Chapter I. Solar and Lunar Eclipses 6 1.1

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ANCIENT RIDDLES OF SOLAR ECLIPSES. Asymmetric Astronomy

Second Edition

By IGOR N. TAGANOV and VILLE-V.E. SAARI

Russian Academy of Sciences Saint Petersburg 2016

Taganov, Igor N., Saari, Ville-V.E. Ancient Riddles of Solar Eclipses. Asymmetric Astronomy. Second Edition – Saint Petersburg: TIN, 2016. – 110 p., 53 ill. Electronic Edition

ISBN 978-5-902632-28-3 © Taganov, Igor N.; Saari, Ville-V.E. 2016

The book examines some of the mysteries of ancient astronomical treatises, for example, known since the Middle Ages the “Wednesday paradox”, and the history of the emergence and spread in the East of the belief that the eclipses of the Sun and the Moon, as well as all the Universe geometry are defined by a single sacred number 108. The calendar cycles of solar eclipses, considered in the book, confirming the old assumption of Indian and Chinese astronomers in 6-8 centuries, show that the probability of a total solar eclipse is larger in the spring and summer months, and the probability of annular eclipse, on the contrary, is larger in the autumn and winter months. Analysis of ancient chronicles of solar and lunar eclipses discovers evidence of gradual deceleration of time, which is confirmed by modern astronomical observations of the orbital movement of the Earth, the Moon, Mercury and Venus. The cosmological deceleration of time is a consequence of the irreversibility of “physical” time, which leads to the fact that all the characteristic time intervals are shorter in the past than in the future. In theoretical cosmology, the use of the concept of decelerating physical time allows to represent the key cosmological parameters of the observable Universe in the form of simple functions of the fundamental physical constants. In the book the astronomical observations that demonstrate various forms of cosmological deceleration of time in deep space, in the solar system and on the Earth are discussed.

Keywords: eclipse, planet orbit, time, time dilation, cosmology discusses

PACS: 95.90.+v, 96.20.-n, 96.30.Dz, 96.30.Ea, 98.80.-k

Other editions:

Таганов И.Н., Саари В.-В.Е. Древние загадки солнечных затмений. Асимметричная астрономия. Санкт- Петербург: ТИН, 2014. ISBN 978-5-902632-14-6.

Taganov, Igor N., Saari, Ville-V.E. Ancient Riddles of Solar Eclipses. Asymmetric Astronomy. – Saint Petersburg: TIN, 2014. ISBN 978-5-902632-15-3.

Kindl Edition (English) ASIN: B012T2FL83. http://www.amazon.com/books Ancient Riddles of Solar Eclipses: Asymmetric Astronomy [Print Replica] Kindle Edition. By Igor Taganov (Author), Ville Saari (Author)

International Interdisciplinary Research Project the “Time Pace”: https://www.timepace.net and www.spiraltime.org

ISBN 978-5-902632-28-3 Electronic Edition

PREFACE Modern astronomers have received from their predecessors the priceless heritage – a detailed description of solar and lunar eclipses observed in the previous epochs. In the late 17th century, this data allowed discovering the acceleration of the Moon, and in the 19th century, the old chronicles permitted to estimate the tidal deceleration of the Earth’s rotation. Nowadays, astronomical chronicles help investigate secular changes of the Earth and Moon orbits. However, the ancient astronomical manuscripts in addition to dispassionate eclipse chronicles contain unusual riddles which are sometimes difficult to interpret scientifically. Astronomers still can not offer any ideas to explain known since the middle Ages “Wednesday paradox” – in Europe, the probability of a solar eclipse on Wednesday almost two times more than, for example, on a Tuesday or Thursday. The book also analyzes the history of the emergence and spread in the East of the belief that eclipses of the Sun and the Moon, as well as all the geometry of the Universe, are defined by a single sacred number 108. The authors qualitatively and quantitatively confirmed the assumption of Indian and Chinese astronomers made in 6-8 centuries that total solar eclipses occur more frequently in the summer months and in the southern regions of the Earth. The calendar cycles of solar eclipses studied in the book show that the probability of a total solar eclipse is larger in the spring and summer months, and the probability of an annular eclipse, on the contrary, is larger in the autumn and winter months. The analyzis of ancient chronicles of solar and lunar eclipses reveals the evidence of gradual deceleration of time, and the observational evidence of this new cosmological phenomenon is discovered in astronomical observations of orbital motion of the Earth, the Moon, Mercury and Venus. Cosmological deceleration of time is a consequence of the irreversibility of the “physical” time, which is registered by our watches, and manifests itself in many “asymmetric” astronomical observations. The use of the “present” standard of time interval leads to linear decrease of the physical and Newtonian intervals ratio in the past and, on the contrary, this ratio linearly increases in the future. Therefore, interpretation of the observations, using the laws of Newton and Kepler with uniform Newtonian time, has led to the emergence of “asymmetric” astronomy in which a retrospective analysis of the motion of celestial bodies does not correspond to observations. The cosmological deceleration of time explains well many observed astronomical phenomena – strange acceleration of the Earth’s rotation, anomalous accelerations of the Moon and planets, as well as systematic discrepancy of isotopic ages of rock samples from the Earth and the Moon. Expressive manifestation of the cosmological deceleration of time is a recently discovered apparent acceleration of the Universe space expansion. In theoretical cosmology, the use of the concept of decelerating physical time allows to represent the key cosmological parameters of the observable Universe in the form of simple functions of the fundamental physical constants. This electronic edition of our monograph, which we present to readers, develops the ideas firstly published in our articles and books:

Taganov, Igor N. Irreversible-Time Physics. – Saint Petersburg: TIN, 2013. ISBN 978-5-902632-12-2. Таганов И.Н. Физика необратимого времени. Санкт-Петербург: ТИН, 2014. ISBN 978-5-902632-16-0.

Taganov, Igor N., Saari, Ville-V.E. Ancient Riddles of Solar Eclipses. Asymmetric Astronomy. – Saint Petersburg: TIN, 2014. ISBN 978-5-902632-15-3. Таганов И.Н., Саари В.-В.Е. Древние загадки солнечных затмений. Асимметричная астрономия. Санкт-Петербург: ТИН, 2014. ISBN 978-5-902632-14-6.

During preparation of this “Second Edition” of our book we have made the necessary corrections and clarifications. Besides in the “Second Edition” we included the new section: “2.4. Description of a movement with decelerating time”.

Readers who became interested in developing of the physics of irreversible time, can visit the regularly updated websites: The Interdisciplinary Research Project “Time Pace”: www.timepace.net and www.spiraltime.org

We thank all those colleagues who have commented on our earlier books and articles and we are grateful to Yu.I. Babenko, Yu.V. Baryshev, A.P. Gagarin, A.L. Gromov, F. Hoyle, J. Masreliez, G. Paturel, A. Sandage, A.G. Shlienov, A.A. Tron, D.A. Varshalovich and A.N. Zemtsov for fruitful discussions, constructive criticism and insightful comments.

Saint Petersburg, 2016 Igor N. Taganov and Ville-V.E. Saari

CONTENTS PREFACE 4 Chapter I. Solar and lunar eclipses 6 1.1. Metaphysics of solar eclipses 16 1.2. The calendar cycles of solar eclipses 38 Chapter II. Cosmological deceleration of time 45 2.1. The art of measuring time 45 2.2. Nonlinear time scales 55 2.3. Cosmological deceleration of physical time 61 2.4. Description of a movement with decelerating time 78 2.5. Cosmological deceleration of time at the Earth and in Solar system. 82 The apparent secular acceleration of the Earth’s rotation 82 Cosmological corrections for the observed accelerations of the Earth and Moon 85 Accelerations of the Mercury and Venus 90 The Pioneer Anomaly mystery 93 Kinetics of the radioactive isotope decays in non-static Universe 96 Registration of the cosmological deceleration of time in laboratories 101 The illusion of Universe’s “accelerating” expansion 103 Literature 105 Physical constants and parameters 107 APPENDIX 108

CHAPTER I. Solar and lunar eclipses To describe the two main types of solar eclipses in modern astronomy the old Latin terms – umbra, antumbra and penumbra are still used (Fig. 1.1). A “partial eclipse” (c. 35 %) occurs when the Sun and Moon are not exactly in line and the Moon only partially obscures the Sun. The term “central eclipse” (c. 65 %) is often used as a generic term for eclipses when the Sun and Moon are exactly in line. The strict definition of a central eclipse is an eclipse, during which the central line of the Moon’s umbra touches the Earth’s surface. However, extremely rare the part of the Moon’s umbra intersects with Earth, producing an annular or total eclipse, but not its central line. Such event is called a “non-central” total or annular eclipse.

Fig. 1.1. Main types of solar eclipses

The central solar eclipses are subdivided into three main groups: a “total eclipse” (c. 27 %) occurs when the dark silhouette of the Moon completely obscures the Sun; an “annular eclipse” (c. 33 %) occurs when the Sun and Moon are exactly in line, but the apparent size of the Moon is smaller than that of the Sun; a “hybrid eclipse” or annular/total eclipse (c. 5 %) at certain sites on the Earth’s surface appears as a total eclipse, whereas at other sites it looks as annular. There are more annular solar eclipses than total because on average the Moon moves too far from the Earth to cover the Sun completely.

Fig. 1.2. “Diamond rosary” (1) and “Diamond ring” (2) during total sun eclipse on November 25, 2011 [http://www.spacetribe.com/]. “Wedding ring” (3) – the annular sun eclipde on May 20, 2012 [/wikipedia.org/]. 6

From ancient times several picturesque effects during solar eclipses are known, for example, “Diamond rosary” (Fig 1.2-1), “Diamond ring” (Fig 1.2-2) and “Wedding ring” (Fig 1.2-3). The “Diamond rosary” and “Diamond ring” now astronomers name the “Baily’s beads” in honor of English astronomer Francis Baily (1774–1844) who first published an explanation of these phenomena – the bursts of light appear around the lunar silhouette when during total sun eclipse the sunlight shines through rugged lunar limb in some places.

Fig. 1.3. The geometry of lunar eclipse. Right part: the lunar eclipse on October 27, 2004 (Eclipse Predictions by Fred Espenak, NASA’s GSFC; http://eclipse.gsfc.nasa.gov/. For earthly observer the Moon daily crosses the sky from east to west, however, with respect to the Earth’s shadow cone and the stars it moves from west to east.

A lunar eclipse occurs when the Moon passes behind the Earth into its shadow (umbra). Lunar eclipses occur when the Sun, Earth and Moon are aligned in syzygy (from the Ancient Greek σύζυγος, suzugos, meaning “yoked together”) – a straight-line configuration of all three celestial bodies (Fig. 1.3). The Moon crosses ecliptic – the plane of the Earth’s orbit around the Sun at positions called “nodes” twice every month and when the Full Moon occurs in the same position at the node, a lunar eclipse can occur. These two nodes allow two to five eclipses per year, parted by approximately six months. An eclipse of the Moon can only take place at Full Moon, and only if the Moon passes through some portion of Earth’s shadow, which composed of two cone-shaped parts, one nested inside the other.

Fig. 1.4. Lunar eclipses: partial (1), total penumbral eclipse (2) and total eclipse (3; the “Blood Moon”).

When the Moon enters into the Earth’s penumbra a partial penumbral eclipse (Fig. 1.4-1) starts with a subtle darkening of the Moon’s surface. When the Moon moves exclusively within the Earth’s penumbra a total penumbral eclipse (Fig. 1.4-2) is visible. Total penumbral eclipses are rare, and when these occur, that area of the Moon which is closest to the umbra can appear somewhat darker

7 than the rest of the Moon. If the entire Moon passes through the umbral shadow, then a total eclipse (Fig. 1.4-3) of the Moon occurs. The type and length of a lunar eclipse depend upon the Moon’s location relative to its orbital nodes. The Moon’s speed through the Earth’s shadow is about one kilometer per second, and total eclipse may last up to more than 100 minutes. However, the total time between the Moon’s first and last contact with the Earth’s shadow is much longer, and could last up to 4 hours. In contrast to a solar eclipse, which can only be viewed from a certain relatively small area of the world, a lunar eclipse may be viewed from anywhere on the night side of the Earth. A lunar eclipse lasts for a few hours, whereas a total solar eclipse lasts for only a few minutes at any given place, due to the smaller size of the Moon’s shadow. A totally eclipsed Moon occurring near Moon’s apogee where its orbital speed is the slowest will lengthen the duration of total eclipse. The inner shadow (umbra) is a region where Earth blocks all direct sunlight from reaching the Moon. The outer shadow (penumbra) is a zone where Earth blocks only part of the Sun’s rays. Though within the umbra the Moon is totally shielded from direct illumination by the Sun, the Moon does not completely disappears as it passes through the umbra because of the refraction of sunlight by the Earth’s atmosphere into the shadow cone. The amount of scattered and refracted light depends on the amount of dust and clouds in the Earth’s atmosphere that influences the color of eclipsed Moon. During an eclipse, the Moon’s disk can take on a dramatically colorful appearance from dark gray or bright orange to coppery-red or dark brown. To describe the color of eclipsed Moon the scale proposed in 1930s by French astronomer André-Louis Danjon (1890–1967) is sometimes used: L=0: Very dark eclipse with Moon almost invisible; L=1: Dark gray or brownish eclipse; L=2: Deep red or rust-colored eclipse; L=3: Brick-red eclipse with the umbral shadow having a bright rim; L=4: Intense copper-red or orange lunar disc having a bright rim and the bluish umbral shadow. A “selenelion” or “horizontal eclipse” occurs when both the Sun and the eclipsed Moon can be observed at the same time, which happens just before sunset or just after sunrise and both bodies will appear just above the horizon at nearly opposite points in the sky. There are a number of high ridges undergoing sunrise or sunset that can see it. Although the Moon is in the Earth’s umbra, the Sun and the eclipsed Moon can both be seen at the same time because the refraction of light through the Earth’s atmosphere causes each of them to appear higher in the sky than their actual geometric positions. The red coloring of eclipsed Moon arises because sunlight reaching the Moon must pass through a long and dense layer of the Earth’s atmosphere, where shorter wavelengths are more likely to be scattered by the air and dust particles, and so by the time the light has passed through the atmosphere, the longer wavelengths dominate. Such scattered light we see as red hues during sunsets and sunrises and it often paints the eclipsed Moon by a reddish color being a cause of the “Blood Moon” phenomenon. Eclipse season is the only time during which the Sun is close enough to one of the Moon’s nodes to allow for an eclipse to occur (Fig. 1.7). During the Eclipse season, whenever there is a Full Moon a lunar eclipse will occur and whenever there is a New Moon a solar eclipse will occur. Each season lasts from 31 to 37 days, returning about every 6 months (173.31 days – half of an Eclipse year). At least two – one solar and one lunar, in any order, and at most three eclipses – solar, lunar, then solar again, or vice versa, will occur during every Eclipse season. Since it is about 15 days between Full Moon and New Moon, if there is an eclipse at the very beginning of the Eclipse season, then there is enough time for two more eclipses. If the last eclipse of an Eclipse season occurs at the very beginning of a calendar year, it is possible for seven eclipses to occur since there is still time before the end of the calendar year for two full Eclipse seasons, each having up to three eclipses. Occasionally 4 total lunar eclipses occur in a sequence with intervals of 6 lunations (the average time for one lunar phase cycle i.e., the synodic period of the Moon, or the period from one New Moon to the next). Such remarkable lunar eclipse series is called the “Tetrad”. Italian astronomer and science historian Giovanni Schiaparelli (1835–1910) noticed that there are eras when Tetrads occur

8 comparatively frequently, interrupted by eras when they are rare and later Dutch astronomer Antonie Pannekoek (1873–1960) explained Tetrad phenomenon and found a period of 591 years. Recently Tudor Hughes explained the Tetrad period variation by secular changes in the eccentricity of the Earth’s orbit and estimated the current period as about 565 years. To describe the regular appearance of Tetrads Belgian astronomer Jean Meeus proposed special astronomical period – the “Tetradia”, which consists of 6984 synodic months and 595 eclipse years (see details in [15]).

Fig. 1.5. Correlation of 2014-2015 Tetrad with the Tetrad, which accompanied the dramatic Gospel events and Jesus Christos crucifixion.

Since the usual red coloring of totally eclipsed Moon the Tetrad of 4 total lunar eclipses sometimes is named the Blood Moon Tetrad. From time to time the Blood Moon Tetrads coincides with feasts of ancient calendars, which are also based on the lunar phase cycles. One of such coincidence is the correlation of 2014-2015 Tetrad (on Gregorian calendar) with the Tetrad, which accompanied the dramatic Gospel events and Jesus Christos crucifixion (Fig. 1.5). The first Blood Moon of 2014 is total lunar eclipse on April 15, which was visible across most of North America, Latin America and the Caribbean, coincides with the first day of Jewish Passover 2014. The second Blood Moon on October 8 coincides with the 1st day of Sukkoth (Tabernacles) 2014, the third on April 4, 2015 coincides with the 1st day of Passover 2015 and the fourth on September 28, 2015 coincides with the 1st day of Sukkoth 2015. In between all these events is a total eclipse of the Sun on March 20, 2015 – the First day of the Jewish month of Nissan, the “leading month” in the Jewish calendar, the month when Israelites were freed from slavery in ancient Egypt. Astronomers found that there was seven Blood Moons on the first day of Jewish Passover and the first day of Sukkoth since beginning of the Christian era and the 2014-2015 Tetrad is the eighth coincidence. The second Astronomer Royal in Britain Edmond Halley (1656–1742) succeeding the first Astronomer Royal John Flamsteed is one of the founders of modern eclipse astronomy. Halley spent most of his time on lunar and comet observations, but was also interested in the problems of gravity, considering the possible derivations of Kepler’s laws of planetary motion. In August 1684, during discussion in Cambridge he persuaded Sir Isaac Newton to publish his calculations based on the 9 inverse square law of gravitation. The first of the three constituent books of Newton’s Philosophiæ Naturalis Principia Mathematica was sent to Halley in spring 1686 and first 300 copies published by Halley at his own financial risk appeared in July 1687. Edmond Halley firstly worked out the historical astronomy methods and started to use them in calculations with comets, in particular, for the orbit of Kirch Comet. In 1705, applying his historical astronomy methods, Halley published Synopsis Astronomia Cometicae, which stated his belief that the comet observations of 1456, 1531, 1607, and 1682 related to the same comet, which he predicted would return in 1758. Halley did not live to witness the comet’s return, but when it did, the comet became generally known as the “Halley’s Comet”. By 1706 Halley had learned Arabic and completed the translation started by Edward Bernard of Books V-VII of Apollonius’s Conics from copies found at Leiden and the Bodleian Library at Oxford. He also completed a new translation of the first four books from the original Greek that had been started by David Gregory. He published these books along with his own reconstruction of Book

VIII in the first complete Latin edition in 1710. In 1718, Fig. 1.6. The Astronomer Royal in Britain he discovered the proper motion of the “fixed” stars by Edmond Halley comparing his astrometric measurements with those given in Ptolemy’s Almagest. The base of the sun eclipse phenomenology is the “Saros cycle”, which has been used from the epoch of ancient Babylonian astronomy to predict eclipses of the Sun and the Moon. This cycle was named the “Saros” (Greek: σάρος) in 1691 by Edmond Halley, who took this word from the “Suda”, a Byzantine lexicon of the 11th century. Though the word “saros” comes from Sumerian number sar (3600) and does not mean “cycle” or “period”, as thought by Halley, it is nonetheless used in modern astronomy. Lunar and sun eclipses could take place when the Earth and the Moon are aligned with the Sun, and the shadow of one body cast by the Sun falls on the other. At Full Moon, when the Moon is in opposition to the Sun, the Moon may pass through the shadow of the Earth, and a lunar eclipse is visible from the night half of the Earth. At New Moon, when the Moon is in conjunction with the Sun, the Moon may pass in front of the Sun as seen from a narrow region on the surface of the Earth and cause a solar eclipse. A sun eclipse does not happen at every New Moon, because the plane of the Moon’s orbit around the Earth is tilted with respect to ecliptic. A lunar eclipse can only occur when the Moon is close to the plane of ecliptic near one of the two its nodes and the Sun is near the opposite lunar node (Fig. 1.7). The inclination of lunar orbit (about 5.15°) is much larger than the average apparent diameter of the Sun (0.53°), much larger than the Moon diameter, as seen from the surface of the Earth (0.525°), and much larger than the shadow of the Earth at the average lunar distance (1.4°). Therefore, at most New Moons the Earth passes too far north or south of the lunar shadow, and at most solar eclipses the apparent angular diameter of the Moon is insufficient to fully obscure the solar disc, unless the Moon is close to perigee. If a solar eclipse occurs at some New Moon, which must be close to a node, then at the next Full Moon the Moon is already more than a day past its opposite node, and may or may not miss the Earth’s shadow. By the next New Moon it is even further ahead of the node, so it is less probable that there will be a solar eclipse. However, about 5 or 6 lunations later (half of an Eclipse year) the New

Moon will be close to the opposite node and the Sun will also move to the opposite node and therefore the geometry of Sun-Earth-Moon system will again be appropriate for one or more eclipses.

Fig. 1.7. Eclipses in the Sun-Earth-Moon planetary system

The time it takes for the Moon to return to a node – the draconic month (27.21 days) is less than the time it takes for the Moon to return to the same ecliptic longitude as the Sun – the synodic month (29.53 days) because during the time that the Moon has completed an orbit around the Earth, the Earth-Moon system have completed about 1 13 of its orbit around the Sun and the Moon has to make up for this in order to come again into conjunction or opposition with the Sun. The orbital nodes of the Moon precess westward in ecliptic longitude, completing a full circle in about 18.5 years, so a draconic month is shorter than a sidereal month (27.32 days) – the time it takes the Moon to return to a given position among the stars. As seen from the Earth, the Sun passes both nodes as it moves along its ecliptic path and the period for the Sun to return to a node is called the “Eclipse year” (346.62 days), which is shorter than a sidereal year (365.24 days) – the time taken by the Earth to orbit the Sun with respect to the fixed stars, because of the precession of the nodes. Any two sun eclipses separated by one Saros consisting of 223 synodic months (18 years 11 days and 8 hours) have very similar characteristics – they occur at the same node with the Moon at nearly the same distance from Earth and at the same time of the year. However, even though the geometry of the Sun-Earth-Moon system will be almost identical after a Saros, the Moon will be in a different position with respect to the fixed stars due to the Moon’s orbit precession. Since the Saros cycle is not equal to a whole number of days, the extra 8 hours shift each succeeding eclipse path on the Earth’s surface by nearly 120 degrees westward. Saros eclipse series returns to the same geographic region every three Saros periods (the “Exeligmos” – 54 years and 34 days). The interrelations between lunar and sun eclipses are determined by the “Sar” cycle – a half of the Saros cycle. After a certain lunar or solar eclipse, after 9 years and 5.5 days (a Sar cycle) an eclipse will occur that is lunar instead of solar, or vice versa, with similar properties. For example, 9 years and 5.5 days after a total solar eclipse a total lunar eclipse will occur. In eclipse astronomy and calendar study, besides Sar, Saros and Exeligmos several other periods with historical names are used. These periods contain the integer numbers of synodic months. The “Metonic cycle” or “Enneadecaeteris” (from Ancient Greek “nineteen years”) was introduced by Greek astronomer Meton of Athens (5th century BCE) as a period of very close to 19 years, which is remarkable for being nearly an integer number of the solar years, Eclipse years and the synodic months. The “Callippic cycle” proposed by Greek astronomer Callippus (c. 370–c. 300 BCE) is an

11 approximate of 76 years and the integer number of synodic months. The Greek astronomer Hipparchus of Nicaea (c. 190–c. 120 BCE) introduced the eclipse cycle that have been named after him in later literature, which closely matches an integer number of synodic months (4267), anomalistic months (4573), years (345), and days (126007). By comparing his own eclipse observations with Babylonian records from 345 years earlier, Hipparchus also verified the accuracy of various eclipse periods that the Chaldean astronomers used.

Table 1.1. Main characteristic periods (approximate), which are used in eclipse astronomy

solar days synodic month Eclipse years tropical years synodic month 29.53 1 draconic month 27.21 0.92 sidereal month 27.32 0.925 anomalistic month 27.55 0.93 Eclipse year 346.62 11.74 1 0.95 lunar year 354.37 12 1.02 0.97 tropical year 365.24 12.37 1.05 1 Sar 3292.66 111.5 9.5 9.015 Saros 6585.32 223 19 18.03 Metonic cycle 6939.69 235 20 19 Inex 10571.95 358 30.50 28.945 Exeligmos 19755.96 669 57 54.09 Callippic cycle 27758.75 940 80 76 Hipparchic cycle 126007 4267 363.53 345 Great Babylonian cycle 161177.95 5458 465 441.29 Tetradia 206241.63 6984 595 564.67

The Moon’s orbit is not a perfect circle but approximates an ellipse and the lunar distance from Earth varies throughout the lunar cycle. The varying distance from the Moon to Earth changes the apparent diameter of the Moon, and therefore influences the probability, duration, and type of an eclipse. The orientation of Moon’s orbit is not fixed and, in particular, the position of the “extreme points line” – the line of the apsides: perigee and apogee, makes a full circle (lunar precession) in about 3233 days (8.85 years). It takes the Moon the anomalistic month (27.55 days) to return to the same apsis. The Moon moves faster when it is closer to the Earth (near perigee) and slower when it is near apogee (furthest distance), thus periodically changing the timing of syzygies by up to ±14 hours (relative to their mean timing), and changing the apparent lunar angular diameter by about ±6%. An eclipse cycle must comprise close to an integer number of anomalistic months in order to perform well in predicting eclipses. During a central sun eclipse, the Moon’s umbra (or antumbra, in the case of an annular eclipse) moves in a roughly west-east direction across the Earth’s surface at the speed of the Moon’s orbital velocity minus the Earth’s rotational velocity. The width of the path of a central eclipse varies depending on the relative apparent diameters of the Sun and Moon. When a total eclipse occurs very close to perigee, the path can be over 250 km wide and the duration of eclipse may be over 7 minutes. Though the length of eclipse path is several thousand kilometers, and an eclipse is visible on the Earth during several hours, there are certain coordinates and time of the “Greatest Eclipse”, which define the site and the instant when the axis of the Moon’s shadow cone passes closest to Earth’s center. Although the Greatest Eclipse differs slightly from the instant of the greatest magnitude and the Greatest Duration for sun eclipses, these differences are small. For lunar eclipses, Greatest Eclipse is defined as the instant when the Moon passes closest to the axis of Earth’s shadow. The coordinates and time of the Greatest Eclipse are individual characteristics of each eclipse and they are used to identify eclipses in databases.

For the Greatest Eclipse the main eclipse characteristics – eclipse Magnitude and Gamma can be evaluated. Eclipse magnitude is the fraction of the Sun’s diameter occulted by the Moon. It is a ratio of diameters and should not be confused with eclipse obscuration, which is a measure of the Sun’s surface area occulted by the Moon. Eclipse Magnitude may be expressed as either a percentage or a decimal fraction (e.g., 80% or 0.80). Gamma is the distance of the Moon’s shadow axis from Earth’s center in units of equatorial Earth radii and at the instant of Greatest Eclipse its absolute value is at a minimum. For lunar eclipses, the Gamma is the Moon’s minimum distance measured with respect to the axis of Earth’s shadow in units of equatorial Earth radii.

Fig. 1.8. The sun eclipse track and Greatest Eclipse on March 29, 2006 (Fred Espenak and Jay Anderson, NASA/TP- 2004-212762 “Total Solar Eclipse of 2006 March 29”).

As an example we can consider the sun eclipse on March 29, 2006, belonging to Saros 139. The narrow band of total sun eclipse began in the Eastern Brazil (“first contact”) then crossed the Atlantic to Africa, where the Moon’s shadow travelled over the northern part of the continent. It went on to cross the Mediterranean Sea to Turkey, after which it passed through part of central Asia to end in Mongolia (“fourth contact”). Partial phases of the solar eclipse were observed by the whole of Europe. Only the south-eastern parts of Africa missed the partial eclipse. At the extreme parts of the eclipse path of totality the duration of “totality” – the maximum phase of a total eclipse during which the Moon’s disk completely covers the Sun, was less than 2 minutes, but it was as long as 4 minutes and 7 seconds in Libya at 9:18 UT (Universal Time), at the moment of Greatest Eclipse. At this moment, the path of totality of the solar eclipse was about 180 kilometers wide. The known history of solar eclipse maps begins in the 17th century, when a path of total solar eclipse crossed northern Europe on August 12, 1654 and German astronomer Erhard Weigel created the earliest known solar eclipse map on the day before the eclipse. In 1676, Johann Christoph Sturm produced an almanac of eclipses, which included a map of two eclipses, one total and one annular, for that year. In 1700 the leading astronomer of the Paris Observatory Giovanni-Dominique Cassini created an eclipse map of the total solar eclipse of 1699. In 1715, Edmond Halley using the Isaac Newton’s theory of gravitation created several eclipse maps of very high accuracy. Later, Halley analyzed observational reports from the 1715 eclipse and produced another map with a refinement of the path of the 1715 eclipse along with the forthcoming 1724 eclipse.

Fig. 1.9. Edmund Halley’s sun eclipse map (1715). Right: the modern eclipse map of predicted sun eclipse (Saros 145) on August 21, 2017 with up to 2.7 minutes of totality. Light blue lines parallel to the dark blue track of totality indicate the degree of partial eclipse to be seen (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by Fred Espenak (NASA's GSFC)”).

By the 20th century, the publication of eclipse maps became routine in most of the major almanacs of the world and continues to this day. In the United States, the U.S. Naval Observatory began publishing eclipse maps in USNO Eclipse Circulars in 1949. In 1970s, started the international program of Lunar Laser Ranging Experiment and “Lunokhod- 1” (1970) was the first of two unmanned lunar rovers landed on the Moon by the Soviet Union as part of its Lunokhod program. Lunochods carried small light reflecting arrays and reflected signals were initially received from Lunokhod-1, but no return signals were detected from 1971 until 2010, due to significant uncertainty in Lunochod locations on the Moon. In 2010, Lunochods were found in Lunar Reconnaissance Orbiter photographs and the retro-reflectors have been used again.

Fig. 1.10. The ranging retro-reflector on the Moon’s surface

Astronauts on the Apollo 11, 14, and 15 missions left more perfect retro-reflectors on the Moon as part of the Lunar Laser Ranging Experiment. The Lunar Ranging Retro-Reflector (LRRR) of Apollo 11 consisted of an array of 100 fused silica cubes, arranged to reflect a beam of light back on a parallel path to its origin. The Apollo 15 reflector had 300 silica cubes. The LRRRs placed on the 14

Moon are aligned precisely so that they face the Earth and scientists from around the world can direct laser beams at the instruments, which reflect beams back to Earth. Such laser locations of the Moon allow precise measurements of distances between the Earth and Moon and the procession of measurement sets gives the accurate estimations of main characteristics of the Moon’s motion in orbit. Starting in the 1970s, the laser locations of the Moon and modern computer technologies were applied to very precise calculations and new maps of solar eclipses. In the 1980’s, the task of publishing detailed eclipse reports and maps in USA was transferred to NASA under the leadership of Fred Espenak of the NASA/Goddard Space Flight Center and Jay Anderson of the Royal Astronomical Society of Canada. Modern theories of the Earth and Moon orbital motions with the value of the Moon’s secular acceleration ( 25.858 arc-sec/cy2) deduced from the Lunar Laser Ranging Experiment allow calculating the eclipse characteristics for the periods of several millennia. In this book we use the “Five Millenniums Canon of Solar Eclipses: –1999 to +3000 (2000 BCE to 3000 CE)” [9], which we henceforth will refer to as “Canon”. An example of information about main characteristics of solar and lunar eclipses from Canon presented in Tables 1.2 and 1.3.

Table 1.2. Main characteristics of 2013-2015 solar eclipses [9] and (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by Fred Espenak (NASA's GSFC)”

Date Time of Saros Type Gamma Magnitude Central Location of Path Greatest duration Greatest width Eclipse Eclipse (km) (UTC) 5/10/ 2013 00:26:20 138 Annular -0.2694 0.954 6m 3s 2.2N;175.5E 173 11/3/2013 12:47:36 143 Hybrid 0.3272 1.016 1m 40s 3.5N;11.7W 58 4/29/2014 06:04:33 148 Annular -1.0000 0.987 — 70.6S;131.3E — 10/23/2014 21:45:39 153 Partial 1.0908 0.811 — 71.2N;97.2W — 3/20/2015 09:46:47 120 Total 0.9454 1.045 2m 47s 64.4N;6.6W 463 9/13/2015 06:55:19 125 Partial -1.1004 0.788 — 72.1S;2.3W —

Table 1.3. Main characteristics of 2014-2015 lunar eclipse Tetrad (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by Fred Espenak (NASA's GSFC)”

Date Time of Greatest Eclipse Saros Type Umbral Partial/Total Duration (UTC) Magnitude 4/15/2014 07:46:48 122 Total 1.291 03h35m/01h18m 10/8/2014 10:55:44 127 Total 1.166 03h20m/00h59m 4/4/2015 12:01:24 132 Total 1.001 03h29m/00h05m 9/28/2015 02:48:17 137 Total 1.276 03h20m/01h12m

Dutch Astronomer George van den Bergh (1890–1961) in his book Periodicity and Variation of Solar (and Lunar) Eclipses introduced the numbering system that is often used for the Saros series. All 8000 solar eclipses from Theodor von Oppolzer’s Canon der Finsternisse (1887) are placed into a two-dimensional matrix, where each Saros series is arranged as a separate column with the eclipses in chronological order. The Saros series columns are staggered so that the interval between any two eclipses in adjacent columns is the “Inex” cycle – 358 synodic months (29 years minus 20 days). Each eclipse has two numbers determining its position in Saros-Inex Panorama (Fig. 1.11-1) and Inex cycle marks the time interval between consecutively numbered Saros series. Within each column of Saros-Inex Panorama is a complete Saros series, which progresses from partial eclipses into total eclipses and back into partials. Each Inex series extends as a graph row, but

15 it gradually bends due to long term period variations as can be seen at Panorama for 61775 solar eclipses, which has been produced by astronomers Luca Quaglia and John Tilley (Fig. 1.11-2).

Fig. 1.11. 1. The fragment of Saros-Inex Panorama. 2. Saros-Inex Panorama for 61775 solar eclipses. 3. Several central solar eclipses from Saros 145 that began with a partial eclipse near the North Pole in 1639 and will terminate with the last eclipse near the South Pole in 3009 (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by Fred Espenak (NASA's GSFC)”. The black circles are locations of Greatest Eclipses

The Saros numbering sequence does not depend on when a series begins or ends and the numbering tends to follow the order in which each series “peaks” – the peak of a series occurs when the umbral shadow axis passes closest to the center of Earth. Since the duration of each series varies up to several hundred years, the first eclipse of a series, which peaks later, can actually precede the first eclipse of a series that peaks earlier. There are approximately forty different Saros series in progress at any one time – as old series terminate, new ones are beginning and take their places. The Saros numbering of lunar eclipses is opposite to that for solar eclipses. Lunar eclipses occurring near the Moon’s ascending node have even Saros numbers and correspondingly, lunar eclipses occurring near the Moon’s descending node have odd Saros numbers. 1.1. Metaphysics of solar eclipses A very rare phenomenon in the planetary mechanics – a total eclipse of the central star in satellite- planet structure in the Sun-Earth-Moon system is determined by its special geometry, which provides a close match of apparent angular sizes  of the Moon and the Sun. Modern astronomers explain the possibility of a total solar eclipse by the fact that the ratio of the Sun diameter to the Moon diameter:

A1  dSM d 400.5 is almost the same as the ratio of the average half-axis of the Earth’s orbit to the average half-axis of the lunar orbit: A2  aEM a 389.3 (Fig. 1.12). The difference between these ratios is only 2.9% and hence:

tg dMMSE a d a 0.53 (1.1) However, in astrology and metaphysics of the East a possibility of total solar eclipses is explained otherwise – by the declaration that the solar eclipse as well as the whole geometry of the Universe determines the unique sacred number 108. Indeed, the ratio of average half-axis of the lunar orbit to the diameter of the Moon: M1  aMM d 110.6 is very close to the ratio of the Sun diameter to the

Earth diameter: M2  dSE d 109.1. The difference between these ratios even less than the difference of A1 and A2 – only 1.4%, and the ratios M1 , M 2 within the accuracy of the ancient astronomical observations could well be estimated by the sacred number 108:

aMMSE d d d 108 (1.2)

Figure 1.12 visually illustrates that the close values of ratios: AA12 provide an approximate similarity of the trapeziums that makes possible the total solar eclipses. However, it is difficult to explain what relation to the geometry of total solar eclipses could have the Earth’s diameter dE , which is included in the ratio M2 and then into relation (1.2).

