1. Introduction

Total solar eclipses have been important in modern Cosmology as they offer the first possibility to test Albert Einstein’s General Theory of Relativity (Einstein 1916). General Relativity predicts that the path of light is bent in a gravitational field, and during a total solar eclipse in 1919 it was possible to observe stars close to the solar disc and to measure a shift in their positions that agreed with the predictions. However, another prediction in Einstein’s theory called “geodetic precession” can be tested by well-documented ancient solar eclipses in combination with the Lunar Laser Ranging (LLR) of the distance to the Moon. A new computer program developed by the author has made it possible to identify ancient solar eclipses back to 2500 BC and to determine the lunar secular acceleration in longitude with high precision. When this value for the secular lunar acceleration in longitude was compared with the lunar radial acceleration measured by LLR, by Kepler’s third law, it was found that there was an extra acceleration in longitude that agreed very well with the prediction from Einstein’s theory.

This use of ancient eclipses has, of course, a different function in modern cosmology than it had for people in the ancient world. The Sun and the Moon were considered as gods in ancient times. An eclipse of these major celestial bodies was a bad omen and in many cases the religious leaders lost their authority because they were accused of not have fulfilled the sacrifices to the gods in a proper way. If, on the other hand, the king and his royal astronomers were able to predict eclipses, they could claim that they had power over the Sun or the Moon. The oldest known example of punishment of the responsible astronomers, who failed in their duty, comes from China. Zhongkang was the fourth King of the Xia Dynasty.

From Zhongkang's fifth year there is the following passage in the "The Punitive Expedition of Yin", a chapter of the Book of Documents that may be regarded as a reference to an eclipse:

This account fits well with the author’s calculations of the solar eclipse in 1961 BC, on October 9, Gregorian calendar, which was total in the Yin territory in the eastern part of the Yellow River valley (Henriksson 2008). In the Sumerian and Old Babylonian collection of omen texts, Enuma Anu Enlil, lunar eclipses could predict the death of the kings and a total solar eclipse (in 1859 BC) was considered to predict the beginning of the Old Babylonian Kingdom, and a total solar eclipse (in 1558 BC) was considered to predict the end of this kingdom (Henriksson 2006).

In cultures without written sources there are pictures that can be interpreted as depictions of solar and lunar eclipses. On the Swedish rock-carvings from the Bronze Age, 1800-500 BC, there exist many examples of images of solar eclipses that can be identified and exactly dated because the Bronze Age people had invented a reference system among the stars with six different types of ships that carried the sun along the ecliptic, the sun's apparent path among the stars (see Figure 1).

The different ships can be identified as constellations constructed from stars along the ecliptic, and the date of an eclipse can be approximately read from the position of the eclipsed sun in relation to one of the six ships. Planets visible during a total solar eclipse have in many cases been depicted as cup-marks. This means that it has been possible to check this interpretation by a direct comparison between the position of the cup-marks in relation to the image of the eclipsed sun and the corresponding computed positions, Henriksson (1999, 2005).

2. Method and Earlier Results

A computer program, devised by this investigator, for the calculation of ancient solar eclipses was completed in June 1985 and has since been successfully tested against all well-defined ancient observations back to 2500 BC. It is mainly based on the theory by Carl Schoch (1931), but with improvements concerning modern astrophysical parameters. All the formulae are expressed in UT (Universal Time) as used by the ancient observers. This means that the slowing down of the rotation of the Earth can be calibrated by a direct comparison with ancient observations. Schoch calibrated the lunar secular acceleration by the very accurate observation, by Timocharis in Alexandria, of a lunar occultation of the bright star Spica in Virgo, in 283 BC, Fig. 1. This means a calibration interval of about 2200 years. An independent critical test of the computer program with the position of the Moon computed in UT and the position of Venus in TT is shown in Fig. 5.

