Tidal disruptions: II. A continuum theory for solid bodies with strength, with applications to the Solar System

We present a comprehensive theory for the breakup conditions for ellipsoidal homogeneous secondary bodies subjected to the tidal forces from a nearby larger primary: for materials ranging from purely fluid ones, to granular rubble-pile gravel-like ones, and to those with either cohesive or granular strength including cohesive rocks and metals. The theory includes but greatly extends the classical analyses given by Roche in 1847, which dealt only with fluids, and also our previous analysis [Holsapple, K.A., Michel, P., 2006. Icarus 183, 331-348], which dealt only with solid but non-cohesive bodies. The results here give the distance inside of which breakup must occur, for both a steadily orbiting satellite and for a passing or impacting object. For the fluid bodies there is a single specific shape (a “Roche Ellipsoid”) that can be in equilibrium at any given distance from a primary, and especially only one shape that can exist at the overall minimum distance (d/R)(ρ/ρp)^1/3=2.455, the classical well-known “Roche limit.” In contrast, solid bodies can exist at a given distance from a primary with a range of shapes. Here we give multiple plots of the minimum distances for various important combinations of body shape, spin, mass density, and the strength parameters characterized by an angle of friction and cohesive strength. Such results can be used in different ways. They can be used to estimate limits on strengths and mass densities for orbiting bodies at a known distance and shape. They can be used to determine breakup distances for passing bodies with an assumed strength and shape. They can be used to constraint physical properties such as bulk density of bodies with a known shape that were known to breakup at a given distance. A collection of approximately 40 satellites of the Solar System is used for comparison to the theory. About half of those bodies are closer than the Roche fluid limit and must have some cohesion and/or friction angle to exist at their present orbital distance. The required solid strength for those states is determined. Finally, we apply the theory to the break up of the SL9 comet at close approach with Jupiter. Our results make clear that the literature estimates of its bulk density depend markedly on unknown parameters such as shape, orientation and spin, and most importantly, material strength characterization.     

Tidal interactions and disruptions of giant planets on highly eccentric orbits

We calculate the evolution of planets undergoing a strong tidal encounter using smoothed particle hydrodynamics (SPH), for a range of periastron separations. We find that outside the Roche limit, the evolution of the planet is well-described by the standard model of linear, non-radial, adiabatic oscillations. If the planet passes within the Roche limit at periastron, however, mass can be stripped from it, but in no case do we find enough energy transferred to the planet to lead to complete disruption. In light of the three new extrasolar planets discovered with periods shorter than two days, we argue that the shortest-period cases observed in the period-mass relation may be explained by a model whereby planets undergo strong tidal encounters with stars, after either being scattered by dynamical interactions into highly eccentric orbits, or tidally captured from nearly parabolic orbits. Although this scenario does provide a natural explanation for the edge found for planets at twice the Roche limit, it does not explain how such planets will survive the inevitable expansion that results from energy injection during tidal circularization.     

Binary asteroid systems: Tidal end states and estimates of material properties

The locations of the fully despun, double synchronous end states of tidal evolution, where the rotation rates of both the primary and secondary components in a binary system synchronize with the mean motion about the center of mass, are derived for spherical components. For a given amount of scaled angular momentum J/J′, the tidal end states are over-plotted on a tidal evolution diagram in terms of mass ratio of the system and the component separation (semimajor axis in units of primary radii). Fully synchronous orbits may not exist for every combination of mass ratio and angular momentum; for example, equal-mass binary systems require J/J′ > 0.44. When fully synchronous orbits exist for prograde systems, tidal evolution naturally expands the orbit to the stable outer synchronous solution. The location of the unstable inner synchronous orbit is typically within two primary radii and often within the radius of the primary itself. With the exception of nearly equal-mass binaries, binary asteroid systems are in the midst of lengthy tidal evolutions, far from their fully synchronous tidal end states. Of those systems with unequal-mass components, few have even reached the stability limit that splits the fully synchronous orbit curves into unstable inner and stable outer solutions.Calculations of material strength based on limiting the tidal evolution time to the age of the Solar System indicate that binary asteroids in the main belt with 100-km-scale primary components are consistent with being made of monolithic or fractured rock as expected for binaries likely formed from sub-catastrophic impacts in the early Solar System. To tidally evolve in their dynamical lifetime, near-Earth binaries with km-scale primaries or smaller created via a spin-up mechanism must be much weaker mechanically than their main-belt counterparts even if formed in the main belt prior to injection into the near-Earth region. Small main-belt binaries, those having primary components less than 10 km in diameter, could bridge the gap between the large main-belt binaries and the near-Earth binaries, as, depending on the age of the systems, small main-belt binaries could either be as strong as the large main-belt binaries or as weak as the near-Earth binaries. The inherent uncertainty in the age of a binary system is the leading source of error in calculation of material properties, capable of affecting the product of rigidity μ and tidal dissipation function Q by orders of magnitude. Several other issues affecting the calculation of μQ are considered, though these typically affect the calculation by no more than a factor of two. We also find indirect evidence within all three groups of binary asteroids that the semimajor axis of the mutual orbit in a binary system may evolve via another mechanism (or mechanisms) in addition to tides with the binary YORP effect being a likely candidate.     

