1. Comment to: "Lunar tidal acceleration, Earth’s deceleration, Delta-T"

I have no problem with the first half of this section, where Peters describes the gravitational interaction between the Earth and the Moon, but when he discusses the calibration of the lunar secular acceleration, I disagree. He claims that the value for the lunar secular acceleration, -26 ± 2 arc seconds/century2 ("/cy²), used by Richard Stephenson, is sufficiently accurate for early epochs back to the 8th century BCE even if this means an extrapolation with this parameter, determined from an analysis of Delta-T ( Δ T) from the transits of Mercury 1677-1973 (Morrison and Ward 1975). In the same solution, the excess motion of the perihelion of Mercury was found to be +41.9 ± 0.5 arc seconds/century ("/cy), which deviates by 1.13 "/cy from the prediction by Einstein's General Theory of Relativity, confirmed by other investigations. These Δ T values were then used by Ward to calibrate the lunar secular acceleration from observations of stellar occultation by the Moon. However, this is a too short time-interval to get a useful determination of the tidal-parabola because we have a strong influence by the non-tidal effect caused mainly by variations in the global temperature during the last 300 years, from the Maunder Minimum to the Modern Global Warming, see Figure 1.

The Lunar Laser Range (LLR) measurements give a very accurate determination of the distance to the Moon, but the position in the sky must be calculated from a very complicated numerical model. An important input parameter is the lunar secular acceleration in longitude and the initial calculations were performed by the value -26 ±2"/cy² used by Stephenson et al. After a series of refined models the value has been improved in DE421 to 25.85"/cy² according to Williams et al. (2008). This value has increased all the time and in Williams et al. (2002) it was -25.71"/cy². I cannot understand how Peters can claim that this is an

independent confirmation of the value for the lunar secular acceleration of the Moon, -26 ±2"/cy² used by Stephenson et al. To me it is a typical result of circular arguments. The earlier values of the lunar secular acceleration 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. It is hard to believe that this a real increase in the lunar secular acceleration during the last 40 years. To me it strongly indicates a systematic error in the numerical model, in my opinion a too low value for the lunar secular acceleration. They should have used Schoch’s value. A solution to this problem can be found at the end of this paper.

Peters continues with a discussion of the increase in the length of the day caused by the tidal braking of the Earth’s rotation and the reconstruction by Stephenson et al., see Figure 2. The dramatic changed in l.o.d. from 1.4 ms/cy to 2.4 ms/cy, around 1000, in Figure 2, was explained in the paper by Stephenson and Morrison (1984) as mainly the result of the socalled glacial rebound after the last Ice Age. The latest corresponding values, mentioned by Peters, are from 1.8 ms/cy to 2.3 ms/cy, but no known geophysical explanation has been found. This effect is in my opinion just a result of Stephenson’s unfortunate choice of -26 ± 2"/cy² as the value for the lunar secular acceleration used in his calibration of Δ T from telescopic observations and timing by astronomical clocks, the oldest with too low precision. The low quality of the data in Figure 2, permits many solutions with corresponding low quality. The red line solution according to Schoch goes directly through the modern data to the Babylonian eclipse observations via the medieval well-defined observations of total solar eclipses. Schoch’s value for l.o.d. was calibrated to be zero in 1900.

The introduction of great non-tidal effects is just a cosmetic way to compensate for the badly determined value -26 ± 2"/cy² for the lunar secular acceleration.

2. Comments to "Henrikson’s observation"

Peters begin this chapter in the following way: "In his paper Henriksson (2010) uses a number of supposed eclipse observations that pre-date the historical observations used by Stephenson, and uses this as an extended observational basis for the lunar ephemeris, and a confirmation of the larger value of the lunar tidal secular acceleration that he uses."

This is a completely correct description if the word "supposed" is removed. Peters attitude to information, that he obviously has no knowledge about, is not what one can expect from an experienced scientist. The general attitude should be to make no statements about issues outside your own area of competence. He is so strongly convinced that I must be wrong that it is not necessary to look at details in my Figure 6, with more than 50, to him unknown, mostly total solar eclipses, because I have tried to prove that it is possible to solve problems that Stephenson has declared as impossible to solve. There exist a great number of well documented total solar eclipses before 700 BCE that Stephenson considers to be the oldest limit for his method. That is mainly the result of his inability to identify the ancient Chinese solar eclipses (Stephenson and Houlden 1986).

