Understanding geosynchronous orbits in an otherwise empty universe

Question

Suppose the universe was completely empty except for a rotating planet with a moon in geosynchronous orbit. How would it be possible to understand why the moon did not fall since there is nothing to compare the rotation of the planet to. Would an observer not see a moon levitating above a planet? How would the observer be able to understand what is happening here?

Answer 1

Maybe the observer would notice the Coriolis force affecting weather patterns on their planet and infer from that that their planet is rotating.

Answer 2

Suppose the universe was completely empty except for a rotating planet with a moon in geosynchronous orbit. How would it be possible to understand why the moon did not fall since there is nothing to compare the rotation of the planet to.

What is there to understand? Mach's ideas (the need for presence of distant external material bodies to justify why inertial systems are preferred) are interesting and stimulative, but they are not accepted as part of standard mechanics. Inertial systems are used in mechanics already because it produces useful and accurate theory; there is no real need to base this usage on the presence of distant external material bodies.

Mechanics explains what happens at all circular orbits, whatever their radius is - when it is below the geosynchronous radius, above it, or right there. We do not need to compare rotation to anything external, the behaviour of the Moon and other falling objects would be enough to determine that there is some force pushing the Moon away from the Earth.

Would an observer not see a moon levitating above a planet? How would the observer be able to understand what is happening here?

Yes, geosynchronous orbit means the moon stays above the same point on Earth's surface. Description of what is happening depends on the frame.

In Earth's frame, since Moon's center is at rest, the Coriolis force is zero, and Moon experiences only two major forces - the gravity force, and the centrifugal force. Since the Moon stays put, these two forces cancel each other. So on the geosynchronous orbit, the centrifugal force has the same magnitude as and opposite direction to the gravitational force. By measuring the distance to the Moon, we could calculate the centrifugal force as minus gravity force, and from $F=m\omega^2 r$, we could determine $\omega$, angular velocity of rotation of the Earth.

In an inertial frame of the common center of mass, both the Earth and the Moon rotate around this center, and this motion is completely explained by action of gravity force, according to Newton's law of gravity.

Answer 3

This is an interesting little question, in that it puts blinders on our observers and asks whether or not we would deduce the same laws as we have found presently.

First off, I would point out that the idea that the moon is captured by a gravitational field is a relatively new theory, as far as society goes. It's credited to Newton, in the 17th century. And, arguably, he got it wrong, as we now know with our advanced measurements that gravity is actually a much more complicated beast than he gave it credit, warping the very fabric of spacetime. Whose to say we have it correct now? Perhaps we will be the ones to have to take our blinders off.

That being said, I think the nickname for Newton's theory is important: the first great unification:

The "first great unification" was Isaac Newton's 17th century unification of gravity, which brought together the understandings of the observable phenomena of gravity on Earth with the observable behavior of celestial bodies in space.

I believe this sort of unification is indeed what you are after. Given that the planet and moon are the only celestial bodies in your universe, the question will be whether the moon's behavior gets unified with the behaviors on the ground. Until that point, there is no reason not to just assume the moon levitates. In our world, this was done by the combined measurements of local systems (on the ground) against celestial systems (such as the path of the planets). In your world, this will have to be done purely with local measurements.

And the rules for rotating motion can indeed be completely discovered locally. Newton's laws of motion can all be fully understood in small laboratory sized experiments. The difference between an "inertial" frame and a rotating frame can be easily understood. Note that here I put "inertial" in quotes. You and I know the planet is rotating, but they don't know it yet. They do know, however, that to the best of their measurements, things that aren't visibly rotating get to use simpler laws of motion. Things that are rotating, such as whisks and centrifuges and Gravitrons (including that last example as something that can be experienced by a human, not just measured in a lab) are easily understood in a rotating frame. From that discovery, it doesn't take long to work out the mathematics for rotating frames.

The next step will be discovering that the planet is rotating. This is a less deductive step. This is the art of science. This requires some scientist to have an itch and start to wonder "I wonder if the ground is truly an inertial frame. We have all of these rotating frames, and we know that a rotating frame approaches inertial as it rotates slower... I wonder if we have a really slow rotating frame." They then might devise something akin to Focault's pendulum, which demonstrates very visibly that the planet is rotating.

