小行星主带上单个行星的质量上限

介绍了N体模拟的Hermite算法,并利用该算法研究了不同质量行星在小行星主带上轨道的演化情况．采用的演化模型是太阳系N体模型(N=7),即把水星、金星、地球的质量加到太阳上,忽略冥王星,同时在小行星主带附近增加一个假想行星,系统演化时间为1亿年．数值模拟显示能够稳定存在于小行星主带上的单个天体的质量上限其量级为10~(25)kg．模拟同时还显示在某些情况下,假想行星与木星之间的低阶共振可以增强系统的稳定性．     

Building the terrestrial planets: Constrained accretion in the inner Solar System

To date, no accretion model has succeeded in reproducing all observed constraints in the inner Solar System. These constraints include: (1) the orbits, in particular the small eccentricities, and (2) the masses of the terrestrial planets - Mars’ relatively small mass in particular has not been adequately reproduced in previous simulations; (3) the formation timescales of Earth and Mars, as interpreted from Hf/W isotopes; (4) the bulk structure of the asteroid belt, in particular the lack of an imprint of planetary embryo-sized objects; and (5) Earth’s relatively large water content, assuming that it was delivered in the form of water-rich primitive asteroidal material. Here we present results of 40 high-resolution (N = 1000-2000) dynamical simulations of late-stage planetary accretion with the goal of reproducing these constraints, although neglecting the planet Mercury. We assume that Jupiter and Saturn are fully-formed at the start of each simulation, and test orbital configurations that are both consistent with and contrary to the “Nice model”. We find that a configuration with Jupiter and Saturn on circular orbits forms low-eccentricity terrestrial planets and a water-rich Earth on the correct timescale, but Mars’ mass is too large by a factor of 5-10 and embryos are often stranded in the asteroid belt. A configuration with Jupiter and Saturn in their current locations but with slightly higher initial eccentricities (e = 0.07-0.1) produces a small Mars, an embryo-free asteroid belt, and a reasonable Earth analog but rarely allows water delivery to Earth. None of the configurations we tested reproduced all the observed constraints. Our simulations leave us with a problem: we can reasonably satisfy the observed constraints (except for Earth’s water) with a configuration of Jupiter and Saturn that is at best marginally consistent with models of the outer Solar System, as it does not allow for any outer planet migration after a few Myr. Alternately, giant planet configurations which are consistent with the Nice model fail to reproduce Mars’ small size.     

Close encounters of a rotating star with planets in parabolic orbits of varying inclination and the formation of hot Jupiters

In this paper we extend the theory of close encounters of a giant planet on a parabolic orbit with a central star developed in our previous work (Ivanov and Papaloizou in MNRAS 347:437, 2004; MNRAS 376:682, 2007) to include the effects of tides induced on the central star. Stellar rotation and orbits with arbitrary inclination to the stellar rotation axis are considered. We obtain results both from an analytic treatment that incorporates first order corrections to normal mode frequencies arising from stellar rotation and numerical treatments that are in satisfactory agreement over the parameter space of interest. These results are applied to the initial phase of the tidal circularisation problem. We find that both tides induced in the star and planet can lead to a significant decrease of the orbital semi-major axis for orbits having periastron distances smaller than 5?C6 stellar radii with tides in the star being much stronger for retrograde orbits compared to prograde orbits. Assuming that combined action of dynamic and quasi-static tides could lead to the total circularisation of orbits this corresponds to observed periods up to 4?C5 days. We use the simple Skumanich law to characterise the rotational history of the star supposing that the star has its rotational period equal to one month at the age of 5 Gyr. The strength of tidal interactions is characterised by circularisation time scale, t _ev , which is defined as a typical time scale of evolution of the planet??s semi-major axis due to tides. This is considered as a function of orbital period P _obs , which the planet obtains after the process of tidal circularisation has been completed. We find that the ratio of the initial circularisation time scales corresponding to prograde and retrograde orbits, respectively, is of order 1.5?C2 for a planet of one Jupiter mass having P _obs ~ 4 days. The ratio grows with the mass of the planet, being of order five for a five Jupiter mass planet with the same P _orb . Note, however, this result might change for more realistic stellar rotation histories. Thus, the effect of stellar rotation may provide a bias in the formation of planetary systems having planets on close orbits around their host stars, as a consequence of planet?Cplanet scattering, which favours systems with retrograde orbits. The results reported in the paper may also be applied to the problem of tidal capture of stars in young stellar clusters.     

