Could Solar Radiation Pressure Explain 'Oumuamua's Peculiar Acceleration?

On 2017 October 19, the first interstellar object in the solar system, 'Oumuamua (1I/2017 U1), was discovered by the Panoramic Survey Telescope and Rapid Response System 1 (Pan-STARRS1) survey. It has a highly hyperbolic trajectory (with eccentricity e = 1.1956 ± 0.0006) and pre-entry velocity of ${v}_{\infty }\approx 26\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (Meech et al. 2017). Based on the survey properties and the single detection, Do et al. (2018) estimated the interstellar density of objects like 'Oumuamua or larger to be n ≈ 2 × 10¹⁵ pc⁻³, 2–8 orders of magnitude larger than expected by previous theoretical models (Moro-Martin et al. 2009). The large variations in its apparent magnitude, and the non-trivial periodicity of the lightcurve, suggest that 'Oumuamua is rotating in an excited spin state (tumbling motion) and has an extreme aspect ratio of at least 5:1 (Drahus et al. 2018; Fraser et al. 2018). This is an unprecedented value for previously known asteroids and comets in the solar system. Belton et al. (2018) have shown that if 'Oumuamua rotates in its highest rotational energy state, it should be extremely oblate (pancake-like).

Recently, Micheli et al. (2018) reported the detection of non-gravitational acceleration in the motion of 'Oumuamua, at a statistical significance of 30σ. Their best fit to the data is obtained for a model with a non-constant excess acceleration that scales with distance from the Sun, r, as Δa ∝ r⁻², but other power-law index values are also possible. They concluded that the observed acceleration is most likely the result of cometary activity. Yet, despite its close solar approach of r = 0.25 au, 'Oumuamua shows no signs of any cometary activity, no cometary tail, and no gas emission/absorption lines were observed (Fitzsimmons et al. 2017; Jewitt et al. 2017; Knight et al. 2017; Meech et al. 2017; Ye et al. 2017). From a theoretical point of view, Rafikov (2018) has shown that if outgassing was responsible for the acceleration (as originally proposed by Micheli et al. 2018), then the associated outgassing torques would have driven a rapid evolution in 'Oumuamua's spin, incompatible with observations.

If not cometary activity, what can drive the non-gravitational acceleration observed? In this Letter we explore the possibility of 'Oumuamua being a thin object accelerated by solar radiation pressure, which would naturally result in an excess acceleration Δa ∝ r⁻².¹ However, for radiation pressure to be effective, the mass-to-area ratio must be very small. In Section 2 we derive the required mass-to-area ratio (or effective thickness), and find (m/A) ≈ 0.1 g cm⁻², corresponding to an effective thin sheet of thickness w ≈ 0.3–0.9 mm. We explore the ability of such an unusually thin object to survive interstellar travel, considering collisions with interstellar dust and gas (Section 3), as well as to withstand the tensile stresses caused by rotation and tidal forces (Section 4). Finally, in Section 5 we discuss the possible implications of the unusual requirements on the shape of 'Oumuamua.

Micheli et al. (2018) had shown that 'Oumuamua experiences an excess radial acceleration, with their best-fit model

$$\begin{eqnarray}&&{\rm{\Delta }}a={a}_{0}{\left(\displaystyle \frac{r}{\mathrm{au}}\right)}^{n}\end{eqnarray} \tag{ 1 }$$

with n = −2

$$\begin{eqnarray}&&{a}_{0}=(4.92\pm 0.16)\times {10}^{-4}\ \mathrm{cm}\,{{\rm{s}}}^{-2}.\end{eqnarray} \tag{ 2 }$$

The value for a₀ is averaged over timescales much longer than 'Oumuamua's rotation period.

An acceleration of this form is naturally produced by radiation pressure

$$\begin{eqnarray}&&P={C}_{R}\displaystyle \frac{{L}_{\odot }}{4\pi {r}^{2}c},\end{eqnarray} \tag{ 3 }$$

where L_⊙ is the solar luminosity, c is the speed of light, and C_R is a coefficient of order unity that depends on the object's composition and geometry. For a sheet perpendicular to the Sun-object vector C_R = 1 + , where is the reflectivity. For a perfect reflector  = 1, and C_R = 2, whereas for a perfect absorber  = 0 and C_R = 1.

