Origin of 1I/'Oumuamua. II. An Ejected Exo-Oort Cloud Object?

For this back-of-the-envelope calculation, we need an estimate of the number of objects that each star could potentially contribute.

Given the lack of observational constraints, our assumption regarding the number of objects in the exo-Oort clouds is based on solar system observations. In the solar system, long-period comets are thought to originate in the Oort cloud and be launched into the inner solar system due to perturbations by the Galactic tide, or by encounters with stars or giant molecular clouds, together with subsequent perturbations by the planets. From the flux of long-period comets, one can estimate the number of objects in this cloud to be 10¹¹–10¹², with (1–5) · 10¹¹ being the most likely value (Brasser & Morbidelli 2013 and references therein). As Brasser et al. (2010) pointed out, this estimate is uncertain because the flux of long-period comets with q < 4 au is only 2–3 per year, so an error of only 1 comet per year can significantly affect the result. In this study, we assume that the Oort cloud contains ∼10¹² objects with diameters ≥2.3 km.

Following Hanse et al. (2018), we assume that the number of bodies larger than 2.3 km for a given exo-Oort cloud is similar to that of the solar system's Oort cloud scaled to the mass of the parent star, in our case ${10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)$. We base this assumption on the following considerations.

The models of Brasser et al. (2010) found that the Oort cloud formation efficiency is similar at a wide range of Galactocentric distances. It will, however, depend on the environment and the planetary architecture of the system. Regarding the latter, Figure 1 in Wyatt et al. (2017) shows that the parameter space to form an Oort cloud is quite restricted, and in the solar system, is populated by Uranus and Neptune. Their figure also shows that the parameter space to have a population of ejected bodies (that never become part of the Oort Cloud) is well-populated by known exoplanets—Jupiter, in the case of the solar system—so it may be that the presence of this gas giant resulted in a relatively depleted Oort Cloud. With this in mind, on the one hand, given the restricted parameter space to form an exo-Oort cloud, assuming that all stars harbor exo-Oort clouds is likely an overestimate. On the other hand, given that the parameter space for ejected bodies is well-populated by known exoplanets, and that these objects would contribute to the background population under consideration (regardless of whether or not they become trapped in an Oort cloud before being completely ejected from the system), assuming all planet-bearing stars contribute about ${10}^{12}\left(\displaystyle \frac{{\text{}}{M}_{* }}{{M}_{\odot }}\right)$ objects might be a reasonable order-of-magnitude estimate. We will go one step further and assume that all stars contribute, irrespective of whether or not they are planet hosts. This latter assumption likely makes our back-of-the-envelope estimate an upper limit.

The above estimate assumes that the size distribution of the objects ejected by other systems is similar to that of the solar system, which is of course unknown. Because the ejection processes are independent of size, another approach would be to estimate the mass that could be ejected per star. That was the approach taken in Moro-Martín (2018) when exploring protoplanetary disks as a potential origin of 1I/'Oumuamua, and the observational constraint in that case was the available mass of solids in protoplanetary disks. In this study, we are working with a complementary observational constraint: the number of long-period comets and inferred population of the solar system's Oort cloud.

Given the above considerations, we can already calculate an upper limit to the number of ejected objects with sizes equal to or larger than 1I/'Oumuamua's contributed by stars in the 1–8 M_⊙ mass range. This would be be given by

$$\begin{eqnarray}&&{N}_{\mathrm{ejected}}^{\max }={\int }_{1{M}_{\odot }}^{8{M}_{\odot }}\xi ({M}_{* }){\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}{10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){{dM}}_{* }.\end{eqnarray} \tag{ 1 }$$

In Equation (1), $\xi ({M}_{* })$ is the initial mass function, for which we adopt the one derived from Kroupa et al. (1993) that finds that the number density of stars per pc³, out to ∼130 pc from the Sun, within the mid-plane of the galaxy, and with stellar masses between M${}_{* }$ and M${}_{* }$ + dM_* (in units of M_⊙), is given by¹

$$\begin{eqnarray}\begin{array}{rcl}n({M}_{* }) & = & \xi ({M}_{* }){{dM}}_{* }\quad \mathrm{with},\\ \xi ({M}_{* }) & = & 0.035{M}_{* }^{-1.3}\quad \mathrm{if}\,0.08\leqslant {M}_{* }\lt 0.5\\ \xi ({M}_{* }) & = & 0.019{M}_{* }^{-2.2}\quad \mathrm{if}\,0.5\leqslant {M}_{* }\lt 1.0\\ \xi ({M}_{* }) & = & 0.019{M}_{* }^{-2.7}\quad \mathrm{if}\,1.0\leqslant {M}_{* }\lt 100.\end{array}\end{eqnarray} \tag{ 2 }$$

The term ${\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}{10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)$ in Equation (1) corresponds to the cumulative number of objects with sizes equal to or larger than 1I/'Oumuamua's, assuming an equilibrium power law size distribution with index q = 3.5, n(r) ∝ r^−3.5, and adopting for 1I/'Oumuamua the intermediate effective radius of 80 m from Drahus et al. (2018).