In the mainstream theory used today the formulae are expressed in the so-called Ephemeris Time (ET), a time flow proportional to the motion of the planets in their orbits, and after 1955 Atomic Time. This combined time scale is called Terrestrial Time (TT). The advantage with this technique is that the formulae are simpler, but the disadvantage is that the UT must be reconstructed from calculations of ancient observations. The transformation between TT and UT requires a set of useful timed records made by ancient observers because the time difference ΔT = TT — UT must be calculated and it is almost impossible to avoid circular arguments.

The most extensive study of ΔT has been performed by Richard Stephenson (1997). His analysis is based on the value for the secular acceleration of the Moon of 26 ±2 "/cy² determined from the telescopic observations of transits of Mercury 1677-1973, analysed by Morrison and Ward (1975). The oldest of these observations have low quality and there is no motivation for an extrapolation by a factor of 7 back to the ancient observations, based on a calibration determined from this 296-year interval.

During the last 25 years the use of terrestrial time has dominated as the time basis in all professional or commercially available computer programs for the calculation of eclipses. Unfortunately it has difficulties already at about 500 AD and is completely useless for observations made before 700 BC.

This computer program has been proved useful at least back to 3000 BC with errors of just a few minutes, caused by quasi-periodic non-tidal effects, Fig. 5.

In this method and computer program Carl Schoch’s values have always been used for the sidereal lunar secular acceleration, n = -29.68 ±0.04 arcseconds/century2, and for the braking of the rotation of the earth, 36.28 ±0.05 seconds/century² (Schoch 1931). The error limits are estimated from the observed deviations during 4500 years.

In 1985 a test of the computer program started with a comparison with the total solar eclipse in Babylon in 136 BC and a Chinese solar eclipse record, the so-called "double dawn" eclipse in Zheng, during the Zhou Dynasty, which could be dated to 899 BC. These results and identifications of several total solar eclipses depicted on Swedish rock-carvings from the Bronze Age were presented in 1996 at the Oxford V Symposium in Santa Fe, Henriksson (2005). After successful identification of the two important total solar eclipse records in Babylon, separated by 301/300 years, with the total solar eclipses in 1859 BC and 1558 BC, Fig. 7, it was possible to date the Old Babylonian Kingdom, the Old Assyrian Kingdom, the Old Hittite Kingdom and the 13th-20th dynasties in Egypt, presented in papers at the SEAC 2002 Conference in Tartu, Henriksson (2006) and at the SEAC 2004 Conference in Kecskemet, Henriksson (2007). At the Oxford 8 and SEAC 2007 Conference in Klaipeda an identification of the Chinese solar eclipses in 2075 BC, 2072 BC, 2071 BC, 1961 BC and 899 BC was presented, which can date the Xia, Shang and Western Zhou dynasties (Henriksson 2008).

There exist no original documents from the earliest dynasty, Xia, but much later chronicles mention important solar eclipses during the reigns of the first kings. An official Chinese group of archaeologists, astronomers, and historians started the Xia-Shang-Zhou Chronology Project (2000) to establish a historical reference frame. Astronomers lead by Ciyuan Liu, made new calculations with different values for the braking of the Earth’s rotation, Liu, Liu and Ma (2003). In this paper there is a critical discussion of the different interpretations of the ancient texts that mention possible solar eclipses. When the author compared his result with the historically possible alternative solutions for the ancient Chinese solar eclipses, suggested by Liu and his group, there was always one solution in common. The most exact information about ancient solar eclipses can be found on two separate cuneiform tablets (Sachs 1974), in the British Museum, which tell us about a total solar eclipse in Babylon, April 15th (Julian Calendar) in 136 BC, with the time given for three different phases of the eclipse. The difference between the time recorded in the cuneiform texts and the author’s computed time is 0 ±2 minutes. This is an excellent proof of the correctness of the parameters used in the computer program.

