Perturbations of Celestial Bodies

deviations of the actual trajectories of celestial bodies from the trajectories these bodies would follow if they were interacting with only one body.

The paths of motion in the two-body problem are the so-called conic sections—ellipse, parabola, and hyperbola. Motion along a conic section may be regarded as a first approximation under the condition that one of the attracting masses considerably surpasses all the others in size. Thus, for example, the motion of the planets around the sun in the solar system may be regarded, in the first approximation, as motion along elliptical orbits. The mutual perturbations of planets in this case are small and may be computed by means of series expansions in powers of small parameters (analytical methods) or by the numerical integration of equations of motion (numerical methods). Customarily accepted as small parameters are the masses of planets, expressed in units of the sun’s mass, as well as the eccentricities and inclinations of their orbits. The terms of the series are called perturbations or irregularities in the motion of celestial bodies and take the form At^m, where m = 1, 2, … , and, A sin (αt + β). Terms of the first form are called secular perturbations, and those of the second form are known as periodic perturbations. The coefficients A include the mass of planets to various positive powers as a factor and therefore have small magnitude. The perturbations that contain masses of the planets to the first power are called perturbations of the first order; those to the second power, perturbations of the second order; and so forth. In constructing a theory of the motion of large planets, one must consider perturbations of the second order and certain perturbations of the third order. Among periodic perturbations, special attention must be given to those whose coefficient α in the argument of the trigonometric function is very small. Since the period of perturbation is equal to 360°/α, then with small α the period of the corresponding perturbation is very large in comparison with the period of the planet’s own revolution around the sun; such perturbations are said to be of long period.

Perturbations in the motion of celestial bodies, including artificial ones, may be caused by the attraction of other celestial bodies, deviation of the shape of these bodies from spherical, the resistance of the environment within which the motion is taking place, a change in the mass of the body in the course of time, the pressure of light, and for other reasons. In the case of binary stars, perturbations are caused by the attraction of other close stars as well as by the general gravitational field of the galaxy. The determination of the perturbations of celestial bodies is a cumbersome problem in computation. Thus, for example, in the theory of the moon’s motion, set forth by E. Brown, the solar perturbations in the formula from which the moon’s longitude is determined, contain 312 trigonometric terms. High-speed electronic computers have been used successfully to calculate perturbations by means of previously set series expansions and to obtain the trigonometric series themselves based on given elements of orbits of celestial bodies. By numerically integrating the equations of motion, the perturbation coordinates of celestial bodies may be directly obtained; then the problem of calculating the perturbations disappears (Cowell’s method). The theory of the perturbed motion of celestial bodies is the main subject of celestial mechanics.