Earth’s Axial tilt and Inclination tilt are related

When I started my exploration my initial thesis was based on the assumption the Axial tilt & Orbital Inclination tilt were related to each other. Initially I thought they were just the same kind of value but expressed in a different manner. That turned out to be wrong. So I made a lot of drawings, notes and calculations that ended up into nothing. A lot of blood, sweat and tears were shed☹

My thesis about there BEING A RELATION between those two was however NOT wrong. It took me quite some time to figure out how exactly but in the end I was finally able to get all numbers working together.

In the Excel sheet in TAB “Chapter 4 Obliquity” you can find the calculations behind the relation between the Axial tilt and the Orbital Inclination tilt.

1. Current Obliquity (which is actually Axial tilt + Inclination tilt)

   As mentioned above, in the dummy universe model the Axial tilt & Orbital Inclination tilt are related to each other.

   The heliocentric model however uses the terms ”Axial tilt” and “Obliquity” to describe the same: The angle between Earth's rotational axis and its orbital axis. To make a distinction of what is really happening I propose to use the term “Axial tilt” to describe the angle between Earth's rotational axis and its orbital axis (just like in the heliocentric model) but to use the term “Obliquity” to describe the RESULT of the Axial tilt and Orbital Inclination.

   In the above mentioned Excel TAB I have added the OBLIQUITY formula’s of Chapront et al (opens in a new tab). (2002) and J. Laskar (opens in a new tab).

   Based upon these two formula’s the J2000 value of OBLIQUITY was ~23.43929083°. The J2000 value has however a startdate of 1 January 2000 12:00 UTC. The start of the 3D model is however 21 June 2000 12:00 UTC, so if you add another (almost) half a year you will come to a number of 23.4392298806° which is just in line with both formula’s.

   So in the dummy universe model AND in the heliocentric model on 21 June 2000 12:00 UTC, the OBLIQUITY was 23.4392298806°.

   The obliquity is however defined by both an Axial tilt factor AND an Inclination tilt factor in the dummy universe model.

   So let’s first have a look at the current Inclination tilt value and cycle to break down how much of the 23.4392298806° is determined by Axial tilt and how much is determined by Orbital Inclination tilt.

2. Inclination cycle

   The Wikipedia page about orbital inclination (opens in a new tab) only mention the CURRENT inclination value of ~1.57° and the Wikipedia page about Earth (opens in a new tab) also only mentions the more specific ~1.57869° as the CURRENT value. I could not find any graph to show what is happening on a longer term with the orbital inclination of Earth.

   I could however find one source paper with a combination of the eccentricity and inclination figures across time. See below graph (opens in a new tab) coming from this paper (opens in a new tab) for more details. The eccentricity part of the graph is wrong (?) but it mentions the Orbital Inclination to increase across time with a specific period.

   

   NOTE: The eccentricity is wrong in this figure because in other sources it mentions it is decreasing. See e.g. the graph on the Wikipedia page (opens in a new tab) but there are many other sources why it is really decreasing like below graph on page 12 coming from this paper (opens in a new tab).

   

   In the next chapter I will dive into the Eccentricity a bit more

   The reason why I still think the Orbital Inclination is correctly shown in this paper is because the period is more or less correct (~100K) and the maximum and minimum degrees are in line with my expectations. I do not know how this graph exactly came to be, why there are almost no sources or historic measurements available about Orbital Inclination tilt, but I take it as a correct source. Maybe someone with some more specialistic knowledge can really confirm this graph.

   I have modified the picture to show you what is different in the dummy universe model.

   

   As can be seen, there are three differences in the dummy universe model compared to the numbers in the top graph: The Inclination moves from 0 to ~2.511 degrees, with a mean value of 1.2555°, is repeating the same pattern, and the cycles are a little longer (a 101,920 year cycle).

   Although there are some small differences between the two pictures, we can also confirm the dummy universe model values for Inclination tilt are more or less in line with the heliocentric theories.