As we mentioned above (in Section 2.1), this estimate would likely be an upper limit because it assumes all stars contribute, irrespective of whether or not they are planet hosts. It also assumes that, if exo-Oort clouds form, they eventually are entirely depleted due to the clearing processes described in Section 1. The resulting estimate is ${N}_{\mathrm{ejected}}^{\max }$=1.6 · 10¹³ pc⁻³.

We can now compare the above estimate to that inferred from the detection of 1I/'Oumuamua. For the latter, we adopt the number density distribution derived by Do et al. (2018) of ${N}_{r\geqslant R}$ = 0.21 au⁻³ ∼2 · 10¹⁵ pc⁻³, an estimate that assumes the objects are isotropically distributed, and adopts for 1I/'Oumuamua an absolute magnitude of H = 22.1, a nominal phase function with slope parameter G = 0.15, a velocity at infinity, v_∞ = 26 km s⁻¹, and a detection frequency of 1 per 3.5 yr (assumed to be the survey's lifetime). Do et al. (2018) noted that the presence of a size distribution leads to an approximately 40% uncertainty, while the existence of inefficiencies in the detection process leads to a cumulative number density (4/3–3/2)× their estimated value.

For comparison, other authors² estimate that the number density is 0.1 au⁻³ = 8 · 10¹⁴ pc⁻³ (Jewitt et al. 2017; Fraser et al. 2018), 0.012–0.087 au⁻³ = 1–7 · 10¹⁴ pc⁻³ (Portegies Zwart et al. 2018), 0.012 au⁻³ = 1 · 10¹⁴ pc⁻³ (Gaidos et al. 2017), or >0.006 au⁻³ = 4.8 · 10¹³ pc⁻³ (lower limit from Feng & Jones 2018). We adopt the cumulative number density estimate from Do et al. (2018) because, unlike the other studies, it is derived based on a careful calculation of the PanSTARRS detection volume. However, as mentioned above, we note that there are uncertainties in their inferred value.

Taking in consideration that our estimate is likely an upper limit, and noting it is approximately two orders of magnitude smaller than the inferred value based on 1I/'Oumuamua's detection, this already leads to the main conclusion of this study: 1I/'Oumuamua is unlikely to be representative of a background population of ejected exo-Oort cloud objects from 1 to 8 M_⊙ stars. This conclusion is only as strong as the assumptions about the masses and size distributions of exo-Oort Clouds; it also assumes that the size distribution of the ejected population is similar to that of the solar system and that it follows an equilibrium power law with index q = 3.5, n(r) ∝ r^−3.5. For further discussion of this result, the reader is referred to Sections 5 and 6.

In Sections 3 and 4, we go beyond this back-of-the-envelope calculation and discuss in more detail the expected contribution from stars in the 0.08–8 M_⊙ mass range to the population of interstellar objects, taking into account the effect of the different perturbing forces and its dependence on the stellar mass, Galactocentric distance, and evolutionary state. In Sections 4 and 5, we also consider a wide range of possible size distributions for the ejected objects, based on solar system observations and on accretion and collisional models. These more detailed calculations reinforce the main conclusion of the study.

The effect of the perturbing forces mentioned above will depend on the semimajor axis of the orbiting body. Also in this case, given the lack of observational constraints, our assumption regarding the extent of the exo-Oort clouds is based on solar system observations. In the solar system, as we mentioned above, long-period comets are thought to originate in the Oort cloud. Based on long-period comet observations, it is estimated that the solar system's Oort cloud has an isotropic distribution with perihelion q ≳ 32 au and inner and outer semimajor axes of ${a}_{\min }^{\mathrm{OC}}$ ∼ 3 · 10³ and ${a}_{\max }^{\mathrm{OC}}$ ∼ 10⁵ au, respectively.

