Quaternary Science Revtews, Vol 10, pp 297-317, 1991
Prmted m Great Britain All rights reserved
0277-3791/91 $0 00 + 50
© 1991 Pergamon Press plc
INSOLATION VALUES FOR THE CLIMATE OF THE LAST 10 MILLION YEARS
A. Berger and M.F Loutre
lnstttut d'Astronomte et de Gdophystque G Lema#re, 2, Chemtn du Cyclotron, BI348 Louvam-la-Neuve, Belgtque
New values for the astronomical parameters of the Earth's orbit and rotation (eccentricity, obhqulty and precession) are
proposed for paleochmatle research related to the Late Miocene, the Phocene and the Q u a t e r n a r y T h e y have been obtained
from a numerical solution of the Lagranglan system of the planetary point masses and from an analytical solution of the Polsson
equations of the E a r t h - M o o n system The analytical expansion developed in this paper allows the direct determination of the
main frequencies with their phase and a m p h t u d e Numerical and analytical comparisons with the former astronomical solution
B E R 7 8 are performed so that the accuracy and the interval of time over which the new solution is valid can be estimated The
corresponding insolation values have also been computed and compared to the former ones This analysis leads to the conclusion
that the new values are expected to be reliable over the last 5 Ma In the time domain and at least over the last 10 M a m the
frequency domain
INTRODUCTION
Research in the astronomical theory of paleochmates
involves four main steps (Berger, 1988)"
1 The theoretical computation of the long-term variations of the Earth's orbital parameters and related
geometrical lnsolatlons
2 The design of climatic models to transfer the
insolation into climate
3 The collection of geological data and their interpretation in terms of chmate
4 The comparison of these proxy data to the simulated
chmatlc variables
This paper will focus only on the first point
The energy available at any given latitude dp on the
Earth, on the assumption of a perfectly transparent
atmosphere and of a constant solar output, is a singlevalued function of the semi-maJor axis, a , of the Earth's
orbit (the ecliptic), its eccentricity, e, its obliquity (the
tilt of the equator on the ecliptic), e and of the
longitude of the perihelion measured from the moving
vernal equinox, 6) (Berger, 1978a,b). The eccentricity
ts a measure of the shape of the Earth's orbit around
the Sun It changes the mean distance from the Earth to
the Sun and therefore the total amount of energy
received by the Earth The geographical and seasonal
pattern of this insolation depends on e and on the
climatic precessional parameter, e sin 6), that describes
how the precession of the equinoxes affects the
seasonal configuration of the Earth-Sun distance
The first calculations of these parameters date back
to the 19th century (Le Verrier, 1855, see Berger, 1988
for a review) Mllankovitch (1941) was however the
first to complete a full astronomical theory of the
Pleistocene ice ages, computing the orbital elements
and the subsequent changes in the insolation and
climate (Imbrle and Imbrle, 1979, Berger and Andjehc,
1988) In the late 1960's judicious use of radioactive
dating and other techniques gradually clarified the
details of the Quaternary time scale, better instrumental methods came on the scene using oxygen
isotope as ice age relics, ecological methods of core
interpretation were perfected, global climates of the
past were reconstructed and climate models became
available (see for example Berger, 1990, for a review of
the significant steps made to improve the astronomical
theory of paleochmates over the last 20 years) With
these improvements in dating and in interpreting the
geological data in terms of paleochmates, it became
necessary to investigate more critically the computation
of the astronomical elements (Berger, 1984) and of
appropriate Insolation parameters (Berger and Pestlaux, 1984) This has allowed us to test first, the astronomical theory in the frequency (Hays et a l , 1976,
Berger, 1977b, Imbrle et a l , 1989, Berger, 1989b) and
in the time domain (Berger et a l , 1990) and, further, to
calibrate the Quaternary (ImbrIe et a l , 1984, Martinson et a l , 1987) and Pllocene (Shackleton et a l , 1990)
time scales
A first tmprovement to the Mllankovitch solution
came in the 1950's from Brouwer and van W o e r k o m
(1950) and later from Sharaf and Boudnlkova (1967)
and A n o h k et al (1969) But a serious step forward was
made with the analytical solution by Bretagnon (1974)
for the planetary point masses and the calculation of e,
e and e sin 6) by Berger (1976, 1977a) which lead to
his 1978 solution (Berger, 1978a,b), referred to here as
BER78 This solution was assumed to provide valuable
information over the last 1 5 Ma in the time domain
and over a much longer period in the frequency domain
(Berger, 1984) The next significant improvement was
related to the numerical Integration made by Laskar a
few years ago (Laskar, 1986, 1988) This calculation
was at the origin of a new astronomical solution
calculated by Berger et al. (1988) and used for extending the validity of the paleochmatic parameters and
insolation over the last 5 to l0 million years (Berger
and Loutre, 1988) It is the purpose of this paper to
297
298
A Berger and M F Loutre
present the final and most accurate version ot this
solutton
In o r d e r to appreciate the t m p r o v e m e n t of the
accuracy m the c o m p u t a t i o n of the astronomical parameters of the E a r t h ' s orbit and rotation, it ts necessary
to introduce some elementary notions of celestial
m e c h a m c s T w o systems will have to be considered
O n e wdl deal with the m o t i o n of nine planetary p o m t
masses a r o u n d the Sun, the o t h e r will constder the
rotatton of the Earth as a result of the luni-solar
attraction
P L A N E T A R Y SYSTEM
Gahlean Frame of Reference
E v e r y two particles in the universe attract each o t h e r
with a force that is dtrectly proportional to the product
of their masses, and reversely p r o p o r t i o n a l to the
square of the dtstance b e t w e e n t h e m , such is the
N e w t o m a n law of gravitational attractton A p p l y m g
this law to the case of N celestial bodies of the solar
system (the Sun and the planets), we are able to express
the equattons of m o t i o n
the equations of motion can also be written as
m?~ :
where V Us denotes the ~ector
0 v~
0 xs
O zj
Hehocentrtc Coordinate System
Instead ot referrmg the posttlon of the N p a m c l e s in
a G a h l e a n frame of reference, they will be referred to
with regard to one of them (the Sun) C o n s e q u e n t l y ,
the radius vector of the planet PI is given by 9---~= %
r I - rs (Ftg 1) F r o m e q u a t i o n (1) we have
iv
G m , rj,
rI =
Gms
#,
,~'
4,
Gm, rs,
y
G m s rsj
+
3
Gm,mjrj--~,
N
r~]
Y~
] = 1.
l~=l
N
(1)
F3
Subtracting them, we obtain
II
where r~ d e n o t e s the radius vector of the particle P~,
with regard to the fixed origin, rj,~s the._~distance vector
between P~ and P, and is equal to r, - r . the distance r~,
is equal to r,s, m, ts the mass of P,, G ~s the Gausstan
gravitational constant derived f r o m Kepler's third law
T h e conservation ot linear m o m e n t u m , whtch can be
o b t a m e d by a d d m g all the N equations (1) together,
tells us that the centre of mass of the N particles moves
uniformly m a straight line I n d e e d , we have
Z
!
rss
+
r5
mjrs =
(3)
m r (VU~)
m~r~ = 0
1
and tt we defme the position, t~, of thts centre ot mass
by
_____)
N
G (m~ + ms) 9~
pj = -
+
Gm, rj,
E
£I
'~
Z
( ; m , {2,
.........
(4)
l
In e q u a t i o n (4), the hrst term ot the right hand side
represents the actton ol the Sun on PI. the second
represents the action ot the o t h e r planets on P / a n d the
thtrd one can be considered as a perturbatton due to the
choice of the reference trame, as it represents the
action of the planets (except P~) on the Sun (1 c the
new origin of the coordinate system)
mtr~
I = I
._.._)
e~
we obtain t, = 0
T h e r e f o r e , assuming that the centre ot mass ot the
system is taken as the origin it does not change the
equations of m o t i o n (1) D e f m m g a force luncUon U~
for each particle P~
N
u,
:
,//
g
r~
~
(.~l?l t
5".
t =
~1
P"
t2)
1
lq
s
~s
o
FIG l Radm~ vector ol the bull and ol the planets with respect to
the centre of mass IO) ol the planetary ~vstcm
Insolation Values for the Last 10 Ma
A s in t h e G a h l e a n f r a m e
function can be defined:
of reference,
299
a force
echptlc
G ( m s + mj)
V1 =
+ Rj
(5)
o/
w i t h Rj, t h e d i s t u r b i n g f u n c t i o n g i v e n b y
R, =
~
-77}7
-
Gm,
1 )
-
+
,,,
plane
(6)
"70
Accordingly, the equations of motion take the same
f o r m as (3)
---,
ouj
mJoj=mJ
U)
ooj
T h i s e q u a t i o n m a y also b e w r i t t e n in t h e x - c o o r d i n a t e .
d2x,
-
dt 2
8U,
- 1 ~< I ~< 9 (8 if P l u t o is e x c l u d e d )
d xj
(8)
w i t h s i m i l a r e q u a t i o n s in y a n d z
Keplenan Elements
A t r a n s f o r m a t i o n f r o m t h e c o o r d i n a t e s (x 1, yj, zj) a n d
v e l o c i t y c o m p o n e n t s (xj, ~/, zj) i n t o to t h e 6 osculanng
e l e m e n t s ( s e m i - m a J o r axis, a; e c c e n t r i c i t y , e; i n c l i n a tion t of the orbit on the reference plane; longitude of
t h e a s c e n d i n g n o d e , if2; l o n g i t u d e o f t h e p e r l h e h o n , ~t,
a n d m e a n l o n g i t u d e , ~,, r e c k o n e d f r o m t h e o r i g i n o f
t i m e , Fig. 2) gives rise to t h e L a g r a n g e e q u a t i o n s (6
t i m e s t h e n u m b e r of p l a n e t s ) r e l a t i n g all t h e o r b i t a l
elements of the planets together and describing their
motion around the Sun (Brouwer and Clemence,
1961)
da
2
FIG 2 Position of the Earth (E) around the Sun (S) In astronomy,
it Is usual to define an orbit and the posltkon of the body describing
that orbit by six quantities called the elements Three of them define
the orientation of the orbit with respect to a set of axes, two of them
define the size and the shape of the orbit (a and e respectively), and
the sixth ( w i t h t i m e ) defines the p o s i t i o n of the body within the orbit
at that time In the case of a planet moving In an elliptic orbit about
the Sun, it is convenient to take a set of rectangular axes in and
perpendicular to the plane of reference, with the origin at the centre
of the Sun The x-axis may be taken towards the ascending node N,
the y-axis being In the plane of reference and 90° from x, while the zaxis IS taken to be perpendicular to this reference plane so that the
three axes form a right-handed coordinate system Yo the reference
point from where the angles are measured As the reference plane is
usually chosen to be the echptlc at a particular fixed date of reference
(named epoch of reference in celestial mechanics, Woolard and
Clemence, 1966), 7o is, m such a case, the vernal equinox at that
fixed date (the vernal equinox is also referred to as the First Point of
Aries indicating the position of the Sun when It crosses the celestml
equator from the austral to the boreal hemisphere) P is the
perihelion, g2 the longitude of the ascending node, to, the argument
of the penhehon, Jt = f2 + to the longitude of the perihelion, t the
lnchnatlon, v the true anomaly, k = ~t + v the longitude of the
Earth in its orbit
dzt
OR
(1 - - e 2 ) 1/2
-
dt
na
0
de
OR
ena 2
ena 2
dt
(1 -
e2)I/2
O 3x
dt
[1--(1-e2)
I/2]
ena 2
O
dt
-
n a 2 (1 -
OR
1
e2) 1/2 sin i
i9f2
tan ~-
na 2 (1 dQ
dt
na 2 (1
-
+
e2) 1/2
1
OR
e2) 1/2 sin t
d t
_ _
2
OR
na
8a
na 2 (1
dR
ena 2
[1
-
l
-
Ol
e 2 ) 1/2
+
n
(1 -- e2) 1/2
OR
(9)
d!
de
d~,
-
tan 5 +
dt
(1 - - e 2 ) 1/2
!
