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A&A
Volume 697, May 2025
Article Number A87
Number of page(s) 11
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/202453290
Published online 08 May 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

Specific angular momentum (j = J/M) and mass (M) are two of the most fundamental properties describing any physical system, including galaxies (e.g. White 1984; Fall & Efstathiou 1980; Romanowsky & Fall 2012; Cimatti et al. 2019). Angular momentum in galaxies is thought to be acquired by gravitational tidal torques before virialisation, according to the tidal torque theory (Peebles 1969; White 1984). This theory predicts a scaling relation for the specific angular momentum of the dark matter haloes of the form j DM ∝ M DM 2 / 3 $ j_{\mathrm{DM}} \propto M_{\mathrm{DM}}^{2/3} $ (Fall et al. 1983; Shaya & Tully 1984; Heavens & Peacock 1988). Baryonic matter is expected to be subjected to the same tidal torques as the dark matter haloes since during the linear stage of structure formation the dark matter and primordial gas are still well mixed.

Observationally, Fall et al. (1983) first showed that both early- and late-type galaxies follow a scaling law in the stellar angular momentum (j*) versus stellar mass (M*) plane, called the Fall relation. Even though the normalisation for spiral galaxies is higher than that for early types, both classes follow a relationship of the form j * ∝ M * α $ j_{\ast} \propto M^{\alpha}_{\ast} $, with α ≈ 0.6. Later studies with more and better data confirmed the results of Fall et al. (1983), generally finding a power law with a slope of around 0.5–0.6 (Romanowsky & Fall 2012; Fall & Romanowsky 2018; Posti et al. 2018; Mancera Piña et al. 2021a), consistent (within the uncertainties) with the expectations from tidal torque theory for the dark matter haloes.

In addition to the stellar component, the angular momentum of the cold gas is also vital for galaxy evolution, but has been studied significantly less. In recent years, the neutral atomic hydrogen (H I) jHI − MHI relation has started to be studied using resolved interferometric observations of late-type massive and dwarf galaxies (Cortese et al. 2016; Chowdhury & Chengalur 2017; Kurapati et al. 2018; Mancera Piña et al. 2021a,b). This jHI − MHI relation is found to be significantly steeper than the Fall relation, jHI ∝ MHI (Kurapati et al. 2018, 2021; Mancera Piña et al. 2021a). In contrast, the picture for the molecular gas (H2) is largely incomplete due to the scarcity of deep and high-resolution CO data. In fact, H2 is often neglected when building the baryonic j − M relation, j bar ∝ M bar 0.6 $ j_{\mathrm{bar}}\propto M_{\mathrm{bar}}^{0.6} $ (e.g. Elson 2017; Kurapati et al. 2018; Murugeshan et al. 2020; Mancera Piña et al. 2021a). Obreschkow & Glazebrook (2014) estimated jH2 for a small sample of 16 nearby galaxies from the THINGS survey (Walter et al. 2008). However, they did not quantify the shape of the jH2 − MH2 relation, and their measurements show large scatter, which casts doubts on the existence of a relation.

Considering that angular momentum regulates galaxy sizes, morphologies, and gas content (Fall et al. 1983; Mo et al. 1998; Romanowsky & Fall 2012; Pezzulli & Fraternali 2016; Mancera Piña et al. 2021b; Hardwick et al. 2022; Elson 2024) and that molecular gas is the primary fuel for star formation (Kennicutt 1998; Bigiel et al. 2008), lacking a quantification of the jH2 − MH2 relation is a significant gap in our attempts to understand the physical properties of the interstellar medium (ISM) in galaxies through their evolutionary pathways (Lilly et al. 2013; Lagos et al. 2017; Catinella et al. 2018; Tacconi et al. 2020; Saintonge & Catinella 2022). Moreover, since processes such as gas accretion, feedback, mergers, and dynamical friction can dramatically alter a galaxy’s angular momentum distribution (Teklu et al. 2015; Stevens et al. 2018; Sweet et al. 2020), quantifying the molecular j − M relation can also offer important constraints for testing and refining models and simulations of galaxy evolution.

In addition, characterising the jH2 − MH2 relation at z = 0 is a crucial benchmark for high-redshift observations. To date, only j* has been studied up to z ≈ 1 − 2 (see Marasco et al. 2019; Sweet et al. 2019; Bouché et al. 2021; Espejo Salcedo et al. 2022; Mercier et al. 2023); there is no consensus on its evolution. The molecular phase is vital in order to study the gas counterpart since H I can only be detected in emission up to z ≲ 0.2 (Gogate et al. 2020; Ponomareva et al. 2021). For example, thanks to ALMA, it is now becoming common to trace the molecular gas kinematics and distribution at z ≳ 3 (e.g. Rizzo et al. 2020, 2023; Rowland et al. 2024). To benchmark the angular momentum content in those young galaxies and their evolution through time, determining the local jH2 − MH2 is imperative. In this paper, we exploit recent deep high-resolution molecular gas observations of 51 local disc galaxies from the PHANGS–ALMA survey to characterise the jH2 − MH2 relation for the first time.

This paper is organised as follows. Section 2 describes our galaxy sample, and Sect. 3 outlines the methods for determining the specific angular momentum. Our results are presented in Sect. 4 and Sect. 5. Finally, we summarise our conclusions in Sect. 6. Throughout the text, we adopt a ΛCDM cosmology with H0 = 70 km s−1 Mpc−1, Ωm = 0.3, and ΩΛ = 0.7.

2. Galaxy sample

Our sample was drawn from the PHANGS-ALMA survey (Leroy et al. 2021a,b), mapping CO J = 2 → 1 emission, hereafter CO(2 − 1), in 90 nearby (d ≲ 20 Mpc) star-forming galaxies at ∼1″ resolution. As detailed below, in selecting our sample, we included galaxies with available CO rotation curves and surface density profiles derived by the PHANGS-ALMA collaboration.

