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1.
Keiji Ohtsuki 《Icarus》2004,172(2):432-445
We examine the rotation of a small moonlet embedded in planetary rings caused by impacts of ring particles, using analytic calculation and numerical orbital integration for the three-body problem. Taking into account the Rayleigh distribution of particles' orbital eccentricities and inclinations, we evaluate both systematic and random components of rotation, where the former arises from an average of a large number of small impacts and the latter is contribution from large impacts. Calculations for parameter values corresponding to inner parts of Saturn's rings show that a moonlet would spin slowly in the prograde direction if most impactors are small particles whose velocity dispersion is comparable to or smaller than the moonlet's escape velocity. However, we also find that the effect of the random component can be significant, if the velocity dispersion of particles is larger and/or impacts of large particles comparable to the moonlet's size are common: in this case, both prograde and retrograde rotations can be expected. In the case of a small moonlet embedded in planetary rings of equal-sized particles, we find that the systematic component dominates the moonlet rotation when m/M?0.1 (m and M are the mass of a particle and a moonlet, respectively), while the random component is dominant when m/M?0.3. We derive the condition for the random component to dominate moonlet rotation on the basis of our results of three-body orbital integration, and confirm agreement with N-body simulation. 相似文献
2.
Keiji Ohtsuki 《Icarus》2006,183(2):384-395
We examine rotation rates of gravitating particles in low optical depth rings, on the basis of the evolution equation of particle rotational energy derived by Ohtsuki [Ohtsuki, K., 2006. Rotation rate and velocity dispersion of planetary ring particles with size distribution. I. Formulation and analytic calculation. Icarus 183, 373-383]. We obtain the rates of evolution of particle rotation rate and velocity dispersion, using three-body orbital integration that takes into account distribution of random velocities and rotation rates. The obtained stirring and friction rates are used to calculate the evolution of velocity dispersion and rotation rate for particles in one- and two-size component rings as well as those with a narrow size distribution, and agreement with N-body simulation is confirmed. Then, we perform calculations to examine equilibrium rotation rates and velocity dispersion of gravitating ring particles with a broad size distribution, from 1 cm up to 10 m. We find that small particles spin rapidly with 〈ω2〉1/2/Ω?102-103, where ω and Ω are the particle rotation rate and its orbital angular frequency, respectively, while the largest particles spin slowly, with 〈ω2〉1/2/Ω?1. The vertical scale height of rapidly rotating small particles is much larger than that of slowly rotating large particles. Thus, rotational states of ring particles have vertical heterogeneity, which should be taken into account in modeling thermal infrared emission from Saturn's rings. 相似文献
3.
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. 相似文献
4.
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. 相似文献
5.
We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies. 相似文献
6.
We present a new formulation of the viscosity in planetary rings, where particles interact through their gravitational forces and direct collisions. In the previous studies on the viscosity in self-gravitating rings, the viscosity consists of three components, which are defined separately in different ways. The complex definitions make it difficult to evaluate the viscosity in N-body simulation of rings. In our new formulation, the viscosity is expressed in terms of changes in orbital elements of particles due to particle interactions. This makes the expression of the viscosity simple. The new formulation gives a simple way to evaluate the viscosity in N-body simulation. We find that for practical evaluation of the viscosity of planetary rings, only energy dissipation at direct inelastic collisions is needed.For tenuous particle disks (i.e., optically thin disks), we further derive a formula of the viscosity. The formula requires only a numerical coefficient that can be obtained from three-body calculation. Since planetesimal disks are also tenuous, the viscosity in planetesimal disks can be also obtained from this formula. In a subsequent paper, we will evaluate this coefficient through three-body calculation and obtain the viscosity for a wide range of parameters such as the restitution coefficient and the radial location in rings. 相似文献
7.
