The Mystery of Gravity (3): Kepler's problem
Let us forget all the beautiful formulas for a while, and concentrate on the phenomenology of orbits and gravity.
Orbits are elliptical. No need to argue about that. But what about the explanation given? The model, as I have noted previously, is that of a sling or a weight attached to a rope being rotated uniformly.
The problem with such a model is that it is more Copernican than Newtonian. The motion described is perfectly circular while orbits are, once again, elliptical.
A perfectly circular orbit would by the way make much more sense logically. Here is why.
In an elliptical orbit, the satellite gets first further and further from Earth, and then starts approaching earth again, describing what we see as an ellipse.
My question is: what makes it come back? 
Gravity is supposed to diminish by the square of the distance, so Earth would have no way of recalling an object to a nearer distance, while the straight line dictated by inertia should get the satellite even farther away from Earth. There is therefore no reason for the satellite to finish its orbit, or for the moon to keep rotating around earth, or earth around the sun...  But they all do. Something is not right here, and it is called gravity.
Let us remember that the elliptical orbit is an empirical fact born out of the tedious and meticulous observations of Tycho Brahe who himself thought that the heavenly bodies were moved by an internal force, or pushed by it as it were. Newton externalized this force in the form we know as gravity. He also offered a geometrical explanation of this phenomenon, using his own calculus or fluxions.
Observations came first, mathematics later. 
It happens that the elliptical orbit can be easily explained by the existence of two foci. If you take a string with both ends attached, and put the string over two nails driven on a board a certain distance from each other, and around a pencil, you will be able to draw an eliptical shape containing both nails, and the total distance from one nail to any point, plus the distance from the other nail to any other point on the ellipse, will be constant. Which is not surprising really, since it is the same string we are using each time. Newton, just like Kepler, was probably inspired by this fact when he built his geometrical model. The area described by the motion of the earth around the sun remains constant all along its orbit.
This image is not really helpful though. Where would we get our second focus when dealing with suns, planets and satellites? That is where mathematical imagination comes to the rescue. Turn the second focus into an abstract point, or, if dealing with big objects, put it in the object itself, making the latter the carrier of two foci, the center of gravity being one of the two.
Our string drawing an ellipse was constrained by two very real nails, and the ellipse therefore was seen as the direct result of the pen's motion under those conditions. This seems quite natural to us, just like we are not surprised when our compass draws a perfect circle on paper. But imagine drawing an ellipse with a single nail, and hearing from mathematicians that it is completely understandable. After all, the fact that the second nail was an abstract nail should not be problem, we just have to look over the fact that it is not really there, in the flesh as it were.
It becomes even more surprising when we are shown two objects rotating elliptically around such a non-existent center, while at the same time describing each it own elipsis.
Here is a very appropriate quote by Kepler: "A mathematical point, whether or not it is the centre of the world, can neither effect the motion of heavy bodies nor act as an object towards which they tend. Let the physicists prove that this force is in a point which neither is a body nor is grasped otherwise than through mere relation." Johannes Kepler "New Astronomy", p.54)
If Kepler had wanted to be consistent, he would have asked himself what physical part played the role of the second focus. Apparently he did not, and in fact Newton didn't either. Even better, nobody until now, as far as I know, has ever stood still by the question what this second focus could be.
I see two reasons for that. First, Newton did a great job in interpreting mathematically Kepler's results. And that seemed enough to justify this approach. Second, the principle of the conservation of energy was introduced, as conservation of momentum in the Principia, later in its more general form of the First Law of Thermodynamics.
Suddenly, the fact that orbiting planets returned to their source was not strange anymore. Unless a satellite reached a certain speed, the escape velocity, it would always come back to its origin. Which was exactly what we could see happening in the sky. Even comets with incredibly large orbits, like comet Haley, return periodically back to the vicinity of the Earth.

I must admit that I find the argumentation still not really convincing. First we learn that the horizontal and the vertical components of motion are independent of each other: the sun attracts more strongly earth when both are close to each other, making it fall faster toward the sun, without changing its horizontal speed. At least, that is what is supposed to happen. But Kepler and Newton teach us that earth at that moment is moving faster than before. In fact, because it is covering the same area at any moment in time, it has to move faster when close to the sun because of the elongated shape of the area in those moments. This is in complete accordance with the idea of the speed of the pen, and the shape of the string as shown by an ellipse with two nails or foci.
But where is the second focus? Without it logic is gone and all we have is a very rational principle that seems to be magically invoked to make the whole more palatable. What makes orbits fall under the group of conservative motions even without a second focus to explain not only the difference in speed, but also the fact that a satellite always resumes its orbit?
No reason is given, or at least, I could find none. We are expected to be satisfied with the blind application of Newton's mathematical description and the First Law of Thermodynamics.
We certainly could do worse since after all they seem to do a very good job at predicting orbits and other physical processes on earth and in space. But then, Ptolemy's astronomy did also a very good job for more than 15 centuries.

Dark Matter Matters
Dark matter would be an ideal candidate for the second focus. Unluckily for us, it seems that there is none to find within the solar system, since it is said to be concentrated mostly around galaxies or cluster of galaxies. Which is too bad. I find the idea suspiciously like that of an improved ether, but it would have presented a rational alternative to the problem of the second focus.
Who knows, maybe physicists will come up with an even more extraordinary invention in the years to come. After all, imagination has never been more appreciated as since Einstein.