Wagner Research Laboratory |
The following article appeared in the March 1999 issue of Physics
Essays. This article presents a valid basis for the idea that the Universe is
self organizing which has been recognized by many. This article especially emphasizes
how the solar system was formed and how it remains stable. These problems have
never really been solved before but this article provides reasonable answers.
This paper was submitted to Physics Essays in May 1994. Much of the material
in this paper was first published in a book entitled W-Waves and a Wave Universe
in 1991. This article posted with permission from Physics Essays. Click
here for more about the journal Physics Essays.
ABSTRACT. The hypothesis that longitudinal standing
waves in the dark matter-vacuum medium are responsible for placing the planets
suggests an equation that describes the location of the orbits of the planets
and their satellites. This equation also suggests a wave velocity equation as
well as a solution to a wave equation. The equation also fits into a picture
where the wave velocity is proportional to the reciprocal of the square root
of the density of the medium, as with sound waves. It is postulated that the
density of dark matter outside the sun is proportional to the reciprocal of
the distance from the sun's center squared. Evidence for the hypothesis is presented,
a wave equation confirms the hypothesis, and implications are discussed.
KEY WORDS. Dark matter, standing waves, solar cycle,
planets, sun, satellites, universe, solar system, inertia.
1. INTRODUCTION
Presently there is an extensive search going on to identify
dark matter. Is it composed of WIMPS, or hot or cold dark matter, or ordinary
matter which cannot be seen readily, or is it something else? I quote Scott
Tremaine in Physics Today: "At present we must admit with some embarrassment
that we don't know what most of the universe is made of"^{1}.
It is proposed that the properties of dark matter may shed
light on the following outstanding problems in physics. I quote or summarize
from Physics Today articles:
The above material suggests that a model that explains more
about the complex organization and provides for long term stability of the solar
system (as well as the universe) and some of the unique aspects of life would
be welcome. For the past 11 years I have been publishing data and theory on
unique slow moving waves that I first identified in 1988. Perhaps these waves
provide important answers to some of the problems of the solar system and the
universe that have not been satisfactorily explained. What I am about to present
appears to complement rather than supplant the known mechanisms for solar system
development and organization.
2. EARLY DEVELOPMENTS
In 1988 I first reported finding slow moving longitudinal waves
in plants (velocities near 1 m/s), in ion filled porous materials, and in the
space surrounding these materials^{4-13}. I called the waves W-waves
because they were first found in live wood. Standing waves in solid matter appeared
to move charge something like sound moves dust in a Kundt's tube. The variations
in charge density were indicated by periodic potential differences as high as
one volt. Since charge is one of the few things free to move in solid matter
this was considered more of a mechanical rather than an electrical effect but
this permits observation of wave effects in some solids. The vertical and horizontal
wave velocities in plants appear to be different providing a reference for the
gravitropism of plants and perhaps indicating a cosmological connection. The
special time of flight methods used to measure W-wave velocities are described
in a book published in 1995 (p.19) and in a later paper^{9,13}. I published
experimental results, demonstrating that these waves exist, from analysis of
charge organization and other features in both plants and ion filled porous
materials. The experiments indicate that the reported waves appear to be an
all pervading major organizing factor for plants, perhaps all life, and perhaps
the solar system as well.
Materials and plants were placed in closed heavy gauge aluminum
shields (and in a mine 300 m underground) with probes to monitor voltages. The
probe output was fed to low noise, high gain, battery operated amplifiers contained
within the shields and the amplifier outputs were monitored on an external low
frequency spectrum analyzer. The typical plant spectra observed contained the
most usually observed above ground spectra with integral multiples of 1.6 Hz
(as well as other typical harmonic series like integral multiples of 0.6 &
2.666.. Hz). It is unknown, as yet, what produces these unique frequencies which
I designate eigen frequencies.
