# Extending the Range of Milankovic Cycles and Resulting Global Temperature Variations to Shorter Periods (1–100 Year Range)

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^{2}

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## Abstract

**:**

## 1. Introduction

- (1)
- The first is associated with Kepler’s laws. In the case of a central field and an elliptical orbit, for the orbit to be closed, it is necessary and sufficient that the orbit’s angular change after n revolutions be of the form $\Delta \phi =2\pi m/n$, where m is the number of full revolutions necessary for the planet to recover its initial position. There are only two central fields in which $\Delta \phi $ is a rational fraction of $2\pi $, ensuring closed orbits, that is fields in ${r}^{2}$ and $1/r$, the latter being the case of our solar system (cf. [22]).
- (2)
- The second involves the joint effects of the Moon and Sun. Let us quote d’Alembert ([23], p. 14): “Enfin, l’inclinaison de l’axe terrestre au plan de l’ecliptique doit modifier aussi l’action du Soleil; car selon que cet axe sera différemment incliné, il fera à chaque point de l’ecliptique un angle différent avec la ligne qui joint les centres de la Terre et du Soleil; par conséquent la quantité et la loi de l’action du Soleil, dépend de l’inclinaison de l’axe, et c’est aussi ce que l’analyse apprend.”

- (3)
- The Sun containing 99% of the total mass of the solar system, [24] shows that the planet’s revolution about the Sun produces an additional precession of about 3.8” per century, or a period of some 33 million years.
- (4)
- Because the Sun is actually a huge rotating mass, there is an additional relativistic component of precession, with a period on the order of 5.8 million years [25].

## 2. The Data: Temperature, Pole Motion, and Solar Ephemerids

#### 2.1. Mean Global Temperatures

**HadCrut**data: HadCrutv from 1870 to 2000 ([26], https://crudata.uea.ac.uk/cru/data/crutem1, accessed on 2 June 2022); HadCrut2 from 1856 to 2006 ([27], https://crudata.uea.ac.uk/cru/data/crutem2, accessed on 2 June 2022); HadCrut3 from 1850 to 2014 ([28], https://crudata.uea.ac.uk/cru/data/crutem3, accessed on 2 June 2022); HadCrut4 from 1850 to 2021 ([29], https://crudata.uea.ac.uk/cru/data/crutem4/, accessed on 2 June 2022) and HadCrut5 from 1850 to 2022 ([30], https://crudata.uea.ac.uk/cru/data/temperature/, accessed on 2 June 2022). Figure 1a shows all the data, and Figure 1b their Fourier transforms. There are rather significant differences between the data series, for instance between 1940 and 2020 in HadCrut3 (yellow curve) vs. HadCrut5 (blue curve). Differences become larger after 1950, to the point that HadCrut3 has a plateau after 2000 when HadCrut5 grows linearly since 1960. We have already worked on these data sets [31] and pointed out these differences [32], Figure 4. These differences of course result in differences in the Fourier spectra of Figure 1. As a result, the dominant spectral peak shifts from 60 to 80 year, a topic discussed in several papers [32,33,34,35].

#### 2.2. Solar Ephemerids

**IMCCE**, http://vo.imcce.fr/webservices/miriade/?forms, accessed on 2 June 2022). We do not present a figure with the raw data: the Earth orbit’s eccentricity is so small that an annual oscillation since 1846 would transform into an unreadable quasi-sinus with 176 oscillations on the 15 cm (or so) width of the figure, that is 14 oscillations per cm.

#### 2.3. Rotation Pole and Length of Day

**IERS**, https://www.iers.org/IERS/EN/DataProducts/EarthOrientationData/eop.html, accessed on 2 June 2022). They consist in the couple of coordinates (${m}_{1}$, ${m}_{2}$) of motion of the rotation pole

**PM**(see [16,36]) and the length of day

**lod**(e.g., [12]). We have selected data set EOP C01 IAU19801. Figure 2a,b respectively show the evolution of the couple (${m}_{1}$, ${m}_{2}$) since 1946 and of

**lod**since 1962. We have used the semi-annual

**lod**data provided by Stephenson and Morrison ([37]) for the period 1832–1997, combined with the

**IERS**data, resulting in the mean curve between 1832 and 2022 shown in Figure 2c (cf. [38]).

## 3. Extraction and Analysis of the Trends and Annual Oscillations

#### 3.1. Methods: SSA

**iSSA**) and we will now do the same for the rotation, temperature and ephemeris time series presented in the previous section. We refer the reader to these papers and to the Golyandina and Zhigljavsky’s book ([41]) for the

**SSA**method, to [42] for properties of the Hankel and Toeplitz matrices that it uses, and to [43] for the singular value decomposition algorithm

