Long term solutions for the insolation quantities of the

Jacques Laskar

IMCCE/CNRS, Observatoire de

A.C.M Correia, A. Fienga, M. Gastineau, F. Joutel, B. Levrard, H. Manche, P. Robutel Astronomical theory of paleoclimates

• Adhémar, 1845 • Croll, 1890 • Pilgrim, 1904 • Milankovitch, 1920, 1941 • Hays, Imbrie, Shackleton, 1976 0.06 0.05 0.04 0.03 0.02 excentricité 0.01 0 4.5 4 3.5 3 2.5 2 1.5 1 inclinaison (degré) 0.5 0 -2 -1.5 -1 -0.5 0 0.5 1 temps (Ma) Precession ( ) and obliquity (X = cos ")

1 H = X2 " 2

d @H = = X0 8 dt @X > > dX @H <> = = 0 dt @ > > :> 3k2 C A m m = M + 2 3 3 3 2  C aM a 4 5 Planetary perturbations 1 H( ; X) = X2 + 1 X2 sin( t +  + ) 2 k k k q Xk Resonance  X = cos " =   0 0 k  50:47"=year  35 500 30 450 25 400 20

insolation (w/m2) 350 obliquyité (degrés) 15

-1 - 0.5 0 -1 - 0.5 0 temps (millions années) temps (million années)

Lourens et al, 2001

GTS2004 :Gradstein, Ogg, Smith, 2004 Astronomical calibration (2004)

GTS2004 :Gradstein, Ogg, Smith, 2004 Astronomical calibration (2004)

Astronomical calibration (in project)

GTS2004 :Gradstein, Ogg, Smith, 2004 Insolation solutions for the Earth

Orbital Solution Precession equations

Pilgrim, 1904 Le Verrier, 1856 Milankovitch, 1920, 1941 +Hill, 1897 Sharaf & Budnikova, 1967 ytical Brouwer & Van Woerkom, 1950 Vernekar, 1972 anal Bretagnon, 1974 Berger, 1978

Laskar et al, 1993 Laskar, 1988, 1990 (-20 Ma -> + 10 Ma) Quinn, Tremaine, Duncan, 1991 (-3Ma -> + 3Ma) Laskar et al, 2004 umerical

n (-250 Ma -> + 250 Ma orb: ~ 40 Ma pre : ~ 20 Ma)

New Challenge : Orbital solution over ~ 60 Myr

Improve the accuracy by 2 orders of magnitude ! Development of a New planetary ephemeris

INPOP

(see Agnès Fienga’s presentation) Interactions in La2004, DE405, INPOP

• Newtonian ( planets ↔ planets, planets ↔ asteroïds (300), asteroïds ↔ asteroïds (5))

• Relativistic corrections (planets, asteroïds) • Non-spherical body ↔ point mass

• Sun (J2) ↔ Planets

• Earth (J2)↔ (Moon, Sun, , Jupiter)

• Moon (J2, C21, C22, S21, S22) ↔ (Earth, Sun, Venus, Jupiter)

• Deformation of extended bodies (tides) ↔ point mass

• Earth (Sun, Moon) ↔ (Moon, Sun, Venus, Jupiter)

• Moon (Spin,Earth, Sun) ↔ (Earth, Sun, Venus, Jupiter)

• Earth Shape ↔ Moon shape (torque exerted by the Moon) (La2004)

