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).     

Near-parabolic cometary flux in the outer solar system

The flux of near-parabolic comets in the outer-planetary region is estimated on the presumption that the major planets and the galactic tide control the dynamics of comets. It is found that the flux of the Oort cloud comets (semi-major axis > 20000 AU) is similar to the case of a strong comet shower derived on the presumption that the galactic tidal force were not operative. On the other hand, the flux of comets with semi-major axes <- 20000 AU is found to be an increasing function of q (perihelion distance) until q reaches 20 AU, while for a 45000 AU it is a rapidly increasing function for q 12 AU. In other words, for comets of the inner extension of the Oort cloud the planetary perturbation acts as a strong barrier for them to penetrate into the inner planetary region.     

Evolution of the Oort cloud angular momentum: Numerical simulation

This paper considers the evolution of a flat svarm of cometary bodies (under the effect of the passage of stars), initially moving in one direction along the circular orbits with radii 1.4×10⁴<r<2×10⁴ AU and along elliptic orbits with semi-major axes 5×10³<a<1×10⁴ AU and with perihelia within 50<q<100 AU. Numerical simulation shows that the original flat belt of comets is thermalizing. Its root-mean-squarez-coordinate grows withr. A cometary cloud forms with a dense flattened inner core and a rarefied halo (the Oort cloud proper). The value =N _core/N _halo varies within a wide range (up to the order of magnitude) depending on the model used (N _core andN _halo are the numbers of comets in the core and the halo, respectively).The hypothesis of a massive Oort cloud (Marochniket al., 1988) implies that the Oort cloud should have a large angular momentum. This paper employs numerical simulation to calculate Oort cloud models to which the initially flat located at the periphery of the solar nebula rotating cometary swarms is evolving in time. The loss of the initial angular momentum over the time of the Oort cloud evolution is not large.     

The Role of Giant Molecular Clouds in the Evolution of the Oort Comet Cloud

We estimated the gravitational influence of giant molecular clouds passing near the Solar system on the orbital evolution of Oort cloud comets. We performed a comparative analysis of the accuracies of the following two methods of allowance for the perturbations from giant molecular clouds: the impulse approximation and numerical integration. The impulse approximation yields fairly accurate estimates of the change in the energy of Oort cloud comets and the probability of their ejection under the influence of a molecular cloud if the path of the Solar system does not cross its boundary and if the molecular cloud may be treated as a point perturbing mass. The comet survival probability in the Oort cloud depends significantly on the internal structure of the perturbing molecular cloud and the impact parameter of the encounter. The most massive injection of comets into the planetary region and their ejection from the Oort cloud take place if the Solar system passes through a giant molecular cloud composed of several high-mass condensations. In this case, most of the comets injected into the planetary region were initially comets of the inner Oort cloud (a 10^–4 AU) with high orbital eccentricities.     

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.     

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.     

Mass and orbit estimation of Planet X via a family of comets

The crucial assumption of this paper is, that the observed clustering of aphelion distances of intermediate-period comets in the 70–90 AU range is due to the influence of a tenth planet, called Planet X. We contribute to the search for Planet X a new and extended evaluation of a family of comets assumed to be Planet X's family of comets.By averaging the aphelion distances of comets that belong to a transplutonic family of comets, we get Planet X's semi-major axis a _x = (83.0 ± 5.3) AU. The comets' orbits also yield the upper limit of the planet's orbital eccentricity e _x - 0.019. If this planet played an important part in sending quasi-periodic comet showers to the inner solar system, we can calculate its orbital inclination i _x = 46 .1 ± 3 .6. By distributing all planets' masses into the heliocentric, torus-like zones, in which they were formed, we get the density distribution of the primordial solar nebula. Extrapolating this distribution we find the mass of the planet M _x = (5.1_–2.4 ^+3.6 M _Earth. A few plausible assumptions (e.g. Uranus and Neptune perturbations being caused by Planet X) lead to Planet X's actual location with declination and eccliptic longitude being = 57 ± 17 and = 54 ± 34 , respectively (1989.5 position). In addition, we give Planet X's apparent brightness dependent on its unknown albedo. All those properties and predictions are more or less in agreement with earlier work on Planet X.     

