FUNDAMENTAL MASS–SPIN–MORPHOLOGY RELATION OF SPIRAL GALAXIES

This study uses all 16 spiral⁴ galaxies of THINGS (Walter et al. 2008), for which stellar and cold gas surface densities have been published by Leroy et al. (2008). This sample, shown in Figure 2 (left) and Table 1, offers the highest quality data to date for a detailed measurement of j_* ≡ J_*/M_*, $j_{\rm H\,\scriptsize {{I}}}\equiv J_{{\rm H}\,\scriptsize {I}}/M_{ {\rm H}\,\scriptsize {I}}$, and $j_{\rm H_2}\equiv J_{\rm H_2}/M_{\rm H_2}$ in spiral galaxies. The sample covers stellar masses from 10⁹ M_☉ to 8 × 10¹⁰ M_☉ and Hubble types T from Sab to Scd. Figure 1 shows the 16 galaxies in the (M_b, T)-plane on top of the distribution of galaxies in the H i Parkes All Sky Survey (HIPASS; Barnes et al. 2001) with resolved morphologies and K-band based stellar masses (494 galaxies, see Meyer et al. 2008). This figure reveals that the 16 galaxies nicely represent the majority of spiral galaxies detected in a typical 21 cm/optically limited survey.

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Table 1. Properties of the 16 Spiral Galaxies Studied in This Paper

NGC	Type	β	R*	Rflat	V	Mb	M*	Mgas	$M_{ {\rm H}\,\scriptsize {I}}$	$M_{\rm H_2}$	jb	j*	jgas	$j_{\rm H\,\scriptsize {{I}}}$	$j_{\rm H_2}$
			(kpc)	(kpc)	(km s−1)	lg (M☉)					lg (kpc  km s−1)
628	Sc	0.04	2.3	0.8	217	10.27	10.1	9.8	9.7	9.0	3.07	2.98	3.23	3.28	2.98
925	SBcd	0.05	4.1	6.5	136	10.16	9.9	9.8	9.8	8.4	3.01	2.94	3.09	3.10	2.95
2403	SBc	0.02	1.6	1.7	134	9.91	9.7	9.5	9.5	7.3	2.85	2.62	3.04	3.08	2.61
2841	Sb	0.10	4.0	0.6	302	10.88	10.8	10.1	10.1	8.5	3.53	3.40	3.91	3.94	3.40
2976	Sc	0.00	0.9	1.2	92	9.18	9.1	8.4	8.3	7.8	2.05	2.03	2.11	2.21	1.93
3184	SBc	0.02	2.4	2.8	210	10.41	10.3	9.7	9.6	9.2	3.09	3.03	3.25	3.32	2.97
3198	SBc	0.03	3.2	2.8	150	10.41	10.1	10.1	10.1	8.8	3.24	2.97	3.45	3.49	3.02
3351	SBb	0.14	2.2	0.7	196	10.44	10.4	9.4	9.2	9.0	2.94	2.91	3.10	3.27	2.75
3521	SBbc	0.16	2.9	1.4	227	10.81	10.7	10.1	10.0	9.6	3.14	3.06	3.38	3.46	2.98
3627	SBb	0.22	2.8	1.2	192	10.62	10.6	9.4	9.0	9.1	2.84	2.84	2.82	2.93	2.77
4736	Sab	0.32	1.1	0.2	156	10.32	10.3	9.0	8.7	8.6	2.37	2.34	2.63	2.86	2.36
5055	Sbc	0.17	3.2	0.7	192	10.91	10.8	10.2	10.1	9.7	3.30	3.18	3.59	3.69	3.08
5194	SBc	0.09	2.8	0.8	219	10.66	10.6	9.8	9.5	9.4	3.19	3.18	3.25	3.12	3.36
6946	SBc	0.10	2.5	1.4	186	10.62	10.5	10.0	9.8	9.6	3.06	3.02	3.17	3.32	2.91
7331	SAb	0.16	3.3	1.3	244	10.99	10.9	10.2	10.1	9.7	3.38	3.35	3.52	3.59	3.30
7793	Scd	0.01	1.3	1.5	115	9.69	9.5	9.2	9.1	(8.6)	2.49	2.43	2.61	2.66	2.40

Notes. The specific angular momenta j were calculated as described in Section 2.3. The bulge mass fractions β were computed as explained in Section 2.2 and illustrated in Figure 14. All other values have been copied from Table 4 in Leroy et al. (2008), inferring $M_{\rm H_2}$ of NGC 7793 from its SFR as described in Section 2.2 and using $M_{\rm gas}=M_{ {\rm H}\,\scriptsize {I}}+M_{\rm H_2}$ and M_b = M_* + M_gas. As explained by Leroy et al., the scale radius R_* and the rotation parameters R_flat and V represent fits to the stellar surface density Σ_*(r)∝exp (− r/R_*) and H i velocity profile v(r) = V[1 − exp (− r/R_flat)]. Measurement uncertainties are not shown in this table, but they are plotted as error bars in the figures of Sections 1–5 and accounted for in all results.

