CH_3OH as a user-friendly density probe: Calibration and beyond

Almost all the physics of star formation critically depends on the number density of the molecular gas involved. However, the methods to estimate this keystone property often rely on very uncertain assumptions about the geometry of the molecular fragment, or depend on overly simplistic, uniform models, or require time-expensive observations to simultaneously constrain the gas temperature as well. An easy-to-use method to observationally derive the number density that is valid under realistic conditions is conspicuously absent, causing an evident asymmetry in how accurately the volume density is estimated, and how often dedicated tracers are used, compared to the gas temperature. To fill this gap, we propose and calibrate a versatile diagnostic tool based on methanol spectral lines that greatly simplifies the inference of molecular number density. Methanol is abundant in both cold and hot gas, and has a dense spectrum of lines, which maximises observational efficiency. It can therefore be applied to a wide variety of scales, from entire clouds to protostellar discs, and both in our Galaxy and beyond. Moreover, this tool does not need to be tailored to the specific source properties (such as distance, temperature, and mass). We construct large grids of clump models and perform radiative transfer calculations to investigate the robustness of different line ratios as density probes with different assumptions, also in the presence of density and temperature gradients. We find that the line ratios of the ($2_K-1_K$) band transitions around $96.7$ GHz are able to fully constrain the average number density along the line of sight within a factor of two-three in the range ∼ 5 cm^ . The range can be extended down to a few times $10^3 cm^ $, when also using line ratios from the ($5_K-4_K$) and/or ($7_K-6_K$) bands, around 241.7 and 338.1 respectively. We provide the reader with practical analytic formulas and a numerical method for deriving volume density and its uncertainty from observed values of the line ratios. Thanks to our calibration of line ratios, we make the estimate of the number density much simpler, with an effort comparable or inferior to deriving excitation temperatures. By providing directly applicable recipes that do not require the creation of a full large velocity gradient model grid, but are equally accurate, we contribute to offsetting the disparity between these two fundamental parameters of the molecular gas. Applying our method to a sub-sample of sources from the ATLASGAL TOP100 we show that the material in the clumps is being compressed, and this compression accelerates in the latest stages.