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Background: In stars, carbon is produced exclusively via the $3\alpha$process, where three $\alpha$ particles fuse to form $^{12}$C in the excitedHoyle state, which can then decay to the ground state. The rate of carbonproduction in stars depends on the radiative width of the Hoyle state. Theradiative width can be deduced by combining three separately measuredquantities, one of which is the $E0$ decay branching ratio. The $E0$ branchingratio can be measured by exciting the Hoyle state in the $^{12}$C$(p,p')$reaction and measuring the pair decay of its Hoyle state and first $2^+$ state.  Purpose: To reduce the uncertainties in the carbon production rate in theuniverse by measuring a set of proton angular distributions for the populationof the Hoyle state ($0^+_2$) and $2^+_1$ state in $^{12}$C in $^{12}$C$(p,p')$reactions between 10.20 and 10.70 MeV, used in the determination of the $E0$branching ratio of the Hoyle state.  Method: Proton angular distributions populating the ground, first $2^+$, andthe Hoyle states in $^{12}$C were measured in $^{12}$C(p,p') reactions with asilicon detector array covering $22^\circ<\theta<158^\circ$ in 14 energy stepsbetween 10.20 and 10.70 MeV with a thin ($60\ \mu$g/cm$^2$) $^{nat}$C target.  Results: Total cross-sections for each state were extracted and thepopulation ratio between the $2^+_1$ and Hoyle state determined at each energystep. By appropriately averaging these cross-sections and taking their ratio,the equivalent population ratio can be extracted applicable for any thick$^{12}$C target used in pair-conversion measurements.  Conclusions: We present a general data set of high-precision $^{12}$C$(p,p')$cross-sections that make uncertainties resulting from the population of the$2^+_1$ and $0^+_2$ states by proton inelastic scattering negligible for anyfuture measurements of the $E0$ branching ratio in $^{12}$C.

High precision proton angular distribution measurements of $^{12}$C(p,p') for determination of the $E0$ decay branching ratio of the Hoyle state