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The data collected by Leroy et al. (2008) comprises multi-wavelength maps from different surveys: kinematic H i maps at a mean resolution of 11'' (∼400 pc) and 5 km s⁻¹ from THINGS (Walter et al. 2008), far-ultraviolet (FUV) maps of 56 resolution from the space-based Galaxy Evolution Explorer (GALEX) Nearby Galaxies Survey (Gil de Paz et al. 2007), 24 μm and 3.6 μm infrared (IR) data with a resolution of ⩽6'' from the space-based Spitzer Infrared Nearby Galaxies Survey (SINGS; Kennicutt et al. 2003), CO(1 → 0) maps of 7'' resolution from the Berkeley-Illinois-Maryland Association (BIMA) Survey of Nearby Galaxies (BIMA SONG; Helfer et al. 2003), and CO(2 → 1) maps of 11'' resolution from the HERA CO Line Extragalactic Survey (HERACLES; Leroy et al. 2009).

From these data (Leroy et al. 2008, see Appendices A–E therein) computed radial surface density profiles Σ(r) as a function of radius r at a resolution of ∼400 kpc, degrading the raw resolution where necessary. Atomic gas densities $\Sigma _{{\rm H}\,\scriptsize {{I}}}$ were computed from the integrated intensity maps of the 21 cm emission line. Molecular gas densities $\Sigma _{\rm H_2}$ were estimated from the CO(2 → 1) maps, except in the case of NGC 3627 and NGC 5194, where CO(1 → 0) maps were used instead. These estimates rely on a constant CO-to-H₂ conversion factor X_{CO(1 → 0)} = 2 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ with an additional correction of 1.36 to include helium, and a fixed line ratio I_{CO(2 → 1)} = 0.8I_{CO(1 → 0)}. Stellar mass densities Σ_* were inferred from the 3.6 μm continuum maps. These maps were first reduced to median radial profiles to minimize the contribution of hot dust and polycyclic aromatic hydrocarbon emission near star-forming regions. The median 3.6 μm profiles were then converted to Σ_*(r) by adopting an empirical K-to-3.6 μm calibration and a constant K-band mass-to-light ratio of $\Upsilon _\ast ^K=0.5\, {M}_{\odot }/{L}_{\odot,K}$, neglecting local variations of a factor ∼two between young and old stellar populations. Star formation rate (SFR) surface densities Σ_SFR, used to complete missing H₂ data (see below), were derived from a combination of FUV and far-IR (FIR) 24 μm continuum maps to capture directly visible and dust-obscured star formation (Appendix D of Leroy et al. 2008).

This paper uses the surface density profiles published by Leroy et al. (2008) up to the following variations. First, surface densities $\Sigma _{{\rm H}\,\scriptsize {{I}}}(r)$ were re-derived from the H i intensity maps (Walter et al. 2008), because the $\Sigma _{{\rm H}\,\scriptsize {{I}}}(r)$ published by Leroy et al. are restricted to ⩾1 M_☉ pc⁻². From the H i maps most $\Sigma _{{\rm H}\,\scriptsize {{I}}}(r)$ can be measured down to about 10⁻² M_☉ pc⁻². Using these extended data, it turns out that limiting $\Sigma _{{\rm H}\,\scriptsize {{I}}}$ to ⩾1 M_☉ pc⁻² decreases $J_{{\rm H}\,\scriptsize {I}}$ and $j_{\rm H\,\scriptsize {{I}}}$ by about 20% and 10%, respectively. These percentages improve to 1% and 0.1% if densities down to 10⁻¹ M_☉ pc⁻² are included, thus motivating the use of the full data. Second, where CO-based H₂ surface densities are missing, they are estimated using an inverted star-formation law $\Sigma _{\rm H_2}=t_{\rm H_2} \Sigma _{\rm SFR}$, where $t_{\rm H_2}=1.9\times 10^9\, {\rm yr}$ is the effective H₂ depletion time found by Leroy et al. (2008). This method is used to infer the total H₂ mass of NGC 7793, the full functions $\Sigma _{\rm H_2}(r)$ of NGC 628/925/2403/2841/7793, as well as large-r parts of $\Sigma _{\rm H_2}(r)$ in the other galaxies. Third, the densities Σ_*(r) and $\Sigma _{\rm H_2}(r)$ are extrapolated beyond the maximal radii R_max, to which they were measured or estimated. The extrapolations use an exponential profile Σ₀exp (− r/R), with parameters Σ₀ and R fitted to the data on the range r ∈ [R_max/2, R_max]. The extrapolated parts are shown as dashed lines in Figure 16 (left). We emphasize that completing H₂ data from SFRs and extrapolating r beyond R_max has no effect on the conclusions of this paper. This post-processing only affects $j_{\rm H_2}$, j_*, and j_b by ∼10% allowing these values to converge to the 1% level (see Section 2.3).