8R
(1
-
e2)
1/
- -
+
8e
t a n -7-
OR
na 2 (1 -- e2) 1/2
8t
However, these equations possess some inconvenient
f e a t u r e s for o r b i t s w i t h s m a l l e c c e n t r i c i t i e s a n d / o r s m a l l
lnchnatlons" the appearance of the eccentricity and of
sin t in t h e d e n o m i n a t o r o f t h e e x p r e s s i o n s for & t / d t
a n d d • / d t l e a d s to s e r i o u s p r o b l e m s r e l a t e d to s m a l l
d e n o m i n a t o r s w h e n e a n d t a p p r o a c h z e r o . A s all
p l a n e t a r y o r b i t s he a l m o s t e x a c t l y in t h e s a m e p l a n e
3IX)
A
B e r g e r and M F
l.outrc
and differ only shghtly from circles, it Is thus desirable
to use a modified form of these equations by settmg
=
=
=
=
11
k
p
q
e sin
e cos
sin t
sin l
Jl
Jt
sin if2
COS
2
OR
dt
na
Ok
~R
- -
2na 2 (1 -
eel u2
qh
8R
2na 2 (1 -
dX
2
OR
dt
na
Oa
(I
-
Ok
e 2 ) I/2
e'l ~-
(It
.....
dh
dt
(1 -
e2) I'-~ OR
na 2
h (1 -
kp
e 2 ) 1/2
dk
e 2) t/2
-
dt
k
(1
-
aq
8 R
na ~
ah
e2) 1/2
rta 2 [1 + (1 -
OR
e 2 ) I/2]
hp
2ha 2 (1 -
c3p
hq
2na 2 (1 -
OR
e 2 ) 1/2
@
dt
c3X
OR
e 2 ) I/2
e 2) v2
m~
(I I
~ ~
[I t
I
~
dR
2na 2 (1 -
e 2 ) 1/2
pl~
e 2 ) 1/2
e 2 ) 1/2
+
Oh
(ll)
4na 2 (1 -
e 2 ) 1/2
q
OR
2ha 2 (1 -- eel 1/2
Ok
t
m I
1 +
m,
- ,/,-
,,,, t ¢
+
+ ¢
,
¢,
tit
-
dt
~1
ttt
Ok
dq
I1t
I
where n / represents the m e a n moUon ot P, ( 2 x / l , . 7 I
being the planet's period of revolution), mj ~s the mass
t
of P/ relative to the mas,~ ot the Sun (m~ = l?l
.57)
and
~R
-
~1
f
. . . . .
1 +
OR
ph
2na 2 (1
I-t = (;nls
O ~.
- -
2na 2 (1 -
~1
with /x gwen by the third law of K e p l e r
8q
II-
p
-I+
Ok
OR
OR
( p - - + q - - )
t,_' ) l :
,9 p
,9 q
m~
Oq
OR
4na z (1 -
OR
In order to obtain the long periodic terms, only the
long period part of the disturbing f u n c n o n R is retained
( B r e t a g n o n , 1974, 1984, Berger, 1984) and e x p a n d e d in
powers of the two following small p a r a m e t e r b
1 T h e r a n o of the masses of the planets to that o!
the Sun
2 T h e i r e c c e n t n c m e s (el and m c h n a t l o n s (t) on the
reference plane
This system (11) might be solved numerically or
analyUcally But in any case. due to the complexity o1
R . th~s d~sturbmg t u n c n o n will have to be truncated
For e x a m p l e , following M d a n k o w t c h (1941). B r o u w e r
and van W o e r k o m (1950). D z l o b e k (1963) and Bretagnon (1974). R can be e x p a n d e d to the second degree ol
the variables h. k . p. q In that case for a planet P~
disturbed by the seven others (Pluto is excluded). R
takes the following torm
dR
eel I/2
(1
I
e:)'-']
OR
--+k
0 It
+
Op
kq
. n a ~ (1
n,c~ [I + (1
OR
2ha 2 (l -
OR
2ha "~ (1 -
e:) 1''-
na 2[1 + (1 - e~) u21 8X
Ok
+
Oh
(10)
So that the equaUons (9) b e c o m e
da
qk
4A,, (p
+ q-t + P-,
~- qT) +
aR
B. (kfl, + tbh ,) + 8AI, (%q, + lbP,)
Op
where A j,, BI, anti (), are functions ot %, -
(1~')
II
In such a case an analytical solution for (11) can be
found With (13). (11) indeed b e c o m e s
301
Insolatmn Values for the Last 10 Ma
Oh,
EARTH-MOON SYSTEM
O, t] (2A,,kj + Bj,k,)
dt
ok,
dt
-
O, t] (2A,,hj + Bj,h,)
Z
14: I
d•l
dt
dhj
After having computed the motion of the planetary
point masses around the Sun, the method by Sharaf and
Boudnlkova (1967) for the E a r t h - M o o n system can be
used to obtain an analytical expansion for the long-term
variations of the other two variables involved in the
astronomical theory of paleocllmates (Fig. 3): the
precession in longitude (V) involved in the calculation
of 6) and the obliquity of the ecliptic or tilt (e), i.e. the
inclination of the ecliptic of a particular date on the
equator of that date. The PoIsson equations for the
E a r t h - M o o n system provide the long-term variations
of the lunl-solar precession in longitude Vf and of the
inclination e/- of the equator on the mean ecliptic of
epoch (Woolard and Clemence, 1966, Lleske et al.,
1977). These equations expanded to the second degree
In eccentricity and inclination can be written as
14: I
-
O, t] (2Al,qj - Aj,q,)
Z
t~= I
- +
[/, t] (2Al,pj - Aj,p,)
dt
where
nl(lpml
[/, l] -
for t > 1
l+mj
nlm~
[/, t] -
dt
l+mj
In the Lagrange method, the second degree terms of
the quantities h, k, p, q of the so-called long period part
of the disturbing function are retained. The resolution
of the system of the differential equations thus obtained
gives the Lagrange solution:
m
h =
~
P cos ef ~ ,
for t < 1.
N, sIn(s,t + 6, + apf)
l
1
P sin ef
2
y.
t
N 2 sm 2(s,t + 6, + V])
(16)
_
/5 sin rs
y,
t
y~
] > t
N,N/ sin[(s, + sj)t + b, + 6j + 2apf]
M, sln(g,t + [3~)
/, = 1
(14)
m
k =
y~
k=
Mx cos(gkt + ~k)
1
E¢
I
p =
N
s
Eco
Z
Nk Sln(skt + 6~)
k= 1
(15)
q =
Nk cos(skt + iSk)
k=
1
where m = n -- 8
Long period terms of higher degree can then be
introduced in the Lagrange equations Their solution
takes the same form as (14) and (15), with more terms
(m > 8 and n > 8). In that way, solutions are accurate
to the first or to the second power with respect to their
masses and to the first or to the third degree with
respect to the planetary e's and t's, respectively if terms
of power equal to or higher than 2 or 3 of their masses
and terms of degree equal to or higher than 3 or 5 In e's
and t's are neglected in the disturbing function (12 and
13) used to write the Lagrange equations
Eq
FIG 3 Precession and obliquity y the vernal equinox of date, Yo
the vernal equinox of reference, 7,,~,~ = Vt the luni-solar precession in longitude, V the general precession m longitude which
provides the longitude of the moving perihelion dJ through d~ = ~
+ V, e! the inchnation of the equator of date on the echptlc of
reference and e the obliquity (Berger, 1984), as we are interested by
the long-term variations of the astronomical elements, their shortterm variations are removed and Y and Yo are more adequately
referred to as mean vernal equinox
302
A Berger and M F Loutre
+ ~
dvJ
dt
_
1
p cose t
3
~.
15
3
2
Y~
1
where h and ~ are constants o1 m t e g r a n o n , and k and
!~ are gwen by
~ . M,M, c o s ( 1 3 , - 13,
t
b"M,~ ,M, sml(g, - g,)t + 15, - 1111 (19)
N2 -
l>t
k : I' cos h
(20)
Po
+ 3 15 cosef
+
15
,- iM, M, cosl(g,- g,)t + 13,-[~,]
P cosE t (cote r -
tanef)
I~ = 15 cosh
-
y~ N, cos(~,t + ~Pt)
N~
+
7
a~ -
5
1
--
a7(a,-
(t~
l)tan 2
2
1
_ __ /5 cos~ t
2
-3P
cose t
1
'
--
/ > I
N, Nt cos[(s, + ~,)t + 6, + 6, + 2~Pd
Z
1
N 2 cos2(s,t + ~, + q~t)
f'°
P
3
"]
(21)
y~
,I M,M, cos(13, - 13,
'1
y~ N,N~ cosl(~s, - ,fit + O, - 011
~]
- P c°sct Z
~]
I
l > t
(17)
where P ts the so-called precesstonal constant of
N e w c o m b (/6 = 54 "9066) and P,, = 17 "3919
Clearly (16) is d e p e n d e n t upon the solunon of the
system (15) m p and q. w h e r e a s (17) d e p e n d s upon both
(14) and (15) and t h e r e f o r e revolves the e l e m e n t s ot the
expansion of (h, k) and (p, q) T h e s e equations (16)
and (17) are solved o r d e r by o r d e r with respect to the
small p a r a m e t e r s (eccentricity and m c h n a n o n ) which
leads to
T h e a, u'. h and b"s ,ire c o m p h c a t e d / u n c t i o n s o1 h, k.
s. g, P,, and P
T h e value o1 ~i' and ~ arc related to qJl and ~I
through the spherical triangle f~. y~, 7 of H g 3
where f2 is the ascending node of the e c h p n c of date on
the e c h p n c of epoch. '~,~ ~s the d l r e c n o n of the mtersecnon of the e q u a t o r ot date w~th the ecllpnc of epoch.
and y the m e a n equinox of date (Berger. 1978a) So.
q, and c can be d e t e r m i n e d by
!
