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Planetary Correlation of the Giza Pyramids ‒ P4 Program Description

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Abstract and Figures

For more than 4500 years people have been fascinated by the pyramids of Giza. Now, there are very strong hints for a correlation between the three great pyramids in Giza and the three inner planets of our solar system: Mercury, Venus, and Earth. This assumption originally was based on three equations, connecting the sizes of the pyramids and planets. Later on, the spatial arrangement of the pyramids and planets could precisely be correlated on the basis of the French planetary theory VSOP87 and by coordinate transformations in three-dimensional space. Because of planetary motion, the positions match only at one or a few points in time. The final outcome includes the fact that in the year 3088 the four inner planets, Mercury to Mars, will stand in a fourfold conjunction within a minimum angular range (ecliptic longitudes) of 3.4°, combined with a simultaneous Mercury transit in front of the solar disk. Furthermore during the last decades, scientists have been searching with high-tech methods for undetected chambers in the Cheops Pyramid. An interesting aspect of the planetary correlation is that it precisely defines a “Sun position” and a “Mars position” inside the Cheops Pyramid. Are these two locations candidates for a new (secret) chamber? The probability that all these findings are based on combined accident is close to zero. The question, whether the Ancient Egyptians could have had this knowledge, is not discussed. Beside the description of the planetary correlation, this report is a user manual for the P4 computer program, enabling the corresponding astronomical calculations. For a better verifiability, it contains a listing of the entire Fortran source code.
Content may be subject to copyright.
Planetary Correlation
of the Giza Pyramids
P4 Program Description
Hans Jelitto
1. edition September 2014
2. edition June 2015. Beside minor revisions, the book title is rearranged,
section 4.7.4 about transit series has been added, the P4 source
code has been slightly worked over, and two more publications
of the author [3, 4] can be downloaded via provided links.
Copyright © 2014, 2015 Hans Jelitto
This work “Planetary Correlation of the Giza Pyramids – P4 Program
Description” (p4-manual-06-2015.pdf), which means text, calculations,
results, and figures, with the following exceptions:
– Figure 3
– Figure 12
– Equations (52) to (66)
– the whole P4 source code in the Appendix
is released under the creative commons license:
(CC) BY-NC-SA 4.0.
See also: http://creativecommons.org/licenses/by-nc-sa/4.0/
and: http://creativecommons.org/licenses/by-nc-sa/4.0/legalcode
For the figures 3, 12, and the equations (52) to (66) to calculate delta-T,
it has to be checked whether a permission from other authors or copy-
right owners is required. For the use of the P4 source code in the Appen-
dix (identical to the file “p4.f95”), of the executable program files “p4-32,”
“p4-64,” and “p4-4-64” and of all associated files, listed in Table 1, except
“p4-manual-06-2015.pdf” (being described above), more information is
provided at the end of this manual in the section “Use of P4 program/
Further Copyrights.”
Hans Jelitto, Ewaldsweg 12, D-20537 Hamburg, Germany
Hamburg, June 2015
This work is dedicated
to my parents
Karl and Käthe Jelitto
Preface
A correlation between the pyramids of Giza and the inner planets
of our solar system has been found. This manual is not only a
user guide for the P4 computer program regarding this correla-
tion, but it also provides some basic information about the techni-
cal and theoretical background, including archaeological, mathe-
matical, and astronomical aspects. Further details and several
other related results, which are not included here, are presented
in the book “Pyramiden und Planeten” (in German). A subsequent
book (in preparation) will provide more details about the results
given here. However, we tried to include all the necessary infor-
mation so that the reader can work properly with the manual and
the program. This manual is intended for scientists and for any-
one, who is interested in the secret of the pyramids.
For a basic overview of the planetary correlation, it is sufficient to
read chapter 1 (introduction), sections 3.1.1–3.1.3, 4.6.3, 4.10,
and chapter 5 (summary). Related lecture videos of the author on
YouTube (English subtitles) can be found with the search items
“pyramiden planeten jelitto.” For the essential ideas of the calcu-
lations, but not in the programming itself, chapter 4 provides the
underlying basic concepts.
Additionally, the appendix contains the entire source code of the
program, which is provided mainly for programmers. When print-
ing the manual, and if the source code is not needed, the corre-
sponding double pages 83–136 can be omitted. If possible, the
printout should be in color and double-sided if the printer supports
that feature. (So, an adequate ring binder can be made.)
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. General technical information . . . . . . . . . . . . 4
2.1 Data files and other related programs . . . . . . . . . . . . . . . 5
2.2 How to start the program . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. Program features . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1 Quick start options (1) – (15) . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 Pyramid positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.2 Chamber positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.3 Planetary conjunctions and transits . . . . . . . . . . . . . . . 14
3.2 Quick start options for the book tables . . . . . . . . . . . . . 17
3.2.1 Book 1 “Pyramiden und Planeten” . . . . . . . . . . . . . . . . 17
3.2.2 Book 2 (in preparation) . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2.3 Special test option (999) . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Detailed options (0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.1 Planetary positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.2 Linear constellations and transits . . . . . . . . . . . . . . . . . . 20
3.3.3 VSOP theory versions . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.4 VSOP coordinate systems . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.5 Transit options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.6 Calendar systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.7 Time systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3.8 Mapping of planets and chambers . . . . . . . . . . . . . . . . . 22
3.3.9 Search method for the dates . . . . . . . . . . . . . . . . . . . . . 22
3.3.10 “Sun position” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.11 Computation of free “Sun position” . . . . . . . . . . . . . . . . 23
3.3.12 Vertical coordinate of pyramid positions . . . . . . . . . . . . 23
3.3.13 The z-coordinate of chamber positions . . . . . . . . . . . . . 24
3.3.14 Datum plane for Earth's surface . . . . . . . . . . . . . . . . . . 24
3.3.15 Specification of timing . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.16 Tolerance in degree or percent . . . . . . . . . . . . . . . . . . . 25
3.3.17 Syzygy with simultaneous transit . . . . . . . . . . . . . . . . . 26
3.3.18 Polarity (orientation of planetary orbits) . . . . . . . . . . . . 26
3.3.19 Complexity of output . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.20 Mode of program output . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Some program outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Option 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 Quick start option 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4.3 Quick start option 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.4 Quick start option 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.5 Quick start option 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4.6 Quick start option 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.7 Quick start option 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4.8 Book option 250 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.9 Book option 381 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.10 Book option 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4.11 List of quick start options . . . . . . . . . . . . . . . . . . . . . . . . 40
4. Technical and theoretical basis . . . . . . . . . . 42
4.1 Positions at the Giza plateau . . . . . . . . . . . . . . . . . . . . . . 42
4.1.1 Positions of pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Positions of chambers . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 VSOP87 – planetary positions . . . . . . . . . . . . . . . . . . . . . 44
4.2.1 VSOP87 full version . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2.2 VSOP87 short version . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.3 Orbital elements and Kepler's equation . . . . . . . . . . . . 46
4.2.4 Accuracy of the theory . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Relation between pyramid and planet positions . . . . . . 48
4.3.1 1-dimensional comparison . . . . . . . . . . . . . . . . . . . . . . 50
4.3.2 2- and 3-dimensional comparison . . . . . . . . . . . . . . . . . 50
4.4 Two fit programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.1 FITEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4.2 Ringfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Coordinate transformation of planetary orbits . . . . . . . . 53
4.6 “Celestial positions” at the Giza plateau . . . . . . . . . . . . . 54
4.6.1 ”Sun position” by system of linear equations . . . . . . . . . 54
4.6.2 ”Sun position” by coordinate transf. and FITEX . . . . . . . 55
4.6.3 Additional “planetary positions” . . . . . . . . . . . . . . . . . . . 56
4.6.4 Geographical coordinates . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Syzygy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7.1 Planetary conjunctions . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7.2 Transit phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.7.3 Position angles of transit . . . . . . . . . . . . . . . . . . . . . . . . 62
4.7.4 Transit series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.8 Universal time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.9 Computational changes from P3 to P4 . . . . . . . . . . . . . . 69
4.9.1 Decimal year . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.9.2 Position tolerance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.9.3 Algebraic sign of X5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.9.4 Date of constellations 13 and 14 . . . . . . . . . . . . . . . . . . 71
4.10 Further specific features . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.10.1 Matching coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.10.2 Obliquity of the ecliptic . . . . . . . . . . . . . . . . . . . . . . . . 73
4.10.3 The riddle of midwinter . . . . . . . . . . . . . . . . . . . . . . . . 74
4.10.4 “Sun position” and concrete platform . . . . . . . . . . . . . 75
4.10.5 “Secret chambers” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5. Summary and epilogue . . . . . . . . . . . . . . . . . . . 79
Appendix P4 Source Code . . . . . . . . . . . . . . . . . . . . . . 82
Main program (basic information, modules) . . . . (double pages)1
– Declarations and initializations . . . . . . . . . . . . . . . . . . . . . . . 10
– 1. main program loop (pyramid and chamber positions) . . . 16
– 2. main program loop (“free search”) . . . . . . . . . . . . . . . . . . 22
– 3. main program loop (conjunctions and transits) . . . . . . . . 24
– format statements and program end . . . . . . . . . . . . . . . . . . 31
Subroutines and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
– Program input –
– “inputdata” (manual input) . . . . . . . . . . . . . . . . . . . . . . . 33
– “inputfile” (reads/writes quick start parameters) . . . . . 43
– “chambers” (changes allocation of chambers) . . . . . . . . 43
– “pchange” (interchange of two chambers) . . . . . . . . . . 43
– “pcheck” (reads and verifies input) . . . . . . . . . . . . . . 44
– “emes” (error message) . . . . . . . . . . . . . . . . . . . . . 44
– Time and dates –
– “konst” (check for constellations 1−14) . . . . . . . . . . 45
– “ephim” (converts different time formats) . . . . . . . . . 45
– “akday” (converts JDE k-number) . . . . . . . . . . . . 45
– “delta_T” (calculates T = TT − UT) . . . . . . . . . . . . . . 46
– “jdedate” (conversion JDE calendar date) . . . . . . . 47
– “sdint” (step function) . . . . . . . . . . . . . . . . . . . . . . . 49
– “weekday” (calculates day of the week) . . . . . . . . . . . . 49
– Astronomy –
– “vsop1” (VSOP87 short version) . . . . . . . . . . . . . . . 49
– “vsop2” (call of VSOP87 full version) . . . . . . . . . . . . 49
– “vsop3” (orbital elements and Kepler's equation) . . 50
– “transit” (computes transit phases) . . . . . . . . . . . . . 51
– “sepa” (separation Sun–planet) . . . . . . . . . . . . . . . 55
– “pos_angle” (position angle during transit) . . . . . . . . . . . 56
– “tserie” (serial number of transit) . . . . . . . . . . . . . . . 57
– “VSOP87X” (VSOP87-subroutine, upgrade) . . . . . . . . . 58
– Coordinates/ positions –
– “kartko” (converts to Cartesian coordinates) . . . . . . 62
– “relpos” (compares pyramid and planet positions) . . 62
– “sonpos” (computes “Sun position” in Giza area) . . . 63
– “invert” (inverts 33 matrix) . . . . . . . . . . . . . . . . . . . 67
– “rotmat” (applies rotational matrices) . . . . . . . . . . . . 67
– “translat” (translation of positions) . . . . . . . . . . . . . . . 68
– “mastab” (changes scale of coordinate system) . . . . 68
– “transfo” (transforms Cartesian coordinates) . . . . . . 68
– “kugelko” (converts to spherical coordinates) . . . . . . . 70
– “aphelko” (“aphelion position” in Giza area) . . . . . . . . 70
– “plako” (additional “planetary positions” in Giza) . . 71
– “geoko” (geographical coordinates) . . . . . . . . . . . . . 74
– “geokar” (geozentric Cartesian coordinates) . . . . . . . 75
– “reduz” (reduces angle to mean value) . . . . . . . . . . 75
– “memo” (keeps numbers in memory for later use) . . 75
– Program output –
– “info” (general information) . . . . . . . . . . . . . . . . . 75
– “titel1” (prints title of output) . . . . . . . . . . . . . . . . . . 75
– “titel2” (prints additional heading lines) . . . . . . . . . 77
– “tabe” (prints table head) . . . . . . . . . . . . . . . . . . . . 79
– “elements” (prints elements of planetary orbits) . . . . . . 81
– “linie” (draws a line in the output) . . . . . . . . . . . . . 82
– “zwizeile” (prints intermediate heading in table) . . . . . 82
– “comtime” (determines computation time) . . . . . . . . . . 83
– “endzeile” (prints summary at end of output) . . . . . . . . 83
– “save_ser” (creates the file “inser-2.t”) . . . . . . . . . . . . . 84
– Fit routines –
– “vsop1tr” (transit, speed of light, VSOP short v.) . . . . 85
– “vsop2tr” (transit, speed of light, VSOP full v.) . . . . . . 85
– “fitmin” (fit routine, two different algorithms) . . . . . . 86
– “ringfit” (circle method to calculate roots) . . . . . . . . 89
– “sekante” (secant method to calculate roots) . . . . . . . 90
– “FITEX” (FITEX: main subroutine of fit program) . . . 90
– “FIT1” (FITEX: minimization of a function f(x)) . . . 98
– “INVATA” (FITEX: inversion of a product matrix) . . . . 100
– “LILESQ” (FITEX: linear least squares problem) . . . . 103
Use of P4 program / Further Copyrights . . . . . . . . . . . . . . . . . . . . . . 137
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
1. Introduction
The purpose of the P4 program is to perform astronomical calculations with respect to the planets
of our solar system and the three pyramids of Giza (Fig. 1). P4 is based on the French planetary
theory VSOP87 [1, 2] (see more below). The fundamental idea is that a correlation exists among
the three inner planets and the three pyramids in Giza. The first papers about this hypothesis were
published in the Austrian journal Grenzgebiete der Wissenschaft (in German) in 1995 [3, 4]. About
two years earlier, the development of the program P3 began, allowing for the mathematical compa-
rison of pyramid positions and planetary positions. Because of three equations (see section 3.1.1)
that define the size of each pyramid, it seems that the Cheops Pyramid (Great Pyramid), the Chef-
ren Pyramid, and the Mykerinos Pyramid represent the planets Earth, Venus, and Mercury, respec-
tively. Furthermore, the pyramid positions correlate with the planetary positions. Because the plan-
ets are moving all the time, their arrangement and distances between each other change continu-
ously. This implies that the geometric arrangements of pyramids and planets match for only one or
a few points of time. Such dates were found, depending on the mathematical approach and further
boundary conditions. So, among other things, the program calculates the dates when Mercury,
Venus, and Earth stand in a constellation according to the arrangement of the Giza pyramids (see
Fig. 1 and [5, p. 95]). The data in Fig. 1 were measured by Sir W. M. F. Petrie [6, 6a]. Excellent
geographical maps, reproducing the pyramids in Egypt, are available, for example, in Cairo [7, 8].
1
Figure 1: Alignment of the Giza pyramids with measured data of W. M. F. Petrie [6, 6a] (distances in meters). Two num-
bers are slightly corrected: The large diagonal is 936.16 m instead of 936.19 m, and one angle is 31° 55' 13'' instead of
34° 10' 11'' (explanation in [5, p. 96; 6, p. 125]). The relative elevations stem from S. Perring (see: [9, part IV, map 1]).
Detailed information is provided in the drawings of Maragioglio and Rinaldi [9]. The angles were calculated from the orig-
inal distances, given in inches (1 inch = 2.54 cm).
The archaeological state of knowledge is that the three great pyramids in Giza were built by the
Egyptian pharaohs Chufu, Chaefre, and Menkaure within the 4th dynasty. In addition to these Egyp-
tian names, the Greek names are Cheops, Chefren, and Mykerinos. In the archaeological chronol-
ogy, the 4th dynasty is dated roughly between the years 2600 and 2480 BC [10, vol. I, p. 970]. (“BC”
means “before Christ.”) On the other hand, in 1987 and 1994 it was reported that the age of sev-
eral buildings of the Old Kingdom, including the pyramids of Giza, was determined independently
with the “accelerator mass spectrometry” (AMS) [11, 12], ordered by the ETH Zürich in Switzer-
land. This is a modern variant of radiocarbon dating in which a particle accelerator is used to deter-
mine the amount of radioactive 14C-isotopes. The result is that, for example, the Cheops Pyramid
has to be dated between the years 3030 BC and 2905 BC with a probability of 95 %. This is a
discrepancy of approximately 400 years! Because it is impossible to shift the chronology of the
pharaohs by 400 years, the reader should keep this point in mind (see details in [5, pp. 361 ff.]).
