Anomalistic Month Research Articles

From Babylonian lunar observations to Floquet multipliers and Conley–Zehnder indices

In 1878, Hill found numerically, in his limiting case of the restricted three-body problem, the so-called Hill’s lunar problem, a planar direct periodic orbit with a period of one synodic month. By virtue of the spatial system’s invariance under a symplectic involution, whose fixed point set corresponds to the planar problem, we can assign to Hill’s orbit planar and spatial Floquet multipliers and planar and spatial Conley–Zehnder indices. We show that these have deep astronomical significance because, on the one hand, we relate the anomalistic month to the planar Floquet multipliers and the planar Conley–Zehnder index. On the other hand, we relate the draconitic month to the spatial Floquet multipliers and the spatial Conley–Zehnder index. Knowledge of this lunar month dates back to the Babylonians, who lived until around 500 BCE. In order to determine the indices, we analyze analytically the bifurcation procedure of the fundamental families of planar direct and retrograde periodic orbits (traditionally known as families g and f) from the rotating Kepler problem for very low energies.

Reassessing evidence of Moon–Earth dynamics from tidal bundles at 3.2 Ga (Moodies Group, Barberton Greenstone Belt, South Africa)

AbstractPast orbital parameters of the Moon are difficult to reconstruct from geological records because relevant data sets of tidal strata are scarce or incomplete. The sole Archean data point is from the Moodies Group (ca 3.22 Ga) of the Barberton Greenstone Belt, South Africa. From the time‐series analysis of tidal bundles from a well‐exposed subaqueous sand wave of this unit, Eriksson and Simpson (Geology, 28, 831) suggested that the Moon’s anomalistic month at 3.2 Ga was closer to 20 days than the present 27.5 days. This is in apparent accordance with models of orbital mechanics which place the Archean Moon in a closer orbit with a shorter period, resulting in stronger tidal action. Although this study’s detailed geological mapping and section measuring of the site confirmed that the sandstone bed in question is likely a migrating dune, the presence of angular mud clasts, channel‐margin slumps, laterally aggrading channel fills and bidirectional paleocurrents in overlying and underlying beds suggests that this bedform was likely located in a nearshore channel near lower‐intertidal flats and subtidal estuarine bars; it thus carries risk of incomplete preservation. Repeated measurements of foreset thicknesses along the published traverse, measured perpendicular to bedding, failed to show consistent spectral peaks. Larger data sets acquired along traverses measured parallel to bedding along the 20.5 m wide exposure are affected by minor faulting, uneven outcrop weathering, changing illumination, weather, observer bias and show a low reproducibility. The most robust measurements herein confirm the periodicity peak of approximately 14 in the original data of Eriksson and Simpson (Geology, 28, 831). Because laminae may have been eroded, the measurements may represent a lower bound of about 28 lunar days per synodic month. This estimate agrees well with Earth–Moon dynamic models which consider the conservation of angular momentum and place the Archaean Moon in a lower orbit around a faster‐spinning Earth.

On the Solar Climate of the Moon and the Resulting Surface Temperature Distribution

