All Figures

$\begin{figure}\par \includegraphics[width=7.7cm,clip]{figures_pdf/1335fig1} \end{figure}$ Figure 1: Conservation of integrals. a) Relative variation of the total energy of the system versus time (in Myr) from -250 Myr to +250 Myr. b) The same after correction of the secular trend due to the tidal dissipation in the Earth-Moon system. c) Relative variation of the normal component of the total angular momentum.

In the text

$\begin{figure} \includegraphics[width=7.7cm,clip]{figures_pdf/1335fig2} \end{figure}$ Figure 2: Normalized repartition of the energy numerical error for 1000 yr steps, over the whole integration, from -250 Myr to +250 Myr. Practically superposed to this curve is the computed normal distribution given by Eqs. (4) and (5).

In the text

$\begin{figure} \includegraphics[width=7.7cm,clip]{figures_pdf/1335fig3} \end{figure}$ Figure 3: Fundamental planes for the definition of precession and obliquity. E_{q_t} and E_{c_t} are the mean equator and ecliptic at date t. E_c₀ is the fixed ecliptic at Julian date J2000, with equinox $\gamma _0$ . The general precession in longitude $\psi$ is defined by $\psi =\Lambda -\Omega$ . $\omega~$ is the longitude of the node, and i the inclination. The angle $\varepsilon$ between E_{q_t} and E_{c_t} is the obliquity.

In the text

$\begin{figure} \includegraphics[width=7.7cm,clip]{figures_pdf/1335fig4} \end{figure}$ Figure 4: Differences La2003-DE406 over the full range of DE406 (-5000 yr to +1000 from J2000). The units for semi-major axis (a) are AU, arcsec for mean longitude (l) and longitude of perihelion (perihelion). The eccentricity (e) and the inclinations variables ( $q=\sin~(i/2)\cos~ (\Omega), p=\sin~(i/2)\sin~ (\Omega)$ ), where i and $\Omega$ are the inclination and node from the ecliptic and equinox J2000.

In the text

$\begin{figure}\par \includegraphics[width=13cm,clip]{figures_pdf/1335fig5} \end{figure}$ Figure 5: Eccentricity of the Earth over 25 Myr in negative time from J2000. The solid line stands for the present solution La2004, while the dotted line is the eccentricity in the La93 solution (Laskar et al. 1993). The differences of the two solutions becomes noticeable after 10 Myr, and significantly different after 15-20 Myr.

In the text

$\begin{figure}\par \includegraphics[width=13cm,clip]{figures_pdf/1335fig6} \end{figure}$ Figure 6: Inclination (in degrees) of the Earth with respect to the fixed ecliptic J2000 over 25 Myr in negative time from J2000. The solid line stands for the present solution La2004, while the dotted line is the eccentricity in the La93 solution (Laskar et al. 1993). The differences of the two solutions become noticeable after 15 Myr, and significantly different after 20 Myr.

In the text

$\begin{figure}\par \includegraphics[width=7.6cm,clip]{figures_pdf/1335fig7} \end{figure}$ Figure 7: Differences in obliquity (in degrees) La2004-La93(1, 1) versus time (in Myr) over 20 Myr from present a). In La2004 both orbital and precession models have been changed. In b) only the precession model has been changed, while in c) only the orbital motion has been changed. It is thus clear that the main change in La2004 arise from the change of precession model.

In the text

$\begin{figure}\par \includegraphics[width=7.6cm,clip]{figures_pdf/1335fig8} \end{figure}$ Figure 8: Differences in climatic precession ( $e\sin \omega$ , where $\omega = \varpi + \psi$ is the perihelion from the moving equinox) La2004-La93(1, 1) versus time (in Myr) over 20 Myr from present a). In La2004 both orbital and precession models have been changed. In b) only the precession model has been changed, while in c) only the orbital motion has been changed. It is thus clear that the main change in La2004 arise from the change of precession model.

In the text

$\begin{figure}\par \includegraphics[width=8.5cm,clip]{figures_pdf/1335fig9} \end{figure}$ Figure 9: Variation of the secular frequencies g_1-9 from -250 to +250 Myr. The frequencies are computed over 20 Myr for g_1-4 and over 50 Myr for g_5-9, after transformation of elliptical elements to proper modes (Laskar 1990).

