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The structure of these pages is as follows:

## 1. What is a calendar?

A calendar is

1. a method for grouping days into larger periods, for example weeks, months, and years.
2. a representation of a division of days, for example on paper.

Humanity has used calendars for thousands of years, because calendars are useful if you want to make appointments, or if you have to date astronomical observations to discover important periodicities, and also if you need to know when periodic things return, such as the weekend, the farmers' market, the harvest, the summer vacation, halloween, your birthday, or the annual tax statement filing date.

The smallest division of a calendar is the day: the period after which the Sun returns to about the same place in the sky. A day is itself divided into smaller parts, but those are usually not considered to be part of the calendar. Calendars and smaller divisions of time are part of time keeping.

To fix a calendar for all times, you need the following parts:

• rules to determine when a day starts. Some calendars have days starting at midnight, and others at midday, or at sunset.
• rules to find the periods (month, year) that a particular day belongs to. You can do that, for instance, by fixing the lengths of the periods (Gregorian, Julian: "January always has 31 days"; Gregorians, Julian, Islamic: "every year has 12 months"), or by specifying their beginning (Islamic: "the month begins on the evening that the crescent moon is visible for the first time after a New Moon"; modern Gregorian: "the year starts on 1 January").
• a first day (the epoch) ― the first day of the first year. This first day is often selected to commemorate an important historical or mythical event, such as the creation of the world, the foundation of the capital city, the beginning of the rule of the leader, or an important religious event.
• a rule for naming periods before the first day. What do you call the year before year 1 of the calendar?

## 2. Which main types of calendars are there?

Most calendars that are used in the world try to follow the Sun or the Moon or both.

There are also calendars that follow neither the seasons nor the phases of the Moon. An example of such a calendar is the Julian day count (not to be confused with the Julian calendar).

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It is possible in principle to devise a calendar that is based on which constellations are visible at night, but I do not know of any people that have done so. It would be difficult to make such a calendar, because to be able to tell what date it is from the location of constellations in the night sky you have to know the time very accurately, and accurate determination of time at night was until a few centuries ago a lot more difficult than measuring the date from observations of the Sun or Moon.

There is a connection between the seasons and which constellations are visible then, but that connection shifts in the course of time, because of the precession of the equinoxes.

The 12 astrological signs of the zodiac are each equally large, each last a twelfth part of the year (i.e., about 30 days), are tied to the seasons, and have the names of astronomical constellations through which the Sun passed during those seasons about 2500 years ago. Because of the precession, the seasons and stars (and constellations) have now shifted relative to each other over about one sign, so the Sun is now no longer in the (astronomical) constellation of the Ram when the Sun is in the (astrological) sign of the Ram.

## 3. Why are leap years needed?

The number of days in a tropical year (a year of the seasons) is not a whole number, so a calendar that has the same number of days each year will get out of step with the seasons. A calendar with 365 days per calendar year (like the Egyptian calendar of long ago) gets out of step with the seasons by about a day every four years, and a better fitting calendar with a constant number of days per year does not exist. To keep the calendar in line with the seasons in the long run, one must adjust the number of days in some calendar years. A calendar year with an extra day in it is called a leap year and the extra day is called a bissextile day (or leap day).

The number of days in a synodical month (a full cycle of the phases of the Moon) is not a whole number, either, so a calendar that has the same number of days in each month will get out of step with the phases of the Moon, so months must have varying numbers of days if the calendar is to keep in line with the phases of the Moon.

The number of synodical months in a tropical year is not a whole number, so a calendar with the same number of synodical months for each calendar year will get out of step with the seasons, and calendar years must have varying numbers of months to stay in tune with the seasons in the long run. The extra months that are inserted once in a while are called embolistic months.

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It seems logical to add special days (such as a bissextile day) at the end of the year, as is done in the Egyptian calendar and Central-American calendars. In the Julian and Gregorian calendars the bissextile day is always added to February, which is the second month of the year, but it seems that a very long time ago when the bissextile day was introduced, February was the last month of the year.

