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Numerical Solution of the Planetary Alignment Equations

Emilio Spedicato
Department of Mathematics, University of Bergamo Piazza Rosate 2,
24129 Bergamo, Italy - emilio at unibg.it
Zhijian Huang
Presently at Computer Science Department, University of Utah, Salt
Lake City.

Abstract

Starting from Talbott, several mythologists have argued that during
the initial phase of human history the planetary system of the Sun was
radically different than now, the planets being aligned during their
revolution around the Sun, with Saturn in a dominating position (the
polar model). In this paper we study numerically the planetary
alignment equations introduced by Grubaugh. The equations constitute a
nonlinear underdetermined system of m-1 equations and m variables (the
distances of the planets from the Sun). The system is homogeneous of
degree -2, hence if x is a solution so is x for any nonzero . Suitable
variable transformations are used to force an assigned planetary
configuration. After setting one of the variables (actually the
distance of the Earth from the Sun) to a given value, the obtained
determined system is solved by the Newton method, which converges in
few iterations from several considered starting points. Computed
parameters of interest are discussed in their relation with the polar
model. Overall the numerical results are basically in agreement with
that model.

1 Introduction

Analysis of the invariant elements in essentially all religions and in
the oldest mythologies worldwide has led several mythologists, notably
Talbott (see The Saturn Myth, 1980), Cardona (see for instance Let
There Be Light, Kronos III, 3, 1978) and Cochrane (see for instance
The Birth of Athena, Aeon II, 3, 1990), to introduce the idea that
during a remote time in the human experience the planetary system was
radically different than now. Such ancient planetary configuration,
called the polar model, some of whose features are recalled below,
evolved into the present state after probably violent and dramatic
events not too many millennia ago. This approach can be seen as a
development of ideas already expressed even if not in a systematic way
by Immanuel Velikovsky in his seminal monograph Worlds in Collision
(Mc Millan, 1951). From an analysis of mainly biblical sources Patten
(see Catastrophism and the Old Testament, 1986) has also introduced a
different, albeit not so radically different, planetary scenario for
the period from about 9500 BC to the year 701 BC. According to Patten
during such a period Mars was moving in an elliptical orbit and every
54 years it approached the Earth with catastrophical effects
(megacatastrophes, including the Flood, happening at longer intervals
depending on the relative positions of the largest planets). Leaving
to a future paper the attempt to conciliate the polar model with the
Patten scenario, here we concentrate on the mathematical study of a
feature of the polar model, namely the alignment of the planets along
a single line during their solar revolution. The following are in fact
basic elements of the polar model:
* the Sun was not visible (obviously, from the part of the Earth
where man was living)
* Saturn was the dominating object in the sky. It loomed large and
his position in the sky appeared fixed
* Venus appeared centered in the middle of Saturn and very luminous;
it was often referred to as the eye of Saturn
* Mars appeared centered in the middle of Venus, but with variable
angular size; it was of dark reddish colour and was often referred
to as the pupil of the eye
* no other planets, in particular Jupiter, were in the record.

The above features indicate the following physical state of the
planetary system:
* a synchronous revolution of the visible planets, including the
Earth, possibly along circular orbits, with the exception of Mars
* the fixed position of Saturn can be explained if it is assumed
that the Earth revolved around the Sun keeping the same face
towards the Sun (as the Moon does with respect to the Earth). The
Earth's axes of revolution and of rotation would coincide and
would be orthogonal to the ecliptic plane. The "day" of the Earth
would be equal to the year. Introducing a rotation axis of the
Earth lying in the ecliptic plane, as envisaged for instance in
Grubaugh (Aeon, III, 3, 1993), would require difficult to explain
non gravitational forces to provide the torque necessary to
maintain Saturn in a fixed position in the sky
* the nonvisibility of the Sun can be explained by the fact that
living conditions on the terrestrial hemisphere facing the Sun
were impossible for man (here we conjecture that the idea of
"hell" as an underworld and a fiery place came from some knowledge
of the climate on the hemisphere facing the Sun). Hence man lived
on the less hot hemisphere facing the external planets
* nonvisibility of Mercury is explained as in the case of the Sun
* nonvisibility of the planets Jupiter, Uranus, Neptune can be
explained either by their irrelevance, being much farther away, or
by their location directly behind Saturn, if the planetary
alignment extended also to them, or finally by their absence. It
is tempting to hypothesize here that Jupiter may have been a late
comer and the actual agent of the great perturbation which
eventually tranformed the polar planetary configuration into the
present one.

In this paper we only consider the equilibrium equations governing the
planetary alignment, under the following assumptions:
* the planets are actually aligned
* the orbits are circular
* the only relevant forces are the gravitational ones
* the planets are spheres with the same mass and size as now
* their order in the alignment is given
* the presence of possible satellites of the planets is ignored.

