Encyclopedia of Marine Geosciences

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Astronomical Frequencies in Paleoclimates

  • André BergerEmail author
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DOI: https://doi.org/10.1007/978-94-007-6644-0_215-2

Keywords

Orbital Element Summer Solstice Winter Solstice Precessional Period Spring Equinox 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Definition

The long-term variations of climate display periods characteristic of three astronomical parameters which are the eccentricity (which fixes the shape of the Earth’s orbit), obliquity (the tilt of the equatorial plane on the plane of the Earth’s orbit around the Sun), and climatic precession (a measure of the distance from the Earth to the Sun at the summer solstice). Their main periods of variations are 400 and 100 kyr for eccentricity, 41 kyr for obliquity, and 23 and 19 kyr for precession.

Introduction

As in the astronomical theory of paleoclimates the glacial-interglacial cycles are of primary interest, this entry focuses on the long-term variations of the astronomical parameters which are involved in the calculation of the energy received by the Earth from the Sun (here called incoming solar radiation or insolation) at time scales of tens to hundreds of thousands of years.

These are the eccentricity, e, obliquity, ε, and climatic precession, \( e\ \sin \tilde{\omega} \), \( \tilde{\omega} \) being the longitude of the perihelion.

The full spectral characteristics of these astronomical elements and the resulting insolation changes date back only to the 1970s. Although the precessional period of 21,000 years was well known since Adhémar (1842) at least, Milankovitch and his contemporaries did not seem to be much interested in these astronomical periods (Berger, 2012). Milankovitch (1920), like Emiliani (1955) 35 years later, estimated only their mean values by counting the number of peaks from the curves that Milankovitch calculated numerically, leading to about 92,000, 40,000, and 21,000 years for e, ε, and \( e \sin \tilde{\omega} \), respectively. These periods were confirmed 20 years later when Berger (1973) completed his calculation of the long-term variations of precession, obliquity, and eccentricity. Besides its high accuracy over the Quaternary, the Berger calculation provided, for the first time, a full list as well as the origin of the periods characterizing the theoretical expansion of e (with periods of 413,000, 95,000, 123,000, 99,000, 131,000, and 2,305,000 years in decreasing order of the term’s amplitudes), of ε (with periods of 41,000, 53,600, and 29,700 years), and of \( e \sin \tilde{\omega} \) (with periods of 23,700, 22,400, 18,900, and 19,200 years) (see Berger, 1978, and for a slightly improved solution Berger and Loutre, 1991). Among these periods, those around 400,000, 2,300,000, and 54,000 and mainly of around 23,000 and 19,000 years were new, their existence not having been previously suspected.

In their science paper, Hays et al. (1976) used a spectral analysis technique which they applied to geological records and to the numerical values of the astronomical parameters calculated by Brouwer and Van Woerkom (1950) and Vernekar (1972) and found also 125,000 and 96,000 years for e, 41,000 years for ε, and 23,000 and 19,000 years for precession.

Orbit of the Earth Around the Sun and Its Axis of Rotation

The plane of the Earth’s elliptical orbit around the Sun is called the ecliptic. The Sun is not located at the center of the ellipse but rather at one of the foci, a fact that was demonstrated in the seventeenth century by the German astronomer Johannes Kepler (1571–1630) whose three laws are:
  • Kepler’s first law. Each planet moves in an ellipse with the Sun at one focus.

  • Kepler’s second law. The radius vector from the Sun to any one planet describes equal areas in equal times.

  • Kepler’s third law. For any two planets, the squares of the periods are proportional to the cubes of the semimajor axes of their orbits.

As the Earth travels counterclockwise around its orbit each year, it is sometimes nearer to and sometimes farther away from the Sun. Today, the Earth reaches the point in its orbital path known as perihelion, the point at which it is the closest to the Sun, on or about 3 January. On or about 4 July, it reaches aphelion, the point farthest from the Sun. At aphelion the distance between the Earth and the Sun, which is in average roughly 150 million kilometers, is about 5 million kilometers greater than it is at perihelion.