The frequency and intensity of comet showers from the Oort cloud

We simulate the Oort comet cloud to study the rate and properties of new comets and the intensity and frequency of comet showers. An ensemble of ∼10⁶ comets is perturbed at random times by a population of main sequence stars and white dwarfs that is described by the Bahcall-Soneira Galaxy model. A cloning procedure allows us to model a large ensemble of comets efficiently, without wasting computer time following a large number of low eccentricity orbits. For comets at semimajor axis a = 20,000 AU, about every 100 myr a star with mass in the range 1M_⊙−2M_⊙ passes within ∼10,000 AU of the Sun and triggers a shower that enhances the flux of new comets by more than a factor of 10. The time-integrated flux is dominated by the showers for comets with semimajor axes less than ∼30,000 AU. For semimajor axes greater than ∼30,000 AU the comet loss rate is roughly constant and strong showers do not occur. In some of our simulations, comets are also perturbed by the Galactic tidal field. The inclusion of tidal effects increases the loss rate of comets with semimajor axes between 10,000 and 20,000 AU by about a factor of 4. Thus the Galactic tide, rather than individual stellar perturbations, is the dominant mechanism which drives the evolution of the Oort cloud.     

Long-Period Comets and the Oort Cloud

Long-period (LP) comets, Halley-type (HT) comets, and even some comets of the Jupiter family, probably come from the Oort cloud, a huge reservoir of icy bodies that surrounds the solar system. Therefore, these comets become important probes to learn about the distant Oort cloud population. We review the fundamental dynamical properties of LP comets, and what is our current understanding of the dynamical mechanisms that bring these bodies from the distant Oort cloud region to the inner planetary region. Most new comets have original reciprocal semimajor axes in the range2 × 10^-5 < 1/a_orig < 5 × 10^-5AU^-1. Yet, this cannot be taken to represent the actual space distribution of Oort cloud comets, but only the region in the energy space in which external perturbers have the greatest efficiency in bringing comets to the inner planetary region. The flux of Oort cloud comets in the outer planetary region is found to be at least several tens times greater than the flux in the inner planetary region. The sharp decrease closer to the Sun is due to the powerful gravitational fields of Jupiter and Saturn that prevent most Oort cloud comets from reaching the Earth’s neighborhood (they act as a dynamical barrier). A small fraction of ～10^-2 Oort cloud comets become Halley type (orbital periods P < 200 yr), and some of them can reach short-period orbits with P < 20 yr. We analyze whether we can distinguish the latter, very ‘old” LP comets, from comets of the Jupier family coming from the Edgeworth-Kuiper belt.     

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.     

Few Comments on the Relation Between the Initial Proto-planetary Disc Model and the Oort Cloud Formation

A fraction of small bodies from the once existing proto-planetary disc was ejected, by the giant planets, to large heliocentric distances and start to build the comet Oort cloud. Considering four models of initial proto-planetary disc, we attempt to roughly map a dependence between the initial disc’s structure and some properties of the Oort cloud. We find that it is difficult to construct the proto-planetary disc if (i) the amount of heavy chemical elements in Jupiter and Saturn is as high as currently accepted and (ii) the total mass of the minimum-mass solar nebula is assumed to be lower than $\approx0.05\,\hbox{M}_{\odot}.$ The behaviour of the Oort cloud formation does not crucially depend on the initial disc model. Some differences in its structure are obvious: since the cloud is known to be filled mainly by Uranus and Neptune, the efficiency of its formation is higher when the initial amount of particles in the Uranus-Neptune region is relatively higher. A significantly large number of Jupiter Trojans in our simulation appears, however, only in the case of the initially non-gapped disc, with the particles situated also close to the Jupiter’s orbit.     

