In this thread, I'll be deriving an alternative form of Newton's famous
Law
of Universal Attraction and demonstrating that it's perfectly possible to achieve identical results with this alternative derivation. However,
there is one very important difference ... the need to invoke, or use, his
Gravitational
Constant is completely eliminated. We therefore end up with an equation that calculates the force of attraction between an orbiting body and the
body being orbited exactly as does Newton's equation yet does so without having to rely on mysterious "gravitational constants" to make the
equation work.
Finally, I'll attempt to show the underlying basis of this Gravitational constant that appears not only in Newton's equation but in many others and
how it can be derived quite simply as a result of nothing more sophisticated than one body orbiting another.
Mathematically, Newton's concept essentially states that every particle in the universe attracts every other particle in the universe with a force
which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
This concept, or Law of Universal Attraction is expressed by this equation:
and here's an image to help get a more visual 'feel' for what this law is saying:
The only term in the above equation that may cause some confusion is the
G term that represents what's known as the
Gravitational
constant and basically was a scaling factor that had to be introduced to make the Law of Universal Gravitation work. Newton found it necessary to
provide a "gravitational constant" so his results could be manipulated into the correct dimension of "force units".
Thus, if the gravitational constant (G) is removed from Newton's equation so it looks like this
the results from this equation are no longer in force dimensions.
So, even though Newton had no idea of where G comes from or what it represents, it's very apparent that the primary purpose for Newton needing to
invent and include this gravitational constant (G) was to "manipulate or fudge" the otherwise incorrect results his equation would have given and
convert them into the correct dimensions of force.
Ok, the above is a brief introduction and explanation regarding the Law of Universal Attraction and it's dependence on the gravitational constant G.
In the following sections, I'll be demonstrating how it's possible to remove entirely the reference to G thereby producing an alternative form of
Newton's equation that still produces the correct answers.
I'll start by making my 1st postulate that
[Color=Yellow]for each and every primary body (e.g. the sun, earth, mars, jupiter, etc) that has a secondary body (or bodies) orbiting it,
there exists a unique NUMERIC CONSTANT that is entirely specific to that primary body and it's associated orbiting bodies.
In other words, I'm stating that there is a unique numeric constant associated with the sun and the planets orbiting it and there is another unique
constant associated with each planet and the moons orbiting that planet.
This "orbiting" constant is the result of the product of the distance (the radius) between the orbiting body and the primary body being orbited,
multiplied by the square of the orbiting body's velocity. If we represent this "orbiting" constant by the letter "k", we have the following
equation:
To show that such an "orbiting" constant actually does exist, I've generated a number of tables showing each primary body and every orbiting body
associated with that primary. In other words, the sun and it's planets ... Earth and it's moon (plus The ISS and Hubble) ... Mars and it's two
moons ... Jupiter and it's moons ... etc, etc
Note, in the following tables, the very minor discrepancies in "orbiting" constants within each system I believe are due solely to minor published
inaccuracies in mean radii and mean velocities.
As is clearly obvious, each primary body and it's associated orbiting body's has a
unique and identifiable "orbiting" constant. The
implication here is that any orbiting body that has a radius and velocity that produces the correct "orbiting" constant for the system that it's a
part of, is therefore in a
STABLE orbit. The corollary is that if the calculated "orbiting" constant value is not the
correct one for the system, then that body's orbit is
UNSTABLE.
Using the Earth as the primary body, we notice that both the ISS and Hubble have been placed in
stable orbits around the
Earth and their respective "orbiting" constants therefore match that of the moon's "orbiting" constant.
The above leads me to my 2nd postulate
within every primary body/orbiting body system, stable orbits can ONLY exist when the individual "orbiting constants" match
the "orbiting constant" specific to the system they inhabit. This is analogous to electrons in an atom being able to exist only in well defined and
stable energy levels.
Ok, moving on ...
Usually, when we need to calculate the velocity of a body that's orbiting another (usually much larger) body, the following equation might be
used.
It basically states that the velocity of an orbiting body is dependent on the combined masses (M) of the primary body and the orbiting body, the
distance (r) separating the two bodies as well as the necessary inclusion of yet again, Newton's Gravitational constant (G).
Now this is a very interesting equation because it allows us to rearrange it's terms to arrive at another equation which has a surprising and
unexpected, yet very direct link or relationship to the "orbiting constants" that I spent so much time discussing earlier on.
So starting with the original velocity equation, we rearrange terms
The end result is an equation that gives the product of G and M in terms of a velocity (v) and a radius (r).
Continued next post ...