2
y
1(,,
+
k),
+
+
-I
I
h -
t/=
~ a,N, cos [(s, + k)t + 6, + ct l
- y~ y . (,,N,N, cos[( % ~ ,, + 2k)t + ~, + 0, + 2 . ]
I
- y~ a,,N~ cos 2 [(s, + k)t + 0, + cq
I
I "; 1
-
y.
t
I
-
y
~
t
a,jN,Njcos[(s, + s, + 2k)t + 6, + ~, + 2tt]
/
t
+ ~_~ G,N,
~
t
C;&,N, cos l(s, - b) t + 6, - ~,] (22)
q) = I~t + (J
1>1
- y
y.
a I) N,N, cos[(~, -
~,)t + 6, -
6,1
(18)
+ Z
y ~ b,N, sln[(6,
G,,N- sm 2 [(s, + k)t + ~), + ,t]
l
+ k)t + i3, + otl
?
+ Z
+ y.. h,,N~ sm 2[(~, + k)t + 6, + ~tl
Z
t
(;qNtN,
+ Z
Z
t
1>¢
+ y~ bqN, Ntsln[(s, + s, + k)t + b, b, + 201]
+ y~
t
+ y~ b,IN,
N , sm[(s, - sl)t + ~, - b,]
t,]
sln[(s,
+
3, +
2k)t + a,
+
-
+
i~, +
2-1
j > r
l
l,]
.]
I
l "> !
~p! = (~t + ct +
sm [(s, + kit + 3, +
t
where
t!
-
0,1
y~ GIIM,M , s,n [(g, - grit + 13, - 13J (23)
/>1
Insolation Values for the Last 10 Ma
e* = h -
~
N2t I 1
a,
(a,-
1) t a n h
303
e0 = 23°4458
ap0 = 1°3960
+
dV
- -
l
1
(2a,
4
1) cot h I
.!
(24)
The C, C', G and G " s are complicated functions of the
a, a', b, b"s and h
The general precession m longitude and the obliquity
can, therefore, be written m the same expansion form
as the orbital elements given by (14) and (15)
e = e* +
~
dt
At cos(v,/ + ~,)
(25)
V = kt + a + ~ , S, sln(~,t + o,)
(26)
l
I
A,, ~,,, ~,, S,, ~, o~ are given by identification of (25)
to (22) and (26) to (23). The constants h, k, !~, e* and
a are computed by solving the set of equations (22)
and (23) corresponding to the initial conditions and by
using the relaUons between them: (20), (21) and (24)
(see the next section for illustration)
LONG-TERM VARIATIONS OF THE EARTH'S
ORBITAL ELEMENTS, PRECESSION AND
OBLIQUITY
The 1978 Solutton
The method stated above has been used by Berger
(1977b, 1978a,b) to give the analytical expressions of
and calculate the astro-lnsolation parameters currently
used for paleochmate reconstruction In these papers,
the orbital elements h, k, p, q have been computed
from Bretagnon's (1974) analytical solution which takes
into account all the long period terms of the disturbing
function up to the fourth degree in e's and t's and the
short period terms (secular part) of the disturbing
function that give, to the second power of the masses,
important long period terms in the solution For this
contribution of the short period terms, all the terms to
the third degree in e's and t's which lead to a significant
modification of the frequencies have been kept This
leaves Bretagnon with solution (14) for (h, k) which has
24 terms and (15) for (p, q) which contains 17 terms
All these terms have been ordered and regrouped by
Berger so that the final expressions (14) and (15)
contain only 19 and 15 terms respectively The amplitude, frequency and phase of all these terms are given
in Table 1 of Berger (1976) The development of e and
V have been obtained by using the method developed
by Sharaf and Boudnlkova (1967) which allows the
terms to the second degree in the eccentricity of the
Earth's orbit to be included To reach a sufficient
accuracy, Berger (1978a) has kept 240 and 411 terms
respectively m (25) and (26) The constants of integration k, h and ct were deduced from the initial
conditions for 1950 0, the reference plane being
of 1850.0
(27)
= 50 "2686
o
In addition, for this solution, k m (18) and (19) was
given by the same expression as for !~, it means that k
was not gwen by (20) but rather by (21), according to
the expansion by Sharaf and Boudnikova (1967) who
used a mixed procedure of integration step by step (see
Berger et al., 1988 for more detads)
This computation leads to the following values
e*
h
ct
k (=~)
=
=
=
=
23°320 556
23°394 901
3°392 506
50 "439 273
(28)
and the amplitudes, phases, and frequencies of (14),
(15), (26) and (25) were also published in Berger
(1978b) with a computer programme available in
Berger (1978a).
This solution, labelled here as BER78, was compared with many previous ones (Berger, 1977a). Moreover, a sensitivity analysis to the number of terms kept
in the disturbing function and to the order of masses
lead to the conclusion that this solution would be quite
accurate over the last 1 5 Ma in the time domain and
the frequencies acceptable for a much longer period
(Berger, 1984). The insolatlons have the same accuracy
(Berger and Pestlaux, 1984) because the formula used
to compute them was without approximation (Berger,
1978a,b), they were used to validate the astronomical
theory (e g. Kutzbach, 1985, Kutzbach and Guetter,
1986; Prell and Kutzbach, 1987, Saltzman et al., 1984;
Berger et al., 1990) and/or calibrate the geological data
(Imbrle et al, 1984, 1989, Martlnson et al., 1987)
The 1990 Solutzon
This former astro-lnsolatlon solution (BER78) will
be compared in this paper to a new solution (BER90)
built up from the Laskar's numerical solution for (e, ~x,
l, if2) (Laskar, 1986 and 1988) and according to the
procedure developed m a previous section of this paper
for e and e sin 6)
The main characteristics of BER90 can be summarlzed as follows, the Lagrange equations giving the
secular evolution of the planets are expanded up to the
second order of the masses and the fifth degree In e's
and fs, including lunar and relativistic contributions
This secular system is Integrated over 30 mdllon years
( - 1 0 to +20 million years) A modified Fourier
analysis is then performed to fit a quasi-periodic
function to the numerical values obtained The elements of the Earth's orbit are referred to the mean
ecliptic and the mean equinox of 2000 0 To allow a
comparison with the solution BER78, the time origin
adopted has been moved to 1950 0 The constants of
integration as well as the values of the disturbing
planetary masses (Table 1) used in the Laskar's
304
A
Berger and M F
TABLE 1 Values ot the planeta D masses used b~¢ Brctagnon m 1974
and m 1982 (this i~ the ratm of the solar mass to the mass of each
planet)
Plam.ts
Bretagnon
{19741
Loutrc
U s i n g t h e initial c o n d m o n s t o r [9511 0 r e t e r r e d to t h e
r e t e r e n c e p l a n e o f 2000 0
~,,
q~.
Bretagnon
(1982)
we d e d u c e d
Mercury
Venus
Earth + Moon
Mars
Jupiter
~,aturn
Llranu~
Nt_ptunc
6,000,000
408,500
328,900
~ 099,01R1
1.1)47 35~
z; 498
22.869
It; 314
6 023 600
408 523
328,900
~ 0t)g.71(l
I 047 355
~, 498 500
22 860
19 314
computahon
arc t h o s e u s e d by B r e t a g m m tot hi',
V S O P 8 2 s o l u t i o n ( B r e t a g n o n , 19825 T h e V S O P g 2
s o l u t i o n , r e p r e s e n t i n g the s e c u l a r varldtlOll ~, o | t h e
p l a n e t a r y o r b i t s , t,, m a d e ot t h e p e r t u r b a t i o n s d e v e l o p e d u p to t h e t h i r d p o w e r o f t h e m a s s e s l o t all t h e
p l a n e t s a n d u p to t h e sixth p o w e r for t h e f o u r o u t e r
p l a n e t s It also c o n t a n l ~ t h e p e r t u r b a t i o n s ot t h e M o o n
onto the Earth-Moon system
A c c o r d i n g to t h e s e n s l t w l t y a n a l y s i s m a d e b 3, B c r g c r
(1977a5, thls i m p r o v e m e n t m t h e p h m e t a r y m a s s e s
r c l a t w e to t h o s e u s e d m t n e s o l u t u m B E R 7 8 is not
e x p e c t e d to s l g m t t c a n t l y c h a n g e t h e n u m e r i c a l ~,dues ol
the s o l u t i o n It is t h e a c c u r a c y with w h m h the p e r t u r b m g l u n c t l o n is k n o w n w h i c h m o s t h i n f l u e n c e s t h e
accul,tcv ol t h e s o l u t i o n ot t h e p l a n e t , u y p o i n t m a s s e s
',vMern
T h e m e t h o d b} S h a r a l a n d B o u d n l k m a (19671 (xcc
p r e v i o u s s e c t i o n ) has t h e n b e e n u s e d to o b t a i n all
e x p a n s i o n l o r t h e o h h q u t t } (22) a n d t h e p r e e e ~ s u m (23)
c o r r e s p o n d i n g to t h e o r b i t a l e l e m e n t s g w e n by L a s k a r
M o r e o v e r , a c c o r d i n g to (205 a n d 1215, the c o n s t a n t ~ k
and I~ a r e n o w c a l c u l a t e d t h r o u g h d i f f e r e n t e x p r e s s u m ~
l e a d i n g to e x p a n s t o n s (225 a n d (23) w h i c h a l e stnctl~ to
the s e c o n d d e g r e e ~ t t h r e s p e c t to t h e E a i t h ' s e c c e n t ~ v
t l h In such a c a s e . h a n d ~t can bc c o n s ~ d m e d ,l~ t h c
onl 3, t w o constant,, o f i n t e g r a t i o n , k a n d I~ b e i n g
~ o m p u t e d b~/ (20) a n d 121), w h i l e the m~t~al c o n d i t i o n
~ 23'~4458
_-: 0°6982 4
t h e t o l l o w m g c o n s t a n t s ot n l t c g r a t ~ o n
h
u
= 23 399 935
=- 1 600 753
F r o m t h e m , we c o m p u t e d
k
I~
=
-
50 "390 811
5(I "417 262
~ 333 410
Brt_tagnon-Bt.rger
Laskar-Berget
Label
BER78
BER90
131)
,h i,
. i e
T h e initial v a l u e ol
t h e d e r l v a t w e ol (26)
tit
computed
at t = 0
dq,
5O "273 147
dt
, ,~
can be c o m p a r e d
to t h e initial v a l u e g i v e n n3 (271
dq,
5tj "2681~
dt
gl~ m g t h e a c c m ac 3 o t t h e t . o m p u t a t l o n a t . o m p l e t e out
ot p h a s e ot tile c h n l a t l c pret.ess~on will o c c u r onl 3 m a
t i m e s p a n ol m o r e t h a n 1411 M a
PALEOCLIMATIC
SOLUTION
K n o w i n g the e x p a n . ~ , m ol tile aMrononllt_al e l e m e n t s
c, l. zr. ~2 (14 a n d 15). ~ (25) a n d q~ (26) ( T a b l e s 3 4,
5 . 7 ) . It is p o s s i b l e to ~ a l e u l a t e t h e e x p a n s i o n ol all the
astlo-nlso]atlon palameter', A', t h e e c c e n t r i c 1 D c a n b e
c o m p u t e d t h r o u g h e- - h: 4- /, 2 t h e e x p a n s i o n t o r e 2 Is
g l v e | l by
e 2 = ~., M 2, + ~
~
w n h Z, =
2M,M, cos(7.
