# Interstellar planetesimals: potential seeds for planet formation?

Amaya Moro-Martín  Space Telescope Science Institute, 3700 San Martin Dr.,
Baltimore, MD 21218  Physics and Astronomy, Center for Astrophysical Sciences,
Johns Hopkins University, Baltimore, MD 21218  Colin Norman  Space Telescope
Science Institute, 3700 San Martin Dr., Baltimore, MD 21218  Physics and
Astronomy, Center for Astrophysical Sciences, Johns Hopkins University,
Baltimore, MD 21218

###### Abstract

We investigate the trapping of interstellar objects during the early stages of
star and planet formation. Our results show a very wide range of possible
values that will be narrowed down as the population of interstellar objects
becomes better characterized. When assuming a background number density of
2$\cdot$1015 pc-3 (based on 1I/’Oumuamua detection), a velocity dispersion of
30 km/s and an equilibrium size distribution, the number of interstellar
objects captured by a molecular cloud and expected to be incorporated to each
protoplanetary disk during its formation is O(109) (50 cm–5 m), O(105) (5 m–50
m), O(102) (50 m–500 m), O(10-2) (500 m–5 km). After the disk formed, the
number of interstellar objects it could capture from the ISM during its
lifetime is 6$\cdot$1011 (50 cm–5 m), 2$\cdot$108 (5 m–50 m), 6$\cdot$104 (50
m–500 m), 20 (500 m–5 km); in an open cluster where 1% of stars have undergone
planet formation, these values increase by a factor of O(102–103). These
trapped interstellar objects might be large enough to rapidly grow into larger
planetesimals via the direct accretion of the sub-cm sized dust grains in the
protoplanetary disk before they drift in due to gas drag, helping overcome the
meter-size barrier, acting as ”seeds” for planet formation. They should be
considered in future star and planet formation models and in the potential
spread of biological material across the Galaxy.

ISM: general – local interstellar matter – minor planets, asteroids: general –
planet-disk interactions – planets and satellites:formation – protoplanetary
disks

##  1 Introduction

Stars form from the collapse of dense regions of molecular clouds. A natural
byproduct of this process is the formation of protoplanetary disks, where,
according to the leading theory of planet formation, planet formation takes
place (Hayashi 1981; Shu et al. 1987; Hartmann 2008). The dust particles in
these disks, approximately 0.1 $\mu$m in size, are strongly coupled to the
gas, resulting in small relative velocities and collisional energies that,
together with ”sticky” microphysical processes (like van der Waals and
electromagnetic forces), result in their efficient collisional growth into cm-
sized bodies (Weidenschilling 1977, 1980; Birnstiel et al. 2011); these
pebbles eventually grow into km-sized planetesimals and the processes by which
these can grow into planetary embryos and planets, via collisions and
gravitational interactions, is fairly well understood(Armitage 2010, and
references therein). However, the intermediate stage by which cm-sized pebbles
grown into km-sized planetesimals poses several challenges, known as the
meter-sized barrier. As the cm-sized particles grow and become less coupled to
the gas, their relative velocities and collisional energies increase,
resulting in collisions that, rather than leading to efficient growth, lead to
inefficient sticking, bouncing, or fragmentation (Zsom et al. 2010); in
addition, because the particles are still coupled to the gas, they experience
gas drag that results in short inward drift timescales. Weidenschilling (1977)
found that a 1 m-sized particles with a density of 3 g$\cdot$cm-3 located at a
radial distance of 5 AU in a solar nebula has a drift lifetime of
O(10${}^{2})$ years. This short drift timescales limits significantly their
lifetime in the disk and, consequently, their opportunity to grow to sizes
unaffected by gas drag (Birnstiel et al. 2012). In addition, the aeolian
erosion that small planetesimals experience in the presence of gas can
constitute a significant barrier for the growth of metre-size objects in
protoplanetary discs (Rozner et al. 2020) and could contribute to a change of
the planetesimal mass function and particle concentration at a unique Stokes
number (Grishin et al. 2020).

Several mechanisms have been proposed to alleviate the meter-sized barrier:
the effect of the gravitational collapse of the solid component in the disk
(Youdin & Su 2002; Youdin & Goodman 2005) and the formation of over-dense
filaments in the disk fragmenting gravitationally into clumps of small
particles that collapse and form planetesimals (Johansen et al. 2007); the
effect of the velocity distribution of the dust particles that allows for the
existence of low-velocity collisions that favor the growth of a small fraction
of particles into larger bodies (Windmark 2012); the presence of pressure
maxima that can trap dust grains (Johansen et al. 2004); the effect of the low
porosity (Okuzumi et al. 2009a) and the electric charging of dust aggregates
(Okuzumi 2009b) to favor sticking over fragmentation; and collisional growth
between particles of very different masses colliding at high velocity (Booth
et al. 2018).

The detections of 1I/’Oumuamua and 2I/Borisov, the first interstellar
interlopers, have pointed out to the existence of an additional factor that
could help overcome the challenges that the meter-sized barrier imposes on
planetesimal and planet formation models: the seeding of the disks with
planetesimals captured or incorporated from the interstellar medium. Grishin
et al. (2019) studied the scenario in which interstellar objects are captured
by the protoplanetary disk due to gas drag, while Pfalzner & Bannister (2019)
studied how, during the cloud-formation process, interstellar objects could be
incorporated to the molecular cloud in the same way as the interstellar gas
and dust, and from there could be incorporated to the the star-forming disks.

In this work, we study the trapping of interstellar objects by gas drag in
several environments characteristic of the star- and planet-formation
processes, including a molecular cloud, a stellar kernel and a protoplanetary
disk. We improve upon Pfalzner & Bannister (2019) work by considering the
effect of gas drag in the trapping of the interstellar objects. In Section 2
we describe the trapping environments and characteristic incoming velocity; in
Section 3 we calculate the probability to capture interstellar objects as a
function of the environmental parameters and the properties of the objects,
taking into account the distribution of their incoming velocities. In Section
4, we use this capture probability to derive an expression for the total
number of captured planetesimals, that we improve in Section 5 taking into
account the trajectory and orbital ellipticity of the particle as it traverses
the environment. In Section 6 we estimate the total number of interstellar
objects that could be captured in the molecular cloud environment and
incorporated to each stellar system, and in Section 7 we do the same for the
pre-stellar kernel and protoplanetary disk environments. In Section 8, we
consider the case of an open cluster and in Section 9 we discuss the results.

In these calculations, we adopt two different assumptions regarding the origin
of the interstellar objects and their expected background density: (1) a
background density of interstellar objects of $N_{\rm R\geqslant R_{\rm
Ou}}=2\cdot 10^{15}{\rm pc}^{-3}$, with $R_{\rm Ou}\sim$ 80 m, inferred from
1I/’Oumuamua’s detection (Do et al. 2018); (2) a lower value of mtotal =
9.5$\cdot$1027 g$\cdot$pc-3 corresponding to the background density that would
be expected from the ejection of planetesimals from protoplanetary disks
(Moro-Martín 2018).

##  2 Trapping environments and incoming velocities

To explore the effect of the gas drag, we start by considering a uniform
density, spherically symmetric molecular cloud and proto-stellar core kernel,
the latter similar to those recently resolved by ALMA (Caselli et al. 2019).
Table 1 lists their physical properties, namely their number density, $n_{\rm
env}$, Temperature, $T_{\rm env}$, cloud radius $R_{\rm env}$ and estimated
lifetime, $\tau_{\rm env}$. We also consider a more dense and compact
environment with gas densities and temperatures characteristic of the mid-
plane of a minimum mass solar nebula (MMSN) at 30 au (Hayashi 1981) and a
protoplanetary disk (Chiang & Goldreich 1997). For the disk case we assume a
uniform density and that the swept up column is that along the scale height. A
detailed study that takes into account the disk structure is given by Grishin
et al. (2019).

The mean velocity dispersion of molecular clouds in the Galaxy is in the range
1–10 km/s (see e.g. Fig. 5 in Dobbs et al. 2019), while the velocity
distribution of young stars in the galaxy is characterized by a dispersion
$\sigma_{R}\sim$ 30 km/s (Mackereth et al. 2019). In the solar neighborhood,
the rotational frequency of the Galaxy is similar to that of the spiral arms
(Gerhard 2011; Nguyen et al. 2018) since the solar neighborhood is near
corotation. This means that, even if we were to assume that star and planet
formation preferentially takes place in the spiral arms (which is strictly
only true for high-mass stars), we can ignore the streaming velocity of the
stars.

Because of corotation, and also because low-mass star formation happens in
molecular clouds in-between arms, let us assume that there is no streaming
velocity and that $u$, the characteristic incoming velocity of interstellar
objects as they enter the environment under consideration, is in the 1–30 km/s
range, given by the velocity dispersion mentioned earlier. The situation would
be different closer to the Galactic center because at $\lesssim$ 4 kpc, the
the streaming velocity itself would be $\sim$ 100 km/s (Nguyen et al. 2018)
and would need to be taken into account.

Table 1: Environments under consideration. Environment | $n_{\rm env}$ | $T_{\rm env}$ | $R_{\rm env}$ | $\tau_{\rm env}$ | $u$
---|---|---|---|---|---
| (cm-3) | (K) |  | (Myr) | (km/s)
Molecular cloud | 102 | 10 | 15 pc | 30 | 1–30
Pre-stellar kernel | 106 | 6.5 | 1500 AU | 1aaThe pre-stellar core kernel is expected to be short-lived, with a lifetime of the order of its free-fall timescale (André et al. 2014). The disk lifetime is 1–10 Myr; here we assumed 1 Myr. | 1–30
MMSN-30aubbGas density and temperature characteristic of the mid-plane of a minimum mass solar nebula at 30 au (Hayashi1981). | 8$\cdot$1010 | 50 | 30 AU | 1aaThe pre-stellar core kernel is expected to be short-lived, with a lifetime of the order of its free-fall timescale (André et al. 2014). The disk lifetime is 1–10 Myr; here we assumed 1 Myr. | 1–30
DISK-30auccGas density and temperature characteristic of the mid-plane of a protoplanetary disk at 30 au, based on Chiang & Goldreich (1997), following Grishin et al. (2019). See Section 7 for discussion; this disk is thicker than the one considered for the MMSN case. | 5$\cdot$109 | 50 | 30 AU | 1aaThe pre-stellar core kernel is expected to be short-lived, with a lifetime of the order of its free-fall timescale (André et al. 2014). The disk lifetime is 1–10 Myr; here we assumed 1 Myr. | 1–30

##  3 Gas drag and Capture Rates

The acceleration that an interstellar object will experience due to gas drag
is

$a_{\rm D}={1\over 2m_{\rm p}}C_{\rm D}\pi r_{\rm
p}^{2}\rho_{env}u^{2}={3C_{\rm D}\rho_{env}u^{2}\over 8\rho_{\rm p}r_{\rm
p}},$ (1)

where $\rho_{\rm env}$ is the mass density of the environment ($\rho_{\rm
env}=n_{\rm env}m_{0}$, with $m_{0}$ = 3.34$\cdot$10-24 g for for H2), and
$m_{\rm p}$, $r_{\rm p}$, $\rho_{\rm p}$, and $u$ are the mass, radius,
density, and incoming velocity of the interstellar object, respectively.
$C_{D}$ is the gas drag coefficient, a function of the Mach number, $M=u/c$,
and the Reynolds numbers, $R_{\rm e}$, that can be approximated by

$\begin{split}C_{\rm D}\simeq\left(\left({24\over R_{\rm e}}+{40\over
10+R_{\rm
e}}\right)^{-1}+{0.23~{}M}\right)^{-1}+{(2-w)M\over(1.6+M)}+w,\end{split}$ (2)

where $w$ = 0.4, if $R_{e}<2\cdot$105, and $w$ = 0.2, if $R_{e}>2\cdot$105
(Adachi et al. 1976; Tanigawa et al. 2014). Figure 1 shows the gas drafg
coefficient as a function of the Reynolds number and the Mach number,
calculated with Equation 2. In Appendix A and Table A1 we calculate, for each
environment and for a given interstellar object incoming velocity, the Mach
number, the Reynolds number, and the Stokes number. The latter is the ratio of
the stopping time to a dynamical time; interstellar objects with $Stk<<$ 1 are
expected to be trapped by gas drag, while those with $Stk>>$ 1 are expected to
escape (see discussion in Appendix A).

Figure 1: Gas drag coefficient as a function of the Reynolds number and the
Mach number, calculated using Equation (2).

To calculate the capture rates we start by assuming, as in Grishin et al.
(2019), that the relative velocity $u$ is a constant through the passage so
that the energy dissipated by the particle as it crosses the environment is
given by

$\Delta E=-{1\over 2}\int{C_{\rm D}(u)\pi r_{\rm p}^{2}\rho_{env}u^{2}}{\rm
d}x\approx-{C_{\rm D}(u)\pi r_{p}^{2}\over 2}\Sigma_{env}u^{2},$ (3)

where $u$ is the relative velocity and we let $u^{2}=u_{0}^{2}+u_{\rm
esc}^{2}$ (see Grishin et al. 2019, Equation (4)). Here, $u_{\rm
esc}=\sqrt{2GM\over R_{\rm env}}$; for the molecular cloud we assume that $M$
is the mass of the cloud but for the pre-stellar core kernel and the
protoplanetary disk we will assume a solar mass at the center. Capture takes
place when the energy loss is larger than the particle’s kinetic energy at
infinity, $\Delta E>E_{0}={1\over 2}m_{p}u_{0}^{2}$. This leads to the capture
condition,

$\begin{split}{3\over 4}{C_{\rm D}(u_{0})\Sigma_{\rm
env}\over\rho_{p}r_{p}}(1+{u_{\rm esc}^{2}\over u_{0}^{2}})>1,\\\ \end{split}$
(4)

which is a condition on the particle’s size.

It is a significant assumption that the velocity-dependent $C_{D}$ can be
taken outside the energy integral and evaluated at a specific velocity. In
Appendix B we derive the energy loss condition when one does not make this
assumption; we discuss two limiting cases: a circular orbit grazing the cloud
and a harmonic oscillator orbit passing through the center of the cloud. It is
also a significant assumption that the relative velocity $u$ is a constant
through the passage. In Section 5 we improve upon this assumption by deriving
the capture rate taking into account the change in the relative velocity as
the particle traverses the environment.

###  3.1 Geometric regime

In the geometric regime, ${u_{\rm esc}^{2}\over u_{0}^{2}}<<1$, the trajectory
of the particle follows a straight line and the column it sweps up is
$\Sigma_{\rm env}=2\rho_{\rm env}\sqrt{R_{\rm env}^{2}-b^{2}}$, where $b$ is
the impact parameter. Following Equation (LABEL:capcondgen), in this case the
condition for capture is

${3\over 2}{C_{\rm D}(u_{0})\over\rho_{p}r_{p}}\rho_{\rm env}\sqrt{R_{\rm
env}^{2}-b^{2}}>1,$ (5)

leading to a capture condition for the impact parameter of

$b(r_{p})<b_{c}(r_{p})=\sqrt{R_{\rm env}^{2}-\left({2\rho_{p}r_{p}\over
3C_{\rm D}(u_{0})\rho_{\rm env}}\right)^{2}}.$ (6)

Following Grishin et al. (2019), the capture probability is

$f_{c}(r_{p})=\left({b_{c}(r_{p})\over b_{\rm max}}\right)^{2}={1\over b_{\rm
max}^{2}}\left(R_{\rm env}^{2}-\left({2\rho_{p}r_{p}\over 3C_{\rm
D}(u_{0})\rho_{\rm env}}\right)^{2}\right),$ (7)

where $b_{\rm max}$ is the maximal impact parameter for an effective close
encounter; in the geometrical regime, $b_{\rm max}=R_{\rm env}$ and the
capture probability is independent of the particle’s velocity (except for the
$u$ dependence in the drag coefficient).

###  3.2 Gravitational focusing regime

In the gravitational focusing regime, ${u_{\rm esc}^{2}\over u_{0}^{2}}>>1$,
for simplicity we will assume the particle enters in a parabolic trajectory.
In the grazing limiting case, the particle is trapped in a circular orbit of
radius $R_{\rm env}$, with swept up column of $2\pi R_{\rm env}$. In the
radial orbit limiting case, the particle traverses the cloud and then reverses
its course, with a swept up column density of 4$R_{\rm env}$. In general, the
orbit is elliptical and is a harmonic oscillator in $r^{2}$ (see Harter Soft
Web Resources for Physics, Chapter 5). Therefore, we parameterize the swept up
column density as $\Sigma_{\rm env}=2\xi\pi R_{\rm env}\rho_{\rm env}$, where
$\xi$ is O(1). The capture condition based on energy loss is then

$\begin{split}{3\over 2}{C_{\rm D}\pi\xi R_{\rm env}\rho_{\rm
env}\over\rho_{p}r_{p}}{u_{\rm esc}^{2}\over u_{0}^{2}}>1,\\\ u_{0}^{2}<u_{\rm
0,c}^{2}={3\over 2}{C_{D}\pi\xi R_{\rm env}\rho_{\rm
env}\over\rho_{p}r_{p}}u_{\rm esc}^{2}.\end{split}$ (8)

Henceforth, we will evaluate the above expression at $\xi$ = 1.