What Peters consider as "supposed" eclipses are clearly described phenomena in the sky such as an unexplained darkness in the middle of the day and sometimes that stars were visible during the day. These are real solar eclipse observations by intelligent people. If the year of rule for a king is mentioned, it is possible to establish an exact chronology. The problem for the users of the method by Stephenson is that it is dependent on information about the time of the eclipse. If this information does not exist, the eclipse is classified as "supposed" and is not a historical event. My method based on the theory by Schoch is so well calibrated and stable that it is enough to know an approximate time of the year or, if it is a total solar eclipse, it is enough to know if it was in the morning, in the middle of the day or in the afternoon. Sometimes there may be several hundreds of years between such rare events and there is no risk for confusion. If a combination of planets or a particular constellation was mentioned as visible during the eclipse, it is very unlikely that the identification is wrong. All the solar eclipses plotted in my Figure 6, in the Journal of Cosmology, are well documented and it is enough to select three of these eclipses to define a unique tidal parabola. This material is really very powerful.

However, it was not possible to discuss all of my successfully identified solar eclipses so I mentioned only some of the most important cases such as the two not earlier identified total solar eclipses in the beginning and at the end of the Old Babylonian Kingdom. Peters writes concerning my identification of these two important total solar eclipses observed in Babylon: "On this, the assyrologist, prof. H. Hunger had the following comment (on the HASTRO-L mailing list):" see Peters text.

This was my answer to prof. Hermann Hunger:

I did not get any reply to this email from professor Hunger, which means that he did not found any important errors or maybe he even considered my results as interesting. It is typical for the debate technique used by Peters that he only quoted the email by Hunger and not my answer nor did he mentioned the absence of a reply to my answer. He is not interested in an objective investigation of all relevant sources, because for him it seems to be more important to neutralize my strong arguments to prevent a discussion of the weak points in his own favourite hypothesis.

Peters first classified all the eclipses earlier than 700 BCE in my Figure 6 as "supposed", but after the email by Hunger and my response to his email he now admits that: "Moreover, the month in the Babylonian calendar (Abu) seems to match, and as Henriksson shows in his fig. 7, the eclipse took place in the constellation of Virgo, next to the constellation of Libra referred to in the text." This means that he accepts my calculation and identification of the eclipse described in the Babylonian text. However, this single total solar eclipse is enough to prove that my method is correct.

Peters continue: "Finally, despite Henriksson’s conjecture to the opposite (at the end of his sections 3), it is possible to compute these ancient eclipses and put them in Mesopotamia, using the modern value for the lunar tidal acceleration of around -26"/cy/cy, and Stephenson’s corresponding curve for DeltaT. Espenak (2009), in his Five Millenium Catalogue of Solar eclipses, predicts that the shadow track for the eclipses –1858-05-15 and –1557-09-11 ran over Mesopotamia. He uses the ELP2000 with new parameters from Chapront et al. (2002) from LLR." It must have been a surprise to Peters that these "supposed" solar eclipses could be confirmed by the most advanced computer programs available and became "real" eclipses. In Peters original attack on my paper in the Journal of Cosmology the Subject on [HASTROL] was "Computational error". I think that it is time now for Peters to regret this statement. However, my calculations have been performed without any correction for unknown nontidal effects, but in the computer program by Espenak it is necessary to include a great correction for non-tidal effects determined by Stephenson from eclipses back to 700 BCE. This correction curve is then extrapolated about 1000 years beyond its last point. There exist also different correction curves and it seems not to be possible in advance to know which of these curves is the best. It seems to be necessary to consult me first.

Stephenson has a very good understanding of all problems with calculations of ancient eclipses and he has pointed out the importance of eclipses close to the horizon. It is therefore very difficult to understand why he not has understood that his own method is falsified by some well-known ancient solar eclipses. Peters writes: "In his article Henriksson specifically addresses two other eclipse reports: the "double dawn" eclipse frame the Bamboo Annals, and the eclipse predicted by Thales according to Herodotus. Stephenson c.s. have studied and discussed these potential observations in detail (Stephenson 1992; Stephenson & Fatoohi 1997). They rejected the former, and did not use the latter (Stephenson 1997 p. 343) because it was not sufficiently specific on the date and location. Nonetheless Henriksson’s claim (in his fig. 8) that ‘other modern computer programs’ can not calculate Thales’ eclipse is false; check e.g. Espenak’s Catalogue for –584-05-28."

I only admit that my statement in fig. 8 is false if it can be proved that Espenak in his calculation has chosen the correct curve by Stephenson for the non-tidal effect. Espenak’s positive result is contradicted by Stephenson’s opinion that it was not sufficiently specific on the date and location. This is a very important eclipse that has been used to test methods to calculate eclipses since the time of Ptolemy. Simon Newcomb claimed proudly that it could be calculated by his new method, but P. Neugebauer discovered a severe calculation error that shifted the total zone of totality away from the correct area. Carl Schoch used this eclipse in his calibration and localized an area between Lake Tatta and the River Halys as the most probable area and the year was given in relation to the first Olympiad and known within narrow limits, so there exists no alternative solution. This means that Stephenson must have realized that his method was falsified. Espenak solved this dilemma by a suitable choice of curve for the non-tidal effect.