Fig. 1.12. The geometry of solar eclipse

In Eastern world, the number 108 is considered sacred by several religions, including Hinduism, Jainism and Buddhism. In Sanskrit alphabet, which is used in sacred Vedic texts there are 54 letters and each letter has masculine and feminine (Shiva and Shakti) symbolic significances. Therefore, the letters of Sanskrit alphabet have totally 54 × 2 = 108 symbolic meanings. The Upanishads – philosophical texts completing the Vedic corpus are classically counted as 108 and are divided into four categories according to the particular Veda to which each of them belong. Some Hinduism traditions assert that there are 108 deities each having 108 names and there are 108 paths to God. The well known bas-relief carving at the famous Angkor Wat temple in Cambodia relates the Hindu story of a serpent being pulled back and forth by 108 gods and asuras (demons) – 54 gods pulling one way, and 54 asuras pulling the other, to churn the Ocean of milk in order to produce Amrita – the elixir of immortality. In front of Angkor Wat temple gate stand 108 statues of 54 gods and 54 demons, which represent the sacred churning of the Ocean. Many Hinduism traditions use the Sri Yantra formed by nine interlocking triangles that surround and radiate out from the central point (bindu), the junction point between the physical universe and its unmanifest source (Fig. 1.13). Four triangles with the apices upwards represent Shiva and the Fig. 1.13. The Sri Yantra with 54 marmas Masculine principle. Five triangles with the apices downward, symbolize female embodiment Shakti and therefore Sri Yantra represents the union of Masculine and Feminine Divine. The nine triangles of various sizes intersect with one another and on the Sri Yantra there are 54 marmas where three lines intersect. Each intersection has masculine and feminine (Shiva and Shakti) qualities and therefore there are totally 54 × 2 = 108 special points of the symbolic Universe. According to Ayurveda, these 108 points correspond to pressure points in the body, where consciousness and flesh intersect to give life to the living being. The Sri Yantra also represents the Heart Chakra, where 108 body energy lines intersect. 17

The Lankavatara Sutra has a section where the Bodhisattva Mahamati asks Buddha 108 questions and another section where Buddha lists 108 statements of negation. The Sanskrit word translated as “statement” is pada, which also means “foot-step” and therefore many Buddhist temples have footways with 108 steps. In Tibet, China and Japan, at the end of the year, a bell is chimed 108 times in many Buddhist temples to finish the old year and welcome the new one with each ring representing one of 108 earthly temptations a Buddhist must overcome to achieve nirvana. The Hinduist and Buddist ritual rosaries (mala) have 108 beads and the Meru – a larger bead that is not tied in the sequence of the other beads. It is the guiding bead, the one that marks the beginning and end of the mala. The Chinese Buddhists and Taoists use a 108 bead rosary (su-chu), which has three dividing beads, so the rosary is divided into three parts of 36 each. The Tibetan Buddhist malas include the Guru-bead and 108 beads symbolizing the most important words of the Buddha writen in the sacred texts of Kangyur consisting of 108 volumes. Traditionally, yoga students stop at the 109th Meru-bead, flip the mala around in their hand, and continue reciting their mantra as they move backward through the beads. The Meru-bead represents the summer and winter solstices, when the Sun appears to stop in its course and reverse directions. In the yoga tradition using a 108-beads mala is a symbolic way of connecting ourselves with the cosmic cycles governing our Universe. If the yogin is able to be so calm in meditation as to have only 108 breaths in a day, the Enlightenment will come. In modern number theory, the number 108 is an abundant semi-perfect number, which divides the twin primes: 107,108,109 and has 12 positive divisors {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108}. It is a refactorable number (tau-number) since it is divisible by the number of its devisers: 108 : 12 = 9. The digital representation of number 108 depends on the used counting base:

108(10) 423 (5)  154 (8)  90 (12)  30 (36) . In decimal counting system the number 108 is a Harshad number (Niven number), which is an integer divisible by the sum of its digits: 108/(1+0+8=9)=12, and has a rare feature to keep constant the sum of digits in three successive divisions by 2: 108 (1+0+8 = 9) : 2 = 54 (5+4 = 9) : 2 = 27 (2+7=9) : 2 = 13.5 (1+3+5 = 9). However, the number theory cannot explain the mysterious role of quantity 108 in the geometry of Sun-Earth-Moon system governing the solar eclipses and to clear up the old puzzle we shall consider the ancient metaphysics of West and East. About the principles and symbolism of ancient Hellenistic metaphysics we know from the writings of Pythagoreans – many prominent Greek and Latin philosophers, including Parmenides and Empedocles [5, 8, 13, 34]. Philolaus (c. 470–c. 385 BCE) is considered by many scholars as most informed Pythagorean author. Pythagoreans considered themselves as followers of legendary Ionian Greek philosopher and mathematician Pythagoras of Samos (supposedly 6th century BCE). Pythagoras wrote nothing down and all the stories about his life and philosophy were written down centuries after he lived, so no reliable information is known about this enigmatic sage. A number of Gnostics even considered Pythagoras not as historical philosopher but as Greek god and placed his monuments next to statues of Hermes. Metaphysical doctrine of the Pythagoreans was the assertion that the evolution of Universe, the movement of planets, the world history and fates of people are governed by several general metaphysical principles, which can be only represented by numbers and geometrical figures. The Pythagoreans were convinced of the existence of one God – the First Principle, the root cause of being and the Universe with the Unit (Monad) as numerical symbol. They did not allow the existence of any other visual image of God, believing that God’s essence cannot be explained by words. Monad was seen by them as the beginning of all things, the equivalent of the Infinite, the absolute number that exists in itself and manifests itself through itself. All forms of life are only certain aspects of the Monad and all the numbers are only combinations and relations of Unit. In Pythagorean metaphysics the Harmony was the main principle of the incarnation of God into the world and the only way to create world order and all objects from the Monad. For the Pythagoreans

Harmony was primarily harmonic movement and the best embodiment of world Harmony, beside geometric and arithmetic proportions, considered the harmony of music.

Fig. 1.14. Pythagoras (Capitoline Museums, Rome) and the reconstruction of Tetraxis

The key to the whole metaphysical doctrine of the Pythagoreans was Pythagorean mathematics, which is amazing and indivisible unity of the symbolic, numerical and geometric images. Aristotle not sharing the whole philosophy of the Pythagoreans, however, carefully studied and highly appreciated the philosophical method of the Pythagoreans, devoting many pages to it in his writings. Criticisms of Aristotle explain the meaning a numeric symbolism for the Pythagoreans [34]:

“Catching up on math, they imagined that its principles underlie all in the world... Certain combination of numbers was in their eyes, neither more nor less than justice; other combination of numbers gives the mind and the reason; another combination spawns the fortune and so on ... The Pythagoreans did not separate numbers from the things. They believe that a number is the fundamental principle and substance of objects, their essence and power ... All Pythagoreans consider elements of a number as material properties, as these elements are found in all things and form all the world ... believing that everything in the world are copies of numbers”.

Thus, the Pythagoreans gave numbers not only ontological status, but even a status of ethical categories. The special importance attached by Pythagoreans to numbers as a system of symbols reflects their inherent distrust that metaphysical principles can be explained by words of spoken language. The general structure of Pythagorean metaphysics appears as the triad: In Pythagorean mathematics, which was both a symbolic language and the main method of their Fig. 1.14-A. The triad of Pythagorean philosophy, the first digits of number series (1, 2, 3, 4, 5) metaphysical categories were not considered numbers – they were the Ultimate

Symbols representing the main metaphysical principles. The Monad (1) – First Principle and the root cause of the Universe was regarded as the embodiment of the metaphysical Principle of structural integrity and global interdependence – any number multiplied by One is itself, and any number divided by One is itself.

“Pythagoras held that one of the first principles, the Monad, is god and the good, which is the origin of the One, and is itself intelligence; but the undefined duad is a deity and the evil, surrounding which is the mass of matter” (Aetius).

Believing that nothing exists without a center, Pythagoreans considered the center of a circle as the geometrical symbol of Monad – the center is the Source and it is beyond understanding, it is unknowable, but like a seed, the Center will expand and will fulfill itself as a circle. By contemplating itself, the circle is able to become a set of circles with each circle sharing the center of the other (Fig. 1.15).

Fig. 1.15. Pythagorean metaphysical symbols The Duad (2) – a symbol of the metaphysical Principle of universal dualism, Pythagoreans considered as the relation between the Finite (odd) and Infinite (even), which projects its “feminine” essence into all even numbers: “In relation to all numbers Duad is as much mother as the Monad is a father”. Of all shapes, the circle is the parent of all following shapes and its first transformation gives two circles side by side building a foundation for all numbers. The Duad, represented by a line connecting the centers of two circles is the door between the One and the Many (Fig. 1.15). Aristotle recorded that the Pythagoreans, treating the Principle of universal dualism, used the following ten “coordinates” [34]:

male – female good – evil right – left light – darkness odd – even finite – infinite straight – curved single – multiple square – round resting – moving

Here the left column includes the most common in the ancient world graphic symbols of dualism, and the right column represents philosophical oppositions, built on the juxtaposition of polar categories. In the Middle Ages and the Renaissance, metaphysical Principles of the entirety and the universal dualism formulated in the first stanza of “Emerald Tablet” (Tabula Smaragdina), known in Arabic translations from 6-8 century, became a kind of “Ten Commandments” in Hermetic philosophy, alchemy and astrology (Newton’s translation):

Verum, sine mendacio, certum et This true without lying, certain & most verissimum: Quod est inferius est sicut true: That which is below is like that quod est superius, et quod est superius which is above & that which is above is est sicut quod est inferius, ad like that which is below to do the perpetranda miracula rei Unius. miracles of One – only thing.

Et sicut res omnes fuerunt ab Uno, And as all things have been & arose meditatione Unius, sic omnes res natae from One by the mediation of One: so ab hac Una re, adaptatione. all things have their birth from this one thing by adaptation.

The Triad (3) in philosophy of Pythagoreans was a metaphysical symbol of synthesis, Creation, as the process of incarnation of God (Monad) into a man and the Universe by Harmony, and is often illustrated by the “Creation Formula”: 1(God) + 2(Harmony) = 3(Triad) The Pythagoreans taught that everything in the world is threefold, has three main qualities and can be represented by a triangular diagram: “Behold the triangle and you are close to solving the problem – all things consist of triads”. The Triad was considered the prototype of a typical number that has the “beginning”, the “middle” and the “end” and is already capable of, as well as all the other numbers, except the Ultimate Symbols (1 and 2), increase the number by multiplying more than by addition. The number 3 is the only number for which the sum and the product of previous numbers are equal: 1+2+3 = 1×2×3. The triangle containing smallest area within the perimeter considered a geometrical symbol of the Triad (Fig. 1.15). The Tetrad (4) in the Pythagorean metaphysics was a symbol of Harmony, considered as the first transformation of the Duad, representing its metaphysical essence. Four is the number that can be formed by addition or by multiplication of Duads: 4 = 2+2 = 2×2. The Tetrad is the first “genuine” even number being therefore the first “female” number and its geometrical symbol was a square or a rhombus (Fig. 1.15). To construct the geometrical symbol of Tetrad, Pythagoreans drew the lines connecting the centers and the intersecting points of the two circles or, when a circle is drawn along a line that connects the two centers, a perfect shape of a square exists within the circle. Pythagoreans claimed that the legendary Pythagoras studying monochord – the instrument with one string and movable frets, discovered an inverse relationship between length of the string and the tone of her sound. That’s why in the Pythagorean schools in addition to study of arithmetic, geometry and astronomy, all students were engaged in a lot of music, learning the principles of Harmony. The base of the Pythagorean Harmony was “tetra chord” literally meaning “four strings”, originally in reference to harp-like instruments such as the lyre or the kithara. After the discovery of the fundamental harmonic intervals (octave, fourth and fifth), the first systematic division of the octave proposed by Pythagoreans was the arithmetical set, allowing for 1:1 = Unison, 2:1 = Octave, 3:2 = Fifth, 4:3 = Fourth within the octave. This Pythagorean diatonic scale adopted Plato (423/428– 348/347 BCE) in the construction of the World Soul in his Timaeus. The pentad (5) in the metaphysics of the Pythagoreans regarded as a symbol of universal structural transformation in the process of creation. The geometrical symbol of a Pentad was the Pythagorean pentagram which was drawn with two points up (Fig. 1.15) and represented the concept of Pentemychos (“five recesses” or “five chambers”) allegedly introduced by the legendary Pythagoras’ teacher and friend Pherecydes of Syros. The Pythagoreans believed that each point of the pentagram represents an Element: Fire, Air, Water, Earth and Idea (Ether). In the interpretation of Pentad as a universal Principle of structural evolution Pythagoreans used the Pythagorean Theorem ( a2 b 2 c 2 ) for a right-angle triangle with sides determined by the Ultimate Symbols ( ab1, 2 ): 122 2 5 . This relation determines the Golden ratio:  (a  c ) b  (1  5) 2  1.618... that was already known in Babylon and ancient Egypt. A Golden triangle is an isosceles triangle, in which the smaller side is in Golden ratio with its adjacent side, has the vertex angle  5 36 and is the only triangle to have its three angles in 2:2:1 proportion. Each corner in Pythagorean pentagram is a Golden triangle. The Golden ratio symbolizes the law of structural invariance – in every stage of a development, the same pattern repeats itself. Every entirety consisting of parts is itself likewise part of a larger entirety. The Golden ratio also corresponds to the principle of order – the underling obeys the orders of his master to the same extent that the master obeys the requirements of the entirety. In living nature, there are plenty of sequences where every member stands to the preceding member in a

Golden proportion or its derivatives. This observation and the fact that the Golden ratio appears after the transformation of masculine and feminine Ultimate Symbols forced Pythagoreans to consider the Pentad as a main symbol of Life and fertility. Because of it the Greeks and later the Arabs named the Pythagorean Theorem the “Theorem of the married women” or the “Bride theorem”. Another confirmation of the Pentad’s “fertile” essence Pythagoreans considered the ability of the number 5 “to spawn” – in the number, obtained by multiplying 5 by an odd (“male”) number, always presents the figure 5. The Decade (10) is the Monad times the Duad times the Pentad (1×2×5 = 10). Since any number times ten is similar to any number times one, it is similar to the Monad; however, the number is brought to a higher level of a hierarchy. The Decade was a crown of Pythagorean metaphysics represented by the “Universe’ Formula”: 1 2  3  4  10 (1.3) This symbolic formula describes a transformation of initial unity (1) into the following unity of higher level (10), which is considered as the basic process that forms the hierarchical structure of the Universe. Many thousand kilometers from Greece in the Far East the Chinese philosopher Lao-Tse in the 6th century BCE formulated almost the same ideas in his “Tao Te Tsin” poem:

“Tao engenders one, one engenders two, two engenders three, and three engenders all beings. All beings contain Ying and Yan, are filled with Tse, and exist in harmony”.

Hermetic philosophers since Middle Ages believed that the mysterious Pythagorean Tetraxis provides a clue for interpretation of the Universe’ Formula (1.3). Tetraxis was the main sacral emblem of Pythagoreans who even took the oath, mentioning it: ”I swear on Those Who gave Tetraxis to our souls”. Pythagoreans maintained that this emblem represents the whole perfection and all harmonious relations governing the Universe. No detailed description, nor picture of Tetraxis are available to date, probably because the emblem always appeared in the focus of Pythagorean doctrines and was only handed down to the initiated Pythagorean followers. Gnostics

who worshipped Pythagoras and gathered the remnants of Fig. 1.16. The different triangular emblems his teaching with patience, repeatedly tried to reconstruct published with the name “Tetraxis” Tetraxis, only succeeding in conceiving it as some triangular ten-terms emblem. The unabated interest in the mysterious Tetraxis of Gnostics and Neo-Platonists including Proclus, was paralleled by the Cabbalists who also used a ten-parts emblem of the Sephiroth Tree as their theosophical symbol. Cabbalistic treatises show various triangular emblems by the name of Tetraxis (Fig. 1.16). It seems that Pythagoreans, who had thorough knowledge of the geometrical symbolism of numbers, would not fail to discover the potential of combinatorial representation of a triad as a composition of three triads, symmetrically converging to a single center. Especially since such emblem appears a possible geometrical representation of the structurally minimal theory or mathematical model basing on a triad of initial premises (Taganov 2009 [33]). Indeed, a minimum of terms in a system is three – two symbols and one relation. The minimum number of consequences of the three assumptions, if each combination of assumptions has only one consequence, is seven. Therefore, total number of consequences in a theory representing a system

22 and holding the property of minimality is ten. Specifying the consequences of a system of three assumptions А, В and С as: A→a,…;AB→ab,…;ABC→abc; one can form the geometrical representation of the structurally minimal theory as the emblem at Fig. 1.17.

Fig. 1.17. The structurally minimal theory in the form of Tetraxis

One argument for this interpretation of Pythagorean Tetraxis is found in similar geometrical structures appearing as mandatory elements in most of Eastern cosmological Yantras and Mandalas. Tetraxis in the form of Fig. 1.17 is employed in esoteric Hermetic and alchemy treatises describing different stages in synthesizing the Philosopher’ stone, and the hidden symbolism of colors in medieval art [33]. The followers of Pythagoras, dividing the numbers into even and odd, distinguished many types of composite numbers, which had, in their view, different quality and different degrees of the “perfection”. Particular emphasis was placed on the “super-perfect” numbers (6, 28, 496,...), which are equal to the sums of their integer parts. The “friendly” numbers were those pairs of numbers, one of which is a sum of divisors of the second number, for example, 220 and 284. Pythagoreans named “independent” the numbers without common devisors. Number 36 Pythagoreans considered as “sacred”, on the grounds that it could be represented as the sum of the cubes of the three Ultimate Symbols: 36 = 13+23+33. Pythagoreans discovered many hidden geometric symbols in numbers that could be represented by figures of counting stones. For example, the numbers 1, 3, 6, 10, ... Pythagoreans named “triangular” numbers; 1, 4, 9, 16 ... – “square”; 5, 12, 22 ... – “pentagonal”. The Pythagoreans taught that every entity and the Universe as a whole, have two aspects: “multiplicity” – the changing characteristics relevant to the constituent parts of a thing, and the “magnitude” – the permanent feature, the “density” of a thing. Multiplicity, in turn, has internal and external aspects. Arithmetic deals with the internal aspect of the multiplicity, and the music with the external aspect. External and internal aspects of the “magnitude” of things and the Universe, explore respectively, geometry and astronomy. The Pythagoreans declared arithmetic the mother of all sciences, proving it by the fact that geometry, music and astronomy depend on arithmetic. Philosophical heritage of the Pythagoreans was very influential during many centuries and has been commented by nearly all major philosophers. Recorded firstly by Philolaus, it developed in the writings of his followers, for example: Diocles of Cnidus, Echecrates, Eurytus, Hippasus, Phanto of Phlius and Xenophilus. Theophrastus, who led Lyceum after the death of Aristotle, gathered and commented all legends about teachings of Pythagoras and writings of his first followers. Almost all the major Greek geometers and astronomers considered themselves Pythagoreans. Even Plato after studying the works of Pythagoreans in Italy began to teach that the division of the One leads to the

23 numbers and “ideas” are filled with numbers. The “ideas” of numbers themselves he put between One and the Reason. The doctrines of the Pythagoreans still serve as a source of rewarding ideas and insightful allegories for the European culture.

The metaphysical idea of planets, stars and celestial phenomena influencing the history and human beings became a base of different astrology systems, which have emerged almost simultaneously in the second half of the first millennium BCE in different cultures around the world – in Greece, Persia, India and China (see e. g. [4, 11]). The widespread tales about wise ancient “Chaldean” and Egyptian astrologies are only legends. For second millennium BCE historians have only several fragments of the compilations of celestial omens. As late as the 6th century BCE an error of almost an entire month was made by the prominent Babylonian astronomers in the attempts to determine through calculation the beginning of a certain past year. The Babylonian theory of the ecliptic representing the course of the Sun through the year, divided among twelve constellations of zodiac, as has now been definitely proved, appeared only in the 6th century BCE. The ancient Babylonian and Persian astrology was originally exclusively mundane, being geographically oriented and specifically applied to the welfare of the state and the king as the governing head of the nation. The concept of individual horoscope as the “birth chart” was developed in Greece, probably by Pythagoreans, and we can analyze the first known Greek individual horoscope dated around 410 BCE. Plato formulated in his Timaeus the first metaphysical concept of astrology: “Time is the flowing Image of the Eternal … and the Planets are the instruments of Time”. After conquers of Alexander the Great the Hellenistic astrology enriched by the ideas of Persian and Indian astrologies gave birth to Ptolemaic astrology and Hermetic astrology. In Greek-Egyptian Ptolemaic astrology the Greek methodological substrate is welded with the Persian astrological legacy that was preserved in Egyptian temples since the 6th century BCE when Egypt was conquered by Persians. The expressive example of Ptolemaic astrology is so-called Dendera Zodiac dated around 50 CE (Fig. 1.22-1). Hermetic astrology was developed by Gnostics and early Christians on the base of Hellenistic system of horoscopes that was welded with some ideas from Persian and Indian astrologies. The Hermetic astrology, effectively explaining the “Gospel star”, which the Magi (astrologers) were following to the Christ cradle, by unusually bright conjunction of Jupiter and Saturn in Pisces (Fishes) that occurred three times in 7 BCE, became the acknowledged science even in Christian Byzantine Empire. The books by Claudius Ptolemy (c. 90–c. 168) and manuscript by Iamblichus (255–330) that incorporates the Hermetic astrology and mystery teachings of the Egyptians and Assyrians into the Neo-Platonic world outlook are the monuments of early European astrology. After the 7th century Muslim conquest of Alexandria – the center of Hermetic astrology, many astrological manuscripts including “The Great Treatise” (Almagest) and “Four books” by Claudius Ptolemy were translated into Arabic. Soon the centers of learning in astrology were set up in Baghdad and Damascus, and the second Abbasid Caliph of Baghdad Al Mansur (754–775) established a major observatory and library in the city, making it the leading world’s astronomical and astrological centre. In the 770s, Caliph al-Mansur has the Indian Siddhanda translated into Arabic, which together with Hermetic astrological manuscripts became a base of the Moslem astrological tradition. Albumasur (Abu Ma’shar; 805–885) was one of the most influential Persian astrologers and his treatise Introductorium in Astronomiam became one of the first astrological books to find its way in translation through Spain into Europe in the Middle Ages. The golden age of European astrology became the Renaissance (14th–17th centuries) when tenths of prominent astronomers has greatly enriched the astrological methodology. Among astrology theorists and masters shined Paracelsus (1493–1541), Tycho Brahe (1546–1601), Francis Bacon (1561–1656), Galileo (1564–1642), Johannes Kepler (1571–1630) and Elias Ashmole (1617–1692). Throughout the Renaissance, astrology was considered a scholarly tradition and it was included into cultural, political and academic contexts, being connected with other scholarly studies, such as astronomy, alchemy, meteorology and medicine.

Though Indian and Chinese astrologies were developing on the base of national cultures and mythologies the exchange by ideas and calculation algorithms during many centuries has led to creation of the universal astrological methodology. All astrologies most often consist of systems of horoscopes purporting to explain aspects of an individual’s personality and predict future events in his life based on the positions of the Sun, Moon, and other celestial objects at the time of his birth. The creation of a horoscope (natus, natal chart, birth chart) consists of the following main stages.

Fig. 1.18. First two stages of the horoscope creation: 1. The helio-centric view of 7 planets in Solar system. 2. The geocentric view of 7 planets in astrological zodiacal coordinate system. (The first minute of III millennium in Western Europe: Greenwich (UK) 0E00, 51N29; 0:01 UT, 1/1/2001).

I. The first stage of the horoscope creation is the same in all astrologies – it is the calculation of planet coordinates for a particular moment of time. The modern astrologers plot out a picture of the Solar system as the helio-centric view (Fig. 1.18-1). In old times to calculate the coordinates of only 7 time-honored planets – Sun, Moon, Mercury, Venus, Mars, Jupiter and Saturn – the complex Ephemerides tables were used. Modern ephemerides are computed electronically from mathematical models of the motion of astronomical objects and the Earth using, for example, the VSOP87 analytical planetary theory. Besides traditional six planet coordinates, modern Western astrologers as a rule calculate celestial coordinates of Uranus, Neptune, Pluto, Chiron and sometimes lunar nodes and coordinates of several constellations and bright stars. II. The second stage of the horoscope creation is a transformation of the Solar system’s helio- centric view into the geo-centric image, which shows how planet positions appear when seen from the Earth as centre (Fig. 1.18-2). Though since Copernicus and Kepler it has been acknowledged that the Earth moves around the Sun and not vice versa, the astrology universe is Earth-centered that symbolically assumes that the planets move around the Earth. Astrologers declare that this does not contradict the known laws of physics, which only describe the visible material causes of things, but astrology is concerned with general metaphysical principles that describe the hidden, proper causes of things and events. The first astrological coordinate system, as in the scientific planetary astronomy, is based on the use of the zodiac – a circle of twelve 30° divisions of celestial longitude that are centered upon the ecliptic – the apparent path of the Sun across the celestial sphere over the course of the year. The zodiac is an ecliptic coordinate system, which takes the position of the Sun at vernal equinox as the origin of longitude. In astrology the twelve divisions of zodiac are called Signs and have rich symbolic and metaphysical meaning.

Fig. 1.19. The astrological qualification of zodiacal Signs. The positive and negative Signs are marked by white and black colors correspondingly, with letters C, F, M indicating Cardinal, Fixed and Mutable qualities of Signs. For Elements connected with Signs the ancient Hermetic symbols (Fire, Air, Water and Earth: ) are used. Red lines – traditional interpretations of the astrological Houses.

Viewed holistically, the universe of astrology is not only a unity in itself, it is a unity of unities. The entirety (symbolic number – 1) of a zodiacal Sign is reinforced by its specific “ruling planet” – for Aries it is Mars, for Taurus it is Venus and so on (Fig. 1.19). The metaphysical Principle of universal duality (symbolic number – 2) requires differentiating of the two types of Sign – a positive, masculine, active Sign refers to any of the six odd-numbered Signs (Aries, Gemini, Leo, Libra, Sagittarius or Aquarius). The six even-numbered Signs are considered as negative, feminine and passive. The metaphysical Principle of the triadic representation of unity (symbolic number – 3) is introduced by three astrological “qualities” – “Cardinal”, “Fixed” and “Mutable” Signs (C, F, M in Figs. 1.18-2, 1.19). The Cardinal Signs initiate the year seasons, when the Sun enters each of these Signs, bringing a change of season. For example, the Sun’s passage through Aries starts the spring in the northern hemisphere, and the autumn in the southern hemisphere. Fixed Sign holds a season in continuation, and the Mutable Signs prepare the seasonal changes ahead. The qualities of Signs are found in the writings of Ptolemy, and seem to have come out of early Greek astrology. The metaphysical Principle of the triadic representation of unity allows considering the zodiac unity as a composition of three quadruplicities (symbolic number – 4) – groups of four Aristotelian Elements: Fire, Air, Water and Earth. Each grouping by quality has all the four Elements and that means that for each Element, there are Cardinal, Fixed and Mutable Signs within that group too (Fig. 1.18, 1.19). The metaphysical qualification of zodiacal Signs corresponds to the Universe’ Formula (1.3). III. The third stage of the horoscope creation introduces the second astrological coordinate system – the system of Houses. The geocentric view of planets in astrological zodiacal coordinate system (Fig. 1.18-2) represents the planet positions relative the Earth as a whole, and does not depend on the location of an observer at the Earth’s surface. It corresponds to horoscope’s modern

26 representation in western popular media that is usually reduced to so-called “sun-sign astrology”, which considers only the zodiac sign of the Sun at an individual’s date of birth. The introduction of House system in astrology allows not only analyzing of the planets movement through Signs of the zodiac but also planets placement in Houses – twelve spatial divisions of the sky that depend on the observer’s location (or birth) at the Earth. All House divisions in astrology have certain features in common: in most the ecliptic is divided into 12 Houses and astronomically determined Ascendant (AC, Eastern horizon) marks the beginning of the first House; the opposing Descendant (DC, Western horizon) marks the beginning of the seventh house. The “quadrant House systems” also use the Midheaven (MC, Medium Coeli) as the beginning of the tenth House and the Nadir (IC, Imum Coeli) as the beginning of the fourth House. In astrology, a “cusp” (from the Latin “spear”) is the imaginary arrow marking a border between Signs or Houses. The solar disc having a diameter of about half a degree in some moments can be situated simultaneously in two adjacent Signs or Houses. To exclude such uncertainty the “Sun cusp” is usually considered coinciding with the Sun center. In astrology there are now more than twenty developed House systems. However, all these House systems can be divided into only two groups – space-based and time-based systems. The space-based systems use the direct geometrical divisions of zodiac to form the House charts. Contrary to it the time-based systems firstly divide the time equivalent of zodiac into 12 parts and then calculate the corresponding geometrical zodiacal positions of evaluated time-intervals. In the most ancient space-based “Sign-House system”, which was used already in the Hellenistic tradition of astrology, the astronomically determined for date, location and precise time of a horoscope Sun cusp marks the Sign that completely coincides with the first House starting the numeration of Houses. The Ascendant designates the rising sign, and the first House begins at zero degrees of the zodiac Sign in which the Ascendant falls, regardless of how early or late in that Sign the Ascendant is. The next Sign after the ascending Sign then becomes the 2nd House, the sign after that is the 3rd House, and so on, with the Houses being 30° each. However this simple system contradicts the “quadrant concept” of astrology because the Midheaven, which has an important symbolic meaning, can appear anywhere between the 8th and 11th Houses because of a varying angle between the Ascendant and Midheaven in high latitudes. Nevertheless the Sign-House system was used by many early traditions of Medieval European astrology and became the base of Indian astrology. The most popular House systems in western astrology are the time-based Placidus system and its modification introduced in the 1960s by the German astrologer Walter Koch. The Placidus system is named after the Italian Benedictine monk and astrologer Placidus de Titis (1603–1668), who popularized its use during the 17th century. It is quite possible that this system was invented by Ptolemy and in the 12th century Hebrew astrologer Abraham Ibn Ezra introduced it into European astrology. The Placidus House system uses astronomically defined Ascendant and Midheaven as the cusps of the 1st and 10th houses, respectively (Fig. 1.20-1). The semi-diurnal arc between these points represents a half-daytime and the duration (time-interval) of semi-diurnal arc can be found. Adding one-third of this time to the sidereal time of considered horoscope gives the cusp of the 12th house, and adding two-thirds of this time gives the cusp of the 11th house. Similarly, the time of the semi- nocturnal arc, the half-night from Nadir to Ascendant, is trisected and subtracted from the time coordinate of considered horoscope to give the cusps of the 2nd and 3rd houses. The time-based Placidus system corresponds to the two-hourly “watches” of ancient astrologers, who numbered the constellations in the order that the stars within them rose to the Ascendant during the twelve watches of the 24-hour period. Each of the Placidus houses represents two planetary hours, the first starting at dawn with the Ascendant.

Fig. 1.20. 1. The horoscope with Placidus House system (red lines and numbers). Aspects: green line – square, blue line – trine). 2. The south-Indian style horoscope. Red numbers are Houses in the “Sign-House system”. (The first minute of III millennium in Western Europe: Greenwich (UK) 0E00, 51N29; 0:01 UT, 1/1/2001).

Although Placidus algorithm is simple in concept, the mathematical calculations behind it is complex, with cusp positions adjusted in a complicated technique based on the use of hour cycles. In old times the specially calculated tables like Raphael’s Tables of Houses were used to find the zodiacal positions of Placidus Houses. Though the Placidus system perfectly corresponds to the “quadrant concept” of astrology, it has a serious flaw. The semi-diurnal and semi-nocturnal arcs are not the same everywhere on the Earth, but vary according to latitude. At polar latitudes some degrees of the Ecliptic never touch the Horizon and therefore, there is no semi-diurnal or semi-nocturnal arc to trisect, and a House chart cannot be constructed at all using the Placidus system for latitudes greater than 66°N or 66°S. During the astrology history many metaphysical arguments allowing every House system to be perceived as most appropriate were considered. However, the every choice of a certain Hause system reveals the astrologers preference to emphasize Time or Space. To avoid this Hamlet question and to match the modern relativistic concept of united spacetime, Argentinean astrologers Wendel Polich and Nelson Page worked out in 1960s the “Topocentric House system”, which combines time and space divisions forming the House chart. One can see that the House systems follow the same metaphysical principles as the zodiac Signs qualification. The entirety (symbolic number – 1) of a House chart is divided in accordance with the Principle of universal duality (symbolic number – 2) into an upper horizon section and lower horizon section. There are 4 angles of a House chart – Ascendant (east), Descendant (west), Midheaven (south), and Immum Coeli (north), which are trisected in accordance with the Principle of triadic representation of unity (symbolic number – 3). In astrology, the nature and type of the relationship between planets in a particular horoscope depends on the geometric angles between the planets – the “aspects”. In horoscopes, planets are seldom found to be exactly aspecting one another and the “orb” indicates the distance, in degrees, from the exact aspect at which that kind of relationship between the planets will begin to be felt. The astrology traditions usually take into account 12 aspects with 30° differences but the following four aspects are considered as most important: the “conjunction” ( 08); “square” (90 5 ); “trine” (120 5 ) and “opposition” (180 8 ). Planets can be seen together in the sky and then gradually separating and changing an aspect. For example, the Moon is in conjunction with the Sun at New Moon. The Moon moves ahead of the Sun, and about seven days later it is 90° from the Sun, forming the First Quarter, known in astrology as 28 the “outgoing square aspect”. Then seven days later again it reaches Full Moon when it is 180° or in opposition to the Sun in the sky. Some seven days later it forms the Last Quarter or “incoming square aspect” and then finally, after a total of about 28 days, it conjoins the Sun again to form the next New Moon. The conjunctions and trines are traditionally considered to be soft, harmonious and auspicious aspects, producing co-operation of planets. The oppositions and squares are usually considered to be difficult, inharmonious and adverse aspects, producing tension and contradictions of planet influences. The aspects of the Ascendant and Midheaven play an important role in astrology, while a planet without major aspects is considered to be very “pure” in its influences. As the solar and lunar eclipses occur when the Sun and Moon are in the same Sign, an eclipse astrological interpretation is defined by the qualities of its zodiacal Sign and corresponding ruling planet:

“The first and most potent cause of such events lies in the conjunctions of the Sun and Moon at eclipse and the movements of the stars at the time” (Ptolemy).

The “Lord of the eclipse” is that planet, which has most relationships by aspect to the degree of the eclipse. In individual horoscopes, eclipses are important when within a degree of a conjunction or opposition to the Sun, Moon or Ascendant. Historically, because the eclipses are caused by a shadow passing across the horoscope “Light” – the Sun or Moon, they were considered as indications of negative and harmful events, suggesting problems for countries, cities and kings. The most common way to interpret an eclipse in ancient astrology was as an omen portending negative trends or occasions, for example, eclipses in Fire Signs predict the death of eminent people, discords and wars; Earth Signs portend agricultural problems; Air Signs foreshadow storms, famine and diseases; Water Signs warn of floods and rebellions. The old chronicles tell many stories about ill-fated astrologers that were beheaded for not alerting their sovereigns in advance about eclipses.