For an object of mass m and area A the acceleration would be

$$\begin{eqnarray}\begin{array}{rcl}a & = & \displaystyle \frac{{PA}}{m}=\left(\displaystyle \frac{{L}_{\odot }}{4\pi {r}^{2}c}\right)\left(\displaystyle \frac{A}{m}\right){C}_{R}\\ & = & 4.6\times {10}^{-5}{\left(\displaystyle \frac{r}{\mathrm{au}}\right)}^{-2}{\left(\displaystyle \frac{m/A}{{\rm{g}}{\mathrm{cm}}^{-2}}\right)}^{-1}{C}_{R}\ \mathrm{cm}\,{{\rm{s}}}^{-2}.\end{array}\end{eqnarray} \tag{ 4 }$$

Comparing Equations (1) and (4) we find that the requirement on the mass-to-area ratio is

$$\begin{eqnarray}&&\left(\displaystyle \frac{m}{A}\right)=(9.3\pm 0.3)\times {10}^{-2}\ {C}_{R}\ {\rm{g}}\,{\mathrm{cm}}^{-2}.\end{eqnarray} \tag{ 5 }$$

For a planar body with mass density ρ, this translates into a requirement on the body's thickness

$$\begin{eqnarray}&&w=\displaystyle \frac{m}{A\rho }=(9.3\pm 0.3)\times {10}^{-2}{\rho }_{0}^{-1}\ {C}_{R}\ \mathrm{cm},\end{eqnarray} \tag{ 6 }$$

where ρ₀ = ρ/(10⁰ g cm⁻³). Typically, ρ₀ ≈ 1–3, giving a thin sheet of 0.3–0.9 mm thickness. Other geometries are also possible, and are discussed in Section 5. The force exerted by the solar wind on a solid surface is negligible compared to that of the solar radiation field and is neglected hereafter.

The observed magnitude of 'Oumuamua constrains its area to be A ≈ 8 × 10⁶α⁻¹ cm², where α is the albedo (Jewitt et al. 2017). This corresponds to an effective radius ${R}_{\mathrm{eff}}\equiv \sqrt{A}/\pi \,=16{\alpha }^{-1/2}$ meters. For our estimation of the mass-to-area ratio, this area translates into a mass of m ≈ 740 (C_R/α) kg.

Next we explore the implications of impacts with interstellar dust grains and gas particles, in terms of momentum and energy transfer. We obtain general requirements for the object's mass-to-area ratio, or alternatively, for the maximum interstellar distance that can be traveled before it encounters appreciable slow-down or evaporation.

3.1. Momentum Transfer: Slow-down

An object with a cross sectional area A traveling a distance L through the interstellar medium (ISM) would accumulate an ISM gas mass of

$$\begin{eqnarray}&&{M}_{\mathrm{gas}}=A{{\rm{\Sigma }}}_{\mathrm{gas}}=1.4{m}_{p}\langle n\rangle {LA},\end{eqnarray} \tag{ 7 }$$

where Σ_gas is the accumulated mass column density of interstellar gas, m_p is the proton mass, $\langle n\rangle$ is the mean proton number density averaged along the object's trajectory, and the factor 1.4 accounts for the contribution of helium to the mass density of the ISM. For trajectories that span Galactic distances the contribution of the solar system to the accumulated column is negligible. The contribution of accumulated dust to the momentum transfer is also negligible, because the typical dust-to-gas mass ratio in the Galaxy is ≈1/100.

Once M_ISM approaches the object's mass, m, the momentum of the traveling object will decrease by a significant amount. The requirement M_ISM/m ≪ 1 translates into a maximum allowed value on the object's mass-to-area ratio, giving

$$\begin{eqnarray}\begin{array}{rcl}{\left(\displaystyle \frac{m}{A}\right)}_{\min ,{\rm{p}}} & = & {{\rm{\Sigma }}}_{\mathrm{gas}}=1.4{m}_{p}\langle n\rangle L\\ & = & 7.2\times {10}^{-3}\langle n{\rangle }_{0}{L}_{0}\ {\rm{g}}\,{\mathrm{cm}}^{-2}.\end{array}\end{eqnarray} \tag{ 8 }$$

In the last equality we normalized to the typical values $\langle n{\rangle }_{0}\,=\langle n\rangle /({10}^{0}\ {\mathrm{cm}}^{-3})$ and L₀ = L/(10⁰ kpc).

