Origin and evolution of the long-period comets

We consider the changes of cometary perihelion distances as a process of diffusion in the value of q, due to perturbations by stars. We find more comets at large q values than is observed. This suggests that a large number of long-period comets is not observed.     

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.     

On the dynamic connection of comets with Uranus

We consider the connection with Uranus for: (1) 945 near-parabolic comets (the period P > 200 years, the perihelion distance q > 0.1 AU), (2) 1277 Kreutz comets (P > 200 years, q < 0.01 AU), and (3) 414 short-period comets (P < 200 years). It turns out that none of near-parabolic comets passed through Uranus’s activity sphere, none of the Kreutz comets approach Uranus closer than 11 AU, and only two short-period comets, C/2006 U7 and C/2006 F2, could have a close approach to Uranus during 5000 years.     

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.     

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.     

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.     

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.     

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.     

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).     

Relationship of comets and planets

We analyze our earlier data on the numerical integration of the equations of motion for 274 short-period comets (with the period P<200 yr) on a time interval of 6000 yr. As many as 54 comets had no close approaches to planets, 13 comets passed through the Saturnian sphere of action, and one comet passed through the Uranian sphere of action. The orbital elements of these 68 comets changed by no more than ±3 percent in a space of 6000 yr. As many as 206 comets passed close to Jupiter. We confirm Everhart’s conclusion that Jupiter can capture long-period comets with q = 4–6 AU and i < 9° into short-period orbits. We show that nearly parabolic comets cross the solar system mainly in the zone of terrestrial planets. No relationship of nearly parabolic comets and terrestrial planets was found for the epoch of the latest apparition of comets. Guliev’s conjecture about two trans-Plutonian planets is based on the illusory excess of cometary nodes at large heliocentric distances. The existence of cometary nodes at the solar system periphery turns out to be a solely geometrical effect.     

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.     

Are There Many Inactive Jupiter-Family Comets among the Near-Earth Asteroid Population?

We analyze the dynamical evolution of Jupiter-family (JF) comets and near-Earth asteroids (NEAs) with aphelion distances Q>3.5 AU, paying special attention to the problem of mixing of both populations, such that inactive comets may be disguised as NEAs. From numerical integrations for 2×10⁶ years we find that the half lifetime (where the lifetime is defined against hyperbolic ejection or collision with the Sun or the planets) of near-Earth JF comets (perihelion distances q<1.3 AU) is about 1.5×10⁵ years but that they spend only a small fraction of this time (∼ a few 10³ years) with q<1.3 AU. From numerical integrations for 5×10⁶ years we find that the half lifetime of NEAs in “cometary” orbits (defined as those with aphelion distances Q>4.5 AU, i.e., that approach or cross Jupiter's orbit) is 4.2×10⁵ years, i.e., about three times longer than that for near-Earth JF comets. We also analyze the problem of decoupling JF comets from Jupiter to produce Encke-type comets. To this end we simulate the dynamical evolution of the sample of observed JF comets with the inclusion of nongravitational forces. While decoupling occurs very seldom when a purely gravitational motion is considered, the action of nongravitational forces (as strong as or greater than those acting on Encke) can produce a few Enckes. Furthermore, a few JF comets are transferred to low-eccentricity orbits entirely within the main asteroid belt (Q<4 AU and q>2 AU). The population of NEAs in cometary orbits is found to be adequately replenished with NEAs of smaller Q's diffusing outward, from which we can set an upper limit of ∼20% for the putative component of deactivated JF comets needed to maintain such a population in steady state. From this analysis, the upper limit for the average time that a JF comet in near-Earth orbit can spend as a dormant, asteroid-looking body can be estimated to be about 40% of the time spent as an active comet. More likely, JF comets in near-Earth orbits will disintegrate once (or shortly after) they end their active phases.     

The activities of comets related to their aging and origin

Two indices have been developed for the purpose of comparing the natures of various classes of comets. The first is the Activity Index (AI), measuring the inherent magnitude increase in brightness from great solar distances to maximum near perihelion. The second, or Volatility Index (VI), measures the variation in magnitude near perihelion. Tentative determinations of these two indices are derived from observations by Max Beyer over more than 30 years for long-period (L-P) and short-period (S-P) comets near perihelion and from other homogeneous sources. AI determinations are made for 32 long-period (L-P) comets and for 14 short-period (S-P). The range of values of AI is of the order of 3 to 10 magnitudes with a median about 6. An expected strong correlation with perihelion distance q, is found to vary as q ^–2.3. Residuals from a least-square solution (AI) are used for comparing comets of different orbital classes, the standard deviation of a single value of AI is only ±1^m.1 for L-P comets and ±1^m.2 for S-P comets.Among the L-P comets, 19 of period P larger than 10⁴ years yield AI = 0^m.27 ± 0^m.25 compared to 0^m.39 ± 0^m.26 for 13 of period between 10² years and 10⁴ years. This denies any fading with aging among the L-P comets. Also no systematic change with period occurs for the VI index, leading to the same conclusions. Weak correlations are found with the Gas/Dust ratio of comets. No correlations are found between the two indices, nor of either index with near-perihelion magnitudes or orbital inclination.The various data are consistent with a uniform origin for all types of comets, the nuclei being homogeneous on the large scale but quite diverse on a small scale (the order of a fraction of kilometer in extent). Small comets thus may sublimate away entirely, leaving no solid core, while huge comets may develop a less volatile core by radioactive heating and possibly become inactive like asteroids after many S-P revolutions about the Sun. When relatively new, huge comets may be quite active at great solar distances because of volatiles from the core that have refrozen in the outer layers.     

