  Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group
$F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which
is a finite-dimensional infinite simplicial complex with some simplicial faces
missing. In this paper, we construct complete geodesic metrics and complete
piecewise smooth Riemannian metrics on $X_n$ which are invariant under
$\mathrm{Out}(F_n)$. One key ingredient is the identification of metric graphs
with tropical curves and the use of the tropical Jacobian map from the moduli
space of tropical curves to the moduli space of principally polarized tropical
abelian varieties.
