  We compute an asymptotic expression for the Arakelov self-intersection number
of the relative dualizing sheaf of Edixhoven's minimal regular model for the
modular curve $X_0(p^2)$ over $\mathbb{Q}$. The computation of the
self-intersection numbers are used to prove effective Bogolomov conjecture for
the semi-stable models of modular curves $X_0(p^2)$ and obtain a bound on the
stable Faltings height for those curves in a companion article
arXiv:1802.06968.
