The star Sirius in ancient Egypt and Babylonia

Introduction

In ancient Egypt two different calendars were used: one solar and one lunar. Egyptians realised early on that the first appearance of the star Sirius in the morning sky - its so called heliacal rising - coincides with the beginning of the inundation of the Nile. The Egyptian solar or civil year consisted of 12 months with 30 days and 5 additional, so called epagomenal days, thus it totalled 365 days. The astronomical solar year is ¹/₄ of a day longer. This means that the astronomical and the civil year drifted apart by approximately one day in four years. Thereby, also the heliacal rising of Sirius was affected by this drift. The timespan in which the heliacal rising of Sirius stays on the same day in the Egyptian calender is called a tetraëteris or quadrennium. A coincidence of civil and astronomical New Years' Day was called apokatastasis by Greek authors, and the period from one apokatastasis to the next is known as the Sothic period.

Sothic period:

Censorinus and other early authors report that Sirius falls behind in the civil calendar by one day in four years due to the fact that the civil year is about ¹/₄ of a day shorter than the astronomical year. After a so called Sothic Cycle, which lasts 1460 astronomical or 1461 civil years, the heliacal rising of Sirius occurs again on New Years' Day, on I akhet 1. In the past, many attempts have been made to pin down the first year of a Sothic period and eventually to determine the absolute date of the implementation of the Egyptian calendar. But the length of a Sothic period is not constant, therefore a schematical calculation with a displacement of one day in four years is not reproducing the real situation in the sky. Nevertheless, the actual length of a Sothic period does not deviate more than 8 years from the assumed 1460 years. Thus it is not surprising that not a single author in antiquity mentioned a so called triëteris (a cycle where the Egyptian date changes already after three years), and it is even questionable if this difference was ever detected in antiquity.

A few observations of Sirius are handed down to us from ancient Egypt and these data can in principle be used for chronological purposes. Thus it is worth investigating Sirius and its phenomena in the course of the year and the uncertainties of such calculations for times far in the past.

Sirius and its phenomena in the course of the year

In the course of the year, Sirius is visible at different phases of the night sky. Figure 1 shows the various phenomena of Sirius for the site Memphis in the year 239 BC (year of the so called Canopus decree) for an assumed arc of vision of 10° and an apparent altitude above the horizon of 3° ^[1]. On the vertical axis hours are plotted (0^h designates midnight), on the horizontal axis months and days are listed.

The lower curve depicts the times of sunset, the upper curve the times of sunrise. The green colored region between the two curves designates the dark time. The three dashed lines show the times of the rising, culmination and setting of Sirius for each day of the year. Eight points on these lines deserve further attention; Klaudios Ptolemaios called them Phaseis. For these points we can find data in the ancient literature. The set of phenomena begins with the cosmic rising of Sirius on July 7^th. On this day, Sirius rises together with the Sun and it is still invisible. It is not until July 19^th that Sirius rises early enough to become observable at dawn: this epoch is called its heliacal rising. Sirius withdraws more and more from the Sun and it is visible during the last hours of the night in the eastern sky. On November 23^rd Sirius sets when the Sun rises. This epoch is called the true cosmic setting and it can not be observed directly. A few days later, on November 28^th, the so called apparent cosmic setting can be observed at dawn. On December 9^th/10^th Sirius crosses the meridian at midnight. Thereafter, the second part of its visibility period starts. Sirius rises earlier in the evening every day and its apparent acronychal rising, which is his last observable rising at dusk, occurs on December 29^th. On January 3^rd it rises exactly when the Sun sets. This phenomenon is called its true achronychal rising. In May, Sirius is visible only during dusk and on May 12^th its last observable setting occurs which is called the heliacal setting. Thus Sirius' circle of visibility is closed. On May 23^rd, Sirius sets together with the Sun (true acronychal setting), which can not be observed directly. From May 12^th until July 19^th Sirius remains invisible.