The first program version was named P3 because of the 3 great pyramids in Giza and the 3
planets Mercury, Venus, and Earth. It was used for computing the astronomical tables in the book
“Pyramiden und Planeten” [5]. After this book was published in 1999, another correlation was
found, namely between the planetary positions and the chamber positions in the Great Pyramid
with an unexpected connection between both correlations. This led to an extension of the program
P3 with several other options. The new program name is P4 because it is an upgrade of P3 and
includes the fourth planet Mars. P4 covers all features of P3, the processing speed has been
optimized, and the application is much easier. The results, which cannot all be provided in this
manual, are described in detail in the subsequent book, here named “Book 2” [13]. Unfortunately,
until now all publications are in German. As a first remedy, this description is written in English.
The comparison of the arrangements is performed mathematically by a coordinate transformation.
An interesting point of this correlation is that, by using the transformation of the planetary arrange-
ment, the position of the Sun can be precisely transferred to the pyramid area (Fig. 2). This means
that we have a “Sun position” at the Giza plateau. Furthermore, the positions of the chambers
define another “Sun position” inside the Cheops Pyramid. In the following, “Sun position” is written
in quotations because here we do not refer to the real Sun but to the corresponding position in the
pyramid area. Later, we will also find a “Mars position” in the Cheops Pyramid.
Figure 2: Schematic representation of the Earth's surface around the Giza pyramids and the orbits of the three inner
planets Mercury, Venus, and Earth, after adaption of pyramid and planetary positions. The geometric arrangement of
constellation number 12 (section 3.4.3) looks very similar. Because of different inclinations, the orbits are slightly tilted
against each other. This fact is neglected in the drawing but is taken into account in the calculations.
2
In Fig. 3 the remarkable system of chambers and corridors in the Cheops Pyramid is given, which
also plays a major role in the correlation between pyramids and planets. The names of the cham-
bers, like “King's chamber” and “Queen's chamber,” originate from classical archaeology and are
based on the explanation that the pyramids were tombs of the pharaohs. At first glance, this
explanation seems reasonable because it is written in countless books. However, this interpretation
is not necessarily correct because a mummy has never been found in an Egyptian pyramid!
Numerous mummies of kings and queens have been discovered, but all of them were found in hid-
den tombs in the desert, like in the Valley of the Kings. More about this is provided in [5].
3
Figure 3: Chamber and corridor system inside the Great Pyramid, as seen from the south (a) and from the
east (b). The figure is based on drawings of Maragioglio and Rinaldi [9, part IV, maps 3–7] (arrangement of (a)
and (b) like in Ref. [14] of R. Stadelmann).
Some of the boundary conditions for the comparison of pyramid and planetary positions are the
following: The “Sun position” can be fixed by placing the “Sun” on the Earth’s surface exactly on
the center line 726 m south of the Mykerinos Pyramid (see Fig. 4). The “Sun position” can also be
free in the two horizontal coordinates and would be fixed only to the Earth’s surface by adapting
the planetary positions, or it can be free in all 3 dimensions. In order to get a better idea, an exam-
ple of the two systems “pyramids” and “planets,” using a 3-dimensional fit, is shown in Fig. 2. The
two planes, “Earth’s surface” and “ecliptic plane” (plane of the Earth orbit), are not coplanar but
tilted against each other. (More about this is given in section 4.10.2.)
Furthermore, the P4 program computes the dates of “linear constellations” of the celestial bodies
Mercury, Venus, Earth, Mars, and the Sun, which means that the planets have nearly the same
ecliptic longitude. These “linear arrangements” of celestial bodies (conjunction and opposition) are
called “syzygy.” In addition, the exact geocentric transit phases, when Mercury or Venus passes
the Sun's disk, can be determined. All options and parameters of the P4 program are provided in
chapter 3.
The calculations were performed as accurately as possible. Strong emphasis was put on the use of
the most recent and precise scientific data. This refers to the astronomical data and computations
as well as to the archaeological data. Concerning the exact dimensions of the Cheops-Pyramid,
the latest results were not always the most accurate one. The reason is that due to weathering
effects the measurement conditions at the Pyramid about 100 years ago were partly better than
today. This is discussed in detail in Ref. [5, pp. 249–255]. However, the corresponding small differ-
ences are important only for the exact shape but not for the size of the Cheops-Pyramid and do not
have any effect on the results in this manual.
2. General technical information
The astronomical calculations are based on the planetary theory VSOP87, developed by P. Bretag-
non and G. Francou at the Bureau des Longitudes, Paris (today: IMCCE, Institut de mécanique
céleste et de calcul des éphémérides) [1, 2]. VSOP means “Varations Séculaires des Orbites
Planétaires” and 87 is the year of publication (1987). The files for the VSOP87 theory can be down-
loaded from the FTP server of the IMCCE homepage: ftp://ftp.imcce.fr/pub/ephem/planets/vsop87/.
A multi-parameter fit program “FITEX” (fit experiment) [15, 16], included in P4, was developed by
G. W. Schweimer, Zyklotron-Laboratorium, KfK (Kernforschungszentrum Karlsruhe, today: KIT,
Karlsruher Institut für Technologie). For calculating calendar dates, an algorithm from the book
“Astronomical Algorithms” by J. Meeus [17] is converted into a subroutine of P4. The conversion of
“terrestrial time” (TT) to “universal time” (UT) is performed by using T = TT UT, calculated by
F. Espenak and J. Meeus (“NASA Eclipse Web Site,” Polynomial Expressions for Delta-T). The P4
program, all subroutines, and other programs from the author are written in Fortran. The whole
package of programs, and all associated files, can be downloaded from the author's homepage:
www.pyramiden-jelitto.de/downloads.html. Note: Through the provided links, many of the given
references can also be downloaded from the Internet. For details of the theoretical basis, see
chapter 4.
The previous program (P3) was originally developed with the IBM Professional Fortran 77 Com-
piler (Version 1.0, Ryan-McFarland) using the SPF/PC editor and the Windows operating system.
Later on, we switched to the GNU-Fortran compiler g77 with Ubuntu Linux, and then to GNU
gfortran. Of course, it is possible to use other Fortran compilers and other operating systems.
It would also be of interest to port the program to languages like C, C++, or Java. However,
because the architectures of these programming languages are quite different to Fortran, it is prob-
ably easier to write a new program. In addition, it would be a good test of the results if the calcula-
tions were performed independently and based, for example, on a theory other than VSOP87.
4
2.1 Data files and other related programs
Table 1 contains a compilation of all files belonging to the astronomical program P4. A few com-
ments about other available programs, and more information about the files in Table 1, are pro-
vided in the following. In this section, program, catalogue, and file names are highlighted in blue.
5
Table 1: All 32 (36) files of the P4 program: program, text, and data files (download from the author's homepage).
File Brief description
p4.f95 Fortran source code, (p4-4.f95, p4-4.pdf: parallelized version, see footnote on page 82)
p4-32 Executable program file for a 32-bit system (can also be used on a 64-bit system)
p4-64 Executable program file for a 64-bit system, (p4-4-64: parallelized version, 4 threads)
p4-32.sh This shell-script clears the screen display and starts p4-32.
p4-64.sh Shell-script – as above – starts p4-64, (p4-4-64.sh: starts p4-4-64)
p4-manual-
06-2015.pdf
User manual with details of the P4 program and main aspects of the planetary correlation
concerning the Giza pyramids (this text)
README Notice to the theory planetary solutions VSOP87 from Bretagnon and Francou
vsop87.doc Technical information about the VSOP87 theory from Bretagnon and Francou
out.txt Output file (if it does not exist, it will be created by the program with the corresponding option)
inedit.t Ancillary input file (can be used to create a new set of parameters for inparm.t)
inparm.t This file contains all input parameters for the quick start options 1 to 15, for Tables 39 to 51 in
[5], and for Tables 17 to 33 and 35 to 37 in [13].
inpdata.t Parameters for FITEX and coordinates of pyramid and chamber positions in Giza
inserie.t Dates of transit series for Mercury and Venus (used only at program start)
invsop1.t Shortened VSOP87D data for the planets Mercury to Mars, typewritten manually from:
J. Meeus, “Astronomical Algorithms” [17, pp. 381 ff.]
invsop3.t Polynomial representation of orbital elements, derived from VSOP82 and taken from:
J. Meeus, “Astronomical Algorithms” [17, pp. 200 ff.]
VSOP87A.mer Mercury: VSOP87A, heliocentric rectangular coordinates, ecliptic J2000.0
VSOP87A.ven Venus: . . . '' . . .
VSOP87A.ear Earth: . . . '' . . .
VSOP87A.mar Mars: . . . '' . . .
VSOP87A.jup Jupiter: . . . '' . . .
VSOP87A.sat Saturn: . . . '' . . .
VSOP87A.ura Uranus: . . . '' . . .
VSOP87A.nep Neptune: . . . '' . . .
VSOP87A.emb Earth-Moon barycenter: . . . '' . . .
VSOP87C.mer Mercury: VSOP87C, heliocentric rectangular coordinates, dynamical equinox
VSOP87C.ven Venus: . . . '' . . .
VSOP87C.ear Earth: . . . '' . . .
VSOP87C.mar Mars: . . . '' . . .
VSOP87C.jup Jupiter: . . . '' . . .
VSOP87C.sat Saturn: . . . '' . . .
VSOP87C.ura Uranus: . . . '' . . .
VSOP87C.nep Neptune: . . . '' . . .
The files README and vsop87.doc in Table 1 provide details about the theory versions of VSOP87
and are given directly by the authors Bretagnon and Francou (download from the homepage of
IMCCE, link provided previously). The file out.txt contains the results after running P4, if the output
parameter is not set otherwise. The next six files in Table 1, beginning with in... are input files,
necessary to run P4. In the file inparm.t, all parameter sets for the quick start options are com-
piled. File inedit.t is a combined input-output file. During each run, all input parameters are stored
at the end of this file. This ancillary file helps to create new parameter sets, which can be added as
new quick start options to the file inparm.t. In this case, the subroutine “inputdata” in p4.f95 has to
be properly adapted. The input parameters in the file inedit.t can also be edited manually and are
adopted by the program with the quick start option 999. This allows for testing new parameter sets.
The file inpdata.t contains parameters for the subprogram FITEX as well as the exact coordinates
of the pyramid chambers and of the pyramids themselves. In inserie.t, several dates (JDE) are
listed to determine the serial numbers of the first Mercury or Venus transits, found after program
start. In invsop1.t are the shortened parameter series of the VSOP87D version, taken from [17,
pp. 381 ff.]. The file invsop3.t contains coefficients for polynomials of third degree for the elements
of planetary orbits, deduced from the version VSOP82 [17, pp. 200 ff.]. All remaining files from
VSOP87A.mer to VSOP87C.nep represent full versions of the planetary theory [1, 2] with a very
high accuracy. They are also available from the FTP server of the IMCCE homepage.
This paragraph provides some general information about the other programs used within the
present pyramid research; the P4 and TOPO programs are new and used in reference [13]. TOPO
calculates the exact volume of the Earth, including the volume of all ice and land masses. All other
programs, including P3, are used and described in the first book [5]. This includes the programs
FORM, SEKAN, PYT, and 7916, which enable geometric calculations concerning the shapes of the
three pyramids of Giza, especially their casing angles. The program DATUM-2 converts the time
system “Julian Ephemeris Day”
1 into a calendar date and is based on an algorithm from the book
by Jean Meeus [17, p. 63]. The program SKYGLOBE [18] is a “planetarium” simulation of the sky
and shows the celestial bodies like stars, planets, Sun, and Moon, as well as Milky way and con-
stellations for every date and location on Earth. It was written by Mark A. Haney as shareware and
is available at http://astro4.ast.vill.edu/skyglobe.htm. In this project, it has been applied only to
check the “ORION correlation” propounded by R. Bauval and A. Gilbert [19]. When the positions
and proper motions of the corresponding stars were taken into account for a quantitative analysis
of the “ORION theory,” large errors and deviations were found. On the one hand, Bauval and
Gilbert were the first to correlate the pyramids with celestial bodies. On the other hand, their
ORION hypothesis did not pass the test [5, pp. 157 ff., 349 ff.]. The analysis in [5] is based on the
Star Catalogue PPM (Positions and Proper Motions) [20, 21].
For those who are interested, the text, formulas, and most figures, including the book cover, were
created using Ubuntu with OpenOffice (now LibreOffice), Inkscape, and GIMP.
2.2 How to start the program
The P4 program does not need any installation. After downloading and unpacking the files, the
easiest way is to store all of them in the same folder (directory), which can be named “P4,for
example. It is assumed now that the operating system is Linux, because the P4 program was
finally developed on the Linux distribution Ubuntu. If another operating system is installed, it is
normally necessary to compile the source code p4.f95 again. In the case of a Windows system,
p4.f95 can be compiled with a Windows compliant Fortran 95 compiler (e.g., ifort or gfortran with
MinGW or Cygwin). Other possibilities are to create a Linux partition beside Windows, to use a
Linux life CD like Knoppix, or to apply “VirtualBox.” Special characters are not used in the program
output, so for character encoding, the Unicode UTF-8 or ISO 8859-15 can be applied.
1 In order to keep consistency with [5] and with the notation of Meeus [17], JDE (“Julian Ephemeris Day” or “Julian Day”)
was used, based on terrestrial time (TT). Today, JD respectively JD(TT) has the same meaning.
6
After creating the folder, we open a terminal in Ubuntu (classic Gnome) through the menus “Appli-
cations Accessories Terminal.” (In most cases, a terminal window width of 80 characters is
sufficient; only one option needs a line length of 148 characters – see section 3.3.5.) In the follow-
ing, all texts on the monitor screen like commands, menus, input data, and program results are
printed in blue (not program names and file names). If, for instance, the folder has the name “P4”
and is located in the path ~/Desktop/P4$, we type the following commands at the command line
in the terminal: cd Desktop/P4 . Now we are in the right folder. The sign “” denotes the return
key. To start the program on a 64-bit computer system, we type ./p4-64.sh , which clears the
screen, and the start menu appears. Another possibility is to start P4 directly with ./p4-64 with-
out clearing the screen. If the program does not start type chmod +x p4-64* . In the case of a
new compilation of the source code with GNU Fortran, use the command gfortran -static -O3
-Wall p4.f95 . For a 32-bit system, the files p4-32.sh or p4-32 have to be used. (Replace 64
by 32 in the above commands.) That's it! In the next chapter we will see how to proceed.
3. Program features
After having typed the start command, the main menu appears on the monitor:
-----------------------
PLANETARY CORRELATION
Program P4, June 2015
-----------------------
pyramids of Giza chambers, Great P. transits, syzygy
----------------------------------------------------------------------
3D Mer. at aph. (1) 3D Mer. at per. (6) Mercury tr. (11)
2D Mer. at aph. (2) Keplers equ. (7) Venus tr. (12)
const. 12, 3088 (3) const. 12, 3088 (8) syzygy, 3 pl. (13)
1.5 days, 3088 (4) 1.5 days, 3088 (9) syzygy, 4 pl. (14)
near aphelion (5) F minimized (10) TYMT-test (15)
----------------------------------------------------------------------
info (111) detailed options (0) (0..15 or book options) : _
The date in the title indicates the last update of the program. In the table we find three different
categories. The first options, (1) to (5), belong to the pyramids in Giza, the options (6) to (10) have
to do with the chambers in the Great Pyramid, and the options (11) to (15) represent planetary
conjunctions. The latter case includes different astronomical events: The three inner planets or the
four inner planets of our solar system stand in conjunction (syzygy). Additionally, if Mercury or
Venus are in conjunction with the Sun, it happens sometimes that they pass in front of the solar
disc, which is called a “transit.” However, before confusion arises, the astronomical relationships
are explained in more detail in the following sections.
