These are the class notes for the paper i presented in Victor Szebehely’s
Theory of Orbits 388P at the University of Texas, Austin. I received an
A for the class, based solely on the presentation. Dr. Szebehely loved
my ideas. He said “All you need to do is add time to your calculations,
and you will get your name lit up in lights.” I actually had time as part
of the analysis but did not include that part in the talk. Nobody else
ever saw this paper. It is the same project I presented to MG Niles
Fulwyler et all at White Sands in 1984, and evaluated by NASA in
1982.
A System of Celestial Mechanics
William H. Clark II, P.E.
Abstract
This is a proposal of a new dynamic presentation of the movement of the major bodies in the Solar System. Two separate functions are generated analytically for this purpose. The first establishes a new plane of the solar system, about which the motion of each planet is shown to be sinusoidal. This is as would be expected from current theory, which indicates uniform accelerated movement in the mathematical presentation. The second function developed in the paper is of a cycloid. This describes exactly the irregular, elliptical shape of the orbits.
The two functions turn out to be only one. The first one, that defines a plane, can be thought of as a line attached to two “wheels” or anomalies, one at the sun and the other near the asteroid belt. These define pivots or “foci” to cause the sinusoidal motion of the line, which, in moving through 360°, generates a plane. Attaching this line slightly off tangent in the main pivot at the sun generates the prolate cycloid of the second function. The image traced in 3D space by this system is a circular wave front, with a linear singularity at the origin and the two “wheels” define distinct event horizons – one near the sun, the other in the area of the asteroids.
In evaluating the pertinence of such a new perspective, a measure of validity is how the system fits into current ideas. The new system helps to explain the axis of rotation of the planets accurately, which existing systems cannot do. It also indicates the exact degree of ellipticity of the orbits of all the planets, and could even be interpreted to identify the earth-moon binary as a significant part of the dynamical system. The asteroids are also an important part of the analysis, as they coordinate precisely with the outer planets in the transformation at the event horizon from the inner to the outer planets, both of which display distinct, unique characteristics.
The coordinates used to make this analysis were obtained from a Basic computer program. The solution is a general one, with the emphasis on developing a rational basis for the form of the functions themselves, their theoretical implications and possible mathematical inter-relationships. These are confirmed, and interesting fundamental possibilities are developed, suggesting a structural basis for gravity as emanating from beyond the extent of the solar system.
In a more abstract sense, the functions do not exhibit a distinct singularity, so are a kind of regularization of the equations of motion for celestial bodies. In fact, the aspect of the functions at the origin (e.g. the Sun), where an anomaly is expected, indicate that the relation alters radically at the singularity and is consequently consistent with the existing body of mathematical knowledge on such singularity phenomena, and their associated event horizons.
W.H.Clark Celestial System 2/6/97 page 2
Introduction
The data used to evaluate the positions of the planets in the Solar System were obtained from a published computer program, modified to output data in the coordinates of the celestial sphere. This computer routine, which used standard values to establish the elliptical and spatial characteristics of the orbits, was used to generate a single set of data: the positions of each planet at increments of ten degrees of heliocentric longitude, throughout the course of its revolution around the sun. This data is provided in Table A1, (in the Appendix) using the ecliptic as the heliocentric equator.
These points for each of the planets were then analyzed to determine a least squares fit at each of the ten degree increments of heliocentric longitude, or pi/16 radians. The slope and intercept of each line were then calculated, as indicated in Table A2. These values (which have a correlation coefficient of .80 to .65) indicate that both the slope varies as a sinusoidal function. A line fixed at one end rotating around 360° defines a flat plane in three-dimensional space.
The analysis also indicates a similar sinusoidal motion of the y-intercept, at the origin: the optimal fit to the line has a sinusoidal motion at its origin. This is as might be expected, at a mathematical singularity, i.e. a variance in the uniform plane at the origin. This origin is located at the geometric center of the solar system which, since the sun moves in a small orbit of its own, is the natural center for any universal coordinate system.
The geometric equivalent of this system is a line attached to two rotating wheels, one at the origin to cause the y-intercept to move with uniform sinusoidal motion and another located at about the asteroids to cause the sinusoidal motion to define the plane.
W.H.Clark Celestial System 2/6/97 page 3
(The amplitudes of the two patterns – and thus the radii of the wheels – are .05 Au and .22 Au, respectively). Rotating this “assembly” around the heliocentric sphere at a constant rate, while also rotating each wheel, generates the plane thus defined.
Though the plane is perfectly flat (with the exception of some expected distortion toward the center) there is an intense turbulent area at the origin and a strong, fully enclosed stationary wave front at the asteroids. It is interesting to plot how the waves behave at this point.
First, consider two situations: one in which there is a uniform plane that cuts through the center of the solar system, without the dimple caused by the motion of the y-intercept. Superimpose upon this pattern the one with the puckered zone. There is a circle where these two patterns intersect, a localized event horizon. Outside this circle, the plane of the solar system is quite uniform. Inside of it, there is a well-defined turbulence. The movement of the intercept up and down in a sinusoidal motion, against the fixed and stable plane, traces out a helical pattern along the axis; mechanically, it performs exactly the motion that makes a top whirl by depressing its springed axis. Consequently, it is expected that the sun would be under a very strong influence not only to rotate itself, but also to rotate in a small orbit about the singularity.