The same dominant frequencies were also found by measuring
many thousands of typical spacings between adjacent plant structures (internodal
spacings). These spacings were assumed to be half wavelengths of standing wave
patterns. The spacings were then converted to frequency by using experimentally
determined wave velocities. Distributions were taken which demonstrated that
plant spacings are quantized with the same unique frequencies apparent as measured
with the spectrum analyzer. Vertical velocities were found to be usually larger
than horizontal velocities in plant material which apparently provides a reference
for the plant's response to gravity. Ratios of vertical to horizontal velocities
were confirmed by comparing averages derived from plant internodal spacings^{12,13}
as well as by direct measurement.
Recent experiments seem to suggest that electromagnetic sources
excite a unique set of slow moving wave modes in the surrounding medium. W-waves
may be produced by electromagnetic interaction with the dark matter-vacuum medium
which may include ordinary matter. Excited W-wave frequencies apparently are
characterized more by the medium rather than the electromagnetic source. Some
forms of matter such as live plants appear to be wave guides for W-waves. The
modes excited electromagnetically seem to be largely independent of the electromagnetic
exciting frequency. 60 Hz, 26 khz, 400 khz, 1270 khz, and other sources have
been tested at this laboratory. W-waves also are likely excited by other forms
of energy such as those found in the sun.
Some have suggested that one might describe W-waves by Maxwell's
equations. Maxwell's equations resulted from the observation of the macroscopic
behavior of pure electromagnetic waves and probably would not apply to W-waves.
It is the author's opinion that Maxwell's equations do not represent ultimate
fundamental physics but only what humans have observed so far. I place W-waves
in a class with inertia which arises from the vacuum and humans generally only
speculate about. The vacuum characteristics are also probably relevant to W-waves
so I talk about the dark matter-vacuum medium. There seems to be quantum like
behavior associated with W-waves.
It was found that the distributions of plant internodal spacings
seem to be different when grown in the presence of certain magnitudes of 60
Hz electromagnetic fields arising from an electric power substation (See chapter
nine of the book Waves in Dark Matter^{9}). A semiconductor detector,
in the vicinity of electronic equipment operated at 60 Hz, driving a low frequency
spectrum analyzer usually indicates a large amplitude presence of typical eigen
frequencies including 26.7 Hz and 80 Hz for example. These frequencies as well
as 60 and 120 Hz are also found in the spacing spectra of plants (derived from
plant internodal spacings) far away from 60 Hz sources and in fossil plants.
The latter peaks are usually of small amplitude (the author analyzed thousands
of spacings from fossil plants in the fall of 1989. Most of this work remains
unpublished.)
If one floats materials (like styrofoam particles or plastic
beads) on water in the presence of electromagnetic sources it appears that the
floatant tends to collect in concentric equally spaced circles (at nodes apparently)
around an approximately cylindrical source like an operating vacuum tube. A
6L6 vacuum tube oscillating at 400 Khz at about 30 watts produced 9 cm spaced
circles. 9 cm is the half wavelength for 26.7 Hz suggesting that the waves causing
the effect are traveling in air at 480 cm/s as was found earlier^{5}.
Is it possible that the waves travel, perhaps in some cases, at the same velocity
in both air and water since later experiments suggest that multiple velocities
are possible^{13}?
Different sources produce somewhat different dominant wavelengths.
It was possible to find similar spacings, with an air medium, using a semiconductor
detector attached to an optical table moving toward or away from a vacuum tube
source. A mechanical method also showed that the circles seemed to be produced
by forces pushing from both directions to keep an instrument on location.
3. EFFECTS OF W-WAVES ON SPACE
Experiments like the latter may indicate that energy is being
absorbed from electromagnetic waves to produce the observed effects in the surrounding
space. The produced wavelengths, frequencies, and velocity are much different
than the corresponding quantities for the producing sources. The resultant wavelengths,
frequencies, and velocity correspond to similar quantities in plants. The experiments
mentioned could lead to tests proving that the cosmic red shift is due to photon
energy degradation rather than the big bang.
I now assume that many bodies, if not all, in the universe
oscillate with the slow moving waves discussed. I assume W-waves from an oscillating
sun penetrate everything in the solar system and the absorbed energy oscillates
in everything. Dark matter and the vacuum may be the basic medium permeating
every material object and "empty space". Ordinary matter seems to have large
effects on the W-waves traveling through it.