2.1. Rotation curves

We used the rotation curves derived by Lang et al. (2020) for the PHANGS-ALMA sample, obtained using the commonly employed tilted-ring model of the observed velocity field (Rogstad et al. 1974; Bosma 1978; Begeman 1987). Lang et al. (2020) provided high-resolution (150 pc) rotation curves for 67 galaxies with ordered rotation.

To improve the radial coverage of the kinematics of our galaxy sample, we supplemented the CO rotation curves with literature H I rotation curves when available. H I extends well beyond the optical and CO emission of galaxies, making it an effective tracer for galaxy dynamics at the outer radii; we note that H I and CO are expected to corotate, and this has been corroborated with observations in nearby galaxies (Bacchini et al. 2020; Laudage et al. 2024). Specifically, we supplemented the CO rotation curves of NGC 2903, NGC 3351, NGC 3521, and NGC 3627, NGC 1365, NGC 3621, NGC 4535, and NGC 4536, with H I rotation velocities (also from tilted-ring models) from Di Teodoro & Peek (2021) and Ponomareva et al. (2016). In addition, we included H I rotation curves for three galaxies without available CO rotation curves, bringing the total sample to 70 galaxies. We obtained the H I rotation curves for NGC 253 from Mancera Piña et al. (2022), NGC 300 from Mancera Piña et al. (2021a), and NGC 7793 from Bacchini et al. (2019). Where needed, we corrected the rotation curve data to match the distance and inclination reported in Lang et al. (2020) to ensure homogeneity.

2.2. Gas surface densities

We used the molecular gas surface density profiles provided by Sun et al. (2022) for 66 of our 70 galaxies. The molecular gas surface density profiles (ΣH2) were derived from the integrated CO(2-1) line intensity I CO ( 2 − 1 ) $ I_{\mathrm{CO}}^{(2-1)} $ via

Σ H 2 M ⊙ pc − 2 = α CO ( 1 − 0 ) M ⊙ pc − 2 1 R 21 I CO ( 2 − 1 ) K km/s , $$ \begin{aligned} \frac{\Sigma _{\text{H}_2}}{M_{\odot } \, \text{ pc}^{-2}}&= \frac{\alpha _{\text{CO}}^{(1-0)}}{M_{\odot } \, \text{ pc}^{-2}} \frac{1}{R_{21}} \frac{I_{\text{CO}}^{(2-1)}}{\text{ K} \text{ km/s}}, \end{aligned} $$(1)

where R21 = 0.65 is the adopted CO(2–1) to CO(1–0) line ratio (den Brok et al. 2021; Leroy et al. 2022), and α CO ( 1 − 0 ) $ {\alpha_{\text{CO}}^{(1-0)}} $ is the CO-to-H2 conversion factor, for the CO(1-0) line.

The quantity α CO ( 1 − 0 ) $ \alpha_{\mathrm{CO}}^{(1-0)} $ remains challenging to constrain precisely due to its sensitivity to local conditions such as gas density, temperature, and metallicity, which vary significantly across galactic environments (Schinnerer & Leroy 2024). The database of Sun et al. (2022) provides four alternative prescriptions for the PHANGS-ALMA sample, considering various factors that impact α CO ( 1 − 0 ) $ {\alpha_{\text{CO}}^{(1-0)}} $.

  1. The fiducial conversion factor used by Sun et al. (2022), which follows a metallicity-dependent calibration first introduced by Sun et al. (2020, see also e.g. Accurso et al. 2017; Schinnerer & Leroy 2024):

    α CO, S20 = 4.35 Z ′ − 1.6 M ⊙ ( K km s − 1 pc 2 ) − 1 . $$ \begin{aligned} \alpha _{\text{CO,} \text{ S20}}&= 4.35 \, Z^{\prime -1.6} \, M_{\odot } \, (\text{ K} \text{ km} \text{ s}^{-1} \, \text{ pc}^2)^{-1}. \end{aligned} $$(2)

  2. The commonly used Galactic average value (see Bolatto et al. 2013; Sandstrom et al. 2013):

    α CO, MW = 4.35 M ⊙ ( K k m s − 1 p c 2 ) − 1 . $$ \begin{aligned} \alpha _{\text{CO,} \text{ MW}}= 4.35 \, M_{\odot }\, (\text {K km s}^{-1}\, pc^{2}) ^{-1}~. \end{aligned} $$(3)

  3. A calibration that takes into account both metallicity and line intensity from the numerical work by Narayanan et al. (2012):

    α CO, N12 M ⊙ pc − 2 ( K km s − 1 ) − 1 = 8.5 Z ′ − 0.65 min [ 1 , 1.5 × ( 〈 I CO ( 2 − 1 ) 〉 K km s − 1 ) − 0.32 ] . $$ \begin{aligned} \frac{\alpha _{\text{CO,} \text{ N12}}}{M_{\odot } \, \text{ pc}^{-2} \, (\text{ K} \text{ km} \text{ s}^{-1})^{-1}}&= 8.5 \, Z^{\prime -0.65} \, \min \left[1, \, 1.5 \times \left( \frac{\left\langle I_{\text{CO}(2-1)} \right\rangle }{\text{ K} \text{ km} \text{ s}^{-1}} \right)^{-0.32} \right]~. \end{aligned} $$(4)

    Here the dependence on the line intensity follows from the effect of varying gas temperature and velocity dispersion in galaxies.

