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Astronomical Solutions over Geological Time

Periodic Reporting for period 2 - AstroGeo (Astronomical Solutions over Geological Time)

Reporting period: 2022-05-01 to 2023-10-31

According to Milankovitch’s theory (Milankovitch, 1941), part of the climatic changes of the past is due to the variations of the insolation on the surface of the Earth resulting from the deformation of its orbit resulting from the gravitational disturbances of the other planets. These variations can be found in the stratigraphic records accumulated over several million years (Ma). The correlation between the geological data and the calculations of celestial mechanics is sufficiently well established so that since 2004, the geological time scales of the most recent periods are established from the astronomical solutions developed at IMCCE (Laskar et al., 2004). Since then, a major effort has been devoted to extend this astronomical calibration over ever longer periods covering the entire Cenozoic, up to 66 Ma. In these works, the astronomical solution is used to calibrate the geological data.

However, extending this work is difficult because celestial mechanics does not allow us to retrace the planetary orbits beyond 60 Ma due to the chaotic nature of the movement of the planets, as discovered by Jacques Laskar thirty years ago. The AstroGeo project aims to overcome this predictability horizon, imposed by the laws of gravitation by using ancient geological data as an additional constraint in obtaining astronomical solutions. The feasibility of the AstroGeo project was demonstrated with the analysis of sediment data from the Newark basin that allowed to recover the state of the solar system 200 Ma ago (Olsen et al., 2019).

AstroGeo will more generally benefit from the very numerous collaborations established for three decades by Jacques Laskar with geologists around the world, with in particular, in France, the ISTEP laboratory, partner of IMCCE as part of the National Agency for Research AstroMeso project started in October 2019.
WP1: Astronomical solutions.
The solar system motion is chaotic which prevents from obtaining a precise solution of its orbital evolution beyond 60 Ma ( Laskar, 1989, Laskar et al, 2011). Beyond that time one need to rely on statistical methods. We performed a statistical analysis of 120 000 orbital solutions of the secular model of the Solar System ranging from 500 Myr to 5 Gyr. We obtained the marginal probability density functions of the fundamental secular frequencies using kernel density estimation. These results will be essential for future geological frequency estimates.
In order to better understand the chaotic diffusion in the solar system, one needs to make statistical studies that require a large amount of realizations. For this, it is impossible to use direct numerical integrations that are much too time consuming. We have devised a simplify model that is very realistic and very fast. It is based on a model where the inner planets are integrated using the Gauss averaging method, and where the outer planet system is modelled as a quasiperiodic motion. Using this model, we have explicited the main resonances that are at the origine of the chaotic behavior of the system. We have confirmed that the resonance 2(g4-g3)- (s4-s3) (Laskar, 1990), is one of the leading ones for the origine of the chaotic behavior of the solar system. This term is essential for geologist as it modulates the Earth's eccentricity at a present 2.4 Ma period.

WP2 : INPOP. INPOP is the high precision planetary ephemerides that is directly adjusted to all planetary observations, from Earth observatory and space missions. This is the basis of the elaboration of a long term solution as La2004 ( Laskar et al, 2004). It is only through this adjustment that we can be confident that the model we are integrating backward in time is a real representation of the evolution of the true solar system. The INPOP model is constantly updated in order to take into account the best available observations. In 2021, we released the INPOP21a version (Fienga et al, 2021). The major points of improvement of INPOP21a relative to INPOP19a (Di Ruscio et al, 2020) are
i) the addition of 2 years of Mars Express data and 1 year of Juno normal points,
ii) the use of new Uranus ground based optical observations reduced with the Gaia DR3,
iii) the modification of the dynamical model for the Kuiper belt object perturbations,
iv) the first determination of the Sun oblateness including the Lense-Thirring effect.

WP3 : Tides and evolution of the Earth-Moon system
The Earth is a deformable body subject to inner, surface, and outer forces acting together to adjust its mass distribution. This mass distribution is quantified by the dynamical ellipticity, and its evolution is largely unknown over geological timescales. As this parameter plays an important role in the evolution of the Earth's rotational motion, its uncertainty propagates to long term solutions of the Earth's orientation. To minimize this uncertainty, we have computed a solution of the Earth's dynamical ellipticity over the past 50 Myr. We do so by combining oceanic proxies of glacial volume with the proper mathematical formalism to reconstruct a self-consistent history of the glacial and oceanic loading. As the Earth's response to this loading is highly dependent on its viscosity, we performed a parametric study and constrained the possible scenarios of dynamical ellipticity evolution. Combined with mantle convection and the tidal response, our findings add a missing puzzle piece toward a complete history of the dynamical ellipticity, and consequently valid extended rotational solutions.
WP3: Tides and evolution of the Earth-Moon system

In 2022 ,we have fulfilled the objectives of WP3 beyond all expectations. We have provided the first coherent model for the tidal evolution of the Earth-Moon system since its formation (Farhat, Auclair-Desrotour, Boué, Laskar, 2022).
Since the work of Georges Darwin, 1880, it is known that due to the tidal interaction between the Earth and the Moon, the Earth’s spin slows down and the Moon goes away. This has been measured first with ancient eclipses, and since 1969 with Lunar Laser ranging, reaching extreme accuracy at 3.83 cm/year. The age of the Moon is now also very well determined at 4.425 Ga. But it has also been realized, more than fifty years ago, that the simple tidal model of Darwin, starting with the present tidal rate, leads to a collision of the Moon with the Earth at about 1.4 Ga, incompatible with the known age of the Moon. In our model, we deliberately avoided to take into account the geological data as we wanted a model that would be free from any circular reasoning. We considered an analytical model, similar to (Webb, 1982), but more refined, with a hemispheric continent that evolves backward in time towards a global ocean, taking into account the plate tectonic reconstruction in the first Ga. This induces a non-trivial analytic treatment that takes into account the continental drift. Still, this model has only two parameters, as in (Webb, 1982), or (Tyler, 2021). An exploration of the parameter space and a search for the best fit to the present rate of tidal dissipation, and the age of the Moon leads to a single solution, fitting these constraints extremely well.
Moreover, this single solution agrees very well with most of the relevant geological data. Two points are of particular interest as they are obtained with cyclostratigraphic at 1.4 Ga, and 2.46 Ga. The agreement with those data points is amazing.
AstroGeo project
Past evolution of the Earth-Moon distance (in Earth radius) vs time ( in Ga)