That the singularity zone is not situated at the sun indicates that more than the gravity of the sun is at work, and that a theoretical possibility might be rationalized there:
The implication is that the dynamic center of the solar system is not the sun but a point about which it too moves. This implies that the main foci of the planetary orbits is not the sun but anther point which is fixed in two-dimensional space, but
W.H.Clark Celestial System 2/6/97 page 4
which a line in three dimensions. This is defined as the end point of the line, which by its movement defines the heliocentric plane in space.
This is consistent with the Newtonian system and Kepler’s laws, which describe the orbital shapes but do not explain the inclination of the plane of the orbits, and other unique characteristics of each major body’s motion in three dimensional space.
The second, outer wheel suggests another hypothesis:
An intense, localized force is needed to explain how such a band of small bodies as the asteroids can exist without being dispersed over virtually infinite time by the various forces extant on the solar system, forces which over time are infinitely stronger than the forces of gravity holding the planets in place.
This “tube” exists in the space in the vicinity of the asteroid belt, identified geometrically as the outer wheel in the mechanism defining the plane. This, of course, implies there must be some physical significance and manifestation to be realized by the plane defined above. Otherwise, to attribute some significance other than a geometric curiosity is folly. This significance will be clearly identified later in this paper.
Further analysis of the data in Table A3 shows that the position of the each planet in its revolution around the sun is itself sinusoidal, with the movement being practically equidistant above and below the heliocentric equatorial plane. This introduces another hypothesis:
For all the planets to show sinusoidal motion with respect to a common plane implies that all of the elliptical shaped orbits are sections from a single conic section; i.e. with the same axis but with different heights. This implies the second
W.H.Clark Celestial System 2/6/97 page 5
foci of all the planets are closely related in 3D space; in fact, such that all are equal in four-dimensional space (3D + time).
Physically, all of the orbits of the major planets are conic sections of similar cones (e.g. the same axis of rotation, but different height), which implies a focus exterior to the bounds of the solar system; or, at least, a focus beyond the limits of the normal plane(s) of motion. Attaching the line off center on the second wheel can more accurately represent this motion. Thus generating a second “foci” in the traditional sense. This idea is treated geometrically in the following section.
The mechanical analogy of this system supports many of the above propositions. First, the elliptical shape of the planets’ orbits can be represented by a spring with a weight (the planet) on one end, and fixed at the other. The oscillation is between the semimajor axis and the perihelion. The spring constant is analogous to the eccentricity of the orbit. The supposed movement of the “fixed” end of this mechanism complicates the analysis, because another mass moving with simple harmonic motion exists there: a single point from which the entire plane is defined. Yet there is not only no mass there, there are no definable points to attach the spring to. Again, as with the conic sections of the foci, a strong force exterior to the solar system is implied. Or another way to possibly interpret the situation is as a three-body problem with two small bodies subjected to a much stronger force from a third. Also the mechanical analogy suggests a possible means to regularize the motion through modeling:
Where physical phenomena can be separated into dependent, but recognizable yet independent patterns of motion, these can be modeled by mechanical or electronic
W.H.Clark Celestial System 2/6/97 page 6
components with the same operating characteristics, so that the net output from the assembly is an exact solution to the otherwise unsolvable problem.
In the case of singularities in celestial mechanics, such modeling of phenomena using electronic components will make it possible to perform accurate calculations much more swiftly than the stepwise integration used to regularize singularities at present.
The anomalies
Before extending the analysis into even further conjectures, it is helpful to first establish a reasonable theoretical basis for the progress so far, on the basis of current theory. To date, there are two accepted and well-known anomalies in orbital theory: the true anomaly and the eccentric one. It is instructive to realize that, when the true anomaly is used as the dependent variable in an analysis, the differential equation of motion becomes sinusoidal.
Whereas the true anomaly correlates to the sinusoidal motion of the common axis, the eccentric anomaly correlates to sinusoidal motion on a higher order. Just as the heliocentric plane established above represents the basis from which the first order differential equation happens, the origin of the solar system appears to be the logical basis for the second order system. Instead of a plane (i.e. a singularity with respect to the plane, since it realizes only two possible dimensions instead of three) the basis is a line.
Another way to compare the two systems is to acknowledge that the true anomaly is based at the geometric center of the ellipse (as the first wheel) while the eccentric anomaly is at the foci (as the outermost, second wheel).
W.H.Clark Celestial System 2/6/97 page 7
The System Wave
Elaborating on the sinusoidal nature of each planet’s movement with respect to the celestial plane defined above, it is instructive at this point to introduce a second frame of reference: the planet’s axis of rotation. This axis is known to always point to the same spot in the celestial sphere, throughout its revolution around the sun. During this movement, the planet moves evenly above and below the ecliptic plane. (To simplify this part of the analysis, it is done with respect to the ecliptic.)
These characteristics can be conveniently represented if the planet is considered to be associated with a longitudinal wave. If the axis of rotation is always tied to this sinusoidal wave, then as the planet moves about the sun the elliptical shape of the orbit causes the planet to move closer and farther away from the sun. As illustrated in Figure 1, if this motion happens on a sinusoidal wave, the axis will always point in the same direction.