The modern Indian astrology (Jyotisha), which developed mainly in second half of the first millennium CE, uses the same metaphysical and methodological principles as the predecessors of Western astrology – ancient Babylonian and Greek astrologies. However, comparing with Western traditions, Indian astrology includes much more national mythology and Hindu religious concepts. In Vedic mythology (II mill. BCE) Aditi (Limitless) is a feminized form of supreme god Brahma and associated with the primordial substance. Aditi is celestial mother of every existing form and being, associated with the essence of space and with mystic Logos. She is mother of the gods (Adityas) and all zodiacal spirits from which the heavenly bodies were born. Aditi flies across the sky on a cock, which symbolizes strength and honor. Her symbolic weapons include the Trishul and a sword. In Hinduism, Adityas refers to the offspring of Aditi and often the word “Aditya” is used in the singular form to mean the sun-god Surya that has hair and arms of gold and travels across sky in the chariot pulled by seven horses. However Bhagavata Purana lists total 12 Adityas as twelve sun- gods, which are the aspects of Lord Vishnu in the form of sun-god, and in each month of the year it is a different Aditya who shines. The Sun before sunrise is often called Savitr – the Vedic sun-god, a personification of the divine influence and vivifying power of the Sun and after sunrise until sunset it is called Surya. After the end of the Vedic period, Savitr disappeared as an independent deity from the Hindu pantheon but in modern Hinduism his name still often occurs, for example, in the famous “Gayatri Mantra”, which is also known as Savitri because of this. The moon-god Chandra of Hinduism is the late transformation of the ancient Vedic lunar deity Soma, which is one of Adityas and therefore the “brother” of sun-gods Savitr and Surya. The Soma name refers to the Vedic sacred drink of immortality made of the plant juices and thus makes the moon-god also the lord of plants and vegetation. There is a legend telling that Chandra was born in

29 the Ocean of Milk that the gods were churning for millennia in order to create Amrita – the drink which confers immortality upon the gods. The fair-haired youth with a club and a lotus in his hands Chandra rides his moon-chariot pulled by ten white horses across the sky every night. The Hinduism worked out the most detailed and useful mythology of eclipses, which greatly influenced not only Eastern but also European astrology and astronomy, by introduction of lunar node concept. Hindu astrology considers the planets and some other cosmic phenomena as Graha – cosmic “influencers” on the history and all living beings. The Navagraha (“nine influencers”) includes the Sun, the Moon, which are considered as planets, Mars, Mercury, Jupiter, Venus, Saturn, as well as positions in the sky – Rahu (north or ascending lunar node) and Ketu (south or descending lunar node).

Fig. 1.21. Rahu and Ketu – the demons of ascending and descending lunar nodes

Rahu, the demon of ascending lunar node is the head of the Demon Snake that swallows the Sun or the Moon causing eclipses. According to legend the asura Rahu in the form of giant Demon Snake drank some of the divine nectar Amrita. But before the nectar could pass his throat, Mohini (the female avatar of god Vishnu) cut off his head. The head, however, remained immortal and is called Rahu, while the rest of the Demon Snake’s body became Ketu – the demon of descending lunar node. It is believed that Rahu’s immortal head occasionally swallows the Sun or the Moon causing eclipses and then the Sun or Moon passes through the opening at the neck, ending the eclipse. It is believed that Rahu and Ketu had a tremendous impact on the Earth’s creation and still seriously influence human lives and fates. In later Iranian and Arabic mythology the ascending node Rahu and the descending node Ketu become the head and the tail of the dragon Al-Djawzahr. Rahu and Ketu as a part of mathematical astronomy were introduced into China during the Tang dynasty (618–907 CE), but with modified meaning. While Rahu was retained in the sense of the lunar ascending node, Ketu was used as a designation for lunar apogee. Grahas are the “cosmic influencers” that affect the auras and minds of human beings and each Graha carries a specific energy quality, which is described in an allegorical form through its scriptural and astrological references. Each of the graha has associations with various categories, such as Colors, Metals, Gemstones, Body parts etc. Humans are capable to tune themselves to the chosen energy of a specific Graha through Samyama – the combined practice of concentration and meditation. It is belived that the cosmic influence, which people receive, contains different energies coming from different celestial bodies. When the believer repeatedly utters a Mantra he is tuning to a particular frequency and this frequency establishes a contact with the cosmic energy and drags it into his body, influencing his present and future. In order to mitigate the negative effects of some planet in the birth chart or to provide more potency to a planet that is in an exalted state, believers undertake pilgrimage to designated Navagraha temples. The traditional arrangement of the Navagraha, for example in Agama Prathishta astrological practice, sets Surya on the central place, Chandra on Surya’s east, Rahu on north-west and Ketu in the north-east. The Navagraha images are generally found in all important Shaivite 30 temples where they are invariably placed in a separate hall, on a pedestal usually to the north-east of the sanctum sanctorum. The indivisible unity of Hindu astrology and mythology of Hinduism supports the everlasting popularity of astrology in India and South East Asia. Millions of people would approach astrologers when they have problems and ask them on how to overcome their troubles by performing certain rituals, which involve worship of Navagraha to overcome ill planet effects. The initial “sun sign” Hindu horoscope (Rashi chart) displays the planet positions inside 12 Signs (Rashi) of zodiac using old-style rectangular chart instead of modern Western circular structure (Fig. 1.20-2). However, in addition to classical planets of Western astrology the Indian astrology also considers positions of ascending lunar node (Rahu) and descending lunar node (Ketu). The Indian astrology uses the sidereal zodiac, where stars are considered to be the fixed background against which the motion of the planets is measured that is different from the tropical zodiac used in Western astrology, where the motion of the planets is measured against the position of the Sun on the spring equinox. After two millennia, as a result of the precession of the equinoxes, the origin of the ecliptic longitude shifted by about 22 degrees. The difference between the Hindu and the Western zodiacs is currently around 24 degrees, increasing by 1.4 degree per century. Therefore, in Western astrology the planets fall into the following Sign, as compared to their placement in the Hindu sidereal zodiac, about two thirds of the time. To introduce the Houses, Indian astrology uses as a rule the ancient “Sign-House system” where the borders of Signs and Houses coincide. However, Indian astrologers employ more than 15 House charts (Varga) derived from initial Rashi chart with different numerations and symbolic interpretations. The center of a heavenly body can be used as a designator (Lagna) of the first House. For example, the Udaya Lagna corresponds to the rising Sun, or Ascendant Sign as first House. The Chandra Lagna corresponds to the first House counted from the sign of the Moon location. However, there are the House charts with the specially calculated numerations, for example, the Nama Lagna corresponds to the first House, which is determined by the numerology of the person’s name. In Eastern astrology, beside Sign and House coordinate systems the third frame of reference is used – the structure of lunar mansions, which divide the zodiac into 27 or 28 sectors relative to fixed stars. The concept of the lunar mansions are considered to be Babylonian in origin and later it was adapted by Egyptian calendar system and by Greek-Egyptian Ptolemaic astrology in the form of 36 “decans”, which can be seen at historical Dendera zodiac (Fig. 1.22-1). In the Chinese astrology 28 lunar mansions are divided into four groups related to the year seasons. The 27 or 28 Nakshatras, the wives of the moon-god Chandra, born by the great king Daksha, one of the sons of Lord Brahma, are the Indian form of the lunar mansions with the names related to the most prominent constellations in each zodiacal sector (Fig. 1.22-2). Beginning at zero Aries, each Nakshatra comprises about 13.33 degree of the ecliptic, covered by the Moon in a 24 hour day. The 27 Nakshatras are divided into 3 cycles (paryaya) and the seven planets together with Rahu and Ketu have the rulership assigned to each Nakshatra in paryaya. The start of each paryaya is always ruled by Ketu. The numeration of Nakshatras develops eastwards starting from a point on the ecliptic precisely opposite the star Spica (Chitra) – the blue giant Alpha Virginis, the brightest star in the constellation Virgo. The lunar mansions play an important role in Indian astrology explaining the influence of the Moon, bright stars and some constellations on the historical events and people fates. The Nakshatra in which an eclipse occurs is “spoiled” for the next six months after the eclipse, and to diminish the negative influence of an eclipse it is common for people to fast for 10 hours before eclipse and then perform a havan (special fire ritual) in the middle of the eclipse.

Fig. 1.22. 1. The sculptured Dendera zodiac with 36 “decans” of ten days each (c. 50 CE, Egypt; now at the Louvre, Paris). 2. The 27 Nakshatras of Indian astrology with important bright stars

The Shulba Sutras (from Sanskrit sulba – “cord, rope”) that are considered to be appendices to the Vedas contain geometry related to fire-altar construction and are the most important sources of knowledge of Indian mathematics from the Vedic period (I mill. BCE). The different fire-altar shapes were associated with different gifts from the gods: “He who desires heaven is to construct a fire-altar in the form of a falcon… those who wish to destroy existing and future enemies should construct a fire-altar in the form of a rhombus”. The need to manipulate different fire-altar shapes led to the creation of the pertinent Vedic mathematics. For example, the “Baudhayana Shulba Sutra” (c. 800 BCE) describes the construction of geometric shapes such as squares and rectangles and the geometric area-preserving transformations: a square into a rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a circle, and transforming a circle into a square. The altar construction also led to an estimation of the square root of 2, using of Pythagorean triples and knowledge of irrationality and irrational numbers. The religious Indian texts of the Vedic period provide evidence for the use of large numbers from 102 (sata) to 1012 (paradha). The mystical properties of numbers were considered spiritually powerful that consequently led to their incorporation into religious texts. Treatises of Jain1 mathematicians of the second half of the first millennia BCE are the links between the mathematics of the Vedic period and that of the Classical period. Jain mathematicians investigated the first powers of numbers like squares and cubes, which enabled them to define simple algebraic equations, and started to use the concept of zero. Probably, Jain mathematicians firstly found the expressive “magic” representations of ancient sacred numbers 9, 12, 108 as the hyperfactorials of metaphysical Ultimate Symbols 1, 2, 3:

1 9  32 12 13  2 2  3 1 108 11  2 2  3 3 (1.4) The so called Classical period (400–1600) is the golden age of Indian mathematics when mathematics was included in the “astral science” (Jyotiḥsastra), which consisted of three sub- disciplines: mathematical sciences (ganita), horoscope astrology (hora) and divination (samhita). During this period the achievements of Indian mathematicians had spread to the all East Asia, the Middle East, and eventually to Europe. The astronomical symbolism of the sacred number 108 started to spread in the East after the works of prominent Indian astronomer and mathematician Aryabhata (476–550) and his followers. The mathematical part of Aryabhata’s main work named by later commentators the Aryabhatiya

1 Jainism is a religion and philosophy that predates its most famous exponent Mahavira (6th century BCE) who was a contemporary or predecessor of Gautama Buddha. 32 covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series, and a table of sines with the word kha (“emptiness”) to mark “zero” in tabular arrangements of digits. In the first chapter of his book Aryabhata described a system of numerals based on Sanskrit phonemes, which attributes a numerical value to each syllable possible in Sanskrit phonology – from ka = 1 up to hau = 1018. Probably, he also firstly used the astrological interpretation of the sacred number 108 – the total quantity of planet positions in all Signs of zodiac: 108 9 12 (1.5) The followers of Aryabhata often referred to his book as Arya-shatas-aShTa – literally, “Aryabhata’s 108”, because there are 108 verses in the text written in the style typical of sutra literature, in which each line is an aid to memory for a complex system. The numerological methods of Indian astrology, which are abundant contrasting the European tradition, are based on the interpretation of the symbolic formula (1.5) and its graphical representation (Fig. 1.23).

Fig. 1.23. The magic square of 9 planets at the center of 12 zodiacal Signs

Figure 1.23 shows the traditional zodiacal and lunar House coordinate systems of Indian astrology with nine planets placed in the central “magic square”. The positions of planets in square grid correspond to the Hindu customs with the Navagraha typically placed in a single square with the Sun (Surya) in the center and the other planet deities surrounding Surya. In central magic square, red numbers corresponding to planets, form an arrangement, where the numbers in each row, and in each column, and the numbers in the forward and backward main diagonals, all add up to the same number – the “magic constant” equal to 15. The 3×3 magic square has been a part of sacred rituals in India since Vedic times and still is today, for example, the well-known “Ganesh yantra” and the 10th-century 4×4 magic square in the Parshvanath Jain temple in Khajuraho. The authority of number 108 became highest when it was discovered that this number is tightly connected with the Golden ratio and defines the geometry of the simplest polygon, which can exist as a regular star – the pentagram of Pythagorean metaphysics. The interior angle of a convex regular n-gon is n 180 360 n that gives for pentagon ( n  5) the sacred number n 108 degrees.

The number 108 is not a member of classical Fibonacci series of numbers that starts with two predetermined terms and each term afterwards is the sum of the preceding two terms: [0, 1] 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377… However, the number 108 is a member of so-called “tetranachi” series of numbers that starts with four predetermined terms, each term afterwards being the sum of the preceding four terms: [0, 0, 0, 1] 1, 2, 4, 8, 15, 29, 56, 108, 208, 401… The “tetranacci constant” – the ratio toward which adjacent tetranacci numbers tend is 1.92756…that differs from the Fibonacci constant (Golden ratio)  1.61803… Nevertheless, the tight connection of the number 108 with Golden ratio demonstrates the analysis of pentagon geometry. As firstly demonstrated Euclid in his Elements (c. 300 BCE) a regular pentagon is constructible using a compass and straightedge, either by inscribing one in a given circle or constructing one on a given edge. In a regular pentagon, all sides are equal in length and each interior angle is 108°. A regular pentagon has central angles of 72° that is a half of Fibonacci number 144. The diagonals of a regular pentagon d are in Golden ratio to its sides s: ds 1,61803….(Fig. 1.24-2)

Fig. 1.24. 1. The Golden Triangle (in red) in the proportions of Leonardo da Vinci. 2. Geometric parameters of a pentagon

Angle determining a length of pentagon side is 2 5  72  , and angle defining pentagon diagonal is: 2 4 5  144  . Side length of a pentagon is: s 2sin( 2) 2sin( 5) and the diagonal length is: d 2sin( ) 2sin( 5) . The ratio of the diagonal to the side for a pentagon is: ds sin sin( 2). This formula can be transformed by using the formula for a double angle: sin 2sin(  2)cos(  2) . The transformation yields: ds 2cos( 2) and after substitution of  25, we get for a pentagon: ds2cos( 5)  1,61803...  . Formula  2cos( 5) and its inverse:  5arccos( 2) determine the exact relation for two irrational numbers  and  . Using the formula: 2cos  eeii , we can find the exact relation for three irrational numbers , and e:

2cos( 5) eeii55   (1.6) The metaphysics traditionally considered numbers and not as just numbers but as symbols of the rational order of the Universe, an imminent natural law, a life giving force behind all things, 34 the universal structure governing and permeating the world. From old times mathematicians were looking for the laconic fractions that represent these irrational numbers with the maximal exactness, for example:  22 7 3.14285714 (Archimedes, 3 century BCE);  377 120 3.14116667 (Aryabhata, 5 century);  355 113 3.14159292 (Zu Chong Zhi, 5 century). Following this tradition we can transform the relations (1.6) into the laconic approximate formula for  ,  and e, which gives a correct value for its first five decimal digits: (e ) 7 5 1.40001384… (1.7) The answer to the question formulated at the beginning of this chapter – “What is the connection of the sacred number 108 and the diameter of the Earth with a solar eclipse?” we can find in the ancient Indian Shulba Sutras, dealing with the sacred geometry of Hindu temples. In these treatises one can read: “Take a pole, mark its height, and then remove it to a place 108 times its height. The pole will look exactly of the same size as the Moon and the Sun”.

Fig. 1.25. Number 108 in the geometry of Sun-Earth-Moon system

And indeed the angular size of a pole at a distance of 108 of its height is:  arctg(1/108) 9.26 103 rad 0.53 , that coincides with the mean angular size of the Sun and Moon in the sky (Fig. 1.25). This means that in the geometry of the Sun-Earth-Moon planetary system the number 108 approximately determines not even two, but three relations of characteristics (see Table. 1.4). Analysis of relations (1.1) and (1.2) shows that they are both consequences of the formula (1.8).

Table 1.4. Unusual coincidences in the geometry of the Sun-Earth-Moon system (millions km)

Sun (S) Earth (E) Moon (M) a (orbit half-axis) 149.6 0.384

d (diameter) 1.392 0.0127 0.0035 ddSE109.6

adES 107.5 adMM109.7 3  dSEMMES a d a d d1/108 9.26 10 rad 0.53 (1.8)

Of course, the estimations of visible angular sizes of the Sun and Moon by (1.8) are only approximate, because the angular dimensions of these celestial bodies are changing due to the ellipticity of the Earth and Moon orbits: S 0.533  0.009 and M 0.525  0.035 . Moreover, the ratio adMM109.7 is valid only for a relatively short historical period. While the distance between the Sun and its planets is fairly stable, the distance between the Earth and the Moon is subject to steady and ultimately very sizable changes. The distance from the Earth to the Moon due to energy losses in tidal friction increases by about 3.8 cm per year. This means that billions of years ago the Moon was much closer to the Earth and its apparent angular size was larger. The modern favored scientific hypothesis for the formation of the Moon is the “giant impact hypothesis” (Big Splash), which suggests that the Moon was formed out of the debris left over from a collision between young Earth and some other heavenly body. One scenario is the collision 35 between the Earth and a planet the size of Mars (George Darwin, 1898; Daly, 1946; Hartmann & Davis, 1975). Another scenario proposes that the Moon and the Earth were created together in a giant collision of two planetary bodies that were each five times the size of Mars (NASA Lunar Science Institute, Robin M. Canup, 2012). The Big Splash hypothesis explains the Earth’s spin and Moon’s orbit having similar orientations; the Moon samples indicating that the surface of the Moon was once molten and the Moon’s relatively small iron core; lower Moon density compared to the Earth and the identical stable isotope ratios of lunar and terrestrial rocks. The Big Splash could initially form a triple system – the Earth with a twin pair of moons, surrounded by a common silicate vapor atmosphere, and the Earth-Moons system could became homogenized by convective stirring. The second smaller moon could be formed in a Lagrange point of the large moon and after tens of millions of years, as the two moons migrated outward from the Earth, solar tidal effects would have made the Lagrange orbit unstable, resulting in a slow-velocity collision of both moons that explains the thickened crust of the Moon’s Far side. The resulting lunar mass irregularities could subsequently produce a gravity gradient that resulted in “tidal locking” of the Moon so that today, only the Moon’s Near side remains visible from Earth. Several billion years ago the rotation rates of the Earth and Moon were larger than today and the Moon was closer to Earth. The mathematical models of the Earth’s rotation estimate the LOD (Length of a Day) as about 6-8 hours at a time 4 billion years ago. The gravitational interaction between Sun, Earth and Moon accompanied by the tidal energy dissipation steadily decreases rotation of the Earth and increases the semi-axis of the Moon’s orbit. In our time the moon and sun’s gravity add about 1.7 milliseconds to the length of a day each century. It appeared that the rate of earth’s rotation in the distant past can be estimated by paleontological studies2.

Fig. 1.26. 1. Proterozoic stromatolites from Bolivia. 2. Nonlinear dependence of LOD (hours) on time. (“Ma” for megaannus – a unit of time equal to one million years; blue circles for fossil coral estimations, red circles for stromatolites estimations)

Periodic growth structures – lines, bands, and rings are preserved in the skeletons or hard parts of some organisms – stromatolites, fossil bivalves and corals. These growth indicators show the number of days in a year at the time when these organisms lived. Measurements of daily layers and yearly patterns in fossil corals from 180 to 400 million years ago show year lengths from 381 to 410 days (Wells, 1963; Scrutton, 1970; Eicher, 1976). The oldest radiologically dated material with well- defined fossil corals revealed that 600 million years ago the day would have been 3.3 hours shorter, corresponding to 424 days in a year.

2 Pannella, G.,. Paleontological evidence on the Earth’s rotational history since the early Precambrian // Astrophysics and Space Science (1972) 16; 212-237. Rosenberg, G. D. and S. K. Runcorn (eds.),. Growth Rhythms and the History of the Earth's Rotation. New York, 1975. Schopf, J.W. (ed.),. Earth's Earliest Biosphere. Its Origin and Evolution. Princeton, New Jersey: Princeton University Press, 1983. 36

Stromatolites, which form yearly patterns much like rings in trees, are other indicators of the Length of Day. These layered bio-chemical structures were formed in shallow water by microorganisms, especially by blue-green cyanobacteria that use water, carbon dioxide and sunlight to create their food. Due to intense ultra violet radiation during the day, the stromatalitic bacteria restrict their activity to night time. Fine grain sediments are bound together by excretions adhered atop one another by the bacteria’s flagella, forming layers that correspond to periods of high activity. The investigation of fossil stromatolites from China showed that one billion years ago, a year was composed of about 516 days and each day would have been on the order of 17 hours long (Pannella et al., 1968; Mohr, 1975; Williams, 1997). Figure 1.26 demonstrates the paleontological estimations of the LOD for past epochs. At the current rate of retreating of the Moon from the Earth about 3.8 cm per year the increase of 5 semi-axis of the lunar orbit is approximately: aTM 0.384 3.8  10  (million km), and the increase of apparent angular size of the Moon is determined by the approximate formula:

5  dMM a0.0035 (0.384 3.8  10  T ) (1.9) In this formula, T (Ma – million years) is negative for the past epochs of Earth’s history. Calculations by formula (1.9) show that, for example, two billion years ago, the apparent size of the Moon was noticeably larger than the Sun (MS0.65  0.53 ), and from Earth surface could be observed only total or partial solar eclipses. In the next two billion years, the Moon’s disk will be noticeably smaller than the Sun (MS0.43  0.53  ), and terrestrial observatories will not be able to observe total eclipses of the Sun.

Fig. 1.27. The Sun and the Moon – the ancient Biblical symbols. 1. Hartmann Schedel, Liber chronicarum. Nuremberg: Koberger, 1493. 2. From Sphæra Mundi; Joannes Sacro Bosco, Venice, 1482.

Eastern sacred number 108 does not play noticeable role in Christian religious philosophy; however European medieval astrologers found another riddle that still amazes modern astronomers. The Sun and the Moon are ancient Biblical symbols, and in the Middle Ages astronomers and astrologists asserted that the days of a week are marked by special astrological signs to remind the sacred days of Creation. In particular, they claimed that on Wednesdays (the middle day of Creation) there are much more sun eclipses, than on Saturdays (the day of God’s rest). During long time, this claim of medieval astronomers was considered as an astrological folklore. However, in 2000s a Belgian astronomer Jean Meeus analyzed the statistics of more than 3500

37 predicted sun eclipses visible in Berlin, Madrid and Moscow for the years 1CE–3000CE [16], and found the expressive calendar cycles. The probabilities of solar eclipses visible from these European capitals dramatically differ from the random probability 1 7 0.143 resulting in the set – (0,17;0,1;0,19;0,1;0,18;0,12;0,14) for days of a week starting from the Monday. Figure 1.28 demonstrates this astonishing statistics.

Fig. 1.28. Probabilities of sun eclipses on different days of a week in Berlin, Madrid and Moscow, which dramatically differ from the expected random probability of 1/7

These varying probabilities show that at least in Western Europe a largest number of solar eclipses seem to occur on a Wednesday though Monday and Friday appear to have eclipse probabilities nearly as high. Modern astronomy proposing arbitrary grouping of days into seven day weeks have no idea what causes the long-standing “Wednesday paradox” with its strange distribution of solar eclipses as the week cycles. 1.2. The calendar cycles of solar eclipses The second half of first millennium of our era was the golden age of Eastern astronomy. Aryabhata (476–550), which we already mentioned, was the first in the row of great Indian scientists of the Classical age3. He was born around 476 in central India and probably became the head of the newborn Nalanda University and its observatory. Nalanda University that is about 100 kilometers south-east of Patna remained a leading centre of learning in India from the fifth century to twelfth century. Aryabhata is the author of several treatises on mathematics and astronomy, many of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy was extensively referred to in the Eastern literature and has survived to modern times. In the astronomical part of his treatise Aryabhata asserted that the Earth spins on its axis, the Earth moves around the Sun and the Moon rotates round the Earth. He believed that the Moon and planets shine by reflected sunlight, insightfully assuming that the orbits of planets are ellipses. Aryabhata described a geocentric model of the Solar system, in which the Sun and Moon are each carried by epicycles – the motions of the planets are governed by two epicycles, a smaller manda (slow) and a larger sighra (fast). Aryabhata probably recognized the general metaphysical Principle of relativity, which asserts that the essence of physical laws does not depend on the employed coordinate system. He not only uses geocentric model but writes about the position of a planet in relation to its movement around the Sun. He estimates radiuses of the planetary orbits in terms of the radius of the Earth orbit around the Sun evaluating planetary periods of rotation around the Sun.

3 India’s first satellite “Aryabhata” launched by Russian space rocket on April 19, 1975 and the lunar crater Aryabhata are named in Aryabhata’s honor. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the Aryabhata Research Institute of Observational Sciences (ARIES) near Nainital, India. 38

Instead of the astrological tradition in which eclipses were caused by demons Rahu and Ketu, Aryabhata firstly explained eclipses in terms of shadows cast by and falling on Earth. He considered the role of moving lunar nodes and estimated the size and extent of the Earth’s shadow. Aryabhata’s computational methods improved by his followers was so accurate that 18th-century scientist Guillaume Le Gentil, during a visit to Pondicherry in India, found that the Indian computations of the duration of the lunar eclipse of 30 August 1765 to be short only by 41 seconds whereas his own charts were long by 68 seconds. Aryabhata estimated the Earth’s circumference with the error of only 0.2 % and the length of the sidereal year with the error less than one second.

Fig. 1.29. 1. Indian mathematician and astronomer Aryabhata (476–550). 2. Chinese astronomer and engineer Yi Xing (683–727)

The astronomical works of Aryabhata and his followers significantly influenced the Eastern astronomy and in particular were commented and included in the Chinese astrology encyclopedia the Treatise on Astrology of the Kaiyuan Era (Da Tang kai yuan zhan jing) compiled by the head editor Gautama Siddha, astronomer and astrologer of Indian descent and numerous scholars from 714 to 724 during the Kaiyuan era of Tang Dynasty. The book, also known as the Kaiyuan Star Observations, contains approximately 600 000 words in 120 chapters (juan). By the tenth century, copies of the manuscript began to disappear and it was said that only in the 17th century Buddhist scholar Cheng Ming Shan found a copy of the original work in the stomach of an old Buddha sculpture [7]. This monumental treatise begins with a discussion of the structure of Heaven and Earth, discusses divinations relating to Heaven, Earth, Sun, Moon, the “five stars”, 28 constellations; meteors, “parasitic” stars, nova and comets, “guest” stars and “shooting” stars. Most importantly, the work collected the results of the observations, theories, and prognostications of earlier astronomers and astrologers prior to the Tang dynasty. Among such diverse astronomical information in the book there is a section describing works of Chinese astronomer, the Buddhist scholar Yi Xing (born Zhang Sui, 683–727), which is known as an author of astronomical celestial globe featured a clockwork escapement mechanism, the first in a long tradition of Chinese astronomical clockworks. The celestial globe was made in the image of the round heavens and on it were shown the lunar mansions in their order, the equator and the degrees of the heavenly circumference. In the early 8th century, Yi Xing beside of his engineering work was in charge of a terrestrial-astronomical survey established by the Tang court in order to obtain new astronomical data that would aid in the prediction of solar eclipses. In addition to the well-known Saros cycle in the Treatise on Astrology of the Kaiyuan Era one can find a hint at the irregular seasonal and geographical frequency of solar eclipses. Commenting the eclipse studies of Aryabhata, Yi Xing wrote the following:

“The Moon consists of water, the Sun of fire, the Earth of earth, and the Earth’s shadow of darkness. The merciful Moon shadows the fierce Sun more often in summer and in southern lands”.

In 2006, F. Espenac and J. Meeus confirmed the seasonal irregularity of sun eclipses by statistical analysis [9]. However, this unusual type of sun eclipse variability was not investigated in detail by planetary astronomy until recently (Taganov&Saari, 2014 [31]). Using the dates and coordinates of the Greatest Eclipses in database of Canon one can evaluate the probabilities of all types of eclipses for all calendar months: pei()() N n ei N n ei and for different geographical sites: pel n el n e . Here and henceforth, the index “e” denotes the evaluations based on the Canon database; N is the calendar number of a month; nNei () is the quantity of i-type eclipses in the N-th month; nei is the total quantity of i-type eclipses during considered epoch; nel is the quantity of central eclipses with the Greatest Eclipse at latitude l for total set of eclipses ne . As demonstrates the statistical analysis of sun eclipses in Canon the calendar distributions (amid months) of all eclipses, – central, partial and hybrid eclipses are accidental. The relative frequencies or probabilities of these eclipses for any month on average correspond to: p 1 12 0.0833. More exactly these probabilities correspond to: pN( ) (0.0849;0.0821;0.0773) (1.10) This formula accounts for 31, 30 and 28.25 days in a month. For example, for the February: p(2) 28.25 365.25 0.0773. All statistically estimated probabilities at Figs. 1.30, 1.32 are normalized (  pei 1). ()i

Figure 1.30 demonstrates the close coincidence of the evaluated probabilities pNeC () of all central eclipses (filled diamonds) with theoretical probabilities pN() (broken green curve 1 near straight dotted line p 1 12 0.0833). In contrast to this almost uniform eclipse distribution the probabilities of total and annular eclipses demonstrate the expressive calendar cycles (Fig. 1.30). In spring-summer half year (March– August) the cycles predict 58 % of all total and 42 % of all annular eclipses with the vice versa eclipse proportion in the autumn-winter half year. To make the quantitative description of eclipse calendar cycles it is necessary to consider the dynamics of “Saros series”. The position of the Sun and the Moon conjunction shifts by about 0.48º with respect to the Moon’s nodes every Saros, and this gives rise to a series of eclipses (Saros series) that may last 1226 to 1550 years and is comprised of 69 to 87 eclipses, of which about 40 to 60 are central (i.e., total, hybrid or annular). If the Moon were in a circular orbit moving a little closer to the Earth, and in the same orbital plane, there would be total solar eclipses every single month, corresponding to the uniform calendar eclipse distribution. However, the elliptical orbit of the Moon around Earth is inclined about 5.1º to Earth’s elliptical orbit around the Sun, and the Moon’s orbit crosses the ecliptic at two nodes that are 180º apart and slowly regress by 19.35º per year with complete cycle 18.6 years. The solar eclipse will be visible from some place on the Earth only if New Moon takes place within the range 15.39º– 18.59º (about 10º–12º for central eclipses) of a node, which happens at two Eclipse seasons approximately six months (173.3 days) apart. These orbital characteristics define the dynamics of Saros series.

Fig. 1.30. Calendar cycles of total (filled circles) and annular (contoured circles) sun eclipses for epoch of 3500 years: 2000 BCE–1500 CE with 4887 central eclipses. Broken green line 1 is the probability estimations by (1.10); blue curve 2 is the probability estimations by (1.15); red curve is the probability estimations by (1.14). Here pi is the eclipse probability and N is the standard calendar number of a month except December with N  0.

The Saros series begins, for example, when the New Moon appears 17º–18º east of the node. The Moon’s shadow passes about 3500 km south of Earth and a partial eclipse with small magnitude will be visible from high southern latitudes. One Saros later, the Moon’s shadow passes around 250 km closer to the Earth’s geometric center and a partial eclipse of larger magnitude will result. This is the start of the initial phase of the Saros series, which lasts about 2.5 centuries with 6 to 25 partial eclipses. The peak phase of a Saros series lasts for 7 to 10 centuries with 39 to 59 central eclipses visible almost every Saros and moving gradually northward. In the middle of the peak phase, central eclipses of long duration occur near the equator. The last central eclipse of the peak phase takes place at high northern latitudes. The final phase of Saros series lasts about 2.5 centuries with 6 to 24 partial eclipses at high northern latitudes with successive smaller amplitude. The considered Saros series ends near North Pole 12 to 15 centuries after it began near the opposite South Pole.

Fig. 1.31. Sun eclipse paths of Saros 136 for 1937–2081 years (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by Fred Espenak (NASA's GSFC)”.

A good example is the eclipse series of Saros 136 that is currently producing the longest total solar eclipses of the 20th and 21st centuries and totally will produce 71 eclipses over 1262 years in the following order: 8 partial, 6 annular, 6 hybrid, 44 total, and 7 partial. Figure 1.31 shows nine solar eclipses from Saros 136 for the years 1937 through 2081. The westward shift about 120° of each eclipse path is a consequence of the extra 8 hours in the length of the Saros period. The 41 northward shift of each path is due to the progressive motion of the Moon with respect to its descending node at each eclipse. During one Saros on average 40 solar and 29 lunar eclipses can be observed. These eclipses belong to 19–21 Saros series, moving from the South Pole, and to 19–21 Saros series, moving from the North Pole. These series are at different phases of their evolution, but majority of series (about 65 %) is at the peak phase, producing central solar eclipses. The distribution of Greatest Eclipses over the Earth’s surface is an important geographical characteristic of sun eclipses, which was not investigated in planetary astronomy until now. Statistical analysis of the Canon reveals the expressive dependence of the probabilities of latitudinal coordinates of the Greatest central eclipses on the latitude (Fig. 1.32). Almost 40 % of the Greatest central eclipses are positioned in the tropics (23.5ºS–23.5ºN) and more than 70 % in the area between 45ºS and 45ºN.

Fig. 1.32. 1. The dependence of latitudinal coordinate probabilities of the Greatest central eclipses pel (filled circles) on latitude l (degrees with negative values for southern hemisphere) for the epoch of 2000 years: 2000 BCE–1 CE with 2915 central eclipses. Solid curve is the theoretical probability estimations by (1.11). 2. The insolation distribution on the Earth’s surface.

The place of the possible Greatest sun eclipse is always positioned near the intersection of the Earth’s surface with the line, connecting the centers of the Earth and the Sun. Projections of the possible Greatest Eclipse are distributed randomly on the frontal plane that is perpendicular to the line, connecting the centers of the Earth and Sun, and corresponding uniform statistical density grows with the increase of considered epoch. The Earth’s rotation averages the longitudes of possible greatest sun eclipses forming uniform longitudinal distribution of sun eclipses. However, the projection of the uniform Greatest Eclipse density in the frontal plane onto the Earth’s curved surface is heterogeneous. As the angle increases between the direction connecting the centers of the Earth and Sun and the coordinate vector of considered meridian, the average Greatest Eclipse density in the frontal plane is reduced in proportion to the cosine of the latitude l. This is the same effect as the decrease of average insolation4 with the growth of the incidence angle. Therefore, the average latitudinal density of central eclipses is nearly proportional to the average insolation, and can be described by the following relation:

pll 0.194 cos (1.11)

4 “Insolation” (solar irradiation) is the total amount of solar radiation energy received on a given surface area during a given time. 42

The amplitude in (1.11) was estimated by the least square method from the database of Canon, and this relation estimates the central eclipse probabilities with relative error less than 2.5 %. Figure 1.33, which shows the paths of total and annular solar eclipses in the period 2000–2020., visually demonstrates a greater frequency of eclipses in the equatorial and sub-equatorial latitudes.

Fig. 1.33. The paths of total, annular and hybrid solar eclipses in the period 2000–2020. Blue paths – total eclipses, red paths – annular and hybrid eclipses (http://eclipse.gsfc.nasa.gov/ “Eclipse Predictions by Fred Espenak (NASA’s GSFC)”.

