1I/2017 'Oumuamua-like Interstellar Asteroids as Possible Messengers from Dead Stars

Tidally disrupted planetoids should naturally have a spread of initial semimajor axes a, especially when one accounts for the whole population of the WDs. Thus, to properly evaluate the efficiency of ISA production in planetary TDEs one needs to account for the distribution of the planetoid semimajor axes dN/da. Here, we use a very simple model in the form of a truncated power law:

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{da}}={C}_{a}{a}^{1-\alpha },\,\,\,a\lt {a}_{o},\end{eqnarray} \tag{ 8 }$$

so that the "surface density" of objects undergoing TDEs ${(2\pi a)}^{-1}\,{dN}/{da}\propto {a}^{-\alpha }$. Here a_o is the highest semimajor axis, with which refractory planetesimals can arrive into the tidal sphere of the WD. Given that we are interested in refractory objects, such as 'Oumuamua, a_o can be roughly associated with the post-MS location of the former iceline in a protoplanetary disk of the WD progenitor (a₀ ∼ (5–10) au, depending on the progenitor mass). The pre-disruption semimajor axis a of planetoids experiencing strong scattering by a giant planet at a distance r can vary between r/2 and infinity.

Note that the distribution (8) does not necessarily reflect the radial distribution of the surface density of solids in a given circum-WD planetary system, simply because the efficiency of launching planetoids into orbits leading to TDEs is a function of a (Frewen & Hansen 2014). Equation (8) is intended to merely represent a simple model for the semimajor axis distribution of planetoids ending up in TDEs, statistically averaged over the whole population of the WDs.

According to Equation (4), the presence of an outer edge at a_o sets a lower limit R_min^o on the size of planetoids that can have at least some of their tidal debris ejected to infinity:

$$\begin{eqnarray}&&{R}_{\min }^{o}={R}_{\min }({a}_{0})=\displaystyle \frac{{r}_{T}^{2}}{2{a}_{o}}.\end{eqnarray} \tag{ 9 }$$

Population of objects with $R\lt {R}_{\min }^{o}$ does not produce any unbound debris. Instead, all fragments resulting from tidal disruption are eventually accreted by the WD.

Planetoids with size $R\gt {R}_{\min }^{o}$ do produce some unbound debris, but only the objects with the pre-disruption semimajor axes a_min < a < a_o, where

$$\begin{eqnarray}&&{a}_{\min }(R)=\displaystyle \frac{{r}_{T}^{2}}{2R}={a}_{o}\displaystyle \frac{{R}_{\min }^{o}}{R}.\end{eqnarray} \tag{ 10 }$$

Based on these considerations, we can compute the efficiency ${f}_{\mathrm{ej}}^{o}(R)$ of unbound mass production in tidal disruption of objects with size R and semimajor axis distribution (8) as

$$\begin{eqnarray}&&{f}_{\mathrm{ej}}^{o}(R)=\displaystyle \frac{{\int }_{{a}_{\min }}^{{a}_{o}}{f}_{\mathrm{ej}}(a,R)\tfrac{{dN}}{{da}}{da}}{{\int }_{0}^{{a}_{o}}\tfrac{{dN}}{{da}}{da}}={\tilde{f}}_{2}\left(\displaystyle \frac{R}{{R}_{\min }^{o}}\right),\end{eqnarray} \tag{ 11 }$$

$(R\gt {R}_{\min }^{o})$ where function ${\tilde{f}}_{2}$ is defined by Equation (40). Note that integration in the denominator of Equation (11) runs from a = 0 to a = a_o since (in the zeroth order approximation) objects from the full range of semimajor axes can be tidally disrupted. On the other hand, for a given size R, production of unbound debris is possible only for a > a_min, hence the integration limits in the numerator.

We plot ${f}_{\mathrm{ej}}^{o}(R)$ as a function of $R/{R}_{\min }^{o}$ for several values of α in Figure 1. It is clear from this figure that lower (positive) values of α, which put more refractory mass at larger distance from the WD, result in considerably higher ejection efficiency (per unit mass in objects of a given size processed in TDEs): ${f}_{\mathrm{ej}}^{o}$ is more than three times higher for α = 0.3 than for α = 1.8. This is because R_min(a) is lower for more distant planetoids, driving ${f}_{\mathrm{ej}}^{o}(R)$ closer to 50% for belts with more mass at high a, see Equations (6)–(7). Also, ${f}_{\mathrm{ej}}^{o}$ grows with R relatively slowly (although much faster for low values of α), reaching 50% only for very large, planetary-scale planetoids.

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In practice one needs to account not only for spatial distribution of tidally destroyed objects, but also for their size distribution. We start by computing the overall efficiency of ejection produced by a belt of minor objects with a continuous mass spectrum; a more sophisticated and realistic model of a planetary system is explored next in Section 2.3. We model the size distribution of planetoids undergoing TDEs as a truncated power law

$$\begin{eqnarray}&&\displaystyle \frac{{dN}}{{dR}}={C}_{R}{R}^{-\beta },\,\,\,R\lt {R}_{\mathrm{cut}},\end{eqnarray} \tag{ 12 }$$

where R_cut is the upper size cutoff. This model is motivated by the observed size distributions of the populations of minor objects in the solar system—asteroid and Kuiper belts. The belt size distribution (12) is schematically illustrated in Figure 3 with β ≈ 3.5, as appropriate for the collisional equilibrium spectrum derived by Dohnanyi (1969).

In our calculation we again assume that planetoids of all sizes are spatially distributed according to Equation (8). Given the finite semimajor axis extent of this distribution, production of unbound fragments in TDEs is possible only if ${R}_{\mathrm{cut}}\gt {R}_{\min }^{o}$ (otherwise all planetoids are accreted with 100% efficiency). Whenever this condition is fulfilled, the fraction of mass ejected in TDEs marginalized over the power law distribution (12) is given by

$$\begin{eqnarray}&&{f}_{\mathrm{ej}}^{\mathrm{sm}}=\displaystyle \frac{{\int }_{{R}_{\min }^{o}}^{{R}_{\mathrm{cut}}}{f}_{\mathrm{ej}}^{o}(R)\tfrac{{dN}}{{dR}}{R}^{3}{dR}}{{\int }_{0}^{{R}_{\mathrm{cut}}}\tfrac{{dN}}{{dR}}{R}^{3}{dR}}={\tilde{f}}_{3}\left(\displaystyle \frac{{R}_{\mathrm{cut}}}{{R}_{\min }^{o}}\right)\end{eqnarray} \tag{ 13 }$$

(note that the integration in the denominator runs between 0 and R_cut as planetoids of all sizes can be involved in TDEs), where

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{\mathrm{cut}}}{{R}_{\min }^{o}}\approx 2.5\left(\displaystyle \frac{{R}_{\mathrm{cut}}}{{10}^{3}\,\mathrm{km}}\right)\left(\displaystyle \frac{{a}_{o}}{5\,\mathrm{au}}\right){\left(\displaystyle \frac{{M}_{0.6}}{{\rho }_{3}}\right)}^{-2/3},\end{eqnarray} \tag{ 14 }$$

see Equations (2) and (9). The explicit expression⁴ for the function ${\tilde{f}}_{3}$ is given by Equation (41).

