Topic: How to Devise a Different Calendarial System

[arakish]

Onaviu-Ondero 7 Year Synod

Here is a nice one for y’all’s at me.  This shows the re-calculations after I made changes to Onaviu’s day, and Ondero’s day and orbit about Onaviu.  As I played around with these calculations, I thought of posting it since it may give others ideas on how to design a calendarial system for their world that is different than our Earth's.

FYI: I tend to convert all the periods into total number of seconds, sometimes making the divisions a bit easier.  Sometimes…

Earth Standard Year = 31,557,600 seconds.

Onaviu Orbital Period = 49,766,400 seconds (this is Onaviu’s year).

Onaviu Rotational Period = 100,800 seconds (this is Onaviu’s day).

Onaviu Rotational Period = 28 hours, each hour = 3600 seconds (Earth’s = 24 hours).

Onaviu Orbital Period = 493.714285… local days; or 493 days, 20 hours (exactly).

Onaviu Orbital Period = 576 Earth days.

This leads to a nasty synodical year cycling.  Calculating it out, I got that Onaviu’s year has a seven year synodical cycle as listed below.

Year 1 = 493 days
Year 2 = 494 days
Year 3 = 494 days
Year 4 = 493 days
Year 5 = 494 days
Year 6 = 494 days
Year 7 = 494 days

And then it starts over again.  Every 7 years will cycle as listed above.  This is to account for the extra 20 hours each year.  Thankfully, it does not have a longer synodical period.  Now you may want to see how this worked.  Remember, each year you are adding 20 extra hours.  Also, remember, each year has a base of 493 days (see above).  Furthermore, remember, Onaviu’s day = 28 hours.

Year 1 = 20 hours, do NOT add an extra day;		493 days
Year 2 = 40 hours, add an extra day, leaving 12 hours;		494 days
Year 3 = 32 hours, add an extra day, leaving 4 hours;		494 days
Year 4 = 24 hours, do NOT add extra day since this is less than 28 hours;		493 days
Year 5 = 44 hours, add an extra day, leaving 16 hours;		494 days
Year 6 = 36 hours, add an extra day, leaving 8 hours;		494 days
Year 7 = 28 hours, add an extra day, leaving 0 hours, resetting the 7yr cycle		494 days

Yuck!  Nasty!  But I can work with it.


Now for the kicker that threw a monkey wrench into the gears.  At least it seemed to at first until I sat down and played with the numbers.

Ondero, Onaviu’s moon, has periods as listed below.

Ondero Rotational Period = 259,200 seconds.

Ondero Orbital Period = 5,443,200 seconds (this is the important one).

Ondero Orbital Period = 21 Ondero days.

Ondero Orbital Period = 54 Onaviu days.

The biggest problem with attempting to devise a calendar is that 493 only factors to (17·29) and 494 factors to (2·13·19).  Note: The “·” dot is the multiplication dot.

A 17, 19, 2, 13, and 19 just do not evenly create any kind of months or weeks.  Kind of like our calendar of 365 factoring to (5·73) and 366 factoring to (2·3·61).  Now you know why some months have 31, some 30, and one 28 or 29 days.

However, I did find something interesting about this system.

Seven Onaviu years = 348,364,800 seconds.  Divide this by Ondero’s Orbital Period of 5,443,200 seconds, and you get 64.  Exactly 64.  This means the people could still base a calendar on Ondero's synodical period, but they will also have to keep up with the offset of 7 and 8 days through the seven year cycle.  Nine Ondero cycles = 486 days.

Thus, Onaviu’s calendar is still going to be screwed up.  But, it does at least have a relatively simple pattern of 7.  As said earlier, at least it was not a longer period.

Using the program Celestia (http://celestia.space/), I found that eclipses from Ondero also cycle in the same seven year cycles.  For four years, there is an eclipse from Ondero (varies in its north-south position), then there are three years where there are no eclipses.  I still have not figured out in which of the years the eclipses occur in the seven year cycle above.  Even Ondero being eclipsed by Onaviu occurs within this same seven year period.  Care to guess how?

Additionally, on the seventh year of the 7yr-cycle, you will have ten full moons.  For the other six years, you will have nine full moons.

Of course, being an amateur astrophysicist, with some background in Orbital Mechanics, I know that synodical periods such as this NEVER work out evenly.  Even our calendarial system is not as exact as we think it is.  Thus, even the above for my campaign world is not that exact.  In fact, I have already calculated that there is an offset of about 36 seconds every seven years.  Now I have to figure out how to compensate for that. 

Guess that is what I get for changing the numbers…

rmfr

[jdale]

>In fact, I have already calculated that there is an offset of about 36 seconds every seven years.  Now I have to figure out how to compensate for that. 

The traditional, real method that people used was "get it close enough, and then insert some extra days when it gets too far off." Even our current, modern calendar has some error in it that no one agrees how to fix. An error of 36 seconds per seven years amounts to 18 minutes over the course of two centuries, but unless you have an accurate timepiece, no one even notices. Just adjust your clock at dawn on the solstice (or whatever reference point you want to use) and don't worry about it; that's how people would solve it in the real world. It's not until you start accumulating weeks of error that it starts being noticeable with planting season. Perfection is not realistic!

Of course if your setting includes highly accurate timepieces and astronomical measurements, that may not hold up as well.

[arakish]

> An error of 36 seconds per seven years amounts to 18 minutes over the course of two centuries...

Rats. I missed the decimal.  Should have been 3.6.


> Even our current, modern calendar has some error in it that no one agrees how to fix.

This is due to Earth's orbital period being 365.25636... solar days each day = 86,400 seconds.  IIRC, the correct rule for Leap Years is that a Leap Year is NOT a Leap Year if the number is evenly divisible by 100, unless it is also evenly divisible by 400.  If I am correct, this means that the year 2000 was a Leap Year, but the years 2100, 2200, 2300 will NOT be a leap year.  Of course, I won't have to worry about it.


> The traditional, real method that people used was "get it close enough, and then insert some extra days when it gets too far off."

And I can see where this will work even for my world.  Going back into Celestia, I calculated that the actual True Onaviu Orbital Period = 49,766,400.5118857 seconds (max precision is 7 decimal places).  That amounts to having to add an extra day to the year once every 9,722,170.398587 years.

Thus, I do not think I shall have to worry about that any time soon.

rmfr

P.S.: > Of course if your setting includes highly accurate timepieces and astronomical measurements, that may not hold up as well.

And no matter how accurate, they shall still have some inaccuracies.  This is due to the orbital perturbations from the constant tug-of-war of all the celestial objects orbiting the star.  Even neighboring stars light-years away.

[arakish]

Phooey.  Went and screwed up again.  Did the calculations from the wrong numbers.  It is about every 200,000 years that an extra day would need to be added.  Much smaller, but still not that worrisome.

rmfr

P.S.: Further refined the number to 196,920 years.  Still no worry.

Done during lunch break....