Depletion of satellite atoms in a collisionless exosphere by radiation pressure

Hydrogen atoms in Keplerian orbits about a planet are dynamically perturbed by solar Ly α radiation. These perturbations are examined here by analyzing the rates of change of the classical orbital elements, with rather different conclusions from those drawn by Bertaux and Blamont (1973) from numerical integration of sample orbits. There are three main effects: high inclination orbits with eccentricities $\text{e ? 0.4}$ are forced toward the ecliptic plane within a few weeks; the perigees of direct [or retrograde] orbits drift rapidly (i.e., in a few days) toward stable positions roughly westward [or eastward] of the planet; satellite orbits in or near such a stable point rapidly lower their perigees and the satellite's life is ended by a collision in the atmosphere. Thus there are effects tending to diminish the number of highly eccentric orbits with distant apogees in all six principal directions (N, S; Sun, anti-Sun; E, W). The various lifetimes are compared for a sample of initial elements.     

Chaotic behavior and the origin of the 31 Kirkwood gap

The sudden eccentricity increases discovered by J. Wisdom (Astron J.87, 577–593, 1982) are reproduced in numerical integrations of the planar-elliptic restricted three-body problem, verifying that this phenomenon is real. Maximum Lyapunov characteristic exponents for trajectories near the $\text{3}\text{1}$ commensurability are computed both with the mappings presented in Wisdom (1982) and by numerical integration of the planar-elliptic problem. In all cases the agreement is excellent, indicating that the mappings accurately reflect whether trajectories are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the $\text{3}\text{1}$ commensurability, both in the planar-elliptic problem and to a more limited extent in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the $\text{3}\text{1}$ Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.     

Asteroids with large secular orbital variations

As the classical linear theory of secular perturbations for asteroids is known not to be adequate for computing the perturbations of asteroids with high eccentricities and/or inclinations, a seminumerical method to calculate the secular perturbations by including higher-degree terms in the disturbing function has been developed. It is here applied to asteroids with small values of $\text{(1 ? e}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{cos}\text{i}$, since the secular variations as well as their deviations from the results derived by the classical linear theory are generally large for such asteroids. It is found that the arguments of perihelion for five of the numbered asteroids are librating around 90 or 270°. For asteroids with $\text{(1 ? e}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>cos</mtext><mtext>\; i</mtext>}}$ less than 0.85 the results of the secular variations are tabulated. Also the stability of such orbits is discussed by comparing the orbital properties of short-periodic comets with them. Generally speaking, orbits of the asteroids are more stable than those of the short-periodic comets, and asteroids with librating arguments of perihelion are more stable than those with circular coplanar orbits although their orbital elements are changed more by secular perturbations.     

On the original distribution of the asteriods. I.

The depletion of an initially uniform distribution of asteroids extending form Mars to Saturn, caused by the gravitational perturbations of Jupiter and Saturn, is calculated by numerical integration of the asteroid orbits. Almost all (about 85%) the asteroids between Jupiter and Saturn are ejected in the first 6000 years Most of the asteroids between the $\text{2}\text{3}$ Jupiter resonance (4.0 A.U.) and Jupiter are ejected in the first 2400 years with the exception of the stable librators (e.g., the Hilda group). Interior to the $\text{2}\text{3}$ resonance the depletion was small, and interior to the $\text{1}\text{2}$ resonance (3.3 A.U.) no asteroids were ejected in the first 2400 years.     

The range of validity of the two-body approximation in models of terrestrial planet accumulation: I. Gravitational perturbations

Gravitational perturbations in semimajor axis, eccentricity, and inclination resulting from close planetesimal encounters (near 1 AU) out to 10 Tisserand sphere of influence radii were calculated by two- and three-dimensional numerical integration. These are compared with the results of treating the encounter as a two-body problem, as is customary in Monte Carlo calculations of orbital evolution and in numerical and analytical studies of planetary accumulation. It is found that for values of $\text{(}\text{V}\text{V}{}_{\mathrm{e}}\text{) ? 0.35 (V =}\text{relative velocity}\text{, V}{}_{\mathrm{e}}\text{=}\text{escape velocity of largest body}\text{)}$, the two-body body approximation fails to describe the outcome of individual encounters. In this low-velocity region, the two-body “gravitational focusing” cross section is no longer valid; “anomalous gravitational focusing” often leads to bodies on distant unperturbed trajectories becoming close encounters and vice versa. In spite of these differences, average perturbations given by the two-body approximation are valid within a factor of 2 when $\text{V}\text{V}{}_{\mathrm{e}}\text{> 0.07}$. In this same velocity range the “Arnold extrapolation,” whereby a few very close encounters are used to estimate the effect of many more distant encounters, is found to be a useful approximation.     