3.1 Quick start options (1) – (15)
Normally, about 10 to 15 different parameters have to be fixed before the astronomical calculation
starts. These parameters determine, for example, the kind of astronomical event, the used VSOP-
version, the coordinate system, the mode of calculation for the “Sun position,” the time period to be
examined, the complexity of the output, and so on. In order to avoid this, the general quick start
options (1) to (15) start the program with predefined parameters just after having typed a short
number. For example, typing 12 makes the program calculate all Venus transits for the years
from 1500 AD to 4000 AD. (“AD” means “Anno Domini” or “after Christ.”) The program output is:
7
TRANSITS OF VENUS
(geocentric transit phases, terrestrial time TT)
< option 12 >
VSOP87C, comb. search, ecliptic of date, all Venus transits
Period (years) from 1500.00 to 4000.00, Jul./Greg. calendar
co/p date/ time: I II nearest III IV sep["]a S
===============================================================================
26. May 1518 22:32: 2 22:49:14 1:59:45 5:10:16 5:27:28 -505.3 3
23. May 1526 16:17:38 16:38: 9 19:14:43 21:51:18 22:11:49 666.7 5
---- (Greg. cal.) -------------------------------------------------------------
v 7. Dec. 1631 3:53:17 5: 2:16 5:20:49 5:39:22 6:48:20 939.3/ 6
4. Dec. 1639 14:58: 4 15:16:26 18:26:47 21:37: 8 21:55:29 -523.6/ 4
6. June 1761 2: 2:20 2:20:35 5:19:30 8:18:25 8:36:40 -570.4 3
3. June 1769 19:15:49 19:34:52 22:25:36 1:16:20 1:35:23 609.3 5
9. Dec. 1874 1:49:12 2:18:56 4: 7:22 5:55:49 6:25:33 829.9/ 6
6. Dec. 1882 13:56:41 14:17:10 17: 5:54 19:54:38 20:15: 7 -637.3/ 4
8. June 2004 5:14:47 5:34:13 8:20:49 11: 7:24 11:26:51 -626.9 3
6. June 2012 22:10:56 22:28:53 1:30:43 4:32:33 4:50:30 554.4 5
11. Dec. 2117 0: 2:31 0:25:39 2:52: 8 5:18:38 5:41:46 723.6/ 6
8. Dec. 2125 13:19:29 13:43:10 16: 5:49 18:28:28 18:52: 8 -736.4/ 4
11. June 2247 8:51:10 9:12:31 11:42:27 14:12:24 14:33:45 -691.3 3
9. June 2255 1:17:39 1:34:41 4:47:36 8: 0:31 8:17:33 491.9 5
13. Dec. 2360 22:47:17 23: 7:30 1:58:44 4:49:57 5:10:10 625.7/ 6
10. Dec. 2368 12:44:56 13:15:24 15: 0:28 16:45:33 17:16: 0 -836.4/ 4
12. June 2490 12: 1:48 12:25:17 14:39:42 16:54: 7 17:17:36 -741.1 3
10. June 2498 4:12: 4 4:28:32 7:48:35 11: 8:38 11:25: 6 442.7 5
16. Dec. 2603 21:14:54 21:33: 8 0:44:29 3:55:49 4:14: 3 517.1/ 6
v 13. Dec. 2611 12:36:50 13:40:18 14: 6: 9 14:31:59 15:35:27 -934.8/ 4
15. June 2733 15:45: 8 16:13:18 18: 0:58 19:48:39 20:16:49 -808.3 3
13. June 2741 7:17: 8 7:33: 5 11: 0:24 14:27:43 14:43:40 385.6 5
17. Dec. 2846 20:24:29 20:41:44 0: 5:13 3:28:41 3:45:55 432.1/ 6
v 14. Dec. 2854 -- -- 13:14:26 -- -- -1026.7/ 4
16. June 2976 18:54: 5 19:27:43 20:53: 7 22:18:32 22:52: 9 -850.5 3
14. June 2984 10:10:33 10:26: 9 13:58:46 17:31:23 17:46:59 336.3 5
-> 18. Dec. 3089 19: 1:49 19:18:10 22:53:36 2:29: 2 2:45:23 320.6/ 6
v 20. June 3219 22:31:18 23:28: 6 0: 0: 6 0:32: 6 1:28:55 -908.1 3
17. June 3227 13: 3:37 13:18:56 16:55:19 20:31:43 20:47: 2 293.4 5
20. Dec. 3332 18:14:30 18:30:23 22:12: 4 1:53:44 2: 9:38 235.5/ 6
v 22. June 3462 1:48:43 -- 2:46:32 -- 3:44:19 -948.1 3
19. June 3470 15:51:28 16: 6:35 19:46:41 23:26:48 23:41:55 247.9 5
23. Dec. 3575 17: 7:58 17:23:32 21:10:32 0:57:31 1:13: 5 131.5/ 6
v 24. June 3705 -- -- 5:35:19 -- -- -989.3 3
21. June 3713 18:30:27 18:45:25 22:27:21 2: 9:18 2:24:17 215.2 5
c 25. Dec. 3818 16:23: 6 16:38:31 20:27:15 0:15:58 0:31:22 41.1/ 6
24. June 3956 21:17:37 21:32:30 1:16:53 5: 1:17 5:16:10 175.2 5
===============================================================================
Computed constellations: 11092 ("/" means ascending node)
Tested planet. passages: 1564
Detected transits : 37
Centr./grazing transits: 1 / 6 CPU-time 0: 0: 0.728 -- end of run.
The general appearance of the printed output is always similar. The first line shows the title of the
program run followed by the second line, giving some basic information. In the third line we find the
number of the selected option, which is often a quick start option. Two up to five lines follow, provid-
ing the remaining information in a brief form so that it is later possible to understand what has been
calculated. These two up to five lines include the following data: the theory version of VSOP, the
astronomical coordinate system, some data about the planets, pyramids or chambers, the time
period, the allowed angular range (e.g., the range of the ecliptic longitudes), and other information.
8
In principle, there are two kinds of output of different magnitudes. At first, each astronomical event,
like a transit, is written down in a single line as provided in the previous table. This kind of output is
useful to get an overview when large time periods are investigated and when many planetary
constellations are found. The other possibility is to characterize every astronomical event by much
more information in several lines. At the end of the output, one or more lines give a summary of
the program run. This includes, for example, the number of calculated and detected astronomical
events as well as the CPU-time in hh:mm:ss.sss. More information about these different kinds of
output is provided in section 3.4.
The following provides an example of an extended output for each found constellation. In order to
illustrate it and avoid an output, which is too long, the time limits (3000 to 3200 AD) are chosen in
such a way that only one planetary constellation is found and printed. The calculation is performed
with a simple approach by iteratively solving Kepler's equation. In this program run, the main con-
dition is that the four planets Mercury, Venus, Earth, and Mars stand in a straight line. It means that
they have almost the same ecliptic longitude, which is called a conjunction or syzygy. The first line
of numbers in the table contains the information that is also shown in a short program output. The
additional lines provide the orbital elements for all eight planets. For more details, see sections 3.3
and 3.4. (This conjunction seems to be an important event with respect to the Giza pyramids.)
PLANETS IN A LINE (SYZYGY)
(angular range of eclipt. longitudes dL minimized, JDE)
< option 0 >
"Keplers equation", ecliptic of date, linear c. Mercury to Mars
Period (years) 3000.00 to 3200.00 (c2) angular range: 5.0000 deg
co k JDE year dt[days] Lm-Lv Lm-Le Lm-Lma dLmin
===============================================================================
12 4519 2849066.03400 3088.376 -13.729 -3.366 -2.601 0.0 3.366
-------------------------------------------------------------------------------
pla. mean long. a [AU] eccentr. asc.node incl. per. per.[AU]
-------------------------------------------------------------------------------
Mer 218.24880 0.38710 0.20585 61.26192 7.02274 94.43121 0.30741
Ven 237.78863 0.72333 0.00626 86.53534 3.40547 146.69081 0.71880
Ear 236.06015 1.00000 0.01624 --- 0.00000 121.70696 0.98376
Mar 244.75076 1.52368 0.09438 57.96608 1.84469 356.11360 1.37988
Jup 319.97784 5.20261 0.05021 111.62426 1.24399 31.99903 4.94137
Sat 46.22049 9.55489 0.05166 123.19406 2.44653 114.53500 9.06126
Ura 312.51632 19.21845 0.04601 79.86019 0.78595 189.20812 18.33424
Nep 177.48520 30.11039 0.00906 143.80990 1.66784 63.69138 29.83766
===============================================================================
Computed constellations: 1052
Number of syzygies : 1 CPU-time 0: 0: 0.008 -- end of run.
The 15 quick start options, representing typical program runs, are specified in more detail. The
mode, in which the parameters are defined individually one after the other, can be entered with the
option “0.” A detailed description of all corresponding menus is provided in section 3.3. It is antici-
pated here, that there are many more quick start options than 15. The additional quick start options,
having three digits, are intended to reproduce the results in the tables of the two books [5, 13].
For a better understanding, we must mention that in Ref. [5] 14 different dates and associated
planetary constellations within the period 13,000 BC to 17,000 AD were analyzed in detail, depend-
ing on the geometrical approach, when comparing pyramid positions with the positions of the
planets. These constellations were numbered 1 to 14. Five of them were more significant than the
others, but later it became clear that the constellation with the number 12 is the most important one
[13].
9
3.1.1 Pyramid positions
One of the main results of the first book [5] is that the three inner planets correlate with the three
pyramids of Giza. More precisely, the Cheops Pyramid represents the planet Earth, the Chefren
Pyramid represents Venus, and the Mykerinos Pyramid represents Mercury. The book describes
how the correlation between pyramids and planets was discovered. Three basic equations were
found that define the sizes of the pyramids. The maximum relative error of these equations is
0.2 %. From recent systematic studies, it came out that the relative uncertainty of the first equation
is approximately 0.001 % [13] ! With S being the base length of the pyramid, V the volume, Q the
aphelion distance (largest distance to the Sun), and c the speed of light, these equations are as
follows:
SCheops
c1s =VEarth
VSun
,
VCheops
VChefren
=VEarth
VVenus
,
SCheops
SMykerinos
=QEarth
QMercury
(1) (3)
(Cheops Pyramid) (Chefren Pyramid) (Mykerinos Pyramid)
10
Figure 4: Correlation between the inner three planets of our solar system and the three
pyramids of Giza. The positions are each projected vertically into the main plane. Mercury
is placed exactly at the aphelion. For better visibility, the Sun is magnified by a factor 6 in
relation to the planetary orbits, and the planets by a factor 500 [5].
Option 1: 3D Mer. at aph. (1) (Compare with main menu at beginning of chapter 3.)
“3D” means 3-dimensional calculation: The comparison of the positions of pyramids and planets is
performed by considering all 3 dimensions. The vertical position of a pyramid is given here by its
center of mass, which is located at a quarter of the pyramid's height. The date is restricted in the
way that Mercury is always placed at the aphelion of the orbit, having the largest distance to the
Sun. The investigated time period are the years 13,000 BC to 17,000 AD. The results of the
VSOP87 theory become less precise, if the date proceeds thousands of years into the past or into
the future. Nevertheless, an estimate of the accuracy [2] (see also section 4.2.4) shows that within
the given years it is by far sufficient for our purpose. In the P4 program, the dates for the application
of VSOP87 are mostly restricted to the described time period. It means that start and end dates
can be chosen freely within that period, but cannot exceed those limits.
The detected dates are listed in a table, where every date is represented by one line. Special dates
are marked at the beginning of the line with the number of the corresponding constellation. These
numbers, 1 to 14, indicate certain planetary constellations, which are defined and described prima-
rily in [5]. The output table using this option is also given in [5, p. 346, upper part of Table 50]. For
more details, see sections 3.3 and 3.4.2.
Option 2: 2D Mer. at aph. (2)
This calculation is similar to that of option 1 with the distinction that the calculation is restricted to
2 dimensions. This means that the positions of the pyramids are projected into the horizontal plane
of the Earth's surface. Accordingly, the positions of the planets are projected into the main plane,
given by the plane of the Earth's orbit, respectively. Therefore, the vertical coordinates are not
taken into account (see also [5, Table 45 on p. 327]).
Option 3: const. 12, 3088 (3)
This option calculates all relevant quantities for the constellation 12. This planetary constellation at
May 31, 3088, 6:19:09 a.m. (TT, terrestrial time) represents the most relevant event out of the 14
constellations concerning the pyramid positions in Giza. Mercury is placed again at the aphelion.
Notice that in the Eq. (3) (above Fig. 4), the aphelion distance QMercury appears. Additionally, the
heliocentric coordinates of all planets from Mercury to Neptune for this special date are trans-
formed to coordinates at the Giza plateau (see also [13, Table 26 in appendix A2]). The program
output is given in section 3.4.3 (see also [13, chapter 4]).
Option 4: 1.5 days, 3088 (4)
In this case a time scan around the date of constellation 12 (pyramid positions) is created. The po-
sitions of the planets are given in time steps of one hour beginning 18 hours before and ending 18
hours after the date of constellation 12 (therefore “1.5 days”). So, the slow change of all important
parameters can be followed easily when times pass through the main moment. Compare with [13,
Table 24 in app. A1] and see also section 3.4.4.
Option 5: near aphelion (5)
This search for planetary constellations represents the pyramid positions in Giza without the
restriction that Mercury is placed at the aphelion. It came out that the planets Mercury, Venus, and
Earth are in line with the pyramid positions only when Mercury is placed not too far away from its
aphelion position. So, in order to keep the computation time short, the constellations are firstly
checked with Mercury in the aphelion. If the agreement of the positions is good enough, Mercury is
placed outside (but near) the aphelion position (short version VSOP87). At the beginning of each
line, “F” means relative error ≤ 0.5 %; “M” means error of scale factor ≤ 2 %; and “>>>” means both
errors ≤ 0.1 %. The errors and especially the theoretical scale factor “M” are described in [5].
11
3.1.2 Chamber positions
Interestingly, 44 days before the “pyramid date” of constellation 12, the planets Mercury, Venus,
and Earth represent the arrangement of the chambers in the Great Pyramid. At this moment,
Mercury is placed exactly at the perihelion of its orbit, the nearest point to the Sun. The correlation
between planets and chambers can be seen in Fig. 5. Notice that the “chamber constellation” also
defines a “Mars position” within the Great Pyramid above the King's chamber. Additionally, the “Sun
position” could be the place of another (secret) chamber. For detailed information and exact coordi-
nates of the new locations, see section 3.4.8. Between the two dates of chamber and pyramid
positions, the five celestial bodies (Sun, Mercury, Venus, Earth, and Mars) are placed nearly in a
straight line. This “linear constellation” (syzygy) is examined in section 3.1.3.
12
Figure 5: Cross-sectional area of the Great Pyramid (Cheops Pyramid) as seen from the east [13]. Common representa-
tion of the known original chamber system in the pyramid, the arrangement of the pyramids themselves, and the planeta-
ry orbits. The time span between the constellations of chamber and pyramid positions is 44 days (red paths), which is
half of Mercury's orbit. On the right side of the “Sun” Mercury is placed at perihelion, on the left side at aphelion. The
whole figure corresponds to constellation 12. All “planetary positions” can be calculated with P4 using the option 380.
The configurations in the figure are roughly true to scale. For the exact positions, use the calculated coordinates.
The origin of the coordinate system is placed at the vertical middle axis of the east wall of the
Queen's chamber on the level of the pyramid base. The x-axis points to the north, the y-axis points
upward (compare with Fig. 5), and the z-axis points to the east. The quick start options, which have
been implemented as main examples, are the following:
Option 6: 3D Mer. at per. (6)
The calculation is analog to option 1. The positions of the pyramids are replaced by the positions of
the chambers in the Great Pyramid, and Mercury is always located at its perihelion. The investi-
gated time period lasts again from 13,000 BC to 17,000 AD.