In Figure 2 we plot the ejection efficiency ${f}_{\mathrm{ej}}^{\mathrm{sm}}$ for a planetoid belt with the power law size distribution (12), as a function of the upper size cutoff of the size spectrum R_cut. Comparing with Figure 1, one can see that the ejection efficiency drops significantly when a spectrum of planetoid sizes is considered. This is because the size distribution (12) contains a certain amount of mass in objects with sizes below ${R}_{\min }^{o},$ which produce no unbound fragments. Also, even for R_cut substantially higher than ${R}_{\min }^{o},$ belt members with $R\sim {R}_{\min }^{o}$ still have low ejection efficiency, driving ${f}_{\mathrm{ej}}^{\mathrm{sm}}$ down.

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Shallow size distributions (lower values of β) result in higher ${f}_{\mathrm{ej}}^{\mathrm{sm}}$, as such size spectra have more mass in the largest objects with R ∼ R_cut, for which the ejection efficiency is highest. Also, as in Figure 1, higher values of α (i.e., less mass at large semimajor axes a ∼ a_o) result in considerably lower ejection efficiency.

Finally, we evaluate the efficiency of ISA production in TDEs sourced by the more realistic planetary systems. We consider a composite model for the mean population of tidally destroyed planetoids, in which a belt of minor objects with size distribution (12) is supplemented with a number of "planets" bigger than R_cut, but still significantly smaller than massive perturbers that do the scattering, see Figure 3. This population of big objects with radius R_lg > R_cut has a total mass M_lg and we will assume that it dominates the TDE mass budget. In other words, M_lg ≳ M_sm, where M_sm is the total mass of minor planetoids (power law part of the size spectrum, see Equation (12)) that end up in TDEs. At least based on the solar system experience, in which the asteroid belt co-exists with a population of terrestrial planets, one expects M_lg ≫ M_sm—the mass of our asteroid belt is just 10⁻³ of the combined mass of the terrestrial planets. For simplicity, we also assume the bulk density of "planets" to be the same as for small planetoids.

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Given that essentially all dynamical processes emplacing planetoids onto low angular momentum orbits (scattering by giant planets, Lidov–Kozai cycles, etc.) are insensitive to the planetoid mass, this model should provide a good description of a rather general planetary system architecture.

We also assume that, in a statistical sense (i.e., averaged over many circum-WD planetary systems), the spatial distributions of both the high and low mass planetoids are the same and are well represented by Equation (8). The number of big objects per planetary system does not need to be integer and should be treated as representing the mean over many systems.

Using the results of Sections 2.1–2.2 we can then write the overall efficiency of mass ejection in the framework of this composite model as

$$\begin{eqnarray}{f}_{\mathrm{ej}}^{\mathrm{comp}} & = & \displaystyle \frac{{f}_{\mathrm{ej}}^{\mathrm{sm}}{M}_{\mathrm{sm}}+{f}_{\mathrm{ej}}^{o}({R}_{\mathrm{lg}}){M}_{\mathrm{lg}}}{{M}_{\mathrm{sm}}+{M}_{\mathrm{lg}}}\\ & = & \displaystyle \frac{{M}_{\mathrm{sm}}{\tilde{f}}_{3}({R}_{\mathrm{cut}}/{R}_{\min }^{o})+{M}_{\mathrm{lg}}{\tilde{f}}_{2}({R}_{\mathrm{lg}}/{R}_{\min }^{o})}{{M}_{\mathrm{sm}}+{M}_{\mathrm{lg}}}.\end{eqnarray} \tag{ 15 }$$

We plot the ejection efficiency ${f}_{\mathrm{ej}}^{\mathrm{comp}}$ in Figure 4 as a function of M_lg/M_sm—mass ratio of the "planetary" and belt populations. We vary different parameters of the model away from their fiducial values ((α, β) = (0.3, 2.4), R_cut = 600 km, R_lg = 3000 km; ${R}_{\min }^{o}=400$ km) to understand how they affect the outcome. Comparing with Figure 2, one can see that adding a population of massive planetoids to the planetary system considerably increases ejection efficiency. This is not surprising, since f_ej rapidly increases with the size R of a disrupted object, see Figure 1. As a result, big planetoids completely dominate ${f}_{\mathrm{ej}}^{\mathrm{comp}}$ for M_lg/M_sm ≳ 1. This is shown in Figure 4 as the rather strong sensitivity of the ejection efficiency to the size of "planets" R_lg and only weak sensitivity to the characteristics of the belt of minor objects (e.g., R_cut).

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As before (see Sections 2.2–2.3), higher values of α and β (less mass in massive, distant objects) result in a considerable reduction of ejection efficiency—even when most of the mass is in large objects (${M}_{\mathrm{lg}}/{M}_{\mathrm{sm}}\gtrsim 10$) ${f}_{\mathrm{ej}}^{\mathrm{comp}}$ can barely reach ∼10%.

The results of this section demonstrate that tidal disruptions of planetoids by WDs can eject substantial amounts of mass into interstellar space. This is despite the fact that the disrupted planetoids are initially gravitationally bound to the WD, which somewhat lowers the efficiency of the mass ejection in these events compared to the TDEs of initially unbound stars by supermassive black holes (Lacy et al. 1982; Rees 1988; Lodato et al. 2009). Nevertheless, mass-weighted ejection efficiency of ∼30% can be easily achieved, provided that the population of the disrupted objects contains most of the mass in large (several 10³ km) objects. Asteroid belts are far less efficient at producing unbound fragments in tidal disruptions of their members. Moreover, objects with sizes ≲400 km starting on semimajor axes ≲5 au get fully accreted in TDEs, leaving no unbound fragments.

Thus, efficient production of refractory, 'Oumuamua-like ISAs in planetary TDEs by WDs (with efficiency of ≳30%) requires that massive, planetary-scale objects dominate the mass budget of planetoids involved in TDEs. This will be important when discussing the observational constraints in Section 5.

We start by adopting a simple, quasi-static picture, in which at every distance r < r_T from a WD the characteristic size of a fragment R_f(r) is given by the maximum size of an object that can still sustain tidal stresses at this separation; in other words, we assume a one-to-one relation between R_f and r in the form ${R}_{f}(r)=D(r)$. As r decreases during the debris transit through the tidal sphere, so does D(r), until the pericenter at $r=q=a(1-e)$ is reached. After the pericenter passage tidal stress starts decreasing and fragmentation stops. In this picture, beyond the pericenter the TDE debris should have a size spectrum with most of the mass concentrated around D(q). Variation of q between R_wd and r_T for different TDEs then results in a spread of values of D(q), shaping the size spectrum of ISAs over multiple TDEs.

