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A
Complete Correction to and Explanation of BODE'S
LAW
by
Miles Mathis
Si
quid novisti rectius istis, Candidus imperti; si non, his
utere mecum
Abstract: This is not “an
improved version” of the law, this is the FULL mechanical
solution. In short, Bode's law, or my correction to it, is just a
simple extension of the orbital equation a = v^{2}/r.
First, I show the sequence is a function of the square root of
two, rewriting Newton's orbital equation to prove it. Then, I
show a further correction to that sequence by importing the
charge field into the equations. This shows that the planets
orbit where they do due to both the orbital (gravity) equation
and the charge field. Yes, the planet sequence is a direct
outcome of the unified field. The old Bode sequence failed mainly
because it failed to include charge interactions between the
planets. In solving these charge interactions, I am forced to do
several fivebody (the four big planets + the Sun) unified field
problems. Using highschool algebra and only a few lines of math,
I am able to solve these 5body problems to any precision
required, resolving the Bode sequence down to almost no margin of
error.
First
published November 20, 2009
Bode's
Law, also known as the Titius/Bode Law, is one of the most famous
unexplained laws in the Solar System. Some have said that solving
it would be like solving Goldbach's
Conjecture or Fermat's Last Theorem. I think it is
considerably more important than either, since it is fundamental
physics, rather than just a puremath brain teaser. Which is to
say that I hopped up and down longer and higher when I solved
this one than I did when I solved Goldbach's Conjecture. Maybe
now a few more people will look closely at my GC solution.
The
first mention of the law is from 1715, so this one has been
sitting around without a solution for almost 300 years. Since
that time, the form of the “law” has been:
a
= n + 4 where n = 0, 3, 6, 12, 24, 48....
Wikipedia
glosses for us the standard model explanation:
There
is no solid theoretical explanation of the Titius–Bode law,
but it is probably a combination of orbital resonance and
shortage of degrees of freedom.
The
first explanation is just a guess, but it is a bad guess since
orbital resonances have been given to gravity, but no one has
ever shown a mechanical cause of any “gravitational
resonance.” Resonances cannot be caused by gravity, and no
one in history has shown that they can. I will show that the
failure of Bode's law on outer planets is caused by a charge
“resonance”, but this resonance has little to do with
the main series of numbers and nothing to do with gravity. The
second explanation from Wiki is an even worse guess, since it
turns the truth on its head. What this editor means by “shortage
of degrees of freedom” is that the math falls into this
alignment because it can't do otherwise. But the failure of the
standard model to explain Bode's Law is actually caused, in part,
by a shortage of degrees of freedom in another sense. Current
celestial mechanics lacks an entire field, since it hasn't
incorporated the foundational E/M field into its equations. I
have shown that the foundational E/M field or charge field is
already in Kepler's
and Newton's equations,
and this field may be called a mathematical degree of freedom.
The equations of
celestial mechanics lack the required complexity, so they can
only be pushed after the fact, as
with Laplace.
Even worse than these two wild guesses
is what comes next at Wiki.
However,
Astrophysicist Alan Boss states that it is just a coincidence,
and the planetary science journal Icarus
no longer accepts papers attempting to provide 'improved'
versions of the law.
Well, may Alan
Boss take a flying leap into a pit of cankered pions and may
Icarus catch a
malign meteor in the teeth. If this isn't a case of science being
shut down by fiat, I don't know what would be. These people and
institutions should be shunned by all real scientists and
thinkers, and I recommend that Icarus
be put into bankruptcy, by readers refusing to read it. I have
never read it and never will. If these editors will be
intellectually bankrupt, then they should be financially bankrupt
as well.
Unfortunately, science is now controlled by this
sort of person, and it isn't just Icarus
that is the problem. All of mainstream physics is now like this.
We have “scientists” that have stated and published
opinions prejudicially on things they know nothing about, and
they don't seem to understand that this is unscientific. Alan
Boss can't explain this phenomenon and he wants to be sure no one
explains it while he is working on something else. Infantile. He
probably throws a fit when his wife works on the crossword puzzle
when he is out of the room. Physics has been taken over by very
small people.
Since I am the one solving these
longstanding problems, it is not for me to follow their policies.
The ignorant do not set rules for the wise. If any of these
journals wants to publish real papers instead of fake papers,
they had better change their attitudes. They seem to believe it
is my loss, not being able to publish with them, but it is their
loss. If Einstein had not published with Annalen
der Physik, would it have been his loss
or theirs? Annalen
gets a historical mention only because of Einstein.
Remember
that, Icarus.
Signed, Apollo.
In physics, as in all else, problem
solvers are primary and publishers are secondary. Publishers are
just administrators and committees and gatekeepers. Historically
and scientifically, they are of no import. Publishers can refuse
to publish, but they cannot stop the spread of information.
