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Universitat Tubingen, Institut fur Astronomie und Astrophysik, Abteilungen: Theoretische Astrophysik und Geschichte der Naturwissenschaften, Auf der Morgenslelle 10, 72076 Tubingen, Germany
 

Abstract. The use of some Arabic medieval solar and lunar eclipse records for the determination of secular changes in the Earth's rotation is critically reviewed. The published results derived from these data suggest a non-uniform decrease in the Earth's rotation rate over the last 27 cy. There is, however, up to this day no sound physical explanation for the deduced 'non-tidal oscillations', with an apparent period of about 1500 yr and a semi-amplitude of some 4 ms in the l.o.d., which overlayed to a constant secular tidal change in the Earth's rotation rate produce a net non-uniform deceleration of the Earth's rotation. In this paper we discuss a set of observations, which were executed by professional Arabic astronomers. We show by our analysis the way in which the non-uniform deceleration of the Earth's rotation was constructed. A correct reading of the Arabic medieval observations shows that they do not contradict a secular constant.decrease in the Earth's rotation rate of nearly -4.6×10^-22 rad s^-2. This value is in accordance with other similar ones derived from ancient eclipse records and from satellite tracking data.

Up to this day there is no better approach for the determination of long-term (or secular) changes in the Earth's rotation than the evaluation of medieval and ancient observational records of solar and lunar eclipses. Even accurate telescopic observations from the last three centuries does not allow the precise determination of such a small change in the length of the day (l.o.d.) of approximately 1.7 ms cy^-1. This secular effect is made indistinct by much larger decade fluctuations in the Earth's rotation rate (Kane and Trivedi, 1990).

The conditions for a solar eclipse depend on the precise location of the observer on the Earth, and on the positions of the Sun and Moon in space. If we use the uniform Earth's rotation rate implicitly contained in a day of 86400 SI s in a computation of the eclipse path for a medieval solar eclipse, the derived eclipse conditions will not correspond with the observed ones. The derived eclipse path will be found to be shifted westwards from the place of observation by an amount proportional to the latitude of the respective path element. The angular difference in geographical longitude, Dl, for which one has to correct the calculations in order to make them coincide with the observation, is proportional to Int(w(t) dt), where w(t) is the instantaneous sidereal Earth's rotation rate. For a constant secular deceleration of the Earth's rotation Dl, or DT = Dl/1.002738 s, will increase quadratically.

In the above argument, we implicitly have assumed that ths Moon's notion, which has a highly sensible influence on the computation of solar eclipses, is known accurately, hi fact, it is as a result of modem measuring techniques like lunar laser ranging, a well known parameter since the middle of the last decade. Studies about the secular change of the Earth's rotation rate before this time were encumbered with the difficulty that one had to separate two highly correlated quantities from a single set of observational data: the Earth's rotation and the secular tidal acceleration of the Moon.

The secular change of the Earth's rotation rate has two main components. They are often called the tidal and the non-tidal secular change. They have opposite signs. The tidal is the larger one by a factor of ca. 4. The reason for the tidal secular change - in a strongly idealized model for the net tidal effect - is that tidal dissipation causes a displacement of the tidal bulge of the Earth relative to the Earth-Moon direction. The tidal bulge exerts a secular torque, which expands the lunar orbit while decelerating the Earth's rotation. The tidal secular change safely can be assumed to be constent for the whole historical period (Brosche and Sundermann, 1982). An extensive deglaciation event at the end of the Pleistocene, some 10000 years ago, seems to be responsible for the non-tidal secular change. Solid ice layers with a thickness of 3 km and a mass of 10¹⁹ kg above northern Canada and Scandinavia melted at this time. The result were a redistribution of water (global sea-level rise of ~ 80 m) and an enormous discharge of load in the deglaciated zones. The changing deformation of the Earth, which was produced in response to this event, caused arid causes a steady variation of the inertia tensor, which in turn induces a change in the Earth's rotation rate [Sabadini and Peltier (1981) and Yoder et al. (1983)].

Though the advances within the field of geophysics have been enormous, the relevant increase in the understanding of the historical observational record has been very slight. The editing of the Arabic sources is difficult and requires strong interdisciplinary collaboration. A review of the work completed up to date is discouraging and shows very clearly that a major improvement in this matter will not take place in the near future. Therefore, astronomers and physicists had to find their own way without the help of historians of science. But not one of them was really trained for this demanding task. The major work was done by R.R. Newton (1970-1984) and F.R. Stephenson together with L.V. Morrison and S.S.Said (1984-1995). The main difference between the two approaches was that Newton attempted a separation of the two highly correlated quantities, the Earth's rotation and the Moon's motion, while Stephenson et al. determined the changes in the Earth's rotation rate using a constant value for the secular tidal acceleration of the Moon (n^∙_tidal = -26.0 arcsec cy^-2). In their new study Staphenson and Morrison (1995) conclude that a long-term fluctuation in the l.o.d. with a semi-amplitude of ~4 ms and a period of ~1500 yr may exist.