Dobrovolskis (1990) carried out an exploration of the breakup conditions for bodies with non-zero internal strength, the results of which were also summarized in Davidsson (1999) in the form of r(D) relations for the minimum distance, at which objects with different material properties and size D can still sustain tidal stress. In our simple model, these results allow us to uniquely relate the characteristic (mass-bearing) size of the resultant fragments to the periastron distance, q, of the original orbit of a parent planetoid, i.e.,

$$\begin{eqnarray}&&q=q({R}_{f}),\end{eqnarray} \tag{ 16 }$$

where we identified ${R}_{f}=D(q)$. It is important that this relation is independent of either the mass of the parent planetoid, or its original semimajor axis—only the periastron distance matters in setting the final fragment size.

In this simple, quasi-static picture the size spectrum dN_f/dR_f averaged over many TDEs can be computed as

$$\begin{eqnarray}&&\displaystyle \frac{{{dN}}_{f}}{{{dR}}_{f}}\approx \displaystyle \frac{{dq}}{{{dR}}_{f}}{\int }_{0}^{\infty }{\left(\displaystyle \frac{R}{{R}_{f}}\right)}^{3}\displaystyle \frac{{d}^{2}N}{{dRdq}}{dR},\end{eqnarray} \tag{ 17 }$$

where ${d}^{2}N/{dRdq}$ is the distribution of tidally disrupted planetoids per unit radius R, and per unit periastron distance q, while the factor ${(R/{R}_{f})}^{3}$ gives the number of fragments with size R_f spawned by a planetoid of size R.

If the mechanism driving planetoids into the low-periastron orbits is the strong scattering by massive perturbers (e.g., a giant planet), which to zeroth order occurs roughly isotropically (i.e., with uniform distribution of the post-scattering velocity component perpendicular to the radius vector to the WD), then planetoids should have roughly uniform distribution in l² around l = 0, where $l=\sqrt{{{GM}}_{\star }a(1-{e}^{2})}$ is the specific angular momentum. This means that highly eccentric planetoids enroute to tidal disruption should have roughly uniform distribution of the periastron distance q. The same conclusion was reached by Katz & Dong (2012) in their study of secular Lidov–Kozai evolution (Kozai 1962; Lidov 1962) leading to WD collisions in triple systems—a completely different scenario resulting in highly eccentric orbits, which has also been invoked to explain atmospheric pollution of WDs (Petrovich & Muñoz 2017; Stephan et al. 2017). Based on these considerations, it is legitimate to approximate

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}N}{{dRdq}}\approx {r}_{T}^{-1}\displaystyle \frac{{dN}}{{dR}},\end{eqnarray} \tag{ 18 }$$

since the periastron distance varies in the range R_wd < q < r_T and ${R}_{\mathrm{wd}}\ll {r}_{T}$ for WDs. Using this result as well as the independence of the $q({R}_{f})$ relation upon the parent planetoid size, we find that, roughly,

$$\begin{eqnarray}&&\displaystyle \frac{{{dN}}_{f}}{{{dR}}_{f}}\approx \displaystyle \frac{3}{4\pi }\displaystyle \frac{{M}_{\mathrm{TDE}}}{\rho {r}_{T}}\,{R}_{f}^{-3}\displaystyle \frac{{dq}}{{{dR}}_{f}}\end{eqnarray} \tag{ 19 }$$

for ${R}_{f}^{\min }\lt {R}_{f}\lt {R}_{f}^{\max }$ and zero outside this range. Here ${R}_{f}^{\min }={R}_{f}({R}_{\mathrm{wd}})$ is the minimum fragment size, which is reached when the pericenter q distance gets close to the WD surface—at this location the tidal stress is highest and only the smallest (internally strongest) fragments can survive. Similarly, ${R}_{f}^{\max }={R}_{f}({r}_{T})$ is the size of the largest objects that can survive tides at r_T due to their internal cohesion. Also, ${M}_{\mathrm{TDE}}={\int }_{0}^{\infty }M({dN}/{dR}){dR}$ is the total mass in planetoids that have undergone TDEs. Note that dN_f/dR_f depends only on the full mass of disrupted planetoids M_TDE and not on their initial size spectrum dN/dR (as long as $R\gtrsim {R}_{f}^{\max })$. The dependence (19) of the size spectrum of TDE fragments (equivalent to the size spectrum of ISAs) on R_f is fully contained in ${R}_{f}^{-3}{dq}/{{dR}}_{f}$.

We now illustrate the implications of this result using a simple prescription for the material properties of a putative planetoid assumed to be composed primarily of iron. This choice does not reflect any expectations about the composition of 'Oumuamua and is selected because of associated mathematical simplicity. The qualitative picture remains the same in the case of rocky planetoids, see Section 3.1.2.

3.1.1. Iron Planetoids

For a planetoid composed primarily of iron (ρ ≈ 8 g cm⁻³) Davidsson (1999) suggests that the tidal splitting distance, which we associate with q, is

$$\begin{eqnarray}&&q({R}_{f})\approx 0.85{\left(\displaystyle \frac{{M}_{\mathrm{wd}}}{\rho }\displaystyle \frac{{p}_{0}({R}_{f})}{S}\right)}^{1/3},\,\,\mathrm{iron},\end{eqnarray} \tag{ 20 }$$

where S is the shear strength (S ≈ 3 × 10³ bar for iron) and the central pressure p₀ is related to the fragment size R_f via

$$\begin{eqnarray}&&{p}_{0}({R}_{f})=\displaystyle \frac{2}{3}\pi G{\rho }^{2}{R}_{f}^{2}.\end{eqnarray} \tag{ 21 }$$

These relations imply that $q({R}_{f})\propto {R}_{f}^{2/3}$, such that

$$\begin{eqnarray}&&\displaystyle \frac{{{dN}}_{f}}{{{dR}}_{f}}\propto {R}_{f}^{-10/3}.\end{eqnarray} \tag{ 22 }$$

Setting q = R_wd in Equation (20) we find

$$\begin{eqnarray}{R}_{f}^{\min } & \approx & 0.88{\left(\displaystyle \frac{S}{{{GM}}_{\mathrm{wd}}\rho }\right)}^{1/2}{R}_{\mathrm{wd}}^{3/2}\\ & \approx & 350\,{\rm{m}}\,{M}_{0.6}^{-0.5}{\left(\displaystyle \frac{{R}_{\mathrm{wd}}}{{10}^{-2}{R}_{\odot }}\right)}^{1.5}.\end{eqnarray} \tag{ 23 }$$

This estimate shows that an R = 10³ km iron asteroid passing very close to the WD surface, at q ∼ R_wd (which according to Equation (18) should happen in several per cent of planetoid TDEs), will disintegrate into at least several ×10⁹ fragments of size ${R}_{f}^{\min }$.

Next, setting q = r_T we find from Equation (2)

$$\begin{eqnarray}&&{R}_{f}^{\max }\approx 0.95{\left(\displaystyle \frac{S}{G{\rho }^{2}}\right)}^{1/2}\approx 250\,\mathrm{km}.\end{eqnarray} \tag{ 24 }$$

This is the size of fragments produced in TDEs of planetoids just barely entering the Roche sphere of a WD. Figure 5 schematically illustrates the mass spectrum of fragments (red dashed curve) predicted by the quasi-static model for iron planetoids.

