We report on the measurement of the beam asymmetry $Sigma$ for the reactions $vec{gamma}prightarrow peta$ and $vec{gamma}p rightarrow peta^{prime}$ from the GlueX experiment, using an 8.2--8.8 GeV linearly polarized tagged photon beam incident on a liquid hydrogen target in Hall D at Jefferson Lab. These measurements are made as a function of momentum transfer $-t$, with significantly higher statistical precision than our earlier $eta$ measurements, and are the first measurements of $eta^{prime}$ in this energy range. We compare the results to theoretical predictions based on $t$--channel quasi-particle exchange. We also compare the ratio of $Sigma_{eta}$ to $Sigma_{eta^{prime}}$ to these models, as this ratio is predicted to be sensitive to the amount of $sbar{s}$ exchange in the production. We find that photoproduction of both $eta$ and $eta^{prime}$ is dominated by natural parity exchange with little dependence on $-t$.
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