Directly observable wave effects are seen on Saturn. When Voyager
took photographs of the North pole of Saturn a hexagonal pattern in a jet indicates
standing waves. Calculations suggest that the frequency involved is a harmonic
of a fundamental oscillating frequency of the planet. The stability of the zonal
jets on Jupiter and Saturn may also be partly due to standing wave effects^{14}
from oscillating slow moving modes of the planet and its surface.
Physicists on both the West and East coast have found charge
organization in materials such as rocks (petro-voltages) in constant environment
shielded containers but left the data unpublished (except for internal reports
which I obtained) because of the lack of a theory. Often coincident pulses or
other unique coincident wave forms were outputted simultaneously from shielded
materials at different locations. It is well known that plant rhythms don't
change whether they are growing on the earth's surface or deep underground under
a steady light. If plants are operated by waves, as I find, then these waves
penetrate matter everywhere. All the findings thus far seem to indicate all
pervading wave action. I reasoned that the solar cycle might be produced by
oscillations of similar waves^{9,10}. If the solar cycle is produced
by this species of waves, then they also are also very slow moving waves in
the sun's matter (near 1 m/s) as found in plants and elsewhere.
It may be that many objects in the universe oscillating with
these slow moving waves are surrounded with standing waves that in the case
of the solar system, for example, had something to do with the placement of
the planets and the satellites and rings of the planets. A simple equation was
found by linear regression and experiment that describes the placement of the
planets^{9,10}:
(1)
where r is the distance from the center of the sun, r_{0}
is the sun's effective radius (which depends on a star's composition when the
satellites are placed). N is an integer for a particular planet. Notice that
equation (1) can also be written as 1.6pln(r/r_{0})=Np.
Using a 5 % larger radius then the present sun's radius for r_{0} (7.3
x 105 km) and N=7 and 8 give the radii of Mercury and Venus's orbits almost
exactly, and Neptune's orbit (N=14), within 2.4 %, considering that they had
circular orbits initially. Probably collisions and other disturbances changed
most of the orbits from circular to elliptical while they were forming or afterward.
All the other planet orbital radii including that for Ceres (using mean value
circular orbits) then fit within 14 percent or better (except for Pluto and
Earth) using N= 9, 10, 11, 12, 13, 14, and 15. The locations of satellites of
planets are also described very well by this equation with proper choices for
r_{0}^{9,10} (see Tables I, II, and III and the discussion section
and other publications^{9} for reasons for variations). Others have
found more complex equations that work for the sun's planets but none that work
for satellites of planets (from unpublished work). The idea that equation (1)
is so simple makes it attractive.
4. A WAVE EQUATION
If equation (1) holds it implies that the velocity of the proposed
waves increases as they move away from the sun. The following velocity equation
is suggested by equation (1):
(2)
where v is the velocity and v_{0} is a constant (on
the sun the radial W-wave velocity at the surface). The type of behavior hypothesized
here suggests that longitudinal waves are involved and that perhaps standing
waves are produced by reflections due to density and velocity gradients, for
example. There is the possibility that the observed phenomena could be due to
some other minimum energy phenomena that is not recognized yet.
Several equations follow from equations (1) and (2). For example
if equations (1) and (2) are correct one can solve for N from (1) and substitute
it into (2) with the result that:
(3)
This simple result can now be substituted for v in the following
simple wave equation:
Equation (1) gives the location of nodes (at r=r_{0}
or greater) so it follows that the locations are also described by sin(1.6pln(r/r_{0}))
with the zeros of the function located on nodes. The latter suggests a standing
wave like solution for (4). We now assume a solution for (4)(using spherical
co-ordinates) of the form F=F_{0}f(r)e^{iwt}
and substitute it into (4) with w=w_{0}.
The resultant differential equation is then solved for f(r). A solution is f(r)=C_{1}r^{-1/2}
sin(r_{0}w_{0}/v_{0} ln r/r0)
for r>r_{0}. C_{1} is a constant. f(r) has been simplified
somewhat from the original solution (see the appendix). Notice that if r_{0}w_{0}/v_{0}
is equated to 1.6p (from above; note that 1/0.625=1.6),
v_{0} comes out to be 1.25 m/s using the mean solar cycle period to
determine w_{0}. In the past I have usually
used approximately 1 m/s as the mean velocity of W-waves traveling radially
in the sun. The 1.25 m/s is also calculated using an integration in another
source on page 91^{9}.