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3.1.2. Rocky Planetoids

3.1.3. Implications of the Simple Model

The results of Sections 3.1.1–3.1.2 demonstrate that, even in the framework of a simple model of tidal disruption outlined in Section 3.1, planetary TDEs naturally provide the possibility of producing 'Oumuamua-size objects: our estimates of ${R}_{f}^{\min }$ for both iron and rocky fragments are rather close to the estimated 'Oumuamua's dimensions of ∼10² m. At the same time, the size spectrum of fragments predicted by the simple, quasi-static model has most of the mass in big objects, see Equation (22) and Figure 5. Steep fragment size distributions have been previously found in the tidal disruption simulations of Debes et al. (2012), although they had rather low resolution. Since our model also predicts ${R}_{f}^{\max }\gg {R}_{f}^{\min },$ this again raises the "large ψ_m" issue mentioned when discussing our estimate (1) for ρ_ISA: only a small fraction of mass in ISAs would then be contained in 'Oumuamua-like objects because ${\psi }_{m}\gg 1$ (see the red ψ_m symbol in Figure 5). Indeed, integrating Equation (19) over the appropriate parts of the fragment size spectrum, one can estimate the ratio between the amount of mass in small objects with size ${R}_{f}^{\min }\sim {R}_{\mathrm{Ou}}$ and the full mass of ISAs contained mainly in large bodies with size $\sim \,{R}_{f}^{\max }$ as

$$\begin{eqnarray}&&{\psi }_{m}\approx \displaystyle \frac{q({R}_{f}^{\max })}{q({R}_{f}^{\min })}=\displaystyle \frac{{r}_{T}}{{R}_{\mathrm{wd}}}\sim {10}^{2}.\end{eqnarray} \tag{ 25 }$$

Thus, this simple model for the TDE fragment mass spectrum predicts 10² times more mass in ISAs than can be inferred from a single detection of 'Oumuamua.

However, as we show next, the quasi-static model is too simplistic and the estimate (25) is too high.

One of the most dramatic outcomes of stellar disruptions by supermassive black holes is a strong vertical (orthogonal to the midplane) compression of a star into a short-lived, dense, pancake-like configuration at some point within the Roche sphere (Carter & Luminet 1983). The vertical collapse of a star with the initial radius R_⋆, and mass M_⋆, into such a flattened object is caused by the component of the black hole gravity normal to the orbital plane, which cannot be balanced by the pressure forces inside the Roche sphere. Prior to the maximum compression (when the density and pressure at the center become high enough to resist vertical collapse) vertical motion towards the midplane has characteristic velocity (Carter & Luminet 1983; Stone et al. 2013)

$$\begin{eqnarray}&&{v}_{z}\sim \displaystyle \frac{{r}_{T}}{q}{\left(\displaystyle \frac{{{GM}}_{\star }}{{R}_{\star }}\right)}^{1/2},\end{eqnarray} \tag{ 26 }$$

significantly exceeding the escape velocity from the surface of the original star for deep plunges with the periastron distance $q\ll {r}_{T}$. Homologous compression at such high velocity can have dramatic consequences, including detonation of the whole star (Carter & Luminet 1983).

Our quasi-static picture of the planetoid TDE presented in Sections 3.1–3.1.3 completely overlooks this highly dynamic aspect of the disruption phenomenon. An obvious difference with the stellar disruption case is the weak compressibility of a tidally destroyed planetoid. However, this is unlikely to eliminate the main features of the dynamic vertical compression picture. Indeed, to zeroth order one may consider a tidally destroyed planetoid as a rubble pile composed of fragments with sizes ${R}_{f}\sim D(r)$ at each point in orbit. Uncompensated vertical gravity of the WD will initiate the collapse of individual elements of this rubble pile towards the midplane of the orbit, which is better described as a granular flow. A good analogy would be the collapse of a pyramid of billiard balls on a pool table once the rack holding its base in place is suddenly removed.

Lack of compressibility may result in additional lateral (parallel to the orbital plane) expansion of the sheared rubble pile, but this will not prevent high-speed convergent motion of the fragments near the midplane. Moreover, conversion of the vertical kinetic energy $\sim {v}_{z}^{2}$ into that of the in-plane motions is unlikely to significantly affect our estimate (3) of the energy spread ΔE. Indeed, using Equations (2), (3), and (26) one can show that ${v}_{z}^{2}/{\rm{\Delta }}E\sim {r}_{T}R/{q}^{2}$, where in (26) we replaced R_⋆, M_⋆ with R, M. As a result, vertical energy can exceed ΔE only for $q\lesssim {({r}_{T}R)}^{1/2}$, which is ≲0.1r_T even for largest planetoids; i.e., ${v}_{z}^{2}\gtrsim {\rm{\Delta }}E$ in at most several per cent of TDEs, given the flat distribution (18) of q. However, even in these rare events most of the vertical energy will still be dissipated inelastically (rather than contribute to in-plane motions) as we describe now.

Head-on collisions of fragments comprising the two hemi-spheres of the original planetoid, which are vertically collapsing towards the orbital midplane at speeds indicated by Equation (26), will result in catastrophic destruction of debris objects and fragmentation cascade down to small sizes. Indeed, even if Equation (26) overestimates the vertical velocity by a large factor, v_z is still so high that it would easily exceed the escape speed (which scales $\propto {R}_{f}$) from the surface of any fragment by a considerable margin, leading to catastrophic disruptions.

This short-duration burst of fragmentation will likely cease, or at least slow down, when the fragment size becomes small enough for the material strength to start opposing the collisional splitting (since the minimum velocity necessary for catastrophic disruption increases with decreasing size in the strength-dominated regime, Housen & Holsapple 1990). In the absence of a good model of either the granular flow-like vertical collapse or the ensuing fragmentation cascade, we hypothesize that the final size distribution of fragments resulting from a planetoid TDE should peak around the size at which the collision velocity leading to catastrophic disruption is minimized. Using the results of Stewart & Leinhardt (2009), this final size of the surviving fragments can be estimated as (Rafikov & Silsbee 2015)

$$\begin{eqnarray}&&{R}_{f}^{\mathrm{dyn}}\sim (0.1-1)\,\mathrm{km},\end{eqnarray} \tag{ 27 }$$

depending on the material properties of the colliding objects.

Needless to say, this estimate is rather uncertain, even though it is supported by physical arguments. A more accurate calculation of ${R}_{f}^{\mathrm{dyn}}$ should be possible via direct simulations of tidal disruptions of asteroid-like objects (Debes et al. 2012; Movshovitz et al. 2012; Veras et al. 2014a). Such simulations could (1) show whether the overall dynamic picture as presented here is correct, (2) determine the magnitude of the convergent vertical motions deep inside the Roche sphere of the WD for comparison with Equation (26), and (3) provide input for calculating collisional evolution of individual "rubble particles" comprising a tidally disrupted planetoid with the ultimate goal of determining the characteristic fragment size ${R}_{f}^{\mathrm{dyn}}$ resulting from a TDE.

