ANGULAR MOMENTUM REGULATES ATOMIC GAS FRACTIONS OF GALACTIC DISKS

In the case of $v(r)={v}_{{\rm{max}}}\forall r\geqslant 0$, Equation (5) reduces to $\kappa =\sqrt{2}\;{v}_{{\rm{max}}}/r$ and hence Equation (2) becomes

$$\begin{eqnarray}&&{Q}_{{\rm{atm}}}=\sqrt{2}\;q\;{x}^{-1}{e}^{x}.\end{eqnarray} \tag{ 6 }$$

Here, q is a dimensionless "global disk stability parameter," originally introduced by Obreschkow & Glazebrook (2014, hereafter OG14), defined as

$$\begin{eqnarray}&&q\equiv \displaystyle \frac{j\sigma }{{GM}},\end{eqnarray} \tag{ 7 }$$

where $j\equiv | {\boldsymbol{J}}| /M$ is the baryonic specific angular momentum of the disk. In a constantly rotating exponential disk, $j=2{r}_{{\rm{disk}}}{v}_{{\rm{max}}}$.

Equation (6) is plotted in Figure 1 (dotted lines) for four typical values of q. Equation (4) can be solved analytically: if $q\geqslant 1/(\sqrt{2}e)$, the disk is entirely stable (${Q}_{{\rm{atm}}}\geqslant 1\;\forall x\geqslant 0$), hence ${f}_{{\rm{atm}}}(q)=1;$ this is a model limitation discussed in Section 3.1. If $q\lt 1/(\sqrt{2}e)$, then ${Q}_{{\rm{atm}}}(x)$ intersects the ${Q}_{{\rm{atm}}}=1$ line twice (see Figure 1). The inner and outer intersection points are ${x}_{\mathrm{core}}=-{W}_{0}(-\sqrt{2}q)$ and ${x}_{\mathrm{hole}}=-{W}_{-1}(-\sqrt{2}q)$, where W₀ and ${W}_{-1}$ are branches of the Lambert W function. As we shall see (Section 2.2), the core stability zone at $x\lt {x}_{\mathrm{core}}$, which only makes a minor mass contribution (up to 26%, but often much smaller), is an artifact of assuming a constant rotation velocity. It disappears for most realistically rising rotation curves. We therefore chose to remove this zone from ${f}_{{\rm{atm}}}$, simplifying Equation (4) to ${f}_{{\rm{atm}}}(q)={\int }_{{x}_{\mathrm{hole}}}^{\infty }\;x\;{e}^{-x}{dx}$. Hence,

$$\begin{eqnarray}&&{f}_{{\rm{atm}}}=(1+{x}_{\mathrm{hole}}){e}^{-{x}_{\mathrm{hole}}}.\end{eqnarray} \tag{ 8 }$$

In the limit $q\to {0}^{+}$, Equation (8) yields the Taylor approximation ${f}_{{\rm{atm}}}=\sqrt{2}q+{ \mathcal O }({q}^{2})$, explaining the approximate proportionality between j/M and ${f}_{{\rm{atm}}}$ found in Obreschkow et al. (2015a). Better first-order approximations are found in log-space. In the vicinity of q = 0.1, this approximation truncated to ${f}_{{\rm{atm}}}\leqslant 1$ reads

$$\begin{eqnarray}&&{f}_{{\rm{atm}}}=\mathrm{min}\{1,2.5{q}^{1.12}\}.\end{eqnarray} \tag{ 9 }$$

This approximation (thick gray line in Figure 2) yields errors below 15% in ${f}_{{\rm{atm}}}$ for ${f}_{{\rm{atm}}}\in (0.01,0.6)$.

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An immediate aside from this derivation is the testable prediction for the inner H i radius. We expect that disks transition from the star+H₂ -dominated part to the H i-dominated part at the radius ${r}_{\mathrm{hole}}={r}_{{\rm{disk}}}{x}_{\mathrm{hole}}$, where

$$\begin{eqnarray}&&{x}_{\mathrm{hole}}=-{W}_{-1}(-\sqrt{2}q)=-1-{W}_{-1}(-{f}_{{\rm{atm}}}/e).\end{eqnarray} \tag{ 10 }$$

This radius only exists if $q\lt 1/(\sqrt{2}e)$. Otherwise, disks are expected to be H i-dominated at all radii. We leave the test of this side result to future work.

