Why is interstellar object 1I/2017 U1 (‘Oumuamua) rocky, tumbling and possibly very prolate?

Why is interstellar object 1I/2017 U1 (‘Oumuamua) rocky, tumbling and possibly very prolate? Abstract The recently discovered first interstellar object 1I/2017 U1 (‘Oumuamua) has brightness that varies by a factor of 10, a range greater than that of any Solar system asteroid, a spectrum characteristic of type D asteroids, and no evidence of evaporating volatiles, contrary to expectation for exo-Oort clouds. ‘Oumuamua is possibly the first example of the proposed ‘Jurads’, objects depleted in volatiles and ejected from planetary systems during the post-main sequence evolution of their parent stars. I suggest that heating by the star’s giant stage fluidized a precursor object as well as driving off any volatiles, causing it to assume the Jacobi ellipsoidal shape of a self-gravitating incompressible liquid. The collision that produced the inferred tumbling motion may have occurred thousands of years after the formation of 1I/2017 U1 ‘Oumuamua. Jacobi ellipsoids have a unique relation among rotation rate, density and axial ratio. The inferred axial ratio ⪆5 suggests a lower bound on the density of 1.6 g cm−3, apparently excluding an icy interior unless it is almost entirely frozen CO2. ‘Oumuamua may be related to accreting objects that pollute white dwarf atmospheres and that may make Soft Gamma Repeaters. stars: neutron, white dwarfs, minor planets, asteroids: individual: 1I/2017 U1 ('Oumuamua) 1 INTRODUCTION The recently discovered interstellar object (ISO) 1I/2017 U1 (‘Oumuamua) is remarkable for its hyperbolic solar orbit (Mamajek 2017; Meech et al. 2017), its 10-fold brightness modulation, greater than that of any Solar system asteroid (Drahus et al. 2018; Fraser et al. 2018; Meech et al. 2017), and the absence (with tight upper bounds) of evaporating volatiles. The light curve is not strictly periodic but rather indicates tumbling (Drahus et al. 2018; Fraser et al. 2018). Although ISOs expelled from exoplanetary systems have been predicted for many years (Hansen & Zuckerman 2017), they have been predicted to be overwhelmingly cometary, evaporating volatiles and emitting dust during close approaches to the Sun like that of 1I/2017 U1 ‘Oumuamua that approached within 0.25 au (Meech et al. 2016). Thus, while the discovery of an ISO was not unexpected, its properties require explanation. The brightness modulation of 1I/2017 U1 ‘Oumuamua suggests an extraordinary axial ratio and indicates a novel origin. The very slight (≲100 erg cm−3) strength required to support the less prolate non-equilibrium shapes of rotating Solar system bodies of similar size indicates that this ISO had an extraordinary history during which its strength was reduced to essentially zero. Its unexpected absence of volatiles is also remarkable, and these two facts point towards a hypothesis of its origin and history. Here, I discuss the possible origin of 1I/2017 U1 ‘Oumuamua in the post-main sequence stage of a planetary system (Veras 2016; Hansen & Zuckerman 2017; Raymond et al. 2018). The increasing luminosity of a post-main sequence star will evaporate volatiles from its satellites. Even those in Oort cloud like orbits will pass close enough to the star to be heated if they are perturbed on to nearly parabolic orbits, which is necessary for them to be expelled by gravitational interaction with massive planets or to collide with other small objects. The rapid mass-loss of white dwarf formation or the essentially instantaneous mass-loss of neutron star formation in a core-collapse supernova will unbind many of the small bodies of the exoplanetary system. It will also put massive exoplanets on elliptic orbits in which they will interact gravitationally with each other and with the small bodies, expelling some and putting others on the low angular momentum orbits required to explain the ‘pollution’ of white dwarf atmospheres (Jura & Young 2014; Farihi 2016; Mustill et al. 2018; Veras, Shannon & Gänsicke 2017; Xu et al. 2017) by heavier elements. Additional mechanisms of expulsion of small bodies operate in binary systems (Stephan, Naoz & Zuckerman 2017). One consequence of heating is the loss of volatile ices, possibly explaining the absence of volatiles in 1I/2017 U1 ‘Oumuamua. Evaporating volatiles fluidize beds of refractory particles. There is evidence for this (at the lower levels of heating occurring in Solar system comets) in images obtained