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In the standard courses of classical mechanics inertial reference frame is defined as a frame with respect to which every free material point, i.e. not interacting with other bodies or fields, moves uniformly.

Then the first Newton law says that inertial frames do exist.

I realize that the notion of inertial frame is an idealization. As far as I understand, the meaning of the first Newton law is that for a given problem one can find a frame which can be considered as inertial with sufficient accuracy.

To make the first Newton's law useful, one should be able to apply it in concrete situations. If I understand correctly, for Newton himself one of the main examples of interest was the planetary motion, he tried to explain Kepler's laws (based on Tycho Brahe's measurements). As far as I understand Kepler (or Brahe?) used the Copernicus frame to describe the planetary motion, i.e. the frame where the Sun is at the origin.

Usually in textbooks the Copernicus frame is claimed to be inertial for planetary motion. Why?? How one can check the definition? I have no idea how to find sufficiently many free bodies in this frame and verify that they move uniformly. The gravity forces of all planets acting on such bodies should be negligible. How it could be done practically?

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For the Sun to be an inertial frame, there would need to be no external forces acting on it. Actually it is the centre of mass of the Solar System (with a set of axes that do not rotate with respect to the "fixed stars") that is approximately an inertial frame, and other stars are far enough away that the forces they exert on the Solar System have only small effects over a lifetime or two.

If the centre of mass of the Solar System is inertial then internal forces within the Solar System (mainly gravity) explain the motions observed. This is what we find (to a good approximation).

Note that the centre of mass of the Solar System is always close to the Sun, and is sometimes inside it.

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  • $\begingroup$ +1 but you should be more precise: the center of mass is a point. To define a reference frame you should also fix a frame of axes at the center of mass. As a matter of fact, this frame is at rest with the "fixed stars". Furthermore, the sun approximatively coincides with the center of mass if mass being very large with respect the other bodies of the system. $\endgroup$
    – Valter Moretti
    Sep 13, 2020 at 8:55

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