The assumption of a constant rotation velocity v brings the caveat of a diverging epicyclic frequency κ as $r\to 0$, making the galaxy unrealistically stable at its center ($x\lt {x}_{\mathrm{core}}$). This divergence disappears if we account for the linear increase of the velocity, $v\propto r$, in the center. Such linear behavior is naturally provided by the rotation curve model (Boissier et al. 2003),

$$\begin{eqnarray}&&v(r)={v}_{{\rm{max}}}(1-{e}^{-r/{r}_{{\rm{flat}}}}),\end{eqnarray} \tag{ 11 }$$

where ${r}_{{\rm{flat}}}$ is the kinematic radius and ${v}_{{\rm{max}}}$ the maximal rotation velocity reached asymptotically as $r\to \infty$. Note that adopting similar rotation curve models (e.g., Courteau 1997) with the same asymptotic behavior leads to very similar results. Our reason for choosing Equation (11) is that, following Equation (8) in OG14, the specific angular momentum then takes the simple form

$$\begin{eqnarray}&&j=2{r}_{{\rm{disk}}}{v}_{{\rm{max}}}[1-{(1+a)}^{-3}],\end{eqnarray} \tag{ 12 }$$

where $a\equiv {r}_{{\rm{disk}}}/{r}_{{\rm{flat}}}$. Using this expression for j, and substituting Equation (11) into (5), Equation (2) becomes

$$\begin{eqnarray}&&{Q}_{{\rm{atm}}}=\sqrt{2}{{qx}}^{-1}{e}^{x}\displaystyle \frac{\sqrt{1-{e}^{-{ax}}}\sqrt{1+({ax}-1){e}^{-{ax}}}}{1-{(1+a)}^{-3}}.\end{eqnarray} \tag{ 13 }$$

Here, q remains as defined in Equation (7), but j is now calculated via Equation (12). As expected, Equation (13) reduces to Equation (6) in the limit of ${r}_{{\rm{flat}}}\to 0$, i.e., $a\to \infty$, corresponding to a constant rotation. The functions ${Q}_{{\rm{atm}}}(x)$ of Equation (13) are shown in Figure 1 for a = 1 and a = 3. These two values approximate dwarf galaxies and massive spirals, respectively (e.g., Figure 5 in OG14).

We cannot find explicit closed-form solutions for Equation (4) when ${Q}_{{\rm{atm}}}$ is taken from Equation (13), but precise numerical solutions can easily be obtained. The solutions for a = 1 and a = 3 are shown in Figure 2 (solid and dashed black lines). Interestingly, these solutions are very similar to that of a constant rotation curve. The approximate solution of Equation (9) (gray line in Figure 2) remains valid for all rotation curves, making it a universal approximation.

We have seen that ${f}_{{\rm{atm}}}$ only depends on q, but what is its physical interpretation? It turns out that q is a global analog to the local Toomre stability parameter, it depends only on global quantities and is a mass-weighted average of ${Q}_{{\rm{atm}}}$ (not the total Q). For $\langle {Q}_{{\rm{atm}}}\rangle ={\int }_{0}^{y}{Q}_{{\rm{atm}}}x\;{e}^{-x}{dx}$ with $y\equiv {r}_{{\rm{max}}}/{r}_{{\rm{disk}}}$, we obtain

$$\begin{eqnarray*}&&\langle {Q}_{{\rm{atm}}}\rangle \\ =\sqrt{2}q\cdot \left\{\begin{array}{ll}y & \mathrm{if}\ {\text{}}v\ \mathrm{const}.,\\ \tfrac{{\displaystyle \int }_{0}^{y}{dx}\sqrt{1-{e}^{-{ax}}}\sqrt{1+({ax}-1){e}^{-{ax}}}}{1-{(1+a)}^{-3}} & \mathrm{otherwise},\end{array}\right.\end{eqnarray*}$$

meaning that $\langle {Q}_{{\rm{atm}}}\rangle \propto q$ within any radius ${r}_{{\rm{max}}}$. A variation of this proportionality relation was used as an Ansatz to explain the j/M–morphology relation in large disk galaxies (OG14) and to argue that massively star-forming disks owe their clumpy structure to low j more than to high gas fractions (Obreschkow et al. 2015b).

Figure 2 compares the model of ${f}_{{\rm{atm}}}(q)$ to nearby disk galaxies from different surveys. We stress that the model (gray line) is not a fit and has zero free parameters.

