The Climate Evolution of Earth's Distant Future and implications for eta Earth

Earth's climate has changed in many ways over the past 4+ gigayears (Gyr), while mostly sustaining temperate conditions via volatile cycling.  This is remarkable given that the Sun's luminosity has changed by almost 30% in 4 Gyr.  The Earth's rotation rate has also changed by a factor of nearly 2 due to the receeding of the moon from Earth and changing bathymetry affecting tidal dissipation.  The climate of deep time future Earth (+1-3 Gyr) has seldom been explored, but one can use Earth's distant past to help inform us. The Sun's luminosity will continue to increase, while the moon's orbit will continue to grow affecting tidal dissipation in whatever bathymetry the Earth has in the future.  Using the ROCKE-3D climate model and VPlanet orbital dynamics components we attempt to model the future climate of Earth and how it might inform us about similar worlds orbiting nearby stars. For example, in one modeled dynamical scenario 1.9Gyr into the future the Earth's length of day (LoD) will increase to 46 days, while it's obliquity will approach zero. The global mean surface temperature (GMST) will only be 7.6C. If we choose a less dissipative scenario we find a LoD=1.5 days, an obliquity of 27.5, and a GMST=40C!  Will Earth eventually enter a moist and then a runaway greenhouse, or will it remain a temperate world until the Sun's red giant phase engulfs it in another 5 gigayears?  We will attempt to provide some answers to these questions.