Methods of Calculating Maya Calendar Dates

Julian Days
Western Calendars
Proleptic Gregorian Calendar
Correlation Constant
Considerations for Calculating Astronomical Events
Lunar Calculations
Venus Calculations
Eclipses
Solstices and Equinoxes
Zenith Passage Days

Julian Days

Astronomers describe a time of observation as a number of days and a fraction of a day from a fundamental epoch. They chose noon January 1st, -4712 Greenwich Mean Time. The Julian day starts at noon because they were interested in things that were visible at night. The number of days and fraction of a day elapsed since this time is a Julian day. The whole number of days elapsed since this time is a Julian day number. In the upper left of a window with a Mesoamerican calendar inscription you will see a Julian day number. Calendar conversions are based on Julian days. A source calendar date is converted to a Julian day and the Julian day is converted to the target calendar.

Western Calendars

In 46 BC Julius Caesar adopted the convention of having three years of twelve months of approximately 30 days each to make a year of 365 days and a leap year of 366 days. This made the length of the civil year 365.25 days, close to the length of the solar year of 365.2422 days. This is the Julian calendar. In spite of this by 1582 there was an appreciable discrepancy between the winter solstice and Christmas and the spring equinox and Easter. Pope Gregory XIII, with the help of Italian astronomer Aloysius Lilius (Luigi Lilio), reformed this system by abolishing the days October 5 through October 14, 1582. This brought the civil and tropical years back into line. He also missed three days every four centuries by decreeing that centuries are only leap years if they are evenly divisible by 400. So for example 1700, 1800, and 1900 are not leap years but 1600 and 2000 are. This is the Gregorian calendar. This program uses the Julian/Gregorian calendar. Dates before 46 BC are converted to the Julian calendar. This is called the proleptic Julian calendar. Astronomical calculations will return a year zero and years before that are negative numbers. This is astronomical dating. There is no year zero in historical dating. In historical dating the year 1 BC is followed by the year 1 so for example, the year -3113 (astronomical dating) is the same as 3114 BC (historical dating). You can use the preferences panel to display years before the year one as astronomical or historical dates.

Calendar reform was adopted at different times in different countries and the Orthodox church never accepted Gregorian calendar reform and still uses the Julian calendar.

Chac will display whether the date you are looking at is in the Julian or Gregorian calendar.

Proleptic Gregorian Calendar

Many books about the Maya and many of the computer programs you can get to do Maya calendar conversions use a calendar called the proleptic Gregorian calendar. In this fictitious calendar all dates before the start of the Gregorian calendar are revised as if the Gregorian calendar had been in use before its adoption in October of 1582. This is how they calculate the date of August 11, 3114 BC for long count 13.0.0.0.0, which actually occurred on September 6, -3113 using the GMT correlation. Converting dates to a calendar that would not be invented for hundreds or thousands of years is historically erroneous. Using the proleptic Gregorian calendar calendar has added a great deal of complexity and confusion to an already extremely arbitrary and revised calendar (ours not the Maya) and to the study of the Maya calendar. The proleptic Gregorian calendar isn't used for anything except Maya calendar conversions. It's amazing that some Mayanists are living in their own world with their own special historically incorrect version of the western calendar. The proleptic Gregorian calendar precludes the use of these conversions for any studies which include other disciplines such as history or astronomy because historians and astronomers use the Julian/Gregorian calendar. When J. Eric S. Thompson was trying to derive the correct correlation constant during the pre-computer days it was easier for him to do the calculations using the proleptic Gregorian calendar because the known correlated dates were before Gregorian calendar reform. He was aware that what he was doing was historically incorrect and he carefully pointed this out in Maya Hieroglyphic Writing by referring to Gregorian dates as N.S. (New System) and Julian dates as O.S. (Old System). Nevertheless this is probably why the proleptic Gregorian calendar is still used by mayanists today. The astronomical algorithm books I have, don't even acknowledge the existence of the proleptic Gregorian calendar but I found an algorithm online and added a panel to Chac to convert between it and the real calendar. This will allow one to check the conversions of other programs, books and online references. In the proleptic Gregorian calendar the days of the week are revised as well. In the Julian/Gregorian calendar Wednesday October 3, 1582 is followed by Thursday, October 14. In the proleptic Gregorian calendar October 3 is a Sunday.