From angular momentum conservation, we have that

$b^{2}u_{0}^{2}=a^{2}u_{\rm esc}^{2}(a)={2GM_{\rm env}a},$ (9)

giving a maximum limit on the angular momentum of the trapped object. To be
trapped we have that $a<R_{\rm env}$, leading to ${b^{2}u_{0}^{2}\over
2GM_{\rm env}}<R_{\rm env}$. The capture condition on the impact parameter is

$\begin{split}b(r_{p})^{2}<b_{c}(r)^{2}={2GM_{\rm env}R_{\rm env}\over
u_{0}^{2}}={u_{\rm esc}^{2}\over u_{0}^{2}}R_{\rm env}^{2}.\end{split}$ (10)

Following Grishin et al. (2019), the weighted capture probability is given by

$f_{c}=\int\int_{D,c}{2b\over b_{\rm max}^{2}}{u^{3}\over
2\sigma^{4}}e^{-{u^{2}\over 2\sigma^{2}}}dbdu,$ (11)

(see their Equation 16). The capture probability is weighted with an
additional $u$ factor to take into account that faster planetesimals have
higher encounter rates than slower ones. Here, $\sigma$ is the velocity
dispersion of the velocity distribution. $b_{\rm max}$ is the maximal impact
parameter for an effective close encounter, that in the gravitational focusing
regime is $b_{\rm max}=R_{\rm env}{u_{\rm esc}\over u_{0}}$, where in this
calculation we are now assuming ${u_{\rm esc}\over u_{0}}>$1\. $D_{c}$ is the
integration domain that results in capture, i.e., the integral over $u$ goes
up to $u_{\rm 0,c}$ (Equation (8)). The domain for impact parameter is
determined by the angular momentum condition, and the domain for the velocity
$u_{0}$ is determined by the energy condition in the $b-u_{0}$ plane. In this
case, the domain is clear because the spherical cloud is finite, unlike the
infinite disk in Grishin et al. (2019). The limiting value of $b$ is given by
Equation (10), and the limiting value of $u_{0}$ is given by Equation (8).

We then have,

$\begin{split}f_{c}(r_{p},a)={1\over b_{\rm max}^{2}}\int^{u_{\rm
0,c}}_{0}du{u^{3}\over 2\sigma^{4}}e^{-{u^{2}\over
2\sigma^{2}}}\int_{0}^{b_{c}}2bdb={1\over b_{\rm max}^{2}}\int^{u_{\rm
0,c}}_{0}du{u^{3}\over 2\sigma^{4}}e^{-{u^{2}\over 2\sigma^{2}}}R_{\rm
env}^{2}{u_{\rm esc}^{2}(a)\over u^{2}}={R_{\rm env}^{2}\over b_{\rm
max}^{2}}{u_{\rm esc}^{2}\over 2\sigma^{2}}\int^{{u_{\rm 0,c}}^{2}\over
2\sigma^{2}}_{0}dte^{-t}\\\ \end{split}$ (12) $f_{c}(r_{p})={R_{\rm
env}^{2}\over b_{\rm max}^{2}}{u_{\rm esc}^{2}\over
2\sigma^{2}}\left[1-e^{-{{u_{\rm esc}^{2}}\over\sigma^{2}}{3\over 4}{C_{D}\pi
R_{\rm env}\rho_{\rm env}\over\rho_{p}r_{p}}}\right].$ (13)

If we define an ensemble average Safronov number
${\bar{\Theta}_{s}}={u_{esc}^{2}\over\sigma^{2}}$, then

$f_{c}(r_{p})={R_{\rm env}^{2}\over 2b_{\rm
max}^{2}}{\bar{\Theta}_{s}}\left[1-e^{-{\bar{\Theta}_{s}}{3\over 4}{C_{D}\pi
R_{\rm env}\rho_{\rm env}\over\rho_{p}r_{p}}}\right].$ (14)

For small arguments of the exponential in Equation (14), we can simplify the
capture rate to

$\begin{split}f_{c}(r_{p})={3\over 8}\pi C_{D}{R_{\rm env}^{2}\over b_{\rm
max}^{2}}{\bar{\Theta}_{s}}^{2}{R_{\rm env}\rho_{\rm
env}\over\rho_{p}r_{p}}={9\over 32}{C_{D}\over b_{\rm
max}^{2}}{\bar{\Theta}_{s}}^{2}{M_{\rm env}\over\rho_{p}r_{p}}.\end{split}$
(15)

Note that the capture rate is for sufficient dissipation in one orbital period
at $R_{\rm env}$. In the case of a constant density cloud, that orbital period
is the same across the cloud. Inner orbits have less dissipation because the
circumference is smaller for one orbital period.

It is important to note that in the calculations above, in the gravitational
focusing case $C_{D}$ is evaluated at $u_{\rm esc}$ while in the geometric
case $C_{D}$ is evaluated at $u_{0}$. This makes it difficult to generate a
formula interpolating neatly between the two cases.

##  4 Total number of captured planetesimals

The total number of interstellar objects that can be trapped will be given by
$N(r_{p})=f_{c}(r_{p})\cdot N_{\rm enter}(r_{p})$, where $f_{c}(r_{p})$ is the
weighted capture probability calculated above, so that

$N(r_{p})={R_{\rm env}^{2}\over b_{\rm max}^{2}}{u_{\rm esc}^{2}\over
2\sigma^{2}}\left[1-e^{-{{u_{\rm esc}^{2}}\over\sigma^{2}}{3\over 4}{C_{D}\pi
R_{\rm env}\rho_{\rm env}\over\rho_{p}r_{p}}}\right]{N_{\rm enter}(r_{p})}.$
(16)

$N_{\rm enter}(r_{p})$ is the number of interstellar objects of radius $r_{p}$
that enter the cross-section $\pi b_{\rm max}^{2}$, $N_{\rm
enter}(r_{p})=n_{\rm ISO}(r_{p})\pi b_{\rm max}^{2}\langle u\rangle\tau_{\rm
env}$. Here, $b_{\rm max}$ is the maximal impact parameter for an effective
close encounter, $n_{\rm ISO}(r_{p})$ is the number density of interstellar
objects of radius $r_{p}$, $\tau_{\rm env}$ is the lifetime of the cloud, i.e.
the timescale over which encounters can occur, and $\langle
u\rangle=\sqrt{8\over\pi}\sigma$ is the mean velocity of the incoming
interstellar objects, where $\sigma$ is the velocity dispersion. From Equation
(16), we then have that,

$\begin{split}N(r_{p})={\sqrt{2\pi}}n_{\rm ISO}(r_{p})\tau_{\rm env}R_{\rm
env}^{2}{u_{\rm esc}^{2}\over\sigma}\left[1-e^{-{{u_{\rm
esc}^{2}}\over\sigma^{2}}{3\over 4}{C_{D}\pi R_{\rm env}\rho_{\rm
env}\over\rho_{p}r_{p}}}\right],\end{split}$ (17)

where for small arguments of the exponential we can simplify the total number
of captured planetesimals to

$\begin{split}N(r_{p})={3\pi\over 4}{\sqrt{2\pi}}{C_{D}}n_{\rm
ISO}(r_{p})\tau_{\rm env}R_{\rm env}^{2}u_{\rm esc}\left({u_{\rm
esc}\over\sigma}\right)^{3}{R_{\rm env}\rho_{\rm
env}\over\rho_{p}r_{p}}.\end{split}$ (18)

To get to this expression for $N(r_{p})$ we assumed in the derivation that the
relative velocity $u$ is a constant through the passage. In the following
section we improve upon this assumption by deriving the capture rate and the
total number of captured planetesimals taking into account the change in the
relative velocity as the particle traverses the environment.

##  5 Improved calculation of the total number of captured planetesimals
taking into account the trajectory and orbital ellipticity of the particle

A uniform density environment, as the one that we are considering here, has a
harmonic oscillator gravitational potential and in this potential the orbit of
the particle is elliptical and is a harmonic oscillator in $r^{2}$ (see Harter
Soft Web Resources for Physics, Chapter 5), $r^{2}={E\over
m_{p}\Omega^{2}}(1+ecos(2\Omega t))$, where $\Omega^{2}={4\over 3}\pi
G\rho_{\rm env}$ and the eccentricity is $e=\left(1-{\Omega^{2}L^{2}\over
E^{2}}\right)^{1/2}$. The energy is $E={1\over 2}m_{p}u^{2}+{1\over
2}m_{p}\Omega^{2}r^{2}$ and from this expression we can solve for the
velocity, $u^{2}=2{E\over m_{p}}-\Omega^{2}r^{2}={E\over m_{p}}(1-ecos(2\Omega
t))$. This allows to calculate the energy dissipated by the particle as it
crosses the environment, taking into account the change in $u$,

$\Delta E=-{1\over 2}\int{C_{\rm D}(u)\pi r_{\rm p}^{2}\rho_{env}u^{2}}u{\rm
d}t=-{1\over 2}{C_{\rm D}(u_{\rm esc})\pi r_{\rm
p}^{2}\rho_{env}\int_{0}^{T/2}u^{3}}{\rm d}t,$ (19)

where we integrate over half a period, with $T={2\pi\over\Omega}$, and we
approximate CD by evaluating it at $u_{\rm esc}$ so it can be taken outside
the integral. Using the expression for $u^{2}$ derived above, and taking into
account that $\left({E\over m_{p}}\right)^{1/2}\approx u_{\rm esc}$ and
$u_{\rm esc}\cdot{T\over 2}$ is half the perimeter of the elliptical orbit,
$u_{\rm esc}\cdot{T\over 2}\approx 2aE(e)$, we have

$\begin{split}\int_{0}^{T/2}u^{3}{\rm d}t=\left({E\over
m_{p}}\right)^{3/2}\int_{0}^{T/2}(1-e{\rm cos}(2\Omega t))^{3/2}{\rm
d}t=\left({E\over m_{p}}\right)^{3/2}{1\over 2\Omega}\int_{0}^{2\pi}(1-e{\rm
cos}\theta)^{3/2}{\rm d}\theta\\\ =\left({E\over m_{p}}\right)\left({E\over
m_{p}}\right)^{1/2}{T\over 2}{1\over 2\pi}\int_{0}^{2\pi}(1-e{\rm
cos}\theta)^{3/2}{\rm d}\theta=\left({E\over m_{p}}\right)u_{\rm esc}{T\over
2}f_{3}(e)=\left({E\over m_{p}}\right)2aE(e)f_{3}(e)\end{split}$ (20)

where $\theta=2\Omega t$, E(e) is the complete elliptic integral of the second
kind, and $f_{3}(e)={1\over 2\pi}\int_{0}^{2\pi}(1-e{\rm cos}\theta)^{3/2}{\rm
d}\theta$. Hereafter we will use $\Psi(e)\equiv E(e)\cdot f_{3}(e)$ that for
$e$ ranging from 0 to 1 is $\Psi(e)$ = 1.20–1.56. In the calculations in
Sections 6 and 7 we use $\Psi(e)$ = 1.20. The energy dissipated by the
particle as it crosses the environment is then

$\Delta E=-C_{\rm D}(u_{\rm esc})\pi r_{\rm p}^{2}\rho_{\rm
env}a\Psi(e)\left({E\over m_{p}}\right).$ (21)

Assuming as before that the relative velocity is $u^{2}=u_{0}^{2}+u_{\rm
esc}^{2}$, where $u_{\rm esc}=\sqrt{2GM_{\rm env}\over R_{\rm env}}$ (see
Grishin et al. 2019, Equation (4)), we have that ${E\over m_{p}}={1\over
2}\left(u_{0}^{2}+u_{\rm esc}^{2}\right)$ so that

$\Delta E=-C_{\rm D}(u_{\rm esc})\pi r_{\rm p}^{2}\rho_{\rm
env}a\Psi(e){1\over 2}\left(u_{0}^{2}+u_{\rm esc}^{2}\right).$ (22)

Capture takes place when the energy loss is larger than the particle’s kinetic
energy at infinity, $\Delta E>E_{0}={1\over 2}m_{p}u_{0}^{2}$. This leads to
the capture condition,

$\begin{split}{3\over 4}{C_{\rm D}(u_{\rm esc})a\rho_{\rm
env}\Psi(e)\over\rho_{p}r_{p}}\left(1+{u_{\rm esc}^{2}\over
u_{0}^{2}}\right)>1,\\\ \end{split}$ (23)

which is a condition on the particle’s size that is similar to that in
Equation (LABEL:capcondgen) when $a\sim R_{\rm env}$. The capture condition in
the gravitational focussing regime, equivalent to Equation (8), is then

$\begin{split}u_{0}^{2}<u_{0,c}^{2}={3\over 4}{C_{\rm D}(u_{\rm
esc})a\rho_{\rm env}\Psi(e)\over\rho_{p}r_{p}}u_{\rm esc}^{2}.\\\ \end{split}$
(24)

This trapping condition can only be satisfied when ${u_{\rm esc}^{2}\over
u_{0}^{2}}>>1$, i.e. low velocity particles with high Safronov numbers. Our
average over the incoming interstellar object distribution takes into account
this stringent trapping condition. Note that the average Safronov numbers that
appear are the result of the gravitational focussing and the phase space
factors at the low end of the velocity space distribution of the interstellar
objects.

Their weighted capture probability, equivalent to Equations (14) and (15), is

$\begin{split}f_{c}(r_{p})={R_{\rm env}^{2}\over 2b_{\rm
max}^{2}}{\bar{\Theta}_{s}}\left[1-e^{-{\bar{\Theta}_{s}}{3\over
8}{C_{D}R_{\rm env}\rho_{\rm env}\Psi(e)\over\rho_{p}r_{p}}}\right]\sim{9\over
64\pi}{C_{D}\over b_{\rm max}^{2}}{\bar{\Theta}_{s}}^{2}{M_{\rm
env}\Psi(e)\over\rho_{p}r_{p}}.\end{split}$ (25)

The total number of interstellar objects that can be trapped, equivalent to
Equations (16) and (17), is

$\begin{split}N(r_{p})={R_{\rm env}^{2}\over b_{\rm max}^{2}}{u_{\rm
esc}^{2}\over 2\sigma^{2}}\left[1-e^{-{{u_{\rm
esc}^{2}}\over\sigma^{2}}{3\over 8}{C_{D}R_{\rm env}\rho_{\rm
env}\Psi(e)\over\rho_{p}r_{p}}}\right]{N_{\rm enter}(r_{p})}\\\
={\sqrt{2\pi}}n_{\rm ISO}(r_{p})\tau_{\rm env}R_{\rm env}^{2}{u_{\rm
esc}^{2}\over\sigma}\left[1-e^{-{{u_{\rm esc}^{2}}\over\sigma^{2}}{3\over
8}{C_{D}R_{\rm env}\rho_{\rm env}\Psi(e)\over\rho_{p}r_{p}}}\right]{N_{\rm
enter}(r_{p})},\end{split}$ (26)

that for small arguments of the exponential becomes

$\begin{split}N(r_{p})={3\over 8}{\sqrt{2\pi}}{C_{D}}{R_{\rm env}\rho_{\rm
env}\Psi(e)\over\rho_{p}r_{p}}n_{\rm ISO}(r_{p})\sigma\pi R_{\rm
env}^{2}\tau_{\rm env}\bar{\Theta}_{s}^{2}={9\over
32}{\sqrt{2\pi}}{C_{D}}\Psi(e){n_{\rm
ISO}(r_{p})\sigma\over\rho_{p}r_{p}}M_{\rm env}\tau_{\rm
env}\bar{\Theta}_{s}^{2}.\end{split}$ (27)

In the expression above, $n_{\rm ISO}(r_{p})\sigma\pi R_{\rm env}^{2}$ is the
rate at which interstellar objects cross the environment, that multiplied by
$\tau_{\rm env}$ gives a total number. The other factors take into account
gravitational focussing and the trapping efficiency due to gas drag. The
latter provides a factor of $R_{\rm env}\rho_{\rm env}$, which allows to
calculate something similar to a dust-to-gas mass ratio but with the mass of
the trapped objects (see Figure 6),

$\begin{split}{N(r_{p})m_{p}\over M_{\rm env}}={3\over
8}{\sqrt{2\pi}}{C_{D}}\Psi(e)\pi r_{p}^{2}n_{\rm ISO}(r_{p})\sigma\tau_{\rm
env}\bar{\Theta}_{s}^{2}.\end{split}$ (28)

We can also calculate the fraction of interstellar objects that are trapped,
out of the total number that are in the cloud at any given time,

$\begin{split}{N(r_{p})\over n_{\rm ISO}(r_{p}){4\over 3}\pi R_{\rm
env}^{3}}={9\over 32}{\sqrt{2\pi}}{C_{D}}\Psi(e){\sigma\tau_{\rm env}\over
R_{\rm env}}{\rho_{\rm env}R_{\rm
env}\over\rho_{p}r_{p}}\bar{\Theta}_{s}^{2}.\end{split}$ (29)

We will now evaluate Equations (26), (28) and (29) using the environmental
parameters ($R_{\rm env}$, $\rho_{\rm env}$, $u_{\rm esc}$, $\tau_{\rm env}$)
from Table 1 (assuming the cloud is made of H2 gas) and the gas drag
coefficient $C_{D}$ calculated in Section A and listed in Table A1. Regarding
the velocity dispersion, $\sigma$, interstellar object observations are too
limited to be able to constrain their velocity distribution and how it
compares to stars-forming-regions. As discussed in Section 2, Mackereth et al.
(2019) found that for the young stars in the Galaxy with ages $<$ 2 Gyr, their
velocity dispersions are $\sigma_{R}\sim$ 30 km/s and $\sigma_{z}\sim$10 km/s.
Because these young stellar systems are thought to be the main source of
interstellar planetesimals, we adopt a velocity dispersion of $\sigma$ = 30
km/s. Note that this underestimates the width of the velocity distribution of
the interstellar objects, ignoring their ejection velocity from their young
parent stellar systems, expected to be in the 1–10 km/s range.

