Orbital evolution and accretion of protoplanets tidally interacting with a gas disk: II. Solid surface density evolution with type-I migration

This paper investigates the surface density evolution of a planetesimal disk due to the effect of type-I migration by carrying out N-body simulation and through analytical method, focusing on terrestrial planet formation. The coagulation and the growth of the planetesimals take place in the abundant gas disk except for a final stage. A protoplanet excites density waves in the gas disk, which causes the torque on the protoplanet. The torque imbalance makes the protoplanet suffer radial migration, which is known as type-I migration. Type-I migration time scale derived by the linear theory may be too short for the terrestrial planets to survive, which is one of the major problems in the planet formation scenario. Although the linear theory assumes a protoplanet being in a gas disk alone, Kominami et al. [Kominami, J., Tanaka, H., Ida, S., 2005. Icarus 167, 231-243] showed that the effect of the interaction with the planetesimal disk and the neighboring protoplanets on type-I migration is negligible. The migration becomes pronounced before the planet's mass reaches the isolation mass, and decreases the solid component in the disk. Runaway protoplanets form again in the planetesimal disk with decreased surface density. In this paper, we present the analytical formulas that describe the evolution of the solid surface density of the disk as a function of gas-to-dust ratio, gas depletion time scale and semimajor axis, which agree well with our results of N-body simulations. In general, significant depletion of solid material is likely to take place in inner regions of disks. This might be responsible for the fact that there is no planet inside Mercury's orbit in our Solar System. Our most important result is that the final surface density of solid components (Σd) and mass of surviving planets depend on gas surface density (Σg) and its depletion time scale (τdep) but not on initial Σd; they decrease with increase in Σg and τdep. For a fixed gas-to-dust ratio and τdep, larger initial Σd results in smaller final Σd and smaller surviving planets, because of larger Σg. To retain a specific amount of Σd, the efficient disk condition is not an initially large Σd but the initial Σd as small as the specified final one and a smaller gas-to-dust ratio. To retain Σd comparable to that of the minimum mass solar nebula (MMSN), a disk must have the same Σd and a gas-to-dust ratio that is smaller than that of MMSN by a factor of 1.3×(τdep/1 Myr) at ∼1 AU. (Equivalently, type-I migration speed is slower than that predicted by the linear theory by the same factor.) The surviving planets are Mars-sized ones in this case; in order to form Earth-sized planets, their eccentricities must be pumped up to start orbit crossing and coagulation among them. At ∼5 AU, Σd of MMSN is retained under the same condition, but to form a core massive enough to start runaway gas accretion, a gas-to-dust ratio must be smaller than that of MMSN by a factor of 3×τdep/1 Myr.     

The gravitational instability in the dust layer of a protoplanetary disk:: axisymmetric solutions for nonuniform dust density distributions in the direction vertical to the midplane

The gravitational instability in the dust layer of a protoplanetary disk with nonuniform dust density distributions in the direction vertical to the midplane is investigated. The linear analysis of the gravitational instability is performed. The following assumptions are used: (1) One fluid model is adopted, that is, difference of velocities between dust and gas are neglected. (2) The gas is incompressible. (3) Models are axisymmetric with respect to the rotation axis of the disk. Numerical results show that the critical density at the midplane is higher than the one for the uniform dust density distribution by Sekiya (1983, Prog. Theor. Phys. 69, 1116-1130). For the Gaussian dust density distribution, the critical density is 1.3 times higher, although we do not consider this dust density distribution to be realistic because of the shear instability in the dust layer. For the dust density distribution with a constant Richardson number, which is considered to be realized due to the shear instability, the critical density is 2.85 times higher and is independent of the value of the Richardson number. Further, if a constant Richardson number could decrease to the order of 0.001, the gravitational instability would be realized even for the dust to gas surface density ratio with the solar abundance. Our results give a new restriction on planetesimal formation by the gravitational instability.     

The full set of gas giant structures: I: On the origin of planetary masses and the planetary initial mass function

Context

Current planet search programs are detecting extrasolar planets at a rate of 60 planets per year. These planets show more diverse properties than was expected.

