# Accurately Measuring Sidereal Years For Calculating Tropical Year

I decided that a sidereal year in my world is about 435.126 days long. It would follow that leap years occur every 8 and 125 years (with a double leap year every 1000 years!), but I hadn't though about whether my civilization would be able to measure the length of a sidereal year with that much accuracy.

This leads me to the following question: How advanced does a civilization need to be in order to predict the length of a sidereal year accurately for the purpose of determining the length of a tropical year?

We can use Earth as an example instead of my world since Earth is much more well known and and we have measured Earth's year to greater precision of 365.25636 days long with rules concerning leap years every 4, 100, 400 years to compensate.

As a follow up, how accurate could a civilization in a rRnaissance-like time period measure a sidereal year? This part concerns my world and we can assume that the civilization in question is advanced enough to tell time based on the movement of the stars, know the world is round, observe retrograde motion of other planets, and assert that the world is round.

EDIT: I would prefer an answer other than "wait and see," to measure the length of a sidereal year. I'm looking for a method that attempts to calculate it for calendar making purposes.

• First question is: Which precision do you need? The Julian calendar come from the time of old Rome and was precise to more than a day for century; to get 5 decimals you probably need atomic clocks and the Relativity thoery. Commented Aug 27, 2015 at 22:42
• The difference is easily measured just by eye, since it is cumulative. Even if they can't measure the 1/8 day, they can just wait 10 years for their calendar to be more than a day off. Commented Aug 27, 2015 at 22:44
• @SJuan - the drift of the calendar was known thousands of years before Caesar applied the Julian 'fix'.- The Egyptian Sothic cycle was the period for the rising of a star to move through an entire year. They just didn't care to fix it. The Romans had a 'fix' before Julius, it just was for the high priest to insert an extra month to get things back in line. Commented Aug 27, 2015 at 22:46
• If memory serves, our calendars, and thus our leap years, are based on the tropical year. I mention it because a civilization has no need to handle leap days in sidereal years until it has a reason to concern itself with the difference between tropical and sidereal years. Commented Aug 27, 2015 at 22:47
• @SJuan76 It's difficult for me to quantify a decimal precision, but our calendars estimate a tropical year to be about 365.2425. I'm looking for something along the lines which addresses how well a civilization can determine the years necessary that a leap year would occur. Commented Aug 27, 2015 at 23:34

You are really comparing the tropical year to the solar day. The Gregorian calendar aims to keep the equinoxes on the same day of the year, and succeeds very well. Hipparchus had a good value for the correction needed to the Julian calendar in the 2nd century BCE. Prehistoric people could detect equinoxes and solstices, so getting the relation to the solar day is just counting over a long period.

If you want the sidereal year, again long observations are your friend. You just have to define the constellations so you know which stars are which, then look at midnight for many days (years). The ancient Greeks, with their epicycles for the planets, clearly had the level of observation required.

• I have selected this answer as the use of equinoxes and predicting where on the calendar they would land would reveal and offsets over the years, achieving the sort of results I desire. Thank you for the help! Commented Aug 31, 2015 at 11:47

There is no need to calculate the days to such precision, and it's just not efficient. There's some complex math and technology needed for that precision that wouldn't exist in your renaissance-like time period.

# What time measurements am I using?

To begin, realize that if the your answer to this question involves seconds, minutes, etc., these terms need to be defined.

On Earth, a "second" is defined as 9.19 billion oscillations of the outermost electron between two electron energy levels of a cesium-133 atom. Let's call this measurement an "EO".

Would your renaissance-esque civilization be able to calculate an EO?

Let's take a look at your calculated sidereal year. You defined a sidereal year as:

435.126 days long

Again, the term "day" is ambiguous. On Earth, that would be 86400 seconds (7.94 trillion EO).

Now assume that you are using the EO in your world. To precisely calculate to 0.126 days, you need to be able to calculate to milliseconds; one thousandth of a second.

# Who sets the standards for time?

In your world, who/what is keeping track of the time? Is there a global standard of time like we have here on Earth? Who/what is dedicating its time to sustaining this time-keeping?

Note that here on Earth, our global standard is a manufactured product that has to be tweaked every so often. This results in our TAI standard (that is used for scientific purposes) being different than our UTC standard.

As of 30 June 2015 when the last leap second was added, TAI is exactly 36 seconds ahead of UTC. The 36 seconds results from the initial difference of 10 seconds at the start of 1972, plus 26 leap seconds in UTC since 1972.

There's also the concept of time zones.

A good example would be in the 18th and early 19th century. If you were to travel, you would have to reset your watch at every stop along the way; a tedious task. The creation of time zones meant that a large area would all be on the same time, so if you traveled from San Francisco to Chicago, you would only need to change your watch twice.

# Where does time begin?

There's a lot to consider here.

• Where do days begin/end? months (if you use them)? years?
• How are the years being counted? Starting from zero? one? a negative?
• Would you use a likeness of a BC/AD system? If so, what events define where one ends and the other begins?

Oh, and here's a fun one.

In the theory of relativity, time dilation is a difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from a gravitational mass or masses.

Not only is this a incredibly tough problem, it brings to mind the issue of miscommunication between people in your world.

Unlike today, a renaissance-like time period wouldn't be able to instantly share data like we can today with the Internet. It could take days, months, or years for them to communicate across the globe; leading to unavoidable inaccuracies in their data.

In conclusion, you either need to create your own time system (and maybe universe), or figure out a reasonable way for your world to calculate time to the millisecond. Don't forget that there's a whole lot of thought that needs to be put into creating a new time system.

I hope you found what you're looking for. Good luck!

The only reason for time is so that everything doesn't happen at once. -Albert Einstein

• Nice write-up, but here on Earth the second wasn't defined in relation to cesium (or anything else of that nature) until 1967. Prior to that, as far back as circa 1000, it was defined as 1/86,400 of a mean solar day. Given that Earth civilizations were able to define and use the second well before our own Renaissance, based on astronomical observations just like the question proposes, it actually IS feasible -- almost certain -- that a precise-enough definition of the second could exist in this world. Even before that, though, the precision of our ancient astronomers is quite staggering... Commented Aug 28, 2015 at 17:36
• "Who sets the standards for time?" -- There is another factor to consider here. Until fast travel became possible, which roughly coincides with the invention of the steam engine and train (as opposed to horse carriage), there was for many purposes no "standard" time. "Noon" was simply the point in time when the sun was highest on the sky in the day in that particular location. This led to even fairly nearby towns in the east-west direction to have different local time. Time zones, with large swatches of geography having the same time, are a quite modern invention.
– user
Commented Aug 28, 2015 at 19:46
• Time zones completely slipped my mind! Good catch. @MichaelKjörling I've added it too the section. Commented Aug 28, 2015 at 19:50