A Monte Carlo investigation of Jovian perturbations on short-period comet orbits

We present statistical distributions of Jovian perturbations on short-period comet orbits resulting from accurate numerical integrations. Our sample of 60, 000 cometary orbits with low inclinations and random orientations is characterized by perihelia between 0 and 7 AU and aphelia between 4 and 13 AU. The perturbations considered are those experienced because of Jupiter's gravitation per orbital revolution by the comets. Regularization and accurate step-length control in the numerical integration gives statistical results appreciably different from those computed by Rickman and Vaghi (1978). Their use of a crude method of integration led to erroneous results for close encounters. Strong asymmetries of the $\text{δ(}\text{1}\text{a}\text{)}$ distributions, in particular for the extreme tails, are observed for perihelion- or aphelion-tangent orbits. These orbits are also shown to experience the strongest energy perturbations on the average. Some results concerning the perturbations of Tisserand parameters are indicated. The perturbation distributions for the angular elements are described and discussed. The role of the minimum distance from Jupiter as an indicator of perturbations is investigated.     

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.     

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.     

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.     

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.     

Saturn's rings. I. Optical thickness of rings A,B, D and structure of ring B

A photometric study of high-resolution (～0″.3) plates of Saturn taken at the Lowell Observatory in 1943 and 1945 is presented. N-S scans were taken over both the planet and rings. The excess brightness due to the planet seen through the rings is found by taking the difference between the central meridian (CM) scans and scans displaced by 5″.7. Adopting a value for the albedo of the planet, it is possible to obtain the optical thickness, τ_CM(r). In particular, for the regions of maximum brightness in rings A and B, we find τ_CM(I_{A max}) = 0.38 ± 0.11 and τ_CM(I_{B max}) = 0.61 ± 0.11. Observations by Barnard made in 1890 show evidence of ring D, recently discovered by Guerin (1969). The value for the optical thickness of this ring is τ_D(I_{D max}) = 0.03 ± 0.01. Ring B exhibits a pronounced (7–10%) decrease in brightness from the extremity of the major axis to the CM. After considering several possible explanations, we conclude that the ring particles are nonspherical and are in synchronous rotation around the planet with their long axis toward it. The mean value for the ratio of major to minor axis for the particles at 15″ is $\text{(a/b)}\text{? 1.08}$. Because of the shape and orientation of the particles, the optical thickness at the extremity of the major axis and at the CM are different for any saturnicentric latitude B ≠ 90°. Under these circumstances, only a minimum value for τ at the extremity can be derived.     

Families of periodic orbits in the general three-body problem for the Sun-Jupiter-Saturn mass-ratio and their stability

Several families of planar planetary-type periodic orbits in the general three-body problem, in a rotating frame of reference, for the Sun-Jupiter-Saturn mass-ratio are found and their stability is studied. It is found that the configuration in which the orbit of the smaller planet is inside the orbit of the larger planet is, in general, more stable.We also develop a method to study the stability of a planar periodic motion with respect to vertical perturbations. Planetary periodic orbits with the orbits of the two planets not close to each other are found to be vertically stable. There are several periodic orbits that are stable in the plane but vertically unstable and vice versa. It is also shown that a vertical critical orbit in the plane can generate a monoparametric family of three-dimensional periodic orbits.     

Hyperion: Collisional disruption of a resonant satellite

Hyperion is an irregularly shaped object of about 285 km in mean diameter, which appears as the likely remmant of a catastrophic collisional evolution. Since the peculiar orbit of this satellite (in $\text{4}\text{3}$ resonance locking with Titan) provides an effective mechanism to prevent any reaccretion of secondary fragments originated in a breakup event, the present Hyperion is probably the “core” of a disrupted precursor. This contrasts with the other, regularly shaped small satellites of Saturn, which, according to B.A. Smith et al. [Science215, 504–537 (1982)], were disrupted several times but could reaccrete from narrow rings of collisional fragments. The numerical experiments performed to explore the region of the phase space surrounding the present orbit show that most fragments ejected with a relative velocity $\text{?0.1}\text{km}\text{/}\text{sec}$ rapidly attain chaotic-type orbits, having repeated close encounters with Titan. Ejection velocities of this order of magnitude are indeed expected for a collision at a velocity of ～ 10 km/sec with a projectile-to-target mass ratio of the order of 10^?3; similar effects could be produced by less energetic but nearly grazing collisions. Such events are not likely to displace the largest remnant (i.e., the present Hyperion) outside the stable region of the phase space associated with the resonance, but could be responsible for the large amplitude of the observed orbital libration.     