Giant planet formation

The accumulation of giant planets involves processes typical for terrestrial planet formation as well as gasdynamic processes that were previously known only in stars. The condensible element cores of the gas-giants grow by solid body accretion while envelope formation is governed by stellar-like equilibria and the dynamic departures thereof. Two hypotheses for forming Uranus/Neptune-type planets — at sufficiently large heliocentric distances while allowing accretion of massive gaseous envelopes, i.e. Jupiter-type planets at intermediate distances — have been worked out in detailed numerical calculations: (1) Hydrostatic gas-accretion models with time-dependent solid body accretion-rates show a slow-down of core-accretion at the appropriate masses of Uranus and Neptune. As a consequence, gas-accretion also stagnates and a window is opened for removing the solar nebula during a time of roughly constant envelope mass. (2) Gasdynamic calculations of envelope accretion for constant planetesimal accretion-rates show a dynamic transition to new envelope equilibria at the so called critical mass. For a wide range of solar nebula conditions the new envelopes have respective masses similar to those of Uranus and Neptune and are more tightly bound to the cores. The transitions occur under lower density conditions typical for the outer parts of the solar nebula, whereas for higher densities, i.e. closer to the Sun, gasdynamic envelope accretion sets in and is able to proceed to Jupiter-masses.     

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

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

Origin of short period comets

Reasons for interest in the origin of short-period comets and the difficulties of computing their long-term dynamcal evolution are reviewed. The relative advantages of a source region in an extended inner core of the Oort cloud or a compact comet belt just beyond the planetary system are finely balanced, and it is premature to consider the problem solved. A complication is that some comets belonging to the Jupiter family may be part of a time-dependent system, possibly the remains of a giant comet such as Chiron which could have been part of the system 104 yr ago. The origin of short-period comets plays a pivotal role in many areas of solar system science: planet formation, the source of water (possibly life) on the terrestrial planets, the cratering record on the terrestrial planets and satellites of the outer planets, and the environmental impact posed by massive bodies and their decay products in the Earth's near-space environment.     

Tidal gravitational forces: The infall of “new” comets and comet showers

The tidal gravitational field of the Galaxy directed into the galactic plane changes the angular momentum of comets in the Oort cloud. For comet orbits with semimajor axis greater than 2 × 10⁴ AU, the change of angular momentum in one orbit is sufficient to bring comets from the Oort cloud into the visible region, causing the infall of “new” comets. The limiting size orbit is weakly dependent on the angle between the major axis of the comet orbit and the galactic plane. The flux of comets into the inner Solar System caused by the galactic tidal field will be continuous and nearly isotropic. This effect appears to exclude any determination of the trajectories of passing stars by analysis of the angular distribution of new comets. The production of intense comet showers by the tidal field of a solar companion or of an interstellar cloud is considered. We show that the direction of a solar companion cannot be found from the present distribution of observable comets. The frequency of comet showers induced by encounters with interstellar clouds is found to be much lower than that from passing stars, and the tidal fields of interstellar clouds are not strong enough to cause comet showers of sufficient intensity to result in Earth impacts.     

Grains accretion processes in a protoplanetary nebula: Effects of the presence of liquid drops

The temporal behaviour of the mass distribution function of iron solid and/or liquid grains is studied in the inner regions of the solar nebula during its gravitational collapse. At distances from the centre of the nebula equal to 0.2 AU, 0.5 AU and 0.7 AU a mass limit for the accretionary process equal, respectively, to 17.6 g, 5.2 g and 3×10^–3 g is obtained in a reasonable time interval. The importance of a drop phase during the growth of the particles and of the sedimentation of the grains during the disk phase is underlined.     