Since oscillating bodies are being considered it seems appropriate
to think of r_{0} as a wavelength or half wavelength. The average velocity
of the waves in planets is assumed to be approximately proportional to the reciprocal
of the square root of the planet's mean density. Equation (1) is assumed to
hold when satellites or planets are placed. Subsequent changes in radius may
have occurred with large temperature changes, with added mass, etc. Satellites
may have been added as mass was being added to a planet. Several possible scenarios
are discussed elsewhwere^{9}. If a planet or star vary in mean density
the wave medium outside the planet or star would also vary similarly in density
because of the different total masses. For a given diameter a less dense planet
would oscillate at a higher frequency than a more dense planet. Less dense bodies
(for example Saturn is less dense than Jupiter) hold less of the surrounding
dark matter medium with a resulting larger velocity for the waves traveling
within and to and from the body. Thus equation (1) holds for both Jupiter and
Saturn but Saturn is about one half as dense as Jupiter. The satellite locations
seem to indicate that both Jupiter and Saturn were close to the same radius
(approx. 80,000 km) when they were very hot and most of their satellites were
placed^{9}.
An approximate expression for the frequency of oscillation
of the sun and other bodies apparently is: 1.19/(r_{0}Öd)
where d is the relative mean density (with water as 1.0) (note that the sun's
radius in meters is just equal in magnitude to 22.2 years (the apparent solar
cycle period) in seconds). 1.19 is the square root of the sun's mean density.
I applied this expression to about 50 main sequence stars, using actual radii
for r_{0}, with considerable variation in radius and relative mean density.
The resulting values were nearly the same for the oscillation frequencies of
all the stars used in the calculations. Two departures from the same value may
be explainable in terms of changes in the wave velocity function as a function
of star density. Note that others have used the reciprocal of the square root
of the mean density in discussing variable star oscillations^{15}. The
relative mean density of the sun is 1.41, of Jupiter 1.33, and of Saturn 0.69.
5. DARK MATTER DENSITY FUNCTION
If one assumes that the hypothesized waves behave like sound
waves in ordinary matter then the velocity of the waves varies inversely as
the square root of the density of the medium. Setting equation (2) equal to
the reciprocal of the square root of the density (multiplied by a constant and
using N=1.6 ln r/r_{0} from equation (1)) yields the following for the
density function:
(5)
where d is the density and C is a constant. The density of
dark matter in the solar system apparently is so small that it doesn't appear
to have any consequence as far as most classical physics is concerned but again
we may be dealing with something perhaps as elusive as inertia, for example,
which is a very large effect.
The density function (equation (5)) is just the density function
one can derive simply or others have reported for dark matter around galaxies(for
r much larger than r_{0}). This is concluded because outer stars move
at constant velocity in orbits about the centers of galaxies. The density of
dark matter, in the outer periphery of the Milky Way, for example, apparently
varies as the reciprocal of the square of the distance from the galactic center
for large r_{0}^{1,16}. It should be pointed out that the actual
distribution of dark matter around the sun is unknown. Equation (1) may not
hold for very large N and r but I would still expect W-wave velocities to become
very large far away from ordinary matter.
6. DISCUSSION
In the past Bode's law has been found useful in locating the
planets (see astronomy and elementary physics texts). For Bode's law the series
0, 3, 6, 12, 24, 48, 96, 192, 384, and 768 is used to initiate the calculation.
Next 4 is added to the numbers giving 4, 7, 10, 16, 28, 52, 100, 196, 388, and
772. If earth's distance from the sun is 1 AU this can be used for the 10 in
the series. Then Mercury is at 3.8, Venus 7.2, Earth 10, Mars 15.2, Ceres is
at 27, Jupiter is at 52, Saturn is at 95.5, Uranus is at 192, Neptune is at
301, and Pluto is at 396 (note that equation (1) does much better than Bode's
law for Mercury, Venus, Neptune, and Pluto). One can see that there is a reasonable
fit in Bode's formula for the first 8 planets. Bode's empirical recipe was found
to work reasonably for these planets but the results lead nowhere because no
theory is suggested. The formula helped locate outer planets early in the development
of astronomy, however. Bode's law suggests a rapidly increasing velocity wave
function as the waves get farther from the sun if waves are involved in planet
placement.