I
g,t + Jl,
[ ABLE 2 Chalactt_rNms ol the 2 dlllt.renl soluHons lol the. oiNtal cKm~_nt,, ~1~.
obhqmtv and the pret.es'qon
Author
(30)
the v a l u e o l k, I~ a n d
I
for dq~ ~s u s e d to test tile a c c m ac~, o l t h e ~ . o m p u t a t u m
dt
(29)
~\c~-urat \
1,?,~.k t t. nt_t_
cl~.glct,
(e, I)
otdLr
masses
~ q, ~
l podl
[ thpll~
~
q'
2
2
9'
2
lt}q()
lt~qO
18511
2l)(}l)
' Order ot the expansion ol the equations (tht. ord~.r ol th,. solution might be high,.1 )
~" Berger has computed t ~, [rom the (e. ~) and 0 if2) system ol P,retagmm
' Berger has computed t tp lrom tht. (e ~t) and (t. (,2) system ol L,isk,u
a Degree of the expansum of (t. "q~)with rtspeet to the Earth's eceentnclt,, act.ordulg
to the formulae by Sharal and Boudmko,,a (19671
X,}
305
Insolation Values for the Last 10 Ma
T A B L E 3 A m p h t u d e s , m e a n rates, phases and periods of the 5 largest a m p h t u d e m the trigonometrical
expansion of the (e, Jr) system (equations 14)
Mean Rate ("/year)
Amphtudes
BER78
1
2
3
4
5
0 018608
0 016275
0 013007
0009888
0003367
BER90
0
0
0
0
0
018970
016318
012989
008136
003870
BER78
BER90
4
7
17
17
16
4
7
17
17
5
20721
34609
85726
22055
84673
Phase (°)
BER78
24898
45549
92249
37774
56847
28
193
128
320
99
Period (years)
BER90
6
8
3
2
3
30
199
151
309
77
6
7
7
8
0
BER78
BER90
308043
176420
72576
75259
76929
305014
173831
72311
74578
232738
The labels B E R 7 8 and B E R 9 0 correspond respectwely to Bretagnon (1974) and Laskar (1988) from
which the developments of e, n, l, f2 originate However, the n u m b e r s gwen here are not those
published by these authors because both soluhons have been assigned to the same standard
astronomical epoch of reference (origin of time is 1950 0) The sign of the a m p h t u d e ol terms 3 and 5 m
column B E R 7 8 has been changed relatively to the values given in Berger (1978a, b) m agreement with
the change of the phase by 180 ° This has been done to allow an easier comparison between the
solutions
T A B L E 4 Amplitudes, m e a n rates, phases and periods of the 5 largest a m p h t u d e terms m the trigonometrical
expansion of the (l. 2 ) system (equations 15)
Mean Rate ("/year)
Amphtudes
1
2
3
4
5
Phase (o)
BER78
BER90
BER78
BER90
0 027672
0 020040
0 012076
0 007609
0005083
0 027538
0 015973
0 010306
0008047
0 005695
0 0
- 1 8 82930
- 5 61094
- 17 81877
- 6 77103
00
- 1 8 85013
-560436
- 17 76134
- 7 05274
BER78
106
248
12
277
305
Period (years)
BER90
2
5
0
4
0
107
245
17
287
143
BER78
6
5
2
2
8
BER90
-68829
230977
72732
191404
-68752
231248
72967
183758
The labels B E R 7 8 and B E R 9 0 correspond respectively to Bretagnon (1974) and Laskar 11988) from which the
developments of e, ;t, t, Q originate However, the n u m b e r s gwen here are not those published by these
authors because both solutions have been assigned to the same standard astronomical epoch of reference
( o n g m of time is 1950 0)
T A B L E 5 A m p h t u d e s , m e a n rates, phases and periods of the 5 largest amplitude terms m the
trigonometrical expansion of the general precession (26)
A m p h t u d e s (")
1
2
3
4
5
Mean Rate ("/year)
BER78
BER90
7391
2555
2022
1973
1240
5911
3597
2865
2691
2217
02
15
76
65
23
BER78
4
2
5
7
7
31
32
24
0
31
60997
62050
17220
63672
98378
Phase (o)
BER90
BER78
31 54068
004374
004782
32 62947
0 09238
251
280
128
168
292
Period (years)
BER90
9
8
3
1
7
247
230
45
288
352
2
4
3
8
0
BER78
BER90
41000
39730
53615
2035441
40521
41090
29630307
27101064
39719
14029011
T h e sign of the a m p h t u d e of term 4 in column B E R 7 8 has been changed relatively to the values gwen m
Berger (1978a, b) m a g r e e m e n t with the change of the phase by 180° This has been done to allow an
easier comparison between the solutions
The expansion for e can then be o b t a m e d , defining
m 2 and a, through.
e = m [1-0.25
M,
m2=
Z
we finally obtain (Berger, 1978a)"
k
M2I' a , m
+
~
t
e~ c o s y k -
0.25 ~
k
and
y~
y . b2
~
1>
a,a1 cos (7,, - )~j) =
t
Y~
k
bk cos Yk
--0.5
b 2 cos2yk
k
~
k
y,
l>k
bkbt
cos(vk
+
Yt)
306
A Berger and M F Loutre
-05
y
y
h
/~/,
+ 0 125 ~
/,
+0375
~
Z
+0375
Z
e=e'
l>h
h
Z
b~b, cos(2ya - 7:)
+ 0 375 y .
X
b,b~
X
X
X
/,
/ >
/~ m
>
cos(27: -
t = ~' -~ y ~ A ,
l
7: + 7,,,) + cos (7a -
~': -
7,,,))]
(32)
with
e~ = b~ + 0 3 7 5 b 3 + I) 75 b~
y~
b7
14h
O n the other hand. the longitude
m e a s u r e d t r o m the equinox of date
+ tp The precessmn (tp). given by
written m short as ~i' = I~t + .
represents the periodic part ot
precession then b e c o m e s
ol the perlhehon
is given by 6) = Jr
(23). can also be
+ 6~, where ?,~,
~, The cllmatm
= e sm (at + (lit + t~) + 6al,)
= e sin (~ + lit + ~l) COS ~q,
+ e cos (at + lit + (~) sm ~xl,
With e q u a t m n s (14) and (23). and h m l t m g the expansion to the second o r d e r m M, and N, the chmatlc
precession can be written as
M,sml(g, + li)t + 13, + "1
/
t
- - G,N,M:sln
y.
l
I
[(s, + g, + k + li)t + ,b, + 1;, + 2.1
+ y~
y~
I
t
[(3,
-
,~,',
+
q,)
(34)
1
-- G,N,M, sin
c o s ( y / + C.,)
e . E,. k,. q),. P , . . ,
Ya/
+ cos (Ta + 7: - Y,,,) + cos (7a
+ y~
P, s i n ( s t
1
l>
1>/,
e sin 6, = y
/:, ~oqT/ + dO,)
c sin 6) = ~ ]
b,b~ cos(2y: + y,)
babtb,,, (cos (Ya + Y/ + 7,,,)
e sin tb
+ y,
I
/,
+ l ) 75
She classical astro-msolatton elements (e, e sm ~,)
and ~ - - already gwen m (25)) can therefore be
written In the same expansion lorm as for h, k, p and q
given by (14) and (15)
b~b, cos(2y~ + y,)
Z
h
v,)
b h cos 37a
~
/,
+0375
b,b, cos(v,
ii, arc gwen by ldentlhcatlon ot
(34) to (32) and (33)
The amplitudes, trequencles, phases ot (14). (15) and
(34) are gwen in Bergm (1978a. b) for B E R 7 8 . as [or
B E R 9 0 , the most important terms of (34) are given in
Tables 6 . 7 and 8 of this paper The m a m advantages of
such d e v e l o p m e n t s , m addition to providing the numerical values of the elements, are that they allow an
m t e r c o m p a r l s o n with the previous solution(s) (see next
section) and also give directly the most important
trequencms ot these lundamental parameters
It is important to stress that the hmJted n u m b e r ol
terms given in Tables 3 to 8 does not allow an accurate
c o m p u t a t i o n of the respective elements, m a n y more
terms have been used tor the c o m p u t a t i o n of (34) Let
us r e m e m b e r that the values related to e. Jr. t. £2 are
associated with an analytical expression used by Laskar
to ht the numerical values he obtained from a
numerical lntegratum ol the Lagrange equations ! o
obtain a g o o d fit, 80 terms (the t~rst ot which ,ire given
m Tables 3 and 4) had lo be kept m the trlgonometrlc,iI
expressions (14) and (15) which normally lead to a large
n u m b e r of terms m the analytical expressions (34) for
lespectlvely the eccentricity, the chmatlc precession
p a r a m e t e r and the obilqmty (Berger and Loutre 1990)
R e - a s s e m b h n g all these terms m such ,l wa~, thal all
the trequencles would be different from term to lerm.
ordering them to immediately have the most ~mpottant
ones and analysing the accuracy ot the numerical v,tlues
o b t a i n e d from (34) ac~.oldmg to the n u m b e r ol tcrms
kept m each ol the cxp.lnsu)ns lead to the lollowlng
conclusions
I-or the o b h q u l t y troln the 6480 terms ol (22). 6320
ha~ e dfllercnt arguments and 704 have ,in a m p h t u d e
largel lhan ()"1, le,ldlng to devmtlons generall~ less
than 0 °01)05. 89 w~th an a m p h t u d e larger than "v'
a h e a d 5 lead to devlallons generally less than 0 '(11
H)I tile dunatlC ple~.esslon a m o n g the 12880 telnls.
1522 h,lve an a m p h t u d e larger than 10 ". leading to
de~ mtk)n', less than 1 "'~ lor m less than 0 ()()()5 lot c
and less than 7 e 10 ~ 1o~ tile chmat~c precessum
With the 92 tern> ha~uag an amplitude larger than
1() a the pleclslOn ~s not ',cry much lower It reaches
Ill ' tor the chtnatl~ prc~.ess|on and 2 ~' tor ~;~
Insolation Values for the Last 10 Ma
307
.
