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Mathematical Relationships of Solar System Bodies
revealed using the Single Body Breakup Hypothesis indicate
radical rethinking required of early Earth models
by T. Frank Lee, M.App.Sc., B.Sc., Dip.C.E., F.Aus.I.M.M. (www.franklee-geologist.com)
The diagrams shown below give the sequence of formation of
our Solar System. Only the larger bodies are considered; bodies
smaller than Europa being excluded from the discussion.
The steps of the sequence given by the diagrams displayed are
laid out in the deductive style. However, the sequence was
determined using the reverse of this – by induction – i.e., from
present to past.
The breakup of the impurely stratified, ellipsoidal “brown body”
to form protoJupiter and Saturn (Steps 2 and 3) obeyed the
Jacobi First Bifurcation Point (refined by Maclaurin and by
Darwin) for a rotating, prolate, ellipsoidal mass of gas.
Separation into two parts occurs when the ellipsoid a-axis is 1.5x
(closely 1.232x) the radius of a sphere containing the same
volume. Because the body was not a pure gas but had a highly
density-stratified core, separation was not theoretically perfect
and the interior stratified heavy matter of protoJupiter had to
readjust to maintain ‘force after’ = ‘force before’ in the closedforce system. This caused collapse inwards of outer matter,
resulting in a central impulsive force which ejected some of the
internal matter. These two mechanisms alternated until all, or
almost all, internally displaced matter was ejected (Steps 3 to 9).
Note the frequency of the numbers 1.23, 1.19, and 0.724
( = 0.9816, = 1.38-1).
These numbers (1.36 and 1.37 in place of 1.38) occur as ratios
throughout the Universe; from the spacing of galaxies down to
the spacing of an atom’s electrons.
1.19 x 1.23 x 0.724-1 = 2.022 = 1.19244.
(1.19 x 1.23 x 1.37 gives 2.)
orbitS2
MJ
–––
= –––––––
MS
orbit J2
orbit = distance from Sun
T. Frank Lee • © 2008 • www.franklee-geologist.com
Poorly differentiated protoJupiter.
Mercury / Venus / Earth = a, satellites = b + c, Neptune = d, Uranus = e, Jupiter = f
Change in ratio of deuterium to hydrogen:
Venus > Mars > Earth > Neptune > Uranus > Jupiter/Saturn
Mercury / Venus /
Earth (a)
T. Frank Lee • © 2008 • www.franklee-geologist.com
pdt = M.v / π.R2. v = orbit speed about Jupiter
(For Mars, vMars – vpJ about Sun.)
vpJ = 12.9km.sec-1
Recent study has shown the “silicate”
string of bodies formed with the centre
of the large composite body coincident
with that of protoJupiter.
T. Frank Lee • © 2008 • www.franklee-geologist.com
At the start of final internal breakup –
At the start of final internal breakup –
the formation of the lesser planets.
the formation of the lesser planets.
(22.24˚)
(22.24˚)
(91.83˚)
(91.83˚)
T. Frank Lee • © 2008 • www.franklee-geologist.com
SOME COMMENTS
1. A T-Tauri effect would have taken place between Steps 1 - 2.
Thus it is to be expected that a (faint) disc of some sort would
develop about the Sun. The T-Tauri mechanism together with
centrifuge separation of matter within the molecular (?) cloud
gives a simple explanation for the Solar System’s 18O being
uniquely higher (+40%) than that of our galaxy.
2. As the “brown” body core passed out of the molecular (?)
cloud (Step 1 - 2) there would have been a rapid increase in the
cloud’s moment of inertia factor, leading to collapse of the cloud
to form a Sun having angular momentum less than that of the
total of its satellites (the planets). Thus the unexpected
phenomenon for our Sun.
3. In step 9, at the start of the ejection Venus blocked the
faster-moving Mercury, stopping Mercury’s rotation and slowing
that of Venus. Mercury rolled around Venus and so acquired a
rotation related to that of Venus and a virtually vertical axial tilt.
Using the orbit speed formula given below (see 6), the
measured v’s of Venus and Mercury, and the mechanism
described above permits a formula to be developed to calculate
the rotation of Neptune.
4. Ejection of each of the planet and satellite groups of bodies
would have been accompanied by a “rain” of “spatter”,
comparable to a volcano. This “spatter” would be the meteors
and comets of the Solar System. Their compositions should thus
be dominantly those of the larger bodies. A brief study of these
bodies, including their isotopes’ variations, suggest this is so.
5. The orbit distance of protoJupiter from the Sun was
determined using the formula ∑(m/d2) = ∑(m)/D2, where m =
mass of each Solar System body and d its orbit distance from the
Sun. The value of D is 5.3 AU. Orbit speed is 12.9 km.sec-1.
The impulse ejection mechanism for the lesser planets and
satellites as given in the diagrams must obey Newton’s Third
Law, i.e., mN/(dN – 5.3)2 + mU/(dU – 5.3)2 must closely equal
mMy/(5.3 – dMy)2 + mV/(5.3 – dV)2 + mE+Mn/(5.3 – dE)2 +
mMs/(5.3 – dMs)2. (Pluto and Ceres have been ignored.)
Calculation shows such to be the case.
6. Force being related to angular momentum, it follows that the
orbit speeds of the planets about the protoJupiter position
should be related to the speed of protoJupiter. And they are. The
orbit speeds (v) of the planets obey the formula v = 12.9 x 1.23n
for n = +6 (Mercury), +5, +4, +3, +2 (edge Asteroid belt), +11/2
(Ceres), +1 (edge Asteroid belt), 0, - , -11/2 (Saturn), -2 (Chiron),
-3, -4, -5 (Pluto), -6 (edge of Kuiper Belt?)
