"God does not care about our mathematical difficulties. He integrates empirically"  Albert Einstein ( 1879  1955 )
Proposition: The Kepler Laws allowed Newton to establish his Law of Universal Gravitation without which Newton would have been at a considerable loss!
Johannes Kepler's empirical foundation for Newton's Law of Universal Gravitation
circa 1610 Johannes Kepler portrait  artist unknown
According to Kepler's 1st Law ( Planetary Law of Ellipses ) the planets move in elliptical orbits with the sun at one the foci:
Prior to Johannes Kepler ( 1571  1630 ), Copernicus ( Polish: Mikolaj Kopernik, 1473  1543 ) supposed that planetary orbits were approximate circles which indeed are ellipses of a special kind. In any event, because circles are special cases of generalized ellipses, any derivations based upon Kepler's 1st Law ( Planetary Law of Ellipses ) for ellipses must also hold true for circles.
Kepler's 2nd Law ( Equal Areas in Equal Times ) states that a planet's motion maps out equal areas in equal times for a radius vector, , drawn from the sun  foci to the planet. This implies that every planet traverses its "assumed circle" with constant velocity, , but with a centripetal acceleration, , directed inwards towards the center of the assumed circle as follows:
Assuming small incremental changes in ,
,
we have the following ratios:
Also another derivation:
Kepler's 3rd Law ( Harmonic Law ) mathematically formalized the data painstakingly collected earlier by Tycho Brahe ( 1546  1601 ) who was born in Skane, Denmark [ now in Sweden ] by clearly stating the following ratio which has the same value for all planets:
That is, total time period is proportional to the 3/2 power of radius for all planets written as follows:
But we also know from Kepler's 2nd Law ( Equal Areas in Equal Times ) that
This then provides one of the first and most important empirical proofs of Newton's Law of Universal Gravitation since planetary centripetal acceleration solely depends upon the distance of the planet from the sun!!
Derivation of Newton's Universal Law of Gravitation based upon Kepler's equations
Sir Isaac Newton ( 1643 – 1727 )
Of course, planets do not accelerate and fall into their center bodies due to a countervailing centrifugal force and Newton's Law of Inertia.
Newton's genius shone when he made his 2nd Law of Motion both famous and universal:
on earth to apply to planetary bodies and the sun as follows:
Hence, here is a further interpretation of the force which maintains a planet's orbit about the sun and which is directly dependent upon the planet's distance from this central body.
Moreover, the universality of Newton's concept of forces acting upon bodies did not just allow for the relative mass of the sun to effect a planet's orbital motion, but equally that there exists a mutual force arising from a planet's mass and hence its reciprocal effect upon the sun. This can all be expressed as follows:
Because of these reciprocal and mutual effects of forces involved  i.e., action vs. reaction  we achieve the following:
This says that there is some universal value, , for both the sun and any planet and therefore for any body whatsoever in the solar system. This universal factor of proportionality, , is called the gravitational constant and was finally and fairly accurately implicitly determined [ within 1% ] in 1797 by British Henry Cavendish ( 1731  1810 ) whose original intent was to calculate earth's density relative to water. Rather, Cavendish misinterpreted his discovery of the gravitational constant as earth's density; nevertheless, the torsion balance apparatus conceived and built earlier by then deceased John Mitchell ( Reverend, geologist and English natural philosopher, 1724  1793 ) was crated and sent on to Cavendish which allowed him to complete his 1797 experiment to within 1% accuracy of the modern gravitational constant calculation! Cavendish published his results in Philosophical Transactions of the Royal Society of London, Vol. 88 (1798), pgs. 469526.
Continuing ...
Or, in words
.
Therefore using slightly more explicit words,
Examples
§ Example 1).^{∗} Two suspended masses, one large and the other smaller, are separated from each other as follows:
The mutual gravitational force of attraction is therefore:
§ Example 2).^{∗} Two suspended celestial bodies, one large and the other smaller, are separated from each other as follows:
The mutual gravitational force of attraction is therefore:
Variation of Gravitational Force of Attraction
§ Example 3).^{∗} An American astronaut lands on a distant exoplanet whose surface gravity force in terms of acceleration is 3.5 m/sec2 and whose radial distance to its center is 2,300 km.
Hence, find the force of acceleration of gravity at an altitude of 500 km above the exoplanet's surface as follows:
and at the planet's surface
Finally, the astronaut will experience gravity acceleration of the following amount:
Now assume that our American astronaut weighs 150 lbs at earth's surface, he/she will have weight
on this exoplanet's surface.
At 500 km above the planet surface, our American astronaut will have weight
^{∗}note: these examples are used in the future upcoming Relativity Science Calculator Mac application
Newton's Laws Produce Universal Law of Conservation of Energy and Confirm Galileo's Law of Falling Bodies
And by applying a bit of integral calculus to Newton's laws, the Universal Law of Conservation of Energy will be clearly evinced:
Notice that in the equations
and
that we have here a mathematical demonstration of Galileo Galilei's Law of Falling Bodies since the final velocities and hence time of any falling body to a certain distance in a vacuum depends solely upon the gravitational force of the attracting body and has absolutely nothing to do with the mass of the falling body itself!!
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