Every astronomer knows that Johannes Kepler decided that ellipses were a better fit to the orbits of the planets than anything that could be done with a reasonable number of circles. In this he a was a true scientist, able, if unwilling (and he wouldn’t be the last), to discard a treasured idea because it wouldn’t fit the facts. As Thomas Huxley said – “The great tragedy of science – the slaying of a beautiful hypothesis by an ugly fact.”.
Kepler was the owner of shoulders on which Isaac Newton acknowledged that he stood, along with those of Galileo Galilei. Newton showed that ellipses are natural consequences of the inverse-square law of attraction, provided that a uniform sphere attracts as if its mass were concentrated at a point. Newton proved that using his own calculus. The sphere can vary in density with depth, but it mustn’t vary with latitude or longitude. In fact, some bodies do vary in this way, and it leads to perturbations in low orbits, as people discovered with early lunar orbiters. The sphere must be a true sphere; any oblateness due to revolution will also cause perturbations.
The family of curves known as conic sections has other members – the hyperbolae, the parabolae, and the circles. A parabola may be thought of as the limit of an ellipse as the position of one focus tends to infinity. A circle can be regarded as the limit of an ellipse as one focus tends to coincidence with the other. The hyperbolae correspond to objects that have more than the escape velocity for their current position, and are probably quite rare now that the Solar System is well settled, but it is quite possible, though unlikely, that occasional sling-shot effects, or even a succession of perturbations, may take a long-period comet beyond the limit, and some stars in globular clusters are probably ejected, doomed thereafter to roam the universe alone. But given that the hyperbola is theoretically infinite, the odds of observing the finite part in the solar system must be small. It depend on how often these orbits are created.
The parabola is a kind of boundary between the ellipses and the hyperbolae, like an ellipse with the empty focus at infinity. It can be ruled out completely on statistical grounds; for a given perihelion, there is an infinite number of ellipses and an infinite number of hyperbolae, but only one parabola. So the probability of finding a parabolic orbit is zero. Even if one existed, the next tiny perturbation from the planets would turn it into an elliptical or a hyperbolic orbit.
The circle, likewise, is infinitely improbable, because if we consider an object at perihelion, there is an infinite number of ellipses that fit outside the circle, and if we instead call that point the aphelion, there is another infinite set of ellipses inside the circle. But unlike the parabola, the circle can be approached, if not quite attained, by the action of physical processes. Tidal interactions between a planet and a moon, for instance, can slowly change the orbital shape, and so the circular orbit can be approached asymptotically. But small perturbations from other planets, or from departures from uniform sphericity of the objects, can prevent the ideal ever being attained, as with the objects in Saturn’s rings.
It seems a safe bet, then, to say that all orbits are ellipses, and an even safer one to say that they are all perturbed ellipses. The diagrams below illustrate the points discussed above.