Given a mass-to-area ratio, the maximum travel distance is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\max ,{\rm{p}}} & = & \displaystyle \frac{1}{1.4{m}_{p}\langle n\rangle }\left(\displaystyle \frac{m}{A}\right)\\ & = & 14\langle n{\rangle }_{0}^{-1}{\left(\displaystyle \frac{m}{A}\right)}_{-1}\ \mathrm{kpc}.\end{array}\end{eqnarray} \tag{ 9 }$$

In the second equation we denoted (m/A)₋₁ ≡ (m/A)/(10⁻¹ g cm⁻²). Figure 1 shows the results from Equation (9) as a function of (m/A). The dashed vertical line indicates our constraint on (m/A) for 'Oumuamua (Equation (5)). Evidently, 'Oumuamua can travel Galactic distances before encountering appreciable slow-down.

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3.2. Energy Transfer: Collisions with Dust Grains

Collisions with dust grains at high velocities will induce crater formation by melting and evaporation of the target material. Because the typical time between dust collisions is long compared to the solidification time, any molten material will solidify before the next collision occurs, and thus will only cause a deformation of the object's surface material, not a reduction in mass. On the other hand, atoms vaporized through collisions can escape and thus cause a mass ablation.

We would like to estimate the minimum mass-to-area ratio required for the object to not lose significant fraction of its mass upon dust-grain collisions. Let m_d be the colliding dust-grain mass, and ϕ the fraction of the kinetic energy that is converted into vaporization of the object's body. The total number of vaporized atoms per collision is then

$$\begin{eqnarray}&&{N}_{v}=\displaystyle \frac{{m}_{v}}{\bar{m}}=\phi \displaystyle \frac{{m}_{d}{v}^{2}}{2{U}_{v}},\end{eqnarray} \tag{ 10 }$$

where m_v is the mass of vaporized material in the object, $\bar{m}$ is the mean atomic mass of the object, and U_v is the vaporization energy. Although highly simplistic, this analysis captures the results of the detailed theoretical model of Tielens et al. (1994) and the empirical data from Okeefe & Ahrens (1977). A good match to the numerical results is obtained for ϕ = 0.2.

Over a distance L there will be many collisions. Adopting the conservative assumption that for each collision all the vaporized material escapes to the ISM, we can account for all of the collisions along a path-length, L, by replacing m_d in Equation (10) with the total accumulated dust mass

$$\begin{eqnarray}&&{M}_{d}={{\rm{\Sigma }}}_{\mathrm{dust}}A={{\rm{\Sigma }}}_{\mathrm{gas}}{\varphi }_{{dg}}A=1.4{m}_{p}\langle n\rangle L{\varphi }_{{dg}}A,\end{eqnarray} \tag{ 11 }$$

where Σ_dust is the dust column density and ϕ_dg is the dust-to-gas mass ratio. This gives

$$\begin{eqnarray}&&\displaystyle \frac{{m}_{v}}{\bar{m}}=\phi \displaystyle \frac{1.4{m}_{p}\langle n\rangle L{\varphi }_{{dg}}{{Av}}^{2}}{2{U}_{v}}.\end{eqnarray} \tag{ 12 }$$

Requiring that the total vaporized mass not exceed half of the object's mass, we obtain a constraint on the minimum mass-to-area ratio of the object

$$\begin{eqnarray}\begin{array}{rcl}{\left(\displaystyle \frac{m}{A}\right)}_{\min ,{\rm{m}}} & = & 1.4{m}_{p}\langle n\rangle L\left(\displaystyle \frac{\phi {\varphi }_{{dg}}\bar{m}{v}^{2}}{{U}_{v}}\right)\\ & = & 3.1\times {10}^{-4}\langle n{\rangle }_{0}{L}_{0}\left(\displaystyle \frac{{\varphi }_{-2}{\bar{m}}_{12}{v}_{26}^{2}}{{U}_{4}}\right)\ {\rm{g}}\,{\mathrm{cm}}^{-2}.\end{array}\end{eqnarray} \tag{ 13 }$$