Arc of vision (arcus visionis)

The arc of vision or arcus visionis of a star is defined as the difference in altitude between the star and the Sun at the moment when the star is observed at the horizon, calculated without the effect of refraction. The light of the star has to pass through the Earth's atmosphere before reaching the observer, and during its passage a ray of light suffers a change in direction owing to refraction. The amount depends on the physical characteristics of the atmosphere and on the altitude of the star. Thus the astronomical refraction is defined as the arc between apparent and true altitude of the star. The amount of refraction is greatest at the horizon. Refraction causes that the apparent altitude of a star is greater than its true altitude. Therefore the arc of vision is a parameter which can be calculated from any observation; it depends on the brightness of the star and on the angular distance between the rising points of the star and the Sun. Sirius is the brightest star in the sky and the angular distance between its rising point and the one of the Sun has decreased with time. This means, in ancient times Sirius should have been visible already at a smaller arcus visionis values than today. Typically the arc of vision is calculated without refraction and with the assumption that the star is visible at an altitude of 0°, i.e. at the horizon, and the Sun 8° (9°, 10°, 11° respectively - depending on the adopted arcus visionis) below the horizon. This classical definition of the arcus visionis with an altitude of the star of 0° is caused by the fact, that in this special case the formulae simplify considerably, an important aspect in the pre-computer era. However, for several reasons it is impossible to observe Sirius in Egypt at an altitude of 0°:

Extinction of rays is strong near the horizon and highly dependent on the transparency of the atmosphere, i.e. highliy dependent on clouds, haze, dust or sand.
Very rarely the horizon is really flat without any hills, mountains or dunes. A ridge with about 200 metres in altitude in a distance of 20 kilometers causes an elevation of the horizon of about 0.6°.

A realistic value for a successful first sighting of Sirius after its period of invisibility is an apparent altitude of 2° to 3° above the horizon, wheras the effect of refraction should be taken into account. In the following I will always denote that angle between Sun and star as arc of vision for which the star has an apparent height of 2° or 3° and the Sun 6° (7°, 8°, 9° respectively) below the horizon. This is in contradiction to the classical definition of the arcus visionis, but reflects the true constraints in the sky. The values of the arcus visionis given here can be simply converted into "classical" values of the arcus visionis by subtracting the value of refraction. The corresponding standard values of refraction are 0.2° at an altitude of the star of 3° and 0.3° at an altitude of 2°.

Calculations

The calculation of heliacal risings and settings of Sirius for antiquity is subject to various uncertainties:

Sirius is a star close to the Sun with relatively large proper motion. The proper motion of a star is its angular change in position over time and it must be taken into account.
The arc of vision is not constant and can be calculated only after an observation was made. Thus calculations should be performed for a range of values of the arc of vision.
Earth's rate of rotation decreases with time. The resulting time difference, called ΔT, which sums up to about 12 hours in 2000 BC and its uncertainty (about 2 hours in 2000 BC) must be accounted for.

The heliacal risings and settings of Sirius between 3000 BC and 2000 AD were calculated and all parameters were systematically varied within the acceptable limits to test the influence of various parameters. For the calculation of the coordinates of Sirius, its coordinates for the year 2000 BC were used as starting point ^[2]. Precession, nutation, the change of the inclination of the ecliptic in the course of time and the proper motion of Sirius were incorporated. For the calulation of the positions of the Sun two different ephemerides were tested. 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 ^[3]. For comparison, the solar coordinates were also calculated using the VSOP2000 theory^[4].

For the computation of ΔT, the formulae of Espenak were used^[5], and for the estimation of the uncertainty of these values, the formula of Huber^[6]. The ΔT values, which 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 35^m	±5^m
500	1^h 55^m	±2^m20^s
1000	26^m 10^s	±55^s

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, which will be published in 2011 as origin of the data. R. Gautschy, "Der Stern Sirius in Ägypten", Zeitschrift für Ägyptische Sprache und Altertumskunde 178, Vol. 2, 2011, 116-131.