Following Hanse et al. (2018), we scale the inner and outer edge of a given exo-Oort cloud to the Hill radius of its parent star in the Galactic potential, such that its inner and outer semimajor axes are

$$\begin{eqnarray}\begin{array}{rcl}{a}_{\min }^{\mathrm{exo} \mbox{-} \mathrm{OC}} & \sim & {a}_{\min }^{\mathrm{OC}}\cdot \,\displaystyle \frac{{R}_{* }}{{R}_{0\odot }}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\displaystyle \frac{{M}_{{\rm{G}}}({R}_{0\odot })}{{M}_{{\rm{G}}}({R}_{* })}\right),\\ {a}_{\max }^{\mathrm{exo} \mbox{-} \mathrm{OC}} & \sim & {a}_{\max }^{\mathrm{OC}}\cdot \,\displaystyle \frac{{R}_{* }}{{R}_{0\odot }}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\displaystyle \frac{{M}_{{\rm{G}}}({R}_{0\odot })}{{M}_{{\rm{G}}}({R}_{* })}\right),\end{array}\end{eqnarray} \tag{ 3 }$$

where R_* and R_0⊙ are the Galactocentric radii of the parent star and the Sun, respectively, and ${M}_{{\rm{G}}}({R}_{* })$ and M_G(R_0⊙) are the Galactic masses contained within those radii. Following this scaling law, Table 1 lists the inner and outer exo-Oort cloud radii for 1 M_⊙ and 8 M_⊙ stars at three different Galactocentric distances (4, 8.5, and 12 kpc), along with the z-component of their Hill ellipsoid (the latter taken from Figure 3 in Veras et al. 2014). Given these assumed exo-Oort cloud radial extents, in Sections 3.2, 3.3, and 3.4, we address the effect of the perturbing forces under consideration.

Table 1.  Regimes of Influence of the Different Exo-Oort Cloud Clearing Processes (in au)^a

	1 M⊙	1 M⊙	1 M⊙	8 M⊙	8 M⊙	8 M⊙
	4 kpc	8.5 kpc	12 kpc	4 kpc	8.5 kpc	12 kpc
Inner boundaryb (${a}_{\min }^{\mathrm{exo} \mbox{-} \mathrm{OC}}$)	1.7 · 103	3 · 103	3.9 · 103	3.4 · 103	6 · 103	7.7 · 103
Outer boundaryb (${a}_{\max }^{\mathrm{exo} \mbox{-} \mathrm{OC}}$)	5.7 · 104	1 · 105	1.3 · 105	1.1 · 105	2 · 105	2.6 · 105
Hill radius (z-comp.)	9.4 · 104	1.5 · 105	2.1 · 105	1.9 · 105	3.1 · 105	4.1 · 105
Galactic tide (ψ = 0.02)	3.3 · 104 (79.9%)	5.5 · 104 (83%)	7.8 · 104 (77.5%)	1.6 · 105 (0%)	1.1 · 105 (83%)	1.6 · 105 (77.5%)
Galactic tide (ψ = 0.001)	1.2 · 104 (98.9%)	2.1 · 104 (99%)	2.9 · 104 (98.8%)	2.5 · 104 (98.9%)	4.2 · 104 (99%)	5.8 · 104 (98.8%)
Galactic tide (ψ = 0.7)	1.2 · 105 (0%)	1.9 · 105 (0%)	2.6 · 105 (0%)	2.3 · 105 (0%)	3.8 · 105 (0%)	5.1 · 105 (0%)
Post-MS mass loss (ψ = 0.02)	2.5 · 102 (100%)	2.5 · 102 (100%)	2.5 · 102 (100%)	2.7 · 101 (100%)	2.7 · 101 (100%)	2.7 · 101 (100%)
Post-MS mass loss (ψ = 0.001)	3.4 · 101 (100%)	3.4 · 101 (100%)	3.4 · 101 (100%)	4.1 (100%)	4.1 (100%)	4.1 (100%)
Post-MS mass loss (ψ = 0.7)	2.8 · 103 (100%)	2.8 · 103 (100%)	2.8 · 103 (100%)	3.2 · 102 (100%)	3.2 · 102 (100%)	3.2 · 102 (100%)
Stellar encountersc (MS)	79 (100%)	2.5 · 102 (100%)	5.9 · 102 (100%)	1.4 · 103 (100%)	4.3 · 103 (100%)	1.0 · 104 (100%)
Stellar encountersc (TPAGB)	5.5 · 103 (99.9%)	1.7 · 104 (99.5%)	4.1 · 104 (97%)	1.3 · 104 (99.9%)	4.2 · 104 (99.1%)	9.9 · 104 (94.4%)

Notes.