Option 7: Keplers equ. (7)
Here, the planetary positions are not calculated with the short or the full version of VSOP87. In-
stead, the positions are determined with the orbital elements by solving Kepler's equation (section
4.2.3). The orbital elements are derived from the VSOP82 theory [17, pp. 197 ff.], and the tran-
scendental equation of Kepler is solved numerically. The other boundary conditions are similar to
option 6. This method does not have the accuracy of the full version of VSOP87, but it has the
advantage that the calculation is rather fast. When the same time period of 30,000 years is investi-
gated, 124,558 constellations are calculated and checked, and the overall computation time is less
than 1 second. Moreover, it is a good test of the other results (see also section 3.4.5).
Option 8: const. 12, 3088 (8)
This computation is analog to option 3. Only the positions of the pyramids are replaced by the posi-
tions of the chambers in the Great Pyramid, and Mercury is placed at its perihelion (section 3.4.8).
The exact date is April 17, 3088, 6:41:13 a.m. (TT, terrestrial time). Now, the planetary positions
are all transformed to the coordinate system of the Great Pyramid. With a deviation of 4.2°, the
transformed ecliptic plane (plane of the orbits) is almost parallel to the central vertical plane in the
pyramid, oriented in north–south direction (x-y-plane). The corresponding origin of this coordinate
system can be seen in Fig. 5 on the ground level of the pyramid, as described previously. The z-
axis, which is not shown, points perpendicularly out of the drawing plane. From this calculation it
was determined that the “Mars position” is also placed inside the Great Pyramid about 40 meters
above the King's chamber (see Fig. 5 and [13, section 4.5, Tab. 25 in app. A2]).
Option 9: 1.5 days, 3088 (9)
Analogously to option 4, this is a 36-hour time scan around the “date of the chambers.” This con-
stellation also got the number 12 because it is closely related to the “date of the pyramids.” The
time difference of 44 days is very short, compared to astronomical scales.
Option 10: F minimized (10)
The results here are similar to those of option 5, but the algorithm is more sophisticated. For the
time period from the year 3500 BC to the year 6500 AD, the planetary positions are compared with
the chamber positions and the date is not restricted in any way. This means that Mercury can be
placed anywhere on its orbit. For each date, where the positions match with each other and the
found relative error is below a certain value (0.25 %), this error is minimized and the constellation
is counted only, if the minimized error is smaller than another limit (0.05 %). The result is 38 dates
within the investigated 10,000 years when these conditions are met. Of course, other boundary
conditions imply that ultimately only one date is left (see option 8, and also [13, Tab. 20 in app.
A2]).
13
3.1.3 Planetary conjunctions and transits
Conjunction means either that two or more celestial bodies have almost the same position in the
sky, or that, for example, two or more planets have the same ecliptic longitude. The latter case can
be seen in Fig. 6. The figure shows the correct dimensions of the orbits in 3088, when the four
planets Mercury, Venus, Earth, and Mars together with the Sun are aligned nearly in a straight line.
As mentioned before, such an arrangement is called “syzygy,” being a generic term of “conjunction”
and “opposition.”
From time to time, Mercury and Venus pass in front of the Sun's disc, which is called a transit. In
Fig. 7 the typical lapse of time is shown for the Venus transit in the year 2012. Here we take Venus
instead of Mercury because of the recent Venus transit, which was a rare event.
14
Figure 6: Approximate true-to-scale representation of the orbits of Mercury, Venus, Earth, and Mars around the Sun. On
May 17, 3088, the four planets and the Sun are positioned nearly in a straight line (syzygy), followed by a Mercury transit.
The places p, a, and K represent perihelion, aphelion and ascending node, and M1 and M4 are the orbital centers of Mer-
cury and Mars. For better visibility, the planets and the Sun are drawn bigger than they would normally look like, if they
were drawn to scale. The dates are given in TT (terrestrial time). The range of ecliptic longitudes dL (or L) is only 3.4°.
The interesting point is that the two events “the 4 planets and Sun in a straight line” and a “transit
of Mercury or Venus” normally do not take place simultaneously. Within the given 30,000 years
(from 13,000 BC to 17,000 AD), this happens only six times, if we fix the maximum angular range
of the ecliptic longitudes to 5°. This means that the coincidence of the given syzygy and a transit of
Mercury or Venus happens only an average of every 5,000 years. This happens exactly between
the two dates of the “chamber constellation” and the “pyramid constellation,” which are separated
by 44 days! Figure 5 shows that within this period of time the four planets and the Sun form nearly
a straight line. About one day later, Mercury passes the solar disc (for more details see [13]).
Option 11: Mercury tr. (11)
The contact dates of all Mercury transits are calculated for the years 2950 to 3200. Additionally, the
minimum separation between Mercury and Sun, the case of ascending or descending node, and
the serial number are given. For the transit series of Mercury and Venus see also section 4.7.4 and
[22, pp. 7–13]. The period includes the year 3088, which is labeled automatically with the number
12. In the given time span, 34 Mercury transits are registered (section 3.4.6).
Option 12: Venus tr. (12)
All Venus transits with their four contact points (phases) and minimum separation are listed for the
years 1500 to 4000. This time period is larger than for Mercury because Venus transits occur less
frequently than Mercury transits. Between the years 1500 and 3000, and with a period of roughly
120 years, two Venus transits occur, following each other with a time difference of 8 years. (The
time limits of this option are chosen in such a way that all results are displayed on one monitor
screen.) If we do not have a full transit but a grazing transit, the corresponding line gets a “v” at the
beginning. (See also the first program output in section 3.1.) The same is true for a grazing Mer-
cury transit, but instead the line gets an “m”. In the previous time period of option 11, there are by
chance no grazing transits of Mercury.
Option 13: syzygy, 3 pl. (13)
This option yields linear constellations (syzygy) of the three planets, Mercury, Venus, and Earth,
together with the Sun. The condition for the syzygy is that the ecliptic longitudes of all three planets
match within an angular range of dL = 5°. The investigated time period is 2900 to 3300. If a transit
of Mercury or Venus also occurs during the syzygy event (within a few hours or a few days), the
beginning of the line in the table gets an “M” or a “V” for a full transit of Mercury or Venus or an “m”
or “v” for a corresponding grazing transit. It might also happen during such a “linear constellation”
that both a Mercury and a Venus transit occur, so that the line is indicated with both letters like, for
example, “MV”. This happens only three times between the years 13,000 BC and 17,000 AD,
assuming all ecliptic longitudes within a range of 5°. Note that this does not mean a simultaneous
transit, because both transits might have a time difference of a few hours or a few days. If a syzygy
is near a known constellation within a certain time limit (10 orbital periods of Mercury 880 days),
the corresponding line is marked with a small arrow “->” [13, Tables 27, 28]. However, for transits in
the remote future or remote past, the precision of VSOP87 has to be considered (section 4.2.4).
Option 14: syzygy, 4 pl. (14)
Now, Mars is also included. This means that the program searches for “linear constellations” of
Mercury, Venus, Earth, Mars, and the Sun. The condition is that the ecliptic longitudes of all four
planets are again placed within the angle, meaning a fourfold planetary conjunction (syzygy).
This happens very rarely. So, the whole time period from 13,000 BC to 17,000 AD is checked. The
coincidence of the given syzygy together with a transit occurs only six times, which means an aver-
age of every 5,000 years (see section 3.4.7).
15
Option 15: TYMT-test (15)
This option is mainly a test to check the processing speed. “TYMT” means Ten thousend Years
Mercury Transit.” The transits of Mercury (geocentric phases) are calculated for the years 3000 BC
to 7000 AD. Using an Intel Core i5-3210M processor (2.5 GHz, 8 Gbytes, dual channel), the TYMT-
test needs 46.0 seconds. During the 10,000 years, 31,520 passages of Mercury along the Sun are
tested and 1,340 transits are found. The results are calculated with the full version of VSOP87.
Interestingly, on the used 64-bit system the 32-bit version of P4 runs faster (43.2 s) than the 64-bit
version. Even higher speed can be obtained, e.g., by parallelizing the processing and using multi-
ple threads. A corresponding program version already exists (see footnote in the appendix on page
82). About 20 years ago, without optimization of the software and using the computer hardware at
that time, the TYMT-test would have needed about 1 month of computation time.
The CPU-time directly depends on the speed of the processor. More “GHz” means less CPU-time.
A criterion, which is more or less independent of the clock frequency and a better measure of the
software efficiency, would be the product of frequency and CPU-time: 2.5 GHz · 43.2 s = 108 GHz·s.
This is just a number and means the number of clock cycles necessary for the whole computation.
We can call it 108 Gc (gigacycles) which is 108 · 109 cycles. In P4, only the CPU-time is provided.
16
Figure 7: Venus transit on June 5−6, 2012, as seen from the center of the Earth (geocentric phases).
The positions 1 to 4 are the geocentric contact points (phases) and “m” represents the place of minimum
separation between Venus and the center of the Sun. The size of Venus and the Sun are drawn to scale
as seen from the Earth. N shows the north direction on the celestial sphere. The direction from E (east)
to W (west) is the direction of the apparent motion of the Sun due to the rotation of the Earth. The angles
of the contact points were calculated with P4 (compare with Meeus [22, p. 48]).
3.2 Quick start options for the book tables
Most of the astronomical tables in the two books [5] and [13] can be reproduced by additional quick
start options, called “book options.” These options are not shown in the main menu, but they can
be found easily. All book options have three digits. The first two digits represent the number of the
table and the last digit indicates the section of the table. For example, Table 39 in the first book [5,
p. 319] consists of three parts, one placed above the other. These parts can be reproduced by the
options 390, 391, and 392, meaning the digits 39 plus one digit 0, 1, or 2 for the different parts. If a
table has only one part, like Table 45 [5, p. 327], a zero has to be appended and the corresponding
option is 450.
3.2.1 Book 1 “Pyramiden und Planeten”
If the Tables 39 to 51 in [5] are reproduced with the P4 program the program output is not always
identical to the tables. In some cases, the program output is much larger, which means that in the
book only the important quantities are printed.
In Table 50, the correlation between pyramids and planets is checked, in which Mercury is fixed to
the aphelion of the orbit and the “Sun position” in the pyramid area is free in all 3 dimensions. The
latter aspect is described in more detail in sections 3.3.10–3.3.12 and 4.6. There are three possi-
bilities to define the vertical position of a pyramid: It can be the center of mass of the pyramid, the
middle of the pyramid base, or the top of the pyramid. The first two cases are presented in Table
50 and can be calculated with the options 500 and 501. The constellations with the pyramid top as
the vertical coordinate had been omitted in the table because there were no significant new results.
Nevertheless, this case can be computed by using the option 502.
In Table 51, the correlation between pyramids and planets is again investigated. Not only is the
“Sun position” on the Giza plateau free in all 3 dimensions, but the date is also free, which means
that the dates are not restricted to the aphelion passages of Mercury. The first book searched for
the constellations with the short version of VSOP87, and the relative error F''pos was minimized by
repeatedly starting the full VSOP87 version by hand using the P3 program. In P4, these results are
calculated automatically with a fit-subroutine and the VSOP87 full version (see also section 3.4.10).
Here, the results sometimes differ in the last digit from those in the book [5] because in [5] the
relative error was minimized by adapting the point of time manually. The search routine from quick
start option (5) uses the short version of VSOP87. Later, this routine was also implemented for the
full version of VSOP87. If the reader wants to check the results in Table 51 with this different
search method, it can be done with the options 517, 518, and 519 (see also section 3.3.16).
3.2.2 Book 2 (in preparation)
Tables 17–33 and 35–38 in Ref. [13] can be reproduced using the corresponding quick start
options as described previously. Table 20, for instance, indicates quick start option 200. The tables
in book 1 (numbers 39–51) and those of book 2 (numbers 17–38) have no overlap. Thus, the op-
tions can be used without addressing explicitly book 1 or book 2.
3.2.3 Special test option (999)
Let us assume that a special parameter setting is used and several runs have to be done by
changing only one parameter. Then it is not convenient to set all other parameters each time by
hand, as described in section 3.3. Instead, it is easier to use the input-output-file “inedit.t.” If the
reader opens this file using an editor, they will see two sections: 1 and 2. An example of the con-
17
tent of inedit.t is provided below. Section 1 (big arrow) is read by the P4 program, if the quick start
option 999 is used, and can be edited by hand. CAUTION! Not all combinations of parameters are
possible and these parameters are not checked by the program when using option 999. The under-
lined parameters (see below) can be changed within their allowed values without any problem
(section 3.3). For other parameters, their modes of operation and interdependencies must be
known.
Section 2 is always overwritten with the presently used parameter values when the program is
started with options other than “999”. Therefore, it is possible to compare the parameters of a
current run (section 2) with the parameters in section 1. In the lines above section 1, the text
should not be changed or deleted because for reading the parameters by the program, the number
of lines must always be the same, and otherwise the original text may be lost.
Example of the content of inedit.t
-----------------------------------------------------------
( User input and current input from program P4 )
-----------------------------------------------------------
( The input data in field 1) can be edited by the )
( user and are read by the program with the option )
( "999". The input data in field 2) are written by )
( the program at each run and can be used for com- )
( parison. The manual input by the user in field 1) )
( allows for the creation of input data to be copied )
( into the file "inparm.t." Number and position of )
( the lines in this file must not be changed! )
-----------------------------------------------------------
Parameter names of the values further below
-----------------------------------------------------------
XXXXX ipla ilin imod imo4 ikomb
XXXXX lv ivers itran isep iuniv
XXXXX ical ika iaph iamax step
XXXXXXXXXXXXX ison ihi irb ijd
XXXXXXXXX zmin zmax ak zjde1
XXXXX dwi dwikomb dwi2 dwi3
X nurtr iek io iout
-----------------------------------------------------------
1) Input to edit (999) - CAUTION: No check of parameters!
===========================================================
3 1 1 0 1
1 3 1 3 1
2 0 1 0 0.00000
5 0 1 15
1900.00000 2100.00000 0.00000 0.00000
0.000 0.000 0.000 0.000
1 1 1 2
===========================================================
2) Last used input (all options except 999)
-----------------------------------------------------------
2 4 3 0 0
1 3 1 1 1
2 1 2 0 24.00000
1 0 1 15
-13000.00000 17000.00000 0.00000 0.00000
1.850 0.000 0.000 0.000
1 3 1 2
-----------------------------------------------------------
*************************** END ***************************
18
When using the option “999” the parameters from field 1 are taken by the program. The para-
meters in fields 1 and 2, provided here, are arbitrary. The parameter names beside the big arrow
correspond to the numbers in fields 1 and 2. If the functionality of a given parameter is not known,
see section 3.3. If the information in this section is not enough, the Fortran source code p4.f95,
listed in the appendix, provides more information. Unfortunately, most comments are in German.
Note: The parameters in inedit.t are not checked with respect to correct input. So, the user should
follow the hints, given above.
3.3 Detailed options (0)
In contrast to the quick start options, the single parameters in the program can also be set individu-
ally one after the other, each given by its own menu. In the main menu on page 7 (last line) we find
detailed options (0). So, the option to get into this modus is 0”. In a program run, not all
combinations of the parameters are meaningful. Those that are not allowed are not presented.
Sometimes a reduced menu is shown. If a number is typed, which is not offered in the menu, an
error message appears. So, the program start with the option “0is controlled by the program and
protected against any false input. It follows a brief overview of menus and options.
1. Planetary positions : pyramids, chambers in Cheops Pyramid, linear constellations (syzygy).
2. Linear constellations and transits : transits of Mercury/Venus, conjunction of 3 or 4 planets.
3. VSOP theory versions : short or full VSOP version, combination of both, planetary elements.
4. VSOP coordinate systems : ecliptic of epoch (VSOP87C, VSOP87D), J2000.0 (VSOP87A).
5. Transit options : equal ecliptic longitudes, nearest separation, transit phases, position angles.
6. Calendar systems : Julian/Gregorian calendar or only Gregorian calendar.
7. Time systems : terrestrial dynamical time (JDE, TT), universal time (UT).
8. Mapping of planets and chambers : assignment of Mercury, Venus, Earth to the chambers.
9. Search method for the dates : Mercury passages at aphelion, perihelion, date not restricted.
10. “Sun position” : south of Mykerinos or Chefren Pyramid, “Sun position” free.