What are the odds of that happening? Pretty darn good I'd say. Scientists are curious beasts. But for more compelling evidence, I'd turn to Relativity. The application of Newtonian physics was relatively easy to see and feel. Discovering relativity was more nuanced. The story is much longer than I give it credit here, and its a very interesting topic. However, if I can point out a key experiment, the Michaleson-Morley experiment. This is one of the influential experiments that lead us towards relativity by trying to tease out a curious thread in electromagnetics: the laws we found suggest there is a "preferred" frame where electromagnetic activity occurs. But what might that frame be? Aether theory was one such theory, and Michaelson and Morley set out to measure predictions made by that theory. As it turns out, they were wrong. Their measurements were almost completely unexplainable by Aether theory.* Resolving these peculiar results was one of the fundamental purposes of relativity.

And I point this out because they weren't trying to prove relativity. There was no relativity to prove at that time. They were trying to prove a different theory, and the results didn't align with that one. If I may take a step away from credible sources for a moment, and quote a comic I love (edit: the comments pointed out that the original source for the quote is Isaac Asimov):

Great scientific discoveries usually sound more like "Hmm, that's funny." than like "Eureka!"

The last step, once we have discovered that our planet is rotating, using local experiments, is the final unification you seek. Unifications like this are not easy. Changes in our understanding of the world are tricky: see the fight over geocentric vs. heliocentric models that consumed the careers of scientists. However, eventually someone would notice that the distance to the moon would be perfect for balancing out the effects of gravity based on its rotation rate. Our evidence for that is, once again, relativity. If you look at what the evidence for relativity initially suggested, it was just the presence of a Lorentz Boost. The novelty of relativity was that it suggested the Boost was due to a fundamentally different way of thinking about spacetime.

And, in that last paragraph, I hid what I think is actually the hardest part of discovering that the moon is no different than anything else. They have to notice that the math works out perfect for balancing gravity and rotating frame effects. Once that is discovered, Occam's razor will quickly suggest that the simplest model (the unified one) is the best. But therein lies a problem. I quietly assumed that they would figure out Newton's law of Universal Gravitation, and in particular the fact that gravity falls off at a rate of $\frac 1 {r^2}$. This is relatively easy to deduce, compared to other challenges here, with calculus and multiple celestial bodies to measure. But we only have 2 datapoints to work with: gravity on the ground, and something about the moon. There's no reason to assume anything except the obvious constant of acceleration for gravity if you're only measuring the same thing everywhere on the ground.

So I feel this singular issue defines the primary limit on this crucial unification. You have to discover that gravity drops off by $r^{-2}$ as you go further from the center of the planet. So you must be able to move meaningfully up. Whether this is by hiking up a mountain or by flying in an aeroplane, you won't discover the inverse-square law of gravity without different altitudes to measure at.

So how hard is it to measure gravity like this? It can be done with any normal scale, albeit you have to have the precision to pull it off. A scale will read lower at higher altitudes, and the inverse-square law will become visible.

How much precision is needed is beyond the scope of your question. From your comment on the other answer, "On the other hand the planet may be a barren rock," it sounds like you are willing to assume worst case scenarios to make this as hard as possible. Spring scales which could measure this were really first made popular by the variant invented by Richard Salter in 1770. Precision had to go up from there. It was in the 1940s that we developed enough electrical capabilities to make scales based on load cells. I don't have a good sense as to when that 1940's technology reached the level you would need (too many unknowns in the planet), but modern gravimeters can detect not only this $r^{-2}$ variation we need as a precursor to unification, but they can read the fine nuances that lead to maps thousands of times more accurate than this particular task would entail.

I would argue that there will be valid reasons for these gravimeters to exist, for the same reasons I give for relativity, and doubly so because the measurement of weight is heavily associated with money, and people will have an interest in what happens with money.

So in the end, I believe they would discover that the planet is rotating, and that the moon's position is due to being in geostationary orbit -- a unification of local and celestial physics. But it would take a long time. The primary limit would be the development of an inverse-square law of gravity. Of all of the steps, that would be the most difficult. However, I do believe scientists would have a reason to develop the gravimeters needed to detect this, and once they do so, the remaining dominos would fall reasonably quickly (compared to how long it took to invent the graivimeters)

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*. This statement about Aether theory isn't entirely true, but its true enough for the narrative here. There are many stack exchange questions on this experiment and its consequences detailing its impact and the path towards relativity if you wish to learn the full story.