Aims

We try to get an overview of possible gas giant (proto-) planets for a full range of orbital periods and stellar masses. This allows the prediction of the full range of possible planetary properties which might be discovered in the near future.

Methods

We calculate the purely hydrostatic structure of the envelopes of proto-planets that are embedded in protoplanetary disks for all conceivable locations: combinations of different planetesimal accretion rates, host star masses, and orbital separations. At each location all hydrostatic equilibrium solutions to the planetary structure equations are determined by variation of core mass and pressure over many orders of magnitude. For each location we analyze the distribution of planetary masses.

Results

We get a wide spectrum of core-envelope structures. However, practically all calculated proto-planets are in the planetary mass range. Furthermore, the planet masses show a characteristic bimodal, sometimes trimodal, distribution. For the first time, we identify three physical processes that are responsible for the three characteristic planet masses: self-gravity in the Hill sphere, compact objects, and a region of very low adiabatic pressure gradient in the hydrogen equation of state. Using these processes, we can explain the dependence of the characteristic masses on the planet’s location: orbital period, host star mass, and planetesimal accretion rate (luminosity). The characteristic mass caused by the self-gravity effect at close proximity to the host star is typically one Neptune mass, thus producing the so-called hot Neptunes.

Conclusions

Our results suggest that hot Jupiters with orbital period less than 64 days (the exact location of the boundary depends on stellar type and accretion rate) have quite distinct properties which we expect to be reflected in a different mass distribution of these planets when compared to the “normal” planetary population. We use our theoretical survey to produce an upper mass limit for embedded planets: the maximum embedded equilibrium mass (MEEM). This naturally explains the lack of high mass planets between 3 and 64 days orbital period.     

Simulations of dense planetary rings: IV. Spinning self-gravitating particles with size distribution

Previous self-gravitating simulations of dense planetary rings are extended to include particle spins. Both identical particles as well as systems with a modest range of particle sizes are examined. For a ring of identical particles, we find that mutual impact velocity is always close to the escape velocity of the particles, even if the total rms velocity dispersion of the system is much larger, due to collective motions associated to wakes induced by near-gravitational instability or by viscous overstability. As a result, the spin velocity (i.e., the product of the particle radius and the spin frequency) maintained by mutual impacts is also of the order of the escape velocity, provided that friction is significant. For the size distribution case, smaller particles have larger impact velocities and thus larger spin velocities, particularly in optically thick rings, since small particles move rather freely between wakes. Nevertheless, the maximum ratio of spin velocities between the smallest and largest particles, as well as the ratio for translational velocities, stays below about 5 regardless of the width of the size distribution. Particle spin state is one of the important factors affecting the temperature difference between the lit and unlit face of Saturn's rings. Our results suggest that, to good accuracy, the spin frequency is inversely proportional to the particle size. Therefore, the mixing ratio of fast rotators to slow rotators on the scale of the thermal relaxation time increases with the width of the particle size distribution. This will offer means to constrain the particle size distribution with the systematic thermal infrared observations carried by the Cassini probe.     

Rotation rate and velocity dispersion of planetary ring particles with size distribution: I. Formulation and analytic calculation

We derive an equation for the evolution of rotational energy of Keplerian particles in a dilute disk due to mutual collisions. Three-dimensional Keplerian motion of particles is taken into account precisely, on the basis of Hill's approximation. The Rayleigh distribution of particles' orbital eccentricities and inclinations, and the Gaussian distribution of their rotation rates are also taken into account. Performing appropriate variable transformation, we show that the equation can be expressed with two terms. The first term, which we call collisional stirring term, represents energy exchange between rotation and random motion via collisions. The second term, which we call rotational friction term, tends to equalize the mean rotational energy of particles with different sizes. The equation can describe the evolution of rotational energy of Keplerian particles with an arbitrary size distribution. We analytically evaluate the rates of stirring and friction for the random kinetic energy and rotational energy due to inelastic collisions, for non-gravitating particles in a dilute disk. Using these results, we discuss equilibrium states in a disk of spinning, non-gravitating Keplerian particles.     