11. Computation of free “Sun position” : free in 2 or 3 dimensions, 3D calc. with SLE or FITEX.
12. Vertical level of pyramid positions : pyramid base, center of mass, top of pyramid.
13. The z-coordinate of chamber positions : east wall, west wall, spatial middle of each chamber.
14. Datum plane for Earth's surface : projection on plane of Earth, Mercury, or Venus orbit (2D).
15. Specification of timing : number of constellation (1–14), k-number, years, Julian Day.
16. Tolerance in degree or percent : tolerance/angular range of ecliptic longitude, relative error.
17. Syzygy with simultaneous transit : all planetary conjunctions, only with simultaneous transit.
18. Polarity (orientation of planetary orbits) : view from ecliptic north, south or both options.
19. Complexity of output : normal output, extended output.
20. Mode of program output : output only on monitor, monitor + file, special output, exit.
In the following sections the menus are described one by one in the order of their appearance
during program start. Each menu is given at the beginning in blue. Note: Not all menus appear at
program start, depending on the kind of computation. On the right side of each menu, the corre-
sponding internal parameter is given, like for example: “ipla”.
19
3.3.1 Planetary positions
Constell. pyr.(1), chamb.(2), lin.(3) : (internal: ipla)
(1) planetary constellation of Mercury, Venus, Earth = positions of the three pyramids in Giza
(2) planetary constellation of Mercury, Venus, Earth = positions of the three chambers in Great P.
(3) “linear constellations” (syzygy, transit)
There are three main categories. The positions of the planets are compared with (1) the positions
of the three pyramids in Giza, or (2) with the system of the three chambers in the Great Pyramid.
Option (3) investigates when the planets build a planetary conjunction and “linear constellation,”
respectively, or a Mercury or Venus transit. The origin of the coordinate system for option (1) is
middle of base area of Mykerinos Pyramid, x-axis points to the north, y-axis points to the west, and
z-axis points upward; for option (2), middle axis of the Queen's chamber in its east wall on the level
of the pyramid base, x-axis points to the north, y-axis points upward, and z-axis points to the east.
3.3.2 Linear constellations and transits
Tr. Mer.(1), Ven.(2), 3-co.(3), 4-co.(4) : (internal: ilin)
(1) transits of Mercury
(2) transits of Venus
(3) triple conjunction of the three planets Mercury, Venus, and Earth
(4) fourfold conjunction of the four planets Mercury, Venus, Earth, and Mars
The “linear constellations” (above option lin.(3)”) are subdivided into the four given menu
points. Options (1) and (2) are clear. Option (3) means a syzygy of the planets Mercury, Venus,
Earth, including the Sun, and option (4) a syzygy of Mercury, Venus, Earth, Mars, and the Sun.
When running option (3), it becomes clear that Mercury and Earth always have the same ecliptic
longitude. This seems reasonable, but it is not put into the program as a boundary condition; it just
comes out as a result. With option (4), three different cases are observed. Either the ecliptic longi-
tudes of Mercury and Mars are identical, those of Mercury and Earth, or those of Venus and Mars.
In principle, there are other combinations of two out of four planets, but other solutions do not exist.
If one thinks about the problem, it becomes clear why. (If p is the number of planets, then the num-
ber of cases, different pairs of planets with equal longitude, is N = (p–1)·(p–2)/2 with p ≥ 3.)
3.3.3 VSOP theory versions
VSOP87 combi. (1), short version (2),
Kepl. equ. (3), full version (4) : (internal: imod)
(1) combination of the short and full version of the VSOP87 theory
(2) short version of the VSOP87 theory, Meeus [17, pp. 381 ff.]
(3) planetary elements, polynomials of third degree [17, pp. 200 ff.] and solving Kepler's equation
(4) full version of the VSOP87 theory, Bretagnon and Francou [1, 2]
Option (3) (solving Kepler's equation) is the fastest algorithm, but it has the lowest accuracy. Opt-
ion (2) (short VSOP87 version) is not so fast, but it has a higher precision. Option (4) (full VSOP87
version) has the highest precision, but it takes the most computation time. The option (1) (combi-
nation of short and full VSOP87 version) is fast and yields the same high precision as (4). So, the
recommendation is: option (1) for larger time periods and option (4) for single constellations.
20
3.3.4 VSOP coordinate systems
System ecl. of epoch (1), J2000.0 (2) : (internal: lv)
(1) ecliptic of epoch (dynamical equinox)
(2) standard system J2000.0 (ecliptic of Jan. 1, 2000, 12:00, TT, or JDE = 2451545.0)
The two options are the two applied coordinate systems for the VSOP87 theory. The short
VSOP87 version is provided only with the ecliptic of epoch (1). The VSOP87 full version and the
orbital elements for solving Kepler's equation are given for both systems.
3.3.5 Transit options
Date equ.L.(1), nearest (2), phases (3)
phases and position angles (4) : (internal: isep)
(1) transit check at equal ecliptic longitudes (planet, Earth), finite speed of light not considered
(2) transit check at minimum separation (nearest approach), finite speed of light not considered
(3) geocentric transit phases, as seen from Earth
(4) geocentric transit phases and position angles (the output needs 148 characters line width)
The first two options are each calculated for a fixed moment. Thus, the travel time of light is not
taken into account. These options were written during an early stage of the program development
and now serve for test purposes. Option (3) yields the true geocentric transit phases 1 to 4 as well
as the minimum separation by considering the finite speed of light. In option (4), the position angles
on the solar disk and the semidiameters of the Sun and planet are also calculated. In addition,
central transits (minimum separation < semidiameter of planet) are labeled with “C” (geocentric
central transit) and “c” (central transit, seen from some place on Earth). Option (4) is the only
option that needs 148 instead of 80 characters line width on the computer monitor.
3.3.6 Calendar systems
Calendar Jul./Greg. (1), only Greg. (2) : (internal: ical)
(1) Automatic choice of Julian and Gregorian calendar
(2) Gregorian Calendar for all times
In the first option, the Julian calendar is used for the years 4712 BC to 1582 AD and the Gregorian
calendar for all other times. It does not make sense to use the historical Julian calendar before
4712 BC because in this distant past the calendar gets more and more out of hand, and there are
no historical events to apply this calendar. In contrast, the Gregorian calendar in these past times
is in much better agreement with the seasons. Option (2) means that only the Gregorian calendar
is used for all times. The calendar menu is presented also when no calendar dates are calculated.
This has to do with the fact that the decimal year, displayed in all outputs, is slightly different for
both calendars.
3.3.7 Time systems
Time system JDE/ TT (1), UT (2) : (internal: iuniv)
(1) JDE (Julian Ephemeris Day, equal to JD) and TT (terrestrial time), respectively
(2) UT (universal time)
21
JDE and TT are identical time scales with a constant length of days. UT takes into account the
deceleration of the Earth's rotation due to tide friction, so that from time to time a leap second is
introduced (in UTC). Because the slowing down of the Earth's rotation cannot be predicted precise-
ly, the terrestrial time (TT) is the accurate measure. With option (2), TT can be transferred to UT by
using the equations for ΔT = TT – UT of F. Espenak and J. Meeus [23, 24] (see section 4.8).
3.3.8 Mapping of planets and chambers
Planets E-V-M (1), E-M-V (2), V-E-M (3),
V-M-E (4), M-E-V (5), M-V-E (6) : (internal: ika)
(1) Earth – Venus – Mercury (4) Venus – Mercury – Earth
(2) Earth – Mercury – Venus (5) Mercury – Earth – Venus
(3) Venus – Earth – Mercury (6) Mercury – Venus – Earth
The three planets each correspond in the given sequence to the King's chamber, the Queen's
chamber, and the subterranean chamber (rock chamber) in the Cheops Pyramid. Option (1) is the
case that actually makes sense. The other options are added as a test and for the sake of com-
pleteness.
3.3.9 Search method for the dates
Passage aph./per. area of aph./per. free
(1) (2) (3) (4) (5) : (internal: iaph)
For options (3) and (4) it follows
Steps per Mercury passage : (internal: iamax)
Step width (hours, real) : (internal: step)
(1) date: Mercury passage at aphelion
(2) date: Mercury passage at perihelion
(3) time interval around aphelion passage of Mercury
(4) time interval around perihelion passage of Mercury
(5) date completely free (within a given epoch)
Note: There are similar input menus in which not all of the options (1) to (5) are given, but the
meaning of the numbers is always the same. In option (1), only the dates are tested, when Mercury
passes the aphelion. Option (2) means the same for perihelion. In options (3) and (4), those con-
stellations are tested that are in a given time interval around the aphelion and perihelion passage
of Mercury, respectively. This could be, for example, an interval starting 7 days before and ending 7
days after each aphelion passage with equal time steps of (for instance) 12 hours. In this case, it
means that 14 · 2 + 1 = 29 dates are checked for each aphelion passage. In option (5), the date is
totally free. So, during the search the time increases in automatically chosen time steps, and if a
promising constellation is found, the relative error is minimized by an automated fit procedure.
For the options (3) and (4), two additional input lines (see above) ask for a specific time interval for
each aphelion or perihelion passage. First, it requires the number of steps per Mercury passage;
and secondly, the step width in hours is required. In the given example, the number of steps would
be 28 and the step width would be 12 (hours). When searching for “linear constellations” (syzygies)
with the short VSOP87 version or the “planetary elements” version (Kepler's equation), the follow-
ing input line allows for a search with fixed time steps:
22
Step width [hrs] (min.-search 0.) (real) : (internal: step)
Thus, in the case that the overall interval dL of ecliptic longitudes of the corresponding planets falls
below a certain limit (e.g., below 5°) the program calculates all of the following constellations in the
given steps (e.g., in “1 hour” steps) until dL again exceeds the previously given limit. If the input 0.0
is given as the step width, the time steps are automatically chosen, and if dL decreases below the
given limit (like 5°), the date is optimized by minimizing dL. This case of automatically minimizing
dL is always used in the combined search with the short and full VSOP87 version.
3.3.10 “Sun position”
Sun pos. Myk.(1), Chefr.(2), free (3) : (internal: ison)
(1) “Sun position” fixed 726 m south of the center of the Mykerinos Pyramid
(2) “Sun position” fixed 963 m south of the center of the Chefren Pyramid
(3) “Sun position” free
In options (1) and (2), the “Sun position” south of Mykerinos Pyramid and south of Chefren Pyramid
means that the “Sun position” is placed exactly on the north−south middle axis of the correspond-
ing pyramid. The given distances were determined geometrically on the basis of figures like Fig. 1
or 8 (yielding the angles
1 and
2 in Fig. 8).
In option (3), the “Sun position” at the Giza plateau is not fixed, and the calculation of it has to be
further specified by the following menu. Note that not fixed” does not mean “not defined.” The
“Sun position” at the Giza plateau is defined exactly by the positions of the three planets Mercury,
Venus, and Earth, when considering all 3 dimensions mathematically.
3.3.11 Computation of free “Sun position”
Sun 2D (1), 3D/SLE (2), 3D/FITEX (3) : (internal: ison2)
(1) “Sun position” free in the 2 horizontal dimensions (meaning “restricted to the Earth's surface”)
(2) “Sun position” free in 3 dimensions, calculation with a System of Linear Equations (SLE)
(3) “Sun position” free in 3 dimensions, calculation with coordinate transformation and FITEX
When the planetary and pyramid constellations are adapted to each other by comparing the coordi-
nates or by coordinate transformation, the “Sun position” can be predefined or not.Predefined”
means that it is restricted in the vertical dimension. This option (1) was applied mainly for the
constellations 1 to 11. Options (2) and (3) are two different ways of calculating the “Sun position” in
3 dimensions on the Giza plateau south of the pyramids and also in the Great Pyramid (for details,
see section 4.6 and [5, app. A16]). In the case of a rather small relative error between pyramid and
planet configuration, both mathematical methods yield the same result and the same “Sun position,”
respectively. In the case of the chambers, the option “2D (1)” does not exist.
3.3.12 Vertical coordinate of pyramid positions
z-coord. base (1), C-M (2), top (3) : (internal: ihi)
(1) z-coordinate at level of pyramid base
(2) z-coordinate at level of center of mass of the pyramid
(3) z-coordinate at top of pyramid
23
When fixing the pyramid positions in 3 dimensions, it is necessary to determine the height level of
the positions. Three alternatives are given. The center of mass of a pyramid (option 2) can be
shown to be located at a quarter of the pyramid height. Options (1) and (2) create mostly the same
or similar results, whereas option (3) also generates other constellations.
3.3.13 The z-coordinate of chamber positions
Wall east (1), middle (2), west (3) : (internal: ihi)
(1) center of east wall of each chamber
(2) spatial middle of each chamber
(3) center of west wall of each chamber
When fixing the chamber positions in 3 dimensions, it is necessary to fix the east−west location (z-
coordinate) of the positions. Because only the east walls of all three chambers are located in the
same vertical plane, but not the west walls, three alternatives are also given here.
3.3.14 Datum plane for Earth's surface
Coord. ecl.(1), Mer.(2-4), Ven.(5) : (internal: irb)
(1) projection plane is ecliptic plane (plane of Earth orbit)
(2−4) projection plane is plane of Mercury orbit
(5) projection plane is plane of Venus orbit
For the 2-dimensional calculation (section 3.3.11, option (1)), the planetary positions are projected
vertically into one plane, as also the pyramid positions are projected vertically onto the Earth's
surface. Three different planes can be tested, defined by the orbits of Earth, Mercury, and Venus,
respectively. The change from the Earth's orbit (heliocentric coordinate system, VSOP87C) to a
system based on the Mercury or Venus orbit is performed with rotational matrices (see section 4.5
and [5, app. A15, pp. 328 ff.]). For Mercury, three different combinations of matrices are available
(options 2–4), all yielding the same result. They were used for test purposes during the develop-
ment of the program.
3.3.15 Specification of timing
Constell. (1..14), k-No. (15), JDE (0) : (internal: ijd) or
Constell. (1..14), years (15), JDE (0) : (internal: ijd)
(1–14) dates of the constellations 1 to 14 as given in [5, p. 315, Tab. 38]
(15) k-number (integer number of Mercury passages through aphelion or perihelion) or
(15) time period in years, specified in the menu lines following below
(0) JDE (Julian Ephemeris Day or Julian Day)
The numbers 1 to 14 belong to the planetary constellations 1 to 14 with the given dates. With these
options, only one constellation is calculated. The k-number in option (15) counts the passages of
Mercury through its aphelion or perihelion (see section 3.3.9, options (1) and (2)). The numbers
start with zero after the beginning of the year 2000. Before that date, the numbers are negative.
The format of the k-number is “real” (number with decimal point). Normally, the number (not the for-
mat) is an integer (the digits after the decimal point are zero), but it does not need to be an integer.
24
The latter case means that the date is not the passage through aphelion or perihelion, but some-
where between both moments. The optionyears (15) allows us to check the dates in a given
time period. Here, the result normally consists of several planetary constellations. The last option
JDE (0) enables us to specify directly a JDE number so that the constellation of this moment is
calculated.
k (real): (internal: ak)
or from year (real): (internal: zmin)
until year (real): (internal: zmax)
or JDE (real): (internal: zjde1)
The Julian Ephemeris Day can be any date, just like the k-number, which implies that the relative
error between alignment of pyramids (chambers) and planets can be very large.
3.3.16 Tolerance in degree or percent
Tolerance ecl. long. Venus, Earth (real) : (internal: dwi) or
Max. F-pos at aphelion/ per. (real) [%] : (internal: dwi) or
Tolerance ecl. long. VSOP short (real) : (internal: dwi)
" " " VSOP full (real) : (internal: dwikomb) or
Max. F-pos VSOP short ver. (real) [%] : (internal: dwi)
" " VSOP full ver. (real) [%] : (internal: dwikomb) or
Max. F-pos, VSOP short, start fitmin [%] : (internal: dwi)
" " VSOP short, final range [%] : (internal: dwikomb) or
Ang. range of eclipt. longitude (real) : (internal: dwi) or
Ecl. angular range, VSOP short v. (real) : (internal: dwi)
" " " , VSOP full v. (real) : (internal: dwikomb)
The accuracy between the theoretical arrangement of the planets given by the positions of the
pyramids, by the positions of chambers, or by a linear constellation and the current positions of
the planets is either measured in degree (difference in ecliptic longitude) or in percent (relative error
Fpos, F'pos, F''pos, see sections 4.3.1, 4.3.2 and also [5]). The upper limit of these quantities has to
be inserted as real number. A larger upper limit yields a larger number of detected constellations.
