... 8

 As a reminder, ring i is centered on $r_i \equiv i\ {\rm kpc}$ and extends between $(i-0.5)\ {\rm kpc}$ and $(i+0.5)~ {\rm kpc}$.

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... midpoint

As in Fig. 4, the midpoint of a box is defined as the point at middle radius, r=r_i, and middle longitude, $l = (l_{\rm min} + l_{\rm max}) /2$.

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