Table 1
| Planet | Semimajor Axis | Inclination of Axis | X |
|---|---|---|---|
| Mercury | 5.8 E7 | 0.0 degrees | 1.2 E7 km |
| Venus | 1.1 E8 | R179 | 6.7 E6 |
| Earth | 1.5 E8 | 23.5 | 2.5 E6 |
| Mars | 2.3 E8 | 25.2 | 2.1 E7 |
| Jupiter | 7.8 E8 | 3.2 | 3.8 E7 |
| Saturn | 1.4 E9 | 26.8 | 7.8 E7 |
| Uranus | 2.9 E9 | R98 | 13.2 E7 |
| Neptune | 4.5 E9 | 29.0 | 3.7 E7 |
| Pluto | 5.9 E9 | 90.0 | 1.5 E9 |
The important data used in the analysis is listed in Table 1. It was obtained from the Handbook of Chemistry and Physics. The semimajor axis is an approximation of the planets’ average orbital distance from the sun. A value x, shown in the third column, is added or subtracted from the semimajor axis to give the perihelion and aphelion of the
W.H.Clark Celestial System 2/6/97 page 8
orbit. These are the maximum and minimum distances of the planet from the sun, respectively.
The next task is to find the wavelength of this “system wave.” By simple trial and error a nominal length of 23.3 E6 km was found to give reasonable results. That is, the largest value that would still permit the planets’ movements and attitude to be rationalized in the longitudinal framework proposed.
Table 2
| Planet | Division by 23.3 E6 | Inclination of orbit to the ecliptic | Maximum distance from the ecliptic |
|---|---|---|---|
| Mercury | 2 + 1.1 E7 km | 7.0 degrees | 7.0 E6 km |
| Venus | 4 + 1.7 E7 | 3.4 | 6.5 E6 |
| Earth | 6 + 1.0 E7 | 0.0 | – – |
| Mars | 9 + 2.0 E7 | 1.9 | 7.6 E6 |
| Apollo Group | 9 + 7.8 E6 | – | – |
| Belt Asteroids | 16 + 2.2 E6 | – | – |
| Trojan Group | 33 + 3.6 E6 | – | – |
| Jupiter | 33 + | 1.3 | 17.0 E6 |
| Saturn | 60 + 2.0 E6 33 + 1.1 E7 | 2.5 | 61.0 E6 |
| Uranus | 124 + 1.0 E7 | 0.7 | 35.0 E6 |
| Neptune | 193 + 3.1 E6 | 1.8 | 141 E6 |
| Pluto | 253 + 5.1 E6 | 17 | 172 E6 |
The second column of Table 2 lists the aspect of each planet upon this wave – the remainder of division by the proposed wavelength. This value will be used to situate each planet at the proper location upon the wave.
Notice the range of values for axial inclination in Table 1. These cannot all be accounted for by a simple sinusoidal wave, because of the nearly perpendicular aspects of Uranus and Pluto. There is also the anomaly of retrograde (reverse) rotation for Venus and Uranus to explain.
W.H.Clark Celestial System 2/6/97 page 9
Again, by trial and error, another wave pattern – albeit closely related to a sinusoidal wave – a prolate cycloid is found to be the ideal solution. In two dimensions this waveform is formed by the movement of a point on the surface of a circle as the circle rolls along a plane. In three dimensions it is a helical pattern, a common function in nature. Thus, the prolate cycloid is not an unlikely conceptualization of gravitational flux.
Cochoids of Nicomedes
Another possible presentation of the “system wave” is a family of curves with the parametric equations:
x = a + cos t & y = a tan t + sin t (1)
These generate a two-part curve: a loop similar to the cycloid traced out, and a shell that can best describe the movements of the outer planets.
This solution has some unique possibilities in the analysis of orbits. The distance is a sinusoidal function, so there can be no singularities because of division by a function that can become zero. This may help to regularize certain orbital analyses. Another possibility is that half of the loop is not the gravity that we know in current Newtonian theory, but is a second, stronger form. This is consistent with the analysis thus far, since certain aspects of orbits that are seemingly rationalized evidence stronger action than a straight linear force – e.g. axial inclinations and orbital inclination.
Gravity
Figure 3 is a sketch of the planets situated upon a either of the two parts of the cycloid, according to the average orbital distances shown in Table 4. The axis is shown as an arrow pointing according to the right hand rule for the planets’ rotation. The
W.H.Clark Celestial System 2/6/97 page 10
positions of the various planets on the waveforms will now be evaluated to see how reasonable the fit is.
First, the motion of Mars moves uniformly on either side of an inflection point in the wave, its axis inclined at about the right amount. Jupiter does the same thing on the lower wave, which arches less to correspond with the lesser axial inclination of Jupiter’s axis. Uranus, also on the lower wave, moves through an entire wavelength, with an eight degree axial inclination off the vertical, slightly greater than Jupiter – since it traverses more of the wave.
Next, Saturn on the lower wave traverses two wavelengths, with a much greater axial inclination. The center point of the wave can be thought of as an inflection point, motivating the body to rotate in order to maintain rotational stability – a two dimensional vista of the singularity at the sun discussed earlier. It is interesting to note that all four planets (with the possible exception of Pluto) have almost the same rotational period – about ten hours. This would imply some similarity in the wave analogy, and it is there in the form of the inflection of the wave traveled.
Back to the upper waveform, there is Mercury shown atop the loop, It moves back and forth on the waveform the distance of its ellipticity, maintaining its axis pointed in the same direction. Although the planet is known to have an axial inclination of zero, as the innermost planet it is close to the event horizon, where the wave may be inflected so that the pattern traced is less exaggerated. The geometry of that area, given what has already been established, would likely fit this expectation.