Equation (1) not only works well to locate the planets, taking
into account early circumstellar matter^{9} which would tend to decrease
the wave velocity, but it works well to locate the satellites of gaseous planets
as well (e.g. see Tables I, II, and III). We do not know the initial conditions
when the satellites of planets were positioned but if equation (1) is correct
a reasonable early solar system history results^{9}. Equation (1) does
not describe the locations of all the sun's planets exactly perhaps because
matter was distributed around the solar system in a different manner when the
satellites and planets were placed. There probably was much circumstellar ordinary
matter still present.
Non-uniformly distributed matter (including dark matter) would
have a large effect on planet placement in a wave operated system. Equation
(1) does describe the locations of Mercury and Venus exactly because the solar
wind had cleared the region of the less dense matter when they were placed.
Wave amplitudes would also be larger closer to the sun source because of the
spread of the waves through an area increasing as r^{2}. The author
of reference 3 suggests that the accretion of volatiles on the giant planets
implies that they must have formed before the solar nebula was completely dissipated^{3}.
The presence of excess matter (increased density regions departing from the
distribution assumed by equation (1)) in space would tend to decrease the wave
velocity thus producing closer spacings. For example Mars and Earth may have
been gaseous planets in a residual circumstellar matter ring or disk whose density
varied with distance from the sun. One would expect the intermediate planets
to be affected most by primordial excess circumstellar matter because such matter
would most likely be located in their vicinity.
The special initial conditions permitted the formation of what
now appears to be an extra planet from equation (1), the earth (of course there
is also the possibility of a planet being injected into the system). When the
circumstellar matter between the planets finally dissipated it left Earth and
Mars more or less where they are today with other planets not fitting equation
(1) exactly (see Table I). I would still expect the system to be stable, however,
because the ever present standing wave pattern would tend to stabilize the system.
One cannot say at this time what the early matter distribution was. Assume an early effective radius (r_{0} of 7.30 x 10^{5} km since this seems to produce a rather good fit taking into account excess matter during placement.
Planet | Position (km) | Calculated Position (km) | N |
Mercury | 5.79 x 10^{7} | 5.80 x 10^{7} | 7 |
Venus | 1.08 x 10^{8} | 1.08 x 10^{8} | 8 |
Earth | 1.50 x 10^{8} | ||
2.02 x 10^{8} | 9 | ||
Mars | 2.28 x 10^{8} | (Effect of early excess matter? (see text)) | |
Ceres | 4.14 x 10^{8} | 3.78 x 10^{8} | 10 |
Jupiter | 7.78 x 10^{8} | 7.06 x 10^{8} | 11 |
Saturn | 1.43 x 10^{9} | 1.32 x 10^{9} | 12 |
Uranus | 2.87 x 10^{9} | 2.47 x 10^{9} | 13 |
Neptune | 4.50 x 10^{9} | 4.61 x 10^{9} | 14 |
Pluto | 5.90 x 10^{9} | 8.61 x 10^{9} | 15 |
Table I. A comparison of the actual
distances r of the planets from the sun's center with the calculated distances
using the orbit equation given above. N is an integer. Note that the middle
planets don't fit as well as the inner or outer planets as one might expect
if one takes into account the extra circumstellar matter that was present when
the planets were placed. Circumstellar matter would tend to reduce the wave
velocity and thus change the planet spacings (see the text).
