How large is a disk -- what do protoplanetary disk gas sizes really mean?

It remains unclear what mechanism is driving the evolution of protoplanetary disks. Direct detection of the main candidates, either turbulence driven by magnetorotational instability or magnetohydrodynamical disk winds, has proven difficult, leaving the time evolution of the disk size as one of the most promising observables able to differentiate between these two mechanisms. But to do so successfully, we need to understand what the observed gas disk size actually traces. We studied the relation between $R_{\rm CO,\ 90\%}$, the radius that encloses 90% of the $^{12}$CO flux, and $R_c$, the radius that encodes the physical disk size, in order to provide simple prescriptions for conversions between these two sizes. For an extensive grid of thermochemical models we calculate $R_{\rm CO,\ 90\%}$ from synthetic observations and relate properties measured at this radius, such as the gas column density, to bulk disk properties, such as $R_c$ and the disk mass $M_{\rm disk}$. We found an empirical correlation between the gas column density at $R_{\rm CO,\ 90\%}$ and disk mass: $N_{\rm gas}(R_{\rm CO,\ 90\%}) \approx 3.73\times10^{21}(M_{\rm disk}/\mathrm{M}_{\odot})^{0.34}\ \mathrm{cm}^{-2}$. Using this correlation we derive an analytical prescription of $R_{\rm CO,\ 90\%}$ that only depends on $R_c$ and $M_{\rm disk}$. We derive $R_c$ for disks in Lupus, Upper Sco, Taurus and DSHARP, finding that disks in the older Upper Sco region are significantly smaller ($\langle R_c \rangle$ = 4.8 au) than disks in the younger Lupus and Taurus regions ($\langle R_c \rangle$ = 19.8 and 20.9 au, respectively). This temporal decrease in $R_c$ goes against predictions of both viscous and wind-driven evolution, but could be a sign of significant external photoevaporation having truncated disks in Upper Sco.