1. Lunar Tidal Acceleration, Earth's Deceleration, and Delta-T

Actual records of lunar observations have been preserved since late Babylonian and ancient Chinese times (dateable from about 8th century BCE). Both civilizations recorded lunar and solar eclipses; the latter in particular help define the mutual postion of Sun, Moon and Earth with high accuracy. This milennia-long baseline is of great potential value for calibrating ephemerides and detecting long-term trends, cosmological or otherwise.

While these astronomical observations involving the Moon can be very useful, the lunar ephemeris is the hardest to compute because even in its simplest form it is three-body problem which can not be exactly solved. Yet already Halley suspected in 1695, based on classical eclipse observations, that the Moon has a secular acceleration that could not be explained by Newton's solution for the lunar orbit. Delaunay and others proposed that much of this could be explained by the tides. See Kushner (1989) for a detailed account.

A qualitative description is as follows: the Moon raises a tide in the Earth's oceans, but this bulge does not only follow the Moon in its monthly orbit around the Earth, but is also dragged along by the rotation of the Earth. As a consequence, the global tidal bulge runs ahead somewhat of the Moon. This mass gives an additional pull to the Moon in its orbit: it is accelerated, but being in a gravitationally bound orbit this means that the Moon moves into a higher orbit, away from the Earth, and gets a smaller angular velocity. On the other hand, conservation of angular momentum requires that the Earth's rotation slows down: the tides work as a brake on the rotating Earth.

The problem has been that the Earth's rotation has been used as the best available and most stable clock for timing astronomical observations (Universal Time, UT). So with lunar observations timed in UT, it is very hard to disentangle lunar tidal acceleration and terrestial deceleration. That the Earth slows down was convincingly demonstrated by Spencer-Jones in 1939. Subsequently, it was decided to use the time variable in the best available ephemerides as astronomical clock: ephemeris time (ET); its current incarnation is Terrestial Time (TT). The accumulated difference between these time scales is called DeltaT = ET - UT. See Britton (1992) Appendix 1 for an account covering the first half of the 20th century. Since 1955 very stable atomic clocks have been available, which are used to maintain the time scale International Atomic Time (TAI) calibrated to have the same rate as ET: as far as we can tell TAI and TT have a constant offset.

Studies to determine the lunar tidal acceleration have continued, with varying results: because all historical observations were timed against the Earth's rotation, it is very difficult to separate lunar tidal acceleration from terrestial deceleration. One study by Morrisson and Ward (1975) used observations of transits of Mercury over the solar disk over almost 3 centuries, next to lunar stellar occultation timings, to independently determine DeltaT, and to establish a value for the lunar tidal acceleration: -26"/cy/cy . One of the most prolific authors on the subject, prof. F.R. Stephenson, has consistently used this value for his lunar ephemerides and used it to reconstruct the history of the Earth's rotation and the DeltaT curve using all available and reliable historic sources. His book gives an (almost) definitive account (Stephenson 1997), and Morrison and Stephenson's DeltaT values are commonly accepted, e.g. in the annual Astronomical Almanac (p.K8). For updated values for the earliest historic past, see Morrison and Stephenson (2004, 2005).

Since the lunar landing missions left retro-reflectors on the surface of the Moon, it has been possible to measure the Moon's position (specifically, its distance) with unprecedented centimeter accuracy through Lunar Laser Ranging (LLR); see e.g. Dickey et al. (1994). The best theory to fit these observations is incorporated into the numerical integrations known as Development Ephemerides of NASA's Jet Propulsion Laboratory. Among others, to account for the tidal acceleration, they add a somewhat simplified global tidal model to the Earth's gravitational field; see Chapront et al. (2002) section 8 . The secular recession rate and tidal contribution to the acceleration of the Moon can be derived from these fits. After about one lunar nodal cycle (18 years) of observations, these fits stabilized: the most recent public ephemeris, DE421, found a value of -25.85"/cy/cy for the lunar tidal acceleration and 38.14 mm/yr for the distace (Williams et al. 2008 p.7), very close to the value from Morrison and Ward as used by Stephenson c.s.

