Empirical Temperature Measurement in Protoplanetary Disks

Accurate measurement of temperature in protoplanetary disks is critical to understanding many key features of disk evolution and planet formation, from disk chemistry and dynamics, to planetesimal formation. This paper explores the techniques available to determine temperatures from observations of single, optically thick molecular emission lines. Specific attention is given to issues such as inclusion of optically thin emission, problems resulting from continuum subtraction, and complications of real observations. Effort is also made to detail the exact nature and morphology of the region emitting a given line. To properly study and quantify these effects, this paper considers a range of disk models, from simple pedagogical models, to very detailed models including full radiative transfer. Finally, we show how use of the wrong methods can lead to potentially severe misinterpretations of data, leading to incorrect measurements of disk temperature profiles. We show that the best way to estimate the temperature of emitting gas is to analyze the line peak emission map without subtracting continuum emission. Continuum subtraction, which is commonly applied to observations of line emission, systematically leads to underestimation of the gas temperature. We further show that once observational effects such as beam dilution and noise are accounted for, the line brightness temperature derived from the peak emission is reliably within 10-15% of the physical temperature of the emitting region, assuming optically thick emission. The methodology described in this paper will be applied in future works to constrain the temperature, and related physical quantities, in protoplanetary disks observed with ALMA.