In the dynamic picture of the planetary TDE presented here, most of the planetoid mass gets converted into objects with a rather narrow spread in sizes around $\sim {R}_{f}^{\mathrm{dyn}}$ (schematically represented by the blue curve in Figure 5). Given that $\sim {R}_{f}^{\mathrm{dyn}}$ is very close to the dimensions of 'Oumuamua, this implies that the resultant size spectrum of the unbound fragments would have

$$\begin{eqnarray}&&{\psi }_{m}\sim 1.\end{eqnarray} \tag{ 28 }$$

As a result, the estimate (1) should then properly reflect the spatial mass density of ISAs in the Galaxy.

The highly dynamic nature of the disruption event, followed by substantial collisional evolution of its products, may also provide important clues for understanding the highly elongated shape and the rotation state of the 'Oumuamua (Drahus et al. 2018).

The results of this section demonstrate that planetary TDEs should be efficient at producing fragments with sizes comparable to 'Oumuamua's dimensions. Even the simple quasi-static model presented in Section 3.1 successfully produces objects with sizes in the 0.1–1 km range, although the overall amount of mass ending up in such fragments is predicted to be rather small, around 1% (formally resulting in high ψ_m ∼ 10²).

A more refined dynamic model (Section 3.2), accounting for the vertical collapse of a rubble pile, into which the planetoid turns inside the WD Roche sphere, predicts substantial collisional grinding of the TDE products (which may also be relevant for explaining the unusual shape and rotation of the 'Oumuamua). We hypothesize that collisional evolution caused by convergent vertical motions will convert most of the original planetoid mass into objects with sizes around 0.1–1 km (see Figure 5). Below this size scale the internal strength of the fragments becomes important, making further collisional fragmentation of the disruption products difficult.

A very important outcome of this dynamic planetary TDE model is the expectation of ${\psi }_{m}\sim 1$ for the size distribution of fragments that it predicts. The implications of this prediction will be further discussed in Section 5, when comparing the existing observational constraints on the metal accretion by the WDs with the inferred space density of the ISAs.

Mamajek (2017) found that prior to its passage through the inner solar system 'Oumuamua had rather low velocity relative to the LSR, around 10 km s⁻¹ in the azimuthal direction, with negligible vertical and radial Galactocentric velocities. This motion appears somewhat unusual in light of the expectations outlined above, which suggest that ISA velocities of up to 30–40 km s⁻¹ in the LSR frame may be typical. At the same time, for a population with a given velocity ellipsoid the probability of drawing an object with a particular velocity relative to the LSR peaks at zero velocity.

Nevertheless, the observed kinematic state of 'Oumuamua would be more likely if it originated from a population with low velocity dispersion. This observation may hold important clues for the origin of this ISA, since it would imply that this object (1) has originated from a dynamically cold Galactic source and (2) is relatively young. Regarding the latter, if 'Oumuamua had spent a long (≳Gyr) time unbound in the Galaxy, its epicyclic motion would be excited by random gravitational perturbation to levels (on average) exceeding its measured space motion. However, if its youth is the consequence of its short lifetime (e.g., against fading or some other decay mechanism) then the production rate of the ISAs should be increased correspondingly (Gaidos 2018) to maintain their observed abundance ρ_ISA given by Equation (1). This is likely to be challenging. A similar possibility is that we are catching 'Oumuamua at the very beginning of its long life in interstellar space, which is, again, a low probability event.

In light of these arguments, we prefer to think that 'Oumuamua's slow Galactic motion simply reflects the peak probability of drawing an object from a Schwarzschild velocity distribution in the LSR frame (Binney & Tremaine 2008), and that this ISA does in fact belong to the old Galactic population. Any model for its origin, in which 'Oumuamua spends ≳Gyr in interstellar space, should arrive at this conclusion. Our idea of the planetary TDE is no exception in this regard.

The results of this section demonstrate that the unbound fragments resulting from planetary TDEs get ejected with characteristic speeds of 10–30 km s⁻¹, depending primarily on the mass of a disrupted planetoid, with more massive objects producing faster ejecta. The mass-weighted velocity distribution of the ejecta (30) is found to be slightly weighted towards higher velocities. It should be convolved with the velocity distribution of the parent WDs (which are dynamically hot) to give a full representation of the ejecta kinematics in the Galactic frame. As a result, we deduce that Gyr-old ISAs should have the velocity dispersion ∼(30–40) km s⁻¹ with respect to the LSR, similar to the old stellar populations.

In light of these findings, we interpret the observed low velocity of 'Oumuamua with respect to the LSR as simply reflecting the highest probability of picking an object from a Schwarzschild velocity distribution. Subsequent detections of the ISAs should find a significant fraction of them to be fast, dynamically hot objects. This conclusion should hold regardless of the origin mechanism of the ISAs, as long as they spend a long time in interstellar space.

ISA ejection in TDEs is highly anisotropic and clustered in phase space, leading to slow phase mixing of the ejecta across the Galaxy. As a result, spatial distribution of the ISAs can exhibit significant fluctuations in density.

A very fortunate feature of planetary material processing by WDs is that we can actually characterize it observationally. Atmospheres of tens of per cent of all WDs are known to be polluted with metals (Farihi 2016), and the most plausible mechanism for this pollution is accretion of the planetary material in the TDEs (Debes & Sigurdsson 2002; Jura 2003), an idea strongly supported by the recent discovery of the disintegrating objects near the WD 1145+017 (Vanderburg et al. 2015). The amount of mass accreted by an average WD during its lifetime can be calculated if one (1) is able to measure the accretion rate ${\dot{M}}_{Z}$ of the refractory material by the WDs and (2) has some idea of how this rate varies as the WD ages.

Wyatt et al. (2014) used an unbiased sample of spectroscopically observed WDs to infer the statistics of accretion rates in different evolutionary stages. Since their sample is relatively large (hundreds of WDs) and unbiased, the observed instantaneous values of ${\dot{M}}_{Z}$ are expected to reflect reasonably well the time-averaged picture of metal accretion by the WDs (but see below for caveats). We use their results for the cumulative distribution of ${\dot{M}}_{Z}$ for different sub-samples to arrive at an estimate of the mean ${\dot{M}}_{Z}\approx 5\times {10}^{8}$ g s⁻¹ for all WDs with ages of 100–500 Myr. It should be mentioned that the determination of this mean value is a non-trivial procedure riddled with various observational biases (Farihi 2016).

On the other hand, theoretical calculations of the long-term dynamics of planetary delivery into the tidal radius of the WD (Mustill et al. 2018), supported by observations (Hollands et al. 2018), suggest that ${\dot{M}}_{Z}$ decays exponentially with a characteristic timescale of ≈1 Gyr. Normalizing this relation to the results of Wyatt et al. (2014) in the 100–500 Myr age bin and integrating ${\dot{M}}_{Z}$ over the lifetime of the WD (several Gy), we arrive at the following observationally motivated estimate of the full (integrated over lifetime) accreted planetary mass of

$$\begin{eqnarray}&&{\rm{\Delta }}{M}_{\mathrm{acc}}\sim 0.003\,{M}_{\oplus }\,\,{\rm{per}}\,{\rm{WD}}.\end{eqnarray} \tag{ 35 }$$

All this mass must have been accreted in the TDEs of a kind that we envisage in this paper. A similar, although less sophisticated exercise has been carried in Hansen & Zuckerman (2017), who arrived at an estimate of ΔM_acc, which is three times higher than (35), primarily because they assumed ${\dot{M}}_{Z}$ to remain at a constant level of 3 × 10⁸ g s⁻¹ over 5 Gyr.

