Galactic perturbations on nearly-parabolic cometary orbits

The effect of galactic perturbations on long-period comet orbits is examined via numerical and analytical means. Relations are found between a comet's initial perihelion position and the positions of succeeding perihelia. It was found that the galactic effects were strongest on the comets initially at galactic latitudes close to 40°. In such cases the galactic perturbations caused the orbit to become almost circular before becoming nearly parabolic again. This effect allows comets with semimajor axes of about 25 000 AU to make only a few passages through the inner solar system in a time interval of 10⁹yr. Thus the galactic field is an important factor in the evolution of long-period comet orbits. The observed distribution of perihelia of long-period comets indicates that galactic effects have been active.     

The magnitude distribution, perihelion distribution and flux of long-period comets

The mass distribution and perihelion distribution of long-period comets are re-assessed. The mass distribution index is found to be 1.598±0.016 , indicating that the distribution is somewhat steeper than was obtained by previous analyses of an amalgam of all the available historical data. The number of long-period comets that have orbital perihelion distances, q , that fall in a specific q to q +d q range is found to be independent of q . It is also noted that the flux of long-period comets to the inner Solar system has remained constant throughout recorded history.
The number of long-period comets, , per 1-au interval of perihelion distance, per year, brighter than H , entering the inner Solar system is found to be given by log₁₀ =−2.607+0.359 H . It is therefore estimated that, for example, about 0.5, 30 and 2000 long-period comets with absolute magnitudes brighter than 0, 5 and 10 respectively pass the Sun on orbits with perihelion distances less than 2.0 au, every century.     

Orientation-dependent effects in Oort's theory of comet origin II. Anisotropies in the distribution of long-period comet orbits

The statistical significance of anisotropies in the distribution of orbital orientations among the long-period and nearly parabolic comets is evaluated. It is suggested that these anisotropies are not the result of observational selection effects. A numerical model for the distribution of orbital orientations is constructed, based on Oort's theory of comet origin and the assumption that the observed anisotropies are caused by multiple planetary perturbations over the course of many perihelion passages. The model, which is restricted to comets with peri-helion distances less than 0.3AU, does not predict any significant anisotropies.     

Orbital evolution of long-period comets I. Trivariate Monte Carlo simulation

The mean orbital evolution of long-period comets for 16 representative initial orbits to short-period comets is calculated by a Monte Carlo method. First, trivariate perturbation distributions of barycentric Kepler energy, total angular momentum, and its z component in single encounters of comets with Jupiter are obtained numerically. Their characteristics are examined in detail and the distributions are found to be simple, symmetric, and easy to handle. Second, utilizing these distributions, we have done trivariate Monte Carlo simulations of the orbital evolution of long-period comets, with special emphasis on high-inclination orbits. About half of the 16 initial orbits are traced up to 5000 returns. For each of these orbits, the mean values of semimajor axis, perihelion distance, and inclination; their standard deviations, survival, and capture rates; as well as time scales of orbital evolution are calculated as functions of return number. Survival rates of the initial orbits with high inclination (～90°) and small perihelion distance (～1–2 AU) have been found to be only two or three times smaller than those of the main-source orbits of short-period comets established quantitatively by Everhart. The time scales of orbitsl evolution of the former, however, are nearly 10 times longer than the latter. There is a general trend that, for smaller perihelion distance, the survival efficiency becomes higher. The results of this paper should be considered a basis for a succeeding paper (Paper II) in which the physical lifetime of comets will be determined, and a comparison with the orbital data will be done.     

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.     

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.     

Planetary impact probabilities for long-period comets

Planetary impact probabilities for long-period (near-parabolic) comets are determined by averaging Öpik's equations over inclination and perihelion distance for each planet. These averaged values compare well with the results of more elaborate Monte Carlo calculations. The impact probabilities are proportional to the square of the normalized capture radius of each planet, which in turn is a function of the planet's radius and mass, so that the major planets have the highest impact probabilities. Encounter velocities have an average value of $\text{3}\text{1}\text{2}$ times the planetary orbital velocity but the most probable encounter velocities are slightly higher than this for the terrestrial planets and slightly lower for the major planets. Comparison of the impact probabilities with the cratering record, corrected for gravity and velocity effects, indicates that long-period comets may account for 3 to 9% of the observed large crattes (diameter > 10 km) on the terrestrial planets. The inclination and perihelion properties of the impact probabilities obtained from numerical averaging provide a simple method for determining the impact probabilities for nonuniform distributions. The perihelion distribution of long period comets from J. A. Fernandez ((1981) Astron. Astrophys.96, 26–35) results in a crater production rate quite similar throughout the solar system, unlike that of a uniform perihelion distribution.     

Possibilities of using extreme observations to estimate cometary nucleus sizes

Not considering very rare in situ measurements of cometary nuclei, observations of comets at large heliocentric distances are the only direct source of our knowledge on their sizes. Observations of a cometary nucleus in pure reflected sunlight, at the time when coma is absent, are the way in which the nucleus size can be estimated. Probabilities that extreme observations represent non—active stages of cometary nuclei and also reliability of derived cometary nucleus sizes are investigated. Statistical analysis is based on a sample of 2842 photometric observations of 67 long-period comets observed at large heliocentric distances. For any long-period comet, there is a probability of 2:3 that the sizes derived on the basis of observations at extreme distances are in good agreement with the real nucleus sizes. For new comets in Oort's sense the probability is 3:4 independent of investigated arcs of orbits. For old comets a chance to estimate correct sizes is 1:2 but on the pre-perihelion arc only 1:3. It is also demonstrated that a premature start of activity prior to perihelion or a longer fading after perihelion is more frequent than a short-time isolated activity at large heliocentric distances.     