Figure 1 displays the various elements of the Earth’s orbit. The ellipse drawn through the perihelion, P, the Earth, E, and the aphelion, A, represents the orbit of the Earth around the Sun.
Fig. 1

The Earth (E)’s orbit around the Sun (S). The astronomical elements are defined in the manuscript. a is the semimajor axis of the elliptical orbit and b its semiminor axis, P is the perihelion, A is the aphelion, SE is spring equinox, SS is summer solstice, FE is fall equinox, WW is winter solstice, SQ is the perpendicular to the plane of the ecliptic, S NP is a parallel to the axis of rotation of the Earth describing the cone of precession, ω is the longitude of the perigee, \( \tilde{\omega} \) is the longitude of the perihelion relative to the moving vernal point γ, λ is the longitude of the Earth in its orbit, υ is the true anomaly, and ε is the obliquity

The Earth’s eccentricity, e, which is a measure of its elliptical shape, is given by \( \sqrt{a^2-{b}^2}/a \), where a is the semimajor axis and b the semiminor axis. The vernal point, γ, also called the First Point of Aries, is the position of the Sun in the sky seen from the Earth at the time of the spring equinox, SE, i.e., in Fig. 1 when the Earth crosses the celestial equator from the austral to the boreal hemisphere. It is the origin from which the ecliptical longitudes are measured. The winter (ww) and summer (ss) solstices and γ are shown at their present-day positions. In such a heliocentric system, γ is in the direction of the fall equinox, FE. In practice, observations are made from the Earth and the Sun is considered as if it were revolving around the Earth. In such a case, the system of reference becomes geocentric instead of heliocentric, and the vernal point is in the direction of the spring equinox.

The principal cause of the seasons is not the lack of uniformity of the apparent annual motion of the Sun but rather the tilting of the Earth’s axis at an angle, ε, presently 23°27′ away from a vertical drawn to the plane of the orbit. ε is therefore also the inclination (tilt) of the equator on the ecliptic and is called the obliquity. Seasons occur because the orientation of that axis of rotation remains approximately fixed in space as the Earth revolves about the Sun in 1 year. Each season begins at a particular point in the Earth’s orbit. Today these points are reached roughly on 20 or 21 March (spring equinox also called vernal or March equinox), 21 or 22 June (summer solstice also called June solstice), 22 or 23 September (fall equinox also called autumnal or September equinox), and 21 or 22 December (winter solstice also called December solstice). Their respective longitudes differ by a multiple of 90° with the longitude at spring equinox being zero by definition.

In the Northern Hemisphere, winter solstice marks the beginning of winter because the North Pole is tipped farthest away from the Sun on that day, making it the shortest day of the year in the entire Northern Hemisphere. Six months later, at summer solstice, the Earth reaches the point at which summer begins in the Northern Hemisphere. At this point, the North Pole is tipped toward the Sun, making the day of the summer solstice the longest day of the year in the Northern Hemisphere. From geometrical consideration, it can be shown that the Sun reaches the zenith at the June-summer solstice at noon of the true solar time (a time similar to the time read on a sundial) at the latitude now of 23°27′ which defines the tropics of Cancer; it reaches the December-winter solstice at the latitude now of −23°27′ which defines the tropics of Capricorn. Between the tropics, the Sun reaches the zenith at noon of the true solar time twice a year. North of the northern polar circle (which latitude is 90° −23°27′ = 66°33′), at the summer (winter) solstice, the day (night) is 24-h long. At the winter solstice, the Earth is today near perihelion (this approximate coincidence has nothing to do with the beginning of the calendar year).

At the spring and fall equinoxes, the two poles are equidistant from the Sun. On these dates, the number of daylight hours equals the number of hours of darkness at every point on the globe. These two points on the Earth’s orbit are therefore known as the equinoxes (the term equinox is indeed derived from the Latin aequus (equal) and nox (night)). In the Northern Hemisphere, the spring/vernal/March equinox marks the beginning of spring and the fall/autumnal/September equinox the beginning of fall. In the Southern Hemisphere the seasons are reversed.