The Earth's rotation is monitored these days by VLBI of radio-telescopic observations of very distant quasars, quite independent of solar system dynamics and the Moon's orbit in particular. It is clear that the Earth's rotation is erratic on all time scales (see e.g. Dickey 1995). One of the results of Stephenson c.s. is that the long-term deceleration of the rotation of the Earth is smaller than would be expected from the corresponding observed lunar tidal acceleration, considering conservation of angular momentum. Morrison and Stephenson (2001 p.262) find a change in the length of day (l.o.d.) of only about +1.8 ms/cy instead of the expected 2.3 ms/cy, and an equivalent parabolic evolution of DeltaT of only about 32s/cy/cy (Morrison & Stephenson 2004 p.332) instead of the expected 42s/cy/cy (Stephenson 1997 pp.37,38 and p.516).

Apparently something is accelerating the Earth's rotation. The most likely candidate is socalled glacial rebound: during the ice age the ice caps around the poles depressed the continental crust into the Earth's mantle. Since the melting of these glaciers, the Earth's crust is still not in equilibrium: the polar regions are still rising. Given that mass, volume, and density of the Earth do not change, this implies a net motion of mass within the Earth from the equator to the poles. This leads to a smaller value for the moment of inertia of the Earth, and conservation of angular momentum then causes the Earth to spin faster (see e.g. Stephenson & Morrison 1995 pp. 197,198). Indeed satellite observations of the Earth's gravity field have shown changes in the Earth's dynamical form factor J2, and this has been included into the latest IAU precession model (Bourda & Capitaine 2004, sections 4&5).

Besides these long-term trends, there are excursions in the DeltaT values with periods of decades and millennia, but the historical observations are too scattered and inaccurate to model them or identify a physical cause.

2. Henrikson's Observations

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 acceleraton that he uses.

In particular, he uses two eclipses, in 1859 BC and 1558 BC. In discussion, it became apparent that he uses an interpretation of a particular omen (foretelling), from the old Assysrian text known as Enuma Anu Enlil. This is a standard collection of omens of various origins, organized by main subject, and codified sometime in the second half of the second millennium BCE. In the translation of van Soldt (1995):

On this, the assyrologist prof. H. Hunger had the following comment (on the HASTRO-L mailing list):

Henriksson's interpretation that the text refers to a solar eclipse may be valid, considering that apparently a later Babylonian commentator thought the same (Hunger mentioned that the comments were added not before 8th cy. BCE). However, it is not an observation report (such as known from much later clay tablets), and there is no unambiguous absolute date.

Henriksson argues that in the relevant period (first half of 2nd millennium BCE) there have been only two total solar eclipses visible in Mesopotamia, in -1858 and -1557, so any mention of a solar eclipse must have referred to these. Moreover, the month in the Babylonian luni-solar 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. But mind that the omen does not mention any year and is not linked to any other dated historical record; indeed the comment (which mentions the month) is in future tense and does not refer to a past event. Henriksson's claim in his section 2 that he established the absolute chronology for the 2nd millennium BCE is insufficiently founded.

Finally, despite Henriksson's conjecture to the opposite (at the end of his section 3), it is possible to compute these ancient eclipes and put them into Mesopotamia, using the modern value for the lunar tidal acceleration of around -26"/cy/cy, and Stephenson's corresponding curve for DeltaT. Espanak (2009) in his Five Millennium Catalog of Solar Eclipses, predicts that the shadow track for the eclipses of -1858-05-15 and -1557-09-11 ran over Mesopotamia. He uses the ELP2000 with new parameters from Chapront et al. (2002) from LLR.

This shows at least that these eclipses can not be used to discriminate between Henriksson's and Chapront's values for the lunar tidal acceleration, so can not be used to improve the lunar ephemeris. The reason appears to be that the shadow track runs largely West-East, so a wide range of DeltaT values – both from Henriksson and from the trend curve of Stephenson c.s. – will put a solar eclipse in Mesopotamia.