Some requirements of a colliding comet source of gamma ray bursts

Colliding comets in the Solar System may be an important source of gamma ray bursts. The spherical gamma ray comet cloud required by the results of the Venera Satellites (Mazets and Golenetskii, 1987) and the BATSE detector on the Compton Satellite (Meeganet al., 1992a, b) is neither the Oort Cloud nor the Kuiper Belt. To satisfy observations ofN(>P _max) vsP _max for the maximum gamma ray fluxes,P _max > 10^–5 erg cm^–2 s^–1 (about 30 bursts yr^–1), the comet density,n, should increase asn a ¹ from about 40 to 100 AU wherea is the comet heliocentric distance. The turnover above 100 AU requiresn a ^–1/2 to 200 AU to fit the Venera results andn a ^1/4 to 400 AU to fit the BATSE data. Then the masses of comets in the 3 regions are from: 40–100 AU, about 9 earth masses,m _E; 100–200 AU about 25m _E; and 100–400 AU, about 900m _E. The flux of 10^–5 erg cm^–2 s^–1 corresponds to a luminosity at 100 AU of 3 × 10²⁶ erg s^–1. Two colliding spherical comets at a distance of 100 AU, each with nucleus of radiusR of 5 km, density of 0.5 g cm^–3 and Keplerian velocity 3 km s^–1 have a combined kinetic energy of 3 × 10²⁸ erg, a factor of about 100 greater than required by the burst maximum fluxes that last for one second. Betatron acceleration in the compressed magnetic fields between the colliding comets could accelerate electrons to energies sufficient to produce the observed high energy gamma rays. Many of the additional observed features of gamma ray bursts can be explained by the solar comet collision source.     

Mean free paths and diffusion coefficients for energetic protons at small heliodistances calculated using Helios 1 and 2 data

Pitch angle scattering of energetic particles (100 MeV) in the interplanetary medium are studied using Helios 1 and 2 magnetometer and plasma data during 1976 near the minimum of solar activity. An IMF configuration was used in the computer experiments which allowed the pitch angle diffusion coefficient, D and hence the parallel mean free path, to be determined. The radial mean free path was found to vary as _r r ^-0.9 between 0.4 and 1 AU, but between 0.3 and 0.4 AU it decreases significantly. To reconcile our value of _r at 1 AU, lying between 0.01 and 0.02 AU, with the average prompt solar proton event profile, an increasing value of _r at lower radial distances would be required.     

Uranus and Neptune interior models

This paper is concerned with the interior structure of Uranus and Neptune. Our approach is three-fold. First, a set of three-layer models for both Uranus and Neptune are constructed using a method similar to that used in the study of the terrestrial planets. The variations of the mass density (s) and flattening e(s) with fractional mean radius s for two representative models of Uranus and Neptune are calculated. The results are tabulated. A comparison of these models shows that these two planets are probably very similar to each other in their basic dynamical features. Such similarity is very seldom seen in our solar system. Secondly, we check the conformance between the theoretical results and observational data for the two planets. And thirdly, the 6th degree Stokes zonal parameters for Uranus and for Neptune are predicted, based on the interior models put forward in this paper.     

The scattered disk population as a source of Oort cloud comets: evaluation of its current and past role in populating the Oort cloud

We have integrated the orbits of the 76 scattered disk objects (SDOs), discovered through the end of 2002, plus 399 clones for 5 Gyr to study their dynamical evolution and the probability of falling in one of the following end states: reaching Jupiter's influence zone, hyperbolic ejection, or transfer to the Oort cloud. We find that nearly 50% of the SDOs are transferred to the Oort cloud (i.e., they reach heliocentric distances greater than 20,000 AU in a barycentric elliptical orbit), from which about 60% have their perihelia beyond Neptune's orbit (31 AU<q<36 AU) at the moment of reaching the Oort cloud. This shows that Neptune acts as a dynamical barrier, scattering most of the bodies to near-parabolic orbits before they can approach or cross Neptune's orbit in non-resonant orbits (that may allow their transfer to the planetary region as Centaurs via close encounters with Neptune). Consequently, Neptune's dynamical barrier greatly favors insertion in the Oort cloud at the expense of the other end states mentioned above. We found that the current rate of SDOs with radii R>1 km incorporated into the Oort cloud is about 5 yr⁻¹, which might be a non-negligible fraction of comet losses from the Oort cloud (probably around or even above 10%). Therefore, we conclude that the Oort cloud may have experienced and may be even experiencing a significant renovation of its population, and that the trans-neptunian belt—via the scattered disk—may be the main feeding source.