Binding Energy

The conics (the orbits solution of the 2-body problem) were found by Newton, who invented the infinitesimal calculus to solve this precise problem. But to determine in which orbit the body will lie, it is necessary to introduce another concept: the binding Energy of the object.
First of all, it is necessary to understand that no physical process can happen without spending some energy (heating a house, walking, etc). In other words, energy can be seen as a veritable money, with different currency that can be changed.

In particular, the binding energy E is a measure of how much a body is tied to another : it corresponds to the work necessary to bring the two bodies at an infinite distance. This energy can be negative if the two bodies are gravitationally bound (as in the case of closed orbits, such as the circle or the ellipse) or positive if the two bodies are not bound, and the orbit is opened.
The value of the binding energy E is inversely proportional and of opposite sign of the major axes. This means that a very little semiaxis corresponds to a strongly negative energy and therefore to a very bound orbit.