Development of a comet as it pursues its orbit

The properties of cometary dust-swarms in almost parabolic long-period orbits are examined. In general their self-gravitation is stronger than the solar disruptive influence for all except the relatively small part of the orbit within planetary distances during which the sun dominates by so great a factor that the individual particles of the swarm pursue independent orbits apart from the possibility of collisions between them. At aphelion the internal relative speeds of particles are only a few centimetres per second, but at and near perihelion they may rise to the order of a kilometre per second. For purely dynamical reasons the extent of the swarm in directions perpendicular to the orbital motion will strongly diminish as perihelion is approached and correspondingly increase thereafter, while the dimension along the orbit will change in direct proportion to the orbital velocity. Every particle must cross through the median orbital plane near perihelion, and collisions between a proportion of the particles will occur at speeds capable of fragmenting them into myriads of smaller dustparticles, also heating them at and near the colliding elements of their surfaces. Increase of reflected sunlight will result and also release of material in gaseous form by solar plus collisional heating. Sufficiently finely divided dust particles will be driven out of the comet by radiation-pressure to form a dust-tail, while suitable gaseous compounds if present will be driven out to give a gas-tail. For Sungrazing comets, complete gasification must occur at and near perihelion, and very considerable extension along the orbit. Such comets would recondense to small solid particles on receding again from the Sun. The effect of passage of the solar system through interstellar gas-clouds is shown to be capable of substantially affecting the angular momentum of a comet about the Sun, thus accounting for the existence of comets with high values of perihelion-distance. This same process would enable cometary particles to adsorb interstellar gases at their surfaces and regenerate their gas-content. The mass-loss by a comet at each return strongly indicates, that comets cannot have originated at the same time as the planets, a result further supported by the rapid expulsion of entire comets through purely dynamical action of the planets. That the quiescent structure of comets consists of a vast widely spaced swarm of minute dust-particles receives circumstantial support from the highly varied and peculiar properties long since recorded for numerous comets. These properties exhibit such erratic diversity as to make clear that only a theory involving considerable range of essential parameters can be capable of accounting for them adequately.     

New edition of the catalogue of short-period comets

The Institute of Theoretical Astronomy in St. Petersburg and the Astronomical Institute in Bratislava are preparing a new edition of the Catalogue of short-period comets. This edition will be supplemented by short-period comets discovered after the year 1983 and comprises some new features, e. g. the evolution of orbital elements between the years 1750 and 2050, and the perihelion passages of comets within the 1994–2050 years. A new method has been employed for the determination of nongravitational parameters from the osculating elements of a comet based on all its observed returns.The method has been tested on the comets P/Comas Solá and P/Forbes with all returns, except the last one. The results have been compared with the osculating elements of the last return and those used in the old edition of the Catalogue of short-period comets. The new method enables a good prediction of osculating elements for the future, at least for the next return.     

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.     

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.     

From the Oort cloud to observable short-period comets – I. The initial stage of cometary capture

We investigate the first stage of the dynamical evolution of Oort cloud comets entering the planetary region for the first time. To this purpose, we integrate numerically the motions of a large number of fictitious comets pertaining to two samples, both with perihelion distances up to 5.7 au and random inclinations; the first sample is composed of comets whose orbits have at least one node close to 5.2 au, while the second is not subject to this constraint. We examine the orbits when the comets come to aphelion after their first perihelion passage within the planetary region, and find that there is a clear statistical dependence of the energy perturbations on the Tisserand parameter. There appear to be two main processes, of comparable importance, governing the shortening of semimajor axes to values of less than 1000 au, i.e. planetary close encounters, especially with Jupiter, and indirect perturbations due to the shifting of the motion from barycentric to heliocentric and back; the former process mostly affects comets crossing the ecliptic at about 5.2 au, or on low-inclination orbits, while the latter mostly affects comets of small perihelion distance. This last result may help to understand the relative paucity of Halley-type comets with perihelion distances larger than about 1.5 au.     

Close encounters of nearly parabolic comets and planets

An overview is given of close encounters of nearly parabolic comets (NPCs; with periods of P > 200 years and perihelion distances of q > 0.1 AU; the number of the comets is N = 1041) with planets. The minimum distances Δ_min between the cometary and planetary orbits are calculated to select comets whose Δ_min are less than the radius of the planet’s sphere of influence. Close encounters of these comets with planets are identified by numerical integration of the comets’ equations of motion over an interval of ±50 years from the time of passing the perihelion. Close encounters of NPCs with Jupiter in 1663–2011 are reported for seven comets. An encounter with Saturn is reported for comet 2004 F2 (in 2001).     

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-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 young long-period comet family of Saturn

The distributions of long-period comets with respect to the minimum distance Δ between their orbits and the orbit of Saturn or Jupiter, constructed by Konopleva using data up to 1972, exhibit a sharp peak at Δ<0.5 au for the Saturnian family, while being fairly monotonic for Jupiter. Hence, in view of the appreciable eccentricity of Saturn's orbit and the rotation of its perihelion longitude with a period of 47 kyr, the conclusion was drawn by Drobyshevski that the objects belonging to this peak are young (10 kyr).
Similar distributions constructed using more recent data show less pronounced differences between one another. Analysis of the distributions for various epochs shows that the initially noted difference is due to observational selection, being inherent to brighter comets. Since on average the cometary activity fades with age, the conclusion that the Saturnian family comets, forming the peak at Δ<0.5 au, are young is all the more substantiated. The question concerning the origin of these comets, which in all likelihood were ejected over a period of a decade from deep inside the Saturnian sphere of influence , is still open. The only self-consistent hypothesis that we see now is that of their appearance as a result of an explosion of the electrolysed ice envelope of Titan. We encourage the development of other explanations.     

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.     

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.     

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.