Binary near-Earth asteroid formation: Rubble pile model of tidal disruptions

We present numerical simulations of near-Earth asteroid (NEA) tidal disruption resulting in bound, mutually orbiting systems. Using a rubble pile model we have constrained the relative likelihoods for possible physical and dynamical properties of the binaries created. Overall 110,500 simulations were run, with each body consisting of ∼1000 particles. The encounter parameters of close approach distance and velocity were varied, as were the bodies' spin, elongation and spin axis direction. The binary production rate increases for closer encounters, at lower speeds, for more elongated bodies, and for bodies with greater spin. The semimajor axes for resultant binaries are peaked between 5 to 20 primary radii, and there is an overall trend for high eccentricity, with 97% of binaries having e > 0.1. The secondary-to-primary size ratios of the simulated binaries are peaked between 0.1 and 0.2, similar to trends among observed asteroid binaries. The spin rates of the primary bodies are narrowly distributed between 3.5- and 6-h periods, whereas the secondaries' periods are more evenly distributed and can exceed 15-h periods. The spin axes of the primary bodies are very closely aligned with the angular momenta of the binary orbits, whereas the secondary spin axes are nearly random. The shapes of the primaries show a large distribution of axis ratios, where those with low elongation (ratio of long and short axis) are both oblate and prolate, and nearly all with large elongation are prolate. This work presents results that suggest tidal disruption of gravitational aggregates can make binaries physically similar to those currently observed in the NEA population. As well, tidal disruption may create an equal number of binaries with qualities different from those observed, mostly binaries with large separation and with elongated primaries.     

Tidal dissipation at arbitrary eccentricity and obliquity

Expressions for tidal dissipation in a body in synchronous rotation at arbitrary orbital eccentricity and obliquity are derived. The rate of tidal dissipation for a synchronously rotating body is compared to that in a body in asymptotic nonsynchronous rotation.     

Modeling stresses on satellites due to nonsynchronous rotation and orbital eccentricity using gravitational potential theory

The tidal stress at the surface of a satellite is derived from the gravitational potential of the satellite's parent planet, assuming that the satellite is fully differentiated into a silicate core, a global subsurface ocean, and a decoupled, viscoelastic lithospheric shell. We consider two types of time variability for the tidal force acting on the shell: one caused by the satellite's eccentric orbit within the planet's gravitational field (diurnal tides), and one due to nonsynchronous rotation (NSR) of the shell relative to the satellite's core, which is presumed to be tidally locked. In calculating surface stresses, this method allows the Love numbers h and ?, describing the satellite's tidal response, to be specified independently; it allows the use of frequency-dependent viscoelastic rheologies (e.g. a Maxwell solid); and its mathematical form is amenable to the inclusion of stresses due to individual tides. The lithosphere can respond to NSR forcing either viscously or elastically depending on the value of the parameter , where μ and η are the shear modulus and viscosity of the shell respectively, and ω is the NSR forcing frequency. Δ is proportional to the ratio of the forcing period to the viscous relaxation time. When Δ?1 the response is nearly fluid; when Δ?1 it is nearly elastic. In the elastic case, tensile stresses due to NSR on Europa can be as large as ∼3.3 MPa, which dominate the ∼50 kPa stresses predicted to result from Europa's diurnal tides. The faster the viscous relaxation the smaller the NSR stresses, such that diurnal stresses dominate when Δ?100. Given the uncertainty in current estimates of the NSR period and of the viscosity of Europa's ice shell, it is unclear which tide should be dominant. For Europa, tidal stresses are relatively insensitive both to the rheological structure beneath the ice layer and to the thickness of the icy shell. The phase shift between the tidal potential and the resulting stresses increases with Δ. This shift can displace the NSR stresses longitudinally by as much as 45° in the direction opposite of the satellite's rotation.     