Evolution of comet orbits under the perturbing influence of the giant planets and nearby stars

The orbital evolution of 500 hypothetical comets during 10⁹ years is studied numerically. It is assumed that the birthplace of such comets was the region of Uranus and Neptune from where they were deflected into very elongated orbits by perturbations of these planets. Then, we adopted the following initial orbital elements: perihelion distances between 20 and 30 AU, inclinations to the ecliptic plane smaller than 20°, and semimajor axes from 5 × 10³ to 5 × 10⁴ AU. Gravitational perturbations by the four giant planets and by hypothetical stars passing at distances from the Sun smaller than 5 × 10⁵ AU are considered. During the simulation, somewhat more than 50% of the comets were lost from the solar system due to planetary or stellar perturbations. The survivors were removed from the planetary region and left as members of what is generally known as the cometary cloud. At the end of the studied period, the semimajor axes of the surviving comets tend to be concentrated in the interval 2 × 10⁴ < a < 3 × 10⁴ AU. The orbital planes of the comets with initial $\text{a ≧ 3 × 10}{}^4\text{AU}$ acquired a complete randomization while the others still maintain a slight predominance of direct orbits. In addition, comet orbits with final a < 6 × 10⁴AU preserve high eccentricities with an average value greater than 0.8 Most “new” comets from the sample entering the region interior to Jupiter's orbit had already registered earlier passages through the planetary region. By scaling up the rate of paritions of hypothetical new comets with the observed one, the number of members of the cometary cloud is estimated to be about 7 × 10¹⁰ and the conclusion is drawn that Uranus and Neptune had to remove a number of comets ten times greater.     

Numerical experiment applicable to the latest stage of planet growth

A three-dimensional numerical model was developed with the goal of studying limited dynamical problems relevant to the latest stage of planet growth in the accretion theory. A small number of large protoplanets (～ Moon size) of different masses, moving around the Sun, are considered. The dynamical evolution and growth of the population is studied under mutual gravitational perturbations, accretion, and collisional fragmentation processes. Gravitational encounters are treated exactly by numerical integration of the N-body problem. Outcomes of collisional fragmentation are modeled according to the results of R. Greenberg et al. (1978, Icarus, 35, 1–26). In the present work, we consider 25 protoplanets with uniform mass distribution in the range 2 × 10²⁵?4 × 10²⁶ g on heliocentric orbits in the Earth zone. These bodies are initially confined to a small volume of space to permit gravitational perturbations by close approaches and collisions within a finite length of integration time. The dynamical evolution of the swarm is followed for four different sets of initial ranges in semimajor axis, eccentricity, and inclination: Δa=0.01, 0.02, 0.04, 0.08 AU; Δe= 0.005, 0.01, 0.02, 0.04; Δi=0°3, 0°6, 1°2, 2°4. Among other results, it is found that average eccentricities and inclinations evolve toward a steady state such that $\text{i}\text{?}\text{1}\text{2}\text{,}\text{e}$; it is also found that, whatever the initial conditions, the population evolves toward a quasi-equilibrium relative velocity distribution corresponding to a Safronov parameter value $\text{θ?10}$. Moreover, the growth process of the growing planet presents very similar behavior in the four cases considered, except for the time scale of evolution, which increases with the initial range of orbital elements. Earlier works of this kind have been presented by L.P. Cox and J.S. Lewis (1980, Icarus, 44, 706–721) and by G.N. Wetherill (1980b, In Geol. Soc. Canad. Spec. Publ., p. 20), although a number of differences exist between the three approaches.     

