The Sun's gravitational attraction is the main force acting on each planet, but there are much weaker gravitational forces between the planets, which produce perturbations of their elliptical orbits; these make small changes in a planet's orbital elements with time. The planets which perturb the Earth's orbit most are Venus, Jupiter, and Saturn. These planets and the sun also perturb the moon's orbit around the Earth—Moon system's center of mass. The use of mathematical series for the orbital elements as functions of time can accurately describe perturbations of the orbits of solar system bodies for limited time intervals. For longer intervals, the series must be recalculated.
Today, astronomers use high-speed computers to figure orbits in multiple body systems such as the solar system. The computers can be programmed to make allowances for the important perturbations on all the orbits of the member bodies. Such calculations have now been made for the Sun and the major planets over time intervals of up to several tens of millions of years.
As accurately as these calculations can be made, however, the behavior of celestial bodies over long periods of time cannot always be determined. For example, the perturbation method has so far been unable to determine the stability either of the orbits of individual bodies or of the solar system as a whole for the estimated age of the solar system. Studies of the evolution of the Earth-Moon system indicate that the Moon's orbit may become unstable, which will make it possible for the Moon to escape into an independent orbit around the Sun. Recent astronomers have also used the theory of chaos to explain irregular orbits.
The orbits of artificial satellites of the Earth or other bodies with atmospheres whose orbits come close to their surfaces are very complicated. The orbits of these satellites are influenced by atmospheric drag, which tends to bring the satellite down into the lower atmosphere, where it is either vaporized by atmospheric friction or falls to the planet's surface. In addition, the shape of Earth and many other bodies is not perfectly spherical. The bulge that forms at the equator, due to the planet's spinning motion, causes a stronger gravitational attraction. When the satellite passes by the equator, it may be slowed enough to pull it to earth.