The solar climate of our Moon is analyzed using the results of numerical simulations and the recently released data of the Diviner Lunar Radiometer Experiment (DLRE) to assess (a) the resulting distribution of the surface temperature, (b) the related global mean surface temperature Ts>, and (c) the effective radiation temperature Te often considered as a proxy for Ts> of rocky planets and/or their natural satellites, where Te is based on the global radiation budget of the well-known “thought model” of the Earth in the absence of its atmosphere. Because the Moon consists of similar rocky material like the Earth, it comes close to this thought model. However, the Moon’s astronomical features (e.g., obliquity, angular velocity of rotation, position relative to the disc of the solar system) differ from that of the Earth. Being tidally locked to the Earth, the Moon’s orbit around the Sun shows additional variation as compared to the Earth’s orbit. Since the astronomical parameters affect the solar climate, we predicted the Moon’s orbit coordinates both relative to the Sun and the Earth for a period of 20 lunations starting May 24, 2009, 00:00 UT1 with the planetary and lunar ephemeris DE430 of the Jet Propulsion Laboratory of the California Institute of Technology. The results revealed a mean heliocentric distance for the Moon and Earth of 1.00124279 AU and 1.00166376 AU, respectively. The mean geocentric distance of the Moon was 384792 km. The synodic and draconic months deviated from their respective means in a range of -5.7 h to 6.9 h and ±3.4 h, respectively. The deviations of the anomalistic months from their mean range between -2.83 d and 0.97 d with the largest negative deviations occurring around the points of inflection in the curve that represents the departure of the synodic month from its mean. Based on the two successive passages of the Sun through the ascending node of the lunar equator plane, the time interval between them corresponds to 347.29 days, i.e., it is slightly longer than the mean draconic year of 346.62 days. We computed the local solar insolation as input to the multilayer-force restore method of Kramm et al. (2017) that is based on the local energy budget equation. Due to the need to spin up the distribution of the regolith temperature to equilibrium, analysis of the model results covers only the last 12 lunations starting January 15, 2010, 07:11 UT1. The predicted slab temperatures, Tslab, considered as the realistic surface temperatures, follow the bolometric temperatures, Tbol, acceptably. According to all 24 DLRE datasets related to the subsolar longitude øss, the global averages of the bolometric temperature amounts to Tbol=201.1k± 0.6K. Based on the globally averaged emitted infrared radiation of FIR>=290.5W·m-2± 3.0W·m-2 derived from the 24 DLRE datasets, the effective radiative temperature of the Moon is Te, M>=Tbol>1/4=271.0k± 0.7K so that Tbol>≅0.742Te, M. The DLRE observations suggest that in the case of rocky planets and their natural satellites, the globally averaged surface temperature is notably lower than the effective radiation temperature. They differ by a factor that depends on the astronomical parameters especially on the angular velocity of rotation.

Supermoon Drives Beach Morphological Changes in the Swash Zone

AbstractThe effect of the supermoon, which appears much larger than a normal full moon, on the morphological changes of a sandy beach was investigated by analyzing a 25‐year daily observation data set of beach morphology in the swash zone. The beach morphology fluctuated in two cycles related to the supermoon: the semisynodic month cycle (from full moon to new moon, and vice versa) and the anomalistic month cycle (from perigee to perigee). The supermoon makes the erosion of the upper swash zone more likely by generating a larger tidal range. The high‐water‐level contour positions observed during supermoons retreated significantly, although there were no trends in the change for all the observations. Erosion due to high waves coincident with a supermoon can be expected to be more severe in the upper swash zone; including the supermoon effect in beach morphodynamic models can improve coastal management.

On a New Analytic Theory of the Moon's Motion II: Orbit and Length of Months

The differential equation in polar coordinates of the Moon's orbit is outlined from the first-order approximation to the Lagrange equations of the Sun-Earth-Moon system expressed with relative coordinates and accelerations. The orbit of the Moon calculated this way is similar to Clairaut's modified orbit and has better parameters than those previously published. An improvement to this orbit is proposed based on theoretical arguments. With help of this new orbit, the variations in the draconic, synodic and anomalistic months are also computed showing very good agreement with observations.

The Final Date of the Antikythera Mechanism

The Antikythera mechanism is a mechanical astronomical instrument that was discovered in an ancient shipwreck (from about 60 bce) at the beginning of the twentieth century. A consensus does not exist on the question of whether the mechanism was built shortly before the shipwreck or significantly earlier. Nevertheless, there is an emerging consensus among scholars about the epoch of the back dials of the mechanism: the Saros dial would start at 27 April 205 bce and the Metonic dial four synodic months earlier, that is, at 25 August of the same year. Using these two starting dates together with the positions that some pointers still show in the extant fragments of the mechanism I calculate the final date of the mechanism (i.e. the date that it showed when it was last cranked, some time before the shipwreck) as approximately 5 March 193 bce or one anomalistic month earlier.

