Gravitational accretion of particles in Saturn's rings

Gravitational accretion in the rings of Saturn is studied with local N-body simulations, taking into account the dissipative impacts and gravitational forces between particles. Common estimates of accretion assume that gravitational sticking takes place beyond a certain distance (Roche distance) where the self-gravity between a pair of ring particles exceeds the disrupting tidal force of the central object, the exact value of this distance depending on the ring particles' internal density. However, the actual physical situation in the rings is more complicated, the growth and stability of the particle groups being affected also by the elasticity and friction in particle impacts, both directly via sticking probabilities and indirectly via velocity dispersion, as well as by the shape, rotational state and the internal packing density of the forming particle groups. These factors are most conveniently taken into account via N-body simulations. In our standard simulation case of identical 1 m particles with internal density of solid ice, ρ=900 kg m⁻³, following the Bridges et al., 1984 elasticity law, we find accretion beyond a=137,000-146,000 km, the smaller value referring to a distance where transient aggregates are first obtained, and the larger value to the distance where stable aggregates eventually form in every experiment lasting 50 orbital periods. Practically the same result is obtained for a constant coefficient of restitution εn=0.5. In terms of rp parameter, the sum of particle radii normalized by their mutual Hill radius, the above limit for perfect accretion corresponds to rp<0.84. Increased dissipation (εn=0.1), or inclusion of friction (tangential force 10% of normal force) shifts the accretion region inward by about 5000 km. Accretion is also more efficient in the case of size distribution: with a q=3 power law extending over a mass range of 1000, accretion shifts inward by almost 10,000 km. The aggregates forming in simulations via gradual accumulation of particles are synchronously rotating.     

N-Body Simulations of Planetesimal Evolution: Effect of Varying Impactor Mass Ratio

We present results from direct N-body simulations of collisions between gravitational aggregates of varying size as part of a study to parameterize planetesimal growth in the Solar System. We find that as the ratio of projectile to target mass departs from unity, the impact angle has less effect on the outcome. At the same time, the probability of planetesimal growth increases. Conversely, for a fixed impact energy, collisions between impactors with mass ratio near unity are more dispersive than those with impactor mass ratio far from unity. We derive an expression for the accretion probability as a function of mass ratio. For an average mass ratio of 1:5, we find an accretion probability of ∼60% over all impact parameters. We also compute the critical specific dispersal energy Q*_D as a function of projectile size. Extrapolating to a projectile size of 1 m with a 1-km target, we find Q*_D=10³−10⁴ J kg⁻¹, in agreement with several other collision models that use fundamentally different techniques. Our model assumes that the components of each gravitational aggregate are identical and indestructible over the range of sampled impact speeds. In future work we hope to incorporate a simple fracture model to extend the range of applicable speeds and we plan to implement our results in a large-scale planetesimal evolution code.     

Particle accretion and dissipation simulator: Collisional aggregation of icy particles

As a part of a project to study the cometary particle environment, a self-consistent ballistic rigid body particle colliding simulator, capable of producing fractal aggregates of icy particles modelled by hard or soft (metaball) spheres has been developed, to be used for creating test particles in future statistical studies of the cometary coma-particle interaction. Instead of a predefined sticking probability, actual surface interactions are modelled and a method for calculating internal stresses and fragmentation of an aggregate during the accretion phase is described. Simulations suggest that taking fragmentation into account has two major consequences, increase in the fractal dimension of resulting particles and a noticeable change in the mass spectrum of an ensemble.     

