CO isotopolog line fluxes of viscously evolving disks

Context. Protoplanetary disks are thought to evolve viscously, where the disk mass - the reservoir available for planet formation - decreases over time as material is accreted onto the central star over a viscous timescale. Observations have shown a correlation between disk mass and the stellar mass accretion rate, as expected from viscous theory. However, this happens only when using the dust mass as a proxy of the disk mass; the gas mass inferred from CO isotopolog line fluxes, which should be a more direct measurement, shows no correlation with the stellar mass accretion rate.Aims. We investigate how (CO)-C-13 and (CO)-O-18 J= 3-2 line fluxes, commonly used as gas mass tracers, change over time in a viscously evolving disk and use them together with gas disk sizes to provide diagnostics of viscous evolution. In addition, we aim to determine if the chemical conversion of CO through grain-surface chemistry combined with viscous evolution can explain the CO isotopolog observations of disks in Lupus.Methods. We ran a series of thermochemical DALI models of viscously evolving disks, where the initial disk mass is derived from observed stellar mass accretion rates.Results. While the disk mass, M-disk, decreases over time, the (CO)-C-13 and (CO)-O-18 J= 3-2 line fluxes instead increase over time due to their optically thick emitting regions growing in size as the disk expands viscously. The (CO)-O-18 3-2 emission is optically thin throughout the disk for only for a subset of our models (M-* <= 0.2 M-circle dot and alpha(visc) >= 10(-3), corresponding to M-disk (t = 1 Myr) <= 10(-3 )M(circle dot)). For these disks the integrated (CO)-O-18 flux decreases with time, similar to the disk mass. Observed (CO)-C-13 and (CO)-O-18 3-2 fluxes of the most massive disks (M-disk greater than or similar to 5 x 10(-3) M-circle dot) in Lupus can be reproduced to within a factor of similar to 2 with viscously evolving disks in which CO is converted into other species through grain-surface chemistry with a moderate cosmic-ray ionization rate of zeta(cr) similar to 10(-17) s(-1). The (CO)-O-18 3-2 fluxes for the bulk of the disks in Lupus (with M-disk not less than or equal to 5 x 10(-3) M-circle dot) can be reproduced to within a factor of similar to 2 by increasing zeta(cr) to similar to 5 x 10(-17) -10(-16 )s(-1), although explaining the stacked upper limits requires a lower average abundance than our models can produce. In addition, increasing zeta(cr) cannot explain the observed (CO)-C-13 fluxes for lower mass disks, which are more than an order of magnitude fainter than what is predicted. In our models the optically thick (CO)-C-13 emission originates from a layer higher up in the disk (z/r similar to 0.25-0.4) where photodissociation stops the conversion of CO into other species. Reconciling the (CO)-C-13 fluxes of viscously evolving disks with the observations requires either efficient vertical mixing or low mass disks (M-dust less than or similar to 3 x 10(-5) M-circle dot) being much thinner and/or smaller than their more massive counterparts.Conclusions. The (CO)-C-13 model flux predominantly traces the disk size, but the (CO)-O-18 model flux traces the disk mass of our viscously evolving disk models if chemical conversion of CO is included. The discrepancy between the CO isotopolog line fluxes of viscously evolving disk models and the observations suggests that CO is efficiently vertically mixed or that low mass disks are smaller and/or colder than previously assumed.