The estimate (35) is orders of magnitude lower than the mass in ISAs per WD given by Equation (34). At face value, this discrepancy represents a serious blow to the idea that planetary tidal disruptions by the WDs can provide a significant source of ISAs. Indeed, the results of Section 2.1 imply that mass ejected in TDEs should be further lowered compared to ΔM_acc by a factor of ${\bar{f}}_{\mathrm{ej}}/(1-{\bar{f}}_{\mathrm{ej}})\sim 0.4$ (for the mean ejection efficiency⁶ f_ej = 0.3), meaning that only ${\bar{f}}_{\mathrm{ej}}{(1-{\bar{f}}_{\mathrm{ej}})}^{-1}{\rm{\Delta }}{M}_{\mathrm{acc}}/{\rm{\Delta }}{M}_{\mathrm{Ou}}\,\approx 0.01 \%$ of ISAs can be explained by tidal disruptions of planetoids by the WDs.

However, there are strong reasons to believe that the estimate (35) likely represents a lower limit on ΔM_acc and the actual accreted mass is much higher. Indeed, the statistics of observed ${\dot{M}}_{Z}$ in Wyatt et al. (2014) are heavily dominated by WDs with the highest values of ${\dot{M}}_{Z}$, around 10¹⁰ g s⁻¹. Also, we know a number of metal-rich WDs, which accrete at even higher rates, in excess of 10¹¹ g s⁻¹ (Bergfors et al. 2014). Adding just several such objects to the sample of Wyatt et al. (2014) would increase the estimate (35) by an order of magnitude. Thus, the determination of ΔM_acc is heavily affected by the small number statistics at the highest ${\dot{M}}_{Z}$.

What this means is that the current observations of ${\dot{M}}_{Z}$ are likely to be revealing to us only the tip of an iceberg,⁷ and the highest ${\dot{M}}_{Z}$ events resulting from the accretion of the largest ("planet"-scale) planetoids that dominate the budget of accreted mass are so rare that our current WD sample is simply too small to catch them in action.

The maximum size of a planetoid that is sampled by current observations of ${\dot{M}}_{Z}$ can be estimated by measuring the total mass in refractory elements contained in the outer convective envelopes of actively accreting WDs. According to Farihi et al. (2010), the largest amounts of atmospheric Ca measured in a handful of objects correspond to the accreted mass equivalent to the mass of a $R={R}_{\mathrm{obs}}^{\max }\approx 300$ km asteroid (similar to Vesta). This can be considered as the maximum size of objects, to which our measurements of ${\dot{M}}_{Z}$ are sensitive because of the rarity of accretion events involving even more massive objects (which at the same time deliver more mass to the WD, boosting ΔM_acc). Note that ${R}_{\mathrm{obs}}^{\max }$ is below our estimate (4) for ${R}_{\min }^{o}$ (see Figure 3), meaning that the currently observed ${\dot{M}}_{Z}$ in metal-rich WDs derive from planetary TDEs, which did not eject any material into interstellar space.

We now describe a (model-dependent) procedure, by which we account for the biases mentioned in Section 5.1 to allow a more meaningful relation to be established between the observed ΔM_acc and the ejected mass M_ej.

As in Section 2.3, we assume that the planetoid population destined for TDEs can be represented, in a statistically averaged sense, as a combination of a belt of minor planets (asteroids) with the size spectrum given by the Equation (12) and total mass M_sm plus a population of large objects with size ${R}_{\mathrm{lg}}\gt {R}_{\mathrm{cut}}$ and total mass M_lg.

As we saw in Section 5.1, so far observations of the WD metal pollution have been telling us only about the properties of relatively small objects. For that reason we will assume that planetoids with size $\lesssim \,{R}_{\mathrm{obs}}^{\max }$ belong to the belt of minor planets, i.e., ${R}_{\mathrm{obs}}^{\max }\lt {R}_{\mathrm{cut}}$. In other words, the manifestations of TDEs involving only a part of the full mass spectrum (illustrated by the gray region in Figure 3) are observable at present.

The total "observable" mass contained in this part of the size spectrum is

$$\begin{eqnarray}&&{M}_{\mathrm{obs}}={M}_{\mathrm{sm}}{\left(\displaystyle \frac{{R}_{\mathrm{obs}}^{\max }}{{R}_{\mathrm{cut}}}\right)}^{4-\beta }\lt {M}_{\mathrm{sm}}.\end{eqnarray} \tag{ 36 }$$

We will also assume that the ejection efficiency is zero for $R\lt {R}_{\mathrm{obs}}^{\max }$, i.e., that the "observable" planetoids are small enough to get fully accreted. This is consistent with our estimate ${R}_{\mathrm{obs}}^{\max }\approx 300$ km $\lt \,{R}_{\min }^{o}$ and Equations (4) and (9).

Using the results of Section 2.3 we can then estimate the ratio between the true ejected mass M_ej and the observable mass M_obs for such a system as

$$\begin{eqnarray}\displaystyle \frac{{M}_{\mathrm{ej}}}{{M}_{\mathrm{obs}}} & = & \displaystyle \frac{{f}_{\mathrm{ej}}^{\mathrm{comp}}({M}_{\mathrm{lg}}+{M}_{\mathrm{sm}})}{{M}_{\mathrm{obs}}}={\left(\displaystyle \frac{{R}_{\mathrm{cut}}}{{R}_{\mathrm{obs}}^{\max }}\right)}^{4-\beta }\\ & & \times \left[\displaystyle \frac{{M}_{\mathrm{lg}}}{{M}_{\mathrm{sm}}}{\tilde{f}}_{2}\left(\displaystyle \frac{{R}_{\mathrm{lg}}}{{R}_{\min }^{o}}\right)+{\tilde{f}}_{3}\left(\displaystyle \frac{{R}_{\mathrm{cut}}}{{R}_{\min }^{o}}\right)\right],\end{eqnarray} \tag{ 37 }$$

where we used Equation (36) and ${f}_{\mathrm{ej}}^{\mathrm{comp}}$ given by Equation (15).

In Figure 7 we plot M_ej/M_obs for fixed ${R}_{\mathrm{obs}}^{\max }=300$ km, ${R}_{\min }^{o}=400$ km (corresponding to a₀ = 5 au, see Equation (9)) and different values of the upper cutoff R_cut, radius of the planetary-scale planetoids R_lg and the mass ratio between the populations of "planets" and minor objects M_lg/M_sm. One can see that the mass ejected in TDEs involving planetary systems with massive objects ("planets") can easily exceed the mass in metals probed by the WD spectroscopic observations by ≳10², provided that the total mass in these big bodies processed through TDEs is ≳10² times larger than the mass in small, asteroid-like objects. For reference, masses of the asteroid belt and terrestrial planets in the inner solar system are such that M_lg/M_sm ∼ 10³.

