Redefining $Q$ for multi-component discs of stars and gas

We point out a fundamental mismatch in the $Q$ stability parameter for Galactic discs: Toomre's $Q = 1$ defines the boundary between axisymmetric stability/instability, while simulations, observations, and theoretical expectations apply $Q$ in the region $Q > 1$ as a measure for spiral activity (e.g. swing amplification), for which $Q$ has not been designed. We suggest to redefine $Q$ to keep $Q = 1$ as the stability boundary, but to equally yield a consistent map between $Q$ and the maximum swing amplification factor. Using the Goldreich-Lynden-Bell formalism, we find that particularly the $Q$ for gas discs has been mismatched, and should be redefined to close to the square of the traditional definition. We provide new formulations of $Q$ for simple, two-component, and multi-component discs, including a discussion of vertically extended discs, providing a simple iterative formula for which we also provide code. We find $Q \approx 1.58$ for the Solar Neighbourhood under our definition, closer to results from simulations. We compare the Milky Way and M74, showing that, consistent with observations, the theory suggests a higher $m$ number for the Milky Way (arguing against a 2-arm pattern) for stellar-dominated patterns. Gas instability arises at much smaller scales ($m \gtrsim 10$), and we link both M74's gas pattern and local spurs in the Milky Way to this gas instability rather than stellar spiral arms.