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The gravity force is inversely proportional to the square of the distance between the two bodies. Why is the exponent in this formula exactly equal to 2, rather than (say) 1.99997054 or 2.00061833? How was the parameter 2.00000000 determined? Via empirical evidence (data science) or from theoretical principles? Or is it not exactly equal to 2? And if not, does it change over time? Or is it slightly different in different parts of our universe?

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It is from theoretical priciples. Newton's law for gravitational attraction. It is derived from Gauss Law.

General relativity fixed this value slightly upper value from 2.00000 . Mercury perihelon shift was correctly estimated by General Relativity. It was one of first tests of GR. This shift is because of this value. The actual value may change because of Sun gravitational field and GR should be used for actual calculations.

Here are two answers to my question, posted on Google+

  • In brief, Newton's inverse square law for gravitation was a generalisation of Kepler's laws, which in turn were based on observations of planetary orbits (in the sense that they were a nice, simple mathematical model which fit the data to within the limits of observation at the time). It was of coursepossible that the "true" exponent was marginally different from 2, but the inverse square law was by far the simplest model which fit all the data (this is Occam's Razor in action). But Newton's inverse square law is in fact wrong! General Relativity gives a far more accurate description of a wider range of gravitational phenomena, successfully describing things which where wholly unknown in Newton's day (e.g. the bending of light by gravity, precession of the perihelions 

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