When the option “time interval around aphelion or perihelion” (section 3.3.9, options (3) and (4))
combined with a (large) time period is used, like 2000 BC to 4000 AD, then this indicates a “special
search.” At first, only passages through aphelion or perihelion with a maximum allowed error “dwi”
are printed. If the current relative error is below another threshold “dwi2,” a time interval of a few
hours or days around this aphelion (perihelion) is also tested by scanning the interval in small time
steps. Then, only constellations beyond aphelion or perihelion passage are printed, if their relative
error, resp. angle, is below a third threshold “dwi3.” Therefore, the corresponding input line (provid-
ed again) is followed by two additional lines:
Max. F-pos at aphelion/ per. (real) [%] : (internal: dwi)
" " consider without printing [%] : (internal: dwi2)
" " print beyond aphelion/per.[%] : (internal: dwi3)
So, when checking a constellation with this search, there are three limits (relative errors), and the
last limit is normally small compared to the other ones. A typical parameter set is: dwi = 3 %, dwi2
= 5 %, and dwi3 = 0.2 % (compare with quick start option 5). In combination with the VSOP87 short
25
version, the latter search was used to create Table 51 in [5] with the previous program version (P3).
The “fine adjustment” was done manually with the full VSOP87 version. Now, the search with quick
start option 5 can be performed with the VSOP87 full version, too. Furthermore, the ”fine adjust-
ment” is automatically done when using an automated minimum search with respect to dwi (section
3.3.9 with step width 0.0; see also example in section 3.4.10).
3.3.17 Syzygy with simultaneous transit
All conjunctions (1), only transits (2) : (internal: nurtr)
(1) all “linear constellations” (syzygies)
(2) only “linear constellations” with associated Mercury or Venus transit
Option (1): When searching for “linear constellations” with three or four planets, all constellations
are printed that meet the given condition, meaning that the range of ecliptic longitudes is smaller or
equal to a given angle dL (e.g., 5°). Constellations, which are accompanied by a Mercury or Venus
transit within a few hours or a few days, are marked with an “M” or aV.” If it is a grazing transit,
the line is marked with “m” or “v”. In the second option, the detected “linear constellations” are prin-
ted only when there is an associated transit of Mercury and Venus. In this case, constellations with-
out a transit are skipped, which reduces the size of the output.
3.3.18 Polarity (orientation of planetary orbits)
View from ecliptic North (1), South (2) : (internal: iek) or
View from eclipt. N (1), S (2), N/S (3) : (internal: iek)
(1) looking from ecliptic north
(2) looking from ecliptic south
(3) looking from ecliptic north and south
When comparing the pyramid and planetary positions in 2 dimensions, the pyramid positions are
projected onto the Earth's surface, and the planetary positions, for example, onto the plane of the
Earth's orbit. In this case, there are two possibilities when looking onto the planets. We can look
from the ecliptic north (option (1)) or from the ecliptic south (option (2)). One figure is the mirror-
inverted configuration of the other one. So, it makes a difference when comparing with the pyramid
positions, where we always look from above the pyramids. Finally, option (3) simply combines both
options (1) and (2). Thus, all constellations found in (1) and (2) are given in (3).
3.3.19 Complexity of output
Output normal (1), extended (2) : (internal: io)
(1) one line for each detected constellation
(2) several lines for each detected constellation
In option (2), the size of the output for each date depends on other parameters. The orbital ele-
ments of all planets can be obtained only by using also the option: Kepl. Equ. (3).
3.3.20 Mode of program output
Mon.(1), file (2), special (3), exit (4) : (internal: iout) or
Monitor (1), mon. + file (2), exit (4) : (internal: iout)
26
(1) output on monitor
(2) output on monitor and written into the file “out.txt”
(3) like (2) but with some output quantities being replaced (also special output for constellation 12)
(4) cancels program start
With option (1), the results are written only onto the computer monitor. When using option (2), all
results are additionally written into the file “out.txt.” This file is overwritten after each program run if
option (2) or (3) is used. So, in order to save the latest results, the created file “out.txt” must be
renamed. Option (3) means that some parameters like
JDE Julian Ephemeris Day
eerror code, when calculating the “Sun position” with FITEX (“0” means “no error”)
it number of iterations when using FITEX
are replaced by
dt [days] time difference to the next aphelion (perihelion) passage
X5 tilt angle between the Earth's surface and the transformed ecliptic plane
Mscaling factor between alignment of planets and pyramids (chambers)
The latter quantities allow for the reproduction of some tables in Ref. [13]. For constellation 12, in
combination with some certain parameter settings, option (3) calculates all “planetary positions” in
the Giza area as provided in Figs. 5 and 12 (book options 380 and 381). When typing the parame-
ters individually for the “special” output of constellation 12, only the internal parameters ipla, imod,
ivers, and ihi can be varied. With option (4), the program start is canceled. The program start and
the running program can be terminated at any time by typing <Ctrl>-C or <Strg>-C, respectively.
Detailed technical information about the subroutines VSOP87X, FITEX, and all other program parts
can be found in the FORTRAN source code p4.f95 (see appendix) and to some extent in chapter 4.
Additional mathematical details of the astronomical calculations are provided in [5, app. A14–A16].
3.4 Some program outputs
Some basic information concerning the program output is provided already in section 3.1. With the
following selected options and corresponding results, most of the expressions and parameters in
the table head of the program output are explained. The output is always printed in blue and calcu-
lated with the 64-bit version of P4. Section 3.4.11 provides a compilation of all quick start options.
3.4.1 Option 0
When starting the program with the option 0,” the parameters can be determined individually in
several menus, described previously and appearing one after the other. In the following example,
the transits of Mercury in front of the Sun are calculated for the years between 1800 and 2130. The
input numbers are underlined.
Input: Constell. pyr.(1), chamb.(2), lin.(3) : 3
Tr. Mer.(1), Ven.(2), 3-co.(3), 4-co.(4) : 1
VSOP87-version full v.(1), short v.(2) : 1
Date equ.L.(1), nearest (2), phases (3)
phases and position angles (4) : 3
Calendar only Greg. (1), Jul./Greg. (2) : 2
Time system JDE/ TT (1), UT (2) : 1
from year (real): 1800
until year (real): 2130
27
Output normal (1), extended (2) : 1
Monitor (1), mon. + file (2), exit (4) : 2
Output: TRANSITS OF MERCURY
(geocentric transit phases, terrestrial time TT)
< option 0 >
VSOP87C, comb. search, ecliptic of date, all Mercury transits
Period (years) from 1800.00 to 2130.00, Jul./Greg. calendar
co/p date/ time: I II nearest III IV sep["]a S
===============================================================================
9. Nov. 1802 6:14:20 6:16: 1 8:58:36 11:41:15 11:42:57 60.9/ 4
12. Nov. 1815 0:18:43 0:20:45 2:33:28 4:46:14 4:48:16 556.1/ 2
5. Nov. 1822 1: 0:53 1: 4:17 2:24:47 3:45:19 3:48:43 -838.8/ 8
5. May 1832 9: 0:31 9: 4: 0 12:25:23 15:46:40 15:50: 9 484.7 7
7. Nov. 1835 17:32:56 17:34:44 20: 7:49 22:40:58 22:42:46 -336.4/ 6
8. May 1845 16:20:41 16:24:23 19:36:49 22:49:10 22:52:52 -547.2 5
9. Nov. 1848 11: 5:28 11: 7:10 13:47:39 16:28:13 16:29:56 163.0/ 4
12. Nov. 1861 5:18:34 5:20:51 7:19:35 9:18:22 9:20:38 657.9/ 2
5. Nov. 1868 5:25:43 5:28:19 7:14:10 9: 0: 3 9: 2:39 -735.1/ 8
6. May 1878 15:12:57 15:16: 6 18:59:52 22:43:30 22:46:39 287.3 7
8. Nov. 1881 22:16:59 22:18:44 0:57:13 3:35:47 3:37:31 -231.8/ 6
10. May 1891 23:54:24 23:59:24 2:21:37 4:43:47 4:48:46 -753.6 5
10. Nov. 1894 15:56:31 15:58:16 18:34:42 21:11:12 21:12:57 266.2/ 4
14. Nov. 1907 10:23:56 10:26:38 12: 6:52 13:47: 7 13:49:48 758.6/ 2
7. Nov. 1914 9:57:23 9:59:37 12: 3:23 14: 7:11 14: 9:25 -630.7/ 8
8. May 1924 21:44: 9 21:47:10 1:41:22 5:35:25 5:38:26 84.6 7
10. Nov. 1927 3: 2:30 3: 4:12 5:45:57 8:27:46 8:29:28 -128.7/ 6
m 11. May 1937 -- -- 8:59:41 -- -- -955.5 5
11. Nov. 1940 20:49:22 20:51:11 23:21:32 1:51:56 1:53:45 368.5/ 4
14. Nov. 1953 15:37:44 15:41:23 16:54:17 18: 7:11 18:10:50 861.8/ 2
6. May 1957 23:59:53 0: 9:54 1:14:46 2:19:37 2:29:37 907.3 9
7. Nov. 1960 14:34:31 14:36:32 16:53:27 19:10:25 19:12:27 -527.9/ 8
9. May 1970 4:20:11 4:23:13 8:16:50 12:10:20 12:13:22 -114.1 7
10. Nov. 1973 7:48:13 7:49:54 10:32:58 13:16: 7 13:17:49 -26.4/ 6
13. Nov. 1986 1:44: 5 1:46: 0 4: 7:57 6:29:57 6:31:52 470.5/ 4
6. Nov. 1993 3: 7:14 3:13:10 3:57:31 4:41:53 4:47:49 -926.7/ 10
m 15. Nov. 1999 21:16:48 21:32:36 21:41:57 21:51:19 22: 7: 6 963.0/ 2
7. May 2003 5:14:16 5:18:44 7:53:28 10:28: 9 10:32:37 708.3 9
8. Nov. 2006 19:13:17 19:15:10 21:42: 9 0: 9:13 0:11: 6 -422.9/ 8
9. May 2016 11:13:36 11:16:48 14:58:33 18:40:11 18:43:22 -318.5 7
11. Nov. 2019 12:36:43 12:38:24 15:20:57 18: 3:35 18: 5:16 75.9/ 6
13. Nov. 2032 6:42:26 6:44:30 8:55:22 11: 6:17 11: 8:22 572.1/ 4
7. Nov. 2039 7:19:30 7:22:44 8:48: 4 10:13:25 10:16:39 -822.3/ 10
7. May 2049 11: 5:22 11: 8:54 14:25:43 17:42:26 17:45:58 511.8 9
9. Nov. 2052 23:55:24 23:57:12 2:31:31 5: 5:55 5: 7:42 -318.7/ 8
10. May 2062 18:18:36 18:22:13 21:38:53 0:55:27 0:59: 4 -520.5 7
11. Nov. 2065 17:26:32 17:28:15 20: 8:20 22:48:30 22:50:13 180.7/ 6
14. Nov. 2078 11:45: 6 11:47:26 13:43:36 15:39:48 15:42: 8 674.3/ 4
7. Nov. 2085 11:45:39 11:48:12 13:37:22 15:26:34 15:29: 7 -718.5/ 10
8. May 2095 17:24: 2 17:27:12 21: 8:40 0:50: 0 0:53: 9 309.8 9
10. Nov. 2098 4:38:48 4:40:32 7:19:52 9:59:17 10: 1: 1 -214.7/ 8
12. May 2108 1:43:45 1:48:26 4:19:33 6:50:38 6:55:18 -724.7 7
14. Nov. 2111 22:19:23 22:21: 8 0:56:51 3:32:38 3:34:24 283.3/ 6
15. Nov. 2124 16:54: 5 16:56:54 18:32:40 20: 8:28 20:11:17 778.9/ 4
===============================================================================
Computed constellations: 14035 ("/" means ascending node)
Tested planet. passages: 1041
Detected transits : 44
Centr./grazing transits: 0 / 2 CPU-time 0: 0: 1.544 -- end of run.
28
More details about the output format is provided in section 3.4.6. This is one example of fixing the
parameters individually. In the following, the results of some quick start options are presented.
3.4.2 Quick start option 1
Output: PLANETS IN ALIGNMENT WITH THE PYRAMIDS OF GIZA
(Mercury at aphelion)
< option 1 >
VSOP87C, comb. search, ecliptic of date, "Sun" free 3D, C-M, FITEX
Ecl. N and S, years -13000.00 to 17000.00 (c2), tolerance F <= 1.50/ 1.00 %
con k year Lm-Lv Lm-Le e it x-Sun y-Sun z-Sun dr P F[%]
===============================================================================
-59934-12435.214 9.349 13.126 0 85 -588.5 370.0 310.5 5.2 * 0.434
-59768-12395.233 13.603 20.167 0 87 -570.7 227.1 397.3 8.1 0.716
-55865-11455.188 12.728 18.358 0 89 -552.5 298.7 394.0 8.0 * 0.697
-35839 -6631.888 -10.931 -15.051 0 63 -411.5 557.6 -273.4 10.5 0.889
-31770 -5651.861 -7.629 -9.869 0 82 -482.4 571.5 -169.5 8.1 0.668
-27867 -4711.714 -8.557 -11.707 0 75 -457.9 570.4 -209.9 4.6 0.379
-23798 -3731.708 -5.250 -6.516 0 65 -526.9 564.6 -99.3 8.6 0.697
-712 1828.517 -19.311 -31.682 0 114 -606.7 -146.1 -182.7 8.1 * 0.803
450 2108.387 10.444 17.871 0 76 -676.8 153.5 282.4 7.9 0.677
3191 2768.562 -20.247 -33.537 0 116 -596.5 -184.6 -143.6 6.5 * 0.662
12 4519 3088.413 13.779 23.191 0 46 -667.5 21.3 272.4 0.8 0.069
7260 3748.588 -16.937 -28.312 0 129 -648.7 -110.6 -136.6 6.7 * 0.642
11163 4688.633 -17.875 -30.163 0 115 -636.4 -148.3 -107.7 5.5 * 0.540
12491 5008.485 16.189 26.748 0 72 -660.7 -70.6 221.9 9.3 0.866
===============================================================================
Computed constellations: 124559 (P: polarity, * view from ecl. south)
Detected constellations: 14 CPU-time 0: 0: 2.412 -- end of run.
After tolerance F the two given values belong to the VSOP87 short and full version. After the
limiting year dates, the parameter (c2) means that both the Julian and the Gregorian calendar as
well as the corresponding decimal years are used within their time periods; (c1) means that the
Gregorian calendar is applied for all times. The parameters in the header line just above the table
are as follows:
con number of constellation 114 or arrow “->,” if date is not far from a known constellation
k number of Mercury aphelion (or perihelion) passage (see section 3.3.15)
year decimal year (astron. counting, which means that the year 0 exists; see section 4.9.1)
Lm-Lv difference of heliocentric longitudes of Mercury and Venus
Lm-Le difference of heliocentric longitudes of Mercury and Earth
e error code from FITEX, “0” means “no error,” more information in the source code (app.)
it number of iterations when using FITEX for calculating the “Sun position” in 3D
x-Sun x-coordinate of the “Sun position” in meters at the Giza plateau (y-Sun, z-Sun analog)
dr accuracy of “Sun position” in the pyramid area in meters (see section 4.9.2)
P polarity: the star “*” represents the view from the ecliptic south (“no star”: ecliptic north)
F[%] relative accuracy of comparing pyramid (chamber) positions with planetary positions
In the subroutine “FITEX,” the error code “e” is named “KE.” In references [5] and [13], the relative
deviation F is also named Fpos, F'pos, or F''pos, depending on the way of calculation. These quanti-
ties are always the relative error when comparing the pyramid positions with the planetary posi-
tions. The options “1” and “500” are identical (compare with [5, Table 50]).
