post-capture
inclination distribution (with $i$ measured from the ecliptic plane) of
captured objects at speeds $v_{\infty}$ = 0.5, 1, and 2 km/s. (Center)
Gaussian kernel density estimate of the post-capture semi-major axis
distribution of captured objects for $v_{\infty}$ = 0.5, 1, and 2 km/s.
(Bottom) Relative fraction (in logarithmic scale) of the post-capture
eccentricity distribution of captured objects for $v_{\infty}$ = 0.5, 1, and 2
km/s. Note that these curves represent probability distribution functions;
there will be fewer captures in the high-speed case than in the low-speed
case.

##  5 Applications

###  5.1 Velocity Averaged Cross Sections

This paper determines the velocity dependent cross section
$\sigma(v_{\infty})$, which can be fit with a function of the form

$\sigma(v_{\infty})={\sigma_{0}\over u^{2}(1+u^{2})^{2}}\qquad{\rm
where}\qquad u\equiv{v_{\infty}\over v_{\sigma}}\,.$ (31)

The capture rate for rocky bodies by our solar system is given by

$\Gamma=n_{R}\langle\sigma v_{\infty}\rangle\,,$ (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

$\langle\sigma
v_{\infty}\rangle=\int_{0}^{\infty}v_{\infty}f(v_{\infty})\sigma(v_{\infty})dv_{\infty}\,,$
(33)

where $f(v_{\infty})$ is the distribution of encounter velocities of the rocky
bodies.

The distribution of relative speeds $f(v_{\infty})$ depends on the
environment. In the solar birth cluster, $f(v_{\infty})$ is determined by the
processes that eject the rocky bodies from their original planetary systems.
In general, the clusters are not sufficiently long-lived for the rocks to
attain a thermal distribution of speeds. Instead, they are expected to retain
the velocity distribution resulting from the ejection process. If ejection
occurs through scattering interactions with giant planets, then
$f(v_{\infty})$ takes the approximate form

$f(v_{\infty})={4v_{\infty}/v_{p}\over(1+v_{\infty}^{2}/v_{p}^{2})^{3}}\,,$
(34)

where the velocity scale $v_{p}^{2}\approx GM_{\ast}/a_{p}$, where $a_{p}$ is
the semi-major axis of the planet that scatters the rocks (e.g., see Moorhead
& Adams 2005 for a derivation).

Note that the distribution (34) is normalized over the entire interval
$0\leq{v_{\infty}}\leq\infty$. In practice, the distribution will have a
maximum value determined by the escape speed from the planets that scatter the
rocky bodies. Notice also that the full distribution will be a convolution of
the distribution of ejection speeds from each planet that scatters rocky
material. As an approximation, we consider only a single distribution and
interpret the velocity scale $v_{p}$ as a typical value. As a result, $v_{p}$
is expected to be comparable to the orbit speed of outer planets, i.e.,
$v_{p}\sim 10$ km/s. Finally, we are assuming that equation (34) corresponds
to the distribution of relative speeds between the rocks and the 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

$\langle\sigma v_{\infty}\rangle=\sigma_{0}{v_{\sigma}^{3}\over
v_{p}^{2}}\int_{0}^{\infty}{4du\over(1+\eta^{2}u^{2})^{3}(1+u^{2})^{2}}=\sigma_{0}{v_{\sigma}^{3}\over
v_{p}^{2}}I(\eta)\,,$ (35)

where $u=v_{\infty}/v_{\sigma}$ (as before), we have defined $\eta\equiv
v_{\sigma}/v_{p}$, and where the second equality defines the integral function
$I(\eta)$. The dimensionless function $I(\eta)$ can be evaluated to obtain

$I(\eta)=\pi{(1+\eta)(1+3\eta)+3\eta^{3}/4\over(1+\eta)^{4}}\,.$ (36)

Note that $I\to\pi$ in the limit $\eta\to 0$, and in practice $\eta\sim 1/10$.
As a result, a good approximation for the capture cross section takes the form

$\langle\sigma v_{\infty}\rangle\approx\pi\sigma_{0}{v_{\sigma}^{3}\over
v_{p}^{2}}\,.$ (37)