One of the most important astronomical Arab sources, az-Zij al-Kabir al-Hakimi  from ibn Yunus (d. A.D. 1009) has not met with a critical edition and a serious commentary. The zij exists as a fragmentary copy. The observations preserved in it, among them the most important compilation of solar and lunar eclipses of the epoch [Leiden, MS 1957 (Cod. Or. 143), Chp. 4-6], were used successfully more than two centuries ago to determine the secular tidal acceleration of the Moon. Caussin published in 1804 the Arabic text of these first chapters together with his French translation (Caussin, 1804). The printed Arabic text represents the-only serious preliminary study, but his French translation is not precise enough and sometimes misleading. Although this publication is by no means a critical edition, it was the basis of the modem astronomical and geophysical studies.

On the other hand we have the astronomical treatise az-Zij as-Sabi' from Al-Battani (b. before 858; d. A.D. 929), which, due to the monumental work done by C.A.Nallino(Nallino, 1899-1907), is the best edited and commented zij [Arabic with Latin translation and notes of the astronomer G. Schiaparelli. For a general view of this zij and many others see Kennedy (1956) and also Nallino (1944)]. Only one Arabic copy exists, which according to Nallino was written about A.D. 1100 (Escorial MS). Al-Battani has preserved in his zij two solar and two lunar eclipse observations. These are probably the ones that helped Halley around the year 1693 to discover the secular (tidal) acceleration of the Moon.

In the following section, we will discuss 5 observations of al-Battani and ibn Yunus. The discussion of the other 36 solar and lunar eclipses recorded in their zijes, has been done in Dalmau (1993) and Dalmau et al. (1997). The few but representative observations discussed in this paper clearly demonstrate the unirained and incorrect use of the eclipse records by the authors mentioned above. It is also the main purpose of our discussion to emphasize the problems inherent in the interpretation of historical records and their prospective solution by critical editions.
 
 

Al-Battam was undoubtedly one of the greatest Islamic medieval astronomers [see Hartner (1970) for an extensive description of his work and his life]. The theoretical framework of his zij was strongly Ptolemaic. Al-Battani used his eclipse observations to demonstrate that Ptolemy's statements about the diameters of the Sun and the Moon, determined by Ptolemy with the help of two lunar eclipses, were untenable. In the 5^th Book, Chp. XIV, of the Almagest  we read: "... the Sun's diameter always subtends approximauly the same angle, there being no noticeable difference due to distance, but that ths Moon subtends the same angle as the Sun only when it is at its greatest distance from the Earth ... [Toomer (1984), p. 252]". According to this passage, annular solar eclipses were not possible.

Ptolemy determined a lower limit for the Moon's diameter using two lunar eclipse observations, in which the Moon was near its apogee. The method used by Ptolemy was based on two points: 1. On the knowledge obtained through the observation of the time and the magnitude of the maximal eclipse phase (according to him the middle of the eclipse); 2. On the knowledge, obtained by his lunar theory, of the Moon's distance in latitude at the middle of the eclipse from the northern limit of the Moon's orbit. This is obtained for a given time from the tabulated mean parameters of the Moon's motion (mean position in longitude, mean anomaly, mean northpoint distance and mean elongation). Ptolemy's proof was correct and convincing [see Toomer (1984), pp. 252-254]. Therefore, in order that al-Battani's refutation could also be convincing, it had to rely, 1. on better observations of the time and magnitude for the middle of the eclipse, and 2. on the mean parameters of the Sun and the Moon's motion as tabulated in his zij [Al-Battam uses for his proofs, departing from Ptolemy's argumentation, the latitude of the Moon at the moment of the apparent conjunction for solar eclipses, and of the latitude at the moment of opposition for lunar eclipses, moments that he calls "the middle of the
eclipse"].