The remaining factor that we need to evaluate Equations (26), (28) and (29) is
the number density of interstellar planetesimals, $n_{\rm ISO}(r_{p})$, that
we discuss below.

###  5.1 Number density of interstellar planetesimals

We now calculate $n_{\rm ISO}(r_{p})$ for planetesimals in the size ranges of
50 cm–5 m, 5 m–50 m, 50 m–500 m, 500 m–5 km, and 5 km–50 km, that we will
refer to as $N_{\rm R_{1}<r_{p}<R_{2}}$, with the corresponding $R_{1}$ and
$R_{2}$ values. We will consider two scenarios.

In the first case, described in Section 5.1.1, we assume that the number
density of interstellar planetesimals larger than 1I/’Oumuamua’s is $2\cdot
10^{15}{\rm pc}^{-3}$, based 1I/’Oumuamua’s detection and following Do et al.
(2018).

However, Do et al. (2018) estimate requires an ejection of about 50 M⊕ of
solids per stellar system, higher, for example, than Adams & Spergel (2005)
estimate of $\gtrsim$ M⊕ per star (see also Grishin et al. 2019). Indeed, Do
et al. (2018) estimate is higher than what the protoplanetary reservoirs can
provide (Raymond et al. 2018; Moro-Martín 2018). Other mechanisms that will
contribute to the number density of interstellar objects beyond the
protoplanentary disk stage include planet-planet scattering (Adams & Spergel
2005) and the release of exo-Oort cloud objects due to stellar-mass loss
during the post-main sequence evolution (Moro-Martín 2019a). However, the
contribution from these latter mechanisms ultimately depend on the initial
overall circumstellar reservoir. To take into account all these potential
contributors, in Section 5.1.2 we adopt for the number density of interstellar
objects the value that would be expected from the ejection of the entire
circumstellar disk reservoir, following Moro-Martín (2018).

####  5.1.1 Number density of planetesimals inferred from the detection of
1I/’Oumuamua

Under this scenario, we adopt the cumulative number density of interstellar
planetesimals inferred from 1I/’Oumuamua’s detection, $N_{\rm R\geqslant
R_{\rm Ou}}=2\cdot 10^{15}{\rm pc}^{-3}$ (Do et al. 2018), with $R_{\rm
Ou}\sim$ 80 m. From this value we can estimate the number density of
interstellar objects in the size ranges considered, $N_{\rm
R_{1}<r_{p}<R_{2}}$, to be used as $n_{\rm ISO}(r_{p})$, by adopting the size
distribution used in Moro-Martín et al. (2009),

$\begin{split}n(r)\propto r^{-q_{1}}~{}{\rm if}~{}r_{min}<r<r_{b}\\\
n(r)\propto r^{-q_{2}}~{}{\rm if}~{}r_{b}<r<r_{max}\\\ q_{1}={\rm
2.0,~{}2.5,~{}3.0,~{}3.5}\\\ q_{2}={\rm 3,~{}3.5,~{}4,~{}4.5,~{}5}\\\
r_{b}={\rm 3~{}km,30~{}km,~{}90~{}km}\\\ r_{max}\approx~{}{\rm
1000~{}km~{}and}~{}r_{min}\approx~{}{\rm 1}~{}\mu{\rm m}.\\\ \end{split}$ (30)

The above distribution is based on solar system observations and on accretion
and collisional models (Bottke et al. 2005 ; Kenyon et al. 2008; Bernstein et
al. 2004; Fuentes, George & Holman 2009; Fraser & Kavelaars 2009; Lamy et al.
2004). The results for $N_{\rm R_{1}<r_{p}<R_{2}}$ are shown in Figure 2 as a
function of the adopted value for $q_{1}$, with the different colors
corresponding to the different size ranges. For the 5 km–50 km size range, the
results also depend on $q_{2}$, and $r_{b}$ (as illustrated by the dotted and
the dashed lines). As expected, the number density of planetesimals depend
sensitively on the parameters chosen for the size distribution, lying within
the ranges shown in the first row of Table 2. The values corresponding to an
equilibrium size distribution are also shown in Table 2. Details on the
calculations can be found in Appendix C.

Figure 2: Number density of interstellar objects in different size ranges,
$N_{\rm R_{1}<r_{p}<R_{2}}$, as a function of the adopted value for the power-
law index $q_{1}$, assuming that the cumulative number density of interstellar
planetesimals with radius similar or larger than 1I/’Oumuamua is $N_{\rm
R\geqslant R_{\rm Ou}}=2\cdot 10^{15}{\rm pc}^{-3}$ (Do et al. 2018), with
$R_{\rm Ou}\sim$ 80 m. The different colors correspond to the different size
ranges as labeled. For the 5 km–50 km size range, the results also depend on
$q_{2}$ and $r_{b}$, with the line types corresponding to the following power-
law parameters: $q_{\rm 2}$ = 2 and $r_{\rm b}$ = 3 km (solid), $q_{\rm 2}$ =
3.5 and $r_{\rm b}$ = 3 km (dashed-dotted), $q_{\rm 2}$ = 6 and $r_{\rm b}$ =
3 km (dashed), $q_{\rm 2}$ = 2–6 and $r_{\rm b}$ = 90 km (dotted).

####  5.1.2 Number density of planetesimals in the interstellar medium
expected from protoplanetary disk ejection

Under this scenario, we assume that the interstellar medium is populated with
planetesimals ejected from circumstellar protoplanetary disks around single
stars and wide binaries, and from circumbinary protoplanetary disks around
tight binaries. In terms of the expected mass density of planetesimals, Moro-
Martín (2018) found that 2.7$\cdot$1027 g$\cdot$pc-3 would be expected to
originate from circumbinary disks around tight binaries111Following Jackson et
al. (2018), this calculation assumes a binary fraction of 0.26, a disk-to-
stellar system mass ratio of ${M_{disk}\over M_{sys}}$ = 0.1Msys, that 10% of
this material migrates in, crossing the unstable radius at which point objects
are ejected, and a gas-to-dust ratio of 100:1 (for details see Section 3 of
Moro-Martín 2018)., and 6.7$\cdot$1026 g$\cdot$pc-3 or 6.8$\cdot$1027
g$\cdot$pc-3 from single stars and wide binaries222These two estimates of the
contribution from single stars and wide binaries assume that 74% stars are in
this category, a disk-to-stellar mass ratio of ${M_{disk}\over M_{*}}$ = 0.01,
and a gas-to-dust ratio of 100:1. The 6.7$\cdot$1026 g$\cdot$pc-3 estimate
corresponds to the assumption that only planet bering contribution, adopting a
planet fraction of 3% for K2–M stars and 20% for A-K-K2 stars. The
6.7$\cdot$1027 g$\cdot$pc-3 estimate corresponds to the assumption that all
stars contribute. For details see Section 3 of Moro-Martín (2018).. In this
latter case, the overall contribution from single and binary stars would
result in a total mass density of planetesimals in the interstellar medium of
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3. Adopting this overall contribution and
the size distribution in Equation (30), we can estimate the number of
interstellar objects in the size ranges considered. The details of this
calculation can be found in Appendix D. The results for $N_{\rm
R_{1}<r_{p}<R_{2}}$ are shown in Figures 3–5. The number density of
planetesimals depend sensitively on the parameters chosen for the size
distribution, lying within the ranges shown in the third row of Table 2. If we
were to adopt a disk-to-star mass ratio of 0.1 instead of 0.01 (which could be
consistent with the disk observations given the uncertainties in the dust
opacities), the values in Table 2 would increase by a factor of 10, as the
total mass density of interstellar objects would increase by a factor of 10,
i.e. mtotal = 9.5$\cdot$1028 g$\cdot$pc-3 (the corresponding results are shown
in the fourth row of Table 2).

Figure 3: Number density of interstellar objects in different size ranges,
$N_{\rm R_{1}<r_{p}<R_{2}}$, as a function of the adopted value for the power-
law index $q_{1}$, assuming that the total mass density of planetesimals in
the interstellar medium is mtotal = 9.5$\cdot$1027 g$\cdot$pc-3, from Moro-
Martín (2018; see Section 5.1.2). The different colors correspond to the
different size ranges as labeled and the different line types to two different
values of the power-law index $q_{2}$, $q_{2}$=2 (solid lines) and $q_{2}$ = 6
(dotted lines). For this panel, we adopt rb = 3 km. Details on how these
values are derived can be found in Appendixes C and D. Figure 4: Same as
Figure 3 for rb = 30 km. Figure 5: Same as Figure 3 for rb = 90 km.

##  6 Total number of captured planetesimals: Molecular cloud

###  6.1 Previous study

Pfalzner & Bannister (2019) argue that during the molecular formation process,
the number density of interstellar objects inside the molecular cloud could be
enhanced with respect to that in the interstellar medium ($10^{15}$ pc-3, from
Do et al. (2018)) by the same factor as the gas ($10^{4}$–$10^{6}$), and that
this could lead to a number density as high as $10^{19}$ pc-3. They also argue
that because in the molecular cloud clumps that lead to star formation the gas
is further enhanced by a factor of $10^{2}$–$10^{3}$, the interstellar objects
could experience a similar enhancement factor, resulting in a number density
of interstellar objects that could be as high as $10^{22}$ pc-3.

In spite of the arguments above, for the number density of interstellar
objects inside the cloud clumps, Pfalzner & Bannister (2019) adopt the
baseline interstellar value of $10^{15}$ pc-3 (from Do et al. 2018). This is
not necessarily a conservative assumption for two reasons: first, the value of
$10^{15}$ pc-3 is uncertain because it is based on only one detection, assumes
an isotropic distribution and, as we mentioned in Section 5.1, it is difficult
to find reservoirs that can account for such a high number density of
interstellar objects; second, as Pfalzner & Bannister (2019) point out,
interstellar objects will have a velocity distribution but only those in the
low-velocity range will be incorporated to the cloud (they point out that if
interstellar objects have a velocity distribution resembling that of young
stars $<$ 2 Gyr, assumed to be their primary source, only $\sim$ 1–4% will
have relative velocities $<$ 10 km/s).

Pfalzner & Bannister (2019) assume that the volume of the clumps, i.e. the
volume of space in the molecular cloud that each star draws material from, is
$V={4\over 3N}\pi R_{\rm cluster}^{3}$, where $N$ is the number of stars in
the cluster and $R_{\rm cluster}^{3}$ is the radius of the cluster, with N =
100–5000 and $R_{\rm cluster}$ of 0.3–0.4 pc, and 1–1.2 pc,
respectively333This assumes an average stellar mass of 0.5 M⊙, a star-
formation efficiency of 10–30%, and a mass-radius relationship following
log($M_{\rm cluster})=3.42\pm 0.01+(1.67\pm 0.025){\rm log}R_{\rm cluster}$.
They also assume that only 10–30% of the material contained within that volume
contributes to the stellar system formation, drawing an analogy with the the
star-formation efficiency444For example, the Core to Disks Spitzer Legacy
survey allowed to compare the mass in young stellar objects to the cloud mass
and found that, over the last 2 Myr, corresponding to the lifetime of the IR
excess emission this survey was sensitive to, the star-formation efficiency
ranges from 3–6%, leading to 15–30% over the lifetime of the cloud (Evans et
al. 2009).. They further assume that only 1% the interstellar material that
contributes to the stellar system formation ends up incorporated to the
protoplanetary disk, escaping accretion onto the star, drawing an analogy of
what happens to the gas as the protoplanetary disk forms (resulting in a disk-
to-star mass ratio of 0.01). Finally, then assume that only 10% of the
interstellar objects incorporated to the disk survive further loss due to
processes like erosion or volatization.

Pfalzner & Bannister (2019) find that approximately 106 objects of
1I/’Oumuamua’s size could be incorporated to each stellar system as a result
of the star formation process, while in the larger size range their estimate
is $\sim$ 104–105 for objects with a diameter of $\sim$1 km, and $\sim$ 103
for a diameter of $\sim$100 km. They note that these values could be 104–106
times higher if in the molecular cloud forming process, the interstellar
objects were to increase their concentration with respect to the interstellar
medium by the same degree as the gas.

###  6.2 Our estimate

####  6.2.1 Number of interstellar objects expected to be captured by the
entire molecular cloud

As Pfalzner & Bannister (2019) point out, their results rely on the assumption
that the interstellar objects have small velocities relative to the molecular
clouds. To overcome this caveat, in our approach we will not assume that the
number density of interstellar objects that are trapped inside the molecular
cloud is initially the same as in the interstellar medium. At any given time,
there will be interstellar objects crossing the cloud, but not necessarily on
bound orbits and with small relative velocities. Instead, using the formalism
developed above, we calculate the number of interstellar objects that could be
captured by gas drag inside the molecular cloud. These objects will indeed
have small relative velocities. As discussed in Section 4, the number of
interstellar objects that cross each environment during its lifetime is
$N_{\rm enter}(r_{p})=n_{\rm ISO}(r_{p})\pi b_{\rm max}^{2}\langle
u\rangle\tau_{\rm env}$. The capture probability of the interstellar objects
has been calculated in Sections 3 and 5, where the condition for capture is
given by how the energy dissipated by gras drag compares to the initial
kinetic energy of the objects, following Grishin et al. (2019). This led to
Equation (26) that we will now use, where for $n_{\rm ISO}(r_{p})$ we adopt
the values calculated in Section 5.1 and listed in Table 2 as $N_{\rm
R_{1}<r_{p}<R_{2}}$ for each of the size ranges. The molecular cloud
properties adopted are those in Table 1. Table 3 lists the resulting number of
interstellar objects that are expected to be captured by gas drag in a
molecular cloud, for different assumptions of their background number density.
The resulting values show a wide range because of the uncertainty in the size
distribution of the interstellar objects. The values corresponding to an
equilibrium size distribution (q1 = $q_{2}$ = 3.5) are also shown in the
table. The calculation assumes a value of $C_{D}$ = 2. However, this is an
approximation because $C_{D}$ depends on velocity. In this particular case,
Table A1 shows that $C_{D}$ can be as high as 11 for the slowest particles
($\sim$ 0.1 km/s). Because the number of objects captured is approximately
proportional to $C_{D}$ (see Equation 27), this means that the number of
particles captured in those size ranges will be higher because of the enhanced
contribution from the particles in the very low-velocity end due to their
increased $C_{D}$, but the enhancement will be modest.

####  6.2.2 Number of captured interstellar objects expected to be
incorporated to each protoplanetary disk at formation

The values listed in the top part of Table 3 correspond to the number of
objects captured in the molecular cloud as a whole. We are now interested in
estimating how many might be incorporated to each protoplanetary disk at
formation. As Pfalzner & Bannister (2019), we assume that a molecular cloud
leads to the formation of a cluster of stars. The volume of space in the
molecular cloud that each star draws material from is $V={4\over 3N}\pi R_{\rm
cluster}^{3}$, where $N$ is the number of stars in the cluster and $R_{\rm
cluster}$ is the radius of the cluster, with N = 100–5000 and $R_{\rm
cluster}$ of 0.3–0.4 pc, and 1–1.2 pc, respectively. As Pfalzner & Bannister
(2019), we further assumed that 30% of the material contained within that
volume contributes to the stellar system formation (30% star-formation
efficiency), that 1% of the interstellar objects incorporated to the systems
end up in the disk (based on the typical disk-to-star mass ratio of 0.01,
assuming the interstellar objects behave similar to the gas), and that 10% of
the objects incorporated to the disk survive erosion or volatization. Under
this scenario, we get that the number of interstellar objects that could be
incorporated to each protoplanetary disk in its process of formation is a
factor of ${3\cdot 10^{-4}\over N}\left({R_{\rm cluster}\over R_{\rm
env}}\right)^{3}$ smaller than the number of interstellar objects trapped in
the entire cloud, listed in the top part of Table 3. These factors are
3$\cdot$10-12 and 2$\cdot$10-12, for N = 100 and 5000, respectively. The
results for N = 100 are listed in the bottom part of Table 3 and will be
discussed in Section 9.1. For comparison, if we assume an overall star
formation efficiency of 1% for the molecular cloud, i.e. that 1% of the cloud
mass is converted into stars, the number of interstellar objects trapped per
solar mass would be the values listed for $N(r_{p})$ at the top of Table 3
multiplied by a factor of ${10^{-2}\over M_{\rm env}(M_{\odot})}=1.4\cdot
10^{-7}$, where $M_{\rm env}=6.9E4M_{\odot}.$

##  7 Total number of captured planetesimals: pre-stellar kernel and
protoplanetary disk

We now describe another two capture scenarios that are relevant during the
early stage of star and planet formation: the trapping of interstellar objects
by gas drag in a pre-stellar core kernel and in a protoplanetary disk. We
follow the same calculation we did above but for the corresponding
environmental parameters listed in Table 1. The estimated number of
interstellar objects expected to be trapped in these environments is listed in
Tables 4 and 5, assuming a gas drag coefficient of $C_{D}$ = 2. Table A1
indicates that for the pre-stellar kernel environment and for the range of
particle sizes and velocities considered, $C_{D}$ = 2–9, while for the
protoplanetary disk environment (listed under the entry DISK-30AU in Table A1)
it can be up to 23 for the slowest particles ($\sim$ 0.1 km/s) with radius 50
cm–50 m; because $N(r_{p})\propto C_{D}$ (Equation 27), this means that the
number of particles captured in those size ranges will be higher from an
enhanced contribution from the particles in the very low-velocity end, due to
their increased $C_{D}$, but the enhancement will be modest.

