Last and first sightings of the lunar crescent

Introduction

If societies used a lunar-based calendar for time-reckoning, a new month usually started when the new lunar crescent could be observed for the first time after new moon. Ancient Egypt is exceptional in this respect; there, the new month started with the invisibility of the old lunar crescent.

For the old or new lunar crescent to be observable, three conditions must be fulfilled:

the lunar crescent must have a minimal width to be bright enough,
the lunar crescent must have a minimal altitude above the horizon, otherwise extinction of Earth's atmosphere renders it invisible,
the Sun must be far enough below the horizon.

Since the time of the Babylonians and maybe already earlier, people tried to find criteria that are decisive for the first sighting of the lunar crescents after new moon. This problem is not solved definitely as of today^[1]. A wide spread usage was the attempt to predict the first sighting by means of the age of the Moon. Such a simple rule was an unreliable criterion because the lunar orbit is inclined relative to the ecliptic and because the orbital speed of the Moon is variable. Depending on the season of the year, this implies that the Moon can have quite different altitudes above the horizon and differences in azimuth between the Sun and the Moon at a certain age of e.g. 28 hours.

The Length of the Lunar Cycle

The notion "lunar month" refers to a full orbit of the Moon with respect to some fixed reference point. The so called lunation designates the timespan of a full orbit of the Moon around the Earth with respect to the Sun. This is the time from one new Moon to the next, which is called synodic period of the Moon. The synodic period varies significantly; the mean length of the lunation of 29 days, 12 hours and 44 minutes is taken as the synodic month. The length of a lunation today can vary between 29 days, 6 hours, 32 minutes and 29 days, 19 hours and 59 minutes. In former times the variation was greater, in the future it will further decline. The reason is that the eccentricity of Earth's orbit declines with time.

The lunations are longest when the Moon moves slowest and the Earth fastest. In that case the Moon is close to its apogee (= point most distant from the Earth), and the Earth close to its perihelion (= point closest to the Sun).
The lunations are shortest when the Moon moves fastest and the Earth slowest. In that case the Moon is close to its perigee (= point closest to Earth), and the Earth close to its aphelion (= point most distant from the Sun).
Today, lunations are usually longer from October to April than during the other half of the year.

Figure 1 shows the length of 235 lunations starting with february 2001 BC. This timespan equals a so called Metonic Cycle of 19 years. After one Metonic Cycle, the same lunar phase repeats on the same day of the calendar. Red triangles mark instances when a lunar month of 29 days repeats four times; green triangles those instances when a lunar month of 30 days repeats five times. From this figure it is obvious that such cumulations occur close to each other in time. The first occurrence of four lunar months with 29 days in a row in this plot lasted from February until May 1986 BC, followed by five lunar months with 30 days in a row from August until December 1986 BC. Then again, from February until May 1985 BC, occurred four lunar months with 29 days in a row.

Since 2001 BC only five lunar months with 30 days or four lunar months with 29 days can follow each other. Between 2001 BC and 2000 AD five lunar months with 30 days in a row took place 12 times. The occurrence of this phenomenon is not equally spaced in time: between 588 BC and 887 AD there was none, in the 2^nd millenium BC on the other hand 6 times (1986, 1950, 1800, 1764, 1578 and 1128 BC). Four lunar months with 30 days in a row are nothing special; this phenomenon occurred 484 times within 4000 years. Four lunar months with 29 days in a row took place 42 times until 65 BC, and never since. Three lunar months with 29 days in a row are nothing special; this phenomenon occurred 644 times within 4000 years.

Figure 2 shows the length of the lunations starting in February 2001 BC for 236 years until March 1765 BC. Each of the 6 panels contains about 39.5 years. Again, red triangles mark instances when a lunar month of 29 days repeats four times; and green triangles those instances when a lunar month of 30 days repeats five times. The following points and various periods are evident:

There is a strong periodic pattern that repeats about every 111 lunations (almost 9 years), which is the time required for the lunar orbital perigee to advance eastward 360° with respect to the Earth orbital perihelion. As Earth's mean eccentricity is declining with time, the amplitude of the tallest peaks will diminish and at the same time the amplitude of the lesser peaks will progressively increase.
The shortest period is nicely discernible in Figure 1; a strong periodic pattern that repeats about every 14 lunations, which is due to the cyclic eastward advance of the lunar orbital perigee.
Less prominent is a weak periodic pattern that repeats about every 2277 lunations (almost 184 years), which is the time required for the lunar orbital nodes to regress westward 180° with respect to the Earth orbital perihelion.
The maximum negative peaks occur when Earth is near its aphelion and thus moving slowest, and when the lunar conjunction is near Moon's perigee and the Moon thus moving fastest.
The maximum positive peaks occur when Earth is near its perihelion and thus moving fastest, and the lunar conjunction is near Moon's apogee and the Moon thus moving slowest.
The shortest negative peaks occur when Earth is near its aphelion and thus moving slowest, and the lunar conjunction is nearly midway between perigee and apogee and the Moon thus moving with near average velocity.
The shortest positive peaks occur when Earth is near its perihelion and thus moving fastest, and the lunar conjunction is nearly midway between perigee and apogee and the Moon thus moving with near average velocity.

For more details, I refer to the following excellent articles and webpages:

J. Meeus, More Mathematical Astronomy Morsels, Richmond 2002, 19-31 (explains all factors which influence the length of a lunar month; More Mathematical Astronomy Morsels).
J. Meeus, More Mathematical Astronomy Morsels, Richmond 2002, 201-205 (explains the long-period variations of Earth's orbit; More Mathematical Astronomy Morsels).
F. R. Stephenson &L. Baolin, On the Length of the Synodic Month, The Observatory 111, 1991, 21-22 (excellent short summary; online available here).
I. Bromberg, University of Toronto, The Length of the Lunar Cycle.

Crescent sighting criteria

Babylonia:
It is without doubt that a Babylonian sighting-criterion for first and last visibility of the lunar crescent existed. This is proven by many entries in the diaries which were not observed but calculated quantities. However, the often-quoted (see e.g. lately Odeh^[18]) Babylonian sighting-criterion, that the new lunar crescent can be observed if the difference in altitude between Moon and Sun exceeds 12°, i.e. if the timelag between sunset and moonset is greater than 48 minutes, is obviously wrong. Fatoohi et al.^[19] have already pointed at this fact. Different procedures with varying complexity for the calculation of first and last visibility are preserved from Babylonia. However, the later and more elaborate techniques did not substitute the earlier simplier methods: both types of prediction coexisted. The easiest rule one can find in text TU 11: the lunar crescent will become visible if the lagtime between moon and sun was at least 10° which corresponds to 40 minutes^[20]. Starting with the Seleucid Era more complex procedures including amongst others contributions due to the elongation of the moon were developed^[21].

Fotheringham:
In the middle of the 19^th century AD, the astronomer Julius Schmidt collected observations of first and last moon sightings - both positive and negative - in Greece. In 1910, Fotheringham published an article in which he analysed Schmidt's data and those of other observers^[2]; he concluded that the lunar crescent is visible if the difference in altitude between Sun and Moon amounts to 12° or more. The necessary minimal difference in altitude slightly decreases if the difference in azimuth is greater than 0°.

Maunder:
Maunder criticised Fotheringham for his drawing the limit by means of the negative observations^[3]. Fotheringham gives the following equation for the computation of the minimal altitude of the Moon:
H_min = 11° - (5. + azimuth) * azimuth * 0.01

Schoch:
Schoch published planetary tables in 1927 which contained also tables with criteria for the first sighting of the lunar crescent after new moon^[4]. Shortly afterwards he revised these tables and after his death they were published in Neugebauer's book^[5].

Comparison of the criteria:

difference in azimuth	Fotheringham	Maunder	Schoch 1927	Schoch 1930
minimal difference in altitude between Sun and Moon
0°	12°	11°	10.7°	10.4°
5°	11.9°	10.5°	10.3°	10°
10°	11.4°	9.5°	9.6°	9.3°
15°	11°	8°	7.6°	8°
20°	10°	6°	-	6.2°

Bruin:
Frans Bruin chose a new ansatz in 1977^[6]. He calculated the necessary minimal brightness of the Moon in order to be observable at a certain sky brightness. Bruin provides a purely graphic solution of the problem. One has to know the width of the Moon and its altitude above the horizon at sunset to determine whether the Moon will be visible or not. In addition, Bruin gives the time for a best possible sighting.

Schaefer:
Schaefer picked up Bruin's ansatz^[7]; he tried to incorporate many parameters such as the transparency of the atmosphere, the site's altitude, the geographical latitude, the temperature, the relative humidity, the aerosol content and the time of the year. He defined a quantity R, which constitutes a logarithmic measure of the Moon's visibility. Schaefer's criterion did not establish itself and today the calculations of the quantity R are no longer available.

