A review of modern concepts of the evolution of cometary orbits for the time span of some million years is presented. These concepts are based on the hypothesis of the existence of the Oort cloud, the theory of diffusion developed by Shtejns, and the work by Kazimirchak-Polonskaya concerning the influence of the major planets in transforming cometary orbits. All these ideas are currently being developed with the aid of rigorous mathematical methods by cometary researchers in various parts of the world.

The cometary observation service is surveyed. Absolute magnitudes and other physical characteristics are given for the comets that appeared during 1965–1970. The general laws of cometary brightness, in particular the secular decrease in brightness, are discussed. Remarks are made about the standardization of cometary photometry.

The study of cometary brightness variations can be a good method for determining conditions in interplanetary space. In this work we compare the light curves of comets with curves showing variations in the geomagnetic field at times when the comets are near the ecliptic plane. We have used photometric observations made of 29 comets between 1881 and 1937. It is shown that an increase in the brightness of a comet is associated with an increase in the geomagnetic activity index and consequently with the influence of the solar corpuscular streams and solar wind.

The cometary observing programme at the Crimean Astrophysical Observatory is described.

The programme of cometary observations at the Nice Observatory is described. Observations are made with the double astrograph by the Trépied-Metcalf method, offset motions of up to half a degree per hour being possible. Five reference stars are measured, and the reductions are made by the least-squares dependence method.

The new short-period comet Churyumov-Gerasimenko, discovered by the authors on plates taken by the Kiev University cometary expedition to Alma-Ata in September 1969, was systematically photographed with fast telescopes at Byurakan and Alma-Ata until March 1970. Measurements were made of the photographic magnitude of the photometric nucleus, as well as of the photographic and photovisual integral magnitudes. The variations in nuclear magnitude were found to be well correlated with changes in the total sunspot area. The integral photometric parameters are Hy = 11.91±0m.54 and n=4.0±0.8 (in the photographic spectral region). Deviations of the tail axis from the prolonged radius vector were considerable. A spectrogram shows the continuum and emission of CN, C2 and C3 in the head, the continuum and a single emission (perhaps CO+) in the tail.

I agree entirely that there is a need for such a service. As the proposal has come from the Institute for Theoretical Astronomy, is it intended that the service should be organized from there? Consideration should be given to having subsidiary centres with identical information available on request – as for the double-star service.

Present and future requirements concerning ephemeris prediction for comets (particularly the short-period comets) are discussed, with reference to both positions and magnitudes.

A series-solution method for highly-eccentric perturbed orbits using a modified form of Hansen's method of partial anomalies is presented. Series in Chebyshev polynomials in the eccentric anomaly of a comet and the mean anomaly at an epoch of a planet provide a theory valid to first order with respect to the masses. The first-order solution becomes a reference solution about which higher-order perturbations are obtained by the method of successive approximations. The first-order solutions are valid approximations for long durations of time, whereas the higher orders are valid only over the interval of time that is selected for the Chebyshev expansions. The method is somewhat similar to Encke's method of special perturbations except that for each successive interval of time perturbations about a first-order solution are calculated instead of perturbations about a conic solution.

The main points of the method of partial anomalies are described. The method makes it possible to obtain analytical solutions for comets moving in orbits with high eccentricities. The method may be applied, in particular, to systems in which the orbits of the disturbed and disturbing bodies have points equidistant from the Sun. The method is applied to the calculation of the general perturbations by Saturn on P/Tuttle.

When, in consequence of a close approach, a comet of the Jupiter group changes its osculating semimajor axis from a > 1 to a < 1 (á = 1 for Jupiter's orbit), or vice versa, then the normal case is that of an abrupt change from one side of Jupiter's orbit to the other one. Under special conditions, however, temporary capture into satellite or ‘Trojan’ status is possible. P/Slaughter-Burnham, the first known comet in temporary 1:1 resonance with Jupiter, sheds some light on the requirements for Trojan captures. In consideration of the recent finding that the Trojan ‘cloud’ around L4 contains probably as many as 700 Trojans brighter than magnitude 20.9, it is suggested that at least some comets of the Jupiter group may have originated among these accumulations around L4 and L5.

The theory of the libration motion of a body close to a commensurability in mean motions with Jupiter is briefly described, and the importance of such motions in that they enable the body to avoid close approaches to Jupiter is discussed. By means of numerical integrations of the equations of motion some of the work that has been done on librations in orbits of low eccentricities is extended to higher eccentricities, so that the results can be applied to cometary orbits. By making some approximations the results are expressed in a very simplified form which makes it possible to compare the sizes of libration regions. This is done for six interior commensurabilities, namely the 3:1, 2:1, 5:3, 3:2, 7:5 and 4:3 commensurabilities.

An intermediate orbit is developed for studying the hyperbolic jovicentric motion of a comet in the vicinity of Jupiter, where the effect of Jupiter's oblateness may become significant. A comparison is made with some of the results obtained by other investigators on the motions of specific periodic comets.

In this work we propose an approach to the calculation of secular perturbations on a comet, valid for any eccentricity (less than unity) and any inclination (not zero). The effect of several planets is discussed, but we suppose that there are no significant close approaches to any planet.

The Taylor formula is used directly in a method of numerical integration of the n-body problem of celestial mechanics; the derivatives in the expansion of the coordinates are calculated successively at each integration step according to the generalized Steffensen rule. The proposed method is the most precise of all numerical methods based on the predetermined part of the Taylor series. The method is used with a variable number of derivatives at each integration step and also with a variable step. The cumulative error in the coordinates increases more slowly in our method than in any other. We can apply the method to the study of the motion of a comet or minor planet, taking into account the perturbations by eight major planets; the method allows for the simultaneous integration of a great number of objects of zero mass.

At the Institute for Theoretical Astronomy we have formed a library containing about 120 standard computer programmes. They include calculation of rectangular coordinates and velocities from elements, Lagrangian interpolation, and orbit improvement by the Eckert-Brouwer method. For the investigation of the motions of the minor bodies of the solar system we have constructed a programme for integration with a variable step and making allowance for perturbations by all the major planets and for nongravitational effects. The calculation of the perturbations is carried out using Herrick's vector parameters by the method of variation of arbitrary constants. The programme has been used for studying the motion of P/Encke.

A series of standard programmes has been developed for numerical integration by Cowell's method of the differential equations of motion of minor bodies. A variable step is used, and perturbations by eight major planets and nongravitational effects are taken into consideration. Further programmes have been constructed as part of a general attempt at ITA to produce numerical theories of cometary motion. They include the reduction of observations, the comparison of the observations with theory and the improvement of orbits. The programmes make it possible to calculate (O–C) residuals of up to 2000 observations simultaneously.