In the protoplanetary disk calculation above we assumed that the swept up
column is that along the scale height so instead of using $R_{\rm
env}\rho_{\rm env}$ in Equation (26) we use $h\rho_{\rm env}$. Following
Grishin et al. (2019), for the mass surface density we adopted the flared disk
model in Chiang & Goldreich (1997), $\Sigma_{g}(r)=\Sigma_{g,0}({r\over
au})^{-{3\over 2}}$, with $\Sigma_{g,0}=2\cdot 10^{3}$ g cm-2 (close the the
value $\Sigma_{g,0}=2.7\cdot 10^{3}$ g cm-2 from Hayashi 1981). The number
density that was used to calculate $C_{D}$ (listed as DISK-30AU in Table A1)
is derived from $\rho_{g}(r)={\Sigma_{g}(r)\over\sqrt{2\pi}h}$ (Chiang &
Goldreich 1997), where $h$ is the scale height, $h=\sqrt{c^{2}r^{3}\over
GM_{*}}$ and $c$ is the sound speed (Shakura & Sunyaev 1973). The temperature
is given by $T_{\rm env}={280\over\sqrt{r}}({L\over L_{\odot}})^{1\over 4}$.
At 30AU, we have $\rho=1.7\cdot 10^{-14}$ g cm-3 ($n=5\cdot 10^{9}$ cm-3),
$T_{\rm env}\sim$ 50 K and $h\sim$ 3 au.

Table 2: Expected number density of interstellar objects, $N_{\rm R_{1}<r_{p}<R_{2}}$, for different assumptions of $n_{ISO}$ and adopting the size distributions in Equation (30). |  | 50cm–5m | 5m–50m | 50m–500m | 500m–5km | 5km–50km
---|---|---|---|---|---|---
N${}_{\rm R\geqslant R_{Ou}}$ = 2$\cdot$1015 pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 3$\cdot$1017–1024 | 3$\cdot$1016–1020 | 3$\cdot$1015–1016 | 1012–3$\cdot$1014 | 5$\cdot$107–3$\cdot$1013
|  $q_{1}$=$q_{2}$=3.5 | 6$\cdot$1020 | 2$\cdot$1018 | 6$\cdot$1015 | 2$\cdot$1013 | 6$\cdot$1010
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8$\cdot$109–2$\cdot$1020 | 8$\cdot$108–3$\cdot$1017 | 8$\cdot$107–3$\cdot$1014 | 9$\cdot$103–1012 | 0.4–109
mtotal = 9.5$\cdot$1028 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8$\cdot$1010–2$\cdot$1021 | 8$\cdot$109–3$\cdot$1018 | 8$\cdot$108–3$\cdot$1015 | 9$\cdot$104–1013 | 4–1010

Table 3: Number of interstellar objects expected to be captured by gas drag in a molecular cloud, for different assumptions of $n_{ISO}$ and adopting the size distributions in Equation (30)aaIn the calculation of $N(r_{p})$, we adopt as $r_{p}$ the smallest particle radius in the corresponding size bin.. |  | 50cm–5m | 5m–50m | 50m–500m | 500m–5km | 5km–50km
---|---|---|---|---|---|---
Trapped by the entire molecular cloudbbSee discussion in Section 6.2.1.
RangeccIndicates the range of $C_{D}$ values corresponding to the smallest particle radius in the corresponding size bin, for velocities of 0–30 km/s (from the Table A1 entry corresponding to the molecular cloud). of $C_{D}$ |  | 2–11 | 2–11 | 2–11 | 2–11 | 2–11
Results for $C_{D}$ = 2 |  |  |  |  |  |
N${}_{\rm R\geqslant R_{Ou}}$ = 2$\cdot$1015 pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8$\cdot$1016–3$\cdot$1023 | 8$\cdot$1014–3$\cdot$1018 | 8$\cdot$1012–3$\cdot$1013 | 3$\cdot$108–8$\cdot$1010 | 103–8$\cdot$108
|  $q_{1}$=$q_{2}$=3.5 | 2$\cdot$1020 | 6$\cdot$1016 | 2$\cdot$1013 | 6$\cdot$109 | 2$\cdot$106
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 2$\cdot$109–6$\cdot$1019 | 2$\cdot$107–8$\cdot$1015 | 2$\cdot$105–8$\cdot$1011 | 3–3$\cdot$108 | 10-5-3$\cdot$104
mtotal = 9.5$\cdot$1028 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 2$\cdot$1010–6$\cdot$1020 | 2$\cdot$108–8$\cdot$1016 | 2$\cdot$106–8$\cdot$1012 | 3$\cdot$101–3$\cdot$109 | 10-4-3$\cdot$105
Incorporated from the cloud to each protoplanetary disk in the process of disk
formationddSee discussion in Section 6.2.2.
N = 100 |  |  |  |  |  |
N${}_{\rm R\geqslant R_{Ou}}$ = 2$\cdot$1015 pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 2$\cdot$105–9$\cdot$1011 | 2$\cdot$103–9$\cdot$106 | 2$\cdot$101–9$\cdot$101 | 9$\cdot$10-4–2$\cdot$10-1 | 3$\cdot$10-9–2$\cdot$10-3
|  $q_{1}$=$q_{2}$=3.5 | 6$\cdot$108 | 2$\cdot$105 | 6$\cdot$101 | 2$\cdot$10-2 | 6$\cdot$10-6
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 6$\cdot$10-3–2$\cdot$108 | 6$\cdot$10-5–2$\cdot$104 | 6$\cdot$10-7–2 | 9$\cdot$10-12–9$\cdot$10-4 | 3$\cdot$10-17–9$\cdot$10-8

Table 4: Same as Table 3 for the pre-stellar core kernel. |  | 50cm–5m | 5m–50m | 50m–500m | 500m–5km | 5km–50km
---|---|---|---|---|---|---
RangeaaIndicates the range of $C_{D}$ values corresponding to the smallest particle radius in the corresponding size bin, for velocities of 0–30 km/s (from the Table A1 entry corresponding to the pre-stellar core kernel). of $C_{D}$ |  | 2–9 | 2–9 | 2–9 | 2–9 | 2–9
Results for $C_{D}$ = 2 |  |  |  |  |  |
N${}_{\rm R\geqslant R_{Ou}}$ = 2$\cdot$1015 pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 3$\cdot$106–1013 | 3$\cdot$104–108 | 3$\cdot$102–103 | 10-2–3 | 5$\cdot$10-8–3$\cdot$10-2
|  $q_{1}$=$q_{2}$=3.5 | 6$\cdot$109 | 2$\cdot$106 | 6$\cdot$102 | 2$\cdot$10-1 | 6$\cdot$10-5
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8$\cdot$10-2–2$\cdot$109 | 8$\cdot$10-4–3$\cdot$105 | 8$\cdot$10-6–3$\cdot$101 | 9$\cdot$10-11–10-2 | 4$\cdot$10-16–10-6
mtotal = 9.5$\cdot$1028 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8$\cdot$10-1–2$\cdot$1010 | 8$\cdot$10-3–3$\cdot$106 | 8$\cdot$10-5–3$\cdot$102 | 9$\cdot$10-10–10-1 | 4$\cdot$10-15–10-5

Table 5: Same as Table 3 for the protoplanetary disk at 30 AUaaAs discussed in
Section 7, in this order-of-magnitude estimate, the properties of the
environment are those characteristic of the protoplanetary disk at 30AU, with
$n=6\cdot 10^{13}$ cm-3 and $h=3\cdot 10^{14}$ cm. We assume that the swept up
column is that along the scale height of the disk, $h$, as discussed in
Section 7..

|  | 50cm–5m | 5m–50m | 50m–500m | 500m–5km | 5km–50km
---|---|---|---|---|---|---
RangeaaAs discussed in Section 7, in this order-of-magnitude estimate, the properties of the environment are those characteristic of the protoplanetary disk at 30AU, with $n=6\cdot 10^{13}$ cm-3 and $h=3\cdot 10^{14}$ cm. We assume that the swept up column is that along the scale height of the disk, $h$, as discussed in Section 7. of $C_{D}$ |  | 2-23 | 2–23 | 2–23 | 2–16 | 2–6
Results for $C_{D}$ = 2 |  |  |  |  |  |
N${}_{\rm R\geqslant R_{Ou}}$ = 2$\cdot$1015 pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 3$\cdot$108–1015 | 3$\cdot$106–1010 | 3$\cdot$104–105 | 1–3$\cdot$102 | 5$\cdot$10-6–3
|  $q_{1}$=$q_{2}$=3.5 | 6$\cdot$1011 | 2$\cdot$108 | 6$\cdot$104 | 2$\cdot$101 | 6$\cdot$10-3
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8–2$\cdot$1011 | 8$\cdot$10-2–3$\cdot$107 | 8$\cdot$10-4–3$\cdot$103 | 9$\cdot$10-9–1 | 4$\cdot$10-14–10-4
mtotal = 9.5$\cdot$1028 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 8$\cdot$101–2$\cdot$1012 | 8$\cdot$10-1–3$\cdot$108 | 8$\cdot$10-3–3$\cdot$104 | 9$\cdot$10-8–1$\cdot$101 | 4$\cdot$10-13–10-3

bbfootnotetext: Indicates the range of $C_{D}$ values corresponding to the
smallest particle radius in the corresponding size bin, for velocities of 0–30
km/s (from the Table A1 entry corresponding to DISK-30AU).

Table 6: Same as Table 5 for the protoplanetary disk at 30 AUaaAs discussed in
Section 7, in this order-of-magnitude estimate, the properties of the
environment are those characteristic of the protoplanetary disk at 30AU, with
$n=6\cdot 10^{13}$ cm-3 and $h=3\cdot 10^{14}$ cm. We assume that the swept up
column is that along the scale height of the disk, $h$, as discussed in
Section 7. in an open cluster environment.

|  | 50cm–5m | 5m–50m | 50m–500m | 500m–5km | 5km–50km
---|---|---|---|---|---|---
RangeaaAs discussed in Section 7, in this order-of-magnitude estimate, the properties of the environment are those characteristic of the protoplanetary disk at 30AU, with $n=6\cdot 10^{13}$ cm-3 and $h=3\cdot 10^{14}$ cm. We assume that the swept up column is that along the scale height of the disk, $h$, as discussed in Section 7. of $C_{D}$ |  | 2-23 | 2–23 | 2–23 | 2–16 | 2–6
Results for $C_{D}$ = 2 |  |  |  |  |  |
N${}_{\rm R\geqslant R_{Ou}}$ = 2$\cdot$1015 pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 2$\cdot$1013–6$\cdot$1019 | 2$\cdot$1011–6$\cdot$1014 | 2$\cdot$109–6$\cdot$109 | 6$\cdot$104–2$\cdot$107 | 3$\cdot$10-1–2$\cdot$105
|  $q_{1}$=$q_{2}$=3.5 | 4$\cdot$1016 | 1013 | 4$\cdot$109 | 106 | 4$\cdot$102
mtotal = 9.5$\cdot$1027 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 5$\cdot$105–1016 | 5$\cdot$103–2$\cdot$1012 | 5$\cdot$101–2$\cdot$108 | 6$\cdot$10-4–6$\cdot$104 | 3$\cdot$10-9–6
mtotal = 9.5$\cdot$1028 g$\cdot$pc-3 |  $q_{1}$=2–3.5; $q_{2}$=3–5 | 5$\cdot$106–1017 | 5$\cdot$104–2$\cdot$1013 | 5$\cdot$102–2$\cdot$109 | 6$\cdot$10-3–6$\cdot$105 | 3$\cdot$10-8–6$\cdot$101

bbfootnotetext: Indicates the range of $C_{D}$ values corresponding to the
smallest particle radius in the corresponding size bin, for velocities of 0–30
km/s (from the Table A1 entry corresponding to DISK-30AU).

Table 7: Results from Grishin et al. (2019)aaDerived from Grishin et al. (2019) Figure 3 left, corresponding to an equilibrium size distribution ($p$=11/6) and a number density of interstellar objects of $2\cdot 10^{15}~{}{\rm pc}^{-3}$ (shown in their figure as transparent blue and green, for a field and cluster environments, respectively). These values are to be compared to those in Tables 7 and 8 corresponding to $N_{\rm R\geqslant R_{\rm Ou}}$ = 2$\cdot$1015 pc-3 and an equilibrium size distribution, $q_{1}$ = $q_{2}$ = 3.5. | 50cm | 5m | 50m | 500m | 5km | 50km
---|---|---|---|---|---|---
Field | 3$\cdot$1012 | 2$\cdot$109 | 8$\cdot$105 | 8$\cdot$102 | 1 | 10-3
Cluster | 1$\cdot$1014 | 2$\cdot$1011 | 9$\cdot$107 | 105 | 102 | 10-1

##  8 The case of an open cluster environment

###  8.1 Increased capture probability in an open cluster environment

The capture efficiency of interstellar objects by a stellar system increases
when considering stars in an open cluster, a common environment for stellar
birth (Adams & S9cmpergel 2005). This is because the relative velocities of
the stars in the cluster are smaller than the relative velocities of the stars
in the field. Belbruno et al. et al. (2012) found that, when considering
chaotic, nearly-parabolic orbits, the transfer probability of solids between
the stars in an open cluster is many orders of magnitude higher than
previously thought, allowing for the exchange of solid material in significant
quantities between the young solar system and nearby planetary systems while
the Sun was still embedded in the birth cluster. Belbruno et al. (2012)
focused on the implications this has on the lithopanspermia hypothesis. Here,
we address whether this exchange of interstellar planetesimals, ejected from a
stellar system where planetesimal formation has already taken place, could
provide potential seeds for planet formation in a nearby stellar system in the
cluster.

The characteristic escape velocities for the chaotic, nearly parabolic
trajectories considered in Belbruno et al. (2012) are $\sim$0.1 km/s,
resulting in incoming planetesimal velocities dominated by the relative
velocities of the stars in the cluster, assumed to be $u\sim$ 1 km/s. However,
under the scenario that we are considering in this paper, the typical ejection
velocities of the objects, that will subsequently become interstellar, are
significantly higher, $u\sim 6.2\pm 2.7$ km/s (Adams & Spergel 2005). In the
calculation of the expected number of captured objects using Equation (26),
instead of the previously assumed velocity dispersion of $\sigma$ = 30 km/s,
we will use $\sigma$ = 6 km/s.

###  8.2 Number density of interstellar objects in an open cluster

We now estimate the expected number density of interstellar objects in an open
cluster, $n_{\rm ISO}(r_{p})^{OC}$, to be used as $n_{\rm ISO}(r_{p})$ in
Equations (26) and (27), by taking into account the number density of stars in
the cluster, their initial mass function, and the number of planetesimals
expected to be ejected from each star (a function of the stellar mass). We
assume that the cluster consists on N = 100–5000 members, as clusters with
this range of sizes are the birthplaces of a large fraction of stars in the
Galaxy (Lada & Lada 2003). For the cluster radius we adopt Rcluster =
1pc$\left({N\over 300}\right)^{1/2}$ (Adams 2010), leading to an average
number density of stars of $n={3N\over 4\pi R_{cluster}^{3}}$, resulting in
122 pc-3 and 18 pc-3, for a cluster with N=100 and 5000 members, respectively.

The overall contribution from all the stars in the open cluster to the number
density of interstellar planetesimals can be estimated following the same
calculation as in Section 5.1.2 for $N_{\rm R_{1}<R<R_{2}}$, shown in Figures
3–5 and listed in Table 2, but scaling it to take into account that the
initial mass function to be used needs to agree with the estimated average
number density of stars in the cluster, as calculated above. The estimate in
Section 5.1.2 adopted the following initial mass function from from Kroupa et
al. (1993),

$\begin{split}n(M_{*})=\xi(M_{*})dM_{*}~{}~{}~{}~{}{\rm
with,}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
\xi(M_{*})=0.035M_{*}^{-1.3}~{}~{}~{}~{}{\rm if~{}0.08}\leq M_{*}<0.5\\\
\xi(M_{*})=0.019M_{*}^{-2.2}~{}~{}~{}~{}~{}{\rm if~{}0.5}\leq M_{*}<1.0\\\
\xi(M_{*})=0.019M_{*}^{-2.7}~{}~{}~{}~{}{\rm if~{}1.0}\leq
M_{*}<100,\end{split}$ (31)

where $\xi$ is the number density of stars per pc3 and the stellar mass is in
units of M⊙. This leads to scaling factors of $f^{\rm OC}=8.9\cdot 10^{2}$ for
N = 100 and $f^{\rm OC}=1.3\cdot 10^{2}$ for N = 5000. We then have that the
number density of interstellar objects in the open cluster is $n_{\rm
ISO}(r_{p})^{OC}=n_{\rm ISO}(r_{p})\cdot f^{\rm OC}$.