Venus is not much farther from the zone where the entire pattern may be warped because of the close coincidence of the two waveforms.
W.H.Clark Celestial System 2/6/97 page 11
The Earth-Moon system is the most striking confirmation of the system. They are unique in the solar system, the Moon being the largest satellite with respect to its host in the whole solar system. This may be a necessary prerequisite for matter to exist in dynamic stability at the cusp of the wavefront as shown. In which case Earth occupies the upper portion of the loop, the moon the lower, in its own orbit. The moon’s orbital plane is inclined at five degrees to the Ecliptic, as is indicated by the inflection of the lines (versus the much higher inflection of the upper cusp, indicative of Earth’s 23° axial inclination).
This assumes that the placement of the minor planets on the inner wave is not only unique, but that every portion of the wave must be occupied for overall stability of the solar system. Each planet occupies a special location, which is not shared by any other. (Refining the system wavelength should resolve the overlap of Venus and Mars, although this may hint of the retrograde rotation of Venus – needed to occupy a common stable energy state. Likewise with Uranus on the other wave, between the Jupiter and Saturn orbital states.) The only empty areas are at the beginning of the inner planet wave.
The Asteroids occupy this area. Even with this rough approximation, notice that the location of the Belt, Trojan and Apollo Groups of asteroids align closely with the average distances of the planets Saturn, Neptune and Pluto, respectively. This bilateral symmetry identifies a transition from the inner planets to the outer planets, and the respective waves by which each is identified. This is a logical location for the outer wheel of the mechanism tracing the sinusoidal motion of the line tracing the plane of the
W.H.Clark Celestial System 2/6/97 page 12
solar system. It it is an almost necessary means to tie the two waves together in the dynamics established thus far.
In review, this analogy has shown remarkable insights into the dynamics of the solar system:
- The range of the inner planets elliptic orbits fill up the entire wavelength, with insignificant overlapping
- The position of each planet is closely aligned to its axial inclination
- The outer planets occupy increments of half or full wavelengths, on the lower portion of the waveform
- The asteroid groups and the average position of the outer planets occupy the same positions, but on the cusp’d and the lower waveforms respectively
This perspective thus fits nicely in a two-dimensional presentation. The third dimension is already calculated, in the form of movement with respect to the new system plane. These figures are represented in Table 3.
The most noticeable characteristic of the analysis is that the midpoint of each planet’s motion is about 0.05 Au below the heliocentric plane calculated by the least squares fit.* This fit was normalized to be symmetric about the known origin of the solar system, or the center of the heliocentric sphere. In order for the pronounced symmetry of both systems to be realized, that is:
- the close symmetry of the least squares fit to a sinusoidal slope and a sinusoidal motion of the y-intercept, and
- the uniform displacement of the midline of the sinusoidal motion of the planets about this plane,
*This matches the y-intercept in Table 2.
W.H.Clark Celestial System 2/6/97 page 13
then a two-part pattern such as that envisioned most closely realizes both conclusions. This is a strong confirmation of the use of the parametric equations proposed. In affect, the defining plane of motion then becomes the bottom half of the equation (1) wave for the inner planets. Thereafter the presumably weaker (since it has a greater definition in space) cusp’d wave diminishes in influence upon the planets, and they drift away from this system, perhaps using systems of moons to define a localized “cusp” that moves within the planets’ frame of reference.
Table 3 Motion of the planets with respect to the new heliocentric plane
| Planet | Movement w.r.t. the system plane (Au) | Centerline | Radius (1/2 range) |
|---|---|---|---|
| Mercury | -0.30 + 0.21 = 0.51 | -0.05 | 0.26 |
| Venus | -0.27 + 0.17 = 0.44 | -0.05 | 0.22 |
| Earth | -0.19 + 0.17 = 0.26 | -0.06 | 0.13 |
| Mars | -0.19 + 0.09 = 0.28 | -0.05 | 0.14 |
| Jupiter | -0.10 + 0.02 = 0.12 | -0.04 | 0.06 |
| Saturn | -0.30 + 0.17 = 0.47 | -0.06 | 0.23 |
| Uranus | -0.77 + 0.74 = 1.51 | -0.03 | 0.75 |
| Neptune | -1.21 + 1.21 = 2.42 | 0.00 | 1.21 |
| Pluto | -1.10 + 1.60 = 1.70 | -0.25 | 0.85 |
Back to the mechanical analogy with springs, the indication is that the motion of the planets with respect to the heliocentric plane is the result an organized force. That is, if the spin of the planet is the angular force with respect to the heliocentric plane here – the further from this plane, the more inclined the axis of rotation must be for dynamic stability.
W.H.Clark Celestial System 2/6/97 page 14
Mathematics
In closing, it is appropriate to confirm the data with function (1). First, develop the radius r in the cartesian coordinate system
r^2 = x^2 + y^2 (2)
~ x^2
therefore, r ~ x ~ a + cos t (3)
This can be substituted into the function for y,
y = a tan t + sin t
= sin t [(a/cos t) + 1]
= sin t cos t (a + cos t)
where the above value for x can be substituted
= r sin t cos t (4)
which, using a trigonometric identity becomes
y = ½ sin 2t
Thus the identity of the curve becomes the two terms
x = a + cos t & y = ½ sin 2t (5)
These correspond to (1) the numerical analysis, which showed the motion of the planets with respect to the plane to be sinusoidal in the two dimensions; (2) the graph, which showed the x-coordinate to be a sinusoidal motion and the y-coordinate or motion perpendicular to the plane to be sinusoidal as well; and (3) the mechanical models, which indicate simple harmonic motion in both instances.