External perturbations on distant planetary orbits and objects in the Kuiper disk

We investigate the potential importance of molecular cloud and stellar perturbations on the orbits of Pluto and more distant (hypothetical) planets up to 500 AU from the Sun. It is found that stellar and molecular cloud-core perturbations are of roughly equal importance. It also is found that the likelihood of substantial perturbations on Pluto is not insignificant, and that numerous substantial stellar and molecular cloud perturbations are likely to have influenced the orbits of any planets beyond 200 AU. These perturbations may contribute to a prevalence of moderate eccentricities and inclinations for planets beyond the orbit of Neptune, and may be a characteristic of distant planetary orbits in other solar systems. Given the recent discovery of chaotic behavior in Pluto's orbit (Sussman and Wisdom 1988), the effects of external perturbations on the long-term stability of Pluto's orbit warrant continued study.     

Synthetic Proper Elements for Outer Main Belt Asteroids

For the orbits with low to moderate inclination and eccentricity, in the asteroid main belt, the analytically computed proper elements are accurate to a level very close to the best result achievable by any analytical theory. This fundamental limitation results from the infinite web of resonances and because of the occurrence of chaotic motions. Still, there are some regions of the belt in which these proper elements are of degraded accuracy, thus preventing a reliable definition of asteroid families and detailed studies of the dynamical structure. We have used a different method to compute asteroid proper elements, following the approach introduced in the LONGSTOP project to describe the secular dynamics of the major outer planets. By applying purely numerical techniques, we produced so-called synthetic proper elements for a catalog of 10,256 asteroids with osculating semimajor axes between 2.5 and 4.0AU.The procedure consisted of simultaneous integration of asteroid and planetary orbits for 2Myr, with online filtering of the short-periodic perturbations. The output of the integration was spectrally resolved, and the principal harmonics (proper values) extracted from the time series. For each asteroid we have also tested the accuracy and stability in time of the proper elements, and estimated the maximum Lyapunov Characteristic Exponent to monitor the chaotic behaviors. This provided information on the reliability of the data for each orbit, in particular allowing to select 1,852 cases for an extended integration (10Myr) of the orbits showing instability. The results indicate that for more than half of the cases the proper elements have a time stability improved by more than a factor 3 with respect to the elements computed by the previous analytical theory. But of course there are also unstable cases for which the proper elements are less accurate and reliable, the extreme examples being 23 orbits exhibiting hyperbolic escape from the solar system. This form of escape from the asteroid belt could be responsible for a significant mass loss over the age of the solar system.     

Long-period comet flux in the planetary region: Dynamical evolution from the Oort cloud

This study analyzes the evolution of 2 × 10⁵ orbits with initial parameters corresponding to the orbits of comets of the Oort cloud under the action of planetary, galactic, and stellar perturbations over 2 × 10⁹ years. The dynamical evolution of comets of the outer (orbital semimajor axes a > 10⁴ AU) and inner (5 × 10³ < a (AU) < 10⁴) parts of the comet cloud is analyzed separately. The estimates of the flux of “new” and long-period comets for all perihelion distances q in the planetary region are reported. The flux of comets with a > 10⁴ AU in the interval 15 AU < q < 31 AU is several times higher than the flux of comets in the region q < 15 AU. We point out the increased concentration of the perihelia of orbits of comets from the outer cloud, which have passed several times through the planetary system, in the Saturn-Uranus region. The maxima in the distribution of the perihelia of the orbits of comets of the inner Oort cloud are located in the Uranus-Neptune region. “New” comets moving in orbits with a < 2 × 10⁴ AU and arriving at the outside of the planetary system (q > 25 AU) subsequently have a greater number of returns to the region q < 35 AU. The perihelia of the orbits of these comets gradually drift toward the interior of the Solar System and accumulate beyond the orbit of Saturn. The distribution of the perihelia of long-period comets beyond the orbit of Saturn exhibits a peak. We discuss the problem of replenishing the outer Oort cloud by comets from the inner part and their subsequent dynamical evolution. The annual rate of passages of comets of the inner cloud, which replenish the outer cloud, in the region q < 1 AU in orbits with a > 10⁴ AU (～ 5.0 × 10^?14 yr^?1) is one order of magnitude lower than the rate of passage of comets from the outer Oort cloud (～ 9.1 × 10^?13 yr^?1).     