Interplanetary dust particles near Jupiter

We calculate the expected counting rate of a flat micrometeoroid detector of finite sensitivity passing in hyperbolic orbit near a planet. We assume that the distribution of particle sizes, s, can be expressed as a power law spectrum of index p, i.e. dN(s) = Cs^?pds, and also that the particles encounter the sphere of influence of the planet with a certain speed v_∞. The results of the calculations are then compared with the results returned by Pioneer 10 in its flyby of Jupiter. The observed increase in impact rate near Jupiter can be completely explained in terms of gravitational “focusing” of particles which are in heliocentric orbits; i.e., they are not in orbit about Jupiter. The absolute concentration of particles near the orbit of Jupiter is of the same order as at 1 AU: the exact ratio being a function of particle speed and spectral index. Data from one flyby are insufficient to determine a unique value for both the spectral index, p, and the particle velocity, v_∞, but limits can be set. For reasonable encounter speeds (corresponding to eccentricities and inclinations of dust particles experienced near the Earth), the particles near Jupiter are characterized by a spectrum of index p ～ 3. The spectral index which best fits the data increases with increasing encounter speeds.     

Planet mass stability margin of two-planetary systems

The Lagrange stability of the Sun-Jupiter-Saturn and 47 UMa two-planetary systems at a time scale of 10⁶ yr was studied using the method of averaging. When the masses of Jupiter and Saturn increase by 19 times, these planets can closely converge. The study of Lagrange stability in the case of successive mass increase allows for the obtainment of upper estimates of possible masses of extrasolar planets. Close approachs in the 47 UMa system are possible when minimal masses increase by 38 times. approachs are revealed when analyzing osculating elements; in averaged elements, approachs are absent. Resonant properties of six extrasolar two-planetary systems where the outer planet is less massive than the inner one are studied. The values of semi-major axes of planet orbits in HD 82943 and HD 73526 systems lie in a narrow resonant zone; in 47 UMa, μ Ara and HD 108874 systems lie in a wide resonant zone. In the HD 12661, the system resonances of a lower order were not revealed.     

A Jupiter fragmented comet: Cause of the K/T boundary record

The extended period of mass extinctions around the K/T boundary correlating with extraterrestrial amino acids in the sediment record constitutes strong evidence of a cometary cause. The input of extraterrestrial matter over 10⁵ yr supports the hypothesis of a giant comet, fragmented into subcomets on close encounter with Jupiter, and subsequently perturbed into Earth-crossing orbits. Copious amounts of dust were emitted via this and possibly successive fragmenting encounters, and via normal cometary evaporation. The dynamics of dust from the disintegrating comet fragments favours retention in Earth-crossing orbits of the sub-micron fraction of organic composition. The shroud of dust accreted in the Earth's upper atmosphere varied with time and imposed climatic stresses that caused species extinctions over 10⁵ yr. While the iridium peak in the sediments coincides with the Chicxulub crater impactor, other iridium detail suggests that some of the impactor material was reinjected into space and in part re-accreted by Earth from the interplanetary orbits.     

