414 EARTH/Orbital Variation (Including Milankovitch Cycles) results of a frequency analysis, which was run over a 10-My-long interval to better resolve the position of individual peaks The peaks correspond to the frequencies given in Table Obliquity The obliquity (tilt) e of Earth’s axis with respect to the orbital plane (see Figure 1) is defined by the angle between Earth’s spin vector s and that of the orbital plane n, and can be computed as cos e ¼ n s, using unit vectors As the inclination and orientation of the orbital plane vary, the obliquity is not constant, but oscillates due to the interference of the precession frequency p and the orbital elements si As shown in Table 3, if the variation in obliquity is approximated by quasiperiodic terms, the result is a strong oscillation with a period of approximately 41 ky, with additional periods around 54 and 29 ky The $41-ky period arises from the simultaneous variation in Earth’s orbital inclination, given by s3, and the precession of Earth’s spin direction, expressed by p Table also shows that the obliquity signal contains contributions from the gi as well as the si fundamental frequencies, due to their combined effect on the change of the orbital plane normal The present day obliquity of approximately 23.45 has varied between $22.25 and $24.5 during the past million years The main climatic effect of variations in Earth’s obliquity is in control of the seasonal contrast The total annual energy received on Earth is not affected, but the obliquity controls the distribution of heat as a function of latitude, and is strongest at high latitudes It is important to note that all of the obliquity frequency components contain the precession constant p Due to tidal dissipation, the frequency of the precession constant p has been higher in the past, a fact that can be shown from geological observations, such as ancient growth rings in corals, and from tidal Table Six leading terms for Earth’s obliquitya Term Frequency ( 00 year 1) Period (ky) Amplitude p ỵ s3 p ỵ s4 p ỵ s3 ỵ g4 g3 p ỵ s6 p ỵ s3 g4 ỵ g3 p ỵ s1 31.613 32.680 32.183 24.128 31.098 44.861 40.996 39.657 40.270 53.714 41.674 28.889 0.0112 0.0044 0.0030 0.0029 0.0026 0.0015 a Principal obliquity frequency components analysed over the past My The frequency terms gi and si refer to those given in Table Data from Laskar J (1999) The limits of Earth orbital calculations for geological time scale use Philosophical Transactions of the Royal Society of London, Series A, Mathematical, Physical and Engineering Sciences 357(1757): 1735 1759 laminations Figure illustrates the variation in obliquity over 1.2 million years The oscillation is dominated by a $41-ky period cycle, and a variation in amplitude is also observed This variation is due to beats arising from the presence of additional $29- and $54-ky periods, which are just visible in the frequency analysis shown on the right-hand side of Figure Climatic Precession The precession of Earth’s spin axis has a profound effect on Earth’s climate, because it controls the timing of the approach of perihelion (the closest approach to the Sun) with respect to Earth’s seasons At present, perihelion occurs on the January, close to the winter solstice With respect to the stars, the precessional movement of Earth’s spin axis traces out a cone shape with a period of $25.8 ky However, due to the precession of the perihelion within the orbital plane, the period of precession, measured with respect to the Sun and the seasons, is shorter The motion of the perihelion is not steady but is caused by a superposition of the different gi frequencies For this reason, the precession of the equinoxes with respect to the orbital plane lurches with a superposition of several periods around $19, 22, and 24 ky The effect of the precession of the equinoxes on the amount of solar radiation received by Earth also depends on the eccentricity If the eccentricity is zero (i.e., the orbit of Earth follows a circle), the effect of the precession of the equinoxes is also zero From a climatic point of view, the eccentricity and longitude of the perihelion combine to create what is termed the ‘climatic precession’, defined as e sin(o), where ˜ e sin(o) ˜ is the longitude of perihelion from the moving equinox This means that the climatic precession index is modulated in amplitude by variations in Earth’s eccentricity A quasiperiodic approximation of the climatic precession time series reveals the contribution from different frequency components, as shown in Table Figure Earth’s obliquity over time (1.2 million years) and fre quency analysis for a 10 My time span The peaks in the fre quency analysis correspond to the frequencies given in Table 3; the numbers over the peaks represent the periods (in thousands of years)