Orbital Elements of a Planet

In astronomy, it is usual to define an orbit and the position of the body describing that orbit by six quantities called elements. As shown in Fig. 2, three of these elements (Ω, ω′, i) define the orientation of the orbit with respect to a set of axes, two of them define the size and the shape of the orbit (a and e, respectively), and the sixth, υ (with time), defines the position of the body within the orbit at that time. In the case of a planet moving in an elliptical orbit around the Sun, it is convenient to take a set of orthogonal axes in and perpendicular to the plane of reference, xoy, with the origin, o, at the center of the Sun or at the barycenter of the planetary system. The z-axis is taken to be perpendicular to this reference plane, so that the three axes form a right-handed orthogonal coordinate system. The reference point from which the angles are measured is labeled γ 0. As the reference plane is usually chosen to be the ecliptic at a particular fixed date of reference (called the epoch of reference, to distinguish it from any date of the past or the future), γ 0 is, in such a case, the vernal point at the epoch of reference which is taken to be 1950 CE in the Berger calculation. The point where the orbit cuts the reference plane with an increasing z-coordinate is called the ascending node, N; Ω, the longitude of that ascending node, is measured in the reference plane from γ 0; P is the perihelion; ω′ is the argument of the perihelion, an angle measured anticlockwise along the orbit from the ascending node; π is the longitude of the perihelion measured from the reference vernal point, γ 0, and is equal to the sum of Ω and ω′; i is the inclination of the orbital plane relative to the fixed reference plane; υ is the true anomaly, the angle measured anticlockwise between the perihelion and the Earth’s position; and λ is the longitude of the Earth in its orbit measured from the spring equinox at a specific date. Note that υ and λ fix the Earth on its orbit, respectively, from the perihelion and from the spring equinox at a specific date (Fig. 1). In the orbital plane, the distance, r, from the Earth to the Sun, normalized to a, is given by the equation of the ellipse:
Fig. 2

The position of the Earth E around the Sun S given by six quantities called the elements (see the manuscript for definitions). N is the ascending node, γ 0 is the vernal point at the epoch of reference, Ω (the angle γ 0 SN) is the longitude of the ascending node, ω′ (the angle NSP) is the argument of the perihelion P, i is the inclination of the ecliptic on the plane of reference, and v is the true anomaly

$$ \rho =\frac{r}{a}=\frac{1-{e}^2}{1+e \cos \upsilon } $$

The perihelion distance is consequently given by a(1 – e) and the aphelion distance by a(1 + e), which leads to the difference between the two being proportional to 2ae.

Long-Term Variations of the Orbital Elements

Due to the mutual attraction of the Sun, the planets, and the Moon, both the orientation and eccentricity of the ecliptic are changing with time. So is the axis of rotation of the Earth and therefore the equator. As a consequence, both the positions of the perihelion and of the vernal point relative to a fixed frame of reference are changing with time. The longitude of the perihelion, \( \tilde{\omega} \), relative to the moving vernal point, γ, is shown in Fig. 1. It is equal to π + ψ, where the annual general precession in longitude, ψ, describes the clockwise absolute motion of γ along the Earth’s orbit relative to the fixed stars, and the longitude of the perihelion, π, describes the anticlockwise absolute motion of the perihelion relative to the fixed stars.

To compute the long-term variations of the orbital elements, Newton’s law of gravitational attraction is applied to the planetary system. This leads to the Lagrange equations, i.e., a system of equations the number of which is equal to six times the number of planets. These equations provide the time evolution of the six orbital elements of each planet. They relate all the orbital elements of the planets between them together and describe their motion around the Sun. However, these equations possess some inconvenient features for orbits with small eccentricities and inclinations, both of which appearing in the denominators of some terms. It is therefore desirable to use a modified form of these equations by setting:
$$ \begin{array}{l}h=e \sin \pi \hfill \\ {}k=e \cos \pi \hfill \\ {}p= \sin i \sin \Omega \hfill \\ {}q= \sin i \cos \Omega \hfill \end{array} $$
If approximations are introduced in the development of the disturbing function (which expresses the mutual attraction of the Sun and the planets), the long-term behavior of h, k, p, q are given by
$$ \begin{array}{l}h={\displaystyle \sum_k{M}_k \sin \left({g}_kt+{\beta}_k\right)}\hfill \\ {}k={\displaystyle \sum_k{M}_k \cos \left({g}_kt+{\beta}_k\right)}\hfill \end{array} $$
(1)
$$ \begin{array}{l}p={\displaystyle \sum_k{N}_k \sin \left({s}_kt+{\delta}_k\right)}\hfill \\ {}q={\displaystyle \sum_k{N}_k \cos \left({s}_kt+{\delta}_k\right)}\hfill \end{array} $$
(2)