Asteroids in cometary orbits

Orbital integrations are presented for a total of 14 asteroids with perihelia inside 1.7 AU and with aphelion distances in excess of 4 AU, 10 of which were discovered in 1979–1984. The integrations were normally extended over approximately ±1000 years in a three-body model (Sun-Jupiter-asteroid). The effects of uncertainties of starting orbits are not treated in this work, and as far as the real asteroids are concerned, the results should be regarded mostly as preliminary indications. A wide variety of orbital evolutions is found, and some of them evidently belong to the cometary, chaotic type. Three such cases are identified with certainty (1983 SA, 1983 XF, and 1984 BC) and two or three more with various degrees of likelihood. An asteroidal motion is found for the well-observed object 1979 VA. A stable libration around the $\text{2}\text{1}$ resonance is found for 1981 FD, which obviously adds to the Griqua group. A long-lasting libration around the $\text{5}\text{3}$ resonance performed by 1982 YA is probably unstable. Temporary librations are also found for 1983 SA ($\text{4}\text{3}$ resonance) and 1983 XF ($\text{2}\text{1}$ resonance), but these objects appear to transit into irregular motions with close approaches to Jupiter (less than 0.01 AU for 1983 XF). A very rapid large-amplitude ω libration around 90° is found in the future motion of 1983 VA. If this will indeed occur for the real asteroid, the object will oscillate with a period of only 750 years between a main-belt orbit of very high inclination and a low-inclination Apollo-type orbit.     

On the Origin of the Last Meteorite Parent Bodies

The existence of gaps in the perihelion distribution of the orbits of multikilometer-sized asteroids that approach the orbits of terrestrial-group planets is confirmed. This property of the orbits of large asteroids suggests the existence among them of a family of last meteorite parent bodies. Astrophysical data were considered for S-asteroids of the Main Belt and those that approach terrestrial planets. The u–x color index, which is related to the position of the absorption band at 950 nm in the asteroid spectra, was chosen for a qualitative comparison of the surface composition of these asteroids (identical or differing composition). The u – x color-index distributions were analyzed statistically according to the perihelion (q) and mean heliocentric distances (a) of the S-asteroids. It is shown that these distributions are -shaped, peaking at q 1.8 AU and a 2.2 AU. The wings of the distributions can be approximated by linear regressions. A comparison of the u – x color-index mean values for S-asteroids in the regions of the Earth and Mars and of the Main Belt prompts the conclusion that the last meteorite parent bodies in the vicinity of the orbits of the Earth and Mars come primarily from various regions of the asteroid Main Belt.     

Geopotential harmonics of order 15 and even degree,from changes in orbital eccentricity at resonance

When a satellite orbit decaying slowly under the action of air drag experiences 15th-order resonance with the Earth's gravitational field, so that the ground track repeats after 15 rev, the orbital eccentricity may suffer appreciable changes due to perturbations from the gravitational harmonics of order 15 and even degree (16, 18, 20…). In this paper the changes in eccentricity at resonance for six satellites in near-circular orbits at inclinations between 56 and 90° have been analysed to derive 11 pairs of equations linking the harmonic coefficients of order 15 and (even) degree l, $\text{C}{}_{\mathrm{l,15}}\text{and}\text{S}{}_{\mathrm{l,15}}$ in the usual notation. These equations (together with eight constraint equations) are solved to give:

     

14.

A model of the galactic tidal interaction with the oort comet cloud     

John J. Matese  Patrick G. Whitman 《Celestial Mechanics and Dynamical Astronomy》1992,54(1-3):13-35

We consider a model of the in situ Oort cloud which is isotropic with a random distrihution of perihelia directions and angular momenta. The energy distribution adopted has a continuous range of values appropriate for long-period (>200 yr) comets. Only the tidal torque of the Galaxy is included as a perturbation of comet orbits and it is approximated to be that due to a quasi-steady state distribution of matter with disk-like symmetry. The time evolution of all orbital elements can be analytically obtained for this case. In particular, the change in the perihelion distance per orbit and its dependence on other orbital elements is readily found. We further make the assumption that a comet whose perihelion distance was beyond 15 AU during its last passage through the Solar System would have orbit parameters that are essentially unchanged by planetary perturbations. Conversely, if the prior passage was inside 15 AU we assume that planetary perturbations would have removed the comet from the in situ energy distribution accessible by the galactic tide. Comets which had their perihelia changed from beyond 15 AU to within 5 AU in a single orbit are taken to be observable. We are able to track the evolution of 10⁶ comets as they are made observable by the galactic tidal touque. Detailed results are obtained for the predicted distribution of new (0 < 1/ < 10^–4 AU^–1) comets. Further, correlations between orbital elements can be studied. We present predictions of observed distributions and compare them with the random in situ results as well as with the actual observed distributions of class I comets. The predictions are in reasonable agreement with actual observations and, in many cases, are significantly different from random when perihelia directions are separated into galactic northern and southern hemispheres. However the well-known asymmetry in the north-south populations of perihelia remains to be explained. Such an asymmetry is consistent with the dominance of tidal torques today if a major stochastic event produced it in the past since tidal torques are unable to cause the migration of perihelia across the latitude barriers ±26°.6 in the disk model.     

