An estimate is made of the maximum range over which the motion of a particle about the Sun and in the galactic field is stable in Hill's sense. An equation fitting the Hill surface is found, with allowance made for the fact that the galactic potential differs from that of a point mass. The volume enclosed by the surface is 3.35 cubic parsecs, the greatest distance from the Sun being 1.42 parsecs, this being along the line to the centre of the Galaxy.

By means of the Oort-Schmidt definition of ‘new’ comets, Sekanina's catalogue of original and future orbits, and the 2σ and 3σ rules, we have compiled a list of 33 ‘new’ comets. We draw some conclusions concerning the size of the cometary cloud.

It is assumed that the accumulation of small, independent, random perturbations in the reciprocal semimajor axis of the orbit of a comet follows a normal distribution law whose standard deviation is a function of the inclination and perihelion distance and that for nongravitational forces it is a function of perihelion distances only; it is also assumed that secular accelerations do not change into decelerations, and vice versa. The standard deviations given by diffusion theory are in good agreement with the mean values of nongravitational impulses obtained from calculations from short overlapping arcs. The mean lifetime of a comet is found to be one hundred revolutions. To explain why many more near-parabolic comets are actually discovered than are theoretically expected the existence of comets of very short lifetimes must be accepted.

Numerical methods are used to establish the fraction of solar radiation utilized in the sublimation of ice on the surface of a cometary nucleus. An estimate can then be made of the lifetime of the nucleus. For a nucleus of radius 2 × 105 cm in an orbit of semilatus rectum 0.6 AU the lifetime is 300 revolutions. Comparison with diffusion theory reveals a significant deficiency of the theory in the case of comets with small reciprocal semimajor axes.

A number of problems concerning the motions and orbital statistics of comets are suggested for numerical discussion.

Single, random, close encounters of long-period comets with Jupiter are studied. In contrast to earlier work, Jupiter's orbit is taken to be elliptical, but this has no effect on the rate of capture into short-period orbits and neither does it influence the distribution of longitudes of perihelion.

Evolution of short-period cometary orbits under the influence of close and moderately close approaches to Jupiter is studied. We have restricted the discussion to orbits in Jupiter's orbital plane and have neglected distant perturbations by Jupiter. The general evolution is a diffusion towards orbits of larger semimajor axes and a loss of comets, mainly due to ejection along hyperbolic orbits. We have found that the dynamical evolution of the orbits severely alters the assumed initial distribution of orbits during a time of some hundred years. We also tried to obtain an idea of the importance of close approaches as compared to the effect of moderate approaches to Jupiter for the evolution of the orbits. This was done by calculating the evolution of a group of comets twice, first considering the effect of perturbations up to a distance of 1.0 AU from Jupiter and then to a distance of 2.0 AU. The statistical results of the two calculations are in general agreement, indicating that close approaches to Jupiter mainly determine the evolution of the short-period comets.

A new classification for short-period comets is proposed – one based on minimum approach distances to the respective planets, rather than on aphelion distances.

The equations of motion of many of the comets of the Jupiter and Saturn families have been integrated for a period of 400 yr. Several examples illustrate the role of the major planets, especially Jupiter, in the evolution of these orbits: in such cases comets may be captured by Jupiter from Saturn's family or ejected from Jupiter's family into Saturn's, and so on; the possible escape of a comet along a hyperbolic orbit and the capture by Jupiter of a comet from a hyperbolic orbit are also demonstrated. Fictitious comets of the Uranus and Neptune families are also studied. It is concluded that a combination of diffusion theory and capture theory is in good agreement with the observations and makes it possible to consider both long- and short-period comets as part of a unified cometary complex.

The evolution of the solar system is surveyed, it being presumed that the Sun, Jupiter, and Saturn formed rather quickly and essentially with the composition of the original collapsing cloud of dust and gas. Just as the refractory material of the cloud is considered to have formed into planetesimals, from which the terrestrial planets collected, so is the icy material supposed to have produced comets, or cometesimals, from which Uranus and Neptune (and to some extent Saturn and Jupiter) were built up. The presence of a residual belt of comets beyond the orbit of Neptune is discussed, analysis of possible perturbative effects on P/Halley indicating that the total mass of such a belt at 50 AU from the Sun could not now exceed the mass of the Earth.

The processes in interstellar space, as well as the structure of nebulae – including the numerous small condensations around newly formed stars – suggest a relationship to the formation of the cometary cloud surrounding the Sun. Study of carbonaceous chondrites has revealed that in the earliest stage of the solar system's existence frequent periods of brief heating occurred, leading to the formation of very complex organic compounds and also of chondrules of crystalline structure. This was most probably a consequence of cometary collisions. The Tunguska phenomenon illustrates some consequences of a similar collision.

It has become evident that comets and other small bodies are indications of eruptive evolution processes occurring in many of the planetary bodies of the solar system. The total number of near-parabolic comets moving in the solar system is 1011 to 1012, but as many as 10 to 15 percent of them are leaving the solar system with hyperbolic velocities. Taking into account also the number of short-period comets that degenerate into asteroids and meteor streams, we have estimated the total number of comets formed during the lifetime of the solar system as 1015 to 1016 (and total mass 1029 to 1031 g). The investigation of comets and other small bodies enables us to evaluate the scale of the processes of cosmic vulcanism and the tremendous internal energy of the planets, that energy being derived from the initial stellar nature of planetary material.