Answer 4

The first thing to establish is whether or not the moon would fall or not. Mach's principle suggests it would fall because his principle says that rotation is only possible when there are distant bodies to rotate relative to. When Einstein was formulating General relativity, he considered Mach's principle, but ultimately, Einstein quietly rejected the principle and did not incorporate the idea into General Relativity. It can be seen that the Schwarzschild solution for a non-rotating black hole and the Kerr solution for a rotating black hole do not require distant background bodies to establish whether rotation is occurring or not. They are both vacuum solutions. Newton considered the idea of rotating a bucket of water to argue for the notion of rotation being absolute. When the water in the bucket is rotating, it forms a curved top surface and climbs up the walls. Newton attributes the reaction of the water to the absolute nature of rotation and centrifugal force. Indeed, it is hard to imagine that if instead of rotating the water in the bucket, we rotated everything in the Universe besides the water in the bucket, the only result would be the curvature of the water in the bucket.

The question states there is nothing in the Universe besides the planet and the moon, but it does also state there are observers. Assuming these are human-like observers, they would eventually notice that they fall back down when they jump into the air and deduce there is some sort of force attracting them downwards. Eventually, they would wonder why the moon does not fall straight down. If they tie a rock to a rope and swing it about their heads, they would eventually figure out the existence of centrifugal forces. In the absence of rocks and rope, they could hold the hands of a small observer, swing the child around themselves in circles, and notice the child rise up off the ground. If there is a smart one amongst them like our Newton, they would eventually figure out that centrifugal force can oppose gravitational force, and this could explain why the moon does not fall straight down if they assume the planet and the moon must be rotating at the same rate as each other.

Answer 5

Rotating reference frames are different from inertial reference frames because if you’re rotating in a vacuum you feel centrifugal force (unless you’re a zero-dimensional point particle, which you’re not). In a non-rotating observer’s reference frame, this equates to a constant centripetal force being applied to your extremities to prevent your extremities from inertially continuing along a straight path and separating from the rest of your body. This phenomenon does happen if you’re spun fast enough, but only because you have finite tensile strength.

The point here is that you can distinguish that you’re rotating without having an external reference. For example, if there’s weather on the planet, you’d be able to identify the Coriolis effect; from a high enough altitude, you might be able to see a tidal bulge; if you have a good estimate of the planet’s mass, you might be able to notice the defect in surface gravity due to the planet’s rotation; when you launch into orbit, you’ll notice that launching in the direction of rotation costs less $\mathrm{d}v$ to get into a stable orbit than launching in the opposite direction.

Geosynchronous orbits are orbits such that the rotation period of the primary is equal to the orbital period of the satellite. It doesn’t matter if there are background stars or not - you can still put an object into orbit with a particular velocity and altitude so that it stays over one point on the planet’s surface.

Answer 6

To understand why the moon doesn't fall newton first questioned, why should it even fall?, what is so special between moon and earth that the moon should fall?, then he later discovered that there exists a force that the earth applies onto the objects present on it. Shouldn't this force be mediated to the moon and hence it should fall? then he later studied about the uniform circular motion. from which he understood why the moon doesn't fall (it is also because the earth is circular, when the moon falls the surface beneath it curves more leading it to fall more then curving more then it goes on forever). now you see the question "why doesn't the moon fall?" had got very less to do with the objects in the outer space while discovering the gravitational force was easy, it can be done in an isolated empty universe too (which has only one gravitationally massive body) and similarly one could discover the laws governing the uniform circular motion just by examining the objects that are already on earth.

Answer 7

This is an addendum to existing answers.

The main point is that whereas Mach's thinking was stimulating, the evidence is that it does not describe our universe. General relativity does. And in G.R. a non-inertial frame such as a rotating one can be defined and discerned without reference to the distant stars or the rest of the universe. Existing answers have said this. What I want to add is the following detail.

It is possible for a local observer to determine whether some reference frame around them (e.g. one made of solid steel struts, or one stabilized by radar ranging, or something like that) is inertial. What they can't do (without looking at the distant stars) is know whether or not that frame has a net rotation relative to the distant stars.

The inhabitants of the planet in the question can tell that a frame rotating with the planet is not inertial. It will therefore be no great mystery that the moon does not fall. They can easily find an inertial frame and they will deduce (or observe) that in such a frame the moon is accelerating.