3.2.1. Introduction

Fundamental to what we feel are some of the potential breakthroughs that may be realized by a Pangean coring transect is the record of Milankovitch cyclostratigraphy preserved in the Triassic-Jurassic basins. Because the underlying celestial mechanics is relatively arcane, but necessary for the understanding of the science issues, and the geological record has potential to help constrain celestial mechanical issues, a short review of the relevant concepts is presented here. More in-depth treatments, from which this is derived, can be found in Laskar (1990, 1999), Laskar et. al. (1993), Berger et. al. (1992), Berger and Loutre (1990, 1994) and Hinnov (2000).

Figure Simplified geometry of the Earth's orbit for paleoclimate. The orbit of the Earth is shown here in a simplified perspective drawing. The horizontal gray plane contains the Earth's orbital plance at an arbitary date and comprises the reference plane. Click for a more detailed view. Abbreviations are: prec., general precession (wobble) of the Earth's rotational axis; obliq., obliquity of the Earth's axis (tilt); I, inclination of the plane of the Earth's orbit relative to the reference frame; P, point of perihelion. Click on the image to see a higher resolution image.

Origin of the Important Climatic Frequencies

Studies of the sedimentary record of orbital change has depended largely on comparison of data from the geological record with a target model, derived from celestial mechanics, often a time series of the amount of sunlight reaching the top of the atmosphere at a particular location (insolation) or a representation of some general aspect of celestial mechanics directly related to insolation (e.g. precession index). The Earth's axis of rotation and the figure of the Earth's orbit are perturbed by the gravitational attraction of the other bodies in the Solar System with an amplitude related to the mass of the bodies and their distance from the Earth. These perturbations produce cycles in insolation, which in turn effects climate. Models of the behavior these perturbations in frequency and time are generated by a three-step process.

1) The analysis of the mutual gravitational interactions and time behavior of the components of the Solar System is an n-body problem and its behavior though time cannot be solved analytically. Instead is usually studied by numerical integration (Quinn et al., 1991), or with a combination of analytically and numerical methods as in Laskar et al. (1993). The numerical integration of the bodies of the Solar System is difficult and requires minimization of error of the observational data for the present as well as considerable finesse in setting up and running on supercomputers.

2) The results of the numerical integration are described by Fourier analysis and summarized as a series of frequencies, phases, and amplitudes representing the periodic behavior of each of the planets (Table These frequencies are split into two classes representing the behavior of the planets in two orthogonal views (Fig. The "g" frequencies are those of the planets viewed downward on a fixed plane of the orbit of the Earth (at an arbitrary date) with its specific orbital orientation and shape. These frequencies reflect the change in the eccentricity and orientation of the figure of the orbits. The other, the "s" frequencies, are those viewed parallel to the reference plane (Fig. and reflect changes in the inclinations of the planes of the planetary orbits relative to that plane. It is the rocking of the plane of the Earth's orbit relative to the reference frame that produces the so-called "obliquity cycle" that averages 41 ky. For the Earth, the other main frequency that is important is that of the precession of the Earth's axis. This is the familiar wobble of the axis caused by the pull of both the Moon and the Sun (with similar amplitudes) on the equatorial bulge of the Earth. Relative to the reference plane the Earth's axis describes a circle with a period of 25,700 years. Described as frequency (1/25,700) this is the precessional constant (p).

3) Combinations of sums the g and s frequencies with p (i.e. p + gi and p + si) yield the various frequencies of "climatic precession" and "obliquity". While the precession constant is (1/25,700), these addition of the relatively small g frequencies to the relatively large p frequency results is a slightly larger frequency, the inverse of which is a slightly smaller period. 

or name
of cycle
Mercury g1 5.596 4.318E-06 231600  
Venus g2 7.456 5.753E-06 173800  
Earth g3 17.365 1.340E-05 74630  
Mars g4 17.916 1.382E-05 72340  
Jupiter g5 4.249 3.278E-06 305000  
Earth s3 -18.851 -1.455E-05 68750  
Mars s4 -17.748 -1.369E-05 73020  
---- g4+s4-s3=g19 16.813 1.297E-05 77090  
E 404 ka g2-g5 3.207 2.475E-06 404100  
E 2.4 Ma g4-g3 0.551 4.250E-07 2352900  
E 962 ka g1-g5 1.348 1.040E-06 961700  
E 697 ka g2-g1 1.860 1.435E-06 697000  
E 95 ka g4-g5 13.667 1.055E-05 94800  
E 124 ka g4-g2 10.460 8.071E-06 123900  
E 99 ka g3-g5 13.116 1.012E-05 98800  
E 131 ka g3-g2 9.909 7.646E-06 130800  
E 105 ka g4-g1 12.319 9.505E-06 105200  
E 139 ka g19-g2 9.357 7.220E-06 138500  
E 99 ka average 13.036 1.006E-05 99400  
E 131 ka average 9.909 7.646E-06 131100  
E 115 ka average 11.064 8.674E-06 115300  