To study correlations between angular momentum and galaxy morphology, the latter is quantified using the stellar mass fraction β in the bulge. In this paper, "bulge" generically refers to any central stellar over-density without further specifying the nature of the component. In the present sample, these bulges are mostly flattened pseudo-bulges (Kormendy & Fisher 2008), nine of which include a bar component. For each galaxy, β is calculated by fitting Σ_*(r) with a model composed of an exponential function for the disk and a Sérsic profile (Sérsic 1963) for the bulge, as described in Appendix A. The resulting values of β are listed in Table 1. The standard errors inferred from resampling are about 0.02.

In the approximation of a flat galaxy with circular orbits, the norm of the angular momentum relative to the center of gravity can be written as

$$\begin{equation} J = \left|\int dM \mathbf {r}\times \mathbf {v}\,\right| = 2\pi \int _0^\infty dr\,r^2\,\Sigma (r)\,v(r), \end{equation} \tag{ 1 }$$

where dM is the mass element, r is the position vector from the center of gravity, v is the velocity vector, v(r) is the norm of v at r = |r|, and Σ(r) is the azimuthally averaged mass surface density of the considered baryonic component. The specific angular momentum is

$$\begin{equation} j \equiv \frac{J}{M} = \frac{\int _0^\infty dr r^2 \Sigma (r) v(r)}{\int _0^\infty dr r \Sigma (r)}. \end{equation} \tag{ 2 }$$

Computing J and j from axially averaged density and velocity profiles allows the outskirts (to r ≈ 14R_*) with low pixel signal-to-noise to be reliably included, but the use of axially averaged surface densities Σ(r) does not, in fact, assume or require Σ to be axially symmetric.

The integral of Equation (2) is evaluated numerically, while correcting for the inclination of the galaxy as detailed in Appendix B. The integrals are evaluated out to the maximal observed H i radius R_{HI, max} ≈ 14R_*. The only exception is NGC 5194—the Whirlpool Galaxy—where the upper bound of the integral is restricted to 14 kpc to suppress the contributions of the interacting close companion NGC 5195 and associated stripped material. Equation (2) is applied to the different baryonic surface densities, resulting in distinct values of j_X ≡ J_X/M_X for all the baryons (j_b), stars (j_*), atomic gas ($j_{\rm H\,\scriptsize {{I}}}$), molecular gas ($j_{\rm H_2}$), and atomic and molecular gas together (j_gas). These values are listed in Table 1.

All measurements of j assume that the baryonic material orbits at the circular velocity v(r) of the H i gas. This assumption of co-rotation between H i, H₂, and stars is justified in the rotation supported parts of the galaxy, where v(r) is dictated by the local gravitational force. In the dispersion supported stellar bulge, however, the stellar rotational velocity is generally smaller than that of the H i disk. For example, in the Andromeda galaxy (M31), the bulge rotation at r ≈ 0–15 kpc (about 50 km s⁻¹, Dorman et al. 2012) is five times smaller than the disk rotation inferred from H i (about 250 km s⁻¹, Unwin 1983). One might thus suspect that the values of j_* presented here overestimate the real values. This effect is nonetheless negligible in late-type galaxies. In fact, even when using H i velocities v(r) for all stars, the stellar angular momentum J_* of the bulge (according to the bulge–disk decomposition of Equation (A1)) only accounts for 0.3% of the total J_* on average. The most extreme bulge contributions are found in NGC 4736 (1.3%), NGC 3627 (0.6%), and NGC 5055 (0.6%). Thus, the contribution of the angular momentum of the stellar bulge to j_* is smaller than the statistical measurement uncertainties of a few percent for j_* (see the following). Bulges nonetheless affect j_* ≡ J_*/M_* through their mass, which takes values up to 0.32 of the total stellar mass M_* in the present sample.