The movement of the Earth and the Moon in elliptical orbits cyclically changes the apparent sizes of the Sun and the Moon. The Magnitude of an eclipse that is the ratio of the apparent size of the Moon to the apparent size of the Sun is maximal (total eclipse) when the Moon is near its perigee and when the Earth approaches its farthest distance from the Sun in July. Quite the opposite, the magnitude of the eclipse is minimal (annular eclipse) when the Moon is near its apogee and when the Earth approaches its closest distance to the Sun in January. Therefore, the relative frequency of total eclipses is larger in summer months and the relative frequency of annular eclipses is larger in winter months. The calendar cycles of total and annular eclipses represented in Fig. 1.30 are results of statistical interference of many different Saros series with raised probabilities of total eclipses in summer months and raised probabilities of annular eclipses in winter months. This qualitative analysis is confirmed by the fact that probability differences peT()()() N  p eT N  p N and

peA  p eA ()() N  p N appear to be proportional to the sun declination  that depends on a year’s season:

peT()() N a T N paA   a A () N (1.12) These proportionalities can be used to obtain the semi-empirical quantitative description of considered calendar cycles. Since the eccentricity of the Earth orbit is quite low, it can be approximated to a circle, and  is approximately given by the following expression [6]:  sin[2 365  (284  n )] . In this formula  is the maximal declination on summer solstice in

43 the northern hemisphere: 23.45 ( 0,41 rad.) and n is the day number of a year starting from n 1 for January 1. We can use this formula with the more convenient form, defining the day number nN365 12 by the calendar number N of a month: (NCN )  sin(2  365    6  ) (1.13) In this formula N is the standard calendar number of a month except December with N  0. Using the least square method, we can estimate for database of Canon the optimal values of parameters C,, aTA a and then from (1.12, 1.13) we can derive the following semi-empirical formulae:

pT ( N ) p ( N )  0.018  sin(2 365  C  6  N ) (1.14)

pA ( N ) p ( N )  0.013  sin(2 365  C  6  N ) (1.15) In this formulae as in (1.13) N is the standard calendar number of a month except December with N  0.

As a measure of divergence of theoretical estimates pi by (1.14, 1.15) and Canon evaluations pei we can use r.m.s. (root mean square) value of the relative differences: si[( p ei  p i ) p ei ]  100(%) . In the Canon for epoch of 1200 years: 1601 CE–2800 CE with 1682 central eclipses the Gregorian calendar is used, and the optimal value of calendar parameter in (1.14, 1.15) is C  240 that provides: sT 1.08% ; sA  2.54% . For epoch of 3500 years: 2000 BCE–1500 CE (Fig. 1.30) with 4887 central eclipses, where only Julian calendar is used for eclipse dates, the optimal value of calendar parameter in (1.14, 1.15) is

C  297 , and sT 1% ; sA  0.45%. In Canon, the Julian calendar is used for eclipse dates before October 15, 1582 and then onwards the eclipse dates correspond to Gregorian calendar. For such mixed use of calendars the optimal value of calendar parameter in (1.14, 1.15) is C  284 , and for the eclipse statistics for the epoch of 5000 years: 2000 BCE–3000 CE with 7129 central eclipses the divergences of theoretical estimates

(1.14, 1.15) and the Canon evaluations are: sT  0.74%; sA  0.51%. Calendar cycles of total and annular sun eclipses are results of statistical interference, summation of manifestations of the three main cyclic movements – the Earth and the Moon movements in elliptical orbits and the slow regress of lunar nodes by 19.35º per year with complete cycle 18.6 years. The medieval Indian and Chinese astronomers were right in their intuitive guesses – the total eclipse probabilities are larger in summer months (Fig. 1.30) and in southern regions (Fig. 1.32). This short section of our book does not pretend to the complete quantitative description of considered phenomena. In particular, the next step could be the representation of empirical amplitudes in (1.11, 1.14, 1.15) as functions of the Earth’s and the Moon’s orbital parameters.

CHAPTER II. Cosmological deceleration of time In the era of space exploration in the second half of the 20th century, the accuracy and reliability of observational and computational methods of planetary astronomy greatly increased. In particular, in the mid-1970s Morrison and Ward, using observations of Mercury transits, estimated the 2 kinematic acceleration of the Moon: nM  26  2 as/cy (arc second per century per century). Using 2 artificial satellite observations Christodoulidis, et al. (1988) obtained nM  25.27  0.61 as/cy and 2 Williams and Dickey (2003) deduced nM 25.7 as/cy from Lunar Laser Ranging data. The results from Lunar Laser Ranging research show that the Moon’s mean distance from Earth is increasing by 3.8 cm per year (Dickey, et al., 1994) and the corresponding acceleration in the Moon’s ecliptic 2 longitude is nM 25.858 as/cy (Chapront, et al., 2002). This is the value that NASA has adopted in lunar ephemeris calculations [http://eclipse.gsfc.nasa.gov/]. From these estimations of lunar acceleration and the conservation of angular momentum in the Earth-Moon system it may be calculated that the rate of increase in the Length of a Day (LOD) due 2 to tides alone is 2.3 ms/cy. Assuming a value nM 26 as/cy Stephenson and Morrison (1995 [23] investigated a wide variety of eclipse observations – mainly from ancient Babylon, ancient and medieval China and medieval Arab world. This study has estimated much lesser rate of increase in the LOD 1.7 ms/cy. To explain the difference of these two estimations (0.6 ms/cy), which significantly exceeds the possible calculation error, we shall consider the history and traditions of time measuring in astronomy. 2.1. The art of measuring time Reliability of quantitative interpretation of mathematical models and theories in Natural Sciences depends on the efficiency of measurement of parameters and constant values included in such symbolic structures. The measurement methodology (metrology) is valuable not only as applied science but also as theoretical methodology because a measurement procedure is the very stage of analysis that links empirical survey with theoretical model construction. Each new symbolic model or theory should meet requirements of the measurement methodology commonly employed in various parts of Natural Sciences. Contemporary measurement methods used in Natural Sciences resulted from historical traditions of different fields of science and technology but not from some general measurement methodology. Most precise and reliable contemporary measurement techniques employ an ancient comparison principle based on the use of common standards. Length and mass standards appeared a very long time ago, and the archeology has samples of such standards of five-millennium age or even more. For a long time length standards were anthropomorphous – enough to think about the “cubit”, a measurement unit used in the old Russia, or the still existing “foot”. The first systems of measurement units formed in Babylon and ancient Egypt. In Babylon the measurement system was based on the “cubit”, the distance between the elbow joint and the end of the thumb of an outstretched arm (54.4 cm). The Babylon cubit consisted of two “spans”; each of the spans being a maximum distance between the palm thumb and little finger (27.2 cm). In its turn each span consisted of ten “fingers” identified as the average width of the male thumb (2.7 cm). Ancient Egypt employed approximately the same system of length units. In Babylon, beginning from the third millennium BCE land surveyors used the “foot” (32.6 cm) to measure distances; each foot equaled to 12 “fingers”, and two feet made one “step” (65.2 cm). In ancient Greece the length of the “foot” (  ) was established as 29.7 cm, and in the Roman Empire an equivalent measuring unit was “pes” (30.5 см). The “step” length in Greece was 77 cm, and the “pass”, its Roman equivalent, had the length of 74 cm.

Some of the ancient weight measurement units reflect our ancestors’ effort to use natural standards, for example, the “carat” (0.2 g – the weight of one seed from a pod of the carob tree), “rati” (0.125 g – the weight of 8 rice or barley grains in India). The smallest unit of length in Babylon was she (0.25 cm) and the smallest unit of weight was “grain” (about 45 milligrams) – correspondingly the length and the weight of barleycorn. In addition to the carat and grain the most commonly used units of weight were the Assyrian sickle (16.8 g), mina (50 sickles – 840 g) and talent (60 minas – 50.4 kg). The value of these main units of weight varied slightly from country to country. At the beginning of the Christian era, Romans widely used “libra” or “pondo” equal to 327.2 grams. After the fall of the Roman Empire Europeans used modified Roman measuring units until up to the 18th century. There were various projects providing for introduction of new measurements that were examined from time to time. For example, in medieval France discussed a new unit of length equal to the length of one side of a honeycomb. Gradually a variety of measures used in European countries became incredibly wide. Thus in the early 1800s the Swiss canton Vaud alone employed 8 units of length, 8 units of weight, 23 different measures of fruit and over 30 measures of liquids. Units of time were established much later than units of length and weight. In the Biblical times tradesmen who couldn’t allow mistakes in measuring of length and weight still used to divide the day approximately into four parts and the night into three “guard shifts”. It was only after the Babylon captivity (6th century BCE) after the Babylon calendar was adopted in Palestine when local people began to get used to the division of the day and night into 12 parts that is “hours”. However, in the Roman Empire during several centuries nighttime still was divided into 4 “guard shifts” of three hours each. Babylon astronomers used to divide the day and the night into 12 hours as early as the second millennium BCE by analogy with the 12 months in a year, which was convenient for scale marking of armillary spheres measuring positions of stars and planets. The same base number (12) was used to divide hours into minutes. However, the Babylonian time units found its way to Europe only at the beginning of the Christian era. For over two millenniums our civilization lived without “seconds”, and only alchemists began to employ seconds approximately from the 13th century, using as the second prototype an interval between two heartbeats.

Fig. 2.1. Stonehenge (Wiltshire, England; supposedly 3000–2000 BCE) [http://fr.wikipedia.org/]

Measuring of time intervals is principally different from static measurements of length and weight because it involves selection and use of a “standard motion”. To define the time scale different forms of motion could be used, and therefore the time metrology is ambiguous. A concrete form of standard motion is characterized by an individual and permanent relation between its spatial and time parameters that, as a rule, is a result of a certain law of nature. Thus, introduction of a time scale requires, firstly, selection of one specific standard motion from out of many. Secondly, it is

46 necessary to build a special device modeling a chosen motion and monitoring all its phases. Such time devices are traditionally referred to as “watches”. Probably most ancient astronomical clocks are megalithic observatories the ruins of which have been found in many places in France, England, Scandinavia and are plentiful in Asia. The Stonehenge in England is a well preserved majestic structure of the kind. Yet no reliable evidence of the construction time of such megalithic observatories has been found, and they still remain an unsolved archeological problem. There is no doubt, however, that such stone calendars were created several millenniums before the Christian era. In ancient times calendars were of both social and economic importance. The calendar was the basic system defining not only the farming works but also the essence of religious and magic rituals. Calendar time calculus is based on astronomic observation of the Earth and Moon movements governed by the Kepler third law. Therefore, the relation between specific time intervals t and space parameter L in the calendar time calculus is: t kL32 (2.1) Here t stands for periods of the Earth or Moon rotation, L denotes major orbit radiuses and k stands for proportionality coefficient. Undoubtedly, builders of megalithic structures were concerned about the problem of precise astronomical estimation of time intervals less than 24 hours. Not without reason in the center of the Stonehenge there is a stone “gnomon”, the sun-dial obelisk. The most ancient descriptions of sun- dials in Egypt were found in texts dating back to the time of Pharaoh Tutmos III (16th century BCE). Museum collections include a lot of ancient sun-dials, however probably the oldest sun-dial is a carved stone from the time of Pharaoh Mernepht (13th century BCE) found in Palestine. In case of ancient sun-dials they used to tell the time not by direction of the shadow but by the length of shadow. In Egypt many high ancient obelisks have a special marking around them to tell the time by the length of their shadows. As written by Herodotus, the Greeks learned to build sun-dials from Babylon. In his writing architect Mark Vitruvius Pollion (1st century BCE) described over thirty various designs of antique sun-dials. In particular, he writes that the designer of the then most accurate sun-dial “arachne” (“spider”) was Eudoxus of Cnidus (c. 408–355 BCE), author of the first cosmological model that was used later by Aristotle. Sun-dials were popular in all countries through the end of the 17th century, and their design was constantly improving. In spite of the complexity of sun-dial construction so that to make them usable in all seasons of year and in different latitudes, the specific 24-hour time interval in sun-dials defined by the constant angular velocity of the Earth: t2 R ( M 5 E )12 where R, M, E stand respectively for radius, mass and kinetic energy of the Earth’s rotation. Therefore, the relation for characteristic time and space intervals of sun-dials is: t kL (2.2) The water-clock (clepsydra) is as old as the sun-dial. The well-preserved water chronometer from the time of Amenchotep III (15th–14th centuries BCE), found in Karnack in Egypt is even older than the most ancient Palestinian sun-dial. It was at the time of King Ashurbanipal when they began to export Babylonian water-clocks to many Mediterranean countries. It was not before long when the clepsydra became an indispensable instrument to astronomers and philosophers. Not without reason, the first mentioning of the clepsydra is associated with a description of the Empedocles philosophy. A water clock is a device which measures time by the regulated flow of water into (inflow type) or out from (outflow type) a vessel where the amount of water is then measured. The bowl-shaped outflow device is the simplest form of a water clock and is known to have existed in Babylon and in Egypt around the 16th century BCE. In Babylon, astronomers used water clocks of the outflow type with cylindrical vessel having a hole in its side near the base. An example of the water clock use by 47 ancient astronomers in Babylon is the record of the total solar eclipse of April 15, 136 BCE written on two clay tablets in British Museum [23]:

“(Year) 175, month XII. The 29th (day), at 24 time-degree after sunrise, solar eclipse; when it began on the south-west side, in 18 time-degree of daytime it was entirely total; Venus, Mercury and the Normal Stars were visible; Jupiter and Mars, which were in their period of invisibility were visible in that eclipse. It threw off (the shadow) from south-west to north-east; 35 time-degree (duration) of onset, maximal phase and clearing”.

In this record, times of the various phases of the sun eclipse expressed by the references of water clock – “time-degree” – a unit equal to 4 minutes. The Greeks and Romans advanced water clock design including the inflow clepsydra with an early feedback system, gearing, and escapement mechanism, which were connected to pointer and resulted in improved accuracy. Further improvements of water clock were made in Byzantium, China and Arab world. It took a long time for the water clock to find its way to Europe. The story about creation of a clepsydra for King Theodorich by Boeceum (6th century) and the tale about luxurious clepsydra presented by King Charles the Great (Charlemagne) to Caliph Harun al-Rashid in the 9th century are the first evidences of the clepsydra appearance in Europe. The first European water clock with complex gears was the astronomical clock created by Giovanni de Dondi in 1365 and in the end of 14th century most European observatories, as well as wealthy citizens in Italy, France and England had water-clocks.

Fig. 2.2. Water clocks (engravings; 18–19 cent.). 1. The French water clock of the 18th century. 2. Ambassadors of Charlemagne present clepsydra to caliph Harun al-Rashid.

The water clock was the most accurate and commonly used timekeeping device in observatories for millennia, until it was replaced by more accurate pendulum clocks in the 17th century. Clepsydras were used in most European countries up to the end of the 17th century, and before the end of the 16th century neither observatories no alchemic laboratories could do without it. Astronomers and alchemists preferred the mercury clepsydras before water-clocks. Famous Ticho de Bragge, the lucky astrologer of Danish King Frederick, used a mercury clepsydra of own design. This time-device was especially accurate, as its daily error did not exceed three minutes having the relative instability around 2 103 . To the sound of falling heavy mercury drops Ticho de Bragge was 48 putting down valuable observational data of planet movements creating his priceless archive that helped Kepler to formulate the laws of planetary motion. The equation (2.2) where the coefficient k depends on the volume velocity of the water or mercury flow from a calibrated orifice describes the relation between time intervals t and space scale grades L for water and mercury clocks. The pendulum-clock era began at the time of alchemy flowering, and the first data about using the pendulum to tell the time are found in the writings by Ibn Unis, an 11th century Arabic alchemist. Studies of Leonardo da Vinci’s heritage kept in Madrid and the Ambrosiani Library in Milan revealed that Leonardo dedicated much time to invent the pendulum time-regulating devices. However, the priority of invention of the pendulum-clock theory and effective pendulum constructions belongs to Galileo Galilei (1564–1642) and Christian Huygens (1629–1695). The memoir Horologium Oscillatorium published by Huygens in 1673 in Paris was the first comprehensive treatment of the theory and practice of the pendulum-clock design. Huygens presented the geometrical derivation of the formula for the mathematical pendulum period: t2 ( L g )12 and proved that oscillations of the real, “physical” pendulum can be described by the same formula if the length L is understood as “effective” length of the material, physical pendulum. In spite of the theory success, necessary accuracy of the pendulum-clocks could only be achieved by effective devices supporting the movement of the pendulum and mechanisms for counting of the pendulum oscillations. Beside a pendulum – the harmonic oscillator used as timekeeping element, the pendulum-clocks include “escapements” – the clock-rate control devices that transfer energy to the timekeeping elements to compensate friction, and allow the number of pendulum oscillations to be counted. First Huygens’ pendulum clock with the “verge escapement” had an error of 1 minute a day, and his later refinements of escapement reduced his clock’s errors to less than 15 seconds a day. The work of Christian Huygens and Robert Hook (1635–1703) aiming at invention of the clock- rate control device based on the torsion pendulum in the 1670s resulted in the design of the “anchor escapement” that significantly improved the accuracy of pendulum clocks. More than two hundred clock-rate control systems, now buried in oblivion, were invented between 1750 and 1850. Even Pierre Baumarche, author of the Barber of Seville and the Marriage of Figaro, invented his own smart “virgule” clock system. Significant progress in the art of the pendulum-clock construction started in the course of search for a reliable solution of the geographical longitude measurement. Accurate determination of latitudes and longitudes is vital for the success of navy and merchant marine expeditions. In case of latitude determination, ship masters knew how to do it quite accurately as early as the 15th century by the position of the pole above the horizon, but the determination of longitude at sea remained a serious problem for most shipmasters up to the end of the 18th century. Being much concerned about sea expedition failures governments of Portugal and Spain in the 16th century, the Netherlands in the 17th century, England and France in the 18th century offered large awards for a solution of the problem of the longitude determination Many skillful horologists and venerable scientists including Christian Huygens and Robert Hook worked hard to invent the reliable marine chronometer-timekeeper. The English inventor John Harrison began his design work in 1726 but successfully tested marine chronometer only past 36 years. After a lot of drudgery and the final test of the chronometer held with personal support of King George III eventually at the end of his life John Harrison was awarded the highest prize by the English Parliament in 1773. However, Harrison’s timekeeper was too complex and costly to set up the engineering ideology of marine chronometers for a long time. The gorgeous marine clock presented in 1766 to Louis XV by the French horologist Pierre Le Roy became the prototype of all mechanical marine chronometers achieving a precision of 0.1 second per day (106 ), which is accurate enough to locate a ship’s position within 1.5 kilometer after a month’s sea voyage.

In terms of the pendulum clock accuracy, the highest achievement is the clock with electromagnetic support of pendulum oscillations invented in 1921 by British engineer William Shortt in collaboration with horologist Frank Hope-Jones. Shortt clock kept time with two pendulums – a master pendulum swinging in a vacuum tank and a slave pendulum in a separate clock, which was synchronized to the master by an electric circuit and electromagnets. The slave pendulum was attached to the timekeeping mechanisms of the clock, leaving the master pendulum virtually free of external disturbances. Relative instability of this horologic masterpiece did not exceed 108 and was less than the relative instability of the Ticho de Bragge mercury clepsydra by five orders. The Shortt clock was the first clock to detect tiny seasonal changes in the Earth’s rotation rate and until the middle of the 20th century such clocks were used to keep exact time in metrology standard institutes and observatories.

Fig. 2.3. 1. The pendulum clock of William Shortt. 2. The modern NPL-CsF2 cesium fountain clock operated by the National Physical Laboratory (UK).

Comparison of formulae determining oscillation periods of pendulums having different designs, leads to the general relation between space and time parameters for all pendulum clocks: t kL12 (2.3) All above described watches may be called “mechanical clocks” because the motions of clock mechanisms are governed by the laws of classical mechanics and the apparent generalization of Eqs. (1.1–1.3) looks as follows:

i tii k L i 1 2;1;3 2 (2.4) In the second half of the 20th century the progress of quantum physics resulted in creation of new time devices that were fundamentally different from mechanical clocks. In accordance with (2.4) the work of all mechanical clocks is determined by the principal spatial characteristic of their mechanism. In new time systems that can be called “quantum clocks”, specific time intervals are determined not by spatial but by time characteristics of one or another quantum process. First model

50 time systems of the new type were piezo-quartz clock and molecular ammonia clock created by G. Lionse in 1953. In piezo-quartz clock he used a quartz generator, and the clock main parameter was the cycle (or frequency) of its oscillations. The main characteristic of ammonia time device is the cycle (or frequency) of emission at quantum transition of an electron between two nearby energy levels in an ammonia molecule. Resonance absorption of radio waves was used as an indicator of compliance with this standard reference frequency. Very soon the relative instability of first quantum clocks was lowered from 107 to 108 . The next stage in the development of quantum time devices was the creation of cesium watches, the main characteristic of which is the emission oscillation period corresponding to the electron transition between two close energy levels in the cesium-133 atom. The first model of the cesium time-device in the National Physical Laboratory in England had the instability of 109 and subsequently was lowered down to 1012 . The frequency uncertainty of modern NPL-CsF2 cesium fountain clock operated by the National Physical Laboratory (UK) is about 2 1016 . The NIST-F2 cesium fountain clock operated since April 2014 by the National Institute of Standards and Technology, and serving as a new U.S. civilian time and frequency standard has the frequency uncertainty close to 1 1016 that corresponds to the error less than one second in about 300 million years. The quantum clocks contrary to pendulum clocks besides the oscillating timekeeping element have internal frequency standards. The clocks that use the electromagnetic spectrum of atoms as frequency standards for their timekeeping elements usually are named the “atomic clock”. The timekeeping elements of many atomic clocks are electronic oscillators operating at microwave frequency. Many atomic clocks keep track of time by a crystal oscillator, which is an electronic oscillator circuit that uses the mechanical resonance of a vibrating crystal of piezoelectric material to create an electrical signal with precise frequency. A good crystal oscillator has the short term stability a few parts in 1013 during several seconds but in the long term test, the “aging” can move off frequency by parts in 109 per week or even per day. Therefore the oscillator of atomic clock is arranged so that its frequency-determining components include an element that can be controlled by a feedback signal. The feedback signal keeps the oscillator tuned in resonance with the frequency of the internal frequency standard, for example, the electronic transitions of cesium or rubidium atoms. A number of methods exist for utilizing hyperfine atomic transitions as internal frequency standard in atomic clocks. By tradition, the hardware that is used to probe the atoms is called the “physics package”. Most investigated are the following internal frequency standards: Atomic beam standard, Atomic gas cell standard, Active maser standard, Fountain standard and Ion trap standard. Some atomic clock systems use external frequency standards. For example, the atomic clock with rubidium standard periodically corrected by the Global Positioning System receiver. This achieves high short-term accuracy, with long-term accuracy equal to the national time standards. The accuracy of an atomic clock mainly depends on two factors. The first factor is temperature of the sample atoms in internal frequency standard – colder atoms allow longer probe times. For example, in the NIST-F2 cesium fountain clock now operated by the National Institute of Standards and Technology (USA), the vertical flight tube of frequency standard is chilled inside a container of liquid nitrogen at –193 C that significantly reduces measurement errors. The second factor is the frequency and intrinsic width of the electronic transition used in frequency standard – higher frequencies and narrow lines increase the standard precision and therefore the optical frequency standards are more prospective in terms of accuracy. The measurement of frequency in an optical frequency standard is similar to that in atomic clocks with the exception that, instead of a microwave oscillator, a stable laser probes a narrow-band optical transition in an ensemble of atoms or a single ion. Breakthroughs in optical frequency measurement, laser stabilization and laser manipulation of quantum absorbers have greatly increased the accuracy of optical atomic frequency standards. Optical clocks outperform the best cesium clocks with respect 51 to relative frequency stability. In 2013, two experimental optical lattice clocks containing about 104 atoms of strontium-87 were able to stay in synchrony with each other at a precision of at least 2 1016 . Another optical clocks using about ytterbium atoms cooled to 10 microkelvin and trapped in an optical lattice showed the relative uncertainty less than 2 1018 .

Fig. 2.4. Reduction of the mean uncertainty of time measuring (logarithmic ordinate scale) in the last thousand years of our history

The long history of fighting for accuracy of standards abounds with bloody episodes when a dishonest merchant could easily lose his arm or head because of cheating when using measuring instruments. Traditionally special attention was attached to weights, standards of mass, as people used them to measure such valuable goods as precious stones, gold, ivory, spices. In Indian markets one can still see medieval weights with personal seals of Shahs. A monument to development of the contemporary mass standard is the appearance of the standard “kilogram” in Europe. At the end of the 18th century the kilogram as a unit was defined as mass of a cubic decimeter of distilled water at 4 degrees centigrade in vacuum. In 1799, the standard kilogram was created – a platinum cylinder of 39 mm in diameter. This standard was kept in the National Archives of France and used to be called the “Archive Kilogram”. At the beginning of the 19th century in the course of precise measurement they established that the mass of a cubic decimeter of distilled water at 4 degrees centigrade in vacuum weighs by 0.028 grams less than the “Archive Kilogram”. Nevertheless, in 1872 by resolution of a special international conference the “Archive Kilogram” was approved as the international standard of mass. In 1889, the 1st General Conference on Weights and Measures adopted a decision about making the new standard of the unit of mass (“kilogram”) in the form of a cylinder of platinum-iridium alloy to be kept by the International Bureau of Weights and Measures in Sevres, near Paris. Duplicates of the Sevres standard were acquired by many countries. Russia bought the duplicate number 12 that was later used perfectly well in the USSR. Now this standard can be seen at the Mendeleyev Research Institute of Metrology in St. Petersburg. It turned out, however, that this symbol of constancy also changes and its mass slowly grows, probably due to oxidation and gas absorption: in 1889 its mass exceeded the mass of the Sèvres standard by 0.068 mg, and in 1948 by 0.085 mg. History of the international standard of length began in 1664 when Christian Huygens offered to define one third of the second-pendulum length at specified geographical latitude as the international unit of length. In 1670, French astronomer Gabriel Mouton suggested that one minute of the Earth meridian arc should be defined as the unit of length. During the French Revolution a special board of

52 scientists headed by Pierre-Simon Laplace submitted a proposal to the French National Assembly to settle on the length of 1/10 000 000 of a quarter of the meridian passing through Paris, as the unit of length. Measuring the arc of the meridian from Barcelona, Spain, to Dunkirk, France, eventually yielded a value for the new unit to be called the “meter”, and its platinum standard was made in the form a bar of 25 mm width and 4 mm thick. In 1799, the French National Assembly approved the platinum standards of the meter and the kilogram that were handed in the Archive of the French Republic. In 1889, the 1st General Conference on Weights and Measures approved the new standard for the meter. The new standard was a platinum X-shape bar having the length of 102 cm. The distance between the marks at the ends of the bar was approved as the international standard for the “meter”. The progress in the time measuring after invention of the atomic clock in the 1950s stimulated specialists in metrology to redefine the whole concept of the standards of length and time. At this time metrologists remembered that popular time and length measuring of our ancestors was based on the conception of the time and space unity. This approach was not purely philosophical as our ancestors used it to create special units and methods for length measuring. For example, the unit called “stadium”, a very popular ancient unit for measuring long distances, was defined as the distance covered by a warrior in arms in prompt steps while the sun was rising above the horizon (about two minutes). In Egypt the stadium was equal to 174.5 m, in Babylon 194 m, in Greece and Rome 185.1 m и 185 m respectively. Eventually this old concept of the system tie between time and space provided the base for contemporary metrology standards for measuring the length and the time. In accordance with the resolution of the 13th International Conference on Weights and Measures (Paris, 1967) all time scales used in science and technology must be based on the “atomic second” adopted by the conference. The atomic second value is determined as equal to 9 192 631 770 cycles of the radiation corresponding to the electron transition between two selected energy levels of the 133Cs atom that is not disturbed by external fields. The 17th General Conference on Weights and Measures redefined the meter as “the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second”. This standard is very close to the procedure of the “stadium” definition in Ancient world. Thus contemporary standards of basic units for measuring the length and time are interrelated and, in essence, are relativistic as they use “natural” standard of velocity presupposing the universal constancy of the speed of light. Formation of a universal system of units in the 19th century was accompanied by negotiations about acknowledgement of the metric system of measurement units on the international scale. The metric system was legally established in France in 1840 and from that time onward its progress through the world has been steady. In 1875 representatives of 17 countries became signatories to the Metric Convention that provided for establishment of the International Bureau of Weights and Measures. In 1889 under resolution of the 1st General Conference of Weights and Measures metric units got the status of the international measuring units. An important event in the history of natural sciences in the late 19th century was the introduction of universal international time scale. In 19th century, high precision pendulum clocks were constructed and gradually more accurate and stable time unit LOD (Length Of a Day) came into use. After Simon Newcomb’s introduction of a precise theory of Earth’s orbital motion and his construction of the “mean equatorial sun”, “Greenwich mean time” has been adopted since 1884 as an International standard later called UT – “Universal Time”. However, in 1939 the director of the Royal Observatory in Greenwich, the president of the International Astronomical Union Sir Harold Spencer Jones, convincingly demonstrated that the LOD is not an ideal unit as a temporal reference because long-term astronomical observations had shown that the average length of a day (LOD) is subject to the frequent millisecond fluctuations.

There are both “external” and “internal” causes of the instabilities of LOD (see e. g. [14, 17, 23]). The most important external factors of the LOD variations are lunar and solar tides in the oceans and variable tidal stresses in the Earth’s crust, which together with the atmospheric currents arising from uneven heating, cause a permanent increase in the LOD about 2.3 milliseconds per century (ms/cy). Internal mechanisms of LOD variations are more diverse. There are short-term variations of the local atmospheric density due to the constantly changing wind flow directions that lead to annual variations of LOD. Variations in the electromagnetic interaction of the core with the Earth’s mantle may lead to changes in LOD with periods of decades and even centuries. The influence of fluctuations of sea level associated with global climate change may affect the value of LOD over several centuries. The long process of post-glacial isostatic compensation of the Earth’s crust form is probably of particular importance. The lifting of the parts of land, which were under the ice has been going on for many thousands years and should lead to a change in the Earth’s moment of inertia and, therefore, to alteration of LOD. The development of the concept of “Ephemeris Time” widely recognized by the world scientific community was one of most important result of Sir Spencer Jones’ works. Ephemeris Time (ET) – is an ideally uniform time scale considered as an independent variable in the differential equations of the theories of celestial body motions in the Solar system. According to the decisions of the VIII International Astronomical Union Assembly ET was introduced in 1952. Later the term “Ephemeris Time” was substituted by TT – “Terrestrial Time” without any significant changes in its definition. The standard for the LOD in the ET-system was taken as a mean value of LOD in UT-system over the period between 1750 and 1890 for the standard epoch of 1820. As we already mentioned, in 1967 the XIII Conference of the International Committee for Weights and Measures changed the definition of a “second” from astronomical to the procedure defined by the cesium atomic clock. Since this transition to a time measurement independent of the Earth’s rotation was made, it became necessary to regularly coordinate this “atomic” time with the mean solar time (UT). Prior to this transition the time defined by the rotation of the Earth was adjusted to compensate for uneven rotation of our planet. The modern “International Atomic Time” (TAI) is a high-precision atomic time scale based on the notional passage of time on the Earth’s geoid. TAI as a time scale is a weighted average of the time kept by over 200 atomic clocks in over 50 laboratories worldwide, which use the definition of the SI “atomic second”. All the clocks are compared using GPS signals and two-way satellite time and frequency transfer. Such averaged time scale is far more stable than any clock would be alone. However, since the reference clocks are on average well above sea level, the TAI slowed down by about 10−12 due to gravitational time dilation. TAI is the basis for “Coordinated Universal Time” (UTC), which is used for civil timekeeping all over the Earth’ s surface, and for “Terrestrial Time” (TT), which is used for astronomical calculations. The rotation of the Earth is now monitored by a special institution – “International Earth Rotation Service” (IERS). This service constantly determines the difference between the time measured by atomic clocks and the time determined by the rotation of the Earth. When a significant discrepancy between these two determinations of the day has accumulated a “leap second” is inserted to reduce the discrepancy and to make the difference between the readings of atomic clocks (TAI) and world time (UTC) no more than 0.9 seconds. Since 30 June 2012 when the last leap second was added, TAI has been exactly 35 seconds ahead of UTC. The 35 seconds results from the initial difference of 10 seconds at the start of 1972, plus 25 leap seconds in UTC since 1972. A definition of the “Terrestrial Time” standard was adopted by the International Astronomical Union (IAU) in 1976 at its XVI General Assembly, and later named “Terrestrial Dynamical Time” (TDT). In 1991, in “Recommendation IV of the XXI General Assembly”, the IAU redefined TDT, also renaming it “Terrestrial Time”. TT is a theoretical time scale and its material realization is TAI, estimates of which are provided in real time by several institutions. Because of the historical

54 difference between TAI and ET when TT was introduced, the TAI defines TT thus: TT = TAI + 32.184 s. The international scientific communities considers now a new precision time scale based on observations of an ensemble of pulsars. This new “pulsar time scale” will serve as an independent instrument of computing TT, and it may eventually be useful to identify defects in TAI. However, for the analysis of planet observations in the past and estimations of the secular perturbations in their motion the ET- and UT-time scales are still the only tools. 2.2. Nonlinear time scales The methodological base of Natural Sciences is the definition of time scale and time measuring since the gross aim of Natural Sciences is the investigation of present in order to predict the future and to explain the past. During all the history Natural Sciences use the discontinuous time scales composed from primary time intervals t that are formed by a “clock” counting periods of some cyclic change, which may be either the changes of a natural phenomenon or of an artificial device. In the time scale of mercury clepsydra the primary time interval corresponds to a single mercury drop and in pendulum clock it is a part of the pendulum oscillation period. In atomic clock the primary time interval is defined by the periodicity of feedback signals that hold the timekeeping crystal oscillator tuned in resonance with the internal frequency standard, for example, the electronic transitions of cesium atoms. In experimental optical clocks the primary time intervals are defined by ultra short pulses of light. The accuracy of a time scale depends on two factors – the relative uncertainty of primary time intervals and the accuracy of forming a time scale from the set of primary time intervals. In spite of impressive progress of accuracy of primary time intervals their short term tests do not guarantee the long term permanence of composite time scale. The artificial uniform Newtonian time scale formed from a set of primary time intervals with high short term accuracy is not the reliable representation of the “physical” time because it says nothing about the “Arrow of time” and difference between past and future. The past and future are different in many ways that reflect the “Arrow of time” – we remember last year, not next year; there is evidence of the past but not of the future; our present actions affect the future and not the past; causes always precede their effects. The Arrow of time is revealed in the way physical processes tend to go over time, and that way is the direction toward disarray, the direction toward equilibrium, the direction toward higher entropy – we can ring a bell but never un- ring it; we can burn a letter but never un-burn it. Many physicists think that the amalgamation of the Universe’s irreversible processes produces the cosmic Arrow of time. It is well accepted that the entropy increase can account for the fact that we remember the past but not the future, that effects follow causes and that we grow old and never young. There is no doubt that time seems to pass and there are two primary theories about Time flow or Time pace – according to the “dynamic” theories, the Time pace is objective, a feature of mind- independent reality of the Universe. The dynamic theories imply that the Time flow is a matter of events changing from being indeterminate in the future to being determinate in the present and past. Time pace is the process when events becoming determinate, so some dynamic theories speak of Time flow as “temporal becoming”. The critics of dynamic theories suppose that there is some objective feature of our brains that causes us to mistakenly believe we are experiencing a flow of time. We have different perceptions at different times and anticipations of experiences always happen before memories of those experiences, but the Time pace itself is not objectively real. Regardless of how we analyze the metaphor of Time flow, time seems to flow in the direction of the future, the direction of the Arrow of time. The main modern metaphysical disagreement is about whether time intervals and events have non-relational, absolute properties of “pastness”, “presentness”, and “futurity”. Does an event have or not have the property of, say, “pastness” independent of the event’s relation to us and our 55 temporal location? After discovery of the cosmological redshift phenomenon astrophysicists already worked out a method of quantitative estimation of “pastness” of photons in spectra of remote cosmic objects. It seems that the method of estimation of “pastness” and “futurity” of time intervals will also be found in the nearest future. The 20th century physics theoretically and experimentally investigated the phenomenon of “time dilation”. In the special theory of relativity, time dilation is an actual difference of elapsed time between two events as measured by observers moving relative to each other with the constant 2 2 1 2 velocity V:  KKt;   (1  V c ) . In geometrical theories of gravitation and, in particular, in the general theory of relativity, the time dilation is registered when an observer compares two time 2 1 2 scales in the areas with different gravitational potential  :  GGt;   (1  2  c ) ;    GM R . The effect of time dilation arises neither from technical construction of the clocks nor from the fact that signals need time to propagate, but from the inner nature of spacetime itself. The relativistic time dilation formulae can be considered as simple linear examples of more general model of “physical” time:   Ft( ;...). Medieval Islamic and Christian philosophers adopted the ancient Zoroastrian and Biblical idea that time is “linear” and that the Universe was created at a definite moment in the past. In 17th century Newton advocated linear time when he represented time mathematically by using a continuous straight line, firstly introduced in the 14th century by French philosopher Nicholas Oresme (c. 1320/1325–1382). In 19th century Europe, the idea of linear time became dominant in both science and philosophy. The discovery of time dilation in 20th century does not change the idea of linear time because the time dilation formulae are equal to linear transformations of uniform linear Newtonian time scale. However, in 1947 Austrian mathematician Kurt Gödel (1906–1978) discovered solutions to the equations of Einstein’s gravity field equations that allowed closed loops of proper time (closed time-like curves)5. Each event in such loop lies in its own causal history and these “causal loops” allow one to go forward continuously in time until he arrive back into his past. It is said that Gödel being a close friend of Einstein have given his paradoxal solution to Einstein as a present for his 70th birthday, and these “rotating Universes” that would allow time-travels to the past have caused Einstein to have doubts about his own general relativity theory. The Gödel’s solution is equivalent to the introduction of a specific nonlinear time scale:   F( t , r ;...). Later scientific discussions revealed that another nonlinear time scales, though not so paradoxal, were already considered by Dutch astronomer Willem de Sitter (1917): 2 2 1 2  (1 r R )  t and by British cosmologist Edward Milne (1933):  t0log( t t 0 ) t 0 . Nonlinear models of physical time can reflect an objective difference between the past ( t ) and the future ( t ): F( t ;...)  F (  t ;...), which sets the direction of the Arrow of time, and is one of the possible definitions of the irreversibility of physical time. Newtonian constant and uniform time scale t is an argument and internal standard of uniformity in such models, allowing one to determine the Time pace of physical time: d dt F( t ;...) , that may not be permanent. The debates about possible variability of the time pace has been going on for centuries. Aristotle already considered an idea about the connection between the perception of time pace and human mind cognition and registration of information. We can read his following parable [33]:

“If there is no change in our mind’s perception, or if we do not notice them, then it seems to us that time has stopped. Thus, in one fable residents of Sard slept at a time when the hero performed his feats. Waking up, they tied the last moment of their staying awake with the awakening moment

5 Gödel, K. An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation // Rev. Mod. Phys. (1 July 1949) 21, 447. 56 and the interval between these moments became erased due to lack of any sensation. Time does not exist without alterations and without motion”.

