One of the first great discoveries of modern physics was Newton writing down the equation for the force of gravity between two massive objects. The discovery was monumental not because it was complicated, but because it was profound. You can state the law in plain English very briefly. “The gravitational force between two objects is equal to their masses multiplied together, divided by the square of the distance between them, and multiplied by a particular universal constant to get the overall scale right.”
This is the way things are, classically. But as far as we know there’s nothing forcing this to be true. Why not have a universe with the same gravitational law modified to read “divided by the fourth power of the distance between them”? It would be pretty weird and a lot of nice theoretical properties would vanish into the ether. Gauss’ law for gravitation would be totally borked, for one. But there’s nothing logically prohibiting it.
It might not lead to a very interesting universe. A universe without gravity, for instance, would seem to be one without stars and planets. No stars and planets means no people and very possibly no intelligent life at all, and so sometimes the so-called Anthropic Principle is invoked to explain why the laws of physics are so rich. If they weren’t (the argument goes), we wouldn’t be here to see them. I’m rather unimpressed by this, but it’s an interesting idea. But what about our universe with modified gravity obeying a fourth-power force law?
Are orbits even possible? Check. It’s still an attractive central potential, and so it’s possible to pick a distance and velocity such that the force is exactly equal to that which is necessary to keep things orbiting in a circle. Which is all well and good, but is the orbit stable? Maybe any arbitrary perturbation will send the planet spiraling out into the distance, or crashing into the sun? This is a trickier question. A bit of a time crunch prevents me from writing out all the math (though we’ll do it shortly), but in fact the answer is “no, it isn’t”. A circular orbit under a 1/r^4 force law will not stay circular. It isn’t stable.
Which means the earth couldn’t orbit the sun, which means we wouldn’t be here to think about it. Which means we don’t live in a universe with a 1/r^4 gravitational force law. As in fact we don’t.