29
3.4.3 Quick start option 3
Output: PLANETS IN ALIGNMENT WITH THE PYRAMIDS OF GIZA
(Mercury at aphelion)
< option 3 >
VSOP87C (2005) full ver., ecliptic of date, "Sun" free 3D, C-M, FITEX
Ecl. N and S, constellation 12, JDE = 2849079.76330, year = 3088.41 (c2)
date (Gregor.,TT) = 31. May 3088, 6:19: 9, Thursday
===============================================================================
con k year Lm-Lv Lm-Le e it x-Sun y-Sun z-Sun dr P F[%]
-------------------------------------------------------------------------------
Lm Bm Rm Lv Bv Rv Le Be Re
xm ym zm xv yv zv xe ye ze
xv-xm xe-xm yv-ym ye-ym zv-zm ze-zm rel. deviation
===============================================================================
12 4519 3088.413 13.779 23.191 0 46 -667.5 21.3 272.4 0.8 0.069
-------------------------------------------------------------------------------
274.2350 -3.8355 0.466784 260.4560 0.3611 0.725141 251.0441 0.0001 1.010140
0.465739 0.000000-0.031224 .704258-.172709 .004569 0.928519-0.397789 0.000001
0.238520 0.462780-0.172709-0.397789 0.035794 0.031226 0.06939575 %
-------------------------------------------------------------------------------
ascending node (M/V/E/Ma): 61.262371 86.535685 --- 57.966374
inclination i (M/V/E/Ma): 7.022736 3.405473 0.000000 1.844689
perihelion pi (M/V/E/Ma): 94.431801 146.691325 121.707611 356.114290
transl. X1, X2, X3; del-t: -0.465804 -0.000042 0.031160 0.000 days
Euler angl. X4, X5, X6; M: -45.993868 24.468218 43.897290 97644154.
===============================================================================
===============================================================================
pla. x[AU] y[AU] z[AU] L B r[AU] Lm-L dev.
-------------------------------------------------------------------------------
Mer 0.034394 -0.464467 -0.031224 274.2350 -3.8355 0.466784 0.0000 0.0000
Ven -0.120230 -0.715090 0.004569 260.4560 0.3611 0.725141 13.7790 1.5890
Ear -0.328133 -0.955359 0.000001 251.0441 0.0001 1.010140 23.1909 1.7809
Mar -0.742601 -1.384620 -0.003376 241.7944 -0.1231 1.571191 32.4406 ---
Jup 3.659951 -3.600495 -0.044977 315.4692 -0.5019 5.134280 -41.2342 -6.4502
Sat 6.950958 6.246050 -0.394537 41.9425 -2.4175 9.353321-127.7075 87.2925
Ura 14.561780-13.283778 -0.228606 317.6278 -0.6645 19.711836 -43.3928 ---
Nep-30.177931 0.905335 0.497313 178.2816 0.9437 30.195604 95.9534 ---
===============================================================================
Celestial pos. in Giza body x[m] y[m] z[m] dr[m]
-------------------------------------------------------------------------------
Local coordinates Sun -667.49 21.30 272.36 0.77
of the "planets" Mercury -0.12 -0.09 16.24 0.15 <
(pyramid positions) Venus 385.58 -239.89 33.43 0.30 <
Earth 739.06 -574.51 23.93 0.14 <
Mars 1373.30 -1232.84 34.00 0.96
Jupiter 4545.10 4889.86 -3044.47 5.07
Saturn -10104.25 10738.27 -928.40 10.57
Uranus 18521.60 19497.37 -12553.34 20.57
Neptune -1354.80 -43610.38 15633.42 31.99
===============================================================================
("<" exact deviation dr) CPU-time 0: 0: 0.084 -- end of run.
Information above the first solid line of the tables:
VSOP87C (2005) full ver., ecliptic of date describes the VSOP version
”Sun” free Sun position” at the Giza plateau free
3D calculation of “Sun position” in 3 dimensions
30
C-M vertical coordinate “z” of pyramid positions at the center of mass of each pyramid
FITEX calculation of “Sun position” by coordinate transformation and fit program FITEX
Ecl. N and S view on ecliptic plane not fixed because of 3-dimensional calculation
The parameters below the first solid line are identical to those in section 3.4.2. The quantities be-
low the dashed line mean the following:
Lm Bm Rm heliocentric spherical coordinates of Mercury
Lv Bv Rv Le Be Re heliocentric spherical coordinates of Venus and Earth
xm ym zm Cartesian coordinates of Mercury; x-axis through Sun and Mercury aphelion
xv yv zv xe ye ze analog Cartesian (rectangular) coordinates for Venus and Earth
xv-xm ... difference of Cartesian coordinates for comparison with pyramid positions
rel. deviation the relative accuracy or error F (resp. Fpos, F'pos, or F''pos)
ascending node ecl. longitude when the planet moves through ecl. plane from south to north
inclination i tilt angle between planes of planetary orbit and Earth's orbit
perihelion pi ecl. longitude for location of shortest distance between planet and the Sun
(M/V/E/Ma) Mercury, Venus, Earth, and Mars
transl. X1, X2, X3 translation coordinates of planetary positions in 3D when using FITEX
del-t time difference between current date and next aphelion/perihelion passage
Euler angl. X4, X5, X6 three angles for rotation of planetary configuration when using FITEX
M scale factor, calculated with M = 1 AU/ X7 (AU = Astronomical Unit)
The next table in the output contains the Cartesian and spherical coordinates of all planets from
Mercury to Neptune. Lm–Lis the difference in ecliptic longitude between Mercury and the corre-
sponding planet, which can be used for a comparison with the accordant angle in the pyramid
area. The quantity dev. is the deviation of Lm–Lin degree to the angles given by the pyramid
positions. The first three angles, belonging to the positions of Mercury, Venus and Earth, are quite
clear (compare with Fig. 8). The deviations for Jupiter and Saturn are based on the positions of the
pyramid at Abu Rawash (Jupiter) and the pyramid area in Abusir (Saturn) [5, Fig. 70, p. 150].
These pyramid locations are very near to the (transformed) orbits of Jupiter and Saturn, after
coordinate transformation of all planetary positions in the solar system with respect to the pyramids
of Giza.
The last table shows the local coordinates of the Sun and all planets after coordinate transforma-
tion to the pyramid area in Giza. The origin of the coordinate system is located in the center of the
base area of the Mykerinos Pyramid. The x-axis points to the north, the y-axis points to the west,
and the z-axis points upward. The quantitydr[m]in the last column is the accuracy of the calcu-
lated “Sun position” and the “planetary positions.” For the calculation of “dr” and for the meaning of
<,” see section 4.9.2 and also [13, Table 25]. The remarkable positions of “Sun” and “Mars” are
highlighted.
3.4.4 Quick start option 4
Output: PLANETS IN ALIGNMENT WITH THE PYRAMIDS OF GIZA
(Mercury near aphelion)
< option 4 >
VSOP87C (2005) full ver., ecliptic of date, "Sun" free 3D, C-M, FITEX
Ecl. N and S, constellation 12, JDE = 2849079.76330, year = 3088.41 (c2)
Special search (interval), step number = 36, step width = 1.000 hour(s)
con k year Lm-Lv Lm-Le e it x-Sun y-Sun z-Sun dr P F[%]
( JDE dt[h] X5 M/10^7 h-Sun " " " " " " )
===============================================================================
31
12 4519 3088.413 13.779 23.191 0 46 -667.5 21.3 272.4 0.8 0.069
2849079.01330 -18.0 25.67 9.649 20.77 -674.6 50.3 272.1 1.6 0.145
2849079.05497 -17.0 25.61 9.656 20.78 -674.3 48.7 272.2 1.5 0.136
2849079.09664 -16.0 25.55 9.662 20.80 -673.9 47.0 272.3 1.4 0.126
2849079.13830 -15.0 25.48 9.668 20.82 -673.5 45.4 272.3 1.3 0.117
2849079.17997 -14.0 25.42 9.674 20.83 -673.1 43.8 272.4 1.2 0.108
2849079.22164 -13.0 25.35 9.681 20.85 -672.7 42.2 272.5 1.1 0.098
2849079.26330 -12.0 25.29 9.687 20.86 -672.3 40.6 272.5 1.0 0.089
2849079.30497 -11.0 25.22 9.693 20.88 -671.9 39.0 272.5 0.9 0.081
2849079.34664 -10.0 25.16 9.700 20.89 -671.6 37.4 272.6 0.8 0.072
2849079.38830 -9.0 25.09 9.706 20.90 -671.2 35.8 272.6 0.7 0.064
2849079.42997 -8.0 25.02 9.712 20.91 -670.8 34.1 272.6 0.6 0.057
2849079.47164 -7.0 24.96 9.719 20.92 -670.4 32.5 272.6 0.6 0.051
2849079.51330 -6.0 24.89 9.725 20.94 -670.0 30.9 272.6 0.5 0.047
2849079.55497 -5.0 24.82 9.732 20.95 -669.5 29.3 272.6 0.5 0.045
2849079.59664 -4.0 24.75 9.738 20.96 -669.1 27.7 272.5 0.5 0.046
2849079.63830 -3.0 24.68 9.745 20.96 -668.7 26.1 272.5 0.5 0.049
2849079.67997 -2.0 24.61 9.751 20.97 -668.3 24.5 272.5 0.6 0.054
2849079.72164 -1.0 24.54 9.758 20.98 -667.9 22.9 272.4 0.7 0.061
2849079.76330 0.0 24.47 9.764 20.99 -667.5 21.3 272.4 0.8 0.069
2849079.80497 1.0 24.40 9.771 21.00 -667.1 19.7 272.3 0.9 0.078
2849079.84664 2.0 24.33 9.778 21.00 -666.7 18.1 272.2 1.0 0.088
2849079.88830 3.0 24.25 9.784 21.01 -666.2 16.5 272.1 1.1 0.098
2849079.92997 4.0 24.18 9.791 21.01 -665.8 14.9 272.0 1.2 0.108
2849079.97164 5.0 24.11 9.798 21.02 -665.4 13.3 271.9 1.3 0.119
2849080.01330 6.0 24.03 9.804 21.02 -664.9 11.7 271.8 1.4 0.130
2849080.05497 7.0 23.96 9.811 21.02 -664.5 10.1 271.7 1.6 0.141
2849080.09664 8.0 23.89 9.818 21.03 -664.1 8.5 271.6 1.7 0.152
2849080.13830 9.0 23.81 9.824 21.03 -663.6 6.9 271.4 1.8 0.164
2849080.17997 10.0 23.74 9.831 21.03 -663.2 5.3 271.3 1.9 0.175
2849080.22164 11.0 23.66 9.838 21.03 -662.8 3.7 271.1 2.1 0.187
2849080.26330 12.0 23.58 9.845 21.03 -662.3 2.2 271.0 2.2 0.199
2849080.30497 13.0 23.51 9.852 21.03 -661.9 0.6 270.8 2.3 0.211
2849080.34664 14.0 23.43 9.859 21.03 -661.4 -1.0 270.6 2.4 0.223
2849080.38830 15.0 23.35 9.865 21.03 -661.0 -2.6 270.4 2.6 0.235
2849080.42997 16.0 23.28 9.872 21.03 -660.5 -4.2 270.2 2.7 0.247
2849080.47164 17.0 23.20 9.879 21.03 -660.1 -5.8 270.0 2.8 0.259
2849080.51330 18.0 23.12 9.886 21.02 -659.6 -7.3 269.8 3.0 0.272
===============================================================================
CPU-time 0: 0: 0.076 -- end of run.
This is a time scan around the aphelion passage of Mercury [13, Tab. 24] in the year 3088 AD.
Theoretical and (almost) ideal values are highlighted (see explanations in [13]). The new param-
eters are described as follows:
Special search (interval) search with Mercury near to the aphelion position
step number number of time steps in the time interval for each aphelion passage
step width width of time steps in hours
The first of the two rows just above the solid line at the beginning of the table is identical to that in
section 3.4.2. It belongs to the very first row of numbers in the table, which gives some quantities
for the date of the aphelion passage. The second of the two header rows belongs to all other rows
in the table. It contains some new parameters:
dt[h] time difference in hours to the middle of the time interval (aphelion passage)
X5 tilt angle X5 between Earth's surface and the transformed plane of the Earth's orbit
M/10^7 scale factor between positions of planets and pyramids (divided by 107)
h-Sun height of the transf. “Sun position” above the southern horizon in degree, as seen
from the “Mercury position” (see Figs. 2, 18, and [5, Fig. 151], and [13, Fig. 13])
32
3.4.5 Quick start option 7
Output: PLANETS IN ALIGNMENT WITH THE CHAMBERS OF THE CHEOPS PYRAMID
(Mercury at perihelion)
< option 7 >
"Keplers equation", ecliptic of date, E-V-M, "Sun" south of sub. cham.
Ecl. N and S, years -13000.00 to 17000.00 (c2) angular range: 1.8500 deg
con k JDE year Lm Lm-Lv Lm-Le del1 del2 P
===============================================================================
-56861 -2550434.93231 -11694.956 230.093 98.696 114.324 1.736 -1.186 *
-54691 -2359541.44361 -11172.307 237.794 -98.170-115.408 -1.210 0.102
-51783 -2103726.57489 -10471.910 248.141 95.639 116.330 -1.321 0.820 *
-42816 -1314905.41676 -8312.191 280.237 -96.743-113.723 0.217 1.787
1 -38913 -971561.04515 -7372.146 294.297 -97.721-115.558 -0.761 -0.048
-36005 -715746.17643 -6671.749 304.808 95.922 116.021 -1.038 0.511 *
-27038 73074.98170 -4511.932 337.411 -96.272-113.791 0.688 1.719
2 -23135 416419.35331 -3571.906 351.692 -97.244-115.611 -0.284 -0.101
-20227 672234.22203 -2871.523 2.367 96.199 115.701 -0.761 0.191 *
-16324 1015578.59364 -1931.497 16.743 95.214 113.840 -1.746 -1.670 *
-11260 1461055.38017 -711.848 35.477 -95.807-113.768 1.153 1.742
7 -8352 1716870.24889 -11.465 46.276 97.477 117.232 0.517 1.722 *
3 -7357 1804399.75177 228.177 49.978 -96.781-115.572 0.179 -0.062
-4449 2060214.62050 928.560 60.818 96.494 115.370 -0.466 -0.140 *
12 4518 2849035.77863 3088.293 94.434 -95.370-113.661 1.590 1.849
8 7426 3104850.64735 3788.690 105.397 97.792 116.894 0.832 1.384 *
4 8421 3192380.15023 4028.338 109.155 -96.348-115.453 0.612 0.057
11329 3448195.01896 4728.735 120.159 96.809 115.031 -0.151 -0.479 *
12324 3535724.52184 4968.383 123.931 -97.326-117.242 -0.366 -1.732
9 23204 4492831.04581 7588.851 165.409 98.118 116.555 1.158 1.045 *
5 24199 4580360.54870 7828.499 169.223 -95.949-115.263 1.011 0.247
27107 4836175.41742 8528.896 180.391 97.132 114.689 0.172 -0.821 *
28102 4923704.92030 8768.544 184.219 -96.933-117.041 0.027 -1.531
10 38982 5880811.44428 11389.013 226.310 98.444 116.222 1.484 0.712 *
39977 5968340.94716 11628.660 230.180 -95.586-115.017 1.374 0.493
42885 6224155.81588 12329.058 241.512 97.455 114.354 0.495 -1.156 *
43880 6311685.31876 12568.705 245.396 -96.577-116.788 0.383 -1.278
54760 7268791.84274 15189.174 288.099 98.766 115.911 1.806 0.401 *
55755 7356321.34562 15428.822 292.026 -95.262-114.737 1.698 0.773
58663 7612136.21434 16129.219 303.521 97.771 114.045 0.811 -1.465 *
59658 7699665.71723 16368.866 307.461 -96.255-116.505 0.705 -0.995
===============================================================================
Computed constellations: 124558 (P: polarity, resp. view on ecliptic)
Detected constellations: 31 CPU-time 0: 0: 0.148 -- end of run.
