Equinoctial Full Moon Models and Non-Gaussianity: Portuguese Dolmens as a Test Case more

In press: Rappenglueck, Rappenglueck and Campion (eds.), 2011. ‘Astronomy and Power’ (British Archaeological Reports) EQUINOCTIAL FULL MOON MODELS AND NON-GAUSSIANITY: PORTUGUESE DOLMENS AS A TEST CASE FABIO SILVA (FABPSILVA@GMAIL.COM) SOPHIA CENTRE FOR THE STUDY OF COSMOLOGY IN CULTURE, UNIVERSITY OF WALES TRINITY SAINT DAVID & AHRC CENTRE FOR THE EVOLUTION OF CULTURAL DIVERSITY, INSTITUTE OF ARCHAEOLOGY, UNIVERSITY COLLEGE LONDON Abstract: There is a growing body of evidence supporting Equinoctial Full Moon alignments in megalithic Iberia where previously these were being interpreted as solar. The author’s own work surveying Neolithic dolmens in Central Portugal is now shown to exhibit Spring and Autumn Full Moon alignments for different riverine regions. Even though these Equinoctial Full Moon events have been well defined, modeling them with simulated data to obtain theoretical predictions for alignment distributions can be trickier. Some issues, like their inherent non-gaussianity, are identified and discussed. Updated as well as new theoretical models for possible Equinoctial Full Moon alignments are also presented. The introduced models will also permit, using statistics on a high number of measurements, to discern meaning within Equinoctial Full Moon alignments by identifying whether there was a preference for alignments to eclipsed full moons or not. Keywords: megalithic astronomy, lunar alignments, spring full moon, autumn full moon, megalithic equinox, Neolithic, Portugal Introduction Since C. Marciano da Silva introduced the Spring Full Moon (SFM henceforth) concept to account for alignments to the so-called “megalithic equinox” (Da Silva, 2004) a growing number of Neolithic tombs have been shown to be aligned to this astronomical event, and some also to its Autumnal counterpart. Dolmens in Alentejo and, more generally in Southwest Iberia were the first to be shown to be oriented towards the SFM (Da Silva, 2004). Using Bayesian analysis, it was found that the megalithic enclosures of Alentejo are also very likely to be oriented towards the Autumn Full Moon (Pimenta et al, 2009). Recently, César González and Belmonte (2010) have done a systematic statistical analysis of most available data for monuments in the Iberian Peninsula and found that the SFM model of Da Silva, for Alentejan and other neighbouring groups, was indeed a better fit than the solar models. Equinoctial Full Moons (EFMs throughout), to generalize the term, are temporal events in which the rise (and set) positions of both full moon and sun have swapped places. In summer the sun rises at a northeastern azimuth (in the northern hemisphere), travels high in the sky and sets in a northwestern azimuth. The full moon, on the other hand, rises at a southeastern azimuth, travels low in the night sky and sets at a southwestern azimuth. In winter the sun rises in the southeast and travels low in the sky, whereas the full moon rises northeast and travels high in the sky. The two celestial bodies changed their “position” at the Autumn Full Moon (AFM henceforth) and will change again at the SFM. In essence, the sun and full moon change their place, relative to the celestial equator, at an EFM. This effect is more pronounced the closer to the equator it is observed, as the east-zenith-west line (the celestial equator) is a clear-cut division of the abode of each of the two luminaries depending on the season. In this paper, current EFM distribution models available in the literature are critically reviewed. These have been treated as bell-shaped when there is no evidence for it, whether a priori or a posteriori. A proper treatment of simulation results is introduced in the first section. In section two these distributions are compared to data on Neolithic dolmens of central Portugal, and alignments to both SFM and AFM are identified. The question of meaning is discussed in section three, where the eclipsed EFM hypothesis is put to the test. Because Portuguese dolmen data suggests megalith builders were doing no such selection, ethnographic evidence for yearly EFM rites and myths is briefly introduced. A full treatment of the subject is left for a future work, now in progress. A table with the range and peak values for all the presented distributions is given and the importance and application of the paper’s main points are discussed in the last section. EFM Modeling EFM DECLINATION DISTRIBUTIONS By simulating moonrises for a period of 110 years for the current epoch, Da Silva (2004) obtained a theoretical distribution for the SFM with a minimum at 85º of azimuth, an average value of 97º.3, and