When you get a Maya calendar program try to convert the date October 5, 1582 into the Maya calendar. If the program doesn't report an error it is either converting Julian dates to proleptic Gregorian dates or failing to check user input.

Correlation Constant

The Maya and western calendars are correlated by using a Julian day number of the starting date of the current creation – 13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.13.0.0.0.0 4 Ajaw, 8 Kumk'u. This is referred to as a correlation constant. The generally accepted correlation constant is the Modified Thompson 2 – Goodman, Martinez, Thompson – GMT correlation of 584,283 days. Using the GMT correlation the current creation started on September 11, -3113 (Julian, astronomical dating) or 3114 BC (Julian, historical dating).

The evidence for the GMT correlation is historical, astronomical and archaeological:

Historical: Calendar Round dates with a corresponding Julian dates are recorded in Diego de Landa's Relación de las cosas de Yucatán, the Chronicle of Oxcutzkab and the books of Chilam Balam. Oxcutzkab and de Landa record a date that is a Tun completion. These support the GMT correlation. The fall of the Aztec Empire, Tenochtitlan, occurred on August 13, 1521. A number of different chroniclers wrote that this was a Tzolk'in (called the Tonalpohualli in Nahuatl) of 1 Snake. Various post-conquest scholars such as Bernardino de Sahagun and Diego Duran recorded Tonalpohualli dates with a calendar date (The Aztecs used a 365 day calendar that was revised during the post-classic period and does not correspond to the Haab'). The Annals of the Cakchiquels contains numerous Tzolk'in dates correlated with European dates. Recinos gives 2 Tihax (Etz'nab') as September 10, 1541 (Julian). On this date the old city of Guatemala, founded at the foot of the Volcan de Agua was destroyed. After a period of heavy rain there was an earthquake and then a lahar. Many indigenous Guatemalan communities, principally those speaking Ixil, Mam, Pokomchí and Quiché and many groups in Veracruz, Oaxaca and Chiapas, Mexico, keep the Calendar Round today. These are all consistent with the GMT correlation.

Astronomical: The Maya recorded astronomical phenomena on stelae, temple inscriptions and codices. The GMT correlation does an excellent job of matching lunar data in the supplementary series (See Lunar Calculations). The Dresden Codex contains an eclipse table which gives eclipse seasons when the Moon is near its ascending or descending node and an eclipse is likely to occur. Dates converted with the GMT correlation agree closely with this almanac, confirming the GMT correlation (See Eclipses). The Dresden Codex contains an almanac of the heliacal phenomena of Venus. Using the GMT correlation the heliacal risings are very close to those that are calculated with modern astronomical algorithms (See Venus Calculations).

Archaeological: Various items that can be associated with specific Long Count dates have been isotope dated. In 1959 the University of Pennsylvania carbon dated samples from ten wood lintels from Tikal. These were carved with a date equivalent to 741 using the GMT correlation. The average carbon date was 746 +/- 34 years.

If a proposed correlation only has to agree with one of these lines of evidence there could be numerous other possibilities. Astronomers have proposed many correlations for example: Lounsbury, Fuls, et. al. and Bohm and Bohm.

Here is a list of some constants that have been suggested, that the program recognizes and will name:

If the program names the correlation rather than displaying its number you can still see how many days it is by looking at the small options text field in the lower left corner of an inscription window.