by the Rosetta spacecraft of Comet 67P/Churyumov–Gerasimenko(European Space Agency 2017). Regions near the centre of the cometary nucleus, near the minimum of its gravitational potential, are covered by fine material, in contrast to the rough craggy appearance of the rest of the body. This is plausibly the result of flow of fluidized particulates to the minimum of the gravitational potential. A body heated by the intense radiation of a red giant star may be entirely fluidized and the residual refractory material assume the shape, a Maclaurin spheroid or, if it has more angular momentum, a Jacobi ellipsoid, of a rotating self-gravitating incompressible liquid. The purpose of this paper is to investigate the consequences of this explanation of the unprecedentedly prolate shape of 1I/2017 U1 ‘Oumuamua. This hypothesis implies a lower density limit that likely excludes the possibility (Fitzsimmons et al. 2018) of an icy interior protected from solar heating by an insulating mantle. Finally, the inference that exoplanetary systems may be disrupted by the evolution of their stars and that white dwarf pollution may be caused by single bodies as massive as ∼1022 g (Xu et al. 2017), or possibly even more massive (Jura et al. 2009), is consistent with the explanation of Soft Gamma Repeater outbursts as the result of accretion of such minor-planet-sized bodies on to neutron stars (Katz, Toole & Unruh 1994). 2 GEOMETRY Jacobi ellipsoids are the equilibrium state of uniformly rotating strengthless homogeneous incompressible fluids with too much angular momentum to be (oblate) Maclaurin spheroids. Even were their specific angular momentum to be increased, strengthless Jacobi ellipsoids will not break up (spinning mass off their extremities) provided they have time to relax to their equilibrium shape. Extremely elongated Jacobi ellipsoids may be unstable to ‘pear-shaped’ (actually, more like egg-shaped) ℓ = 3, ‘dumb-bell’ ℓ = 4 and higher instabilities (Eriguchi, Hachisu & Sugimoto 1982; Hachisu & Eriguchi 1982) but the non-linear development of these instabilities is not understood, so here I assume Jacobi ellipsoids. The 10-fold modulation of the brightness of 1I/2017 U1 ‘Oumuamua would imply, in a naïve model that assumes a prolate spheroid, Lambert’s Law reflectivity with uniform albedo and oppositional geometry (the actual solar angle during photometric observations was about 20°), an axial ratio of 10:1 if viewed in the plane of rotation, and greater for other viewing angles (Meech et al. 2017). However, Lambert’s law is not valid for Solar system asteroids. Modelling that assumes 1I/2017 U1 ‘Oumuamua is described by the scattering properties of known Solar system asteroids (Drahus et al. 2018; Fraser et al. 2018) sets a lower bound on the axial ratio of about 5:1. No upper bound on the axial ratio can be obtained because the angle between the line of sight and the rotation axis is unknown, but a lower bound of about 65° can be set on the angle if arbitrarily large axial ratios are considered (Drahus et al. 2018). Assuming a plausible upper bound on the density of 3 g cm−3 leads to an upper bound on the axial ratio of about 8 ( Section 3), from which a somewhat more stringent lower bound on the angle may be found. Uniformly rotating (very quickly enforced by friction) self-gravitating incompressible fluids of uniform density are oblate Maclaurin spheriods (with no modulation of their scattered light) if their angular momentum is low, or triaxial Jacobi ellipsoids if they have higher angular momentum (Chandrasekhar 1969; Tassoul 1978). For large specific angular momenta, the two smaller semi-axes b and c of the Jacobi ellipsoids converge, as shown in Fig. 1, and the ellipsoid approaches a prolate spheroid. Figure 1. View largeDownload slide Semiprincipal axes a, b and c of Jacobi ellipsoids as a function of the normalized angular momentum $${\cal L} \equiv L/\sqrt{GM^3r_0}$$ where r0 ≡ (abc)1/3 (Chandrasekhar 1969). At higher angular momentum $${\cal L} \gtrsim 0.5$$b → c. For lower angular momenta ($${\cal L} \lt 0.3$$, outside the range of the figure) a = b > c and the body is an oblate Maclaurin spheroid. $$a/\sqrt{bc}$$ is the axial ratio used in Fig. 2. Figure 1. View largeDownload slide Semiprincipal axes a, b and c of Jacobi ellipsoids as a function of the normalized angular momentum $${\cal L} \equiv L/\sqrt{GM^3r_0}$$ where r0 ≡ (abc)1/3 (Chandrasekhar 1969). At higher angular momentum $${\cal L} \gtrsim 0.5$$b → c. For lower angular momenta ($${\cal L} \lt 0.3$$, outside the range of the figure) a = b > c and the body is an oblate Maclaurin spheroid. $$a/\sqrt{bc}$$ is the axial ratio used in Fig. 2. 