For the Berger (1978) solution, the amplitude M k and N k , the mean rates g k and s k , and the phases β k and δ k were calculated from Bretagnon (1974). Their numerical values are available in Berger (1973 and 1978) and also in Tables 1 and 2 of Berger and Loutre (1990). For the Berger and Loutre (1991) solution, Laskar (1988) was used leading to slightly different values of the orbital elements for periods of time longer than 1 million years (Berger and Loutre, 1992) but not changing our line of argument. Tables 3 and 4 of Berger and Loutre (1991) give the numerical values of the amplitudes, mean rates, and phases in decreasing order of the magnitude of the five most important terms for the two solutions Berger (1978) and Berger and Loutre (1990).

Table 1 provides the periods associated with the five largest amplitudes in decreasing order of magnitude for Eqs. 1 and 2. The values of i for Eq. (1) and j for Eq. (2) are those corresponding to the ordering traditionally used in celestial mechanics. Although there is not a one-to-one relationship between these periods and the individual planets, it can be shown that their origin is, to some extent, associated to one particular planet.
Table 1

Periods related to the frequencies g and s of the most important terms in the series expansion of (e, π) and (i, Ω)

 

Frequencies g

 

Frequencies s

i

Period

Planet

j

Period

Planet

5

308,043

Jupiter

5

  

2

176,420

Venus

3

68,829

Earth

4

72,576

Mars

1

230,977

Mercury

3

75,259

Earth

4

72,732

Mars

1

249,275

Mercury

2

191,404

Venus

   

6

49,339

Saturn

Because of the importance of the mass of Jupiter, the frequency (s 5) associated with that planet is equal to 0. This explains why the spectrum of i is dominated by two periods around 70 kyr and two of about 210 kyr, those characterizing Eq. 2. If the invariable plane (plane perpendicular to the total angular momentum) is taken instead of the ecliptic of the epoch, the term involving s 5 is excluded and the spectrum of the inclination on that plane is dominated by periods of 98, 107, and 1,300 kyr coming from the resonances between the Earth and Mercury, Mars and Venus, and Earth and Mars respectively. The periods of about 100 kyr are close to the periods of 100 kyr of eccentricity but totally different, and they are not associated with the 100 kyr of the geological records (Berger et al., 2005).

Long-Term Variations of Obliquity and Precession

After having calculated the motion of the planetary point masses around the Sun, the Poisson equations for the Earth-Moon system were used to compute the long-term variations of two astronomical parameters which, with the eccentricity, play a fundamental role in the long-term seasonal and latitudinal variations of insolation. These two parameters are obliquity, ε and \( \tilde{\omega} \) the longitude of the perihelion relative to the moving vernal point.

Although the hypothesis that the planets attract each other as if the mass of every one of them was concentrated in their respective center of mass is valid for the calculation of the orbital elements, the flattening of the planets has a perceptible effect on their rotation. The Moon’s gravitational attraction on the Earth causes two small bulges directed toward the Moon and away from it in the ocean and on the solid Earth. Dissipative processes cause a lag in the tidal response, and a torque is exerted that does not vanish when averaged over an orbital period of the Moon. The consequence of this torque is a change in the Earth’s angular momentum or, equivalently, to an increase of the length of the day by about one thousandth of a second in 100 years. At the same time, the bulges slow the Moon down in its orbital motion and lead to an increase in the Earth-Moon distance of the order of a few centimeters per year (400 millions years ago, the length of the day was about 22 h, the year consisted of about 400 days, and the Moon must have been 4 % closer to the Earth).

Viewed by an observer on the Earth near the North Pole, the stars appear to trace out concentric circles, the center of which defines the celestial North Pole, the extension of the Earth’s rotational axis in the sky. This celestial North Pole currently lies close to the star Polaris. As already noted by Hipparchus in about 120 BCE, the rotational axis is observed to move slowly and to trace out a cone, clockwise, with a half-angle of about 23.5° around the pole of the ecliptic, a motion which takes about 25,700 years to go full circle around the heavens. This steady motion of the rotational axis in space is called the astronomical precession of the Earth (or general precession in longitude). This is due to the inclination of the major axis of the oblate Earth to the ecliptic. Consequently, the net gravitational force on the Earth due to the Sun exerts a torque which attempts to draw the equator into the plane of the ecliptic, but the spinning of the Earth resists this; instead the torque causes motion of the spin axis about the pole of the ecliptic. This observed precession results from the sum of the solar and lunar torques (because of the large mass of the Sun and the proximity of the Moon) plus a rather minor contribution arising from the other planets. In addition, the complex interplay of the solar and lunar orbits induces small oscillations in the secular precessional motion of the rotational axis; these oscillations are known as forced nutations. The principal nutation term arises from a 19-year periodicity in the inclination of the Moon’s orbit, but these nutations are not considered in the long-term variations of the astronomical parameters discussed here because of their small magnitude.