3. Unsolved Problems in the Calibration of Ephemeris Time

Stephenson and Morrison (1984) give a general review of the problems involved in the computation of ancient eclipses. However, the value -26.0 (arcseconds/century²), used as the lunar secular acceleration in the Atlas of historical eclipse maps, East Asia 1550 B.C.-A.D. 1900 (Stephenson and Houlden 1986), has unfortunately not been successful, as no reliable identifications with solar eclipses in the old Chinese texts have been possible. In "Observations and Predictions of Eclipse Times by Early Astronomers" by John Steele (2000), Steele tries to identify ancient eclipses by a computer program designed by F. R. Stephenson. He writes on page 15-16: "These programs are based upon the solar ephemeris of Newcomb (1895) and a corrected version of the lunar ephemeris designated j=2 (IAU 1968), incorporating a lunar acceleration of – 26 "/cy² as determined by Morrison and Ward (1975) from analysis of the transits of Mercury". His values for ΔT are taken from Stephenson and Morrison (1995). These values are obtained from a spline fit of both timed and untimed eclipse observations made in various cultures. However, a simple parabola is expected from the tidal braking, but this does not satisfy all the constraints of the data, as is discussed in Stephenson and Morrison (1995). The introduction of a spline curve is an attempt to correct for systematic errors caused by wrong values of ΔT and n.

There is no way to avoid circular arguments and the interval is so short that non-tidal effects are incorporated in the parabolic fit of ΔT, Fig. 3. Stephenson et al. extrapolate back to the ancient eclipses via low quality lunar eclipses. Great deviations are interpreted as unknown non-tidal effects (Morrison and Stephenson 2002). In another recent study by Huber and de Meis (2004) the number of known possible lunar eclipses in the late Babylonian texts has been increased from 172 in 1973 to 269 and the corresponding known possible solar eclipses from 32 to 90. However, these authors admit that it is not a trivial task to distinguish between predicted and observed eclipses. They believe that the best way to determine ΔT is from lunar eclipses. In my opinion, the results from this method have so far not been very useful because of the low precision in the lunar eclipse observations. Furthermore, if it is hard to judge whether these lunar eclipses are genuine observations or theoretical predictions, it seems questionable to me if this new material is a major step forward.

The value of n = -26 "/cy² is fixed in modern computer programs, but the user has an option to change ΔT! This violates the principle of conservation of angular momentum in a closed system. It is written on page K8 in the Astronomical Almanac: "To calculate the value of ΔT for a different value of the tidal term (n '), add -0.000091(n '+26)(year-1955)² seconds to the tabulated values of ΔT." In a paper by Xu Huaguan et al. (1996) there is a table with values of n determined by two modern methods such as satellite measurements of the tides made during 1977 and 1978 and LLR-measurements between 1969-1987. The values of n from the satellite measurements are -27.3 ±5.2, -27.4 ±3.0 and -25.0 ±3.0 "/cy² and the results from LLR are -24.6 ±5.0, -23.8 ±4.0, -25.3 ±1.2 and -25.4 ±0.1 "/cy². The last value comes from the analysis by Xu Huaguan et al. (1996) of LLR-data during 1969-1987. The uncertainty is obviously underestimated in this paper as Williams et al. (2002) got n = -25.71"/cy² from much better LLR-data and an almost twice as long time interval. The most recent value, n = -25.85"/cy², can be found in Williams et al. (2008), determined from a total of 16,601 ranges measured between March 16, 1970 to December 27, 2007. The author predicts that in about five years the measured value of n by this method will be very close to -26.0 "/cy² because this value is already included in the numerically integrated ephemerides necessary to compute the position of the Moon. It is, in my opinion, impossible to avoid circular arguments by trying to determine n by this method, and the ancient eclipse observations do not support the value used.

Steel (2000) avoids a discussion of several famous early total solar eclipses such as that predicted by Thales from Miletos, in 585 BC, and reported by Herodotus to have taken place during a battle, Fig. 8. That eclipse had earlier been used as a fundamental test of the formulae and methods for calculations of ancient eclipses. Frustrated modern eclipse calculators who cannot calculate this total solar eclipse, very famous during antiquity, have convinced themselves that it probably was a total lunar eclipse!