In his article Henriksson specifically addresses two other eclipse reports: the "double dawn" eclipse from the Chinese 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; Stephenson 2008). 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. the catalog of Espenak & Meeus (2009) for -584-05-28.

Stephenson (2008) formulated four criteria that a historic record must meet to be useable for studying the lunar orbit and Earth's rotation:

3. Henrikssons' Value for the Lunar Tidal Acceleration

Henriksson states he uses a computer program he wrote in 1985, based on a study by Schoch published in 1931. The value for the lunar tidal acceleration is -29.68"/cy/cy, and the corresponding rate for DeltaT is +36.28s/cy/cy (his section 2). Moreover, he uses UT as his time variable rather that ET. Schoch wrote his article before the introduction of ephemeris time and had to use UT. He built on the then-current Brown theory for the motion of the Moon, but fitted the acceleration parameter to a single supposed lunar occultation of Spica observed by Timocharis in 283 BCE (see below). Incidentally, Stephenson has used the Improved Lunar Ephemeris (designated j=2) that is also derived from Brown's theory.

Henriksson in his section 2 writes how he prefers using UT because it allows direct comparison of ancient observations with his theory, whereas ephemerides in ET require a way to establish DeltaT to interpret observations timed (essentially) in UT. But because one needs to know the lunar tidal acceleration to derive DeltaT from eclipses and occultations, Henriksson accuses previous authors of circular reasoning, and also states that the method is unuseable before about 700 BC (because there are no reliable observations before that time that allow to determine DeltaT).

However Henriksson apparently does not recognize that UT obviously is not a stable time scale and it is impossible to formulate an accurate physically sensible ephemeris in it. For example, according to the theory of tidal acceleration, the Earth must be slowing down at a steady rate, which requires us to add a leap second to our clocks every year and a half or so in order to stay in sync with the actual rotation angle. However, from the beginning of 1999 to the end of 2005 no leap seconds at all were necessary because the Earth was speeding up. This fact is quite independent from any lunar observations. In these years, there has been no indication (from LLR) that the Moon was slowing down. There are non-tidal forces working on the Earth that affect DeltaT but not the lunar tidal acceleration. This must also have been the case in the past.

Indeed besides a slower than anticipated rate, Stephenson finds erratic excursions from a smooth parabola in DeltaT, to which he fitted a series of cubic spline curves. Henriksson in his section 3 states that this is a failed attempt to correct for systematic errors caused by using wrong values for DeltaT and for the tidal acceleration. However using a wrong value for the tidal acceleration can only change the quadratic term in DeltaT, and can not cause, nor be corrected by, higher-order terms. Instead, when using UT as time variable, lunar and planetary ephemerides must reflect the erratic behaviour of the Earth's rotation. The fact that the supposed observations fall neatly on a smooth parabolic curve in Henriksson's fig.6 is an indication that his ancient observations are fictional.

Later in his section 3 Henriksson derides a correction formula due to Morrison and Stephenson and annually quoted in the Astronomical Almanac, to correct their DeltaT values for any revision of the lunar tidal acceleration: Henriksson states that this would violate conservation of angular momentum. He fails to recognize that, just because derivation of DeltaT and the lunar tidal acceleration are interdependent when based on lunar occultation and eclipse observations timed in UT, the derived DeltaT values should be corrected if you use an improved ephemeris with another value for the lunar tidal acceleration.

Moreover he does not address the issue that his preferred values for the lunar tidal acceleration and the rate of DeltaT (from Schoch), do leave angular momentum missing: with his larger value for the lunar tidal acceleration, the quadratic parameter in DeltaT should be even bigger than the 42 or 44 s/cy/cy predicted by Christodoulidis (Stephenson 1997 pp.37,38).

