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Imagine there is only the sun and the earth. We could chose our reference frame in such a way that the angular speed around the sun is zero. In a $xy$-plane the $x$-axis always points to the earth. If we'd choose our reference frame like this centrifugal and gravitational forces would no longer be in equilibrium alas it is wrong. But what does define how that reference system has to be aligned? From our defined laws about equilibrium we know how it should be but what is it that defines that it is like that? Is it the mass distribution of the stellar system, if yes, which physical law defines that?

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  • $\begingroup$ I'm confused. In a two-entity model of the sun and the Earth, what is to stop the sun from orbiting the Earth? It's false to say gravitational forces would not be in equilibrium - there would still be the same relation between the sun and the Earth (with the sun as the larger body), even if the Earth was held to be stationary. Without any other body in the universe to refer to, it would be impossible to determine which one was moving (indeed, stationary rotation on their axes would be indistinguishable from an orbit - we only perceive an orbit because of the presence of the wider heavens). $\endgroup$ – Steve Jan 25 '18 at 6:35
  • $\begingroup$ @Steve: I meant to draw a picture where the bodies are not orbiting each other but rather rest on the x-axis and the earth is slightly oscillating on this axis (due to radius not being constant). In that case forces are not in equilibrium. $\endgroup$ – squanto773 Jan 25 '18 at 8:11
  • $\begingroup$ But in that case we would simply define the forces involved differently. For example, it would be apparent (as now) that there is an ideal distance at which the forces are balanced - as the earth oscillates (although, again, without the presence of the heavens, it would be impossible to determine whether it was the sun or the Earth oscillating), if the oscillation-towards happens too quickly, then a repulsive counter-force is generated. If the oscillation-away happens too slowly, then an attractive counter-force is generated - like a ball bouncing on the ground. It all still reconciles. $\endgroup$ – Steve Jan 25 '18 at 21:08
  • $\begingroup$ Thanks for asking this question by the way. The notion that the Earth is not moving (much) relative to the sun - but is oscillating around a fixed distance - has given me an invaluable insight into relativity - for, in relativity, is is impossible to say that the Earth is orbiting the sun, unless you employ the cosmic background as a reference frame. In a two-entity model (such as the one you conceive), the concept of an "orbit" becomes meaningless and undefinable (at least, it becomes observationally equivalent to, and indistinguishable from, the rotation of the sun on its axis). $\endgroup$ – Steve Jan 26 '18 at 4:06
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If you're inside a windowless spaceship, and you have little thrusters you can position to apply angular acceleration to the craft, you could keep changing your angular velocity a little at a time and seeing how various experiments come out. You could connect two masses to either end of a spring, and then see what the equilibrium length of the spring is in various orientations (masses at rest in the spaceship). You would find one orientation in which the equilibrium length was shortest. That is the axis of rotation. With your thrusters, you apply some torque about this axis. If you find that the spring is more stretched afterward, then apply some torque in the opposite direction. After a few iterations, you reach a state where the length of the spring is the same regardless of orientation. Now you're in an inertial reference frame, and you found it without looking at the stars.

In short, one way to define an inertial reference frame is to say that it is a frame in which the laws of mechanics may be applied without fudging by adding in fictitious forces.

The question of whether the frame you find by the process just described depends in some way on the fixed stars or not is difficult to answer definitively, and, to my mind anyway, has more to do with how we choose to think about things than with determining the outcome of any experiment. We can't, for instance, do the experiment of starting the entire universe rotating to check whether frames fixed relative to the stars would still be inertial. If we somehow could, and we did, and we found that yes, such frames are still inertial, then how would we know we had indeed spun the universe? How could we measure the rotation rate?

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  • $\begingroup$ The question of whether the frame you find by the process just described depends in some way on the fixed stars or not is kind of metaphysical. Not really. This is just Mach's principle, and Mach's principle is not metaphysical. There are physical theories that don't obey Mach's principle (such as general relativity) and others that do a better job of obeying it (such as Brans-Dicke gravity). $\endgroup$ – Ben Crowell Jan 24 '18 at 22:56
  • $\begingroup$ @BenCrowell I will change the wording. $\endgroup$ – Ben51 Jan 24 '18 at 23:43
  • $\begingroup$ @BenCrowell: Thanks for pointing out Mach's principle. Actually this is all my question is about. "Why should your arms be pulled away when the stars are whirling? Why should they be dangling freely when the stars don't move?" I just ment to asked the same question but with an other model namely sun and earth. $\endgroup$ – squanto773 Jan 25 '18 at 7:45
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Well, we generally want our reference frame to be an inertial one. And one that follows the Earth rotating around the sun isn't one. So yeah, things look funky in that non-inertial frame. In the non-empty universe we use the stars as our reference, since motion with respect to them is so slow as to be negligible over a human lifetime.

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  • $\begingroup$ Thanks for mentioning the inertial reference frame. The question is, what is the rule to set that reference frame? Yes, we can tell from the equilibrium condition how it should be, but what "outside force" is governing that it is like that? $\endgroup$ – squanto773 Jan 25 '18 at 8:15
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The choice of z-axis seems fairly obvious: it should be perpendicular to the plane of Earth's orbit (a.k.a the ecliptic plane). You may then think picking an x-axis is arbitrary, but it's not. Because the rotational axis of the Earth is tilted (by 23.4 degrees), there is a preferred choice called the vernal line. It is the line created by the ecliptic plane and Earth's own equatorial plane (recall your highschool geometry: a line is formed when two planes intersect). This vernal line aligns itself with the sun twice a year: once at the spring and fall equinoxes. It is natural to want to choose an x-axis that aligns with the vernal equinox because it makes coordinate transformations easier.

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Most astronomers use this type of coordinate system aligned with the vernal line and the ecliptic plane. When you are not interested at where the Earth is at, such as a space mission to Mars or Jupiter, then a distant star is typically used to define an x-axis, and the z-axis is kept as the ecliptic plane.

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