The various hypotheses on the origin of comets are surveyed, with particular reference to the phenomenon of comet-streaming, as revealed by statistical analysis of the directions of the perihelia of cometary orbits. The difficulties of reconciling comet-streaming with any of the hypotheses of cometary origin is pointed out. It is shown that in the case of the interstellar hypothesis these difficulties may be overcome by supposing that the velocities of incoming comets are decreased through the accretion of material from an Oort cloud of solar-system comets. An estimate is made of the total number of interstellar comets in the Galaxy.

The evolution of the orbits of 19 asteroids of particular interest has been studied over the interval 1660–2060, perturbations by Venus to Pluto being taken into account. Information was obtained about the encounters with Venus, the Earth, and Mars. A few approaches of Hidalgo to Jupiter were noted. In distinction to the orbits of short-period comets, the orbits of the 19 asteroids are stable throughout the 400-yr interval.

The long-term evolution of the orbits of 944 Hidalgo, 1036 Ganymed and 1134 Kepler is investigated.

By means of the Gauss-Halphen-Goryachev method secular orbital variations have been studied for a period of 4000 yr (from −50 to +3950). The method has been applied to the periodic comets Halley, Brorsen-Metcalf, Pons-Brooks, Westphal, Olbers, Neujmin 1, and Encke. Investigations have demonstrated that the use of the method for most comets is not expedient as it does not allow for the possibility of approaches to the major planets and thus does not reflect the real evolution of the cometary orbits. Application of the method over the interval of 2000 years back from the epoch 1950.0 for the planets 279 Thule, 1162 Larissa, 1180 Rita, and 1202 Marina, as well as from the epoch 1850.0 for 1 Ceres has given adequate results and has displayed the stability of these orbits. Study of the secular perturbations on the Leonids over the interval of 4000 yr has confirmed the stability of that meteor stream. By means of the same method 14 minor meteor streams were investigated, and their orbits also proved to be stable. The availability of the various systems of osculating elements has permitted us to estimate for the first time the possibility of the encounter of those streams with the Earth over a 4000-yr period. For further details see Galibina (1970a, 1970b, 1971).

The Halphen-Goryachev method has been used for estimating the elements of the orbits of the Quadrantid and δ Aquarid meteor streams 4000 yr back from the epoch of osculation. Three meteor orbits were selected for each stream: the mean orbit, based on data by Shelton (1965) and by Jacchia and Whipple (1961), and also two variants. Two cases were discussed: in the first case only the perturbations by Jupiter were taken into account, and in the second those by Jupiter and Saturn. Designating the initial epoch of osculation as the year zero, we compared our results for the Quadrantids with those by Hamid and Youssef (1963) at the final epoch (the year −4000). The disagreements in the elements i, ω, and Ω were 3°, 8°, and 1°, respectively (in absolute value). In comparing the mean orbits of the two streams it was noticed that between the years −800 and −900, and also between −2300 and −2400, the eccentricities and inclinations to the ecliptic were practically identical. However, the minimum differences in Ω and ω were 174° and 153° between −800 and −900, and 15° and 30° between −2300 and −2400. Consequently, if only secular perturbations are taken into account we cannot obtain a positive answer to the question of connection between the Quadrantid and δ Aquarid streams during the 4000-yr interval. The orbital changes caused by the Poynting-Robertson effect will be negligible for the meteors discussed and can be disregarded.

A model of the interplanetary dust medium that includes two subsystems, a spherical component and a flat one, has been proposed by the author as a result of analysis of radar meteor orbits. The disintegration of short-period comets is the main source of the flat dust cloud, while the disintegration of long-period comets is the main source of the spherical cloud. Assuming that dust particles fall into the Sun due to the Poynting-Robertson effect, one can estimate the intensity of ejection of dust from long-period comets necessary for the maintenance of the observed density of interplanetary dust. It has been found that the long-period comets should eject between 7 × 1014 and 2 × 1015 g of meteoric dust per year.

We study the secular evolution of the orbit described by a meteoric particle moving under the influence of planetary perturbations and the Poynting-Robertson effect. Using a simplified model, this study is made with only one disturbing planet. We present preliminary results on the long-term variation of the osculating elements of the orbit.

The increase in the width of a meteor stream with time is usually regarded as due to differences in the secular variations of the slightly different orbits of individual meteoroids. However, the ordinary differential planetary perturbations acting on different parts of an elliptical ring of meteoroids have a more important effect. Model calculations are presented for a filament-like stream that makes close approaches to Jupiter. They confirm the major importance of deformations of the stream by direct planetary perturbations that act even on a filament-like stream of zero thickness. The disturbed part of the stream takes the form of a bend, dispersing along the initial orbit and producing a significant decrease in the spatial density of the meteoric particles.