Henriksson repeatedly claims (in his abstract, sections 3, 4, and 5) that all modern ephemerides use a wrong value for the tidal acceleration pre-set on the value of -26"/cy/cy derived by Morrison and Ward. He contradicts himself since he also quotes different values for the tidal acceleration from various successive numerical integrations. The fact is that these values are not fixed but are the result of fitting tidal force parameters to LLR observations, and are completely independent from any value of DeltaT - unlike older observations of lunar occultations and eclipses timed in UT. There is no circular reasoning involved in modern lunar ephemerides. So after more than two lunar nodal periods of LLR observations we can say with confidence that the present tidal acceleration in the Moon's motion is close to -26"/cy/cy, which happens to be close to the value derived by Morrison and Ward from observations of the Moon and Mercury over the past three centuries; and that Schoch's old value of close to -30"/cy/cy is wrong. Henriksson states that this present-day value for the lunar tidal acceleration, established over a short span of time, can not be extrapolated to the historic past, which would also be true for the longer span of telescopic observations. This has been a matter of concern and is addressed by e.g. Stephenson 1997 p.36 . The tidal exchange of angular momentum in the Earth-Moon system has been active ever since the oceans formed, and its rate most likely did not change much after the rise of ocean levels after the melting of the glaciers of the ice age about 10,000 years ago. The rotation rate of the Earth, and therefore DeltaT, is much more sensitive to sea level changes and re-distribution of mass, as already discussed. Henriksson may have a point that historical global temperature changes are reflected into the observed rotation rate; also see Reijs (2006) for a discussion and fit. Henriksson claims (in his abstract and his section 4) that he corrected for non-tidal forces. He does not describe what these are, how he models them, or how he corrects for them. He only quotes a slight modification for Schoch's value for the lunar acceleration −29.65"/cy/cy, which he appears not to have used in his computer program). This is odd, since the non-tidal forces should affect only the Earth's rotation rate and DeltaT, and not the lunar motion. Also it is known that the non-tidal forces are not constant, so a simple modification of a quadratic curve (whether for DeltaT or the lunar motion) for all historic times will not do. And indeed that is exactly what Stephenson and others have found.

4. The occultation of Spica in 283 BCE

Henriksson, following Schoch, calibrates his lunar ephemeris against an observation by Timocharis from 283 BCE, transmitted in the Almagest; in the translation by Toomer (1984) VII 3 H29,H30 (p. 336 in Toomer's edition):

Henriksson computes the re-appearence of Spica after an occultation just after Moonrise, as seen from Alexandria (ca. 31ºN, 30ºE); see his fig.4 . In discussion with Henriksson, the author with several others (prof. Paul Hirose, Victor Reijs) has tried to reproduce this event. The surprising result is that not the position of the Moon appears to make the difference, but that Henriksson puts Spica 4'W and 6'S of where Reijs and others find it.

Hirose used the HIPPARCOS coordinates and proper motion for Spica at J2000 from SIMBAD (see link), and a self-written program called Tinyac (see link). It is based on the latest SOFA routines (see link) implementing the IAU 2006 precession and 2000A nutation standards. The author can confirm Hirose's coordinates for the star using his own routines, and Reijs finds similar results with the up-to date Swiss Ephemeris (see link). For the Moon, Hirose (and SwissEphem) use the DE406 (Standish 1998), the latest long-term ephemeris from JPL from 1998, which has an effective lunar tidal acceleration of -25.826"/cy/cy (Chapront et al. 2002 Table 7).

For DeltaT the author made a third-order Besselian interpolation from the values in the table of Morrison and Stephenson (2004), and applied a correction of -79s to be consistent with the lunar tidal acceleration value in the DE406, using the correction formula from Morrison and Stephenson listed in the Astronomical Almanac (p.K8).

Henriksson took his J2000 star coordinates and proper motion from the Bright Star Catalogue, which differ only little from SIMBAD. The Moon's position he computes with his implementation of Schoch's expressions. See Table 1.