The eclipses were observed in the cities ar-Raqqa and Antakiya. The first of them was al-Battani's main working site and also the reference place of his zij (ar-Raqqa j = 36°). Al-Battani does not make any comments about how or by whom the observations were carried out. Also, he does not comment how he determined the time of the middle of the eclipses. In the following, we present the results of our calculations of the eclipses according to al-Battani's tabulated values and motion theory (only for ar-Raqqa). The procedures to calculate the times and the magnitudes of lunar and solar eclipses are given in Chapters LIII and LIV of al-Battam's zij. The procedure for each eclipse calculation consists of some 35 steps. Starting point of the calculations are the tabulated values of the mean sun, the solar anomaly (in Ptolemaic sense), the position of the solar apogee and the numbers of days elapsed since the reference epoch of the zij (Du l-Qarnayn). Al-Battani's procedure is straightforward leaving no room, at any step, for any kind of assumptions; that means, a medieval astronomer wanting to control al-Battani's refutation of Ptolemy's statements about the diameters of the Sun and the Moon, would have obtained, following al-Battani's rules, the same results as we have obtained. It must be emphasized that al-Battam has not reported any of his own calculations regarding the times. As additional information -just to compare al-Battani's lunar theory with the modem one implicitly contained in the ephemeris DE102/LE51 of the Jet Propulsion Laboratory [see: Newhall et al. (1983)] - we have also computed some eclipse parameters using DE102. In the following only some important values are given. Al-Battam's descriptions of the observations are given in quotation marks [see also Nallino (1907), pp. 56-57]:

1. Lunar eclipse A.D. 883, Jul 23. (ar-Raqqa): The time of the true opposition was 07:55:00 p.m. mean local time. The Sun was in Leo 3°45'. From al-Battani's
 
 

Equation of Time (E.q.T.), we derive for this solar position: 3°4'; or 00:12:16 in time units. The local true time of the true opposition was 08:07:16 p.m. (equinoctial hours). In good agreement with the time given by al-Battam: "by a little amount more than 8 equinoctial hours after noon". According to al-Battani's zij, this lunar eclipse should have been total, with a magnitude of 17.25 digits (digits = twelves of the diameter) [modem computations using DE102 leads to 11.5 digits; according to Ptolemy: 11.5 digits], but from the observational record we read: "little more than a half and a third of the diameter".

2. Solar eclipse A.D. 891, Aug 8. (ar-Raqqa): The time of true conjunction was 00:50:24 p.m. mean local time. The Sun was in Leo 19ÿ32'. From al-Battani's E.q.T. we derive for this solar position: 3°16'32"; or 00:13:04 in time units. The true local time of the true conjunction was 01:03:28 p.m. (equinoctial hours). The apparent conjunction was, as stated by al-Battam, 00:07:30 after the time of the true conjunction, that is 01:10:58 p.m. (equinoctial hours) or 01:03:01 p.m. in seasonal (unequal) hours for ar-Raqqa (using day-hours of 01:07:34). In good agreement with the time given by al-Battani: "one seasonal hour after noon". According to Ptolemy, the magnitude should have been greater than 3/4.

3. Solar eclipse A.D. 901, Jan 23. (ar-Raqqa): The time of true conjunction was 21:19:29 p.m. mean local time. The Sun was in Aquarius 8°7'. From al-Battam's E.q.T., we derive for this solar position: 0°14'46"; or 00:00:59 in time units. The local true time of the true conjunction was then 21:20:28 p.m. (equinoctial hours). The apparent conjunction was, as stated by al-Battam, 50 min before the time of the true conjunction, that is 20:30:28 p.m. The time is counted from the noon of the previous day of observation; this means 03:29:32 before the actual noon. In perfect agreement with the time given by al-Battam: "less than 31/2 hours [equinoctial hours] before noon". In Figure 1, we have plotted the eclipse path with lines of equal magnitude. The diagram was calculated using DT = 0 s. The plotted magnitudes are digits of the diameter. The observed magnitude in Antakiya was little more than the half part of the Sun. In ar-Raqqa, the magnitude was " in the appearance" less than 2/3. From Figure 1, we can see that the difference of the observed magnitudes is, for any value of DT, not so high as the records insinuate [the varying of DT means that one has to shift the eclipse path eastwards in longitude with DT = Dl/1.002738 s]. According to Ptolemy, the eclipse should have been total.

4b. Lunar eclipse A.D. 901 Aug 2. (ar-Raqqa): The time of the true opposition was 15:21:28 p.m, mean local time. The Sun was in Leo 14°20'. From al-Battani's E.q.T., we derive for this solar position: 3°9'20" or 00:12:37 in time units. The local true time of the true opposition was then 15:34:05 p.m. (equinoctial hours). This is in perfect agreement with the time given by al-Battani: "15 1/3 + 1/4 equinoctial hours after noon, approximately". According to al-Battani's tables, this lunar eclipse should have been total (he observed: ~ 12 digits), with a magnitude of 18.0 digits [DE102: 12.8 digits].