Here we defined the normalized parameters, φ₋₂ = φ_dg/10⁻², ${\bar{m}}_{12}=\bar{m}/(12{m}_{p})$, as appropriate for carbon-based materials (e.g., graphite or diamond); U₄ = U/(4 eV), as appropriate for typical vaporization energies (e.g., for graphite, U_v = 4.2 eV); and v₂₆ = v/(26 km s⁻¹), the velocity at infinity of 'Oumuamua. For a given mass-to-area ratio, the maximum allowed distance before significant evaporation is

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\max ,{\rm{d}}} & = & \displaystyle \frac{1}{1.4{m}_{p}\langle n\rangle }\left(\displaystyle \frac{{U}_{m}}{\phi {\varphi }_{{dg}}\bar{m}{v}^{2}}\right)\left(\displaystyle \frac{m}{A}\right)\\ & = & 330\langle n{\rangle }_{0}^{-1}\left(\displaystyle \frac{{U}_{4}}{{\varphi }_{-2}{\bar{m}}_{12}{v}_{26}^{2}}\right){\left(\displaystyle \frac{m}{A}\right)}_{-1}\ \mathrm{kpc}.\end{array}\end{eqnarray} \tag{ 14 }$$

For our constrained value for the mass-to-area ratio, 'Oumuamua can travel through the entire galaxy before a significant fraction of its mass is evaporated. Evaporation becomes important at higher speeds. Comparing Equations (8) and (13) we find that only for speeds above

$$\begin{eqnarray}\begin{array}{rcl}{v}_{\mathrm{crit}} & = & \sqrt{\displaystyle \frac{2{U}_{m}}{\bar{m}\varphi \phi }}\\ & = & 130{\left(\displaystyle \frac{{U}_{4}}{{\varphi }_{-2}{\bar{m}}_{12}}\right)}^{1/2}\ \mathrm{km}\,{{\rm{s}}}^{-1},\end{array}\end{eqnarray} \tag{ 15 }$$

vaporization dominates over slow-down.

3.3. Energy Transfer: Collisions with Gas Particles

When an object travels at a high speed, collisions with atoms in the ISM can potentially transfer sufficient energy to produce sputtering. This process was studied in the context of dust grains in hot shocks (Tielens et al. 1994). For an object traveling at a velocity v, over a distance L through the ISM, the total number of sputtered particles is

$$\begin{eqnarray}&&{N}_{s}=\displaystyle \frac{{m}_{s}}{\bar{m}}=2{AL}\langle n\rangle {Y}_{\mathrm{tot}},\end{eqnarray} \tag{ 16 }$$

where ${Y}_{\mathrm{tot}}={\sum }_{i}{Y}_{i}{x}_{i}$ is the total sputtering yield (which depends on the kinetic energy), summed over collisions with different species (i.e., H, He, and metals), and x_i is the abundance of the colliding species relative to hydrogen.

The minimal mass-to-area ratio below which half of the object's mass will be sputtered is

$$\begin{eqnarray}\begin{array}{rcl}{\left(\displaystyle \frac{m}{A}\right)}_{\min ,{\rm{s}}} & = & \bar{m}\langle n\rangle {{LY}}_{\mathrm{tot}}\\ & = & 6.2\times {10}^{-5}{\bar{m}}_{12}\langle n{\rangle }_{0}{L}_{0}{Y}_{-3}\ {\rm{g}}\,{\mathrm{cm}}^{-2}.\end{array}\end{eqnarray} \tag{ 17 }$$

In the second equality we normalized to Y_tot = 10⁻³, corresponding to kinetic energies E ≈ 30–100 eV, corresponding to v ≈ 40–70 km s⁻¹. For lower speeds, as that of 'Oumuamua, the yield is even lower, further decreasing the value of (m/A)_min,s. At higher speeds, the yield increases but typically remains below 0.01 (Tielens et al. 1994); thus, at any velocity vaporization and slow-down remain the dominating processes limiting the allowed distance that an object can travel through the ISM.

Cosmic-rays are expected to cause even less damage. Although their energy density is comparable to that of the ISM gas, they deposit only a very small fraction of their energy as they penetrate through the thin object.

A thin object can be torn apart by centrifugal forces or tidal forces if its tensile strength is not sufficiently strong. Typical values for the tensile strengths of various materials are shown in Table 1. Next, we calculate whether centrifugal or tidal forces can destroy 'Oumuamua.