**SVD**).

- Step 1: embedding step

**SSA**, consists in projecting the one-dimensional time series in a multidimensional space of series ${\mathcal{X}}_{N}$ such that vectors ${X}_{i}={({x}_{i},\dots ,{x}_{i+L-1})}^{t}$ belong to ${\mathcal{R}}^{L}$ where $K=N-L+1$. The parameter that controls the embedding is L, the size of the analyzing window, an integer between 2 and $N-1$. The Hankel matrix has a number of symmetry properties: its transpose ${\mathbf{X}}^{t}$, called the trajectory matrix, has dimension K. Embedding is a compulsory step in the analysis of non-linear series. It consists formally in the empirical evaluation of all pairs of distances between two offset vectors, delayed (lagged) in order to calculate the correlation dimension of the series.

- Step 2: Decomposition in singular values—
**SVD**

**SVD**of non-zero trajectory matrix

**X**(dimensions $L\times K$) takes the shape:

**X**can then be represented as a simple linear sum of elementary matrices

**X**${}_{i}$. If all eigenvalues are equal to 1, then decomposition of

**X**into a sum of unitary matrices is:

**X**($d=\mathrm{rank}\phantom{\rule{4pt}{0ex}}\mathbf{X}=max\left\{i\right|{\lambda}_{i}>0\}$)

**SVD**allows one to write

**X**as a sum of d unitary matrices, defined in a univocal way.

**X**be a suite of L lagged parts of (

**X**and ${X}_{1},\dots ,{X}_{K}$) the linear basis of its eigenvectors. If we let:

**SVD**is a very good choice for the analysis of the embedding matrix, since it provides two different geometrical descriptions.

- Step 3: reconstruction

**SSA**. In order to regroup the unit matrices, one divides the set of indices {i${}_{1}$,…,i${}_{d}$} into m disjoint subsets of indices $\{{I}_{1},\dots ,{I}_{m}\}$.

**X**${}_{I}$ that regroups indices I can be written:

**X**,

**X**${}^{\left(1\right)}$ and

**X**${}^{\left(2\right)}$ be the embedding matrices of series $\chi $, ${\chi}^{\left(1\right)}$ and ${\chi}^{\left(2\right)}$. These two sub-series are separable (even weakly) in Equation (6) if there is a collection of indices $\mathcal{I}\subset \{1,\dots ,d\}$ such that ${\mathbf{X}}^{\left(1\right)}={\sum}_{i\in \mathcal{I}}{\mathbf{X}}_{i}$, respectively, if there is a collection of indices such that ${\mathbf{X}}^{\left(1\right)}={\sum}_{i\notin \mathcal{I}}{\mathbf{X}}_{i}$.

**SVD**components can be summarized by the decomposition into several elementary matrices, whose structure must be as close as possible to a Hankel matrix of the initial trajectory matrix (this is true on paper only, in reality things are much more difficult).

- Step 4: the diagonal mean, also known as the hankelization step

**Y**gives:

**SSA**(

**iSSA**). Since the relation (6) is linear, we can iterate the decomposition. We start with a small value of L (we are looking for the longest period) that we increase until getting a quasi-Hankel matrix (step 1 and 2). We then extract the corresponding lowest frequency component that it subtracted from the original signal. We again increase the value of L to find the next component (shortest period). The algorithm stops when no pseudo-cycle can be detected or extracted. In this way, we scan the series from low to high frequencies.

#### 3.2. Methods: Wavelets

**iSSA**, we have compared some of the

**iSSA**results with those obtained with the method of continuous wavelets, that is widely used in the literature. Figure 3b shows the scalogram (i.e., the continuous wavelet transform) of

**lod**since 1962 (Figure 2b). We have selected a Morlet wavelet. Between the dashed red lines are the ordinates of the wavelet coefficients associated with the 1 year oscillation. Given the property of information redundancy of the wavelet kernels, one can extract the ridge in the scales covered by the 1 year periodicity and reconstruct the signal [44]. The (wavelet) reconstructed annual component of lod is shown as the black curve in Figure 3a. The comparison is good but not perfect: the modulation of the iSSA curve (in red) is smoother than that of the wavelet reconstruction. The similarity of the two curves in Figure 3a together give us confidence that

**iSSA**can correctly extract the annual components of the three time series introduced in Section 2.

## 4. The Lissajous Diagrams

**lod**.

## 5. Discussion

**lod**), we have seen that the celestial positions of solstices moved significantly in the past 180 years. To our knowledge, this is the first time one evidences in the observational data what [23] called the apparent precession of solstices (that is seen from Earth) and that Milanković (part 2, chapter 2) worked on, but on much longer time periods of millions of years. We recalled in the introduction the basic equation from Milanković’s thesis (1920) that links time variations of heat received at a given location on Earth to solar insolation, known functions of the location coordinates, solar declination and hour angle, with an inverse square dependence on the Sun–Earth distance. We can let W play the role of the heat/energy term in Equation (1). The goal is to translate the drift of solstices as a function of distance to the Sun into the geometrical insolation theory of [3].