For comparison, we can determine the velocity averaged cross section for the
scenario where the cluster rocks are virialized and have the same (Maxwellian)
velocity distribution as the stars.555In general, we expect the stars to reach
virial equilibrium much faster than the rocky ejecta. The stars start their
cluster trajectories with sub-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 to slow them down. Moreover, the stellar virialization starts as
soon as stars form, whereas the planet formation and the subsequent ejection
of rocks occurs many Myr later. In this limit,
$\langle\sigma{v_{\infty}}\rangle$ can be written in the form

$\langle\sigma v_{\infty}\rangle=\sigma_{0}{v_{\sigma}^{2}\over
s}\sqrt{2\over\pi}\int_{0}^{\infty}{v_{\infty}dv_{\infty}\over
s^{2}}{\exp[-v_{\infty}^{2}/2s^{2}]\over(1+v_{\infty}^{2}/v_{\sigma}^{2})^{2}}\,,$
(38)

where $s$ is the velocity dispersion of the distribution. Note that the value
of $s$ for the distribution of relative speeds is larger than the value
$s_{0}$ for the velocity distribution of the stars in the clusters
($s=\sqrt{2}s_{0}$). Here we define the variable $w\equiv v_{\infty}/s$ and
the parameter $\xi=s/v_{\sigma}$, so that

$\langle\sigma v_{\infty}\rangle=\sigma_{0}{v_{\sigma}^{2}\over
s}\sqrt{2\over\pi}\int_{0}^{\infty}wdw{\exp[-w^{2}/2]\over(1+\xi^{2}w^{2})^{2}}\equiv\sigma_{0}{v_{\sigma}^{2}\over
s}\sqrt{2\over\pi}J(\xi)\,,$ (39)

where the second equality defines the integral function $J(\xi)$. The exact
form for $J(\xi)$ can be found. If we define
$\mu=1/(2\xi^{2})=v_{\sigma}^{2}/(2s^{2})$, then

$J(\xi)=J(\mu)=\mu\left[1-\mu\,{\rm e}^{\mu}\,E_{1}(\mu)\right]\,,$ (40)

where $E_{1}(x)$ is the exponential integral (Abramowitz & Stegun, 1972). In
the limit $\xi\to 0$ ($\mu\to\infty$), the function $J=1$; in the opposite
limit $\xi\gg 1$ ($\mu\to 0$), $J=\mu=1/(2\xi^{2})$. A good working
approximation for the cross section of equation (39) thus has the form

$\langle\sigma v_{\infty}\rangle\approx\sigma_{0}{v_{\sigma}^{4}\over
s}{\sqrt{2/\pi}\over v_{\sigma}^{2}+2s^{2}}\,,$ (41)

which is exact in the limits and has a relative error less than $\sim 20\%$
over the entire range $0\leq\xi\leq\infty$. One can use equations (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 Sun remained in its birth cluster. The
capture rate is given by equation (32), with velocity-averaged cross section
specified through equations (39) and (40). The rocky bodies will have a
distribution of sizes $g(R)$, which is defined here such that

$n_{R}=\int_{0}^{\infty}g(R)dR\qquad{\rm
and}\qquad\rho_{R}=\int_{0}^{\infty}g(R)m(R)dR\,,$ (42)

where $m(R)$ is the mass of the rock as a function of its size. With these
definitions, the capture rate $\Gamma$ can be converted into a mass accretion
rate given by

${\dot{M}}=\rho_{R}\langle\sigma v_{\infty}\rangle\,,$ (43)

where $\rho_{R}$ is the mass density of the cluster in the form of rocks.
Given that each planetary system in the cluster is expected to eject a few
Earth masses of rocky material (e.g., Rice & Laughlin 2019), the density
$\rho_{R}$ is given by

$\rho_{R}={\alpha M_{\oplus}N_{\ast}\over V}=\alpha M_{\oplus}n_{\ast}\,,$
(44)

where $\alpha$ is a dimensionless factor of order unity and $n_{\ast}$ is the
number density of stars. For completeness, note that the inclusion of icy
planetesimals will increase this density estimate. In any case, the total mass
in rocky bodies captured by the solar system during its cluster phase can be
written in the form

$(\Delta M)_{R}=\alpha
M_{\oplus}\left[\int_{0}^{\infty}n_{\ast}dt\right]\langle\sigma
v_{\infty}\rangle\equiv\alpha M_{\oplus}\langle
n_{\ast}\rangle\tau\langle\sigma v_{\infty}\rangle\,.$ (45)

The final equality defines the mean density of the cluster, where $\tau$ is
its effective lifetime. A number of studies have found upper bounds on the
product $\langle{n_{\ast}}\rangle\tau$ by requiring that the solar system is
not overly disrupted, including considerations 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 indicates that the product is bounded by
$\langle{n_{\ast}}\rangle\tau\,\raise 1.29167pt\hbox{$<$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,$ $2\times 10^{4}$ pc-3 Myr.