TABLE 6 Amphtudes, mean rates, phases and periods of the 5 largest amphtude terms m the
trigonometrical expansion of chmauc precession (second equation m 34)
Mean Rate ("/year)
Amphtudes
1
2
3
4
5
BER78
BER90
BER78
BER90
0 018608
0 016275
0 013007
0009888
0003367
0 018970
0 016318
0 012989
0008136
0 003870
5464648
57 78537
68 29654
67 65982
67 28601
54
57
68
67
55
66624
87275
33975
79501
98574
Phase (°)
Period (years)
BER78
BER90
BER78
BER90
32 0
197 2
131 7
323 6
102 8
32 2
201 3
153 4
311 4
78 6
23716
22428
18976
19155
19261
23708
22394
18964
19116
23149
The sign of the amphtude of terms 3 and 5 m column BER78 has been changed relatwely to the values
given m Berger (1978a, b) m agreement with the change of th~ phase by 180° This has been done to
allow an eas~er comparison between the solutions
TABLE 7 Amphtudes, mean rates, phases and periods of the 5 largest amphtude terms m the
trigonometrical expansion of obhqmty (equation 25)
Amphtudes (")
BER78
1
2
3
4
5
-2462
-857
-629
-414
-311
22
32
32
28
76
Mean Rate ("/year)
BER90
-1969
-903
-631
-602
-352
00
50
67
81
88
BER78
31
32
24
31
44
60997
62050
17220
98378
82834
BER90
31
32
32
24
30
54068
62947
08588
06077
99683
Phase (o)
Period (years)
BER78
BER90
BER78
BER90
251 9
280 8
128 3
292 7
15 4
247 14
288 79
265 33
129 70
43 20
41000
39730
53615
40521
28910
41090
39719
40392
53864
41811
TABLE 8 Amphtudes, mean rates, phases and periods of the 5 largest amphtude terms m the
trigonometrical expansion of eccentricity (first equation of 34)
Amphtudes
1
2
3
4
5
Mean Rate ("/year)
Phase (°)
Period (years)
BER78
BER90
BER78
BER90
BER78
BER90
BER78
BER90
0 011029
0008733
0007493
0006724
0 005812
0 011268
0008819
0007419
0005600
0004759
3 13889
13 65006
10 51117
13 01334
9 87446
3 20651
13 67352
10 46700
13 12877
9 92226
165 2
99 7
294 5
291 6
126 4
169 2
121 2
312 0
279 2
110 1
412885
94945
123297
99590
131248
404178
94782
123818
98715
130615
The sign of the amphtude of terms 2 and 3 m column BER78 has been changed relatwely to the values
given in Berger (1978a, b) m agreement with the change of the phase by 180° This has been done to
allow an easier comparison between the solutions
F o r t h e e c c e n t r t c l t y as t h e s e r i e s e x p a n s i o n is s l o w l y
c o n v e r g e n t a n d t h e n u m b e r o f t e r m s Is h u g e , t h e
accuracy of the numerical values for e can be very
p o o r tf u n c o n t r o l l e d t r u n c a t i o n s a r e m a d e . K e e p i n g
all t h e t e r m s ( 1 1 4 7 9 ) f o r w h i c h t h e a m p h t u d e is l a r g e r
t h a n 4 × 10 - 6 l e a d s t o a d e v i a t i o n o f a b o u t 2 × 10 - 6
F o r t h e g e n e r a l p r e c e s s i o n in l o n g i t u d e t h e 8 4 9 2 t e r m s
w i t h a n a m p h t u d e l a r g e r t h a n 0 "01 g w e r i s e t o a
d e v l a t t o n l e s s t h a n 8 × 10 - 5 d e g r e e W i t h 2 3 9 4 t e r m s ,
t h e d e v t a t l o n ~s g e n e r a l l y less t h a n 6 × 10 - 3 d e g r e e
W i t h 2 6 2 t e r m s w h o s e a m p l i t u d e s a r e l a r g e r t h a n 50",
it Is o f t h e o r d e r o f 0 °2
Therefore, m order to avoid any loss of accuracy by
having to limit the expansions in order to provide the
expressions (34) with an acceptable, managable numb e r o f t e r m s , o n l y t h e n u m e r i c a l v a l u e s o f e, e, e s i n
TABLE 9 Value of the different constants m the
development of the astro-cllmatlc elements for
BER78 and BER90 (to = 1950 0)
BER78
e* (°)
k ('7year)
R ('7year)
(o)
23
50
50
3
and insolation for the
available upon request
320
439
439
392
556
273
273
506
BER90
23 333 410
50 390 811
50 417 262
1600753
l a s t 10 m l l h o n y e a r s will b e
from the first author
I N T E R C O M P A R I S O N OF THE ASTROPALEOCLIMATIC PARAMETERS
First,
let us point
out that even
if t h e p l a n e s
of
3118
A Bergcr and M F Loutr~
reterence for the solunon BER78 and BER90 are not
the same (1850 II and 2000 01 the comparison between
the two solutions for the obhqmty and chmanc precession ks entirely vahd as the values calculated fm
paleochmatm research are instantaneous (1 e they refer
to the reference planes of the date and not ot the
epoch)
The accuracy ot the solution depends essentially
upon the accuracy and the number ot terms kept in the
p e r t u r b a n o n / u n c t i o n (Berger, 1976, 19841 In the case
of BER90, It also depends on the numerical process
used to obtain the analytmal development ot the
elements the F o u n m analysis for the Earth's orbital
elements (h, k, p and q) was limited to 811 terms giving
rise to an accuracy ot about 0 1% tot the eccentricity
and 1 5% for the lnchnatlon, but this accuracy does not
depend on time and affects only the values ,it timer
dose to the present-day (Laskar, 19881
A nalyu~ al ( omparI son
Tables 3 to 8 provide, for BER78 and BER90, the
characteristics of the 5 largest terms m the orbital
systems (e, rt) and (t, g2) as well as in ~. ll,, e sin d)
and e The trequencles wdl provide automatically the
spectra of the astro-lnsolatlon parameters, as needed m
the vahdanon process of the astronomical theory (c g
h n b n e et a l , 1984, Berger, 1989a, b)
For the obliquity the number ot terms for the
solution BER9() ks much larger than for BER78 for
BER90 there are 149 terms for whmh the amphtude ts
larger than 1" whereas there are only 47 for BER78
The 4 hrst terms ot BER90 have to be compared
respectively with term numbers 1,2, 4, 3 of BER78, the
5th term does not have any corresponding term m
BER78 Comparison o1 the two solutions shows only
weak dflferences m the frequencies tot the most
unportant terms, the differences in the trequencles
generate differences m the periods ot the order of a
tew tens to hundreds of years, they amount to 200 years
tor the 53,864 yr-perlod but are generally less lmpoltant tot the other terms On the contrary, the amphtudes m BER90 are significantly different lrom those
o1 BER78 -1969" for the first term ot BER90 against
- 2462' tor the con espondlng term m BER78 This
dltterence of 20% ks slightly compensated for b~ thc
appearance of a new 41,000-yr term (number 5 in
BER90), the 29,000-yr term ot BER78 ranking only 6
m BER90 with an amplitude of - 2 0 6 " against -312"
Differences can even reach 50% for the 4th term ot
BER90 but are only a few percent lot the 3rd one 'ks
tor the phase, the differences are less than 30 ° fin the
important terms
The comparison ot ~p between BER78 and BERg0 ks
more complicated Very large periodicities (about 311
Ma) appear m BER90 with important amphtudes
These periods are characteristics ot the exmence ol
almost commensurable characteristic trequencms It
the~ are omitted, the second and third terms ot BER78
can be compared to terms 4 and 7 respectively m
BER90, the frequencies and phases being m good
agreement and thmr respective amphtudes quite ~omparable
As was also the case [oi the obhqulty, the numbel ol
terms m the expansion of the chmatm precessum is
larger R)r BER90 than tot BER78 In BER90 there arc
110 terms lor which the amphtude is larger than ~, Y
10 ~, whereas there ,ire only 46 in BER78 As therc arc
many trequencms close to each other, it ks not very ~ as~
to fred the terms which have to be compared Nevertheless, the 5 first terms ol BER90 can be compared
respectively with term numbms l, 2, 3, 4 6 ot BER78
The 5th term of BER78 corresponds to the 9th term ol
BERg0, which Dyes more weight to the 23,0011-~i
period m BER90 This docs not preclude a good
agreement between the 2 solutions the periods generally &tier only by a tcw tens to a few hundreds ot
years, the amplitudes ale qmte close to each other (the
absolute difference ts only 3 7 × 10 -a for the first term,
I c a relative dflferencc ot about 2%) but for ~ome
terms the dff|erence is more important (the amphtudc
ot the 9th term of BER90 corresponding to a lq,000-y!