These long-term variations of ε and ψ can be expressed analytically as is the case for h, k, p, q:
$$ \varepsilon =\varepsilon *+{\displaystyle \sum_i{A}_i \cos \left({\gamma}_it+{\varsigma}_i\right)} $$
(3)
$$ \psi =kt+\alpha +{\displaystyle \sum_i{S}_i \sin \left({\xi}_it+{\sigma}_i\right)} $$
(4)
ε*and α are constant of integration and k is the precessional constant. For the Berger (1978) solution, their numerical values are
$$ \begin{array}{l}\varepsilon *=23{}^{\circ}320556\hfill \\ {}\alpha =3{}^{\circ}392506\hfill \\ {}k=50^{{\prime\prime} }439273\hfill \end{array} $$
The amplitudes, mean rates, and phase of Eqs. 3 and 4 are given in Berger (1978). Some more detailed analytical developments are given in Berger and Loutre (1991) with for the most important terms:
$$ \begin{array}{l}{\gamma}_i={\xi}_i={s}_j+k\hfill \\ {}{\varsigma}_i={\sigma}_i={\delta}_j+\alpha \hfill \end{array} $$

Long-Term Variations of Eccentricity, Obliquity, and Climatic Precession

The incoming solar radiation changes from day to day due to the Earth’s elliptical motion around the Sun. But there are other changes of interest related to the planetary system and the Sun’s interior. In particular, the total solar energy received by the whole Earth over one full year varies, by a very small amount (Berger, 1977; Berger and Loutre, 1994), because the mean Earth-Sun distance varies in relation to changes in the shape of the Earth’s orbit around the Sun (the eccentricity, e). The solar output (the so-called solar constant) and the opacity of the interplanetary medium are also changing, but their effects remain difficult to prove at our time scales.

In addition, the seasonal and latitudinal distributions of insolation have also long-term variations which are related to the orbit of the Earth around the Sun and to the inclination of its axis of rotation. These involve three well-identified astronomical parameters (Fig. 1): the eccentricity, e; the obliquity, ε; and the climatic precession, \( e \sin \tilde{\omega} \), a measure of the Earth-Sun distance at the summer solstice. \( \tilde{\omega} \), the longitude of the perihelion, is a measure of the angular distance between the perihelion and the vernal equinox that are both in motion. In a geocentric system, the angle ω is the longitude of the perigee and its numerical values are obtained by adding 180° to \( \tilde{\omega} \) (Fig. 1). The present-day value of e is 0.016. As a consequence, although the Earth’s orbit is very close to a circle, the Earth-Sun distance, and consequently the insolation, varies by as much as 3.2 % and 6.4 %, respectively, over the course of 1 year. The obliquity, which defines our tropical latitudes and polar circles, is presently 23°27′. The longitude of the perihelion is 102°, which confirms that the Northern Hemisphere winter occurs when the Earth is about the closest to the Sun.