Planets in the asteroid belt

Abstract— The main asteroid belt has lost >99.9% of its solid mass since the time at which the planets were forming, according to models for the protoplanetary nebula. Here we show that the primordial asteroid belt could have been cleared efficiently if much of the original mass accreted to form planetsized bodies, which were capable of perturbing one another into unstable orbits. We provide results from 25 N‐body integrations of up to 200 planets in the asteroid belt, with individual masses in the range 0.017–0.33 Earth masses. In the simulations, these bodies undergo repeated close encounters which scatter one another into unstable resonances with the giant planets, leading to collision with the Sun or ejection from the solar system. In response, the giant planets' orbits migrate radially and become more circular. This reduces the size of the main‐belt resonances and the clearing rate, although clearing continues. If ～3 Earth masses of material was removed from the belt this way, Jupiter and Saturn would initially have had orbital eccentricities almost twice their current values. Such orbits would have made Jupiter and Saturn 10–100x more effective at clearing material from the belt than they are on their current orbits. The time required to remove 90% of the initial mass from the belt depends sensitively on the giant planets' orbits, and weakly on the masses of the asteroidal planets. 18 of the 25 simulations end with no planets left in the belt, and the clearing takes up to several hundred million years. Typically, the last one or two asteroidal planets are removed by interactions with planets in the terrestrial region     

A former asteroidal planet as the origin of comets

The idea of a missing planet between Mars and Jupiter has been with us since the formulation of the Titius-Bode law. The discovery of the asteroid belt in that location led to speculation about a planetary breakup event. Both ideas remained conjectures until Ovenden's finding in 1972, from which it could be derived that the mass of the missing planet was about 90 Earth masses and that its breakup was astronomically recent. Apparently much of that mass was blown out of the solar system during the disruption of the planet. Because of the action of planetary perturbations, only two types of orbits of surviving fragments could remain at present-asteroid orbits and once-around very-long-period elliptical orbits. Objects in the latter type of orbit are known to exist-the very-long-period comets. A large number of these are on elliptical trajectories with periods of revolution of 5 million years; yet they are known to have made no more than one revolution in an orbit passing close to the Sun. By direct calculation it is possible to predict the distribution of the orbital elements of objects moving on long-period ellipses which might have originated in a breakup event in the asteroid belt 5 million years ago. The comet orbits have the predicted distribution in every case where a measure is possible. Some of the distribution anomalies, such as a bias in the directions of perihelion passage, are statistically strong and would be difficult to explain in any other uncontrived way. In addition, a relative deficiency of orbits with perihelia less than 1 AU indicates that the comets must have had small perihelion distances since their origin, rather than that they have been perturbed into small perihelion orbits from a distant “cloud” of comets by means of stellar encounters. The comet orbital data lead to the conclusion that all comets originated in a breakup event in the asteroid belt (5.5±0.6) × 10⁶ years ago. Asteroid and meteoritic evidence can now be interpreted in a way which not only is supportive but also provides fresh insights into understanding their physical, chemical, and dynamical properties. Particularily noteworthy are the young cosmic-ray exposure ages of meteorites, evidence of a previous high-temperature/pressure environment and of chemical differentiation of the parent body, and compositional similarities among comets, asteroids, and meteorites. Certain “explosion signatures” in asteroid orbital element distributions are likewise indicative. Tektites may also have originated in the same event; but if so, there are important implications regarding the absolute accuracy of certain geological dating methods. Little is known about possible planetary breakup mechanisms of the requisite type, though some speculations are offered. In any case, the asteroid belt is an existing fact; and the arguments presented here that a large planet did disintegrate 5 million years ago must be judged on their merits, even in the absence of a suitable theory of planetary explosions.     

Orbital resonance in a dissipative medium

Orbital resonances tend to force bodies into noncircular orbits. If a body is also under the influence of an eccentricity-reducing medium, it will experience a secular change in semimajor axis which may be positive or negative depending on whether its orbit is exterior or interior to that of the perturbing body. Thus a dissipative medium can promote either a loss or a gain in orbital energy. This process may explain the resonant structure of the asteroid belt and of Saturn's rings. For reasonable early solar system parameters, it would clear a gap near the 2:1 resonance with Jupiter on a time scale of a few thousand years; the gap width would be comparable to the Kirkwood gap presently at the location in the asteroid belt. Similarly, a gap comparable in width to Cassini's division would be cleared in Saturn's rings at the 2:1 resonance with Mimas in ～10⁶ yr. Most of the material from the gap would be deposited at the outer edge of ring B. The process would also affect the radial distribution of preplanetary material. Moreover, it provides an explanation for the large amplitude of the Titan-Hyperion libration. Consideration of the effects of dissipation on orbits near the stable L₄ and L₅ points of the restricted three-body problem indicates that energy loss causes particles to move away from these points. This results explains the large amplitude of Trojan asteroids about these points and the possible capture of Trojan into orbit about Jupiter.     