You can set the correlation constant in Chac to whatever you want (within arbitrary limits currently #defined as 200,000 - 1,000,000). If you use a constant other than the GMT the program will display the correlation constant text field in red and show a skull and crossed bones icon to point out that you are using an incorrect correlation:

name or number of days  
correlation

Here is an example of a correlation problem: The Spanish conquered the Mexican capitol, Tenochtitlan, on Tuesday, August 13, 1521. The Mexicans told the conquistadors that the Tzolk'in (Tonalpohualli in Nahuatl) was Ce Coatl (1 Snake). This is right if you use the GMT correlation and the Julian calendar. If you use the proleptic Gregorian calendar you have to revise the conquest date to August 23 in your imaginary calendar. If you don't use the GMT correlation you won't calculate this as 1 Snake. This is an example of why you should not use the proleptic Gregorian calendar and why you should use the GMT correlation.

Considerations for Calculating Astronomical Events

Most astronomical algorithms are calculated by means of a formulae containing powers of the time (T, T², T³, ...). Such polynomial expressions are only valid if the values of T are not too large. Meeus uses the eccentricity of the orbit of Uranus as an example:

e = 0.463 812 2 — 0.000 027 293 T + 0.000 000 078 9 T²

Where T is the time measured in Julian centuries (35,625 days) from the beginning of the year 2000. For values of T much greater than 30 or less than -30 the expression is no longer valid. And for T = -3307 the formula would give e = 1 and so in the year -328,790 the orbit of Uranus would be parabolic, originating outside of the solar system. Another example of this is the VSOP 87 theory which uses polynomials with thousands of terms. This theory allows one to calculate the positions of the planets with very great accuracy.

The Earth's rotation is generally slowing down, however this occurs with unpredictable irregularities. For this reason Universal time varies from Dynamical Time (measured with atomic clocks). The difference, ΔT, is expressed as ΔT = TD - UT. The value of ΔT for times before there were atomic clocks is deduced from astronomical phenomena. ΔT is well-constrained by the times of observed eclipses back to about the year 900. Also it is not a great correction. Before this it is not well-constrained. Approximations I have contain an uncertainty of about two hours from calculated times in the year -3,000. It is also large - about 0.9 days. For example see Meeus and Espenak. This won't make big difference in calculations going back to the Maya classic period but for events in the distant past this will be significant.

Issues like these demonstrate the difficulty in studying astronomical events in the distant past. Modern astronomical algorithms are only reliable for calculating the times of events for a few thousand years.

Lunar Calculations

Thanks to Copernicus we know that the Earth orbits the Sun and the Moon orbits the Earth. For this reason modern astronomers refer to the conjunction of the Sun and Moon (the time when the Sun and Moon have the same ecliptic longitude) as the new Moon. Mesoamerican astronomy was observational not theoretical. The people of Mesoamerica didn't understand the Copernican nature of the solar system. They had no theoretical understanding of the orbital nature of the heavenly bodies. Some authors analyze the lunar inscriptions based on this modern understanding of the motions of the Moon but there is no evidence that the Mesoamericans did. For the people of pre-columbian Mesoamerica there were two ways to observe the new moon:

1. The first evening when one could look to the west after sunset and see the thin crescent Moon. This is referred to as the Palenque system. Even today it can be difficult to predict this and it's important because for example the feast of Ramadan starts on this day. Given our modern ability to know exactly where to look, when the crescent Moon is favorably located, from an excellent site, on rare occasions, using optical instruments, observers can see the and photograph the crescent Moon less than one day after conjunction. Generally, most observers won't see the new Moon with the naked eye until the first evening when the lunar phase day is at least 1.5. If you choose this option in the Chac preferences panel the lunar cycles will be based on the rule that the new Moon is the first day when the lunar phase day is at least 1.5 at six in the evening in time zone -6 (the time zone of the Maya area). Using this approach and the GMT correlation, the program agrees well with classic inscriptions, for example: An inscription at the Temple of the Sun at Palenque records that on Long Count 9.16.4.10.8 there were 26 lunar days completed in a 30 day lunation. This Long Count is also the entry date in the Eclipse table in the Dresden Codex.