4. A Test of Kepler’s Third Law

Williams and Dickey (2002) have been aware of an unexpected problem during their analysis of the LLR-measurements and write on their page 3: "The eccentricity rate presents a mystery. Lunar tidal dissipation has a significant influence on this rate, and the total rate should be the sum of Earth (1.3 x 10-11/yr) and Moon (-0.6 x 10-11/yr) effects. However, we find it necessary to solve for an additional anomalous eccentricity rate of (1.6 ±0.5) x 10-11/yr (Williams et al. 2001) when fitting the ranges. The anomalous rate is equivalent to an additional 6 mm/yr decrease in perigee distance. Note that the integrated ephemerides (and DE series) do not include the anomalous eccentricity rate."

All known interactions with the Moon have been taken into account including the nongravitational perturbation of the radial component from solar radiation pressure and relativistic precession of the geodesic first determined by Willem de Sitter (1916). They used the value of 1.90 "/cy² for the relativistic precession of the geodesic, calculated by Chapront- Touzé and Chapront (1983). Precession rates as small as 0.001 "/yr were taken into account. The position of the Moon has been computed from the numerically integrated ephemerides DE403 (Standish et al., 1995) and DE336 which use n = -26.0 ±2"/cy² considered to be supported by the analysis of the ancient eclipses by Stephenson and Morrison (1984). However, this is a great mistake because non-tidal effects, caused by the global warming since 1680, have disturbed the calibration during the too short interval 1677-1973. The author has found a correlation between ΔT-values published by Martin (1969) and Stephenson et al. (1984) and the yearly mean temperatures by Moberg et al. (2005). The Earth rotates faster when the temperature is low and slower when the temperature is high, as today. These ephemerides should instead have been calculated with n = -29.68 ±0.04 "/cy² determined by Carl Schoch (1931) from a direct analysis of ancient solar eclipses, and verified by the author back to at least 2500 BC. An even better value is n = -29.65 ±0.04 "/cy² obtained by the author after correction for non-tidal effects.

The elliptical motion of two bodies is described by:

where r is the distance between the two bodies, a is the semi-major axis of the ellipse, e is the eccentricity and E is the eccentric anomaly. At perigeum E = 0, and after derivation with respect to time we get

Williams et al. (2002) have already solved for the yearly increase of a, and da/dt is assumed to be equal to zero. The second least square solution was interpreted as an anomalous eccentricity rate of (1.6 ± 0.5) x 10-11/yr, which according to (2) gives a decreasing perigee distance dr/dt = 6.15 mm/yr, with a = 384399 km and e = 0.0549 and da/dt = 0, in good agreement with the 6 mm/yr given by Williams et al. (2002).

An alternative explanation for this anomaly is, in my opinion, that a too small value of the lunar secular acceleration in longitude has been used in the calculations of the motion of the Moon and therefore an additional radial increase da/dt can be expected. By putting the anomalous dr/dt = 0 into (2) we can calculate an additional yearly increase in the distance to the Moon of da/dt = 6.51 mm/yr. If this correction is added to the result, 37.9 mm/yr, from the first least square solution by Williams et al. we get 44.41 mm/yr as the result from the LLR. The published uncertainty was ±0.7 mm/yr for the first least square solution of the LLRmeasurements, but in an email from Williams, sent on April 07, 2004, he writes that the expected error in the published value 37.9 mm/yr, "is no worse" than ±0.2 mm/yr.

After derivation of Kepler’s third law for two-body motion with respect to time we get: where n is the mean motion of the Moon. This formula can be used to calculate the radial acceleration of the Moon that corresponds to the directly observed secular lunar acceleration -29.65 "/cy², from ancient eclipses. The result is da/dt = 43.86 mm/yr. Williams et al. (2002) have also taken into account the gravitational attraction from the Sun, which means that the result from (3) should be divided by 1.0028, which gives the final result da/dt = 43.73 mm/yr from the value for n by this investigator.Williams et al. The difference between the measured radial acceleration from LLR and the corresponding value from the lunar secular acceleration of the longitude from ancient eclipses da/dt (LLR - eclipses) is +0.68 ±1.92 (mm/yr). The expected theoretical deviation in da/dt (LLR - eclipses) is ±0.0 (mm/yr) for Einstein - de Sitter. The precession of the local geodesic in the Earth-Moon system of references, according to the Einstein - de Sitter Cosmology, is included in the sidereal secular lunar acceleration determined from the analysis of ancient solar

eclipses. The corresponding radial lunar acceleration had been added to the LLRmeasurements in the calculations.

