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).     

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 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.     

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.     

Galactic tidal perturbations of long-period comets and the distribution of the inclinations

Perturbation of the perihelion distance q of long-period comets by the galactic tidal force is calculated using Cowell's method. It is shown that the maximum perturbation is suffered by those with i (inclination) close to 50 ~ 60 and not by those with i close to 90 , contrary to the prediction of the first order perturbation theory. The dependence of the perturbation of q upon i is compared with the distribution of the inclinations of observed long-period comets and it is shown that the later is not consistent with an isotropic cloud of comets perturbed by the galactic tid alone. A close stellar encounter is unlikely to be an external disturbance. It is argued that giant molecular cloud is the most likely mechanism of the external disturbances.     

Long-term effects of the Galactic tide on cometary dynamics

We introduce a model for integrating the effects of Galactic tides on Oort cloud comets, which involves two procedures, according to the values of the osculating semi-major axis a and eccentricity e. Ten simulations of the dynamics of 10⁶ comets over 5 Gyr are performed using this model. We thus investigate the long-term effects of the Galactic tide with and without a radial component, the effects of the local density of the Galactic disk, and those of the Oort constants. Most of the results may be understood in terms of the integrability or non-integrability of the system. For an integrable system, which occurs for moderate semi-major axes with or without radial component, the dynamics is explained by periodic variation of the cometary perihelion, inducing the depletion of the outer region of the Oort cloud, a constant flux from the inner region after 500 Myr, and the quick formation of a reservoir of comets with argument of perihelion near 26.6°. When the system is non-integrable, the efficiency of the tide in reducing the cometary perihelion distance is enhanced both by replenishing the Oort cloud domain from which comets are sent toward the planetary system, and by reducing the minimal value that the perihelion distance may reach. No effects of varying the Oort constants were observed, showing that the flat rotation curve is a satisfactory approximation in Oort cloud dynamics.     

End-states of short-period comets and their role in maintaining the zodiacal dust cloud

Physical lifetimes and end-states of short-period comets are analysed in connection with the problem of the maintainance of the zodiacal dust cloud. In particular, the problem of the comet-asteroid relationship is addressed. Recent studies of the physical properties of Apollo-Amor asteroids and short-period comets (e.g., Hartmann et al., 1987) show significant differences between them, suggesting that they are distinct classes of objects. A few percent of the active SP comets might become asteroidal-like bodies in comet-type orbits due to the buildup of dust mantles. The remainder probably disintegrate as they consume their volatile content so their debris can only be observed as fireballs when they meet the Earth. Unobservable faint SP comets — i.e., comets so small (m 10¹⁴ g) that quickly disintegrate before being detected, might be a complementary source of dust material. They might be completely sublimated even at rather large heliocentric distances (r - 3 AU). Yet the released dust grains can reach the vicinity of the Sun by Poynting-Robertson drag. The mass associated with unobservable SP comets with perihelion distances q 3 AU might be comparable to that computed for the sample of observed SP co-mets with q 1.5 AU. It is concluded that SP comets (from the large to the unobservable small ones) may supply an average of several tons/sec of meteoric matter to the zodiacal dust 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.     

Dynamical evolution of comets and the problem of cometary fading

Possibilities to explain the observed 1/a-distribution are discussed in the light of improved understanding of the dynamical evolution of long-period comets. It appears that the fading problem applies both to single-injection and continuous-injection models. Although uncertainties due to nongravitational effects do not allow detailed results to be drawn from the observed 1/a-distribution at small perihelion distance q, that for q 1.5 AU shows that a constant fading probability cannot explain all the features of the observed distribution. Assuming that comets can reappear following a period of fading, values for the assumed constant fading and renewal probabilities, and the total cometary flux have been estimated for q > 1.5 AU.     