15.

2-Gyr Simulation of the Oort-cloud Formation II. A Close View of the Inner Oort cloud after the First Two Giga-years     

Giuseppe Leto  Marián Jakubík  Tomáš Paulech  Luboš Neslušan  Piotr A. Dybczyński 《Earth, Moon, and Planets》2009,105(2-4):263-266

We simulate the formation of the Oort cloud (OC) till the age of 2 Gyr starting from an initial disc of planetesimals made by 10 038 test particles. The results on the outer part of the distant comet reservoir are reported by Neslu?an et al. (this issue). Here we deal with the evolution of the population and structure at 2 Gyr of the complementary inner part of the Oort cloud. The dynamical evolution of the massless test particles was followed via the numerical integration of their orbits. We considered the perturbations produced by four giant planets assuming they have their current orbits and masses, as well as the perturbations caused by the Galactic tide and passing stars. The efficiency of the formation of inner OC is found to be very low: only about 1.1% of all considered particles ended in this part of the OC. At 2 Gyr, the dynamics of the inner cloud is mainly governed by the dominant z-term of the Galactic tide. The number density of the bodies is proportional to the heliocentric distance, r, as r ^?3.53. The directional distribution of orbits is still strongly inhomogeneous. There are large empty regions in the space angles around the Galactic Equator points with the galactic longitude 90 and 270° (non-rotating frame), or there are only few bodies having the ecliptical latitude higher than +60° or lower than 60°. A strong concentration of objects at the Ecliptic is apparent up to ≈1,000 AU, with a possible—but still not proved—extension to ≈1,500 AU. Beyond r ≈ 6,000 AU, bodies directly above and below the Sun, with respect to the Ecliptic, are absent.     

16.

On the presumed capture origin of Jupiter's outer satellites     

T.A. Heppenheimer 《Icarus》1975,24(2):172-180

The problem of the origin of Jupiter's outer satellites is treated within the framework of the theory of capture through collinear libration points. Lower bounds for the satellites' semimajor axes are found from a corrected rederivation of Bailey's capture theory. Upper bounds are found from a new derivation of the stability limit for satellites, based on Floquet stability theory.It is shown that if the bodies had near-zero relative velocity when passing the libration point, direct orbits would lie outside retrograde orbits, which is not the case for Jupiter. It is found that the dimensions and distributions of the direct group are well explained by libration-point capture with $\text{Jupiter's mass}\text{=}\text{1}\text{1730}$ solar mass, which is interpreted as indicating capture soon after Jupiter's formation. But ad hoc assumptions are required for this capture model to explain the retrograde group. It is concluded that the direct and retrograde groups may have had different mechanisms of origin.     

17.

Exospheric perturbations by radiation pressure: II. Solution for orbits in the ecliptic plane     

Joseph W. Chamberlain 《Icarus》1980,44(3):651-656

An earlier paper gave solutions for the mean time rates of change of orbital elements of satellite atoms in an exosphere influenced by solar radiation pressure. Each element was assumet to beahve independently. Here the instantaneous rates of change for three elements $\text{(e, ω,}\text{and}\text{θ = ω + Ω)}$ are integrated simultaneously for the case of the inclination i = 0. The results (a) confirm the validity of using mean rates when the orbits are tightly bound to the planet and (b) serve as examples to be reproduced by the complicated numerical solutions required for arbitrary inclination. Strongly bound hydrogen atoms perturbed in Earth orbit by radiation pressure do not seem a likely cause of the geotail extending in the anti-Sun direction. Instead, radiation pressure wil cause those particles' orbits to form a broad fan-shaped tail and to deteriorate into the Earth's atmosphere. Whether loosely bound H atoms are plentiful enough to create the geotail depends on their source function versusr; that question is beyond the scope of this paper.     