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.     

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:

     

15.

Ionospheric despatch centre in Europe     

《Physics and Chemistry of the Earth, Part C: Solar, Terrestrial & Planetary Science》1999,24(4):355-357

     

16.

Detection of the Yarkovsky effect for main-belt asteroids     

《Icarus》2004,170(2):324-342

     

17.

An efficient parallel algorithm for O(N2) direct summation method and its variations on distributed-memory parallel machines     

《New Astronomy》2002,7(7):373-384

     

18.

Extrasolar Planetary Systems     

Ksanfomaliti  L. V. 《Solar System Research》2000,34(6):481-495

The discovery of planetary systems around alien stars is an outstanding achievement of recent years. The idea that the Solar System may be representative of planetary systems in the Galaxy in general develops upon the knowledge, current until the last decade of the 20th century, that it is the only object of its kind. Studies of the known planets gave rise to a certain stereotype in theoretical research. Therefore, the discovery of exoplanets, which are so different from objects of the Solar System, alters our basic notions concerning the physics and very criteria of normal planets. A substantial factor in the history of the Solar System was the formation of Jupiter. Two waves of meteorite bombardment played an important role in that history. Ultimately there arose a stable low-entropy state of the Solar System, in which Jupiter and the other giants in stable orbits protect the inner planets from impacts by dangerous celestial objects, reducing this danger by many orders of magnitude. There are even variants of the anthropic principle maintaining that life on Earth owes its genesis and development to Jupiter. Some 20 companions more or less similar to Jupiter in mass and a few infrared dwarfs, have been found among the 500 solar-type stars belonging to the main sequence. Approximately half of the exoplanets discovered are of the hot-Jupiter type. These are giants, sometimes of a mass several times that of Jupiter, in very low orbits and with periods of 3–14 days. All of their parent stars are enriched with heavy elements, [Fe/H] = 0.1–0.2. This may indicate that the process of exoplanet formation depends on the chemical composition of the protoplanetary disk. The very existence of exoplanets of the hot-Jupiter type considered in the context of new theoretical work comes up against the problem of the formation of Jupiter in its real orbit. All the exoplanets in orbits with a semimajor axis of more than 0.15–0.20 astronomical units (AU) have orbital eccentricities of more than 0.1, in most cases of 0.2–0.5. In conjunction with their possible migration into the inner reaches of the Solar System, this poses a threat to the very existence of the inner planets. Recent observations of gas–dust clouds in very young stars show that hydrogen dissipates rapidly, in several million years, and dissipation is completed earlier than, according to the accretion theory, the gas component of such a planet as Jupiter forms. The mass of the remaining hydrogen is usually small, much smaller than Jupiter's mass. However, the giant planets of the Solar System retain a few percent of the amount of hydrogen that should be contained in the early protoplanetary disk, creating difficulties in understanding their formation. A plausible explanation is that gravitational instabilities in the protoplanetary disk could be the mechanism of their rapid formation.     

19.

Urey prize lecture: Chaotic dynamics in the solar system     

《Icarus》1987,72(2):241-275

     

20.

Relative crater production rates on planets     

William K. Hartmann 《Icarus》1977,31(2):260-276

Dynamical histories of planetesimals in specified orbits, calculated by Wetherill (1975) and others, have estimates of relative numbers of impacts on different planets. These impact rates, $\text{F}$, are converted to crater production rates, F, by means of tables developed in this paper. Conversions are dependent on impact velocity and surface gravity. Crater retention ages can then be derived from (crater density)/(crater production rate). Such calculations of impact rates and their histories give the only basis, independent of sample dating, for establishing absolute geologic histories of the planets, contrary to published implications that this can be done by comparison of photos alone. A survey of the results, from orbits of interplanetary objects studied to date, indicates that the terrestial planets have crater production rates within a factor ten of each other, and that planet's crater retention ages can probably be determined with a factor of ±3. Further calculations of orbital histories of additional interplanetary bodies are suggested to put photogeologic analyses from spacecraft imagery on a firmer basis.Applications to Mars, as an example, using least-squares fits to crater-count data, suggest an average age of 0.3 to 3 b.y. for two types of channels. The Tharsis volcanics are found to be slightly younger than the channels (strongly confirmed by photomorphology since they are not cut by channels) and Olympus Mons is about 0.06 to 0.6 b.y. old, contrary to recent assertions that Olympus Mons is 2.5 b.y. old and most Martian volcanic provinces older than 3 b.y. Data strongly support the hypothesis that Martian channels formed in a fluvial climate that persisted on Mars until the Tharsis volcanism caused a change in the Martian obliquity state, as outlined by Toon, Ward, and Burns (1977).     

