Bounding hadronic uncertainties in $c\to u$ decays

Time-dependent CP asymmetry measurements in $D\to h^+h^-$ decays, where $h=\pi$ or $\rho$ can be used to constrain the angle $\beta_c$ of the $cu$ unitarity triangle up to theoretical uncertainties. Here we discuss the theoretical uncertainty from penguin contributions that can be mitigated through the use of isospin analyses. We show that uncertainty from penguin pollution on a measurement of $\beta_c$ (or alternatively the mixing phase) in $D^0\to \pi^+\pi^-$ ($\rho^+\rho^-$) decays is $2.7^\circ$ ($4.6^\circ$). We also comment on the applicability of this method to $D^0\to\rho\pi$ decays for which measurements of weak phases with a precision below the one degree level may be possible.