The H$α$ specific angular momentum of dwarf galaxies

Abstract: The relationship between a galaxy's specific angular momentum $j$ and its mass $M$, parameterised by $j \propto M<sup>α$ 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 $10<sup>{9}$ $M_{\odot}$. However, whether or not it holds for the low-mass dwarf galaxies, especially given their varied morphologies, remains uncertain. Here we use H$α$ observations of 49 star-forming dwarf galaxies from the SH$α$DE survey, as well as 20 high-mass `control' galaxies, to investigate the stellar $j_{<em>}$-$M_{</em>}$ relation down to masses below $10<sup>{6}$ M$<em>{\odot}$. We find that the star-forming dwarf galaxies follow the same $j</em>{<em>}$-$M_{</em>}$ relation as high-mass disk-like galaxies, with $α= 0.53 \pm{0.4}$, 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$α$DE survey. We find that the simulated dwarf galaxy population follows the extrapolated $j_{<em>}$-$M_{</em>}$ relation with a flattening and significant scatter towards lower $j_{<em>}$ values. Following the evolution of these galaxies, we find that dwarf galaxies experience a gradual loss in $j_{</em>}$ with time, while high-mass galaxies experience a sudden jump in $j_{<em>}$ before settling into a stable state. This dynamical evolution leads to a redshift dependence in $α$, with $α= 0.45$ at $z = 2$ and 0.55 at $z = 0$. Despite the apparent simplicity in the present-day $j_{</em>}$-$M_{*}$ relation over a wide mass range, the evolution of $j$ leading to this relation is complex and dynamic.

• The paper demonstrates that star-forming dwarf galaxies follow the same stellar specific-angular-momentum–mass relation as larger disks, with slope α = 0.53 ± 0.04 across five orders of magnitude down to 10⁵⋅⁷ M☉.
• Using high-resolution Hα integral-field spectroscopy of 49 dwarfs and 20 control galaxies, the study finds no elevated dwarf angular momentum, challenging earlier H I-based results while identifying higher j⋆ among galaxies with higher specific star-formation rates.
• The paper’s IllustrisTNG analysis reveals divergent histories: some galaxies rapidly form stable, high-angular-momentum disks near 10⁸–10⁹ M☉, while others remain irregular, though the physical driver and possible low-mass simulation effects remain unresolved.

Overview and motivation

The Fall relation — the scaling between a galaxy's stellar specific angular momentum $j_*$ and stellar mass $M_*$, $j_* \propto M_*^{\alpha}$ — is one of the most robust structural scaling relations in galaxy evolution, and its slope is thought to originate in the tidal torquing of dark matter halos. While well established for galaxies above roughly $10^{9}\,M_{\odot}$, its extension into the dwarf regime has been contested: several HI-based studies report dwarfs lying above an extrapolation of the high-mass relation, with a suggested transition mass near $10^{9.1}\,M_{\odot}$. Deeley et al. address this question with a new tracer, Hα emission, using the SHαDE survey of 49 star-forming dwarf galaxies observed with FLAMES/GIRAFFE on the VLT, supplemented by 20 higher-mass SAMI control galaxies. The key methodological advantage is spectral resolution: SAMI's resolution ($R \approx 4263$) is generally insufficient to measure velocity dispersions in dwarfs, whereas SHαDE was designed specifically to resolve the kinematics of low-mass systems.

The observational sample and measurement of $j_*$

The SHαDE dwarfs span $10^{5}$–$10^{8.5}\,M_{\odot}$ and were selected on Hα flux ($> 5\times10^{-16}$ erg s$M_*$0 cm$M_*$1 Å arcsec$M_*$2), ensuring measurable ionised-gas kinematics; the control sample spans $M_*$3–$M_*$4. Specific angular momentum is computed from Hα-derived velocity fields and continuum-derived exponential-disk mass maps, using the spaxel-weighted sum $M_*$5, extrapolated with a disk model to 3 effective radii ($M_*$6). Convergence is verified by fitting exponential curves to cumulative $M_*$7 profiles: 85 per cent of galaxies attain at least 80 per cent of their extrapolated total within the observational footprint. Monte Carlo realisations of the velocity maps, masses, and redshifts quantify the uncertainties, revealing occasionally bimodal posteriors for irregular galaxies driven by degeneracy between inclination and intrinsic rotation. The thin-disk deprojection assumption is tested by varying intrinsic thickness $M_*$8 up to 0.35 and by using optically derived ellipticities; the fitted slope is stable within uncertainties.