  4. Following Bolatto et al. (2013), a calibration considering a dependence on the molecular cloud surface density and total baryonic surface density:

    α CO, B13 M ⊙ pc − 2 ( K km s − 1 ) − 1 = 2.9 exp ( 40 M ⊙ pc − 2 Z ′ 〈 Σ mol, pix 〉 ) ( Σ total 100 M ⊙ pc − 2 ) − γ $$ \begin{aligned} \frac{\alpha _{\text{CO,} \text{ B13}}}{M_{\odot } \, \text{ pc}^{-2} \, (\text{ K} \text{ km} \text{ s}^{-1})^{-1}}&= 2.9 \exp \left( \frac{40 \, M_{\odot } \, \text{ pc}^{-2}}{Z\prime \left\langle \Sigma _{\text{mol,} \text{ pix}} \right\rangle } \right) \left( \frac{\Sigma _{\text{total}}}{100 \, M_{\odot } \, \text{ pc}^{-2}} \right)^{-\gamma } \end{aligned} $$(5)

    with γ = { 0.5 , if Σ total > 100 M ⊙ pc − 2 0 , otherwise . $$ \begin{aligned} \text{ with} \quad \gamma&= {\left\{ \begin{array}{ll} 0.5,&\text{ if} \Sigma _{\text{total}} > 100 \, M_{\odot } \, \text{ pc}^{-2} \\ 0,&\text{ otherwise} \end{array}\right.}. \end{aligned} $$(6)

    Here Σmol, pix gives the molecular cloud surface density and Σtotal the total surface density (see Sun et al. 2022).

The difference in αCO between the different calibrations often differs by more than the typical uncertainties quoted in the above references. To incorporate these different prescriptions to obtain αCO, we adopted the following approach. For each galaxy, at each radius, we combine the available αCO, MW, αCO, S20, αCO, B13, and αCO, N12 to construct a master conversion factor. The master conversion factor is derived by averaging the four prescriptions, and we adopt the standard deviation as uncertainty. The master αCO is then used in Eq. 1 to derive our final ΣH2(R) profile, with uncertainties determined through standard error propagation. Combining the calibrations into a master αCO is a conservative approach that manages to capture realistic uncertainties better; this is demonstrated in the right panel of Fig. 1, where the resulting surface density profiles for the various α CO ( 1 − 0 ) $ \alpha_{\mathrm{CO}}^{(1-0)} $ calibrations are shown for IC 5273.

thumbnail Fig. 1.

Example of the functional forms fitted to the rotational velocities and molecular gas surface density profiles (in this case for the galaxy IC 5273). Left: Observed rotation curve (grey markers with error bars) and the best-fit model (solid blue curve) with its 1σ and 2σ uncertainties represented by the shaded region. Right: Molecular gas surface density profile assuming our master αCO conversion factor (grey markers with error bars). The best-fit model is shown as a solid red curve, with its 1σ and 2σ uncertainties indicated by the shaded region. Profiles derived using alternative CO-to-H2 conversion factors (S20, MW, N12) are overplotted for comparison (B13 was not available for IC 5273). As can be seen, our profile represents a good compromise between the different calibrations and incorporates realistic uncertainties.

From our 66 galaxies we also excluded NGC 1512, NGC 2566, NGC 2775, NGC 4569, and NGC 4826 since their surface density profiles are too irregular to allow a robust quantification of their angular momentum. Similarly, the rotation curve of NGC 5068 is too compact and shows no signs of flattening, making it unsuitable for our purposes. Considering this, we ended up with a final sample of 60 galaxies, spanning the mass ranges 109 ≲ M*/M⊙ ≲ 1011, 108 ≲ MH2/M⊙ ≲ 1010, and 10−2 ≲ MH2/M* ≲ 10−0.75. In Table A.1 we present the final sample with their Hubble types, stellar masses, and distances adopted from Leroy et al. (2021b).

3. Computing j

In this section we describe how we computed the specific angular momentum for our galaxy sample by exploiting their rotation curves and surface density profiles. For a disc, the specific angular momentum within a radius R is described by

j ( < R ) = J ( < R ) M ( < R ) = 2 π ∫ 0 R R ′ 2 Σ ( R ′ ) V ( R ′ ) d R ′ 2 π ∫ 0 R R ′ Σ ( R ′ ) d R ′ , $$ \begin{aligned} j(< R) = \frac{J( < R)}{M( < R)} = \frac{2\pi \int _{0}^{R} R^{\prime 2} \Sigma (R\prime ) V(R\prime ) \, \mathrm{d}R\prime }{2\pi \int _{0}^{R} R\prime \Sigma (R\prime ) \, \mathrm{d}R\prime }\, , \end{aligned} $$(7)

where V(R) is the rotation curve and Σ(R) the surface density profile. For a typical disc galaxy, most of the angular momentum can be measured by tracing the kinematics and mass surface density out to R ≥ 2Re (Romanowsky & Fall 2012; Posti et al. 2018; Mancera Piña et al. 2021a), where Σ(R) becomes small and V(R) has flattened. Measuring j from insufficiently extended data can lead to its underestimation (see Sect. 3.3). Exploiting the empirical facts that rotation curves are flat at large radii (Bosma 1978; Begeman 1987; de Blok et al. 2008; Kuzio de Naray et al. 2008) and gas surface densities decay exponentially (Bigiel et al. 2008; Wang et al. 2016), we extrapolate the observed data to capture all j. For this, we fit Σ(R) and V(R) with functional forms.

