Not really. You can rotate at any speed you want. Rotate at too low a speed, and the satellite falls out of the sky. Rotate too fast, and the satellite is flung away. Rotate at the right speed, and the centrifugal force balances the gravitational force, and you have a nice stable orbit.
Yes, thank you, you’ve captured my meaning exactly. And the word “centrifugal” was intentional, and is appropriate in the context of a balancing force to the centripetal force of gravity acting on the satellite. I won’t quibble with @Pleonast about the preferred Newtonian explanation of an orbit, but the one I gave is certainly a valid one, and was meant to illustrate the contrast with the GR description.
What if it’s a 3-body problem…?
There’s a key word in this sidebar: “circular”. Replacing that with “curved” that would be another way to clarify the initial statement. (It’s obvious that we all know what we’re talking about. I’m only nitpicking so that novice physicists don’t start along paths that will lead to future confusion.)
Even more incentive to pick an inertial frame. Co-rotating frames are mostly used when all the observations are made from one object (like the Earth). Does anyone actually do orbital mechanics from the frame of reference of the satellite?
When an accelerometer is sitting on top of a building it reads a constant 1G upward (it’s measuring the deviation from free fall motion, and relative to free-fall motion, an accelerometer on the top of a building is moving upwards.
What do I know about astronomy to judge whether this approach is wise, but e.g. let’s grab the random (old) book Theory of Orbits: The Restricted Problem of Three Bodies by Victor Szebehely. Chapter 1 describes a simplified, illustrative problem with two bodies orbiting their center of mass (located at the origin) along circles. I won’t copy every single equation, but the coordinates of the masses m_1 and m_2 are (X_1,\,Y_1) and (X_2,\,Y_2),
\begin{align}
X_1 &= b \cos nt^*,\quad X_2 = -a\cos nt^*\\
Y_1 &= b \sin nt^*,\quad Y_2 = -a\sin nt^*
\end{align}
where n is their common angular velocity, t^* is time, etc. An infinitesimally small mass m_3 orbits in the same plane according to
d^2X/d{t^*}^2 = \partial F/\partial X,\quad d^2Y/d{t^*}^2 = \partial F/\partial Y,
where F = k^2(m_1/R_1 + m_2/R_2). (k is supposed to be Gauss’s gravitational constant).
So far, so good, but F=F(X,Y,t^*) depends on time, so now a rotating coordinate system (\bar x, \bar y) is introduced in which m_1 and m_2 are fixed, and introducing dimensionless variables x,\,y,\,t,\,r_1,\,r_2,\,\mu_1,\,\mu_2 the equations of motion of the satellite end up having the form
\begin{align}
\ddot x - 2\dot y &= \Omega_x,\\
\ddot y + 2\dot x &= \Omega_y.
\end{align}
Now you can analyse the function \Omega = \Omega(x,y), work out where the Lagrange points are, find families of solutions, whatever.
Later on (Chapter 10) the third particle is allowed to move in three dimensions, and elliptic orbits (hence a coordinate system which rotates with variable angular velocity) are considered, but the same basic scheme is used.
There are lots of fun theories of gravity, almost all of which are nonsense, but still interesting to think about.
One of these is the idea that there is a constant flux of lightweight particles zooming in all directions. Most of them pass through an object, but sometimes they give a little bump in the direction of travel.
An object far away from anything else will be bumped equally on all sides and not move. But if you put two massive objects next to each other, they will shadow each other slightly. Each object will block some of the particles from hitting the other. And so each will have a little bit extra force pushing towards each other. As they get closer, the angular extent of the shadow gets larger, and so even more particles are shadowed, increasing the force further until they collide.
In practice, you can’t make the numbers work out. Plus everything should immediately get white-hot from the energy released in the inelastic collisions. But still, kinda fun.
James Hogan used that idea in The Two Faces of Tomorrow, I think
I believe the term is pseudo-gravity.
Particles are “attracted” to each other by falling into each other’s mutual “shadow”.
For instance, the particles zooming around must have some average rest frame. And an object moving at any speed relative to that average rest frame would feel an effective drag force from them.
Do photons have an average rest frame?
An individual photon doesn’t have a rest frame. But any collection of photons, so long as they’re not all going in exactly the same direction, does. For instance, there’s a rest frame of the cosmic microwave background radiation, which we can detect because it’s slightly redshifted in one direction and slightly blueshifted in the opposite direction. And there is, in fact, effectively a “drag” force from the cosmic microwave background radiation, though it’s only noticeable for subatomic particles moving through the background at extremely high speeds.
Is it actually a single rest frame, though? It is a frame of reference that can be used as a reference, but it’s no more inherent than the reference frame of photons leaving any surface. The fact that that surface was everywhere makes it interesting, and certainly something to study, but does it make it a special reference frame?
It would be noticeable to macroscopic objects moving at those speeds, too, it’s just that macroscopic objects don’t generally move that fast.
As I said earlier, I find the visualization of space-time being drawn in by mass to be the most accurate and useful way of understanding gravity. This describes how a gravitational field would work much better than the bowling ball on a trampoline, IMO.
Now, at the risk of overextending the analogy, if we quantize the field and introduce virtual gravitons into the mix we end up with a situation similar to what @Dr.Strangelove described, but instead of lightweight particles, they are massless gravitons, only carrying the inertia of the field, which ends up dragging along anything they interact with, which is any particle with mass or energy.
Does that make any sense?
It’s a single reference frame, given any point in the Universe to observe it from. Other points in the Universe will see a different (though similar) CMB, each with their own average rest frame.
As to whether it’s a “special reference frame”, that depends on how you define “special”. The laws of physics themselves work the same in that reference frame as they do in any other. But there are some phenomena that do depend on one’s speed relative to that reference frame, not because of the laws of physics, but just because of where stuff happens to be.