A note on the tidally induced potential of a satellite in eccentric orbit

A planetary body moving on an eccentric orbit around the primary is subject to a periodic perturbing potential, affecting its internal mass distribution. In a previous paper (Rappaport et al., 1997, Icarus 126, 313), we have calculated the periodic modulation of the gravity coefficients of degree 2, for a body on a synchronous orbit. Here, the previous analysis is extended by considering also non-synchronous orbits, and by properly accounting for the apparent motion of the primary due to the non uniform motion along the elliptical orbit. The cases of Titan and Mercury are briefly discussed.     

Tidal pattern instabilities on multi-moon planets

The equilibrium tide-generating forces in the lunar orbital plane of a planet of radius R are calculated for the case of N moons of mass Mi orbiting the planet at instantaneous polar coordinates (Di, αi). For the case of a single moon, there are only two high tides. For the case of two moons, it is found that there can exist a critical lunar orbital distance at which two high tides become unstable with respect to formation of three high tides. Bifurcation diagrams are presented which depict how the angular positions of the high and low tides on the planet vary with the lunar distances and lunar separation angle. Tidal stability diagrams, which illustrate the stability regions for various tidal patterns as a function of lunar distances and lunar separation angle, are presented for various values of D₂/D₁ and M₂/M₁. Generally speaking, the aforementioned tidal instability, and hence the propensity for formation of three high tides on a two-moon planet, exists over a significant range of lunar distances and separation angles provided that M₂/M₁∼³(D₂/D₁). For the case of N>2 moons, the tidal stability diagram becomes more complex, revealing a diversity of potential tidal patterns.     

Tidal heating in Enceladus

The heating in Enceladus in an equilibrium resonant configuration with other saturnian satellites can be estimated independently of the physical properties of Enceladus. We find that equilibrium tidal heating cannot account for the heat that is observed to be coming from Enceladus. Equilibrium heating in possible past resonances likewise cannot explain prior resurfacing events.     

Subsurface oceans and deep interiors of medium-sized outer planet satellites and large trans-neptunian objects

The detection of induced magnetic fields in the vicinity of the jovian satellites Europa, Ganymede, and Callisto is one of the most surprising findings of the Galileo mission to Jupiter. The observed magnetic signature cannot be generated in solid ice or in silicate rock. It rather suggests the existence of electrically conducting reservoirs of liquid water beneath the satellites' outermost icy shells that may contain even more water than all terrestrial oceans combined. The maintenance of liquid water layers is closely related to the internal structure, composition, and thermal state of the corresponding satellite interior. In this study we investigate the possibility of subsurface oceans in the medium-sized icy satellites and the largest trans-neptunian objects (TNO's). Controlling parameters for subsurface ocean formation are the radiogenic heating rate of the silicate component and the effectiveness of the heat transfer to the surface. Furthermore, the melting temperature of ice will be significantly reduced by small amounts of salts and/or incorporated volatiles such as methane and ammonia that are highly abundant in the outer Solar System. Based on the assumption that the satellites are differentiated and using an equilibrium condition between the heat production rate in the rocky cores and the heat loss through the ice shell, we find that subsurface oceans are possible on Rhea, Titania, Oberon, Triton, and Pluto and on the largest TNO's 2003 UB₃₁₃, Sedna, and 2004 DW. Subsurface oceans can even exist if only small amounts of ammonia are available. The liquid subsurface reservoirs are located deeply underneath an ice-I shell of more than 100 km thickness. However, they may be indirectly detectable by their interaction with the surrounding magnetic fields and charged particles and by the magnitude of a satellite's response to tides exerted by the primary. The latter is strongly dependent on the occurrence of a subsurface ocean which provides greater flexibility to a satellite's rigid outer ice shell.     

