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:
l 1 ¼ g 2 À g 5
l 2 ¼ g 4 À g 5
l 3 ¼ g 4 À g 2
l 4 ¼ g 3 À g 5
l 5 ¼ g 3 À g 2
For obliquity, the main periods are 41 and 54 kyr, coming, respectively, from
g 1 ¼ s 3 þ k
g 3 ¼ s 6 þ k
For climatic precession, the main periods are 23.7,
22.4, 18.98, and 19.16 kyr with
a 1 ¼ g 5 þ k
a 2 ¼ g 2 þ k
a 3 ¼ g 4 þ k
a 4 ¼ g 3 þ k
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:
l 1 ¼ a 2 À a 1
l 2 ¼ a 3 À a 1
l 3 ¼ a 3 À a 2
l 4 ¼ a 4 À a 1
l 5 ¼ a 4 À a 2
As a consequence,
l 3 ¼ l 2 À l 1
l 5 ¼ l 4 À l 1
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 Figure 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.
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
ASTRONOMICAL FREQUENCIES IN PALEOCLIMATES
31
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:
l 1 ¼ g 2 À g 5
l 2 ¼ g 4 À g 5
l 3 ¼ g 4 À g 2
l 4 ¼ g 3 À g 5
l 5 ¼ g 3 À g 2
For obliquity, the main periods are 41 and 54 kyr, coming, respectively, from
g 1 ¼ s 3 þ k
g 3 ¼ s 6 þ k
For climatic precession, the main periods are 23.7,
22.4, 18.98, and 19.16 kyr with
a 1 ¼ g 5 þ k
a 2 ¼ g 2 þ k
a 3 ¼ g 4 þ k
a 4 ¼ g 3 þ k
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:
l 1 ¼ a 2 À a 1
l 2 ¼ a 3 À a 1
l 3 ¼ a 3 À a 2
l 4 ¼ a 4 À a 1
l 5 ¼ a 4 À a 2
As a consequence,
l 3 ¼ l 2 À l 1
l 5 ¼ l 4 À l 1
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 Figure 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.
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
ASTRONOMICAL FREQUENCIES IN PALEOCLIMATES
31