Probabilistic galeeactic dynamics - I. The Sun and GJ 710 with Monte Carlo, linearized, and unscented treatments

Abstract

Deterministic galactic dynamics is impossible due to a lack of precise knowledge of the initial conditions and galactic potential. Instead of treating stellar orbits deterministically, we integrate not only the mean but also the covariance of a stellar orbit in the Galaxy. As a test case, we study the probabilistic dynamics of the Sun and the star GJ 710 which is expected to cross the Oort Cloud in 1.3 Myr. We find that the uncertainty in the galactic model and the Sun's initial conditions are important for understanding such stellar close encounters. Our study indicates significant uncertainty in the solar motion within 1 Gyr and casts doubt on claims of a strict periodic orbit. In order to make such calculations more practical, we investigate the utility of the linearized and unscented transformations as two efficient schemes relative to a baseline of Monte Carlo calculations. We find that the linearized transformation predicts the uncertainty propagation as precisely as the Monte Carlo method for a few million years at least 700 times faster. Around an order of magnitude slower, the unscented transformation provides relative uncertainty propagation to a very high precision for tens of millions of years. The first order linearized transformation provides an efficient method which works to Gyr time-scales for small initial uncertainty problems and for propagation over hundreds of million years for larger initial uncertainty problems.