3 DENSITY Jacobi ellipsoids have a unique relation among their semi-axes, density and rotation rate that can be used to establish bounds on their density from the measured rotation rate and the axial ratio inferred from their light curve. Alternatively, assumed bounds on their density can be used to constrain their geometry. Several slightly different values of the rotation rate have been reported (Drahus et al. 2018; Meech et al. 2017; Bolin et al. 2018; Fraser et al. 2018). Adopting a period of 7.55 h (Drahus et al. 2018), roughly the mean of these values, leads to the relation between axial ratio and density shown in Fig. 2. The lower bound on the axial ratio of about 5 fitted (Drahus et al. 2018; Fraser et al. 2018) to the light curve implies a minimum density of 1.6 g cm−3. This is inconsistent with all plausible icy materials except CO2, whose density is close to this value; even the density of CO2 is inconsistent unless we happen to lie almost exactly in the plane of rotation, the axial ratio is less than inferred from the light curve or the rotation period is significantly longer than the adopted value. This argues against the hypothesis (Fitzsimmons et al. 2018) that an icy core is protected against heating and evaporation by an insulating non-volatile mantle. Figure 2. View largeDownload slide Relation between density and axial ratio (defined as $$a/\sqrt{bc}$$) of a Jacobi ellipsoid with rotation period 7.55 h (Chandrasekhar 1969). The empirical lower bound of 5 on the axial ratio implies a lower bound of 1.6 g cm−3 on the density. Figure 2. View largeDownload slide Relation between density and axial ratio (defined as $$a/\sqrt{bc}$$) of a Jacobi ellipsoid with rotation period 7.55 h (Chandrasekhar 1969). The empirical lower bound of 5 on the axial ratio implies a lower bound of 1.6 g cm−3 on the density. 4 TUMBLING: THE EVOLUTIONARY HISTORY The tumbling motion of 1I/2017 U1 ‘Oumuamua (Drahus et al. 2018; Fraser et al. 2018) implies that it is not rotating only around its axis of greatest moment of inertia, but is a state of higher rotational energy. It requires at least minimal elastic strength to maintain. Tumbling must have been the result of collision after the epoch of fluidization because any viscous, plastic or frictional flow would lead to very rapid relaxation to periodic rotation about the axis of greatest moment of inertia (the lowest energy state). The required strength is difficult to estimate (we have only the crudest hint of the rotational dynamics), but is less than the extremely small central pressure p. For high axial ratios a ≫ b ≈ c (Fig. 1) and the ellipsoid can be approximated as an infinite cylinder of radius b: \begin{equation*} p \approx \pi G \rho ^2 b^2 \sim 20\ \rm{erg\,\,cm}^{-3} \end{equation*} (1) for a density ρ ∼ 3 g cm−3 and radius b ∼ 30 m. Tumbling implies that the body is not exactly a prolate spheroid, but is at least slightly triaxial; prolate spheroids may ‘cone’ around their long axes, but do not tumble, a process that produces aperiodic light curves (Sher 1971). For a mean axial ratio $$a/\sqrt{bc} \ge 5$$, as indicated by the depth of the light curve (Drahus et al. 2018; Fraser et al. 2018), b and c differ by $$\le 10\hbox{ per cent}$$. The collision that made 1I/2017 U1 ‘Oumuamua tumble may have made it triaxial by asymmetrically removing material. Fitzsimmons et al. (2018) indicate a thermal diffusivity D ∼ 10−4 cm2s−1. This implies a cooling time for an object with b ≈ c ∼ 30 m \begin{equation*} t_{\mathrm{ conduction}} \sim {b^2 \over D} \sim 10^{11}\ \text{s}. \end{equation*} (2) In order for a collision to have produced tumbling, it must have occurred after 1I/2017 U1 ‘Oumuamua was no longer fluidized, either because the volatiles were exhausted or because it cooled below volatilization temperatures. The red giant stages of stellar evolution are longer than tconduction so a body of these dimensions would have heated throughout. If 1I/2017 U1 ‘Oumuamua was completely fluidized and did not recover mechanical strength while still hot but depleted of volatiles, then at least thousands of years (and possibly much longer) must have elapsed from the end of the red giant stage of the parent star and its intense heating to the collision that made it tumble; 