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The five

**HadCrut**mean global temperatures since 1850. (

**a**) The five mean global temperature data sets

**HadCrut1**to

**HadCrut5**from 1850 to the present maintained by the Hadley Center for Climate Prediction and Research (see text). (

**b**) The 5 Fourier spectra of the 5 data sets in Figure 1a.

**Figure 2.**Pole motion and length of day from IERS data. (

**a**) Evolution of the couple (${m}_{1}$, ${m}_{2}$) since 1846 (

**IERS**data). (

**b**) Evolution of lod since 1962 (

**IERS**data). (

**c**) Black curve: lod semi-annual data since 1832 (from [37]); gray: daily lod data since 1962 from

**IERS**; red curve: The median of the trend for these two combined data sets.

**Figure 3.**Extraction of oscillatory component by continues wavelet transform: an example. (

**a**) Black curve: annual component of

**lod**extracted by the method of continuous wavelets; Red curve: annual component of

**lod**extracted by the

**iSSA**method. (

**b**) Scalogram of lod since 1962. The wavelet cone of influence is symbolized by the gray area. The red dashed lines enclose the wavelet coefficients corresponding to the 1 year period.

**Figure 4.**Annual components extracted from pole path and

**lod**. (

**a**) The annual couple (${m}_{1}$, ${m}_{2}$) of polar motion coordinates extracted by

**iSSA**. (

**b**) The annual component of

**lod**extracted by

**iSSA**.

**Figure 5.**Drift of equinoxes and solstices since 1846. (

**a**) Lissajous diagram for the couple of rotation pole motion coordinates (${m}_{1}$, ${m}_{2}$) as they vary with Sun–Earth distance, i.e., as a function of time. (

**b**) Projection of the drift of equinoxes and solstices from 1846 to the present in the (${m}_{1}$, ${m}_{2}$) plane.

**Figure 6.**Drift of equinoxes and solstices since 1846 with the new parameters ${m}_{1}^{*}$ and ${m}_{2}^{*}$. (

**a**) Lissajous diagram for the couple of coordinates (${m}_{1}^{*}$, ${m}_{2}^{*}$) as they vary with Sun-Earth distance, i.e., as a function of time. (

**b**) Projection of the drift of equinoxes and solstices from 1846 to the present in the (${m}_{1}^{*}$, ${m}_{2}^{*}$) plane.

**Figure 7.**Time evolution of winter and summer solstices. (

**a**) Time evolution of winter and summer solstices for component ${m}_{1}^{*}$. (

**b**) Time evolution of winter and summer solstices for component ${m}_{2}^{*}$.

**Figure 8.**Trend, one year and sixty years components extracted from

**HadCrut**curves. (

**a**) The first

**iSSA**components (trends) of the five

**HadCrut**Center temperature series introduced in Section 2.1, and their median shown in black among the 5 trends and alone above as an inset. (

**b**) Annual iSSA components of the same series as in Figure 8a. (

**c**) Sixty year

**iSSA**components of the same series as in Figure 8a.

**Figure 10.**(

**top**) In red, the derivative of the

**iSSA**trend of temperature; in green and blue, the inverse square of the drift of solstices. (

**bottom**) A phase quadrature has been applied to the solstices curves above, that is a backward translation of 15 year (=60 years/4).

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**MDPI and ACS Style**

Lopes, F.; Courtillot, V.; Gibert, D.; Le Mouël, J.-L.
Extending the Range of Milankovic Cycles and Resulting Global Temperature Variations to Shorter Periods (1–100 Year Range). *Geosciences* **2022**, *12*, 448.
https://doi.org/10.3390/geosciences12120448

**AMA Style**

Lopes F, Courtillot V, Gibert D, Le Mouël J-L.
Extending the Range of Milankovic Cycles and Resulting Global Temperature Variations to Shorter Periods (1–100 Year Range). *Geosciences*. 2022; 12(12):448.
https://doi.org/10.3390/geosciences12120448

**Chicago/Turabian Style**

Lopes, Fernando, Vincent Courtillot, Dominique Gibert, and Jean-Louis Le Mouël.
2022. "Extending the Range of Milankovic Cycles and Resulting Global Temperature Variations to Shorter Periods (1–100 Year Range)" *Geosciences* 12, no. 12: 448.
https://doi.org/10.3390/geosciences12120448