Stephenson and Said (1989) analyzed the observations assuming that the only unknown parameter was the clock error DT arising from irregularities in the Earth's rotation. For the solar eclipse of the year A.D. 901 (in ar-Raqqa) they determined DT = 1840 sand for the lunar eclipse of the same year (in ar-Raqqa) they deduced DT = 420 s. Despite the fact that they have not considered the unusual definition of al-Battani's "middle of the eclipse", to explain this large difference in DT for observations done in the same year they have to assume either an observational error or a scribal mistake in the range of 24 minutes (using the correct "middle of the eclipse" the error is of some 20 minutes). This means, that the recorded times in al-Battani's zij cannot be correct (if one does not want to assume the catastrophe theory for DT - change of 20 minutes in 1 year - or an incorrect gravitational theory in DEI 02).

The agreement between our calculations and al-Battani's observations (see Table I) is obvious and reveals the true reason for the large difference in DT found by Stephenson and Said (1989). The tables in Ptolemy's Almagest and also those in al-Battani's zij require a broad observational foundation, as shown by the chapters leading to their construction here and there; and the lunar or solar eclipses mentioned above form no part of this foundation. As we know from ibn al-Qifti, al-Battani wrote two recensions of his zij. The extant Arabic manuscript must be a copy of the second recension, for it contains the lunar eclipse of August 2. A.D. 901. The first recension of the zij could not have contained this eclipse [and probably not the solar eclipse of the same year], because Thabit ibn Qurra, an elder contemporary of al-Battani, wrote before he died on February A.D. 901 some comments about one of the last chapters of the zij (Hartner, 1970). The zij was already published at the time of the eclipse(s). Despite these facts, the observed times are consistent with al-Battam's solar and lunar theory, suggesting that the numerical data have been manipulated.

To gain more clarity, future studies will have to reexamine the whole numerical context. Further, it is clear from above arguments that for the time being the observations listed in Table I cannot be used to determine DT values. Stephenson and Said (1989) by having used them (all weighted with 1.0) introduced a large bias in their statistical analysis, because al-Battani's observations are the ones in their data set, which produce the lowest DT-values.

Newton (1984), starting from his general mathematical analysis, discards them and argues that the error must come from recording and not from observation. He concludes that the observations, specially the lunar ones, may have been forced to agree with a preconceived theory.

Ibn Yunus observed a partial solar eclipse on December 13th, A.D. 977 in the mosque of Qarafa in the vicinity of today's Cairo. This is the first solar eclipse of his zij, that he himself has witnessed. If we compare it with the other eclipse records contained in his zij, we find something unusual, a list of the names of 8 persons, which precedes the observational part of the record. Five of them were instructed on the event which soon was to happen, but were not specialized in astronomical matters. The other three came together with him to the place of observation. Ibn Yunus does not comment, why he has preserved those names in an astronomical treatise. Perhaps he wanted them as witnesses or had another social motive. We give here the translation of the record:

"And the group waited for the beginning of this eclipse, and it sensibly began to appear, when the altitude of the Sun was greater than 15 degrees and less than 16.

And the opinion of those, who were present, was in agreement, that the eclipsed part of the Sun's diameter was about eight digits, and this is of the area of its circle less than seven digits. And its clearance (of the Sun) was complete, when its altitude was greater than 33 degrees, with about a third of a degree, according to that, what I [Ibn Yunus] have determined for myself about its altitude. And those, who were present, were in agreement about the completion of clearance (of the Sun);" [Caussin (1804), pp. 179_12-15 + 181_1-3; MS 1057, (Cod. Or. 143), p. 110 _4-16]

The description of the observation in the first part of the record does not allow an accurate interpretation of the observed eclipse magnitude. The moment when "it sensibly began to appear" was not the astronomical beginning of the eclipse. Ibn Yunus gives for this moment only a vague measurement of the altitude of the Sun. This shows clearly that he either was not interested in making a better description of this eclipse phase, or that he simply could not make a better measurement of the observed phase (not of the altitude!). It must be pointed out that he is not saying: the beginning of the eclipse has occurred at this altitude of the Sun. He is saying: at the moment, when "it sensibly began to appear" the Sun's altitude was "greater than 15 degrees and less than 16". Very different the description of the end of the eclipse: "and its clearance was complete". The described moment is the moment of the end or just after the end. Only for the second determination of the Sun's altitude does he make a personal statement --using his authority--, that he himself has determined the altitude to be 33 degrees and (more) by about a third of a degree.