1I/2017 U1 ‘Oumuamua must have remained in the fossil planetary system surrounding the former red giant, where the density of solid bodies was high enough to make collision likely, for at least that long before being expelled. The dynamical processes that expel small bodies also put other small bodies on collision orbits with their parent stars. This is consistent both with the inference that polluted white dwarfs accrete solid bodies for ∼109 yr and with the active lifetimes of Soft Gamma Repeaters of a few thousand years after the supernovæ in which their neutron stars were born. 5 DISCUSSION The argument of this paper can be summarized as follows: The fact that 1I/2017 U1 ‘Oumuamua is apparently so prolate (more than any Solar system asteroid) implies that it was once strengthless (unlike Solar system asteroids, which were never fluidized). A strengthless period in its past may possibly have been the result of evaporative fluidization of even a small volatile fraction by the luminous post-main sequence stage of its parent star. Complete loss of volatiles in the distant past explains the absence of any coma or other evidence of evaporation after its close passage by the Sun. Its equilibrium Jacobi ellipsoidal figure when fluidized was frozen in after the volatiles were lost because there was no force to disrupt it. The relation among density, axial ratio and rotation rate of Jacobi ellipsoids permits a lower bound on density to be inferred. Present tumbling requires a collision after solidification, although the strength required to support tumbling (equation 1) is small. A significant probability of collision implies that the body remained in its parent exoplanetary system for at least its cooling time (equation 2). A number of mechanisms have been proposed for its subsequent expulsion, but expulsion was not an immediate dynamical consequence of the end, gradual (formation of a planetary nebula) or instantaneous (supernova), of the parent star’s red giant stage; they are not the subject of this paper. The data contain rich implications that were not expected when ISOs were first hypothesized: Some of the light curves of 1I/2017 U1 ‘Oumuamua (Meech et al. 2017) are asymmetric about their minima, rising more slowly than they fell. This may be explained as a shadowing effect if the rotation is prograde with respect to its orbital motion about the Sun (an accident with 50 per cent probability of occurrence for an ISO). The extraordinary and unanticipated axial ratio of 1I/2017 U1 ‘Oumuamua, combined with the absence of volatiles, indicates its origin in the exoplanetary system of a star that had passed through the red giant stage. The combination of the rotation period with the inferred large axial ratio sets a lower bound on the density of 1I/2017 U1 ‘Oumuamua that excludes an icy composition, even one under a non-volatile crust that might be consistent with the observed absence of a coma. The inference that 1I/2017 U1 ‘Oumuamua is tumbling sets further constraints on its history: it likely remained in a region of comparatively high density of solid objects for at least thousands of years after it formed. The discovery of an interstellar interloper confirms expectations from the study of ‘polluted’ white dwarfs that post-main sequence evolution drastically changes the orbits of planets and planetesimals, putting some on nearly radial orbits penetrating within the Roche radius of the star and unbinding others, and suggests that analogous processes may produce Soft Gamma Repeater outbursts. The absence of volatiles and extreme axial ratios will be correlated in ISOs because they have the common cause of intense heating and fluidization. However, not all ISOs need these properties. Tumbling will be common because expulsion is more likely from densely populated exoplanetary systems and because post-main sequence stellar evolution disrupts them. ACKNOWLEDGEMENTS I thank B. Hansen and B. Zuckerman for useful discussions. REFERENCES Bolin B. T. et al. , 2018 , ApJ , 852 , L2 CrossRef Search ADS Chandrasekhar S. , 1969 , Ellipsoidal Figures of Equilibrium . Yale Univ. 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D. , Klein B. , Jura M. , 2017 , ApJ , 836 , L7 https://doi.org/10.3847/1538-4357/836/1/7 CrossRef Search ADS © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society: Letters Oxford University Press

Why is interstellar object 1I/2017 U1 (‘Oumuamua) rocky, tumbling and possibly very prolate?