Wiki also tells us this:
Dubrulle
and Graner have shown that powerlaw distance rules can be a
consequence of collapsingcloud models of planetary systems
possessing two symmetries: rotational invariance (the cloud and
its contents are axially symmetric) and scale invariance (the
cloud and its contents look the same on all length scales), the
latter being a feature of many phenomena considered to play a
role in planetary formation, such as turbulence.
There
it is again, the power law used to fudge a very squishy answer. I
have already shown in many papers, including my
paper on Laplace, how the power law is used to push bad
equations to fit data, by using infinite series to assign terms
to errors. These errors are called “remaining inequalities”
or something like that, and a lot of fancy math is used to drop
or add terms. But since I will show that Bode's law can be
explained without any sort of calculus, with 9th grade algebra,
all this talk of collapsing clouds is just further nebulosity.
These are the types of papers the mainstream likes to publish:
papers full of pompous impossibilities like scale invariance
(clouds are density fields and cannot be scale invariant) and
incomprehensible and undefined equations. The mainstream physics
paper is little more than an institutional efflux, a career
propellant adding another antilogical pollutant to the infinite
stream of modern absurdities.
We have been diverted into
these ridiculous solutions simply because the mainstream is
afraid of the correct solution. The correct solution concerns my
unified field, which adds charge to gravity in the equations of
celestial mechanics. The mainstream can't bear this solution
because it destroys all their old gravityonly math that goes
back to Laplace and Lagrange. My unified field destroys not only
the equations of Laplace, it destroys entire newer fields like
perturbation theory and chaos theory. They don't want to correct
all the field equations, and they don't want anyone to do it for
them, so they hunker down and protect what they have any way they
can. This despite the fact that they have decades of conspicuous
data showing the presence of a very large charge field not only
in the Solar System but in the galaxy and universe as well (and
more coming in daily). To see mainstream data in conspicuous
agreement with my unified field and my analysis here, you may go
to Wikipedia and type in "Heliospheric
Current Sheet".
As you will now see, the solution
to this problem is so simple that it makes three centuries of
physicists and mathematicians look like bumblers. I looked at
that sequence of numbers above for about half a minute before I
saw it was based on the square root of 2. The “law”
has been in the wrong form since the beginning, and so no one was
able to see the proper sequence.
Currently, the sequence
goes like this:
4, 7, 10, 16, 28, 52....
But it
should be written as
4, 5√2, 7√2, 11√2,
20√2, 36√2....
Which can be written as
2^{2} (2^{2}
+ 1)√2 (2^{2}
+ 1 + 2)√2 (2^{2}
+ 1 + 2 + 2^{2})√2 (2^{2}
+ 1 + 2 + 2^{2} +
3^{2}+ 4^{2})
√2
If we want to express this with Mercury as 1,
then we just divide by 4.
2^{2}
/2^{2} [(2^{2}
+ 1)√2]/2^{2} [(2^{2}
+ 1 + 2)√2]/2^{2} [(2^{2}
+ 1 + 2 + 2^{2})√2]/2^{2} [(2^{2}
+ 1 + 2 + 2^{2}+
3^{2})
√2]/2^{2}
Which
expands to:
2^{2}
/2^{2} √2
+ (1/2^{2})√2 √2
+ (1/2^{2})√2
+ (2/2^{2})√2 √2
+ (1/2^{2})√2
+ (2/2^{2})2^{2}
+ (2^{2}/2^{2})√2 √2
+ (1/2^{2})√2
+ (2/2^{2})2^{2}
+ (2^{2}/2^{2})√2
+ (3^{2}/2^{2})√2
Which
simplifies to:
1 (5/4)√2 (7/4)√2
(11/4)√2 (20/4)√2
You will say,
“Great, you expressed Bode's Law in terms of √2. So
what?” Well, the sowhat is that it ties directly into my
correction to Newton's equation a = v^{2}/r.
I have shown that the equation should read a = v^{2}/2r,
since our current expression of the orbital velocity is not a
velocity. Yes, a = v^{2}/r
works if v = 2πr/t, but 2πr/t isn't a velocity. It is a
curve over a time, which isn't a velocity. It is just a heuristic
ratio that we like because it is easy to measure. But since the
orbit curves, it must be an acceleration, and that acceleration
is expressed by the equation,
a_{orb}
= 2√2πr/t
If we use that acceleration along with
the corrected equation, a = v^{2}/2r,
we get the same relationship we have now with a = v^{2}/r
and v = 2πr/t. That is where the √2 comes from. And that
is why our current orbital equations work despite being faulty.
They are consistently faulty, so we don't see the faults except
when we are in need of mechanics like I am doing now. Our current
heuristics doesn't prevent us from doing engineering, but it does
prevent us from doing foundational mechanics like this, or from
solving simple problems like explaining Bode's law.