5. A Modified Theory of Gravity

Dvali et al. (2003a) write on page 2: "Recent observations suggest that the Universe is accelerating on the scales of the present cosmological horizon [1]. This indicates that, either there is a small vacuum energy (or some 'effective' vacuum energy), or that conventional laws of Einstein gravity get modified at very large distances, imitating a small cosmological constant [2, 7, 8]. The first possibility is unnatural in the view of quantum field theory1, since the required value of vacuum energy ~ 10^-12 GeV⁴ is unstable under quantum corrections.

This unnaturalness goes under the name of the Cosmological Constant Problem. In this respect the second approach, of modifying gravity in far infrared can be more promising since it is perturbatively stable under quantum corrections. The unnaturally small value of the vacuum energy is replaced by the idea that laws of conventional gravity break down at very large distances; beyond a certain crossover scale r_c. The value of r_c is perturbatively-stable ie. it does not suffer from cut-off sensitive corrections experienced by vacuum energy."

For a radius of crossover of 6 Gpc (1.85 x 10²⁸ cm) the expected cosmological precession is -0.516 "/cy² according to formulas (13), (14) and (15) in Dvali et al. (2003a). This corresponds to an additional lunar radial acceleration of +0.76 mm/yr. The observed value +0.68 mm/yr is very close to this value, but the expected value 0.0 mm/yr from the theory of Einstein - de Sitter lies also within the error limits of ±1.92 mm/yr, from the second least square analysis by Williams et al. (2002). The standard deviation ±1.92 mm/yr, is calculated from the uncertainty in the anomalous eccentricity rate ±0.5 x 10-11/yr, and include probably errors caused by the unfortunate use of n = -26.0 ±2"/cy² in the numerical integrations by Standish et al. (1995). This anomalous eccentricity rate has not been included in the latest numerical integration DE421, because its source is still unknown Williams et al. (2008).

If the systematic errors are properly corrected for in the second solution by Williams et al. (2002), the theory by Dvali et al. (2003a) may be considered to be in somewhat better agreement with the observations than the theory by Einstein - de Sitter. However, the precession of the geodesic predicted by the General Theory of Relativity by Albert Einstein in 1916 and specified in detail by Willem de Sitter in 1916, is clearly demonstrated to be within the limits of error in this investigation.

6. Discussion

This investigation has confirmed the precession of the geodetic in the Earth-Moon-system, predicted in Einstein’s General Theory of Relativity, with a deviation of 1/3 sigma, and a deviation only 1/17 sigma from the preliminary parameter in the string theory of Dvali et al. (2003a). LLR measurements have been used in combination with the author’s improved value for the lunar secular acceleration in longitude of -29.65 ±0.04 "/cy², in complete agreement with the total solar eclipses back to at least 2500 BC.

Stephenson and his different co-authors have used a too small correction for the braking of the rotation of the Earth and the corresponding acceleration of the longitude of the Moon. Therefore the analysis by Williams and Dickey (2002) of the LLR measurements has failed to confirm the precession of the geodesic in the Earth-Moon system predicted by de Sitter from Einstein’s General Theory of Relativity. The deviation is three sigma and significant. When the result from the new Apache Point Observatory Lunar Laser-ranging Operation (APOLLO) in New Mexico is analyzed, with a numerical model based on the author’s value for the lunar secular acceleration, -29.65"/cy², it might be possible to judge if Einstein’s theory is good enough to explain the observations or whether more advanced theories will be needed. If for instance the string theory of Dvali (2003b) can be confirmed then we may not need the enigmatic Dark Energy.

Acknowledgments: I want to thank Dr Brady Caldwell, Department of Astronomy and Space Physics, Uppsala University and Associate Professor Mary Blomberg, Department for Archaeology and Ancient History, Uppsala University, for their comments and corrections of my English.