Bīrūnī's Examination of the Path of the Centre of the Epicycle in Ptolemy's Lunar Model

Ptolemy's complete lunar model in Almagest V produces a large variation of distance of the centre of the epicycle, between its maximum at mean conjunction and opposition to the mean position of the Sun and its minimum at mean quadrature, so that the resulting path of the centre of the epicycle about the Earth is an oval figure. In Figure 1, with the Earth at T, the centre of the epicycle L moves uniformly about T on an eccentric of radius R with centre E and apogee A, through the mean elongation from the mean position of the Sun S in the direction of increasing longitude, while the apsidal line of the eccentric rotates in the opposite direction through the same mean elongation. Consequently, L reaches the apogee A and perigee B of the eccentric twice in each mean synodic month, which produces the oval path, farthest from the Earth at mean conjunction and opposition, closest at mean quadrature. The Moon M moves on the epicycle of radius r in the direction opposite to the motion of L through the mean anomaly, completed in an anomalistic month, uniformly with respect to the mean apogee F, which has an 'inclination' (pmsneusis) toward a point P, opposite to the direction of F from the Earth and with the same eccentricity, e = PT = TE.1 The true apogee G lies on the line TEG from the Earth. Our concern here is the path described by the variable distance of the lunar epicycle from the Earth p = TL. At mean conjunction and opposition, p = EA + ET=R + e and at mean quadrature p = EA - ET = R - e. The figure has been drawn using historical values due to Muhyi al-Din al-Maghrib! (d. 1283) of the Maragha Observatory in north-western Iran, where R + e = 60, R = 51,e = 9, and r = 5;12.2 Thus at conjunction and opposition p = R + e = 60 and at quadrature p = R - e = 42.Ptolemy, as Pedersen remarks, always conceives the motion of the lunar epicycle centre as a circular motion around the moving centre of the deferent. He never asks for the orbit described by the lunar epicycle centre relative to the centre of the Earth.3 Abu al-Rayhan al-Biruni (973-1048) in his al-Qanun al-mas'udi VII.7.14 considers this problem in the form of question and answer. The translation of the relevant passage is as follows:Q: What [shape] does its [i.e., the Moon's] epicycle centre describe by this motion [i.e., according to Ptolemy's model of a movable eccentric]?A: If it is assumed that the Sun is at rest and if the lunar epicycle centre is at its orb's apogee in its [mean] conjunction or opposition [to the Sun] and is at the perigee in its [mean] quadrature, it will describe a rounded rectangular shape by its motion. It might be thought that it [i.e., the path of the epicycle centre] is an ellipse of the [right circular] conic or cylindrical sections. [But,] it is not so.Take the Moon's orb

Detection of cosmogeophysical periodicities by the statistical analysis of intervals between the dates of birth of genetic relatives

A statistical analysis of time intervals between the dates of birth of genetic relatives has been carried out on the basis of 33 family trees. Using the Monte Carlo method, a significant departure of the distribution of birthdays from random results is detected relative to two long-period solar harmonics known from the theory of the Earth tides, i.e., a solar elliptical wave (Sa) with a period of an anomalistic year (365.259640 days) and a solar declinational wave (Ssa) with a period of half of the tropical year (182.621095 days). Further research requires larger statistical samples and involves clarifying the effect of long-period lunar harmonics, i.e., an lunar elliptical wave (Mm) with a period of an anomalistic month (27.554551 days) and a lunar declinational wave (Mf) with a period of half of a tropical month (13.660791 day), as well as the impact of important lunar and solar tides of time intervals with periods of half (14.765294 days, the interval between syzygial tides at new and full moon) and a whole (29.530588 days) synodic month. It is known that the periodic compression and stretching of the Earth’s crust at the time of the tides by means of the piezoelectric effect lead to the generation of long-period electric oscillations with periods corresponding to the harmonics of the theory of the Earth tides. The detection of these harmonics in connection with biological processes will make it possible to determine the impact of regular cosmogeophysical fluctuations (tidal waves) on the processes in the biosphere.