W.H.Clark Celestial System 2/6/97 page 15
Future Research
There are several productive avenues to be followed henceforth, tests of the system herein proposed.
- Evaluate the apparent warping of space at the center of the solar system, to determine if the line traced by light would be bent to the extent established by Relativity Theory and by observation. The geometry of this area would indicate that the affect on the light would vary, depending on the direction in the heliocentric sphere from which it comes.
- Trace the actual paths of the planets, known by observation and relate these paths to the ideal waveforms. The inflections of the waveforms needed to conform to the actual paths should be related in some way to the other characteristics of their motions such as rate of rotation and period of revolution.
- Establish the event horizon and its affect upon this region of space.
Conclusion
A popular analogy for presentation of orbital theory is to depict Newton or Kepler standing at the seashore looking for some nice shells for his collection. This brief paper has shown just this sort of presentation. The gravity wave is much the same as waves breaking at sea, and the prolate cycloid has the characteristic of the shell of a mollusk. This is as chaos theory says it should be, natural phenomena imitating natural patterns at other scales. It remains to be seen exactly how exact these ideas are, but it is hoped that enough credence exists that we will at least be able to see a little farther out to sea.
W.H.Clark Celestial System 2/6/97 page 16
References
- Adventures in Celestial Mechanics, by Victor G. Szebeheley The University of Texas Press, Austin
- Amateur Astronomy, Patrick Moore, W.W. Norton Company, NY 1968
- Calculus, Early Transcendentals ,3rd Edition, James Stewart, Brooks/Cole Publishing Company
- Celestial Basic, Astronomy on your Computer, Eric Burgess, Cybex
- Elementary Differential Equations and Boundary Value Problems, 5th Edition, by William C. Boyce and Richard C. DiPrima, John Wiley & Sons Inc.
W.H.Clark Celestial System 2/6/97 page 17
Table 1A Distance of each planet from the sun at ten-degree increments
| Long. | Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune | Pluto |
|---|---|---|---|---|---|---|---|---|---|
| 0 | 0.3557 | 0.72659 | 1.98552 | 1.401 | 4.9579 | 9.53 | 20.061 | 29.864 | 48.002 |
| 0.0963 | 0.34189 | 0.72578 | 1.78684 | 1.4172 | 4.9534 | 9.4259 | 20.005 | 29.831 | 48.791 |
| 0.3927 | 0.32995 | 0.72487 | 1.58841 | 1.4375 | 4.9584 | 9.3262 | 19.918 | 29.807 | 49.221 |
| 0.589 | 0.3202 | 0.7239 | 1.3975 | 1.4611 | 4.9729 | 9.2347 | 19.803 | 29.793 | 49.275 |
| 0.7854 | 0.31302 | 0.72291 | 1.2216 | 1.4871 | 4.9961 | 9.1548 | 19.663 | 29.789 | 48.951 |
| 0.9817 | 0.30869 | 0.72193 | 1.0673 | 1.5145 | 5.0274 | 9.0897 | 16.505 | 29.795 | 48.261 |
| 1.1781 | 0.30737 | 0.72101 | 0.94063 | 1.5422 | 5.0653 | 9.0418 | 19.335 | 29.812 | 47.233 |
| 1.3744 | 0.30911 | 0.72017 | 0.84641 | 1.5693 | 5.1085 | 9.013 | 19.159 | 29.838 | 45.905 |
| 1.5708 | 0.31386 | 0.71946 | 0.78828 | 1.5946 | 5.1554 | 9.0044 | 18.983 | 29.872 | 44.328 |
| 1.7671 | 0.32141 | 0.71889 | 0.76847 | 1.6172 | 5.2041 | 9.0163 | 18.816 | 29.914 | 42.564 |
| 1.9635 | 0.3315 | 0.71849 | 0.78774 | 1.6361 | 5.2527 | 9.0483 | 19.662 | 29.961 | 40.679 |
| 2.1598 | 0.34371 | 0.71828 | 0.84536 | 1.6508 | 5.2994 | 9.0991 | 18.528 | 30.012 | 38.747 |