As the exact observing site is unknown in most historical Sirius sightings, the heliacal risings and settings of Sirius were calculated for Alexandria, Memphis, Illahun, Theben, Elephantine and Babylon and the arc of vision has been varied between 8° and 11°. It should be noted that the actual altitude of the Sun below the horizon in the files in the following table can vary by almost one degree.

Example:
elevation of Sirius of +2° and arc of vision of 8°: all altitudes of the Sun between -6.0 and -6.999999° fulfill this criterion
elevation of Sirius of +2° and arc of vision of 9°: all altitudes of the Sun between -7.0 and -7.999999° fulfill this criterion, and so on.

The downloadable data in the following table contain in the column "download data heliacal rising and setting" for each site and chosen arc of vision:

The date of the heliacal rising of Sirius in the Gregorian (Julian prior to 1582) Calendar,
the time in Greenwich mean time when the heliacal rising of Sirius could be observed best (to obtain local time simply add 2 hours),
the apparent altitude of the Sun,
the date of the heliacal rising of Sirius in the Egyptian Calendar,
the date of the heliacal setting of Sirius in the Gregorian (Julian prior to 1582) Calendar,
the time in Greenwich mean time when the heliacal setting of Sirius could be observed best (to obtain local time simply add 2 hours),
the apparent altitude of the Sun,
the date of the heliacal setting of Sirius in the Egyptian Calendar,
number of days of Sirius' invisibility,
the value of ΔT in seconds,
the value of the uncertainty of ΔT in seconds.

From the downloadable data in the column "download tetraëteris" one can extract the location of a triëteris (epochs when the heliacal rising of Sirius falls on the same day in the Egyptian calendar for only three years) and of a penteteris (epochs when the heliacal rising of Sirius falls on the same day in the Egyptian calendar for five years). These tables contain the following data:

The starting year of a tetraëteris or triëteris/penteteris,
the Egyptian date,
the corresponding Gregorian (Julian prior to 1582) date in july/august of the heliacal rising of Sirius.

Example:
-2118 2 peret 15 18/18/17/17
-2114 2 peret 16 18/18/17
-2111 2 peret 17 18/18/18/17/17

The first row shows an example where the heliacal rising of Sirius fell on 2 peret 15 in the Egyptian Calendar and in the years -2118 and -2117 on July 18^th in the Julian Calendar, in the years -2116 and -2115 on July 17^th. This is an example for a tetraëteris.
The second row shows a triëteris, i.e. the heliacal rising of Sirius fell on the same day in the Egyptian Calendar (2 peret 16) only on three successive years (-2114, -2113 and -2112). The corresponding Julian dates are July 18^th and July 17^th in -2112.
The third row is an example for a penteteris: for five years, from -2111 until -2107, the heliacal rising of Sirius took place on 2 peret 17. In the Julian Calendar, this phenomenon took place on July 18^th in -2111 to -2109, and on July 17^th in -2108 and -2107.

site	altitude above horizon	arc of vision	download data heliacal rising and setting	download tetraëteris
Alexandria	2°	8°	here	here
	3°	8°	here	here
	2°	9°	here	here
	3°	9°	here	here
	2°	10°	here	here
	3°	10°	here	here
	2°	11°	here	here
	3°	11°	here	here
Memphis	2°	8°	here	here
	3°	8°	here	here
	2°	9°	here	here
	3°	9°	here	here
	2°	10°	here	here
	3°	10°	here	here
	2°	11°	here	here
	3°	11°	here	here
Illahun	2°	8°	here	here
	3°	8°	here	here
	2°	9°	here	here
	3°	9°	here	here
	2°	10°	here	here
	3°	10°	here	here
	2°	11°	here	here
	3°	11°	here	here
Theben	2°	8°	here	here
	3°	8°	here	here
	2°	9°	here	here
	3°	9°	here	here
	2°	10°	here	here
	3°	10°	here	here
	2°	11°	here	here
	3°	11°	here	here
Elephantine	2°	8°	here	here
	3°	8°	here	here
	2°	9°	here	here
	3°	9°	here	here
	2°	10°	here	here
	3°	10°	here	here
	2°	11°	here	here
	3°	11°	here	here
Babylon	2°	8°	here	here
	3°	8°	here	here
	2°	9°	here	here
	3°	9°	here	here
	2°	10°	here	here
	3°	10°	here	here
	2°	11°	here	here
	3°	11°	here	here

