 with a force that is inversely proportional to the square of the distance of separation r. This 1/r2 term causes the earth to describe an ellipse around the sun. The presence of other planets and some other effects introduce an additional but small force component ef. On the long haul, this tiny force causes the shape of the earth's orbit to change. where m is the earth's mass¹. 5.1.2 Unperturbed system behavior: neglecting small forces The smallness of the term ef in (5.1) suggests that it only has a small impact on the earth's behavior. In studying the earth's motion, we are therefore tempted, at least for the time being, to forget about this perturbation term. Hence, we use (5.3) as the force in the equation of motion (5.2). As shown in Fig. 5.1, this force causes the earth to describe an elliptic orbit around the sun. This behavior can be interpreted as a periodic oscillation with a period that equals a single year. The orbit drawn in Fig. 5.1 is by no means unique. This non-uniqueness should be understood as the existence of a multitude of steady-state solutions, each satisfying (5.2) for E = 0. If the earth is set off with different initial conditions, different orbits result, either located in a different plane or with a different shape. Fig 5.2 shows a number of different orbits, each solving (5.2) for E = 0. Characterizing the earth's (possible) behavior therefore requires us to find all possible solutions to (5.2) for E = 0. Mathematically, for E = 0, the equation (5.2) is called unperturbed equation of motion. This equation models the behavior of the system's core. In the current example, the core is associated with the earth, the sun and the 1/r2 attracting force between them. The steady-state solutions that solve the unperturbed equations of motion are called the states of the core system. Characterizing the core system behavior involves finding This is a somewhat simplified representation of the problem. A more accurate problem statement can be found in [Hest87]. 5.1 THE STORY BEHIND THE MATH earth