3.1. Rotation curve fitting

We considered two models to extrapolate V(R): an arctan function (Courteau 1997) and a more flexible multi-parameter function (Rix et al. 1997). The functions are given, respectively, by

V rot ( R ) = V 0 2 π arctan ( R / r t ) , $$ \begin{aligned} V_{\text{rot}}(R)&= V_0 \frac{2}{\pi } \arctan (R / r_{\rm t}), \end{aligned} $$(8)

V rot ( R ) = V t ( 1 + r t R ) β [ 1 + ( r t R ) ξ ] 1 / ξ . $$ \begin{aligned} V_{\mathrm{rot} }(R) = V_{t} \frac{\left( 1 + \frac{r_{\rm t}}{R} \right)^{\beta }}{\left[ 1 + \left( \frac{r_{\rm t}}{R} \right)^{\xi } \right]^{1/\xi }}. \end{aligned} $$(9)

Here V0 is the asymptotic velocity reached at an infinite radius, and rt is the turnover radius between the rising and the outer flat part of the rotation curve. The Vt parameter Eq. (9) is a scale velocity regulating the amplitude of the rotation curve. The two additional parameters, ξ and β, regulate the rise and flattening of the rotation curve.

For simplicity, we first fitted the rotation curves with the arctan function, finding satisfactory fits for most cases. For five galaxies (NGC 300, NGC 1365, NGC 2903, NGC 3521, and NGC 3627) with complex kinematics the multi-parameter function works better. In practice, we retrieved posterior distributions for the best-fitting parameters using a Markov chain Monte Carlo (MCMC) routine with the Python package emcee (Foreman-Mackey et al. 2013), assuming flat priors on the fitting parameters. Our models successfully reproduced the observed kinematics, as illustrated in Fig. 1 for one representative galaxy.

3.2. Surface density profile fitting

To fit ΣH2(R), we considered a polyexponential function, which has been shown to effectively capture the complex behaviour of the gas surface densities in nearby galaxies (Bacchini et al. 2019; Mancera Piña et al. 2022). The polyexponential profile is given by

Σ H 2 ( R ) = Σ 0 e − R / R Σ ( 1 + c 1 R + c 2 R 2 + c 3 R 3 ) , $$ \begin{aligned} \Sigma _{\text{H}_2}(R)&= \Sigma _{0} e^{-R/R_{\Sigma }} \left(1 + c_1 R + c_2 R^2 + c_3R^3\right), \end{aligned} $$(10)

where Σ0 is the central surface density; RΣ is the scale radius; and c1, c2, and c3 are polynomial coefficients. For some profiles (especially those with sharp peaks or strong declines), one polyexponential profile is not flexible enough to reproduce all the observed surface density features. In those cases, we fit instead a combination (sum) of two polyexponential profiles. Similar to our approach with the rotation curves, we used emcee to determine the best-fitting parameters. An example of the fit to a representative ΣH2(R) profile is shown in Fig. 1.

3.3. Specific angular momentum

We adopted a Monte Carlo approach to compute the specific angular momentum and its uncertainties. Specifically, we generated a distribution of jH2 values by calculating jH2 for 100 000 random MCMC realisations of both the rotation curves and surface density profiles fitting parameters. For each realisation, we integrated the angular momentum profiles (i.e. Eq. (7)) up to 50 kpc.

Our integration limit exploits the extrapolated profiles described in the previous section and is a choice to ensure that we obtained a converged estimation of jH2. We note that the exact value of the upper limit has no impact on our results since ΣH2(R)≈0 well before our integration limit. Nevertheless, galaxies for which jH2(< R) is not sufficiently converged at the limits of the data, typically galaxies without flattening of the rotation curve, will have a larger dependence on the functional forms used in the extrapolation. We define a convergence factor ℛ to quantify the degree of dependence of our results on our extrapolation, which allows us to avoid over-reliance on the extrapolated data. ℛ is defined as the ratio of jH2(< R) evaluated at the extent of the rotation curve data to the extrapolated jH2(< R) (see e.g. Mancera Piña et al. 2021a). We establish a threshold such that galaxies with ℛ < 0.7 are excluded from the derivation of the molecular j − M relation (see below). In Appendix C, we examine the impact of the minimum required convergence factor on our results, but we emphasise already that our results below are robust against sensible variations of ℛmin. Applying the convergence cutoff results in a final sample of 51 galaxies with converged jH2 profiles. Table A.1 (A.2) lists the jH2 and MH2 for our converged (non-converged) galaxies1. These values and their uncertainties (which also account for distance uncertainties as reported in Table A.1) correspond to the medians and 1σ uncertainties obtained with our Monte Carlo realisations, as described above.

4. The molecular j − M relation

4.1. Shape and dependences

In Fig. 2 we show the distribution of our converged sample in the jH2 − MH2 plane. Our analysis reveals a clear relation, with a particularly tight scaling law for MH2 ≳ 109 M⊙, and with increasing scatter below this mass. We fit the observed distribution with a power law of the form

log ( j H 2 kpc km s − 1 ) = α [ log ( M H 2 / M ⊙ ) − 9 ] + β , $$ \begin{aligned} \log \left( \frac{j_{\mathrm{H}_2}}{\text{ kpc} \text{ km} \text{ s}^{-1}} \right)&= \alpha [\log (M_{\mathrm{H}_2} / M_{\odot }) - 9] + \beta ~, \end{aligned} $$(11)

where α denotes the slope of the relation and β the intercept. We include the orthogonal intrinsic scatter (σ⊥) following Bacchini et al. (2019) and Mancera Piña et al. (2021a). We find the best-fitting parameters to be α = 0.53 ± 0.04, β = 2.62 ± 0.02, with an orthogonal intrinsic scatter σ⊥ = 0.11 ± 0.02. In Fig. 2 we also show the best-fitting relation and its intrinsic scatter2. The slope of the molecular relation aligns closely with that of the j*–M* relation (α ≈ 0.5 − 0.6, e.g. Romanowsky & Fall 2012; Fall & Romanowsky 2018; Posti et al. 2018; Mancera Piña et al. 2021a), whereas it is less steep than the jHI − MHI relation (α ≈ 0.8 − 1, Kurapati et al. 2021; Mancera Piña et al. 2021a).

thumbnail Fig. 2.