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This result is relatively insensitive to either the size of the big objects, or the upper cutoff of the spectrum of the asteroid-like objects. It is quite sensitive to both spatial and size distributions of the disrupted objects (indices α and β). However, even for the rather unfavorable case (α = 1.8, β = 3.4) shown in Figure 7, the ejected mass still exceeds the observable mass by ∼10 when the big, thousand-km objects dominate the mass of the planetary system (M_lg/M_sm ≳ 10²).

These considerations allow us to estimate the relative contribution (by mass) of the planetary TDEs to the observed ISA population as

$$\begin{eqnarray}\displaystyle \frac{{M}_{\mathrm{ej}}}{{\rm{\Delta }}{M}_{\mathrm{ISA}}} & = & \displaystyle \frac{{M}_{\mathrm{ej}}/{M}_{\mathrm{obs}}}{{\rm{\Delta }}{M}_{\mathrm{ISA}}/{\rm{\Delta }}{M}_{\mathrm{acc}}}\\ & \approx & 0.3\,{\psi }_{m}^{-1}{p}_{0.2}^{1.5}{n}_{15}\,\displaystyle \frac{{M}_{\mathrm{ej}}/{M}_{\mathrm{obs}}}{{10}^{3}},\end{eqnarray} \tag{ 38 }$$

where we identified M_obs with ΔM_acc and used estimates (34) and (35). The results of Section 3 suggest that ψ_m ∼ 1 for the population of ISAs produced in planetary TDEs, which is a critical ingredient. Then, Equation (38) implies that such events can potentially account for a significant share—up to ∼30%—of 'Oumuamua-like ISAs, provided that (1) 'Oumuamua's albedo is not much lower than 0.2 and (2) metal accretion by the WDs is dominated on average by big, planetary-scale (several 10³ km) planetoids, and that M_ej/M_obs ≳ 10³.

The latter implies a significant abundance of planetary-scale planetoids, which raises an issue of the observability of the consequences of their TDEs. To assess this question we compare the number of such planetoids N(R_lg) with the number of planetoids $N(\sim {R}_{\mathrm{obs}}^{\max })$ of radius comparable to the maximum "observed" size ${R}_{\mathrm{obs}}^{\max }$ in our composite model of the planetary system, see Section 2.3. Assuming that the mass spectrum of small planetoids (12) is dominated by the largest objects, i.e., β < 4, we can relate $N(\sim {R}_{\mathrm{obs}}^{\max })$ to the number of planetoids near the upper size cutoff $N(\sim {R}_{\mathrm{cut}})$ as $N(\sim {R}_{\mathrm{obs}}^{\max })/N(\sim {R}_{\mathrm{cut}})\approx {({R}_{\mathrm{obs}}^{\max }/{R}_{\mathrm{cut}})}^{1-\beta }$. Since also $N({R}_{\mathrm{lg}})/N(\sim {R}_{\mathrm{cut}})\,\approx ({M}_{\mathrm{lg}}/{M}_{\mathrm{sm}}){({R}_{\mathrm{cut}}/{R}_{\mathrm{lg}})}^{3}$, one can relate

$$\begin{eqnarray}&&\displaystyle \frac{N({R}_{\mathrm{lg}})}{N(\sim {R}_{\mathrm{obs}}^{\max })}\approx \displaystyle \frac{{M}_{\mathrm{lg}}}{{M}_{\mathrm{sm}}}{\left(\displaystyle \frac{{R}_{\mathrm{cut}}}{{R}_{\mathrm{lg}}}\right)}^{3}{\left(\displaystyle \frac{{R}_{\mathrm{obs}}^{\max }}{{R}_{\mathrm{cut}}}\right)}^{\beta -1}\\ &&\quad \approx \,0.4\,\displaystyle \frac{{M}_{\mathrm{lg}}/{M}_{\mathrm{sm}}}{{10}^{3}}{\left(\displaystyle \frac{5}{{R}_{\mathrm{lg}}/{R}_{\mathrm{cut}}}\right)}^{3}{\left(\displaystyle \frac{{10}^{3}{\rm{km}}}{{R}_{\mathrm{cut}}}\right)}^{2.5},\end{eqnarray} \tag{ 39 }$$

where we set β = 3.5 and ${R}_{\mathrm{obs}}^{\max }=300$ km.

This estimate demonstrates that the number of large planetoids can be comparable to the number of largest objects that are currently observed to be engulfed by the WDs (for M_lg/M_sm = 10³, which according to Figure 7 and Equation (38) is needed for planetary TDEs to contribute substantially to the population of ISAs). Given that we have seen ∼1 of the latter kind of objects, it is plausible that the expansion of the spectroscopically studied sample of the WDs by a large factor may result in a detection of a system that underwent a TDE involving an Earth-sized object relatively recently (although the proper interpretation of the nature of such an object may be an issue, given the high metal abundance in its atmosphere).

Spectroscopic observations of metal-polluted WDs suggest that, on average, they accrete ΔM_acc ∼ 0.003 M_⊕ of refractory material over their lifetime. However, this estimate is based on observations of systems that have accreted only relatively small, asteroid-like objects in the recent past. It completely misses the much larger amount of mass infrequently delivered by big, planetary-scale objects: such events are too rare to be observed at the moment given our limited sample of objects.

We correct for this observational bias by considering a simple model for the architecture of a typical planetary system feeding the WD accretion, and demonstrate that the estimate (35) should be boosted by a factor ${M}_{\mathrm{ej}}/{M}_{\mathrm{obs}}\sim 10\mbox{--}{10}^{3}$, provided that the mass accreted in the planetary objects (≳3000 km) exceeds that in asteroid-like bodies by a factor ≳10–10³.

Including this bias, our calculations show that planetary TDEs can potentially account for tens of per cent of 'Oumuamua-like ISAs, provided that (1) ${\psi }_{m}\sim 1$ (as suggested by the results of Section 3) and (2) the accretion of refractory elements by the WDs is dominated by tidal disruptions of large, planetary-scale objects (with M_lg/M_sm ∼ 10³, like in the solar system).

We now discuss other ideas proposed for the origin of 'Oumuamua-like ISAs. Many of these suggestions (Laughlin & Batygin 2017; Portegies Zwart et al. 2018; Trilling et al. 2017; Raymond et al. 2018) appeal to direct ejections of large amounts of mass from the forming planetary systems as a result of gravitational scattering by massive perturbers (giant planets).

There are two obvious issues with this suggestion. First, the non-volatile appearance of 'Oumuamua restricts its origin to within a few au from the star, inside the iceline. It is well known (e.g., Petrovich et al. 2014; Morrison & Malhotra 2015) that directly scattering planetoids into interstellar space from this range of separations is extremely difficult, even for very massive planets. A lot of planetary material can be ejected from the outer regions (Raymond et al. 2012), but it will be volatile-rich, unlike 'Oumuamua.