^aObjects orbiting at semimajor axis larger than the listed values would be subject to perturbations due to the process under consideration that could lead to escape. Shown in parenthesis is the fraction of the corresponding exo-Oort cloud that would lie inside a given regime of influence. ^bCalculated from Equation (3) assuming M_G(4 kpc) = 2.1 · 10¹¹ M_⊙, M_G(8.5 kpc) = 3.7 · 10¹¹ M_⊙, and M_G(12 kpc) = 4.8 · 10¹¹ M_⊙, in agreement with the Galactic gravitational potential derived by Watkins et al. (2018) corresponding to the sum of the galactic nucleus, bulge, disk, and halo. ^cClosest stellar distances derived by Veras et al. (2014) using a Galaxy model that closely reproduces observations of all the local stellar kinematics. Note that these closest stellar distances have not been derived directly from the initial mass function described in Section 2.

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Regarding the perturbation of post-main sequence mass loss, Veras et al. (2011, 2014) and Veras & Tout (2012) improved upon previous calculations by removing the traditional "adiabatic" approximation³ and solving instead the full variable-mass, two-body problem, assuming isotropic mass-loss and adopting a realistic multiphasic nonlinear, mass loss prescription (important because the strength and duration of the post-main sequence mass loss strongly affect the results in the solar-mass range). Depending on the stellar properties that determine the strength and timescale of the post-main sequence mass loss, they found that this mass loss can result in the ejection of Oort cloud objects.

For stars in the 0.8–0.9 M_⊙ mass range, Veras et al. (2011) found that, even though they lose half of their mass in their post-main sequence stage, this mass loss is not strong enough to eject bodies from their exo-Oort clouds. For stars in the 1–8 M_⊙ mass range, they found that their mass loss is strong enough to guarantee the ejection of a fraction of the Oort cloud-like bodies. For more massive stars, 8–20 M_⊙, the sudden mass loss experienced during the supernova explosion would also trigger the ejection of orbiting bodies at Oort-cloud distances. However, given the uncertainty of whether these latter stars might harbor exo-Oorts (among other things because of the exo-Oort cloud formation timescale), in this study we do not consider the contribution of potential exo-Oort clouds around stars in this high-mass range. To conclude: under our assumptions, and based on Veras' results, the dominant contribution of post-main sequence mass loss to the ejected population of exo-Oort cloud object would come from 1 to 8 M_⊙ stars.

For stars in the 1–8 M_⊙ mass range, Figure 3 in Veras et al. (2014) shows the regions where the dynamical evolution of their orbiting bodies would be under the influence of post-main sequence mass loss (black lines), together with the two other processes under consideration: Galactic tide (brown lines), and stellar encounters (pink lines). They calculated these regions by comparing the orbital timescale of the orbiting body to the timescales of these three different dynamical processes. In their figure, the degree to which post-main sequence mass loss and the Galactic tide affect the dynamics of the orbiting objects are measured by the nondimensional index ψ, defined as the ratio of the orbital timescale of the object to the timescale of the process under consideration. Veras et al. (2014) points out that ψ = 0.02 is the standard value; smaller values of ψ (0.001) would correspond to adiabaticity transitions for objects that are sensitive to small changes in orbital behavior (like comets with high eccentricity), while larger values of ψ (0.7) would correspond to higher thresholds for nonadiabatic orbital change. Depending on the value of ψ, the system becomes nonadiabatic to different extents and from this information one can estimate whether bodies orbiting at a given semimajor axis range could be subject to ejection⁴ (Veras et al. 2014).

To see how post-main sequence mass loss and the two other potential exo-Oort cloud clearing processes may affect the exo-Oort clouds, and considering the unknown orbital evolution of the Sun in the Galaxy, we extract three vertical slides from Figure 3 in Veras et al. (2014), corresponding to three different Galactocentric distances: 4, 8.5, and 12 kpc. This assumes the stars are in circular orbits around the Galactic center and takes into account that stars in this range of Galactocentric distances, encompassing the possible migration of the Sun in the Galaxy, could have contributed to the population of ejected exo-Oort cloud objects encountering the solar system. The information extracted from these slides is summarized in Table 1, which lists, among other quantities, the adiabaticity boundaries corresponding to post-main sequence mass loss of stars in the 1–8 M_⊙ mass range for different vales of the ψ index (0.02 in solid line; 0.001 in dashed–dotted line; 0.7 in dashed line). A body orbiting at a semimajor axis beyond these boundaries would be subject to become dynamically unstable. Ejection, however, is not guaranteed, as collision with the star is a possible outcome in some cases (Veras et al. 2014). The location of the adiabaticity boundary corresponding to post-main sequence mass loss is independent of Galactocentric distance and is closer to the star in the case of more massive stars, making their exo-Oort cloud objects more prone to escape.