Lucien Froidevaux 《Icarus》1981,46(1):4-17
The variation in infrared equilibrium brightness temperature of Saturn's A, B, and C rings is modeled as a function of solar elevation B′ with respect to the ring plane. The basic model includes estimates of minimum and maximum interparticle shadowing in a monolayer approximation. Simple laboratory observations of random particle distributions at various illumination angles provide more realistic shadowing functions. Radiation balance calculations yield the physical (kinetic) temperature of particles in equilibrium with radiation from the Sun, Saturn, and neighboring particles. Infrared brightness temperatures as a function of B′ are then computed and compared to the available 20-μm data (Pioneer results are also briefly discussed). The A and B rings are well modeled by an optically thick monolayer, or equivalently, a flat sheet, radiating from one side only. This points to a temperature contrast between the two sides, possibly due to particles with low thermal inertia. Other existing models for the B ring are discussed. The good fit for the monolayer model does not rule out the possibility that the A and B rings are many particles thick. It could well be that a multilayer ring produces an infrared behavior (as a function of tilt angle) similar to that of a monolayer. The C ring brightness increases as B′ decreases. This contrast in behavior can be understood simply in terms of the low C ring optical depth and small amount of interparticle shadowing. High-albedo particles (A?0.5) can fit the C ring infrared data if they radiate mostly from one hemisphere due to slow rotation or low thermal inertia (or both). Alternatively, particles isothermal over their surface (owing to a rapid spin, high inertia, or small size), and significantly darker (A?0.3) than the A and B ring particles, can produce a similar brightness variation with ring inclination. In any case, the C ring particles have significantly hotter physical temperatures than the particles in the A and B rings, whether or not the rings form a monolayer. 相似文献
8.
Yoshiyuki Kawata 《Icarus》1983,56(3):453-464
Models of Saturn's rings based on the classical multilayer assumption have been studied in the infrared. Thermal energy balance of Saturn's rings is treated rigorously by solving the infrared radiative transfer equations. It was found that a homogeneous multilayer model is incompatible with the observed infrared brightness variation of the A and B rings, although it can fit that of the C ring. The alternative inhomogeneous multilayer model with dark particles within a bright haze of small icy particles is presented in order to satisfy the available infrared data of the A, B, and C rings. The results based on the inhomogeneous multilayer model may be summarized as follows: The observed infrared brightness data of the three rings are explained in terms of the different optical thickness without having significant differences in the ring-particle properties, such as albedo, spin rate, and sizes. But each ring contains a different amount of bright haze particles and their concentration within the rings depends on whether or not dark particles emit radiation mostly from one hemisphere (slow rotator and/or low thermal inertia). If a dark particle is an isothermal radiator, the possible ranges of A1 and A2 for all three rings are given by 0.9 ? A1 ? 0.95 and 0.0 ? A2 ? 0.15, where A1 and A2 are the bolometric bond albedos of a bright haze and a dark particle, respectively. The possible ranges of the optical thickness ratio X of the dark particle layer to the total ring layer for the rings A, B, and C are given by 0.65 ? X ? 0.75, 0.8 ? X ? 0.9, and 0.8 ? X ? 1.0, respectively. If a dark particle is a slow rotator, we obtain 0.9 ? A1 ? 0.95 and 0.0 ? A2 ? 0.4 for all three rings. The ranges of X for the rings A, B, and C are given by 0.35 ? X ? 0.7, 0.65 ? X ? 0.9, and 0.35 ? X ? 1.0, respectively. In this paper, for the first time, a consistent model is presented which is applicable to all three rings from the multilayer point of view. 相似文献
9.