Jupiter's Radius | 71,500 km | 83,700 km effective. |
(71500 km | ||
Orbit equation: | later radius) | |
r=r_{0}(exp(0.625N)) | Multiply by 1000 below. |
Satellite | Position | Calculated Position | N |
Metis | 128 | 134 (see text) | |
Adrastea | 129 | From later radius | |
Contact | DR. Wagner | for | reprint |
Amalthea | 182 | 156 | 1 |
Thebe | 222 | (250) | |
Contact | DR. Wagner | for | reprint |
Io | 422 | 292 | 2 |
(466) | |||
Europa | 671 | 546 | 3 |
Ganymeade | 1,070 | 1,020 | 4 |
Callisto | 1,880 | 1,905 | 5 |
Missing | 3,559 | 6 | |
Satellites? | 6,649 | 7 | |
Contact | DR. Wagner | for | reprint |
Leda | 11,094 | ||
Himalia | 11,480 | ||
Lysithea | 11,720 | 12,422 | 8 (Split orbit?) |
Elara | 11,737 | ||
Contact | DR. Wagner | for | reprint |
Ananke | 21,200 | ||
Carme | 22,600 | ||
Pasiphae | 23,500 | 23,208 | 9 |
Sinope | 23,700 |
Table II. A comparison of the actual
distances (r) of the satellites of Jupiter from the planet center with the derived
distance using the orbit equation above. Again, as with the sun, excess matter
(and other factors such as collisions) present when the satellites were placed
likely caused differences. It is likely that Jupiter was larger when the outer
satellites were placed but wave effects tended to keep the closest satellites
in wave orbits as the size decreased (or the inner satellites were placed after
the planet cooled). Thus I show placement using both the present radius and
a larger effective radius for the outer orbits (see the text).
Saturn's radius: 60,400 | 83,000 km effective |
(r_{0}) Multiply by 1000 below. |
Satellite | Position | Calculated Position | N |
Atlas | 138 | ||
Promethius | 139 | ||
Pandora | 142 | 155 | 1 |
Epimetheus | 151 | r=r_{0}(exp(0.625N)) | |
Janus | 151 | (orbit equation) | |
Mimas | 186 | ||
Enceladus | 238 | ||
Tethys | 295 | 290 | 2 |
Telesto | 295 | ||
Calypso | 295 | ||
Dione | 377 | ||
Helene | 377 | ||
Rhea | 527 | 541 | 3 |
Titan | 1,222 | 1,011 | 4 (Matter influence?) |
Hyperion | 1,481 | 1,889 | 5 |
Iapetus | 3,561 | 3,529 | 6 |
(Missing satellite?) | 6.593 | 7 | |
Phoebe | 12,952 | 12,318 | 8 |
Table III. A comparison of the actual distances of the
satellites of Saturn from Saturn's center with distances calculated using the
orbit equation given. Note that there are small differences but the presence
of other matter and catastrophic events such as collisions over the course of
a long time would be expected to make differences. Otherwise it appears there
is an obvious correlation. Notice the missing satellite at 3559 km for Jupiter.
The corresponding satellite (Iapetus) is present for Saturn. This is additional
evidence the proposed wave theory is correct.
In considering tables I, II, and III, in the wave theory, one
would expect the satellites or planets closest to the planet or sun to fit equation
(1) the best due to the large amplitude of the waves closest to the source (or
oscillator). I thus chose the effective radius (r_{0}) to fit this condition.
According to Table I the apparent effective radius of the sun (taking into account
the wave velocity within the oscillating body) appears to be close to 7.30 x
10^{5} km. This radius is slightly larger than the present radius as
might be expected. The first two planets fit almost exactly even though the
first one is at N=7. Anything located at an N smaller than 7 would apparently
have been blown away by the solar wind.
In Table II, for Jupiter, the first 3 satellites don't fit
well with the large radius apparently because the radius was considerably larger
because of the planet's high temperature when the outer satellites were placed.
If one uses the present radius the first three satellites fit well which is
consistent with the idea that these satellites were placed after the planet
had cooled down. Another possible explanation that fits in with the wave hypothesis
is that the closer three satellites tended to be forced in closer to fit the
proper orbits as the planet cooled because the amplitude of the wave action
is larger closer to the planet.
In table III, for Saturn, the effective radius that I chose
by trial and error seems to work well for most of the satellites. The effective
radius chosen suggests Saturn was also very hot when its satellites were placed
and considerably larger as one would expect for a gaseous planet. Notice in
table 2 for Jupiter there is a satellite missing at 3559 kilometers. The corresponding
satellite is present for Saturn (Iapetus). The ideas here again suggest that
the wave hypothesis is correct.