You
will say, “2√2πr/t still looks like a curve over a
time to me.” But it isn't because I
have also shown what π is in that equation: for that
equation to work, π must be an acceleration itself. We have
thought that π was just a bald number, with no dimensions, but
π has the dimensions of m/s^{2}
(as I have shown elsewhere exhaustively). This gives the orbital
acceleration the dimensions m/s^{3}.
The orbital acceleration is actually the summation of three
separate velocities during any dt, so it is rigorously defined as
a variable acceleration. A velocity is a Δs, a constant
acceleration is a ΔΔs, and a variable acceleration is
a ΔΔΔs. That is, a variable acceleration is the
third derivative
of distance.
All this has been ignored historically, so
we have heuristic equations that have been hiding a lot of
important information. Our orbital mechanics, like our celestial
mechanics, has been a compressed engineer's math, instead of a
theorist's math.
You will say, “I still don't get
it. How does the √2 help us solve this?” It helps
because, via my corrected orbital equation, it tells us that
Bode's law comes right out of the orbital equation.
Look
at my expression of the series above. We have the square law from
Newton's orbital equation, but it is an additive square law. Each
planet is not only orbiting the Sun, it is also orbiting inner
planets. The Earth's relative distance is given by this equation:
r = [(2^{2}
+ 1^{2} +
2^{2})√2]/2^{2}
In that equation, the Earth is the third term in
parentheses, Mercury is the first, and Venus is the second. So we
are being told that the Earth's orbit is added on top of the
other two. The Earth is orbiting them as well as the Sun. You
have to include the inner orbits in the equations for the outer
orbits. Very simple, right? In this way, the corrected Bode
equation is an analogy of the Pauli exclusion principle. But
unlike with Pauli, here we have the mechanics in full view right
in front of us. The Earth cannot share Venus' orbit, because the
Earth's term must be added to the term of Venus. The Earth is
excluded from the second orbit, because each term is added, as a
distance, in a set pattern. “Two squared” cannot be
at the same distance as “one squared”, since the
numbers determine the distance.
And it is not just the
math that provides the exclusion. As I have said elsewhere, math
is just an expression of the mechanics. Math cannot be a cause.
The mechanical cause of this math is the exclusion provided in
the unified field by the charge component of that field. Gravity
cannot exclude. Only the charge field can exclude. The Earth
cannot inhabit a different part of Venus' orbit because the two
bodies have different charge fields. The Earth has a greater
charge field than Venus, due to greater mass and density, so the
Sun keeps it at a greater distance. The Sun's charge field and
the Earth's charge field actually meet, in space, physically,
particle to particle.
Another way to say all that is to
just look at the equation a = v^{2}/r.
We don't even have to look at my correction to see how Bode's law
comes right out of that equation.
a =v^{2}/r
so r = v^{2}/a
Which tells us that the radius is determined by the
square of the orbital velocity. But that can apply to any radius.
Any radius will give us an orbital velocity, so how does the Sun
pick specific orbits for the planets? It does so with equations
like this
r = [(2^{2}
+ 1^{2} +
2^{2})√2]/2^{2}
If you drop √2, you see the similarity of the two
equations. The squared terms in parentheses are actually orbital
“velocities” (really, orbital accelerations). Bode's
law is just an expansion of the orbital equation, but it gives us
the relationship of one orbit to the other. Each orbit is some
factor of two of the other orbits. And each outer orbit must
orbit all inner orbits. The simple exclusion is in a simple
mathematical relationship.
If
that is all I had to say, this paper wouldn't have been terribly
convincing to the hardliners, I admit. I have shown the law as a
function of the square root of two, but unless you were impressed
and convinced by my correction to Newton's equation (and that is
unlikely), this will not seem very momentous. The reason this
spins out into a good story of its own is that expressing Bode's
law with the square root of two allows us to correct it. We have
been told that Bode's law fails with Neptune and Pluto, but
predicts the other orbits pretty well. I will now tell you
exactly why Bode's law fails with Neptune and Pluto.
Bode's
law fails with Neptune and Pluto because Bode's law is in the
wrong form. It mimics the right progression by a sort of
accident. This is the only possible way that Alan Boss can be
seen as correct. Bode's law, as written, is nearly correct only
by accident. It is a mathematical coincidence that it follows the
right progression for the inner planets. Titius and Bode and the
rest just matched a simple math to the data, without any
mechanics, and their math is not complex enough to fit the real
mechanics. It wasn't even transparent enough for us to see that
it was coming straight out of the orbital equation a =v^{2}/r.
I have solved that problem, but I still have to show that,
although my corrected “law” is superior to Bode's, my
failures can be corrected. Yes, the greatest difference between
my law and Bode's is that I can show you why the planets outside
Jupiter fail to fit the pattern. I will give you the pattern,
show you the variance, then show you the correction to the
variance, solving the problem completely.