Eccentric Extrasolar Planets: The Jumping Jupiter Model

Most extrasolar planets discovered to date are more massive than Jupiter, in surprisingly small orbits (semimajor axes less than 3 AU). Many of these have significant orbital eccentricities. Such orbits may be the product of dynamical interactions in multiplanet systems. We examine outcomes of such evolution in systems of three Jupiter-mass planets around a solar-mass star by integration of their orbits in three dimensions. Such systems are unstable for a broad range of initial conditions, with mutual perturbations leading to crossing orbits and close encounters. The time scale for instability to develop depends on the initial orbital spacing; some configurations become chaotic after delays exceeding 10⁸ y. The most common outcome of gravitational scattering by close encounters is hyperbolic ejection of one planet. Of the two survivors, one is moved closer to the star and the other is left in a distant orbit; for systems with equal-mass planets, there is no correlation between initial and final orbital positions. Both survivors may have significant eccentricities, and the mutual inclination of their orbits can be large. The inner survivor's semimajor axis is usually about half that of the innermost starting orbit. Gravitational scattering alone cannot produce the observed excess of “hot Jupiters” in close circular orbits. However, those scattered planets with large eccentricities and small periastron distances may become circularized if tidal dissipation is effective. Most stars with a massive planet in an eccentric orbit should have at least one additional planet of comparable mass in a more distant orbit.     

Model consideration of the bombardment event of the asteroidal belt by the planetesimals scattered from the Jupiter zone

The temporal evolutions of the planetesimals scattered from the Jupiter zone for different masses of the proto-Jupiter [(a) 0.1 and (b) 1.0 of the present mass] are investigated. Due to the combined effects of the orbital evolution of the planetesimals and the elimination of these projectiles either via impact capture or injection into escape velocity by the outer planets, the whole scattering process lasts about 10⁸ yr for case (a) and about 10⁷ yr for case (b). The longer time scale may be a good estimate for the accretion time interval of Jupiter while the shorter one (10⁷) gives the upper time limit of the late heavy-bombardment epoch of the terrestrial planets due to planetesimals scattered from the Jupiter zone. The limiting value of the encounter velocity U at the end of the scattering process is ≈0.6. Consideration of the collisional interaction of these projectiles with the asteroids indicates that the corresponding bombardment effect could be rather appreciable. Also, the asteroids on the inner edge of the main asteroid belt would have been bombarded more severely than those on the outer edge. From this point of view, the structure of the asteroidal belt could be affected significantly not only by Jupiter's gravitational perturbation effect but also by its early scattering process.     

Change in the orbital period of a binary system due to dynamical tides for main-sequence stars

We investigate the change in the orbital period of a binary system due to dynamical tides by taking into account the evolution of a main-sequence star. Three stars with masses of one, one and a half, and two solar masses are considered. A star of one solar mass at lifetimes t = 4.57 × 10⁹ yr closely corresponds to our Sun. We show that a planet of one Jupiter mass revolving around a star of one solar mass will fall onto the star in the main-sequence lifetime of the star due to dynamical tides if the initial orbital period of the planet is less than P _orb ≈ 2.8 days. Planets of one Jupiter mass with an orbital period P _orb ≈ 2 days or shorter will fall onto a star of one and a half and two solar masses in the mainsequence lifetime of the star.     