Spin rates of small moonlets embedded in planetary rings: I. Three-body calculations

We investigate the spin rates of moonlets embedded in planetary rings, subject to collisions with surrounding small particles, using three-body integrations including friction and spins. All successive impacts of the particle with the moonlet are followed, including a possible sliding phase after the initial inelastic rebounds. Two methods for treating impacts, (1) as instantaneous velocity changes and (2) using an impact force model, are applied after Salo (1995, Icarus 117, 287). Conducting a series of integrations with various initial summed spin velocity of the moonlet and the particle, we determine the equilibrium spin rate for which the averaged torque vanishes. This equilibrium spin rate corresponds to the final spin rate of the moonlet if the moonlet is much larger than the surrounding particles; it also corresponds to the mean spin rate for a ring composed of identical particles. We find that the equilibrium spin rate is enhanced by sliding orbits as compared with the spin rate determined by considering only the first impacts of the particles with the moonlet. If the random velocities of incident particles are small enough, the resulting equilibrium spin rate of the moonlet can be larger than the synchronous rotation rate, for r_p∼1, where r_p denotes the sum of radii of the colliding pair normalized by their mutual Hill radius. In this special case aggregates without internal strength may become rotationally unstable. However, the equilibrium spin rate decreases with increasing random velocity, and aggregates are always rotationally stable in the more likely case where the relative velocities are comparable to the mutual escape velocity. We also compare our results with the mean spin rates found in previous N-body simulations, and find a good agreement for optically thin rings; however the spin rates for optically thick rings are significantly larger than those predicted by our three-body calculations.     

From discs to planetesimals: Evolution of gas and dust discs

I review the processes that shape the evolution of protoplanetary discs around young, solar-mass stars. I first discuss observations of protoplanetary discs, and note in particular the constraints these observations place on models of disc evolution. The processes that affect the evolution of gas discs are then discussed, with the focus in particular on viscous accretion and photoevaporation, and recent models which combine the two. I then discuss the dynamics and growth of dust grains in discs, considering models of grain growth, the gas–grain interaction and planetesimal formation, and review recent research in this area. Lastly, I consider the so-called “transitional” discs, which are thought to be observed during disc dispersal. Recent observations and models of these systems are reviewed, and prospects for using statistical surveys to distinguish between the various proposed models are discussed.     

The jovian anticyclone BA: I. Motions and interaction with the GRS from observations and non-linear simulations

A study of the dynamics of the second largest anticyclone in Jupiter, Oval BA, and its red colour change that occurred in late 2005 is presented in a three part study. The first part, this paper, deals with its long-term kinematical and dynamical behaviour monitored since its formation in 2000 to September 2008 using ground-based observations archived at the public International Outer Planet Watch (IOPW) database. The vortex changed its zonal drift velocity from 1.8 m s⁻¹ in the period 2000-2002 to 0.8 m s⁻¹ in 2002-2003, and to 2.5 m s⁻¹ since late 2003. It also migrated southwards by 1.0 ± 0.5° in latitude between 2000 and 2004, remaining afterwards at an almost fixed latitude position. During the period 2000-2007, the oval also changed its triangular-like shape to a more symmetrical one. No latitudinal change was found in the months before the development of a red annulus in its interior. The colour change took place in less than 5 months in 2005-2006 and no red colour feature was observed to have been present or entrained by BA months before the annulus development. After detailed examination of the four encounters between BA and GRS that took place during this 9 year period, we did not detect any noticeable change in its drift rate or in apparent structure associated with the encounters at cloud level. Also, the area of BA did not significantly change in this period. Additionally, we found that BA displays a long-term oscillation of ∼160 days in its longitude position with peak to peak amplitude of 1.2°. Numerical experiments using the global circulation model EPIC reproduce accurately the shape, connecting it to its latitude migration, and morphology of the oval and confirm that no strong interaction between BA and the GRS is possible at least in the current situation.     

