Constraining $^{30}$P($p$,$\gamma$)$^{31}$S for nova nucleosynthesis by measuring low-energy $^{31}$Cl $\beta$-delayed proton decays*

Reducing uncertainties in the $^{30}$P($p$,$\gamma$)$^{31}$S reaction rate is important for understanding nucleosynthesis in ONe novae. This thermonuclear rate influences the isotopic ratios and chemical abundances of nova ejecta, constrains peak explosive temperatures, and provides insights into mixing between the underlying white dwarf and hydrogen-rich material from the donor star. The reaction proceeds primarily through proton capture into narrow resonance states lying just above the proton-emission threshold in $^{31}$S. We determined the intensity of a very weak proton branch for a single, potentially dominant resonance by measuring $^{31}$Cl $\beta$-delayed proton decays using the GADGET system. We report one of the smallest proton branching ratios ever measured for such low-energy resonances. Combined with a theoretical lifetime, we a report a lower limit on the total reaction rate.