How accurate are the measurements of j? Let us first discuss statistical uncertainties. By repeating the computations of j via Equation (2) with random Gaussian variations of Σ(r) matching their r-dependent measurement uncertainties, it turns out that such uncertainties affect j by less than 0.1%. These errors are negligible relative to those associated and v(r). In computing v(r) via Equation (B4), the inclination-dependent deprojection factor C(φ, i) introduces an uncertainty in the normalization of v(r) of 2%–4% for the given inclination uncertainties. A second order uncertainty can result from non-circular orbits, as a non-circular velocity component v_⊥, perpendicular to the circular orbit, can perturb the measurement of the circular component v(r). To estimate the magnitude of this effect, we generated 10⁵ mock galaxies, inclined at 51° (the average inclination of the present sample), with constant v(r) = V₀ = 200 km s⁻¹, and a non-circular dipole component of amplitude v_⊥ = 10 km s⁻¹, typical for spiral galaxies (e.g., Beauvais & Bothun 1999). For every mock galaxy, the orientation of the non-circular component in the plane perpendicular to the circular motion was chosen randomly. For each mock galaxy, we then recovered a circular velocity V from the line-of-sight component v_z (see Figure 15), assuming only circular orbits. The resulting values V are centered on V₀, but scattered with a standard deviation of 4 km s⁻¹ (2%). Hence, the statistical error introduced when assuming circular orbits in the presence of a realistic non-circular component is approximately 2%. Another source of statistical uncertainty is associated with the finite maximal observing radii R_max. In fact, due to the r² term in the angular momentum integral, non-detected, low-density material in the outer (r > R_max) regions contributes more significantly to J than to M. Thus the question, to what extent j converges within r ⩽ R_max, requires careful examination. Figure 2 (right) shows the cumulative specific angular momenta $j(r)\equiv \int _0^r dr^{\prime }\,r^2\Sigma (r^{\prime })v(r^{\prime })/\int _0^r dr^{\prime }\,r\Sigma (r^{\prime })$ of stars (for H i and H₂ see Figure 16, right). To assess how well these functions have converged, a model for their extrapolation beyond R_max is needed. Upon assuming an exponential disk Σ(r)∝exp (− r/R) rotating at a constant circular velocity V, j(r) becomes

$$\begin{equation} j(r) = \left[2+\frac{(r/R)^2}{1+r/R -\exp (r/R)}\right]R \,V. \end{equation} \tag{ 3 }$$

Explicit fits of Equation (3) to the measured j(r) predict that the measured j have converged at the 1% level for j_b, j_*, and $j_{\rm H_2}$, and at the 10% level for $j_{\rm H\,\scriptsize {{I}}}$ and j_gas. Details and exceptions are given in Appendix B.

The measurements of j might also be subject to systematic errors. Errors in light-to-mass conversions (i.e., luminosity-to-stellar mass and CO-to-H₂) equally affect J and M, thus canceling out in j. Only variations of these conversions within a galaxy can affect j. This might be significant for the CO-to-H₂ conversion, which can vary along r due to a metallicity gradient. A few available measurements for NGC 5194 (Arimoto et al. 1996) suggest that the H₂/CO ratio increases by a factor ∼two on two exponential scale radii. Accounting for this variation increases $j_{\rm H_2}$, j_gas, and j_b in NGC 5194 by about 20%, 10%, and 2%, respectively. Similar changes might apply to other galaxies in the sample. However, because the CO-to-H₂ conversion remains uncertain (Obreschkow & Rawlings 2009c), this paper maintains the constant value of Leroy et al. (2008). Other errors can result from a breakdown of the flat disk model in the case of disturbed or warped galaxies, but in the present sample such effects are negligible based on visual inspection. Distance errors affect j linearly. The 16 galaxies considered here have Hubble flow distances (Table 1 in Walter et al. 2008) on the order of 10 Mpc with expected uncertainties around 5% that are partially correlated.