Note the large difference in the star's computed position. The program of Henriksson contains precession expressions from Bretagnon, taken from Laskar (1986, Table 8); he has shared these with the others in the discussion, and we found no numerical errors. The difference in general precession (Henriksson and Bretagnon use the 1976 IAU constant which is about 0.3"/cy larger than the 2006 constant used by SOFA) adds up to only 7" and can not explain the offset, and a re-computation by Hirose using Bretagnon's precession yields expected results far away from Henriksson's position. Annual aberration and nutation each contribute less than half an arc minute. In the subsequent discussions Henriksson reported a programming error (an uninitialized variable which gets a random value in the program as compiled on his current computer), and we assume this causes his deviant results. Different programs yield results compatible with Hirose's, so we accept these as correct.

So the author is confident, on basis of the best presently available ephemerides and procedures, that there was no occultation of Spica seen from Alexandria on 9 Nov. 283 BCE. Moreover, the modern ephemeride is fully consistent with Ptolemy's report of Timocharis' observation, which may refer to a conjunction rather than an occultation. Because this was not an occultation, Timocharis' observation can not set strict limits on the lunar acceleration anyway.

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. The obvious answer must be that Schoch adjusted his theory to match this particular observation. Any flaws in the original Brown theory have been compensated by adjusting a single parameter (the acceleration in longitude) to this single observation (and one eclipse observation attributed to Hipparchos for the solar acceleration; see Britton 1992 p.166 for a critical discussion). This also means that Henriksson can not use this observation to validate his value for the tidal acceleration, because that would introduce circular reasoning. Also note that the time variable in Schoch's theory is UT, so an effective acceleration different from the modern ephemerides measured in TT is to be expected – DeltaT is partly included in Schoch's expressions. As I argued above, UT is irregular and should not be used for a physically sensible ephemeris, and in addition there are non-tidal contributions to DeltaT so these can not be modeled by adjusting the lunar acceleration in UT.

The conclusion must be that the Schoch/Henriksson value for the lunar tidal acceleration is wrong also for the historic past. It is based on an erroneous treatment of too few and already dubious observations. There is no reason to keep using an obsolete lunar theory and ignore 80 years of subsequent improvements in theory and observations. The lunar acceleration as measured for the present with LLR, and the DeltaT values compatible with it, reproduce all real or supposed observations brought forward by Henriksson in his article. So it is reasonable to use the present-day value for studying the historic record, and reject Schoch/Henriksson's larger value.

5. Eccentricity Rate

Williams and Dickey (2002) have reported that fitting LLR observations to the DE model leaves unexplained residuals in the eccentricity rate of the lunar orbit: Williams expresses this as EQUIVALENT to 6 mm/yr in the perigee distance. Henriksson attempts to use this to support his higher value for the lunar tidal acceleration. In his section 4 he starts an analysis of an expression for an undisturbed elliptic orbit, apparently in a misguided attempt to link the eccentricity rate to the acceleration.

First, the eccentricity rate describes the change in the eccentricity, i.e. the SHAPE of the lunar orbit. The tidal acceleration changes the mean orbital velocity of the Moon and the mean distance, i.e. the SIZE of the orbit. These are independent orbital parameters. Williams (2002) established the anomalous eccentricity rate, and also gave the equivalent effect on perigee distance as an example because ranging was the topic of the conference report (Williams, personal communication).

Now Henriksson analyses this using an expression for an elliptic orbit:

This gives the radial distance (r) of an orbiting mass (the Moon) from a central mass (the Earth) as a function of the so-called eccentric anomaly (E), which itself is a transcedental function of time. This is valid for two masses moving under their mutual gravitational interaction without other disturbing forces; or if you like an instantaneous "osculating" orbit that can be derived for a mass with some initial position and velocity moving in some complex gravitational field. Henriksson evaluates the equation at perigee (E=0), and takes the derivative to time: his Eq.2:

This analysis is inappropriate on several accounts. The variable r is the instantaneous distance between the two bodies that varies from moment to moment, and not the perigee distance (typically designated q) which is constant in an elliptic orbit. Henriksson's Eq.2 is an expression for the continuous change in distance between the two bodies, but evaluated at perigee, when d(r)/dt is 0 by definition: indeed the expression (2) evaluates to 0 because both d(a)/dt and d(e)/dt are 0 for an elliptic orbit.

