A dynamical study on the habitability of terrestrial exoplanets - I. Tidally evolved planet-satellite pairs

We investigate the obliquity and spin period of Earth-Moon-like systems after 4.5 Gyr of tidal evolution with various satellite masses (m(s) = 0.0025m(p) - 0.05m(p), where m(p) is the planet mass) and initial planetary obliquity (epsilon(0) = 0 degrees-175 degrees), and discuss their relations to the habitability of the planet. The satellite initially orbits in the planet's equatorial plane at similar to 4 planetary radii and the planet's initial rotation period is 5 h. The other tidal parameters are modelled after the Earth and Moon and we keep the satellite on a circular orbit. We find three possible outcomes: either (i) the system is still evolving, such as our own, (ii) the system is in the double synchronous state, with the planet's obliquity at either 0 degrees or 180 degrees, or (iii) the satellite has collided with the planet. The case (iii) occurs for initial planetary spins in the range epsilon(0) similar to 60 degrees-120 degrees. For other epsilon(0), the satellite survives. The transition between case (i) and (ii) is abrupt and occurs at slightly larger satellite mass (m(s) similar to 0.02m(p)) than the lunar mass. For higher masses the system is in the double synchronous state and the final planetary spin periods (P-p) are longer than 96 h. We also discuss the habitability of the planet in each case. We suggest that cases (ii) and (iii) are less habitable than case (i). Using results from models of giant impacts and satellite accretion, we found that the systems that mimic our own, i.e. with rotation period 12 < P-p < 48 h and current planetary obliquity epsilon(p) < 40 degrees or epsilon(p) > 140 degrees only represent 14 per cent of the possible outcomes. This estimate may only be reliable to within factors of a few, depending on how the probability is evaluated. Elser et al. conclude that the probability of a terrestrial planet having a heavy satellite is 13 per cent. Combining these results suggests that the probability of ending up with a system such as our own is of the order of 2 per cent.