Relations between A_V and τ_V. Left panel: τ_V inferred from Eq. (11) for the targets plotted against log₁₀(). The heavy dashed curve shows the true solution, and the lighter weight solid curve shows not identity, but rather y  = log₁₀(τ_V)  =  x − log₁₀(1.086)+log₁₀(2), where x  =  log₁₀(). The factor of 2 takes into account that only probes the extinction along the line of sight, i.e., the front of the foreground screen, and the other factor is from Eq. (9). The difference of the two quantities τ_V and is negligible at low τ_V, but becomes significant for ≳0.3 − 0.5 mag. At  > 1 mag, τ_V “runs” away, relative to A_V. Right panel: Residuals of τ_V − 2 /1.086 versus log(A_V). Although difficult to appreciate in the left panel, 34% of the data points have residuals > 0.05, marked by the horizontal dashed line.