the longitude of the Earth in its orbit measured from the
spring equinox at a specific date. Note that u and l fix
the Earth on its orbit, respectively, from the perihelion
and from the spring equinox at a specific date (Figure 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:
r ¼
r
a
¼
1 À e
2
1 þ e cos u
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, e
o , relative to the
moving vernal point, g, is shown in Figure 1. It is equal
to p + c, where the annual general precession in longitude,
c, describes the clockwise absolute motion of g along the
Earth’s orbit relative to the fixed stars, and the longitude of
the perihelion, p, 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:
h ¼ e sin p
k ¼ e cos p
p ¼ sin i sin O
q ¼ sin i cos O
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
h ¼
X
k
M k sin g k t þ b k
ð
Þ
k ¼
X
k
M k cos g k t þ b k
ð
Þ
ð1Þ
p ¼
X
k
N k sin s k t þ d k
ð
Þ
q ¼
X
k
N k cos s k t þ d k
ð
Þ
ð2Þ
For the Berger (1978) solution, the amplitude M k and
N k , the mean rates g k and s k , and the phases b k and d 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-toone relationship between these periods and the individual
planets, it can be shown that their origin is, to some extent,
associated to one particular planet.
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).
Astronomical Frequencies in Paleoclimates, Table 1 Periods
related to the frequencies g and s of the most important terms
in the series expansion of (e, p) and (i, O)
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
28
ASTRONOMICAL FREQUENCIES IN PALEOCLIMATES
spring equinox at a specific date. Note that u and l fix
the Earth on its orbit, respectively, from the perihelion
and from the spring equinox at a specific date (Figure 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:
r ¼
r
a
¼
1 À e
2
1 þ e cos u
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, e
o , relative to the
moving vernal point, g, is shown in Figure 1. It is equal
to p + c, where the annual general precession in longitude,
c, describes the clockwise absolute motion of g along the
Earth’s orbit relative to the fixed stars, and the longitude of
the perihelion, p, 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:
h ¼ e sin p
k ¼ e cos p
p ¼ sin i sin O
q ¼ sin i cos O
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
h ¼
X
k
M k sin g k t þ b k
ð
Þ
k ¼
X
k
M k cos g k t þ b k
ð
Þ
ð1Þ
p ¼
X
k
N k sin s k t þ d k
ð
Þ
q ¼
X
k
N k cos s k t þ d k
ð
Þ
ð2Þ
For the Berger (1978) solution, the amplitude M k and
N k , the mean rates g k and s k , and the phases b k and d 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-toone relationship between these periods and the individual
planets, it can be shown that their origin is, to some extent,
associated to one particular planet.
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).
Astronomical Frequencies in Paleoclimates, Table 1 Periods
related to the frequencies g and s of the most important terms
in the series expansion of (e, p) and (i, O)
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
28
ASTRONOMICAL FREQUENCIES IN PALEOCLIMATES