7. CONCLUSIONS, OBSERVATIONS, AND IMPLICATIONS
Some of the possible implications, ramifications, and results of the above simple calculations, experiment, and discussion include:
8. OTHER IMPLICATIONS AND SUMMARY
We may not understand completely how these waves are related
to the gravitational constant but these waves manifest themselves in many different
ways in the solar system as discussed. These include in the spacings of the
moons and planets and in the rings of the planets. These waves may be a decisive
force in determining the stability of the solar systems. These waves are just
as important today as in the past, however, planetary collisions with other
objects, for example, have obscured some of the basic results of the fundamental
wave action.
The wave hypothesis that I have presented may solve a multitude
of problems for the universe as a whole. It may provide an answer as to why
the universe is apparently excessively lumpy for the present extant theory to
explain properly. Large bubbles may exist in space because large amplitude oscillations
in dark matter have forced baryonic matter to the periphery. Oscillations at
high frequencies may produce the high temperatures observed in gas clouds, and
may have something to do with the x-ray background. The wave hypothesis appears
to provide a simple solution as to why rings of gaseous planets don't appear
to dissipate (continuous stabilization) even without shepherd moons present
and why sometimes there is a ring rather than a satellite. Perhaps the ring
structure of a gaseous planet mirrors the internal structure of the planet etc.^{9,10}.
I believe, because the wave solutions are so simple compared to other proposed
solutions, that the wave hypothesis provides a very viable alternative to other
hypotheses. Additionally it provides continuous solar system stabilization which
a correct theory would need to provide. The simple solutions and stabilization
that the wave hypothesis provides are strong arguments in favor of waves in
the dark matter-vacuum medium providing organization and stabilization for the
solar system. Otherwise chaos would probably have destroyed the solar system
long ago (or it should have never developed) as has been calculated using chaos
theory^{21}.
APPENDIX: MORE ON THE SOLUTION OF EQUATION (4):
The differential equation in r resulting from substituting
the proposed solution F=F_{0}f(r)e^{iw}0^{t}
into the wave equation in spherical co-ordinates is r^{2}f"(r)+2rf'(r)+C_{2}f(r)=0.
C_{2} is a constant and equals (r_{0}w_{0}/v_{0})^{2.}.
The given differential equation is a common differential equation with complex
roots here. A general solution is:
where k_{1} and k_{2} are constants. The sine
and cosine solutions are discussed, for example, on pages 259 and 260 of "Elementary
Differential Equations and Boundary Value Problems" by Boyce and DiPrima, Wiley
1996, 6th ed. To evaluate C_{2} use the period (T_{0}) of the
solar cycle as the number of seconds in 22.2 years and the fact that r_{0},
the radius of the sun in meters, is equal in magnitude to 22.2 years in seconds
(remember that w_{0}=2p/T_{0}). This
leaves (2p)^{2} in the numerator of C_{2}
and using v_{0} as approximately 1 m/s gives C_{2}=39.5 thus
C_{2}>>1/4 under the square root. Neglecting the 1/4 results in
about a 0.3 % error in the square root. ln (r/r_{0}) is substituted
for ln r because ln (r/r_{0}) can be written as (ln r -ln r_{0})
with ln r_{0} a constant not affecting the validity of the solution.
These considerations together with known data produce the simplified solution
to the wave equation in the text with the r part of the solution given by: f(r)=C_{1}r^{-1/2}sin(r_{0}w_{0}/v_{0}
ln r/r_{0}).
ACKNOWLEDGMENTS
I wish to thank Dr. Robert Zimmermann of the University of
Oregon physics department for the long discussions we had on the subject of
waves in dark matter. I also wish to thank Dr. Kemble Yates and Dr. Art Clemons
of the Southern Oregon University mathematics department for their suggestions.
I thank Eric Moret of the Oregon State University physics department and the
Southern Oregon University foreign language department for translating the abstract
into French. I am grateful to my wife Claudia for reading and checking the manuscript.
REFERENCES
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