Disentangling protoplanetary disk gas mass and carbon depletion through combined atomic and molecular tracers

The total disk gas mass and elemental C, N, O composition of protoplanetary disks are crucial ingredients for our understanding of planet formation. Measuring the gas mass is complicated, since H$_2$ cannot be detected in the cold bulk of the disk and the elemental abundances with respect to hydrogen are degenerate with gas mass in all disk models. We present new NOEMA observations of CO, $^{13}$CO, C$^{18}$O and optically thin C$^{17}$O $J$=2-1 lines, and use additional high angular resolution Atacama Large Millimeter Array millimeter continuum and CO data to construct a representative model of LkCa 15. The transitions that constrain the gas mass and carbon abundance most are C$^{17}$O 2-1, N${_2}$H$^+$ 3-2 and HD 1-0. Using these three molecules we find that the gas mass in the LkCa 15 disk is $M_\mathrm{g}=0.01 ^{+0.01}_{-0.004} M_{\odot}$, a factor of six lower than estimated before. The carbon abundance is C/H = ($3 \pm 1.5) \times10^{-5}$, implying a moderate depletion of elemental carbon by a factor of 3-9. All other analyzed transitions also agree with these numbers, within a modeling uncertainty of a factor of two. Using the resolved \ce{C2H} image we find a C/O ratio of $\sim$1, which is consistent with literature values of H$_2$O depletion in this disk. The lack of severe carbon depletion in the LkCa 15 disk is consistent with the young age of the disk, but contrasts with the higher depletions seen in older cold transition disks. Combining optically thin CO isotopologue lines with N$_2$H$^+$ is promising to break the degeneracy between gas mass and CO abundance. The moderate level of depletion for this source with a cold, but young disk, suggests that long carbon transformation timescales contribute to the evolutionary trend seen in the level of carbon depletion among disk populations, rather than evolving temperature effects and presence of dust traps alone.