Dynamical evolution of the Oort cloud: I. A Monte Carlo simulation

We have studied the long-time dynamical evolution of a population of comets surrounding the Solar System at a large distance. Orbital changes are caused by random passing stars. We first emphasize the need for a new simulation because of the lack of completeness of previous analytical and numerical studies. Then the solar neighborhood is modeled by a sphere of 1 pc in radius, which stars cross at random in direction and distance. The geometry of the encounters allows us to compute the impulse gained by the star and the Sun, in the context of an impact approximation. Then we determine the change of orbital elements for a population of comets and follow the evolution of the frequency distribution for the five Keplerian elements. Clouds are selected in such a way that we test the two main hypotheses for the origin of the Oort cloud, and also the regions of stability in an aphelion-eccentricity diagram. We show that stellar perturbations randomize the cloud and prevent one from inferring the initial cloud configuration from the current distribution. Clouds are depleted by the diffusion of comets into the planetary regions, where they become planet-influenced comets or are ejected from the Solar System. The diffusion of aphelion toward interstellar regions proves to be the major source of cometary loss. Direct ejection to hyperbolic orbits amounts to 9% of the originally population over the age of the Solar System. Finally the current and original cloud populations are estimated at 1.8 × 10¹² and 2 × 10¹³ comets and we discuss these results.     

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).     

Star passages through the Oort cloud

Stars passing through the Oort cloud eject comets to interstellar space and initiate showers of comets into the planetary region. Monte Carlo simulations of such passages are performed on a representative distribution of cometary orbits. Ejected comets generally lie along a narrow tunnel “drilled” by the star through the cloud. However, shower comets come from the entire cloud, and do not give a strong signature of the star's passage, except in the inverse semimajor axis distribution for the shower comets. The planetary system is likely not experiencing a cometary shower at this time.     

Star passages through the Oort cloud

Stars passing through the Oort cloud eject comets to interstellar space and initiate showers of comets into the planetary region. Monte Carlo simulations of such passages are performed on a representative distribution of cometary orbits. Ejected comets generally lie along a narrow tunnel drilled by the star through the cloud. However, shower comets come from the entire cloud, and do not give a strong signature of the star's passage, except in the inverse semimajor axis distribution for the shower comets. The planetary system is likely not experiencing a cometary shower at this time.     

Some comments on the Oort cloud

The arguments used by Lyttleton to prove the nonexistence of the Oort cloud are reviewed, and it is shown that Oort's hypothesis remains consistent with observation. The 1950 model of the cloud cannot be correct and, by use of the results from a number of more recent papers, an improved model is described and compared with observations. It is emphasized that comparison of the predictions of theory with observations should concentrate on thea-distribution, as the 1/a-distribution masks much of the detailed structure of the theory. An order of magnitude argument is given which shows that 20% of so-called new comets have passed through the planetary system before, and the implications of this to the statistics of near-parabolic comet orbits are briefly investigated.     

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.     

Orbital element distributions in the Oort cloud

In this paper we consider orbital element distributions for comets moving on admissible orbits in the Oort cloud and distributions for some functions that depend on the orbital elements. Moreover, we find the probability of an event that an arbitrarily chosen admissible orbit belongs to the set (r) of orbital elements and the distribution of circular velocities in the cloud.     

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.     

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.     

Time-dependent injection of Oort cloud comets into Earth-crossing orbits

The effect of close stellar encounters in modulating the influx rate of Oort cloud comets is investigated. In particular, it is shown that comet showers intense enough to be reflected in crater statistics can be produced at intervals of 80 million years or so, provided we are dealing with an Oort cloud consisting of a heavy core of comets. In this case, there is found a strong predominance of incoming comets from the sky zone where the perturbing star makes its closest approach. We have also performed numerical simulations of the time evolution of comet showers or bursts. From this numerical study, a long tail of residual shower comets is found to follow the major event with an intensity (as compared with the intensity of the shower as its peak) of ∼10⁻² after 20—30 million years. Our results thus suggest that residual shower comets may be clustered mainly on certain sky areas and observable at practically any time given the lasting effects of a shower. This might explain some of the observed clustering of aphelion points of long-period comets.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.     

The statistical effects of galactic tides on the Oort cloud

This report is a comment on two papers by Matese and Whitman (1989, 1992). We discuss here the applicability of uniform probability densities for the orbital parameters of the Oort cloud comets.     

The statistical effects of galactic tides on the Oort cloud

This report is a comment on two papers by Matese and Whitman (1989, 1992). We discuss here the applicability of uniform probability densities for the orbital parameters of the Oort cloud comets.     