a maximum at 110º, for the latitude of Évora, Portugal. As noted by that author a similar distribution for the AFM can be constructed from first principles. This is shown in Fig 1 in blue. Pimenta et al (2009), using the Alcyone Ephemeris software package, simulated both SFM and AFM moonrises for a period of about 428 years in the current epoch, for which there is absolute confidence in the lunar orbital parameters. This distribution includes a much bigger sample of years than the previous one, which means there is greater confidence in its statistical significance. It also includes a bigger sample of standstill cycle, which is of relevance as will be shown. These authors have kindly provided us with the results of this simulation, which will henceforth be used. NON-GAUSSIANITY IN LUNAR MODELS Pimenta et al use the simulated distributions as priors to test megalithic enclosure alignment data from south Portugal using Bayesian analysis. They have, however, obtained the average and standard deviation of the simulated declinations to model a Gaussian curve (the black dashed line in Fig 1), using it then as a prior to test the data against. Da Silva has also said that the SFM’s azimuths ‘exhibit a bell shaped distribution’ in his recent paper (2010). But there is no a priori reason why this distribution should be Gaussian shaped. This is an event that involves both solar and lunar mechanics that combine in a very non-linear way. This is made evident by the fact that such Full Moons don’t occur on the same date (on a solar calendar) every year. This is why a distribution is needed to account for the range of possibilities, as well as the differences in likelihood. Such distribution needs not be bell-shaped (Gaussian) as it is being driven by the celestial mechanics and not observations. Comparison of AFM distributions 1 FS (2011) Pimenta et al (2009) Da Silva (2004) Relative normalized frequency 0.8 0.6 0.4 0.2 0 10 5 0 5 10 15 20 Declination (degrees) Figure 1. Declination histograms simulated for Autumn Full Moonrise. The approach described in the text is represented by the black solid curve. Pimenta et al’s (2009) normal distribution is represented by the dashed curve, and the original binned histogram from Da Silva (2004)’s paper on the Spring Full Moon is represented in blue. Each simulated moonrise should instead be treated as a theoretical prediction for a value behind an observation. This amounts to saying that, if our goal is to get a moonrise distribution that is to be compared to measured histograms, the errors inherent in any observation need to be taken into account. A bell shaped distribution should be created for each individual value coming out of the simulation. The declination value would be the peak of the distribution, which would have a spread (standard deviation) directly related to the observational errors. All these individual bell-shapes would then be added to get the total distribution, which needs not be Gaussian. For astronomical events where there is a unique declination at any given epoch, for example the winter solstice sunrise, this amounts to a single Gaussian. But other more complex events, like EFM’s, would exhibit more complicated distributions. The inherent error that is reflected by the standard deviation, has to be estimated and should reflect both simulational uncertainty, observational error, choice of lunar limb, refraction issues as well as alignment issues that arise when using rough stones (see the discussion on the aptly title “problem of precision” in Appendix A of Silva, 2010). The shape of the EFM distribution depends heavily on this value. For too high values the distribution will “fatten” and flatten out, whereas for too small values it will not be “smooth”. For the rest of the paper a standard deviation of one degree of declination was chosen, which is high enough to accommodate Schaefer & Liller (1990) uncertainty in the declination due to refraction issues alone. In the EFM case this has to be done for each declination value obtained from the simulation. This is shown, for the AFM moonrise, as the black curve in Fig 1. There is a considerable difference between this curve and Pimenta et al’s Gaussian curve (the dashed black one): the distribution’s peak is shifted by a degree. This is because the original distribution is not bell shaped but has a small bump on the side. A similar effect occurs for SFM, even though the peak’s shift is smaller in that case. It is feasible, now, to fit the actual curve using Gaussian fitting techniques. It turns out that it can be fitted to good accuracy using two to five Gaussians, which would fully describe each EFM distribution. However, it is still not obvious if