The equilibrium equations, which are essentially those considered by
Grubaugh (Aeon, III, 3, 1993), are solved numerically using the Newton
method. It appears that at least in a sizable region of the variable
space they have a unique solution. We also compute related quantities
of interest in the polar model, as angles of visibility, duration of
the Earth year, ratios of gravitational tidal forces, giving some
discussion of their significance.

2 The Grubaugh equations for the planetary alignment

Suppose that we have a system of m planets with masses M1,..., Mm
revolving around the Sun on circular orbits, with the same period and
along a same line. Let their distances from the Sun be x1,..., xm. Let
Vi be the modulus of the velocity of the ith planet. Then syncronicity
and circularity of the orbits imply for every i, j:

Vi/ xi = Vj/xj. (1)

Under the given assumptions the gravitational force acting on the ith
planet must be equal to the centripetal force, hence we obtain, with
M0 the Sun mass, x0 = 0 and G the gravitational constant, for
i=1,...,m

Cancelling Mi in (2) and using (1) to express V2i in terms of, say,
the Earth velocity VE and distance xE we obtain, for i=1,..., m

Equations (3) are a system of m equations in the m+1 variables xi and
VE. We can get VE from one of the equations, for instance from the
Earth's equation, obtaining a system of m-1 equations for the m
variables xi. Such equations have the following form, with E the index
of the Earth in the given planetary sequence

Notice that G does not appear in (4) and that if we write the above
system as F(x) = 0 , then F( x) = F(x)/ 2, hence the system is a
homogeneous one of degree -2. Therefore if x is a solution, so is x
for any nonzero . Notice also that the system can be put in the
following matrix form

Zu = Su, (5)

where u is the vector of the planetary masses, u = (M1,..., Mm)T, Z is
skewsymmetric and S is rank one. Hence the system can be viewed as a
special case of a generalized eigenvalue problem, where the
eigenvector u is given.

3 Solving the equation numerically

Equations (4) have been solved numerically for three planetary
configurations. Configuration A consists of the planets Earth, Mars,
Venus and Saturn in this order; configuration B has Mercury between
Sun and Earth; configuration C has Jupiter beyond Saturn. In order to
obtain a solution respecting the given configuration we use a change
of variables. For configuration A let xE, xMS, xV and xS be the
distances of Earth, Mars, Venus and Saturn from the Sun. Then we
define variables y1,..., y4 by relations

xE = y21, (6)
xMS = xE + y2 = y21 + y22, (7)
xV = xMS + y23 = y21 + y22 + y23, (8)
xS = xV + y24 = y21 + y22 + y23 + y24. (9)
In a similar way the distance from Jupiter to the Sun is written as xJ
= xS + y25, while for the distance xME of Mercury we use the following
transformation which forces 0 xME xE

xME = [ y26 /(1 + y26)] xE. (10)

By fixing a value of one of the variables, say xE equal to unity, the
system becomes determined and can be solved numerically, using one of
the standard methods (we do not consider using algorithms for
underdetermined nonlinear systems, like the ABS methods developed by
Spedicato and Huang, Optimization Methods and Software 5, 1995). We
have chosen the Newton method, with the Jacobian matrix approximated
by finite differences (it would not be difficult however to evaluate
it analytically). The stepsize for the approximation of the
derivatives was taken as = min {10-6, ||ri||}, ||ri|| being the
Euclidean norm of the residual vector, ri =[ Z(xi) - S(xi)]u. This
choice for guarantees, in exact arithmetic, Q-quadratic rate of
convergence.

Initial values must be given to the variables to start Newton method.
We used five different starting points. For configuration A we took xS
= xE + k/5 xE, k=1,...,5.
We took xV = xE +( k/5 xE) / 3, xMS = xE + ( k/5 xE) / 4. The
computations were done on a compatible 386DX2/40 with zero machine
10-20. From all starting points Newton method converged in a few
iterations to the same solution. While this is not a proof that the
system has a unique solution (there are actually multiple solutions in
the transformed space) this however can be taken as an indication that
the solution may be unique in a rather large region of the variables
space.

4 The computed results

Tables 1 to 9 give the numerical results (1 to 3 relate to
configuration A, 4 to 6 to configuration B, 7 to 9 to configuration
C). We first notice that the addition of Mercury makes almost no
difference, so we can only consider configurations A and C. Table 1
gives in the first four columns the distances xE , xMS , xV , xS for
different values of x1= xE taken as a parameter. Notice that the
equations have been individually solved despite the fact (which was
not noticed initially) that once the solution for a given x1 is known
then the solution for x1 is just times the solution for x1. Columns
headed by RES and IT give the final residual Euclidean norm and the
number of iterations required by Newton method to achieve the given
stopping criterion (||r|| 10-8). Notice that the estimated (by LINPACK
routines) condition number of the final Jacobian, given in Tables 2,
5, 8 under the heading RCOND, depends generally on x1 and may affect
the number of iterations (it might also affect the dynamical stability
of the solutions, to be investigated in a later work). Table 2 gives,
under the headings ALPHA2, ALPHA3, ALPHA4, the angles in degrees under
which Mars, Venus and Saturn would be viewed from the Earth. Under VT
we give the velocity of revolution of the Earth, measured in
astronomic units per (present) year. Under YT we have the
corresponding length of the year, in terms of present years, its
inverse under 1/YT giving how many times shorter than presently would
be the corresponding Earth "year". In Table 3 we give the ratios of
the gravitational tidal forces of Saturn over Sun (under TPS), of
Saturn over Mars (under TPMS), of Saturn over Venus (under TPV) and of
Sun over Mars (under TPO). The following Tables are similarly to be
interpreted, noticing that x(0) gives the distance of Mercury to the
Sun, x(5) and ALPHA5 are the distances of Jupiter to the Sun and its
visibility angle from the Earth.