If we take $v_{\sigma}$ = 0.5 km/s, $v_{p}$ = 10 km/s, and $\sigma_{0}$ =
$2\times 10^{5}$ au2, then the velocity averaged capture cross section becomes
$\langle\sigma{v_{\infty}}\rangle$ $\approx$ 800 au2 km/s. With this cross
section, the total mass in captured rocky material from equation (45) is about
$(\Delta M)_{R}\sim 10^{-3}M_{\oplus}$. 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 $\sim 1\%$ (Brasser et al., 2006), so we would
expect $\sim 10^{-5}M_{\oplus}$ 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), then some fraction of the material
will leave the cluster during its first crossing. The high speed tail of the
velocity distribution will thus be de-populated. In practice, however, most of
the capture events arise from the low-speed portion 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 the rocky material to encounter the solar
system with a velocity distribution comparable to that of the field stars,
i.e., a Gaussian distribution with $s\sim 40$ km/s. In this case, the velocity
averaged cross section $\langle\sigma{v_{\infty}}\rangle\approx 0.08$ au2
km/s. If we also assume that each planetary system ejected the same mass in
rocks during its formative phases, then the density of rocky material will be
proportional to the stellar density (we are thus assuming negligible losses).
As a result, the product $\langle{n_{\ast}}\rangle\tau\sim 460$ pc-3 Myr, and
the expected mass in captured rocks is about $(\Delta M)_{R}\approx$ $2\times
10^{-9}M_{\oplus}$. Using the approximate retention rate of $1\%$, we would
expect only $\sim 2\times 10^{-11}M_{\oplus}$ of these rocks to remain in the
inner Oort cloud. This inventory of captured alien material from the field is
exceedingly small, roughly the equivalent of 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 ordinary solar 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 Oort cloud, so that the retention fraction
could be smaller than assumed here (most of the captured interstellar bodies
initially have Jupiter-crossing orbits, whereas the planetesimals in the Oort
cloud could have different origins). In any case, most of the captured objects
will be ejected, and some will eventually collide with the Sun. Although these
calculations provide working order-of-magnitude estimates, in forthcoming work
we will refine these projections by numerically investigating the long-term
behavior of the captured bodies from this work.

##  6 Conclusions

This paper has revisited the problem of capturing interstellar objects on
initially hyperbolic trajectories into bound states. Using an ensemble of 500
million numerical fly-by simulations, the main result of this study is the
determination of our solar system’s capture cross section as a function of
encounter speed (see Figure 2). The resulting capture cross section shows the
power-law velocity dependence $\sigma\sim v_{\infty}^{-2}$ in the limit of low
speeds and the dependence $\sigma\sim v_{\infty}^{-6}$ in the limit of high
speeds. The capture cross section $\sigma(v_{\infty})$ over the entire range
of asymptotic speeds can be fit with the function given in equation (31).

This paper also presents an analytic treatment of the capture problem using
the approximation of matched conics and the (inverse) gravitational slingshot
effect (Section 2). These arguments show that capture by both close encounters
with the Sun (Section 2.2), and by close encounters with a giant planet
(Section 2.3), have the same nearly velocity dependence as that seen in the
numerical simulations (namely $\sigma\sim v_{\infty}^{-2}$ at low speeds and
$\sigma\sim v_{\infty}^{-6}$ 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 with
Jupiter dominate. Close encounters with the other giant planets contribute to
the cross section, but do not dominate the dynamics. More specifically, for
the particular case of $v_{\infty}$ = 1 km/s, capture events due to close
encounters with Jupiter are $\sim 100$ times more likely than captures due to
Saturn. The frequency of close encounters with Uranus and Neptune are smaller
(than for Saturn) by an additional factor of $\sim 100$.

