Thermal conductivity of porous aggregates.

$\mathit{Context.}$ The thermal conductivity of highly porous dust aggregates is a key parameter for many subjects in planetary science; however, it is not yet fully understood. $\mathit{Aims.}$ In this study, we investigate the thermal conductivity of fluffy dust aggregates with filling factors of less than $10^{-1}$. $\mathit{Methods.}$ We determine the temperature structure and heat flux of the porous dust aggregates calculated by $N$-body simulations of static compression in the periodic boundary condition. $\mathit{Results.}$ We derive an empirical formula for the thermal conductivity through the solid network $k_{\rm sol}$ as a function of the filling factor of dust aggregates $\phi$. The results reveal that $k_{\rm sol}$ is approximately proportional to ${\phi}^{2}$, and the thermal conductivity through the solid network is significantly lower than previously assumed. In light of these findings, we must reconsider the thermal histories of small planetary bodies.