4.1. Rotation

'Oumuamua's lightcurve shows periodic modulations on an order of 6–8 hr. Ignoring the tumbling motion, let us estimate the tensile stress originating from the centrifugal force. The largest stress is produced if the object is elongated such that the longest dimension is perpendicular to the rotation axis. We denote this dimension as d. Considering the object as being made of two halves, each located with a center of mass at a distance d/4 from the rotation axis, and ignoring self-gravity, a radial force of magnitude

$$\begin{eqnarray}&&F=\displaystyle \frac{1}{2}m{{\rm{\Omega }}}^{2}\displaystyle \frac{d}{4},\end{eqnarray} \tag{ 18 }$$

will be exerted on each half. The associated tensile stress is

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{rot}} & = & \displaystyle \frac{1}{4}\rho {d}^{2}{{\rm{\Omega }}}^{2}\\ & = & 0.25\ {\rho }_{0}\ {d}_{4}^{2}\ {{\rm{\Omega }}}_{-4}^{2}\ \mathrm{dyne}\,{\mathrm{cm}}^{-2},\end{array}\end{eqnarray} \tag{ 19 }$$

where d₄ ≡ d/(10⁴ cm), Ω₋₄ ≡ Ω/(10⁻⁴ s⁻¹). This is much smaller than typical tensile strengths of normal materials, and even of that of the comet 67P/Churyumov–Gerasimenko (see Table 1). Thus, even when self-gravity is ignored, 'Oumuamua can easily withstand its centrifugal force.

4.2. Tidal Forces

The tidal force will be maximal if the long dimension of the object is parallel to the Sun-object vector. Again, modeling the object as consisting of two halves as in Section 4.1, the difference in the gravitational force experienced by the far and near ends of the object is

$$\begin{eqnarray}&&\delta F\approx \displaystyle \frac{1}{4}d\displaystyle \frac{{{GM}}_{\odot }m}{{r}^{3}},\end{eqnarray} \tag{ 20 }$$

where r is the distance of the center of mass from the Sun. The associated tensile stress,

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\mathrm{tid}} & \approx & \displaystyle \frac{1}{4}\rho {d}^{2}\displaystyle \frac{{{GM}}_{\odot }}{{r}^{3}}\\ & = & 9.9\times {10}^{-7}\ {\rho }_{0}\ {d}_{4}^{2}{\left(\displaystyle \frac{r}{\mathrm{au}}\right)}^{-3}\ \mathrm{dyne}\,{\mathrm{cm}}^{-2}.\end{array}\end{eqnarray} \tag{ 21 }$$

Even at perihelion (r = 0.25 au), the tensile stress is negligible.

The critical distance below which tidal forces dominate over centrifugal is

$$\begin{eqnarray}&&{R}_{\mathrm{tid}}={\left(\displaystyle \frac{{GM}}{{{\rm{\Omega }}}^{2}}\right)}^{1/3}=3.4{{\rm{\Omega }}}_{-4}^{-2/3}{\left(\displaystyle \frac{M}{{M}_{\odot }}\right)}^{1/3}\ {R}_{\odot }.\end{eqnarray} \tag{ 22 }$$

Thus, unless 'Oumuamua encountered an extremely close approach to a star in its past, it is unlikely that tidal forces played any significant role.

We have shown that the observed non-gravitational acceleration of 'Oumuamua may be explained by solar radiation pressure. This requires a small mass-to-area ratio for 'Oumuamua of (m/A) ≈ 0.1 g cm⁻². For a planar geometry and typical mass densities of 1–3 g cm⁻² this gives an effective thickness of only 0.9–0.3 mm, respectively. For a material with lower mass density, the inferred effective thickness is proportionally larger. We find that although very thin, such an object can travel over galactic distances, maintaining its momentum and withstanding collisional destruction by dust grains and gas, as well as centrifugal and tidal forces. For 'Oumuamua, the limiting factor is the slow-down by accumulated ISM mass, which limits its maximal travel distance to ∼10 kpc (for a mean ISM particle density of ∼1 cm⁻³). Our inferred thin geometry is consistent with studies of its tumbling motion. In particular, Belton et al. (2018) inferred that 'Oumuamua is likely to be an extremely oblate spheroid (pancake), assuming that it is excited by external torques to its highest energy state.