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Abstract

Abstract The recently discovered first interstellar object 1I/2017 U1 (‘Oumuamua) has brightness that varies by a factor of 10, a range greater than that of any Solar system asteroid, a spectrum characteristic of type D asteroids, and no evidence of evaporating volatiles, contrary to expectation for exo-Oort clouds. ‘Oumuamua is possibly the first example of the proposed ‘Jurads’, objects depleted in volatiles and ejected from planetary systems during the post-main sequence evolution of their parent stars. I suggest that heating by the star’s giant stage fluidized a precursor object as well as driving off any volatiles, causing it to assume the Jacobi ellipsoidal shape of a self-gravitating incompressible liquid. The collision that produced the inferred tumbling motion may have occurred thousands of years after the formation of 1I/2017 U1 ‘Oumuamua. Jacobi ellipsoids have a unique relation among rotation rate, density and axial ratio. The inferred axial ratio ⪆5 suggests a lower bound on the density of 1.6 g cm−3, apparently excluding an icy interior unless it is almost entirely frozen CO2. ‘Oumuamua may be related to accreting objects that pollute white dwarf atmospheres and that may make Soft Gamma Repeaters. stars: neutron, white dwarfs, minor planets, asteroids: individual: 1I/2017 U1 ('Oumuamua) 1 INTRODUCTION The recently discovered interstellar object (ISO) 1I/2017 U1 (‘Oumuamua) is remarkable for its hyperbolic solar orbit (Mamajek 2017; Meech et al. 2017), its 10-fold brightness modulation, greater than that of any Solar system asteroid (Drahus et al. 2018; Fraser et al. 2018; Meech et al. 2017), and the absence (with tight upper bounds) of evaporating volatiles. The light curve is not strictly periodic but rather indicates tumbling (Drahus et al. 2018; Fraser et al. 2018). Although ISOs expelled from exoplanetary systems have been predicted for many years (Hansen & Zuckerman 2017), they have been predicted to be overwhelmingly cometary, evaporating volatiles and emitting dust during close approaches to the Sun like that of 1I/2017 U1 ‘Oumuamua that approached within 0.25 au (Meech et al. 2016). Thus, while the discovery of an ISO was not unexpected, its properties require explanation. The brightness modulation of 1I/2017 U1 ‘Oumuamua suggests an extraordinary axial ratio and indicates a novel origin. The very slight (≲100 erg cm−3) strength required to support the less prolate non-equilibrium shapes of rotating Solar system bodies of similar size indicates that this ISO had an extraordinary history during which its strength was reduced to essentially zero. Its unexpected absence of volatiles is also remarkable, and these two facts point towards a hypothesis of its origin and history. Here, I discuss the possible origin of 1I/2017 U1 ‘Oumuamua in the post-main sequence stage of a planetary system (Veras 2016; Hansen & Zuckerman 2017; Raymond et al. 2018). The increasing luminosity of a post-main sequence star will evaporate volatiles from its satellites. Even those in Oort cloud like orbits will pass close enough to the star to be heated if they are perturbed on to nearly parabolic orbits, which is necessary for them to be expelled by gravitational interaction with massive planets or to collide with other small objects. The rapid mass-loss of white dwarf formation or the essentially instantaneous mass-loss of neutron star formation in a core-collapse supernova will unbind many of the small bodies of the exoplanetary system. It will also put massive exoplanets on elliptic orbits in which they will interact gravitationally with each other and with the small bodies, expelling some and putting others on the low angular momentum orbits required to explain the ‘pollution’ of white dwarf atmospheres (Jura & Young 2014; Farihi 2016; Mustill et al. 2018; Veras, Shannon & Gänsicke 2017; Xu et al. 2017) by heavier elements. Additional mechanisms of expulsion of small bodies operate in binary systems (Stephan, Naoz & Zuckerman 2017). One consequence of heating is the loss of volatile ices, possibly explaining the absence of volatiles in 1I/2017 U1 ‘Oumuamua. Evaporating volatiles fluidize beds of refractory particles. There is evidence for this (at the lower levels of heating occurring in Solar system comets) in images obtained by the Rosetta spacecraft of Comet 67P/Churyumov–Gerasimenko(European