Moonlets and clumps in Saturn's F ring

Cassini UVIS star occultations by the F ring detect 13 events ranging from 27 m to 9 km in width. We interpret these structures as likely temporary aggregations of multiple smaller objects, which result from the balance between fragmentation and accretion processes. One of these features was simultaneously observed by VIMS. There is evidence that this feature is elongated in azimuth. Some features show sharp edges. At least one F ring object is opaque and may be a “moonlet.” This possible moonlet provides evidence for larger objects embedded in Saturn's F ring, which were predicted as the sources of the F ring material by Cuzzi and Burns [Cuzzi, J.N., Burns, J.A., 1988. Icarus 74, 284-324], and as an outcome of tidally modified accretion by Barbara and Esposito [Barbara, J.M., Esposito, L.W., 2002. Icarus 160, 161-171]. We see too few events to confirm the bi-modal distribution which Barbara and Esposito [Barbara, J.M., Esposito, L.W., 2002. Icarus 160, 161-171] predict. These F ring structures and other youthful features detected by Cassini may result from ongoing destruction of small parent bodies in the rings and subsequent aggregation of the fragments. If so, the temporary aggregates are 10 times more abundant than the solid objects. If recycling by re-accretion is significant, the rings could be quite ancient, and likely to persist far into the future.     

A model for accretion of the terrestrial planets

One possible origin of the terrestrial planets involves their formation by gravitational accretion of particles originally in Keplerian orbits about the sun. Some implications of this theory are considered. A formal expression for the rate of mass accretion by a planet is developed. The formal singularity of the gravitational collision cross-section for low relative velocities is shown to be without physical significance when the accreting bodies are in heliocentric orbits. The distribution of particle velocities relative to an accreting planet is considered; the mean velocity increases with time. The internal temperature of an accreting planet is shown to depend simply on the accretion rate. A simple and physically reasonable approximate expression for a planetary accretion rate is proposed.     

The influence of grain rotation on the structure of dust aggregates

We study the effect of rotation during the collision between dust aggregates, in order to address a mismatch between previous model calculations of Brownian motion driven aggregation and experiments. We show that rotation during the collision does influence the shape and internal structure of the aggregates formed. The effect is limited in the ballistic regime when aggregates can be considered to move on straight lines during a collision. However, if the stopping length of an aggregate becomes smaller than its physical size, extremely elongated aggregates can be produced. We show that this effect may have played a role in the inner regions of the solar nebula where densities were high.     

Models of Jupiter's growth incorporating thermal and hydrodynamic constraints

We model the growth of Jupiter via core nucleated accretion, applying constraints from hydrodynamical processes that result from the disk-planet interaction. We compute the planet's internal structure using a well tested planetary formation code that is based upon a Henyey-type stellar evolution code. The planet's interactions with the protoplanetary disk are calculated using 3-D hydrodynamic simulations. Previous models of Jupiter's growth have taken the radius of the planet to be approximately one Hill sphere radius, RH. However, 3-D hydrodynamic simulations show that only gas within ∼0.25RH remains bound to the planet, with the more distant gas eventually participating in the shear flow of the protoplanetary disk. Therefore in our new simulations, the planet's outer boundary is placed at the location where gas has the thermal energy to reach the portion of the flow not bound to the planet. We find that the smaller radius increases the time required for planetary growth by ∼5%. Thermal pressure limits the rate at which a planet less than a few dozen times as massive as Earth can accumulate gas from the protoplanetary disk, whereas hydrodynamics regulates the growth rate for more massive planets. Within a moderately viscous disk, the accretion rate peaks when the planet's mass is about equal to the mass of Saturn. In a less viscous disk hydrodynamical limits to accretion are smaller, and the accretion rate peaks at lower mass. Observations suggest that the typical lifetime of massive disks around young stellar objects is ∼3 Myr. To account for the dissipation of such disks, we perform some of our simulations of Jupiter's growth within a disk whose surface gas density decreases on this timescale. In all of the cases that we simulate, the planet's effective radiating temperature rises to well above 1000 K soon after hydrodynamic limits begin to control the rate of gas accretion and the planet's distended envelope begins to contract. According to our simulations, proto-Jupiter's distended and thermally-supported envelope was too small to capture the planet's current retinue of irregular satellites as advocated by Pollack et al. [Pollack, J.B., Burns, J.A., Tauber, M.E., 1979. Icarus 37, 587-611].     

