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deda
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Currently the Earth has two extreme distances from the sun: Apogee and Perigee (one minimal, one maximal). Shouldn’t there be four extreme distances between the sun and the earth: pair of opposite minimums and pair of opposite maximums? Both the sun and the Earth have mass. So, there must be center of mass between them. I find the center of force at the same position. The system can be balanced only if that center is immovable. Statically, the condition for balance makes the ratio of their forces inverse the ratio of their distances from that center. The center also must be between them. The angle of rotation must be same for both and its makes the two oscillate around same center. Assume their trajectories are perfect circles. Now let's double the force at one point. It will be same as if we added only half of the force at that side and opposite half on the other side. What matters is that we added force only in one direction. Its component with normal direction remained the same. So the trajectory must have pair of opposite minimal and maximal extremes i.e. the trajectory must be an ellipse with same center as the center of force.
The way Kepler solves the trajectory using Newton’s laws makes the forces strangely variable, which I find hard to explain because of absence of balance. Here is an example proving that Kepler’s solution doesn’t hold: Dig a tunnel through the Earth all the way from the North Pole to the South Pole. Drop one object from distance H. The extreme distance on the other side will be -H from the center of earth. Now repeat the whole thing again but this time input additional force at the drop point (you would prefer me to say drop it with initial speed). The extreme distance on south will be bigger than the initial one north but on return the extreme distance north will equalize with the south one. It’ll only seem as if you’ve dropped it from bigger distance without the additional force input. Point made.
The way Kepler solves the trajectory using Newton’s laws makes the forces strangely variable, which I find hard to explain because of absence of balance. Here is an example proving that Kepler’s solution doesn’t hold: Dig a tunnel through the Earth all the way from the North Pole to the South Pole. Drop one object from distance H. The extreme distance on the other side will be -H from the center of earth. Now repeat the whole thing again but this time input additional force at the drop point (you would prefer me to say drop it with initial speed). The extreme distance on south will be bigger than the initial one north but on return the extreme distance north will equalize with the south one. It’ll only seem as if you’ve dropped it from bigger distance without the additional force input. Point made.