In the current
Bode list, Jupiter's number is predicted to be 52 and Saturn's is
102. That would be expressed by me as 36√2 and 72√2.
Unfortunately, Saturn doesn't fit my pattern. In my pattern,
Saturn should be (36 + 5^{2})√2
= 61√2. Then Uranus would be (61 + 6^{2})
= 97√2 and Neptune would be (97 + 7^{2})
= 146√2. Or, if we take Mercury as 1:
1 (5/4)√2 (7/4)√2
(11/4)√2
(20/4)√2 (36/4)√2 (61/4)√2 (97/4)√2 (146/4)√2 (210/4)√2
As a first approximation, that would be the series I
would predict with my math. But my series diverges from Bode's
series at Saturn. It fails at Saturn rather than Neptune. Why?
Before I tell you, let me show you that my series already
matches the data as well or better than Bode's series. I predict
an orbit for Venus of 1.024 x 10^{8}.
The actual orbit is 1.075 x 10^{8}.
An error of 4.77%. Bode's law predicts an orbit for Venus of
1.034 x 10^{8},
an error of 3.8%. I predict an orbit for the Earth of 1.433, an
error of 4.2%. Bode's law predicts an orbit for the Earth of
1.478, an error of 1.2%. I predict an orbit for Mars of 22.52, an
error of 1.18%. Bode's law predicts an orbit for Mars of 23.64,
an error of 3.73%. I predict an orbit for Ceres of 4.095, an
error of 1.04%. Bode's law predicts 41.37, an error of .02%. I
predict an orbit for Jupiter of 73.7, an error of 5.32%. Bode's
law predicts 76.83, an error of 1.3%.
My average error
for the first six planets is 2.75%. Bode's average error is 2.1%.
For Saturn I predict 124.9, an error of 12.85%. Bode
predicts 147.8, an error of 3.11%. For Uranus, I predict 198.6,
an error of 30.97%. Bode predicts 289.6, an error of .66%. For
Neptune, I predict 298.9, an error of 33.6%. Bode predicts 573.3,
an error of 27.3%. For Pluto, I predict 430, an error of 27.2%.
Bode predicts 1,141, an error of 93%.
So, for Saturn and
Uranus and Neptune, Bode is better. For Pluto, I am better. My
average error for the four outer planets is 26%. Bode's average
error is 31%. But my errors are grouped much better, since they
go from 13 to 34, a deviation of 10.5 points from my mean. Bode
has a deviation of 46 points. I have no complete misses, while
Bode's prediction for Pluto can be called a complete miss.
Just
as a matter of statistics, my equations for the ten planets are
better than Bode's. My average error is 12.1%, while Bode's is
15.2%; and my deviation is much less. Even so, I admit that my
correction is not completely convincing at this stage. A 33%
error on Neptune is not impressive, for instance, unless I can
show the cause of the error. So I will now show how my errors can
be taken down to almost nothing, with simple mechanics. The first
thing to notice is that Jupiter causes the predictions to fail.
With Bode's law, this was not the case. Bode's predictions are
just as good from Jupiter to Uranus as they are for the planets
inside Jupiter. Bode's prediction for Saturn is very good, and
for Uranus it is astonishingly good. But my predictions go from
an average error of 2.7% inside Jupiter to an average of 26%
outside. This will prove to be a plus for my series, since
Jupiter IS the physical cause of this variance. Bode should
have found a variance beyond Jupiter, and he didn't. Beyond that,
I will show that Bode's match on Uranus was just a fluke.
Actually, you can see the variance beyond Jupiter even
without doing any math. A passing glance at the Solar System
would tell you that Jupiter is a dividing line. Inside Jupiter,
most things are caused by the Sun. Outside Jupiter, most things
are caused by Jupiter. The fields outside Jupiter cannot be the
same as those inside.
This became ever clearer to me in
my recent papers on axial
tilt, where I did the charge field calculations for the outer
planets. These four planets determine the unified field
variations in the entire System, and cause the bulk of the tilts,
both inside and outside Jupiter's influence. And, due to its
mass, Jupiter's charge influence is crucial.
If we look
at the four Jovians and list their charge field densities with
Uranus as 1, we get Neptune as 1.523, Saturn as 3.544, and
Jupiter as 22.84. The charge fields of all the other planets are
negligible compared to those four (although Pluto will play a
small part in the solution below). The charge field plays only a
small part in the unified field inside Jupiter, which is why my
predicted numbers work until we are past Jupiter. The outer
planets all perturb eachother strongly, as a matter of charge,
and so we shouldn't find orbital distances that can be predicted
without the charge component of the Unified Field. What is the
easiest way to show this, with the simplest math?
We must
first look again at my errors for the Jovians in my predictions,
above. What would it take to prove my errors were not accidents?