In the absence of dynamical simulations that take into consideration the details of the stellar encounters, Veras et al. (2014) studied the regime in which this effect might be important by comparing the semimajor axis of the orbiting body to the closest expected encounter distance over a given timescale, where for the latter they consider the main sequence lifetime (labeled MS) and the thermally pulsating AGB lifetime (labeled TPAGB). Table 1 lists these values, corresponding to 1 M_⊙ and 8 M_⊙ stars at Galactocentric distances of 4, 8.5, and 12 kpc. These results indicate that the closest expected encounter distance has a strong Galactocentric distance dependence and that close flybys may be common during both stages of stellar evolution, implying that exo-Oort cloud erosion due to stellar flybys is an important clearing process that needs to be considered. For the Sun, for example, this distance extends into the scattered disk/inner solar system region during the main sequence stage.

Regarding the strength of the effect (measured by the index ψ in the other two processes considered), in the case of stellar encounters, Veras et al. (2014) pointed out that it is not possible to predict it without detailed dynamical simulations because it depends strongly on the unknown orientation of the stellar collision. Hanse et al. (2018) carried out detailed dynamical simulations to estimate Oort cloud erosion and the possibility of capturing exo-comets during close encounters with other stars harboring similar Oort clouds. They assumed three different possible orbits of the Sun in the Galaxy (migrating inward, migrating outward, and not migrating), leading to different stellar encounter sequences. They found that, over the lifetime of the Sun, the Oort cloud will lose 25%–65% of its mass due mainly to stellar encounters with stars in the 0.9–8.1 M_⊙ mass range (the Galactic tide not playing a significant role); they also found that the transfer of comets happens only during the extremely rare, relatively slow (≤0.5 km s⁻¹), close (≤10⁵ au) flybys, and are often lost shortly after the encounter, due to the Galactic tide or consecutive encounters with other stars, leading to a fraction of only ∼10⁻⁵–10⁻⁴ of captured exocomets in the Oort cloud (see Levison et al. 2010).

Table 1 also shows the location of the adiabaticity boundaries corresponding to the Galactic tide. They often lie inside the exo-Oort clouds being considered, indicating this process also contributes to their erosion. However, given that the timescale for the Galactic tide to affect the dynamical evolution of an Oort cloud-like object is on the order of a Gyr (Veras et al. 2011), whereas stellar encounters and post-main sequence mass loss operate in a much faster timescale, and given that the boundaries corresponding to these two other processes are significantly closer to the star,⁵ the latter two are expected to dominate the clearing.

As Table 1 indicates, the two processes expected to dominate the exo-Oort clouds clearing are: (1) post-main sequence mass loss, for the stars in the 1–8 M_⊙ mass range that have reached this stage of stellar evolution; and (2) stellar encounters, for the stars that are still on their main sequence. It becomes critical, therefore, to estimate how many stars in the mass range considered have reached the end of their main sequence. In Section 4, we estimate this value and the expected contribution to the population of ejected exo-Oort cloud objects from the two main clearing processes mentioned above.

Based on the discussion in Section 3, we calculate the contribution of post-main sequence mass loss of stars in the 1–8 M_⊙ mass range to the cumulative number density of ejected exo-Oort cloud objects with sizes equal or larger than 1I/'Oumuamua's, ${N}_{\mathrm{exo} \mbox{-} \mathrm{OC}}^{\mathrm{PMS}}$, using this expression:

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\mathrm{exo} \mbox{-} \mathrm{OC}}^{\mathrm{PMS}} & = & {\displaystyle \int }_{1{M}_{\odot }}^{8{M}_{\odot }}\xi ({M}_{* }){f}_{\mathrm{PMS}}({M}_{* })\\ & & \times \ \left(\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\right){10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){f}_{\mathrm{eject}}^{\mathrm{PMS}}({M}_{* }){{dM}}_{* },\end{array}\end{eqnarray} \tag{ 4 }$$

where the different terms are described below.