| 2.3562 | 0.3576 | 0.71825 | 0.9391 | 1.6606 | 5.3424 | 9.1668 | 18.42 | 30.065 | 36.841 |
| 2.5525 | 0.37262 | 0.71843 | 1.06537 | 1.6651 | 5.38 | 9.2488 | 18.34 | 30.118 | 35.036 |
| 2.7489 | 0.3882 | 0.71879 | 1.21932 | 1.6641 | 5.4108 | 9.3419 | 18.293 | 30.168 | 33.399 |
| 2.9452 | 0.40373 | 0.71923 | 1.39501 | 1.6578 | 5.4337 | 9.4426 | 18.28 | 30.215 | 31.995 |
| 3.1416 | 0.41863 | 0.72001 | 1.58572 | 1.6463 | 5.4476 | 9.5469 | 18.302 | 30.255 | 30.877 |
| 3.3379 | 0.43231 | 0.72082 | 1.7841 | 1.6301 | 5.4521 | 9.651 | 18.358 | 30.288 | 30.088 |
| 3.5343 | 0.44425 | 0.72173 | 1.98253 | 1.6098 | 5.4471 | 9.7507 | 18.445 | 30.312 | 29.658 |
| 3.7306 | 0.454 | 0.7227 | 2.17339 | 1.5862 | 5.4326 | 9.89422 | 18.56 | 30.326 | 29.604 |
| 3.927 | 0.46118 | 0.72369 | 2.34934 | 1.5602 | 5.4094 | 9.2291 | 18.7 | 30.33 | 29.928 |
| 4.1233 | 0.46551 | 0.72467 | 2.50363 | 1.5328 | 5.3782 | 9.9872 | 18.858 | 30.324 | 30.618 |
| 4.3197 | 0.46683 | 0.72559 | 2.63031 | 1.5051 | 5.3402 | 10.0351 | 19.028 | 30.307 | 31.646 |
| 4.516 | 0.46509 | 0.72643 | 2.72453 | 1.478 | 5.297 | 10.0639 | 19.204 | 30.281 | 32.974 |
| 4.7124 | 0.46034 | 0.72714 | 2.78266 | 1.4527 | 5.2501 | 10.0725 | 19.38 | 30.247 | 34.551 |
| 4.9087 | 0.45279 | 0.72771 | 2.80247 | 1.4302 | 5.2014 | 10.0606 | 19.547 | 30.205 | 36.315 |
| 5.1051 | 0.4427 | 0.72811 | 2.7832 | 1.4112 | 5.1528 | 10.0286 | 19.701 | 30.158 | 38.2 |
| 5.3014 | 0.43049 | 0.72832 | 2.72558 | 1.3965 | 5.1061 | 9.9778 | 19.835 | 30.107 | 40.132 |
| 5.4978 | 0.4166 | 0.72835 | 2.63184 | 1.3868 | 5.0631 | 9.9101 | 19.944 | 30.054 | 42.038 |
| 5.6941 | 0.40158 | 0.72817 | 2.50557 | 1.3822 | 5.0255 | 9.8281 | 20.023 | 30.001 | 43.843 |
| 5.8905 | 0.386 | 0.72781 | 2.35162 | 1.3832 | 4.9947 | 9.735 | 20.07 | 29.951 | 45.48 |
| 6.0868 | 0.37047 | 0.72728 | 2.17593 | 1.3895 | 4.9719 | 9.6343 | 20.083 | 29.904 | 46.884 |
| 0.35557 | 0.72659 | 1.98522 | 1.401 | 4.9579 | 9.53 | 20.061 | 29.864 | 48.002 |
W.H.Clark Celestial System 2/6/97 page 18
Table 1B Distance of each planet above the heliocentric plane at ten-degree increments
| Long. | Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune | Pluto |
|---|---|---|---|---|---|---|---|---|---|
| 0 | -0.09065 | -0.05765 | 0 | -0.02378 | -0.01942 | -0.04007 | -0.01341 | -0.0236 | -2.822 |
| 0.0963 | -0.07293 | -0.0538 | 0 | -0.01932 | -0.01972 | -0.04267 | -0.01239 | -0.02719 | -2.967 |
| 0.3927 | -0.0524 | -0.04789 | 0 | -0.01412 | -0.01925 | -0.04363 | -0.01089 | -0.02974 | -2.998 |
| 0.589 | -0.02987 | -0.04013 | 0 | -0.00837 | -0.01805 | -0.04291 | -0.00898 | -0.03114 | -2.914 |
| 0.7854 | -0.00618 | -0.03083 | 0 | -0.0023 | -0.01615 | -0.04054 | -0.00673 | -0.03135 | -2.718 |
| 0.9817 | 0.01774 | -0.02034 | 0 | 0.00386 | -0.01364 | -0.03661 | -0.00421 | -0.03035 | -2.417 |
| 1.1781 | 0.04098 | -0.00908 | 0 | 0.00987 | -0.0106 | -0.03128 | -0.00153 | -0.02819 | -2.024 |
| 1.3744 | 0.06265 | 0.00254 | 0 | 0.0155 | 0.00715 | -0.02475 | 0.0012 | -0.02494 | -1.552 |
| 1.5708 | 0.08191 | 0.01405 | 0 | 0.02053 | -0.00342 | -0.01726 | 0.00389 | -0.02073 | -1.021 |
| 1.7671 | 0.09802 | 0.02503 | 0 | 0.02478 | 0.00043 | -0.00911 | 0.00644 | -0.01573 | -0.451 |
| 1.9635 | 0.11036 | 0.03505 | 0 | 0.02807 | 0.00427 | -0.00061 | 0.00873 | -0.01012 | 0.136 |