2. The first morning when one could no longer see the waning moon. According to Thompson: "The latter method of counting (disappearance of the old moon) is still current in some Tzeltal, Chol, and Tzotzil villages in Chiapas". This is the method used on Quirigua stela E - 9.17.0.0.0, which records this date as the zero date of a lunation. Alternately, a solar eclipse was visible in Central America two days later on 9.17.0.0.2, so this is probably an eclipse warning. The eclipse table in the Dresden Codex gives warnings of eclipses as the first day when one can't see the Moon. If you select this method for the new moon with the Chac preferences panel, the new moon will be the first morning when one can no longer see the waning moon at six in the morning in time zone -6. This will be approximately three days earlier than method 1.

According to Teeple, the Quirigua stela E inscription is lunar deity two and most inscriptions (but not all, particularly earlier ones) use this system. This is the default lunar deity for Chac. Fuls et al. studied a large number of lunar inscriptions and found that there is no consistent method of determining the start of a lunation or moon number. This is why the user can select any one of the six moon numbers in the Chac preferences.

Venus Calculations

The heliacal phenomena are the first day or the last day when Venus is visible as a morning or evening star. These are calculated as an arcus visionis - the difference in altitude between the body and the center of the Sun at the time of geometric rising or setting of the body, not including the 34 arc minutes of refraction that allows one to see a body before its geometric rise or the 0.266,563,88... degree semidiameter of the sun. Atmospheric phenomena like extinction are not considered. The required arcus visionis varies with the brightness of the body. Because Venus has phases and its diameter varies as it orbits closer to the Sun than the Earth, the brightness is different for different events. Values for calculating the arcus visionis used for Venus are taken from Meeus and Salvo De Meis after Carl Schoch and others:

The rising and setting phenomena of Venus are calculated using methods found in Astronomical Algorithms by Jean Meeus. Meeus gives a shortened version of Pierre Bretagnon and Gerard Francou's VSOP 87 theory with enough terms to produce acceptable accuracy. I use the complete VSOP 87 theory with about 5,000 terms to calculate the heliocentric coordinates of the Earth and Venus. For some calculations Meeus gives a simple low-accuracy method and a high-accuracy method. I use the high-accuracy methods for all calculations. My calculations generally vary from the examples in Meeus by less than an arc second. To calculate the time of rising or setting of a body and its altitude and azimuth, one needs the location of the observer. I use temple 1 at Tikal: 89° 37' 23.31" W - 17° 13' 19.38" N.

The appearance and disappearance of Venus were extremely important to the Maya. The Dresden Codex contains a Venus almanac of these events. The heliacal rising was particularly important. It's complex to calculate the heliacal rising of Venus accurately. One has to be able to calculate:

1. the time of the inferior conjunction of Venus
2. the heliocentric coordinates of Venus and the Earth using the VSOP 87 theory
3. Since one wants the apparent, not the actual position of Venus, he must calculate the time it takes for light to get from Venus to the Earth, adjust the time and recalculate.
4. correct for abberation
5. convert to the slightly different FK5 coordinate system
6. correct for nutation
7. convert from elliptical to equatorial coordinates
8. calculate the rise, transit and set times of Venus with at least one additional iteration
9. convert from right ascension and declination to altitude and azimuth at the observer's location

These calculations will rely on many other astronomical and mathematical algorithms.

This is quite complex, but calculating the time of inferior conjunction is trivial. A method for calculating the time of Venus' conjunctions can be found on pages 249-252 of Meeus. This is probably why some studies of the Dresden Codex use a rule of thumb: that the heliacal rising is four days after inferior conjunction. For some dates this is correct but for some it will be several days off. The reason is that the orbit of Venus is not parallel to the orbit of the Earth. Because its orbital plane is inclined at 3.94 degrees to the ecliptic, its declination (angle between a body and the equator) at inferior conjunction will rarely be the same as the Sun. When this happens there will be a transit of Venus, a once in a lifetime occurrence. The following illustration shows the possible geometry at heliacal rising:

This diagram over-simplifies this because Venus' inferior conjunction and rising aren't exactly coincident. There are two positions of the Sun. In the one on the left the Sun's declination is greater than Venus'. In the other one, its declination is less. The difference between the declination of Venus and the Sun can be about 8.3 degrees. For an observer in Central America the difference in altitude can be about five. It may be in either one of these positions or anywhere in between. In the first example, Venus' altitude is below the sun and it will take several extra days to rise until it has the necessary altitude of 5.7 degrees to be visible as the morning star. In the second case it is already higher than the Sun and might be visible the next day. I used these methods to generate the following table showing the first 20 cycles of 260 heliacal risings of Venus based on the Dresden Codex Venus almanac:

Heliacal risings of Venus from the Venus table of the Dresden Codex
Julian dates are calculated as an observer at the Caracol at Chechen Itza (20, 40' 45.34" N 88, 34' 14.71" W).
Starting Julian date 2062523 - 10.5.6.4.0 1 Ahau entry date:

This table demonstrates several things:

1. Heliacal risings calculated using the GMT correlation are an excellent match for the data in the Dresden Codex. The first 20 dates based on the codex average only 0.1 day from calculated values.

2. The cycle of heliacal risings is irregular, varying by five days. Although the Maya were able to observe the average length of the Venus cycle, they couldn't accurately predict any particular heliacal rising.

3. For the almanac to work they had to devise a method to correct for the drift caused by the difference between the 583.92 day average cycle and the canonical 584 days. Pages 184—196 of The Skywatchers by Anthony F. Aveni (2001), describe the Venus tables in the Dresden Codex and other sources in detail and describe two possible correction schemes from Dresden.

When Venus passes between the Earth and the Sun at inferior conjunction it is close to the Earth. For this reason its position in the sky changes rapidly. Because it's far from the Earth as it passes behind the Sun near superior conjunction, its position in the sky changes little each day. Although it is possible to calculate an arcus visions for its disappearance and reappearance, this is approximate for a naked-eye observer. In addition the length of its disappearance behind the Sun is quite variable. Using the arcus visions method the length of invisibility is from 47 to 62 days, averaging 53.73 days. The 90 day length of invisibility in the Dresden codex Venus almanac is canonical, not astronomical.

Eclipses

The Dresden Codex contains an eclipse table that gives the dates of possible eclipses. Chac calculates the occurances of eclipses as maximum eclipse in 24 hour time starting at midnight in the time zone of the Maya civilization — time zone -6. For some eclipses this will be a different calendar date than it is in UT. The eclipse panel shows the type of the next or last eclipse, as the time of maximum eclipse, not opposition or conjunction, as a Long Count, the number of days to or from the current inscription, the Julian date and a calendar date with an hour and minute in 24 hour time. If there is an eclipse on the date, the inscription's eclipse text field shows the time as maximum eclipse in 24 hour time in time zone -6 and the type of eclipse.

Solstices and Equinoxes

The times of Solstices and Equinoxes are calculated using methods in Astronomical Algorithms by Jean Meeus. These times are only reliable from the year -1,000 to 3,000. They are corrected from UT to time zone -6 (the time zone of the Maya area). This will give a different date for some of them. To convert to UT, add six hours (0.25 Julian days).

Zenith Passage Days

In the tropics the Sun passes directly overhead twice each year. These zenith passage days were very important to the people of Mesoamerica. These are calculated using the method for finding the solstices and equinoxes in Astronomical Algorithms by Jean Meeus. Then the time at noon is incremented or decremented to find the closest day to the day when the Sun's declination at noon is equal to the site's latitude. Local noon is calculated as an integer Julian day number (noon UT) minus 0.25 Julian days (for time zone -6) and corrected to local sun time using the equation of time. The Earth's rotation varies over time due to rotational precession. This causes the equinoxes to precess and this is why the year is needed to find the solstices and equinoxes. The sun is only in the south at noon on the solstices and equinoxes. The rest of the year the Sun is east or west of the meridian. The equation of time is a correction for this. The analemma that one sees on sundials is a graphical representation of this.