Libration of the Moon, Hevelius's Theory, and its Early Reception in England

In first-order approximation, the Moon rotates uniformly in the course of a month, at the same time revolving in her orbit around the Earth with variable angular velocity, according to Kepler's second law. Although the periods of both motions are synchronized, changes in orbital speed cause an observer on Earth to see either the western or the eastern edge of the lunar disk, depending on the conditions. Because of this, the maximum angular shift of surface objects relative to the centre of the Moon is ±7.9°. This is what we call optical libration in longitude. And, moreover, the Moon's rotational axis is not perpendicular to the orbital plane, but tilted away from the normal to the orbital plane by 6°4G, which is the reason why, in the course of the month, we have alternating views of the north and of the south pole of the Moon. This phenomenon, known as optical libration in latitude, leads to the angular shift of surface objects (±6.8° maximum; see Figure 1), in relation to the centre of the Moon. One also needs to consider that observations of the Moon's surface are not conducted from the centre of the Earth, but from its surface, which each day makes one revolution around the Earth's axis. In this way, analogous to the phenomenon of diurnal parallax, we have diurnal libration, also called parallactic libration. Its maximum value is equal to 57'. The three types of libration described here enable us to view not half but 59% of the Moon's surface from the surface of the Earth.1For obvious reasons, the period of libration in longitude is the anomalistic month (the time between successive passages of the Moon through perigee), while in latitude it is the draconic month (the time between successive passages of the Moon through her ascending node). Due to the incongruence of these periods, the manifestations of libration change from month to month. Optical libration in longitude and in latitude can be illustrated as the oscillations of a point on the surface of the Moon that marks the centre of the lunar disk (at perigee, for example), in relation to a line connecting the Earth and the Moon at their centres of mass. In Figure 2, two diagrams based on this model are used to illustrate the variability of optical libration, by showing different paths of the centre of the lunar disc for January 1635 and January 1643. Finally, we need to recognize that optical libration in latitude and longitude has a direct consequence for an observer on the Earth: the changing position of lunar spots relative to the edges of the lunar disk.The First Reports of LibrationAlthough the optical libration of the Moon is a phenomenon that can be seen with the naked eye, its discovery was strictly tied to the development of telescopic observations of the Moon's surface in the first half of the seventeenth century, as well as to the emergence of selenography, a new discipline in the field of celestial cartography2 The oldest preserved source that most likely refers to the phenomenon of libration is a note from December 1611, made by Thomas Harriot (c. 1560-1621) in the course of one of the three lunar observations he made that month. The English mathematician noticed that two features of the Moon's surface (today called Mare Frigoris and Sinus Roris) appeared closer to the edge of the lunar disk than was the case on the map he had drawn sometime earlier (Figure 3).3 Since he did not follow up on his note, we cannot rule out the possibility that he thought his own map to be inaccurate in this respect. Nonetheless, we could consider this to be the first (qualitative) record of libration in latitude. However, the handwritten notes left by Harriot were not discovered until the end of the eighteenth century,4 so most often it is Michael Florentius van Langren (1600-75) and Galileo Galilei (1564-1642) who are given credit for first mentioning lunar libration.The Dutchman van Langren, Royal Cosmographer and Mathematician to King Philip IV of Spain, wanted to create a map of the Moon that could be used to determine terrestrial longitude through the comparison of sunset and sunrise times at defined points corresponding to the surface features of the Moon. …

The Saros cycle: obtaining eclipse periodicity from Newton's laws

The Saros cycle has been known since antiquity and refers to the periodicity of eclipses. It is the least common multiple of three periods: the synodic, the draconic, and the anomalistic months. We show how to obtain these periods from Newton's laws with a precision greater than 0.02% using only the sidereal month and year as references.

