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Chapters05 Obliquity & Inclination

Let me share something that took me quite a while to figure out. When I started my journey my initial thesis was based on the assumption the Axial tilt and 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. It took me quite some time to figure out how exactly they were related but in the end I was finally able to get all numbers working together.

In the Excel sheet in TAB “Chapter 5 Obliquity” you can find the calculations behind the relation between the Axial tilt and the Inclination tilt.

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

    My theory is the Axial tilt and Inclination tilt are related to each other.

    Currently the terms “Axial tilt” and “Obliquity” are used interchangeable to describe the same behaviour: 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 but to use the term Obliquity to describe the RESULT of the Axial tilt and Inclination tilt.

    I have added the OBLIQUITY formula of Chapront et al . (2002) and J. Laskar  in the excel.

    The obliquity is however defined by both an Axial tilt factor AND an Inclination tilt factor. So let’s first have a look at the current Inclination tilt value.

  2. Inclination cycle

    The Wikipedia page about orbital inclination  only mention the CURRENT inclination value of ~1.57° and the Wikipedia page about Earth  also only mentions the more specific ~1.57869° as the CURRENT value, same as this website . The J2000 value has a startdate of 1 January 2000 12:00 UTC. The start of the 3D model is however 21 June 2000 00:00 UTC, so we need to take this into consideration as well.

    It was very hard to find any graph to show what is happening on a longer term with the inclination of Earth.

    I could however find one source paper with a combination of the eccentricity and inclination figures across time. See below graph  coming from this paper  for more details. It seems to come from a solar system model by Varadi Runnegar and Ghil  which mentions Earth’s inclination varies between 0 and about 0.05 radians (or about 3 degrees). Just to make it clear, It is a model and nothing is confirmed. We could however derive the current trend so the values need to be in line with this paper.

    It clearly mentions the Orbital Inclination in the past was higher so we are currently decreasing across time.

    Research graph from Varadi, Runnegar and Ghil showing Earth's orbital inclination variation over millions of years, ranging between 0 and ~3 degrees relative to the invariable plane

    The Orbital Inclination picture in this paper shows the maximum and minimum degrees. I have modified the picture to show what is actually happening.

    Modified graph showing orbital inclination with mean value of 1.4951° oscillating between 0.9311° and 2.0591° in a repeating 99,392-year cycle

    As can be seen, there are three differences compared to the current model:

    • The inclination cycle has a mean value of ~1.4951°
    • The inclination cycle moves from ~0.9311° to ~2.0591°.
    • The inclination cycle is repeating the same pattern with a duration of 99,392 years.

    Although there are some small differences between the two pictures, we can also confirm the values for Inclination tilt are fully in line in the current timeframe.

  3. What are the values for the Axial tilt?

    There are several formulas available for the Axial tilt. The value on 1 January 2000 AD 12:00 UTC was ~23.43929083°.

    Currently there is no mean value known yet. I expect however the MEAN value of the Axial Tilt to be ~23.42723° and oscillating by +/- ~0.564° (above and below).

  4. What are the values for the Inclination tilt?

    The value of Earth’s Inclination to the invariable plane on 1 January 2000 AD 12:00 UTC was ~1.57869°.

    Currently there is no mean value known yet. I expect however the MEAN value of the Inclination Tilt to be ~1.4951° and oscillating by +/- ~0.564°.

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

    Here’s where it gets interesting. Both the Axial Tilt and Inclination Tilt have the same angle of ~0.564°. The result is ~23.42723° +/- 2 * ~0.564°.

    The total result of the Axial and Inclination Tilt resulting in the experienced Obliquity Tilt of Earth compared to the Sun path will therefore oscillate in a period of 298,176 years between ~22.36° and ~24.58°.

    Graph showing combined obliquity effect from axial tilt (23.42723° ±0.564°) and inclination tilt (±0.564° each), resulting in total obliquity oscillating between 22.36° and 24.58° over 298,176 years

    Now that we’ve seen 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  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  and shows the obliquity across time.

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

    Original graph from J. Laskar's paper showing theoretical obliquity variation from 8,000 BC to 12,000 AD, displaying obliquity around 23.44° at 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:

    Graph highlighting limited measurement timeframe around current age (year 2000 AD), showing most historical obliquity data is theoretical modeling rather than actual measurements

    I have updated the picture with the numbers as mentioned above (mean value of ~23.42723° and oscillating by twice ~0.564°) in Green. The data is compared to formulas 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 FOR 1000 OF YEARS around the current age (year 2000 AD). However on the longer term the predictions from J. Laskar differ from the numbers in the Holistic Universe Model.

    Modified Laskar graph with Holistic Universe Model values overlaid in green (mean 23.42723° ±1.128°), comparing against Chapront et al. and Laskar formulas, showing agreement in recent millennia but divergence long-term

    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 Perihelion precession cycle a little more visible (from 8,072 BC to 10,564 AD).