With the capture cross section as a function of velocity specified, the
effective mean cross section $\langle v\sigma\rangle$/$\langle{v}\rangle$ can
be determined for any distribution of encounter speeds. For the case of a
Maxwellian distribution and a power-law distribution 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 $(\Delta M)_{R}$ in the Oort cloud that originates from other planetary
solar systems (Section 5.2). The mass accreted while the Sun lived within its
birth cluster is of order $(\Delta M)_{R}\sim 10^{-5}M_{\oplus}$, 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 research remain. The simulations of
this paper consider the capture of interstellar objects and the resulting
cross sections include all capture events, independent of their residence time
in the solar system as bound objects. Future work should determine how long
captured bodies can remain bound to the Sun, since many such objects are
expected to be ejected from the system or to collide with other solar system
members. The residence time (ejection time) should thus be determined for each
type of orbit displayed by the captured objects. With these results in place,
one can make a refined estimate of the current population of alien objects in
the solar system, along with their expected orbital properties.

We would like to thank David Gerdes, Hsing-Wen Lin, and Larissa Markwardt for
helpful discussions during the preparation of this manuscript. This material
is based upon work supported by the National Aeronautics and Space
Administration under Grant No. NNX17AF21G issued through the SSO Planetary
Astronomy Program and by the National Science Foundation under Grant No.
AST-2009096.

##  Appendix A Upper Bound on Incoming Speed for Capture

In this Appendix, we find upper limits on the asymptotic speed, i.e., the
largest speed for which a rock can be captured 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,

$v_{\infty}^{2}<4Uv\cos\theta-4U^{2}<4Uv\,,$ (A1)

where the second inequality follows because $\cos\theta<1$ and $4U^{2}>0$. The
largest speed $U$ possible for the Sun, relative to the center of mass, is the
orbit speed due to the giant planets, where

$U={m\over M}\left({G(M+m)\over a_{p}}\right)^{1/2}\approx{m\over
M}\left({GM\over a_{p}}\right)^{1/2}\,,$ (A2)

where the second expression is consistent with the ordering approximation and
where we have included the reflex speed due 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

$U<{2m_{J}\over M}\left({GM\over a_{J}}\right)^{1/2}\,.$ (A3)

Thus far, our upper bound has the form

$v_{\infty}^{2}<8{m_{J}\over M}\left({GM\over
a_{J}}\right)^{1/2}v=8{m_{J}\over M}\left({GM\over
a_{J}}\right)^{1/2}\left[v_{\infty}^{2}+{2GM\over R_{\odot}}\right]^{1/2}\,,$
(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

$v_{\infty}^{2}<8{m_{J}\over M}\left({GM\over
a_{J}}\right)^{1/2}\left({2GM\over
R_{\odot}}\right)^{1/2}=8\sqrt{2}{Gm_{J}\over(a_{J}R_{\odot})^{1/2}}\approx(8\,{\rm
km/s})^{2}\,.$ (A5)

To derive a rigorous version of the upper bound, we define two velocity
components

$u\equiv 8{m_{J}\over M}\left({GM\over a_{J}}\right)^{1/2}\qquad{\rm
and}\qquad w\equiv\left({2GM\over R_{\odot}}\right)^{1/2}\,,$ (A6)

so that the limit takes the form

$v_{\infty}^{2}<uw\left[1+{u^{2}\over 4w^{2}}\right]^{1/2}+{u^{2}\over 2}\,.$
(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 of the Sun ($\sim 620$ km/s),
we find $u\ll{w}$. In this limit, the full expression of equation (A7) reduces
to the form $v_{\infty}^{2}<uw$, 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 energy through 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.,

$v_{2}^{2}<2v_{\rm orb}^{2}\,,$ (A8)

where the orbital speed $v_{\rm orb}$ of the planet is a measure of the depth
of the gravitational potential well at the location of the planet (the
location of the encounter). We also assume that the incoming rock obeys
conservation of energy so that

${1\over 2}v_{\infty}^{2}=E={1\over 2}v^{2}-{GM\over r}\approx{1\over
2}v^{2}-v_{\rm orb}^{2}\qquad\Rightarrow\qquad v^{2}=v_{\infty}^{2}+2v_{\rm
orb}^{2}\,.$ (A9)

Using this expression in the result for the post-encounter speed from before,
we obtain

$v_{\infty}^{2}+4U^{2}<4U\cos\theta[v_{\infty}^{2}+2v_{\rm orb}^{2}]^{1/2}\,.$
(A10)