We present lightcurves and analysis for four new monolithic fast-rotating asteroids: 2000 AG6, 2000 DO8, 2000 EB14, and 2000 HB24. Their rotation periods of 4.60, 1.30, 107.47, and 13.05 min place them well below the critical threshold for the rotation rate of strengthless prolate ellipsoids, as we demonstrate. These four objects join the five previously identified fast-rotating asteroids. The sharp segregation in spin rates between these nine objects and asteroids with more typical spin rates is somewhat puzzling. No observed objects larger than about 200 m spin with rates faster than the critical rate for strengthless prolate ellipsoids, while no objects smaller than 200 m have shown spin rates slower than this critical limit. We hypothesize that these small, fast-rotating objects are representative of the building blocks of the “rubble pile” asteroids and are in fact derived from impacts into already existing “rubble piles.” 相似文献
10.
The confining curves in the general three-body problem are studied; the role of the integralc 2 h (angular momentum squared times energy) as bifurcation parameter is established in a very simple way by using symmetries and changes of scale. It is well known (Birkhoff, 1927) that the bifurcations of the level manifolds of the classical integrals occur at the Euler-Lagrange relative equilibrium configurations. For small values of the mass ratio ε=m 3/m 2 both the positions of the collinear equilibrium points and thec 2 h integral are expanded in power series of ε. In this way the relationship is found between the confining curves resulting from thec 2 h integral in the general problem, and the zero velocity curves given by the Jacobi integral in the corresponding restricted problem. For small values of ε the singular confining curves in the general and in the restricted problem are very similar, but they do not correspond to each other: the offset of the two bifurcation values is, in the usual, system of units of the restricted problem, about one half of the eccentricity squared of the orbits of the two larger bodies. This allows the definition of an approximate stability criterion, that applies to the systems with small ε, and quantifies the qualitatively well known destabilizing effect of the eccentricity of the binary on the third body. Because of this destabilizing effect the third body cannot be bounded by any topological criterion based on the classical integrals unless its mass is larger than a minimum value. As an example, the three-body systems formed by the Sun, Jupiter and one of the small planets Mercury, Mars, Pluto or anyone of the asteroids are found to be ‘unstable’, i.e. there is no way of proving, with the classical integrals, that they cannot cross the orbit of Jupiter. This can be reliably checked with the approximate stability criterion, that given for the most important three-body subsystems of the Solar System essentially the same information on ‘stability’ as the full computation of thec 2 h integral and of the bifurcation values. 相似文献
11.
The collinear equilibrium position of the circular restricted problem with the two primaries at unit distance and the massless body at the pointL 3 is extended to the planar three-body problem with respect to the massm 3 of the third body; the mass ratio μ of the two primaries is considered constant and the constant angular velocity of the straight line on which the three masses stay at rest is taken equal to 1. As regards periodic motions ‘around’ the equilibrium pointL 3, four possible extensions from the restricted to the general problem are presented each of them starting with a simple or a doubly periodic orbit of the family α of the Copenhagen category (μ=0.50). Form 3=0.10, μ=0.50 (i.e. for fixed masses of all three bodies) the characteristic curve of the extended family α is found. The qualitative differences of the families corresponding tom 3=0 andm 3=0.10 are discussed. 相似文献
12.
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. 相似文献
13.
In this paper we have studied the locations and stability of the Lagrangian equilibrium points in the restricted three-body problem under the assumption that both the primaries are finite straight segments. We have found that the triangular equilibrium points are conditional stable for 0<μ<μ c , and unstable in the range μ c <μ≤1/2, where μ is the mass ratio. The critical mass ratio μ c depends on the lengths of the segments and it is observed that the range of μ c increases when compared with the classical case. The collinear equilibrium points are unstable for all values of μ. We have also studied the regions of motion of the infinitesimal mass. It has been observed that the Jacobian constant decreases when compared with the classical restricted three-body problem for a fixed value of μ and lengths l 1 and l 2 of the segments. Beside this we have found the numerical values for the position of the collinear and triangular equilibrium points in the case of some asteroids systems: (i) 216 Kleopatra-951 Gaspara, (ii) 9 Metis-433 Eros, (iii) 22 Kalliope-243 Ida and checked the linear stability of stationary solutions of these asteroids systems. 相似文献
14.