and how this would be of use within archaeoastronomy so it won’t be discussed presently. Because the simulations were done for the current epoch (1620–2047 CE) adjustments are needed in order to correct the distribution to any given epoch. According to the latest simulations (Laskar, 1986), already cited by Pimenta et al (2009), the adjustments should be no greater than 0º.4 for 1000 BCE, 0º.6 for 3000 BCE, 0º.7 for 5000 BCE and 0º.8 for 8000 BCE, which should be subtracted for the SFM values, and added to the AFM values (see Table 1). These are only approximations and while proper EFM simulations for such epochs are needed, they are nevertheless good indicators of what to expect. EFM Alignments in Central Portugal A recent survey (Silva, 2010) of Middle to Late Neolithic (c. 4000-3000 BCE) megalithic tombs in a region of central Portugal delimited by the Mondego river to the south, and the Douro river to the north, has shown a scatter in the declinations of the main axial orientations (see Fig 6 in Silva, 2010). The data of this survey was in good agreement with a previous one conducted by Michael Hoskin (1998; 2001) in which it was concluded that the tombs were oriented towards sunrise at the date of their construction (Senna-Martinez et al, 1997). However, this author suggested that this simplistic interpretation presents a narrow view of the available data disregarding outliers that suggest lunar, and even stellar, alignments. Another fact that Hoskin and colleagues’ interpretation could not account for was the preference of Mondego river basin tombs towards negative declination orientations, whereas tombs in all other nearby hydrographical areas, that is in the Vouga, Paiva, Torto and Coa river basins, prefer positive values for their declinations (Silva, 2010). Consolidating the dataset acquired by this author with the one reproduced by Hoskin in his book (2001), declination histograms for the tombs’ orientation can be constructed. Data from the more recent survey was given prevalence, which is to say that data from Hoskin’s survey was only used for tombs that weren’t measured recently. To each measurement was attributed a Gaussian distribution using the measured declination as its mean value and a standard deviation that tried to account for both precision (see appendix A in Silva, 2010) and observers’ errors. Comparing declination values for the 18 tombs that were encompassed by both surveys a value of 3 degrees of declination for the standard deviation was arrived at. This value was given to all measurements in the consolidated dataset. The resulting histograms (Fig 2) have been divided for tombs in the Mondego (Fig 2, top) and other river basins (Fig 2, bottom). The sample sizes (25 and 30 respectively) are enough to give some statistical significance to the analysis of the relative frequencies normalized to the average value. The declination sign preference previously mentioned is now made explicit by the fact that almost no peaks are present in the non-preferred areas of each set of tombs, except for some barely significant peaks with frequency values very close to the average. In other words, the statistically significant peaks for Mondego tombs have indeed negative declinations, and the reverse is true for the combination of Vouga, Paiva, Torto and Coa tombs. Declination Histograms 5 4 3 Vouga, Paiva, Torto and Coa basins Relative Normalized Frequency 2 1 0 Mondego basin 4 3 2 1 0 Measured Data Spring Full Moon Autumn Full Moon sMLS WS smLS 10 0 10 nmLS SS nMLS Declination(degrees) Figure 2. Declination histograms for measured data of the consolidated survey (black lines, see text), for dolmens of the Mondego river basin (bottom), and other local river basins (top). The expected distribution for Spring Full Moonrise and Autumn Full Moonrise are also represented (blue dashed and dotted lines respectively). The highest peak in the Mondego histogram is very close to the expected declination peak for an alignment with a southern minor lunar standstill, and there is as well a plateau close to the declination for the northern minor lunar standstill in the histogram for the other river basins. Alignments with lunar standstills had indeed been suggested previously (Silva, 2010), and seem to be quite common alignments in megalithic monuments (Ruggles 1999, Sims 2006). There is also a barely significant peak at the Winter Solstice declination for tombs in the other basins, another common alignment in megalithic Europe. There are however quite significant peaks in both histograms for values inbetween the mentioned solar and lunar extremes and zero degrees of declination. One of which is, in fact, the highest peak of the distribution. These