The length of days and years in the Dummy Universe Model

The most fundamental astronomical unit of time is the day, measured in terms of the rotation of Earth around his axis. There is, however more than one way to define the day. Usually we think of it 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.

We can also express the length of the year in several different ways. The length of the year 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 the Sun aligns to the same star is called Sidereal year. The sidereal year is currently around ~1,224.5 seconds longer then a Solar year. 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, regressing by about 1 full day every ~58 years. The date of perihelion thus moves completely through the solar year in an EXPERIENCED PERIOD about ~21,000 years.

If you do a search on google on 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 (opens in a new tab).

Additionally the length of the Solar year and Sidereal year are also fluctuation across time. There are a lot of great sources that describe the length of days and years at different Epoch. It is all quite confusing. Why is it that hard to come up with one conclusive answer? If we can’t agree on the length of year, how can we even calculate all orbits?

The current length of a Solar year is decreasing in time and the length of a Sidereal year is expected to be slowly increasing in time according to this site (opens in a new tab) with a downloadable excel sheet. The expected duration of the precession of the equinoxes was therefore longer in the past Epoch. The current rate is around ~25,772 years.

The current heliocentric model doesn’t have a medium a.k.a. average a.k.a. MEAN length. All is fluctuating randomly and there are no limitations. In the past we even had days of 21 hours (opens in a new tab) according to some theory. 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.

Can we perhaps define the length of different types of year with the coin rotation paradox?

1. The coin rotation paradox

   The text on Wikipedia about the coin rotation paradox (opens in a new tab) defines it as followed:

   ”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 (opens in a new tab). You see a coin which is 1/3 the size of the circle it is orbiting. You need to press the 3rd play button. 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. If you look at it again from the start and focus on the black dot at the left side on the smaller circle. Count the number of times you see the black dot again at the left side and that number is 4. So exactly one time extra rotation because it also rotates on its own axis.

   We know quite sure the length of different type of days and therefore can calculate the length of certain types of year with it!

2. The difference between MEAN and CURRENT values (and EXPERIENCES)

   We know the length of different types of years and days are not fixed. There is no single number. In the past the solar year was longer than today’s value. In the currently accepted heliocentric model there is also no mean value available because everything works upon each other without a clear balance. The only thing that holds us together are the gravity forces.

   I do not think that is correct. I believe our solar system is a balanced system and therefore must have some 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 CURRENTLY EXPERIENCED lengths of all precession types, could be totally different than the MEAN lengths of all precession types.

   So if a mean solar year and mean sidereal year lengths are different than today’s value, also the length of the precession of the equinoxes length are different. The CURRENTLY EXPERIENCED length of the precession of the equinoxes is ~25,772 years is in totally different than the MEAN length of the precession of the equinoxes of 23,520 years (more about this number later).

   This does not mean the solar year and sidereal are just experiences. These are really hard values with seconds a year. The resulting EXPERIENCE of a full cycle of precession is however different.

3. The CURRENT length of a Solar year, Sidereal year, Solar day, Sidereal day and Anomalistic year.

   As mentioned above the is no single 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 solar year is decreasing in time.

   According to wikipedia the Solar year (opens in a new tab) 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”

   The CURRENT length of a Solar year is ~365.24219 days a year. With the term CURRENT I mean the length it MORE OR LESS had on date 21 June 2000, 12:00 UTC. This date is the startdate of the 3D model.

   In the 3D model the following lengths are taken as input:

   • CURRENT SOLAR DAY – connected to Solar year = 86,400.0030764764 SI seconds a day

   • CURRENT SIDEREAL DAY – connected to Solar day = 86164.0905328760 SI seconds a day

   • CURRENT STELLAR DAY = EARTH ROTATION DAY = 86,164.0989036905 SI seconds a day

   • CURRENT SOLAR YEAR to keep the calendar in place = 365.242190503538 days a year which is 31,556,925.2595057 seconds a year.

   • CURRENT SIDEREAL YEAR – connected to Solar year = 365.256363098868 days a year which is 31,558,149.7717422 seconds a year.

   • CURRENT ANOMALISTIC YEAR = 365.259633853988 days a year which is 31,558,432.3649846 seconds a year.

   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.

4. The length of a Sidereal year should be related to the length of a mean Sidereal day (but isn’t).

   The description of the Sidereal year (opens in a new tab) can be described as:

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

   So 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. This is the reason why we have the popularly called precession of the equinoxes (opens in a new tab). 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 during the March Equinox (~21 march) on the edge of the constellation of Pisces and Aquarius. You can read a lot about it 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 a stellar day by linking it to the mean 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.