    Graph showing obliquity broken down into axial tilt (blue line) and inclination tilt (red line) components from 12,000 BC to 10,564 AD, with combined effect in green compared to Laskar's formula

    The Axial tilt and 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.

  7. Where is the invariable plane tilted to?

    You might be wondering: where exactly is the invariable plane positioned? The PERIHELION-OF-EARTH orbits the Sun in one Inclination precession period which moves Earth on its path on the invariable plane. Currently Earth is tilted ~1.57869° to the invariable plane. The mean tilt is ~1.4951°.

    The Aries-Pisces point is the highest point and the Virgo-Libra point is the lowest point.

    Diagram showing Earth's tilt relative to the invariable plane, currently at 1.57869° with mean tilt of 1.4951°, indicating Aries-Pisces as highest point and Virgo-Libra as lowest point

    This feature of a mean value is currently not mentioned anywhere. If some astronomer could confirm it, it will be a great find to confirm the correctness of the model.

    So beside our solar system in total being tilted ~60 degrees to the galactic plane, also the PERIHELION-OF-EARTH has a specific tilt on its orbit around the Sun.

  8. Comparing the Obliquity values to the current formulas around 2000 AD

    Let me walk you through the different timescales to show how these values compare.

    First, the broader view spanning ~25,000 years in both directions around our lifetime showing Obliquity. Note how the Laskar and Chapront formulas contradict each other at larger timescales:

    Mid-term graph showing obliquity variations spanning ~25,000 years in both directions around our lifetime, comparing model values with current astronomical formulas. The Laskar and Chapront formulas are contradicting

    At the same timescale, we can also show both Axial tilt and Inclination tilt which RESULTS in the Obliquity:

    Mid-term graph displaying both axial tilt and Inclination combined in one view spanning ~25,000 years in both directions, showing their interrelated behavior

    Zooming in to a few thousand years, we can compare the Obliquity values more precisely with current astronomical formulas:

    Mid-term graph showing obliquity variations over tens of thousands of years, comparing model values with current astronomical formulas
  9. Behaviour of the Inclination tilt around 2000 AD
    Graph showing inclination tilt behavior around year 2000 AD, with values oscillating in a regular pattern over thousands of years
  10. Long term fluctuation of Axial, Inclination & Obliquity

    To show the Obliquity movements along the 298,176-year cycle, here’s the overview:

    Comprehensive overview showing the obliquity fluctuations over the complete 298,176-year Holistic-Year cycle

    The Obliquity cycles have a MEAN duration of 37,272 years, though the actual top-to-top intervals can vary:

    Comprehensive overview showing the obliquity fluctuations over the complete 298,176-year Holistic-Year cycle including the obliquity cycles which on average have a duration of 37,272 years

    Here’s the Inclination movement shown as the absolute value:

    Long-term graph showing inclination tilt effect isolated over the 298,176-year cycle, displaying the inclination component separately

    The same Inclination movement, this time relative to the mean value:

    Long-term graph showing relative inclination tilt variations over the 298,176-year cycle, illustrating the orbital inclination changes measured against different reference frames

    And the Axial movement relative to the mean value:

    Long-term graph showing axial tilt effect isolated over the 298,176-year cycle, displaying the axial precession component separately from inclination

    Finally, here’s the key insight: Axial tilt and Inclination tilt oscillate at the same amplitude (~0.564°) throughout the 298,176-year cycle which RESULTS in the Obliquity cycle:

    Graph comparing axial tilt and inclination tilt around year 2000 AD, showing both oscillating at approximately ±0.564° amplitude
  11. Summary inclination movement

    To close off, this is a picture of the movement of the PERIHELION-OF-EARTH as experienced from Earth. Due to Earth’s movement around the EARTH-WOBBLE-CENTER the eccentricity changes and therefore we experience it as the BLUE pattern.

    Diagram showing PERIHELION-OF-EARTH movement as experienced from Earth, with blue pattern illustrating how Earth's movement around EARTH-WOBBLE-CENTER causes changing eccentricity

    In PURPLE you see the dates where Inclination precession meets Axial precession (see date 1245-12-14).

    In GREEN the current date and Inclination. In year 29,200 AD we will reach the minimum and in year 78,896 AD it is maximum again. This inclination tilt compared to the stars stays fixed.

    NOTE there are only 16 FIXED PURPLE POINTS in the sky compared to the fixed stars where EVERY TIME the inclination precession meets axial precession on its 298,176 year cycle! It is a 16 spoke wheel of time!

    NOTE this does not mean automatically the experienced Inclination precession or Axial precession or Perihelion precession is always the same across time. These are average values. The reason why there could be small differences experienced on Earth is because the Axial and Inclination tilt also work upon each other and there are other smaller movements at play.

Now that we understand how Axial and Inclination tilt work together to create the obliquity we observe, let’s explore another important factor: the Eccentricity effect in the next chapter.

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04 Lenght of Days & Years06 Orbital Eccentricity