E. Perdios 《Astrophysics and Space Science》1993,199(2):185-188
In the framework of the solar system case (with only the larger primary radiating) of the photogravitational restricted three-body problem we compute and present some non-symmetric asymptotic orbits connecting the outer collinear equilibrium pointL
3 with the neighbourhood of one of the triangular equilibrium pointsL
4, 5. Such orbits have not been found previously in the restricted problem. 相似文献
15.
Linda J. Spilker Stuart H. Pilorz John C. Pearl Shawn M. Brooks Scott G. Edgington F. Michael Flasar Cedric Leyrat 《Planetary and Space Science》2006,54(12):1167-1176
In late 2004 and 2005 the Cassini composite infrared spectrometer (CIRS) obtained spatially resolved thermal infrared radial scans of Saturn's main rings (A, B and C, and Cassini Division) that show ring temperatures decreasing with increasing solar phase angle, α, on both the lit and unlit faces of the ring plane. These temperature differences suggest that Saturn's main rings include a population of ring particles that spin slowly, with a spin period greater than 3.6 h, given their low thermal inertia. The A ring shows the smallest temperature variation with α, and this variation decreases with distance from the planet. This suggests an increasing number of smaller, and/or more rapidly rotating ring particles with more uniform temperatures, resulting perhaps from stirring by the density waves in the outer A ring and/or self-gravity wakes.The temperatures of the A and B rings are correlated with their optical depth, τ, when viewed from the lit face, and anti-correlated when viewed from the unlit face. On the unlit face of the B ring, not only do the lowest temperatures correlate with the largest τ, these temperatures are also the same at both low and high α, suggesting that little sunlight is penetrating these regions.The temperature differential from the lit to the unlit side of the rings is a strong, nearly linear, function of optical depth. This is consistent with the expectation that little sunlight penetrates to the dark side of the densest rings, but also suggests that little vertical mixing of ring particles is taking place in the A and B rings. 相似文献
16.
《New Astronomy》2016
This article examines the effects of the zonal harmonics on the out-of-plane equilibrium points of Robe's circular restricted three-body problem when the hydrostatic equilibrium shape of the first primary is an oblate spheroid, the shape of the second primary is an oblate spheroid with oblateness coefficients up to the second zonal harmonic, and the full buoyancy of the fluid is considered. It is observed that the size of the oblateness and the zonal harmonics affect the positions of the out-of-plane equilibrium points L6 and L7. It is also observed that these points within the possible region of motion are unstable. 相似文献
17.
H. Salo 《Earth, Moon, and Planets》1985,33(2):189-200
Numerical simulations of 200 mutually colliding non-identical particles indicate that the equipartition of random kinetic energy is possible only in systems having a narrow distribution of particle masses. Otherwise the random energy is concentrated on heavy particles. The form of the velocity distribution versus particle mass depends also on the elastic properties of the particles, and on the relative importance of the particle size. If the coefficient of restitution is a weakly decreasing function of impact velocity, a large difference in the equilibrium velocities of largest and smallest particles is possible. On the other hand, if the elasticity drops to a low level even in the small velocity regime, the dispersion of velocities is maintained by finite size and differential rotation, and the velocities of smallest particles are, at most, slightly larger than those of the largest ones. The results of simulations are consistent with the predictions of the collisional theory of non-identical particles (Hämeen-Anttila, 1984). The application to Saturn's rings indicates that the geometric thickness of cm-sized particles is of the order of 50 m in the rarefied regions of the rings. Without the gravitational encounters a thickness of about 30 m is derived. These estimations are made by using the latest measurements (Bridges et al., 1984) for the restitution coefficient of icy particles. 相似文献
18.