component ks about hall the corresponding amphtude
of the 5th term of BER781, the difference between the
phases is generally less than 20"
For the time series tol chmanc precession, as ~scll as
tol obhqmty, the difference in amphtudes me compensated tor bv new te~m,, haling more or less smnl,n
trequencles and phases
The analysis ol the c~centnclty Is nmch more
comphcated the number ot telms Is greater tol BER90
than tor BER78 (90 against 42 tor ,ill terms with ,in
amphtude larger than 4 ~ 10 a), the difference m the
periods Increases v~lth their length (the difference is
8,700 years lor the hrst term but only 16"~ years lor the
second) which makes thc comparison term b 5 term
more dehcate Nevertheless, assuming that the hrst
terms etfectwely correspond to each other, the change
m the phases between BER90 and BER78 ale ol the
same order ol magnitude than lor the other elements
(gene! ally about 211° lot the nlost important terms), the
agreement between the amphtudes rem,uns ver~ good
with a difference ol the order ot 10- a (but iI becomes
more nnpoltant --- ol the order ot 10 ~- - tor the othel
terms)
This comparison lcads to the conclusum that the
solutums BER78 and BER90 can only beconlc dlflerent belore 1 5 mflhon years ago (a tew tnnes the
period ol 4011,000 w a r s corresponds to the largest
amphtude tol the eccentricity)
Nlut'lUt l( {ll ( olnpat {~Oll
From the analyncal cxpresslons (34), time ~cnc> can
be generated and compared together, tot example over
the previous 5 mllhon years and the next million years
centred at 19511[) 1he vananons m time ot the
eccentricity the obhqmty and the climatic precession
corresponding to BER90 are presented in Figs 4 to 9 for
a time interval going trom 0 to 6 Ma BP They are
compared to the solunon BER78 tor the time mtmval 0
to 3 Ma BP in Figs 4, ~ and 6
309
Insolation Values for the Last 10 Ma
BERg0
BER78
In BER90, during the last 5 mdhon years (next
Obllqmty
Obhqmty
mdhon years), the eccentricity is seen to vary between
0 000267 (0 001694) and 0 057133 (0 052614) with an
average quasl-pertod of 96,805 (93,100) years Simultaneously, the obhqu~ty of the Earth's orbit has vaned
between 22 °08 (22 °28) and 24 °54 (24 °32) with an
average quasi-period of 41,074 (41,174) years Whde
the chmat~c precession oscillates between - 0 05625
( - 0 05193) and 0.05623 (0 05201) with an average
quasz-penod of 21,000 (21,378) A characteristic
feature of the time evolution of the eccentricity ~s the
-400
almost complete disappearance of the 100,000-year
cycle between 2 4 Ma BP and 2 8 Ma BP (Ftg 6a), as
•-~ -500
well as between 4 4 Ma BP and 4.8 Ma BP (Fig 8a),
leaving only the 400,000-year cycle. The obhqulty is
~ -000
charactensed by very small changes m amphtude
between 3 Ma BP and 35 Ma BP (Fig 7b), and
-700 ~
between 4 Ma BP and 4 5 Ma BP (Fig 8b)
A visual check to Figs 4, 5 and 6 shows that the
solution BER90 is m good agreement with BER78 over
-000
the last 1 5 × 106 years, for the eccentricity, as well as
for the obliquity and chmattc precession They become
-900~
dwergent only before 1 5 × 106 BP The eccentnctty
curves look totally different at the 100 ka t~me scale
-1000
starting 1 5 Ma BP The amphtude of the 100,000 yearcycle disappears in the two solutions leawng only the FIG 4b Comparison between BER78 and BER90 of the long-term
vanatzons of the obhqulty from 1 Ma BP to the present (1950 0
400,000-year envelope but for different t~me intervals
AD)
BRE74
LASS8
BERT8
|*l||||
BER90
|_-*||~||
Eccentricity
Eccentrlclty
BER7B
Precession
8~J88ZS
-10
-
BER90
Precession
8Z88~8
-100
_
oo
-00~
-300
-40C
-400
~ -500
-000
j:
e~
rt~
-0~
-700
-700
4800 ~
-800
-900
-90G:
_,ooo
FIG 4a Comparison between BER78 and BER90 of the long-term
vanattons of the eccentricity from 1 Ma BP to the present (1950 0
A D ) BRE 74 refers to Bretagnon (1974) and LAS 88 to Laskar
(1988)
FIG 4c Comparison between BER78 and BER90 of the long-term
variations of the chmat~c precession from 1 Ma BP to the present
(1950 0 A D )
310
A
BRE74
BER78
Eccentrlmt
[
Loutrt_
I.,A888
BEB90
Eccentnczty
~ !! iI i1 t! i1
~ , , ,o ~ o
-~oooi'* ~'~'7'-
-100~
-llO0
-.oo1/
-1200-~
-12001
-1300
Berger and M
o
o
o
o
o
o
~
-1300
-1400
-1400
-1500
-1500
-~0oo ~--J)
o
-1100
-t3oo
~l
~
BER90
Precession
BERT8
PrecesslOD
-1000~
!
E-~-1600
-1700L
-~001~
-1000
-1000
-1800~
-1900
-1900
_2000~
-2000
I, I
FIG 5a Compar,son between BER78 and BER90 ol the long-term
vanatmns of the eccentr,c~ty from 2 Ma BP to 1 Ma BP
BER78
BERg0
Obhqmty
Obhqmty
ot
FIG 5c Compartson between BER78 and BER90
the long-term
variations ol the chmatte precession trom 2 Ma BP to 1 Ma BP
BRE74
BER78
Eccentricity
LAS88
BER90
Eccentnczty
-200d o © o o o o
~/r'r'f'r' I
-21oo
~
j
-2400
2~ -2500
-2000
-2700
-2f100
---
----
- ,~'--~
<
_200o__<2___
-2000 , I , I , d - ~ r , ' , ~
FIG 5b Comparison between BER78 and BER90 ot the long-term
variations of the obhqmty from 2 Ma BP to 1 Ma BP
l
_~oooL,,,~,~,l,l
F I G 0a Comparison betwe~.n BER78 and BER90 ot the long-term
v a n a u o n s of the eccentnc,tv from 3 Ma BP to 2 Ma BP
311
Insolation Values for the Last 10 Ma
BER78
BEE90
0bhqmty
0bhqulty
LAS88
BER90
Eccentricity
BERg0
~66||
-2000
BER90
Precessson
Obhquzty
8 Z 8 8 8 ~
_2000~ ? ? o o o
-.o, ,. .... i
-210
-3100
-2LOC
-220
-320(
-320©
___) i
-220~
- ~ .
-240C
~.~
-350~
~
-2201
-3800
-2701
-3700 ~
-3700
-2001
-3800
-2900
-3000
--4000 ~ ' ' ' ~ ' I t l t
FIG 6b Comparison between BER78 and BER90 of the long-term
variations of the obliquity from 3 Ma BP to 2 Ma BP
BERTfl
Precessaon
FIG 7 Long-term vanatlons of the eccentricity (a), the obhqulty (b)
and the chmauc precession (c) from 4 Ma BP to 3 Ma BP for BER90
LAS88
BEE90
Eceen~el~
BER90
Precession
??70°°o
-2000 , I ' I '~1 ' I '.
o
o
o
o
o
o
o
BEE90
0bhqul~
-~ ~,~,~,
-210C
-4100 "
i
-22o
-420 " ~
i
-2300 ~
-420
i
-2400
~
-
BEE90
Preeesslon
~
-~0,
~-4500
~ -2800
"
i
-4600
-
~
-2700~i
-4700
-2800
-4800
-290C
-4900
~
-30(
FIG 6c Comparison between BER78 and BER90 of the long-term
variations of the climatic precess]on from 3 Ma BP to 2 Ma BP
FIG 8 L o n g - t e r m v a n a t l o n s o f t h e e c c e n t n c l t y ( a ) , t h e o b h q u l t y ( b )
andthechmatlcprecesslon(c) f r o m 5 M a B P t o 4 M a B P f o r B E R ~
3[2
A Berger and M F Loum_
LAS88
BERg0
BER90
Eccentrlclty
Obhqulty
-5100! ~
BER90
:8~gSggg
Precession
!
-5300
-5500
-5800~
i
-5900
-~ooo~,,, ~
FIG 9 Long-term~a~mtlonx ol the eccentricity(,t) the obhqmt} (hi
and the chmatlc precessum (¢) lrom 6 Ma BP to '~ Ma BP for BER90
this chaiacterlstlc shape, reflecting the 400,00t)-vear
component m BER78 between 1 7 Ma BP and 2 1 Ma
BP, can be round in BER90 between 2 4 Ma BP and 2 8
Ma BP The obhqmty is out of phase b,e one tourth ol a
cycle (10,000 years) at 1 9 Ma BP with BER90 leading
BER78 The first discrepancy between the two solutions for chmatlc precession appears at 1 3 Ma BP but
the hrst significant difference occurs between 1 ~ and
1 7 Ma BP This contrrms the conclusion made in 1984
comparing the solution BER78 with other solutions and
estimating the influence ot the perturbanons not taken
into account (relatlvistm and lunar), Berger (1984) concluded that the tm~e series for the precession, eccentricity and obllqmty can be considered rehable for the
last 1 5 × 10*" years only
In the case of the eccentricity, the drtlerence m phase
between BER78 and BER90 ~s much less than a hall
period over the last 5 Ma For the obliquity, the mare
difference arises between 3 and 3 5 Ma BP during
which the new solution is far less regular than BER78
(r e less slnusoldal) The two solutions for the climatic
precessmn are out of phase by half a mean cycle (10,000
years) at 2 Ma BP, but they recover very rapidly at 2 2
Ma BP the phase difference is again small
THE INSOLATION VALUES
Simulation of the past chmate reqmres the calculation of the daffy or monthly msolatmn instead of, or in
addmon to, the Mdankovltch caloric season's insolation (Berger, 1978c) The daily mid-month or monthly
mean insolation can bc derived lrom a srlnplc but
accurate set of formulae (Berger, 1978a, b) For the
sake ol comparison, the mid-month daily insolation,
defined from a constant increment of the true longitude
ol the Sun, starting at the spring equinox, ,ire computed
lor C,lch 10 degrees ol latitude Let us recall that these
values represent the insolation at around the 2()th ol
each month In the same ~,lx, monthly mean insolation
values, averaged over I(I degree latrtudmal zones will
also bc displayed tot the intercomparlson between
BER78 and BER90
Analysis ol the lnsol,itum ~alues obtained [rotn
BER90 brings some gcneral conclusions insolation >
dominated by precession mainly in thc equatollal
regions but the obhquit3 signal rs remlorced at the
solstlccs and at high l,ttltudcs The role of eecentrieitx
m modulating the precessional component m the
\arlatlon ol insolation is very visible through the
40{I,()00 },car cycle (sec lot example the monthly mean
insolation lor March 2{b-:~0°N (fqgs 12, 13 and 14))
Foi thc last I 5 mllhon years the BER78 and BERO0
insolation values are very snmlar (Figs 10 and 11 ) "1his
conlirms the limit ol \ahdrt~ already given lo~ the.
orbital parameters
l h e same characteristics hold
therelore lor the two solutions over that time sp,m Foi
example tot the last 2()lt,0(}{I }'cars, the most slgnllu.ant
deviations o| the 65°N July mid-month msolatkm trom
the 1950 l} / \ D xaluc (427 W/m n) are round to be
located ,uound 185 ka BP (--28 w/me), 160 ka BP (- q
W/m:) 1-~7 ka BP ~--!i ~ / i n e) 114 ka BP (--~5 P~/
m 2 ) 93 ka BP ( - 6 W/in-), 71} ka BP ( -1 t) W/m e) 41 ka
BP I -- 10 W/m:) and 22 ka BP ( - 9 W/m e) as lar ds thc
negatl~c de'~ldtlOnS arc concerned, and around 197 ka
BP (46 W/m-'), 173 ka BP (54 w/me), 148 ka BP (28 W/
me). 126 ka BP (6(I W/toni, 104 ka BP (48 W/m:) 82 ka
BP (40 W/me), 56 ka BP (34 W/nan), 33 ka BP (15 W/
m -~) and 10 ka BP (43 W/m e) tor the positive deviations
in addition to the analysts of this 65°N July msolaturn, ~t m~ght also be sigmtlcant to compare the
monthly mean msolatum values given by the 2 solutu)ns
lot June over the latitudinal band 80-9(t°N and tor
December 70-80°S, foi March 20-30°N and lor September 1{t-2()°S These lantudes and months are indeed
among those which were retained rn the insolation
climate index (Berger c t a l , 19811 ,is s h o ~ m g a
statistically significant ~_orrelatron with 6 0 Is records
The behaviour ol the curves tot March 20-30°N and
September 10--20°S are very similar and their intercompanson wdl be restricted to the msolanon curxe for
March 20-30°N Globally. the insolation curves lor
March 20°-30°N (Figs 12-14) are very srmdar tot the
two solutions until around 1 4 ~< 10~ BP One ol the
characteristics ot the BER78 curve is a well-marked
beat between 1 7 and 2 1 mllhon years for whmh the
400,000 year 'envelope" corresponds to the d~sappearance of the 100,000 year eccenmcity cycle, and to a
shght damping of the amphtude of obhqulty, and to a
less extent of precession This same feature appears
later ( 2 4 t o 2 8 M a B P )
mBER90 Inamoredetaded
analysis, we can see that until 200 ka BP. there exists
313
Insolation Values for the Last 10 Ma
BER78
BERg0
July 65N
1 u l y 65N
BERT8
March 2 0 - 3 0 N
BERg0
March S0-30N
31i§§~
t~
-____
FIG 10 Comparison between BER78 and BER90 of the long-term
vanatJons of the 65N mid-month July insolation from 1 Ma BP to the
present
FIG 12 Comparison between BER78 and BER90 of the long-term
variations of the monthly mean insolation for March 20N-30N from
1 Ma BP to the present
BERT8
BERg0
BER78
BER90
July 65N
July 65N
March 20-30N
March 20-30N
EII~|J
..^fl§gl~|!