18.

Collisionless Saturnian rings     

F.Curtis Michel 《Planetary and Space Science》1981,29(10):1137-1142

We show that particles orbiting a central body ($\text{i.e.}$, Saturn's rings) can be assembled into one or more dense ($\text{i.e.}$, opaque) independent rings without interparticle collisions taking place despite the inevitable particle oscillations about the ring plane. The resultant apparent bulk motion is a slow “rolling” motion of the ring, as it orbits, the individual rings describing a “helical” motion. Such rings would only evolve due to external perturbations or (slow) internal gravitational perturbations, since the particles need never collide. This picture opens up the possibility of having hollow rings, for example. Moreover, it is possible that an initially uniform disk of randomly moving particles may spontaneously separate into a series of such rings. The consequence would be a striated disk having virtually zero internal viscosity.     

19.

Detection of the Yarkovsky effect for main-belt asteroids     

《Icarus》2004,170(2):324-342

     

20.

Almost rectilinear halo orbits     

K. C. Howell  J. V. Breakwell 《Celestial Mechanics and Dynamical Astronomy》1984,32(1):29-52

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.     

$\text{l}$	109$\text{C}{}_{\mathrm{l,15}}$	109$\text{S}{}_{\mathrm{l,15}}$
16	?13.7 ± 1.3	?18.5 ± 2.7
18	?42.3 ± 1.8	?34.7 ± 3.4
20	10.5 ± 3.1	29.8 ± 5.2
22	?8.6 ± 3.8	?20.2 ± 7.4

A model of the galactic tidal interaction with the oort comet cloud

We consider a model of the in situ Oort cloud which is isotropic with a random distrihution of perihelia directions and angular momenta. The energy distribution adopted has a continuous range of values appropriate for long-period (>200 yr) comets. Only the tidal torque of the Galaxy is included as a perturbation of comet orbits and it is approximated to be that due to a quasi-steady state distribution of matter with disk-like symmetry. The time evolution of all orbital elements can be analytically obtained for this case. In particular, the change in the perihelion distance per orbit and its dependence on other orbital elements is readily found. We further make the assumption that a comet whose perihelion distance was beyond 15 AU during its last passage through the Solar System would have orbit parameters that are essentially unchanged by planetary perturbations. Conversely, if the prior passage was inside 15 AU we assume that planetary perturbations would have removed the comet from the in situ energy distribution accessible by the galactic tide. Comets which had their perihelia changed from beyond 15 AU to within 5 AU in a single orbit are taken to be observable. We are able to track the evolution of 10⁶ comets as they are made observable by the galactic tidal touque. Detailed results are obtained for the predicted distribution of new (0 < 1/ < 10^–4 AU^–1) comets. Further, correlations between orbital elements can be studied. We present predictions of observed distributions and compare them with the random in situ results as well as with the actual observed distributions of class I comets. The predictions are in reasonable agreement with actual observations and, in many cases, are significantly different from random when perihelia directions are separated into galactic northern and southern hemispheres. However the well-known asymmetry in the north-south populations of perihelia remains to be explained. Such an asymmetry is consistent with the dominance of tidal torques today if a major stochastic event produced it in the past since tidal torques are unable to cause the migration of perihelia across the latitude barriers ±26°.6 in the disk model.     

2-Gyr Simulation of the Oort-cloud Formation II. A Close View of the Inner Oort cloud after the First Two Giga-years

We simulate the formation of the Oort cloud (OC) till the age of 2 Gyr starting from an initial disc of planetesimals made by 10 038 test particles. The results on the outer part of the distant comet reservoir are reported by Neslu?an et al. (this issue). Here we deal with the evolution of the population and structure at 2 Gyr of the complementary inner part of the Oort cloud. The dynamical evolution of the massless test particles was followed via the numerical integration of their orbits. We considered the perturbations produced by four giant planets assuming they have their current orbits and masses, as well as the perturbations caused by the Galactic tide and passing stars. The efficiency of the formation of inner OC is found to be very low: only about 1.1% of all considered particles ended in this part of the OC. At 2 Gyr, the dynamics of the inner cloud is mainly governed by the dominant z-term of the Galactic tide. The number density of the bodies is proportional to the heliocentric distance, r, as r ^?3.53. The directional distribution of orbits is still strongly inhomogeneous. There are large empty regions in the space angles around the Galactic Equator points with the galactic longitude 90 and 270° (non-rotating frame), or there are only few bodies having the ecliptical latitude higher than +60° or lower than 60°. A strong concentration of objects at the Ecliptic is apparent up to ≈1,000 AU, with a possible—but still not proved—extension to ≈1,500 AU. Beyond r ≈ 6,000 AU, bodies directly above and below the Sun, with respect to the Ecliptic, are absent.     