The formation of the Oort cloud in open cluster environments

We study the influence of an open cluster environment on the formation and current structure of the Oort cloud. To do this, we have run 19 different simulations of the formation of the Oort cloud for 4.5 Gyrs. In each simulation, the Solar System spends its first 100 Myrs in a different open cluster environment before transitioning to its current field environment. We find that, compared to forming in the field environment, the inner Oort cloud is preferentially loaded with comets while the Sun resides in the open cluster and that most of this material remains locked in the interior of the cloud for the next 4.4 Gyrs. In addition, the outer Oort cloud trapping efficiencies we observe in our simulations are lower than previous formation models by about a factor of 2, possibly implying an even more massive early planetesimal disk. Furthermore, some of our simulations reproduce the orbits of observed extended scattered disk objects, which may serve as an observational constraint on the Sun's early environment. Depending on the particular open cluster environment, the properties of the inner Oort cloud and extended scattered disk can vary widely. On the other hand, the outer portions of the Oort cloud in each of our simulations are all similar.     

Orbits of short period comets captured from the Oort cloud

Oort cloud comets occasionally obtain orbits which take them through the planetary region. The perturbations by the planets are likely to change the orbit of the comet. We model this process by using a Monte Carlo method and cross sections for orbital changes, i.e. changes in energy, inclination and perihelion distance, in a single planet-comet encounter. The influence of all major planets is considered. We study the distributions of orbital parameters of observable comets, i.e. those which have perihelion distance smaller than a given value. We find that enough comets are captured from the Oort cloud in order to explain the present populations of short period comets. The median value of cos i for the Jupiter family is 0.985 while it is 0.27 for the Halley types. The results may explain the orbital features of short period comets, assuming that the active lifetime of a comet is not much greater than 400 orbital revolutions.     

Long-term evolution of Oort Cloud comets: capture of comets

We test different possibilities for the origin of short-period comets captured from the Oort Cloud. We use an efficient Monte Carlo simulation method that takes into account non-gravitational forces, Galactic perturbations, observational selection effects, physical evolution and tidal splittings of comets. We confirm previous results and conclude that the Jupiter family comets cannot originate in the spherically distributed Oort Cloud, since there is no physically possible model of how these comets can be captured from the Oort Cloud flux and produce the observed inclination and Tisserand constant distributions. The extended model of the Oort Cloud predicted by the planetesimal theory consisting of a non-randomly distributed inner core and a classical Oort Cloud also cannot explain the observed distributions of Jupiter family comets. The number of comets captured from the outer region of the Solar system are too high compared with the observations if the inclination distribution of Jupiter family comets is matched with the observed distribution. It is very likely that the Halley-type comets are captured mainly from the classical Oort Cloud, since the distributions in inclination and Tisserand value can be fitted to the observed distributions with very high confidence. Also the expected number of comets is in agreement with the observations when physical evolution of the comets is included. However, the solution is not unique, and other more complicated models can also explain the observed properties of Halley-type comets. The existence of Jupiter family comets can be explained only if they are captured from the extended disc of comets with semimajor axes of the comets   a <5000 au  . The original flattened distribution of comets is conserved as the cometary orbits evolve from the outer Solar system era to the observed region.     

Long-term evolution of Oort Cloud comets: methods and comparisons

Two long-term simulation methods for cometary orbits, a Monte Carlo method and a direct integration method, are compared with each other. The comparison is done in seven inclination and perihelion distance intervals, and shows differences in dynamical lifetime and capture probabilities for the following main reasons. We use a finite energy step approximation in the Monte Carlo method and the method considers only close approaches with the planets. The differences can be taken into account statistically and it is possible to calculate the correction factors for the capture probability and dynamical lifetime in the Monte Carlo method. Both corrections depend on the inclination and on the value of the minimum energy step. The capture probabilities of the short-period comets originating in the Oort Cloud are calculated by the corrected Monte Carlo method and compared with published results.     

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.     

Perihelion evolution of observed new comets implies the dominance of the galactic tide in making Oort cloud comets discernable

For an Oort cloud comet to be seen as a new comet, its perihelion must be moved from a point exterior to the loss cylinder boundary to a point interior to observable limits in a single orbit. The galactic tide can do this continuously, in a non-impulsive manner. Near-parabolic comets, with specific angular momentum , will most easily be made observable. Therefore, to reduce the perihelion distance H must decrease. Since weakly perturbed comets are, in general, more numerous than strongly perturbed comets, we can anticipate that new comets made observable by a weak tidal torque will more likely be first observed when their slowly changing perihelion distances are approaching their minimum osculating values under the action of the tide, rather than receding from their minimum values. That is, defining ΔH_tide as the vector change due to the galactic tidal torque during the prior orbit, and H_obs as the observed vector, the sign S≡Sign(H_obs·ΔH_tide) will more likely be −1 than +1 if a weak galactic tidal perturbation indeed dominates in making comets observable. Using comet data of the highest quality class (1A) for new comets (a>10,000 AU), we find that 49 comets have S=−1 and 22 have S=+1. The binomial probability that as many or more would exhibit this characteristic if in fact S=?1 were equally likely is only 0.0009. This characteristic also persists in other long-period comet populations, lending support to the notion that they are dominated by comets recently arrived from the outer Oort cloud. The preponderance of S=−1 also correlates with weakly perturbed (i.e., smaller semimajor axis) new comets in a statistically significant manner. This is strong evidence that the data are of sufficiently high quality and sufficiently free of observational selection effects to detect this unique imprint of the tide.