8 J. Laskar et al.: Insolation quantities of the Earth

5. Comparison with DE406 Table 2. Maximum difference between La2004 and DE406 over the whole time interval of DE406 ( 5000 yr to +1000 yr with origin Using a direct numerical integrator, our goal is to provide a at J2000); Col. 1: 100 to +10−0 yr; Col. 2: 1000 to +1000 yr; long term solution for the orbital and precessional elements Col. 3: 5000 to +1−000 yr. EMB is the Earth–M−oon barycenter. of the Earth with a precision that is comparable with the − usual accuracy of a short time ephemeris. We have thus com- λ ($$ 1000) pared our solution with the most advanced present numerical × integration, DE406, that was itself adjusted to the observa- 8 60 90 tions (Standish 1998). Over the full range of DE406, that is Venus 12 107 143 from 5000 yr to +1000 yr from the present date, the maxi- EMB 3 28 79 mum −difference in the position of the Earth–Moon barycenter 19 171 277 is less than 0.09 arcsec in longitude, and the difference in ec- Jupiter 5 69 69 centricity less than 10 8. The difference in the longitude of the Saturn 1 4 7 8 − J. Laskar et al.: Insolation quantities of the Earth Moon are more important, as the dissipative models that are Uranus 1 3 4 Neptune 2 7 16 used in the5t.wCooinmtepgarratioisonns arweithsligDhEtly40d6ifferent. They amount Table 2. Maximum difference between La2004 and DE406 over the to 240 arcsec after 5000 years, and the eccentricity difference is whole tPimluetointerval of D3 E406 ( 50009 yr to +100107 yr with origin of the ordeUr osifn2g a d1i0re5c,t bnuutmitesrihcoaul lidntbeegrnaototer,dotuhratgoonalyisthteoapvr-ovide a at J200M0)o;oCnol. 1:13140700 to +108−05 8y7r;3Col. 22:211070204 to +1000 yr; − − − eraged motloionngotfe×rthme sMolouotinonwiflolrhathvee soorbmitealsiagnndifipcraenctesinsiflouneanl ceelements Col. 3:Ea5r0t0h0 to +100010yr. EMB is th8e8Earth–Moo2n38barycenter. of the Earth with a precision that is comparable with the − on the precession and obliquity of the Earth. Table 2 summa- 10 usual accuracy of a short time ephemeris. We have thus com- a (UA 10 ) (15 m) rizes the maximum differences between La2004 and DE406 for λ×($$ 1000) pared our solution with the most advanced present numerical Mercury 4 43× 58 the main orbital elements, over the whole interval of DE406, Mercury 8 60 90 integration, DE406, that was itself adjusted to the observa- Venus 13 23 27 for different time intervals. Venus 12 107 143 tions (Standish 1998). Over the full range of DE406, that is EMB 29 62 62 EMB 3 28 79 from 5000 yr to +1000 yr from the present date, the maxi- Mars 27 76 76 − Mars 19 171 277 6. CompamruimsodniffwerietnhceLian9th3e position of the Earth–Moon barycenter Jupiter 268 470 565 is less than 0.09 arcsec in longitude, and the difference in ec- Jupiter 5 69 69 We have compared in Figs. 5 a8nd 6 the eccentricity and in- SaturnSaturn 743 1 1073 4 1168 7 centricity less than 10− . The difference in the longitude of the clination oMf tohoensecuare larmorseoliumtioponrtaLna9t, 3asanthde thdiessnipeawtivperesenmodtelsso-that are UranusUranus 1608 1 3379 3 3552 4 lution La2u0s0ed4.inOthveertw10o inMteygr,ratiothentswaroe osligrbihttlyal sdoiffluetrieonnts. TLhae9y3amount NeptunNeeptune 3315 2 5786 7 6458 16 and La200t4oa2r4e0naercasrelyc aidfteenrt5ic0a0l0, ayneadrsd,oanndothdeiffececresnitgrinciifitycadnifftleyrence is Pluto Pluto 4251 3 10 603 9 10 603 17 5 over 20 Moyfr.thTehoerdeiffr eorfe2nce1s0in− e, cbcuetnittrsihcoituyldinbcerneoatseedrtehgaut loanrllyythe av- Moon Moon 2313 470 20845 873 252210724 with time,eraamgeodumntointigontoof×athbeouMto0o.n0w2 iilnl htahvee seocmceenstirginciifitycaanfttienrfluence Earth Earth 487 10 3918 88 10 238238 on the precession and obliquity of the Earth. Table 2 summa- 10 20 Myr, about 1/3 of the total amplitude 0.063, while the dif- e (a (1U0A10) 10 ) ference in rtihzeesotrhbeitmalaxinimcluinmatdiioffnerreenaccehsebse≈1twdeeegnrLeea,2t0o04coamndpDarEe406 for × × MercuMryercury 33 4 52 43 144 58 to a maximthuemmvairniaotirobnitaolfe4l.e3mdeengtrse, eosv.eTrhtheesewdhiffoleereinntceersvarlesoufltDE406, VenusVenus 12 13 33 23 103 27 mostly fromforadsimffearlelndtifftimereenincteerivnatlhs.e main secular frequency g 6 EMB EMB 30 29 80 62 98 62 from Jupiter and Saturn that is now g = 28.2450 arcsec/year 6 Mars Mars 62 27 130 76 468 76 (Table 3) 6in.sCteoadmopfar2i8s.o2n20w7iathrcLseac9/y3ear in the previous so- Jupiter 268 470 565 Jupiter 30 95 158 lution. Beyond 20 million years, the differences between the Saturn 