If we take $U=v_{\rm orb}\cos\phi$, this expression can be rewritten in the
form

$v_{\infty}^{4}<8v_{\rm
orb}^{2}v_{\infty}^{2}\cos^{2}\phi(2\cos^{2}\theta-1)+16v_{\rm
orb}^{4}\cos^{2}\phi(2\cos^{2}\theta-\cos^{2}\phi)\,.$ (A11)

We can thus obtain an upper limit on the asymptotic speed for an object to be
captured by taking $\cos\theta=1=\cos\phi$, i.e.,

$v_{\infty}^{4}<8v_{\rm orb}^{2}v_{\infty}^{2}+16v_{\rm orb}^{4}\,,$ (A12)

which leads to the bound

$v_{\infty}<2v_{\rm orb}(1+\sqrt{2})^{1/2}\,.$ (A13)

The fastest orbit is that of Jupiter, where $v_{\rm orb}\approx$ 13 km/s, so
we have the limit $v_{\infty}\,\raise 1.29167pt\hbox{$<$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,40$ km/s.

We thus find that the maximum possible speed (for capture) is larger for the
channel involving close approaches to planets (in particular Jupiter) than for
close approaches to the Sun. On the other hand, planet encounters are expected
to occur much less frequently. Note that these limits do not account for
effects such as radiation pressure or atmospheric drag. It is therefore
possible in principle to capture rocks with larger $v_{\infty}$, but such
events would be exceptionally rare.

##  Appendix B Rock Capture by Circumstellar Disks

Rock capture could also take place due to gas drag if interstellar bodies
enter the solar nebula while it retains its gaseous component. This Appendix
explores the efficacy of this process.

If a rock has speed $v_{0}$ when it enters the gaseous region of the disk, it
will have final speed $v_{f}$ given by

$v_{f}=v_{0}\exp[-\rho A\ell/m]\,,$ (B1)

where $\rho$ is the gas density, $A$ is the rock area, $m$ is the rock mass,
and $\ell$ is the length traveled. If a rock comes from infinity with speed
$v_{\infty}$, then it will have a larger velocity when it hits the disk, so
that $v_{0}$ is given by

$v_{0}^{2}=v_{\infty}^{2}+{2GM\over r}\,.$ (B2)

In order for the rock to lose enough energy to enter into a bound orbit, the
final speed must be less than the limit

$v_{f}^{2}=\left[v_{\infty}^{2}+{2GM\over r}\right]\exp\left[-{2\rho
A\ell\over m}\right]<{2GM\over r}\,.$ (B3)

The capture criterion thus becomes

${2\rho A\ell\over m}>\ln\left[1+{v_{\infty}^{2}r\over 2GM}\right]\,.$ (B4)

If the rock passes through the disk vertically, then $\rho\ell=\Sigma$, where
the surface density for the solar nebula can be written in the form

$\Sigma(r)=\Sigma_{1}\left({1{\rm au}\over r}\right)^{3/2}\,,$ (B5)

where $\Sigma_{1}\approx 3000$ g/cm2. If the rock passes through the nebula at
an arbitrary angle $\theta$, then $\rho\ell=\Sigma/\cos\theta$. We can also
write $A=\pi R^{2}$ and $m=\rho_{R}(4\pi/3)R^{3}$ where $R$ is the size of the
rock. If we work in the low speed limit with $v_{\infty}$ = 1 km/s, and write
$R$ in units km and $r$ in units of au, then the criterion for capture reduces
to the simple form

${36\over\cos\theta}>Rr^{5/2}\,.$ (B6)

For the fiducial case where $\cos\theta=1$, the solar nebula can thus capture
incoming rocks with radii $R\,\raise 1.29167pt\hbox{$<$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,1$ km at radial location $r=1$ au. Somewhat larger
rocks can be captured for typical inclined trajectories (e.g., the limit
becomes $R\,\raise 1.29167pt\hbox{$<$}\kern-7.5pt\lower
3.01385pt\hbox{$\sim$}\,2$ km for $\cos\theta=1/2$). Smaller rocks are readily
captured, and rocks as large as dwarf planets are highly unlikely to be
captured (see also Brasser et al. 2007). For incoming trajectories that are
confined to the plane of the disk, the above treatment must be modified to
include the disk structure (i.e., the above treatment does not apply in the
limit $\cos\theta\to 0$).