Red and yellow points represent the galaxies with the most accurate ($\sim 5 \%$) measurements of j and M, hence the most precise q-values. The j-measurements rely on a pixel-by-pixel integration of the angular momentum using high-resolution H i maps as velocity tracers. Red points show the 16 regular spiral galaxies from THINGS (Walter et al. 2008), for which such precise j-measurements were obtained by OG14. Yellow points represent dwarf galaxies from LITTLE THINGS (Hunter et al. 2012) with j-values measured by Butler et al. (2016). Disk masses M include stellar masses (based on Spitzer data) and H i masses (from VLA). The THINGS disk masses also include H₂ masses, inferred from CO data. No H₂ is included in LITTLE THINGS, but the H₂ /H i ratio is very small ($\ll 0.1$) in such low-mass systems (Leroy et al. 2005; Schruba et al. 2012).

Green points in Figure 2 represent galaxies from the HIPASS blind H i survey (Meyer et al. 2004). Here, we use the subsample (860 galaxies), for which high-fidelity K-band magnitudes where compiled by Meyer et al. (2008). The baryonic disk masses are determined from K-band stellar masses and H i masses. Since H₂ masses are not readily available for this sample, we include a uniform H₂ fraction (${\text{}}{M}_{{{\rm{H}}}_{2}}/{\text{}}M$) of 4%, the mean measured in our 16 THINGS galaxies. The j-values are approximated as $2{r}_{{\rm{disk}}}{v}_{{\rm{max}}}$ (assuming twice the j for H i and correcting via Equation 6 in OG14), where the exponential radius ${r}_{{\rm{disk}}}$ is computed from r-band radii in the optical cross-match catalog (Doyle et al. 2005) and ${v}_{{\rm{max}}}$ is taken to be equal to the H i half-linewidth, corrected for the optical inclination. The galaxies processed in this way exhibit a relatively large ($\sim 0.35\;\mathrm{dex}$) scatter around our model of ${f}_{{\rm{atm}}}(q)$, consistent with the expected large empirical uncertainties. In Figure 2, we therefore limited this sample to a more accurate subsample of 75 objects with certain optical-H i cross-matching (no source confusion), reliable inclination corrections (edge-on galaxies with inclinations larger than 60 degrees), reliable Hubble-flow distances ($\gt 10\;\mathrm{Mpc}$), and a large number of optical resolution elements (effective radius $\gt 20^{\prime\prime}$).

Finally, the blue diamond shows the Milky Way. Here, ${f}_{{\rm{atm}}}$ was taken from Obreschkow et al. (2011) and $j=2{r}_{{\rm{disk}}}{v}_{{\rm{max}}}$ with ${v}_{{\rm{max}}}=220\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and ${r}_{{\rm{disk}}}=3.9\;\mathrm{kpc}$ for all disk baryons (Kafle et al. 2014 find 4.92 kpc for a Miyamoto-Nagai disk, which must be multiplied by 0.82 for an exponential disk).

Overall, Figure 2 demonstrates a remarkable agreement between model and observations. The Spearman correlation between q and ${f}_{{\rm{atm}}}$ is 0.91, and the rms offset between data and model is about $0.2\;\mathrm{dex}$ in ${f}_{{\rm{atm}}}$. This scatter is better than the scatter ($\sim 0.3\;\mathrm{dex}$) of current empirical predictors of ${f}_{{\rm{atm}}}$ based on optical data (Catinella et al. 2012). If we account for the fact that the H i dispersion σ exhibits some intrinsic empirical scatter of about 40% (light gray zone in Figure 2), data and model become statistically consistent. Moreover, the offset of the data is virtually uncorrelated with mass M (Pearson coefficient below 0.1), making it clear that the entire mass dependence of ${f}_{{\rm{atm}}}$ must be contained in the mass dependence of q—an idea developed in Section 3.2.

We performed a critical analysis of the parameter q as a predictor of ${f}_{{\rm{atm}}}$, relative to other candidates. Explicitly, we checked if the overall rotational support, quantified by $\sigma /{v}_{{\rm{max}}}$ is a suitable predictor of ${f}_{{\rm{atm}}}$. We also investigated a model where the disk baryons are atomic if and only if the surface density lies below a threshold ${{\rm{\Sigma }}}_{{\rm{crit}}}$. Both models produce more scatter ($\gt 0.3\;\mathrm{dex}$) and lower Spearman (∼0.7 with clear outliers) than ${f}_{{\rm{atm}}}(q)$ in Figure 2.

The clearest limitation of our model is its prediction that galaxies with $q\gt 1/(\sqrt{2}e)$, typically small dwarf galaxies, are fully atomic (${f}_{{\rm{atm}}}=1$). In reality, such galaxies are never strictly dark, although their baryon mass is dominated by H i (Doyle et al. 2005). This limitation originates directly from the assumption that there are no partially atomic regions in the disk.

