The H$\alpha$ specific angular momentum of dwarf galaxies
The relationship between a galaxy's specific angular momentum $j$ and its mass $M$, parameterised by $j \propto M^\alpha$ and known as the Fall relation, has emerged as a fundamental scaling relation reflecting key physical and morphological properties of galaxies. This relation has been well studied for galaxies with masses above $10^{9}$ $M_{\odot}$. However, whether or not it holds for the low-mass dwarf galaxies, especially given their varied morphologies, remains uncertain. Here we use H$\alpha$ observations of 49 star-forming dwarf galaxies from the SH$\alpha$DE survey, as well as 20 high-mass `control' galaxies, to investigate the stellar $j_{*}$-$M_{*}$ relation down to masses below $10^{6}$ M$_{\odot}$. We find that the star-forming dwarf galaxies follow the same $j_{*}$-$M_{*}$ relation as high-mass disk-like galaxies, with $\alpha = 0.53 \pm{0.4}$, demonstrating that the relation holds across 5 orders of magnitude in mass. We then select a matching sample from the IllustrisTNG cosmological simulation and create mock observations resembling the SH$\alpha$DE survey. We find that the simulated dwarf galaxy population follows the extrapolated $j_{*}$-$M_{*}$ relation with a flattening and significant scatter towards lower $j_{*}$ values. Following the evolution of these galaxies, we find that dwarf galaxies experience a gradual loss in $j_{*}$ with time, while high-mass galaxies experience a sudden jump in $j_{*}$ before settling into a stable state. This dynamical evolution leads to a redshift dependence in $\alpha$, with $\alpha = 0.45$ at $z = 2$ and 0.55 at $z = 0$. Despite the apparent simplicity in the present-day $j_{*}$-$M_{*}$ relation over a wide mass range, the evolution of $j$ leading to this relation is complex and dynamic.