Following Plato and Aristotle many philosophers associated time pace with the movement and processes of physical reality change, arguing that not only the motion is measured by time, but also time by a movement: “they are determined by each other, because time defines the movement being his “number”, and movement defines time” [34]. Democritus (c. 460–c. 370 BC) and Epicurus (342/341–271/270 BC) were founders of antique atomism and creators of first models of discreteness of space and time. Among the Greek philosophers there were many devotees of atomism, which considered the changes of time pace as the result of short disrupts in each movement’s continuity. The problem of the divisibility of time has for a long time been a source for lively discussions of scientists and philosophers. As a monument to these disputes, there are the well-known paradoxes of Parmenides’s disciple the philosopher Zeno of Elea (c. 490–c. 430 BC) “On the competition in the race between Achilles and the tortoise” and “On the impossibility of arrow flight”. Original tractates by Zeno of Elea had not been found, but the stories collected by Diogenes Laertius (2–3 cent.), and Zeno’s ideas as presented by Aristotle, give reason to suppose that Zeno wrote his paradoxes during ancient discussions about the existence of the “atoms of time”, that is, discussing the heuristic models of discrete time. Aristotle did not share concepts of atomism, asserting that the adoption of the existence of atoms of space and time inevitably leads to the assumption of existence of the “atoms of speed and movement” and to the paradoxical situation where the “movement will consist not of movements but of the instant displacements of something motionless” (Aristotle Physics, 6.1 in [33]). Aristotle saw the best way to overcome the Zeno’s paradoxes in the approval of the continuity of the movement. Speaking of continuity and infinite divisibility of motion he essentially attributed these properties to the space and time. The fact that these ideas were not generally acknowledged in ancient Greek philosophy is evidenced by the popularity of Zeno, and the very existence of his Aporias preserved to our days. Medieval philosopher Moshe ben Maimon (Maimonides, Rambam 1135–1204) witnesses in his The Guide for the Perplexed about attempts of rational interpretation of antique models of heterogeneous time and movement by the Arab philosophers of his time:

“If the two bodies seem to us moving at different speeds, the reason for this difference is not more rapid or more slow motion, but that motion which we call slow, is interrupted by intervals of rest more frequently than a movement we call fast”6.

French medieval philosopher Nikolai Otrekure (c. 1299–c. 1369) deserved honor of the public burning of his books at the order of Pope Clement VI because of their innovative ideas. The philosopher, in particular, argued that matter, space and time are made of the indivisible atoms, points and moments, and all the processes of synthesis and degradation of matter occur through the rearrangements of atoms. Resolution of the paradox of motion of bodies at different speeds without discarding an existence of the variable “atoms of time” philosopher saw in the existence of microscopic breaks (morulae) in each movement’s continuity:

“The more morulae in the motion, so it is slower, the less – so it is faster, but if they do not exist at all, it will be extremely fast”7.

6 Maimonides M. The Guide for the Perplexed. NY, 1946, p. 121. 7 Зубов В.П. Николай из Отрекура и древнегреческие атомисты // Труды Института истории естествознания и техники АН СССР (1956) 10, 338–383. 57

The first mathematical model of inhomogeneous time with varying time scale appeared a year after the publication of Einstein’s gravitation equations. In 1917, Dutch mathematician and astronomer Willem de Sitter (1872–1934) had found an unusual solution to Einstein’s equations for a static metric8 – in it the metric coefficient of the time coordinate was a function of distance from the observer: ds2 c 2(1  rRdt 2 2 ) 2  (1  rR 2 2 ) 1 dr 2 (2.5) In the De Sitter world clocks that are further from the observer’s position at r  0 would register 2 2 1 2 slower paces of time than the observer’s clock: t  t0 (1  r R ) . This relation defines the ratio 2 2 1 2 of wavelengths of emitted atomic lines: 0 (1rR ) , which describes the redshift: 22 z 0 12  r R . This part of the “De Sitter effect” could be regarded as “apparent motion” of recession. However, there is a second part of the effect (De Sitter 19339) due to tendency of particles placed into De Sitter spacetime to accelerate. The result when there were no initial negative velocities of test particles, put into the manifold, was that the actual accelerating motion of particles away from the observer just canceled the quadratic effect of redshift leaving an almost linear redshift-distance relation. The non-relativistic character of De Sitter cosmology becomes clear in considering the equation for motion of a photon along the radial world line ( ds  0 in 2.5): c* dr dt  c(1  r 2 R 2 )  const . Hence the speed of light in the De Sitter world decreases with distance from the coordinate origin at r  0 . Any static form of a line element can be transformed into a non-static form by a suitable substitution of new coordinates, which are functionally dependent on the original space-like and time-like coordinates. In 1920s, Lemaitre and Robertson independently discovered coordinates transforming De Sitter static line element (2.5) into non-static form: ds2 c 2 d 2 exp(2 c R )  dl 2 l r(1  r22 R )exp(  ct R )  t  cR2  lg(1  r22 R ) (2.6) The model (2.6) describes non-uniform 4D-expansion of the Universe and corresponds to the following Universe’s equation of state: 00p 0 0 for proper density 00 and pressure p0 . For 00 the Universe radius is determined by the Einstein cosmological constant: R 3 . The model can be regarded as spatially closed if  is positive, as transforming into the open “flat” spacetime of special theory of relativity if  is equal to zero, and as spatially open but curved spacetime if  is negative. With negative pressure the De Sitter model describes an exponentially “inflating Universe”, which is one of the basic concepts of the Big Bang cosmology. The “Kinematic cosmology”, proposed by British astrophysicist and mathematician Edward Arthur Milne (1896–1950) in 1933, also considers the inhomogeneous time scale10. Edward Milne, realizing the fundamental importance of time in Natural Sciences, attempted to present cosmology as purely deductive system based on the methodology of time measuring. According to Milne any mechanical device, which associate events with a scale of ever increasing numbers may serve as a clock. He considered the Universe as a “substratum” formed by a set of moving “fundamental observers” exchanging signals reflected immediately, so that the information exchange for each pair of observers is a single unbroken zigzag signal. All signals contain information of the clock-reading indicating the time of latest reflection-event, so that observers are able to read off one another’s

8 De Sitter, W. // Proc. Akad. Wetensh. Amsterdam (1917 a) 19, 1217. De Sitter, W. // MNRAS (1917 b) 78, 3. 9 De Sitter, W. Astronomical Aspect of the Theory of Relativity. University of California Pub. in Mathematics (Berkeley: Univ. Calif. Press), 1933, 2, No. 8. 10 Milne, E.A. World-Structure and the Expansion of the Universe // Zeitschrift für Astrophysik (1933) 6, 1–95. Milne, E.A. Relativity, gravitation and world-structure, Oxford: Clarendon Press, 1935. 58 clock indirectly. All measurements including space coordinates and velocities were reduced by Milne to timing the signals as they went from one observer to another. The general problem of time-keeping in Milne Universe is – whether some particular observer, which has set an arbitrary clock for himself, can make it congruent with clocks of other observers irrespective of their relative motion? Milne has found that the universal congruence is possible for two time scales – uniform scale t, which he referred to as “atomic” time, and inhomogeneous logarithmic time scale  t0log( t t 0 ) t 0 . Fundamental observer using t-scale views the expanding Universe where other observers retreat from him with velocities that are proportional to their radial distance from an observer. Formally it is the cosmological model of an expanding empty space with uniform time scale and negative curvature of spacetime – for the initial condition: tR0: 0 the Universe’s radius expands with the speed of light: R ct . The total regraduation from t-time to logarithmic  -time of all clocks will transmute a uniformly expanding Milne substratum in flat space into a stationary substratum in hyperbolic space. However, the structure of the substratum and its main laws will remain the same in spite of the regraduation. In particular all atoms must shrink continuously, according to -scale. The motion of a test-particle, which is inertial when described with the use of -scale will no longer appear inertial when it is described using t-scale. The effect of passing from -time to t-time is that inertial motion is replaced by accelerated. When the motion of a test-particle is analyzed with respect to t-scale the result is the emergence of an apparent virtual “force” giving the particle a spontaneous acceleration. In 1950s Russian mathematician Vladimir Fock (1898–1974) demonstrated that every non- relativistic spacetime has relativistic conformal image in Galilean spacetime, which he named “Friedman-Lobachevski space” (FLS)11. Theory of FLS was built on the assumption that the metric interval allows the group of Lorentz transformations and can be represented in the form: ds2 F( S )( dx 2 dl 2 ) S x22 l . A group of homogeneous Lorentz transformations provides isotropy of FLS and the origin of reference frame has no peculiarities and any point of FLS can be transferred to the origin by these transformations. Using the theory of FLS Fock created the relativistic version of Friedmann cosmology. The solution of the Einstein gravitation equations for uniform space distribution of matter with finite mass density determines the radial line element of the spherically symmetric FLS corresponding to Friedmann’s model:

ds2(1  ) 4 ( c 2 dt 2  dr 2 )  t22() r c   const (2.7) The Friedmann-Fock cosmology with the radial line element (2.7) uses the same assumptions as the Friedmann’s model, but unlike it provides a constant speed of light on the radial geodesics ( ds  0 ). This cosmology describes an accelerating 4D-expansion of spacetime with a change of not only the space scale but also the time scale. The redshift dependence on the distance for a luminous object in this cosmology is nonlinear. In 1999, American physicist Johan Masreliez, using Fock’s conformal representation of radial interval, proposed relativistic version of the Steady-State Cosmology – the model of “Scale Expanding Cosmos” (SEC)12. This model uses the radial line element that defines the concurrent growth of the both space and time scales:

11 Fock V. The Theory of Space, Time and Gravitation. London: Pergamon Press, 1959. Фок В.А. Теория пространства, времени и тяготения. Изд. 2-е. М.: ГИ ФМЛ, 1961. 12 Masreliez, C.J. The scale expanding cosmos // Astroph. and Space Sci. (1999) 336; 399–447. Masreliez, C.J. The Progression of Time. Charleston SC, 2012.

ds2exp(2 t T )  ( c 2 dt 2  dr 2 ) (2.8) The relativistic SEC-model describes an accelerating 4D-expansion of the spacetime, introducing for initial condition: t 0: 0 the second time scale  , which is connected with the Newtonian time t by the relation:  T[exp( t T ) 1] T const that corresponds to the redshift: z T exp( t T )  1. Conducted retrospective analysis of cosmological models convincingly demonstrates the internal relation of inhomogeneous time scales with the redshift phenomenon. This allows us to consider the inverse problem – to find inhomogeneous time scale that corresponds to numerous astrophysical studies of the cosmological redshift.

In the methodology of science besides the general Principle of relativity is very important to pay special attention to the Principle of the measurement relativity. All physical characteristics are relative, since their numerical values depend on the measurement methodology and standards used for basic units. The relativity of physical quantities highlights their dimensions, for which special rules specify the use of basic units with the conventional standards. The Principle of the measurement relativity corresponds to the assertion that the laws of nature do not depend on the standards used in the measurements of physical characteristics. Each physical quantity, for example, a length is defined by its numerical value r and dimension []L that are established by the relation:

r[] L R L0 (2.9)

Here R is the absolute length in the selected reference frame, and L0 is the standard of length used in the measurement of R in this frame of reference.

The change of the standard can be represented by the dimensionless scale-factor ai so that the new standard is defined as:

Lii a L0 (2.10) One of the consequences of the Principle of the measurement relativity is the constancy of the ratio of two absolute values of a physical quantity when the standards used in the measurements change. For example, considering the case of length’s measurement with the use of the scale-factor (2.10), we have:

rLrL12[]:[]():()():(): RL 1020 RL  RaL 102012 RaL  RR  const (2.11) The Principle of the measurement relativity unambiguously determines the functional structure of the dimensions of physical quantities in the form of power monomials: []LMTa b c of basic, primary dimensions, such as the fundamental triad {,,}LMT – “space L – mass M – time T”. For example, the dimension of energy in the CGS dimension system is [cm2 g s-2]. A detailed derivation of the dimension formula can be found e. g. in the book13 (pp. 19–21). An important consequence of the Principle of the measurement relativity and relations (2.10, 2.11) is the independence of scale-factor on the physical nature of the standards used for the basic units. For example, in the formulation of the relativistic Principle of constancy of the speed of light there are no restrictions of the methods for measuring the speed of light. It is assumed, in particular, that any speed (including the speed of light) can be measured as by finite macroscopic intervals of length and time {,}rt and with the help of microscopic, quantum characteristics, for example, using the wavelengths and periods of photons {,}t :

13 Седов Л.И. Методы подобия и размерностей в механике. Изд. 10-е. М.: Наука, 1987. 60

drdt()()()() PrNt1  1   P 2  N 2  t  c (2.12) With independence of the reference values of basic units on the physical nature of standards used for scales R00, the following relation for the scale-factor holds:

a r R00 (2.13) Belief in the independence of the reference values of the basic units on the physical nature of the used standards embodied in modern metrology, justifying the use of quantum devices, as standards of international macroscopic “meter” and “second”. 2.3 Cosmological deceleration of physical time In the 1920s, West Slipher, Knut Lundmark, Milton Humason and Edwin Hubble used the same formula for the redshift as modern astronomers do while processing the results of their observations (see e. g. [22, 35]):

z (  0 )  0     0    0  1 (2.14)

Here 0 is the wavelength of the spectral line of reference standard source in the laboratory, corresponding to the observed spectral line  (Fig. 2.5).

Fig. 2.5. The redshifts in spectra of galaxies and quasars

In 1929, Edwin Hubble published his analysis of observations of the galaxy spectra that showed a linear relation of redshifts in the spectra on the distances to observed galaxies:

cz c()  0  0  c    0  Hr (2.15) Equation (2.15), which is called “Hubble law”, has an unusual form – in its left hand side is a function of photon wavelength – a microscopic quantum characteristic, and the right side is the cosmological macroscopic characteristic – the distance from an observer to very distant cosmic object. The difference in the parameter values of the left and right sides in the Hubble law is tens of orders of magnitude. Belgian Roman Catholic priest and astronomer Monsignor Georges Lemaître (1894–1966), which introduced the term “redshift”, used in analysis of Hubble’s observations not the wavelengths of photons but their periods t and performed the following transformation to define the redshift: 61

z( t   tr )/  t r  [() at  at ()]/() r at r daaaadt   (2.16) Here the index “r” marks the time of radiation emission by some cosmic object. For the expanding Universe scale-factors are a()() t a tr for tt r and therefore observed shift in the spectra is “red” i. e. the spectral lines are shifted toward the longer wavelengths in the spectrum. Relations (2.16) lead to a linear dependence of redshift on the distance from an emitting object, assuming that for the relatively small distances one can take dt r c and then: cz a a r  H() t  r a da dt (2.17) From this Lemaitre’s deduction it is seen that he considered (2.17) as an approximate relation with the coefficient (“Hubble factor”) depending on time and obtained by replacing the differentials with finite intervals. The similarity of the formula (2.17) with the relation for the Doppler Effect gave Lemaitre ground to name Hubble phenomenon “apparent Doppler Effect” comparing redshifts with the radiation characteristics of moving objects. This analogy uses the fact that the observable manifestations of non-static state of spacetime in the Universe can be visually illustrated by the Doppler Effect formula:

z()()r        r  r  V c (2.18) The substitution to (2.17) of the apparent “effective” velocity of the Universe space expansion:

Vexp  cz leads to:

Vexp  c  z  a a  r  H() t  r (2.19) The effective velocity in (2.19) can be much greater than the speed of light for distant radiating objects with z 1 that contradicts to the special relativity theory. During the recent years astronomers have discovered cosmic objects with very large values of redshifts: the galaxy IOK-1 in the constellation Coma Berenices with the redshift z 7 and the object GRB 090423 in the constellation Leo with the redshift z 8.2 . Southern European Observatory on Mount Paranal in Chile recently announced the suggested discovery of the object “1916” with the redshift of about 10 near the center of a rich compact cluster of galaxies Abell 1835. The apparent “effective” recession velocities of these distant objects in accordance with (2.19) are far beyond the speed of light. The violation of special relativity assertion that no speed can exceed the speed of light is a consequence of the fact that (2.19) is only approximate formula, since its derivation is based on the use of the non-relativistic Doppler Effect for low velocities and, respectively, for small values of 12 redshift. The relativistic formula for the Doppler Effect has the form: r[(c  V ) ( c  V )] and instead of (2.18) leads to the following formula for the relativistic redshift:

12 z(r   )   (    r )  r  [( c  V )( c  V )]1  (2.20) Iinstead of in (2.19) we obtain a more complicated relativistic formula that determines

Vcexp  for all values of the redshift:

22 Vexp  c [(1  z )  1] [(1  z )  1] (2.21) In (2.17) the Hubble parameter depends on time. Only the assumption that for small changes of the scale-factor and consequently small values of redshift we can use the present value of the Hubble factor: H0 a 0 a 0 transforms (2.17) into the approximate relation:

cz H0 r or z H0 t with t r c (2.22) For the metrological interpretation of the Hubble law, there are several approaches, for example: 1. After transforming the left side of (2.15) with the use of macroscopic classical kinematics to interpret the relation as the base of cosmological models. This approach assumes that all distances in remote parts of the Universe are smaller than in the vicinity of the Earth. In the process of a photon motion from remote emitting object associated with the photon wave “stretches” and, accordingly, astronomers observe the increased characteristic wavelengths of the spectral lines. 2. To interpret and analyze the relation (2.15) using quantum kinematics of photons (Taganov 2003 [24–32]). In this case it is necessary to take into account the inextricable quantum bond between photon wavelength and a period of associated oscillation:  ct. This gives reason to suppose that in remote parts of the Universe not only all distances are smaller, but all the time intervals are shorter than in the Earth’s vicinity. In the first third of the 20th century, when quantum physics just began to develop, cosmologists chose the first approach and following after Georges Lemaitre regarded the Hubble’s law as a confirmation of predicted by general theory of relativity “expansion” of the Universe space. Let us now consider the quantum interpretation of the Hubble law, which was not used by the classical cosmology. In the analysis of the redshift phenomenon should be taken into account that the difference between physical time and Newtonian time is only detectable for large cosmological time intervals. In Micro World, due to quantum uncertainty relations the difference between intervals of physical and Newtonian time is negligible:  t even for processes with high energy. Therefore, in estimation of the local speed of a photon in the microcosm can be used both the Newtonian and physical time: c dr d dr dt .

From (2.13) it follows: rR 00. Substitution of this relation into the motion equation of a photon: dr dt c transforms this equation to the form:

d dt 00 c R (2.23)

In this formula, 0 is the wavelength of a photon at the time of emission by a cosmic object at the beginning of his long journey to the instruments of observers at the Earth. Because of the alleged invariance of the radiation laws in the Universe, this wavelength must match the wavelength of the laboratory standard, which is used by observers.

The solution of equation (2.23) for the initial conditions: t 0:0 is the formula:

 0(1 c R 0  t )   0 (1  Ht ) (2.24)

Parameters HR, 0 in cosmology are usually called, respectively, Hubble’s “constant” and

“radius”. Applying to (2.24) the definition of redshift z 0 1, we obtain the customary form of the Hubble law: z Ht (2.25) Equations (2.24, 2.25), corresponding to the Hubble law for the Newtonian time can be obtained not only for photons, but also for micro particles with nonzero masses. Analysis of the equations of De Broglie and Planck, determining the motion of particles with finite mass, results due to the Principle of the measurement relativity in a linear differential equation of the first order for the redshift, which has a solution corresponding to (2.24, 2.25) (Taganov 2008 [27, 28]). Therefore, for Newtonian time the Hubble law in accordance with the Principle of the measurement relativity is a form of macroscopic description of the uniform motion of micro particles with quantum characteristics determined by the uncertainty relations and the equations of Planck and De Broglie. 63

Classical cosmology employs absolute Newtonian time, which is used as a continuously varying mathematical parameter with invariable and uniform scale. To introduce the “physical” time it is necessary to analyze quantum physical process – the movement of a photon, which is used to determine the distances in observational cosmology.

According to (2.24) the cosmological increase of the initial wavelength of a photon 0 is defined as: 0 (1Ht ) , which after the use of the photon period tc00 has the form:

(1 Ht )  c t0 . For measuring large cosmological distances in this equation can be taken:

dr; t0 dt and then: dr(1  Ht )  cdt . For finite distances, which can be defined as: rc  , t integration of this equation gives: r c  c (1  Ht ) dt and therefore (Taganov 2003 [24-32]): 0

t  (1 Ht ) dt  t  H 2  t 2 (2.26) 0 t H1[(1  2 H ) 1 2  1] (2.27) Equations (2.26, 2.27) determine the relationship of the decelerating physical time  and the Newtonian time t with its invariable and uniform scale, when for the determination of the large cosmological time intervals and distances the motion of photons is used in accordance with the Principle of the constancy of the speed of light and the Principle of the measurement relativity. The relations (2.26, 2.27) can be obtained directly from the formulation of the Principle of measurement relativity (2.13). The scale factor defined by (2.13) allows obtaining for redshift defined as z 0 1 the following formula:

a r R00  1  z  Z (2.28)

The central equality in (2.28) can be considered as the Hubble law if we define the scale R0 as the Hubble radius R0  RH c H : cz Hr and z Ht for t r c (2.29)

For measuring large cosmological distances in the relation (2.28) can be taken: dr; t0 dt and then:

r R0  dr cdt (2.30) dr cdt1  z  Z (2.31)

The use of the relation t0 dt is equal to the setting of uniform time scale t in the rest frame of an observer, which is measured by the quantum watch and can be named “atomic time”. The equation (2.30) formally corresponds to the expanding space, where radial distances and the apparent velocities of observed objects grow with time:

V dr dt  c R0  r  Hr (2.32) Equation (2.32) is equal to the equation dr r Hdt , which determines the exponential dependence of the radial coordinate on time. However, the use of the relativistic condition r ct 2 transforms (2.32) in the equation dr dt cH tdt that has the solution: r cH2  t  r0 . From the Principle of measurement relativity follows a completely different interpretation of the redshift phenomenon. In the mathematical formulation of this Principle (2.28) the central equality

64 r R0 1 z is the “Hubble law”: cz Hr in the coordinate system shifted for the condition: rz0: 0 with the “Hubble constant” H c R0 . In the approximate formulation of the Principle of measurement relativity that follows from (2.30, 2.31): a dr dt  V  R0 c  r the parameter V defines the “apparent” velocity of the Universe’s expansion. Consequently, the redshift phenomenon not at all indicates expansion of the Universe, but is a result of the universal application of the Principle of measurement relativity for measuring both the microscopic characteristics of photons (  ) and giant cosmic distances (r). In 1930s, Edward Milne proposed another interpretation of (2.32). He deduced the proportionality of the velocity and coordinate: V∞r from his Cosmological Principle, which is based on two postulates: 1. All observations of the Universe, which can make an observer in a certain place, are similar to observations of another observer in any other place. 2. Each observer sees himself to be at a center of spherically symmetric isotropic volume. In 1936, Arthur Walker (1909–2001) in his papers on Milne’s kinematic theory14 showed that Milne’s usage of the special relativity theory is an unnecessary restriction, and Milne’s kinematic method does not presuppose the validity of the Lorentz transformations. In 1944, Walker presented a proof demonstrating that the first Milne’s postulate, which can be interpreted as an assertion of the universal space homogeneity, is contained in the second one, which entails the universal space isotropy15. It must be stressed that the Walker’s proof of the sufficiency of the universal isotropy presupposes the existence of a smooth paths between all observers, which cannot be guaranteed in fractal large-scale cosmic structures. The Principle of the measurement relativity and its consequence (2.28), which can be used for interpretation of the observed structures with any geometry, is an alternative method to deduce the proportionality of the velocity and coordinate (2.32): V∞r , which is equivalent to the postulate of the universal space isotropy. In September 1910, at the mathematical and physical section of the 82th meeting of German naturalists and physicians in Königsberg Russian physicist Vladimir Ignatowsky (1875–1942) in his report Some general remarks on the principle of relativity16, demonstrated the ability to create an axiomatic special relativity theory (STR) using only the following assumptions: 1. The assumption of space isotropy. 2. Principle of the kinematic relativity and the equivalence of inertial reference frames. 3. The assumption of the linear form of coordinate transformations between inertial reference frames. In this approach, the Poincare-Einstein postulate of independence of light speed from the motion of its source is not used as the initial assumption and the relativistic velocity addition rule became one of results of the Ignatowsky’s axiomatic STR. The assertion that a photon moving at the speed of light in one reference frame, will move with the same speed in any other frame of reference has lost the status of a postulate, becoming the proven theorem in the axiomatic STR proposed by Ignatowsky. Axiomatic method of Ignatowsky was further developed in 1911 in the article17 of Philipp Frank and Hermann Rothe. This study showed that the most general coordinate transformations between the two inertial frames are linear-fractional functions. These functions can include several parameters. Including of the Principle of relativity into the set of initial assertions introduces two

14 Walker, A.G. // Proc. Lond. Math. Soc. (1937) 42, 90; (1940) 46, 113-154; (1943) 48, 161-179. 15 Walker, A.G. // Journ. Lond. Math. Soc. (1944) 19, 219-229. 16 W. von Ignatowsky. Einige allgemeine Bemerkungen zum Relativitätsprinzip // Verh. d. Deutsch. Phys. Ges. (1910) 12: 788–796. (see also http://synset.com). 17 Philipp Frank und Hermann Rothe. Über die Transformation der Raumzeitkoordinaten von ruhenden auf bewegte Systeme // Ann. der Physik (1911) Ser. 4, Vol. 34, No. 5; 825–855. (Перевод – http://synset.com). 65 parameters with the dimension of speed – the speed of the relative motion of inertial reference systems and the speed of signal propagation in the processes of the coordinate identification in the systems of reference. If additionally an axiom of the space isotropy included, the general coordinate transformations become the Lorentz transformations. To take into account the anisotropy of space it is necessarily to introduce in transformations an additional dimensionless parameter. In the case of using the “absolute time” assumption and, therefore, the concept of absolute “simultaneity” of events, the Lorentz transformations become the Galilean transformations of classical mechanics. In the 20th century, a method of axiomatic creation of the STR without the Poincare-Einstein postulate of the constancy of the speed of light was developed. This idea was firstly introduced in the articles of Vladimir Ignatowsky, Philipp Frank and Hermann Rothe, and was repeatedly rediscovered and discussed (e. g. Ya.P. Terletskii 1965; N.D. Mermin 1986; A.R. Lee, T.M. Kalotas 1975; Achin Sen 1994; S. Nishikawa 1997). Detailed review of the history and the contemporary form of the axiomatic STR may be found in the article by A.K. Guts18 and the book by S.S. Stepanov19. Summarizing the above analysis, we can conclude that the proportionality of the velocity and the radial coordinate (2.32): V∞r, being a consequence of the Principle of measurement relativity, corresponds to the postulate of space isotropy already discussed by several mathematicians and astrophysicists. In addition, this relation under the assumption of the equivalence of all inertial coordinate systems and linearity of coordinate transformations provides universal constancy of the speed of light and determines the use of the Lorentz transformations and relativistic kinematics. Therefore proportionality of the velocity and the radial coordinate (2.32): V∞r can be the basis of relativistic quantum cosmography. The relation (2.30), which is a specific form of the universal isotropy postulate, describes the t dependence of proper distance (i.e. as measured by radar methods) on redshift: r c Z() d . For 0 Z(0) 1 and dt H1 dZ this relation transforms to:

Z r c H  d  c2 H  ( Z 2  1) (2.33) 0 In addition to the atomic time t we can introduce the macroscopic scale of physical time  using the condition of the macroscopic constancy of the speed of light:   rc. For z Ht we can again receive from (2.33) the relations (2.26, 2.27):  (Z22  1) 2 H  t  H 2  t (2.34)

t H1[(1  2 H ) 1 2  1] (2.35) Visual representations of physical processes with the use of potential play an important role in physics, allowing the wide use of general equation: q  , in which the vector characteristic of a process “flux” or “flow” q is determined by the gradient of potential  . This representation is used, for example, in electrodynamics, hydrodynamics and thermodynamics to describe physical “flows’. With the help of (2.34, 2.35) the “flow” of physical time can be presented in the same visual form:

12 q  d dt  t  1  At  (1  2 A  ) (2.36) The flow of physical time is directed from the past with smaller characteristic intervals into the future with larger characteristic intervals.

18 Гуц А.К. Аксиоматическая теория относительности // УМН (1982) 37:2(224); 39—79. 19 Степанов С.С. Релятивистский мир (2012) // http://synset.com 66

Equations (2.34, 2.35) demonstrate progressive increase of physical time intervals  comparing to the scale of constant intervals of Newtonian time t, which is a kind of “expansion” of the physical time. Modern cosmology often uses the term “expansion of space”. However, instead of rather tongue-tied words “expansion of time” seems appropriate to use the more precise term “deceleration of the time pace”. In this book, the term “time pace” is sometimes used. Poincaré, Einstein and Minkowski have already started to apply this term during development of the special theory of relativity. Pace (or “course”, or “flow”) of physical time  1 (s-1) is defined as the parameter inverse with respect to some characteristic time interval  . Increase of the characteristic time interval as is the case in (2.34) corresponds to decrease of magnitude of the time pace, that is, to the “deceleration of physical time”. Decelerating, “expanding” time fundamentally differs from Newtonian time, which is always used in the Natural Sciences as invariable and uniform continuum. Irreversible time, which reveals the cosmological deceleration, in this book is called “physical”. The use of the term “physical time” can be justified by analogy with the term “physical vacuum”, which in modern quantum physics replaced the classical concept of the “emptiness”, as an abstract three-dimensional mathematical space. Direction of the Arrow of time defines the pace of time. There is no direction of the reversible Newtonian time: tt(  )  0 . Orientation of the Arrow of physical time: (tt ) (  )  0 defines an objective difference between the future and the past. In accordance with (2.34) the use of the “present” standard of time interval leads to linear decrease of the physical and Newtonian intervals ratio in the past (negative t and  ):  t12  H  t . On the contrary, this ratio linearly increases for time intervals in the future (positive t and  ):  t12  H  t .

Fig. 2.6. Scales of idealized Newtonian time and decelerating physical time

The degree of the spacetime asymmetry can be determined by the sum of coordinates before and after the inversion: (t  t; r   r ). For example, the asymmetry of the homogeneous Minkowski spacetime with r ct can be defined by the relations: Atr t (  t )  0; A  r ( t )  r (  t )  0 . Thus, the asymmetry of a homogeneous spacetime is zero. The same estimate of asymmetry of metrically inhomogeneous spacetime with the physical time (2.34) and r c () t gives relations:

2 A ( t )  (  t )  Ht  0 (2.37)

2 Ar  c( t )  c (  t )  cHt  0 (2.38) Hence, a characteristic feature of the spacetime with decelerating physical time is its asymmetry. In asymmetrical spacetime the physical time inversion cannot be achieved only by the inversion of Newtonian time: ()()tt   , and this asymmetry of spacetime causes several new physical phenomena. Just as the use of the Principle of relativity introduces into kinematics the fundamental constant of the speed of light, the application of the Principle of the measurement relativity, which defines the

67 group of similarity transformations for the standards of basic units, introduces into kinematics additional fundamental constant – the Hubble parameter. The phenomenon of cosmological deceleration of irreversible physical time is consistent with the general concept of the kinematic time dilation. In special theory of relativity, time dilation is defined by Lorentz transformations for two inertial reference systems. More generally, the time dilation is defined by the coordinate transformations that take into consideration the relative acceleration of reference systems. In the concept of equivalence of inertial and gravitational mass (in general theory of relativity) time dilation is determined by the difference in gravitational potentials of compared frames of reference. The phenomenon of the cosmological deceleration of time is result of comparing the reference systems of different scales – the microscopic frame of reference with quantum kinematics and the macroscopic one used, for example, in cosmology.

