  We herein report a theoretical study of the geometrical structure of porous
dust aggregates formed via ballistic cluster-cluster aggregation (BCCA). We
calculated the gyration radius $R_{\rm gyr}$ and the graph-based geodesic
radius $R_{\rm geo}$ as a function of the number of constituent particles $N$.
We found that $R_{\rm gyr} / r_{0} \sim N^{0.531 \pm 0.011}$ and $R_{\rm geo} /
r_{0} \sim N^{0.710 \pm 0.013}$, where $r_{0}$ is the radius of constituent
particles. Furthermore, we defined two constants that characterize the
geometrical structure of fractal aggregates: $D_{\rm f}$ and $\alpha$. The
definition of $D_{\rm f}$ and $\alpha$ are $N \sim {( R_{\rm gyr} / r_{0}
)}^{D_{\rm f}}$ and ${R_{\rm geo}} / {r_{0}} \sim {\left( {R_{\rm gyr}} /
{r_{0}} \right)}^{\alpha}$, respectively. Our study revealed that $D_{\rm f}
\simeq 1.88$ and $\alpha \simeq 1.34$ for the clusters of the BCCA.
  In addition, we also studied the filling factor dependence of thermal
conductivity of statically compressed fractal aggregates. From this study, we
reveal that the thermal conductivity of statically compressed aggregates $k$ is
given by $k \sim 2 k_{\rm mat} {( r_{\rm c} / r_{0} )} \phi^{(1 + \alpha) / (3
- D_{\rm f})}$, where $k_{\rm mat}$ is the material thermal conductivity,
$r_{\rm c}$ is the contact radius of constituent particles, and $\phi$ is the
filling factor of dust aggregates.