peaks correspond closely, in both peak value and range, to the expected value for alignments with Spring Full Moonrise, for the Mondego tombs, and Autumn Full Moonrise, for tombs in the other nearby basins. This clear-cut difference between the two sets of tombs encourages their independent analysis and might even suggest different purposes or builders for each group. This hypothesis, which gains further credence from architectural and archaeological considerations, will be dealt with in a future work. EFM’s and Meaning The question of meaning is an imposing one in prehistoric archaeoastronomy. Whereas the importance of both solstices and standstills can and has been quickly ascribed to their visual uniqueness, the importance of an Equinoctial Full Moon event hasn’t been considered properly. One is quick to note on the importance of the Spring season, the renewal of nature, etc., and argue the naturalness of a spring rite. However, one question still remains unanswered: why, then, perform the rites at the EFM, instead of at the actual equinox, or the second full moon following the equinox when the spring is well under way? What is so unique about the EFM that makes it a special, meaningful time? BEYOND THE FULL MOON PARADIGM A possible explanation for this is the fact that lunar eclipses occur at these times on very specific years. Whereas usually eclipse dates fall near to the solstices, on standstill years they always occur on the full moon closest to the equinoxes. This is because, at this time, the lunar nodes are close to the equinoxes, which is, in fact, the cause for the extreme lunar declinations that are called standstills (Morrison 1980). The existence of alignments to both EFM and lunar standstills in close proximity and in monuments of the same style and of the same period, as in the Portuguese case above, is suggestive that such a connection might already have existed in the Neolithic mind. For a given place, lunar eclipses are much more common than solar eclipses. This is because when one occurs it is visible during the night across the whole hemisphere. They are also more varied in the sense that an eclipsed full moon can exhibit very distinct colours, ranging from near invisibility to brownish red or bright orange. Whether one postulates that pre-literate societies considered specific times on the lunar cycle, like Dark Moon (Sims 2006; 2007; 2010), or the often mentioned but less ethnographically corroborated Full Moon (Da Silva 2010), or that the very fact of lunar periodicity was the important thing (Lévi-Strauss 1973; 1978), a lunar eclipse, in which the bright Full Moon turns Dark, or red, would be a momentous time. Claus Clausen et al (2008) have explored lunar eclipse alignments for the megalithic sites of Denmark. They have simulated both solar and lunar positions for the Neolithic period and identified the presence of a “fingerprint feature” in the eclipse histograms. This fingerprint, consisting of two peaks, roughly at 100º and 120º of azimuth shows up only for western European longitudes and, then again, only for certain centuries. They have identified this fingerprint in the Danish passage grave data they’ve collected. This technique is, however, not always applicable as, sometimes, one might not have dates accurate to a century, particularly so if dating is reliant only on radiocarbon samples. In fact, one might not even have dates at all, attributing a prehistoric monument to a particular epoch based on architectonical and other archaeological similarities as is the case for most of the Portuguese dolmens, where radiocarbon dating was only possible for a few of them. There is, nevertheless, another way to test the lunar eclipse hypothesis based on the already mentioned fact that they only occur at these declinations on standstill years. From the simulated EFM dataset presented previously, one can select and plot the histogram for the declination of moonrise on standstill years only. Lionel Sims’ work on Stonehenge suggests the concept of standstill year to be far more important than the actual, geocentric, lunar declination extreme, which, he notes, is not even observed (Sims 2007). A standstill year is comprised of the thirteen or so lunations around the declination extreme that would be visible on the upper Grand Trilithon window at Stonehenge as seen from the Heel Stone (Sims 2006). The standstill extremes of declination always peak at the half-moon closest to the equinoxes, the Spring Equinox for a major standstill, and the Autumn Equinox for a minor one (Morrison 1980, Sims 2006). This means that the Full Moon closest to the standstill extremes is an Equinoctial Full Moon by definition, and it would for sure be