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.