a~
FIG 11 Comparison between BER78 and BER90 of the long-term
variations of the 65N mid-month July insolation from 2 Ma BP to
1 Ma BP
FIG 13 Comparison between BER78 and BER90 of the long-term
variations of the monthly mean msolaUon for March 20N-30N from
2 Ma BP to 1 Ma BP
314
A Bcrger and M F Loutrc
BER78
March 20-30N
BERg0
March 20-30N
!|iI~
-2000
-2100
-2~'00
-'~400
-2700
-290(
-300(
FIG 14 Comparison between BER78 and BER90 of the long-term
vanatlons of the monthly mean insolation for March 20N-30N from
3 Ma BP to 2 Ma BP
some tiny differences in amphtudes between the two
solutions (less than 5 W/m 2) Before 40(I ka BP the
amplitude exhibits increasing differences reaching 10
W/m 2, with the amphtude of BER78 somenmes being
larger than in BER90 (400 ka to 830 ka), somenmes
smaller (830 ka to 1100 ka) Before 1 5 Ma BP, the
differences become more significant, in particular
between 1 9 and 2 7 Ma BP At 1 9 Ma BP, BER90
leads BER78 and its amphtude IS smaller, but before
1 95 Ma, BER90 starts to lag behind BER78, at around
2 Ma, the two solutmns are completely out of phase and
they remain out of phase over more or less 100,000
years the amplitude of BER78 being smaller than
BER90 For the next 300,000 years, there are period',
durmg which the two solutions are m phase (2 08-2 10
Ma BP, 2 17-2 22 Ma BP, 2 28-2 33 Ma BP) alternating with periods where BER90 lags behind BER78
Between 2 4 and 2 7 Ma BP the mean quasi-period ot
BER90 (22,150 years) becomes larger than in BER78
(20,570 years) giving rise to pertods during which the
two solutions are m phase (around 2 6 Ma BP) or
totally out of phase (around 2 5 Ma BP) Finally, at
2 7 Ma BP the two solutions are again in phase
The comparison of the soluttons over other lantudJnal zones and months shows stmllar behaviour beats
can be seen in the June monthly mean insolation
averaged over the latitudinal band between 80 and
90°N, but do not occur at the same time for the two
solutions The differences between the amphtudes of
the insolation curves increase with time back in the
past From a few W/m 2 durmg the last 200,000 years to
10 W/m -~ around 700 ka, it leaches 20 W/m e at 1 ~ Ma
BP Belore 13 Ma BP, the two solutions become
dlfterent A characteristic leature of the insolation
around 4 4 Ma is the small variations m amplitude
related to the small value of the eccentricity at that time
(e is almost 0 ,it 4 38 Ma BP) and to the small changes
m precession and obhqulty I'he same feature has a
striking appearance m the 65°N mid-month lnsolanon
values for July (Fig IS)
In the case of the monthly mean values lor Decembe~
70-80S some tiny differences can already be seen
around 1 Ma BP (few W/m -~in amphtude), but they arc
much more perceptible before for example, differences in amplitude hke around 14 Ma BP and
appearance of new relative maxima at l 29 and 1 41 Ma
BP
For the winter latitudes at December 611--70N and
June 50-60S, the amplitude of the variations are very
small and consequently the comparison more difficult
Broadly for December 60-70N the two solutions differ
very few from each other (less than 1 W/m -~) over the
last 900,000 years Between 900 ka and 1 5 Ma, the
differences in amphtude increase and phase lags appear
at different times Betore l 5 Ma, the two solutions can
hardly be compared sometimes they look very similar
(2 4 Ma to 2 7 Ma) while at other times they are totally
different (1 9 Ma to 2 3 Ma)
CONCLUSION
The BER90 solution includes for (e, rr) and (;, g2),
terms depending upon the second power as to the
dlsturbmg masses and on the fifth degree with respect
to the planetary eccentrlcmes and lnchnations
The conclusions drawn lrom tormer comparisons ol
different astronomical solutions for the astro-lnsolatlon
parameters (Berger. 1984) are confirmed
•
The accuracy ol thc solunon depends upon the
accuracy of the constants, the initial condmons
and the expansions themselves
•
The ~,alues ol obhqulty and pietessum are
strongly dependent upon the accurac~ ol the
system (l, ~ ) , mole than upon the accuracy ulth
which the Polsson equations can be solved
•
Back to 1 Ma, the tyro solutions BER7,~ and
BER90 are ver 3 smlflar Between 1 Ma and l 5
Ma rome dltferenccs arise, which are not ~cr~
important, but, earher than that, the two solunon,, become very different, so that for periods
prewous to 1 5 Ma the solution BER90 must be
used when insolation values are needed to lorcc
climate models A prehminary comparison ol
BER9I) with two numerical integrations ot a set
of equations for the dynamics of both the
planetary pomt masses and the E a r t h - M o o n
system (Laskar, personal communication m
Berger et a l , 1988, Quinn el a l , 1991) leads
us to conclude that these values remain reh-
Insolation Values for the Last 10 Ma
BER90
J u l y 65N
BER90
J u l y 65N
E§g~IH
315
able until 5 to 10 M a ago, but most p r o b a b l y not
for periods e a r h e r than 10 Ma, as this seems to
be the limit of validity of the astronomical solution I n d e e d , before 10 Ma, the orbits of the
inner planets look chaotic any two orbits with
n e a r b y initial conditions diverge (Laskar, 1989,
1990)
T h e l n t e r c o m p a r l s o n b e t w e e n B E R 7 8 and
B E R 9 0 shows also that B E R 7 8 might continue
to be used without any p r o b l e m up to 1 Ma BP
H o w e v e r , it IS preferable to B E R 9 0 for the last
glacial-interglacial cycles because of its better
accuracy close to present-day times, the reproduction of the p r e s e n t - d a y conditions f r o m
B E R 9 0 indeed suffers f r o m the fit carried-out by
L a s k a r (1988) to represent ItS numerical values
for h, k, p, q by trigonometrical series Fortunately, this numerical p r o c e d u r e does not
affect the solution outside the time origin
m
FIG 15a-b Long-term variations of the 65N mid-month July
insolation from 3 Ma BP to 2 Ma BP (a) and from 4 Ma BP to 3 Ma
BP (b)
BER90
J u l y 65N
BER90
J u l y 65N
Finally, it is highly significant for the rehabdity
of the solution that three soluttons o b t a i n e d
i n d e p e n d e n t l y for e, e, e sin 6~ all agree over
the last 3 Ma at least M o r e o v e r , use of these
new values has already shown a m u c h better and
m o r e natural fit with the geological records over
the last 5 M a (Shackleton et a l , 1990; Hllgen,
1991)
T h e shght a d v a n t a g e of the analytical p r o c e d u r e used
here for B E R 9 0 , and earlier for B E R 7 8 , allows a
straightforward way to obtain all the frequencies and
related amplitudes and phases which characterize the
astronomical p a r a m e t e r s without having to rely on
spectral techniques at all I m p r o v e m e n t s m a d e not only
in the straightforward numerical integration of the
planetary point masses and E a r t h - M o o n systems, but
also in the analytical p r o c e d u r e s to only deal directly
with the long t e r m variations ( B r e t a g n o n , 1990, Bretagn o n and Simon, 1990), are therefore very encouraging
ACKNOWLEDGEMENTS
m
We very much thank J Laskar for providing us with the numerical
values of his 1988 solution One of the authors (MFL) was supported
by contract CEA BC-4561 of the Commissariat Franqats a l'Energw
Atomique which IS greatly acknowledged Graphics were made by
F Mercier who is warmly thanked, with the graphics package
obtained from the National Center for Atmospheric Research
(USA)
REFERENCES
FIG 15c-d Long-term variations of the 65N mid-month July
insolation from 5 Ma BP to 4 Ma BP (c) and from 6 Ma BP to 5 Ma
BP (d)
Anohk, M V , Kraslnsky, G A and Plus, L J (1969) Trigonometrical theory of the perturbations of major planets (in Russian)
Trudy Institute Theorettcheskol Astronomu Leningrad 14, 1-48
Berger, A (1976) Obhquity and precession for the last 5,000,000
years Astronony and Astrophysics, 51, 127-135
Berger, A (1977a) Long-term variations of the Earth's orbital
elements Celestial Mechanics, 15, 53-74
Berger, A (1977b) Support for the astronomical theory of climatic
change Nature, 269, 44-45
Berger, A (1978a) A simple algorithm to compute long term
316
A
Berger and M F Loutre
variations o1 daily or monthly insolation Contribntion No 18
Instltut d ' A s t r o n o m i e et de Gdophyslque G kemaitrc Llm~erslt~
C a t h o h q u e de Louvdm, kouvaln-la-Neu,,e
Berger, A (1978b) Long-term variations ol ddll'~ msolations and
Quaternary ehmatlc changes Journal o/ Atrnospheru 5~:emes
35(2), 2362-2367
Berger, A (1978c) Long-term variations el calorie insolation
resulUng trom the Earth s orbital elements Quaterna#v RPseart.h
9 139-167
Bcrger A (1981) l h c astronomical theor,, ol pdleodlmatus In
Berger A ( c d ) , Chrnan~ Vanattom and Vartabtht~ Facts and
Fheortes pp 5(11-525 Reldel, Dordrecht Holland
Bcrger A (1984) Accuracy and f r e q u e n o e s stablhtv ol th~ L,lrlh s
orbital elements during the Quaternary In Bergei A l m b n u I
Hays J
Kukla G , Saltzman, B (eds), Mtlanl~olttd~ and
(hmate pp .~-39 Reldel Dordrecht, Holland
Berger A (19881 Mllankovitch Theory and Climate Rel'wns it:
Geophvsus 26(4), 624--687
Berger A (1989a) Fhe spectral characteristics el prc-Quatern,lrs
climatic records an example of the relationship between tht_
astronomical theor,~ and gee-sciences In Berger A Schneider
S and Duplessy J C] (cds), Cbmate and Geoscwmes pp 47-76
Kluwer Academic Publishers, Dordrecht Holland
Berger, A (1989b) Pleistocene climatic vanabihtv at astronomical
Irequencies Quaternar~ hlternattonal, 2, 1-14
Berger, A ( 19901 Astronomical theory of paleochmates and the last
glacial-interglacial cycle Quaternary 3ttente Rel'tews (m press)
Berger, A and Andlehc, 'I P (1988) Mllutin Milanko,<ltch, pert tit.
la th6orie astronomiquc des pal6oehmats Hlstotre et Mcsa#e
Edmon du CNRS, Ill-3, 38.5-402 Paris
Berger, A and Loutre, M F (1988) Nev, insolation ~alues lor thl_
climate ol the last l(J mllhon years Sc Report 1988/1z; Instltut
d ' A s t r o n o m l e et de Geophvsique G Lemaitre Umverslte ( a t h o [ique de Louvam, Louvam-la-Neuve
Bcrger A and Loutre M F (1990) Orlglnc des Irequenc<_s dt.s
dl6ments astronomlques lntervenant dans le cakul dc I'lnsolation
Bulletin de la Classe des ~clem eL A~ademw Rovale de Bel(tque, ~¢
~eru" I11/~), 45-1116
Bcrger A and Pestiau--, P (t984) Accuracy arid st,iblht~, el tht.