###  8.3 Number of captured interstellar planetesimals per protoplanetary disk
in an open cluster environment

The equivalent of Equation (27) in the cluster environment, estimating the
expected number of captured interstellar planetesimals per protoplanetary
disk, is given by

$\begin{split}N(r_{p})={3\pi\over 8}{\sqrt{2\pi}}{C_{D}}n_{\rm
ISO}(r_{p})f^{\rm OC}\tau_{\rm env}R_{\rm env}^{2}u_{\rm esc}\left({u_{\rm
esc}\over\sigma}\right)^{3}{R_{\rm env}\rho_{\rm
env}\Psi(e)\over\rho_{p}r_{p}}.\end{split}$ (32)

The results are listed in Table 6, and are scaled from those in Table 5 by a
factor of $f^{\rm
OC}\left({\sigma_{field}\over\sigma_{OC}}\right)^{3}=500({30\over 6})^{3}$ =
6.25$\cdot$104, where we have assumed $f^{\rm OC}=500$.

##  9 Discussion

###  9.1 Molecular cloud environment

In Section 6, we estimated how many interstellar objects can be trapped by the
molecular cloud. Assuming these objects are uniformly distributed, we followed
Pfalzner & Bannister (2019) to get a rough estimate of how many of these
trapped interstellar objects could be incorporated to each star-forming disk.
We looked at the case of a cloud clump forming a star cluster with N = 100
members. If we adopt a cumulative number density of interstellar objects in
the interstellar medium of $N_{\rm R\geqslant R_{\rm Ou}}$ = 2$\cdot$1015 pc-3
and the size distributions given by Equation (30), we find that the number of
trapped objects for the different particle sizes are: 2$\cdot$105–9$\cdot$1011
(50 cm–5 m), 2$\cdot$103–9$\cdot$106 (5 m–50 m), 2$\cdot$101–9$\cdot$101 (50
m–500 m), 9$\cdot$10-4–2$\cdot$10-1 (500 m–5 km), and
3$\cdot$10-9–2$\cdot$10-3 (5 km–50 km).

For comparison, Pfalzner & Bannister (2019), using the same range of size
distributions, estimate that the number of interstellar objects incorporated
to each disk at formation will be approximately 106 of 1I/’Oumuamua’s size,
$\sim$ 104–105 with a diameter of $\sim$1 km, and $\sim$ 103 for a diameter of
$\sim$100 km. Their results are several orders of magnitude higher than our
estimates. Our approach differs from Pfalzner & Bannister (2019) in that the
number density of interstellar objects that we use in our calculation, for the
different size ranges, corresponds to the objects that have been trapped from
the interstellar medium by the molecular cloud, while Pfalzner & Bannister
(2019) assumes that the number density of interstellar objects inside the
cloud is the same as in the interstellar medium. Their approach relies on the
assumption that the interstellar objects have small velocities relative to the
molecular clouds. We assume a velocity dispersion of 30 km/s and in this case
the relative velocities can be significant. At any given time, there will be
interstellar objects crossing the cloud, but most of them will not be trapped
(see below) and will not necessarily have small relative velocities. However,
we can assume that the small number of objects that are trapped will indeed
have small relative velocities. We can calculate the fraction of interstellar
objects that are trapped, out of the total number that are in the cloud at any
given time, using Equation (29): adopting a lifetime, $\tau_{\rm env}$ = 30
Myr, we get that ${N(r_{p})\over n_{\rm ISO}(r_{p}){4\over 3}\pi R_{\rm
env}^{3}}={9\over 32}{\sqrt{2\pi}}{C_{D}}\Psi(e){\sigma\tau_{\rm env}\over
R_{\rm env}}{\rho_{\rm env}R_{\rm
env}\over\rho_{p}r_{p}}\bar{\Theta}_{s}^{2}={3\cdot 10^{-3}\over r_{p}}$,
where $r_{p}$ is in cm.

However, Dobbs & Pringle (2013) note that the molecular cloud lifetime may be
effectively much longer than the $\tau_{\rm env}$ = 30 Myr value adopted
above. This is because giant molecular clouds may dissolve by breaking up into
small fragments, but these fragments will maintain their molecular cloud
identity during at least a rotation period or even many rotation periods
around the Galaxy, reforming giant molecular clouds in the spiral arms that
will then fragment again after $\tau_{\rm env}\sim$ 30 Myr. In this case, the
number of interstellar objects that are trapped by the cloud fragment can be
significantly increased because of its increased lifetime.

As Pfalzner & Bannister (2019) point out, this order-of-magnitude calculation
relies heavily on the assumption that the interstellar objects behave
similarly to the gas in the cloud- and star-formation processes, but this is
only valid if these processes are dominated by gravitational forces and not by
gas dynamics, because the interstellar objects and the gas are largely
decoupled. In an interesting new study using numerical simulations, Pfalzner
et al. (2021) confirm that the interstellar objects follow the collapse of the
gas in the molecular cloud. They also find that the number of interstellar
objects bound to each future star-forming system is disproportionately higher
for the more massive systems that will eventually lead to subclusters of
stars. In their low-resolution numerical simulations, each interstellar test
particle represents about 1010 interstellar objects, and this does not allow
to follow the fate of smaller ensembles of interstellar objects. Due to the
resolution of the simulations, it is also not possible to model how many
interstellar objects will be bound to solar-type stars. But their very
interesting initial proof-of-concept results indicate that the number of
interstellar objects that might be incorporated to each star-forming disk can
be very diverse, contrary to our naive scenario in which the interstellar
objects captured by the molecular cloud are distributed uniformly among the
stars in the cluster. They find that some star forming systems are very rich
in bound interstellar objects while others are depleted of them. Pfalzner et
al. (2021) simulations assume that the initial distribution of interstellar
objects are not moving relative to the cloud but, as pointed out earlier, the
velocity dispersion can be significant. Future numerical simulations
accounting for the velocity distribution of the interstellar objects should
also consider the population of objects already trapped by the cloud, as
calculated in our study.

###  9.2 Protoplanetary disk environment

Table 5 shows the number of interstellar objects expected to be captured by
gas drag in a protoplanetary disk, for different assumptions of the number
density of interstellar objects and adopting the size distributions in
Equation (30). When adopting a cumulative number density of $N_{\rm R\geqslant
R_{\rm Ou}}$ = 2$\cdot$1015 pc-3 and an equilibrium size distribution ($q_{1}$
= $q_{2}$ = 3.5), we find that the number of captured objects are 6$\cdot$1011
(50 cm–5 m), 2$\cdot$108 (5 m–50 m), 6$\cdot$104 (50 m–500 m), 2$\cdot$101
(500 m–5 km), and 6$\cdot$10-3 (5 km–50 km). These results are lower than
those in Grishin et al. (2019) for $p$ = 11/6 (corresponding also to an
equilibrium size distribution), listed in Table 7 for the field environment.
Our results show that a significant number of interstellar planetesimals can
be incorporated to the protoplanetary disks by gas drag. But they also
indicate that the uncertainty in the expected number of captured objects is
very significant. This is due to the wide range of possible size (mass)
distributions for interstellar objects in Equation (30) (this uncertainty is
reflected in the wide range of values listed in Table 5 under $q_{1}$=2–3.5;
$q_{2}$=3–5), and to the uncertainties in the background number density of
interstellar objects (from a high value of $N_{\rm R\geqslant R_{\rm Ou}}$ =
2$\cdot$1015 pc-3, corresponding to the number density inferred from
1I/’Oumuamua’s detection, to a lower estimate labeled as mtotal =
9.5$\cdot$1028 g$\cdot$pc-3, corresponding to what would be expected from the
ejection of planetesimals from protoplanetary disks). In the future, we will
be able to narrow down these estimates as the population of interstellar
objects becomes better characterized, both observationally and via
simulations, in particular its background number density, size, and velocity
distributions. The simulations of the latter should take into account the
ejection process of the planetesimals from their parent system and how their
long journey in the Galaxy affects their initial velocity, as the interstellar
planetesimal population is subject to size-dependent dynamical heating.

###  9.3 Trapped interstellar objects-to-gas mass ratio

Using Equation (28) we can calculate the mass ratio of the trapped
interstellar objects to the gas in the environment, ${N(r_{p})m_{p}\over
M_{\rm env}}$. Figure 6 shows the results when we adopt an equilibrium size
distribution for the interstellar objects. This is just a zeroth-order
estimate that assumes the interstellar objects are uniformly distributed in
the environment. However, their mixing within the environment will be complex
because of the dynamics (as discussed by Pfalzner et al. 2021 in the case of a
star-forming molecular cloud), because the gas does not have a uniform
density, and because the size and phase space distributions of the
interstellar objects are uncertain, affecting their stokes numbers and
trapping efficiencies.

Figure 6: Trapped interstellar objects-to-gas mass ratio for the molecular
cloud, pre-stellar kernel and protoplanetary disk environments, calculated
using Equation (28) and assuming an equilibrium size distribution for the
interstellar objects.

###  9.4 Implications

The interest of these trapped interstellar objects is that they could have
sizes large enough to overcome the drift barrier, as the maximum drift speed
in the protoplanetary disk is found for bodies around a meter in size,
decreasing significantly for larger sizes: for planetesimals with bulk
densities of 3 g$\cdot$cm-3 located at a radial distance of 5 AU in a solar
nebula, Weidenschilling (1977) found drift timescales of O(10${}^{2})$ yr,
O(10${}^{4})$ yr, O(10${}^{5})$ yr, and O(10${}^{7})$ yr, for particles with
sizes of 1 m, 10 m, 100 m, and 1 km, respectively. The settling timescale into
the disk mid-plane also decreases with size (Weidenschilling 1980; see e.g.
Figure 2 in Bate & Lorén-Aguilar 2017). Both processes can favor the rapid
growth of these trapped planetesimals into larger bodies via the direct
accretion of the sub-cm sized dust grains in the protoplanetary disk, with
their more rapid settling time allowing them to get a boost in accretion
growth as they settle into the mid-plane. This can lead to the growth of
planetesimals by a factor of 106 in mass before the onset of collisions with
other planetesimals in the disk and erosion (that start to become important at
planetesimal sizes of $\sim$100 km, Xie et al. 2010). Mutual collisions
between the trapped interstellar planetesimals can be neglected555This is
because the planetesimals’ growth rate, $t_{\rm
growth}={\bar{m}\over\dot{m}}={\bar{m}\over\rho_{\rm dust}^{\rm disk}\pi
r^{2}\sigma_{\rm dust}^{\rm vel}}$, times the collisional rate of the
planetesimals, $R_{\rm col}=n\pi r^{2}\sigma^{\rm vel}$, indicates that for
mutual collisions to become important, i.e. $R_{\rm coll}t_{\rm
growth}={n\bar{m}\over\rho_{\rm dust}^{\rm disk}}{\sigma^{\rm
vel}\over\sigma_{\rm dust}^{vel}}={N\bar{m}\over m_{\rm dust}^{\rm
disk}}{\sigma^{\rm vel}\over\sigma_{\rm dust}^{vel}}>$ 1, the number of
planetesimals trapped in the disk needs to be $N>{m_{\rm disk}^{\rm
dust}\over\bar{m}}{\sigma_{\rm dust}^{\rm vel}\over\sigma^{vel}}$. Here,
$\bar{m}$, $r$, and $\sigma^{\rm vel}$ are the average mass, radius, and
velocity dispersion of the trapped planetesimals, and $n$ and $N$ are their
number density in the disk and total number, respectively; while $\rho_{\rm
dust}^{\rm disk}$, $m_{\rm dust}^{\rm disk}$, and $\sigma_{\rm dust}^{\rm
vel}$ are the mass density, total mass, and velocity dispersion of dust in the
disk, respectively. For a 1-m sized planetesimal, assuming a dust disk with a
dust mass of 10-4M⊙ and that ${\sigma_{\rm dust}^{\rm
vel}\over\sigma^{vel}}\sim 0.1$, collisions would become important for
$N>10^{22}$, orders of magnitude larger than the estimates in Table 2..

We therefore conclude that the trapping of interstellar objects could
potentially provide seeds for planet formation that help overcome the meter
size barrier and that this possibility should be considered in future planet
formation models. After the disk is seeded, the evolution of the objects is
subject to rapid growth and collisional disruption. The actual origin of the
planets and the small-body population will be a complex combination of these
processes. In the solar system, for example, we have to account for the
formation of a large small-body population of asteroids, comets and Kuiper
belt objects; the Kuiper belt alone is thought to host of the order of 105
objects $>$ 100 km, and long-period comet observations indicate that of the
order of 1012 objects $>$ few km populate the Oort cloud. Planet-formation
models are necessary to assess how many interstellar seeds should be
sufficient to account for the large dynamical range of masses.

As Grishin et al. (2019) and Pfalzner & Bannister (2019) argue, this trapping
mechanism could not provide the first generation of seeds but, because there
is the possibility that it will have an increasingly important role later on,
as the number density of interstellar planetesimals in the Galaxy increases,
it might alleviate the problem of requiring all the planet-forming disks to
have fine-tuned conditions to overcome the meter-sized barrier. In the case of
the open cluster, for example, if we assume that only 1% of the stars in the
open cluster have undergone planet formation, leading to planetesimals
ejection, we get that the expected number of ejected objects that can end up
captured by a neighboring protoplanetary system is 1% of those listed in Table
6. The expected number of trapped objects can remain very significant, i.e. a
small percentage of stars in the cluster that have successfully undergone
planet formation can provide a large number of interstellar seeds.

In this work, we have adopted the number density of interstellar objects based
on observations of 1I/’Oumuamua and based on the estimated mass density
expected from protoplanetary disk ejection. However, we still lack a detailed
understanding of sources and sinks, including dynamical heating, that lead to
the observed number density distributions (in phase space and size) of these
interstellar population; this will be the subject of future work.

Another aspect that needs to be taken into account is that the porosity of the
objects we are considering here can be very significant. In the solar system,
the most primitive dust ”aggregates” have porosities of 60%–70%. More fluffy,
fractal-like objects, like those expected to formed in protoplanetary disks
(Okuzumi et al. 2012; Kataoka et al. 2013) would be trapped more easily
because of their low mass per unit area and therefore our total capture rates
for porous objects may increase by an order of magnitude or more. In the
context of the suggestion that 1I/’Oumuamua might be a very low-density object
(Moro-Martín 2019b), in future work we will consider this issue of the effect
of the porosity in the capture of interstellar objects.

We also need to take into account how the evaporation, processing, accretion
and sputtering of interstellar objects in the different phases of the
interstellar medium can affect the population of planetesimals. This is of
interest regardless of the nature of the objects, but is particularly
important if they are highly porous or are a fragment of H2 ice (Seligman &
Laughlin 2020) or N2 ice (Jackson & Desch 2021), as it has been proposed for
1I/’Oumuamua. And generally, for all interstellar objects originating from
planetary systems, it is important to understand how their ejection and entry
(with a wide range of dynamical histories) can alter their physical properties
(Raymond et al. 2020).

Belbruno et al. (2012) studied the exchange of planetesimals between stars in
an open cluster and the implications on the lithopanspermia hypothesis.
Because in this work we are studying the exchange of planetesimals at a
Galactic level, it is of interest to understand what constraints the processes
mentioned earlier (that can alter the physical properties of the interstellar
objects during ejection, transit and entry) impose on the possibility of
lithopanspermia across the Galaxy.

##  10 Conclusion

An unsolved problem in planet formation theory is how cm-sized pebbles grow
into km-sized planetesimals, as particles approaching a meter size grow
inefficiently due to increased collisional energies; in addition, they have
short inward drift timescales due to gas drag that limit significantly their
lifetime in the disk and their opportunity to grow to sizes unaffected by gas
drag. Several mechanisms have been proposed to alleviate this meter-sized
barrier. In this study we address one that has recently been suggested on the
wake of the discovery of the first interstellar interlopers: the seeding of
the star- and planet-forming environments with interstellar objects captured
from the interstellar medium. These trapped objects could be large enough to
experience a longer inward drift timescale and a higher mid-plane settling
speed, favoring their rapid growth into larger bodies via the direct accretion
of the sub-cm sized dust grains in the protoplanetary disk.

In this work, we estimate the total number of interstellar objects in the 50
cm–50 km size range expected to be trapped by the molecular cloud and later
incorporated to the disk of each forming stellar system, and also those
expected to be trapped by the protoplanetary disk after it formed. Our
calculations consider the uncertainties in the size distribution of the
interstellar objects, and different assumptions regarding their origin and
their expected background density. We also consider the case of the trapping
of interstellar planetesimals by stars inside an open cluster, where the
relative velocities of the stars are small and the exchange of planetesimals
between them can be more effective.

