The orbit of P/Stephan-Oterma has been determined using 128 observations at the two apparitions (1867 and 1942) and taking into account perturbations by Venus to Neptune. The evolution of the comet's orbit has been studied over the interval 1660–2060.

A brief survey is given of past determinations of the masses of the principal planets from analyses of the motions of comets. Some numerical experiments using comets which have close approaches to Jupiter are made. As a result of these experiments, it is concluded that the conventional least squares solution for the correction to the mass of Jupiter is inadequate for comets which have a close approach to Jupiter. It is further concluded that perhaps, in some cases, the apparent presence of nongravitational forces is merely a manifestation of the failure of the conventional orbit correction process to adjust correctly the orbits of objects which undergo very large perturbations, and it also may be a consequence of errors in the adopted planetary masses. It is suggested that the use of partial derivatives obtained through the numerical integration of the variational equations may overcome the difficulties.

Necessary and sufficient conditions are formulated for determining the mass of Jupiter from large perturbations induced in cometary orbits in the sphere of action of Jupiter. A procedure for the investigation has been developed and programmed for an electronic computer. Comparison of heliocentric and jovicentric computations shows that the perturbations on P/Wolf could be determined with great accuracy when this comet passed through Jupiter's sphere of action in 1922. The first attempt has been made to determine the mass of Jupiter using this passage and the observations of the comet in 1925. The resulting value for the reciprocal mass is 1047.345.

An attempt has been made to derive the mass of Jupiter from modern observations of the minor planet 10 Hygiea. The calculations have been made using computer programmes prepared at the Institute for Theoretical Astronomy. From 250 observations 1932 to 1967 we obtained the reciprocal mass 1047.345±0.040 (rms error). After excluding some less reliable observations and allowing for the elliptic aberration terms we obtained 1047.326±0.033. By means of a more sophisticated computer programme and utilizing 100 additional observations, some of them made in 1969, we obtained the value 1047.324±0.023. Our results are compared with recent determinations by other investigators.

The principal features of the motion of Hidalgo over the interval 1400–2900 are described. The possibility that this object is an extinct (or nearly extinct) comet nucleus is discussed. A determination of the mass of Saturn, using observations of Hidalgo during 1920–1964, is presented and compared with other recent determinations.

A common method of determining the mass of a planet is to solve for the orbit of some other object several times with different trial values for the mass in question. It is pointed out that a parabola fitted to the values of the sums of the squares of the residuals may be used to obtain, not only the planetary mass, but also its probable error. Marsden's determination of the mass of Saturn from observations of Hidalgo is used as an example.

It is shown that in a sufficiently small interval there is a uniform distribution of the minimum distances between the orbit of a comet and the orbits of minor planets. It is also shown that there is a uniform distribution of the distance of the planet from the point of closest approach when the comet reaches that point. It is found that P/Daniel may collide with a microplanet of mass 4.05 × 108 g during 100 revolutions. It is possible for one of 60 short-period comets to meet during 100 revolutions a microplanet of mass 9.9 × 109 g and radius 8.6 m. The changes in reciprocal semimajor axis of a comet arising from close encounters with minor planets are also studied but found to be insignificant.

Formulae are obtained for the distribution of molecules in the cometary head, taking into account the conditions of hydrodynamic and free molecular flow in various regions around the nucleus. Experimental data are used to derive physical parameters near the nuclei of comets 1952 III, 1955 V, 1957 III, and 1960 II and the rate of decrease of mass. The possibility of chemical reactions in the region close to the nucleus is discussed. Gas condensation is shown to be a possible cause of dust formation under the conditions existing near the nucleus, and this process may be responsible for the major portion of the cometary dust component. The observed grouping of synchrones in the cometary tail can be explained on the assumption that the nuclear surface comprises two (or more) areas differing essentially in evaporation rate, the amount of matter ejected varying over the rotation period of the nucleus. Charged dust particles are shown to form, with electrons and ions, a common medium, i.e., dust plasma, which can be treated by the same methods used for ordinary plasma. Special investigations appear to be desirable when comets intersect meteor streams.

According to modern data water ice is the main constituent of cometary nuclei. The rate of its evaporation determines the rate of liberation of other constituents, including those that determine the photometric properties of comets. The estimates of the total mass loss per perihelion passage seem to give values about one order of magnitude smaller than those necessary to explain the nongravitational forces as due to the reactive force of material leaving the nuclei. Thus the conventional explanation of the nongravitational forces deserves further study from the point of view of the physical theory of comets.

The probable parent-molecules of radicals such as C3 and N2+ are discussed, and it is concluded that cometary nuclei may contain complicated organic molecules, such as C3H4, CH2N2, and C4H2. It is suggested that these molecules are formed by radiation synthesis in solid phase. In a time interval of order 107 to 109 yr bombardment from cosmic rays would be expected to transform the chemical composition to a depth of 1 m. Solar cosmic rays do not penetrate as far, and as a result the surface layer of the nucleus can be enriched with unsaturated hydrocarbons. After a critical concentration of this explosive material is reached a further burst of solar cosmic rays can initiate an explosion and thus an outburst in the comet's brightness. This mechanism is the only one advanced to date that can explain the synchronism of the energy output over the whole nuclear surface.