| 2.1598 | 0.11847 | 0.04372 | 0 | 0.03028 | 0.00794 | 0.00791 | 0.01069 | -0.00413 | 0.718 |
| 2.3562 | 0.12202 | 0.0507 | 0 | 0.03134 | 0.01131 | 0.01613 | 0.01223 | 0.00203 | 1.273 |
| 2.5525 | 0.12088 | 0.05574 | 0 | 0.03118 | 0.01425 | 0.02373 | 0.01331 | 0.0081 | 1.779 |
| 2.7489 | 0.11509 | 0.05864 | 0 | 0.02983 | 0.01663 | 0.03042 | 0.01388 | 0.01387 | 2.216 |
| 2.9452 | 0.10489 | 0.05929 | 0 | 0.02733 | 0.01838 | 0.03594 | 0.01391 | 0.0191 | 2.569 |
| 3.1416 | 0.09065 | 0.05765 | 0 | 0.02378 | 0.01942 | 0.04007 | 0.01341 | 0.0236 | 2.822 |
| 3.3379 | 0.07293 | 0.0538 | 0 | 0.01932 | 0.01972 | 0.04267 | 0.01239 | 0.02719 | 2.967 |
| 3.5343 | 0.0524 | 0.04789 | 0 | 0.01412 | 0.01925 | 0.04363 | 0.01089 | 0.02974 | 2.998 |
| 3.7306 | 0.02987 | 0.04013 | 0 | 0.00837 | 0.01805 | 0.04291 | 0.00898 | 0.03114 | 2.914 |
| 3.927 | 0.00618 | 0.03083 | 0 | 0.0023 | 0.01615 | 0.04054 | 0.00673 | 0.03135 | 2.718 |
| 4.1233 | -0.01774 | 0.02034 | 0 | -0.00386 | 0.01364 | 0.03661 | 0.00421 | 0.03035 | 2.417 |
| 4.3197 | -0.04098 | 0.00908 | 0 | -0.00987 | 0.0106 | 0.03128 | 0.00153 | 0.02819 | 2.024 |
| 4.516 | -0.06265 | -0.00254 | 0 | -0.0155 | 0.00715 | 0.02475 | -0.0012 | 0.02494 | 1.552 |
| 4.7124 | -0.08191 | -0.01405 | 0 | -0.02053 | 0.00342 | 0.01726 | -0.00389 | 0.02073 | 1.021 |
| 4.9087 | -0.09802 | -0.02503 | 0 | -0.02478 | -0.00043 | 0.00911 | -0.00644 | 0.01573 | 0.451 |
| 5.1051 | -0.11036 | -0.03505 | 0 | -0.02807 | -0.00427 | 0.00061 | -0.00873 | 0.01012 | -0.136 |
| 5.3014 | -0.11847 | -0.04372 | 0 | -0.03028 | -0.00794 | -0.00791 | -0.01069 | 0.00413 | -0.718 |
| 5.4978 | -0.12202 | -0.0507 | 0 | -0.03134 | -0.01131 | -0.01613 | -0.01223 | -0.00203 | -1.273 |
| 5.6941 | -0.12088 | -0.05574 | 0 | -0.03118 | -0.01425 | -0.02373 | -0.01331 | -0.0081 | -1.779 |
| 5.8905 | -0.11509 | -0.05864 | 0 | -0.02983 | -0.01663 | -0.03042 | -0.01388 | -0.01387 | -2.216 |
| 6.0868 | -0.10489 | -0.05929 | 0 | -0.02733 | -0.01838 | -0.03594 | -0.01391 | -0.0191 | -2.569 |
| -0.09065 | -0.05765 | 0 | -0.02378 | -0.01942 | -0.04007 | -0.01341 | -0.0236 | -2.822 |
W.H.Clark Celestial System 2/6/97 page 19
Table 2 Slope and intercept of the line of least squares
| Longitude | Slope | Intercept | Correlation Coeff. |
|---|---|---|---|
| .0000 | -.04 | .23 | -.78 |
| .1963 | -.05 | .25 | -.79 |
| .3927 | -.05 | .25 | -.80 |
| .5890 | -.05 | .25 | -.80 |
| .7854 | -.04 | .24 | -.80 |
| .9817 | -.04 | .22 | -.80 |
| 1.1781 | -.03 | .19 | -.80 |
| 1.3744 | -.03 | .15 | -.79 |
| 1.5708 | -.02 | .11 | -.79 |
| 1.7671 | -.01 | .07 | -.81 |
| 1.9635 | .00104 | .02 | .29 |
| 2.1598 | .01 | -.02 | .65 |
| 2.3562 | .02 | -.06 | .65 |
| 2.5525 | .03 | -.09 | .64 |
| 2.7489 | .03 | -.11 | .62 |
| 2.9452 | .04 | -.13 | .61 |
| 3.1416 | .04 | -.14 | .59 |
| 3.3379 | .05 | -.14 | .58 |
| 3.5343 | .05 | -.15 | .57 |
| 3.7306 | .05 | -.15 | .57 |
| 3.9270 | .04 | -.15 | .58 |
| 4.1233 | .04 | -.15 | .60 |
| 4.3197 | .03 | -.15 | .62 |
| 4.5160 | .03 | -.13 | .65 |
| 4.7124 | .02 | -.10 | .68 |
| 4.9087 | .01 | -.07 | .74 |
| 5.1051 | -.00091 | -.02 | -.24 |
| 5.3014 | -.01 | .02 | -.65 |
| 5.4978 | -.02 | .07 | -.71 |
| 5.6941 | -.03 | .12 | -.74 |
| 5.8905 | -.03 | .17 | -.76 |
| 6.0868 | -.04 | .2 | -.77 |
W.H.Clark Celestial System 2/6/97 page 20
Table 3 Distance of each planet from the new plane at ten-degree increments
| Long. | Mercury | Venus | Earth | Mars | Jupiter | Saturn | Uranus | Neptune | Pluto |
