Electromagnetism or gravitomagnetism?
[See also  http://philpapers.org/post/20174
Halley's "Synopsis of the Astronomy of Comets", 1705, is an ingenious mathematical proof of the parabolic shape of a comet's path and fits those cosmic wanderers right into Newtonian Physics, but it does not explain how such paths are possible. We understand now the why of the famous "hypotheses non fingo", Newton could simply not start to imagine what such an explanation would have looked like. The only thing he could say was that his equations seemed to work. A cosmologist's nightmare.]

Ever since Faraday it has been understood that electricity and magnetism are inseparable. Electric current influences the magnetized needle of a compass, and the motion of a magnet can create an electrical current in a metal coil.
Gravity seems different not only because it always is attractive, but also because it is understood to be the reason why bodies not only move towards each other, but also orbit each other. And that is problematical.
If you spread iron fillings on a sheet covering a magnet, the small particles will move only once and then take their final position. Once they have done that, the whole pattern remains frozen in time until you disturb it.
It seems therefore natural to make a big distinction between (electro)magnetism and gravity.
It would certainly sound pretentious to want to outsmart Newton and his theory of a universal gravitation force, but please allow me at least to ask some questions.
What if the motions of the heavenly bodies where not caused by gravity but by something else?
Certainly not something completely different. After all, it is possible to rotate an object using only magnets, but, and that sounds quite important, not by using one single magnet, unless it is built as a hollow ring in which is fitted the object supposed to rotate.
It would then seem that all a massive star like our sun can do is attract other bodies, but not make them rotate in an orbit around itself. For that we need more bodies influencing each other in such a way that they keep rotating about each other.
Maybe, and I am freeling speculating, the heliocentric model, just like the ancient geoentric one, is much too limited to explain all that happens in our corner of the galaxy and beyond. There is certainly no denying the sun a primordial role in the way all the planets move relative to each other, but that does not mean that they should be considered as passive bodies which are simply answering to the solar directives.
Notice that the Einsteinian solution of spacetime being "deformed" by the mass of stars and planets offers no solution to the question why those bodies are and keep moving in their orbit. In fact, it should make it even easier for the sun to attract all the planets to its surface creating a gigantic lump.
So where is this motion all coming from?
A possible explanation would be that the motion of each heavenly body is determined by all the other bodies in the universe. Maybe that is true, but such an explanation could, maybe, never be made operational and translated into handy equations. It would forever remain a gratuitous metaphysical remark.
Let us go back to our simple picture of a magnet and iron fillings. Textbooks all mention the lines emerging from one pole (south?) and entering the other (north?). They also all rush in emphasizing that those lines should not be considered as a path or trajectory for the objects caught in them. Which is of course the least they can do since nothing really moves along those lines.
This justified caution does not solve the following puzzle though. Why those lines? Why are not all filings simply drawn to the magnet to form an iron heap? In fact, that is exactly what they do if you release all the filings directly above the magnet. So, there is no physical law against it.
The pattern formed by a magnet resembles very strongly the one created by large stars: planets are attracted until a certain distance, and then the attraction suddenly seems to stop.
I have no explanation for that, but it does, vaguely, show why planets have different orbits. It looks like the attrative force is applied only once and is only strong enough to move objects a limited distance towards the magnet.
This is quite an illogical state of affairs: why can't the magnet pull objects on a certain line closer to it if it can do that for objects on lines not only closer but also farther away?
Please remember that the so-called magnetic lines are in fact an idealization of a messy reality. The more iron fillings you throw on the paper the more blurred the lines get. The fillings only form a nice regular pattern when you have just enough of them to react to the magnet without influencing too much each other.
This is also what the theory of electric fields teaches us. There is a force at any arbitrary point of the field. Which means that we shouldn't even be able to distinguish lines from each other and the iron fillings should either be dispersed uniformely or all lie around and immediately close to the magnet.
Magnetic lines seem to add to the obscurity of the analysis more than they help clarify it.
A logical explanation of the influence of a magnet on close objects could be the following: an object is attracted to the magnet, and so doing acquires a repulsive charge which cancels the attractive force, leaving the object in a closer but fixed position.
This would certainly make sense to me, but then I am not a physicist and I would have no idea how to reconcile this idea with current Physics.

The attractiveness of such a principle is that it would explain why there are so many different orbits. It still would not explain why planets and stars are rotating, or moving at all. Maybe we need the Big Bang to explain motion at all.

One thing seems certain to me: the last word has not been spoken on gravity, be it Newtonian or Einsteinian. The universe is still a mystery.