Hubble’s law (2.15) in the form: r c H ()  00  can be regarded as a transformation of the spatial coordinate of the line of sight in the microscopic quantum system of reference into macroscopic system of reference. In the complete transformation of the coordinate system, the appropriate transformation of time should be defined in addition to the transformation of the spatial coordinate. The relation (2.26, 2.34):  t  H2  t2 complements the Hubble law, determining the appropriate transformation of time, when comparing the microscopic quantum systems of reference with macroscopic ones. Substituting (2.35) in the Hubble law: z Ht , we get the Hubble law for physical time: zH(1  2 )12  1 (2.39) Comparison of (2.39) with the relations (2.28) leads to the dependence of scale factor on physical time: a( ) 1  z  (1  2 H )12 (2.40) Equation (2.34) after the use of the Hubble law: z Ht can be transformed to: ( t ) t  Ht 2  z 2 (2.41) This relation shows that one of the interpretations of the cosmological redshift is the doubled relative difference between the physical and Newtonian age of a photon. This characteristic of individual photons does not depend on the distribution of matter in the Universe. Therefore, astrophysicists register the Hubble law with the same precision in the immediate vicinity of the Milky Way with very inhomogeneous distribution of galaxies and in the remote cosmos, where the distribution of matter in average is more homogeneous. The physical dimension of Hubble constant depends on the functional type of the Hubble’s law. In (2.25): z Ht the dimension of Hubble constant is: []HT 1 that is, for example, []H  s-1. In the Hubble’s law (2.29): cz Hr the dimension of the constant is: [][]HVL 1 that is, for example, []H  km/s/Mpc (Mpc – Megaparsec). The ratio between these frequently used dimensions is: 1 km/s/Mpc 3.2 1020 s-1. In our time, in astrophysics and cosmology the most popular estimations of the current value of the Hubble constant are in the range of 60–70 km/s/Mpc. For example, the scientists of the international research project “Planck” have recently published the estimation: H0 = (2.17 0.02) 1018 s-1 ( 67.8 0.77 km/s/Mpc). In the cosmology with decelerating physical time, the Hubble parameter not used as a variable, depending on the time factor of classical cosmology, but it is considered as a fundamental constant, which defines the degree of metrical heterogeneity of un-static Universe’s spacetime.

The development of cosmology with the irreversible physical time allows, in particular, determining the Hubble constant and other key cosmological parameters as a simple functions of the fundamental physical constants (Taganov 2008 [27–29, 32]) Considering the time-scale invariant cosmology we have to remember that for such theory like for covariant formalisms the Hamiltonian H in Hamilton-Jacobi equation for the Principle of least action: HS    0 is equal to zero and one can use the relation: S   0 S const (2.42) The Principle of least action (2.42) in the form: S E const allows using the theory of potentials in relativistic Universe to determine the interactions, for example, the gravitational and electromagnetic forces. From (2.42) with r c; c const follows: E const const r (2.43) From the Principle of least action (2.42): S E const for constant volume V const and energy density: E  EV follows the relation: E  const . The form of the Principle of least action: S pr  r  r  r  const can be considered as a differential equation that has a solution: r2  const  . With this solution, the relation may be represented in the form:

2 E  const E  const r (2.44) This equation, which is one of the possible forms of the Principle of least action, is used in the cosmology with irreversible time as the equation of state for the Universe.

For a 00;;; z    r  c  from (2.34) one can obtain the following relations for the redshift z and the cosmological distance r: zH(1  2 )12  1 (2.45)

r c2 H  [(1  z )2  1] (2.46) Equation (1.60) can be made more exact by analyzing the thermodynamics of relativistic matter in a finite volume (Taganov 2008 [27–29, 32]). If the dependence of the chemical potential  of a substance on the density is of the form:  n the relationship between pressure and density is given by: P   n1 (see e. g.20]). The total energy is: E(3 n  1 5 n  1)  Gm2 r and the gravitational 2 potential energy is: EG  (3 n 5 n  1)  Gm r . For the relativistic matter n 13 and the total energy is equal to zero: E  0 with the gravitational energy: E32 Gm2 r . The total energy of a finite G volume, as follows from these relations, and taking into account the energy equivalent of the mass of 22 matter, is given by: E mc  EU (3 Gm 2 r )  0 . Here EU is an internal energy, including the kinetic energy of the object’s non-relativistic subsystems, which can be estimated as 2 EUG  U2  3 Gm 4 r in accordance with the non-relativistic virial theorem. With these assumptions, this equation takes the form: mc22(3 Gm 4 r ) 0 (2.47)

12 Substituting into (2.47) the transformed relation (2.45): RRH0 (1 2 ) one can get an estimate for the process of growth of the Universe mass:

20 Landau, L.D., Lifshitz, E.M.: Statistical Physics. Pergamon, London,1958. 69

2 1 2 M4 c R0 3 G  (1  2 H ) (2.48) From energy balance (2.47) can be derived the relation for the mass density: 3 m m V34 m r , defined as an extensive parameter for the finite masses and volumes and therefore estimating an average mass density: m 43 c2 r G (2.49)

22 m ()c G r (2.50) There is a great diversity in types of the observed cosmic objects and for many of them their basic characteristics vary in ranges of several orders of magnitude. This diversity can be seen, for example, in the handbook21, where the ranges and mean values of the main types of cosmic objects, calculated from the relatively reliable astrophysical data, are collected. These objects form a large- scale structure of the observable Universe. From the relation (2.49, 2.50) follow proportions for average masses, mass densities and radii for different cosmic objects: mr (2.51)

  r 2 (2.52) Figure 2.7 demonstrates how (2.51, 2.52) coincide with the astrophysical data presented in21.

Fig. 2.7. The comparison of average observed characteristics of different cosmic objects with the estimations (2.51; upper graph) and (2.52; lower graph).

Thus, the simple estimations (2.51, 2.52) derived from the cosmological equation of state (2.47) are in good agreement with astronomical data, in spite of the great diversity in qualitative and quantitative characteristics of different types of cosmic objects. The traditional terms of theoretical cosmology such as “radius of the Universe” and the “age of the Universe” should be treated as a metaphors and it is preferable to use the term “cosmological scales” of the non-uniform spacetime instead of these metaphoric terms. Estimates of the cosmological scales of time (the “age of the Universe”) and distance (“radius of the Universe”) can be obtained using the relation (2.34). The estimate of the Newtonian age of the

21Allen’s Astrophysical Quantities. 4th ed. Ed. Arthur N. Cox. AIP Press&Springer, 1999. ISBN 0-358-98746-0 70

Universe can be determined from the condition   0 for which the equation (2.34) has the form:  02  t  H  t2 . Hence: 1 Ht 2   0 and respectively:

tHp 2 32.15 Gyr (2.53) The velocity of the apparent Universe’s expansion V cz reaches the limit of the speed of light at z 1. For this value of redshift the relation (2.46) has the form: r c  c2 H  [(1  z )2  1] from which for z 1 follows:  1 2H  [(1  1)2  1] and then:

17  THH 3 2  7.618  10 s (24.15 Gyr) (2.54) In (2.53, 2.54) and henceforth in the calculations of the numerical values of cosmological scales we use the theoretical value of Hubble constant (2.61) derived a half page below.

The relation RHH cT together with (2.54) gives the estimation of the cosmological distance scale (“radius of the Universe”):

28 RHH cT 3 c 2 H  2.283  10 cm (2.55) The cosmological scale of mass can be derived from the equation for the total energy of relativistic matter (2.47): mc22(3 Gm 4 r ) 0 . From this energy balance there can be obtained the estimate for the Universe’s mass for rR H :

2 3 56 MHH4 c R 3 G  2 c GH  4.099  10 g (2.56) With the help of (2.56) the average densities of mass, energy and action can be evaluated for the cosmological scales (2.54, 2.55):

2 2 2 30 -3 mH(c  G )  R  4 H 9  G  8.227  10 g сm (2.57)

2 2 2 9 -3 Em c 4 c H 9  G  7.394  10 erg сm (2.58)

29-3 SEH T 2 c H 3  G  5.627  10 erg s сm (2.59)

From the Principle of least action: S E const for constant volume VH  const and (2.59) 2 follows the relation: SH 23c H G const defining stationary action in mega world. Planck equation t2 const can also be represented as an equation of stationary action in the micro world, if we assume the existence of a finite volume vPl  const for the quantum of action:

t vPl2 v Pl const . We may also assume that the quantum of action is defined in the same 3 volume as the elementary charge, that is in the sphere: vrPl 43 e with the classical electron 22 radius: ree e m c . Corresponding action density in the micro world will be: 3 SPl2v Pl  3 8 r e  const . Universality of the Principle of least action allows us to formulate the conditions of dynamic unity of the micro world and the mega world in the form of a condition of universal stationary action density:

2 3 10 -3 SH SPl KSe2 c H 3 G  3 8 r  1.768  10 erg s cm (2.60) This relation allows representing the Hubble constant as a simple function of fundamental constants:

2 3 4 6 3 18 -1 H9 G 16 c ree  (9 G c 16 e )  m  1.969  10 с (61.6 km/s/Mpc) (2.61)

This theoretical value conforms well to astrophysical estimations of this parameter from observations. George Lemaitre estimated the Hubble parameter in 1927 using only several observations of the nearby galaxies and got the value 627 km/s/Mpc. Edwin Hubble estimated this value as 500 km/s/Mpc in 1930s. In 1940s the value of 200 km/s/Mpc was more frequently used by astronomers.

Fig. 2.8. Estimations of the Hubble parameter during 1927–2010 (ordinates in km/s/Mpc; Huchra, J. https://www.cfa.harvard.edu/)

The constant growth of the reliability and precision of the astronomical observations has gradually diminished the systematic and statistical errors in estimations of the Hubble constant. In 1970–1990, the bulk of all published data dealing with determination of redshifts forced researchers to accept the Hubble parameter to be within the range 50–80 km/s/Mpc.

Fig. 2.9. Estimates of the Hubble constant (ordinates in km/s/Mpc), obtained by different methods during the past three decades: diamonds – estimates from the observations of supernovae of Sne Ia type; triangles – estimates from the Tully-Fisher correlation; oblique crosses are calculations based on the observations of globular star clusters; small filled circles are estimates from the study of surface brightness fluctuations and large filled circles mark all other methods (Huchra, J. https://www.cfa.harvard.edu/)

The Hubble parameter estimations of the HKP (Hubble Key Program) research program aimed at the study of Cepheids in galaxies up to 20 Mpc ( z  0.1) gave the values22: 68 6 km/s/Mpc (2000) and 72 8 km/s/Mpc (2001). The recent international research project studying supernovae of the SNe Ia type with high redshifts ( z 0.1 1) gave the value of the Hubble parameter: 65 7 km /s/Mpc [20]. And, finally, worldwide recognized experts in the cosmological constants estimations A. Sandage and G. Tammann recently published the following results for the Hubble constant: 63.2 1.3 km/s/Mpc (Tammann 200623) and 62.3 1.3 km/s/Mpc (Sandage 200624). Figure 2.9 visually demonstrates the convergence of the Hubble constant estimated by different astronomical observations to the theoretical value (2.41) – 61.6 km/s/Mpc.

Equation (2.44): E  const holds for any moment  , due to invariance of the equation of state

(2.47), including cosmological time scale (2.54):  THH 32 . This time scale together with (2.61) allows us defining a constant in (2.44) and to write it in the form:

2 3 6 6 3 9 -3 ET k kT2 c H 3 G  3 8  r e  (3 c 8  e )  m e  5.629  10 erg s cm (2.62) The Planck equation, defining the lower limit of the action 2 , is essentially a laconic statement of the Principle of least action in Micro World. Equation (2.62), which methodologically is very closely linked with the Planck equation, can be regarded as the law of the constancy of action density in the Mega World. Since 1980s the use of orbital and high performance ground base astronomical observations with high spectral resolution, and the improvement of computer technologies, have resulted in a set of new statistical methods for the “three-dimensional” analysis of the distribution of matter in the Universe. The transition from two-dimensional projection of large-scale structure of the Universe on the celestial sphere to the “three-dimensional” picture is performed with estimates of galaxy redshifts with subsequent calculation of their distance according to the Hubble law: r cz H – the “third” coordinate of the “three-dimensional” galaxy distribution. In the 1980s no more than a thousand redshifts of galaxies had been measured, more than a hundred thousand were available in the mid-1990s, and in our time, the number of known redshifts of the galaxies is about several millions. However, it should be kept in mind that the “three dimensional” picture of the galaxy distribution formed with the help of the Hubble law is not a true three-dimensional cross-section of 4-dimensional spacetime of the observable Universe at some moment of time. Hubble law allows us to estimate the remoteness of galaxies along the line of sight only shifted over the time to the past. For example, the distribution of galaxies at a distance of 300 Mpc is seen as what it was about a billion years ago due to the finite speed of light. Therefore, the “three dimensional” distribution of galaxies investigated with modern statistical methods is a set of spherical layers of the three-dimensional distribution of galaxies observed at different successive moments of the Universe history. The use of Hubble law in the statistical analysis of the galaxy distribution to calculate the “third” coordinate involves the use of the relation rc  . If we define the characteristic time in (2.62), 2 applying this relation and use the equality Em c , the (2.62) becomes:

1  1  1  1 -3 mT(2cH 3 G )  r  ( k c )  r  1.878  10  r g cm (2.63)

22 Freedman, W. et al // ApJ. (2001) 553; 47. 23 Tammann, G.A. The Ups and Downs of the Hubble Constant // Rev. Mod. Astron. (2006) 19;1 (arXiv:astroph/0512584v1 23 Dec 2005) 24 Sandage, A. et al. The Hubble Constant: A Summary of the Hubble Space Telescope Program for the Luminosity Calibration of Type Ia Supernovae by Means of Cepheids // ApJ. (2006) 653; 843-860. 73

For the fractals there can be defined with the help of the relation mr D the Hausdorff (3D ) dimension D, which is connected with the mass density by the relation: m  r . Comparing this formula with (2.63) we arrive to the conclusion that the relation (2.63) should generate the fractal distribution of matter in the Universe with the fractal dimension (Taganov 2008 [27–29, 32]): D  2 (2.64) Therefore, the statistical methods for studying the galaxy distributions using the Hubble law to determine one of the coordinates should detect the fractal dimension of large-scale structures in the Universe to be of order D  2 in accordance with (2.63). Indeed, in 1987 Luciano Petronero published his results of the statistical study of galaxy catalogs25, which confirmed the fractal dimension of the cosmic large-scale structure: D 2 0.2 . One of the advantages of fractal statistical methods for studying the large-scale structure of the Universe is the ability to estimate the average mass density for different scales, without determining the masses of all galaxies in the studied volumes. This enables a direct comparison of (2.63) with the observations. Figure 2.10 demonstrates the comparison of the formula (2.63) with the results of statistical estimations of the matter density in galaxy distributions. Different symbols on this figure correspond to the data from different catalogs of galaxies. As can be seen the theoretical formula (2.63; dashed line) sufficiently good fits the catalog data up to observation scales in the order 300 Mpc. Pietronero and his followers found that the value of the fractal dimension does not depend on the spatial scale of the statistical analysis. It turned out that the fractal dimension D 2 characterizes not only the galaxy distributions, but also is valid for the Fig. 2.10. The comparison of the theoretical distributions of galaxy clusters, the distribution of radio formula (1.88) (dashed line) with calculations based on several galaxies catalogs59. galaxies, X-ray sources and quasars and holds at least up to the scale of 1000 Mpc. So, the results of the current studies of the observed large-scale structure of the Universe are in satisfactory agreement with the relation (2.63), which is a consequence of (2.62).

From the very beginning of theoretical cosmology the comparison of theoretical estimates of redshifts and distances to cosmic objects with the data of astrophysical observations was an acknowledged method of testing new cosmological models. In cosmology, the most common way to compare models with observations is the analysis of the “Hubble diagrams” – dependencies of apparent magnitude mz() and “distance module” ()()z m z M (M is the absolute magnitude) on redshift. Modern CDM -model (“Lambda-CDM model”) of classical cosmology emerged in the late 1990s as a “concordance cosmology”, after a period of time when disparate observed properties of the Universe appeared mutually inconsistent, and there was no consensus on the structure of the energy density of the Universe [22, 35]. The model is frequently referred to as the “standard model

25 Pietronero L. The fractal structure of the Universe: correlations of galaxies and clusters and the average mass density // Physica A. (1987)144; 257. 74 of Big Bang cosmology”, since it is the simplest model that provides a reasonably good match to several astrophysical observations. The CDM -model, describing expansion of the Universe space, contains the empirical estimations of cosmological parameters:  0.7;M 0.3 and empirical estimate of the Hubble constant: H 70 km/s/Mpc ( 2.25 1018 s-1). The distance module (measuring distances in Mpc) predicted by the -model is:

z  5lg[(1 zr ) ]  25 r c H [  (1  )3   ] 1 2 d Mpc (2.65)  M  0 To calculate the distance moduli in cosmography, using the concept of decelerating physical time, the estimation does not require any empirical parameters. A theoretical estimate of the Hubble constant (2.61) and formula (2.46) define the following dependence of the distance modulus on redshift (Taganov 2008 [27–29, 32]):  5lgr 25 r c2 H  [(1  z )2  1] Mpc (2.66) The comparison of formulae (2.65, 2.66) with observations of supernovae SNe Ia, quasars and gamma-ray bursts in the redshift range z 0.17 6.6 is displayed at the Fig. 2.8 (details see in [32]).

Fig. 2.11. The comparison of theoretical equations with observations of SNe Ia supernova (red circles), quasars (green triangles) and gamma-ray bursts (diamonds). Curve 1 – (2.66), curve 2 – ΛCDM-model (2.65).

The “r.m.s. scatter” – the root mean square scatter is frequently used in astrophysics and astronomy as a quantitative measure of deviation of observational data from theoretical relations. Formula (2.66; curve 1) has the minimal r.m.s. scatter – 0.48 mag.; r.m.s. scatter for ΛCDM-model (2.65; curve 2) are somewhat bigger – 0.59 mag.

The equation (2.48) predicts a gradual increase of the Universe mass, that is, the existence of processes of synthesis of new matter. The average rate of synthesis of new matter can be estimated 3 47 -1 -3 by cosmological scales (2.54–2.56): QMTVHGHHHH 8 27  10 g s cm . This rate of mass growth means, for instance, that in the whole volume of the Earth during all its history could appear no more than 2 103 gram of hydrogen, not enough to fill a child’s balloon. Yet in the whole Universe this mass growth means the birth of new cosmic objects with the total mass of more than 105 solar masses, that is of the same order as masses of new globular star cluster or a dwarf galaxy, emerging every second.

The estimated characteristics of the Universe mass growth discussed above by no means suggest uniform matter synthesis across the Universe. It seems rather that high-energy processes of matter synthesis occur in relatively few centers, like quasars or active nuclei of massive galaxies. The Principle of least action, used in the cosmology with irreversible time allows determination of the interdependence of key cosmological characteristics of the Universe and the basic processes in the micro world represented by fundamental constants (Table 2.1; see details in [21–23]) Table 2.1. The key cosmological parameters Key cosmological parameters Observations Theoretical estimations (Taganov 2008 [27–29, 32]) Coupling constant of the K2 c2 H 3 G  3 8 r 3  1.768  10 10 micro world and the mega Se erg c cm-3 world (erg c cm-3) Hubble constant (s-1; (Tammann et al, 2006) 2 3 18 -1 H9 G 16 c re  1.970  10 s = 61.6 km/s/Mpc) H 63.2 1.3 km/s/Mpc (Sandage et al, 2006) H 62.3 1.3 -3 Average mass density (g cm )  (5  10)  1030 4H2 9  G  9 2 G 64  c 4 r 6  8.217  10 30 m me -3 g cm Energy density (erg cm-3) and 13 4 2 13 CMB 4.19 10 CMB m e (1  m e m p )  3.931  10 erg CMB temperature (К) -3 14 TKCMB 2.728 0.004 cm TCMB CMB  2.685 K Fractal dimension of the   r1 3 1  1  1 -3 m me3 8cr  r  1.878  10  r g cm cosmic large-scale structures D 2 0.2 D  2

In Table 2.1: gravitational constant G 6.674 108 cm3 g-1 s-2; Planck constant h 2  1.055  1027 erg s; the speed of light in vacuum c 2.998 1010 cm s-1; elementary electrical charge e ( e22.307 10  19 g cm3 s-2); classical 2 2 13 28 24 radius of electron ree e m c 2.818  10 cm; mass of electron me 9.109 10 g, mass of proton mp 1.673 10 g, Stefan-Boltzmann constant  7.566 1015 erg cm-3 K-4.

The theoretical formula for the Hubble constant (2.61) visually demonstrates the inner quantum nature of the Hubble law and relations (2.34), defining decelerating physical time. From (2.61), it follows that in the transition to classical physics when the value of the Planck constant tends to zero (  0), the Hubble constant also vanishes ( H  0), and thus disappear both – the redshift and cosmological deceleration of time pace. According to Plato, Pythagorean discovery of irrational numbers deeply impressed antique philosophers and they vowed not to use numbers in the quantitative research of Nature. Therefore, since the dawn of classic Greek philosophy throughout the Middle Ages and Renaissance, up to the end of 17th century, the main formal language of the young Natural Sciences was geometry, which was based on the rich heritage of antique Greek mathematics. The first attempts to return the numbers in the Natural Sciences were made in the early 14th century by philosophers of Merton College in Oxford, who later became known as “Oxford calculators”. Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbleton with their students replaced the geometrical qualities of Greek physics, at least for motions, by the numerical quantities. The Oxford Calculators first formulated the mean speed theorem: a body moving with constant velocity travels the same distance as an accelerated body in the same time if its velocity is half the final speed of the accelerated body. The ideas of the Oxford Calculators have been further developed by their younger contemporary – philosopher, Bishop of Lisieux, a counselor of King Charles V of France, Nicholas Oresme (c. 1320/1325–1382) who taught at the Sorbonne in Paris. In his Tractatus de configurationibus qualitatum et motuum Oresme considering a quality or accidental form, such as heat, distinguished

76 the intensio (the degree of heat at each point) and the extensio (as the length of the heated rod). These two terms were often replaced by latitudo and longitudo. Oresme conceived the idea of visualizing these concepts by plane figures, approaching to the idea of rectangular coordinates in modern terminology – a length proportional to the longitudo was the abscissa at a given point, and a perpendicular at that point, proportional to the latitudo, was the ordinate. Oresme deduced the equation of the straight line, and thus forestalled Descartes in the invention of analytical geometry.

Fig. 2.12. I. Nicholas Oresme did not differ in his works graphic representations of intervals of time and space, which subsequently led to the neglect in the Natural Sciences of the most important ontological characteristic of time – its irreversibility. II. The first graphic representation of the kinematics of motion in the tractate of Nicholas Oresme, where ordinates represent the time and abscissa – the instant speed.

Oresme applied his concept to the analysis of local motion where the latitudo or intensity represented the speed, the longitudo represented the time, and the area of the figure represented the distance travelled (Fig. 2.12-II). He showed that his method of figuring the latitude of forms is applicable to the movement of a point, on condition that the time is taken as longitude and the speed as latitude; quantity is, then, the space covered in a given time. In his treatises, Oresme not only reproduced the theorem on the kinematics of uniformly accelerated motion, but also considered a thought experiment with mechanical motion in the cabin of a coming ship illustrating the relativity of mechanical movement. These achievements of Nicholas Oresme became well known and were taught in many universities in Europe – at Oxford by William Heytesbury, then, at Paris and in Italy, shortly becoming a basis of the mechanical motion kinematics, which was developed later by Galileo, Descartes, Newton and Leibniz. Representation of time intervals in the form of straight-line segments in geometric diagrams, illustrating the kinematics of mechanical movement, became the first step by which the category of time lost its main ontological attribute – its irreversibility, which distinguishes time from space. Geometrization of time by Nicholas Oresme and his followers was a radical departure from the interpretation of time in Greek philosophy. Any movement, and any process involving the change of physical reality, always have some degree of irreversibility, and this irreversibility is the ontological foundation of the “physical” time in Greek philosophy. Aristotle believed that in time as in any continuous substance there must be some invariant – the “first measure”, the primary moment of time, the instant, which he called “now”: “From the above it is clear that in time there is something indivisible, what we call “now” [34]. The most important characteristics of this “now” is a potential indivisibility and at the same time, the potential continuity, so that this “now” can serve as a boundary between the actually nonexistent “past” and “future”. 77

Introduced by Aristotle minimal time-interval “now” was intended in the first place, being indivisible, to define the movement as unchanging feature of each dynamic process and, secondly, ensuring the continuity of the connection between past and future states of the process, to guarantee the irreversibility of the observed physical processes. The basic method of describing motion by Galileo became the use of the geometrical image of time as a straight line, introduced by Nicholas Oresme. Newton, following Galileo mixed space segments, velocities and geometrical time-intervals in his kinematic diagrams. He introduced the absolute “mathematical” time with the image of a continuous flow in a straight line (“fluxions”). Using time intervals as an equal to space segments in the adopted reference systems provided Newton with effectiveness and visibility of the geometrical description of the mechanical movement. Gradually, the geometric representation of time intervals in the form of straight-line segments has created the image of a reversible time as an infinitely extended straight line that can be followed and measured in both directions. The use of time as one of the coordinate in reference systems made time a specific geometrical parameter with invariable and uniform scale – an independent variable in the equations of motion. This process of geometrization of time was completed in the 20th century by giving to the time a physical dimension of space by the relation: l ct , where c (m/s) is the invariable speed of light, and by the use in the special theory of relativity a square of spacetime interval: ds2 c 2 dt 2 dl 2 with dimension of the square length. Mechanistic determinism of Galileo and Newton, which became the foundation of classical physics, was based on the reversibility of the geometric representation of time introduced by Nicholas Oresme, and assumed that each moving body will return to its original state, if you reverse the time, because the trajectory of a moving material body is uniquely determined by the initial conditions. Until the late 19th century, the use of Newton’s completely reversible absolute time in classical physics did not lead to serious contradictions with the observations. However, in the 20th century, when theoretical physics began to use the combined four-dimensional spacetime continuum, the situation changed – instead of the use of asymmetric four-dimensional continuum with irreversible time a symmetric spacetime not corresponding to the physical reality was introduced. In modern science, including theoretical physics, as in the early classical mechanics, unnatural reversible time of Newton, based on the model of the medieval concept of geometric time by Nicholas Oresme, is still used. This “original sin” of Natural Sciences has unintended consequences and creates a set of paradoxes and methodological problems for science. 2.4. Description of a motion with decelerating time Theoretical physics and, in particular, cosmology and astronomy assume without careful discussion the complete equivalence of absolute space r with an idealized Newtonian time t, and the “physical” space l and time  , the properties of which are investigated by analyzing our observations of nature. However, this may be questioned, in particular, because physics and astronomy use Newtonian time, which is reversible contrary to all the experience of Natural sciences. General mathematical model of the irreversible physical time  , which is characterized by the condition: (tt )   (0)   (  ) can be written in the form:

at() lim 0 a ( ) 1 aa()()   (2.67) These relations for the constant speed of light determine the metric non-uniformity of the physical spacetime. Using the relations: l c; r ct , after dividing both sides of (2.67) by the speed of light, we get:

l a()()  r  a l c  r liml0 a ( l c ) 1 (2.68)

As it follows from this relation, in metrically inhomogeneous spacetime with irreversible time a segment length depends not only on the coordinates of its beginning and end, but also on the size of the segment. Using the differential of (2.68): dl a() dr to form the interval: ds2 c 2 d 2 dl 2 for metrically inhomogeneous spacetime, we obtain the relation: ds2 c 2 d 2  a 2()  dr 2 (2.69) This definition of interval formally coincides with the radial interval used in cosmology for the spherically symmetric “flat” spacetime of “expanding” Universe (see e. g. [22, 35]): ds2 c 2 dt 2  a 2() t  dr 2 (2.70) As was already noticed in 1930s (see e. g. [22, 35]), non-stationary metric with the interval (2.70) does not provide the constancy of the speed of light – for a photon trajectory (ds  0) from (2.70) follows variable speed of light: dr d c a() const . The use of the time-dependent intervals (2.70) in the cosmological models contradicts to the relativistic physics of photons, which asserts the constancy of the speed of light in vacuum. All retrospective reasoning about the Universe’s past based on the use of cosmological models using the metric (2.70) (including “standard” General relativity model of modern theoretical cosmology) cannot be considered reliable because, in particular, the equations of the electromagnetic field and relativistic mechanics suggest universal constancy of the speed of light, which provides only the Minkowski metric with interval: ds2 c 2 dt 2 dr 2 (2.71) Analysis of the model of metrically inhomogeneous physical spacetime leads to interval (2.69) with physical time  , which is able to ensure the universal constancy of the speed of light. However, the interval (2.69) describes not the kinematic Universe expansion, but a kind of the “expansion without movement”, reflecting the peculiarities of irreversible time and metrically inhomogeneous space. In this “steady”, but inhomogeneous spacetime, there are no privileged points in space and the “initial” moment of time. As for the term “expansion of the Universe’s space”, it should be considered as a historical metaphor, born together with the first interpretation of the cosmological redshift as “apparent Doppler Effect”.

The equations for the derivatives of scale-factor a r R0 : da dt a и da d  a may be obtained from (2.69) with the condition of the universal constancy of the speed of light on the trajectories of photons ( ds  0 in 2.71): dr dt adr d  c  const . Hence:

R00 da dt aR da d  c  const or da dt ada d and further minded that: da dt da d  d  dt  da d   a we receive: d dt a (2.72) a a a (2.73) In kinematics with irreversible time the following convenient deceleration parameters can be 22 used: qt   aa a; q   aa a  . Positive values of deceleration parameter correspond to decelerating motions and negative ones to accelerating motions. Using deceleration parameters we can proof the following Kinematic theorem of the Spiral time concept: for deceleration parameters qqt ;  with reversible (t) and irreversible ( ) time the following relation holds (Taganov 2005 [25, 27–29, 32]):

qqt  1 (2.74)

The derivation of (2.74) uses the following transformations: from (2.72): d dt a it follows: a da dt  da d  d dt  a  a (I) and ddtdad() dad22   d  dt  aa  (II). Therefore, 22 2 3 2 2 2 2 a a a aa  and hereinafter: qt   aaa  ( aaaa   ) ( aa  )  q  1.

From (2.74) follows that the moderate ( q 1) deceleration of motion in irreversible physical time appears an accelerating motion ( qt  0 ) in the reversible Newtonian time. The motion with constant speed in the Newtonian time ( qt  0 ) appears the decelerating motion in physical time

( q 1). These effects are similar to changes in the kinematics during transitions between inertial and non-inertial frames of reference.

For the scale-factor: a r R00; R const are valid: arRa0;;;  rRarRa  0  0   rR  0 and one can derive from (2.74) the following relation:

2 2 2 3 r  R0 () r r r r (2.75) If we use for the interpretation of (2.75) the second law of Newton: mr  F , it turns out that in spacetime with physical time per unit mass of the moving body acts a virtual force. For example, from (2.75) follows that the inertial motion with constant velocity r const;0 r in Newtonian time will looks as decelerating motion in physical time, caused by virtual braking force: 2 2 3 F  r   R0 r r . This phenomenon was already discovered in 1930s by Willem de Sitter and Edward Milne for several types of inhomogeneous scales for time26. In the derivation of formulae describing a motion in the asymmetric spacetime with decelerating physical time, one can use the Principle of least action, however, for the sake of visibility, we shall use the relations (2.34) defining the cosmological deceleration of physical time: d dt a 1  Ht  (1  2 H )12 for t 0: 0  t Ht 2 2 (2.76)

Let us start with a definition of velocities: ut  dr dt and u  dr d , for which with the help of (2.76) we obtain the following formulae:

12 ut  drdddtuau     (1  Ht )  u  (1  2 H  ) (2.77)

12 u  drdtdtd   uaut  t(1  Ht )  u t (1  2 H ) (2.78) Hence, in the asymmetric spacetime the values of Newtonian and physical velocities of bodies vary so that at any moment the following relation holds:

12 ut u  a 1  Ht  (1  2 H ) (2.79)

22 Equations (2.77, 2.78) allow determining of the kinetic energies: Ett mu2; E mu 2 , and then the equations of motion in an asymmetric spacetime can be determined using corresponding

Lagrangians, which for conservative systems have the forms: LEUtt;;. Of particular interest is the relation for acceleration:

u du  d (2.80)

For constant initial velocity utt u0 const the equation (2.80) leads to the following formula:

26 De Sitter, W. Astronomical Aspect of the Theory of Relativity. University of California Pub. in Mathematics (Berkeley: Univ. Calif. Press), 1933, 2, No. 8. Milne, E.A. World-Structure and the Expansion of the Universe // Zeitschrift für Astrophysik (1933) 6, 1–95. 80

32 u   ut0  H(1  2 H ) (2.81) From (2.81) follows that the uniform motion of a test body with constant initial velocity utt u0 const without the action of any external forces will start to slow down with a negative acceleration u .