Comparison with ancient observations

For ancient Egypt, I am aware of six reported observations of Sirius, which can in principle be used for chronological purposes. Presumably the most reliable one is the report of a heliacal rising of Sirius from Ptolemaic times. It is mentioned in the so called decree of Canopus from year 9 of Ptolemaios III. Euergetes I. Thus in the 10^th month, on Payni 1^st (corresponds to II shemou 1) the feast of the heliacal rising of Sirius should be celebrated. From the Ptolemaic time, we also know the reference point of the observation: Olympiodorus reports that in greek-roman times the heliacal rising of Sirius in Memphis was decisive. A heliacal rising of Sirius in Memphis in 239 BC was only possible on July 19^th if the arc of vision amounted about 10°.

Another observation of a heliacal rising of Sirius is the so called Censorinus date from 139 AD. Censorinus reports that 100 years before he published his book (239 AD) the heliacal rising of Sirius coincided with the New Year's Day in Egypt which equals Thoth 1^st (I ahket 1). A heliacal rising of Sirius in Memphis in 139 AD was only possible on July 21^st if the arc of vision amounted between 10° and 11°.

The following four ancient observations are not suitable for computations for an arc of vision because the exact absolute years are not known. Thus one can only infer rough absolute dates for the mentioned pharaohs when one assumes some realistic span of values for the arc of vision:

A fragment of a temple diary from year 7 of pharaoh Sewoseret III, which was found at Illahun, contains the prediction of a heliacal rising of Sirius^[7]. The entry of the date was made on III peret 23 which presumably is the arriving day of the letter at Illahun. The date mentioned for the heliacal rising is IIII peret 16. In addition, another fragment from the same temple diary of year 7 survived in which the arrival of the offerings for the feast of the heliacal rising of Sirius is mentioned on IIII peret 17. With an assumed arc of vision between 9° and 11° and an assumed reference point at Memphis/Heliopolis, the heliacal rising took place on IIII peret 16 between 1881 BC and 1868 BC.
In the Egyptian Western Desert at Gebel Tjauti a graffito containing the date of an observation of a heliacal rising of Sirius was found. The inscription is dated into the 17^th dynasty and it mentions that in year 11 of some unknown king the heliacal rising of Sirius took place on II shemou 20^[8]. The text suggests an observation at Gebel Tjauti, probably from the top of the plateau. For an observational point at Gebel Tjauti (close to Theben) and for a value for the arc of vision between 9° and 11°, the heliacal rising took place on II shemou 20 between 1608 BC and 1595 BC.
On the island Elephantine, on a stone belonging to the remnants of the temple of Khnum an inscription says that the feast of the heliacal rising of Sirius took place on Epiphi 28^th (equals III shemou 28). This stone is part of a feast list with informations from the times of pharaoh Thutmose III. Unfortunately the exact year is not given. Thus this date can only give a rough idea about the abolute dating of pharaoh Thutmose III. For an assumed reference point Memphis/Heliopolis and for a value of the arc of vision between 9° and 11°, the heliacal rising of Sirius occured on III shemou 28 between 1474 BC and 1461 BC.
On the verso of the medical papyrus Ebers a famous double calendar was found. In its first two rows a heliacal rising of Sirius is reported in year 9 of pharaoh Amenhotep I on III shemou 9. For an assumed reference point Memphis/Heliopolis and for a value of the arc of vision between 9° and 11° this observation suggests some year between 1550 BC and 1537 BC for the 9^th year of Amenhotep I.