Space Agency 2017). Regions near the centre of the cometary nucleus, near the minimum of its gravitational potential, are covered by fine material, in contrast to the rough craggy appearance of the rest of the body. This is plausibly the result of flow of fluidized particulates to the minimum of the gravitational potential. A body heated by the intense radiation of a red giant star may be entirely fluidized and the residual refractory material assume the shape, a Maclaurin spheroid or, if it has more angular momentum, a Jacobi ellipsoid, of a rotating self-gravitating incompressible liquid. The purpose of this paper is to investigate the consequences of this explanation of the unprecedentedly prolate shape of 1I/2017 U1 ‘Oumuamua. This hypothesis implies a lower density limit that likely excludes the possibility (Fitzsimmons et al. 2018) of an icy interior protected from solar heating by an insulating mantle. Finally, the inference that exoplanetary systems may be disrupted by the evolution of their stars and that white dwarf pollution may be caused by single bodies as massive as ∼1022 g (Xu et al. 2017), or possibly even more massive (Jura et al. 2009), is consistent with the explanation of Soft Gamma Repeater outbursts as the result of accretion of such minor-planet-sized bodies on to neutron stars (Katz, Toole & Unruh 1994). 2 GEOMETRY Jacobi ellipsoids are the equilibrium state of uniformly rotating strengthless homogeneous incompressible fluids with too much angular momentum to be (oblate) Maclaurin spheroids. Even were their specific angular momentum to be increased, strengthless Jacobi ellipsoids will not break up (spinning mass off their extremities) provided they have time to relax to their equilibrium shape. Extremely elongated Jacobi ellipsoids may be unstable to ‘pear-shaped’ (actually, more like egg-shaped) ℓ = 3, ‘dumb-bell’ ℓ = 4 and higher instabilities (Eriguchi, Hachisu & Sugimoto 1982; Hachisu & Eriguchi 1982) but the non-linear development of these instabilities is not understood, so here I assume Jacobi ellipsoids. The 10-fold modulation of the brightness of 1I/2017 U1 ‘Oumuamua would imply, in a naïve model that assumes a prolate spheroid, Lambert’s Law reflectivity with uniform albedo and oppositional geometry (the actual solar angle during photometric observations was about 20°), an axial ratio of 10:1 if viewed in the plane of rotation, and greater for other viewing angles (Meech et al. 2017). However, Lambert’s law is not valid for Solar system asteroids. Modelling that assumes 1I/2017 U1 ‘Oumuamua is described by the scattering properties of known Solar system asteroids (Drahus et al. 2018; Fraser et al. 2018) sets a lower bound on the axial ratio of about 5:1. No upper bound on the axial ratio can be obtained because the angle between the line of sight and the rotation axis is unknown, but a lower bound of about 65° can be set on the angle if arbitrarily large axial ratios are considered (Drahus et al. 2018). Assuming a plausible upper bound on the density of 3 g cm−3 leads to an upper bound on the axial ratio of about 8 ( Section 3), from which a somewhat more stringent lower bound on the angle may be found. Uniformly rotating (very quickly enforced by friction) self-gravitating incompressible fluids of uniform density are oblate Maclaurin spheriods (with no modulation of their scattered light) if their angular momentum is low, or triaxial Jacobi ellipsoids if they have higher angular momentum (Chandrasekhar 1969; Tassoul 1978). For large specific angular momenta, the two smaller semi-axes b and c of the Jacobi ellipsoids converge, as shown in Fig. 1, and the ellipsoid approaches a prolate spheroid. Figure 1. View largeDownload slide Semiprincipal axes a, b and c of Jacobi ellipsoids as a function of the normalized angular momentum $${\cal L} \equiv L/\sqrt{GM^3r_0}$$ where r0 ≡ (abc)1/3 (Chandrasekhar 1969). At higher angular momentum $${\cal L} \gtrsim 0.5$$b → c. For lower angular momenta ($${\cal L} \lt 0.3$$, outside the range of the figure) a = b > c and the body is an oblate Maclaurin spheroid. $$a/\sqrt{bc}$$ is the axial ratio used in Fig. 2. Figure 1. View largeDownload slide Semiprincipal axes a, b and c of Jacobi ellipsoids as a function of the normalized angular momentum $${\cal L} \equiv L/\sqrt{GM^3r_0}$$ where r0 ≡ (abc)1/3 (Chandrasekhar 1969). At higher angular momentum $${\cal L} \gtrsim 0.5$$b → c. For lower angular momenta ($${\cal L} \lt 0.3$$, outside the range of the figure) a = b > c and the body is an oblate Maclaurin spheroid. $$a/\sqrt{bc}$$ is the axial ratio used in Fig. 2. 