A Model for the Common Origin of Jupiter Family and Halley Type Comets

A numerical simulation of the Oort cloud is used to explain the observed orbital distributions and numbers of Jupiter-family (JF) and Halley-type (HT) short-period (SP) comets. Comets are given initial orbits with perihelion distances between 5 and 36 au, and evolve under planetary, stellar and Galactic perturbations for 4.5 Gyr. This process leads to the formation of an Oort cloud (which we define as the region of semimajor axes a > 1,000 au), and to a flux of cometary bodies from the Oort cloud returning to the planetary region at the present epoch. The results are consistent with the dynamical characteristics of SP comets and other observed cometary populations: the near-parabolic flux, Centaurs, and high-eccentricity trans-Neptunian objects. To achieve this consistency with observations, the model requires that the number of comets versus initial perihelion distance is concentrated towards the outer planetary region. Moreover, the mean physical lifetime of observable comets in the inner planetary region (q < 2.5 au) at the present epoch should be an increasing function of the comets’ initial perihelion distances. Virtually all observed HT comets and nearly half of observed JF comets come from the Oort cloud, and initially (4.5 Gyr ago) from orbits concentrated near the outer planetary region. Comets that have been in the Oort cloud also return to the Centaur (5 < q < 28 au, a < 1,000 au) and near-Neptune high-eccentricity regions. Such objects with perihelia near Neptune are hard to discover, but Centaurs with characteristics predicted by the model (e.g. large semimajor axes, above 60 au, or high inclinations, above 40°) are increasingly being found by observers. The model provides a unified picture for the origin of JF and HT comets. It predicts that the mean physical lifetime of all comets in the region q < 1.5 au is less than ～200 revolutions.     

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.     

A note on the total mass of comets in the solar system

The assumption that the very low albedo determined for Halley's comet is typical of all short period comets, taken together with the assumption that the average sizes of long and short period comets are approximately equal, leads to an increase in the total mass of comets in the solar system by almost two orders of magnitude. If gravitational ejection from the Uranus - Neptune zone during the later phases of planet formation is indeed responsible for the classical Oort cloud between 10⁴–10¹⁵ AU, then the mass of comets in this transplanetary region during cosmogonie times has to exceed the combined masses of Uranus and Neptune by over an order of magnitude. Furthermore, if the recent arguments for as many as 10¹⁴ comets in an inner Oort cloud between ~40– 10⁴AU are valid, then the total mass of comets in the solar system approaches 2% of a solar mass.     

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.     

Persistent evidence of a jovian mass solar companion in the Oort cloud

We present updated dynamical and statistical analyses of outer Oort cloud cometary evidence suggesting that the Sun has a wide-binary jovian mass companion. The results support a conjecture that there exists a companion of mass ≈ orbiting in the innermost region of the outer Oort cloud. Our most restrictive prediction is that the orientation angles of the orbit plane in galactic coordinates are centered on Ω, the galactic longitude of the ascending node = 319° and i, the galactic inclination = 103° (or the opposite direction) with an uncertainty in the orbit normal direction subtending <2% of the sky. Such a companion could also have produced the detached Kuiper Belt object Sedna. If the object exists, the absence of similar evidence in the inner Oort cloud implies that common beliefs about the origin of observed inner Oort cloud comets must be reconsidered. Evidence of the putative companion would have been recorded by the Wide-field Infrared Survey Explorer (WISE) which has completed its primary mission and is continuing on secondary objectives.     

Invisible comets on evolutionary track of short-period comets

We systematically surveyed the orbits of short-period (SP) comets that show a large change of perihelion distance (q) between 1–2 AU (visible comets) and 4–5 AU (invisible comets) during 4400 years. The data are taken from Cosmo-DICE (Nakamura and Yoshikawa 1991a), which is a long-term orbital evolution project for SP comets. Recognizing that q is the most critical element for observability of comets, an invisibility factor (f), defined as the ratio of unobservable time span to observable span during 4400 years, is calculated for each of the large-q-change comets. A detection limit for each comet is obtained from the heliocentric distance at discovery and/or the absolute magnitude at recent apparitions. A mean f value for 35 SP comets with 2.9 J (J is the Tisserand's invariant) is found to be 19.8. This implies that for each visible SP comet of this J-range, at every epoch of time, there exist about 20 invisible comets near the capture orbits by Jupiter, under the assumptions of steady-state flux and ergodicity for the SP-comet population.     

The distribution of short-period comet magnitudes

Short-period comets with P 15 yr represent one of the most complete comet samples. The magnitude distribution of these comets was analysed using a maximum likelihood method. The brightness (magnitude) index for the comets with H ₁₀ 11 mag was estimated together with the large sample errors and found to be 0.62 ± 0.09. It was clear that many faint comets with H ₁₀ > 11 mag remain to be discovered. Some of the faint, smaller comets have probably been removed from the distribution altogether.Observational selection was also apparent for the sample of comets with perihelia q < 1.5 AU. It was found that comets satisfying the combined criteria P 15 yr, H ₁₀ 11 mag, q < 1.5 AU probably represent the most complete set of comets available. The brightness index of this sample estimated by maximum likelihood was 0.69 ± 0.14. This translates into a mass distribution index s of 1.69 ± 0.14 indicating that most of the mass is contained in a few of the larger comets rather than spread throughout the smaller ones. This distribution, although modified by mass loss, is most likely to have been produced by a process of particle accretion.     