Out of this six reports of heliacal risings of Sirius from ancient Egypt due to textual inconsistencies (Censorinus report, date from Illahun), fundamental doubts whether the date is actually a date of a heliacal rising of Sirius (Gebel Tjauti date, Ebers date) or due to incomplete information (Elephantine date) only the date in the Canopus decree deserves the designation "anchor for chronology". All other reports can be used to reduce the possible absolute timespan for a pharaoh, but no reduction to four years is possible. Instead one has to account for a timespan of 12 to 14 years - under the assumption that the reference point is known.

To maybe reduce this timespan further one has to look for other observations of Sirius from antiquity. Such observations are included in the books of Klaudios Ptolemaios (2^nd century AD) and Geminos (1^st century BC) in the so called parapegmata. The book Phaseis of Ptolemaios is a calendar which contains dates for heliacal risings and settings for thirty bright stars and dates for more or less constant changes in the weather. Ptolemaios gives dates for heliacal risings and settings of Sirius in Egypt which are summarised in the following table. The table also contains the arc of vision which was computed from these observations:

climate	latitude	Alexandrinian Date	Julian Date	arc of vision	tetraëteris
14 hours	31°	Epiphi 28^th	july 22^nd	12°	22/22/22/21
14 hours	31°	Pachon 17^th	may 12^th	10°	12/12/12/11

Thus from Egypt, four decent observations of Sirius are known from antiquity which can be used for a computation of the arc of vision due to the fact that the point of reference is known:

source	time	arc of vision
Canopus-decree	239 BC	10°
Censorinus	139 AD	10° - 11°
Ptolemaios: rising	2. century AD	11° - 12°
Ptolemaios: setting	2. century AD	9° - 10°

In addition, we have one hint by Dositheos mentioned by Geminos from which a slightly smaller arc of vision between 8° and 9° can be deduced. One must, however, keep in mind that the dating system of Geminos is less accurate than the one in the previously mentioned sources. All informations together suggest a mean value of the arcus visionis of Sirius in ancient Lower Egypt of 10°. This value agrees well with the 9° to 10° that were found during an observing campaign in 1926 in Cairo/Heliopolis^[9]. When excellent weather conditions prevailed, Sirius could be observed at an arc of vision of 9° on the day of its heliacal rising. In case of slight haze and the weather conditions were at an average, the value increased to 10° to 11°. Haze and dust close to the horizon encountered frequently during summer in Egypt, therefore excellent weather conditions can be expected only seldomly. In addition, there are hints that the climatic conditions changed during time. Presumably, the climate was a more humid than today: the river Nile contained more water and the plain of inundation was broader^[10]; this favours haze and the creation of fog both of which are extremely unfavourable for celestial observations close to the horizon.

Bibliography

1 This figure has been plotted in analogy to figure 16 in P.V. Neugebauer, Astronomische Chronologie, Berlin 1929, 151, who had plotted it for a latitude of 50°N and "present time".
2 M.A.C. Perryman et al., The Hipparcos Catalogue, Astronomy &Astrophysics 323, 1997, L49-52.
3 E. M. Standish, JPL Planetary and Lunar Ephemerides, DE405/LE405, Jet Propulsion Laboratory Interoffice Memorandum 312.F, 1998.
4 X. Moisson &P. Bretagnon, Analytical Planetary Solution VSOP2000, Celestial Mechanics and Dynamical Astronomy 80, 2001, 205-213.
5 F. Espenak, Polynominal expressions for deltaT, http://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html
6 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.
7 U. Luft, Die chronologische Fixierung des ägyptischen Mittleren Reiches nach dem Tempelarchiv von Illahun, Wien 1992, 58.
8 J.C. Darnell, Theban Desert Road Survey in the Egyptian Western Desert Vol. 1, Chicago 2002, 49-52.
9 L. Borchardt &P. V. Neugebauer, Orientalische Literaturzeitschrift 29 Nr. 5, 1926, 309-16 und L. Borchardt &P. V. Neugebauer, Orientalische Literaturzeitschrift 30 Nr. 6, 1927, 441-48.
10 P. F. O'Mara, Journal of Near Eastern Studies 62 Vol.1, 2003, 26.

snflogo This work was supported by a Marie Heim-Vögtlin grant of the Swiss National Science Foundation.