Linking the collisional history of the main asteroid belt to its dynamical excitation and depletion

The main belt is believed to have originally contained an Earth mass or more of material, enough to allow the asteroids to accrete on relatively short timescales. The present-day main belt, however, only contains ∼5×¹⁰⁻⁴ Earth masses. Numerical simulations suggest that this mass loss can be explained by the dynamical depletion of main belt material via gravitational perturbations from planetary embryos and a newly-formed Jupiter. To explore this scenario, we combined dynamical results from Petit et al. [Petit, J. Morbidelli, A., Chambers, J., 2001. The primordial excitation and clearing of the asteroid belt. Icarus 153, 338-347] with a collisional evolution code capable of tracking how the main belt undergoes comminution and dynamical depletion over 4.6 Gyr [Bottke, W.F., Durda, D., Nesvorny, D., Jedicke, R., Morbidelli, A., Vokrouhlický, D., Levison, H., 2005. The fossilized size distribution of the main asteroid belt. Icarus 175, 111-140]. Our results were constrained by the main belt's size-frequency distribution, the number of asteroid families produced by disruption events from diameter D>100 km parent bodies over the last 3-4 Gyr, the presence of a single large impact crater on Vesta's intact basaltic crust, and the relatively constant lunar and terrestrial impactor flux over the last 3 Gyr. We used our model to set limits on the initial size of the main belt as well as Jupiter's formation time. We find the most likely formation time for Jupiter was 3.3±2.6 Myr after the onset of fragmentation in the main belt. These results are consistent with the estimated mean disk lifetime of 3 Myr predicted by Haisch et al. [Haisch, K.E., Lada, E.A., Lada, C.J., 2001. Disk frequencies and lifetimes in young clusters. Astrophys. J. 553, L153-L156]. The post-accretion main belt population, in the form of diameter D?1000 km planetesimals, was likely to have been 160±40 times the current main belt's mass. This corresponds to 0.06-0.1 Earth masses, only a small fraction of the total mass thought to have existed in the main belt zone during planet formation. The remaining mass was most likely taken up by planetary embryos formed in the same region. Our results suggest that numerous D>200 km planetesimals disrupted early in Solar System history, but only a small fraction of their fragments survived the dynamical depletion event described above. We believe this may explain the limited presence of iron-rich M-type, olivine-rich A-type, and non-Vesta V-type asteroids in the main belt today. The collisional lifetimes determined for main belt asteroids agree with the cosmic ray exposure ages of stony meteorites and are consistent with the limited collisional evolution detected among large Koronis family members. Using the same model, we investigated the near-Earth object (NEO) population. We show the shape of the NEO size distribution is a reflection of the main belt population, with main belt asteroids driven to resonances by Yarkovsky thermal forces. We used our model of the NEO population over the last 3 Gyr, which is consistent with the current population determined by telescopic and satellite data, to explore whether the majority of small craters (D<0.1-1 km) formed on Mercury, the Moon, and Mars were produced by primary impacts or by secondary impacts generated by ejecta from large craters. Our results suggest that most small craters formed on these worlds were a by-product of secondary rather than primary impacts.     

