The calculation of cometary and planetary orbits, based on a solution of the unperturbed two-body problem, offers a simple and easily understood method of predicting the position of such a body at any desired time. In addition, the analytical description of the motion in terms of conic sections also provides a particularly clear way of describing the relationship between the various orbital elements and the position in the orbit.
There is, however, the disadvantage that the perturbations caused by the major planets cannot be incorporated. Although the gravitational attraction exerted by the planets is generally an order of magnitude smaller than that of the Sun, over a period of many years it can lead to considerable deviations from an unperturbed orbit. The perturbations are particularly significant when a comet or a minor planet repeatedly passes close to one of the major planets. In this case, substantial changes are possible — as may be observed with various periodic comets. Examples are Comet P/Pons-Winnecke (1819 III), whose orbital inclination increased from 10° to 22° between 1819 and 1976 because its aphelion lay close to Jupiter, and Comet P/Wolf (1884 III), whose orbital period increased from 6.8 years to 8.3 years, following an encounter with Jupiter in 1922.
Although the perturbations caused by the major planets to most minor-planet orbits are far smaller than those mentioned in these examples, they still generally prevent satisfactory predictions of the orbits over a period of many years on the basis of a simple Keplerian orbit. This means that to find and identify faint minor planets it is essential to take these perturbations into account in calculating the ephemerides. Basically, there are two different procedures that may be used to do this. In the special perturbations method, we begin with the position and velocity of the minor planet at an initial epoch, and calculate the orbit step by step, using numerical integration, taking the accelerations caused by the Sun and all the planets into account, until we reach the desired end-point. The second possibility, the general perturbations method, consists of expressing the deviations from an unperturbed orbit as a periodic series, which then enables us to calculate the perturbed position at any given time. The advantage of an analytical method is, however, offset by having to devote considerable effort to developing the series expansions, which prevents the method from being applied to the several thousand minor planets. Analytical perturbation theories are therefore generally available for just the major planets (see Chap. 6).
KeywordsState Vector Orbital Element Minor Planet Analytical Perturbation Theory Major Planet
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- Introduction to the theory of multistep methods for the numerical solution of ordinary differential equations and description of the variable order variable step size Adams method DE.Google Scholar
- The most recent orbital elements of minor planets are published annually in the Ephemerides of Minor Planets  of the St. Petersburg Institute of Theoretical Astronomy as well as in various astronomical yearbooks.Google Scholar