Orbital resonances in the inner neptunian system: I. The 2:1 Proteus-Larissa mean-motion resonance

We investigate the orbital resonant history of Proteus and Larissa, the two largest inner neptunian satellites discovered by Voyager 2. Due to tidal migration, these two satellites probably passed through their 2:1 mean-motion resonance a few hundred million years ago. We explore this resonance passage as a method to excite orbital eccentricities and inclinations, and find interesting constraints on the satellites' mean density () and their tidal dissipation parameters (Qs>10). Through numerical study of this mean-motion resonance passage, we identify a new type of three-body resonance between the satellite pair and Triton. These new resonances occur near the traditional two-body resonances between the small satellites and, surprisingly, are much stronger than their two-body counterparts due to Triton's large mass and orbital inclination. We determine the relevant resonant arguments and derive a mathematical framework for analyzing resonances in this special system.     

Embedded star clusters and the formation of the Oort cloud: II. The effect of the primordial solar nebula

This paper deals with Oort cloud formation while the Sun was in an embedded cluster and surrounded by its primordial nebula. This work is a continuation of Brasser et al. [Brasser, R., Duncan, M., Levison, H., 2006. Icarus 184, 59-82], building on the model presented therein, and adding the aerodynamic drag and gravitational potential of the primordial solar nebula. Results are presented of numerical simulations of comets subject to the gravitational influence of the Sun, Jupiter, Saturn, star cluster and primordial solar nebula; some of the simulations included the gravitational influence of Uranus and Neptune as well. The primordial solar nebula was approximated by the minimum-mass Hayashi model [Hayashi, C., Nakozawa, K., Nakagawa, Y., 1985. In: Black, D.C., Matthews, M.S. (Eds.). Protostars and Planets II. Univ. of Arizona Press, Tucson, AZ] whose inner and outer radii have been truncated at various distances from the Sun. A comet size of 1.7 km was used for most of our simulations. In all of our simulations, the density of the primordial solar nebula decayed exponentially with an e-folding time of 2 Myr. It turns out that when the primordial solar nebula extends much beyond Saturn or Neptune, virtually no material will end up in the Oort cloud (OC) during this phase. Instead, the majority of the material will be on circular orbits inside of Jupiter if the inner edge of the disk is well inside Jupiter's orbit. If the disk's inner edge is beyond Jupiter's orbit, most comets end up on orbits in exterior mean-motion resonances with Saturn when Uranus and Neptune are not present. In those cases where the outer edge of the disk is close to Saturn or Neptune, the fraction of material that ends up in the subsequently formed OC is much less than that found in Brasser et al. [Brasser, R., Duncan, M., Levison, H., 2006. Icarus 184, 59-82] for the same cluster densities. This implies that for comets of roughly 2 km in size, the presence of the primordial solar nebula hinders OC formation. A byproduct of some of our simulations are endresults with a substantial fraction of the comets in the Uranus-Neptune scattered disk. A subsequent followup of this material is planned for the near future. In order to determine the effect of the size of the comets on OC formation efficiency, a set of runs with the same initial conditions but different cometary radii have been performed as well, from which it is determined that the threshold comet size to begin producing significant Oort clouds is roughly 20 km. This implies that the presence of the primordial solar nebula acts as a size-sorting mechanism, with large bodies unaffected by the gas drag and ending up in the OC while small bodies remain trapped in the planetary region, in the models studied.     

Long-term evolution of the spin of Mercury: I. Effect of the obliquity and core-mantle friction

The present obliquity of Mercury is very low (less than 0.1°), which led previous studies to always adopt a nearly zero obliquity during the planet’s past evolution. However, the initial orientation of Mercury’s rotation axis is unknown and probably much different than today. As a consequence, we believe that the obliquity could have been significant when the rotation rate of the planet first encountered spin-orbit resonances. In order to compute the capture probabilities in resonance for any evolutionary scenario, we present in full detail the dynamical equations governing the long-term evolution of the spin, including the obliquity contribution.The secular spin evolution of Mercury results from tidal interactions with the Sun, but also from viscous friction at the core-mantle boundary. Here, this effect is also regarded with particular attention. Previous studies show that a liquid core enhances drastically the chances of capture in spin-orbit resonances. We confirm these results for null obliquity, but we find that the capture probability generally decreases as the obliquity increases. We finally show that, when core-mantle friction is combined with obliquity evolution, the spin can evolve into some unexpected configurations as the synchronous or the 1/2 spin-orbit resonance.     

