kepler's planetary orbits
RELATIONSHIP BETWEEN KEPLER'S SINGULAR ELLIPSE
Kepler could not be aware that the Sun was involved in mutual orbit with the planets because that discovery was not made until 100 years after his death. He therefore assumed that the Sun was rigidly fixed in space, and all motion was attributable solely to planetary motion about a fixed Sun.
& TWO CONCENTRIC CIRCULAR ORBITSWorking only with the angular position history (from Brahe's records) Kepler was able to determine that the trajectory of Mars must involve changing radial distance and tangential velocity from a fixed point in space that he falsely assigned to the Sun. He was unaware that the data he was using was affected by the motion of the Sun which resulted in the changes in both radius and velocity he attributed to Mars.
He was able to mathematically fit the trajectory that he believed Mars to travel in an elliptical shape with the fixed Sun located at one of the foci of that ellipse. But the fit did not recognize that the point chosen for the Sun was moving harmonically back and forth between the two foci of that envisioned ellipse. However, lacking that recognition of the motion of his reference point, the equation 'worked' and his declaration was accepted as accurate.
Today we are aware that two celestial bodies involved in mutual rotation, both move in opposition around a common center that we call the 'center of gravity' for the combination of the two bodies. The ratio of the radial distances of the two is inversely proportional to the ratio of their 'masses'. When we apply that knowledge to Kepler's analysis, then the angular - time history that Kepler utilized can be recognized to be a mathematical identity to two concentric circular orbits, with the foci of the ellipse that Kepler perceived being located at the intersection of the lesser radial circular orbit and the major axis of that perceived ellipse.
These orbital relationships are depicted below. For clarity, the eccentricity of the depicted ellipse is much greater than that which actually exists between the Sun and planets.
The top box (red outline) is the equation for the perimeter of an ellipse with fixed focal point as envisioned by Kepler.
The center box (green outline) is the equation for that same ellipse when the point of reference moves harmonically along the major axis on the opposite side of the point (x,y) of interest on the ellipse perimeter.
The lower box (blue outline) is the equation for that same ellipse when the point of reference moves harmonically in both horizontal and vertical directions. In which case the equation is actually that for two concentric circles which both have their center at the center of the ellipse, with radial lengths equal to the minor and major axis of the ellipse.
The red, green, and blue lines shown crossing the ellipse are all equal and each represents the distance between the two celestial objects (Sun and planet) which are in mutual rotation. Had Kepler been aware that two celestial bodies rotate around their joint center of gravity, then he might never have created the concept of one (the planet) moving in an ellipse around the other (his 'fixed' Sun).
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