Yallop:
Yallop tried to merge the ansatz from Bruin with the criteria from Maunder and Schoch^[8]. He adopts the values of Schoch from 1930 for the minimal altitude of the Moon at a certain difference in azimuth, at the same time he resorts to the moment of the best possible sighting and the width of the lunar crescent of Bruin. Yallop accounts for the lunar parallax and the topocentric width of the lunar crescent and introduces a parameter q which describes the threshold for a possible successful sighting. He distinguishes in total six zones for q. For historical purposes only the first two or three are relevant:

A		q >	+0.216	easily visible; ΔH ≥ 12°
B	+0.216	≥ q >	-0.014	visible under perfect conditions
C	-0.014	≥ q >	-0.160	may need optical aid to find the crescent
D	-0.160	≥ q >	-0.232	will need optical aid to find the crescent
E	-0.232	≥ q >	-0.293	not visible even with a telescope; ΔH ≤ 8.5°
F	-0.293	≥ q		not visible, below theoretical limit (=Danjon criterium); ΔH ≤ 8°

This criterion is currently considered to be the best^[8].

Calculations

The calculation of last/first visibilities of the lunar crescent before/after new Moon and of New Moon epochs for times far in the past is subject to several uncertainties:

Earth's rate of rotation decreases with time. The resulting time difference, called ΔT, sums up to about 12 hours in 2000 BC and its uncertainty (about 2 hours in 2000 BC) must be accounted for.
Today astronomers are still working on the problem of refining the prediciton criteria for a successful last/first sighting of the lunar crescent ^[1]. Here Yallop's criterion is used, but only the zones A and B were considered.

New Moon epochs and last/first visibilities of the lunar crescent before/after New Moon were calculated between 2000 BC and 2000 AD. The uncertainty in ΔT was accounted for and two different lunar and solar ephemerides were used. First, the longterm DE406 ephemerides of the Jet Propulsion Laboratory, which enable the calculation of the positions of the Sun, the Moon and of all planets between 3001 BC and 3000 AD ^[9]. For comparison the solar coordinates were also calculated using the VSOP2000 theory^[10] and the lunar coordinates using the ELP/MPP02-theory ^[11]. For the computation of last/first visibilities of the lunar crescent the criterion of Yallop has been used, but only his zones A and B were considered. This criterion is based on geocentric calculations, i.e. the observer is assumed to be in the centre of the Earth. The calculations presented here are topocentric, i.e. the observer is assumed at some geographical latitude φ and longitude λ. Thus it had to be checked whether the Yallop criterion can be used and if so, whether the threshold values between the different zones of visibility have to be adjusted. Comparison with more than 600 modern observations (between 1859 and 2004) and with about 440 reported sightings or non-sightings from ancient Babylonia (between 568 BC and 73 BC) showed, that in most cases the observations can be reproduced^[12]. The threshold values have been adopted to the topocentric calculation based on the calculations with a mean ΔT value as follows:

A		q >	+0.095	easily visible
B	+0.095	≥ q >	-0.135	visible under perfect conditions
C	-0.135	≥ q >	-0.280	may need optical aid to find the crescent

For the computation of ΔT, the formulae of Espenak were used^[13]; the uncertainty of these values were estimated with the formula of Huber^[14]. The ΔT values that were obtained for an assumed secular acceleration of the Moon of -26.0"/cy² were adjusted to the secular acceleration of the Moon corresponding to the ephemerides (-25.826"/cy²).

year	ΔT	uncertainty (ΔT)
-3000	20^h 31^m	±2^h 30^m
-2500	16^h 30^m	±1^h 42^m
-2000	12^h 54^m	±1^h 02^m
-1500	9^h 44^m	±32^m
-1000	7^h 01^m	±11^m
-500	4^h 45^m	±7^m
0	2^h 55^m	±5^m

For a more detailed explanation of the solar and lunar ephemerides see here.

Data download

If you download the following data and use them in a publication, please mention the adress of this website and the following paper as origin of the data: R. Gautschy, "Monddaten aus dem Archiv von Illahun: Chronologie des Mittleren Reiches", Zeitschrift für Ägyptische Sprache und Altertumskunde 178, Vol. 1, 2011, 1-19.

