Neat and fun stuff you are doing. Congratulations!
Some comments that may or may not be of use are below.
On Jan 6, 2008, at 2:17 PM, OrionWorks wrote:
My apologies for this lengthy 6-page post. I didn't feel like
serializing it. Instead I broke it up onto several sections.
Several months ago I initiated personal research into running extended
computer simulations based on the simple rules governing celestial
mechanics, (CM). I'd like to report on some of the more interesting
observations discovered from this extended research.
* TO DECAY OR NOT TO DECAY:
The most familiar Celestial Mechanics (CM) algorithm employs the
inverse square of the distance formula: [F = r * k/r^2)]. In my
simulations I iterated the hell out of the "point" satellite as it
orbits a stationary attractor conveniently position at the (0,0)
origin of my graphic screen. When generating 50,000,000 or more
iterations per simulation occasionally beautiful Mandela-like patterns
can be generated particularly if the orbits are more eccentric
(elliptical) BUT NOT TOO ELLIPTICAL. The mysteries of chaos theory
become apparent. Occasionally beautiful fractal-like patterns can be
introduced into a previously stable orbital system when one allows the
roving satellite to dip too closely inwards towards the main attractor
body. A dangerous game is played at these close approaches as
iterative sampling is in danger of becoming too gross resulting in the
generation of increasingly unstable vectors with each close approach.
I think you can improve your simulation accuracy and/or performance
by varying the time increment depending on velocity and/or
acceleration. When an iteration is completed you can calculate the
delta t using a formula based on the velocity and/or acceleration
which is then available.
Eventually the chaos introduced becomes too great to maintain any form
of orbital stability. Wildly eccentric the orbital patterns eventually
break down resulting in an ejected satellite. In a sense it's like
watching a form of radioactivity decay at work. As previously
mentioned about a month ago on this forum I openly speculated on
whether an obscure branch of chaos theory might be involved in
governing the laws of nuclear radioactivity. This was based on the
fact that quantum mechanics follows discrete "jumps" or states of
existence in a manner somewhat similar to discrete "jump" states that
are generated per computer iteration. Is it possible that the
instability as perceived in these computer simulations capture a
similar mechanism governing the rules of radioactive decay?
* MOND MADNESS:
If you modify the Inverse Square of the Distance formula, where
instead of squaring of the radius portion you keep it at a single
power [F = R * k/r] you will end up generating another kind of stable
orbit, a most curiously looking one at that. Instead of generating
elliptical orbits you can produce stable triad-like patterns involving
perihelion sweeps 120 degrees apart per "swing". IOW, as the satellite
sweeps past the main attractor (aphelion) gaining speed it doesn't
gain speed as fast is it would if the Force was being inversed
squared. As a result the satellite swings past at a more obtuse angle,
120 degrees to be exact. Iterating these orbital formulas on a massive
scale will also occasionally produce interesting patterns,
particularly if you allow the satellite to dip too closely towards the
main attractor body.
Some might find it interesting that recently there has been a lot of
interest in researching this branch of "modified" celestial mechanics.
It is known as the study of Modified Newtonian dynamics, or MOND.
http://en.wikipedia.org/wiki/Modified_Newtonian_dynamics
Research into MOND mechanics is intriguing to some because it seems to
explain more accurately the orbital characteristics of individual
stars orbiting within galaxies. It's important to keep in mind that
MOND influences only begin to become apparent at the vast distances
registered between stars, NOT the shorter distances as measured within
solar systems. Granted, there continues to be considerable debate on
this topic both pro and con but at present the MOND theory is still
being seriously researched.
Of course the most popular theory these days involves postulating the
existence of huge gobs of unseen and unproven "dark matter" (or Mills
hydrinos) to explain the same orbital characteristics of individual
stars in most galaxies. Sometimes it's difficult to determine which
theory is more audacious, the MOND theory, Hydrinos, or the existence
of a whole-lotta unseen unproven Dark Matter.
Adding more sauce to the goose it has recently been confirmed (and
subsequently ignored as expeditiously as possible by mainstream
science) that space probes particularly the Pioneer and Voyager
classes, which have essentially left the confines of our solar system,
have been discovered to be decelerating faster than the inverse square
of the distance would predict. Curiously, when they take MOND CM into
consideration it would seem to do a better job of predicting the
behavior of these far flung artificial satellites.
You might find it interesting that black holes as a source of
negative gravitational mirror matter can explain the form of the MOND
equation as applied to galaxies. See pp. 23 ff:
http://mtaonline.net/~hheffner/FullGravimag.pdf
* ZANY SPIRALING CUBED ORBITS, aka THAT JUST HOW THE EGG ROLLS!
Just for the hell of it I tinkered with the original CM formula
modifying [F = K/r^2] to [F = K/r^3]. I cubed the inverse law – to the
third power. It produced interesting and unexpected results. The most
important discovery was that, at least in my own research, I found
that generating stable orbits turned out to be a MUCH MORE DIFFICULT
animal to tame. Cubed orbital trajectories tend to spiral inwards and
then spiral outwards in multiple 360 degree sweeps, sometimes in an
oscillating in-out pattern. First the satellite spirals inwards
towards the main attractor gaining speed, or velocity, all the way.