N-Body simulations of growth from 1 km planetesimals at 0.4 AU

We present N-body simulations of planetary accretion beginning with 1 km radius planetesimals in orbit about a 1 M_⊙ star at 0.4 AU. The initial disk of planetesimals contains too many bodies for any current N-body code to integrate; therefore, we model a sample patch of the disk. Although this greatly reduces the number of bodies, we still track in excess of 10⁵ particles. We consider three initial velocity distributions and monitor the growth of the planetesimals. The masses of some particles increase by more than a factor of 100. Additionally, the escape speed of the largest particle grows considerably faster than the velocity dispersion of the particles, suggesting impending runaway growth, although no particle grows large enough to detach itself from the power law size-frequency distribution. These results are in general agreement with previous statistical and analytical results. We compute rotation rates by assuming conservation of angular momentum around the center of mass at impact and that merged planetesimals relax to spherical shapes. At the end of our simulations, the majority of bodies that have undergone at least one merger are rotating faster than the breakup frequency. This implies that the assumption of completely inelastic collisions (perfect accretion), which is made in most simulations of planetary growth at sizes 1 km and above, is inappropriate. Our simulations reveal that, subsequent to the number of particles in the patch having been decreased by mergers to half its initial value, the presence of larger bodies in neighboring regions of the disk may limit the validity of simulations employing the patch approximation.     

Conditions for collisional growth of a grain aggregate

Collisional growth of grain aggregates is a critical process in the early stage of planet formation. A collision between grain aggregates is numerically simulated by means of a smoothed particle hydrodynamic code, treating a grain aggregate as a continuum media. A model for mechanical response of a grain aggregate is developed based on published experimental data. Free parameters of the model are the bulk modulus, compressive, shear, and tensile strengths of a grain aggregate, and impact velocity. I have determined three conditions for the growth of an aggregate within the mechanical response model. (1) Compressive strength is the smallest among the three components of strengths. (2) Impact velocity is as low as 4% of the sound speed of an aggregate. (3) Effective restoration of the strengths is necessary due to reconnection between grains followed by compaction of an aggregate. Possibilities of these conditions in the solar nebula are discussed.     

Gravitational instability in the dust layer of a protoplanetary disk: Interaction of solid particles with turbulent gas in the layer

Gravitational instability of the dust layer formed after the aggregates of dust particles settle toward the midplane of a protoplanetary disk under turbulence is considered. A linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation is solved. Turbulent diffusion and the velocity dispersion of solid particles and the perturbation of gas azimuthal velocity in the layer upon the transfer of angular momentum from the dust phase due to gas drag are taken into account. Such an interaction of the particles and the gas establishes upper and lower bounds on the perturbation wavelength that renders the instability possible. The dispersion equation for the layer in the case when the ratio of surface densities of the dust phase and the gas in the layer is well above unity is obtained and solved. An approximate gravitational instability criterion, which takes the size-dependent stopping time of a particle (aggregate) in the gas into account, is derived. The following parameters of the layer instability are calculated: the wavelength range of its subsistence and the dependence of the perturbation growth rate on the perturbation wavelength in the circumsolar disk at a radial distance of 1 and 10 AU. It is demonstrated that at a distance of 1 AU, the gas–dust disk should be enriched with solids by a factor of 5–10 relative to the initial abundance as well as the particle aggregates should grow to the sizes higher than about 0.3 m in order for the instability to emerge in the layer in the available turbulence models. Such high disk enrichment and aggregate growth is not needed at a distance of 10 AU. The conditions under which this gravitational instability in the layer may be examined with no allowance made for the transfer of angular momentum from the gas in the layer to the gas in a protoplanetary disk outside the layer are discussed.     

Features around embedded moonlets in Saturn's rings: The role of self-gravity and particle size distributions

This paper presents the results of N-body simulations of moonlets embedded in broad rings, focusing specifically on the saturnian A ring. This work adds to previous efforts by including particle self-gravity and particle size distributions. The discussion here focuses primarily on the features that form in the background particles as a result of the moonlet. Particle self-gravity tends to damp out features produced by embedded moonlets and this damping is enhanced if the moonlet is simply the largest member of a continuous size distribution. Observable features around an embedded moonlet appear to require that the largest ring particles be no more massive than 1/30 the mass of the moonlet. These results, compared with current and future Cassini observations, will provide insight into the nature of the particle population in the saturnian rings. Some time is also spent analyzing the way in which the background particles cluster around the moonlet. The accretion of small particles onto the moonlet can be limited by disruptive collisions with the largest ring particles in the particle size distribution.     