Long-term evolution of the spin of Venus: II. numerical simulations

We present here the numerical application of the theoretical results derived in Correia et al. (2003, Icarus 163, 1-23) for the spin evolution of Venus since its formation. We explore a large variety of initial conditions to cover the possible formation and evolutionary scenarios. In particular, we pay special attention to the evolutions which cross the chaotic zone resulting from secular planetary perturbations (Laskar and Robutel, 1993, Nature 361, 608-612). We demonstrate that Venus’ axis can be temporarily trapped in a secular resonance with the node of Neptune’s orbit, which can prevent it from being tilted to 180° and will drive it toward 0°. We test several dissipation models and parameters to evaluate their contribution to the planet’s spin history. We confirm that despite the variations in the models, only three of the four final spin states of Venus are possible (Correia and Laskar, 2001, Nature 411, 767-770) and that the present observed retrograde spin state of Venus can be attained by two different processes. In the first scenario (π−), the axis is tilted toward 180° while its rotation rate slows down, while in the second one, the axis is driven toward 0° obliquity and the rotation rate decreases, stops, and increases again in the reverse direction to a final equilibrium value (0−).     

Orbit determination with very short arcs: II. Identifications

When the observational data are not enough to compute a meaningful orbit for an asteroid/comet we can represent the data with an attributable, i.e., two angles and their time derivatives. The undetermined variables range and range rate span an admissible region of Solar System orbits, which can be sampled by a set of Virtual Asteroids (VAs) selected by means of an optimal triangulation [Milani, A., Gronchi, G.F., de' Michieli Vitturi, M., Kne?evi?, Z., 2004. Celest. Mech. Dyn. Astron. 90, 59-87]. The attributable 4 coordinates are the result of a fit and they have an uncertainty, represented by a covariance matrix. Two short arcs of observations, represented by two attributables, can be linked by considering for each VA (in the admissible region of the first arc) the covariance matrix for the prediction at the time of the second arc, and by comparing it with the attributable of the second arc with its own covariance. By defining an identification penalty we can select the VAs allowing to fit together both arcs and compute a preliminary orbit. Two attributables may not be enough to compute an orbit with convergent differential corrections. Thus the preliminary orbit is used in a constrained differential correction, providing solutions along the Line Of Variation which can be used as second generation VAs to further predict the observations at the time of a third arc. In general the identification with a third arc will ensure a well determined orbit, to which additional sets of observations can be attributed. To test these algorithms we use a large scale simulation and measure the completeness, the reliability and the efficiency of the overall procedure to build up orbits by accumulating identifications. Under the conditions expected for the next generation asteroid surveys, the methods developed in this and in the preceding papers are efficient enough to be used as primary identification methods, with very good results. One important property is that the completeness in finding the possible identifications is as good for comparatively rare orbits, such as the ones of Near-Earth Objects, as for main belt orbits.     

Modeling the 3-D secular planetary three-body problem: Discussion on the outer υ Andromedae planetary system

The three-dimensional secular behavior of a system composed of a central star and two massive planets is modeled semi-analytically in the frame of the general three-body problem. The main dynamical features of the system are presented in geometrical pictures allowing us to investigate a large domain of the phase space of this problem without time-expensive numerical integrations of the equations of motion and without any restriction on the magnitude of the planetary eccentricities, inclinations and mutual distance. Several regimes of motion of the system are observed. With respect to the secular angle Δ?, possible motions are circulations, oscillations (around 0° and 180°), and high-eccentricity/inclination librations in secular resonances. With respect to the arguments of pericenter, ω₁ and ω₂, possible motions are direct circulation and high-inclination libration around ±90° in the Lidov-Kozai resonance. The regions of transition between domains of different regimes of motion are characterized by chaotic behavior. We apply the analysis to the case of the two outer planets of the υ Andromedae system, observed edge-on. The topology of the 3-D phase space of this system is investigated in detail by means of surfaces of section, periodic orbits and dynamical spectra, mapping techniques and numerical simulations. We obtain the general structure of the phase space, and the boundaries of the spatial secular stability. We find that this system is secularly stable in a large domain of eccentricities and inclinations.     