We find that the number of trapped interstellar objects can be significant.
For example, when assuming a background number density of 2$\cdot$1015 pc-3
(based on 1I/’Oumuamua’s detection), a velocity dispersion of 30 km/s and an
equilibrium size distribution, the number of interstellar objects captured by
a molecular cloud and expected to be incorporated to each protoplanetary disk
during its formation is 6$\cdot$108 (50 cm–5 m), 2$\cdot$105 (5 m–50 m),
6$\cdot$101 (50 m–500 m), 2$\cdot$10-2 (500 m–5 km). After the disk formed,
the number of interstellar objects that it could capture from the interstellar
medium during its lifetime is 6$\cdot$1011 (50 cm–5 m), 2$\cdot$108 (5 m–50
m), 6$\cdot$104 (50 m–500 m), 2$\cdot$101 (500 m–5 km). The latter estimate
assumes a field environment. In an open cluster environment, if 1% of the
clusters stars have undergone planet formation, the number of interstellar
objects that a neighboring protoplanetary disk in the cluster could capture
during the disk lifetime is a factor of $\sim$600 larger than the field values
quoted above (1% of the values listed in Table 6). As mentioned, these
estimates correspond to an equilibrium size distribution. In this study we
consider a wide range of possible size distributions for interstellar objects
and expected background density. The corresponding results show a wide range
of possible values (listed in Tables 3–6, for the different environments
considered).

In the future, we will be able to narrow down these estimates as the
population of interstellar objects becomes better characterized, both
observationally and via simulations, in particular its background number
density, size, and velocity distributions. This will soon be possible when the
Vera Rubin Observatory starts operating, as it is expected that it will detect
a large number of interstellar objects (could be one per month). Simulations
to understand their distribution (in phase space and size) should take into
account the ejection process of the planetesimals from their parent system and
how their long journey in the Galaxy affects their initial velocity, as the
interstellar planetesimal population is subject to size-dependent dynamical
heating. Future simulations of molecular clouds and star-forming environments
(Pfalzner et al. 2021) and of planet formation should take into account the
presence of a population of trapped interstellar objects. The latter
simulations are necessary to estimate how many of these interstellar seeds
would be necessary to account for the planets and small body populations in
solar and extra-solar planetary systems and to assess if the trapped
interstellar objects can play an important role.

We thank the referee Evgeni Grishin for numerous insightful suggestions that
have considerably improved this study. We thank David Jewitt, Tushar Mittal
and Stu Weidenschilling for stimulating conversations on this subject.

##  Appendix A Effect of the Gas Drag

Table A1 in Appendix A lists the Mach number, $M=u/c$, corresponding to each
environment under consideration and for a given interstellar object incoming
velocity. The sound speed is given by $c=\left(\gamma kT_{\rm
env}/m_{0}\right)^{1/2}$, where $k$ is Boltzmann’s constant and $\gamma$ is
the adiabatic index, adopting a value of 7/5 corresponding to diatomic
molecules. For a given interstellar object radius (in the range of 50 cm to 50
km), Table A1 also lists the corresponding Reynolds number, calculated
following Adachi et al. (1976), $R_{\rm e}={2\rho_{\rm env}{\rm ur_{\rm
p}}\over\mu}$. Here, $\mu$ is the viscosity, $\mu$ = ${\rho_{\rm
env}\rm{c_{m}l}\over 3}$, $l$ is the mean free path of the gas molecules,
$l={1\over n_{\rm env}\sigma}$, $\sigma$ is the gas molecule cross section
($\sigma$ = 2.6$\cdot$10-15 cm2 for H2), and $c_{m}$ is the mean thermal
velocity of the gas molecules. Because the latter is given by
$c_{m}=\left(8kT_{\rm env}/\pi
m_{o}\right)^{1/2}=\left(8/\gamma\pi\right)^{1/2}c$, we have that $R_{\rm
e}=4.4M{r_{\rm p}\over l}=4.4MK^{-1}$, where $K$ is the Knudsen number,
$K=l/r_{\rm p}$, also listed in Table A1.

The gas drag coefficient in Equation (2) can be expressed as a function of
$R_{\rm e}$ and $K$ by using $M={KR_{\rm e}\over 4.4}$, giving

$\begin{split}C_{\rm D}\simeq{24\over R_{\rm e}}\left(\left(1+{5R_{\rm e}\over
3(10+R_{\rm e})}\right)^{-1}+1.25K\right)^{-1}+\left({(2-w)KR_{\rm e}\over
7.04+KR_{\rm e}}+w\right)={C_{1}(R_{\rm e},K)\over R_{\rm e}}+C_{2}(R_{\rm
e},K),\end{split}$ (A1)

where $C_{1}(R_{\rm e},K)$ and $C_{2}(R_{\rm e},K)$ are given by

$C_{\rm 1}=24\left(\left(1+{5R_{\rm e}\over 3(10+R_{\rm
e})}\right)^{-1}+1.25K\right)^{-1}\\\ $ (A2)

and

$C_{\rm 2}={(2-w)KR_{\rm e}\over 7.04+KR_{\rm e}}+w,$ (A3)

and are slowly varying over the range of $R_{\rm e}$ and $K$ considered.
Equations (A2) and (A3) will be used in Appendix B.

###  A.1 Stokes number

Table A1 also lists Stokes number, $Stk$, which is the ratio of the stopping
time, $t_{\rm s}$, to a dynamical time, ${2R_{\rm env}\over u}$ (the crossing
time of the cloud), so that $Stk={t_{\rm s}u\over 2R_{\rm env}}$. The stopping
time is $t_{\rm s}={x_{\rm f}-x_{\rm i}\over u}$, where $x_{\rm f}-x_{\rm i}$
is the distance travelled until the object halts to a stop. This distance can
be calculated as follows. Consider an interstellar object with an incoming
velocity $u$. It experiments an acceleration due to gas drag given by Equation
(1) so its position as a function of velocity is given by

$x(v)=\int{v\over a(v)}{\rm d}v={8\rho_{\rm p}r_{\rm p}\over 3\rho_{\rm
env}}\int{1\over C_{\rm D}(R_{\rm e},K)}{{\rm d}v\over v}.\\\ $ (A4)

Replacing ${{\rm d}v\over v}$ by ${{\rm dR_{\rm e}}\over R_{\rm e}}$, we have
that the distance travelled until the object halts to a stop, i.e. from when
it has a Reynolds number of $R_{\rm e,i}$ until $R_{\rm e,f}$ = 0, is given by

$\begin{split}x_{\rm f}-x_{\rm i}={8\rho_{\rm p}r_{\rm p}\over 3\rho_{\rm
env}}\int\limits_{R_{\rm e,i}}^{R_{\rm e,f=0}}{1\over C_{\rm D}(R_{\rm
e},K)}{{\rm d}R_{\rm e}\over R_{\rm e}}\\\ ={8\rho_{\rm p}r_{\rm p}\over
3\rho_{\rm env}}\int\limits_{R_{\rm e,i}}^{R_{\rm e,f=0}}{1\over{C_{1}(R_{\rm
e},K)\over R_{\rm e}}+C_{2}(R_{\rm e},K)}{{\rm d}R_{\rm e}\over R_{\rm e}}\\\
\simeq{8\rho_{\rm p}r_{\rm p}\over 3\rho_{\rm env}C_{2}(R_{\rm
e,i},K)}\left[{\rm ln}(C_{1}(R_{\rm e},K)+C_{2}(R_{\rm e},K)R_{\rm
e})\right]_{R_{\rm e,i}}^{R_{\rm e,f=0}}\\\ ={8\rho_{\rm p}r_{\rm p}\over
3\rho_{\rm env}C_{2}(R_{\rm e,i},K)}{\rm ln}\left({C_{1}(R_{\rm e,i},K)\over
C_{1}(R_{\rm e,i},K)+C_{2}(R_{\rm e,i},K)R_{\rm e,i}}\right),\end{split}$ (A5)

where the integral has been calculated using Equation (A1) and assuming that
$C_{1}(R_{\rm e},K)$ and $C_{2}(R_{\rm e},K)$ are constant over the range of
$R_{\rm e}$ and $K$ considered.

Using the expression above, the Stokes number is given by

$\begin{split}Stk(R_{\rm e},K)={t_{\rm s}u\over 2R_{\rm env}}={x_{\rm
f}-x_{\rm i}\over 2R_{\rm env}}\\\ \simeq{4\rho_{\rm p}r_{\rm p}\over
3\rho_{\rm env}C_{2}(R_{\rm e,i},K)R_{\rm env}}{\rm ln}\left({C_{1}(R_{\rm
e,i},K)\over C_{1}(R_{\rm e,i},K)+C_{2}(R_{\rm e,i},K)R_{\rm e,i}}\right)\\\
=-{4\rho_{\rm p}m_{o}\over 3\rho_{\rm env}^{2}C_{2}(R_{\rm e,i},K)R_{\rm
env}K\sigma}{\rm ln}\left(1+{C_{2}(R_{\rm e,i},K)\over C_{1}(R_{\rm
e,i},K)}R_{\rm e,i}\right).\end{split}$ (A6)

To express the Stokes number as a function of $R_{\rm e}$ and $K$ we have
replaced $r_{\rm p}$ by ${l\over K}={m_{o}\over K\rho_{\rm env}\sigma}$, where
$m_{o}$ and $\sigma$ are the mass and cross section of the gas molecule,
respectively (assuming H2, $m_{0}$ = 3.34$\cdot$10-24 g and $\sigma$ =
2.6$\cdot$10-15 cm2).

Alternatively, we can express Equation (A6) above as a function of $M$ and $K$
by using $R_{\rm e}=4.4MK^{-1}$ and putting $C_{1}$ and $C_{2}$ in Equations
(A2) and (A3) as a function of $M$ and $K$. We then have

$Stk=-{4\rho_{\rm p}m_{o}\over 3\rho_{\rm env}^{2}C_{2}(M,K)R_{\rm
env}K\sigma}{\rm ln}\left(1+{C_{2}(M,K)\over C_{1}(M,K)}4.4{M\over
K}\right)\\\ $ (A7)

with

$C_{\rm 1}=24\left(\left(1+{5\over 3}{4.4{M\over K}\over 10+4.4{M\over
K}}\right)^{-1}+1.25K\right)^{-1}\\\ $ (A8)

and

$C_{\rm 2}={(2-w)4.4M\over 7.04+4.4M}+w={(2-w)\over{1.6\over M}+1}+w.$ (A9)

Figures 7–9 show the Stokes number, $Stk$, as a function of the Knudsen
number, $K$, calculated using Equations (A7)–(A9) for the different
environments and incoming velocities under consideration. Table A2 shows the
critical object radius, rp (cm), for which the Stokes number reaches the
values of 1 and 0.1. Interstellar objects with $Stk<<$ 1 are expected to be
trapped by gas drag, while those with $Stk>>$ 1 are expected to escape.

Table A1: Effect of gas drag.aa(*) indicates objects with $Stk<1$. (**) indicates objects with $Stk<0.1$.Particles with $Stk<<1$ are expected to get trapped. | rp = 50 cm | rp = 5 m | rp = 50 m | rp = 500 m | rp = 5 km | rp = 50 km
---|---|---|---|---|---|---
Molecular cloud |  |  |  |  |  |
Knudsen number (K) | K=7.69$\cdot$1010 | K=7.69$\cdot$109 | K=7.69$\cdot$108 | K=7.69$\cdot$107 | K=7.69$\cdot$106 | K=7.69$\cdot$105
u = 0.1 km/s, M = 0.42 |  |  |  |  |  |
Reynolds number (Re) | 2.38$\cdot$10-11 | 2.38$\cdot$10-10 | 2.38$\cdot$10-9 | 2.38$\cdot$10-8 | 2.38$\cdot$10-7 | 2.38$\cdot$10-6
Drag coefficient (CD) | 11.08 | 11.08 | 11.08 | 11.08 | 11.08 | 11.08
Stokes number | 3.96$\cdot$102 | 3.96$\cdot$103 | 3.96$\cdot$104 | 3.96$\cdot$105 | 3.96$\cdot$106 | 3.96$\cdot$107
u = 1 km/s, M = 4.16 |  |  |  |  |  |
Reynolds number (Re) | 2.38$\cdot$10-10 | 2.38$\cdot$10-9 | 2.38$\cdot$10-8 | 2.38$\cdot$10-7 | 2.38$\cdot$10-6 | 2.38$\cdot$10-5
Drag coefficient (CD) | 2.60 | 2.60 | 2.60 | 2.60 | 2.60 | 2.60
Stokes number | 2.51$\cdot$103 | 2.51$\cdot$104 | 2.51$\cdot$105 | 2.51$\cdot$106 | 2.51$\cdot$107 | 2.51$\cdot$108
u = 7 km/s, M = 29.10 |  |  |  |  |  |
Reynolds number (Re) | 1.66$\cdot$10-9 | 1.66$\cdot$10-8 | 1.66$\cdot$10-7 | 1.66$\cdot$10-6 | 1.66$\cdot$10-5 | 1.66$\cdot$10-4
Drag coefficient (CD) | 2.07 | 2.07 | 2.07 | 2.07 | 2.07 | 2.07
Stokes number | 5.88$\cdot$103 | 5.88$\cdot$104 | 5.88$\cdot$105 | 5.88$\cdot$106 | 5.88$\cdot$107 | 5.88$\cdot$108
u = 30 km/s, M = 124.74 |  |  |  |  |  |
Reynolds number (Re) | 7.13$\cdot$10-9 | 7.13$\cdot$10-8 | 7.13$\cdot$10-7 | 7.13$\cdot$10-6 | 7.13$\cdot$10-5 | 7.13$\cdot$10-4
Drag coefficient (CD) | 2.01 | 2.01 | 2.01 | 2.01 | 2.01 | 2.01
Stokes number | 8.80$\cdot$103 | 8.80$\cdot$104 | 8.80$\cdot$105 | 8.80$\cdot$106 | 8.80$\cdot$107 | 8.80$\cdot$108
Pre-stellar core kernel |  |  |  |  |  |
Knudsen number (K) | K=7.69$\cdot$106 | K=7.69$\cdot$105 | K=7.69$\cdot$104 | K=7.69$\cdot$103 | K=7.69$\cdot$102 | K=7.69$\cdot$101
u = 0.1 km/s, M = 0.52 |  |  |  |  |  |
Reynolds number (Re) | 2.95$\cdot$10-7 | 2.95$\cdot$10-6 | 2.95$\cdot$10-5 | 2.95$\cdot$10-4 | 2.95$\cdot$10-3 | 2.95$\cdot$10-2
Drag coefficient (CD) | 9.15 | 9.15 | 9.15 | 9.15 | 9.15 | 9.07
Stokes number | 1.01$\cdot$102 | 1.01$\cdot$103 | 1.01$\cdot$104 | 1.01$\cdot$105 | 1.01$\cdot$106 | 1.01$\cdot$107
u = 1 km/s, M = 5.16 |  |  |  |  |  |
Reynolds number (Re) | 2.95$\cdot$10-6 | 2.95$\cdot$10-5 | 2.95$\cdot$10-4 | 2.95$\cdot$10-3 | 2.95$\cdot$10-2 | 2.95$\cdot$10-1
Drag coefficient (CD) | 2.46 | 2.46 | 2.46 | 2.46 | 2.46 | 2.46
Stokes number | 5.87$\cdot$102 | 5.87$\cdot$103 | 5.87$\cdot$104 | 5.87$\cdot$105 | 5.88$\cdot$106 | 5.91$\cdot$107
u = 7 km/s, M = 36.1 |  |  |  |  |  |
Reynolds number (Re) | 2.06$\cdot$10-5 | 2.06$\cdot$10-4 | 2.06$\cdot$10-3 | 2.06$\cdot$10-2 | 2.06$\cdot$10-1 | 2.06
Drag coefficient (CD) | 2.05 | 2.05 | 2.05 | 2.05 | 2.05 | 2.05
Stokes number | 1.30$\cdot$103 | 1.30$\cdot$104 | 1.30$\cdot$105 | 1.30$\cdot$106 | 1.30$\cdot$107 | 1.31$\cdot$108
u = 30 km/s, M = 154.72 |  |  |  |  |  |
Reynolds number (Re) | 8.85$\cdot$10-5 | 8.85$\cdot$10-4 | 8.85$\cdot$10-3 | 8.85$\cdot$10-2 | 8.85$\cdot$10-1 | 8.85
Drag coefficient (CD) | 2.01 | 2.01 | 2.01 | 2.01 | 2.01 | 2.01
Stokes number | 1.91$\cdot$103 | 1.91$\cdot$104 | 1.91$\cdot$105 | 1.91$\cdot$106 | 1.91$\cdot$107 | 1.92$\cdot$108
MMSN-30au |  |  |  |  |  |
Knudsen number (K) | K=9.62$\cdot$101 | K=9.62 | K=9.62$\cdot$10-1 | K=9.62$\cdot$10-2 | K=9.62$\cdot$10-3 | K=9.62$\cdot$10-4
u = 0.1 km/s, M = 0.19 |  |  |  |  |  |
Reynolds number (Re) | 8.51$\cdot$10-3 | 8.51$\cdot$10-2 | 8.51$\cdot$10-1 | 8.51 | 8.51$\cdot$101 | 8.51$\cdot$102
Drag coefficient (CD) | 23.27 | 22.76 | 13.89 | 4.66 | 1.25 | 0.64
Stokes number | 2.36$\cdot$10-2 (**) | 2.53$\cdot$10-1 (*) | 4.03 | 1.27$\cdot$102 | 5.93$\cdot$103 | 2.11$\cdot$105
u = 1 km/s, M = 1.86 |  |  |  |  |  |
Reynolds number (Re) | 8.51$\cdot$10-2 | 8.51$\cdot$10-1 | 8.51 | 8.51$\cdot$101 | 8.51$\cdot$102 | 8.51$\cdot$103
Drag coefficient (CD) | 3.58 | 3.44 | 2.85 | 1.80 | 1.33 | 1.27
Stokes number | 0.19$\cdot$10-1 (*) | 2.01 | 2.57$\cdot$101 | 5.31$\cdot$102 | 1.29$\cdot$104 | 2.26$\cdot$105
u = 7 km/s, M = 13.01 |  |  |  |  |  |
Reynolds number (Re) | 5.96$\cdot$10-1 | 5.96 | 5.96$\cdot$101 | 5.96$\cdot$102 | 5.96$\cdot$103 | 5.96$\cdot$104
Drag coefficient (CD) | 2.16 | 2.14 | 2.07 | 1.91 | 1.83 | 1.83
Stokes number | 0.57$\cdot$10-1 (*) | 5.80 | 6.45$\cdot$101 | 9.63$\cdot$102 | 1.58$\cdot$104 | 2.27$\cdot$105
u = 30 km/s, M = 55.78 |  |  |  |  |  |
Reynolds number (Re) | 2.55 | 2.55$\cdot$102 | 2.55$\cdot$103 | 2.55$\cdot$104 | 2.55$\cdot$105 | 2.55$\cdot$106
Drag coefficient (CD) | 2.03 | 2.01 | 1.97 | 1.96 | 1.95 | 1.95
Stokes number | 9.93$\cdot$10-1 (*) | 9.36 | 1.00$\cdot$102 | 1.32$\cdot$103 | 1.90$\cdot$104 | 2.56$\cdot$105
DISK-30au |  |  |  |  |  |
Knudsen number (K) | K=1.54$\cdot$103 | K=1.54$\cdot$102 | K=1.54$\cdot$101 | K=1.54 | K=1.54$\cdot$10-1 | K=1.54$\cdot$10-2
u = 0.1 km/s, M = 0.19 |  |  |  |  |  |
Reynolds number (Re) | 5.31$\cdot$10-4 | 5.31$\cdot$10-3 | 5.31$\cdot$10-2 | 5.31$\cdot$10-1 | 5.31 | 5.31$\cdot$101
Drag coefficient (CD) | 23.44 | 23.34 | 22.36 | 16.17 | 6.01 | 1.61
Stokes number | 3.74$\cdot$10-1 (*) | 3.76 | 3.93$\cdot$101 | 5.51$\cdot$102 | 1.55$\cdot$104 | 6.85$\cdot$105
u = 1 km/s, M = 1.86 |  |  |  |  |  |
Reynolds number (Re) | 5.31$\cdot$10-3 | 5.31$\cdot$10-2 | 5.32$\cdot$10-1 | 5.32 | 5.32$\cdot$101 | 5.32$\cdot$102
Drag coefficient (CD) | 3.60 | 3.59 | 3.49 | 3.02 | 2.00 | 1.37
Stokes number | 3.03 | 3.05$\cdot$101 | 3.15$\cdot$102 | 3.80$\cdot$103 | 7.01$\cdot$104 | 1.76$\cdot$106
u = 7 km/s, M = 13.01 |  |  |  |  |  |
Reynolds number (Re) | 3.72$\cdot$10-2 | 3.72$\cdot$10-1 | 3.72 | 3.72$\cdot$101 | 3.72$\cdot$102 | 3.72$\cdot$103
Drag coefficient (CD) | 2.16 | 2.16 | 2.15 | 2.10 | 1.94 | 1.84
Stokes number | 9.08 | 9.10$\cdot$101 | 9.22$\cdot$102 | 9.93$\cdot$103 | 1.39$\cdot$105 | 2.30$\cdot$106
u = 30 km/s, M = 55.78 |  |  |  |  |  |
Reynolds number (Re) | 1.60$\cdot$10-1 | 1.60 | 1.60$\cdot$101 | 1.60$\cdot$102 | 1.60$\cdot$103 | 1.60$\cdot$104
Drag coefficient (CD) | 2.03 | 2.03 | 2.03 | 2.02 | 1.98 | 1.96
Stokes number | 1.48$\cdot$101 | 1.48$\cdot$102 | 1.49$\cdot$103 | 1.56$\cdot$104 | 1.96$\cdot$105 | 2.84$\cdot$106