Molecular j − M relation for our final sample of converged galaxies. The circles show our sample, while the crosses show the results of Obreschkow & Glazebrook (2014). The dashed line indicates the best-fitting relation, and the grey band shows its orthogonal intrinsic scatter.

With our best-fitting relation, we examined the jH2 residuals (ΔjH2 − MH2 = jH2 − jH2, fit) as a function of various parameters to identify any underlying secondary dependences. The parameters considered include Hubble type, effective radius (Re), star formation rate (SFR), stellar mass (M*), and molecular gas fraction MH2/M*. We obtained the values for Hubble type, Re, SFR, and M* from Leroy et al. (2021b). First, we note that we find no significant correlation between jH2 and star formation rate (SFR) at fixed MH2. We surmise that this could be due to the narrow range of SFRs in our sample (by selection, the PHANGS-ALMA galaxies lie on the star-forming main sequence) or that the variations in the star formation efficiency of H2 are independent of jH2.

Among all the other parameters, the only significant trend (p-value = 0.003) observed in our residual analysis is a moderate anti-correlation between ΔjH2 − MH2 and MH2/M*. This relationship indicates that galaxies with higher MH2/M* tend to have lower jH2 at a given MH2. Similarly, higher MHI/M* are associated with lower H I specific angular momentum, while the opposite happens in the j* − M* relation, with galaxies with high j* having a high MHI/M* at fixed M* (see Mancera Piña et al. 2021b). We find no clear dependence with Hubble type or Re in our jH2 − MH2 relation. While the main focus of this work is on the H2 relation, we also performed preliminary explorations of the jH2 − M⋆ and its dependences, which we discuss in Appendix D.

4.2. Comparison with previous works

In Fig. 2 we compare our results to the study of Obreschkow & Glazebrook (2014), who analysed 16 nearby spiral galaxies from the THINGS survey (Walter et al. 2008). In their case the molecular gas surface densities were obtained from CO(2–1) maps at 11″ resolution from the HERACLES survey (Leroy et al. 2009), when available, or from CO(1–0) maps at 7″ resolution from the BIMA survey (Helfer et al. 2003) otherwise. The data used by Obreschkow & Glazebrook (2014) was of lower resolution than those exploited in this work; in addition they used a higher line ratio of ICO(2 → 1) = 0.8 ICO(1 → 0) and they adopted the αCO of the Milky Way. For the kinematics, the approaches were also somewhat different since Obreschkow & Glazebrook (2014) derived H I rotation velocities based on a pixel-by-pixel fitting technique (see their Appendix B for details) and assumed co-rotation between H I and H2. As shown in Fig. 2, unlike our results, the sample from Obreschkow & Glazebrook (2014) exhibits greater scatter, biased towards higher JH2 values, and does not show a clear trend. We find that the overlapping galaxies in our samples differ in the jH2 − MH2 plane; these differences are likely attributable to the different αCO and line ratios adopted. For these galaxies (NGC 628, NGC 3351, NGC 3521, NGC 3627, and NGC 7793), the mean (median) difference in jH2 is 0.04 (0.13) dex, with a maximum discrepancy of approximately 0.25 dex. Whereas the difference in jH2 is not systematic towards one direction, for all of the overlapping galaxies (except NGC 7793) Obreschkow & Glazebrook (2014) finds lower MH2 values compared to our values. The mean (median) difference, without taking NGC 7793 into account, is 0.32 (0.28) dex, with a maximum difference of about 0.7 dex. For NGC 7793, Obreschkow & Glazebrook (2014) find a higher value of MH2; however, this is the only galaxy in their sample for which they infer MH2 from the SFR (see their Sect. 2.2) instead of from the CO maps of Leroy et al. (2008).

Overall, our results show the close similarity between the jH2 − MH2 and j* − M* relations and highlight the significance of molecular angular momentum as an important regulator of the interstellar medium. Future work with samples that span a larger range of physical properties, such as the upcoming KILOGAS ALMA survey, will shed light on possible dependences and further refine our understanding of the jH2–MH2 relation. Despite this, our current relation can already be used to test theoretical models, and in the next section we provide some first examples.

5. Testing theoretical models

Angular momentum measurements (to date only j* and jHI) have been used to test and constrain analytic models based on disc stability (e.g. Obreschkow et al. 2016; Romeo 2020; Romeo et al. 2023) as well as more elaborated semi-analytical models and hydrodynamical simulations (e.g. Obreja et al. 2016; Stevens et al. 2016; Lagos et al. 2017; El-Badry et al. 2018; Zoldan et al. 2018; Lagos et al. 2024). Our novel work allows us to exploit the molecular j–M relation for the first time. In this section we compare our results against the expectations of an analytic stability model and a complex semi-analytic model.

First, we compare our findings to the scaling relation driven by disc instability introduced by Romeo (2020) as j i σ ̂ i G M i ≈ 1 $ \frac{j_i \hat{\sigma}i}{G M_i} \approx 1 $, where σ ̂ $ \hat{\sigma} $ is the radial velocity dispersion, properly averaged and rescaled, and i denotes a given mass component. This relation appears to be in good agreement with observations of the H I phase (Mancera Piña et al. 2021a, but see also Mancera Piña et al. 2021b). For H2, we can rewrite the expression by Romeo (2020) as

j H 2 ∝ M H 2 σ ̂ H 2 , $$ \begin{aligned} j_{\mathrm{H}_2} \propto \frac{M_{\mathrm{H}_2}}{\hat{\sigma }_{\mathrm{H}_2}}~, \end{aligned} $$(12)

where σ ̂ H 2 = 0.4 σ ¯ H 2 $ \hat{\sigma}_{\mathrm{H}_2} = 0.4\bar{\sigma}_{\mathrm{H}_2} $, and σ ¯ H 2 $ \bar{\sigma}_{\mathrm{H}_2} $ is the radial average of the H2 velocity dispersion (see Eq. (4) and Sect. 2.3 of Romeo 2020). To match this stability relation to our observed jH2–MH2 relation with a slope α ≈ 0.5, we need the scaling σ ∝ MH20.5 (but see also Romeo et al. (2020) for potential second-order dependences ). Instead, for our data, the dependence is much weaker, σ ∝ MH2γ, with γ ≈ 0.05 − 0.2, as shown by the kinematic measurements by Sun et al. (2022), Rizzo et al. (2024).