If we use for the interpretation of (2.81) the second law of Newton: mu  F , it turns out that in an asymmetric spacetime per unit mass of the moving test body acts a braking force caused by the cosmological deceleration of physical time:

32 f F m  u    ut0  H(1  2 H ) (2.82) For small, comparing to the cosmological scales, observation time it can be taken: 1 2H 1 and then the relation (2.82) simplifies:

f  ut0 H (2.83)

The force f in (2.82, 2.83) is similar to the viscous resistance force when a body moves in a continuous viscous medium – it is proportional to the velocity of a body and has direction opposite to the direction of the body’s motion. Hubble constant plays the role of the “cosmological viscosity” of the asymmetric spacetime. If we consider instead of the motion of a test body the movement of a photon with uct0  then (2.83) takes the form (for the theoretical value of H (2.61):

8 -2 f u    cH  5.9  10 cm s (2.84) As was mentioned, the spontaneous accelerations of bodies and corresponding virtual forces in spacetime with heterogeneous nonlinear time scale were already considered by several scientists, for example, (De Sitter 1933) and (Milne 1935)26. In asymmetrical spacetime the speed of light is constant only in the decelerating physical time: c  c const , and the equation of a photon trajectory has the form: rc  . If the motion of a photon is described using Newtonian time, the speed of light according to (2.77) is variable:

ct  drdt  ddtcr ( )  cd  dt  c  (1  Ht ) (2.85) Let us consider the speed of the apparent space “expansion”, reflecting the specific features of metrically inhomogeneous spacetime. According to the Hubble law for Newtonian time to determine this speed it is necessary to use the relation: Vt  c z cHt . The generalization of this relation can be obtained by differentiating it with respect to time:

dut dt cH (2.86) If we interpret the equation (2.86) with the use of Newton’s second law for Newtonian time, we can say that the velocity of a test body in the metrically inhomogeneous spacetime changes as if on the unit mass of a body acts an additional force. Integrating of (2.86) with the initial condition: t0: utt u 0 gives the equation:

utt u0 cHt (2.87) This equation determines the influence of the inhomogeneous spacetime metric on the velocity of a test body, which appears in the form of superposition of apparent rate of the space “expansion” and independent of this “expansion” initial velocity ut0 of a body. Observational data discussed in this book suggest that all physical processes in a non-stationary Universe evolve in the cosmologically decelerating irreversible time, and that just this time should be used in mathematical models of the large-scale structure of the Universe. This assumption raises a 81 question: how can one be certain of the physical time advantages in astrophysics if there are no methods of direct estimation of the time intervals between astrophysical events? Mathematical models in physics and cosmology can be formulated as the general relations:

F( t ; xi ,...) 0 ( ;xi ,...) 0 (2.88)

Here xi stands for the observable parameters. To exclude time from these mathematical models, one can use integration or relations between time and some depending on time characteristics yt() and y() of physical processes. Using the reverse functions t f() y and  ()y one can represent (2.88) in the form:

F[ f ( y ); y , xi ,...] 0 [ (y ); y , xi ,...] 0 (2.89)

For example, in cosmology the luminosity distance rL can be estimated from apparent magnitudes and the redshifts with an assumption of the speed of light constancy and then: t rL ( z ;...) c and   rL ( z ;...) c [22, 35]. Using such relations, we can obtain from (2.88):

xii f( z ;...) xzii ( ;...) (2.90) Comparing these relations with observational data, we can judge the correctness and usefulness of the concept of irreversible physical time with nonlinear scale. 2.5. Cosmological deceleration of time on the Earth and in solar system The possibility of astrophysical observation of local un-static state of the Universe within the boundaries of our galaxy using cosmological redshift measurements is limited by the accuracy of spectral methods and by Doppler Effect, due to significant peculiar velocities of the Earth and stars in our galaxy. For typical relative star velocities of 100–300 km/s in our galaxy the Doppler shift is (3 10) 104 . Such Doppler shift constrains the cosmological redshift measurements from our galaxy to distances of order 2–5 Mpc where cosmological and Doppler redshifts have one order. A characteristic feature of Doppler distortion of the cosmological redshift is anisotropy of Hubble constant values estimated from observation. For the galaxies in our super-cluster with the center in Virgo constellation the deviation of the Hubble constant estimates from the mean value for some directions is 150 % and more. Anisotropy of the Hubble constant is only getting much less for galaxies in nearby super-clusters in the Leo and Hercules constellations (140 and 190 Mpc respectively). In contrast to cosmological redshift the evidences of non-static state of spacetime in the phenomenon of cosmological deceleration of time do not depend on the movement of cosmic objects. Therefore, the deviation of spacetime from static state in terms of cosmological deceleration of time can be found in the Solar system, in the Earth-Moon system and even on the Earth itself (Taganov 2003 [24–26, 30, 32]). The physical nature of observed manifestations of the cosmological deceleration of time is rooted in the fact that all uniform motions in decelerating physical time analyzed with the use of invariable uniform Newtonian time scale look as accelerating under influence of virtual forces. The apparent secular acceleration of the Earth’s rotation In our papers and books (see e. g. [24–26, 30, 32]) we have already considered the influence of local non-static state of spacetime on the kinematics of the Sun-Earth-Moon system where the most expressive phenomenon of cosmological time deceleration is manifested in the apparent secular acceleration of the Earth’s rotation. A study of the acceleration of the Moon conducted by astronomers made it possible to measure the angular velocity of Earth’s rotation after subtraction of its tidal deceleration by the Moon and

Sun. To the surprise of geophysicists it turned out that after eliminating of all known tidal influences, the angular velocity of the Earth reveals the secular growth with the relative increase   of about 10-7 over the last 2200 years, and from 1700 to 1950 approximately 4 10-8 [14, 17, 23]. In addition, discrepancies between historical (so-called “antique”) and the current data in the estimates of the both acceleration of the Moon and of the Earth’s rotation were discovered. For over three centuries, from the historical articles by Edmond Halley, the antique and medieval eclipse observations have been the only source of information about the long-term secular trends in the kinematics of the Earth-Moon system. From ancient times the Earth’s rotation and the Moon orbital motion were used as standards for the time units. Up to the end of the 18th century the observed time interval between two successive crosses of the local meridian by the Sun, the so-called “Sun day” had remained as the basic unit of time calculation. However, due to the ellipticity of the Earth’s orbit, its inclination to the plane of ecliptic and the deviation of the Earth’s shape from the ideal sphere the actual length of the solar day may differ from the average 24-hour duration in the range of about 30 seconds. The difference between the Ephemeris (ET) and Universal solar (UT) times t is the measure of the time interval determination error (“ephemeris correction”), which originates from irregularities of the Earth’s orbital motion and from fluctuations of the Earth’s rotation. This quantity, the so-called “solar acceleration” in (2.100) is displayed by the equation:

t tET t UT (2.91) In 1994, in one of the Buddhist monasteries near the Lake Baikal was found a fragment of ancient astronomical chronicles of solar eclipses in 2–11 centuries, composed in the medieval Buddhist university of Nalanda in India. It turned out that these very precise and detailed descriptions of historical eclipses systematically differ from modern calculations. Theoretical 2 estimation of ephemeris correction calculated for this astronomical chronicle was: tT41 c with

t measured in seconds and Tc in Julian centuries (36525 Julian days). This theoretical estimation 2 systematically overestimated the values of t calculated from ancient observations: tT31 c . After discussing these strange discrepancies growing for older observations with astronomers in the University of Oxford and with Professor Fred Hoyle, author in 1995 suggested that these differences are evidence of a new physical phenomenon – the cosmological deceleration of time. Analysis of the modern metrological doctrine and the redshift phenomenon with the use of photon quantum kinematics allowed him to derive the formulae (2.34) for physical time, which is actually registered by our watches: In 1997, F.R. Stephenson and W. Dalmau confirmed the effectiveness of the physical time conception and formulas (2.34) in the interpretation of ancient astronomical observations. Stephenson investigated a vast amount of data on the observations of more than 200 sun eclipses in the period preceding the invention of telescope – eclipses observed by astronomers of ancient Babylon, ancient Greece, ancient and medieval China, as well as by the medieval Arab world observatories along with the information about the observations of eclipses in medieval Europe27 [23]. In his analysis, Stephenson adopted the value of the permanent acceleration of the Moon to be: 26 arcsec/cy2 (as/cy2) and calculated the estimations of t considered as the measure of the Earth’s rotational velocity variations. It turned out that the average rate of the LOD growth is about 1.7 ms/cy (milliseconds per century), which is significantly less than the LOD’s rise due to the solar and lunar tides, which is about 2.3 ms/cy. Proposing that there were no significant changes in global sea level during the period under study, it can be assumed that the acceleration of the Moon and the energy dissipated by tidal friction

27 Stephenson, F.R. Historical Eclipses and Earth’s Rotation. Cambridge University Press, 1997. 83 remained constant. Theoretical estimation of the increase of the t value due to the tidal friction with constant lunar acceleration: 26 as/cy2 (dashed curve 1 at the Fig. 2.13) corresponds to the equation [23]:

2 tT42 c (2.92)

Here t is measured in seconds and Tc is in Julian Centuries (36525 Julian Days) up to 1820 in accordance with the ET-system definition. It can be seen that the equation (2.92) (dashed curve 1 in Fig. 2.13) systematically overestimates the values of t calculated from observations. In order to obtain an equation fitting the entire set of calculated estimation markers shown in Fig. 2.13, it should be assumed that in addition to the tidal action of the Moon and the Sun slowing the Earth’s rotations there is some unknown factor that compensates the observed discrepancy in LOD growth with the rate approximately 0.6 ms/cy.

Fig. 2.13. Estimations of t (s.; ordinates) – time determination errors due to Earth’s rotation irregularities calculated from eclipse observations from 700 BCE until 1500 CE. Circles – ancient Babylon observations, filled circles – ancient observations in Greece, crosses – observations of ancient and medieval Chinese astronomers, filled squares – observations of medieval Arabian observatories [23]. The curves 1 and 2 correspond to (2.92) and (2.94).

The cosmological deceleration of time visually explains the discrepancy between the estimation (2.92) and calculations from observations displayed at the Fig. 2.13. In order to account for the cosmological deceleration of time one ought to use the slowing physical time instead of the idealized 2 uniform Newtonian time tUT in accordance with (2.34): UTt UT Ht UT 2. It is possible to obtain the following relation from (2.91, 2.92) under this condition: ttTT UT and then from (2.92):

22 t tTT  UT 42  T c  Ht UT 2 (2.93)

If we take, for example, the estimation of H by (2.61) then we shall have for Tc 1 century 9 2 18 9 2 (tUT 3.156 10 s): HtUT 2 1.97  10  (3.156  10 ) / 2  9.8 and then:

22 t(42  9.8)  Tcc  32.2  T (2.94)

In this equation as in (2.92) t is measured in seconds and Tc in Julian centuries. The solid line 2 at the Fig. 2.13 represents the equation (2.94) and fits well with the set of markers corresponding to the calculations of t made from the historical eclipse observations. It is interesting to mention that W. Dalmau28 after analysis of several Arabic medieval 2 observations of eclipses derived an empirical formula: tT31.5 c , which is very close to (2.94). As was already noticed the observed average decrease of the LOD value after subtraction of tide influences is approximately: 1.7 2.3   0.6 ms/cy. For the 86400 seconds per day and 36525 days in Julian century the average rate of LOD decrease can be evaluated as: d t dt [(1.7  2.3)  1039  36525] (86400  36525)   6.944  10 (2.95) The apparent Earth’s acceleration and corresponding LOD decrease are the manifestations of the cosmological deceleration of physical time (2.34) and: t  t    H2  t 2 . This relation and Hubble constant value (2.61) allow for the theoretical estimation of the LOD decrease rate: d t dt Ht 1.97 1018  86400  36525  6.217  10 9 (2.96)

Fig. 2.14. Fluctuations of the LOD corrected for the tidal influence. The solid line clearly demonstrates the tendency 7 of the LOD decrease: LOD8.64  10  (0.622 Tc ) (ms), according to the theoretical equation (2.96). (From the data of IERC (International Earth Rotation Center Bulletin C 36.) https://oceanography.navy.mil and tycho.usno.navy.mil)

Comparing two estimations (2.95, 2.96) one sees that theoretical estimate (2.96) differs from observational one (2.95) less than in 10 %, which is significantly less than the mean uncertainty of the observational data illustrated at the Fig. 2.14. From the observational estimation (2.95) and theoretical relation (2.96), it is possible to find the estimate of Hubble constant from the observations of day length growth: H 6.944  109 (86400  36525)  2.2  10 18 s-1(68.8 km/s/Mpc) (2.97) Cosmological corrections for the observed accelerations of the Earth and Moon Secular acceleration of the Moon was predicted in 1695 by Edmond Halley, who compared his own observations of the Moon with ancient astronomical data. In his article in Phylosophical Transactions for 1695 Halley declared:

“And I could then pronounce in what Proportion the Moon’s Motion does Accelerate; which that it does, I think I can demonstrate, and shall (God willing) one day, make it appear to the Publick”.

28 Dalmau, W. Critical remarks on the use of medieval eclipse records for the determination of long term changes in the earth’s rotation // Surveys in Geophysics, 1997, 18: 213-223. 85

This brief announcement was Halley’s only published words on the Moon’s acceleration and he never returned to the topic, despite his later extensive investigations of lunar motion. Halley’s sensational announcement of the acceleration of the Moon elicited almost no response from the astronomical community. The first explicit reference to Halley’s hypothesis appeared in the second edition of Newton’s Principia. It should be borne in mind that historically, the term “acceleration” has different meanings in planetary science and physics. The coordinate of the uniformly accelerated body in physics is given 2 by the equation: L u0  t  ut 2  t where u0 is the initial velocity of the body, and ut is “kinematic” 2 acceleration. In astronomy the equation of the type L L0  n  T  w  T is frequently used for describing the motion of the planets. In such equations w is also called the “acceleration” – astronomical one. Comparing the above equations, it is easily seen that the “astronomical” acceleration is just the half of the “kinematic” one. So, for example, if according to the present data the Moon’s “kinematic acceleration” is about 26 as/cy2, then the corresponding “secular astronomical acceleration” of the Moon is 13 as/cy2. The estimations of the Moon’s acceleration made by several astronomers in the 18th century confirmed the guess of Halley: Meyer estimated the Moon acceleration as 9 as/cy2 and Damuaze calculated secular increment of the Moon’s longitude as 10 as/cy2. These are the “astronomical” accelerations, not “kinematic” ones, which are two times larger. In 1754, Immanuel Kant proposed the hypothesis that the gradual slowing down of the Earth’s rotation and the acceleration of the Moon are due to the tides on the Earth. However, Laplace categorically rejected the idea of Kant, since astronomers had not discovered any secular accelerations of the Earth or any other planet by that time. In 1770, the Paris Academy of Sciences announced a competition for the best explanation of the phenomenon of accelerated motion of the Moon. Neither Euler nor Lagrange found an explanation to the acceleration of the Moon and recognition came to the theory of Laplace, which he proposed in 1787. Laplace suggested that the gravitational perturbations from the planets of the Solar System led to a gradual decrease in the eccentricity of Earth’s orbit, which, in its turn, affects the perturbation of the lunar orbit by the Sun, causing the gradual increase of the orbital velocity of the Moon. Laplace demonstrated impressive accuracy of his theory, estimating the acceleration of the Moon as 10.18 as/cy2 .This is almost exactly the same as the observational data in his days. Soon several astronomers calculated their own theoretical estimations of the acceleration of the Moon using gravitational concept of Laplace – Damuaze: 10.7, Plana: 10.6, Pontekulan: 12.24, Hansen: 12.18 (as/cy2). The gravitational theory of Laplace was considered to be the exhaustive explanation for the acceleration of the Moon until 1853. In that year, John Adams, known for his independent from Le Verrier prediction of the existence of Neptune, found errors in the Laplace’s and his followers’ calculations. He corrected the errors of Laplace and found the gravitational acceleration of the Moon to be only 5.7 as/cy2. Adams demonstrated that Laplace’s gravitational theory can explain only part of the observed acceleration of the Moon. Charles Delaunay corrected the errors of Laplace found by Adams and got the theoretical estimate of the gravitational acceleration of the Moon to be 6.18 as/cy2. After a rather lengthy discussion, Plana and Hansen also accepted Adams’s result and agreed that the Laplace gravitational theory can explain only 6.1 as/cy2 of the total observed acceleration of the Moon. This acceleration 2 is sometimes called Laplace-Adams acceleration: nG 6.1 as/cy and it is still used as a theoretical estimate of the acceleration of the Moon related to gravitational perturbations in the Solar system. Adams achievement was not only his improvement of Laplace calculations. He also convincingly demonstrated that the gravitational concept in principle cannot explain the total acceleration of the Moon. John Adams suggested that the acceleration of the Moon is determined not only by gravity, 86 but also by something else and he prophetically said: “This may show the way to important discoveries in physics”. In 1870s, the old idea of Kant about the Earth tides was revived in order to develop a precise theory of the Moon’s motion. In 1865, Charles Delaunay explained the “missing arcseconds” in the angular acceleration of the Moon by tidal effects in the Earth-Moon system. The gravitational forces of attraction created by Sun and Moon form a “tidal wave”, which leads to a local raise of sea levels, and which in the Earth’s crust creates a moving field of stresses. The tidal waves follow slightly ahead the Moon because of relatively fast rotation of the Earth. Tidal waves slow down the Earth’s rotation due to the friction. At the same time, this slowing down of the Earth’s rotation has an effect on the Moon’s motion gradually increasing the axis of Moon’s orbit. Thus, tidal waves slow down the Earth’s rotation and decrease the angular velocity of Moon’s motion. On the one hand, the tidal waves lead to a true slowdown of the Moon’s motion and on the other hand, slowing the Earth’s rotation leads to an apparent acceleration of the Moon’s motion observed from the Earth. If the effect of slowing down of the Earth’s rotation by tidal waves exceeds the true deceleration of the Moon, we will observe an apparent acceleration of the Moon in its orbit. The motion of the Moon along its orbit is occurring in the direction of the Earth’s rotation, and if one takes the positive direction of the Moon’s acceleration to be in the direction of the Moon’s 2 motion then the Laplace-Adams acceleration nG 6.1 as/cy is positive and the acceleration due to tide influences is negative. The gravitational concept of Laplace and tidal mechanism of Kant became the basis of the Moon motion theory (the Hill-Brown theory), developed in 1878–1909 (see, e. g., a modern presentation of this theory in [6]). In the modern version of this theory, the value 6.1 as/cy2 of the total Moon’s acceleration is explained by the gravitational concept of Laplace, and the main part of the rest by the Kant-Delaunay tidal mechanism. Due to the imperfections of the geophysical theory of the Earth it is still not possible to estimate theoretically, firstly, the value of the tidal moment and rate of tidal energy dissipation, and secondly, fluctuations of the Earth’s rotation velocity. Therefore, the Hill- Brown theory is of a semi-empirical nature. According to the modern conceptions the observed acceleration of the Moon is defined by two groups of phenomena: 1. The sum of the gravitational perturbations from the bodies of the Solar system, which form the 2 Laplace-Adams acceleration nG 6.1 as/cy and tidal interactions in the Solar-Earth-Moon system, which is still cannot be calculated precisely. All these perturbations slow down the Moon’s motion producing its actual negative acceleration and the observed growth of its orbit axis. 2. The apparent positive acceleration of the Moon originating from the tidal friction slowing down the Earth’s rotation under the condition of the different velocities of the Moon’s motion and Earth’s rotation. In astronomy, it is common to use equations for the planetary orbital longitude (in arcseconds) of the form:

L L0 nTc (2.98)

In this equation Tc is the Newtonian (ephemeris) time in Julian centuries (36 525 days) from the “fundamental epoch” – the midday on January 0, 1900 (here and below index “c” marks time intervals expressed in Julian centuries); L0 – planet longitude at the initial moment; n – is the average planet motion over century (in arcseconds) corresponding to the laws of Kepler and Newton. The theory of the tidal acceleration of the Moon is based on the Kepler and Newton laws and the conservation laws of the energy and rotational momentum in the Earth-Moon system. For the angular acceleration of the Moon nM and acceleration of the Earth’s rotation E the following equations are used (see, e. g. [6]):

2 nMM3 N mb EMS ()NNI  (2.99)

Here m is the mass of the Moon, b – average semi-axe of the Moon’s orbit, NM and NS – momenta of tidal friction forces acting on Earth correspondingly from Moon and Sun, I MR2 3 Earth’s moment of inertia. The terms “motion of the Sun” and “acceleration of the Sun” are frequently used in the theory of Moon’s motion, since the works by Sir Spencer John for the description of the observed solar relocation on the celestial sphere, which is the reflection of the Earth’s orbital and rotational motions.

The accelerations of the Moon nM and Sun nS observed from the Earth are determined by the relations:

nMMME  n  n E nSSSEESE  n  n   n  E (2.100)

Here nnMS13.37 is the ratio of the average motions of the Moon and the Sun, which is the number of sidereal lunar months in the year and EMn  27.32 is the ratio of the mean Earth’s angular velocity to the mean motion of the Moon, which is the number of days in the sidereal lunar month. In the first part of the 20th century, several astronomers undertook a tremendous amount of work aimed at the determination of the secular lunar acceleration variations. The special feature of these calculations was the use of two groups of astronomical data – the so called “antique” data, approximately since the first millennium BCE to the first centuries CE and “modern” observations systemized approximately over the period 1700–1950. These two epochs of “antique” and “modern” observations are divided by the period of around 2 000 years. In 1920, John Fotheringham published the first calculations of the solar and lunar tidal accelerations using the “antique” astronomical data (Table 2.1). In 1927, Willem De Sitter improved these calculations attracting additional “antique” data (Table 2.2).

Table 2.2. The values of astronomical accelerations in the Earth-Moon system determined by the “antique” and “modern” observations

[2-1] [2-2] [2-3] 2 Antique observations 4.7 5.22 0.30 Moon’s acceleration nM as/ cy Modern observations 11.22 0.5 2 Antique observations 1.5 1.8 0.16 Acceleration of Sun nS as/cy Modern observations 1.23 0.04

References

2-1. Fotheringham, J. Secular accelerations of sun and moon as determined from ancient lunar and solar eclipses, occultation and equinox observations // Mon. Not. Roy. Astron. Soc. (1920) 80, 578. 2-2. De Sitter, W. On the secular accelerations and the fluctuations of the longitudes of the moon, the sun, Mercury and Venus // Bull. Astron. Inst. Netherlands (1927) 4, 21. 2-3. Jones, Sir H. Spencer. The rotation of the Earth and the secular acceleration of the sun, moon and planets // Mon. Not. Roy. Astron. (1939) 99, 541.

As Table 2.2 demonstrates, the difference between the “modern” estimation of the “solar acceleration” and that calculated from antique observations is: 2 nS  n Sa  n Sc  (0.47  0.23) as/cy (2.101) Here and below the indexes “а” and “с” mark correspondingly parameters calculated with historical (“antique”) data and “modern” ones. 88

The difference between the values of the Moon’s acceleration obtained with the “antique” and “modern” data is (Table 2.2): 2 nM  n Ma  n Mc  (6.2  0.7) as/cy (2.102) These specific features of the Earth-Moon system kinematics can be explained if one takes into account the local non stationary character of the spacetime metric (Taganov 2003 [26, 28, 30, 32]). The difference between the Newtonian and decelerating physical time is given by the relation (2.34) with the theoretical value of the Hubble constant (2.61). It is handy to use the Hubble constant with the dimension century-1 (cy-1) for the astronomical calculations, multiplying the value of the Hubble 18 9 9 constant (2.61) by the number of seconds in Julian century: Hc 1.97  10  3.156  10  6.22  10 cy-1. For relatively short periods, for example, for astronomical calculations over the past 250 years, cosmological corrections are smaller than the average observational errors. However, the calculations with “antique” data collected 20 centuries earlier, require the corrections that exceed the observational errors. Therefore, for example, to estimate the “acceleration of Sun” with “antique” data the reduced intervals of physical time in the past should be taken into account. Cosmological correction is determined by subtracting the equation of the orbital motion of the planet with Newtonian time (2.98) from the same equation written with decelerating physical time: 2 L  L0  nTc  n  H c2  T c , which leads to the following relation for the cosmological correction of the planet’s astronomical acceleration:

n  n Hc 2 (2.103) Correction (2.103) for the “solar acceleration”, which is actually the apparent acceleration of the Earth in orbit, has the form:

89 2 nS 1.296  10  (6.22  10 2)  0.4 as/cy (2.104) This value agrees well with (2.101) and corresponding estimation of the Hubble constant from the observed “solar acceleration” (2.101) will be:

9 -1 Hce2  n S n S  7.25  10 cy (71.8 km/s/Mpc) (2.105) Correction (2.103) for the acceleration of the Moon is given by:

99 2 nM 1.74  10  (6.22  10 2)  5.41 as/cy (2.106) The Hubble constant estimation from the Moon’s correction (2.103) and (2.102) will be:

9 -1 Hce2  n M n M  7.13  10 cy (70.6 km/s/Mpc) (2.107) In spite of the about 15% difference of the corrections (2.104, 2.106) and estimations from observations, the coincidence of the presented theory with observations can be considered as satisfactory, especially taking into account that standard deviations of astronomical data about accelerations in the Earth-Moon system usually are not less than 20%. The determination of the reliable estimation of the Moon’s acceleration is still a very difficult task due to significant oscillations of the Earth’s angular velocity (Fig. 2.14). Recent estimations of the Moon’s acceleration obtained by different methods diverge significantly – for example, Robert 2 Newton in 1968 arrived to the value nM  20  3 as/cy on the basis of the analysis of the perturbations of the satellites orbits due to lunar tides. However, Van Flandern in 1970 got 2 nM  52  16 as/cy from the analysis of lunar observations over the period 1955–1969.

It is believed that since 1970s after the installation of laser reflectors on the lunar surface a more reliable method for the quantitative analysis of the components of lunar acceleration appeared. Now 2 NASA recommends the value nM 25.858 as/cy obtained during the laser location sessions, which corresponds to the Moon’s recession from Earth at the average speed of 3.8 cm/year. However, one should keep in mind the possibility of both larger and lower values in near years due to the inevitable variations of the Earth’s rotational velocity. Many astronomers believe that for the reliable estimation of the average lunar acceleration the analysis of lunar observation over the period of at least 2–3 centuries is required. In 1939 Sir Spencer 2 Jones calculated this value to be nM  22.44  1 as/cy using the observations over the period 1677– 1935 and obtained by several methods the secular acceleration of Sun, which he estimated to be 2 nS 1.23 0.04 as/cy (see the Table 2.2). This estimation, which takes into account not only solar observation but also the observation of Mercury, is the most frequently used in modern astronomy. Accelerations of the Mercury and Venus Of course, not only astronomical observations of the Earth and the Moon motions require corrections for the effect of the cosmological deceleration of physical time, and the introduction of corresponding terms to the equations of the planetary motion. The motion of all heavenly bodies should be analyzed taking into account the cosmological deceleration of physical time, which determines the actual moments of their observations. In the early 20th century, the attention of astronomers was attracted to the short-time fluctuations of the longitude of the Moon discovered by Simon Newcomb in 1875. Newcomb published a detailed study of the fluctuations of the Moon’s longitude showing the existence of its periodic oscillations with significant amplitudes and many short-term components with relatively small amplitudes. In 1926, Ernest Brown summarized his studies of the lunar longitude fluctuations and showed that their nature is connected mainly with the short-term variations of the Moon’s velocity and not with the smooth periodic variations. In 1927, Willem De Sitter confirmed Brown’s observations and clearly demonstrated the existence of velocity fluctuations for the Mercury and Venus29. Sir Spencer Johns concluded that the discrepancies between the planets longitudes and their theoretical values have the form of small fluctuations of their accelerations30. To account for the longitude fluctuations the additional term  is added to the theoretical equation (2.98): L L0 nTc transforming it to the equation for the “observable longitude”:

Ltc L0  nT  (2.108) A precise theory of the planetary motions needs to take into account the local non-stationary nature of the spacetime metric (Taganov 2013 [30, 32]). The difference between the time scales of the Newtonian ephemeris time and the decelerating physical time is given by the relation (2.34) with theoretical value of the Hubble constant (2.61). Inserting the correction for the cosmological time 2 deceleration: (THTc c 2 c ) into the equation (2.108) by changing the Newtonian ephemeris time

Tc with the physical time according to (2.34) we get:

2 L  L0  nTc  n  H c2  T c  (2.109) The third term in this equation is equivalent to the small additional acceleration of the planet.

29 De Sitter, W. On the secular accelerations and the fluctuations of the longitudes of the moon, the sun, Mercury and Venus // Bull. Astron. Inst. Netherlands. (1927) 4; 21. 30 Jones, Sir H. Spencer. The rotation of the Earth and the secular acceleration of the sun, moon and planets // Mon. Not. Roy. Astron. (1939) 99, 541. 90

The value of the planet longitude fluctuation is calculated in astronomy with help of the longitude correction dL , which is equal to the difference between the theoretical value (2.98) and observed longitude. To account for the cosmological deceleration of time the theoretical value (2.98) must be compared with the (2.109):

22 dL Lt  L   n  H c2  T c   n  T c (2.110) The analysis of the modern observations of the Mercury and Venus coordinates clearly demonstrates their deviation from the theoretical values according to (2.98) (see e. g. Seidelman et al. (1985, 1986)31,Yao & Smith (1988, 1993)32, Standish & Williams (1990)33, Kolesnik (1995, 1996)34). The detailed studies of the trends in the motion fluctuations for the Mercury and Venus indicate some accelerating factor, which manifests itself in the longitude corrections for both planets 2 to be of the form dL nTc . This form of the correction corresponds well to the third term in the equation (2.109).

Average motions of the Venus nV and Mercury nM (in arcseconds per century), according to the 8 8 laws of Kepler and Newton, differ: nV 2.1 10 and nM 5.4 10 with corresponding cosmological 9 -1 corrections in (2.109, 2.110) with Hc 6.217 10 cy :

89 2 nV n V  H c 2  2.1  10  6.217  10 2  0.66 as/cy (2.111)

89 2 n M n M  H c 2  5.4  10  6.217  10 2  1.68 as/cy (2.112) Figure 2.15 displays the corrections to the theoretical values of Mercury and Venus longitudes 35 dLM and dLV for the observational period 1900–2000 (Kolesnik & Masreliez 2004 ) and theoretical 2 estimations of trends dL    n   Tc according to the equations (2.110) with cosmological corrections (2.111, 2.112).

For the reliable statistical estimations of nV and n M the total observational period 1900–2000 (Fig. 2.15) was divided into N  39 intervals of 2.5 years each that ensures the acceptable statistical error of the order:  1N  1 39 0.16(16%) . The least square estimation 2 2 of the parameters in the equation (2.110) gives: dLVc0.1  0.33  T and dLMc 0.01  0.87  T .

Numerical estimations of nV and n M turn out to be connected with the values of the mean fluctuations  , which behave as random values for the relatively short observational statistical series with   0 for Tc   . To obtain the trend parameters in (2.110) without the influence of the average values of fluctuations it is possible to use the known relation for the short series trends (see, e. g. [3]):

31 Seidelman P. K., Santoro E. J., Pulkkinen K. F. In Dynamical Astronomy. Eds. Szebenhey V., Balazs B. Austin, Texas, 1985; 55. Seidelman P. K., Santoro E. J., Pulkkinen K. F. In Relativity in Celestial Mechanics and Astrometry. Eds. Kovalevsky J. and Brumberg V.A. Kluwer, Dordrecht,1986; 99. 32 Yao Z-G., Smith C. In Mapping the Sky. Eds. Debarbat S., Eddy J.A., Eichhorn H.K., Upgren A.R. Kluwer, Dordrecht, 1988; 501. Yao Z-G., Smith C. In Developments in Astrometry and Their Impact on Astrophysics and Geodynamics. Eds. Muller I.I. and Kolaczek B. Kluwer, Dordrecht, 1993; 403. 33 Standish E.M., Williams J.G. In Inertial Coordinate System on the Sky. Eds. Lieske J.H. and Abalakin V.K. Kluwer, Dordrecht, 1990; 173. 34 Kolesnik Y. B. // A&A (1995) 294, 876. Kolesnik Y. B. In Dynamics, Ephemerids and Astrometry of the Solar System. Eds. Feeraz-Mello S., et. al. Kluwer, Dordrecht, 1996; 477. 35 Kolesnik, Y., Masreliez, C.J. Secular trends in the mean longitudes of the planets derived from optical observations // AJ (2004), 128, No 2; 878. 91

2 n ()() dL Tc (2.113)

2 2 Here  is the r.m.s. deviation, which has the value  (Tc ) 0.28 cy for the adopted uniform division of the abscissa axe. The estimations of nV and n M from (2.113), corresponding to the observations displayed at Fig. 2.15 are presented in Table 2.3

Fig. 2.15. Corrections to the theoretical values of longitude for the Mercury dLM and Venus dLV (in arcseconds) for the observations in 1900–2000. The trends with cosmological corrections (2.111, 2.112) – dashed lines 1: 2 2 dLVc0.2  0.66  T 2: dLMc0.23  1.68  T .

Table 2.3. The comparison of the theoretical values of cosmological corrections to the planet motions with their observational estimations.

Average motion Theoretical cosmological correction Observational estimations of the 2 2 n (as/cy) n  n Hc 2 (as/cy ) average values n (as/cy ) 9 Moon 1.74 10 5.41 6.2 0.7 (2.102) 8 Mercury 5.4 10 1.68 1.62 0.26 Venus 2.1 108 0.66 0.61 0.1 8 Earth 1.296 10 0.4 0.47 0.23 (2.101) 7 Mars 6.89 10 0.21 –

The corresponding estimations of the Hubble constant from the observed accelerations of the Venus and Mercury presented in Table 2.3 are:

9 -1 HVe2 n Ve n V  5,81  10 cy (57.5 km/s/Mpc) (2.114)

9 -1 HMe2 n Me n M  6  10 cy (59.4 km/s/Mpc) (2.115) Gathering together all estimations (2.97, 2.105, 2.107, 2.114, 2.115) one can evaluate the average value of the Hubble constant, calculated from the observed motions of the Solar system bodies: 18 -1 He (2.1  0.2)  10 s ( 65.6 0.7 km/s/Mpc) (2.116) This value corresponds to the estimations obtained from the distant space observations (see Fig. 2.9) and only slightly differs from the theoretical value (2.61). The physical nature of cosmological corrections discussed above is rooted in the fact that all uniform motions in the decelerating physical time turn out to be accelerating when are analyzed with the use of uniform Newtonian time scale.

The Pioneer Anomaly mystery The development of space research with cosmic probes in the second half of the 20th century brought unexpected confirmation of the above discussed features of mechanical motion in decelerating physical time. More than 40 years ago on March 3, 1972 the “Pioneer 10” and then on April 6, 1973 its twin “Pioneer 11” – the first space probes to explore distant planets were launched from Cape Canaveral in the USA. After the “Pioneer 10” passed close to Jupiter completing its research program it headed for the edge of the Solar system along an almost hyperbolic trajectory in the ecliptic plane. In June 1983, “Pioneer 10” crossed the orbit of Pluto, the most distant planet of the Solar system and become the first manmade spacecraft, which entered into the interstellar space. Officially, the research program of the “Pioneer 10” spacecraft was completed on August 31, 1997 when it was 67 a.u. away from the Sun and was leaving Solar system with the speed about 12.2 km/s. Last time the signal from the “Pioneer 10” was received on the end of January 2003 when it was 82 а.u. (more than 12 billion km) away from the Sun. Now this silent envoy of mankind continues its route toward the star Aldebaran – the orange eye of Taurus constellation, and if not damaged by meteors it will in two million years bring to the outskirts of this star the modest message of mankind on the golden plate of its antenna.

Fig. 2.16. The space probes “Pioneer 10” (left) and “Ulysses” (right)

In the late 1990s, the strange phenomenon of a small acceleration of the “Pioneer 10” probe directed toward the Sun and seemingly slightly decreasing its velocity was discovered. Initially, in 8 -2 1994, this anomalous acceleration was estimated as aP10 (8  3)  10 cm s . In 1998, after the detailed analysis of all inner processes, which could influence this value, a new estimation was obtained for both probes [1]:

8 -2 aP10/11 (8.74  1.33)  10 cm s (2.117) It turned out that these strange accelerations do not depend on the probe’s velocities in the range 7.2 12.2 km/sec. At the heliocentric distances less than 20 a.u. the anomalous accelerations are masked by the solar wind and solar radiation pressure forces. In spite of numerous ideas, which already have been suggested in the attempts to explain these accelerations, there is still no progress in this field. The discovery of these anomalous accelerations stimulated the scientific community to continue research of this phenomenon called “Pioneer anomaly”. In 2002, after the thorough checking of observational data the authors of the first publication confirmed their result of the 1998 with the (2.117) estimation of the mean values of anomalous accelerations [1] and gave the additional 8 -2 estimates – “Pioneer-10”: aP10 7.84 10 cm s (observational period 01.1987–07.1998); “Pioneer- 8 -2 11”: aP11 8.55 10 cm s (observational period 01.1987–10.1990). Also, in this paper was

93 presented the estimation of the anomalous acceleration of the space probe “Ulysses”, launched on October 6, 1990:

8 -2 aU (12  3)  10 cm s (2.118) To discover the anomalous accelerations the authors of [1] compared the observed Doppler frequency  obs ()t of the signal sent from the Earth and then redirected back by the space probe with the theoretically calculated one  mod ()t , which was obtained by the mathematical modeling using, of course, the Newtonian time. In their articles, authors of the discovery of “Pioneer Anomaly” write the following system of equations to calculate the anomalous acceleration aP (equations (15) in [1]):

[obs (t ) mod ( t )]    0  (2 a P t ) c (2.119)

mod(t ) 0 {1 [2 u mod ( t ) c ]} (2.120)

Here  0 is the base frequency of the ground station, coefficients 2 appear due to the twofold observational data procedure, minus in the right side of the (2.119) accounts for the opposite direction of the anomalous acceleration relative to the velocity of the space probe. Anomalous accelerations can be logically explained on the basis of the asymmetric spacetime conception with the decelerating physical time (Taganov 2003 [24-32]). To determine  obs ()t and consequently the value of the anomalous acceleration aP in the uniform scale of the Newtonian time one should, firstly, take the variable velocity of light instead of the constant one in accordance with the equation (2.85): ct  c (1  Ht ); c  c  const , secondly, it is necessary to take into account the influence of the non-static space-time on the probe’s velocity as in the relation (2.85): u*()()mod t u mod t c Ht . Combining these modifications we arrive to the following form of the equations (2.119, 2.120):

[obs ()t mod ()] t    0  (2)(1 a P t c  Ht ) (2.121)

mod(t ) 0 {1 [2 u mod ( t ) c ]} (2.122)

obs (t )0 {1  [2( u mod ( t )  cHt ) c (1  Ht )]} (2.123) Substituting (2.122) and (2.123) into (2.121) we get:

aP  cH umod H (2.124) In (2.124) we can neglect the second term due to the large difference between the velocity of light and that of a space probe (7–12 km/s): cH umod H . Using the theoretical value of the Hubble constant from (2.61), one gets the estimation of the “cosmological” acceleration (2.124), which corresponds to the observed anomalous accelerations (2.117, 2.118) within the order of magnitude:

8 -2 aP  cH 5.9  10 cm s (2.125) The physical nature of the abnormal acceleration of space probes is deeply connected with asymmetry and metric heterogeneity of spacetime and associated with the cosmological deceleration of physical time, which always leads to the apparent acceleration when Newtonian uniform time scale is used for the calculations. It should be noted that the theoretical estimation (2.125) is only of the order of magnitude close to the observed abnormal accelerations, which are noticeably bigger than the values given by (2.125). This could be due to the fact that the observed acceleration values (2.94, 2.95) are the sums

94 of the cosmological acceleration (2.125) and accelerations aint connected with the physical processes inside the space probes:

aobs  cH aint (2.126) The research made in recent years confirmed this suggestion. It turned out that there are numerous processes on board of the probes, which can cause their additional accelerations and one of the most important is the asymmetric heat radiation reaction. All the discussed probes are supplied with the relatively powerful radioisotope sources of energy, which use plutonian isotope 238Pu with the half-life period 87.7 years. The Pioneer generator’s power at the beginning of the mission was about 2.5 kWt, and that of «Ulysses» probe even larger, but the actual powers were only

Wp 160 Wt at the «Pioneers» and WU  285 Wt at «Ulysses» due to the relatively low energy efficiency of the radioisotope sources. Recently several attempts were made to find a theoretical estimate of the additional acceleration Fig. 2.17. Comparison of the equation (2.127) with of space probes due to their asymmetric thermal abnormal acceleration estimations radiation. However, the estimates obtained so far are highly variable because of the difficulty in modeling of the heat radiation of the probes with a very complex geometry and multiple sources of thermal energy. So, Vyacheslav Turyshev, a recognized expert on the “Pioneer Anomaly” believes 8 that the heat contribution to the anomalous acceleration does not exceed 30 % that is aint 2.6 10 cm s-2 for “Pioneers”. Other authors36 give significantly higher contribution of the asymmetric effects 8 -2 of thermal radiation: aint (2.3  6.7)  10 cm s . It seems reasonable to suggest the approximate linear model of the abnormal acceleration formation where the heat radiation part is proportional to the electrical power of the probe’s source:

aobs  cH bW (2.127) Figure 2.17 demonstrates the comparison of the relation (2.104; dashed line) for b  0.02 cm s-2 Wt-1 with the estimations (2.117, 2.118). Equation (2.127) fits well with the observations with the r.m.s. deviation for the estimations (2.117, 2.118) to be only 0.4 cm s-2 under the minimal observational uncertainty in (2.117) equal to 1.33 cm s-2. Electrical power resources of the probes gradually reduce due to the depletion of isotope generators and other internal processes. So, according to the telemetry actual electrical power of Pioneer 10 in 1987 was 95 Wt and only 68 Wt in 1998. For gradually decreasing electrical power the equation (2.127) predicts a decrease of the anomalous acceleration.