Table A2: Critical object radius, rp (cm), for gas trapping | Molecular | Pre-stellar | MMSN-30au
---|---|---|---
| cloud | kernel |
| (Stk=1) | (Stk=1) | (Stk=1)
u = 0.1 | 0.1 | 0.5 | 2$\cdot$103
u = 1 | 0.02 | 0.09 | 3$\cdot$102
u = 7 | 0.009 | 0.04 | 9$\cdot$101
u = 30 | 0.006 | 0.03 | 6$\cdot$101
Figure 7: Stokes number, $Stk$, as a function of the Knudsen number, $K$,
calculated using Equation (A7), assuming the conditions of the molecular cloud
environment (see Table 1). The different colored lines correspond to different
incoming velocities of the interstellar object with respect to the cloud. The
top x-axis corresponds to the object’s radius, given by $r_{p}=l/K$, where $l$
is the mean free path of the gas molecules. Interstellar objects with $Stk<<$
1 are expected to be trapped by gas drag, while those with $Stk>>$ 1 are
expected to escape. The dotted lines indicate the critical Knudsen
number/object radius for which $Stk$ = 1 Figure 8: Same as Figure 7 but for
the stellar kernel environment (see Table 1). Figure 9: Same as Figure 7 but
for the MMSN-30au environment, based on Hayashi (1981; see Table 1).

For the molecular cloud and stellar kernel environments, characteristic of the
earliest stages of star and planet formation, all particles in the 50 cm–50 km
radius range have $Stk>>1$, even when considering very low incoming
velocities. This indicates that gas drag is not an efficient mechanism for the
trapping of meter-size particles in those environments.

Another way to see this is to compare the column density of the incoming
interstellar object, ${m\over\pi r^{2}}$, to that of the gas it will swept as
it moves through the cloud, $2R_{\rm env}\rho_{\rm env}$. We find that for
interstellar objects with a radius in the range $r$ = 50 cm–50 km, we have
that ${m\over\pi r^{2}}\sim 7\cdot 10^{1}$–$7\cdot 10^{6}$ g $\cdot$ cm-2.
This compares to $2R_{\rm env}\rho_{\rm env}\sim 3\cdot 10^{-2}$ g$\cdot$cm-2,
for the molecular cloud environment, and $2R_{\rm env}\rho_{\rm env}\sim
10^{-1}$ g$\cdot$cm-2, for the pre-stellar core kernel environment (assuming
in both cases a molecular hydrogen composition). In both cases, the column
density of the gas is significantly lower that that of the object and the
particle has too much momentum to be trapped by gas drag.

At a later stage of the star and planet formation process, the characteristic
gas density and temperature are higher. Tables 2 and 3 and Figure 9 show that,
when considering the densities and temperatures characteristic of the mid-
plane of the minimum mass solar nebula at 30 au (labeled as MMSN-30au in
Tables 1 and 2), interstellar planetesimals with radius $<$ 50 cm would have
$Stk<1$ for all the incoming velocities considered, but $Stk>1$ in the meter-
sized range, except at the slowest incoming velocities.

Here, we are approximating this environments as spherical and having a uniform
density, rather than the structured protoplanetary disks they really are. A
detailed study of the trapping of planetesimals by gas drag during the
protoplanetary disk phase that takes into account their structure is given by
Grishin et al. (2019).

##  Appendix B Capture Conditions

Grishin et al. (2019) estimated the capture conditions by calculating the
energy dissipated by the particle during its passage through the environment.
Capture takes place when the energy loss is larger than the particle’s kinetic
energy at infinity, $\Delta E>E_{0}$, with $E_{0}={1\over 2}m_{p}u_{0}^{2}$.
In their calculation of the energy dissipated, they assumed that the
acceleration that the particle experiments due to gas drag is given by
Equation (1) and adopted a constant value for the drag coefficient. However,
in our calculations in Section A, we are taking into account the velocity
dependency of the gas drag coefficient, given by Equation (A1), so instead of
Equation (1), the gas drag the particle experiments would be given by

${{\rm d}u\over{\rm d}t}=-{\pi r_{p}^{2}\over
2m_{p}}\rho_{env}\left(C_{1}(u)u{Kc\over 4.4}+C_{2}(u)u^{2}\right),$ (B1)

where $c$ is the sound speed and C1(u) and C2(u) are given by Equations A2 and
A3. Defining $\bar{C_{1}}(u,K)=C_{1}(u){Kc\over 4.4}$ and
$\Sigma_{p}={m_{p}\over\pi r_{p}^{2}}$,

${{\rm d}u\over{\rm d}t}=-{1\over
2\Sigma_{p}}\rho_{env}\left(\bar{C_{1}}(u,K)u+C_{2}(u)u^{2}\right).$ (B2)

We then have,

${1\over u}{{\rm d}u\over{\rm d}t}={{\rm d}u\over{\rm d}x}=-{1\over
2\Sigma_{p}}\rho_{env}(x)\left(\bar{C_{1}}(u,K)+C_{2}(u)u\right).$ (B3)

Defining

$\begin{split}y\equiv\bar{C_{1}}(u,K)\int{1\over 2\Sigma_{p}}\rho_{env}(x){\rm
d}x\\\ \end{split}$ (B4)

and

$\begin{split}\eta\equiv{C_{2}(u)\over\bar{C_{1}}(u,K)},\end{split}$ (B5)

we have that

$\begin{split}{\rm d}y={\bar{C_{1}}(u,K)\over 2\Sigma_{p}}\rho_{env}(x){\rm
d}x,\\\ \end{split}$ (B6) $\begin{split}{{\rm d}u\over{\rm
d}y}=-\left(1+{C_{2}(u)u\over\bar{C_{1}}(u,K)}\right)=-\left(1+\eta\right),\\\
\end{split}$ (B7)

and

$\begin{split}-{\rm d}y={{\rm d}u\over 1+\eta u}.\end{split}$ (B8)

Solving for the velocity, $u$, we get

$u=\eta^{-1}\left[Ae^{-\eta\int{\rm d}y}-1\right],$ (B9)

where $A\equiv e^{C\eta}$ and $C$ is a constant of integration. This leads to

$u={\bar{C_{1}}(u,K)\over
C_{2}(u)}\left[Ae^{-C_{2}(u)\int\limits_{0}^{x}{\rho_{\rm env}(x)\over
2\Sigma_{p}}{\rm d}x}-1\right].$ (B10)

The initial condition $x=0$, $u=u_{0}$ gives,

$1+{C_{2}(u_{0})\over\bar{C_{1}}(u_{0},K)}u_{0}=A.$ (B11)

so that

$\begin{split}1+{C_{2}(u)\over\bar{C_{1}}(u,K)}u=\left(1+{C_{2}(u_{0})\over\bar{C_{1}}(u_{0},K)}u_{0}\right)e^{{-C_{2}(u)\over
2\Sigma_{p}}\int\limits_{0}^{x}\rho_{\rm env}(x){\rm d}x}.\end{split}$ (B12)

For small ${C_{2}(u)\over 2\Sigma_{p}}\int\limits_{0}^{x}\rho_{\rm env}(x){\rm
d}x$, we get that

$\begin{split}u={C_{2}(u_{0})\over C_{1}(u_{0})}{C_{1}(u)\over
C_{2}(u)}u_{0}-{1\over
2\Sigma_{p}}\left(\bar{C_{1}}(u_{0},K)+{C_{2}(u_{0})C_{1}(u)\over
C_{1}(u_{0})}\right)\int\limits_{0}^{x}\rho_{\rm env}(x){\rm d}x.\end{split}$
(B13)

Generally, we have

$\begin{split}{\bar{C_{1}}(u,K)+C_{2}(u)u\over\bar{C_{1}}(u_{0},K)+C_{2}(u_{0})u_{0}}=e^{{-C_{2}(u)\over
2\Sigma_{p}}\int\limits_{0}^{x}\rho_{\rm env}(x){\rm d}x},\end{split}$ (B14)

$\begin{split}-{2\Sigma_{p}\over C_{2}(u)}{\rm
ln}\left({\bar{C_{1}}(u,K)+C_{2}(u)u\over\bar{C_{1}}(u_{0},K)+C_{2}(u_{0})u_{0}}\right)=\int\limits_{0}^{x}\rho_{\rm
env}(x){\rm d}x.\end{split}$ (B15)

Here we can assume that $\bar{C_{1}}$ and $C_{2}$ are slowly varying, so that
$\bar{C_{1}}(u,K)\simeq\bar{C_{1}}(u_{0},K)$ and $C_{2}(u)\simeq
C_{2}(u_{0})$, that can be evaluated using Equations (A2) and (A3). The
resulting expression,

$\begin{split}-{2\Sigma_{p}\over C_{2}(u_{0})}{\rm
ln}\left({\bar{C_{1}}(u_{0},K)+C_{2}(u_{0})u\over\bar{C_{1}}(u_{0},K)+C_{2}(u_{0})u_{0}}\right)=\int\limits_{0}^{x}\rho_{\rm
env}(x){\rm d}x.\end{split}$ (B16)

can be solved numerically to calculate the velocity of the particle $u$ as a
function of the mass it sweeps up as it traverses the environment. Capture
takes place when the energy dissipated is larger than the particle’s kinetic
energy at infinity.

We now look at the case of a spherical cloud.

The energy dissipated by the particle as it crosses a spherical cloud of
uniform density, $\rho_{\rm env}$, is

$\begin{split}\Delta E=\int{F_{\rm D}}{\rm d}x=-{1\over 2}\int{C_{\rm
D}(R_{\rm e},K)\pi r_{\rm p}^{2}\rho_{env}u^{2}(x)}{\rm d}x\\\ =-{\pi
r_{p}^{2}\over
2}\rho_{env}\int\left(\bar{C_{1}}(u_{0},K)u+C_{2}(u)u^{2}\right)u{\rm
d}t.\end{split}$ (B17)

where the integral is along the perturbed orbit, and, as we indicated above,
$\bar{C_{1}}$ and $C_{2}$ are slowly varying. The two limiting cases are when
the particle goes through the center of the cloud and when the particle grazes
the cloud in an almost parabolic orbit.