Dual black holes in merger remnants – I. Linking accretion to dynamics

We study the orbital evolution and accretion history of massive black hole (MBH) pairs in rotationally supported circumnuclear discs up to the point where MBHs form binary systems. Our simulations have high resolution in mass and space which, for the first time, makes it feasible to follow the orbital decay of a MBH either counter- or corotating with respect to the circumnuclear disc. We show that a moving MBH on an initially counter-rotating orbit experiences an 'orbital angular momentum flip' due to the gas-dynamical friction, i.e. it starts to corotate with the disc before a MBH binary forms. We stress that this effect can only be captured in very high resolution simulations. Given the extremely large number of gas particles used, the dynamical range is sufficiently large to resolve the Bondi–Hoyle–Lyttleton radii of individual MBHs. As a consequence, we are able to link the accretion processes to the orbital evolution of the MBH pairs. We predict that the accretion rate is significantly suppressed and extremely variable when the MBH is moving on a retrograde orbit. It is only after the orbital angular momentum flip has taken place that the secondary rapidly 'lights up' at which point both MBHs can accrete near the Eddington rate for a few Myr. The separation of the double nucleus is expected to be around ≲10 pc at this stage. We show that the accretion rate can be highly variable also when the MBH is corotating with the disc (albeit to a lesser extent) provided that its orbit is eccentric. Our results have significant consequences for the expected number of observable double active galactic nuclei at separations of ≲100 pc.     

High-resolution simulations of clump–clump collisions using SPH with particle splitting

We investigate, by means of numerical simulations, the phenomenology of star formation triggered by low-velocity collisions between low-mass molecular clumps. The simulations are performed using a smoothed particle hydrodynamics code which satisfies the Jeans condition by invoking on-the-fly particle splitting.
Clumps are modelled as stable truncated (non-singular) isothermal, i.e. Bonnor–Ebert, spheres. Collisions are characterized by M ₀ (clump mass), b (offset parameter, i.e. ratio of impact parameter to clump radius) and     (Mach number, i.e. ratio of collision velocity to effective post-shock sound speed). The gas subscribes to a barotropic equation of state, which is intended to capture (i) the scaling of pre-collision internal velocity dispersion with clump mass, (ii) post-shock radiative cooling and (iii) adiabatic heating in optically thick protostellar fragments.
The efficiency of star formation is found to vary between 10 and 30 per cent in the different collisions studied and it appears to increase with decreasing M ₀, and/or decreasing b , and/or increasing     . For   b < 0.5  collisions produce shock-compressed layers which fragment into filaments. Protostellar objects then condense out of the filaments and accrete from them. The resulting accretion rates are high,     , for the first     . The densities in the filaments,     , are sufficient that they could be mapped in NH₃ or CS line radiation, in nearby star formation regions.     

Co-accretion of the Earth-Moon system after the giant impact: reduction of the total angular momentum by lunar impact ejecta

Though the Moon is considered to have been formed by the so-called giant impact, the mass of the Earth immediately after the impact is still controversial. If the Moon was formed during the Earth's accretion, a subsequent accretion of residual heliocentric planetesimals onto the protoearth and the protomoon must have occurred. In this co-accretion stage, a significant amount of lunar-impact-ejecta would be ejected to circumterrestrial orbits, since the mean impact velocity of the planetesimals with the protomoon is much larger than the escape velocity of the protomoon. Orbital calculations of test particles ejected from the protomoon, whose semimajor axis is smaller than that of the present Moon, reveal that most of the particles escaping from the protomoon also escape from the Hill sphere of the protoearth and reduce the planetocentric angular momentum of the primordial Earth-Moon system. Using the results of the ejecta simulations, we investigate the evolution of the mass ratio and the total angular momentum (Earth's spin angular momentum + Moon's orbital angular momentum) of the Earth-Moon system during the co-accretion. We find that the mass of the protomoon is almost constant or rather decreases and the total angular momentum decreases significantly, if the random velocity of planetesimals is as large as the escape velocity of the protoearth. On the other hand, if the random velocity is the half of the escape velocity of the protoearth, the mass ratio is kept to be almost as large as the present value and the decrease of the total angular momentum is not so significant. Comparing with the results of giant impact simulations, we find that the mass of the protoearth immediately after the Moon-forming impact was 0.7-0.8 times the present value if the impactor-to-target mass ratio was 3:7, whereas the giant impact occurred almost in the end of the Earth's accretion if the impactor-to-target mass ratio was 1:9.     