For the sites Alexandria, Heliopolis, Memphis, Illahun, Abydos, Theben, Abu Simbel, Elephantine and Babylon tables were created, containing, for a mean value of ΔT, epochs of New Moon and last or first visibilities of the lunar crescent in the Julian, the Egyptian and the Babylonian Calendar respectively. If there arose any difference in the date when the uncertainty of ΔT was accounted for, it is marked accordingly in the tables. In such cases one cannot say exactly on which day the last or first visibility occured.

The following downloadable data correspond to version 2 of my program (June 2012). In comparison to version 1 on one hand newer routines for the calculation of precession have been incorporated, the thresholds of the q-values for the sighting criterion have been adopted to the topocentric calculations based on a mean ΔT, and on the other hand a bug in the calculation of the Moon rising and setting times has been removed.

In the following table, the downloadable data contain in the column "download data last (first)" for each site:

The date of the last/first visibility in the Julian (after 1582, the Gregorian) Calendar,
the time in Greenwich mean time when the lunar crescent could be observed best (to obtain local time simply add 2 hours),
the value of ΔT in seconds,
the value of the uncertainty of ΔT in seconds,
the date of the last/first visibility in the Egyptian/Babylonian Calendar,
the time of surise in Greenwich mean time in case of last visibilities, resp. time of sunset in Greenwich mean time in case of first visibilities (to obtain local time simply add 2 hours),
the time of moonrise in Greenwich mean time in case of last visibilities, resp. time of moonset in Greenwich mean time in case of first visibilities (to obtain local time simply add 2 hours),
the value of the Yallop criterion if the lunar crescent falls into zone B,
a code if there arose a difference in the date in positive direction (+1) or in negative direction (-1) when the uncertainty of ΔT was accounted for,
the date of (geocentric) New Moon in the Julian (after 1582, the Gregorian) Calendar,
the time in Greenwich mean time of New Moon (to obtain local time simply add 2 hours).

The downloadable data in the column "q-values" contain the q-values of successive days calculated with a mean ΔT value. If a sighting of a lunar crescent is documented which should not have been possible theoretically according to the tables, one can take a look in the q-value file and get the value of the sighting criterion on the following or preceeding day. If the q-value falls into category C, the observation is not in contradiction to the caclulations. The smallest q-value of an undoubtedly sighted lunar crescent from Babylon is -0.287.

Ort	download data last	download data first	download q-values
Alexandria	here	here	here
Heliopolis	here	here	here
Memphis	here	here	here
Illahun	here	here	here
Abydos	here	here	here
Theben	here	here	here
Abu Simbel	here	here	here
Elephantine	here	here	here
Babylon	here	here	here

Important: The date in the Egyptian calendar in the tables changes at midnight like the Julian/Gregorian date. This must be taken into account when real and computed data are compared!
The given Babylonian dates prior to 747 BC are uncertain, because no regular system of intercalation is known for earlier times. We know from administrative texts that sometimes each of multiple successive years had an intercalary month. It was not possible to take into account such peculiarities here. An intercalary month is added at the end of the Babylonian year if the New Years' Day would fall prior to the vernal equinox. This means, that the deviation of the theoretical Babylonian calendar which is used here from the actually used one can amount several months!

Comparison with observed data from ancient Egypt

It is an unsettled question when exactely an Egyptian day started. Most scholars assume that a new day started at dawn^[15]. Others think that the Egyptian day started at sunrise^[16]. Both groups refer to data cited in the Almagest of Klaudios Ptolemaios which they interpret in different ways^[17].

For the interpretation of observed data this means that if the day began with sunrise, the observation took place at the end of the previous day. If the lunar crescent cannot be observed anymore, a few minutes later a new lunar month starts. Due to the uncertain beginning of the Egyptian day, the Egyptian date in the tables changes at midnight. This has to be taken into account when using these data. In the case of an assumed beginning of the day at sunrise one has to seek for coincidence with the reported daynumber reduced by two in the tables of last visibility.

Example:
If a first lunar day is reported on I peret 7, one has to seek in the table of the last visibilities for the date I peret 5. In the Egyptian calendar, the date changes only after sunrise whereas in the calculated tables it changes already at midnight. As we deal with an observation taking place before sunrise, the old lunar crescent wasn't observed anymore for the first time on I peret 6. Thus the last successful sighting occured on I peret 5.

On the other hand, if the Egyptian day is assumed to start at dawn, the observation took place at the beginning of the new day and the reported Egyptian date equals the one in the table. The day of the first non-sighting corresponds to the first lunar day of the new month. In this case one has to seek for coincidence with the reported daynumber reduced by one in the tables of last visibility.