743 1073 1168 We have compared in Figs. 5 and 6 the eccentricity and in- Saturn 69 107 138 two solutioclinns atiobecnomofetmheosecure nolarticseoalbultioe.nTLha9e s3olauntdiothnse innewobplirqesen- t so- Uranus 1608 3379 3552 Uranus 99 135 144 uity and climlutioaticn Lpar2ecessio004. Ovnerar1e0pMloyttedr, thien tFwigos.or7biatanldso8l.uThtiones La93 Neptune 3315 5786 6458 Neptune 65 177 201 difference ainndoLbal2iq0u0i4tyaroevneerar2ly0idMenytricaaml, oanudntdsotonoat bdoiffuetr2sigdnei-ficantly Pluto 4251 10 603 10 603 grees, whiocvhemr 2e0anMsytrh.aTthtehediffoberlieqnucietsyincyecclceesntorfictihtyeitnwcroeassoelure-gularly Pluto Moon 92 23 215 204 215520 tions becomweithnetaimrlye,oaumt oofunpthinagsetaoftaebrotuhtis0d.0a2te.inThtheedieffcceerenntrciceisty after Moon Earth11 152 487 252 8323918 252 81302238 La2004-La2903Mayrer, pablootutet d1/3ino(fat)h,ewtohtialleaimnp(lbit)udoenly0.t0h6e3p, wrehcieles-the dif- Earth 460 3669 10 8694 ≈ e ( 10 ) ference in the orbital inclination reaches 1 degree, to compare × 10 sion model has been changed and in (c) only the orbital model Mercury s3i3n (i/2) ( 1052) 144 to a maximum variation of 4.3 degrees. These differences result × has been changed. Comparing 7b and 7c shows that most of MercuryVenus 2 12 18 33 88103 mostly from a small difference in the main secular frequency g the differences between the La93 and La2004 obliquity solu- 6 Venus EMB 5 30 55 80 144 98 from Jupiter and Saturn that is now g = 28.2450 arcsec/year tions result from the change in the dissipativ6e model of the Mars 62 130 468 (Table 3) instead of 28.2207 arcsec/year in the previous so- EMB 17 190 414 Earth–Moon system, while the only change of the orbital so- Jupiter 30 95 158 lution. Beyond 20 million years, the differences between the Mars 48 492 2053 lution would lead to a much smaller change of only 0.6 degree two solutions become more noticeable. The solutions in obliq- JupiterSaturn 1 69 11 107 122138 in obliquity after 20 Myr. Similar conclusions can be made for uity and climatic precession are plotted in Figs. 7 and 8. The SaturnUranus 1 99 6 135 100144 the climatidciffpreerecnecsseioinn o(Fbilgiqsu.i8tya–ocv)e.r 20 Myr amounts to about 2 de- UranuNseptune 2 65 5 177 5201 Pluto 92 215 215 grees, which means that the obliquity cycles of the two solu- Neptune 2 6 6 Moon 11 152 252 832 252 832 7. Seculatironfrsebqecuoemnecnieeasrly out of phase after this date. The differences Pluto 5 13 17 La2004-La93 are plotted in (a), while in (b) only the preces- Earth 460 3669 8694 Over long time scales, the determination of longitude of the Moon 5765 97 275 600 789 sion model has been changed and in (c) only the orbital model sin (i/2) ( 1010) planet on its orbit is not very important, and we are more con- Earth 19 202 × 841 has been changed. Comparing 7b and 7c shows that most of Mercury 2 18 88 cerned by the slow evolution of the Earth orbit under secu- the differences between the La93 and La2004 obliquity solu- Venus 5 55 144 lar planetary perturbations. The semi-major axis of the Earth tions result from the change in the dissipative model of the EMB 17 190 414 Earth–Moon system, while the only change of the orbital so- Mars 48 492 2053 lution would lead to a much smaller change of only 0.6 degree Jupiter 1 11 122 in obliquity after 20 Myr. Similar conclusions can be made for Saturn 1 6 100 the climatic precession (Figs. 8a–c). Uranus 2 5 5 Neptune 2 6 6 7. Secular frequencies Pluto 5 13 17 Over long time scales, the determination of longitude of the Moon 5765 97 275 600 789 planet on its orbit is not very important, and we are more con- Earth 19 202 841 cerned by the slow evolution of the Earth orbit under secu- lar planetary perturbations. The semi-major axis of the Earth Remarks on Precession Equations

I J " k w

K MA(t)

J J I l n g j i " EJ2000 h I

N1 Remarks on Precession Equations

I J " k w K Remarks on Precession Equations

I J angular momentum " k w K axis of figure C K˙ = W w K A ^ nearly diurnal motion around w 2 < K >= cosJ w = w + O(J ) J < 0.2!! INPOP K - w

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l b O Time (year) Integration of the angular momentum

•Averaging over the diurnal motion (Boué and Laskar, 2006)

•Takes into account the tidal deformation of the Earth, or post glacial deformation

•Do not depend on the internal model of the Earth

•The precession motions of K and w are the same

Suggestion : Any rotational solution for the Earth should provide also its angular momentum solution !