On the origin and initial temperature of Jupiter and Saturn

A two-stage growth of the giant planets, Jupiter and Saturn, is considered, which is different from the model of contraction of large gaseous protoplanets. In the first stage, within a time of ～3 × 10⁷ years in Jupiter's zone and ～2 × 10⁸ years in Saturn's zone, a nucleus forms from condensed (solid) material having the mass, ～10²⁸ g, necessary for the beginning of acceleration. The second stage may gravitating body, and a relatively slow accretion begins until the mass of the planet reaches ～10 m_⊕. Then a rapid accretion begins with the critical radius less than the radius of the Hill lobe, so that the classical formulae for the rate of accretion may be applied. At a mass m > m₁ ≈ 50 m_⊕ accretion proceeds slower than it would according to these formulae. When the planet sweeps out all the gas from its nearest zone of feeding (m = m₂ ≈ 130 m_⊕), the width of the exhausted zone being $\text{～}\text{built1}\text{3}$ of the whole zone of the planet) growth is provided the slow diffusion of gas from the rest of the zone (time scale increases to 10⁵?10⁶ years and more). The process is terminated by the dissipation of the remnants of gas. In Saturn's zone m₁ > m₂ ≈ 30 m_⊕. The initial mass of the gas in Jupiter's zone is estimated. Before the beginning of the rapid accretion about 90% of the gas should have been lost from the solar system, and in the planet's zone less than two Jupiter masses remain. The highest temperature of Jupiter's surface, ≈5000°K, is reached at the stage of rapid accretion, m < 100 m_⊕, when the luminosity of the planet reaches 3 × 10^?3 L_⊙. This favors an effective heating of the inner parts of the accretionary disk and the dissipation of gas from the disk. The accretion of Saturn produced a temperature rise up to 2000?2400° K (at m ≈ 20?25 m_⊕) and a luminosity up to 10^?4 L_⊙.     

On the Instability of Jupiter's Trojans

We have numerically explored the mechanisms that destabilize Jupiter's Trojan orbits outside the stability region defined by Levison et al. (1997, Nature385, 42-44). Different models have been exploited to test various possible sources of instability on timescales on the order of ∼10⁸ years.In the restricted three-body model, only a few Trojan orbits become unstable within 10⁸ years. This intrinsic instability contributes only marginally to the overall instability found by Levison et al.In a model where the orbital parameters of both Jupiter and Saturn are fixed, we have investigated the role of Saturn and its gravitational influence. We find that a large fraction of Trojan orbits become unstable because of the direct nonresonant perturbations by Saturn. By shifting its semimajor axis at constant intervals around its present value we find that the near 5:2 mean motion resonance between the two giant planets (the Great Inequality) is not responsible for the gross instability of Jupiter's Trojans since short-term perturbations by Saturn destabilize Trojans, even when the two planets are far out of the resonance.Secular resonances are an additional source of instability. In the full six-body model with the four major planets included in the numerical integration, we have analyzed the effects of secular resonances with the node of the planets. Trojan asteroids have relevant inclinations, and nodal secular resonances play an important role. When a Trojan orbit becomes unstable, in most cases the libration amplitude of the critical argument of the 1:1 mean motion resonance grows until the asteroid encounters the planet. Libration amplitude, eccentricity, and nodal rate are linked for Trojan orbits by an algebraic relation so that when one of the three parameters is perturbed, the other two are affected as well. There are numerous secular resonances with the nodal rate of Jupiter that fall inside the region of instability and contribute to destabilize Trojans, in particular the ν₁₆. Indeed, in the full model the escape rate over 50 Myr is higher compared to the fixed model.Some secular resonances even cross the stability region delimited by Levison et al. and cause instability. This is the case of the 3:2 and 1:2 nodal resonances with Jupiter. In particular the 1:2 is responsible for the instability of some clones of the L4 Trojan (3540) Protesilaos.     

A satellite-asteroid mystery and a possible early flux of scattered C-class asteroids

Spectrophotometric measurements of the subset of satellites thought to be of capture origin, orbiting Saturn, Jupiter, and Mars, indicate that they typically (all?) have neutral spectra and low albedo, indicating spectral class C. This finding is puzzling in light of evidence that class-C objects are native only to the outer half of the asteroid belt. How did C-like objects approach, and get captured by, planets? Probably Jupiter resonances scattered a high flux of C-type objects out of the belt and throughout much of the primordial solar system. Such scattering could occur only at the close of planet accretion when extended atmospheres could affect capture. The largest flux of scattered objects came from resonances in the most heavily populated regions of the belt near 2.8 AU: in particular, the 7:3 and 5:2 resonance regions primarily populated by C's. Hence, most captured satellites are C's, not D's, S's, etc. This scenario supports and expands upon the capture scenario involving extended protoatmospheres, proposed by J. B. Pollack, J. Burns, and M. Tauper (1979, Icarus 37, 587–611) and D. Hunten (1979, Icarus 37,113–123).     

