The long-term evolution and initial size of comets 46P/Wirtanen and 67P/Churyumov-Gerasimenko

We present a new method to study the long term evolution of cometary nuclei in order to retrieve their original size. We considered the two cases of comet 46P/Wirtanen and 67P/Churyumov-Gerasimenko. We calculated the past evolution of the orbital elements of both comets over 100,000 years using a Bulirsch- Stoer integrator and over 450,000 years using a Radau integrator, that we combined with a realistic model for the erosion of the nucleus to estimate their original size. We derived that for the long term evolution of the orbital elements of a comet, the main effect is the chaos resulting from close encounters with planets, and this effect is dominant whatever the choice of the integrator or the presence or not of non-gravitational forces. We estimated the dynamical lifetime of comet 46P to 133,000 years and 105,000 years for comet 67P. For the erosion, our model assumes a spherical nucleus made of a macroscopic mixture of pure dust and pure water ice, thermally decoupled. Erosion depends strongly on the active fraction and density of the nucleus. Assuming a density of 300 kg/m-3 and an average active fraction over time of 10%, we obtain an initial radius of 1.3 km for 46P and 2.8 km for 67P. The upper limit is obtained assuming a density of 100 kg/m-3 and an active fraction of 100%, and corresponds to an initial radius to 21 km for 46P and 25 km for 67P. Erosion due to water only occures under 4 AU and is restricted to short periods, !7% of the lifetime of a comet. We will apply this method to several other comets in the future