My errors for the Jovians are 5.32, 12.85, 30.97, and 33.6. If my
theory is correct, then the charge perturbations among the
Jovians should be shown to follow that sequence of numbers. In
other words, if I can show that the charge field interactions of
those four big planets cause
those errors, I can dissolve those errors, solving this problem.
In that case, the orbital distance will be shown to be caused by
two fundamental factors, not just one. The first factor is the
factor I showed above, where each planet orbits the Sun and all
the planets below it. The second factor is the charge interaction
between planets, which adds a variance to the first factor.
As
I have shown in previous papers, the strength of the charge field
of a body relative to surrounding bodies can be calculated by
multiplying the mass of the body times its density. This is
because we seek a charge density, and charge
is dimensionally the same as mass. This is one of the hidden
secrets of physics: the statcoulomb reduces to the same
dimensions as mass, and the Coulomb is just mass per second. I
say that this gives us a relative density, because by multiplying
mass and density, we can get the charge field strength of one
body as a percentage of another body. But we cannot find an
absolute amount of charge that way. To simplify all calculations,
I will find charge effects of one body on another by using
relative numbers instead of absolute numbers. By "relative,"
I just mean one body relative to another. If one body has ten
times the mass of other, I will use the numbers 1 and 10, instead
of the absolute masses.
The only other thing you need to
know is that charge moving out from the Sun acts differently than
charge moving in toward the Sun. The Sun sets the main field
lines in the charge field, and the planets simply inhabit that
main charge field. Any charge that moves between planets must
move on those preset charge lines. In short, charge loses
density as it moves out from the Sun and gains density as it
moves in. Because charge moves in the unified field, it will lose
charge on the way out by the inverse quad (fourth power) while
gaining charge as a straight function of distance on the way in.
So if charge moves from Jupiter to Saturn, say, it falls by the
quad. If it moves from Saturn to Jupiter, it gains as a function
of distance (neither squared nor quad). Although this may seem
counterintuitive at first, I explain the full mechanics of it in
previous papers, and have already confirmed it in my paper on
axial tilt—which uses similar equations to the ones
here.
We will look at Saturn first. We will calculate all
the charge perturbations on Saturn and add them up, to see how
much Saturn is pushed from its predicted orbit. Jupiter has a
charge field (mass times density) that is 6.445 times greater
than that of Saturn. To find the potential difference between
them, we can let the charge fields meet at Saturn. If Saturn's
charge field is 1 at its own surface, Jupiter's charge at
Saturn's surface is 6.445^{1/4}
= 1.593. Since they meet there in vector opposition, we subtract
them, and the result is .593. We then apply that potential
difference in both directions, so the variance at Jupiter is
1.593. But since Saturn is smaller, it will feel a greater force
from the same charge perturbation. It will feel 6.445 x .593 =
3.822. Therefore, Saturn feels 3.822/1.593 = 2.399 times the
variance of Jupiter. That is to say, Saturn is perturbed by the
charge of Jupiter 2.399 times as much as Jupiter is perturbed by
Saturn.
You may need to read that preceeding paragraph
several times. You must understand the field mechanics there, as
well as the math, to understand anything that follows.
It
would be better if you understood the mechanics in that last
paragraph, but if your head is spinning, maybe I can simplify it
for you. As you see, we found at the end that Saturn is perturbed
by the charge of Jupiter 2.399 times as much as Jupiter is
perturbed by Saturn. Could we derive that number more directly?
Well, yes. We can estimate
it straight from mass and surface area, without having to look at
field separations at all. The mass differential between Jupiter
and Saturn is 3.34. The surface area differential is 1.456.
Dividing the first by the second gives us 2.293, which is close
to 2.399. If Jupiter and Saturn existed at the same orbital
distance from the Sun, that would be our perturbation number, you
see. But we need a little more math and mechanics when the two
planets are at different distances from the Sun. We need to take
into account that this perturbation between Jupiter and Saturn is
taking place in a preexisting charge field, where the main lines
are set by the Sun. We need to take into account the fact that
Jupiter's charge is moving out from the Sun and Saturn's charge
is moving in toward the Sun. This is what makes this a multibody
problem, as well as making it a unified field problem. You may
think my math here is difficult, but I encourage you to compare
it to mainstream or historical multibody problems. Not only is
it a huge simplification, but I walk you through every step,
telling you why it is necessary as a matter of mechanics. What
other mathematician in history has done that?
Uranus and
Neptune will also perturb Saturn, but their effects are much
smaller. I will calculate Uranus' variance on Saturn to show
this. Saturn has 3.544 more charge density than Uranus, so if
Uranus' charge is 1, Saturn's is 1.372. But if Saturn's field is
1 (to match its value in the above paragraph), then Uranus' is
.729. Uranus is 2.2 times further away from Saturn than Jupiter
is, so Saturn will feel a variance of only .729/2.2^{4}
= .0311. We can add that to the variance from Jupiter, obtaining
3.822 + .0311 = 3.8531.
