**Newton's Law of Universal Gravitation**

**"**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 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. 469-526*.

Continuing ...

Or, in words

Therefore using slightly more explicit words,

**§ 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 Calculator Mac application

**Newton's Laws Produce Universal Law of Conservation of Energy and Confirm Galileo's Law of Falling Bodies**

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!!