In the text

$\begin{figure}\par \includegraphics[width=8.5cm,clip]{figures_pdf/1335fig10} \end{figure}$ Figure 10: Variation of the secular frequencies s_1-9 from -250 to +250 Myr. The frequencies are computed over 20 Myr for s_1-4 and over 50 Myr for s_6-9, after transformation of elliptical elements to proper modes (Laskar 1990).

In the text

$\begin{figure}\par \includegraphics[width=8cm,clip]{figures_pdf/1335fig11} \end{figure}$ Figure 11: Variation of the semi-major axis of the Earth-Moon barycenter (in AU) from -250 to +250 Myr.

In the text

$\begin{figure}\par \includegraphics[width=14cm,clip]{figures_pdf/1335fig12} \end{figure}$ Figure 12: ( Top) Eccentricity of the Earth from -11 to +1 Myr. The solid curve is the solution La2004. Almost completely hidden behind the solid curve is a dashed curve representing the quasiperiodic approximation of Table 4 with 26 periodic terms. ( Bottom) Inclination of the Earth from -11 to +1 Myr. The solid curve is the solution La2004. The dashed line (almost identical to the solid curve) is the quasiperiodic approximation of Table 5 with 24 periodic terms. On both plots, as the two curves are nearly identical, the difference of the two solutions is also plotted in dotted curve.

In the text

$\begin{figure}\par \includegraphics[width=8cm,clip]{figures_pdf/1335fig13} \end{figure}$ Figure 13: a) Evolution of length of the day for the Earth, in hours, from -250 to +250 Myr. b) Residuals with the fit of the averaged solution with the polynomial expression (41).

In the text

$\begin{figure}\par \includegraphics[width=8cm,clip]{figures_pdf/1335fig15} \end{figure}$ Figure 15: Comparison of the solution of the obliquity La2004 (solid line) with its approximation using Table 7 (dotted line) with 26 periodic terms. The difference of two solutions (+22 degrees) is also plotted.

In the text

$\begin{figure}\par \includegraphics[width=8cm,clip]{figures_pdf/1335fig16} \end{figure}$ Figure 16: Comparison of the solution of the precession La2004 with its approximation from -15 to +2 Myr. The grey curve is obtained after removing uniquely the secular trend $\psi_0+p_0~ t+p_1~ t^2$ . The dark curve are the residual (in radians) after removing the resonant term (Eq. (34)), which is also displayed in solid line.

In the text

$\begin{figure}\par \includegraphics[width=8cm,clip]{figures_pdf/1335fig17} \end{figure}$ Figure 17: Comparison of the solution of the climatic precession of La2004 with its approximation from -11 to +1 Myr. The grey curve is the full climatic precession $e\sin \bar \omega$ . The dark curve are the residual after removing the approximation given by Eq. (37).

In the text

$\begin{figure}\par \includegraphics[width=8.5cm,clip]{figures_pdf/1335fig18} \end{figure}$ Figure 18: a) Evolution of the Earth-Moon semi major axis (in Earth radii) from -250 to +250 Myr. The grey zone is the result of the integration of the full equations, while the black curve is the integration of the averaged equations, as used in the precession computations; b) residuals with the fit of the averaged solution with the polynomial expression (39).

In the text

$\begin{figure}\par \includegraphics[width=8.3cm,clip]{figures_pdf/1335fig19} \end{figure}$ Figure 19: a) Evolution of the precession frequency of the Earth p (in arcsec/yr) -250 to +250 Myr; b) residuals with the fit of the averaged solution with the polynomial expression (40).

In the text

$\begin{figure}\par \includegraphics[width=8.2cm,clip]{figures_pdf/1335fig20} \end{figure}$ Figure 20: Stability of the solution for eccentricity of the Earth. a) Difference of the nominal solution La2004 with stepsize $\tau =5.\times 10^{-3}$ years, and La2004^*, obtained with $\tau ^*=4.8828125\times 10^{-3}$ years; b) difference of the nominal solution with the solution obtained while setting $J_2^{\rm S}=0$ for the Sun (instead of $2\times 10^{-7}$ in the nominal solution).

