The methodology provided by Betagnon and Simon in:

TABLES FOR THE MOTION OF THE SUN AND THE FIVE PLANETS FROM - 4000 TO + 2800

TABLES FOR THE MOTION OF URANUS AND NEPTUNE FROM + 1600 TO + 2800

Pierre Bretagnon and Jean-louis Simon

Service des Calculs et de Mé`canique Cé`

`leste du Bureau des Longitudes`

77, avenue Denfert-Rochereau, 75014 Paris, France. Published by Willmann-Bell, Inc., Richmond, 1986.

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SINGLE AND COMPOUND SYNODIC CYCLES

A real-time synodic function that describes the motions of adjacent planets
depends on the determination of the respective radius vectors, the corresponding
velocities and the corresponding periods of the two planets in question.
In dealing with the relative motions of planets in this manner, however,
it becomes feasible to undertake investigations that were hitherto numerically
impractical. One might, for example combine adjacent synodic cycles, i.e.,
for *four* adjacent planets the first and second synodic cycles may
be combined to form a further synodic with the same technique applied to
the third and fourth synodics and so on until one single synodic difference
function is obtained. All that is required to accomplish this goal are
simultaneous values for the radius vectors of the planets in question.
For the four adjacent superior planets Jupiter, Saturn, Uranus and Neptune
this requirement is readily met by the power series data and formulas provided
by Bretagnon and Simon (1986) outlined in the
previous section.
In other words, with values known for the radius vectors for the four
major superior planets
at any specified point in time, theoretical orbital "periods" are
also available, i.e., for each radius vector R(t) there exists a
theoretical mean orbital period of revolution P(t), and
corresponding orbital velocity, V(t) -- both readily obtained from
Kepler's Third Law in full: P^{ 2} = R ^{3} = V^{ -6}.
Next, with the corresponding Periods determined in this way, the
orbital motions of the four major superior planets may be reduced to
one single variable function SD6 (t) by applying the general synodic formula:

such that Synodic Difference cycle 1 (** SD1** ) is determined from the
relative motion of Jupiter with respect to Saturn,

The mean values (in years) for the six cycles are as follows:

As it turns out, for both the mean periods and the real time variations, synodic difference cycles

SD6 AND SOLAR ACTIVITY CYCLES

The fact that there is a chaotic component associated with the combined motions of the four major superior planets is of some interest, especially when it is remembered that these four adjacent bodies together possess more than

Nevertheless, irrespective of whether the major planets are currently believed to have a minimal effect on the Sun or not, these cycles (especially the

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SD6, SOLAR ACTIVITY, SOLAR ROTATION RATES, AND THE MAUNDER MINIMUM

"ANOMALOUS SOLAR ROTATION IN THE EARLY 17TH CENTURY"

Eddy, John A., et al: Science, 198:824-829, 1977.

ABSTRACT

"The character of solar rotation has been examined for two periods in the 17th century for which detailed sunspot drawings are available: A.D. 1625 through 1626 and 1642 through 1644. The first period occurred 20 years before the start of the Maunder sunspot minimum, 1645 through 1715; the second occurred just at its commencement. Solar rotation in the earlier period was much like that of today. In the later period, the equatorial velocity of the sun was faster by 3 to 5 percent and the differential rotation was enhanced by a factor of 3. The equatorial acceleration with declining solar activity is in the same sense as that found in recent doppler data. It seems likely that the change in rotation of the solar surface between 1625 and 1645 was associated with the onset of the Maunder minimum."

"The character of solar rotation has been examined for two periods in the 17th century for which detailed sunspot drawings are available: A.D. 1625 through 1626 and 1642 through 1644. The first period occurred 20 years before the start of the Maunder sunspot minimum, 1645 through 1715; the second occurred just at its commencement. Solar rotation in the earlier period was much like that of today. In the later period, the equatorial velocity of the sun was faster by 3 to 5 percent and the differential rotation was enhanced by a factor of 3. The equatorial acceleration with declining solar activity is in the same sense as that found in recent doppler data. It seems likely that the change in rotation of the solar surface between 1625 and 1645 was associated with the onset of the Maunder minimum."

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Norm: 2.9 urad s

In 1625: 2.92 urad s^{ -1 }

In 1642: 3.03 urad s^{ -1}

(The Sun as a Star, Ed. Stuart Jordon, NASA, 1981:17)In 1642: 3.03 urad s

THE INFERIOR PLANETS

One of the main advantages of time-series analysis is the ability to investigate complex interactions that are neither indicated nor suspected from mean or extremal values. A case in point concerns the relationships between the three inferior planets Venus, Earth and Mars. Irrespective of whether one considers that Earth presently occupies a* synodic* rather than
*planetary* position between Venus and Mars or not, the synodic cycles
in question are undoubtedly complex. It is well known, for example, that
Earth is in *2 : 1* resonant relationship with Mars; what is
not so well understood, however, is when and where such resonances actually
take place. Here again it is a matter of computing periodic functions,
in this case the sidereal motions of the two planets in question plus the
varying synodic cycle. A plot of the resulting time series data in 1-day
increments from 1990 to 1995 (1828 data points) is shown below:

One of the main advantages of time-series analysis is the ability to investigate complex interactions that are neither indicated nor suspected from mean or extremal values. A case in point concerns the relationships between the three inferior planets Venus, Earth and Mars. Irrespective of whether one considers that Earth presently occupies a

A similar approach may be adopted to include Venus. Here the investigation can be extended to include the determination of fractional constants and whole number multiples such that the synodic and sidereal cycles intersect. An example of the latter is shown in the following graph:

Next, combining all three inferior planets in the same manner including integer and fractional constants we also obtain:

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Lastly, a Lucas-series variant:

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Copyright © 1998. John N. Harris, M.A.(CMNS). Last Updated on July 16, 2004