Now Neptune. Saturn has 2.328
times more charge than Neptune, so if Neptune's charge is 1,
Saturn's is 1.235. But Neptune is 4.69 times as far away from
Saturn as Jupiter is, so Saturn will feel a variance of only
.81/4.69^{4} =
.00167. We add that to the other variances, 3.8531 + .00167 =
3.855.
Saturn has 14,743 times as much charge as Pluto.
So if Pluto's charge is 1, Saturn's is 11.02. But Pluto is 6.84
times further away than Jupiter, so that force drops to (1/11.02)
x (1/6.84^{4}) =
.0000415. This brings the total variance to 3.855.
Now
let us compare that total to the total for Jupiter. We have
already calculated the variance on Jupiter from Saturn. But we
have to add the variances on Jupiter from Uranus and Neptune.
Uranus has 3.544 times less charge than Saturn and is 3.2 times
further away. Which gives us an extra variance of .00269. Neptune
has 2.327 times less charge than Saturn and is 5.688 times
further away. So an extra variance of .000411. Adding those to
1.593 gives us 1.596. Then, 3.855/1.596 = 2.4154. Saturn has
2.4154 times the
variance of Jupiter.
To see if this has filled our margin
of error, we consult the earlier numbers. My error for Jupiter
was 5.32%. My error for Saturn was 12.85%. That is a ratio of
12.85/5.32 = 2.4154.
If we compare the two bolded numbers, we find a perfect match. My
theory has been born out by the numbers.
Before we move
on to find the variance for the other planets, let us pause to
show why Jupiter's variance is 5.32%. I need to show that in
order to finish my proof. I need it because showing Saturn's
relative variance is not enough. Saturn's number depends on
Jupiter's, so I need to prove Jupiter's number in order to set my
baseline for the four big planets. The math is very simple and
can be shown in only a few lines. We already know that Jupiter's
relative charge density is 22.84. We compare that to all the
charges from the Jovians as they exist at
the distance of Jupiter. We can estimate
this by looking only at charge from Saturn, since the other
charges are almost negligible. 22.84 + 3.544^{1/4}
= 24.2. Jupiter's own charge is 94.4% of that total charge,
leaving a difference of 5.6%. That is (roughly) the cause of
Jupiter's variance. Jupiter is mainly perturbed by Saturn.
As
Jupiter is pulled higher, all the Jovians are pulled higher, as a
group. Jupiter sets the baseline and the other Jovians follow.
This is precisely what my math shows. I was able to dissolve the
entire error for Saturn because I found the variance relative to
Jupiter.
Now let us look at the variances on Uranus. If
Uranus' charge is 1, Saturn's charge at Uranus is 3.544^{1/4}
= 1.372. Uranus will feel 3.544 x .372 = 1.318 from Saturn.
Now
Neptune's variance on Uranus. Neptune is larger than Uranus and
outside it, so our previous math is difficult to apply. We will
solve by an easier math. Neptune's charge is 2.327 times less
than Saturn's, so Neptune's variance on Uranus is also 2.327
times less. Saturn's variance on Uranus was 1.318, so Neptune's
is 1.318/2.327 = .5664. But Neptune is 1.128 times further away,
so the variance drops to .5664/1.128^{4}
= .350. Because Neptune is larger and outside, this variance is
negative: .35.
Now Jupiter's variance on Uranus: Jupiter
has 22.84 times as much charge as Uranus, so if Uranus has a
charge of 1, Jupiter has a charge of 2.186. Uranus' variance
relative to Jupiter would be 22.84 x 1.186 = 27.09. But Uranus is
1.454 times further away from Jupiter than from Saturn, so Uranus
only feels a force of 27.09/1.454^{4}
= 6.061.
And, finally, Pluto's variance on Uranus. Uranus
has 4,160 times more charge than Pluto, so if Pluto's charge is
1, Uranus' is 8.03. But Pluto is 2.1 times farther from Uranus
than Saturn is. So, (1/7.03) x (1/2.1^{4})
= .0323.
Adding the four variances gives us 7.061. But now
we have to scale Uranus' variance up to Saturn's. In the
equations for Saturn above, we let Saturn equal 1; here we let
Uranus equal 1. To do this we simply use the variance between
Saturn and Uranus, from above. It was 1.318, so 7.061 x 1.318 =
9.307. Uranus' relative variance is 9.307 and Saturn's was 3.855.
Dividing, we get 2.4143.
So we return to our prediction errors above. The error
for Saturn was 12.85% and for Uranus 30.97%. The ratio is 2.4101.
The bolded numbers again show a very good match. Here we have an
error of .00173. We have brought our Bode series error down from
30.97% to .173%.