New terms and parameters:
"Keplers equation" calculation with orbital elements and solving Kepler's equation
E-V-M mapping of Earth, Venus, Mercury to King's, Queen's, and subterranean
chamber
"Sun" south of sub. cham. “Sun position” fixed, south of subterranean chamber
angular range limit for angular deviations in degree when comparing the positions
del1 del2 angular deviations
1 and
2 in degree between angles of planetary posi-
tions (Lm Lv and Lm Le) and of chamber positions in Cheops Pyramid
This run uses the orbital elements given as polynomials of third degree and solving Kepler's equa-
tion numerically. It needs less than 1 second to check more than 124,000 constellations. The re-
sults are not as precise as calculated with the short and full versions of VSOP87, but the computa-
tion is much faster and all important constellations are found.
33
3.4.6 Quick start option 11
Output: TRANSITS OF MERCURY
(geocentric transit phases, terrestrial time TT)
< option 11 >
VSOP87C, comb. search, ecliptic of date, all Mercury transits
Period (years) from 2950.00 to 3200.00, Jul./Greg. calendar
co/p date/ time: I II nearest III IV sep["]a S
===============================================================================
18. Nov. 2953 12:58:21 13: 0:38 15: 2:35 17: 4:34 17: 6:52 -646.7/ 17
19. May 2963 2:31: 0 2:34:14 6: 6:17 9:38:13 9:41:27 386.9 18
21. Nov. 2966 6: 4:14 6: 5:57 8:48:19 11:30:45 11:32:28 -141.4/ 16
21. May 2976 9:57:19 10: 1:20 12:54:45 15:48: 6 15:52: 6 -634.7 15
24. Nov. 2979 23:58:20 0: 0: 9 2:31:53 5: 3:40 5: 5:30 359.5/ 14
25. Nov. 2992 18:52:55 18:56:30 20:11:23 21:26:17 21:29:53 855.5/ 12
19. Nov. 2999 17:34:26 17:36:30 19:52:29 22: 8:31 22:10:35 -542.5/ 17
20. May 3009 8:48:36 8:51:37 12:39:28 16:27:11 16:30:12 190.9 18
22. Nov. 3012 10:51:51 10:53:33 13:37:29 16:21:31 16:23:13 -36.8/ 16
23. May 3022 17:33:33 17:39:50 19:28:51 21:17:50 21:24: 7 -836.4 15
25. Nov. 3025 4:55:32 4:57:27 7:20:44 9:44: 4 9:46: 0 463.5/ 14
18. Nov. 3032 6:15:14 6:23:43 6:53: 7 7:22:32 7:31: 1 -945.6/ 19
28. Nov. 3038 0:31: 3 0:43:41 0:59:11 1:14:41 1:27:19 958.8/ 12
21. Nov. 3045 22:13: 3 22:14:57 0:41:37 3: 8:21 3:10:15 -436.5/ 17
21. May 3055 15:25:25 15:28:23 19:21:16 23:13:59 23:16:56 -14.3 18
23. Nov. 3058 15:42: 5 15:43:47 18:27:21 21:11: 1 21:12:43 66.7/ 16
26. Nov. 3071 9:56:42 9:58:47 12:10:45 14:22:47 14:24:52 567.6/ 14
19. Nov. 3078 10:18:37 10:22: 5 11:42:28 13: 2:52 13: 6:20 -839.5/ 19
12 18. May 3088 17:10:47 17:16: 8 19:20:59 21:25:48 21:31: 8 796.5 20
22. Nov. 3091 2:54:52 2:56:41 5:31: 3 8: 5:30 8: 7:19 -332.3/ 17
23. May 3101 22: 4:47 22: 7:50 1:54:59 5:41:59 5:45: 1 -212.9 18
24. Nov. 3104 20:34:48 20:36:32 23:17:46 1:59: 5 2: 0:48 172.4/ 16
27. Nov. 3117 14:59:23 15: 1:43 16:58:56 18:56:12 18:58:32 670.4/ 14
20. Nov. 3124 14:42:24 14:45: 3 16:31:23 18:17:46 18:20:24 -735.1/ 19
21. May 3134 22:49:57 22:53:44 1:52:34 4:51:19 4:55: 6 601.2 20
23. Nov. 3137 7:38:45 7:40:30 10:20:16 13: 0: 8 13: 1:53 -226.8/ 17
24. May 3147 4:59: 9 5: 2:27 8:32:13 12: 1:53 12: 5:11 -414.1 18
26. Nov. 3150 1:28:23 1:30: 9 4: 7: 4 6:44: 4 6:45:50 276.0/ 16
28. Nov. 3163 20: 6:36 20: 9:25 21:46:36 23:23:49 23:26:37 773.6/ 14
21. Nov. 3170 19:14:30 19:16:46 21:21:19 23:25:55 23:28:10 -630.3/ 19
21. May 3180 4:52: 1 4:55:16 8:24:42 11:54: 1 11:57:15 405.9 20
24. Nov. 3183 12:26:11 12:27:54 15:10:53 17:53:57 17:55:40 -123.0/ 17
24. May 3193 12: 2:16 12: 6:10 15: 4: 4 18: 1:52 18: 5:45 -612.6 18
26. Nov. 3196 6:22:28 6:24:19 8:54:45 11:25:16 11:27: 6 379.7/ 16
===============================================================================
Computed constellations: 10687 ("/" means ascending node)
Tested planet. passages: 788
Detected transits : 34
Centr./grazing transits: 0 / 0 CPU-time 0: 0: 1.192 -- end of run.
In the header, the expressioncomb. searchmeans “combination search”: The search starts for
each transit with the VSOP87C short version and continues with the full version. Nonetheless, no
central nor grazing transit appears during this time period. The constellation number (12) at the
beginning of the line is automatically generated by the program (subroutine “konst”). This program
run is similar to the book option 310 [13, Tab. 31].
The parameters in the last header line are as follows:
34
co number of constellation (such as 12), “->” means “near to a known constellation”
p partial transit: “m” Mercury, “v” Venus; “c” or “C” central transit (not given here)
date calendar date of middle of the transit, more precisely: minimum separation
I II III IV times of inner and outer contact points, transit phases (see Fig. 7)
nearest moment of nearest approach (min. sep.) between planet and center of the Sun
sep["] minimum separation between planet and the Sun in arc seconds
a ascending node: “slash” (descending node: “no slash”)
S serial number of transit
In our epoch, the passage of Mercury through the ascending node always takes place in Novem-
ber, the passage through the descending node in May. However, the ascending and descending
node (a) for Mercury and Venus are not determined by the given months but independently on the
basis of geometrical considerations.
Each transit of one series is labeled with the same number. In contrast, the absolute value of the
serial numbers are arbitrary. Jean Meeus, for example, did not name each series with a number
but with a letter A, B, C, ... [22, pp. 42 ff.]. Here, we take the numbers used on the NASA/Goddard
Space Flight Center website, webmaster Fred Espenak:
http://eclipse.gsfc.nasa.gov/transit/catalog/MercuryCatalog.html
and http://eclipse.gsfc.nasa.gov/transit/catalog/VenusCatalog.html
(Note: In these links, the transit phases are given in universal time UT.) In order to always get the
same serial numbers S, independently from the starting date of the chosen time period, the first
numbers are taken from the file “inserie.t”. Thus, this file is used only at the beginning of each run.
All other serial numbers are determined during runtime of the program.
3.4.7 Quick start option 14
Output: PLANETS IN A LINE (SYZYGY)
(angular range of eclipt. longitudes dL minimized, JDE)
< option 14 >
VSOP87C, comb. search, ecliptic of date, linear c. Mercury to Mars
Period (years) -13000.00 to 17000.00 (c1), angular r.: 6.00/ 5.00 deg
co tr k JDE year dt[days] Lm-Lv Lm-Le Lm-Lma dLmin
===============================================================================
-62144 -3015259.12387 -12967.601 -38.133 1.757 0.0 0.105 1.757
-61116 -2924752.04882 -12719.801 36.451 -2.558 -3.025 0.0 3.025
-56018 -2476304.86414 -11491.995 15.891 -1.871 -3.100 0.0 3.100
-55699 -2448270.17542 -11415.238 -11.643 2.577 3.490 0.0 3.490
-54830 -2371781.88140 -11205.820 31.286 1.469 0.0 0.900 1.469
-51975 -2120688.28715 -10518.350 -27.612 -1.048 -0.384 0.0 1.048
M -50946 -2030182.60335 -10270.553 -42.389 -0.500 -0.988 0.0 0.988
-48544 -1818813.77667 -9691.845 24.059 4.074 0.0 1.955 4.074
-48225 -1790778.08803 -9615.086 -2.474 3.782 3.234 0.0 3.782
-44501 -1463199.29860 -8718.206 -21.543 2.831 -0.189 2.831 3.019
V -40777 -1135613.28929 -7821.306 -33.392 -2.288 -2.121 0.0 2.288
-39749 -1045106.26361 -7573.506 41.143 -4.656 -3.657 0.0 4.656
-33463 -492135.45751 -6059.523 36.617 -0.732 -0.497 0.0 0.732
-29579 -150537.15060 -5124.259 -38.030 -3.780 -2.775 0.0 3.780
-28046 -15654.33423 -4754.962 -12.227 4.108 3.948 0.0 4.108
-27177 60834.13961 -4545.544 30.882 1.499 0.0 0.676 1.499
-24322 311928.64773 -3858.071 -27.103 2.726 3.541 0.0 3.541
-23293 402434.40305 -3610.274 -41.808 3.463 3.984 0.0 3.984
35
-20891 613802.18908 -3031.569 23.600 4.661 0.0 3.870 4.661
-20572 641837.19908 -2954.812 -3.613 4.509 2.495 0.0 4.509
-19384 746366.77815 -2668.619 18.379 0.226 4.109 0.0 4.109
-16848 969416.60704 -2057.930 -22.063 4.094 0.469 0.0 4.094
-13124 1297003.80471 -1161.026 -32.723 2.174 3.048 0.0 3.048
M -12096 1387510.46824 -913.228 41.449 -2.172 -0.337 0.0 2.172
-5810 1940480.90518 600.754 36.554 0.079 0.707 0.0 0.707
-5650 1954490.74930 639.112 -28.698 4.457 0.0 0.403 4.457
-4621 2044997.03231 886.909 -42.875 4.093 0.0 1.425 4.093
-2955 2191576.37670 1288.230 -20.467 -4.154 0.0 -0.771 4.154
-1926 2282079.86037 1536.020 -37.445 0.453 2.400 0.0 2.400
476 2493450.15754 2114.732 30.475 1.507 -0.232 1.507 1.739
795 2521489.41998 2191.501 7.515 -3.907 0.0 -1.484 3.907
12 M 4519 2849066.01327 3088.376 -13.750 -3.397 -2.605 0.0 3.397
5548 2939566.30702 3336.157 -33.917 3.882 0.0 0.569 3.882
8243 3176650.26922 3985.271 -27.352 -3.312 0.0 -0.379 3.312
8269 3178981.57686 3991.654 16.752 -0.800 1.467 -0.800 2.267
9272 3267156.02956 4233.067 -42.053 -3.820 -0.574 0.0 3.820
15557 3820126.65275 5747.049 41.208 -2.124 0.0 -1.952 2.124
15717 3834138.13217 5785.411 -22.408 -2.064 -3.418 0.0 3.419
15743 3836472.36564 5791.802 24.622 4.659 4.389 0.0 4.659
16746 3924642.67733 6033.204 -38.324 2.520 0.0 2.339 2.520
19441 4161725.56802 6682.315 -32.830 -3.781 0.0 -0.412 3.781
20974 4296612.30826 7051.623 -3.103 -4.188 0.0 -3.368 4.188
M 21843 4373097.00488 7261.031 36.229 -0.135 0.380 -0.135 0.515
24698 4624193.65495 7948.510 -19.615 -0.632 4.214 -0.632 4.846
26915 4819212.51626 8482.453 -28.801 -1.437 -3.049 0.0 3.049
28129 4926065.39990 8775.007 29.292 -0.821 -3.563 0.0 3.563
28448 4954105.08050 8851.777 6.750 -2.902 0.0 -1.249 2.902
32172 5281682.80510 9748.654 -13.384 -0.969 0.0 -0.897 0.969
35922 5611596.79305 10651.928 15.543 -0.971 0.043 -0.971 1.013
36925 5699772.17611 10893.344 -42.331 -3.639 0.0 -2.630 3.639
38113 5804287.97146 11179.498 -34.123 -2.641 -3.546 0.0 3.546
39646 5939173.53429 11548.803 -5.573 -1.199 -3.007 0.0 3.007
43210 6252743.27610 12407.327 41.406 0.134 2.572 0.0 2.572
M 43370 6266755.55351 12445.692 -21.412 1.706 1.352 0.0 1.706
43396 6269087.74498 12452.077 23.576 3.799 2.388 0.0 3.799
44399 6357258.98958 12693.482 -38.437 3.641 1.774 0.0 3.641
49496 6805712.91680 13921.307 35.715 -0.904 -0.924 0.0 0.924
50844 6924245.65229 14245.838 -14.233 3.579 -0.162 3.579 3.742
54568 7251829.78723 15142.733 -27.956 2.517 2.077 0.0 2.517
56101 7386721.26013 15512.054 6.504 -1.307 1.289 -1.307 2.596
===============================================================================
Computed constellations: 150628
Number of syzygies : 60 CPU-time 0: 0: 7.920 -- end of run.
New expressions and parameters:
linear c. linear constellation, syzygy
angular r. max. angular range, first value: short version, second value: full version of VSOP87
co number of constellation
tr transit, “M,” “V”: full transit, “m,” “v”: grazing transit (within a few hours or days)
dLmin minimum angular range dLmin of ecliptic longitudes of all participating planets
The moment of minimum angular range for the ecliptic longitudes of all participating planets does
not need to happen within the period of the planetary transit, but can happen shortly before or after
the transit. So, the time difference between the moment of minimum angular range and transit can
be a few hours or days. The angular range (angular r.) of 6° andin the head lines belong to
the short and the full version of VSOP87. The first number should be larger than the second one
(see also [13, Table 29]). Otherwise, one or a few constellations can be lost.
36
3.4.8 Book option 250
The table [13, Tab. 25] represents all important data when the planets stand in a constellation
according to the chamber arrangement in the Cheops Pyramid. This book option is identical to
option 8. Mercury is placed in its perihelion 44 days before the “pyramid constellation” (option 3).
Two significant locations in the Cheops Pyramid – secret chambers (?) – are highlighted.
Output: PLANETS IN ALIGNMENT WITH THE CHAMBERS OF THE CHEOPS PYRAMID
(Mercury at perihelion)
< option 250 >
VSOP87C (2005) full ver., ecliptic of date, E-V-M, "Sun" free 3D mid., FITEX
Ecl. N and S, constellation 12, JDE = 2849035.77863, year = 3088.29 (c2)
date (Gregor.,TT) = 17. Apr. 3088, 6:41:13, Tuesday
===============================================================================
con k year Lm-Lv Lm-Le e it x-Sun y-Sun z-Sun dr P F[%]
-------------------------------------------------------------------------------
Lm Bm Rm Lv Bv Rv Le Be Re
xm ym zm xv yv zv xe ye ze
xv-xm xe-xm yv-ym ye-ym zv-zm ze-zm rel. deviation
===============================================================================
12 4518 3088.293 -95.595-113.868 0 105 -21.78 -17.38 -8.76 0.20 0.570
-------------------------------------------------------------------------------
94.2332 3.8355 0.307417 189.8280 3.3144 0.720012 208.1012 -0.0001 0.998755
0.306728 0.000000 0.020564-.070079 .715384 .041627-0.404127 0.913342-0.000002
-0.376807-0.710856 0.715384 0.913342 0.021064-0.020566 0.56966279 %
-------------------------------------------------------------------------------
ascending node (M/V/E/Ma): 61.260938 86.534589 --- 57.965443
inclination i (M/V/E/Ma): 7.022735 3.405472 0.000000 1.844689
perihelion pi (M/V/E/Ma): 94.429919 146.689667 121.705528 356.112069
transl. X1, X2, X3; del-t: -0.305872 0.001023 -0.020245 0.000 days
Euler angl. X4, X5, X6; M: -16.990371 -4.182910 50.631409 2316903300.
===============================================================================
===============================================================================
pla. x[AU] y[AU] z[AU] L B r[AU] Lm-L