Libration Force
The Libration Force
Kepler coined the term “libration” for the oscillation of a planet’s distance from the Sun, approaching and receding.
He analyzed the libration for the eccentric-equant model, and found it unexpectedly complicated. Stephenson (p.78):
Many absurdities were involved in supposing that a planet could move, … non-uniformly, about the vacant center of the eccentric, with no guide except the apparent magnitude of the solar disk. Such complicated hypotheses, although designed to yield a perfectly simple eccentric circular path, were not physically credible…
Notice the remarkable thing that Kepler was doing here. He was analyzing motion on an eccentric circle, a model that had been in general use for nearly two millenia, apparently the simplest possible model with any empirical accuracy. He took apart this beautifully simple model and showed that as a physical process (and in the absence of solid spheres) it was really quite complicated, so complicated as to raise doubt about whether it could be real. He had performed so radical a reassessment by interpreting astronomy, for the first time, as a physical science.
Eventually Kepler achieved the elliptical orbit. Seeking a physical explanation, he hit on a magnetic force to produce the libration:
What if all the bodies of the planets are enormous round magnets? Of the earth (one of the planets, for Copernicus) there is no doubt. William Gilbert has proved it.
But to describe this power more plainly, the planet’s globe has two poles, of which one seeks out the sun, and the other flees the sun. So let us imagine an axis of this sort, using a magnetic strip, and let its point seek the sun. But despite its sun-seeking magnetic nature, let it remain ever parallel to itself in the translational motion of the globe…
—Astronomia nova, Chapter 57.
The figure at the top of this post (taken from the Epitome of Copernican Astronomy) shows how it works. (The figure in the Astronomia nova has extra clutter.) Kepler explains:
[When] the strip is at A and E, there is no reason why the planet should approach or recede, since it holds its ends at equal distance from the sun, and would undoubtedly turn its point towards the sun if it were allowed to do so by the force that holds its axis straight and parallel. When the planet moves [counterclockwise] away from A, the point approaches the sun perceptibly, and the tail end recedes. Therefore, the globe begins perceptibly to navigate towards the sun. After E, the tail end perceptibly approaches and the head end recedes from the sun. Therefore, by a natural aversion, the whole globe perceptibly flees the sun…
—Astronomia nova, Chapter 57. [I have changed the letters from C and F to A and E to match the diagram from the Epitome.]
Implicit: the magnetic force weakens with distance, so when the head
is closer to the Sun than the tail, the net force is attractive. And vice versa.
Kepler argued that this scheme gave the force a sinusoidal dependence on the longitude, and showed that this agreed with the libration for an elliptical orbit. Some aspects of this demonstration needed special pleading. Stephenson details the strong and the weak points of the reasoning (pp.110–117).
But: “The theory had one glaring flaw, however. The magnetic axis of the planet had to maintain a constant direction, perpendicular to the apsidal line.” (Stephenson, p.117.) The Earth’s rotational axis doesn’t come close to meeting this requirement. So why should we believe it holds for Mars? Kepler acknowledged the problem:
I will be satisfied if this magnetic example demonstrates the general possibility of the proposed mechanism. Concerning the details, however, I have my doubts. For when the earth is in question, it is certain that its axis, whose constant and parallel direction brings about the year’s seasons at the cardinal points, is not well suited to bringing about this reciprocation… And if this axis is unsuitable, it seems there is none suitable in the earth’s entire body, since there is no part of it that rests in one position while the whole body of the globe revolves in a ceaseless daily whirl about that axis.
As one possible out, Kepler appealed to a planetary mind.
Besides the radial libration, planets have a libration in latitude. This enmeshed the theory in further difficulties. Ever inventive, Kepler devised ad hockery around all these rough spots. But we have a contrast: we can trace a direct path from the whirlpool force to the area law. This cannot be said for Kepler’s libration theory. Kepler’s whirlpool speculations came years before the area law. The libration force came after the elliptical orbit.
There is a reason for this. You can justify the whirlpool force (more or less) using the conservation of angular momentum. Kepler’s libration force has no counterpart in Newtonian physics.
Diagonal Argument 