The Fall relation extended to $M_*$9

The central observational result is that the dwarf galaxies follow the same $j_* \propto M_*^{\alpha}$0–$j_* \propto M_*^{\alpha}$1 relation as high-mass disks, with a fit over the full sample of $j_* \propto M_*^{\alpha}$2 and $j_* \propto M_*^{\alpha}$3 (intrinsic orthogonal scatter 0.35 dex), in close agreement with the disk-galaxy relation of Posti et al. ($j_* \propto M_*^{\alpha}$4). The relation therefore holds across five orders of magnitude in stellar mass, down to $j_* \propto M_*^{\alpha}$5. This continuity is notable given the morphological diversity of the sample: while disk galaxies show smaller scatter about the relation (median offset 0.035, $j_* \propto M_*^{\alpha}$6) than compact or irregular galaxies ($j_* \propto M_*^{\alpha}$7–0.43), all three classes follow the same relation, implying that even disordered, turbulent dwarfs retain a substantial fraction of their halo's original angular momentum in their star-forming components. This contradicts earlier HI-based claims of elevated dwarf angular momenta (Butler et al.; Chowdhury & Chengalur; Kurapati et al.), which the authors attribute to extrapolated rotation curves and to samples biased towards high HI-fraction systems — consistent with the known correlation between gas fraction and $j_* \propto M_*^{\alpha}$8.

Two caveats qualify this result. First, residuals correlate with star formation rate (Spearman coefficient = 0.48, $j_* \propto M_*^{\alpha}$9): high-sSFR dwarfs sit at higher normalisation, mirroring the gas-fraction dependence seen at high masses. The slope is unchanged between high- and low-sSFR subsamples, but the sample contains no quenched dwarfs, so the relation for the quenched population remains untested. Second, $10^{9}\,M_{\odot}$0 is measured from the kinematics of young star-forming regions; if older stellar populations formed under different dynamical conditions, the derived values may differ from the true stellar $10^{9}\,M_{\odot}$1. A comparison against SAMI control galaxies shows SHαDE values systematically higher, attributable to SAMI's larger PSF (median FWHM 2.04 arcsec versus 0.88 arcsec), which blurs velocity gradients.

Mock observations and the simulated Fall relation

To connect observations to theory, the authors construct a matched sample of 512 galaxies from IllustrisTNG50, generate SKIRT radiative-transfer images and SHαDE-like mock Hα flux and velocity maps (weighting star-forming gas particles by their SFR-derived Hα emission, convolved with the instrumental PSF and footprint), and apply the identical analysis pipeline. Above $10^{9}\,M_{\odot}$2, the simulated galaxies reproduce the Posti et al. relation and the observed control sample. Below this mass, however, the simulated population shows increased scatter towards low $10^{9}\,M_{\odot}$3, with the upper envelope continuing along the observed relation — equivalently, a flattening of the relation. Mock measurements recover intrinsic $10^{9}\,M_{\odot}$4 of the star-forming component with a median bias of 0.08 dex and scatter of 0.39 dex, but systematically overestimate the intrinsic all-stellar $10^{9}\,M_{\odot}$5 (median bias 0.44 dex), indicating that the star-forming component has elevated angular momentum relative to the full stellar population, particularly at low mass. No bias is found with inclination, $10^{9}\,M_{\odot}$6, or effective radius.