The evolution of a cometary nucleus depends on the size distribution of the dust content. If all the dust particles are small enough, they will all be thrown off into the coma; the secular decrease in the comet's brightness is due simply to the decrease in the nuclear radius, and the comet eventually disintegrates. On the other hand, if larger grains are present, they will start to form spots of a low-emissivity mineral envelope on the surface of the nucleus; in this case, the decrease in brightness is due to the increase in the area of the screened part of the surface, and the reduction in the nuclear radius is comparatively small. It is shown that after about 90% of the surface has been screened, the rate of screening increases considerably, as smaller particles become trapped between the larger ones; and there will be a corresponding acceleration in the brightness decrease. These nuclei are eventually covered with a solid mineral envelope, and they are transformed into asteroids.

Three methods have been developed for determining the sizes of cometary nuclei. They are all based on the spotted model for an icy nucleus. The first method follows from the relationship between the icy (unshielded) area and the total absolute brightness, although the quantity determined is actually a function of the radius and the shielding coefficient. The second method is based on analysis of the secular brightness decrease and the assumption that it is possible to establish when a new surface layer of dust starts to form; in practice, this method is found to be applicable only in the cases of P/Pons-Winnecke and P/Tuttle. The third method is also based on the secular brightness decrease, and it requires some assumption about the volume fraction of dust in the nucleus.

It is shown that sudden outbursts of comets, or even complete splitting of their nuclei into several parts, are rather frequent phenomena that need to be accounted for in the evolution of cometary orbits. The distribution of the points in which such events occur indicates that the tidal action of the Sun and Jupiter, and also solar radiation, are much more probable causes of these effects than are collisions with the asteroids. Splitting is found to be relatively more frequent for comets moving in hyperbolic orbits. The observed velocities of separation indicate that the disruption of the nucleus provides an effective mechanism for the ejection of the fragments from long-period into hyperbolic orbits, augmenting the loss by planetary perturbations. The outbursts can be responsible for smaller changes in the orbits, which after repetition might produce a quasi-secular variation.

Two types of cometary nuclei are considered: a homogeneous icy nucleus and an icy nucleus covered with a dispersive surface layer, 1 cm thick and having poor heat conductivity. The temperature of the evaporating ice surface of a nucleus rotating in a period of 6 hours about an axis perpendicular to the orbital plane was determined by numerical integration of the equation of heat conduction in the first case and of the Stefan problem in the second. For a nucleus of radius 105 cm and mass of 3.8 × 1015 g the reactive force was found to be about 107 dyn in either case, and the secular change in mean motion is of the order of the observed values. For the homogeneous nucleus the dependence of the reactive force on heliocentric distance is obtained.

The effects of rotation of a cometary nucleus on the character of nongravitational parameters are discussed. It is suggested that the change of a nongravitational acceleration to deceleration (and vice versa) may be related to a precessional motion of the nucleus, or to secular variations in the lag angle coupled with severe orbital modifications.

A study of the nongravitational deviations in the motion of P/Encke suggests that the nucleus of this comet can conceivably be approximated by a core-mantle model, composed of a porous solid core with embedded ices, surrounded by a thick envelope of free ices that gradually sublimates out.

The problems of laboratory simulation of cometary phenomena are discussed. Results are given of experimental investigations of the thermodynamic parameters of H2O ice and of frozen mixtures of electrolytes during sublimation under conditions of high vacuum and low temperature. At a distance of 1 AU from the Sun the equilibrium temperature of H2O ice is shown to be about −75 °C. Conditions are determined for the formation and disintegration of dust matrices during the sublimation of ices containing small dust particles. Dust particles are shown to leave the surface of the ice in the form of conglomerates of primary particles. The outflow velocities of the particles are found to be as high as several metres per second. The results obtained are discussed in relation to cometary phenomena.

Experimental results for the determination of the exhaust thrust developed by vacuum-sublimation products of H2O ice and frozen aqueous electrolytes are described. The contribution of the exhaust thrust to nongravitational effects on comets is discussed.

The theory of planetary accumulation leads quite definitely to the conclusion that the formation of Oort's cometary cloud is the result of ejection of bodies to the outermost parts of the solar system due to encounters with the giant planets during their growth. Uranus and Neptune could have grown to their present dimensions only if the initial mass of solid material in their zones were substantially larger than that of these planets. The relative velocities of the bodies were increased through perturbations by the planetary embryos, and on reaching the escape velocity they would start to be ejected. Our concept of this process differs from that suggested by Öpik by the assumption that Jupiter and Saturn accreted hydrogen, not in solid but in gaseous state, and by the introduction of a more effective mechanism for the interaction with several embryos. In their final stages the embryos ejected amounts of mass an order of magnitude higher than the amounts accreted. Most of the mass was ejected into interstellar space by Jupiter, but the cometary cloud was created mainly by Neptune. The mass of the cloud is estimated to be about three times that of the Earth.

An attempt has been made to estimate the rate of destruction of the Oort cloud by stars passing through it. Two different mechanisms of cloud dispersion have been investigated. Numerical estimates show that the cumulative dispersing mechanism plays the leading role. The lower limit for the half-life of the cloud is 1.1 × 109 yr.