Since the Saturn orbit insertion of the Cassini spacecraft in mid-2004, the Cassini composite infrared spectrometer (CIRS) measured temperatures of Saturn’s main rings at various observational geometries. In the present study, we apply our new thermal model (Morishima, R., Salo, H., Ohtsuki, K. [2009]. Icarus 201, 634-654) for fitting to the early phase Cassini data (Spilker, L.J., and 11 colleagues [2006]. Planet. Space Sci. 54, 1167-1176). Our model is based on classical radiative transfer and takes into account the heat transport due to particle motion in the azimuthal and vertical directions. The model assumes a bimodal size distribution consisting of small fast rotators and large slow rotators. We estimated the bolometric Bond albedo, AV, the fraction of fast rotators in cross section, ffast, and the thermal inertia, Γ, by the data fitting at every radius from the inner C ring to the outer A ring. The albedo AV is 0.1-0.4, 0.5-0.7, 0.4, 0.5 for the C ring, the B ring, the Cassini division, and the A ring, respectively. The fraction ffast depends on the ratio of scale height of fast rotators to that of slow rotators, hr. When hr = 1, ffast is roughly half for the entire rings, except for the A ring, where ffast increases from 0.5 to 0.9 with increasing saturnocentric radius. When hr increases from 1 to 3, ffast decreases by 0.2-0.4 for the B and A rings while no change in ffast is seen for the optically thin C ring and Cassini division. The large ffast seen in the outer A ring probably indicates that a large number of small particles detach from large particles in high velocity collisions due to satellite perturbations or self-gravity wakes. The thermal inertia, Γ, is constrained from the efficiency of the vertical heat transport due to particle motion between the lit and unlit faces, and is coupled with the type of vertical motion. We found that in most regions, except for the mid B ring, sinusoidal vertical motion without bouncing is more reasonable than cycloidal motion assuming bouncing at the midplane, because the latter motion gives too large Γ as compared with previous estimations. For the mid B ring, where the optical depth is highest in Saturn’s rings, cycloidal vertical motion is more reasonable than sinusoidal vertical motion which gives too small Γ. 相似文献
19.
Letr 1,r 2,r 3 be arbitrary coordinates of the non-zero interacting mass-pointsm 1,m 2,m 3 and define the distancesR 1=|r 1?r 3|,R 2=|r 2?r 3|,R=|r 1?r 2|. An eight-dimensional regularization of the general three-body problem is given which is based on Kustaanheimo-Stiefel regularization of a single binary and possesses the properties:
- The equations of motion are regular for the two-body collisionsR 1→0 orR 2→0.
- Provided thatR?R 1 orR?R 2, the equations of motion are numerically well behaved for close triple encounters.
20.
We have calculated the radar backscattering characteristics of a variety of compositional and structural models of Saturn's rings and compared them with observations of the absolute value, wavelength dependence, and degree of depolarization of the rings' radar cross section (reflectivity). In the treatment of particles of size comparable to the wavelength of observation, allowance is made for the nonspherical shape of the particles by use of a new semiempirical theory based on laboratory experiments and simple physical principles to describe the particles' single scattering behavior. The doubling method is used to calculate reflectivities for systems that are many particles thick using optical depths derived from observations at visible wavelengths. If the rings are many particles thick, irregular centimeter- to meter-sized particles composed primarily of water ice attain sufficiently high albedos and scattering efficiencies to explain the radar observations. In that case, the wavelength independence of radar reflectivity implies the existence of a broad particle size distribution that is well characterized over the range 1 cm ? r ? m by n(r)dr = n0r?3dr. A narrower size distribution with is also a possibility. Particles of primarily silicate composition are ruled out by the radar observations. Purely metallic particles, either in the above size range and distributed within a many-particle-thick layer or very much larger in size and restricted to a monolayer, may not be ruled out on the basis of existing radar observations. A monolayer of very large ice “particle” that exhibit multiple internal scattering may not yet be ruled out. Observations of the variation of radar reflectivity with the opening angle of the rings will permit further discrimination between ring models that are many particles thick and ring models that are one “particle” thick. 相似文献