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Background: The $^{15}$O($\alpha ,\gamma$)$^{19}$Ne bottleneck reaction inType I x-ray bursts is the most important thermonuclear reaction rate toconstrain experimentally, in order to improve the accuracy of burst light-curvesimulations. A proposed technique to determine the thermonuclear rate of thisreaction employs the $^{20}$Mg($\beta p\alpha$)$^{15}$O decay sequence. The key$^{15}$O($\alpha ,\gamma$)$^{19}$Ne resonance at an excitation of 4.03 MeV isnow known to be fed in $^{20}$Mg($\beta p\gamma$)$^{19}$Ne; however, theenergies of the protons feeding the 4.03 MeV state are unknown. Knowledge ofthe proton energies will facilitate future $^{20}$Mg($\beta p \alpha$)$^{15}$Omeasurements.  Purpose: To determine the energy of the proton transition feeding the 4.03MeV state in $^{19}$Ne.  Method: A fast beam of $^{20}$Mg was implanted into a plastic scintillator,which was used to detect $\beta$ particles. 16 high purity germanium detectorswere used to detect $\gamma$ rays emitted following $\beta p$ decay. A MonteCarlo method was used to simulate the Doppler broadening of $^{19}$Ne $\gamma$rays and compare to the experimental data.  Results: The center of mass energy between the proton and $^{19}$Ne, feedingthe 4.03 MeV state, is measured to be 1.21${^{+0.25}_{-0.22}}$ MeV,corresponding to a $^{20}$Na excitation energy of 7.44${^{+0.25}_{-0.22}}$ MeV.Absolute feeding intensities and $\gamma$-decay branching ratios of $^{19}$Nestates were determined including the 1615 keV state. A new $\gamma$ decaybranch from the 1536 keV state in $^{19}$Ne to the ground state is reported.The lifetime of the 1507 keV state in $^{19}$Ne is measured to be4.3${^{+1.3}_{-1.1}}$ ps resolving discrepancies in the literature. Conflicting$^{20}$Mg($\beta p$) decay schemes in published literature are clarified.

Doppler Broadening in $^{20}$Mg($\beta p\gamma$)$^{19}$Ne Decay