   As explained in the appendix, there is a small difference of ~8.37ms (=86,164.0989036905 -/- 86,164.09053083288) observable between the time it takes the day on Earth to align with the same star (stellar day) and the time it takes for Earth's rotation period relative to the precessing mean March equinox (Sidereal 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, 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 found.

   You can read for instance this Wikipedia talk section for starters (opens in a new tab). Or this open letter to the INTERNATIONAL ASTRONOMICAL UNION (opens in a new tab) Or this explanation of universal time (opens in a new tab) Or these Astronomical units and constants (opens in a new tab)

   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 (which in my view is the key to solve the mystery).

   The mean rotational period of 86164.09966 seconds a day (0.9997269672 Solar day) can for instance be found in these discussions/papers this old journal (opens in a new tab) this older discussion (opens in a new tab) this old book (opens in a new tab) this old discussion (opens in a new tab) 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 (opens in a new tab) This here (opens in a new tab)

   My conclusion is that we are missing another type-of-day reference. This new type-of-day length we cannot measure DIRECTLY since it is no longer coming back in any new high accuracy measurements but we can conclude it INDIRECTLY.

   ”Spoiler alert: The reason why we cannot measure it directly: Earth is slowly moving along its Earth Precession Orbit (EPO) around CENTER and therefore loses ~9.13ms of time per day.”

   This missing type-of-day is related to the length of a Sidereal year. That might be confusing since we already have a term called “Sidereal day”. However the currently known “Sidereal day” is actually related to the solar year. Let’s therefore call this new type of day the “True Sidereal Rotation Period”.

   By being able to define the length of the “True Sidereal Rotation Period” we should be able to define the length of a Sidereal year. They are directly related to each other.

   Now again, first let’s get back to basics of the geo-heliocentric model to explain the Axial precession.

   Since Earth is orbiting CENTER in a Clock Wise direction AND the Sun orbits the HELION POINT in Counter Clock Wise direction in the dummy universe model, 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 with 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 HELION POINT, because the Earth on its orbit around CENTER is moving in opposite direction to the Sun around the HELION POINT.

   This concept might be hard to grasp so I have created a few picture how we move along on our 23,520 years journey around The Orbit Center. I will come back on this 23,520 year number later in time. For now you can just consider it as the MEAN LENGTH OF A GREAT YEAR.

   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 CENTER slightly changed in a clockwise direction. There is also ALMOST 1 sidereal finished. This ALMOST takes CURRENTLY ~1,224.5 seconds.

   So about ~1,224.5 seconds later, the Sun has shifted a small bit counter clock wise 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.

   I didn’t explain the exact duration of a Great year yet (I will in the next chapter) but I think you will understand if I say, the Sun has 23,520 mean Solar years in a Great Year of 25,519 mean Sidereal years 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.

   Additionally the “True Sidereal Rotation Period” needs to be linked to the “Sidereal year” for the same reason the number of Sidereal days exceed one compared to the number of Solar days according to the coin rotation paradox. We can therefor make a calculation formula: =((“mean Sidereal year in days”/(“mean Sidereal year in days”+1))*86400) = “Mean True Sidereal Rotation Period”

   So if we know the length of the Great year AND we know the length of the Mean Solar year, we should be able to calculate the length of the “Mean True Sidereal Rotation Period” and “Mean Sidereal Year”.

   I will not further detail the exact calculations in this document. That will be even more boring. I have created an Excel sheet TAB “Chapter 2” where you can change the yellow cells and based upon what you fill in there, the numbers are rolling out. Check it for yourself.

   The result however is – if the length of a Great-Year is 23,520 years (which I think it is) – the length of a “Mean Sidereal Year” = 365.257758865990 days a year AND the length of a “Mean True Sidereal Rotation Period” = 86,164.1005605792 SI seconds a day.

   To check: 1/((86400/86164.1005605792)-1) = 365.257758865990

   To check (reversed): =((365.257758865990/(365.257758865990+1))*86400) = 86,164.1005605792

5. The distance of Earth to CENTER

   So because of Earth’s orbit around CENTER we lose ~9.13ms per rotation (CURRENT true rotation period -/- Sidereal day) instead of the ~8.37ms as calculated by comparing the Sidereal day with the stellar day. ~9.13ms per rotation – which is ~3.34 SI seconds per year - however still results in ~1,224.5 SI seconds lost per mean Sidereal year. How can these numbers add up?