Quaternary terrestrial insolation In Berger, A , l m b r l c J tta,,s
J Kukla, G and Saltzman, B (eds) Mtlankovmh and (Ttmate
pp 83-112 Reldel Publ C o m p a n y , Dordrecht Holland
Berger, A , Guiot, J , Kukla G and Pestlaux, P (1981) Long term
•~ariations ot the monthly insolation as related to c h m a n c changes
Geologtsche Rundwhau, 70, 748-758
Berger, A Loutre, M F and Laskar, 1 (1988~ Llnc nou~elle
solution astronomique pour les l(i derniers millions d anndes Sc
Report 1988/14 lnstltut d ' A s t r o n o m l e et de Geophysiquc (,
Lemaitre Unlversitt~ C a t h o h q u e de Louvam Louv,nn-la-Neu~c
Berger, A Gallde H Fichefet, T , Marsiat, I and Fricot Ch
(1991)) "I estlng the astronomical theory with a coupled climate-ice
sheet model Global and Palentarv Change, 3(1/21 113-124
Bretagnon, P (1974) T e r m e s 5_ Iongues pdrlodes dans le systemt.
solaire A~tronom~ and 4strophvslcs 30 141-lS4
Bretagnon P (1982) [heorlc du m o u v e m e n t de I cnsembh, des
planetes Solution VSOP82 Astronomy and Astrophvw¢s !14
278-288
Bretagnon P (1984) Accuracy ol the long term planctar 3 theor~
In Berger A , lmbrle, J , Hays, J Kukla, G , Saltzmdn, B (cds)
Mdanl, ovltch and (hmate, Part I pp 41-53 Reidel Pnbl
Compan~ Dordrecht, ttolland
Bretagnon, P (1990) Methode itdratlve de construction d u n e
theorie g~nerale plandtalre Astronomy and ,tstroph~sus 231
561-570
Bretagnon, P and Simon J -L 119901 Theorlt. general,, du couph.
Jupiter-Saturne par une mdthode it6rahve Astronom~ a#id Astrophysus 239 387-398
Brouwer, D and Clemence, G M (196l) Methods o] (clerical
Me~hamcs Academic Press, New York 598 pp
Brouwer, D and van W o e r k o m , A J J (1950) q h c sccular vandtmns of the orbital elements el the principal planets 4str Palters
Amer Ephem Naut Almanac, Xlll(II), Washington D C 81I(17
D/lobeLk () (196~,) Mathematt¢ul [heorlcs o/ l'lanetarv Alot:ons
Do,,cr Publications Ne~ York (translated by Harrmgton, M
and Husst_y W I ) 294 pp
Hays, I D , lmbrie, J and Shackleton N J ( 19761 Vail,mon~ m the
Earth s orbit Pacemaker el the I,.e Ages ,Silence 194 t 12t-1 [~2
Hllgen F 1t99t) C a h b r a t n m el Gauss to Matuvama sapropt_l
patterns m the Medlterrane,ln to the astrononncal record ,rod
Imphcaton tor the global polarity tnnu scale
h n b r , . J and hnbrie, K P (11~1791 h c ,Iqes 5ollmqthe ~t~sh'#~
Enqov~ New Jersey 224 pl y
lmhric J , Ha'¢s I , M a m n s o n I) (J M d n t v r e , \ Ml\ ,\ (
Morel', J J Plsias N G
Prell, W L and Shackleton N J
(19841 l h e orbital theor; ol Pleistocene chmate support lrom a
revised ( ' h m n o l o g y o1 the marine hr~O record In Bergei /\
lmbric
I
Ilays
I
Kukla (, and Sahzman P, (uds)
MHanhol mh and (/mum, pp 26~J1-~,0';; Reldel D o r d r t c h l I Iotland
hnbrlc I M d n i y r e A and M i x ,\ 11989) ()t.oanie ic~,polis~, to
t)rbil,il lorcin7 in lhc lalc Qtiatolnar'~ Observational and t.xpcrlmcrltal slr,ilegies In Berg(.i A
~chneider, $ dlld 1)uplos<,'~
I(I
reds) (/relate altd (,eos(lellces, pp t21-164 KltlwCl
Academic Publishers, l)ordrechl Holland
kutzb,ich I (1985) Modehng tit paleochmalcs
t d l a m ~ I;t
(;eol?hvslts 2 8 A , 159-b~6
Kutzbach J and Guettel P I I19801 l h e mlluent.t, el dlanglng
orbit<il parameters and surlace boundary conditions on chmatlc
simulations tot the past 18,(11~i years Journal o[ Atniosphe#:t
Sttemes 43{ 16), 1726 17'~l;I
L,isk,:r J (1986l Secular reims el da,,sleal planetary theorl~.s u,qna
the r~-sults el general theory tstronomv and Astrophlsu s 157,
$9-791
Lask,u 1 (1988J 5eeulat ~.xolution I,l the solar svstcnl o~ci 10
nnllions Vedrs IffroHot;,ts a;':d tslrophiw~s 198, 341-162
Ldskar J (1981}) A numerical experiment on the chaotic beh,isnm[
ol the solar w s t e m lValure, 338 237-2~,8
L,iskal
I (1990) The t.haolic lnotion el the solar s)stem
\
numerical estlmale el the silo tll the chaotic zones I~.attts
Lc \ , e i r i c r 1I I I (1854) R u c h c r c h e s a s t r o f l o n n q u c s
[ ()bM'll II[OIIU Imperial de ['a#ls
l#lnah,~ dt
Lleskc I H
lcderlc
I
Prlckc W ,rod M o i a n d o tt flq77)
l=~xpicssiorls hit the precession quantities based upon the IAU
(1976) s y s t e m ol aMroilonllt,l[ c o n s t a n t s 4311on,,:#ni arid ls/,,o +
p/l~,wts 58 1 16
Martlnson, I) (~ PlsldS N tw II<ixs I I) lmbriu 1 , Mooic I (.
and Shacklelon, N I (1987) ~\ge dating and the orbit,it thcorv el
thl. ice ages de,<elopment el a high-resolution tl to ~,00 (i00-vcar
ehionostratlgrdphy Quatetna#~ Researdi, 271l) 1-29
MIlankovllch M (19411 Kanon der Erdbestrahlung und suint.
A n ~ e n d u n g aul d,is Eiszcltcnproblem Royal Serbian %tlentes,
Spe~ pith 1 ¢2, ~e( tton o/Mathemattt al and Natural St :erices, Vol
~,:~ Belgrade, ~33 pp (~( atom ol Insolation and the le~_ ,%gt_
Problem
L n g h s h Trdnsl,mon b~ Israel Program lor Scl~ntlhc
Franslatlon and published lor the U S Department el ( ommur~.t.
and the Natnmal Science Foundation Washington D (.
19i69)
Prell W L and Kutzbach J E (19871 Monsoon ~ariabdit~ o',er tht_
past 150,0t111 years Joarna[ +~[ (,eophvwcal Research 92(D7)
841 I -8425
Ounln I R [ iemainc 8 and l)mlcan M ( 1991 ) ,\ Ihlct_ inllhon
"~t.ar integration el th(. [ a r l h • orbit (in pros',1
Saltzm,irl B , Hansen, A R and Maaseh, K A (1984) lhc lai~.
Ouatern,lr~¢ glaciations ,is the icspons¢ el a three-componenl
teedback system to Earth-Orbital Forcing Journal o[ Alml~vJherll
~t:t'llt¢'S 41(2]). "43811-'4~8')
Shackleton, N J Berger, A ,ind Pcltler, W R (199t1) An alteinati~c
aMrononncal cahbratlon of the h)wer Pleistoeen¢ time SL,Ile based
on O D P site 677 Phtlowq~ht(al Tramat tlons o[ the Rol al ,~,o,:ell'
t dtnbttrteh 81, 251-261
%haral S G and Boudmkos.i N & (1967) Secular "~arl,mon~, el
elements ol the Earth s orbit which influences the climates ol Ihu
gcoloDedl past (m Russian) 7rudv Institute TheorenHleshol
A stronomn, Lemngrad ! 114 ~ 231-261
Woolard E W and Clemenee (~ M (196'5t 5phetual ,Istronom;
Academic Pros,, New Y o r k 45 ~, pp
Insolation Values for the Last 10 Ma
317
A D D I T I O N A L DATA
The disk at the end of this issue contains the orbital and insolation data referred to in this paper The disk is a 3 5
inch double density - - double sides - - 720 KB containing four IBM(~) format A S C I I files
File __ 90 T O P contains the introductory information for the three data files
File 1
90 D A T contains 0-5 Myr BP
•
first column time in ka (negative for the past, origin (0) IS 1950 A . D )
•
second column eccentricity, E C C
•
third column: longitude of perihelion from moving vernal equinox in degree and decimals, O M E G A
•
fourth column obliquity in degree and decimals, O B L
•
fifth column climatic precession, E C C
SIN(OMEGA)
•
sixth column: mid-month insolation 65N for July in W/m e
•
seventh column mid-month insolation 65S for January in W/m e
•
eighth column mid-month insolation 15N for July in W / m e
•
ninth column mid-month insolation 15S for January in W/m 2
File 2
90. D A T contains: 0-1 Ma BP
•
first column time in ka (negative for the past, origin (0) is 1950 A D.)
•
second to eighth column: mid-month insolation 90N, 60N, 30N, 0, 30S, 60S, 90S for December in W/m 2
File 3
90 D A T contains 0-1 Ma BP
•
first column time in ka (negative for the past, origin (0) is 1950 A D )
•
second to eighth column: mid-month insolation 90N, 60N, 30N, 0, 30S, 60S, 90S for June in W/m e