The long-term variations of obliquity are given by Eq. 3
$$ \varepsilon =\varepsilon *+{\displaystyle \sum_i{A}_i \cos \left({\gamma}_it+{\varsigma}_i\right)} $$
For eccentricity, its analytical development can be calculated from
$$ \begin{array}{lll}e=\sqrt{{\left(e \sin \tilde{\omega}\right)}^2+{\left(e \cos \tilde{\omega}\right)}^2}\hfill & \mathrm{or}\ \mathrm{from}\hfill & \sqrt{{\left(e \sin \pi \right)}^2+{\left(e \cos \pi \right)}^2}\hfill \end{array} $$
which leads to
$$ e=e*+{\displaystyle \sum_i{E}_i \cos \left({\lambda}_it+{\phi}_i\right)} $$
(5)
with e* = 0.0287069.
From the definition of \( \tilde{\omega}=\pi +\psi \) and using Eqs. 1, 4, and 5, the climatic precession parameter \( e \sin \tilde{\omega} \) can be expressed into the following trigonometrical expansion as a quasiperiodic function of time:
$$ e \sin \tilde{\omega}={\displaystyle \sum_i{P}_i \sin \left({\alpha}_it+{\eta}_i\right)} $$
(6)
The amplitudes P i , A i , and E i ; the frequencies α i , γ i , and λ i ; and the phases η i , ς i , ϕ i in Eqs. 3, 5, and 6 (Table 2) have been computed by Berger (1978) and Berger and Loutre (1991). These analytical expansions can be used over 1–2 millions years (Berger and Loutre, 1992), but for more remote times, numerical solutions are necessary (Laskar, 1999).
Table 2

Amplitudes, mean rates, phases, and periods of the five largest amplitude terms in the trigonometrical expansion of climatic precession, obliquity, and eccentricity (BER78 refers to Berger, 1978 and is based upon Bretagnon, 1974 and BER90 refers to Berger and Loutre, 1991 and is based on Laskar, 1988)

 

Amplitudes

Mean rate (ʺ/year)

Phase (°)

Period (years)

BER78

BER90

BER78

BER90

BER78

BER90

BER78

BER90

Climatic precession

1

0.018608

0.018970

54.64648

54.66624

32.0

32.2

23,716

23,708

2

0.016275

0.016318

57.78537

57.87275

197.2

201.3

22,428

22,394

3

0.013007

0.012989

68.29654

68.33975

131.7

153.4

18,976

18,964

4

0.009888

0.008136

67.65982

67.79501

323.6

311.4

19,155

19,116

5

0.003367

0.003870

67.28601

55.98574

102.8

78.6

19,261

23,149

Obliquity

1

−2462.22

−1969.00

31.60997

31.54068

251.9

247.14

41,000

41,090

2

−857.32

−903.50

32.62050

32.62947

280.8

288.79

39,730

39,719

3

−629.32

−631.67

24.17220

32.08588

128.3

265.33

53,615

40,392

4

−414.28

−602.81

31.98378

24.06077

292.7

129.70

40,521

53,864

5

−311.76

−352.88

44.82834

30.99683

15.4

43.20

28,910

41,811

Eccentricity

1

0.011029

0.011268

3.13889

3.20651

165.2

169.2

412,885

404,178

2

0.008733

0.008819

13.65006

13.67352

99.7

121.2

94,945

94,782

3

0.007493

0.007419

10.51117

10.46700

294.5

312.0

123,297

123,818

4

0.006724

0.005600

13.01334

13.12877

291.6

279.2

99,590

98,715

5

0.005812

0.004759

9.87446

9.92226

126.4

110.1

131,248

130,615

The sign of the amplitude of terms 3 and 5 in climatic precession and of terms 2 and 3 in eccentricity in column BER78 has been changed relatively to the values given in Berger (1978) in agreement with the change of the phase by 180°. This has been done to allow an easier comparison between the solutions. The five terms given in this table do not allow an accurate computation of the astronomical parameters. More terms are requested. They are available with a computer program in http://www.astr.ucl.ac.be and http://www.elic.ucl.ac.be/modx/elic/index.php?id=83.

Equations 3 and 5 show that ε and e vary quasiperiodically around constant values ε* (23.32°) and e* (0.0287). This implies that, in the estimation of the order of magnitude of the terms in insolation formulae where ε and e occur, they may be considered as a constant to a first approximation. On the other hand, in the insolation formulas used to study past and future astronomical forcing of climate, the amplitude of \( \sin \tilde{\omega} \) is modulated by eccentricity in the term \( e \sin \tilde{\omega} \). The envelope of \( e \sin \tilde{\omega} \) is therefore given exactly by e.

Besides their simplicity and practicability for easy computation, Eqs. 3, 5, and 6 and their derivation allow to explain the origin of the main periods associated with the astronomical theory of paleoclimates (for full details see Berger and Loutre, 1990). The periods calculated in Berger (1978) (but also in Berger and Loutre, 1991) come from the fundamental periods of the orbital elements associated with the g k in Eq. 1 and s k in Eq. 2 and from the period of the astronomical precession associated with k.