In summary, the specific angular momenta have statistical uncertainties of a few percent (3%–5%) for j_b, j_*, and $j_{\rm H_2}$, and ∼10% for $j_{\rm H\,\scriptsize {{I}}}$ and j_gas. Potential systematic uncertainties are estimated to about 10%.

Most measurements of angular momentum in the literature do not have detailed kinematic maps at their disposal. They therefore resort to approximations of j based on global measurements. In this section, we compare typical approximations of j_*, labeled as $\tilde{j_{\ast }}$, against our precision measurements j_*. Since the typical deviations between $\tilde{j_{\ast }}$ and j_* turned out to be much larger than the few percent statistical uncertainties of j_*, the latter can be considered as exact in this comparison.

The most common approximation of j_*, already used by Fall (1983), relies on the flat, exponential disk model of Equation (3). In the limit of r → ∞ this equation reduces to (e.g., Equation (7) in Mo et al. 1998),

$$\begin{equation} \tilde{j_{\ast }}= 2R_{\ast }V, \end{equation} \tag{ 4 }$$

requiring only the exponential scale radius R_* of the stellar disk and the (constant) circular velocity V. Those two parameters can be estimated from other measurements, for instance R_* ≈ 0.6r_e ≈ 0.3r₂₅, where r_e is the "effective radius" containing half the light and r₂₅ is the "isophotal radius" with a B-band surface brightness of 25 mag arcsec⁻². The velocity V can be estimated from the total H i linewidth or from optical linewithds at the radius r₂₅ (or beyond), corrected for turbulence and galaxy inclination. Figure 3(a) (triangles) shows the values $\tilde{j_{\ast }}$ given by Equation (4), normalized to the reference values j_*. Filled triangles use R_* and V derived from the full stellar surface densities Σ_*(r) and deprojected velocity profiles v(r); they are the values R_* and V (Table 1) adopted from Leroy et al. (2008). Open triangles use approximate scale radii, fitted only to r ⩽ 2R_*. In general, this approximation based on the exponential disk model provides remarkably good results. The rms error of $\tilde{j_{\ast }}$ for all 16 galaxies is about 30% (0.10 dex). Using only data within r ⩽ 2R_*, this rms error increases to 40% (0.14 dex).

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Another approximation, introduced by Romanowsky & Fall (2012), builds on the flat disk model with a surface density described by the Sérsic profile Σ(r)∝exp [ − b_n(r/r_e)^1/n] with free parameters r_e and n (n = 1 for exponential disk, n = 4 for a de Vaucouleurs profile). The factor b_n ≈ 2n − 1/3 + 0.009876/n ensures that r_e is the effective radius, $\int _0^{r_{\rm e}}dr\,r\,\Sigma (r)=0.5\int _0^\infty dr\,r\,\Sigma (r)$. Romanowsky & Fall find that j_* is approximated by

$$\begin{equation} \tilde{j_{\ast }}=k_n v_{\rm s} r_{\rm e}, \end{equation} \tag{ 5 }$$

where v_s is the deprojected rotation velocity measured at a radius 2r_e and k_n ≈ 1.15 + 0.029n + 0.062n². To test this approximation the 16 galaxies of this work were fitted with single Sérsic functions, once using the whole profiles Σ_*(r), once artificially restricting them to radii r ⩽ 2R_*, where R_* is again the exponential scale radius given by Leroy et al. (2008). The velocities v_s = v(2r_e) are then taken as the average of the de-projected H i velocity v(r) between 1.9r_e and 2.1r_e. The resulting approximations $\tilde{j_{\ast }}$ are shown in Figure 3(a) (circles). If the Sérsic functions are fitted to the full data (filled circles in Figure 3(a)), that is roughly within r ⩽ 14R_*, the rms error is about 30% (0.11 dex), comparable to the exponential disk model. However, when fitting only within r ⩽ 2R_* (open circles), the rms error heavily increases to 2500% (1.4 dex), with $\tilde{j_{\ast }}$ being systematically larger than j_*. This large error can be traced back to the fact that Sérsic functions fitted to the inner (r ⩽ 2R_*), bulgier part of the galaxy, systematically overestimate the surface density at larger radii by overestimating the index n, as illustrated in Figure 3(b). In conclusion, the Sérsic approximation of Equation (5) is much more prone to errors than the exponential disk approximation of Equation (4).