##  Appendix C Dimensional Analysis

The capture problem has more than one dimensionless field, so that the cross
section of interest cannot be directly determined 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 characterize the systems are thus the masses
$(M,m)$ and the orbital radius of Jupiter $a_{J}$. Note that the eccentricity
of the Jovian orbit is too small to matter, and the radii of the bodies also
do not contribute. The other variables in the problem are the gravitational
constant $G$ and the speed of the incoming rock $v_{\infty}$. Here we want to
determine the cross section as a function of these variables. We thus want to
determine the impact parameter $b$ of the incoming orbits that allow for
capture.

The first estimate for the cross section is thus

$\sigma_{1}=\pi a_{J}^{2}\,.$ (C1)

The remaining (relevant) variables can be combined to construct two
dimensionless fields, i.e.,

$\Lambda_{1}\equiv{GM\over v_{\infty}^{2}a_{J}}\qquad{\rm
and}\qquad\Lambda_{2}\equiv{Gm\over v_{\infty}^{2}a_{J}}\,.$ (C2)

Alternatively, one could define the second field to be the mass ratio $m/M$,
and the quantity $\Lambda_{2}=\Lambda_{1}(m/M)$. At low speeds, the capture
cross section has the form

$\sigma_{2}=\pi a_{J}^{2}\Lambda_{1}\,,$ (C3)

whereas at high speeds the cross section becomes

$\sigma_{3}=\pi a_{J}^{2}\Lambda_{1}\Lambda_{2}^{2}\,.$ (C4)

We can combine all of the above results to write the capture cross section in
terms of the dimensionless fields of the problem:

$\sigma_{4}=\pi
a_{J}^{2}{\Lambda_{1}\Lambda_{2}^{2}\over(1+\Lambda_{2})^{2}}\,.$ (C5)