The main periods of eccentricity in Eq. 5 are 413, 95, 123, 100, and 131 kyr coming, respectively, from the following relationships where the subscripts of g (and s) are the classical ones reported in Table 1:
$$ \begin{array}{l}{\lambda}_1={g}_2-{g}_5\hfill \\ {}{\lambda}_2={g}_4-{g}_5\hfill \\ {}{\lambda}_3={g}_4-{g}_2\hfill \\ {}{\lambda}_4={g}_3-{g}_5\hfill \\ {}{\lambda}_5={g}_3-{g}_2\hfill \end{array} $$
For obliquity, the main periods are 41 and 54 kyr, coming, respectively, from
$$ \begin{array}{l}{\gamma}_1={s}_3+k\hfill \\ {}{\gamma}_3={s}_6+k\hfill \end{array} $$
For climatic precession, the main periods are 23.7, 22.4, 18.98, and 19.16 kyr with
$$ \begin{array}{l}{\alpha}_1={g}_5+k\hfill \\ {}{\alpha}_2={g}_2+k\hfill \\ {}{\alpha}_3={g}_4+k\hfill \\ {}{\alpha}_4={g}_3+k\hfill \end{array} $$
This leads to conclude that the periods characterizing the expansion of e are nonlinear combinations of the precessional periods and, in particular, that the eccentricity periods close to 100 kyr are originating from the periods close to 23 and 19 kyr in precession:
$$ \begin{array}{l}{\lambda}_1={\alpha}_2-{\alpha}_1\hfill \\ {}{\lambda}_2={\alpha}_3-{\alpha}_1\hfill \\ {}{\lambda}_3={\alpha}_3-{\alpha}_2\hfill \\ {}{\lambda}_4={\alpha}_4-{\alpha}_1\hfill \\ {}{\lambda}_5={\alpha}_4-{\alpha}_2\hfill \end{array} $$
As a consequence,
$$ \begin{array}{l}{\lambda}_3={\lambda}_2-{\lambda}_1\hfill \\ {}{\lambda}_5={\lambda}_4-{\lambda}_1\hfill \end{array} $$
which clearly shows that not all the periods of eccentricity are independent. Using the periods associated to the largest amplitude terms in Eqs. 1 and 2, we would conclude that the periods of 413, 95, and 100 kyr are the most fundamental ones (123 and 131 kyr deriving directly from them) with:
  • 413 kyr coming from the resonance between Venus and Jupiter

  • 95 kyr coming from the resonance between Mars and Jupiter

  • 100 kyr coming from the resonance between Earth and Jupiter

For the obliquity, Mars, Earth, and Moon explain the 40-kyr period, Saturn being related to the 54 kyr one. For climatic precession, in addition to the Moon effect, the two periods close to 23 kyr come from Jupiter and Venus and those close to 19 kyr from Mars and Earth.

Instability of the Astronomical Periods

Over the last few million years, eccentricity varies between 0 and 0.06 with an average period of 95 kyr, being slightly shorter over the future (91 kyr). Correlatively, the line of apses (line through P and A in Fig. 1) made a revolution in a fixed reference frame with an average period of 125 kyr but largely varying between 20 and 250 kyr. This dispersion around the average value is related to the existence of very short periods related to the very low values of eccentricity. If these short periods are ignored, the minimum value of the period becomes more realistic (60 kyr).

Figure 3 shows the long-term variations of these three astronomical parameters over the past 400 ka and into the future for the next 100 ka. It shows in particular that the 100-kyr period is not stable in time (Berger et al., 1998), being remarkably shorter over the present day. Actually, the most important theoretical period of eccentricity, 400 kyr, is weak before 1 Ma BP and is particularly strong over the next 400 ka, with the strength of the components in the 100-kyr band changing in the opposite way. It is worth pointing out that this weakening of the 100-kyr period started about 900 ka ago when this period began to appear very strongly in paleoclimate records. This implies that the 100-kyr period found in paleoclimatic records is definitely not linearly related to eccentricity. We are now approaching a minimum of e at the 400-kyr time scale: at 27 ka AP, the Earth’s orbit will be circular. Actually, transitions between successive strong 400-kyr cycles (as it is the case now) are characterized by very small eccentricity and short eccentricity cycles with a low amplitude of variation.
Fig. 3

Long-term variations of eccentricity, climatic precession, obliquity, and insolation at 65°N at the summer solstice from 400 ka BP to 100 ka AP. Time is progressing from right to left (The data are derived from the formula given in Berger (1978))

For precession, the average period is roughly 21.5 kyr with large dispersion varying between 14 and 30 kyr. At the 400-kyr time scale, the amplitude and frequency modulations are inversely related: when the amplitude of precession is small (large), the period is short (long). It is the reverse at the 100-kyr time scale.