Romanowsky & Fall (2012) do not, in fact, use the Sérsic approximation of Equation (5) to estimate j_* of spiral galaxies because they also find the Sérsic fits to be too uncertain. Instead, they adopt a more robust approach that separates the galaxy into an exponential disk and a smaller "classical" bulge with a de Vaucouleurs profile (fixed Sérsic index n = 4). For both components j_* is approximated separately and then recombined. Six of the Romanowsky galaxies are also in the present sample. Their values $\tilde{j_{\ast }}$, plotted in Figure 3 (pink stars), yield an rms error of about 50% (0.17 dex).

A serious concern is that the errors of the approximations $\tilde{j_{\ast }}$ correlate significantly with j_* (albeit not with β), as shown in Figure 4. This correlation applies both to the $\tilde{j_{\ast }}$ calculated via Equation (4) (triangles in Figure 4) and to those determined by Romanowsky & Fall (2012; pink stars). The correlation is best fitted by the dashed line in Figure 4(a), which can be rewritten as

$$\begin{equation} \left[\frac{j_{\ast }}{10^3\,\rm kpc\,{\,{\rm km\,s}^{-1}}}\right]\approx 1.01\left[\frac{\tilde{j_{\ast }}}{10^3\,\rm kpc\,{\,{\rm km\,s}^{-1}}}\right]^{ 1.3}. \end{equation} \tag{ 6 }$$

This non-linearity between j_* and $\tilde{j_{\ast }}$ is traceable to two features. Firstly, the stellar surface density Σ_*(r) systematically deviates from an exponential in such a way that the fraction f_J of stellar angular momentum outside the half-light radius r_e increases with mass. This fraction ranges from about f_J ≈ 75% at M_* = 10⁹M_☉ to f_J ≈ 85% at M_* = 10¹¹ M_☉. To account for the high values of f_J and their variability, the scale radius R_* used in Equation (4) can be fitted on r > r_e rather than on the whole disk. When doing so, the correlation between ${\rm \,lg\,}(\tilde{j_{\ast }}/j_{\ast })$ and  lg (j_*) is reduced by 60%. The remaining 40% are traceable to a systematic variation of the rotation curves v(r) with mass. This can be seen by looking at the fits (Boissier et al. 2003)

$$\begin{equation} v(r)\approx V\left[1-\exp \left(-\frac{r}{R_{\rm flat}}\right)\right] \end{equation} \tag{ 7 }$$

performed by Leroy et al. (2008); their best-fitting parameters V and R_flat are listed in Table 1. The ratio between Leroy's stellar scale radius R_* and R_flat is found to increase with M_*, roughly by a factor of three per dex in M_*, as shown in Figure 5 (see also de Blok et al. 2008). Thus, the normalized rotation curves v(r/R_*)/V increase faster in more massive galaxies—an effect that is also seen in radial variations of the Tully–Fisher (TF) relation in larger galaxy samples (Yegorova & Salucci 2007). We can account for this effect by convolving Equation (7) with an exponential surface density Σ_*(r)∝exp (− r/R_*) in Equation (2). This solves to

$$\begin{equation} \tilde{j_{\ast }}= 2R_{\ast }V\frac{(R_{\ast }+R_{\rm flat})^3-R_{\rm flat}^3}{(R_{\ast }+R_{\rm flat})^3}. \end{equation} \tag{ 8 }$$

Using Equation (8) with R_* fitted to Σ_*(r) on r > r_e completely removes the correlation between j_* and $\tilde{j_{\ast }}/j_{\ast }$ (crosses in Figure 4(a)). In conclusion, $\tilde{j_{\ast }}$ approximated by Equation (4) and Romanowsky & Fall (2012) is offset from j_* via Equation (6) due to a systematic mass dependence of the disk shape and rotation curve.

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Intriguingly, the 16 spiral galaxies turn out to be highly correlated in (M, j, β) space. Figure 6 shows this space in four alternative projections, revealing that the data form a plane in the space spanned by  lg M_b,  lg j_b, and β, where "lg" denotes the base 10 logarithm. Importantly, this M–j–β relation does not seem to be affected by central bars—a feature worth investigating in future studies.⁵ The plane can be expressed as

$$\begin{equation} \beta =k_1{\rm \,lg\,}\left[\frac{M}{10^{10}\,{M}_{\odot }}\right]+k_2{\rm \,lg\,}\left[\frac{j}{\rm 10^3\,\rm kpc\,{\,{\rm km\,s}^{-1}}}\right]+k_3, \end{equation} \tag{ 9 }$$