included in the standstill year. There is then interest in distinguishing between alignments to Equinoctial Full Moons on all standstill years (blue line in Fig 3), major standstill years only (dotted line), or minor standstill years only (dashed line), as opposed to a distribution that doesn’t distinguish standstill from interstandstill years (the black curve), i.e. the one that has been considered so far. These have all been renormalized to unity to better represent their differences. The distribution for interstandstill years only is so similar to the ‘all years’ one that it’s not worth representing at this point. These distributions have very different shapes and all of them exhibit non-gaussian features, like multiple peaks. Such fingerprint features (to borrow the term from Claus Clausen et al) are not discernible in the consolidated dataset of dolmen orientations for Central Portugal (Fig 1), and therefore one can’t distinguish between these models with this dataset and analysis. It seems feasible, though, that a statistical method appropriate to test broader datasets will permit such distinctions to be made. Table 1 shows the declination values for the range and primary peaks of each one of the distributions present in Fig 3, for moonrise on both SFM and AFM. As is clear from the figure some of the distributions have other relevant features like peaks or plateaus. The values for these are: i) SFM, all standstill years, secondary peak at -9º.8; ii) SFM, minor standstill years only, secondary peak at -9º.6; iii) AFM, all standstill years, secondary peak at 5º.3; iv) AFM, major standstill years only, secondary peak at 5º.5; v) AFM, minor standstill years only, secondary peak at -0º.1; vi) AFM, minor standstill years only, tertiary peak at 3º.5. It should be remembered that the values of Table 1 and the ones just mentioned are values for simulations in the current epoch and that they should be shifted accordingly when looking for alignments in the past. Another thing to keep in mind when looking at this table is that, close to the equinoxes, the Full Moon changes its declination two to three degrees in a single night. Hence if one has western orientations that might potentially be EFM moonset alignments one should take such a shift into account (again subtracting these to the SFM values while adding them to the AFM values). The same effect is responsible for small shifts in the distribution at different longitudes. All of these shifts are, naturally, only approximations and proper simulations with further study are needed, but nevertheless this table provides the basis for current work with EFM alignments. 1 0.8 0.6 Autumn Full Moons Relative normalized frequency 0.4 0.2 0 1 0.8 0.6 0.4 0.2 0 15 all standstill years only minor standstill years only major standstill years only Spring Full Moons 10 5 0 5 10 15 Declination (degrees) Figure 3. Declination histograms for different models of both Equinoctial Full Moons. The black solid line includes all 428 years of the simulation, the dash-dotted blue line includes only the standstill years in the simulated period. The red dashed and dotted lines include respectively only minor and major standstill year data. Spring Full Moon Peak -4.0 -3.2 -1.8 -3.5 Autumn Full Moon Peak 3.0 1.8 1.5 7.6 All years All Standstill Major only Minor only Min -15.0 -14.2 -13.4 -14.3 Max 5.9 3.5 3.5 3.4 Min -6.4 -4.2 -4.1 -4.3 Max 14.3 12.6 10.3 12.9 Table 1. Moonrise Declination values for the different EFM distributions presented in the paper. These include values for the minima and maxima of each distribution as well as it’s primary peak. Secondary and tertiary peaks values are found in the text. MYTH, RITUAL AND DEEP STRUCTURE But let us, for the moment and only briefly, entertain the idea suggested by the data from central Portugal, that EFMs were always meaningful, without any yearly discrimination. It turns out that the ethnography of pre-literate societies seems to support this case. Whereas at equatorial latitudes the solar seasons might not be as important, mainly due to the much more influential climatic dry and wet seasons (Lévi-Strauss 1969; 1973) at more temperate latitudes the changing of the seasons might have been observed and infused with meaning. In his exhaustive analysis of Native American mythology, the anthropologist and structuralist Claude Lévi-Strauss analyzed certain myths and rites of an ‘equinoctial’ (his word) character. Such sets of myths are present, in various forms, from Tierra del Fuego to Alaska and the structural analysis of them and associated rites shows an inherent structure that pops out all over the world in rites that happen around the equinoxes (Lévi-Strauss 1973). For