3 DENSITY Jacobi ellipsoids have a unique relation among their semi-axes, density and rotation rate that can be used to establish bounds on their density from the measured rotation rate and the axial ratio inferred from their light curve. Alternatively, assumed bounds on their density can be used to constrain their geometry. Several slightly different values of the rotation rate have been reported (Drahus et al. 2018; Meech et al. 2017; Bolin et al. 2018; Fraser et al. 2018). Adopting a period of 7.55 h (Drahus et al. 2018), roughly the mean of these values, leads to the relation between axial ratio and density shown in Fig. 2. The lower bound on the axial ratio of about 5 fitted (Drahus et al. 2018; Fraser et al. 2018) to the light curve implies a minimum density of 1.6 g cm−3. This is inconsistent with all plausible icy materials except CO2, whose density is close to this value; even the density of CO2 is inconsistent unless we happen to lie almost exactly in the plane of rotation, the axial ratio is less than inferred from the light curve or the rotation period is significantly longer than the adopted value. This argues against the hypothesis (Fitzsimmons et al. 2018) that an icy core is protected against heating and evaporation by an insulating non-volatile mantle. Figure 2. View largeDownload slide Relation between density and axial ratio (defined as $$a/\sqrt{bc}$$) of a Jacobi ellipsoid with rotation period 7.55 h (Chandrasekhar 1969). The empirical lower bound of 5 on the axial ratio implies a lower bound of 1.6 g cm−3 on the density. Figure 2. View largeDownload slide Relation between density and axial ratio (defined as $$a/\sqrt{bc}$$) of a Jacobi ellipsoid with rotation period 7.55 h (Chandrasekhar 1969). The empirical lower bound of 5 on the axial ratio implies a lower bound of 1.6 g cm−3 on the density. 4 TUMBLING: THE EVOLUTIONARY HISTORY The tumbling motion of 1I/2017 U1 ‘Oumuamua (Drahus et al. 2018; Fraser et al. 2018) implies that it is not rotating only around its axis of greatest moment of inertia, but is a state of higher rotational energy. It requires at least minimal elastic strength to maintain. Tumbling must have been the result of collision after the epoch of fluidization because any viscous, plastic or frictional flow would lead to very rapid relaxation to periodic rotation about the axis of greatest moment of inertia (the lowest energy state). The required strength is difficult to estimate (we have only the crudest hint of the rotational dynamics), but is less than the extremely small central pressure p. For high axial ratios a ≫ b ≈ c (Fig. 1) and the ellipsoid can be approximated as an infinite cylinder of radius b: \begin{equation*} p \approx \pi G \rho ^2 b^2 \sim 20\ \rm{erg\,\,cm}^{-3} \end{equation*} (1) for a density ρ ∼ 3 g cm−3 and radius b ∼ 30 m. Tumbling implies that the body is not exactly a prolate spheroid, but is at least slightly triaxial; prolate spheroids may ‘cone’ around their long axes, but do not tumble, a process that produces aperiodic light curves (Sher 1971). For a mean axial ratio $$a/\sqrt{bc} \ge 5$$, as indicated by the depth of the light curve (Drahus et al. 2018; Fraser et al. 2018), b and c differ by $$\le 10\hbox{ per cent}$$. The collision that made 1I/2017 U1 ‘Oumuamua tumble may have made it triaxial by asymmetrically removing material. Fitzsimmons et al. (2018) indicate a thermal diffusivity D ∼ 10−4 cm2s−1. This implies a cooling time for an object with b ≈ c ∼ 30 m \begin{equation*} t_{\mathrm{ conduction}} \sim {b^2 \over D} \sim 10^{11}\ \text{s}. \end{equation*} (2) In order for a collision to have produced tumbling, it must have occurred after 1I/2017 U1 ‘Oumuamua was no longer fluidized, either because the volatiles were exhausted or because it cooled below volatilization temperatures. The red giant stages of stellar evolution are longer than tconduction so a body of these dimensions would have heated throughout. If 1I/2017 U1 ‘Oumuamua was completely fluidized and did not recover mechanical strength while still hot but depleted of volatiles, then at least thousands of years (and possibly much longer) must have elapsed from the end of the red giant stage of the