We infer that models purely based on disc stability, although powerful tools with the virtue of being simple enough to be falsifiable and intuitive, appear to only partially capture the jH2 − MH2 relation. We surmise that there could be different reasons for this shortcoming. On the one hand, numerous other factors beyond stability conditions can influence the distribution of angular momentum in galaxies, such as gas accretion, star formation, feedback, mergers, and dynamical friction (e.g. Pezzulli et al. 2017; Lagos et al. 2017; Stevens et al. 2018; Cimatti et al. 2019). In addition, it could be that the limitations in the model arise from its attempt to globalise the Toomre parameter (a local condition at a given radius) into one single average value (see Romeo 2020 for details). Finally, we also note that the Romeo (2020) relation was calibrated with the data from Obreschkow & Glazebrook (2014), which shows a scattered jH2 − MH2 plane that does not fully conform with our new results.

For the semi-analytic models, we performed a first exploration using SHARK (Lagos et al. 2024). SHARK (v2.0) incorporates a range of physical processes, including halo growth and mergers, gas accretion, chemical enrichment, and stellar and AGN feedback. In addition, SHARK features a direct and independent modelling of j*, jHI, and jH2. SHARK has demonstrated great consistency with the observed scaling relations, such as those between SFR, M*, gas content, specific SFR, and black hole mass, making it a valuable model to compare our findings to.

To compare our jH2 − MH2 relation with SHARK, we selected only SHARK galaxies with M* within ±3σ of our own M* range and B/T < 0.3, ensuring a focus on disc galaxies in a comparable mass regime. Figure 3 presents our comparison, which shows a remarkable agreement between the SHARK predictions and our data within our observed masses and B/T ranges. Overall, the agreement in Fig. 3 suggests that the physical processes incorporated in SHARK (see Lagos et al. 2018, 2024 for details) manage to capture the underlying mechanisms shaping the jH2 − MH2 relation at z = 0.

thumbnail Fig. 3.

Comparison between our measured jH2 − MH2 relation and the prediction of the SHARK semi-analytical model (version 2.0). The SHARK sample was selected to include galaxies with M* and B/T in the range of the observational data. The median distribution of SHARK is shown with a solid curve, and the 1σ uncertainty is given by the purple band.

We note that at the low- and high-mass regimes SHARK predicts a flattening of the relation3. It will be interesting to test this prediction with larger galaxy samples, even though this will be complicated for the low-mass end, given the low metallicity of low-mass galaxies. Additional future promising research avenues include a dedicated study of different hydrodynamical simulations (which will allow us to test whether the physics implementation of SHARK is unique for reproducing the data) as well as the study of the jH2 − MH2 relation at higher redshifts, for which we now provide a local baseline.

6. Conclusions

Exploiting observations from the state-of-the-art PHANGS-ALMA survey, we characterised for the first time the scaling relation between the specific angular momentum of molecular gas (jH2) and molecular gas mass (MH2) for disc galaxies in the local universe. Our analysis reveals a clear power-law correlation in the jH2 − MH2 plane (Fig. 2), which we fit with a power law. The best-fitting relation has a slope of α = 0.53 ± 0.04 and an intercept of β = 2.62 ± 0.02, aligning closely in slope with the well-studied stellar j–M relation and contrasting with the steeper slope observed for neutral atomic gas (H I).

We compared our findings with the predictions from an analytical model based purely on disc stability, and find that it does not recover the jH2 − MH2 completely. On the other hand, compared with predictions from the SHARK semi-analytical model, we find a better agreement (Fig. 3), which supports the model’s portrayal of molecular gas dynamics in galaxy evolution.

Overall, our work provides a valuable tool for upcoming studies of gas dynamics at earlier cosmic times and places new constraints to be reproduced by galaxy formation models and simulations.

1

For consistency, MH2 is estimated by integrating the denominator in Eq. (7). We find great agreement when comparing our values to the CO luminosities reported in Leroy et al. (2021b), with a median difference of 0.03 dex.

2

The inclusion of H I rotation curves does not bias our results for the best-fitting relation; we verify this explicitly in Appendix B.

3

Some models and simulations suggest a similar flattening or break at the low-mass end of the j⋆ − M⋆ relation (Obreja et al. 2016; Stevens et al. 2016), although it has not been seen in observations (Mancera Piña et al. 2021a)

4

The outliers are extreme but do not solely drive the correlation; significant p−values for MH2/M* (p-value = 0.003), log(Re) (p-value = 0.004), and log(Re, H2) (p-value = 0.00005), remain without the tail.