On the presumed capture origin of Jupiter's outer satellites

The problem of the origin of Jupiter's outer satellites is treated within the framework of the theory of capture through collinear libration points. Lower bounds for the satellites' semimajor axes are found from a corrected rederivation of Bailey's capture theory. Upper bounds are found from a new derivation of the stability limit for satellites, based on Floquet stability theory.It is shown that if the bodies had near-zero relative velocity when passing the libration point, direct orbits would lie outside retrograde orbits, which is not the case for Jupiter. It is found that the dimensions and distributions of the direct group are well explained by libration-point capture with $\text{Jupiter's mass}\text{=}\text{1}\text{1730}$ solar mass, which is interpreted as indicating capture soon after Jupiter's formation. But ad hoc assumptions are required for this capture model to explain the retrograde group. It is concluded that the direct and retrograde groups may have had different mechanisms of origin.     

Exospheric perturbations by radiation pressure: II. Solution for orbits in the ecliptic plane

An earlier paper gave solutions for the mean time rates of change of orbital elements of satellite atoms in an exosphere influenced by solar radiation pressure. Each element was assumet to beahve independently. Here the instantaneous rates of change for three elements $\text{(e, ω,}\text{and}\text{θ = ω + Ω)}$ are integrated simultaneously for the case of the inclination i = 0. The results (a) confirm the validity of using mean rates when the orbits are tightly bound to the planet and (b) serve as examples to be reproduced by the complicated numerical solutions required for arbitrary inclination. Strongly bound hydrogen atoms perturbed in Earth orbit by radiation pressure do not seem a likely cause of the geotail extending in the anti-Sun direction. Instead, radiation pressure wil cause those particles' orbits to form a broad fan-shaped tail and to deteriorate into the Earth's atmosphere. Whether loosely bound H atoms are plentiful enough to create the geotail depends on their source function versusr; that question is beyond the scope of this paper.     

Collisionless Saturnian rings

We show that particles orbiting a central body ($\text{i.e.}$, Saturn's rings) can be assembled into one or more dense ($\text{i.e.}$, opaque) independent rings without interparticle collisions taking place despite the inevitable particle oscillations about the ring plane. The resultant apparent bulk motion is a slow “rolling” motion of the ring, as it orbits, the individual rings describing a “helical” motion. Such rings would only evolve due to external perturbations or (slow) internal gravitational perturbations, since the particles need never collide. This picture opens up the possibility of having hollow rings, for example. Moreover, it is possible that an initially uniform disk of randomly moving particles may spontaneously separate into a series of such rings. The consequence would be a striated disk having virtually zero internal viscosity.     

Almost rectilinear halo orbits

Numerical studies over the entire range of mass-ratios in the circular restricted 3-body problem have revealed the existence of families of three-dimensional halo periodic orbits emanating from the general vicinity of any of the 3 collinear Lagrangian libration points. Following a family towards the nearer primary leads, in 2 different cases, to thin, almost rectilinear, orbits aligned essentially perpendicular to the plane of motion of the primaries. (i) If the nearer primary is much more massive than the further, these thin L₃-family halo orbits are analyzed by looking at the in-plane components of the small osculating angular momentum relative to the larger primary and at the small in-plane components of the osculating Laplace eccentricity vector. The analysis is carried either to 1st or 2nd order in these 4 small quantities, and the resulting orbits and their stability are compared with those obtained by a regularized numerical integration. (ii) If the nearer primary is much less massive than the further, the thin L₁-family and L₂-family halo orbits are analyzed to 1st order in these same 4 small quantities with an independent variable related to the one-dimensional approximate motion. The resulting orbits and their stability are again compared with those obtained by numerical integration.