Not only have I solved the problem I set
out to solve, I have found another problem to solve later. Notice
we have the same number between Saturn and Uranus as we had
between Jupiter and Saturn. 2.4143 here and 2.4154 above. That
cannot be a coincidence. We will look at those two variances more
closely in an upcoming paper.
Also, because I have shown
that the charge field fills the gap between prediction and data,
Bode's raw prediction for Uranus must have been a fluke. Bode's
prediction was only .66% wrong, which looked impressive until I
showed how the Unified Field caused the orbital distance
mechanically. Bode's series is based on a straight extension of
the equation a=v^{2}/r,
and I have just shown that cannot work by itself. It cannot work
because it ignores the E/M field entirely. Since Bode's original
math contains no mechanics or mechanical postulates, it must have
achieved the correct number by luck.
Now, the set of
equations for Neptune. We will start with Pluto, as the nearest
planet. Neptune has 6,334 times more charge than Pluto, so if
Neptune's charge is 1, Pluto's is .000158.
Jupiter has 15
times more charge than Neptune, so if Neptune's charge is 1,
Jupiter's is 1.968. So the variance on Neptune is 15 x .968 =
14.52. But Jupiter is 2.647 times further from Neptune than Pluto
is, so we find 14.52/2.647^{4}
= .2956.
Saturn has 2.327 times more charge than Neptune,
so if Neptune's charge is 1, Saturn's is 1.235. The variance on
Neptune is 2.327 x .235 = .547. But Saturn is 2.18 times further
away from Neptune than Pluto is, so its variance drops to
.547/2.18^{4} =
.0241.
Again, rather than compare Neptune to Uranus, we
will solve the perturbation between them by comparing Uranus to
Saturn. We just found Saturn's raw variance on Neptune to be
.547. Since Uranus has 3.544 times less charge than Saturn, it's
variance upon Neptune must be .547/3.544 = .1543. But Uranus is
1.159 times farther away than Pluto, so .1543/1.159^{4}
= .0855. And again, this variance is negative: .0855.
Add
them all up to get .2344. Once again, we have to scale Neptune's
variance up to Saturn's. Saturn is 1.887 times farther from
Neptune than from Uranus, so we can develop the transform like
this: 3.544 x 6.445 x 1.887 x .2344 = 10.1. That is Neptune's
scaled variance. We compare that to Uranus' scaled variance,
which was 9.307. Dividing gives us 1.0855.
According to my series errors, my error for Uranus was
30.97%; and for Neptune, 33.6%. Therefore, Neptune should have
been perturbed 1.0849
times as much as Uranus. The two bolded numbers are a near match
again. The error is .000533. My method has successfully dissolved
the series errors for the Jovians, using very simple math. I
think we can confirm that the charge field fills the variances
extremely well.
Now let us look at the variance on the
Earth. Notice that we are not calculating the Earth as a
percentage of any other planet, as we were doing with the math of
the Jovians. Instead, we are looking for an absolute motion in
the field, as we did with Jupiter above. The Earth has 13.05
times the charge of Mars, so if Mars' charge is 1, the Earth's
variance from Mars is ^{4}√13.05
= 1.9006. The Earth has 1.3 times the charge of Venus, so the
variance from Venus is ^{4}√1.3
= 1.0679. Venus is smaller and lower, so it will pull the Earth
lower, as Uranus does with Neptune. This makes Venus' number
negative. Now the variance from Jupiter. Since Jupiter is larger
and higher, its variance on the Earth will also be negative. We
compare Jupiter's variance on the Earth to Mar's variance on the
Earth. Jupiter has 997 times as much charge, but it is 8.14 times
as far away. 997/8.143^{4}
= .2268. Since Mars' charge was the baseline 1 in this paragraph,
Jupiter's variance is just 1 x .2268. Mercury will also give us a
negative variance. Mercury has 1/14.14 times the charge of Venus
and is 2.213 times farther away from the Earth. That is .00295.
If Mars is 1, Venus is 10.034, so we multiply by ten. The
variance from Mercury is then .0296. We will also include Saturn,
for good measure. Saturn has 154.7 times the charge of Mars and
is 16.64 times farther away. 154.7/16.64^{4}
= .00202. Again, negative, and not completely negligible. Add
them all up. 1.9006  1.0679  .2268  .0296  .00202 = .5743. We
have given the Earth a charge of 13.05 in this math, so we
compare 13.05 to .5743. The total charge at the Earth of the
Earth plus variances is 13.05 + .5743 = 13.624. 13.05/13.624 =
.9578. 1  .9578 = .04215. Or 4.215%. The error from my Bode
series was 4.2%, so we are very close.
Now to
answer some important questions about the math and mechanics. You
will say, “How can Saturn pull Jupiter higher? I thought
you said we didn't have any attractions in your unified field?”
