Dynamics of Planetary Systems Within Star Clusters: Aspects of the Solar
  System's Early Evolution
  Most planetary systems -- including our own -- are born within stellar
clusters, where interactions with neighboring stars can help shape the system
architecture. This paper develops an orbit-averaged formalism to characterize
the cluster's mean-field effects as well as the physics of long-period stellar
encounters. Our secular approach allows for an analytic description of the
dynamical consequences of the cluster environment on its constituent planetary
systems. We analyze special cases of the resulting Hamiltonian, corresponding
to eccentricity evolution driven by planar encounters, as well as hyperbolic
perturbations upon dissipative disks. We subsequently apply our results to the
early evolution of our solar system, where the cluster's collective potential
perturbs the solar system's plane, and stellar encounters act to increase the
velocity dispersion of the Kuiper belt. Our results are two-fold: first, we
find that cluster effects can alter the mean plane of the solar system by
$\lesssim1\deg$, and are thus insufficient to explain the $\psi\approx6\deg$
obliquity of the sun. Second, we delineate the extent to which stellar flybys
excite the orbital dispersion of the cold classical Kuiper belt, and show that
while stellar flybys may grow the cold belt's inclination by the observed
amount, the resulting distribution is incompatible with the data.
Correspondingly, our calculations place an upper limit on the product of the
stellar number density and residence time of the sun in its birth cluster,
$\eta\,\tau\lesssim2\times10^4\,$Myr/pc$^3$.