Eventually the satellite picks up sufficient centripetal velocity to
overcome the inner pull of the attractive cubed force causing the
satellite to begin spiraling outward. However, the inverse cubed law
means that the attractive force becomes weaker more quickly than what
occurs under the traditional inverse square of the distance rule. This
means deceleration of the satellite occurs more slowly. This tends to
result in a greater percentage of escape velocities.
What's interesting about these exotic cubed orbital observations is
that the spiral patterns often mimic what is observed in bubble
chambers. Charged particles spiral inward (or outwards) from a
centered point – this all due to the influence of a strong magnetic
field bending the path of charged particles into a spiral path.
Regarding the spiral-like paths noticed in bubble chambers I'm sure a
different principal of physics is involved rather than my cubed CM^3
formula. Nevertheless, it's interesting to notice the similarities.
Something you might be interested in is applying a 1/r^4 law. That
is the form of attraction between magnetic dipoles, which most nuclei
and particles are. The following gives a sample relativistic
calculation for a special close up magnetically bound version of the
hydrogen atom:
http://www.mtaonline.net/~hheffner/DeflateP1.pdf
A magnetic orbital can become stable (avoiding collapse to a point
that is - they are anything but stable!), despite the 1/r^4 force,
because the force diminishes as the waveforms overlap, i.e. when the
de Broglie wavelengths overlap.
* HYBRIDIZED CM - BREAKING THROUGH THE COULOMB BARRIER:
Perhaps the most interesting observation I made was to introduce a
HYBRID CM formula where I incorporated BOTH the inverse square of the
distance AND the inverse cube
I think it should be the inverse quartic of the distance law for a
more realistic picture.
of the distance into the same
simulation. If you give the inverse square of the distance a
negative/repulsive value while keeping the inverse cube a
positive/attractive value one can generate simulations that seem to
mimic the behavior of the both the COULOMB BARRIER and STRONG NUCLEAR
FORCE, as witnessed within atomic nucleus. First, the negative /
repulsive inverse square of the distance portion influences all
particles forcing them to fly apart, away from the main attractor.
However, when both squared and cubed CM formulas are introduced there
exists a magical no-mans-land where the struggle between attraction
and repulsion negate each other – where the "Coulomb Barrier" resides.
Move our wandering satellite any closer to the main attractor and the
forces of the inverse cube of the distance begin to take over
dramatically. Once the so-called "Coulomb Barrier" is crossed the
satellite has no choice but to be drawn inward with significantly more
attractive force than what the satellite ever experienced repulsively
outside where the inverse square of the distance force reigned
supreme. It would appear that, proportionally speaking, cube
influences are the order of several magnitudes stronger than squared
influences. They also occur on a much tinier scale spatially. Cubed
influences quickly become negligible outside of the influences of the
Coulomb Barrier, where the particle is essentially governed by the
inverse square of the distance formula. Incorporating a hybrid formula
combining both an inversed negative squared and attractive cubed
formula seem to mimic the characteristics associated with the STRONG
NUCLEAR FORCE and the Coulomb Barrier.
I should confess that I did not come up with this zany idea of
combining squared and cubed formulas all on my own. I cannot remember
where I read the article but I could swear there has been some
research into this speculative branch, particularly where researchers
used the cubed formula at tiny sub atomic distances to simulate the
behavior patterns of sub atomic particles.
I posted a lot of stuff along those lines here, except it was the
inverse quartic law, not inverse cubic law, though from the below it
sounds like that was not what you recall.
If memory serves me the
article postulated that within the tiny quantum scale the nature of
"matter" manifests in more than three dimensions, which according to
their POV accounts for the influence of the inverse cubed distance
force. BTW, I'm NOT talking about some variant of STRING THEORY where
more than ten dimensions have occasionally been postulated. The cube
CM influences I'm referring to occur on a much larger scale, the
sub-atomic scale – not in the teeny-tiny super string scale.
I'll conclude with an interesting speculative point regarding how
these inverse cube of the distance orbits might tend to physically
manifest as perceived within our mundane 3-D reality. As best as I can
visualize it cubed influenced orbital paths would not manifest in the
manner familiar to 2-D planar inversed squared disks, or rings. It
seems more likely that cubed orbits would manifest as pulsating
(shells), eggshells to be more exact, when projected into our 3-D
reality, particularly if one introduces an oscillating pattern into
the orbital period. Again, I'm reminded of the popular term "orbital
shells" used to describe electron orbits of atoms. I wonder if a
similar description could be incorporated describing the shape of the
atomic nucleus as well. Research indicates that the actual shapes of
atomic nucleus are anything by spherical. Some species can be oblong,
football shaped, or flattened Jupiter-like shapes. Some can even
sprout a little ring bulge in the middles as if the nucleus had a
hoolahoop. They also tend to be less stable than their spheroid
siblings.
Horace Heffner
http://www.mtaonline.net/~hheffner/
