Thermonuclear supernovae originating from the explosion of a white dwarf accreting mass from a companion star have been suggested as a site for the production of $p$ nuclides. Such nuclei are produced during the explosion, in layers enriched with seed nuclei coming from prior strong $s$ processing. These seeds are transformed to proton-richer isotopes mainly by photodisintegration reactions. Several thousand trajectories from a 2D explosion model were used in a Monte Carlo approach. Temperature-dependent uncertainties were assigned individually to thousands of rates varied simultaneously in post-processing in an extended nuclear reaction network. The uncertainties in the final nuclear abundances originating from uncertainties in the astrophysical reaction rates were determined. In addition to the 35 classical $p$ nuclides, abundance uncertainties were also determined for the radioactive nuclides $^{92}$Nb, $^{97,98}$Tc, $^{146}$Sm, and for the abundance ratios $Y$(${}^{92}$Mo)/$Y$(${}^{94}$Mo), $Y$(${}^{92}$Nb)/$Y$(${}^{92}$Mo), $Y$(${}^{97}$Tc)/$Y$(${}^{98}$Ru), $Y$(${}^{98}$Tc)/$Y$(${}^{98}$Ru), and $Y$(${}^{146}$Sm)/$Y$(${}^{144}$Sm), important for Galactic Chemical Evolution studies. Uncertainties found were generally lower than a factor of two, although most nucleosynthesis flows mainly involve predicted rates with larger uncertainties. The main contribution to the total uncertainties comes from a group of trajectories with high peak density originating from the interior of the exploding white dwarf. The distinction between low-density and high-density trajectories allows more general conclusions to be drawn, also applicable to other simulations of white dwarf explosions.
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