Figure 3 shows ${f}_{{\rm{atm}}}$ as a function of the disk mass M for the same galaxies as in Figure 2. In addition it includes data from the "representative sample" of 760 galaxies at $z\approx 0$, published with the final data release (DR3) of the GASS survey (Catinella et al. 2010, 2013). At stellar masses ${M}_{* }\gt {10}^{10}{M}_{\odot }$ in this sample, the true distribution of H i fractions ($\gtrsim 2 \%$) is sampled uniformly. We obtained K-band magnitudes from 2MASS (Jarrett et al. 2000) and selected the disk-dominated galaxies roughly by retaining only concentration parameters ${r}_{90}/{r}_{50}$ below 2.8 (Park & Choi 2005) in the r-band and excluding galaxies in the red sequence (NUV–$r\gt 4.5$) of the color–magnitude diagram (Figure 4 in Catinella et al. 2013). The resulting sample counts 213 galaxies (including 3 H i-non-detections). It is the only sample in Figure 3 revealing an unbiased mean relation between ${f}_{{\rm{atm}}}$ and M. As in our HIPASS data, we included a constant 4% H₂ fraction in M with negligible implications.

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The scatter in Figure 3 at fixed M is about twice as large as in Figure 2 at fixed q. Moreover, in Figure 3, different data sets are offset to each other in ${f}_{{\rm{atm}}}$, whereas they occupy the same locus in Figure 2, implying that ${f}_{{\rm{atm}}}(q)$ is insensitive to H i selection bias. Particularly interesting are the two galaxies from HIPASS marked as green stars in Figures 2 and 3. These objects are field galaxies that are extremely H i-poor and H i-rich for their mass; however, their H i fraction seems quite normal for their q-values. In other words, these galaxies are simply outliers in their relative amount of angular momentum, probably explainable by their (past) cosmic environment. These results confirm that q is fundamental for setting ${f}_{{\rm{atm}}}$ in disk galaxies.

To explain the mean trend of ${f}_{{\rm{atm}}}$ with M in our framework, we need to express q in ${f}_{{\rm{atm}}}(q)$ as a function of M. Assuming that disk galaxies condense out of scale-free cold dark matter halos, we expect galactic disks to fall on a relation $j={{kM}}^{2/3}$ (OG14; Romanowsky & Fall 2012), where k depends on the halo spin parameter λ (Bullock et al. 2001) and the fractions of retained mass ${f}_{{\rm{M}}}=M/{M}_{{\rm{halo}}}$ and specific angular momentum ${f}_{{\rm{j}}}=j/{j}_{{\rm{halo}}}$. For simplicity we will assume ${f}_{{\rm{M}}}=0.05$ and ${f}_{{\rm{j}}}=1$, typical for Milky Way–like disks, although in general one should vary both to account for the physics of galaxy formation. Nevertheless, we get a remarkably good agreement for the general trend. The halo spin parameters exhibit a skewed distribution of mode $\lambda \approx 0.03$ (Bullock et al. 2001), leading to $k\approx 90$ for j in units of $\mathrm{kpc}\;\mathrm{km}\;{{\rm{s}}}^{-1}$ and M in units of ${10}^{9}{M}_{\odot }$. Substituted into Equation (7), we obtain

$$\begin{eqnarray}&&q=\displaystyle \frac{{{kM}}^{2/3}\sigma }{{GM}}\approx 0.22{\left(\displaystyle \frac{M}{{10}^{9}{M}_{\odot }}\right)}^{-1/3}.\end{eqnarray} \tag{ 14 }$$

for $\sigma =10\;\mathrm{km}\;{{\rm{s}}}^{-1}$ (Section 2). With Equation (9), we then find the approximate mean trend

$$\begin{eqnarray}&&{f}_{{\rm{atm}}}\approx 0.5{\left(\displaystyle \frac{M}{{10}^{9}{M}_{\odot }}\right)}^{-0.37},\end{eqnarray} \tag{ 15 }$$

as long as ${f}_{{\rm{atm}}}\lt 1$. This function is shown as a thick line in Figure 3 with the gray shading representing the large 68% quantile scatter expected from the log-normal λ distribution (Bullock et al. 2001). The agreement of this model with the empirical data is good. All empirical data points, except for those from the GASS survey (purple), come from H i-selected surveys and therefore naturally lie above the mean expectation of ${f}_{{\rm{atm}}}(M)$.