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.     

Maximum non-gravitational acceleration due to out-gassing cometary nuclei

Maximum possible acceleration due to out-gassing from cometary nuclei is calculated for H₂O and CO(N₂) molecules. It is found that the maximum excess velocity at great distance is 0.18 km s^–1 so that excess velocities less than this value are compatible with the non-gravitational acceleration due to non-symmetric out-gassing. On the other hand, Comet 1975q and comet 1955V have excess velocities 0.81 and 0.80 km s^–1 respectively. These comets may be regarded as the candidates for possible interstellar comets.     

Capture of comets from the Oort cloud into Halley-type and Jupiter-family orbits

This paper analyzes the capture of comets into Halley-type and Jupiter-family orbits from the nearparabolic flux of the Oort cloud. Two types of capture into Halley-type orbits are found. The first type is the evolution of near-parabolic orbits into short-period orbits (with heliocentric orbital periods P < 200 years) as a result of close encounters with giant planets. This process is followed by a very slow drift of cometary orbits into the inner part of the Solar System. Only those comets may pass from short-period orbits into Halley-type and Jupiter-family orbits, which move in orbits with perihelion distances q < 13 au. In the second type of capture, the perihelion distances of cometary orbits become rather small (< 1.5 au) during the first stage of dynamic evolution under the action of perturbations from the Galaxy, and then their semimajor axes decrease as a result of diffusion. The capture takes place, on average, in 500 revolutions of the comet about the Sun, whereas in the first case, the comet is captured, on average, after 12500 revolutions. The region of initial orbital perihelion distances q > 4 au is found to be at least as important a source of Halley-type comets as the region of perihelion distances q < 4 au. More than half of the Halley-type comets are captured from the nearly parabolic flux with q > 4 au. The analysis of the dynamic evolution of objects moving in short-period orbits shows that the distribution of Centaurs orbits agrees well with the observed distribution corrected for observational selection effects. Hence, the hypothesis associating the origin of Centaurs with the Edgeworth-Kuiper belt and the trans-Neptunian region exclusively should be rejected.     

Orbital evolution of giant comet-like objects

We investigated by numerical integrations the long-term orbital evolution of four giant comets or comet-like objects. They are Chiron, P/Schwassmann-Wachmann 1 (SW1), Hidalgo, and 1992AD (5145), and their orbits were traced for 100–200 thousand years (kyr) toward both the past and the future. For each object, 13 orbits were calculated, one for the nominal orbital elements and other 12 with slightly modified elements based on the rms residual of the orbit determination and on the number of observations. As past studies indicate, their orbital evolution is found to be very chaotic, and thus can be described only in terms of probability. Plots of the semi-major axis (a) and perihelion distance (q) of the objects treated here seem to cross each other frequently, suggesting a possibility of their common evolutionary paths. About a half of all the calculated orbits showedq- ora-decreasing evolution. This indicates that, at least on the time scale in question, the giant comet-like objects are possibly on a dynamical track that can lead to capture from the outer solar system. We could hardly find the orbits with perihelia far outside the orbit of Saturn (q>15 AU). This is perhaps because the evolution of the orbits beyond Saturn is so slow that substantial orbital changes do not take place within 100–200 kyr.     

Vaporization of comet nuclei: Light curves and life times

We have examined the effects of vaporization from the nucleus of a comet and show that a latitude dependence of vaporization can, in some cases, explain asymmetries in cometary light curves. We also find that a non-uniform distribution of solar radiation over a comet can considerably shorten the vaporization lifetime compared to the results normally obtained by assuming that the nuclear surface is isothermal.Independent of any latitude effects, comets with CO₂-dominated nuclei and with perihelion distances less than 0.5 AU have vaporization lifetimes less than or comparable to their dynamical ejection times. This may explain the observed deficit of comets with small perihelion distances. Similarly comets with CO₂-dominated nuclei and perihelia near Jupiter's orbit have vaporization lifetimes that are shorter than the time for capture into short-period orbits. We suggest, therefore, that at least some new comets are composed in large part of CO₂, while only H₂O-dominated comets, with lower vaporization rates, can survive to be captured into short-period orbits.     

The population of faint Jupiter family comets near the Earth

We study the population of faint Jupiter family comets (JFCs) that approach the Earth (perihelion distances q<1.3 AU) by applying a debiasing technique to the observed sample. We found for the debiased cumulative luminosity function (CLF) of absolute total magnitudes H₁₀ a bimodal distribution in which brighter comets (H₁₀?9) follow a linear relation with a steep slope α=0.65±0.14, while fainter comets follow a much shallower slope α=0.25±0.06 down to H₁₀∼18. The slope can be pushed up to α=0.35±0.09 if a second break in the H₁₀ distribution to a much shallower slope is introduced at H₁₀∼16. We estimate a population of about 10³ faint JFCs with q<1.3 AU and 10<H₁₀<15 (radii ∼0.1-0.5 km). The shallowness of the CLF for faint near-Earth JFCs may be explained either as: (i) the source population (the scattered disk) has an equally very shallow distribution in the considered size range, or (ii) the distribution is flattened by the disintegration of small objects before that they have a chance of being observed. The fact that the slope of the magnitude distribution of the faint active JFCs is very similar to that found for a sample of dormant JFCs candidates suggests that for a surviving (i.e., not disintegrated) object, the probability of becoming dormant versus keeping some activity is roughly size independent.