  We modify Gross's construction of mirror symmetry for $\mathbb{P}^2$ by
introducing a descendent tropical Landau-Ginzburg potential. The period
integrals of this potential compute a modification of Givental's J-function,
explicitly encoding a larger sector of the big phase space. As a byproduct of
this construction, new tropical methods for computing certain descendent
Gromov-Witten invariants are defined.