In the text

$\begin{figure}\par \includegraphics[width=8.5cm,clip]{figures_pdf/1335fig21} \end{figure}$ Figure 21: Stability of the solution for inclination of the Earth. a) Difference of $\sin i/2$ for the nominal solution La2004 with stepsize $\tau =5.\times 10^{-3}$ years, and for La2004^*, obtained with $\tau ^*=4.8828125\times 10^{-3}$ years; b) difference of $\sin i/2$ for the nominal solution with the solution obtained while setting $J_2^{\rm S}=0$ for the Sun (instead of $2\times 10^{-7}$ in the nominal solution).

In the text

$\begin{figure}\par \includegraphics[width=8.5cm,clip]{figures_pdf/1335fig22} \end{figure}$ Figure 22: Difference in obliquity for different tidal dissipation factor (Table 9): a) $\rm La2004 - La2004^{(0.95)}$ ; b) $\rm La2004 - La2004^{(0.90)}$ . In c) and d), the nominal solution of the obliquity is plotted in solid line, while La2004^(0.95) c) and La2004^(0.90) d) are in dotted line. After an adjustment of the time scale, the differences of the solutions are much smaller over 40 Myr: e) $\rm La2004 - La2004^{(0.95)}$ ; f) $\rm La2004 - La2004^{(0.90)}$ .

In the text

$\begin{figure}\par \includegraphics[width=7.7cm,clip]{figures_pdf/1335fig23} \end{figure}$ Figure 23: Resonant arguments (in radians versus time in Myr) $\theta = (s_4-s_3)-2(g_4-g_3)$ ( top) and $\sigma =(g_1-g_5)-(s_1-s_2)$ ( bottom) for different solutions: a) is the secular solution La93 (Laskar 1990; Laskar et al. 1993) while b) is the present nominal solution La2004. The characteristics of the other variants La2004_1.5 c), La2004₀ d), La2004^** e) are listed in Table 9. The arrows indicate the portions of the solutions when the argument (s₄-s₃)-(g₄-g₃) is in libration.

In the text

$\begin{figure}\par \includegraphics[width=8.3cm,clip]{figures_pdf/1335fig24} \end{figure}$ Figure 24: Stability of the g₅ ( top) and g₂-g₅ ( bottom) arguments. The difference (in radians) of the angles related to g₅ and g₂-g₅ from the nominal solution La2004 and an alternate solution La2004_1.5 a), La2004₀ b), and La2004^* c) (Table 9).

In the text

$\begin{figure}\par \includegraphics[width=8.3cm,clip]{figures_pdf/1335fig25} \end{figure}$ Figure 25: 405 kyr term in eccentricity. Maximum difference (in radians) of the argument $\theta _{g_2-g_5}$ of g₂-g₅ in all 6 solutions of Table 9 (La2004, La2004^*, La2004^**, La2004₀, La2004_1.0, La2004_1.5) with respect to the linear approximation $\theta_{g_2-g_5} = 2434''+ 3.200''~ t$ where t is in yr.

In the text

$\begin{figure}\par \includegraphics[width=8.5cm,clip]{figures_pdf/1335fig26} \end{figure}$ Figure 26: Eccentricity of the Earth. Nominal solution La2004 filtered in the interval $[2.2 {\rm ''/yr}, 8 {\rm ''/yr}]$ (solid line), and approximation with a single periodic term (47) (dotted line).

In the text

$\begin{figure} \includegraphics[width=7.7cm,clip]{figures_pdf/1335figA1} \end{figure}$ Figure A.1: Fresnel functions C(x) (solid line) and S(x) (dotted line).

In the text

$\begin{figure}\par \includegraphics[width=7.7cm,clip]{figures_pdf/1335figA2} \end{figure}$ Figure A.2: Obliquity variation (in degree) versus time resulting from the crossing of the resonance in the $\nu _{23}$ term (solid line). In the same plot, the dependence of the precession phase is illustrated. for $\delta =-\pi /4$ the obliquity offset is null (dashed line) while it is maximal for $\delta =+\pi /4$ . The nominal solution (solid line) corresponds nearly to a maximal decrease of the obliquity.

In the text