How Low Can Q Go?

Gravitational instability plays a substantial role in the evolution of galaxies. Various schemes to include it in galaxy evolution models exist, generally assuming that the Toomre $Q$ parameter is self-regulated to $Q_\mathrm{crit}$, the critical $Q$ dividing stable from unstable conditions in a linear stability analysis. This assumption is in tension with observational estimates of $Q$ that find values far below any plausible value of $Q_\mathrm{crit}$. While the observations are subject to some uncertainty, this tension can more easily be relieved on the theoretical side by relaxing the common assumption that $Q\ge Q_\mathrm{crit}$. Based on observations of both $z\sim 2$ disks and local face-on galaxies, we estimate the effect of gravitational instability necessary to balance out every other physical process that affects $Q$. In particular we find that the disk's response to low $Q$ values can be described by simple functions that depend only on $Q$. These response functions allow galaxies to maintain $Q$ values below $Q_\mathrm{crit}$ in equilibrium over a wide range of parameters. Extremely low values of $Q$ are predicted when the gas surface density is greater than $\sim 10^3$ M$_\odot$ pc$^{-2}$, the rotation curve provides minimal shear, the orbital time becomes long, and/or when the gas is much more unstable than the stellar component. We suggest that these response functions should be used in place of the $Q\ge Q_\mathrm{crit}$ ansatz.