On the shapes and spins of “rubble pile” asteroids

We examine the shape of a “rubble pile” asteroid as it slowly gains angular momentum by YORP torque, to the point where “landsliding” occurs. We find that it evolves to a “top” shape with constant angle of repose from the equator up to mid-latitude, closely resembling the shapes of several nearly critically spinning asteroids imaged by radar, most notably (66391) 1999 KW4 [Ostro, S.J., Margot, J.-L., Benner, L.A.M., Giorgini, J.D., Scheeres, D.J., Fahnestock, E.G., Broschart, S.B., Bellerose, J., Nolan, M.C., Magri, C., Pravec, P., Scheirich, P., Rose, R., Jurgens, R.F., De Jong, E.M., Suzuki, S., 2006. Science 314, 1276-1280]. Similar calculations for non-spinning extremely prolate or oblate “rubble piles” show that even loose rubble can sustain shapes far from fluid equilibrium, thus inferences based on fluid equilibrium are generally useless for inferring bulk properties such as density of small bodies. We also investigate the tidal effects of a binary system with a “top shape” primary spinning at near the critical limit for stability. We find that very close to the stability limit, the tide from the secondary can actually levitate loose debris from the surface and re-deposit it, in a process we call “tidal saltation.” In the process, angular momentum is transferred from the primary spin to the satellite orbit, thus maintaining the equilibrium of near-critical spin as YORP continues to add angular momentum to the system. We note that this process is in fact dynamically related to the process of “shepherding” of narrow rings by neighboring satellites.     

The global shape of Europa: Constraints on lateral shell thickness variations

The global shape of Europa is controlled by tidal and rotational potentials and possibly by lateral variations in ice shell thickness. We use limb profiles from four Galileo images to determine the best-fit hydrostatic shape, yielding a mean radius of 1560.8±0.3 km and a radius difference a−c of 3.0±0.9 km, consistent with previous determinations and inferences from gravity observations. Adding long-wavelength topography due to proposed lateral variations in shell thickness results in poorer fits to the limb profiles. We conclude that lateral shell thickness variations and long-wavelength isostatically supported topographic variations do not exceed 7 and 0.7 km, respectively. For the range of rheologies investigated (basal viscosities from 10¹⁴ to ) the maximum permissible (conductive) shell thickness is 35 km. The relative uniformity of Europa's shell thickness is due to either a heat flux from the silicate interior, lateral ice flow at the base of the shell, or convection within the shell.     

A proof that tidal heating in a synchronous rotation is always larger than in an asymptotic nonsynchronous rotation state

In a recent paper, Wisdom [Wisdom, J., 2008. Icarus, 193, 637-640] derived concise expressions for the rate of tidal dissipation in a synchronously rotating body for arbitrary orbital eccentricity and obliquity. He provided numerical evidence than the derived rate is always larger than in an asymptotic nonsynchronous rotation state at any obliquity and eccentricity. Here, I present a simple mathematical proof of this conclusion and show that this result still holds for any spin-orbit resonance.     

Appearance of Saturn’s F ring azimuthal channels for the anti-alignment configuration between the ring and Prometheus

In this article we explore the aspect of the F ring with respect to the anti-alignment configuration between the ring and Prometheus. We focus our attention on the shape of the F ring’s azimuthal channels which were first reported by Porco et al. (Porco, C.C., Baker, E., Barbara, J., Beurle, K., Brahic, A., Burns, J.A., Charnoz, S., Cooper, N., Dawson, D.D., Del Genio, A.D., Denk, T., Dones, L., Dyudina, U., Evans, M.W., Giese, B., Grazier, K., Helfenstein, P., Ingersoll, A.P., Jacobson, R.A., Johnson, T.V., McEwen, A., Murray, C.D., Neukum, G., Owen, W.M., Perry, J., Roatsch, T., Spitale, J., Squyres, S., Thomas, P., Tiscareno, M., Turtle, E., Vasavada, A.R., Veverka, J., Wagner, R., West, R. [2005] Science, 307, 1226-1236) and numerically explored by Murray et al. (Murray, C.D., Chavez, C., Beurle, K., Cooper, N., Evans, M.W., Burns, J.A., Porco, C.C. [2005] Nature 437, 1326-1329) who found excellent agreement between Cassini’s ISS reprojected images and their numerical model via a direct comparison. We find that for anti-alignment the channels are wider and go deeper inside the ring material. From our numerical model we find a new feature, an island in the middle of the channel. This island is made up of the particles that have been perturbed the most by Prometheus and only appears when this satellite is close to apoapsis. In addition, plots of the anti-alignment configuration for different orbital stages of Prometheus are obtained and discussed here.     