Resolution is a persistent concern: galaxies of $10^{9}\,M_{\odot}$7 comprise only a few hundred stellar particles in TNG50. The authors test this by repeating the analysis with TNG100 and TNG300; the downward scatter begins at the same mass regardless of resolution, arguing against a purely numerical origin, though this inference is indirect and the authors explicitly caution that resolution effects cannot be excluded.

Divergent evolutionary histories of $10^{9}\,M_{\odot}$8

Tracing each galaxy's history through the simulation reveals two evolutionary pathways. All galaxies initially lose $10^{9}\,M_{\odot}$9 with time. Galaxies that grow beyond roughly $10^{9.1}\,M_{\odot}$0–$10^{9.1}\,M_{\odot}$1 subsequently undergo a rapid jump to a higher-$10^{9.1}\,M_{\odot}$2 state, accompanied by the collapse of gas into a coherently rotating disk and the formation of a stellar disk; after the transition, $10^{9.1}\,M_{\odot}$3 remains remarkably stable, with most high-mass galaxies settled by roughly 6 Gyr after the Big Bang. Galaxies that never reach this mass range continue losing angular momentum and remain irregular. Fitting the relation at every snapshot yields a redshift-dependent slope: $10^{9.1}\,M_{\odot}$4 at $10^{9.1}\,M_{\odot}$5, reaching its present-day value by $10^{9.1}\,M_{\odot}$6 and stabilising by $10^{9.1}\,M_{\odot}$7.

The physical cause of the transition is not conclusively identified. Mergers are ruled out for many transitioning galaxies, which are isolated; no change in star-formation burstiness is detected at the transition time; and the change in gravitational-potential shape proposed by Hopkins et al. from FIRE simulations is seen in some cases but not correlated across the full sample. The authors favour a combined scenario involving the end of bursty star formation, potential deepening, and a switch from misaligned cold-mode accretion to isotropic cooling from a hot halo, noting the qualitative agreement with the FIREBox analysis of Benavides et al., though the latter finds its disk-formation transition an order of magnitude higher in mass. The consistency of the transition mass range between TNG50 and TNG100 argues against a particle-number threshold, but this remains an open question.

Limitations and open questions

Several limitations bear directly on the interpretation of these results. The observational sample is star-formation selected, so the conclusions apply to star-forming dwarfs only; the relation for quenched dwarfs is unmeasured. The use of Hα kinematics as a proxy for stellar kinematics is validated at high masses but rests on the assumption that young stars trace the older population, which the simulation suggests becomes less accurate for dwarfs. The simulation-side findings — the low-$10^{9.1}\,M_{\odot}$8 flattening and the $10^{9.1}\,M_{\odot}$9 jump — are physically plausible but not demonstrably free of resolution effects, since TNG50 resolves $R \approx 4263$0 galaxies with only hundreds of stellar particles, and the resolution-invariance test across TNG50/100/300 is suggestive rather than decisive. The comparison to HI studies is approximate, as IllustrisTNG does not model atomic/molecular partitioning and the neutral-gas $R \approx 4263$1 is computed with a simplifying weighting. Finally, the mechanism of the $R \approx 4263$2 transition remains undetermined; the authors identify high-resolution observations of galaxies in the $R \approx 4263$3–$R \approx 4263$4 range, combined with resolved HI mapping, as the observations required to resolve it.

Conclusion

This paper demonstrates, using high-spectral-resolution Hα integral-field spectroscopy, that the stellar $R \approx 4263$5–$R \approx 4263$6 relation of star-forming galaxies is continuous from $R \approx 4263$7 down to below $R \approx 4263$8, with $R \approx 4263$9 — resolving prior tension between HI-based studies in favour of a scale-free relation. The result implies that tidal-torque acquisition and baryonic angular-momentum conservation operate independently of halo mass, even for shallow-potential, feedback-dominated, morphologically irregular systems. The accompanying IllustrisTNG50 analysis shows that the simplicity of the present-day relation conceals complex, divergent evolutionary histories, including a rapid transition to high-$j_*$0 disk-dominated states in the $j_*$1–$j_*$2 range whose physical driver remains unidentified.