While our scenario may naturally explains the peculiar acceleration of 'Oumuamua, it opens up the question: what kind of object might have such a small mass-to-area ratio? The observations are not sufficiently sensitive to provide a resolved image of 'Oumuamua, and one can only speculate on its possible geometry and nature. Although periodic variations in the apparent magnitude are observed, there are still too many degrees of freedom (e.g., observing angle, non-uniform reflectivity, etc.) to definitely constrain the geometry. The geometry should not necessarily be that of a planar sheet, but may acquire other shapes; for example, involving a curved sheet, a hollow cone or ellipsoidal, etc. Depending on the geometry our estimated value for the mass-to-area ratio will change (through C_R in Equation (5)), but the correction is typically of order unity.

Known solar system objects like asteroids and comets have mass-to-area ratios orders of magnitude larger than our estimate for 'Oumuamua. If radiation pressure is the accelerating force, then 'Oumuamua represents a new class of thin interstellar material, either produced naturally, through a yet unknown process in the ISM or in protoplanetary disks, or is of an artificial origin.

Considering an artificial origin, one possibility is that 'Oumuamua is a lightsail, floating in interstellar space as debris from advanced technological equipment (Loeb 2018). Lightsails with similar dimensions have been designed and constructed by our own civilization, including the IKAROS project and the Starshot Initiative.² The lightsail technology might be abundantly used for transportation of cargo between planets (Guillochon & Loeb 2015) or between stars (Lingam & Loeb 2017). In the former case, dynamical ejection from a planetary System could result in space debris of equipment that is not operational any more³ (Loeb 2018), and is floating at the characteristic speed of stars relative to each other in the solar neighborhood. This would account for the various anomalies of 'Oumuamua, such as the unusual geometry inferred from its lightcurve (Meech et al. 2017; Belton et al. 2018; Drahus et al. 2018; Fraser et al. 2018), its low thermal emission, suggesting high reflectivity (Trilling et al. 2018), and its deviation from a Keplerian orbit (Micheli et al. 2018) without any sign of a cometary tail (Fitzsimmons et al. 2017; Jewitt et al. 2017; Knight et al. 2017; Meech et al. 2017; Ye et al. 2017) or spin-up torques (Rafikov 2018). Although 'Oumuamua has a red surface color, similar to organic-rich surfaces of solar system comets and D-type asteroids (Meech et al. 2017), this does not contradict the artificial scenario; irrespective of the object's composition, as it travels through the ISM its surface will be covered by a layer of interstellar dust, which is itself composed of organic-rich materials (Draine 2003).

Alternatively, a more exotic scenario is that 'Oumuamua may be a fully operational probe sent intentionally to Earth vicinity by an alien civilization. Based on the PAN-STARRS1 survey characteristics, and assuming natural origins following random trajectories, Do et al. (2018) derived that the interstellar number density of 'Oumuamua-like objects should be extremely high, ∼2 × 10¹⁵ pc⁻³, equivalent to ∼10¹⁵ ejected planetisimals per star, and a factor of 100 to 10⁸ larger than predicted by theoretical models (Moro-Martin et al. 2009). This discrepancy is readily solved if 'Oumuamua does not follow a random trajectory but is rather a targeted probe. Interestingly, 'Oumuamua's entry velocity is found to be extremely close to the velocity of the local standard of rest, in a kinematic region that is occupied by less than 1 to 500 stars (Mamajek 2017).

It is too late to image 'Oumuamua with existing telescopes or chase it with chemical rockets (Hein et al. 2017; Seligman & Laughlin 2018), so its likely origin and mechanical properties could only be deciphered by searching for other objects of its type in the future. In addition to the vast unbound population, thousands of interstellar 'Oumuamua-like space debris are expected to be trapped at any given time in the solar system through gravitational interaction with Jupiter and the Sun (Lingam & Loeb 2018). Deep wide-area surveys of the type expected with the Large Synoptic Survey Telescope⁴ will be particularly powerful in searching for additional members of 'Oumuamua's population of objects.

A survey for lightsails as technosignatures in the solar system is warranted, irrespective of whether 'Oumuamua is one of them.

We thank Manasvi Lingam, Paul Duffell, Quanzhi Ye, and an anonymous referee for helpful comments. This work was supported in part by a grant from the Breakthrough Prize Foundation.