Formation of gas giant planets: core accretion models with fragmentation and planetary envelope

We have calculated formation of gas giant planets based on the standard core accretion model including effects of fragmentation and planetary envelope. The accretion process is found to proceed as follows. As a result of runaway growth of planetesimals with initial radii of ∼10 km, planetary embryos with a mass of ∼10²⁷ g (∼ Mars mass) are found to form in ∼10⁵ years at Jupiter's position (5.2 AU), assuming a large enough value of the surface density of solid material (25 g/cm²) in the accretion disk at that distance. Strong gravitational perturbations between the runaway planetary embryos and the remaining planetesimals cause the random velocities of the planetesimals to become large enough for collisions between small planetesimals to lead to their catastrophic disruption. This produces a large number of fragments. At the same time, the planetary embryos have envelopes, that reduce energies of fragments by gas drag and capture them. The large radius of the envelope increases the collision rate between them, resulting in rapid growth of the planetary embryos. By the combined effects of fragmentation and planetary envelope, the largest planetary embryo with 21M_⊕ forms at 5.2 AU in 3.8×10⁶ years. The planetary embryo is massive enough to start a rapid gas accretion and forms a gas giant planet.     

The web of three-planet resonances in the outer Solar System: II. A source of orbital instability for Uranus and Neptune

The motion of the giant planets from Jupiter to Neptune is chaotic with Lyapunov time of approximately 10 Myr. A recent theory explains the presence of this chaos with three-planet mean-motion resonances, i.e. resonances among the orbital periods of at least three planets. We find that the distribution of these resonances with respect to the semi-major axes of all the planets is compatible with orbital instability. In particular, they overlap in a region of 10⁻³ AU with respect to the variation of the semi-major axes of Uranus and Neptune. Fictitious planetary systems with initial conditions in this region can undergo systematic variations of semi-major axes. The true Solar System is marginally in this region, and Uranus and Neptune undergo very slow systematic variations of semi-major axes with speed of order 10⁻⁴ AU/Gyr.     

From planetesimals to terrestrial planets: N-body simulations including the effects of nebular gas and giant planets

We present results from a suite of N-body simulations that follow the formation and accretion history of the terrestrial planets using a new parallel treecode that we have developed. We initially place 2000 equal size planetesimals between 0.5 and 4.0 AU and the collisional growth is followed until the completion of planetary accretion (>100 Myr). A total of 64 simulations were carried out to explore sensitivity to the key parameters and initial conditions. All the important effect of gas in laminar disks are taken into account: the aerodynamic gas drag, the disk-planet interaction including Type I migration, and the global disk potential which causes inward migration of secular resonances as the gas dissipates. We vary the initial total mass and spatial distribution of the planetesimals, the time scale of dissipation of nebular gas (which dissipates uniformly in space and exponentially in time), and orbits of Jupiter and Saturn. We end up with 1-5 planets in the terrestrial region. In order to maintain sufficient mass in this region in the presence of Type I migration, the time scale of gas dissipation needs to be 1-2 Myr. The final configurations and collisional histories strongly depend on the orbital eccentricity of Jupiter. If today’s eccentricity of Jupiter is used, then most of bodies in the asteroidal region are swept up within the terrestrial region owing to the inward migration of the secular resonance, and giant impacts between protoplanets occur most commonly around 10 Myr. If the orbital eccentricity of Jupiter is close to zero, as suggested in the Nice model, the effect of the secular resonance is negligible and a large amount of mass stays for a long period of time in the asteroidal region. With a circular orbit for Jupiter, giant impacts usually occur around 100 Myr, consistent with the accretion time scale indicated from isotope records. However, we inevitably have an Earth size planet at around 2 AU in this case. It is very difficult to obtain spatially concentrated terrestrial planets together with very late giant impacts, as long as we include all the above effects of gas and assume initial disks similar to the minimum mass solar nebular.     