$\text{l}$	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

Extrasolar Planetary Systems

The discovery of planetary systems around alien stars is an outstanding achievement of recent years. The idea that the Solar System may be representative of planetary systems in the Galaxy in general develops upon the knowledge, current until the last decade of the 20th century, that it is the only object of its kind. Studies of the known planets gave rise to a certain stereotype in theoretical research. Therefore, the discovery of exoplanets, which are so different from objects of the Solar System, alters our basic notions concerning the physics and very criteria of normal planets. A substantial factor in the history of the Solar System was the formation of Jupiter. Two waves of meteorite bombardment played an important role in that history. Ultimately there arose a stable low-entropy state of the Solar System, in which Jupiter and the other giants in stable orbits protect the inner planets from impacts by dangerous celestial objects, reducing this danger by many orders of magnitude. There are even variants of the anthropic principle maintaining that life on Earth owes its genesis and development to Jupiter. Some 20 companions more or less similar to Jupiter in mass and a few infrared dwarfs, have been found among the 500 solar-type stars belonging to the main sequence. Approximately half of the exoplanets discovered are of the hot-Jupiter type. These are giants, sometimes of a mass several times that of Jupiter, in very low orbits and with periods of 3–14 days. All of their parent stars are enriched with heavy elements, [Fe/H] = 0.1–0.2. This may indicate that the process of exoplanet formation depends on the chemical composition of the protoplanetary disk. The very existence of exoplanets of the hot-Jupiter type considered in the context of new theoretical work comes up against the problem of the formation of Jupiter in its real orbit. All the exoplanets in orbits with a semimajor axis of more than 0.15–0.20 astronomical units (AU) have orbital eccentricities of more than 0.1, in most cases of 0.2–0.5. In conjunction with their possible migration into the inner reaches of the Solar System, this poses a threat to the very existence of the inner planets. Recent observations of gas–dust clouds in very young stars show that hydrogen dissipates rapidly, in several million years, and dissipation is completed earlier than, according to the accretion theory, the gas component of such a planet as Jupiter forms. The mass of the remaining hydrogen is usually small, much smaller than Jupiter's mass. However, the giant planets of the Solar System retain a few percent of the amount of hydrogen that should be contained in the early protoplanetary disk, creating difficulties in understanding their formation. A plausible explanation is that gravitational instabilities in the protoplanetary disk could be the mechanism of their rapid formation.     

Relative crater production rates on planets

Dynamical histories of planetesimals in specified orbits, calculated by Wetherill (1975) and others, have estimates of relative numbers of impacts on different planets. These impact rates, $\text{F}$, are converted to crater production rates, F, by means of tables developed in this paper. Conversions are dependent on impact velocity and surface gravity. Crater retention ages can then be derived from (crater density)/(crater production rate). Such calculations of impact rates and their histories give the only basis, independent of sample dating, for establishing absolute geologic histories of the planets, contrary to published implications that this can be done by comparison of photos alone. A survey of the results, from orbits of interplanetary objects studied to date, indicates that the terrestial planets have crater production rates within a factor ten of each other, and that planet's crater retention ages can probably be determined with a factor of ±3. Further calculations of orbital histories of additional interplanetary bodies are suggested to put photogeologic analyses from spacecraft imagery on a firmer basis.Applications to Mars, as an example, using least-squares fits to crater-count data, suggest an average age of 0.3 to 3 b.y. for two types of channels. The Tharsis volcanics are found to be slightly younger than the channels (strongly confirmed by photomorphology since they are not cut by channels) and Olympus Mons is about 0.06 to 0.6 b.y. old, contrary to recent assertions that Olympus Mons is 2.5 b.y. old and most Martian volcanic provinces older than 3 b.y. Data strongly support the hypothesis that Martian channels formed in a fluvial climate that persisted on Mars until the Tharsis volcanism caused a change in the Martian obliquity state, as outlined by Toon, Ward, and Burns (1977).