An important gravitation lesson was given to me in late 1963 as a First year Physics student at the University of Chicago. Just two weeks earlier I had never even seen an Integral Calculus symbol, and now I wound up having to solve multiple Vector Integral Calculus problems as homework. The Professor was teaching us about Newton and his Fluxions (which we call Integral Calculus now). He had just taught us and derived for us Newton's standard equation of gravitational force.

I encourage all readers to examine Newton's beautiful math (Calculus) derivation of his Shell Theorems. (Many internet pages present the correct Integral Calculus involved. Actually, some of those mathematicians, including Wikipedia, make a subtle error in their math, where they try to use Cartesian (x,y,z) coordinates, where Newton had used the correct Spherical coordinates (r,θ,φ).)

The far more interesting lesson was to be next. Clearly, the Professor had decided to demonstrate how brilliant Isaac Newton was, even regarding the Fluxions that he had just invented. The Professor then decided to do the calculation (of Newton's) for a location halfway down inside the Earth. He stated a requirement that the Earth must be uniformly symmetric, although density increase with depth is allowed.

The Professor then decided to use a Spherical Polar coordinate system, as Newton did, and then he set up a triple Integral Calculus problem (of Newton's gravitational force equation mentioned above) in that coordinate system.

We students (me just barely 17 years old then) were given the actual math as homework. U. of Chicago seemed to think that if you could not do the math, you did not belong there. Unfortunately, my High School Physics Teacher was not very good and we had never gotten to ANY of the more complex Chapters in the Physics textbook, so I had a LOT to learn, really fast.

The Professor had given us some suggestions. Possibly the most important of those was to first calculate the Integral Vector Calculus total for the two angular coordinates for a very thin spherical shell centered on the center of the Earth, but for such a shell which was at a radius which was greater than the radius of the chosen location inside the Earth.

The start of the homework problem was initially just a double Vector Integral, for all locations everywhere on that shell, which had delta-r (miniscule) shell thickness. I encourage all readers of this to repeat that homework assignment. Today, such Calculus problems are called Newton's Shell Theorems.

It turns out that, no matter what specific location is chosen inside the Earth, an incremental location right "behind" you is close so the inverse-square law applies, but there is such an enormous area "across" the shell, that the net attractive force Vector from way over there EXACTLY EQUALS the attractive force Vector from that tiny area "behind" you, which is a Force Vector which is in the exact opposite direction. When you do the complete Vector Integrals for that specific shell, Newton found that the Calculus Vector Integral is EXACTLY ZERO.

The next part of that homework was to do a Third Integral, the third coordinate in the radial direction, r, for the series of shells from your location within the Earth up to the surface of the Earth. It was obvious to us students that the Vector Integral of a bunch of zero-amplitude Vectors is zero.

For the student, the net effective mass of the Earth acts as though it is at the Center of the Earth, so the effective gravitational field would have a 4 times factor compared to for us at the surface.

Therefore, there are four mathematical effects which must be calculated, (1) the ignoring of all the mass of the Earth which was at greater radius than you are at; (2) the average density of that portion of the Earth which is within that radius (around 2.5 times as great); (3) the net mass of that portion of the Earth, that is, density times volume or (2.5 * 1/8); and (4) the inverse-square distance of you from the center of the Earth (which is 4 times greater).

For the specific homework problem, the Professor noted that the Core of the Earth is considerably more dense than our Crust, and so the net measured effect of all these factors would be a SLIGHT INCREASE in local gravitational field (1/8 * 4 * 2.5), as you were halfway down inside the Earth as you did an entire trip "Journey to the Center of the Earth" as Jules Verne wrote long ago.

If you continued that trip to the center of the Earth all the way to the Core, there would be no remaining gravitational force Vector there at all!

However, consider Newton's reasoning and math above. Consider the very Core of the Sun, maybe a space the size of the Earth. There is NOT the mass of 330,000 times the Earth pressing down on that Core (which has always been totally accepted as logical for creating the enormous pressure and therefore temperature). Per Newton's Integral Calculus, that simply cannot be true. That portion of the Sun, its very Core, REALLY, only has roughly the weight of ONE Earth gravitationally pressing down on it.

Logic being what it is, this reasoning and math of Newton seems airtight. There MUST BE some other explanation for how the Sun (and all other stars) creates sufficient gravitational pressure and high temperature for Fusion. We necessarily must be very wrong in our understanding of even this basic idea.

Newton WAS right, and a whole lot of Physics students did the homework problems to confirm it.

We THINK that the Sun creates Nuclear Fusion due to the enormous heat of the Core of the Sun being "crushed" by the gravitation of the mass of 330,000 times the Earth. It seems likely to be somehow true, that the heat of many millions of degrees Kelvin is necessary in order to have the Hydrogen atoms crash into each other fast enough to fuse together. But the logic which we try to use to describe how that tremendous heat must exist in the Core of the Sun must certainly be very wrong.

We SHOULD collectively be able to figure out the correct explanation! Let's do it!

But Newton seems to have been right in the logic and math described above, which suggests that the Sun does NOT "crush" its Core to cause the high temperatures there. This seems to suggest that we might have some far better, and possibly simpler, method for creating Fusion.

The same logical errors just discussed regarding stars being able to gravitationally collapse to cause enormous pressure in their Cores, and therefore cause the heating to many millions of degrees Kelvin, must apply even more intensely regarding the current understanding of black holes. We claim that the very center of a black hole is crushed to molecular size, entirely due to the spectacular amount of mass which surrounded it and forced that collapse with all the implications we credit to black holes. That logic MUST be totally wrong!

*Nuclear Physics May be Fairly Simple *

*Nuclear Physics - Statistical Analysis of Isotope Masses*

This presentation was first placed on the Internet in June 2017.

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C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago