The length of days and years

Now let’s talk about something that seems simple but turns out to be surprisingly complex: the length of day and the lenght of years.

Usually we think of the day as the rotation period of Earth with respect to the Sun, called the Solar day. However astronomers also use the term Sidereal day, which is defined as the rotation period of Earth with respect to the stars. The term “day” is however quite confusing in this case since it is not really a day but a period.

Additionally we can also express the length of the year in several different ways.

The length of the year where the Sun moves from equinox to equinox (equivalently, solstice to solstice) is called the Solar year (a.k.a. tropical year).

The length of a year where the Sun aligns to the same star is called a Sidereal year. The Sidereal year is currently around ~1,224.5 seconds longer than a Solar year resulting in an CURRENTLY EXPERIENCED AXIAL PRECESSION PERIOD of about ~25,772 years.

We can also measure the length of the year from perihelion to perihelion, which is called the Anomalistic year. On average, the Anomalistic year is about 25 minutes longer than the Solar year, so the date of perihelion slowly shifts over time, CURRENTLY regressing by about 1 full day every ~57 years. The date of perihelion thus moves completely through the solar year in an CURRENTLY EXPERIENCED PERIHELION PRECESSION PERIOD of about ~21,000 years.

If you do a search on google on all those length of years you will get a lot of results. Measurements from solstice to solstice, equinox to equinox, or from sun-star-alignment to sun-star-alignment (Sidereal year). You read about 365.2425 or 365.2422. Some say it is 365.24219. Others have different numbers.

What is extra confusing is, because the length of a year is not a rounded number, the calendar needs to implement a correction day (a.k.a. leap day) to make sure the solstices/ equinoxes do not run off in time. For some basic knowledge have a look at this site about the Solar year .

Additionally the length of the Solar year and Sidereal year are also fluctuating across time. There are a lot of great sources that describe the length of days and years at different Epoch like this site  with a downloadable excel sheet (epoch_calc_v2012.xls) that helped me a lot. But it is all quite confusing.

1. The difference between MEAN and CURRENT values

   Currently the heliocentric model doesn’t have an average a.k.a. MEAN length of a day or a year. All is fluctuating randomly and there are no limitations. According to some theory in the past we even had days of 21 hours . I don’t think that is correct. In my view that is just running a simulation with some wrong input parameters resulting in wrong output numbers.

   We know the length of different types of years and days are not fixed. There is no single number. We only know that in the past the solar year in days and in seconds was longer than today’s value. I looked at the numbers and around 1246 AD the length of a solar year was ~365.242236 days of 86,400 ephemeris seconds day and currently it is ~365.24219 days and we still calculate with 86,400 ephemeris seconds day. One of the reasons there is no mean value available is because our current theory thinks everything works independently on each other without a clear balance. The only thing that holds us together are the gravity forces.

   Here’s my thesis: our solar system is a balanced system and therefore must have MEAN values for solar year, sidereal year, solar day, sidereal day and anomalistic year.

   If you take this as a principle, the outcome means the current lengths of all precession types could be totally different than the mean lengths of all precession types.

2. The length of years and days used in the model

   As mentioned above there is no single conclusive number for the length of e.g. a Solar year. It can (and will) differ year by year. The only thing we can conclude is THE TREND. So the trend for the last hundreds of years, is for instance the length of the solar year in days and seconds is decreasing in time.

   According to wikipedia the Solar year can be described as:

   “A tropical year or Solar year (or tropical period) is the time that the Sun takes to return to the same position in the sky – as viewed from the Earth or another celestial body of the Solar System – thus completing a full cycle of astronomical seasons. For example, it is the time from vernal equinox to the next vernal equinox, or from summer solstice to the next summer solstice”

   For all items which I mention CURRENT I mean the length it MORE OR LESS had on date 21 June 2000, 00:00 UTC. This is the startdate of the Interactive 3D Solar System Simulation.

   I mostly used the numbers in the Epoch document and have taken the following lengths as input to the Interactive 3D Solar System Simulation:

   • CURRENT SOLAR DAY – connected to Solar year = ~86,400.0030498014 SI seconds a day

   • CURRENT SIDEREAL DAY – connected to Solar day = 86,164.0935657957 SI seconds a day based upon the real length of day (86,164.0905326261 calculated with 86,400 seconds/ day).

   • CURRENT SOLAR YEAR to keep the calendar in place = ~365.242177222969 days a year which is ~31,556,925.2259806 seconds a year calculated with the real length of a solar day.

   • CURRENT STELLAR DAY = ~EARTH ROTATION DAY – connected to Sidereal year = ~86,164.1019386534 SI seconds a day (86,164.0989054839 calculated with 86,400 seconds/ day).

   • CURRENT SIDEREAL YEAR – connected to Solar year and stellar day = 31,558,149.7591855 seconds a year (365.256362953536 days a year calculated with 86,400 seconds/ day)

   • CURRENT ANOMALISTIC YEAR = ~31,558,432.3572673 seconds a year (365.259633764668 days a year calculated with 86,400 seconds/ day)

   All these values are in line with scientific measurements but because measurements can differ year by year, this is not to say it is measured that particular day/year exactly.

   Additionally the current length of a solar year in days and seconds is decreasing in time (and science calculates with ephemeris days of fixed length of 86,400 seconds a day) and the length of a sidereal year in seconds is slowly increasing in time.

   The expected duration of the axial precession was therefore longer in the past Epoch. The current rate is around ~25,772 year

3. The length of years and days can be calculated based upon the movement of the longitude of perihelion.

   Based upon the J2000 value of the longitude of perihelion (~102.95°), and because we know the longitude of perihelion value was 90° around year 1246 AD, we can derive if the current longitude moved faster or slower than the MEAN value.

   Let me explain it with an example for the values of year 2000 AD 21 June 00:00 UTC. According to the formula by J Meeus this value is ~102.94544° (See Astronomical Algorithms. 2nd Ed. Willmann-Bell. 1998 Table 31.A) and according to NASA  this value was 102.94719° on 01-01-2000 12:00 UTC. I have used 102.95° in below example.

   

   As a next step I have visualized the value of ~102.95° is actually the value as experienced on Earth. The actual movement needs to be calculated from the EARTH-WOBBLE-CENTER which is ~0.00308211 AU closer to the PERIHELION-OF-EARTH.

   

   In the final step we can calculate the actual angle taken from the EARTH-WOBBLE-CENTER. On 21 June 00:00 UTC in year 2000 AD the actual movement is already ~105.8° instead of the on Earth experienced ~102.947°.

   

   In the interactive 3D Solar System Simulation the values of both Earth’s longitude of perihelion AND the actual value of the longitude of perihelion are added.

   • Earth’s longitude of perihelion = PERIHELION-OF-EARTH
   • Actual longitude of perihelion = MID-ECCENTRICITY-ORBIT
   

   We can calculate all length of days and years in year 2000 AD based upon the longitude of perihelion value which is moving ~11.44% slower than expected. Slower motion results in an Axial precession movement with a longer length than the MEAN value of ~22,937 years. Therefore the J2000 value was ~25,771.57534 years.

   

4. The length of years and days can be calculated with the help of the coin rotation paradox

   Now here’s something fascinating. Wikipedia defines the coin rotation paradox  as follows:

   “The paradox is related to sidereal time: a Sidereal day is the time Earth takes to rotate for a distant star to return to the same position in the sky, whereas a Solar day is the time for the sun to return to the same position. A year has around 365.25 Solar days, but 366.25 Sidereal days to account for one revolution around the sun.[8] As a Solar day has 24 hours, a Sidereal day has around 365.25/366.25 × 24 hours = 23 hours, 56 minutes and 4.1 seconds.”

   Although the Wikipedia comment is not correct as such about the length being 365.25 days, it shows the logic.

   The coin rotation paradox is quite nicely shown in this great demo . You see a coin which is 1/3 the size of the circle it is orbiting. Press the 3rd play button and watch. At first you only see the smaller circle rotating 3 times around the bigger one. Intuitively this is the expected behaviour. But actually it rotates 4 times: it also rotates around its own axis. Look again from the start and focus on the black dot at the left side of the smaller circle. Count the number of times you see the black dot return to the left side — that number is 4. So exactly one extra rotation because it also rotates on its own axis.

   Here’s the key takeaway: if we know the length of a certain type of day, we can calculate the length of certain types of year with it!

5. The difference between solar year and sidereal year explained

   Let me first explain the difference between a sidereal year and solar year with the help of the coin rotation paradox.

   The description of the Sidereal year  can be described as:

   “The Sidereal year also called a sidereal orbital period, is the time that Earth takes to orbit the Sun once with respect to the fixed stars”

   Since Earth is orbiting EARTH-WOBBLE-CENTER in a clockwise direction AND the PERIHELION-OF-EARTH orbits the Sun in a counter-clockwise direction, AND we have the coin rotation paradox as a rule, the number of Solar years should exceed the number of Sidereal years by exactly 1. In other words, if you look at our solar system from a birds eye view, there is exactly 1 more time you see the Sun orbiting the Earth, compared to the number of times you see the Sun orbits the PERIHELION-OF-EARTH, because the Earth on its orbit around the EARTH-WOBBLE-CENTER is moving in opposite direction to the PERIHELION-OF-EARTH around the Sun.

   This concept might be hard to grasp so I have created a few picture how we move along on our ~22,937 years journey around the EARTH-WOBBLE-CENTER.

   Below you will see the starting point in year 0 of a Great-year. I have taken a start date of June solstice year 2000 AD.

   

   After 1 Solar year, so June solstice 2001 AD, the axis of Earth orbiting the EARTH-WOBBLE-CENTER slightly changed in a clockwise direction. There is also ALMOST 1 Sidereal year finished. This ALMOST takes CURRENTLY ~1,224.5 seconds.

   

   So about ~1,224.5 seconds later, the Sun has shifted a small bit counter-clockwise and is aligned with the same star again showing as 1 Sidereal year being completed. The Solar year has however also grown a bit.

   

   Fast forward to the end of the Great-year cycle, the axis is the same again as the start date of June solstice 2000 AD, but there needs to be exactly 1 Sidereal year less than the number of Solar years.

   

   So the Sun has ~22,937 mean Solar years in a Great Year of ~22,936 mean Sidereal years (exactly 1 less) because of the same reason the number of Sidereal days exceed one compared to the number of Solar days according to the coin rotation paradox. There is no other way.

6. The difference between sidereal day and stellar day can’t explain the difference between solar year and sidereal year directly

   Here’s the puzzle. There is a difference between the length of year from March equinox to March equinox (the returning seasons on Earth so summer stays summer over a long time), and the time for the Sun to align with the same star again. This difference is currently around ~1,224.5 seconds a year and is the reason why we have the popularly called precession of the equinoxes. Every year the Sun is moving a slight bit backwards compared to e.g. the equinox-date, referenced to the fixed stars. Currently the Sun is on the border of the constellation of Pisces and Aquarius during the March Equinox (~21 march). You can read a lot about the “Age of Aquarius” online. Where is this difference between Solar year and Sidereal year coming from?

   I believe the key in solving this issue is to involve the length of the stellar day by linking it to the length of a Sidereal year. This is not a one-on-one relation but it can be derived. So let’s start at looking at the length of day figures as agreed by science.

   According to the official numbers for the sidereal day and stellar day calculated in 86,400 seconds a day, there is a small difference  of ~8.37ms (=86,164.0989036905 -/- ~86,164.0905326) between the time it takes for Earth to align Earth’s rotation period relative to the precessing mean March equinox (Sidereal day) and the time it takes the day on Earth to align with the same star (stellar day).

   The main question is therefore how does ~8.37ms per day - which normally should mean the precession of the equinoxes needs to be ~3.07 seconds a year (8.37ms * ~366.24218) - could turn into ~1,224.5 seconds a year?

   Although there have been many theories, open letters, talks, etc about this difference, so far there is no officially agreed scientific explanation.

   You can read for instance this Wikipedia talk . Or this open letter to the INTERNATIONAL ASTRONOMICAL UNION  Or this explanation of universal time  Or these Astronomical units and constants 

   NOTE: The first link was originally open as a discussion point on Wikipedia – our supposed knowledge base - but in the live version it is gone. I am very pessimistic on the state of the internet nowadays. In the past it was an open space where things could be discussed. Now it is a moderated platform where narratives are guarded . Luckily there is archive.org  but we humans must solve this more structurally.

   In the references provided above, some have made suggestions the real length of stellar day is wrongly calculated. Especially some older books make this reference. They actually provide an old value of stellar day that links it to the Sidereal year directly.

   The rotational period of ~86164.09966 seconds a day (0.9997269672 Solar day) can for instance be found in these discussions/papers this old journal  this older discussion  this old book  this old discussion  Also the specific 1.002 737 803 086 value AND calculation related to the Sidereal year could be found in old science books. see eg  This here 

   So to conclude the value for a stellar day - as it is calculated nowadays - can’t link it to anything directly. Suggestions are made to link it to the sidereal year by increasing its number since ~3.07 * ~366.24218 IS NOT ~1224.5. The length of the stellar day would have to be larger to make that calculation work.

   My thesis to explain 3.07 seconds a year of stellar day-sidereal day differences can become 1224.5 seconds a year difference between solar year and sidereal year is the 3.07 seconds is due to axial precession and since axial and inclination meet each other every perihelion precession we need to calculate the movements in the factor 13/16.

7. The distance of Earth to the EARTH-WOBBLE-CENTER explains the relation of the stellar day to the sidereal year

   The Earth to EARTH-WOBBLE-CENTER is roughly at a distance of 1:324.5 compared to Earth to the Sun. Therefore any second movement will increase the difference between the Solar year and Sidereal year by 324.5/13*16 = ~399.3 times.

   So because of Earth’s orbit around the EARTH-WOBBLE-CENTER we CURRENTLY loses ~8.37ms per rotation (CURRENT Stellar day -/- Sidereal day) which is ~3.07 SI seconds per year. The end result is a difference of 1224.5 seconds between solar year and sidereal year.

   To make the relation between the different types of year & day more clearly visible AND the relation to Earth’s wobble, I have created the following (exaggerated) picture.

   

   The picture is created as seen from Earth.

   • At the start both the Sun and Earth are in position 0. The Sun is moving counter clockwise around Earth in 1 year time period. Earth is moving clockwise on its CURRENTLY EXPERIENCED Precession Orbit in ~25,772 years.

   • One CURRENT Solar year of ~365.242177222969 days later, Earth moved to rotation-position ~366.242177222969, completing a circle of seasons, showing the sun – one Solar year later - in position A

   • A little later (a period of 365.256357127 days a year calculated with the real length of a solar day, Earth moved further up to rotation-position 366.256388864434, and the sun is aligned with the fixed star again, showing the sun – one CURRENT Sidereal year later - in position B. The difference between Sun’s point A and B is known as the Axial precession which has a CURRENT EXPERIENCED duration of ~25,772 years.

   • The ratio of the distance between Earth to the PERIHELION-OF-EARTH (which is the average distance of Earth to the Sun) compared to the distance between Earth and the EARTH-WOBBLE-CENTER should be in a ratio to explain the time difference between the seconds lost between the Solar year and the Sidereal year AND the time difference between the seconds lost between the Sidereal day and the Stellar day in seconds per year.

   This ratio needs to be ~324.5/13*16 = ~399.3. So any one SI second movement between Sidereal day and Stellar day results in a ~399.3 SI seconds increase between solar year and sidereal year.

   

8. The length of an anomalistic year

   The definition for the anomalistic year:

   “The anomalistic year is the time taken for the Earth to complete one revolution with respect to its apsides. The orbit of the Earth is elliptical; the extreme points, called apsides, are the perihelion, where the Earth is closest to the Sun, and the aphelion, where the Earth is farthest from the Sun. The anomalistic year is usually defined as the time between perihelion passages.”

   If you dive a bit deeper , the length of this type of year is related to perihelion precession cycle of 18,636 years.

   In the appendices I have provided the background on this type of precession.

   “The real movement of the Inclination precession compared to the background stars is actually 99,392 years. But since the Axial precession of ~22,937 years is in the opposite direction to the Inclination precession, they meet each other every 18,636 years.”

   So in a period of 18,636 years Earth closest point to the Sun (currently on 3rd of January), will move forward in time to the same date again. The anomalistic year is therefore currently 31,558,432.3572673 SI seconds a year.

   So all numbers have a pattern and are in sync with each other. So e.g. the Solar year is related to the Anomalistic year.

Summary of different types of day/ year

Since it currently quite confusing to link the proper type of day, to the proper type of year, I summed them up.

The recommendation is to realign the length of certain types of years/ days to the length it had in the year 1246 AD when the solstices were aligned with the PERIHELION-OF-EARTH. The same recommendation has been done by Laplace . You can search for “perihelion” in this document to find some background.

• MEAN SOLAR DAY – connected to Solar year = 86,399.5657965675 SI seconds a day

• MEAN SIDEREAL DAY – connected to Solar day = 86,163.6575681587 SI seconds a day

• MEAN STELLAR DAY = mean Earth rotation period – connected to mean Sidereal year = 86,163.6669758971 SI seconds a day

• MEAN SOLAR YEAR = 365.242273019961 days a year.

• MEAN SIDEREAL YEAR – connected to mean Stellar day AND mean Solar year = 31,558,149.6846777 SI seconds a year

• MEAN ANOMALISTIC YEAR = 365.261872819962 days/ 31,558,467.2136858 seconds a year

The movement of all years and days are connected to each other in a perihelion precession cycle of 18,636 years.

All length of days and years are in correspondence with each other. The exact formulas are further detailed in the excel and you can see the results in the Interactive 3D Solar System Simulation.

Now that we understand how the length of days and years connect to each other, let’s look at the obliquity and inclination in the next chapter.