   The way to solve this issue: We know the CURRENT mean distance of the Sun to the HELION POINT: 149,597,870.698828 km. The only way it is possible ~3.34 seconds on Earth turns into ~1,224.5 seconds is Earth moving on its orbit around The Orbit Center at a distance of 149,597,870.698828 km / (1,224.5/ 3.34) (= 366.256363098868) = 408,251.25 km

   So any SI second moved by Earth results in 366.256363098868 SI seconds delay in the Sidereal year.

   To make the relation between the different types of year & day more clearly visible AND the relation to Earth traveling on its orbit around CENTER, I have created the following (exaggerated) picture.

   

   • 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 Earth CURRENTLY EXPERIENCED Precession Orbit in ~25,772 years.

   • One CURRENT Solar year of 365.242190503538 days later, Earth moved to rotation-position 366.242190503538, completing a circle of seasons, showing the sun – one CURRENT Solar year later - in position A

   • A little later (a period of 365.256363098868 days a year), Earth moved further up to rotation-position 366.256363098868, 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 precession of the equinoxes which has a CURRENT EXPERIENCED duration of ~25,772 years.

   • The ratio of the distance between Sun and the HELION POINT (which is the average distance of Earth to the Sun) compared to the distance between Earth and CENTER should be the same ratio as the time difference between the seconds lost between the CURRENT Solar year and the CURRENT Sidereal year AND the time difference between the seconds lost between the Sidereal day and the CURRENT True Rotation Period seconds in a year. If there was no time difference at all, the location of CENTER would have been the same location as Earth itself AND THERE WOULD NOT HAVE BEEN ANY PRECESSION.

   In one Great Year the Sun has orbited exactly one Sidereal year less then Solar years.

6. The mean length of a Solar year

7. The mean length of a Sidereal year

8. The mean length of an anomalistic year

   The definition for the anomalistic year (opens in a new tab):

   ”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. Its average duration is 365.259636 days (365 d 6 h 13 min 52.6 s) (at the epoch J2011.0).”

   If you dive a bit deeper (opens in a new tab), the length of this type of year is related to apsidal precession

   In the appendix I have provided the background on this type of precession

   ”The real movement of the apsidal precession compared to the background stars is actually 101,920 years. But since the axial precession of 23,520 years is in the opposite direction to the apsidal precession, they meet each other around every 19,110 years.”

   So in a period of 19,110 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 ~365.259636 days a year.

   In the dummy universe model the numbers have a pattern and are in sync with each other. So e.g. the Solar year is related to the Anomalistic year. I will explain it in the next chapter but I can already say the length of the apsidal precession is 19,110 years resulting in a mean duration of the anomalistic year of 365.261304124895 days a year.

   I will elaborate more on the exact duration in the next chapter.

9. 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 solar day should ideally be 86,400 SI seconds a day. The length of a SI second is however artificially chosen and corresponds to the length of a solar day around 1900 AD.

   Recommendation is the realign the MEAN solar day length of 86,400 SI seconds a day as it was according in year 1246 AD, which corresponds to a current length of ~86,399.9939227693 SI seconds a day. This requires the length of the SI second to change. The same recommendation has been done by Laplace (opens in a new tab)

   NOTE: All below definitions are based upon a SI second that is connected to the length is was in year 1246 AD.

   • MEAN SOLAR DAY – connected to Solar year = 86,400 SI seconds a day

   • MEAN SIDEREAL DAY – connected to Solar day = 86164.0905578014 SI seconds a day

   • CURRENT STELLAR DAY = EARTH ROTATION DAY = 86,164.0989036905 SI seconds a day

   • MEAN SOLAR YEAR to keep the calendar in place = 365.242229199372 days a year. So the current Gregorian calendar can be improved. More on that later.

   • MEAN TRUE SIDEREAL ROTATION PERIOD (NEW)– connected to mean Sidereal year = 86,164.1005605792 SI seconds a day

   • MEAN SIDEREAL YEAR – connected to Mean True Sidereal Rotation Period (NEW) AND Mean Solar year (NEW) = 365.257758865990 days a year

   • MEAN ANOMALISTIC YEAR = 365.261304124895 days a year

10. Summary picture

    To summarize what we have learned so far about the relation between the different types of rotation and precession cycles and axial/ inclination tilt, I have created the following picture to show the movements.