   NOTE The combinations of the fundamental frequencies
   that produce the periods of the predicted eccentricity cycles
   are listed in order of their amplitudes according to Laskar

Table 1: Origin of present values of Eccentricity cycles based on the fundamental frequencies of the planets.

3.89E-05 25678
g1 + p 4.33E-05 23115
g2 + p 4.47E-05 22373
g3 + p 5.23E-05 19104
g4 + p 5.28E-05 18952
g5 + p 4.22E-05 23684
The average of most of these is 21 ky. The s frequencies are negative numbers that vary less than the g frequencies, and their addition to the p results in frequencies that are somewhat smaller than p. The inverse of the s plus p frequencies are hence larger than p, with most lying close to 41 ky. The g plus p and s plus p frequencies are shown in Table

Combinations of differences of the individual g frequencies result in the frequencies of the "eccentricity cycles". These are the modulators of climatic precession cycles and are algebraically and physically equivalent to "beat" cycles of the various climatic precession cycles, themselves. Thus, the 404 ky eccentricity cycle, very important for this workshop, is derived from g2-g5 = 1/404 = (g2 + p) - (g5 + p). Because they are the difference of very small frequencies, their periods are long relative to those of climatic precession. 

Table Origin of the Cycles of Climatic Precession.

Combinations of differences of the individual s frequencies result in a parallel series of modulators of obliquity that, until the last couple of years (e.g. Lourens and Hilgen, 1997; Hinnov, 2000), have received almost no paleoclimatic attention at all. Although these combinations of g, and p are the most important to the behavior of the Earth climate, there are many more combinations of g and s frequencies together which are needed for a full model of the time and frequency behavior of insolation.

Constraining Celestial Mechanical Chaos with Geological Data

The limit of the precision of the data going into the numerical integration produce large variations in the output over various times scales. This results in a chaotic drift in the values of the g and s frequencies over tens to hundreds of millions of years within a specifiable chaotic region (see Laskar, this report). This chaotic behavior makes it impossible to produce an insolation curve for more than about 20 million years into the past (or future), and indeed makes it impossible to predict the value for the longest period modulators of precession and obliquity. In fact, for 200 million years ago there can be as much as a 40% difference between extreme possibilities in the periods of the beats between some of the combinations of g frequencies within their chaotic zones.

It very important to note that the precession "constant" p also evolves though time as a consequence of tidal and climate friction from the gravitational attraction of the moon, as well as changes the geophysical properties of the Earth itself. However, although this can modeled, geological data are still required for the value of p to be specified with precession in the distant geological past (Neron de Surgy and Laskar, 1997; Berger et al., 1992; Hinnov, 2000).

However, the fact that most of the frequencies thought to be important to paleoclimate studies are derived from combinations of a relatively few fundamental frequencies is a very powerful potential tool for getting celestial mechanical information directly out the geological record. Using a time scale derived by tuning to the most stable of the eccentricity cycles, the 404 ky cycle (see Laskar, this report, and Olsen, this report) the other long eccentricity cycles can be identified and their component g frequencies solved for (e. g. Olsen and Kent, 1999a). The very long length of the Newark basin and other Triassic-Jurassic basin cyclical records makes it possible to see such long cycles, and because of this it is possible to determine the fundamental periods. A similar procedure could be used for the obliquity cycle modulators. Because the eccentricity cycles and obliquity modulators are linked (see Laskar, this report), results from cores recovered in tropical regions where precision is dominant (i.e. the Newark basin cores), make predictions for the obliquity signal that can be tested by cores containing a strong obliquity response, most likely from the high latitudes. The geological record can this constrain celestial mechanical values for the past. Finally, by extending the procedure outlined above, it may be possible to obtain the phases and even amplitudes of the g and s values for the Triassic-Jurassic eventually allowing the construction of an insolation curve for the Earth Mesozoic.