## 4. Which leap year rules are best?

In the course of history, very many different methods have been invented to insert bissextile days and embolistic months. The oldest method was to observe when the calendar was getting out of step with the lunar phases or the seasons, and then insert an extra day or month. With this method, the insertions were somewhat arbitrary, and one could not predict the calendar, which was not very convenient. Eventually, most societies discovered certain periods that almost exactly fit a whole number of days and a whole number of (synodical) months, or days and (tropical) years, or months and years. With these periods, fixed rules could be devised for inserting extra days or months, so the calendar could be predicted far into the future and could yet closely follow the lunar phases or seasons or both.

If you know the lengths of the synodical month and the tropical year accurately enough, then you can find convenient periods to base a solar calendar on. For example, the length of the tropical year is currently 365.2421898 days. A multiple of this number that is as close as possible to a whole number of days forms a good basis for a solar calendar. Suitable multiples (under 1000 years) are:

Table 1: Good Calendars for Tropical Years

 years 1 4 29 33 128 (400) (450) days 365 1461 10,592 12,053 46,751 (146,097) (164,359) slack 6h 45m 34m 11m 25s (3h) (20m) deviation 6h 11m 70s 20s 0.2s (27s) (3s) growth 4 128 1230 4270 430,000 (3200) (31,000)

The "slack" is the (absolute) difference between the listed number of days and the listed number of tropical years. The "deviation" is the (absolute) difference between the average calendar year (based on the stated number of days and years) and the tropical year. The "growth" show after how many calendar years the difference grows to 1 day (if you assume that the length of the tropical year does not change). The values for the approximation 400 years = 146,097 days (used in the Gregorian calendar) and 450 years = 164,359 days (used in the Orthodox calendar) are listed between parentheses because those approximations aren't very good, compared to the approximations based on the much shorter periods of 33 and 128 years.

If you want to make the calendar year such that the beginning of spring is on average on the same day of the year each year (for example, to keep Easter in its proper place), then you must use a year length of 365.2423740 days. Suitable multiples are:

Table 2: Good Calendars for Spring Years

 years 1 4 29 33 (128) (400) (450) 590 days 365 1461 10,592 12,053 (46,751) (146,097) (164,359) 215,493 slack 6h 44m 41m 143s (34m) (72m) (98m) 59s deviation 6h 11m 86s 4s (16s) (11s) (13s) 0.1s growth 4 131 1010 19,900 (5400) (7900) (6600) 900,000

In this case, the approximations of 400 and 450 years are better than that of 128 years, but still worse than that of 33 years.

The period of 4 years forms the basis of the Julian calendar, and the period of 400 years that of the Gregorian calendar. 400 tropical years equal 146,097 days, except for a difference of 3 hours. 400 March-equinox years are equal to 146,097 days, except for a deviation of 72 minutes.

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The above table shows that a calendar based on the correspondence 128 years = 46,751 days runs 0.2 seconds a year out of step with a (season-averaged) tropical year, while the Gregorian calendar runs 27 seconds a year out of step with the tropical year, and that a calendar based on 33 years = 12,053 days runs 4 seconds a year out of step with the spring year, compared to 11 seconds a year for the Gregorian calendar. Yet we use the Gregorian calendar (based on a period of 400 years) and not a calendar based on the period of 128 or 33 years, because

1. changing a calendar that is used in very many countries is very difficult. Changing from the Julian calendar (based on a period of 4 years) to the Gregorian calendar (based on a period of 400 years) was also a slow process: The last countries to make that change did so some 300 years after the first ones did.
2. the difference between the average length of a year in a 128-year calendar and a 400-year calendar is so small that it takes 3200 years for it to grow to 1 day. And for a 33-year calendar the difference takes 13,200 years to grow to 1 day. Why should we worry about that today?
3. whether a 128-year calendar is really better than a 400-year calendar depends on what you are trying to match the calendar to. If you want to follow the average of all seasons, then the 128-year calendar is better than the 400-year calendar, but if you tie the calendar to the ascending equinox (the beginning of spring, in the northern hemisphere), then the 400-year calendar is more accurate than the 128-year calendar.
4. calendar rules for a 128-year calendar or a 33-year calendar are more difficult to use than calendar rules for a 400-year calendar, because 400 is a more round number than 128 or 33 are. It is easier to calculate in your head whether 400 evenly divides a certain year number than whether 128 or 33 evenly divide that year number.

The periods of 33 and 128 years have, as far as I know, not been used for any historical or modern calendar.

Here's a similar table for the correspondence between days and synodical months:

Table 3: Good Calendars for Synodical Months

 months 1 2 15 17 49 850 days 30 59 443 502 1447 25,101 slack 11h 88m 59m 29m 99s 41s deviation 11h 44m 4m 2m 2s 0.05s

A period of 49 months corresponds to 1447 days, with a difference of 99 seconds.

Here's a similar table for the correspondence between synodical months and tropical years:

Table 4: Good Calendars for Synodical Months in Tropical Years

 years 1 2 3 8 11 19 334 months 12 25 37 99 136 235 4131 slack 11d 8d 3d 38h 36h 2h 42m deviation 11d 4d 25h 5h 3h 7m 8s

The period of 12 synodical months yields an ordinary year (without an embolistic month) in most lunar and lunisolar calendars. The period of 99 months was used for a while by the ancient Greeks, but the mismatch with the tropical year runs to more than a day for each 8 years. The period of 235 months is know as the Cycle of Meton (a Greek), though it was already known to the Babylonians and is the basis of the Babylonian and Jewish calendars.

We can also search for periods of a whole number of days that fit both a whole number of synodical months and a whole number of tropical years, with as little slack as possible. The following table shows the best periods under 300,000 days, with an estimate for the slack.

Table 5: Good Calendars for (Ecliptic-Averaged) Tropical Years and Synodical Months

 days 257,861 121,991 237,042 135,870 20,819 months 8732 4131 8027 4601 705 years 706 334 649 372 57 slack 3h 3h 4h 6h 6h ------- -------- -------- -------- -------- -------- days 243,982 101,172 115,051 142,810 223,163 months 8262 3426 3896 4836 7557 years 668 277 315 391 611 slack 7h 7h 7h 7h 8h

Table 6: Good Calendars for March Equinox Years and Synodical Months

 days 257,861 121,991 142,810 278,680 135,870 months 8732 4131 4836 9437 4601 years 706 334 391 763 372 slack 3h 3h 5h 6h 6h -------- -------- -------- -------- -------- ------- days 20,819 243,982 264,801 237,042 156,689 months 705 8262 8967 8027 5306 years 57 668 725 649 429 slack 6h 7h 7h 7h 8h

It appears that it is not possible to find a period that snugly fits whole numbers of days, synodical months, and tropical years --- at least not with periods less than 820 years.

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## 5. Must Calendars Ever Be Adjusted?

The average lengths (over many years) of the tropical year, the synodical month, and the solar day change slowly and not in an entirely predictable way, so no calendar that can be predicted arbitrarily far into the future can stay in step with the seasons or the lunar phases or both for ever.

Because of the tides of the Moon and other causes, the rotation of the Earth around its axis is slowing down gradually, and the Moon is moving further away from the Earth. Because of this, the length of the solar day (the average period after which the Sun crosses the celestial meridian again, seen from the same location) currently increases by 0.0017 seconds per century (with an estimated uncertainty of 3 percent), and the length of the synodical month increases by about 0.038 seconds per century [Stephenson].

The gravity of the other planets influences the orbit of the Earth, but that influence averages to nearly zero in the long run. The semimajor axis of the Earth keeps its length to within 0.003% over 500 million years centered on now (see //adsabs.harvard.edu/abs/2004A%26A...428..261L, Figure 11). That means that the sidereal year says at nearly the same length. The tropical year is like the sidereal year but includes the effect of precession. All in all, the length of the tropical year currently decreases by about 0.53 seconds per century (but that rate of change varies slowly).

The increase in the length of the solar day means that we need to insert a leap second once in a while (and ever more often) to keep our calendar day of normally 86400 seconds long (24 hours of each 60 minutes of each 60 seconds) in step with the rotation of the Earth around its axis. A leap second is always inserted just before midnight, preferably at the end of June or December. In such a case, the second of 23 hours 59 minutes 59 seconds is followed by the second of 23 hours 59 minutes 60 seconds (the leap second), and then by the second 0 hours 0 minutes 0 seconds of the next day. At the date of writing, 11 January 2004, the last time (so far) that a leap second was inserted was at the end of 1998. See //www.boulder.nist.gov/timefreq/pubs/bulletin/leapsecond.htm.

In principle, it is possible that a second must be skipped (a negative leap second), but this has not been necessary so far. See //tycho.usno.navy.mil/leapsec.html for more general information.

The decrease in the length of the tropical year and the increase in the length of the solar day mean that the number of solar days in a tropical year is slowly decreasing.

If you want to keep your predictable solar calendar in step with the seasons, then you'll have to adjust the rules of your calendar eventually, because those rules assume a constant length for the year (measured in solar days). You can also keep the rules the same, but then the calendar will get out of step with the seasons.

Because the calendar only uses whole days, any adjustment to the calendar is at least one day in size, so even a calendar that follows the seasons perfectly on average will occasionally be about half a day ahead of the seasons, or half a day behind. It is therefore not useful to worry about your calendar getting out of step by less than half a day, so you could take half a day as the greatest allowable deviation between the seasons and the calendar. If, for example, the average deviation has increased to +1/2 day, then you can adjust the calendar by 1 day so that the average deviation is −1/2 day and gradually increases to +1/2 day again, when you can make a further adjustment of 1 day, and so on. On average, the seasons will remain on the same calendar dates as they are now. It is not a disaster if you wait longer, but then you'll have to make an adjustment of more than 1 day, or you'll have to accept (part of) the net shift relative to long ago, or both.

We have examples from the past for both options: When the Gregorian calendar was introduced in 1582 in Catholic countries, the calendar was shifted by 10 days to correct for the slow drift of the seasons relative to the Julian calendar over the preceding 1200 years. It seems likely that Christmas is celebrated at the calendar date (25 December) that used to be associated with the midwinter feast (Yule) at the winter solstice in the northern hemisphere, but nowadays that solstice is usually on 21 December. The difference of 4 days has, for unknown reasons, not be corrected when the Gregorian calendar was introduced. If and at what deviation the calendar will be adjusted is a decision for politicians in the distant future.

Not everyone agrees what the Gregorian calendar is supposed to follow. It is clear that it follows "the seasons", but which one, exactly? If you assume that every calendar day has exactly 86400 seconds (so no leap seconds), then the average calendar year (365.2425 days) is now about 27 seconds longer than the average tropical year (365.24219 days) which follows the average of all seasons. However, in book [Steel], Duncan Steel argues that the Gregorian calendar was created to stay in step with the March equinox (the ascending equinox, the beginning of spring in northern latitudes) and the average calendar year is "only" 11 seconds longer than the average March equinox year (of 365.24237 days).

The point is not that one is correct and the other one is not, but that reasonable people can disagree about this issue, because the annual differences are very small. It really doesn't matter that much, unless the small annual difference has added up to a total that is too great.

How can you calculate something like that? If the increase in the length of the day is the same every day, then the total delay relative to the beginning of the first day is equal to the average of the extra length of the first day and of the last day (compared to the standard day), multiplied by the number of days in the period.

For example, assume that every next day is 1 second longer than the previous one, and that you measure for one week. Then the second day gives a delay of 1 seconds (compared to the first one), the third day 2 seconds, the fourth day 3 seconds, and so on, and the seventh day 6 seconds. The total delay accumulated during that week is then 0 + 1 + 2 + 3 + 4 + 5 + 6 = 21 seconds relative to the case where every day were as long as the first one. Seven days with delay are 21 seconds longer in total than seven days that each are as long as the first day. The average of the extra lengths of the first and last day is then (0 + 6)/2 = 3 seconds and the number of days is 7, so the total delay is also equal to 3 × 7 = 21 seconds.

I reported an increase in the length of the day of 0.0017 seconds per century, so 0.017 seconds after 10 centuries (which is equal to 1000 years). The average solar day is already about 0.003 seconds longer than the standard day of 86400 seconds, so the first day of the period of 1000 years is about 0.003 seconds longer than 86400 seconds and the last day of that period will probably be about 0.020 seconds longer than 86400 seconds. The number of days in 1000 Gregorian years is about 365243, so the total delay that is accumulated during the 1000 years is about (0.003 + 0.020)/2 × 365243 = 4200 seconds or 1.2 hours. Because of differences in rounding, this is not exactly equal to the 1.3 hours I mentioned earlier.

These shifts are not exactly proportional to the elapsed time, so you cannot, for example, multiply everything by 10 to get the values for the next 10,000 years. The exact rotation of the Earth over the next 10,000 years is so uncertain that predictions over such a long period of time have little value.

If you think that the calendar should follow the average of all seasons, and if you take a boundary of 1/2 day, then it would be time to omit a leap day (i.e., a negative leap day) in about 13 centuries, but if you think the calendar should keep the March equinox (and Easter) at the same calendar date, then it will take much longer before an adjustment is necessary, perhaps as many as 60 centuries.

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## 6. When Does a Day Start?

Even today a new calendar day does not begin at midnight in all calendars. In the Jewish and Islamic calendars a new day starts at sunset. In the Hindu calendar a new day starts at sunrise. And for the Julian Day Number the calendar day starts at 12 o'clock noon.

In the westen calendars it is nowadays custom to begin a new calendar day at midnight. Various sources say that this custom began only in the 19th century, but they do not explain why. It became necessary to standardize times when transportation by trains began from about 1840, because the train schedules would become very complicated if they had to take the local times of all stations into account.

During the International Meridian Conference in Washington D.C. (USA) in 1884 (//wwp.greenwichmeantime.com/info/conference-finalact.htm), one of the things that were decided by the participating countries was to measure geographical longitudes from Greenwich, to define a universal time for the whole world (then Greenwich Mean Time, nowadays Coordinated Universal Time = UTC) that begins at midnight mean solar time as measured in Greenwich, and to have the calendar day begin at midnight "as soon as possible". Local times (not at sea and not in astronomy) were explicitly excluded from this standard, but I think that it was convenient for countries to apply this existing standard to civil life as well, if a standard became necessary.

Nowadays we also have time zones, areas in which everybody uses the same standard time. This was also convenient for, for example, train schedules and telegraph connections. Time zones were not discussed during the International Meridian Conference, but were defined here and there when that became convenient. The first time zone, Greenwich Mean Time, was instituted in Great Britain on 1 December 1847, because of the trains. The railways in the United States defined time zones in 1883, and within a few years most large cities used them as well. Time zones were enshrined in law in the US only in 1918. By 1929, most large countries had introduced time zones. (//en.wikipedia.org/wiki/Time_zone)

In the Netherlands, it was decided in 1866 that train stations should keep the mean time of Amsterdam. Only in 1909 a time zone was instituted for use in the whole country (and not just for the railways and telegraphs). (//www.phys.uu.nl/~vgent/wettijd/wettijd.htm)

## 7. When does a century start? And a millennium?

A century is a period of 100 years. The period from the year 1965 until the year 2064, inclusive, is therefore a century. People sometimes refer to the "so-manyeth" century, for example the 20th century. Of old, this means the so-manyeth century from the epoch, which is by definition on day 1 of year 1. Therefore, the first century runs from the first day of year 1 through the last day of year 100, and the 21st century from the first day of year 2001 through the last day of year 2100.

A millennium is a period of 1000 years. The period from the year 847 until the year 1846, inclusive, is therefore a millennium. The counting of millennia since the epoch is similar to that of centuries. So, the first millennium ran from year 1 through year 1000.

The first day of the 21st century and the 3rd millennium (in the Gregorian calendar) was 1 January 2001.

Many people celebrated the beginning of the 21st century and the 3rd millennium already on 1 January 2000, and many news media (newspapers, radio, television) joined in. Those people think that the 21st century runs from 2000 through 2099, and not from 2001 throug 2100. It is understandable that the change from 1999 to 2000 was more impressive than the change from 2000 to 2001. If you want to regard 2000 as the first year of the 21st century, then (working backwards) the 1st century gets squeezed: you'll have to accept that either the first year of the 1st century was the year 1 BC (a year before Christ in a century after Christ?), or that it had only 99 years. Both options are ugly.

It seems to me that we can make both groups of people happy if we separate the meanings. We can refer to "the nineties" if we mean (for now) the years 1990 through 1999. We can expand this to "the nineteen hundreds" for 1900 through 1999, and "the twenty hundreds" for 2000 through 2099. With this alternative, the "twenty-first century" and "third millennium" can keep their original meanings, with which they started on 1 January 2001.

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The centuries count back before the era (epoch). The first century AD began at the beginning of the year 1 and continued until the end of the year 100. The century immediately preceding that one is the first century BC and it began at the beginning of the year 100 BC and ended at the end of the year 1 BC, after which came the beginning of the year AD 1. Likewise, the second century BC ran from the beginning of 200 BC until the end of 101 BC.

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## 8. Unequal Month Lengths in the Western Calendar

The ultimate origin of the lengths of the months in de Julian and Gregorian calendars is lost to us; no explanations from the time at which those lengths were set are known. The history of the lengths of the months that is known to us is described on the Historical Calendars Page.

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## 9. When are Calendars Invented?

Inventing a calendar is easy, because a calendar is just a method for giving each day its own name so you can distinguish it from any other day. For example, you can create your own calendar in which the name of each day is equal to the number of days since your birth. Today could be "Melanie Day 6300" in your calendar (if your name is Melanie), and tomorrow could be "Melanie Day 6301".

Every community that has a government that arranges things for the common good (such as the building of roads or the maintaining of levies or the keeping ready of soldiers to defend the country) needs a calendar to be able to make appointments (such as appointments for the collection of taxes to pay for the things that the government arranges).

A government needs civil servants that keep track of things and arrange things and so are not available to work on the land or to go hunting or fishing to get food. There can only be civil servants and an organized government if the farmers and fishermen and hunters produce more food than they need for their own families, so that the civil servants and other government officials can eat, too.

If there is enough to eat, then there can also be researchers that keep track of the sky and the seasons to determine how long a month or a year actually is, to base a calendar on.

You can expect that a community invents a regular calendar only when enough food is produced.

We don't know for sure what the first calendar was. It may be that the first calendar was invented before people could even write. And the oldest calendars we do know of seem to have been old already at the time when the oldest surviving documents were written that mention them. The oldest calendars that I know of are the Egyptian Calendar and the Chinese Calendar, which both date back to about 1500 BC or earlier.

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## 10. Blue Moons

A second Full Moon in a single calendar month is sometimes called a "Blue Moon", probably because "once in a blue moon" means "very seldomly" and a second Full Moon in a single calendar month also does not happen very often. Such a Blue Moon can occur only in solar calendars in which at least some months have 30 days.

If a Full Moon happens close to the beginning or end of a calendar month, then, because of time zone differences, the moment of that Full Moon may fall in one calendar month in some places, but in the next calendar month in other places.

If two Full Moons occur in a single calendar month, then one of them must be close to the beginning of the month and the other one must be close to the end of the month, and for both of them time zone differences may shift them across the month boundary into the previous or the next month, depending on where you are. A particular Blue Moon isn't necessarily a Blue Moon everywhere on Earth, even if everyone follows the same calendar.

Anyway, these calendrical Blue Moons do not look or act any different from regular Full Moons. Something like a Blue Moon doesn't even exist in many other calendars (such as the Muslim and Jewish calendars). So, these Blue Moons really aren't very interesting.

If you want to know when there might be a Blue Moon, then you can look at the list of dates of Full Moons, at the bottom of the Moon table page. Whenever the date of Full Moon jumps from the beginning of the month (2 or 1) to the end of the month (e.g., 31 or 30), there is a chance for a Blue Moon. This happens about once every three years. The next time this will happen is around August 2004. That month may be a Blue Moon, depending on your location.

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## 11. PlanetCalendars

We can define a comparable calendar for every planet, if we use the same set of rules for all of them. Here are the rules that we'll use:

With these definitions, I find the following month lengths for the planet calendars:

I II III IV V VI VII VIII IX X XI XII
Mercury 6.5 8.2 9.8 10.8 10.4 8.9 7.2 5.9 5.1 4.8 4.9 5.5
Venus 18.9 18.9 19.0 18.9 18.8 18.7 18.6 18.5 18.5 18.5 18.6 18.7
Earth 30.5 31.0 31.3 31.5 31.3 30.9 30.4 29.9 29.6 29.4 29.6 30.0
Mars 62.9 67.2 68.5 66.3 61.4 55.8 51.1 48.2 47.4 48.8 52.2 57.3
Jupiter 336 328 329 338 353 371 387 396 395 384 367 350
Saturn 905 955 991 1001 981 937 886 840 812 805 820 856
Uranus 2804 2747 2640 2518 2414 2350 2340 2385 2477 2596 2712 2790
Neptune 4942 4963 5002 5048 5090 5115 5117 5096 5055 5009 4968 4944
Pluto 11972 10438 8201 6324 5121 4531 4457 4884 5888 7567 9772 11666

I II III IV V VI VII VIII IX X XI XII
Mercury 0.04 0.05 0.06 0.06 0.06 0.05 0.04 0.03 0.03 0.03 0.03 0.03
Venus 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16
Earth 30.5 31.0 31.3 31.5 31.3 30.9 30.4 29.9 29.6 29.4 29.6 30.0
Mars 61.2 65.4 66.7 64.5 59.8 54.3 49.7 46.9 46.1 47.5 50.8 55.7
Jupiter 812 794 796 817 853 896 936 958 955 929 888 845
Saturn 2037 2151 2233 2255 2208 2110 1994 1893 1828 1812 1847 1927
Uranus 3903 3824 3676 3506 3360 3272 3258 3320 3448 3614 3776 3885
Neptune 7363 7394 7452 7521 7582 7620 7622 7590 7530 7461 7401 7365
Pluto 1875 1634 1284 990 802 710 698 765 922 1185 1530 1827

For example: Month V is 353 Earth days long, which is the same as 853 Jupiter days.

In this calendar, the ascending equinox (the beginning of the northern spring and the southern autumn) is always at the beginning of month I, the northern solstice (the beginning of the northern summer and the southern winter) at the beginning of month IV, the descending equinox (the beginning of the northern autumn and southern spring) at the beginning of month VII, and the southern solstice (the beginning of the northern winter and southern summer) at the beginning of month X. On Earth, 1 January (from the Gregorian calendar) comes after about the first third of month X, and the beginning of month I corresponds to about 21 March.

You can calculate the planet month that a planet is in using $$λ_\text{Sun}$$ from equation 7 on the Solar Position Calculation Page: The month number $$m$$ is equal to

$$m = \frac{λ_\text{Sun}}{30°} + 1$$

For example, if $$λ_\text{Sun}$$ is equal to 112.3 degrees, then the month number is equal to 112.3/30 + 1 = 4.743, so we're then about three fourths through month IV.

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## 12. Calendar Calculations

On the calculation page about the Julian Day Number I provide formulas for converting dates from and to the Julian Day Number, for various modern and historical calendars.

Some websites that offer conversion between Hebrew and Gregorian Calendars are //hebcal.com, //www.fourmilab.ch/documents/calendar/, and //www.funaba.org/en/calendar-conversion.cgi.

Go to the modern calendars or the historical calendars.

languages: [en] [nl]

//aa.quae.nl/en/antwoorden/kalenders.html;
Last updated: 2021-07-19