Second, direct ejections by giant planets unbind planetoids with equal efficiency regardless of their size. As a result, in this scenario the size spectrum of ISAs should directly reflect the size spectrum of the planetoids in the parent planetary system. This immediately brings back the large ψ_m issue discussed in Section 1: both the observed architecture of our solar system and the current simulations of planetesimal formation (Simon et al. 2016, 2017; Schäfer et al. 2017) finding top-heavy planetoid size distributions with β ≈ 2.8 suggest that ψ_m should be very large, ≳10², and maybe as high as 10⁴ (Raymond et al. 2018, see black ψ_m label in Figure 5 of our paper). The resultant estimate (1) of the ISA abundance would then run into conflict with the availability of refractory mass in forming planetary systems and require unrealistic efficiency of ejecting planetoids. This issue cannot be overcome by adopting a bottom-heavy size distribution of planetoids with most mass in ∼1 km objects, as collisional evolution would wipe out this population on a very short timescale while the WD progenitor is still on the main sequence (Wyatt et al. 2007), resulting in excessive infrared dust luminosities.

Regarding direct ejections and the large ψ_m issue, it is interesting to highlight the existing constraints on the abundance of volatile-rich interstellar comets. Jura (2011) used the upper limits on H abundances in the atmospheres of two He-dominated WDs to argue that the spatial density of O locked in interstellar comets (which could deliver water to these WDs) is below ∼0.5 M_⊕ pc⁻³, regardless of the comet's size. On the other hand, Engelhardt et al. (2017) set a constraint on the abundance of ∼1 km interstellar comets, which is 100 times more stringent than the limit on ISAs (as the former are much easier to detect).

These limits, combined with the constraint (1), make rather plausible the possibility that (1) the space number density of refractory objects is much higher than that of interstellar comets in the size range ∼(0.1–1) km, while (2) the total (integrated over the full size spectrum) space mass density of ISAs is lower than that of interstellar comets. This situation would be a natural consequence of the two different production channels for these populations: interstellar comets can be sourced by direct ejections from the outer regions of the planetary systems and have ψ_m ≫ 1, while ISAs are produced in planetary TDEs by the WDs with ψ_m ∼ 1, as envisaged in this work.

Ćuk (2018) suggested a scenario, in which refractory 'Oumuamua-like objects are first produced in tidal disruptions of planetoids by dense, compact stars (red dwarfs), and then propelled to interstellar space by their binary companions. While this scenario bears some resemblance to what we consider in this work, we find it rather unlikely. First, there is no natural reason for producing large fluxes of low-periastron objects (sourcing TDEs) in planetary systems around red dwarfs, which are stable over many billions of years. On the contrary, in the WD case the mass loss during the post-MS evolution of the progenitor naturally results in the dynamical destabilization of the planetary system (Debes & Sigurdsson 2002). Second, the necessity of having a companion star in a particular range of separations must significantly reduce the efficiency of this ISA production channel. And the direct ejections of fragments in the process of TDEs, i.e., without the assistance of the binary companions (as relied upon in our work) would be inefficient in the red dwarf case, since their icelines lie very close to the star.

Jackson et al. (2018) came up with a scenario, in which a central binary ejects circumbinary minor objects that migrate towards it through the protoplanetary disk. Given that circumbinary planets (and planetary systems) are expected to be rather common (Armstrong et al. 2014; Silsbee & Rafikov 2015), this scenario appears to be rather attractive.⁹ Unfortunately, like many other ideas, this one is affected by the aforementioned large ψ_m issue. Jackson et al. (2018) also find that in their model volatile and non-volatile objects are ejected in roughly equal numbers, which is in tension with the observational constraints of Engelhardt et al. (2017).

Speaking of dead stars, we could not overlook other ISA production scenarios involving stellar remnants, which do not rely on unbound fragment production in planetary TDEs.

Raymond et al. (2018) mention the possibility of exoplanetary Oort Cloud ejections during the post-MS evolution of the intermediate mass stars (Stone et al. 2013; Veras et al. 2014b). However, all the released objects will be volatile-rich and would not contribute to the population of 'Oumuamua-like objects.

Hansen & Zuckerman (2017) suggested another obvious mechanism for injecting refractory minor objects into the interstellar space: direct ejection of asteroid-like bodies orbiting WDs by massive perturbers without entering the Roche sphere of the WD. Indeed, the same massive planets that scatter planetoids into the low-periastron orbits resulting in TDEs should also be capable of scattering planetoids into hyperbolic orbits. Moreover, the efficiency of this process can easily be much higher than scattering into orbits resulting in TDEs, especially for massive (>1 M_J) perturbing planets orbiting on low-eccentricity orbits (Frewen & Hansen 2014). Even for low (Neptune) mass planets Frewen & Hansen (2014) and Mustill et al. (2018) find that direct ejections unbind at least as much mass as gets processed in TDEs.

It is clear that this channel of ISA production should indeed be operating as suggested by Hansen & Zuckerman (2017). However, as with other direct ejection scenarios discussed in Section 6.1, the large ψ_m problem will have a ruinous effect. Even though direct ejections of objects interior of the (radially expanded) iceline can unbind a large amount of refractory mass in objects of all sizes, only a tiny fraction of this mass will be in 'Oumuamua-like objects. Thus, processing even a small mass of planetoids through TDEs, which results in a population of fragments with ψ_m ∼ 1, should still be much more efficient at producing 'Oumuamua-like ISAs.

Supernova (SN) explosions represent another (evolved) candidate for ejecting large amounts of refractory material into interstellar space. The sudden loss of more than half of the initial mass during the SN explosion should unbind all orbiting planetary objects, including refractory asteroids.

Based on nucleosynthesis arguments, Arnett et al. (1989) estimate the number of SNe that took place in the Galaxy over its age to be ∼10⁹. Fukugita & Peebles (2004) provide a similar figure for the number of neutron stars and black holes in our Galaxy, based on IMF and stellar evolution considerations. This is about a factor of 10 less than the number of WDs believed to reside in the Galaxy. Combined with Equation (34), these estimates suggest that to explain the abundance of the ISAs implied by the 'Oumuamua's detection, each SN explosion should be releasing ≈100ψ_m M_⊕ in refractory objects (for 'Oumuamua's albedo of 0.2).

Taken at face value, this amount of ejected mass may not look problematic, as one might expect massive stars ending their lives as SNe to harbor more massive planetary systems than the less massive WD progenitors (although Reffert et al. 2015 found that planetary occurence drops for stars more massive than 3 M_⊙). Given that the solar system contains at least several M_⊕ in refractory elements (in the form of asteroids, terrestrial planets, and, partly, cores of giants), it may not be surprising that a 20 M_⊙ star could be capable of releasing ∼10² M_⊕ of refractory elements when it goes off as an SN.

However, once again, this scenario suffers from the large ψ_m issue, as the ejection of planetary objects during the SN is size-independent, making the ejected mass budget prohibitive. Moreover, it is not even clear if objects in the 'Oumuamua size range (0.1–1 km) survive the explosion. Indeed, ejection of a 10 M_⊙ envelope by the SN would blast a 1 km asteroid orbiting at 5 au with roughly its own mass of ejected gas moving at thousands of km s⁻¹. This can easily result in serious structural damage to such an unlucky asteroid (and could endow its remains with a velocity with respect to the LSR, which is too high).