   Now we have seen the inclination cycle, let’s further break them down for the EXACT numbers of Axial tilt and Orbital Inclination tilt with the help of the Excel.

   Please have a look at the Excel by now and try to understand how these values are related. You can also just have a look at the 3D dummy universe model to confirm what I mentioned is also added in the model. Feel free to play with other settings and see these settings are THE ONLY VALID NUMBERS WHICH ARE IN LINE WITH THE OBSERVATIONS WE SEE AROUND US.

3. What are the EXACT values for the Axial tilt?

   The experienced value of the Axial tilt on 21 June 12:00 UTC was 23°26′21″ (23.4392298806°). The real MEAN value of the Axial Tilt is however 23.2925° and oscillating by 1.2555° (above and below) so from 22.037° to 24.548°.

4. What are the EXACT values for the Inclination tilt?

   The experienced value of the Inclination to the invariable plane on 21 June 12:00 UTC was 1.5786889°. The real MEAN value of the Inclination Tilt is however 1.2555° and oscillating by +/- 1.2555° (above and below) so from 0 to 2.511°.

5. What are the effects on Earth’s obliquity of both Axial tilt and Inclination tilt?

   Both the Axial tilt and Orbital Inclination tilt have an angle of 1.2555°. The result is 23.4392298806° +/- 2 * 1.2555°.

   The total result of the Axial and Inclination Tilt resulting in the experienced Axial Tilt of Earth compared to the Sun path will therefore oscillate in a period of 305,760 years between 20.7815° and 25.8035° resulting in the changes in Earth’s climate in a 305,760 years cycle.

   

   Now we have shown how Axial and Inclination are related to each other on the LONGER TERM values, let’s also plot them on the SHORTER TERM.

6. Obliquity broken down into Axial and Inclination with J. Laskar paper

   The first picture is a coming from the paper “Secular Terms of Classical Planetary Theories Using the Results of General Relativity” as created by Laskar (opens in a new tab) which that shows the (expected) obliquity of the axial tilt in the years 8,000 BC to 12,000 AD.

   This famous picture can be found on a lot of websites (opens in a new tab) and shows the obliquity across time.

   It shows the Obliquity to be ~23.44° around year 2000 AD.

   

   Just to make it clear, this is a theoretic model and there are (off course) no measurements outside the small timeframe around our age (year 2000 AD):

   

   I have added all measurements that are coming out of the 3D dummy universe model based upon the settings as mentioned above (mean value of 23.2925° and oscillating by twice 1.2555°) in Green. The data is compared to formula’s provided by “Chapront et al. (2002)” and “J. Laskar's Mean Obliquity Formula” as added in the Excel sheet. The theoretically numbers of J. Laskar are completely in accordance with the measurements and the numbers in the dummy universe model FOR 1000 OF YEARS around the current age (year 2000 AD). However on the longer term the predictions from J. Laskar differ from the dummy universe model.

   

   The break-down of what is actually happening is added in the next picture. In Blue the Axial Precession. In Red the Inclination precession. In Green the result of both (which was shown as “dotted Green” in the previous picture).

   NOTE: I have changed the horizontal axis in the next picture to the years according to the BC and AD values instead of counting the years from 2000 AD as shown in the previous J. Laskar pictures AND I have added the start year of 12,000 BC instead of 8,000 BC as shown in the previous J. Laskar pictures. I have done this to make the Apsidal precession cycle a little more visible (from 8,309 BC to 10,801 AD).

   

   The Axial tilt & Orbital Inclination tilt are related to each other: They work upon each other. Therefore the currently experienced ~23.44° does not only contain the axial tilt factor but also contains the inclination-tilt factor. The current Axial Tilt value is actually ~23.116041° so ~0.323189° below the experienced axial tilt which is due to the inclination tilt above the mean value of the inclination tilt (0.323189°+ 1.2555° = 1.5786889°).

   In the Excel AND in the 3d dummy universe model you can look at those numbers.

   Let’s also look at the Eccentricity effect of the dummy universe model in the next chapter.