SPH simulations of negative (nodal) superhumps: a parametric study

Negative superhumps in cataclysmic variable systems result when the accretion disc is tilted with respect to the orbital plane. The line of nodes of the tilted disc precesses slowly in the retrograde direction, resulting in a photometric signal with a period slightly less than the orbital period. We use the method of smoothed particle hydrodynamics to simulate a series of models of differing mass ratio and effective viscosity to determine the retrograde precession period and superhump period deficit  ɛ₋  as a function of system mass ratio q . We tabulate our results and present fits to both  ɛ₋  and  ɛ₊  versus q , as well as compare the numerical results with those compiled from the literature of negative superhump observations. One surprising result is that while we find negative superhumps most clearly in simulations with an accretion stream present, we also find evidence for negative superhumps in simulations in which we shut off the mass transfer stream completely, indicating that the origin of the photometric signal is more complicated than previously believed.     

Transfer,loss and physical processing of water in hit-and-run collisions of planetary embryos

Collisions between large, similar-sized bodies are believed to shape the final characteristics and composition of terrestrial planets. Their inventories of volatiles such as water are either delivered or at least significantly modified by such events. Besides the transition from accretion to erosion with increasing impact velocity, similar-sized collisions can also result in hit-and-run outcomes for sufficiently oblique impact angles and large enough projectile-to-target mass ratios. We study volatile transfer and loss focusing on hit-and-run encounters by means of smooth particle hydrodynamics simulations, including all main parameters: impact velocity, impact angle, mass ratio and also the total colliding mass. We find a broad range of overall water losses, up to 75% in the most energetic hit-and-run events, and confirm the much more severe consequences for the smaller body also for stripping of volatile layers. Transfer of water between projectile and target inventories is found to be mostly rather inefficient, and final water contents are dominated by pre-collision inventories reduced by impact losses, for similar pre-collision water mass fractions. Comparison with our numerical results shows that current collision outcome models are not accurate enough to reliably predict these composition changes in hit-and-run events. To also account for non-mechanical losses, we estimate the amount of collisionally vaporized water over a broad range of masses and find that these contributions are particularly important in collisions of \(\sim \) Mars-sized bodies, with sufficiently high impact energies, but still relatively low gravity. Our results clearly indicate that the cumulative effect of several (hit-and-run) collisions can efficiently strip protoplanets of their volatile layers, especially the smaller body, as it might be common, e.g., for Earth-mass planets in systems with Super-Earths. An accurate model for stripping of volatiles that can be included in future planet formation simulations has to account for the peculiarities of hit-and-run events and track compositional changes in both large post-collision fragments.     

Planetesimals to protoplanets – II. Effect of debris on terrestrial planet formation

In this paper, we extend our numerical method for simulating terrestrial planet formation to include dynamical friction from the unresolved debris component. In the previous work, we implemented a rubble pile planetesimal collision model into direct N -body simulations of terrestrial planet formation. The new collision model treated both accretion and erosion of planetesimals but did not include dynamical friction from debris particles smaller than the resolution limit for the simulation. By extending our numerical model to include dynamical friction from the unresolved debris, we can simulate the dynamical effect of debris produced during collisions and can also investigate the effect of initial debris mass on terrestrial planet formation. We find that significant initial debris mass, 10 per cent or more of the total disc mass, changes the mode of planetesimal growth. Specifically, planetesimals in this situation do not go through a runaway growth phase. Instead, they grow concurrently, similar to oligarchic growth. The dynamical friction from the unresolved debris damps the eccentricities of the planetesimals, reducing the mean impact speeds and causing all collisions to result in merging with no mass loss. As a result, there is no debris production. The mass in debris slowly decreases with time. In addition to including the dynamical friction from the unresolved debris, we have implemented particle tracking as a proxy for monitoring compositional mixing. Although there is much less mixing due to collisions and gravitational scattering when dynamical friction of the background debris is included, there is significant inward migration of the largest protoplanets in the most extreme initial conditions (for which the initial mass in unresolved debris is at least equal to the mass in resolved planetesimals).     