Obliquity varies between 22° and 24°5 with a very stable period of 41 kyr, but there is an amplitude modulation with a time duration of about 1.3 Myr. The spectra of both the amplitude and frequency modulations of obliquity display significant power at 171 and 97 kyr. Although this last might look close to the so-called 100-kyr eccentricity period, they are not related (Mélice et al., 2001). At the 1.3-Myr time scale, a large amplitude corresponds to a short period, the reverse being true at the 170-kyr time scale.

While today the winter solstice occurs near perihelion, at the end of the deglaciation, roughly 10 ka BP, it occurred near aphelion. Moreover, because the length of the seasons varies in time according to Kepler’s second law, the solstices and equinoxes occur at different calendar dates during the geological past and in the future. Presently in the Northern Hemisphere, the longest seasons are spring (92.8 days) and summer (93.6 days), while autumn (89.8 days) and winter (89 days) are notably shorter. In 1250 CE, spring and summer have had the same length (as did autumn and winter) because the winter solstice occurred at perihelion. About 4,500 years into the future, the Northern Hemisphere spring and winter will have the same short length and consequently summer and fall will be equally long.

Summary and Conclusions

The most important periods of the three astronomical parameters which drive the long-term variations of climate are 400 and 100 kyr for eccentricity, 41 kyr for obliquity, and 23 and 19 kyr for precession. Although they are often called Milankovitch periods, they are not originating from his work. For example, the 41-kyr obliquity cycle and the average period of climatic precession, 21 kyr, date from the mid-nineteenth century. Among the properties of these astronomical parameters, some are less known and are stressed here again.

The term climatic precession has been introduced in the 1970s to avoid the too often confusion made at these early times with the astronomical precession.

The first theoretical period of eccentricity, about 400 kyr, and the double precessional peaks, 23 and 19 kyr, were only discovered in the early 1970s and play presently a more and more important role in paleoclimatology. The 400-kyr period is particularly strong over the next 400 kyr, whereas the 100 kyr is very weak. Actually, transitions between successive strong 400-kyr cycles are characterized by very small eccentricity. This is happening now and at 27 ka AP, the Earth’s orbit will be circular.

The analytical calculation of the trigonometrical expansions of the astronomical elements shows that the eccentricity periods close to 100 kyr are originating from the periods close to 23 and 19 kyr in precession, and the 400-kyr period is a combination of the two first precessional periods. The periods of eccentricity are therefore not all independent, those of 413, 95, and 100 kyr being the most fundamental ones.

For precession, the average period is roughly 21.5 kyr with a large dispersion varying between 14 and 30 kyr. At the 400-kyr time scale, when the amplitude of precession is small (large), the period is short (long). It is the reverse at the 100-kyr time scale.

Obliquity varies between 22° and 24°5 with a very stable period of 41 kyr, but there is an amplitude modulation with a period of about 1.3 Myr. At this 1.3-Myr time scale, large amplitude corresponds to a short period, the reverse being true at the 170-kyr time scale.

As obliquity and eccentricity vary around constant values (23.32° for obliquity and 0.0287 for eccentricity), these elements may be considered as a constant in a first very rough approximation as compared to the variations of climatic precession. As the amplitude of \( e \sin \tilde{\omega} \) is modulated by eccentricity, the envelope of \( e \sin \tilde{\omega} \) is given exactly by e.

Because the length of the seasons varies in time according to Kepler’s second law, the solstices and equinoxes occur at different calendar dates through time. The meteorological seasons are therefore continuously shifting through the astronomical ones and do not correspond to the same configuration of the Earth relative to the Sun.

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© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Georges Lemaître Center for Earth and Climate Research, Université catholique LouvainLouvain-la-NeuveBelgium