Search for Phobos and Deimos gas/dust tori using in situ observations from Mars Global Surveyor MAG/ER

More than 490 elliptical aerobraking and science phasing orbits made by Mars Global Surveyor (MGS) in 1997 and 1998 provide unprecedented coverage of the solar wind in the vicinity of the orbits of the martian moons Phobos and Deimos. We have performed a comprehensive survey of magnetic field perturbations in the solar wind to search for possible signatures of solar wind interaction with dust or gas escaping from the moons. A total of 1246 solar wind disturbance events were identified and their distribution was examined relative to Phobos, the Phobos orbit, and the Deimos orbit. We find that the spatial distribution of solar wind perturbations does not increase near or downstream of Phobos, Phobos’ orbit, or Deimos’ orbit, which would have been expected if there is significant outgassing or dust escape from the martian moons. Of the 1246 magnetic field perturbation events found in the MGS data set, 11 events were found within 2000 km of the Phobos orbit, while three events were found within 2000 km of the Deimos orbit. These events were analyzed in detail and found to likely have other causes than outgassing/dust escape from the martian moons. Thus we conclude that the amount of gas/dust escaping the martian moons is not significant enough to induce detectable magnetic field perturbations in the solar wind. In essence we have not found any clear evidence in the MGS magnetic field data for outgassing or dust escape from the martian moons.     

A survey for satellites of Venus

We present a systematic survey for satellites of Venus using the Baade-Magellan 6.5 m telescope and IMACS wide-field CCD imager at Las Campanas observatory in Chile. In the outer portions of the Hill sphere the search was sensitive to a limiting red magnitude of about 20.4, which corresponds to satellites with radii of a few hundred meters when assuming an albedo of 0.1. In the very inner portions of the Hill sphere scattered light from Venus limited the detection to satellites of about a kilometer or larger. Although several main belt asteroids were found, no satellites (moons) of Venus were detected.     

The heat flow of Europa

The heat flow from Europa has profound implications for ice shell thickness and structure, as well as for the existence of an internal ocean, which is strongly suggested by magnetic data. The brittle-ductile transition depth and the effective elastic thickness of the lithosphere are here used to perform heat flow estimations for Europa. Results give preferred heat flow values (for a typical geological strain rate of ¹⁰⁻¹⁵ s⁻¹) of 70-110 mW m⁻² for a brittle-ductile transition 2 km deep (the usually accepted upper limit for the brittle-ductile transition depth in the ice shell of Europa), 24-35 mW m⁻² for an effective elastic thickness of 2.9 km supporting a plateau near the Cilix impact crater, and >130 mW m⁻² for effective elastic thicknesses of ?0.4 km proposed for the lithosphere loaded by ridges and domes. These values are clearly higher than those produced by radiogenic heating, thus implying an important role for tidal heating. The ?19-25 km thick ice shell proposed from the analysis of size and depth of impact structures suggests a heat flow of ?30-45 mW m⁻² reaching the ice shell base, which in turn would imply an important contribution to the heat flow from tidal heating within the ice shell. Tidally heated convection in the ice shell could be capable to supply ∼100 mW m⁻² for superplastic flow, and, at the Cilix crater region, ∼35-50 mW m⁻² for dislocation creep, which suggests local variations in the dominant flow mechanism for convection. The very high heat flows maybe related to ridges and domes could be originated by preferential heating at special settings.     