Growth of planetesimals by impacts at ∼25 m/s

We study central collisions between millimeter-sized dust projectiles and centimeter-sized dust targets in impact experiments. Target and projectile are dust aggregates consisting of micrometer-sized SiO₂ particles. Collision velocities range up to 25 m/s. The general outcome of a collision strongly depends on the impact velocity. For collisions below 13 m/s rebound and a small degree of fragmentation occur. However, at higher collision velocities up to 25 m/s approximately 50% of the mass of the projectile rigidly sticks to the target after the collision. Thus, net growth of a body is possible in high speed collisions. This supports the idea that planetesimal formation via collisional growth is a viable mechanism at higher impact velocities. Within our set of parameters the experiments even suggest that higher impact velocities might be preferable for growth in collisions between dusty bodies. For the highest impact velocities most of the ejecta is within small dust aggregates about 500 μm in size. In detail the size distribution of ejected dust aggregates is flat for very small particles smaller than 500 μm and follows a power law for larger ejected dust aggregates with a power of −5.6±0.2. There is a sharp upper cut-off at about 1 mm in size with only a few particles being slightly larger. The ejection angle is smaller than 3° with respect to the target surface. These fast ejecta move with 40±10% of the impact velocity.     

How common are Earth-Moon planetary systems?

The Earth’s comparatively massive moon, formed via a giant impact on the proto-Earth, has played an important role in the development of life on our planet, both in the history and strength of the ocean tides and in stabilizing the chaotic spin of our planet. Here we show that massive moons orbiting terrestrial planets are not rare. A large set of simulations by Morishima et al. (Morishima, R., Stadel, J., Moore, B. [2010]. Icarus. 207, 517-535), where Earth-like planets in the habitable zone form, provides the raw simulation data for our study. We use limits on the collision parameters that may guarantee the formation of a circumplanetary disk after a protoplanet collision that could form a satellite and study the collision history and the long term evolution of the satellites qualitatively. In addition, we estimate and quantify the uncertainties in each step of our study. We find that giant impacts with the required energy and orbital parameters for producing a binary planetary system do occur with more than 1 in 12 terrestrial planets hosting a massive moon, with a low-end estimate of 1 in 45 and a high-end estimate of 1 in 4.     

The Nebular Shock Wave Model for Chondrule Formation: Shock Processing in a Particle-Gas Suspension

We present numerical simulations of the thermal and dynamical histories of solid particles (chondrules and their precursors—treated as 1-mm silicate spheres) during passage of an adiabatic shock wave through a particle-gas suspension in a minimum-mass solar nebula. The steady-state equations of energy, momentum, and mass conservation are derived and integrated for both solids and gas under a variety of shock conditions and particle number densities using the free-molecular-flow approximation. These simulations allow us to investigate both the heating and cooling of particles in a shock wave and to compare the time and distance scales associated with their processing to those expected for natural chondrules. The interactions with the particles cause the gas to achieve higher temperatures and pressures both upstream and downstream of the shock than would be reached otherwise. The cooling rates of the particles are found to be nonlinear but agree approximately with the cooling rates inferred for chondrules by laboratory simulations. The initial concentration of solids upstream of the shock controls the cooling rates and the distances over which they are processed: Lower concentrations cool more slowly and over longer distances. These simulations are consistent with the hypothesis that large-scale shocks, e.g., those due to density waves or gravitational instabilities, were the dominant mechanism for chondrule formation in the nebula.