T. Frank Lee • © 2008 • www.franklee-geologist.com
7. As the present-day axial tilt ratios of the planet couples
Jupiter/Saturn, Uranus/Earth, Neptune/Venus, (Mars/Moon,
Triton/Pluto) are the same as they have been calculated to have
been at separation (Saturn) and ejection, it follows the axial tilts
of the planets are primary features. There can have been NO
tilting by giant impacts or by collision/reconstitution of bodies.
Small impacts possibly caused minor wobble.
inclined from the geometric figure axis. Past workers have
suggested this for Mars and Moon. It is therefore no surprise that
the dipole axial tilts of the following planets are offset, obeying the
ubiquitous formula A = B x Cn. In this case, t = 9.6 x 1.22n, where n
= 0 (Jupiter), 1 (Earth), 2 (Mars), 8 (Neptune), and 9 (Uranus). This
conclusion is of great importance when studying the Earth. See
note E in the Geological Implications section below.
8. Saturn did not suffer ejection. All partly-differentiated
bodies which did, as described above (less Mercury and
Venus?), must almost certainly have had their cores offset to
some extent. That is, the bodies should exhibit asymmetric
structure; the inner core axis of each should be offset and
9. The product of the orbit times of the planets, plus Ceres and
Pluto, about the Sun in Earth years multiplied by 3
= 3x0.241x0.615x1x1.881x4.60x11.86x29.46x84.0x164.8x247.7
= 4.60935 x 109 years – the Age of the Solar System.
SOME GEOLOGICAL IMPLICATIONS
A The radial relationships of Earth to the other “silicate”
bodies (excluding Mercury), to that of Uranus ( = that of Venus to
Neptune), and to planetary axial tilts, plus other evidence, show
that the Earth has suffered NO expansion of any but a trifling
amount over its whole life. The same can be said for its mass.
B The rotation ratios of the planet couples Jupiter/Saturn,
Uranus/Earth, and Neptune/(Venus + Mercury) are the same in
the Present as what they were at their origins. It follows that the
rotation of Earth, on average, has remained the same over its
whole life. But it is known the Moon is slowing Earth’s rotation.
Therefore there must be a mechanism which periodically speeds
up the Earth. One can be suggested. (See web site.)
C Because of Comment 7, i.e., Earth’s tilt has remained
constant over the body’s whole life (excluding some minor
wobble due to impacts), the current suggestion that Proterozoic
glaciation was due to a large-angle change of tilt is invalid.
D Development of the Solar System as shown by the diagrams
leads to the conclusion that the inner core of the Earth is not
dominantly Ni/Fe but a mixture of metastable, high pressure
minerals. Analyses of meteorites can be argued as supporting
this view.
E If Earth has an offset core (suggested in Comment 8) and the
core is a metastable, solid body surrounded by a fluid medium
then the known difference in behaviour of deep Earth seismic
waves in its “eastern” and “western” hemispheres is explained.
It also explains the differences between theoretical and actual
values in other seismic studies that have appeared in print over
the years.
T. Frank Lee • © 2008 • www.franklee-geologist.com
The location and orientation of the inner core was
determined in 1987 during a geological study of the Earth’s
surface structures. The inner core centre was computed to be at
the intersection of a straight line drawn between the two
magnetic poles and the 6° N latitudinal plane. The offset is
towards the Pacific Ocean. (See MAppSc thesis RMIT University,
1989 and web site.)
the continental plates of matter (Sial) were in existence from the
beginning of Earth. They did not form by exudation, and/or
differentiation during Primordial time. There is evidence for this.
It can be said, then, that hypotheses of Primordial Earth based
on the Nebula Theory must be false. This, in turn, casts doubt on
likewise based models of Early Archaean and the development
of life on Earth.
F A core as described in E can produce a magnetic field, the
strength of which will vary over geological time and suffer
intermittent reversals. The main field (there should be and are
two) will be offset and inclined to the figure axis.
I Using published paleomagnetic poles for a single
continental plate and the method of Plate Path Plotting, which is
based on the concept of an offset core See “Gondwana Eight”
pp.523-9: Balkema Press, Rotterdam, 1993 or references above)
the plate can be moved step-by-step back in time from the
present to give latitude, longitude, and orientation at each step.
Plotting has been done for all continents back to the Cambrian.
Looked at from past to present the plates appear to slide over
the curved Earth (Sima), at each step paralleling one of the
vertical principal or stress planes mentioned in Implication G.
The plates kept remarkably constant form and ricocheted from
one another or became attached or broke off when they met. An
example of the plotting is attached.
G The offset and description of the Earth’s inner core given
above allows a triaxial stress figure to be constructed for the
inner Earth, giving the locations of principal axes, principal and
shear planes, and what I term “cones of stress” planes about the
two “horizontal” stress axes. These coincide with megastructures on the Earth’s surface. (See web site.)
H The sum of the areas of Earth’s continental plates equals the
surface area of Mars. This is almost certainly because Mars had
to separate from the differentiated “silicate” core at a
mathematically determined size, leaving a skin of Mars-type
silicate on the core that became part of the following Earth. Thus
The above work is part of a study of the Earth and the Solar
System carried out over 20 years.
by T. Frank Lee, M.App.Sc., B.Sc., Dip.C.E., F.Aus.I.M.M. (www.franklee-geologist.com)