[Physics] background on Newton's gravitation

[cj at mb-soft.com]

An important gravitation lesson was given to me in late 1963 as a
First year Physics student.  Just two weeks earlier I had never even
seen an Integral Calculus symbol, and now I wound up having to solve
multiple 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 Newton's standard equation of
gravitational force (for the surface of the Earth).

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 Polar coordinate system, and then
he set up a triple Integral Calculus problem in that coordinate
system.

We students (me just barely 17 years old) were given the actual math as
homework.  The Professor had given us some suggestions.  Possibly the
most important of those was that to calculate the Integral Vector Calculus
total 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 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.  It turns out
that 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 from way over there EXACTLY EQUALS the attractive
force 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, 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.

What the Professor had taught us, of the Fluxions math which Newton had
done three hundred years ago is that, the strength of the gravitational
field at ANY location inside the (symmetric) Earth was ONLY due to the Mass
of the Earth which happened to be in the Core part of the Earth which
was the size of the sphere of the Earth which was the size of the location
you happened to be at.  For the homework problem the Professor gave
us, of half the radius, that smaller ball which actually was providing
a gravitational field for you, was only 1/8 the volume of the entire
Earth.  The density of the Core of the Earth was much higher than our
Crustal rocks, but gravitation acts at an inverse-square law.  For the
student, the net effective mass of t he Earth acts as though it is at the
Center of the Earth, so the effective gravitational field would have
a 1/4 factor compared to at the surface.

Therefore, there are four 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 did an entire trip
"Journey to the Center of the Earth" as Jules Verne wrote long ago.

I see an entirely different important example of Newton's mathematical analysis.
Consider the Sun or any other star.  We usually describe it as though the whole
WEIGHT of the entire Sun is pressing down onto the very Core of the Sun,
which we then say creates spectacular temperatures of billions of
degrees Kelvin, which is then what we claim causes Hydrogen nuclei to
FUSE into each other to form Helium atoms and which releases the
spectacular amounts of energy that our Sun radiates away and which
keeps us all alive.

But 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.  How in the world could THAT
create enough pressure and therefore temperature of billions of degrees
Kelvin, to initiate and maintain nuclear fusion?

Logic being what it is, this reasoning and math seems airtight.  There MUST
BE some other explanation for how the Sun (and all other stars) create
sufficient 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.

I realize that these comments involve some Integral Vector Calculus which some of you may n ot be familiar.  I only felt it important to present this logic of Newton for some background for your group.  Whether in the Earth or within any star, Newton proved that all of the mass which happened to be at greater radial distance, has no net effect regarding the Vector Force of gravitation.  Personally, I findn this most troubling inside of stars, as the enormous "gravitational weight" which alleged presses down to create enormous pressure and therefore the billions of degrees of Kelvin temperature in the very Core of every star, simply is not possibly true.

Carl Johnson


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