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The three Kepler laws of planetary motion were established by analysis of Tycho’s measurements. The choice of coordinate system is important for Kepler’s laws. For example it is important that the Sun is at the origin. Also the choice of axis is important: for example if one chooses the first axis to be the ray from Sun to Earth, then Earth will stay on this axis all the time and it’s orbit will not be elliptic.

Newton deduced from Kepler’s laws and his classical mechanics his gravity law. On the one hand, Kepler’s laws hold in a specific frame. On the other hand Newton’s mechanics holds in inertial frames only . In all known to me literature it is tacitly assumed that Newton’s mechanics is applicable in Kepler’s frame (well, approximately: Sun is assumed to be infinitely heavy). Thus implicitly it is assumed that the frame used by Kepler is almost inertial. WHY?

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The contrast that you are suggesting is not there.

Kepler's law of areas anticipated the principle of conservation of angular momentum. Kepler was in a position to recognize the law of areas because the orbit data he used described the orbits with respect to a non-rotating coordinate system.

Newton later demonstrated that Kepler's law of areas is a logical consequence of his laws of motion. Newton showed that a law of areas arises for any central force. That is, for any central force the circumnavigating motion of an object wil be such that equal areas will be swept out in equal intervals of time.

So the efforts of Kepler and Newton and Kepler have in common that the planetary motion they used was planetary motion with respect to a non-rotating coordinate system.


The only difference is that Newton regognized that according to the laws of motion the Sun itself must be moving too, with respect to the common center of mass of the Solar system. Newton could infer the mass ratio of the Sun and Jupiter by comparing the periods of the planets of the solar system to the periods of the orbits of the moons of Jupiter. From that Newton could infer the distance of the Sun to the common center of mass of the Sun-Jupiter system

Quote from a page written by Thayer Watkins at San Jose State University:

For the Sun-Jupiter system the magnitude of the Sun's counter orbit is not so insignificant. The mass of the Sun is only about 1050 times as large as that of Jupiter. The radius of Jupiter's orbit around the center of mass of the two bodies is about 484 million miles so the radius of the Sun's counter orbit around their center of mass would be about 462 thousand miles. This is about 28 thousand miles above the surface of the Sun.

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  • $\begingroup$ When you mention non-rotating coordinate system, what do you mean? With respect to what it is non-rotating? I think purely kinematically this notion does not make sense. $\endgroup$
    – MKO
    Commented Apr 17, 2021 at 15:51
  • $\begingroup$ @MKO About non-rotating coordinate sytem: as we know, the orbit of Mars is ellipse-shaped. This ellipse-shape arises because gravity is an inverse square law. The correct ellipse arises if and only if the motion of the planet is described with respect to a non-rotating coordinate system. Another example, the plane of the Equator and the plane of the Earth's orbit intersect. Astronomers know that line of intersection moves very slowly (precession of the equinoxes) but over the span of a human lifetime that line can be regarded as stationary. So: multiple independent measures of non-rotating. $\endgroup$
    – Cleonis
    Commented Apr 17, 2021 at 16:46
  • $\begingroup$ “The correct ellipse arises if and only if the motion of the planet is described with respect to a non-rotating coordinate system.” Again, what do you mean by non-rotating coordinate system? How do you define this notion? $\endgroup$
    – MKO
    Commented Apr 17, 2021 at 18:43
  • $\begingroup$ Tycho measured the positions of stars and planets alike. Kepler analyzed these relative positions, assuming the non-planetary stars to be relatively stationary. That's the coordinate system which Kepler used. Rotating? Maybe, but relative to the motions of planets in our system, not so much. $\endgroup$
    – Bill N
    Commented Apr 18, 2021 at 16:05
  • $\begingroup$ @BillN Some time ago I asked on the astronomy site about the possibility of an intergalactic star. One answer described a peripheral star dislodged from its parent galaxy due to tidal effect from a passing galaxy. Such a star retains its planets. After a couple of billions of years the parent galaxy is no longer visible to the naked eye. Then the Kepler problem is much harder, but not insurmountable. With axial tilt: a planet's year is still just as measurable. On the scale of a human lifetime the equinoxes will effectively be stationary. $\endgroup$
    – Cleonis
    Commented Apr 18, 2021 at 16:33

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