...)

The biggest uncertainties in the derived values of $\eta _0$ are in r, ${\rm d}r$ and $\beta$: e.g., if black body temperatures underestimate the true radius by a factor of 2 and the width of the ring is ${\rm d}r=0.5r$ then the $\eta _0$ values would have to be reduced by a factor of 10; changes to $\beta$ would increase $\eta _0$.

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