Probabilistic galactic dynamics I - the Sun and GJ 710 with Monte Carlo, linearised and unscented treatments

Deterministic galactic dynamics is impossible due to the space-time randomness caused by gravitational waves. 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 linearised and unscented transformations as two efficient schemes relative to a baseline of Monte Carlo calculations. We find that the linearised 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. There exist a variety of problems in galactic dynamics which require the propagation of the orbital uncertainty for more than one or two objects and the first order linearised 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.