## References

  * Abramowitz & Stegun (1972) Abramowitz, M., & Stegun, I. A. 1972, Handbook of Mathematical Functions
  * Adams (2010) Adams, F. C. 2010, ARA&A, 48, 47, doi: 10.1146/annurev-astro-081309-130830
  * Adams & Laughlin (2001) Adams, F. C., & Laughlin, G. 2001, Icarus, 150, 151, doi: 10.1006/icar.2000.6567
  * Adams et al. (2006) Adams, F. C., Proszkow, E. M., Fatuzzo, M., & Myers, P. C. 2006, ApJ, 641, 504, doi: 10.1086/500393
  * Adams (1846) Adams, J. C. 1846, MNRAS, 7, 149, doi: 10.1093/mnras/7.9.149
  * Bate et al. (1971) Bate, R. R., Mueller, D. D., & White, J. E. 1971, Fundamentals of astrodynamics.
  * Batygin et al. (2020) Batygin, K., Adams, F. C., Batygin, Y. K., & Petigura, E. A. 2020, AJ, 159, 101, doi: 10.3847/1538-3881/ab665d
  * Batygin et al. (2019) Batygin, K., Adams, F. C., Brown, M. E., & Becker, J. C. 2019, Phys. Rep., 805, 1, doi: 10.1016/j.physrep.2019.01.009
  * Batygin & Brown (2016) Batygin, K., & Brown, M. E. 2016, AJ, 151, 22, doi: 10.3847/0004-6256/151/2/22
  * Becker et al. (2018) Becker, J. C., Khain, T., Hamilton, S. J., et al. 2018, AJ, 156, 81, doi: 10.3847/1538-3881/aad042
  * Binney & Tremaine (2008) Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition
  * Brasser et al. (2006) Brasser, R., Duncan, M. J., & Levison, H. F. 2006, Icarus, 184, 59, doi: 10.1016/j.icarus.2006.04.010
  * Brasser et al. (2007) —. 2007, Icarus, 191, 413, doi: 10.1016/j.icarus.2007.05.003
  * Brown et al. (2004) Brown, M. E., Trujillo, C., & Rabinowitz, D. 2004, ApJ, 617, 645, doi: 10.1086/422095
  * Hands & Dehnen (2020) Hands, T. O., & Dehnen, W. 2020, MNRAS, 493, L59, doi: 10.1093/mnrasl/slz186
  * Heggie (1975) Heggie, D. C. 1975, MNRAS, 173, 729, doi: 10.1093/mnras/173.3.729
  * Jewitt & Luu (2019) Jewitt, D., & Luu, J. 2019, ApJ, 886, L29, doi: 10.3847/2041-8213/ab530b
  * Kouwenhoven et al. (2010) Kouwenhoven, M. B. N., Goodwin, S. P., Parker, R. J., et al. 2010, MNRAS, 404, 1835, doi: 10.1111/j.1365-2966.2010.16399.x
  * Lada & Lada (2003) Lada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57, doi: 10.1146/annurev.astro.41.011802.094844
  * Lagrange (1776) Lagrange, J.-L. 1776, Mémoires de l’ Académie de Berlin, 199 Oeuves complétes VI 255 (Paris, Gauthier-Villars 1869)
  * Laplace (1799-1825) Laplace, P. S. 1799-1825, Traité de Mécanie Céleste (Paris: Gauthier-Villa)
  * Laplace (1846) —. 1846, in Sur les Comètes, Connaissance des temps
  * Le Verrier (1846) Le Verrier, U. J. 1846, Astronomische Nachrichten, 25, 65
  * Li & Adams (2015) Li, G., & Adams, F. C. 2015, MNRAS, 448, 344, doi: 10.1093/mnras/stv012
  * Li & Adams (2016) —. 2016, ApJ, 823, L3, doi: 10.3847/2041-8205/823/1/L3
  * Lingam & Loeb (2018) Lingam, M., & Loeb, A. 2018, AJ, 156, 193, doi: 10.3847/1538-3881/aae09a
  * Luu & Jewitt (2002) Luu, J. X., & Jewitt, D. C. 2002, ARA&A, 40, 63, doi: 10.1146/annurev.astro.40.060401.093818
  * Meech et al. (2017) Meech, K. J., Weryk, R., Micheli, M., et al. 2017, Nature, 552, 378, doi: 10.1038/nature25020
  * Moore et al. (2020) Moore, N. W. H., Li, G., & Adams, F. C. 2020, ApJ, 901, 92, doi: 10.3847/1538-4357/abb08f
  * Moorhead & Adams (2005) Moorhead, A. V., & Adams, F. C. 2005, Icarus, 178, 517, doi: 10.1016/j.icarus.2005.05.005
  * Morbidelli et al. (2020) Morbidelli, A., Batygin, K., Brasser, R., & Raymond, S. N. 2020, MNRAS, 497, L46, doi: 10.1093/mnrasl/slaa111
  * Mustill et al. (2016) Mustill, A. J., Raymond, S. N., & Davies, M. B. 2016, MNRAS, 460, L109, doi: 10.1093/mnrasl/slw075
  * Namouni & Morais (2020) Namouni, F., & Morais, M. H. M. 2020, MNRAS, 494, 2191, doi: 10.1093/mnras/staa712
  * Parker (2020) Parker, R. J. 2020, arXiv e-prints, arXiv:2007.07890.  https://arxiv.org/abs/2007.07890
  * Pfalzner (2013) Pfalzner, S. 2013, A&A, 549, A82, doi: 10.1051/0004-6361/201218792
  * Pineault & Duquet (1993) Pineault, S., & Duquet, J.-R. 1993, MNRAS, 261, 246, doi: 10.1093/mnras/261.2.246
  * Portegies Zwart (2009) Portegies Zwart, S. F. 2009, ApJ, 696, L13, doi: 10.1088/0004-637X/696/1/L13
  * Rein & Spiegel (2015) Rein, H., & Spiegel, D. S. 2015, MNRAS, 446, 1424, doi: 10.1093/mnras/stu2164
  * Rice & Laughlin (2019) Rice, M., & Laughlin, G. 2019, ApJ, 884, L22, doi: 10.3847/2041-8213/ab4422
  * Sheppard & Trujillo (2016) Sheppard, S. S., & Trujillo, C. 2016, AJ, 152, 221, doi: 10.3847/1538-3881/152/6/221
  * Siraj & Loeb (2019) Siraj, A., & Loeb, A. 2019, ApJ, 872, L10, doi: 10.3847/2041-8213/ab042a
  * Tombaugh (1946) Tombaugh, C. W. 1946, Leaflet of the Astronomical Society of the Pacific, 5, 73
  * Tsiganis et al. (2005) Tsiganis, K., Gomes, R., Morbidelli, A., & Levison, H. F. 2005, Nature, 435, 459, doi: 10.1038/nature03539
  * Valtonen (1983) Valtonen, M. J. 1983, The Observatory, 103, 1
  * Valtonen & Innanen (1982) Valtonen, M. J., & Innanen, K. A. 1982, ApJ, 255, 307, doi: 10.1086/159830