Acknowledgments

We want to thank Gabriele Pezzulli, Luca Cortese, Alessandro Romeo, Michael Fall, Francesca Rizzo, and Danail Obreschkow for valuable discussions, and Paul van der Werf for reading an earlier version of this work. We also thank the anonymous referee for a constructive report that helped improve the paper. PEMP acknowledges the support from the Dutch Research Council (NWO) through the Veni grant VI.Veni.222.364. We thank the PHANGS collaboration for making their data available. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2012.1.00650.S, ADS/JAO.ALMA#2015.1.00925.S, ADS/JAO.ALMA#2015.1.00956.S, ADS/JAO.ALMA#2017.1.00886.L, ADS/JAO.ALMA#2018.1.01651.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. We have used SIMBAD, NED, and ADS services extensively, as well as the Python packages NumPy (Oliphant 2007), Matplotlib (Hunter 2007), SciPy (Virtanen et al. 2020), and Astropy (Astropy Collaboration 2018), for which we are thankful.

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Appendix A: Molecular angular momentum catalogue

In Table A.1 we show galaxy properties and our calculated values for jH2, MH2, and ℛ for our sample of 51 converged galaxies. Additionally, we provide the results for our non-converged sample in Table A.2, but we note that these results are less reliable. In both tables, Hubble types, distances, and stellar masses are from Leroy et al. (2021b).

Table A.1.

Selected galaxy properties.

Table A.2.

Selected galaxy properties non-converged sample.

Appendix B: Including H I rotation curves

To extend the radial coverage of our sample’s kinematics, we supplemented our CO rotation curves with literature H I rotation curves when available (see Sect.2.1). In Fig.B.1 we highlight in the jH2 − MH2 plane (i) the eight galaxies whose CO rotation curves are complemented by H I velocities (orange), (ii) the three galaxies for which only H I data were used (green), and (iii) all remaining galaxies that rely solely on CO rotation curves (pink).

We tested how including H I rotation curves could affect the jH2 − MH2 relation by examining the influence of these galaxies on the best-fit parameters (Table B.1). Any resulting changes in the slope lie within the uncertainties of our fiducial relation, confirming the robustness of our approach and indicating that the inclusion of H I rotation curves does not bias our results presented in Sect. 4.

thumbnail Fig. B.1.

Converged sample of 51 galaxies in the jH2 − MH2 plane. We indicate whether the CO rotation curve (pink), the CO curve supplemented with the H I rotation curve (orange), or only the H I rotation curve (green) was used to determine jH2 for each galaxy.

Table B.1.

Impact of rotation curves on the best-fit parameters to the jH2 − MH2 relation.

Appendix C: Convergence criteria

We obtain the molecular j − M relation by imposing a convergence limit ℛ > 0.7, as detailed in Sec. 3.3. This appendix explores how robust our findings are against our ℛ threshold choice.

Figure C.1 shows the total sample of 60 galaxies in the jH2 − MH2 plane with ℛ indicated by the colour of the markers, the values of the non-converged galaxies are provided in Table A.2. To test the impact of our ℛ threshold, we repeat our fitting procedure but include the galaxies satisfying ℛ > 0.2 (in which our full sample is comprised) and the more restrictive ℛ > 0.9. The figure shows the resulting best-fit relations, and the fitting parameters are provided in Table C.1. Typically, the galaxies with the lowest ℛ lie above the fiducial relation, which may be counterintuitive, but it is likely the result of the functional forms overestimating the velocity at which the rotation curve flattens.

As can be seen from Fig. 2 and Table C.1, all the fitting parameters remain consistent within their uncertainties regardless of the specific convergence criterion employed. Even though the sample size decreases as we impose stricter convergence limits, we still recover slopes and intercepts that are statistically consistent. This all demonstrates the robustness of our approach and results.

thumbnail Fig. C.1.

Total sample of 60 galaxies in the jH2 − MH2 plane, with their convergence factors, as given in Table A.1, indicated by colour. The best-fitting relations to samples with different ℛ thresholds are shown with grey (all data), magenta (ℛ > 0.9), and black (ℛ > 0.7 ) curves.

Table C.1.

Best-fit parameters to the jH2 − MH2 relation, varying the minimum required ℛ.

Appendix D: The jH2 − M* relation

Our analysis also allows us to explore the jH2 content at fixed M*, the dominant baryonic mass component for all our galaxies. The stellar masses are provided in Table A.1, where we adopt an uncertainty of 0.1 dex (Leroy et al. 2021b). As shown in Fig. D.1, except for a few outliers, most of our converged sample follows a clear trend in the jH2 − M* plane, which we parametrise with the relation

log ( j H 2 kpc km s − 1 ) = α [ log ( M ∗ M ⊙ ) − 10 ] + β . $$ \begin{aligned} \log \left( \frac{j_{\rm H_2}}{\text{ kpc} \text{ km} \text{ s}^{-1}} \right)&= \alpha \left[ \log \left( \frac{M_{*}}{M_{\odot }} \right) - 10 \right] + \beta ~. \end{aligned} $$(D.1)

This expression differs from Eq. 11 only in the mass shift (which affects the normalisation β). The best-fitting parameters are α = 0.66 ± 0.08, β = 2.48 ± 0.03, with an orthogonal intrinsic scatter σ⊥ = 0.16 ± 0.02. We have checked that our fit is not significantly affected by the presence of the four outliers (see below): excluding these four galaxies does not affect the slope of either the jH2–MH2 or the jH2–M* relations, indicating that our results are robust.

As for the jH2 − MH2 relation, we look for second dependences in the jH2 − M* plane. In contrast with the jH2 − MH2 relation (where we find an anti-correlation), we find a strong positive correlation with MH2/M* (p-value = 0.00005). Furthermore, we find a significant (p-value = 0.0004) correlation with log(Re), indicating that galaxies with more extended light distribution tend to have higher jH2, as also observed for j* (Mancera Piña et al. 2021a). We show the dependences on log(Re) and MH2/M*, in the top and lower-middle panel of Fig. D.1.