The influence of lunar phenomena on the incidence of emergency cases

It is widely believed that the incidence of specific emergency cases shows clustering during long observation periods. Though there is no scientific proof, many physicians and other emergency staff believe in influences of the moon or the signs of the zodiac. The aim of our retrospective study over 6 years was to evaluate (a) if there are any statistically documented peaks of frequency of emergency cases at all, and (b) if they can be linked to lunar phenomena. We evaluated all three aspects of the moon: The ‘synodic’ moon (=lunar phases), the ‘sideric’ moon (=distance between moon and the earth) and the moon in her relation to the signs of the zodiac (=influence of the zodiac). A total of 11 134 patients entered the study. We found highly significant clusters of emergency calls, mainly for lung disorders. However, neither aspect of the moon showed the slightest correlation with the frequency of emergency calls (sideric month ( P=0.99), synodic month ( P=0.85) and zodiac ( P=0.85)). Trigonometric regression with the period of the anomalistic month ( P=0.173) and with the synodic month ( P=0.28) did not show any influence of the moon on emergency in either cases. Though our retrospective data analysis documented clustering of emergency cases, any influence of the moon and the signs of the zodiac can be definitely ruled out.

Babylonian Lunar Theory ReconsideredZur Entstehung der babylonischen Mondtheorie: Beobachtung und theoretische Berechnung von Mondphasen. Lis Brack-Bernsen

Previous articleNext article No AccessEssay ReviewsBabylonian Lunar Theory Reconsidered Zur Entstehung der babylonischen Mondtheorie: Beobachtung und theoretische Berechnung von Mondphasen. Lis Brack-Bernsen John M. SteeleJohn M. Steele Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Isis Volume 91, Number 1Mar., 2000 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/384630 Views: 1Total views on this site Citations: 1Citations are reported from Crossref Copyright 2000 History of Science Society, Inc.PDF download Crossref reports the following articles citing this article: Book Reviews, Interesting Publications, Treasurer's Report, Notes on Contributors, Forthcoming Articles, HESS Officers, Earth Sciences History 19, no.22 (Jan 2000): 216–250.https://doi.org/10.17704/eshi.19.2.1282qp4336862030

Reliability of lunar orbital periods extracted from ancient cyclic tidal rhythmites

Ancient tidal periods extracted from the geologic record are useful to constrain and calibrate theoretical models of the evolution of the earth-moon system. Analysis of rocks that contain such tidal periodicities, termed cyclic rhythmites, have tended to concentrate on long series and have focused on the more obvious periods. For example, neap-spring periods have been the most widely discussed; such information can be reliably extracted from bedform thickness series measured within laterally accreted tidal bundles and vertically accreted tidal laminae. Other lunar orbital data is also encoded in cyclic rhythmites; however this additional information has been much less extensively applied to earth-moon models. Herein, short segments of well-formed cyclic rhythmites are examined to determine how reliably the various tidal periods can be extracted via harmonic analysis. Rhythmite series analyzed here exhibit well-formed neap-spring tidal periods and also exhibit a prominent fortnightly inequality or a variation in the height of successive spring tides. This inequality is related to the elliptical nature of the lunar orbit (anomalistic month) and it is demonstrated that this period can also be reliably extracted from cyclic rhythmites. Tropical periods, which reflect the changing lunar declination that occurs during a tropical month, have also been reported from cyclic rhythmites. Unlike synodic and anomalistic periods, the apparent tropical periods extracted from some cyclic rhythmites do not always accurately correspond to the expected values. In only a few cases can this period be reliably extracted from short segments of cyclic tidal-rhythmite data. This period is commonly only poorly encoded in rhythmites because: (1) it tends to be masked by higher amplitude synodic periods, and (2) it is not well expressed within predominantly semidiurnal tidal systems. Samples selected from the Proterozoic and Phanerozoic tend to be more common during specific intervals of geologic time. These intervals were characterized by glacio-eustatic flux and, in some cases, by unique continental configurations. Understanding these basic controls on rhythmite occurrences should aid in the discovery of additional examples.