Such a slow drift ( j daobs dt ) of the anomalous accelerations towards the lower values was confirmed37 during the special processing of the observational data of the Pioneers trajectories: 8 -2 8 -2 jP10 (  0.21  0.04)  10 cm s /year and jP11 (  0.34  0.12)  10 cm s /year. Recently the final

36 Bertolami, O., Francisco, F., Gil, P.J.S., Paramos, J. // Phys. Rev. D (2008) 78; 103001. // Space Sci. Rev. (2010) 151; 75. Modelling the reflective thermal contribution to the acceleration of the Pioneer spacecraft // arXiv: 1103.5222v1 [physics.space-ph] (2011). 37 Toth, V.T. Independent analysis of the orbits of Pioneer 10 and 11 // Int. J. Mod. Phys. D (2009) 18; 717. 95 analysis of all available observations of the Pioneer 10 and 11 trajectories confirmed38 the gradual 8 -2 decrease of the abnormal accelerations with the average drift about jP  0.2  10 cm s /year. In conclusion, it should be noted that the discovery of the anomalous accelerations of space probes is an example of the rare case of direct confirmation of the actual asymmetric spacetime measured by high precision special devices, revealing a difference between decelerating cosmological time and the idealized Minkowski symmetric model with invariable Newtonian time. Moreover, the analysis of the nature of this discordance and sufficiently precise theoretical estimate of the anomalous accelerations of space probes are good illustrations of the specific description of motion in spacetime with decelerating physical time. Kinetics of the radioactive isotope decays in non-static Universe Manifestations of the phenomenon of the cosmological deceleration of time can be observed not only in the grand scale of the Universe or the Solar system, but also in microcosm. In particular, the deceleration of physical time with respect to the uniform invariable Newtonian time scale reveals itself in the analysis of the radioactive decay data (Таганов 2003 [24–28, 32]). The decay rate constant of an isotope is proportional to the probability of its unstable nucleus decay, which, in its turn, is inversely proportional to the mean lifetime of the excited nucleus of this isotope. The decay rate constant should be calculated within the concept of decelerating physical time because the real decays occur namely in the physical time and if it is so, then the difference between the actual number of decayed nuclei and number calculated with the uniform Newtonian time scale should grow with the growth of the isotope half-life period. A characteristic feature of the spacetime with decelerating time is its asymmetry. Therefore, there are two scenarios of analysis of the decay kinetics with two different time origins. 1. In a retrospective analysis, the initial reference point of time is our epoch and in accordance with (2.34) the ratio of physical and Newtonian time intervals decreases linearly as we consider more ancient epochs (negative t and  ):  t12  H  t . 2. When considering the kinetics of the isotope decay in the future the time counting off starts from the beginning of the decay process and in the forecast of future events (at positive t and  ) the ratio of physical and Newtonian time intervals, on the contrary, linearly increases:  t12  H  t . Therefore, the radioactive isotope decay in physical time is described by the modified radioactive decay law:

IIIII 2 N N00exp     N exp[    ( t  Ht 2)] (2.128)

I I Here N0 , N are numbers of isotope I nuclei, which have not decayed by the initial and I arbitrary moments,  is the decay rate constant of the isotope measured in the physical time. The difference in the calculation results with the Newtonian and physical time becomes significant only for large time intervals, for example, in the Isotope Geochronology (IG). The methodology of IG in addition to the law of radioactive decay uses the relations:

III III tLe e t ee L  (2.129)

I I Here t and  denote the decay rate constants of the I-th isotope used for the age I I determination, te and  e are the experimental estimations of the sample age in Newtonian time scale I and in “physical” decelerating time scale. Le is the so called “kinetic function” depending on the empirically determined concentrations of the isotopes, which determine the isotope relations forming

38 Turyshev, S.G., Toth, V.T., Ellis, J., Markwardt, C.B. Support for temporally varying behavior of the Pioneer anomaly from the extended Pioneer 10 and 11 data set // Preprint arXiv:1107.2886v1 [gr-qc] 14 Jul 2011. 96 during the main I-th isotope decay. Here and further index “e” denotes the experimental estimations of the parameters. I 39 The kinetic function Le in (2.129) is determined by the specific features of the IG method and I there are many modifications of the “kinetic functions” Le in IG, but what is important is the fact I I that decay rate constants t and  in (2.129) always appear as divisors. In a retrospective analysis in isotope geochronology should be borne in mind that the “age” of a geological sample is essentially negative number, as it is defined by a negative value of time in the past, when started the decay of an isotope, using which the age is determined. In the concept of decelerating physical time all the characteristic intervals in the past are less than at present and in the future. Therefore, the physical duration of the isotope decay due to deceleration of physical time in the past is less than its value in the Newtonian time and accordingly physical age of a geological sample is less than its Newtonian age. For example, to the estimation of the duration of Earth’s past tE 4,5 billions of Newtonian years corresponds approximately  E 3,9 billions of physical years (in accordance with 2.34). In order for a shorter interval of physical time disintegrated the same amount of isotope atoms as I for larger interval of Newtonian time the physical decay constant  should be larger than the Newtonian decay constant  I , therefore in the retrospective analysis of isotope decay always: t II   t . For relatively short time intervals (usually not more than 20 years), during which the experimental determination of the decay rate constants of long-living isotopes are performed it is not possible to take into account the cosmological deceleration of physical time. Therefore, the constants used in the isotope decay calculations, in fact, are somewhat underestimated experimental estimations of the decay rate constants for decelerating physical time. Comparing of the relations (2.128, 2.129) for the Newtonian and decelerating physical time gives:

IIIIIIII IIII ett e  e (1   t  )   e  (   t  1) et e  t  (2.130) This relation demonstrates that the difference between the age estimations in the physical and Newtonian time scales grows with the sample age. Simultaneous estimation of the sample age in Newtonian time scale with two isotopes with (1) (2) different decay rate constants gives two values, as it follows from (2.130) for e  e  e , with the difference:

1  2  1 (1) (2) (1) (2) te t e  t e [1  ( t  ) (   t )] (2.131) From (2.131) follows that for the sample with the only one certain age in the physical time estimations of its Newtonian age made with different isotopes will give different values with the smaller age estimation obtained with the isotope with larger lifetime. It should be noted that the difference in the estimates of the Newtonian age of a sample is not confined to a retrospective analysis of the isotope decay kinetics (for negative values of age), but appears also in the forecast of the isotope decay kinetics, that is, for positive values of the age (for details see [24–28, 32]) . It is more effective to compare isotope age estimations in different time scales made by the IG methods using isotopes with significantly different decay rate constants. The two widely used in IG

39 Faure, G., Mensing, D. Isotopes - Principles and applications. 3rd Edition. J. Wiley & Sons, 2005. ISBN 0-471- 38437-2. Mattinson, J. M., Revolution and evolution: 100 years of U-Pb geochronology // Elements (2013) 9, 53–57. Горохов И.М. Рубидиево-стронциевый метод изотопной геохронологии. М., 1985. 97

238 238 10 -1 methods – (U/Pb)-method (decay rate of U: t 1.55 10 year ) and (Rb/Sr)-method (decay 87 87 11 -1 rate of Rb: t 1.42 10 year ) are just of this type. During last two decades the (Sm/Nd)-method is considered by many geochemists as the most accurate isotope dating method, which even used sometimes as a standard to improve the age estimations received by other isotope methods. The (Sm/Nd)-method is based on detection of radioactive products of the isotope 147Sm decay with the isotope 144Nd as a final product. Decay rate 147 147 12 -1 constant of the Sm is t 6.42 10 year and it is smaller than decay constants of all other isotopes used in IG. One can expect that the Newtonian age of a geological sample determined by this method will be less than the age estimations by the (U/Pb) and (Rb/Sr) methods. It is possible to choose several tens out of hundreds publications where the same samples were analyzed simultaneously by (U/Pb), (Rb/Sr) and (Sm/Nd) methods (Tables A1 and A2 in Appendix). The differences in sample age estimations by (U/Pb), (Rb/Sr) and (Sm/Nd) methods are unmistakably seen especially for the samples older than one billion years. The estimations of the Newtonian ages (Table A1, columns 1, 2 in Appendix) obtained with both (U/Pb) and (Rb/Sr) methods systematically differ with the average divergence of the order 0.14 billion years (6.5 %). The estimations of the Newtonian ages (Table A2, columns 1, 2 in Appendix) obtained with both (U/Pb) and (Sm/Nd) methods systematically differ with the average divergence of 0.13 billion years (8.5 %). These divergences are significantly larger than the mean uncertainty of age determinations in IG – 1-2 %. 238 87 The Newtonian ages of the geological samples from Tables A1, A2 in Appendix: tei , tei and 147 238 87 147 tei allow for the experimental determinations of parameters Lei , Lei and Lei with the help of relation (2.129). Then, using again the (2.129) with the decay rate constants for physical time scale it 238 87 147 is possible to find the values of e ,  e and  e , which minimize the sum of differences between 238 238 87 87 estimated ages in physical time:  [(LLei e )  ( ei e )]  0 . ()i For the data in Tables A1, A2 in Appendix the values of the decay rate constants for physical time evaluated using the gradient descent method of minimization listed in Table 2.3. To estimate the I I sample physical age  e having the experimental isotopic estimation of its Newtonian age te one can IIII use relation (2.107): et e  t  and Table 2.4.

Table 2.4. The decay rate constants of the radioactive isotopes for Newtonian and physical time scales

238U 87Rb 147Sm I -1 10 11 12 t (year ) 1.55 10 1.42 10 6.42 10 I -1 10 11 12  e (year ) 1.806 10 1.55 10 6.87 10

The analysis of physical age estimations demonstrates that the mean discrepancy between physical ages determined by the (U/Pb) and (Rb/Sr) methods is less than 0.0002 billion years (0.01 %) and the mean discrepancy for estimations by the (U/Pb) and (Sm/Nd) methods is less than 0.0002 billion years (0.012 %), so the discrepancies diminish more than in two orders of magnitude, comparing with the differences between Newtonian age estimations. If we use in (2.131) the first index – 1 as an indicator of the (U/Pb)-method and the second index – 2 for the (Rb/Sr) or (Sm/Nd) method then the relation (2.131) takes the following form for the evaluated decay rate constants in physical time scale (Table 2.4):

238 87 238 te t e 0.0632  t e (2.132)

238 147 238 te t e 0.0816  t e (2.133) One can clearly see at the Fig. 2.18 that the difference between estimations of Newtonian age of a geological sample (Tables A1 and A2 in Appendix) is larger for the isotopes with smaller decay rate constants. In spite of the experimental points scattering at the Fig. 2.18 the relative average divergence (divided by the sample age) of the experimental data and the theoretical equations (2.132) and (2.133) does not exceeds 5 %.

Fig. 2.18. The discrepancy between the Newtonian age estimations for geological samples determined by: the (U/Pb) and (Rb/Sr) methods – circles; the (U/Pb) and (Sm/Nd) methods – triangles. Solid lines: 1 – (2.132); 2 – (2.133); broken lines - corresponding trends evaluated by isotopic dating.

The concept of cosmological deceleration of time gives a unique opportunity to estimate the value of the Hubble constant not from the redshifts of distant galaxies, but from the isotopic ages of rock samples from the Earth and the Moon.

Fig. 2.19. Comparison of the isotope age estimations for the geological samples (circles for the (U/Pb)-method, 2 triangles for the (Rb/Sr)-method and squares for the (Sm/Nd)-method) with the theoretical equation: ei  t ei  Ht ei 2 (solid curve) for the theoretical value of the Hubble constant (2.61) 1.97 1018 s-1.

The relation (2.34) allows us to estimate the Hubble constant using the statistics of the geological sample ages measured in Newtonian and physical time scales. For the isotope age estimations 2 equation (2.34) can be written as: ei  t ei  H e t ei 2. Using statistics of Tables A1 and A2 in Appendix one gets the following least squares estimation:

18 -1 He (2.15  0.2)  10 s ( 67.2 6 km/s/Mpc) (2.134)

These estimations are close to the theoretical value of the Hubble constant (2.61): H  61.6 km/sec/Mpc; (1.97 1018 sec-1) and correspond to the commonly used in astrophysics interval: 60 75 km/s/Mpc. At the Fig. 2.19 coordinate axes correspond to the physical and Newtonian ages of the geological 2 samples in billions of years and the solid line – to the equation (2.34): ei  t ei  Ht ei 2 with the theoretical value of the Hubble constant (2.61): 1.97 1018 s-1. The root mean square deviation of the age estimations represented in Tables A1 and A2 in Appendix from theoretical relation (2.34) is less than 5 %.

In 1970s E.K. Herling and G.V. Ovchinnikova40 confirmed the above discussed systematic difference between the rock and mineral age estimations simultaneously determined by (U/Pb) and (Rb/Sr) isotope methods. They noted that in the majority of analyzes the (Rb/Sr)-method gives smaller values of the sample ages, which corresponds to the equations (2.130, 2.131) because the decay rate constant of 87Rb is more than the order of magnitude less than that of 238U. In geochemistry the discrepancy between ages of geological samples obtained by (U/Pb), (Rb/Sr) and (Sm/Nd) isotope methods is usually explained by different processes of chemical elements and their isotopes migrations during the rock metamorphisms (see, e. g.41). It is believed that in ancient rocks isotope migration occurred repeatedly and was particularly intense in the epochs of diastrophic cycles when there was a rise of geo-isotherms, rock deformation and intense motions of solutions and fluids. The phenomenon of (Rb/Sr) isotopic “rejuvenation” geochemists commonly associate with the extraction of significant amount of Rb and Sr from the rocks in the processes of thermal metamorphism with the preferential extraction of the products of radioactive decay, which are less tightly bound to the crystal structure of minerals. Radiogenic 87Sr can migrate faster than the stable 86 Sr, which leads to the deceleration of the 87Sr 86 Sr ratio’s growth in (2.130). Intensive leaching of the Rb by alkaline fluids can also contribute to the emergence of the effect of (Rb/Sr) isotopic “rejuvenation”, which leads to a decrease of radiogenic Sr in the rocks. Perhaps all these processes took place and, in particular, the differences in the estimates of (Rb/Sr)-ages for the same rocks from different sites on the Earth are associated with these processes. The processes of thermal metamorphism, mentioned above, seem to be also among the major causes of the experimental points scattering in Fig. 2.18. However, it seems quite unlikely that a complex set of radiochemical processes, causing the phenomenon of (Rb/Sr) and (Sm/Nd) isotopic “rejuvenation” are somehow accidentally synchronized on the Earth and on the Moon in such a way that it is determined by a single parameter – the Hubble cosmological constant, which determines the degree of non-stationarity of the Universe without any relation to radiochemistry and geochemistry. One additional argument in favor of the assumption that the average effect of (Rb/Sr) isotopic “rejuvenation” is not determined by washing out of the elements with alkaline fluids are the results of isotope chronology of lunar rock samples. It is believed that the thermal metamorphism of the Moon rocks was not influenced by hot solutions and fluids. Nevertheless the overall effect of (Rb/Sr) isotopic “rejuvenation” of the lunar rocks turns out to be on average even more distinct than on the Earth (samples 29–32 in Table A1 in Appendix). The cause of the observed discrepancies between the age estimations of the geological samples from the Earth and Moon obtained with radioisotope methods using isotopes with different decay

40 Герлинг Э.К., Овчинникова Г.В. К вопросу о постоянстве скорости радиоактивного распада // Геохимия (1970) № 8; 891–902. Герлинг Э.К., Овчинникова Г.В. Постоянна ли скорость радиоактивного распада? // Радиохимия (1977) №1; 106–120. 41 Салоп Л.И. Геологическое развитие Земли в докембрии. Л.: Недра, 1982. Dickin, A.P. Radiogenic Isotope Geology. Cambridge University Press, 2005. 100 rate constants is the divergence of the Newtonian and physical time scales. The coincidence of the Hubble parameter estimations for the extragalactic objects with estimate obtained by the isotope chronology methods for the rocks from the Earth and the Moon can be considered as one additional confirmation of the phenomenon of cosmological deceleration of physical time. Registration of the cosmological deceleration of time in laboratories One way to register the cosmological deceleration of time in the laboratory is to investigate the stability of the resonance frequency of a high-frequency oscillator for a large time period. In the last decades, significant progress in the creation of precise frequency standards was attained. In particular, cryogenic sapphire oscillators (CSO), invented in the 1980s, have been used for many years and their characteristics have been studied in many laboratories. The CSO have very stable resonance frequencies and in short time tests they demonstrate frequency variation less than one part per 1015 . This high frequency stability allows the successfully use of CSO in devices with an atomic clock operating near the quantum noise limit. There is a successful experience with CSO testing Lorentz-invariance and stability of the value of the fine structure constant. A recognized leader in the development and study of CSO is a group of engineers and physicists at the University of Western Australia42. The first model of CSO with sapphire crystal with a diameter of 3 cm was tested in 1989, showing a deviation from the resonance frequency of 9.733 GHz less than one part per 1014 in tests lasting from 10 to 300 seconds. In 1998, the second generation of CSO with increased to 5 cm diameter sapphire crystal and resonance frequency 11.932 GHz showed frequency instability (3 4) 1015 in the tests with duration up to 100 seconds. Short- time instability of the third generation of CSO, according to reports in 2006, was reduced to 6 1016 in the test with duration of 20 seconds.

Fig. 2.20. The linear decrease of the resonance frequency of CSO 0 (ordinates – [()]/   00  ; abscissa – test duration in days). A is the tests of CSO (1, 2) in French Space Agency (CNES, Toulouse); B is testing of the CSO (3) in National Measurement Institute (Australia).

Despite the extremely high frequency stability in short-time CSO tests, their long-time tests (months and years) already in 1990s revealed a drift of the resonance frequency. This change in the resonance frequency had a linear dependence on time over the years, and always had the same trend – the resonance frequency decreased (Fig. 2.20).

42 Bize, S., Wolf, P., Abgrall, M. et al. Cold Atoms Clocks, Precision Oscillators and Fundamental Tests // Lect. Notes Phys. (2004) 648; 189. Hartnett, J.G., Locke, C.R., Ivanov, E.N. et al. Cryogenic sapphire oscillator with exceptionally high long-term frequency stability // (2006) arXiv: physics/0608124. 101

The long-time frequency stability of CSO can be affected by various factors: defects of mechanical processing and compilation, the slow change of vacuum in the chamber, hidden defects of sapphire crystal and more. Some engineers have tried to improve the system of fastening sapphire, suspecting that the frequency drift can be caused by a gradual decrease of stresses in the clumps. Especially unusual was the linear dependence of the frequency shift on the time, because, as a rule, the slow relaxation of stresses is characterized by gradually decreasing frequency drift with time. Despite the many and varied changes of the CSO design, the average resonance frequency drift observed in tests has not changed over the past 20 years, and this linear drift even became known as the “natural permanent drift” of CSO. Linear decrease of the frequency in long-time tests of CSO was described by the equation:

[()]/    00  K  with the frequency drift decrement K 1/day (Table 2.5)

Table 2.5. Estimates of the CSO frequency drift decrement [5.1-5.3]

Dates Test duration (years) K 1013 (1/day) CNES (Toulouse) 2003-2005 3 -2.41 CNES (Toulouse) 2004-2005 3 -2.44 Paris observatory 2003-2005 5 -1.28 Paris observatory 2003-2005 5 -1.47 National Measurement 1995-1998; 10 -0.85 Institute (Australia) 2004-2006 Average 1.69 0.71

References

5.1. Tobar, M.E., Ivanov, E.N., Locke, C.R. et al. Long Term Operations and Perfotmance of Cryogenic Sapphire Oscillators // (2006) arXiv: physics/0608202. 5.2. Hartnett, J.G., Locke, C.R., Ivanov, E.N. et al. Cryogenic sapphire oscillator with exceptionally high long-term frequency stability // (2006) arXiv: physics/0608124. 5.3. Bize, S., Wolf, P., Abgrall, M. et al. Cold Atoms Clocks, Precision Oscillators and Fundamental Tests // Lect. Notes Phys. (2004) 648; 189.

With the help of relations: cc ; 00  , the determination of a redshift z 0 1 and the equation (2.39): zH(1  2 )12  1 we can obtain the following equation for the oscillator frequency drift in the concept of cosmological deceleration of time:

12 (  0 )  0  (  0  )  z (1)(12) z H   1 H  (2.135) Thus, the frequency drift decrement of precise oscillators during their long-time tests must match the Hubble constant (2.61). In long-time CSO tests (Table 2.5) the rates of frequency change in different designs of CSO always remained within the range (0.85 2.44) 1013 1/day being in average (1.69 0.71) 1013 1/day, which is very close to the theoretical value of Hubble constant (2.61): H 1.970 1018 s-1 1.7 1013 1/day. In concluding this section, it can be supposed that the highly stable frequency standards on cryogenic sapphire oscillators during several years already register the cosmological deceleration of time in laboratories.

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The illusion of Universe’s “accelerating” expansion The most impressive manifestation of cosmological deceleration of time in distant cosmos is the illusion of accelerating expansion of the Universe. Recently, the scientific community lively discussed the possible evidence of “accelerating expansion” of the Universe, discovered independently by two teams of researchers (Perlmutter, Riess et al. [18, 20, 21]). In order to detect the phenomenon of “accelerating expansion” scientists used estimates of the so-called cosmological parameters M ,  in the standard model of classical cosmology, obtained by processing the results of the study of supernovae SNe Ia explosions. These research projects evaluated the following 0.09 0.05 estimates of the cosmological parameters: Me 0.280.08 [18] and Me 0.290.03 [20]. Relation

M   1 allows for a standard “flat” cosmological model to estimate the value of  and then using the formula: qtM12   to calculate the “deceleration parameter“ for Newtonian time (see e.g. [22, 35]):

qte  0.56  0.11 (2.136) This negative value of the deceleration parameter allowed supposing that in our epoch the Universe expands with acceleration. Unusual image of the “accelerated expansion” of the Universe gradually became familiar, and in 2011, the Nobel Committee awarded the Nobel Prize in Physics to astrophysicists Saul Perlmutter, Brian Schmidt and Adam Riess with the words: “For the discovery of accelerated expansion of the universe through observations of distant supernovae”. There is no doubt that a huge amount of careful observations of supernovae, made by several dozen of astronomers during nearly 20 years, is highly commendable. However, this study did not discover the “accelerating expansion” of the Universe, but became an impressive confirmation of the phenomenon of cosmological deceleration of time. In the concept of decelerating irreversible time, the apparent evidence of “accelerating expansion” of the Universe has an ordinary explanation (Taganov 2005 [25–29, 32]). Using the rules of differentiation and the formula (2.72): d dt a one can obtain the relation between the 2 cosmological deceleration parameter for physical time: q  aa a  and the deceleration parameter 2 qt  aa a for Newtonian time (see the derivation in part 2.4):

qqt  1 (2.137)

From (2.137) follows that the moderate ( q 1) deceleration of the Universe expansion in physical time seems an accelerating expansion ( qt  0 ) in the Newtonian time. Uniform expansion in the Newtonian time ( qt  0 ) corresponds to the decelerating expansion in physical time

( q 1). These coordinate effects are similar to changes in the kinematics during transitions between inertial and non-inertial frames of reference. Using (2.136, 2.137) we can estimate the deceleration parameter in physical time corresponding to observations of supernovae:

qqe te 1  (  0.56  0.11)  1   0.44  0.11 (2.138) Hence, in physical time, which is actually governs the real astrophysical observations, the “expansion” of the Universe is not accelerating, but rather slowing. The work [21] presents an analysis of the kinematics of the alleged evolution of the Universe for the past epochs based on statistical analysis of redshifts in the spectra of supernovae SNe Ia (for z up to 1.55). An equation used for analysis of the Universe evolution has the form of a Taylor series expansion with two-parameter representation of the dependence of deceleration parameter on the 103 redshift. Estimations of kinematic parameters obtained in this article, within the uncertainty boundaries of observations coincide with the estimates of the apparent acceleration of the expansion of the Universe based on Eq. (2.34):  t Ht2 2 , which describes the cosmological deceleration of irreversible time (see e. g. [28]). Hence, the illusion of “accelerating expansion” of the Universe aroused from the fact that for the interpretation of the observations of astrophysical processes developing in decelerating irreversible physical time, researchers used the cosmological model with invariable uniform scale of Newtonian time. The cause of discovered “accelerating expansion” of the Universe, many theorists associate with non-zero Einstein’s cosmological constant, which, in turn, is explained by assumption of an existence of the mysterious “dark energy”. Dark energy is a hypothetical form of energy that invisible fills space and is responsible for the accelerating expansion of our Universe. In the standard CDM -model of classical cosmology, this mysterious dark energy composes nearly three-quarters of the mass-energy of the Universe! Since the “dark energy” is included in the standard model of classical cosmology only to explain the accelerating expansion of the Universe, this ghost will not be slow to disappear as soon as the illusory nature of the “accelerating expansion” of the Universe will be recognized.

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Russian Geographical Society, 23-27 June 2008, Saint Petersburg, Russia. Eds. Yu.V. Baryshev, I.N. Taganov, P. Teerikorpi. Vol. 1,2. St.-Petersburg.: TIN, 2008. ISBN 978-5-902632-06-1. Vol. 1; 264–274. arXiv:0807.4083v1 [physics.gen-ph] 25 Jul 2008. 27. Taganov, I.N. Conception of Quantum Cosmology. In Practical Cosmology. Proceedings of the International conference “Problems of Practical Cosmology”, held at Russian Geographical Society, 23-27 June 2008, Saint Petersburg, Russia. Eds. Yu.V. Baryshev, I.N. Taganov, P. Teerikorpi. Vol. 1,2. St.-Petersburg.: TIN, 2008. ISBN 978-5-902632-06-1. Vol. 2; 68–85. arXiv:0807.4399v1 [physics.gen-ph] 28 Jul 2008. 28. Taganov, Igor N. Quantum Cosmology. Deceleration of Time. St.-Petersburg.: TIN, 2008. ISBN 978-902632-05-4. 29. Taganov, Igor. Das expandierende Universum und die Verlangsamung der Zeit // Nachrichten der Olbers-Gesellschaft Bremen (2009) 225; 4–10. 30. Taganov, Igor. Kosmische Verlangsammung der Zeit im Sonnensystem und auf der Erde // Nachrichten der Olbers-Gesellschaft Bremen (2013) 241; 12–19. 31. Igor Taganov & Ville Saari. Antike Rätsel der Sonnenfinsternisse. Die Kalendarzyklen // Nachrichten der Olbers-Gesellschaft Bremen, 244, Januar 2014. 4-10. 32. Taganov, Igor N. Irreversible-Time Physics. - Saint Petersburg: TIN, 2013. ISBN 978-5-902632- 12-2. Kindl Edition (English) ASIN: B00I214VIG. http://www.amazon.com/books. “Physics of irreversible time: The new natural phenomena”. ISBN-13: 978-3659513541. ISBN-10: 3659513547. LAP LAMBERT Academic Publishing. https://www.lap-publishing.com. «Физика необратимого времени. Hовые явления природы». ISBN-13: 978-3659286674. ISBN-10: 3659286672. LAP LAMBERT Academic Publishing. https://www.lap-publishing.com. 33. Taganov I.N., Kalinitchenko G.A., Jalpachik A.S., Taganova A.V. Para-Realism. Saint- Petersburg: TIN, 2009. ISBN 978-5-902632-10-8. Таганов И.Н., Калиниченко Г.А., Таганова А.В., Ялпачик А.С. Пара-Реализм. Санкт-Петербург: ТИН, 2009. 34. The Works of Aristotle. Transl. under the editorship of W.D. Ross, 12 vols. Oxford: Clarendon Press, (1908–1952). 35. Weinberg, S. Cosmology. Oxford University Press, 2008.

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Physical constants and parameters Gravitational constant G 6.674 1011 N m2 kg-2 (6.674 108 cm3 g-1 s-2) Planck constant h 6.626 1034 J s (6.626 1027 erg s) ( 4.135 1021 MeV s) h 2  1.055  1034 J s (1.055 1027 erg s) (6.581 1022 MeV s) Speed of light in a vacuum c 2.998 1010 cm s-1 2 3.066 10 Mpc/Gyr Elementary charge e 1.602 1019 C ( e 4.803 10 10 g1/2 cm3/2 s-1 e22.307 10  19 g cm3 s-2) 28 Electron mass me 9.109 10 g 24 Proton mass mp 1.673 10 g Fine structure constant  ec23 7.297  10 1 (  137.036) 2 2 13 Classic electron radius ree e m c 2.818  10 cm 2 3 18 -1 Hubble constant (1.61) H9 G 16 c re  1.969  10 s 9 -1 21 (61.6 km/s/Mpc; 6.217 10 cent. ; 6.217 10 Gyr )

1 dine (g cm s-2) 105 N; 1 J 107 erg (g cm2 s-2) 6.241 1018 eV Solar mass ( M ) 1.989 1033 g Astronomical unit (a.u.) 1.496 1013 cm Light-year (ly) 9.461 1017 cm (0.307 pc) Parsec (pc) 3.086 1018 cm ( 2.063 105 a.u.; 3.26 ly; 1 Mpc 3.086 1022 m) Sidereal year (yr) 3.156 107 s (1Gyr 3.156 1016 s)

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APPENDIX In Appendix for the sake of reader’s convenience presented the isotopic estimations of geological sample ages used in figures 2.18 and 2.19. Table A1. The isotopic estimations of the geological sample ages (U/Pb and Rb/Sr methods

Newtonian age Physical age (bill. Geological samples Ref. (bill. years) years) U/Pb Rb/Sr U/Pb Rb/Sr 238 87 238 87 tei tei  ei  ei 1 2 3 4 5 6 1 0,35 0,298 0,300 0,273 Montiogranite. W Corsica. [1] 2 0,38 0,368 0,326 0,337 Gneiss. SW Poland. [2] 3 0,424 0,412 0,364 0,377 Granitoids. NW Spits Bergen. [3] 4 0,432 0,428 0,371 0,392 Granite. SN Ireland. [4] 5 0,512 0,511 0,439 0,468 Granite. N America, Oklahoma. [5] 6 0,64 0,557 0,549 0,510 Intrusive rocks. W Cameroun. [6] 7 0,954 0,884 0,819 0,810 Granite. S Norway. [7] 8 0,991 0,866 0,850 0,793 Granite. S Mexico. [8] 9 1,02 1,004 0,875 0,920 Pegmatite. Canada, Ontario. [9] 10 1,151 1,053 0,988 0,965 Granitoids. Canada. [10] 11 1,28 1,16 1,098 1,063 Gneiss. N America, Adirondack. [11] 12 1,563 1,423 1,341 1,304 Granite. SW Sweden. [12] 1,633 1,356 1,401 1,242 13 1,64 1,6 1,407 1,466 Granite. Finland. [13] 14 1,75 1,65 1,502 1,512 Granitoids. Spits Bergen. [14] 15 1,762 1,687 1,512 1,545 Gneiss. Central Australia. [15] 16 1,8 1,74 1,545 1,594 Granite. N Sweden. [16] 1,87 1,772 1,605 1,623 17 1,854 1,777 1,591 1,628 Pegmatite. SE Sweden. [17] 18 1,876 1,775 1,610 1,626 Metavolcanite. Canada, Ontario. [18] 19 2,14 1,95 1,837 1,786 Granite. Russia, Pechenga. [19] 20 2,696 2,560 2,314 2,345 Granite. Canada, Ontario. [20] 2,737 2,55 2,349 2,336 2,746 2,43 2,357 2,226 21 2,7 2,557 2,317 2,342 Granite. Canada, Manitoba. [21] 2,807 2,674 2,409 2,450 22 2,758 2,704 2,367 2,477 Basalt. Gneiss. Brazil. [22] 23 2,915 2,7 2,502 2,473 Gneiss. SW Greenland. [23] 24 2,995 2,753 Granite-gneiss. Africa, [24] 2,570 2,522 Sierra Leone. 25 3,044 2,97 2,612 2,721 Gneiss. India, Karnataka. [25] 3,175 3,08 2,725 2,822 3,185 3,08 2,733 2,822 26 3,357 3,098 2,881 2,838 Gneiss. W Australia, Mt. Narryer. [26] 27 3,402 3,25 2,920 2,977 Leucotonalite. S Africa, N Natal. [27] 28 3,76 3,622 3,227 3,318 Granite. W Greenland, Jsua. [28] 29 4,0 3,778 3,433 3,461 Sample 68415. Moon. [29] 30 4,25 3,74 3,647 3,426 Basalt. Sample 75075. Moon. [30] 31 4,272 4,0 3,666 3,664 Granite. Moon. [31] 32 5,0 4,575 4,291 4,191 Crystalline rock. Moon. [32] References 1. Rossi, P.C. // R. Acad. Sci., Ser. 2. (1988) 307, 13; 1541–1547. 2. Breemen, O. van, Bowes, D.R., Aftalion, M. et al. // Ann. Soc. ged. pol. (1988) 58; 1–2: 3–19. 3. Balasov, Ju.A. // Polar. Res. (1996) 15, 2; 153–165. 108

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НАУЧНОЕ ИЗДАНИЕ

Доктор физ.-мат. наук Таганов Игорь Николаевич и Саари Вилле-Вели Ейнари

ANCIENT RIDDLES OF SOLAR ECLIPSES. Asymmetric Astronomy Second Edition

ELECTRONIC EDITION

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