The fact that Peters has not mentioned any suitable solution by Espenak to the "double dawn" eclipse means that it is likely that this very critical eclipse is without the range even of his greatest correction curves. That Stephenson has failed is obvious.

My solution can be seen in Figure 3a+b. It is written in the Old Version of the Bamboo Annals that: "During the first year of King Yi the day dawned twice at Zheng." Yi was a King of Western Zhou and the first year of his reign has been dated to between 966 BC and 899 BC by different authors. K. Pang (1987) analyzed this eclipse and dated it correctly to 899 BC, but his calculation of the brightness of the morning sky was not performed correctly Henriksson (2008).

At the Oxford V Conference in Santa Fe 1996 I presented my own calculations of the brightness of the morning sky for the annular solar eclipse on April 11, Gregorian, 899 BC, at 04.49 local mean solar time, ending 2 minutes before sunrise (Henriksson 2005).

3. Comments to "Henriksson’s value for the lunar tidal acceleration"

I have no problems with Peters’ presentation of the general theoretical background in this section, but he contradicts himself when he claims that I have violated these rules in my program. I have pointed out that the 300 year interval used by Morrison & Ward for the calibration of the lunar secular acceleration value -26 ± 2"/cy² is too short because of the significant variation in the non-tidal effects during this short period of time and that it is impossible to separate the tidal and non-tidal effects from the observed Δ T. I have illustrated this problem in Figure 1, with Δ T calculated for the values -26"/cy² and –29.68"/cy² from the formula from the Astronomical Ephemeris that I quoted in section 3 in the Journal of Cosmology. In the method used by Schoch, with a calibration interval of more than 2000 years, the determination of the tidal braking of the Earth’s rotation can easily be separated from at least effects with periods shorter than 500 years in the non-tidal variation of the Earth’s rotation. This is the reason why I believe that Schoch’s determination of the lunar secular acceleration is much more realistic than Morrison and Wards. They have also correctly admitted that their error is ±2"/cy².

When I subtracted Schoch’s tidal parabola, determined from completely different observations during more than 2000 years, from the Δ T values in the Astronomical Ephemeris the total maximum amplitude of the non-tidal terms is less than ± 50 seconds, see the blue curve in Figure 1. These variations correlate very well with the variations in the Earth’s global mean temperature. However, the non-tidal influence is so small that I have never found it necessary to include it in my calculations of ancient solar eclipses. When I finished my computer program in 1985, I made some experiments with different values for the lunar secular acceleration, but the improvements were insignificant and I decided to use the original value by Schoch.

It is sometimes difficult for me to follow the logic in Peters’ text, and it seems as if he also has had problems to understand my text. It is therefore meaningless to refute his critique in detail because he contradicts himself. If his critique of my method is correct it should be impossible for me to perform all the successful solar eclipse calculations I have presented in Figure 6, in the Journal of Cosmology. He also admits that my calculation of the Moon’s position during the conjunction observed by Timocharis in Alexandria, in 283 BCE, is unexpectedly precise. He writes in his section "The occultation of Spica in 283 BCE": "There remains the question how Henriksson and DE406 can give such similar results for the Moon’s position and time of conjunction, considering their difference of 4"/cy/cy in the lunar longitude."

I have a version of my relatively simple computer program that even was possible to run in FORTAN 77 on an ATARI 1040, in 1988, because no complex corrections are needed. This is an example where William Occam’s razor should be used and he wrote: "entities must not be multiplied beyond necessity" (entia non sunt multiplicanda praeter necessitatem), and to quote Isaac Newton: "We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances. Therefore, to the same natural effects we must, so far as possible, assign the same causes."

I think that the best value of the lunar secular acceleration is the value that needs no corrections at all!

4. Comments to "The occultation of spica in 283 BCE"

In this section Peters gives a good description of the observation by Timocharis in Alexandria in 283 BCE. There is also a good presentation of the discussion of my Figure 4. It was Victor Reijs who first tried to re-calculate this figure with a Swiss Ephemeris program and his result was that there was no actual occultation of Spica by the Moon, as in my Figure 4, only a conjunction. After an intense discussion with Victor Reijs, Paul Hirose and Anton Peters we concluded that my calculated declination for Spica was 7' to low. Fortunately, this figure was just my attempt to illustrate the situation observed by Timocharis from Alexandria, used by Carl Schoch to calibrate the secular acceleration of the longitude of the Moon. Schoch’s calibration in 1926 has of coarse nothing to do with my 7' error in the declination of Spica in my Figure 4. After re-reading the text by Schoch I now understand that this was not his final calibration of the lunar secular acceleration, but the starting value in an iterative process with other ancient observations, mostly total solar eclipses. However, I am very thankful to these competent colleagues that this error finally was discovered.

