  Most stars form in star clusters and stellar associated. To understand the
roles of star cluster environments in shaping the dynamical evolution of
planetary systems, we carry out direct $N$-body simulations of four planetary
systems models in three different star cluster environments with respectively
N=2k, 8k and 32k stars. In each cluster, an ensemble of initially identical
planetary systems are assigned to solar-type stars with $\sim 1 M_{\odot}$ and
evolved for 50~Myr. We found that following the depletion of protoplanetary
disks, external perturbations and planet-planet interactions are two driving
mechanisms responsible for the destabilization of planetary systems. The planet
survival rate varies from $\sim 95\%$ in the N=2k cluster to $\sim 60\%$ in the
N=32k cluster, which suggests that most planetary systems can indeed survive in
low-mass clusters, except in the central regions. We also find that planet
ejections through stellar encounters are cumulative processes, as only $\sim
3\%$ of encounters are strong enough to excite the eccentricity by $\Delta e
\geq 0.5$. Short-period planets can be perturbed through orbit crossings with
long-period planets. When taking into account planet-planet interactions, the
planet ejection rate nearly doubles, and therefore multiplicity contributes to
the vulnerability of planetary systems. In each ensemble, $\sim 0.2\%$ of
planetary orbits become retrograde due to random directions of stellar
encounters. Our results predict that young low-mass star clusters are promising
sites for next-generation planet surveys, yet low planet detection rates are
expected in dense globular clusters such as 47 Tuc. Nevertheless, planets in
denser stellar environments are likely to have shorter orbital periods, which
enhances their detectability.
