Quokka-based understanding of outflows (QED) -- IV. Limitations of Hα as an outflow diagnostic

The presence of broad wings in the H$\alpha$ line is commonly used as a diagnostic of the presence and properties of galactic winds from star-forming galaxies. However, the accuracy of this approach has not been subjected to extensive testing. In this paper, we use high-resolution simulations of galactic wind launching to calibrate the extent to which broad H$\alpha$ wings can be used to infer the properties of galactic outflows. For this purpose, we analyse a series of high-resolution wind simulations from the QED suite spanning two orders of magnitude in star formation surface density ($\Sigma_\mathrm{SFR}$). We show that the broad component of H$\alpha$ emission correlates well with the wind mass flux at heights $\sim1$ kpc above the galactic plane, but that the correlation is poor at larger distances from the plane, and that even at 1 kpc the relationship between mass flux and surface brightness of broad H$\alpha$ is significantly sub-linear. The sub-linear scaling suggests that the electron column density in the wind increases systematically with outflow strength, and that the conventional assumption of constant electron density in the wind leads to a systematic overestimate of how steeply mass loading factors depend on $\Sigma_\mathrm{SFR}$. We provide empirical scaling relations that observers can apply to correct for this effect when converting H$\alpha$ measurements to mass outflow rates. Finally, we use synthetic observations of the density-diagnostic $[\mathrm{S_{II}}]\,\lambda\lambda6716,6731$ doublet to show that using this diagnostic only slightly improves estimates of wind outflow rates compared to the naive assumption of constant electron density, and performs significantly worse than the empirical correlation we provide.