5. Comments to "Eccentricity rate"

It is not interesting to discuss any details in Peters’ manipulations of the formulas in this section. He cannot explain the observed anomaly and the result is meaningless. Peters write: "Moreover, the perigee distance is of no interest here but apparently Henriksson has been led astray by the way Williams presented his results." This is true today but it was not clear in 2004 when I wrote most of the text in section 4, in the Journal of Cosmology. The problems with the perigee distance in DE403 are now solved in DE421 (Williams 2008). I interpreted the anomaly in perigee distance as a result of their initial value for the lunar secular acceleration, -26"/cy², because it was not clear to me how they had corrected for the precession of the geodesic, according to Einstein, before they used it in the modified Third Law by Kepler to calculate the LLR-distances and my exercise with the formulas for elliptic motion is no longer relevant. The background is explained below. During February 2004 I finished my investigation of the non-tidal effects on the Earth’s rotation and concluded that they were not significant for the analysis of ancient observations. However, Schoch had, of course, not taken into account the modern values of Δ T, see Figure 1. I made a least square fit to get an improved tidal parabola including data from the ancient solar eclipses and the modern Δ T-values. From this parabola I calculated my improved value - 29.65"/cy² for the lunar secular acceleration. However, the difference from Schoch’s value is so small that it is not necessary to include it in my computer program.

At the end of February 2004 I realized that it was no longer possible to make any further improvements of the lunar secular acceleration and I decided to use Kepler’s Third Law to test if it was compatible with the LLR-measurements of the radial motion of the Moon. I found that the Moon accelerated about -4"/cy² more in longitude than expected from its radial velocity. The deviation was significant and I understood that it might be an effect of Einstein’s Theory of General Relativity (Einstein 1916). When I looked at the relativistic formulas for the motion of the Moon, there was a term –2 x 1.92"/cy² that very nicely matched my deviation -4"/cy² . I thought that this might be an important confirmation of Einstein’s theory and I called Ulf Danielsson, professor in theoretical physics at Uppsala University, to get a professional opinion. We spend the afternoon on Internet and tried to understand the details in the analysis of the LLR-data. It became clear that the value -26"/cy² had initially been used for the lunar secular acceleration, but Einstein’s effect –2 x 1.92"/cy² was already included in the numerical model. After that Danielsson said that you have either confirmed Einstein’s theory or maybe have discovered something even more important. However, it is confusing that the Einstein effect is of the same magnitude as the difference between the two different values for the lunar secular acceleration discussed in this paper. I wrote a series of emails to James Williams, responsible for the LLR-project, to check if I had misunderstood something. This was not the case, and he kindly sent me a lot of important papers he and his colleagues had written, as I mentioned in the Journal of Cosmology. They had taken into account all known great and small effects on the motion of the Moon, and I thought that the difference between -26"/cy² and -29.65"/cy² should cause a significant systematic error that reduced the precision in the final result. To my disappointment everything seemed to be normal, in the beginning of the paper, but then Williams described an unexpected anomaly. I wrote the following about this in the Journal of Cosmology: "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. ′"

After publication of DE421 (Williams 2008) the problems with the perigee distance in DE403 is solved. Williams write on page 7: "When the dissipitation effects from Earth and Moon are added together, the resulting acceleration in longitude is –25.85"/cy² and the semimajor axis rate is 38.14 mm/yr. These derived values depend on a theory which is not accurate to the number of digits given." Unfortunately, I got only seven days to write a reply to Peters’ critique and it was not until during the last day that I read this text in Williams (2008), also referred to by Peters, so I have not been able to take this information into account in the beginning of this paper.

It is now clear that the lunar secular acceleration –25.85"/cy² determined by the LLRmeasurements is only valid in the Earth-Moon inertial system, used in the calculation of the LLR-distances by the modified Third Law of Kepler. Peters, Steele (2000) and others have taken the value, from the different determinations by LLR, of the lunar secular acceleration, as a confirmation of their own favourite value, -26 ± 2"/cy², determined by Morrison and Ward. It must be the determination of this value that Williams considered as "not accurate to the number given". However, the value from LLR must be corrected for Einstein’s precession of the geodesic –2 x 1.92"/cy² = -3.84"/cy² (Nordtvedt 1996) to get the sidereal lunar acceleration used in the calculations of the solar and lunar eclipses.

The sidereal lunar secular acceleration determined by LLR is therefore:

6. Comments to "Conclusions"

All statements by Peters in his "Conclusions" are false, according to my opinion, and he has contradicted himself when different parts of his text are compared.

Acknowledgements I want to thank prof. Paul S. Hirose, Dr Anton R. Peters and Victor Reijs for their discussions and calculations of the precession of Spica during Timocharis observation in Alexandria in 283 BCE. I also want to thank assistant prof. Mary Blomberg for here reading of my manuscript.