Stephenson and Said (1989), without a plausible justification, put the altitude of the Sun at exactly 15.°5 for the first geometrical contact (The first geometrical contact is the moment when the two luminaries are tangential to each other; thus, at this moment the Sun is by no means eclipsed). Further, they put the 4. geometrical contact at exactly 33.°3 [Newton's procedure is nearly the same. (Newton, 1984), p. 114]. They then deduce DT(l.contact) = 1820 s and DT(4.contact) = 1980 s and give to both values the same statistical weight of 1.0 in a least square fit with a parabola. Beyond the fact that it does not make sense to interpret the observed parameters in that manner, they introduce a systematic error by doing this. As one can see from Figure 2, the first and the fourth contact are not statistically independent quantities. DT is always the same for all circumstances of a particular eclipse, and this fact can be used to weight and determine a lower limit of error for the particular eclipse.

We have used, as a working hypothesis, the parabola DT = 31.5s cy^-2 (t - t₀)² with t₀ =A.D. 1800 (w^∙_secular= -4.6×10^-22rad s^-2) and n^∙_tidal = -26.21 arcsec cy^-2 (DE102), which is in. agreement with the modem determination of the net secular change in the Earth's rotation rate and also with ancient determinations of the same value. Thus, we derive for this date: DT = 2130 s. For thisDT value, the altitude of the Sun was at the moment of the first geometrical contact at 14.°25. At the moment when the Sun's altitude was 15.°5, the eclipse magnitude was 1.2 digits

of the diameter. For the fourth geometrical contact, we find the altitude of the Sun to be 33.ÿ0. For this AT, the maximal eclipse magnitude was 7.5 digits.

The eclipse magnitude was at the moment, when the eclipse "sensibly began to appear", nearly 1.2 digits, and this is in agreement with naked-eye observations. The eclipse record says nothing about the observational technique. It is, however, not necessary to speculate: such a retarded apprehension is possible. In the solar eclipse from Jan 24. A.D. 1004, which is the best described in his zij, ibn Yunus has recorded a similar case: the magnitude of the eclipse was at the moment when it was apprehended = 1.5 digits and the altitude of the Sun was 16.°0; but for this eclipse ibn Yunus makes an estimate for 'the beginning' to be as the Sun's altitude was 18.°5.

At the end of this eclipse record, we read that the Sun and the Moon were each near perigee. In the following two eclipse records, we find similar statements. Did ibn Yunus use these three eclipses like al-Battani to determine the diameters of the Sun and the Moon? This question and many others should be solved first, because the evaluation of al-Battani's observations clearly indicate that they have been manipulated. In this case, we must calculate these three eclipses using the zij of ibn Yunus in the manner illustrated in section 3. But it is impossible to do so as long as there is no critical edition of this important medieval source. Already now we may safely state that this eclipse record cannot, by any means (not even a statistical one) be used against the concept of a uniform change in the Earth's rotation rate.
 

Historical observations of eclipses have been the best tools for determining secular changes in the Earth's rotation rate, until today. The analysis of eclipse records given by F.R.Stephenson, L.V. Morrison, S.S. Said and R.R. Newton suggests that there is evidence for a non-uniform decrease in the Earth's rotation rate. In the present paper, w'e have shown on a set of representaive observations how the supposed irregularities in the Earth's rotation were constructed. We have limited ourselves to the discussion of the two main Arabic historical sources written by professional astronomers, because the results obtained by Stephenson et al. have mainly been derived from this two sources (88% of their data set). In our opinion, the authors listed above have not evaluated the Arabic records with the sufficient care. They have taken the numerical data from the sources without considering the context. The statistical method used by them meets by no means the requirements of the subject. The theoretical models and astronomical tables used by the Arabic astronomers are essential for their work and their observations, but they have been completely neglected. For the present, it can be stated that the Arabic medieval observations do not contradict a constant secular change in the rate of the Earth's rotation, its numerical value being nearly -4.6×10^-22 rad s^-2. The exact determination of the secular change in the Earth's rotation will depend on a critical edition with a serious commentary of the zij of ibn Yunus, because it yields the best source of information on the relevant observations.

Acknowledgements

We wish to thank Prof. Dr. Harms Ruder and Prof. Dr. Matthias Schramm (Institut fur Astronomic und Astrophysik, Universitat Tubingen) for support and helpful discussions, Mr. E.M. Standish (Jet Propulsion Laboratory) for providing a copy of DE102/LE51, Prof. Dr. Fuat Sezgin (Institut fiir Geschichte der Arabisch-Islamischen Wissenschaften, Universitat Frankfurt) for providing a copy of Leiden MS 1057 (Cod. Or. 143).
 

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