|---|---|---|---|---|---|---|---|---|---|
| 0 | -0.30643 | -0.25859 | -0.15059 | -0.19774 | -0.0511 | 0.11113 | 0.55905 | 0.94099 | -1.132 |
| 0.0963 | -0.30584 | -0.26751 | -0.16066 | -0.19846 | -0.02205 | 0.17863 | 0.73791 | 1.21441 | -0.778 |
| 0.3927 | -0.2859 | -0.26165 | -0.17058 | -0.19224 | -0.02133 | 0.17268 | 0.73505 | 1.21066 | -0.787 |
| 0.589 | -0.26386 | -0.25394 | -0.18012 | -0.18531 | -0.0194 | 0.16883 | 0.73118 | 1.20854 | -0.7 |
| 0.7854 | -0.23366 | -0.24191 | -0.19114 | -0.18282 | -0.0563 | 0.08565 | 0.53982 | 0.92023 | -1 |
| 0.9817 | -0.18991 | -0.21146 | -0.17731 | -0.15556 | -0.03254 | 0.10698 | 0.55603 | 0.94148 | -0.707 |
| 1.1781 | -0.1398 | -0.17745 | -0.16178 | -0.13386 | -0.04864 | 0.04997 | 0.38854 | 0.67618 | -0.797 |
| 1.3744 | -0.07808 | -0.12585 | -0.12464 | -0.08742 | -0.00389 | 0.09564 | 0.42598 | 0.72021 | -0.325 |
| 1.5708 | -0.02181 | -0.08156 | -0.09423 | -0.05758 | -0.01031 | 0.05283 | 0.27357 | 0.46673 | -0.245 |
| 1.7671 | 0.03123 | -0.03778 | -0.06232 | -0.02905 | -0.01753 | 0.01105 | 0.1246 | 0.21342 | -0.095 |
| 1.9635 | 0.09002 | 0.0143 | -0.02082 | 0.00637 | -0.02119 | -0.03002 | -0.03068 | -0.06128 | 0.073 |
| 2.1598 | 0.13503 | 0.05654 | 0.01155 | 0.03377 | -0.02505 | -0.06308 | -0.1546 | -0.28426 | 0.351 |
| 2.3562 | 0.17487 | 0.09634 | 0.04122 | 0.05813 | -0.03554 | -0.10721 | -0.29617 | -0.53928 | 0.596 |
| 2.5525 | 0.1997 | 0.12419 | 0.05804 | 0.07123 | -0.05715 | -0.16373 | -0.44691 | -0.80545 | 0.818 |
| 2.7489 | 0.21344 | 0.14708 | 0.07342 | 0.0899 | -0.0357 | -0.13984 | -0.42493 | -0.78119 | 1.324 |
| 2.9452 | 0.21874 | 0.16052 | 0.0742 | 0.09102 | -0.06897 | -0.21177 | -0.58732 | -1.0595 | 1.419 |
| 3.1416 | 0.2139 | 0.16885 | 0.07657 | 0.09793 | -0.05849 | -0.20181 | -0.57869 | -1.04661 | 1.727 |
| 3.3379 | 0.20131 | 0.16776 | 0.0608 | 0.08781 | -0.10289 | -0.28988 | -0.75551 | -1.33721 | 1.603 |
| 3.5343 | 0.18019 | 0.1618 | 0.0587 | 0.08363 | -0.10311 | -0.29391 | -0.76137 | -1.33586 | 1.665 |
| 3.7306 | 0.15717 | 0.154 | 0.04133 | 0.07906 | -0.10358 | -0.2992 | -0.76906 | -1.33518 | 1.584 |
| 3.927 | 0.13773 | 0.15188 | 0.05603 | 0.08989 | -0.05023 | -0.20635 | -0.59128 | -1.03187 | 1.671 |
| 4.1233 | 0.11364 | 0.14135 | 0.04985 | 0.08482 | -0.05149 | -0.21288 | -0.60011 | -1.03262 | 1.343 |
| 4.3197 | 0.09502 | 0.13731 | 0.07109 | 0.09498 | 0.00039 | -0.11978 | -0.41932 | -0.73104 | 1.224 |
| 4.516 | 0.0534 | 0.10567 | 0.04826 | 0.07016 | -0.02176 | -0.14717 | -0.44734 | -0.75351 | 0.693 |
| 4.7124 | 0.00888 | 0.07141 | 0.04435 | 0.05041 | -158.1 | -0.08419 | -0.29149 | -0.48421 | 0.43 |
| 4.9087 | -0.03255 | 0.03769 | 0.04198 | 0.03092 | 0.01756 | -0.0215 | -0.13192 | -0.21633 | 0.158 |
| 5.1051 | -0.08996 | -0.01439 | 0.02253 | -0.00679 | 0.02042 | 0.02974 | 0.0292 | 0.05756 | -0.081 |
| 5.3014 | -0.13417 | -0.05644 | 0.00726 | -0.03631 | 0.02312 | 0.07187 | 0.16766 | 0.2825 | -0.337 |
| 5.4978 | -0.18369 | -0.10613 | -0.01736 | -0.0736 | 0.01995 | 0.11207 | 0.31665 | 0.52906 | -0.502 |
| 5.6941 | -0.22883 | -0.15389 | -0.04483 | -0.10971 | 0.01652 | 0.15112 | 0.46739 | 0.77195 | -0.584 |
| 5.8905 | -0.27351 | -0.20681 | -0.09945 | -0.15833 | -0.03679 | 0.09163 | 0.41823 | 0.71467 | -1.022 |
| 6.0868 | -0.29007 | -0.2302 | -0.11296 | -0.17175 | -0.0195 | 0.14943 | 0.58942 | 0.9771 | -0.893 |