Episodic volcanism on Enceladus: Application of the Ojakangas-Stevenson model

The main equations in the paper “Episodic volcanism of tidally heated satellites with application to Io” by Ojakangas and Stevenson [Icarus 66, 341-358] are presented; numerical integration of these equations confirms the results of Ojakangas and Stevenson [Icarus 66, 341-358] for Io. Application to Enceladus is considered. It is shown that Enceladus does not oscillate about the tidal equilibrium in this model by both new nonlinear stability analysis and numerical integration of the model equations.     

Cycloid crack sequences on Europa: Relationship to stress history and constraints on growth mechanics based on cusp angles

The original model developed to explain cycloidal cracks on Europa interprets cycloids as tensile fractures that grow in a curved path in response to the constantly rotating diurnal tidal stress field. Cusps form when a new cycloid crack segment propagates at an angle to the first in response to a rotation of the principal tidal stress orientation during a period of no crack growth. A recent revised model states that a cycloid cusp forms through the creation of a secondary fracture called a tailcrack at the tip of an existing cycloid segment during shearing motion induced by the rotating tidal stress field. As the tailcrack propagates away from the cusp, it becomes the next cycloid segment in the chain. The qualitative tailcrack model uniquely accounts for the normal and shear stresses that mechanically must resolve onto the tip of an existing cycloid segment at the instant of cusp formation. In this work, we provide a quantitative framework and test of the hitherto purely conceptual tailcrack model. We first present a relative age sequence inferred from geologic mapping of multiply cross-cutting cycloids in Europa's trailing hemisphere and place this into the context of the global stress history. The age sequence requires a cumulative minimum of 630° of shell reorientation due to nonsynchronous rotation to account for the observed range of orientations of cycloids of different ages. We determined the back-rotated longitudes of formation of two cycloid chain examples and used mathematical modeling of europan tidal stresses to show that the tailcrack model for cusp formation is not only viable, but places constraints on the overall development of a cycloid chain by controlling the timing of cusp development within Europa's orbit. For all cusps analyzed, the exact ratio of resolved shear to normal stress required to form the cusp angles by a process of tailcracking, as governed by the principles of linear elastic fracture mechanics, is produced at the tip of a shearing cycloid segment during Europa's orbit. Cusp formation occurs after the point in the orbit at which the maximum tensile principal tidal stress occurs, implying that tensile tidal stresses are not directly responsible for cusp development. Instead, cusps develop when a tailcrack forms at the tip of a cycloid segment in response to the highly perturbed stress field induced during concomitant opening and shearing at the tip of the cycloid segment.     

Spin limits of Solar System bodies: From the small fast-rotators to 2003 EL61

Holsapple [Holsapple, K.A., 2001. Icarus 154, 432-448; Holsapple, K.A., 2004. Icarus 172, 272-303] determined the spin limits of bodies using a model for solid bodies without tensile or cohesive strength, but with the pressure-induced shear strengths characteristic of dry sands and gravels. That theory included the classical analyses for fluid bodies given by Maclaurin, Jacobi and others as a special case. For the general solid bodies, it was shown that there exists a very wide range of permissible shapes and spin limits; and explicit algebraic results for those limits were given. This paper gives an extension of those analyses to include geological-like materials that also have tensile and cohesive strength. Those strengths are necessary to explain the smaller, fast-rotating asteroids discovered in the last few years. I find that the spin limits for these more general solids have two limiting regimes: a strength regime for bodies with a diameter <3 km, and a gravity regime for the larger bodies with a diameter >10 km (which is the case covered by the earlier papers). I derive explicit algebraic forms for the dependence of the spin limits on shape, mass density and material strength properties. The comparison of the theory to the database for the spins of asteroids and trans-neptunian objects (TNO's) objects shows excellent agreement. For large bodies (diameter D>10 km), the presence of cohesive and/or tensile strength does not permit higher spin rates than would be allowed for rubble pile bodies. Thus, the fact that the spin rates of all large bodies is limited to periods greater than about 2 h does not imply that they are rubble piles. In contrast, for small bodies (D<10 km) the presence of even a very small amount of strength allows much more rapid spins. Small bodies might then be rubble piles but require a small amount of bonding. Finally, I make some remarks about the application of the theory to the TNO's and large asteroids, and question whether a common assumption by researchers that those bodies must take on relaxed fluid shapes is warranted. If not, then the densities and shapes required by that assumption are not valid. I use 2003 EL61 as a prime example.