Good question. As you can see from my math, I am calculating
charge differentials. Using these, we find that planets are
pushed higher either by larger planets below or smaller planets
above. Conversely, planets are pushed lower by larger planets
above or smaller planets below. With this simple rule, we see
that Jupiter will push Saturn higher, and Saturn will also push
Jupiter higher. They both go higher. But why is that, as a matter
of mechanics? Haven't I said that charge is a straight
bombardment, which would imply that bodies can only repel
eachother via charge? Well, yes, in the simplest case, that is
true. If two bodies aren't already in the field of a third larger
body, that is true. They could only repel eachother. But that
isn't the situation we have here, is it? To look at Jupiter and
Saturn, we must be aware that they exist in the greater field of
the Sun at all times.
I did not calculate absolute charge
above, as I have already admitted. I calculated relative charge,
or a charge differential. I calculated the four Jovians relative
to eachother, leaving the Sun out of it. But if we want to find a
motion in the field relative to the Sun (inward or outward), we
have to bring the Sun back into the question. In the closed
system of Jupiter/Saturn, for example, I found that we have “more
charge out.” It is clear why that would move Saturn out.
But the charge is not just moving out at Saturn, it is moving out
across the whole system. Charge is moving out at Jupiter as well.
If we put that closed Jupiter/Saturn system into the greater
system of the Sun, then charge will be moving out in the entire
vicinity. Even charge below Jupiter will be moving out, because
our J/S system has created a low pressure across the entire J/S
system.
It is very useful to think of charge like wind,
creating low pressure and high pressure. This analogy is much
more useful than the idea of potential or pluses and minuses, in
my opinion, because you can visualize the field blowing from one
place to another. This isn't just a metaphor, it is what is
physically happening. The charge field is a very fine particulate
wind, and it moves from high density to low density. Density and
pressure are the same thing, in this regard. So the charge wind
is blowing Jupiter just as much as it is blowing Saturn, and it
is blowing in the same direction. For this reason, they both go
higher, even without any real pull.
You will say, "Since
you mention that axial tilt paper, why is your math different
there than here? In
those tilt papers you let the outer planet increase as the
charge goes in. Here you don't. For instance, you keep Saturn at
1 in the equations, and take the fourth root of Jupiter's charge.
In the tilt paper you increase Saturn's charge by the distance."
Another good question. All you have to do is notice that
in the tilt papers I am calculating perturbations against the
Sun. Here I am calculating the perturbations against eachother.
There, I have three bodies, with one in the middle. Here, I have
two bodies, and no middle. So although the math looks similar, it
isn't really the same. I use the same relative numbers and the
same charge field mechanics, but here the numbers are relative to
eachother and there the numbers are relative to the Sun.
Conclusion: I have completely
solved Bode's law, and I think we are due for a name change for
that law. Both Titius and Bode failed rather spectacularly to
apply the right equation. The solution is rather simple, as you
see, and there was no reason for this to sit in mothballs for 300
hundred years. Physicists couldn't look at it without scales on
their eyes, since they had bought the “gravity only”
interpretation. Laplace “solved” the perturbation
equations 230 years ago, and no one has had the gumption to look
closely at them since then. Mathematicians failed to solve this,
too, and we may assume it is because they got deflected in about
1820, or 190 years ago, by new maths. They weren't interested in
simple algebra like I do here: they wanted to use curved fields
and infinities and complex numbers and quaternions and lord knows
what else. Actually solving a simple problem of mechanics was
beneath them. It really makes you wonder how anything ever gets
done.
In physics and math, nothing much does get done, as
I have shown. The history of physics and math has not been a
wonderland of brilliance and fast progression; it has been a
shocking wasteland of deflection, misdirection, and complete
incompetence, and it is only getting worse. I expect the response
to my papers to continue to be vicious, since there is nothing
more reactionary than a field of sinecures. It will be like
trying to overthrow the Aristotelians or the French Academy or
any other nest of nepotism and privilege and corruption. But they
had best put on their waders, because the water is high. I am
coming right at them, and I am used to deep currents.
Addendum:
many have asked why these charge photons have not been
discovered. My answer is that they
have. All the photons we already know about are part of the
charge field. The entire electromagnetic spectrum is the charge
field. We do not have to propose new photons, we can use the ones
we already have. I have given all photons mass and radius, so all
photons must cause mechanical forces by contact. This has long
been known (see the photoelectric effect) but not fully
interpreted.
You may now see the simple unified field
equations for the threebody problem (Sun, Earth, Moon) in my
newest paper on Lagrange
Points. Using the charge field, I am able to show that the
Moon is in fact hitting these points of field balance. This
proves my assertions in this paper once more. It also clarifies
the math I do here, since although it is the same field math, it
is even more stripped down and transparent there.
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