###  B.1 Particle crossing the center of the cloud

Because a cloud of constant density has an harmonic oscillator potential well,
the position of the particle is $x(t)=-a{\rm cos}(wt)$, where the harmonic
oscillator period and frequency are $T=\sqrt{{3\pi}\over{G\rho_{\rm
env}}}={{2\pi}\over w}$ and $\dot{x}(t)=aw{\rm sin}(wt)$. The velocity of the
particle is therefore $u(t)=u_{\rm 0}+u_{\rm esc}{\rm sin}(wt)$. We now
calculate how much energy is dissipated in half a cycle to see if the particle
will come back (i.e. if the particle is trapped by the environment). We have
that

$\begin{split}\Delta E=-{\pi r_{p}^{2}\over
2}\rho_{env}\int\left[\bar{C_{1}}(u,K)u+C_{2}(u)u^{2}\right]u{\rm d}t\\\
=-{\pi r_{p}^{2}\over
2}\rho_{env}\int_{0}^{\pi\over\omega}[\bar{C_{1}}(u,K)(u_{\rm 0}+u_{\rm
esc}{\rm sin}(wt))^{2}+C_{2}(u)(u_{\rm 0}+u_{\rm esc}{\rm sin}(wt))^{3}]{\rm
d}t.\end{split}$ (B18)

As above, we can assume that $C_{1}$ and $C_{2}$ are slowly varying, so that
$\bar{C_{1}}(u,K)\simeq\bar{C_{1}}(u_{0},K)$ and $C_{2}(u)\simeq
C_{2}(u_{0})$, leading to

$\begin{split}\Delta E=-{\pi r_{p}^{2}\over
2}\rho_{env}[{\pi\over\omega}(\bar{C_{1}}(u_{0},K)u_{\rm
0}^{2}+C_{2}(u_{0})u_{\rm 0}^{3}+{\bar{C_{1}}(u_{0},K)\over\omega}(4u_{\rm
0}u_{\rm esc}+{\pi\over 2}u_{\rm
esc}^{2})+{C_{2}(u_{0})\over\omega}({3\pi\over 2}u_{\rm 0}u_{\rm
esc}^{2}+6u_{\rm 0}^{2}u_{\rm esc}+{4\over 3}u_{\rm esc}^{3})]\end{split}$
(B19)

and

$\begin{split}{\Delta E\over
E_{0}}=-{\rho_{env}\over\Sigma_{p}\omega}[\pi(\bar{C_{1}}(u_{0},K)+C_{2}(u_{0})u_{\rm
0})+\bar{C_{1}}(u_{0},K)(4{u_{\rm esc}\over u_{\rm 0}}+{\pi\over 2}{u_{\rm
esc}^{2}\over u_{0}^{2}})+C_{2}(u_{0})u_{0}({3\pi\over 2}{u_{\rm esc}^{2}\over
u_{\rm 0}^{2}}+6{u_{\rm esc}\over u_{\rm 0}}+{4\over 3}{u_{\rm esc}^{3}\over
u_{0}^{3}})].\end{split}$ (B20)

Defining $x={u_{\rm esc}\over u_{0}}$, we have that

$\begin{split}{\Delta E\over
E_{0}}=-{\rho_{env}\over\Sigma_{p}\omega}[\pi\bar{C_{1}}(u_{0},K)+C_{2}(u_{0})u_{\rm
0})+\bar{C_{1}}(u_{0},K)(4x+{\pi\over 2}x^{2})+C_{2}(u_{0})u_{0}({3\pi\over
2}x^{2}+6x+{4\over 3}x^{3})].\end{split}$ (B21)

To take into account the effect of the gravitational focusing outside the
cloud, in the expression for $\Delta E$ above, we replace $u_{0}$ by
$u_{0}\sqrt{1+{u_{\rm esc}^{2}\over u_{0}^{2}}}$, leading to

$\begin{split}\Delta E=-{\pi r_{p}^{2}\over
2}\rho_{env}[{\pi\over\omega}(\bar{C_{1}}(u_{0},K)u_{\rm 0}^{2}({1+{u_{\rm
esc}^{2}\over u_{0}^{2}}})+C_{2}(u_{0})u_{\rm 0}^{3}({1+{u_{\rm esc}^{2}\over
u_{0}^{2}}})^{3\over 2}+{\bar{C_{1}}(u_{0},K)\over\omega}(4u_{\rm 0}u_{\rm
esc}({1+{u_{\rm esc}^{2}\over u_{0}^{2}}})^{1\over 2}+{\pi\over 2}u_{\rm
esc}^{2})\\\ +{C_{2}(u_{0})\over\omega}({3\pi\over 2}u_{\rm 0}u_{\rm
esc}^{2}({1+{u_{\rm esc}^{2}\over u_{0}^{2}}})^{1\over 2}+6u_{\rm 0}^{2}u_{\rm
esc}({1+{u_{\rm esc}^{2}\over u_{0}^{2}}})+{4\over 3}u_{\rm
esc}^{3})].\end{split}$ (B22)

Defining $x={u_{\rm esc}\over u_{0}}$ and replacing $\omega=\sqrt{4\pi
G\rho_{env}\over 3}$, we get an expression for ${\Delta E\over E_{0}}$ as a
function of the environment parameters ($\rho_{\rm env}$, $R_{\rm env}$,
$T_{\rm env}$), particle parameters ($r_{p}$, $m_{p}$) and velocity at
infinity, $u_{0}$,

$\begin{split}{\Delta E\over E_{0}}=-{\rho_{env}\over\Sigma_{p}}\sqrt{3\over
4\pi G\rho_{env}}[\pi(\bar{C_{1}}(u_{0},K))(1+x^{2})+C_{2}(u_{0})u_{\rm
0})(1+x^{2})^{3\over 2}+\bar{C_{1}}(u_{0},K)(4x(1+x^{2})^{1\over 2}+{\pi\over
2}x^{2})\\\ +C_{2}(u_{0})u_{0}({3\pi\over 2}x^{2}(1+x^{2})^{1\over
2}+6x(1+x^{2})+{4\over 3}x^{3})].\end{split}$ (B23)

Trapping takes place when the final energy is less than the binding energy to
the cloud,

$\begin{split}{E_{f}\over E_{BE}}={E_{0}+\Delta E\over E_{BE}}={1+{\Delta
E\over E_{0}}\over{E_{BE}\over E_{0}}}={{1+{\Delta E\over E_{0}}\over
x^{2}}<1},\end{split}$ (B24)

so that $-{\Delta E\over E_{0}}>1-x^{2}$. This limiting case where the
particle crosses through the center of the cloud corresponds to maximum
dissipation.

###  B.2 Particle grazing the cloud

We now look at the case of lowest dissipation, where the particle enters the
cloud in an almost parabolic orbit and gets trapped in a circular orbit or
radius $a$. In this case, the energy dissipated is a function of $a$,

$\begin{split}\Delta E=-{\pi r_{p}^{2}\over
2}\rho_{env}\int\left[\bar{C_{1}}(u_{\rm cir})u_{\rm cir}+C_{2}(u_{\rm
cir})u_{\rm cir}^{2}\right]{\rm d}x=-\psi
a\pi^{2}r_{p}^{2}\rho_{env}\left[\bar{C_{1}}(u_{\rm cir})u_{\rm
cir}+C_{2}(u_{\rm cir})u_{\rm cir}^{2}\right],\end{split}$ (B25)

where $\psi$ is of order 1 and depends on the exact nature of the incoming
orbit. When the orbit is just trapped, it moves on an ellipse with orbital
period $T=\sqrt{2\pi\over G\rho}$, independent of the position in the cloud.
The motion is a circular orbit with an epicyclic motion in the square of the
radial coordinate (it is not linear in the radial coordinate; see Harter Soft
Web Resources for Physics, Chapter 5.2). Once the orbit is just trapped as an
ellipse in the cloud, it will continue to dissipate. Since the highest
velocity occurs at the minor axis, the orbit will circularized rather than
become more elongated. For the geometrical image see Figure 5.2.2 and 5.2.1 in
Harter (Soft Web Resources for Physics).

In the geometric regime, when the particle is grazing the cloud, the impact
parameter is $b=a=R_{\rm env}$ and we have that $u_{0}=u_{\rm
cir}=b\sqrt{{4\over 3}G\rho_{\rm env}}$ and $E_{0}={1\over 2}m_{p}u_{\rm
cir}^{2}={1\over 2}m_{p}b^{2}{4\over 3}G\rho_{\rm env}$. There are two
limiting cases for $u_{0}$. The first one is when the parabolic orbit is just
trapped to a circular orbit at the edge of the cloud, $u_{0}=u_{\rm cir}$. The
second limiting case is for small impact parameters, when the orbit oscillates
radially across the cloud; in this case, the circular velocity squared is of
the order of the depth of the potential well and therefore $u$ averaged over
the orbit is of the order of $\chi v_{\rm cir}$, where $\chi$ is of order
unity and depends on the exact nature of the orbit.

$\begin{split}{\Delta E\over E_{0}}=-\psi{3\over
2\Sigma_{p}Gb}[\bar{C_{1}}(\chi u_{\rm cir})\chi u_{\rm cir}+C_{2}(\chi u_{\rm
cir})\chi^{2}u_{\rm cir}^{2}].\end{split}$ (B26)

Trapping takes place when $-{\Delta E\over E_{0}}>1-x^{2}$, where $x={u_{\rm
esc}\over u_{0}}$ and $u_{\rm esc}$ is the escape velocity from the cloud.

In the gravitational focusing regime, $a=q$, where $q$ is the distance of
closest approach. In the parabolic approximation,
$a=q\approx{b^{2}u_{0}^{2}\over 2GM}$. Because of conservation of angular
momentum, we have that $b^{2}u_{0}^{2}=q^{2}u_{\rm cir}^{2}(a)$, where $b$ is
the impact parameter and $u_{\rm cir}=\sqrt{2}{GM\over bu_{0}}$. The initial
energy can then be given as a function of the impact parameter, $E_{0}={1\over
2}m_{p}u_{\rm cir}^{2}=m_{p}\left({GM\over bu_{0}}\right)^{2}$. We then have,

$\begin{split}{\Delta E\over E_{0}}=-\psi{\rho_{\rm env}b^{4}u_{0}^{4}\over
2\Sigma_{p}G^{3}M^{3}}\left[\bar{C_{1}}(u_{\rm cir})u_{\rm cir}+C_{2}(u_{\rm
cir})u_{\rm cir}^{2}\right].\end{split}$ (B27)

Following Equation (B24) above, trapping would take place when $-{\Delta
E\over E_{0}}>1-x^{2}$, i.e.

$\begin{split}\psi{\rho_{\rm env}b^{4}u_{0}^{4}\over
2\Sigma_{p}G^{3}M^{3}}\left[\bar{C_{1}}(u_{\rm cir})u_{\rm cir}+C_{2}(u_{\rm
cir})u_{\rm cir}^{2}\right]>1-{u_{\rm esc}^{2}\over u_{0}^{2}}.\end{split}$
(B28)

To calculate the capture rates, we would need to use the critical capture
condition, $-{\Delta E\over E_{0}}=1-x^{2}$ (i.e. Equation (B28) with an
equality), to get an expression for the critical impact parameter $b$ for
capture, as a function of the environment parameters ($\rho_{\rm env}$,
$R_{\rm env}$, $T_{\rm env}$, M), particle parameters ($r_{p}$, $m_{p}$), and
velocity at infinity, $u_{0}$. However, the dependency of Equation (B28) on
$u_{\rm cir}$ (which depends on $b$) does not allow to solve analytically for
$b$. This is why in Section 3 we used a simplified expression for the drag
force, $F_{\rm D}={1\over 2}C_{\rm D}(u)\pi r_{\rm
p}^{2}\rho_{env}u^{2}={3C_{\rm D}(u)\rho_{env}u^{2}\over 8\rho_{\rm p}r_{\rm
p}}$, assuming, as in Grishin et al. (2019), that the relative velocity $u$ is
a constant through the passage. This makes the drag coefficient also a
constant through the passage and it can be taken out of the energy dissipation
integral. As we discuss in Section 3, this allows to solve for the critical
impact parameter from which, following Grishin et al. (2019), we can calculate
the capture rates.

##  Appendix C Derivation of $N_{\rm R_{1}<R<R_{2}}$ from $N_{\rm R\geqslant
R_{0}}$

If we assume that the size distribution is given by

$\begin{split}n(r)=A_{1}r^{-q_{1}}~{}{\rm if}~{}r_{min}<r<r_{b}\\\
n(r)=A_{2}r^{-q_{2}}~{}{\rm if}~{}r_{b}<r<r_{max},\\\ \end{split}$ (C1)

with A2 = A1 r${}_{b}^{q_{2}-q_{1}}$ (to have continuity at rb), then, for
R${}_{0}<$ rb,

$\begin{split}N_{\rm r\geqslant R_{0}}=A_{1}\left[{1\over
1-q_{1}}[{r_{b}^{1-q_{1}}}-R_{0}^{1-q_{1}}]+{r_{b}^{q_{2}-q_{1}}}{1\over
1-q_{2}}[{r_{max}^{1-q_{2}}}-r_{b}^{1-q_{2}}]\right].\end{split}$ (C2)

Since we know $N_{\rm r\geqslant R_{0}}$, we can solve for A1 and use the
power law above to get $N_{\rm R_{1}<R<R_{2}}$.

The calculation in Section 5.1.1 adopts the cumulative number density of
interstellar objects inferred from 1I/’Oumuamua’s detection, estimated by Do
et al. (2018) to be $N_{\rm R\geqslant R_{\rm Ou}}=2\cdot 10^{15}{\rm
pc}^{-3}$, for which we adopt $R_{\rm Ou}\sim$ 80 m. But 1I/’Oumuamua’s brief
visit left many open questions regarding its nature, like its shape, size,
composition, and internal structure (’Oumuamua ISSI Team et al. 2019, and
references therein; Sekanina 2019; Moro-Martín 2019b) that result in
uncertainties in its origin and the inferred number density distribution of
interstellar objects (Moro-Martín 2018; 2019a). 2I/Borisov, the second
interstellar object that has been detected, has an unquestionable cometary
nature, similar to the planetesimals expected to be ejected from
protoplanetary disks (the majority of which will originate outside the
snowline of their parent systems). We now assess to what degree the detection
of 2I/Borisov can change the inferred number density of interstellar objects
in the ISM and therefore the results listed above for $N_{\rm R_{1}<R<R_{2}}$.
Following Jewitt & Luu (2019), the cumulative number density of interstellar
objects similar or larger than 2I/Borisov would be $N_{\rm R\geqslant R_{\rm
Bor}}={1\over V_{\rm det}}$, where $V_{\rm det}={4\over 3}\pi r_{\rm det}^{3}$
is the detection volume, with $r_{\rm det}$ = 3 AU, the distance from the Sun
at which 2I/Borisov was discovered. Adopting the size distribution in Equation
(30) and Equation (9) in Moro-Martín (2019a), we get that the cumulative
number density of interstellar objects similar or larger than 1I/’Oumuamua is

$\begin{split}N_{\rm R\geqslant R_{\rm Ou}}={3\over 4\pi r_{\rm
det}^{3}}{{1\over-q_{1}+1}\left[r_{b}^{-q_{1}+1}-R_{\rm
Ou}^{-q_{1}+1}\right]+{r_{b}^{q_{2}-q_{1}}\over-
q_{2}+1}\left[r_{max}^{-q_{2}+1}-r_{b}^{-q_{2}+1}\right]\over{1\over-
q_{1}+1}\left[r_{b}^{-q_{1}+1}-R_{\rm
Bor}^{-q_{1}+1}\right]+{r_{b}^{q_{2}-q_{1}}\over-
q_{2}+1}\left[r_{max}^{-q_{2}+1}-r_{b}^{-q_{2}+1}\right]}.\end{split}$ (C3)

Because 2I/Borisov’s cometary nature has made so far the determination of the
size of the nucleus challenging, Figure 10 shows the results as a function of
2I/Borisov’s size, adopting $R_{\rm Ou}$ = 80 m. Jewitt et al. (2020) proposed
that $R_{\rm Bor}$ = 0.2–0.5 km. If we were to adopt this range, we have that
the resulting value for $N_{\rm R\geqslant R_{\rm Ou}}$ would be in the range
(0.2–0.5)$\cdot 10^{15}{\rm pc}^{-3}$ (for q1 = 2) to (3–100)$\cdot
10^{15}{\rm pc}^{-3}$ (for q1 = 5), encompassing the adopted value of $N_{\rm
R\geqslant R_{\rm Ou}}=2\cdot 10^{15}{\rm pc}^{-3}$ that was based on the
detection of 1I/’Oumuamua alone (with the caveat that, given their very
different properties, 1I/’Oumuamua and 2I/Borisov might not belong to the same
population of interstellar comets).

Figure 10: Inferred cumulative number density of planetesimals with sizes
equal or larger than 1I/’Oumuamua’s, $N_{\rm R\geqslant R_{\rm Ou}}$ (adopting
$R_{\rm Ou}$ = 80 m), as a function of 2I/Borisov’s radius, calculated using
Equation (C3). The different colors correspond to different values of the
power-law index $q_{\rm 1}$, with $q_{\rm 1}$ = 2 (red), 2.5 (orange), 3
(yellow), 3.5 (green), 4 (light blue), 4.5 (dark blue), and 5 (pink). The line
types correspond to the following power-law parameters: $q_{\rm 2}$ = 2 and
$r_{\rm b}$ = 3 km (solid), $q_{\rm 2}$ = 6 and $r_{\rm b}$ = 3 km (dashed),
$q_{\rm 2}$ = 2 and $r_{\rm b}$ = 30 km (dotted), and $q_{\rm 2}$ = 6 and
$r_{\rm b}$ = 30 km (dashed-dotted). These results are to be compared to the
value of $N_{\rm R\geqslant R_{\rm Ou}}=2\cdot 10^{15}{\rm pc}^{-3}$ adopted
in Section 5.1.1, based on the detection of 1I/’Oumuamua.

##  Appendix D Derivation of $N_{\rm R_{1}<R<R_{2}}$ from mtotal

We can get the number density of planetesimals with radius R${}_{1}<$ R
$<R_{2}$ from the total mass density of planetesimals by first solving for the
cumulative number density $N_{\rm r\geqslant R0}$ using Equation (17) of Moro-
Martín (2018),

$\begin{split}{N_{\rm r\geqslant R_{0}}=m_{total}9(4\pi\rho)^{-1}}{{1\over
1-q_{1}}[{1\over r_{b}^{3}}-{1\over R_{0}^{3}}({R_{0}\over
r_{b}})^{4-q_{1}}]+{1\over 1-q_{2}}[{1\over r_{max}^{3}}({r_{max}\over
r_{b}})^{4-q_{2}}-{1\over r_{b}^{3}}]\over{3\over
4-q_{2}}\left[\left({r_{max}\over r_{b}}\right)^{4-q_{2}}-1\right]+{3\over
4-q_{1}}\left[1-\left({r_{min}\over
r_{b}}\right)^{4-q_{1}}\right]},\end{split}$ (D1)

assuming R${}_{0}<$ rb, and then using the expressions derived in Appendix C
to get $N_{\rm R_{1}<R<R_{2}}$ from $N_{\rm R\geqslant R_{\rm 0}}$.

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