The Temporal Requirements of Directly Observing Self-gravitating Spiral Waves in Protoplanetary Disks with ALMA

We investigate how the detectability of signatures of self-gravity in a protoplanetary disk depends on its temporal evolution. We run a one-dimensional model for secular timescales to follow the disk mass as a function of time. We then combine this with three-dimensional global hydrodynamics simulations that employ a hybrid radiative transfer method to approximate realistic heating and cooling. We simulate ALMA continuum observations of these systems and find that structures induced by the gravitational instability (GI) are readily detectable when q = Mdisk/M*  0.25 and Router  100 au. The high accretion rate generated by gravito-turbulence in such a massive disk drains its mass to below the detection threshold in ∼104 years, or approximately 1% of the typical disk lifetime. Therefore, disks with spiral arms detected in ALMA dust observations, if generated by self-gravity, must either be still receiving infall to maintain a high q value, or have just emerged from their natal envelope. Detection of substructure in systems with lower q is possible, but would require a specialist integration with the most extended configuration over several days. This disfavors the possibility of GI-caused spiral structure in systems with q < 0.25 being detected in relatively short integration times, such as those found in the DSHARP ALMA survey. We find no temporal dependence of detectability on dynamical timescales.