Dynamical evolution of a cometary swarm in the outer planetary region

The dynamical evolution of bodies under the gravitational influence of the accreting proto-Uranus and proto-Neptune is investigated. The main aim of this study is to analyze the interrelations between the accretion of Uranus and Neptune with other processes of cosmological importance as, for example, the formation of a cometary reservoir from bodies placed into near-parabolic orbits by planetary perturbations and the scattering of bodies to the region of the terrestrial planets. Starting with a mass ratio (initial mass/present mass) of 0.1, Uranus and Neptune acquire masses close to their present ones in a time scale of 10⁸ years. Neptune is found to be the most important contributor of comets to the cometary reservoir. The time scale of bodies scattered by Neptune to reach near-parabolic orbits (semimajor axes a > 10⁴ AU)is about 10⁹ years. The contribution of Uranus was partially inhibited because a large part of the residual bodies of its accretion zone fell under the strong gravitational influence of Jupiter and Saturn. A significant fraction of the bodies dispersed by Uranus and Neptune reached the region of the terrestrial planets in a time scale of some 10⁸ years.     

Dynamics of the Jupiter Trojans with Saturn’s perturbation when the two planets are in migration

In a previous paper (Hou et al. in Celest Mech Dyn Astron 119:119–142, 2014a), the problem of dynamical symmetry between two Jupiter triangular libration points (TLPs) with Saturn’s perturbation in the present configuration of the two planets was studied. A small short-time scale spatial asymmetry exists but gradually disappears with the time going, so the planar stable regions around the two Jupiter TLPs should be dynamically symmetric from a longtime perspective. In this paper, the symmetry problem is studied when the two planets are in migration. Several mechanisms that can cause asymmetries are discussed. Studies show that three important ones are the large short-time scale spatial asymmetry when Jupiter and Saturn are in resonance, the changing orbits of Jupiter and Saturn in the planet migration process, and the chaotic nature of Trojan orbits during the planet migration process. Their joint effects can cause an observable difference to the two Jupiter Trojan swarms. The thermal Yarkovsky effect is also found to be able to cause dynamical differences to the two TLPs, but generally they are too small to be practically observed.     

Planet X and the origins of the shower and steady state flux of short-period comets

In the planet X model periodic comet showers are associated with passages of the planet's perihelion and aphelion points through a primordial disk of comets believed to lie beyond the orbit of Neptune. A strong feature of this model is that the required orbital elements and mass of planet X are consistent with independently predicted values based on the residuals in the motions of Uranus and Neptune. Here we present a more extensive analysis of the model taking into account the fact that only those comets scattered directly into the zones of influence of Saturn and Jupiter can contribute to a shower whose duration is consistent with observation (≲ 15 myr). These requirements impose a minimum planetary inclination of ≈25°, which in turn restricts the semimajor axis to be ≲100 AU. A fraction of the comets scattered directly into the zones of influence of Uranus and Neptune will evolve on time scales of ∼10⁸ years into the steady state flux of short-period comets. We find that the absolute numbers of shower and steady state are comparable and compatible with the known terrestrial cratering rate, assuming the existence of long-lived extinct comet cores. Canonical planet X model parameters, deduced in part from the scattering dynamics analysis, are: semimajor axis ≈80 AU, eccentricity ≈0.3, inclination ≈45°, and mass ≈5m_⊕. An analysis is given which suggests that planet X, in its present orbit, can create the requisite density gradient of comets near perihelion and aphelion during the lifetime of the Solar System. The required inclination of planet X's orbit (≳25°) may explain the failure of previous surveys to discover the planet as its present latitude is not likely to be near the ecliptic. It it exists, the best immediate hope of finding planet X is the ongoing IRAS search in the 100-μm band and the full sky optical survey by Shoemaker and Shoemaker. Independent of the question of periodic comet showers, the existence of planet X and the comet disk can readily explain the origin of the steady state flux of short-period comets over a wide range of parameters.