Example:
If a first lunar day is reported on I peret 7, the lunar crescent was not observed anymore for the first time on I peret 7. Thus the last successful sighting occured on I peret 6.

Bibliography

1 The IOP Moonwatch Programm collects observations of first sightings of the lunar crescent with the aim to refine the sighting criterion. The same aim has the Islamic Crescents' Observation Project.
2 J. K. Fotheringham, On the smallest visible phase of the Moon, Monthly Notices of the Royal Astronomical Society 70, 1910, 527-530.
3 E. W. Maunder, On the samllest visible phase of the moon, Journal of British Astronomical Association 21, 1911, S. 355-362.
4 C. Schoch, Planeten-Tafeln für Jedermann, 1927.
5 P. V. Neugebauer, Astronomische Chronologie, 1929, B17.
6 F. Bruin, The first visibility of the lunar crescent, Vistas in Astronomy 21, 1977, 331-358.
7 B. E. Schaefer, Visibility of the lunar crescent, Quarterly Journal of the Royal Astronomical Society 29, 1988, 511-523.
8 B. D. Yallop, A Method for Predicting the First Sighting of the New Crescent Moon, NAO Technical Note No. 69, 1997.
9 E. M. Standish, JPL Planetary and Lunar Ephemerides, DE405/LE405, Jet Propulsion Laboratory Interoffice Memorandum 312.F, 1998.
10 X. Moisson &P. Bretagnon, Analytical Planetary Solution VSOP2000, Celestial Mechanics and Dynamical Astronomy 80, 2001, 205-213.
11 J. Chapront &G. Francou, The lunar theory ELP revisited. Introduction of new planetary perturbations, Astronomy &Astrophysics 404 (2003), 735-742.
12 The more than 600 modern observations are summarised in M. Sh. Odeh, New Criterion for Lunar Crescent Visibility, in: Experimental Astronomy 18, 2004, 39-64. The Baylonian observations are listed in S. Stern, The Babylonian month and the new moon: sighting and prediction, Journal for the History of Astronomy 39, 2008, 22-30.
13 F. Espenak, Polynominal expressions for deltaT, http://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html
14 P. J. Huber, Modeling the Length of Day and Extrapolating the Rotation of the Earth, 91-104, in: F. Bònoli, S. de Meis & A. Panaino, Astronomical Amusements: papers in honour of Jean Meeus, Mailand 2000.
15 Parker, Krauss, and Spalinger belong to this group: R. A. Parker, The Beginning of the Lunar Month in Ancient Egypt, Journal of Near Eastern Studies 29, 1970, 219; A. Spalinger, Rez. von C. Leitz, Studien zur ägyptischen Astronomie, 1991, Orientalische Literaturzeitschrift 87, 1992, 23-26; R. Krauss, CChEM 4, 2003, 193-195.
16 Leitz and Luft share this opinion: C. Leitz, Studien zur ägyptischen Astronomie, ÄA 49, Wiesbaden 1989, 50-51; U. Luft, Der Tagesbeginn in Ägypten, Altorientalische Forschungen 14, 1987, 3-11.
17 U. Luft, Der Tagesbeginn in Ägypten, Altorientalische Forschungen 14, 1987, 3-11. R. Krauss, Ägypten &Levante 8, 1998, 122-123.
18 M. Sh. Odeh, New criterion for lunar crescent visibility, Experimental Astronomy 18, 2004, 39-64.
19 L. J. Fatoohi, F. R. Stephenson, S. S. Al-Dargazelli, The Babylonian first visibility of the lunar crescent: data and criterion, The Journal for the History of Astronomy xxx, 1999, 51-72.
20 L. Brack-Bernsen & H. Hunger, TU 11. A Collection of Rules for the Prediction of Lunar Phases and of Month Lenghts, SCIAMVS 3, 2002, 3-90, 39 and 66.
21 M. Ossendrijver, Babylonian Mathematical Astronomy: Procedure Texts, 2012, 161-178 and 195-202.

snflogo Diese Arbeit wurde vom Schweizerischen Nationalfonds im Rahmen eines Marie Heim-Vögtlin Stipendiums finanziert.

I thank Alfred Gautschy, Peter J. Huber, Rolf Krauss and Kurt Locher who contributed valuable comments at various stages of this project.