parent star and its intense heating to the collision that made it tumble; 1I/2017 U1 ‘Oumuamua must have remained in the fossil planetary system surrounding the former red giant, where the density of solid bodies was high enough to make collision likely, for at least that long before being expelled. The dynamical processes that expel small bodies also put other small bodies on collision orbits with their parent stars. This is consistent both with the inference that polluted white dwarfs accrete solid bodies for ∼109 yr and with the active lifetimes of Soft Gamma Repeaters of a few thousand years after the supernovæ in which their neutron stars were born. 5 DISCUSSION The argument of this paper can be summarized as follows: The fact that 1I/2017 U1 ‘Oumuamua is apparently so prolate (more than any Solar system asteroid) implies that it was once strengthless (unlike Solar system asteroids, which were never fluidized). A strengthless period in its past may possibly have been the result of evaporative fluidization of even a small volatile fraction by the luminous post-main sequence stage of its parent star. Complete loss of volatiles in the distant past explains the absence of any coma or other evidence of evaporation after its close passage by the Sun. Its equilibrium Jacobi ellipsoidal figure when fluidized was frozen in after the volatiles were lost because there was no force to disrupt it. The relation among density, axial ratio and rotation rate of Jacobi ellipsoids permits a lower bound on density to be inferred. Present tumbling requires a collision after solidification, although the strength required to support tumbling (equation 1) is small. A significant probability of collision implies that the body remained in its parent exoplanetary system for at least its cooling time (equation 2). A number of mechanisms have been proposed for its subsequent expulsion, but expulsion was not an immediate dynamical consequence of the end, gradual (formation of a planetary nebula) or instantaneous (supernova), of the parent star’s red giant stage; they are not the subject of this paper. The data contain rich implications that were not expected when ISOs were first hypothesized: Some of the light curves of 1I/2017 U1 ‘Oumuamua (Meech et al. 2017) are asymmetric about their minima, rising more slowly than they fell. This may be explained as a shadowing effect if the rotation is prograde with respect to its orbital motion about the Sun (an accident with 50 per cent probability of occurrence for an ISO). The extraordinary and unanticipated axial ratio of 1I/2017 U1 ‘Oumuamua, combined with the absence of volatiles, indicates its origin in the exoplanetary system of a star that had passed through the red giant stage. The combination of the rotation period with the inferred large axial ratio sets a lower bound on the density of 1I/2017 U1 ‘Oumuamua that excludes an icy composition, even one under a non-volatile crust that might be consistent with the observed absence of a coma. The inference that 1I/2017 U1 ‘Oumuamua is tumbling sets further constraints on its history: it likely remained in a region of comparatively high density of solid objects for at least thousands of years after it formed. The discovery of an interstellar interloper confirms expectations from the study of ‘polluted’ white dwarfs that post-main sequence evolution drastically changes the orbits of planets and planetesimals, putting some on nearly radial orbits penetrating within the Roche radius of the star and unbinding others, and suggests that analogous processes may produce Soft Gamma Repeater outbursts. The absence of volatiles and extreme axial ratios will be correlated in ISOs because they have the common cause of intense heating and fluidization. However, not all ISOs need these properties. Tumbling will be common because expulsion is more likely from densely populated exoplanetary systems and because post-main sequence stellar evolution disrupts them. ACKNOWLEDGEMENTS I thank B. Hansen and B. Zuckerman for useful discussions. REFERENCES Bolin B. T. et al. , 2018 , ApJ , 852 , L2 CrossRef Search ADS Chandrasekhar S. , 1969 , Ellipsoidal Figures of Equilibrium . Yale Univ. 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Monthly Notices of the Royal Astronomical Society: LettersOxford University Press

Published: May 3, 2018

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