Injection of Oort Cloud comets: the fundamental role of stellar perturbations

We present Monte Carlo simulations of the dynamical evolution of the Oort cloud over the age of the Solar System, using an initial sample of one million test comets without any cloning. Our model includes perturbations due to the Galactic tide (radial and vertical) and passing stars. We present the first detailed analysis of the injection mechanism into observable orbits by comparing the complete model with separate models for tidal and stellar perturbations alone. We find that a fundamental role for injecting comets from the region outside the loss cone (perihelion distance q > 15 AU) into observable orbits (q < 5 AU) is played by stellar perturbations. These act in synergy with the tide such that the total injection rate is significantly larger than the sum of the two separate rates. This synergy is as important during comet showers as during quiescent periods and concerns comets with both small and large semi-major axes. We propose different dynamical mechanisms to explain the synergies in the inner and outer parts of the Oort Cloud. We find that the filling of the observable part of the loss cone under normal conditions in the present-day Solar System rises from <1% for a < 20 000 AU to about 100% for a ? 100 000 AU.     

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.     

Cometary brightness variation and nucleus structure

The Bobrovnikoff and Beyer photometric data for more than 100 comets have been analyzed for intrinsic brightness variations,before andafter perihelion, according to ther ^–n law, wherer is solar distance. The Oort and Schmidt classification of comet age has been extended and applied with Marsden's new determinations of inverse semi-major axis, 1/a, original. All classes of comets withP>25 yr show statistically the same value ofn after perihelion. New comets approach perihelion with smaller values ofn and older comets with increasingly larger values (Table II). For comets ofP<25 yr,n is larger and erratic.A physical interpretation involves the quick loss of a frosting of super-volatile materials from new comets; then, for all comets, the development of an insulating crust after perihelion. The crust also includes globs of meteoroidal and icy material. The crust tends to be purged near perihelion but generally to grow in a spotty fashion with cometary age. The orientation of the axes of rotating comets is shown to be an important unknown factor in cometary brightness variations. A speculation is made concerning the axis of rotation for C/Kohoutek, 1973 XII.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May 1978.This research has been supported by Grant No. NSG 7082 from the U.S. National Aeronautics and Space Administration.     

Radial gradients and anisotropies due to galactic cosmic rays

Numerical calculations have been made of the radial gradients and the anisotropyvector atr=1 AU due to galactic cosmic-ray protons and helium nuclei. The model used assumes transport by convection and anisotropic diffusion, and includes the energy losses due to adiabatic deceleration. The present calculations are for the 1964–65 solar minimum. An important constraint applied ineach case was that the model reproduces the electron modulation known from deductions of the galactic spectrum and observations of the near-Earth spectrum; and also reproduces the near-Earth proton and helium nuclei spectra. The diffusion coefficients have been based upon those deduced from magnetic-field power spectra.The principal aim has been to provide estimates of radial gradients and anisotropies, particularly at kinetic energiesT100 MeV/nucleon, by the complete solution of realistic models. Typical values for protons, obtained with a galactic differential number density (total energy)^–2.5, atT50 MeV are: radial gradient, 25%/AU; radial anisotropy, –0.2%; azimuthal anisotropy, 0.2%. These values change markedly when the galactic spectrum is cut-off or greatly enhanced atT<150 MeV, but the intensity spectrum near Earth remains substantially unchanged.It has been shown that it is possible to obtain negative radial gradients and positive radial anisotropies atT50 MeV for galactic particles and thus to mimic solar sources. The radial gradient for 1964–65 reported by Anderson (1968) and by Krimigis and Venkatesan (1969) are shown to be consistent with the diffusion coefficient deduced from the magnetic-field power spectrum; those reported by O'Gallagher are higher than expected and that for 20T30 MeV protons appears to be inconsistent. More precise data on conditions throughout the solar cavity are required if more definitive gradients and anisotropies are to be determined.