4.1.1. Number Density of Stars that Have Reached the Post-main Sequence Stage: $\xi ({\text{}}{M}_{* }){\text{}}{f}_{\mathrm{PMS}}({\text{}}{M}_{* })$

The term $\xi ({M}_{* })$ is the initial mass function described in Equation (2) (from Kroupa et al. 1993), while the term ${f}_{\mathrm{PMS}}({\text{}}{M}_{* })$ corresponds to the fraction of stars that have reached the post-main sequence stage at the current time. To calculate this fraction, following Kroupa et al. (1993), we assume a constant star formation rate over the age of the Galactic disk (for which we adopt the value of t_disk = 1.2 · 10¹⁰ yr). We also assume that the stars arrive at the main sequence at time t₀ and that their main sequence lifetime is t_ms, with

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}({t}_{0}({M}_{* })) & = & -5.380{M}_{* }+9.0322\,\,\,\,\mathrm{if}\,{M}_{* }\leqslant 0.15\\ \mathrm{log}({t}_{0}({M}_{* })) & = & -0.9398{M}_{* }+8.3543\,\,\,\,\mathrm{if}\,{M}_{* }\gt 0.15,\end{array}\end{eqnarray} \tag{ 5 }$$

and

$$\begin{eqnarray}&&{t}_{\mathrm{MS}}({M}_{* })={10}^{6}\left(\displaystyle \frac{2550+669{M}_{* }^{2.5}+{M}_{* }^{4.5}}{0.0327{M}_{* }^{1.5}+0.346{M}_{* }^{4.5}}\right),\end{eqnarray} \tag{ 6 }$$

with ${t}_{0}({M}_{* })$ and ${t}_{\mathrm{MS}}({M}_{* })$ in years (from Kroupa et al. 1993).

Under the above assumptions, the fraction of stars that have reached the post-main sequence stage at the current time is approximated by

$$\begin{eqnarray}&&{f}_{\mathrm{PMS}}({M}_{* })=\displaystyle \frac{{t}_{\mathrm{disk}}-({t}_{0}({M}_{* })+{t}_{\mathrm{MS}}({M}_{* }))}{{t}_{\mathrm{disk}}}.\end{eqnarray} \tag{ 7 }$$

4.1.2. Cumulative Number of Bodies with Sizes Equal to or Larger than 1I/'Oumuamua's: $\left(\tfrac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\right){10}^{12}\left(\tfrac{{M}_{* }}{{M}_{\odot }}\right)$

As discussed in Section 2.2, for a given exo-Oort cloud, we assume that the number of bodies with diameter larger than 2.3 km is similar to that of the solar system's Oort cloud scaled to the mass of the parent star, ${10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right)$. Multiplying by the factor $\left(\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\right)$, with R₁ = 0.08 km and R₂ = 1.15 km, converts this number to the cumulative number of objects with sizes equal or larger than 1I/'Oumuamua's (adopting for 1I/'Oumuamua the intermediate effective radius of 80 m from Drahus et al. (2018). To calculate this factor, following Moro-Martín (2018), we assume that the size distribution can be approximated as the following broken power law, based on solar system observations and on accretion and collisional models:

$$\begin{eqnarray}\begin{array}{rcl}n(r) & \propto & {r}^{-{q}_{1}}\,\mathrm{if}\quad {r}_{\min }\lt r\lt {r}_{b}\\ n(r) & \propto & {r}^{-{q}_{2}}\,\mathrm{if}\quad {r}_{b}\lt r\lt {r}_{\max }\\ {q}_{1} & = & 2.0,2.5,3.0,3.5\\ {q}_{2} & = & 3,3.5,4,4.5,5\\ {r}_{b} & = & 3\,\mathrm{km},30\,\mathrm{km},90\,\mathrm{km}\\ {r}_{\max } & \approx & 1000\,\mathrm{km}.\end{array}\end{eqnarray} \tag{ 8 }$$

This leads to

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\\ =\,\displaystyle \frac{\tfrac{1}{-{q}_{1}+1}\left[{r}_{b}^{-{q}_{1}+1}-{R}_{1}^{-{q}_{1}+1}\right]+\tfrac{{r}_{b}^{{q}_{2}-{q}_{1}}}{-{q}_{2}+1}\left[{r}_{\max }^{-{q}_{2}+1}-{r}_{b}^{-{q}_{2}+1}\right]}{\tfrac{1}{-{q}_{1}+1}\left[{r}_{b}^{-{q}_{1}+1}-{R}_{2}^{-{q}_{1}+1}\right]+\tfrac{{r}_{b}^{{q}_{2}-{q}_{1}}}{-{q}_{2}+1}\left[{r}_{\max }^{-{q}_{2}+1}-{r}_{b}^{-{q}_{2}+1}\right]},\end{array}\end{eqnarray} \tag{ 9 }$$

with q₁, q₂, and r_b adopting the values listed above, and with R₁ = 0.08 km and R₂ = 1.15 km.