On the Babylonian Lunar Theory: A Construction of Column ? from Horizontal Observations

We demonstrate that column Φ in System A of the Babylonian moon ephemerides can be derived from such horizontal phenomena as were observed and recorded by the Babylonians. Combining four of the so-called ‘Lunar Six’ in such a way that the effects of the oblique ascension are eliminated, we obtain a curve which oscillates, indeed, with the exact period and the approximate amplitude of Φ. Our curve (which we call S) also contains oscillations with the approximate period of the Saros and allows us to find the period relation which is underlying column Φ. Herewith it has been shown for the first time that the length of the anomalistic month can be derived from horizontal observations.

The development and transmission of 248-day schemes for lunar motion in ancient astronomy

Among the achievements of ancient astronomy was the invention of schemes, based on comparatively simple principles, for predicting the positions of the heavenly bodies. For example, as observed against the fixed stars, the Moon moves at a velocity that fluctuates around a mean of roughly 1 3° of longitude per day. Of the components contributing to this fluctuation, or "anomaly", the largest and most readily observed is a simple periodic variation that causes the Moon to move back and forth over a range of about 6° either way with respect to a uniformly-moving "mean Moon". The period of this "first anomaly" is the anomalistic month. Since after one anomalistic month the Moon is at the same distance from the mean Moon as it was at the start, its total motion over any whole number of anomalistic months equals the contemporaneous mean motion. This principle, in various forms, underlies every sophisticated method devised in antiquity for computing the Moon's position at specific times; it reduced the problem to one of describing the longitudinal motion over only a single anomalistic month. Even Ptolemy's discovery of the second lunar anomaly, a function of the Moon's elongation from the Sun, made necessary only an adjustment to the older procedure. A special class of methods, characterized by their use of an anomalistic month of ^ days, is familiar to students of ancient astronomy, thanks especially to Neugebauer's pioneering analyses of most of the pertinent texts, whether Babylonian, Greco-Egyptian, or Indian. Common to all the variants was a simplicity of construction and use, unavoidably entailing an imprecision in the predicted longitudes. Yet a hitherto puzzling passage in Ptolemy, as Toomer has recently shown,1 proves that even such a respected astronomer as Hipparchus employed a scheme of this sort in his computations. My intention here is to survey the evidence for the history of these methods, in such a way that the relationships of the variants may be more readily seen,

On the Interaction Between Tectonic Processes of the Earth and the Moon

At present seismographs are operating on the Moon as well, installed there owing to the successful Apollo missions. However these data are insufficient for detailed statistic investigations. That is why in case of the Moon we are to use indirect indications of its activity, such as the data on transient light phenomena from the catalogues by Miss B. Middlehurst. Among the great number of earthquakes there were chosen only the strong earthquakes (magnitude 6.5) with focuses deeper than 70 km. According to these characteristics 630 earthquakes were selected from 1904 to 1967. In the Middlehurst catalogue during the same period about 370 transient events on the Moon are registrated. A distribution of lunar events on the days of an anomalistic month gives evidence of the influence of the Earth's tidal forces (the Middlehurst effect). It appears that the distribution of earthquakes gives a similar curve. Thus the tidal interaction of the Earth and the Moon establishes certain synchronism in tectonic activity of these planets. The further statistic analysis reveals some more causal relation between the processes of the Earth and the Moon. Strongly pronounced maximum of lunar events is observed with the interval of 2–3 days after the earthquakes and the maximum of earthquakes – with quite the same interval after the lunar events. The peaks of these maxima exceed the mean number of events by a factor 3. The Moon Earth system is the astronomical example of a direct interaction of the processes in the neighbouring celestial bodies.The corresponding experiments, made at the Pulkovo Observatory, confirm the possibility of immediate interactions of irreversible processes due to the change of physical properties of time. Thus we can form a chronology of orogenesis on the Moon judging from the data on the history of the Earth. Tectonic processes of the Earth and the Moon seem to be in such a close interaction as if the Moon were in direct contact with the Earth, i.e. in other words, were its seventh continent. These conclusions give evidence of the extreme importance of regular seismic observations on the Moon.