tokens: Cyan and brown lines represent MI and MG, respectively, with v ∞ ≃ v flat = v tot ( r cut ), where r cut denotes the radii at which the integrals are cut. The inset is a magnification of N range = [0, 10]. Right panel : Δ Q MI − Δ Q MG as a function of radial range. The error bars are smaller than the points. From the left panel of Fig. 3 it is clear that MI and MG drift similarly as the range is decreased. As long as the drifts of MI and MG are approximately equal, the measured Q can be corrected similarly to the case of the acceleration scale in the previous section. The right panel of Fig. 3 shows Δ Q MI − Δ Q MG in the context of varying range. |Δ Q MI − Δ Q MG |≲0.01 ≃ const for N range ≳ 3, meaning that the measured Q can be corrected with Δ Q (range) = Δ Q MI ≃ Δ Q MG for galaxies which fulfill max[ r ] ≳ 3 r d . Table 2 lists Δ Q (range) for different ranges. The approach to the theoretical limit for increasing range is slow for the Kuzmin disk. For this reason Δ Q (range) = 0 is only reached for N range > 55 in this case. The corrections in Table 2 isolate the range effect from the acceleration effect. To mimick the observational situation, however, we have repeated the same exercise considering galaxies with max( g obs ) 0.15 a 0 , applying the corrections in Table 1 . We find that the corresponding corrections due to N range are very similar to those given in Table 2 . Table 2. Range corrections. Notes. The uncertainty on Δ Q (range) is negligible. 2.3. Sampling resolution In order to gauge the effect of the spatial resolution of the observations, Q is calculated as a function of the spacing between sampled points in units of r d . The sampling for each galaxy is performed in the range r j ∈ [0.01, 200] r d . In order to compute Q from a discrete set of points, the integral of Eq. ( 2 ) is discretized viz with where v f ≃ v ∞ being the asymptotically flat velocity which is determined as the arithmetic mean between the chain of points that are within 5% of the arithmetic mean of the two outermost points (similarly to Lelli et al. 2016 ). If the third outermost point is not within 5% of the arithmetic mean of the two outermost points, the galaxy is not assigned an asymptotically flat velocity. The left panel of Fig. 4 shows the arithmetic mean of Q calculated via Eq. ( 7 ) – as predicted by MI (black) and MG (blue) – of the mock data for which max( g obs )≤0.01 a 0 as a function of the spacing between the sampled points in units of r d . As expected, Q approaches the theoretical value for extremely small spacings ( r 0.1 r d ) and progressively increases for larger spacing between the sampled points. Fig. 4. Left panel : arithmetic mean of Q for all mock galaxies with max[ g obs ]≤0.01 a 0 as a function of spacing between sampled points. Black and blue lines represent MI and MG, respectively. Right panel : Δ Q MI − Δ Q MG as a function of spacing between sampled points. From the right panel of Fig. 4 it is clear that the increase for
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