The equilibrium power law size distribution with index q = 3.5, n(r) ∝ r^−3.5, is a specific case of this broken power law where q₁ = q₂ = 3.5. For this particular case, $\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}={\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}$. The values listed in Table 2 correspond to this specific case, while Figures 1 and 2 show the results when adopting the broken power law size distribution in Equation (8).

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Table 2.  Cumulative Number Density of Ejected Exo-OC Objects with Sizes Equal to or Larger than 1I/'Oumuamua^a

	Nexo‐OC (pc−3)
Contribution from 1 to 8 M⊙ stars
Contribution from post-main sequence mass lossb:
${\int }_{1{M}_{\odot }}^{8{M}_{\odot }}\xi ({M}_{* }){f}_{\mathrm{PMS}}({M}_{* }){\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}{10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){f}_{\mathrm{eject}}^{\mathrm{PMS}}({M}_{* }){{dM}}_{* }$	1.0 · 1013
Contribution from stellar encountersc:
${\int }_{1{M}_{\odot }}^{8{M}_{\odot }}\xi ({M}_{* })(1-{f}_{\mathrm{PMS}}({M}_{* })){\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}{10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){f}_{\mathrm{eject}}^{\mathrm{enc}}({M}_{* }){{dM}}_{* }$	1.3· 1012
Upper limitd:
${\int }_{1{M}_{\odot }}^{8{M}_{\odot }}\xi ({M}_{* }){\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}{10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){{dM}}_{* }$	1.6 · 1013
Contribution from 0.08 to 1 M⊙ stars
Contribution from stellar encountersc:
${\int }_{0.08{M}_{\odot }}^{1{M}_{\odot }}\xi ({M}_{* })(1-{f}_{\mathrm{PMS}}({M}_{* })){\left(\displaystyle \frac{0.16}{2.3}\right)}^{-2.5}{10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){f}_{\mathrm{eject}}^{\mathrm{enc}}({M}_{* }){{dM}}_{* }$	1.7 · 1013
Inferred from 1I/'Oumuamua's detection:
	2.0 · 1015

Notes.

^aAssuming a single power law size distribution of index q = 3.5. ^bAssuming for the ejection efficiency, ${\text{}}{f}_{\mathrm{eject}}^{\mathrm{PMS}}$, the values derived from the simulations by Veras et al. (2014; see discussion in Section 4.1.3). ^cAssuming an ejection efficiency ${\text{}}{f}_{\mathrm{eject}}^{\mathrm{enc}}$ = 0.5, based on dynamical simulations by Hanse et al. (2018) for the Sun's Oort cloud (see discussion in Section 4.2.3). ^dSee discussion in Section 2.

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4.1.3. Efficiency of Ejection of Exo-OC Bodies due to Post-main Sequence Mass Loss: ${f}_{{eject}}^{{PMS}}({M}_{* })$

The contribution of stellar encounters of main sequence stars in the 1–8 M_⊙ mass range to the cumulative number density of ejected exo-Oort cloud objects with sizes equal to or larger than 1I/'Oumuamua's, ${N}_{\mathrm{exo} \mbox{-} \mathrm{OC}}^{\mathrm{enc}}$, is given by:

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\mathrm{exo} \mbox{-} \mathrm{OC}}^{\mathrm{enc}} & = & {\displaystyle \int }_{1{M}_{\odot }}^{8{M}_{\odot }}\xi ({M}_{* })(1-{f}_{\mathrm{PMS}}({M}_{* }))\\ & & \times \ \left(\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\right){10}^{12}\left(\displaystyle \frac{{M}_{* }}{{M}_{\odot }}\right){f}_{\mathrm{eject}}^{\mathrm{enc}}({M}_{* }){{dM}}_{* },\end{array}\end{eqnarray} \tag{ 10 }$$

where the different terms are described below.

4.2.1. Number Density of Stars on the Main Sequence: $\xi ({\text{}}{M}_{* })(1-{\text{}}{f}_{\mathrm{PMS}}({M}_{* }))$

4.2.2. Cumulative Number of Bodies with Sizes Equal to or Larger than 1I/'Oumuamua's: $(\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}){10}^{12}(\displaystyle \frac{{M}_{* }}{{M}_{\odot }})$

4.2.3. Efficiency of Ejection of Exo-Oort Cloud Bodies due to Stellar Encounters: ${f}_{{eject}}^{{enc}}({M}_{* })$

The resulting contributions from 1 to 8 M_⊙ stars to the population of ejected exo-Oort cloud objects (calculated using Equations (4) and (10)) are listed in Table 2, assuming for the size distribution a single power law with index q = 3.5. The results corresponding to the broken power law size distributions described in Section 4.1.2 are shown in Figure 1, where in this case we are adding up the contributions of post-main sequence mass loss (Equation (4)) and stellar encounters (Equation (10)). The solid lines correspond to the ejection efficiencies discussed in Sections 4.1.3 and 4.2.3, while the dotted–dashed lines adopt the assumption that all the bodies in the exo-Oort clouds are ejected. The large differences among the values shown in Table 2 and Figure 1 (also Figure 2) are due to the $\left(\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\right)$ factor discussed above.

