Cut a cone with a flat plane. Change the angle of the cut and the shape of the section changes.
Cut parallel to the base and you get a circle. Tilt a little and you get an ellipse. Tilt until the plane matches the slant of the cone and the section stops closing: a parabola. Tilt further and it is a hyperbola.
These four are the conic sections. Apollonius worked them out more than years ago, with nobody supposing they had anything to do with the heavens.
The paths that come out of an inverse square force are these four and nothing else. There is no fifth shape, and no path that fails to be one of the four.
The division into four can be restated in terms of energy rather than the angle of a cut. Take the energy of motion and add the energy of position. The latter is fixed to be infinitely far away, so it is negative nearer in.
If the sum is negative, infinity is out of reach. The body turns back somewhere. That is an ellipse.
If the sum is exactly , the body arrives infinitely far away with no speed left. That is the knife edge of barely failing to return: a parabola.
If the sum is positive, there is speed to spare even at infinity. The body leaves and does not come back. That is a hyperbola.
In polar coordinates with the focus at the origin, the four fit on one line: .
Here is the eccentricity. At it is a circle, below an ellipse, at exactly a parabola, above a hyperbola. The length is the semi-latus rectum, which sets the size.
For of or more there are directions in which the denominator vanishes. In those directions there is no path. That is why a hyperbola does not spread everywhere but reaches out only within an open angle.
There are four names but only one thing being written. Move the single number and the shape passes continuously from one to the next.
Every planet has a small eccentricity and traces a nearly circular ellipse. For anything near the divide, look to the comets.
The comet of Halley has eccentricity . The ellipse is long and thin but closed, so it returns every years: last in , next in .
A comet with an eccentricity very close to , by contrast, grazes the Sun once and departs. Whether the measured value is or decides between a return in ten thousand years and no return at all.
Telling those two apart is not easy. A pass near a planet alters the orbit slightly, and can carry it across the divide.
An eccentricity well above settles the matter. Such a body is not held by the Sun and came from somewhere else.
Oumuamua, found in , had an eccentricity of . It was the first object from outside the solar system ever seen. It was oddly elongated, and what it actually was remains unsettled.
The comet Borisov, in , came in at . At that value the path is nearly straight, and passing the Sun bends it only slightly. It carried a tail, so there was no doubt that it was a comet.
Neither will ever return. That is what an eccentricity above means.