Using Eq.1 with the eccentric anomaly is distracting, Williams' numerical value can more easily be derived directly from the expression for perigee distance (like Williams did himself):

What Henriksson apparently is looking for is an expression that gives the secular change in perigee distance as a function of secular changes in the eccentricity and the mean distance, which themselves would be complex functions of disturbing forces. The Eq.1 for an undisturbed elliptic orbit, in which 'a' and 'e' are constants, can not provide that. He should look for a development of the orbital parameters in a (semi-)analytical lunar theory like the ELP (Chapront & Chapront-Touzé 1983). Moreover, the perigee distance is of no interest here but apparently Henriksson has been led astray by the way Williams presented his results. The relevant parameter of the anomaly that Williams discovered, is a secular change in the eccentricity. Henriksson fails to recognize that the Moon is followed all along its orbit, and that LLR is perfectly capable of separating changes in the shape of its orbit (the eccentricity) from the size of its orbit, the latter of the two being the relevant parameter of tidal acceleration. So Henriksson's attempt to re-interpret the eccentricity rate anomaly as an acceleration is false.

6. Modifications to General Theory of Relativity

Henriksson also repeatedly (in his sections 4, 5, and 6) mentions the geodesic precession, the main relativistic effect in ephemerides. How this well-known effect relates to the observational and theoretical issues he mentions remains unclear in his article, but he seems to associate it with the eccentricity rate as well as the (cosmological and|or tidal) expansion rate of the orbit: which are unrelated, because the geodesic precession has an effect on the ORIENTATION of an orbit, whereas the eccentricity rate changes its shape, and the expansion rate its size as I mentioned before - again Henriksson confuses distinct orbital parameters. He states that LLR has failed to confirm precession of the geodesic, while in fact it has confirmed the effect to beter than 1% (Willams et al. 2004). Incidentally, the JPL ephemerides numerically integrate the relativistic differential equations of motion, and the geodesic precession is implicit in these differential equations.

In his section 5, Henriksson discusses an article by Dvali et al. (2003). They predict an additional cosmological precession of the Moon's perigee of 1.4E−12, apparently in units of radians per revolution. Henriksson quotes this as −0.516”/cy/cy. Apart from the sign (depending on the cosmological branch) and the trivial conversion to arc seconds, Henriksson apparently multiplied Dvali's number by the square of the number of anomalistic months per century, but that unit is wrong: it is not an acceleration.

In any case tidal acceleration does have only a small secondary effect on the perigee (Chapront & Chapront-Touzé 1983 Table 7), the geodesic or cosmological precession is completely irrelevant in this discussion, and its prediction by the General Theory of Relativity can not be confirmed nor contradicted by Henriksson's results even if they were valid.

7. Conclusions

In his article, Henriksson makes false statements about the ability to compute past eclipses and occultations with modern ephemerides based on Lunar Laser Ranging. He gives no reliable observational evidence that would support his larger value for the lunar tidal acceleration. His value is based on an old and obsolete analysis that is essentially based on a single ancient observation that has been wrongly interpreted and wrongly computed. Henriksson's analysis of the Moon's orbit is fatally flawed and irrelevant for cosmological interpretation.

NOTES: The author of this article has had extensive e-mail discussions with dr Henriksson on various aspects of his article (Henriksson 2010) both on an e-mail list (HASTROL@listserv.wvu.edu) and privately, in which dr Henriksson shared more of his data and calculations, some of which will be quoted here.

Acknowledgements The author is very greatful for the contributions of prof. Paul S. Hirose and Victor Reijs, who helped analyze the problems, performed high-accuracy calculations, and reviewed earlier versions of this article; and to prof. H. Hunger for interpretation of the Enuma Anu Enlil text. In addition the author thanks the referees for valuable suggestions. Finally the author also thanks Dr. Göran Henriksson for openly discussing his results and sharing some of his code and intermediate results.

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