In Section 2.3, we argued that our back-of-the-envelope calculation was an upper limit because it assumes that all stars contribute, irrespective of whether or not they are planet hosts, and that all the objects are ejected, irrespective of whether or not they form part of an exo-Oort cloud before being completely ejected from the system. The latter assumption is justified by the models from Veras et al. (2011) that find that, even if the exo-Oort cloud objects remain on stable orbits after the period of post-main sequence stellar mass loss, they may be ejected later on because the Galactic tide will be stronger relative to the star's gravity, shrinking the stable region as the orbits expand, or because the influence from nearby stars will become stronger, leading to more ejections due to stellar encounters.

Another consideration that makes our estimate an upper limit is dynamical heating. Ejected exo-Oort cloud objects will be scattered out of the Galactic disk due to passing stars, giant molecular clouds, or spiral arms, with the smaller objects being more strongly affected because of the conservation of momentum (Feng & Jones 2018). Therefore, it is expected that the objects unbound from their parent stars early in the Galactic disk lifetime would no longer contribute to the population of interstellar objects in the Galactic disk due to dynamical heating. Because we are taking into account the contribution from all ejections, regardless of when the objects became unbound, our estimate is likely an upper limit. This strengthens our conclusion that it is unlikely that 1I/'Oumuamua is representative of a background population of exo-Oort cloud objects ejected from 1 to 8 M_⊙ stars.

As mentioned in Section 3.2, post-main sequence mass loss of stars with masses <1 M_⊙ do not contribute to the population of ejected exo-Oort cloud bodies because their mass loss is not strong enough to unbind these objects. However, given their high number density, stellar encounters of low-mass stars could contribute significantly to the population of ejected exo-Oort cloud bodies if exo-Oort clouds around these stars were to be common. In this regard, Wyatt et al. (2017) argues that these Oort clouds could be populated by objects scattered by low-mass planets (comparable to the Earth) located at 5 au, with the low mass of the star facilitating the scattering of objects encountering the planets. It remains unknown, however, how frequent these exo-Oort clouds may be around low-mass stars, with the higher frequency of planets around these stars possibly facilitating their formation.

Assuming the scenario that they are frequent, we estimate their contribution to the cumulative number density of ejected exo-OC objects with sizes equal or larger than 1I/'Oumuamua by expanding the integral in Equation (10) to include stars with masses 0.08 ≤ M_* < 1 M_⊙. The results are listed in Table 2 and Figure 2. As indicated above, the large differences among the values shown are due to the $\left(\displaystyle \frac{{N}_{r\geqslant {R}_{1}}}{{N}_{r\geqslant {R}_{2}}}\right)$ factor discussed above. As Table 2 and Figure 2 illustrate, the contribution from stellar encounters of low-mass stars, being about twice that of the contribution from post-main sequence mass loss and stellar encounters of 1–8 M_⊙ stars, still could not account for the inferred cumulative number density of interstellar objects based on 1I/'Oumuamua's detection, unless the q₁ index of the size distribution is large (q₁ ∼ 5.0) and we assume all the objects of these exo-Oort clouds are ejected (shown as dotted lines in Figure 2).

However, in the case of low-mass stars, there is another consideration for which to account: the low incoming velocity of I/'Oumuamua, within 3–10 km s⁻¹ of the LSR (Gaidos et al. 2017; Mamajek 2017; Do et al. 2018). This makes a low-mass stellar origin less likely, as these stars tend to have larger dispersion velocities because they tend to be older and because low-mass objects are more affected by dynamical heating (Dehnen & Binney 1998). This kinematic constraint is also relevant when considering the contribution of 1–8 M_⊙ stars, as it includes that of old stars with higher dispersion velocities (as mentioned in Section 5.1).

5.3.1. Flux of Micrometeorites and Meteorites on Earth

5.3.2. Number Density of Free-floating, Planetary-mass Objects from Gravitational Microlensing Surveys