Amplitude Modulation Patterns: The ‘Fingerprint’ of Orbital Cycles
Tidal Dissipation and Dynamical Ellipticity
Chaos in the Solar System
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EARTH/Orbital Variation (Including Milankovitch Cycles) 417 Figure Joint time frequency analysis of an arbitrary mixture of the eccentricity, obliquity, and climatic precession signal to allow a better visual representation of the main climatically im portant orbital variations The amplitude at a particular frequency and time is colour coded, red corresponding to a high relative amplitude A selection of amplitude modulation terms is high lighted (boxes inside figure); these represent the ‘fingerprint’ of an astronomical model The numbers near the fingerprints represent periods (see Table 5) The significance of amplitude modulation cycles is twofold First, if these cycles can be detected in the geological record, they allow the placement of geological data into a consistent framework within these amplitude modulation envelopes, even in the absence of individual cycles and in the presence of gaps The extraction of long amplitude modulation cycles typically requires high-fidelity geological records that are millions of years long Of particular value for the generation of geological time-scales beyond the Neogene is the $400-ky-long eccentricity cycle, because it is considered to be very stable In addition, if the eccentricity signal could be found in the geological data directly, as well as through its modulation of the climatic precession amplitude, it might be possible to evaluate phase lags between the astronomical forcing and the geological record The second significant use of amplitude modulation cycles is that they are related to specific dynamical properties of a given astronomical model These properties are related to the chaotic nature of the Solar System, and potentially allow the use of geological data to provide constraints on the dynamical evolution of the Solar System and astronomical models Tidal Dissipation and Dynamical Ellipticity The mean fundamental orbital frequencies gi and si, as well as the precession constant p, are likely to have changed throughout geological time However, whereas the changes in gi and si have probably been small, the precession constant p is likely to have changed significantly Changes are caused by the effects of energy (tidal) dissipation as well as by redistribution of mass (due, e.g., to waxing and waning ice-caps and mantle convection) Changes in p are related to changes of the length of day, which is reflected in geological and palaeontological records In particular, Earth’s tidal response to the gravitational pull from the Sun and the Moon is not instantaneous This means that the tidal bulge that develops on Earth (and on the Moon), is not aligned with the direction of the Moon’s gravitational pull This pull exerts a torque on Earth, which leads to a gradual decrease in its rotational velocity In addition, conservation of angular momentum leads to an increase in the distance from Earth to the Moon over time, and a change in Earth’s rotational velocity leads to a redistribution of mass on Earth (‘dynamical ellipticity’) The dynamical ellipticity of Earth can also be affected by mantle convection and ice loading These processes modify Earth’s precession constant p, the frequency of which has decreased over geological time Because p is contained in the expressions for obliquity and climatic precession (see Tables and 4), the periods of obliquity and climatic precession also change In 1994, Berger and Loutre estimated possible values for changes of astronomical periods, based on astronomical and geological observations Their results are illustrated in Figure The effects of changing tidal dissipation and dynamical ellipticity values have a large impact on astronomical calculations, and have to be obtained from observation Strictly speaking, astronomical calculations cannot be performed independently of the chosen Earth model, and particularly, there cannot be separate treatment of the Earth–Moon system This is why numerical computations are invaluable Chaos in the Solar System Probably the most significant development of astronomical theory in recent times has been the discovery of the chaotic behaviour of the Solar System by Jacques Laskar in 1990 Laskar established that the dynamics of the orbital elements in the Solar System are not fixed for all times, but rather are unpredictable over tens of millions of years This is due to the non-linear gravitational interaction of the different bodies in the Solar System, which makes it theoretically impossible to calculate the exact movements of celestial bodies from their present-day masses, velocities, and positions over long periods of time This feature poses limits on the use of astronomical theory for the purposes of creating astronomically calibrated