5 Analysis of the numerical results

The following observations follow from an inspection of the Tables:
* in agreement with the Talbott et al. interpretation of the
mythological record planet Saturn revolves quite close to the
Earth. Its distance from the Sun is only about 5% greater than the
Earth distance, if Jupiter is not considered, about 4.5% if
Jupiter is considered. Jupiter would be only about 12% more
distant than Earth and completely invisible, its visibility angle
being smaller than the angles of Mars, Venus and Saturn
* Venus visibility angle is about the half of the Saturn angle, this
value being essentially unaffected by the presence of Jupiter
* Mars visibility angle appears to be about 7% greater than the
Venus angle, and therefore about the half of Saturn angle
* Saturn visibility angle would be greater than the present Moon's
angle for xE less than about 1.5 astronomical units
* for xE = 1 the Earth would revolve around the Sun about 7% slower
than presently in absence of Jupiter, about 14% slower in presence
of Jupiter; for xE = 1/2 the Earth would revolve 2.6 times faster
than presently in absence of Jupiter, 2.4 times faster in presence
of Jupiter
* gravitational tidal force of the external planets would be larger
than the Sun's tidal force, the planets closest to the Earth
having the greatest force. Notice that the ratio of these forces
is independent of the actual Earth distance from Sun, due to the
homogeneity property of equation (4). Notice that the ratio of the
planetary tidal forces over the Sun's tidal force would be
comparable with the present ratio of Moon's over Sun's tidal force
(about 2.2), but the resulting action would be greater if the
Earth kept the same face towards the planets. This fact may have
had remarkable influence on the geological structure of the Earth
(it is tempting to associate the formation of a unique continental
mass, the Pangea, with such a tidal action).

While most of the numerical results appear to be quite favourable to
the polar model, a problem is that Mars, having a greater visibility
angle, would completely cover Venus, the "eye" of Saturn. However
there are two possibilities to remove this problem:
* according to the polar model the visibility angle of Mars was not
fixed but varying; this could be explained either by Mars not
moving along a circular orbit, but along an elliptical one, as
suggested by the stability analysis of Grubaugh, or by the fact
that the atmospheres of Venus and Saturn would periodically and
syncronously expand, increasing their visibility angle, or to the
presence of the Moon, that has been ignored here
* since the numerical results assign to Saturn an orbit much closer
to the Sun than presently, it is quite possible that the
atmospheres of Saturn and likely of Venus were hotter than
presently and consequently more expanded, resulting in a visible
diameter of the planets larger than the present diameter, which
has been considered in the computations. Moreover if the system
collapsed after the catastrophical capture of Jupiter from
possibly the molecular cloud in the Orion region that was crossed
by the solar system not more than a few million years ago, then
substantial loss of mass from Saturn and Venus could be expected,
again leading to a reduction of the diameter.

6 Conclusions

The polar model has been partially investigated under some
assumptions. The numerical results do actually provide a quite
satisfactory validation. Further work is necessary along the following
lines:
* analysis of the dynamical stability of the circular syncronous
orbits (i.e. time integration of the motion equations to verify if
the alignment is stable; this may be true only for a range of
values of xE ).
* analysis of the orbits in presence of a strong perturbation. It is
of particular interest to investigate if the arrival of Jupiter as
a body coming from outside the previous solar system and passing
presumibly close to Saturn and Venus could result in its capture,
with the removal of Saturn to a farther away orbit, the passage of
Venus into the present orbital region and a perturbation of Mars'
orbit into an elliptical one with strong eccentricity. This would
result in the close and catastrophical interactions of Mars with
Earth that have been considered by Patten.
* determination of the distance xE of the Earth to the Sun at the
time of the polar model configuration. This could be estimated, in
addition to the stability argument given before, if mythological
information could be obtained on the relative size of the
visibility angle of the Moon versus the planetary angles, under
the assumption that the Moon-Earth distance has not changed.

Of course the above problems, particularly the last one, are of
considerable mathematical difficulty, both in the modelling and in the
numerical solving, but should be within reach of computation of
present workstations.

Ackowledgements

Thanks are due to Patten and Cardona for epistolarian exchange and
personal discussions which have led to the study of this fascinating
problem.

Work supported by MURST 60% 95 funds.