example, analysing a myth, and related rite, from the Mandan people of the North American Plains he remarks on the inherent change of position of the Sun and Moon. He calls this an “equinoctial concern” because he interprets it, not incorrectly, as a change to happen at the only time when night and day are of equal length: the equinox (Lévi-Strauss 1978). But it is possible, without detracting the eminent anthropologists’ conclusions, that knowledge of EFMs is here made explicit, as the time the sun and moon actually change positions in the sky. In fact, it is possible that EFMs are the ethnographic definition of equinox, besides being the proposed definition for the megalithic equinox. Conclusions By going beyond a Gaussian paradigm, where a peak orientation, and sometimes an associated standard deviation, is chosen to represent the theoretical (or predicted) alignment distribution it has been shown how different the actual distributions can be from their Gaussian approximations. This has tremendous implications when these models are compared with measured orientation data as not only the peaks, but also the shape of the distributions, convey knowledge. To identify an EFM alignment a large sample of orientations might not be needed, as the dolmen data of Fig 2 suggests. But if one wants to go a bit deeper and discern between several models then the shape of the distribution becomes paramount and advanced statistical tools and huge samples will be required. Techniques such as Bayesian Analysis (Pimenta et al, 2009), Cluster Analysis and Principal Component Analysis (González Garcia and Belmonte, 2010), which are primed for this sort of job, will certainly be amongst the first ones to be applied. Evidence for EFMs in the ethnographic records can also be found, but they seem to be in disguise, mostly because EFMs were unknown to the ethnographers who seem to have confused them with the equinoxes. It seems that ethnography, and especially structural analysis, have much to offer archaeoastronomy in general, but particularly in this case. A proper treatment of these aspects is now in preparation. The case of the Portuguese dolmens serves as an example of the presence of wide distributions of declinations characteristic of EFMs that were previously being interpreted as alignments to sunrise/sun-climb. It is feasible that EFMs and possibly other, yet unknown, “distribution-type” events might reshape interpretations of some site orientations and unlock many others throughout megalithic Europe and even beyond. References Clausen, C., Einicke, O. and Kjaergaard, P., 2008. The Orientation of Danish Passage Graves. Acta Archaeologica, 79, 216-229. Da Silva, C. M., 2004. The Spring Full Moon. Journal for the History of Astronomy, xxxv, 1-5. Da Silva, C. M., 2010. Neolithic Cosmology: The Equinox and the Spring Full Moon. Journal of Cosmology, 9, 2207-2216. González Garcia, A. 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A Bayesian Approach to the Orientations of Central Alentejo Megalithic Enclosures. Archaeoastronomy, XXII. Ruggles, C., 1999. Astronomy in Prehistoric Britain and Ireland. New Haven and London, Yale University Press. Schaefer, B., and Liller, W., 1990. Refraction Near the Horizon. Publications of the Astronomical Society of the Pacific, 102, 796805. Senna-Martinez, J. C., López-Plaza, M. S., and Hoskin, M., 1997. Territorio, ideología y cultura material en el megalitismo de la plataforma del Mondego (Centro de Portugal). In O Neolítico Atlántico e as Orixes do Megalitismo. Actas del Coloquio Internacional (Santiago de Compostela, 1-6 de Abril de 1996). Santiago de Compostela: Universidade de Santiago de Compostela, 657-676. Silva, F., 2010. Cosmology and the Neolithic: A New Survey of Neolithic Dolmens in Central Portugal. Journal of Cosmology, 9, 2194-2206. Sims, L., 2006. The ʻSolarizationʼ of the Moon: Manipulated Knowledge at Stonehenge. Cambridge Archaeological Journal, 16 (2), 191-207. Sims, L., 2007. What is a Lunar Standstill? Problems of Accuracy and Validity in ‘The Thom Paradigm’. Mediterranean Archaeology & Archaeometry, Special Issue, 6 (3), 157-163. Sims, L., 2010. Coves, Cosmology and Cultural Astronomy, in N. Campion (ed.), Cosmologies: Proceedings of the Seventh Annual Sophia Centre Conference 2009, 4-28. Cerendigion. Acknowledgements: The author would like to thank Kim Malville, Lionel Sims, Cândido Marciano da Silva, Claus Clausen and David Fisher for insightful discussions during the course of this work, as well as the organizers of SEAC2010 for the opportunity to present and publish it. A special thanks goes to Fernando Pimenta for allowing the use and critical discussion of the data he simulated using Alcyone Ephemeris.