Additionally, we compute Re, H2, the radius within which half of the molecular gas is contained. This quantity, analogous to the stellar effective radius Re, is derived from integrating the cumulative ΣH2 profiles. Similar to Re, we find a robust positive correlation between jH2 and log(Re, H2) with a p-value of 2 × 10−9. This molecular size dependence is seen across our mass regime, but it is particularly evident at log(M*/M⊙)≈10.5, where we observe a tail of outliers exhibiting significantly lower Re, H2 and jH2, indicating that their molecular gas is significantly less extended that for the other galaxies4. The tail is formed by the galaxies NGC 1317, NGC 3626, NGC 4293, and NGC 4457.

Upon inspection, we find that in addition to their very low Re, H2, these galaxies show significantly lower molecular gas fractions, as illustrated in the lower-middle panel of Fig. D.1. We also notice that none of these galaxies is a normal spiral; instead, they are classified as lenticulars (S0) or intermediate spirals (SABa) based on their morphology (bottom panel of Fig. D.1, Hubble types are provided in Table A.1). This was not necessarily expected, as the discs of lenticular galaxies (and we have checked this is the case for our sample) are found to lie on the same j* − M* relation for spirals, as shown by Rizzo et al. (2018), Mancera Piña et al. (2021a), see also discussion in Romanowsky & Fall (2012).

The combination of normal j* but low jH2, Re, H2, and MH2/M*, seen primarily for lenticular or intermediate spirals, is likely related to their formation mechanism, which remains a topic of active study (Deeley et al. 2020). These mechanisms include passive evolution, where cold gas is gradually consumed by star formation (van den Bergh 2009; Laurikainen et al. 2010; Bellstedt et al. 2017), or more violent processes such as mergers and tidal interactions that rapidly strip or deplete the gas (Bekki & Couch 2011; Falcón-Barroso et al. 2015; Querejeta et al. 2015). Whatever the exact formation channel at play, our results suggest that lenticular galaxies, for example spirals, self-regulate to lie on the j* − M* and jH2 − MH2 relations (see also Mancera Piña et al. 2021b) but have gone through a process that altered mostly their gas disc, preferentially removing gas from the outer, high-jH2 regions. As a result, jH2 and MH2 are significantly reduced, while the dominant stellar components remain largely unaffected. These observations suggest a variety of formation paths for lenticulars, or at least diverse evolutionary paths for their molecular gas reservoirs, which may explain the large observed range of jH2 at fixed M* for this galaxy type. This highlights the importance of angular momentum studies in understanding the diversity in galaxy morphology observed in the Universe. However, considering that our sample is predominantly composed of galaxies with minimal bulge components, a more representative sample, including a greater number of lenticular galaxies, would be necessary to draw clearer conclusions.

thumbnail Fig. D.1.

jH2 − M* relation for our converged sample, the dashed line and grey band indicate the best-fitting relation and its orthogonal intrinsic scatter, respectively. The panels highlight the dependence of the relation on the effective radius (top), Re, H2, the radius within which half of MH2 is contained (upper-middle), molecular gas fraction (lower-middle), and morphology (bottom).

All Tables

Table A.1.

Selected galaxy properties.

In the text
Table A.2.

Selected galaxy properties non-converged sample.

In the text
Table B.1.

Impact of rotation curves on the best-fit parameters to the jH2 − MH2 relation.

In the text
Table C.1.

Best-fit parameters to the jH2 − MH2 relation, varying the minimum required ℛ.

In the text

All Figures

thumbnail Fig. 1.

Example of the functional forms fitted to the rotational velocities and molecular gas surface density profiles (in this case for the galaxy IC 5273). Left: Observed rotation curve (grey markers with error bars) and the best-fit model (solid blue curve) with its 1σ and 2σ uncertainties represented by the shaded region. Right: Molecular gas surface density profile assuming our master αCO conversion factor (grey markers with error bars). The best-fit model is shown as a solid red curve, with its 1σ and 2σ uncertainties indicated by the shaded region. Profiles derived using alternative CO-to-H2 conversion factors (S20, MW, N12) are overplotted for comparison (B13 was not available for IC 5273). As can be seen, our profile represents a good compromise between the different calibrations and incorporates realistic uncertainties.
In the text
thumbnail Fig. 2.

Molecular j − M relation for our final sample of converged galaxies. The circles show our sample, while the crosses show the results of Obreschkow & Glazebrook (2014). The dashed line indicates the best-fitting relation, and the grey band shows its orthogonal intrinsic scatter.
In the text
thumbnail Fig. 3.

Comparison between our measured jH2 − MH2 relation and the prediction of the SHARK semi-analytical model (version 2.0). The SHARK sample was selected to include galaxies with M* and B/T in the range of the observational data. The median distribution of SHARK is shown with a solid curve, and the 1σ uncertainty is given by the purple band.
In the text
thumbnail Fig. B.1.

Converged sample of 51 galaxies in the jH2 − MH2 plane. We indicate whether the CO rotation curve (pink), the CO curve supplemented with the H I rotation curve (orange), or only the H I rotation curve (green) was used to determine jH2 for each galaxy.
In the text
thumbnail Fig. C.1.

Total sample of 60 galaxies in the jH2 − MH2 plane, with their convergence factors, as given in Table A.1, indicated by colour. The best-fitting relations to samples with different ℛ thresholds are shown with grey (all data), magenta (ℛ > 0.9), and black (ℛ > 0.7 ) curves.
In the text
thumbnail Fig. D.1.

jH2 − M* relation for our converged sample, the dashed line and grey band indicate the best-fitting relation and its orthogonal intrinsic scatter, respectively. The panels highlight the dependence of the relation on the effective radius (top), Re, H2, the radius within which half of MH2 is contained (upper-middle), molecular gas fraction (lower-middle), and morphology (bottom).
In the text

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