Photometric modeling of Saturn's rings: I. Monte Carlo method and the effect of nonzero volume filling factor

The scattering properties of particulate rings with volume filling factors in the interval D=0.001-0.3 are studied, with photometric Monte Carlo ray tracing simulations combining the advantages of direct (photons followed from the source) and indirect methods (brightness as seen from the observing direction). Besides vertically homogeneous models, ranging from monolayers to classical many-particle thick rings, particle distributions obtained from dynamical simulations are studied, possessing a nonuniform vertical profile and a power law distribution of particle sizes. Self-gravity is not included to assure homogeneity in planar directions. Our main goal is to check whether the moderately flattened ring models predicted by dynamical simulations (with central plane D>0.1) are consistent with the basic photometric properties of Saturn's rings seen in ground-based observations, including the brightening near zero phase angle (opposition effect), and the brightening of the B-ring with increasing elevation angle (tilt effect). Our photometric simulations indicate that dense rings are typically brighter in reflected light than those with D→0, due to enhanced single scattering. For a vertically illuminated layer of identical particles this enhancement amounts at intermediate viewing elevations to roughly 1+2D. Increased single scattering is also obtained for low elevation illumination, further augmented at low phase angles α by the opposition brightening when D increases: the simulated opposition effect agrees very well with the Lumme and Bowell (1981, Astron. J. 86, 1694-1704) theoretical formula. For large α the total intensity may also decrease, due to reduced amount of multiple scattering. For the low (α=13°) and high (α=155°) phase angle geometries analyzed in Dones et al. (1993, Icarus 105, 184-215) the brightness change for D=0.1 amounts to 20% and −17%, respectively. In the case of an extended size distribution, dynamical simulations indicate that the smallest particles typically occupy a layer several times thicker than the largest particles. Even if the large particles form a dynamically dense system, a narrow opposition peak can arise due to mutual shadowing among the small particles: for example, a size distribution extending about two decades can account for the observed about 1° wide opposition peak, solely in terms of mutual shadowing. The reduced width of the opposition peak for extended size distribution is in accordance with Hapke's (1986, Icarus 67, 264-280) treatment for semi-infinite layers. Due to vertical profile and particle size distribution, the photometric behavior is sensitive to the viewing elevation: this can account for the tilt-effect of the B-ring, as dense and thus bright central parts of the ring become better visible for larger elevation, whereas in the case of smaller elevation, mainly low volume density upper layers are visible. Since multiple scattering is not involved, the explanation works also for albedo well below unity. Inclusion of nonzero volume density helps also to model some of the Voyager observations. For example, the discrepancy between predicted and observed brightness at large phase angles for much of the A-ring (Dones et al., 1993, Icarus 105, 184-215) is removed when the enhanced low α single scattering and reduced large α multiple scattering is allowed for. Also, a model with vertical thickness increasing with saturnocentric distance offers at least a qualitative explanation for the observed contrast reversal between the inner and outer A-ring in low and high phase Voyager images. Differences in local size distribution and thus on the effective D may also account for the contrast reversal in resonance sites.     

Corona-like atmospheric escape from KBOs: I. Gas dynamics

We show that for low temperatures (T∼30 K) and small, but non-negligible, gravitational fields the hydrodynamic escape of gas can be treated by Parker's theory of coronal expansion [Parker, E.N., 1963. Interplanetary Dynamical Processes. Interscience Publishers, New York]. We apply this theory to gas escape from Kuiper belt objects. We derive limits on the density and radius of the bodies for which this theory is applicable, and show how the flow depends on the mean molecular weight and internal degrees of freedom of the gas molecules. We use these results to explain the CH₄ dichotomy seen on KBOs [Schaller, E.L., Brown, M.E., 2007. Astrophys. J., 659, L61-L64].