where k₁, k₂, and k₃ are free parameters, fitted to the data using a trivariate regression (Appendix C) that accounts for normal measurement errors in all three dimensions. The best fits are (k₁, k₂, k₃) = (0.34 ± 0.03, −0.35 ± 0.04, −0.04 ± 0.02) if (M, j) = (M_b, j_b), and (k₁, k₂, k₃) = (0.31 ± 0.03, −0.33 ± 0.05, −0.02 ± 0.02) if (M, j) = (M_*, j_*). The intervals denote 68% confidence intervals of the correlated uncertainties. Equation (9) is represented by the plane in Figures 6(a) and (b). The correlation between the measured values of β and those predicted by Equation (9) is surprisingly high, with a Pearson correlation coefficient of 0.95. The reduced χ² of the fit is 0.9; thus the deviations of the data from the fit are entirely accounted for by measurement errors. In other words, the data is consistent with zero intrinsic scatter off Equation (9). Another interesting feature is that Equation (9) is irreducible in the sense that it cannot be explained based on the two-dimensional (2D) relations M–j, M–β, and j–β. This is best seen when projecting the data onto the three planes (crosses in Figure 6(b)). In any of these planes, the reduced χ² (14.4, 11.5, and 18.1) of a linear regression is significantly higher than in 3D.

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When discussing the data in the (M, j)-plane it is convenient to rewrite Equation (9) as

$$\begin{equation} \frac{j}{10^3\,\rm kpc\,{\,{\rm km\,s}^{-1}}}=k \xi (\beta)\left[\frac{M}{10^{10}\,{M}_{\odot }}\right]^\alpha, \end{equation} \tag{ 10 }$$

where ξ(β) = exp [ − gβ] (obtained when exponentiating Equation (9)) is a bulge-dependent scaling factor equal to unity in the case of a pure disk (β = 0). The best-fitting parameters are (k, α, g) = (0.77  ±  0.07, 0.98  ±  0.06, 6.65  ±  1.02) if (M, j) = (M_b, j_b), and (k, α, g) = (0.89  ±  0.11, 0.94  ±  0.07, 7.03  ± 1.35) if (M, j) = (M_*, j_*). Equation (10) is shown as solid lines in Figure 6(c) for different values of β. Interestingly, the exponent α is consistent with α = 1. Upon imposing α = 1, the best fit to Equation (10) is (k, g) = (0.76  ±  0.05, 6.83  ±  0.61) for all baryons and (k, g) = (0.91  ±  0.09, 7.59  ±  0.79) for stars only. This fit is shown as dashed lines in Figure 6(c). Given α = 1, Equation (10) can then be rewritten as

$$\begin{equation} \beta =k_1{\rm \,lg\,}\left[\frac{j M^{-1}}{10^{-7}\,\rm kpc\,{\,{\rm km\,s}^{-1}}\,{M}_{\odot }^{-1}}\right]+k_2 \end{equation} \tag{ 11 }$$

with (k₁, k₂) = (− 0.34  ±  0.03, −0.04  ±  0.01) for baryons and (k₁, k₂) = (− 0.30  ±  0.03, −0.01  ±  0.01) for stars. Equation (11) is shown as dashed lines in Figure 6(d).

The M–j–β relation turns out to be surprisingly similar for all baryons (star+cold gas) and for stars alone (Figure 7). In fact, the fitting parameters for baryons and stars (given below Equation (9)) are consistent within their uncertainties. This is due to the fact that adding cold gas approximately moves the galaxies in the (M, j) plane along lines of constant β (blue lines in Figure 7). In other words, the transition from stars to baryons essentially moves the galaxies inside a fixed M–j–β plane. Formally, the close similarity between the relations M_b–j_b–β and M_*–j_*–β is due to the fact that β varies approximately as $j_{\rm b}M_{\rm b}^{-1}$ coupled with the fact that $j_{\rm b}M_{\rm b}^{-1}\approx j_{\ast }M_{\ast }^{-1}$. The latter equation is possible, as the contribution of cold gas to the baryon angular momentum J_b (about 34% on average) is higher than the contribution of cold gas to M_b (about 23%).

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The similarity between the relations M_b–j_b–β and M_*–j_*–β might break down if dwarf galaxies of much higher gas fractions were included. The precise relationship between angular momentum in stars and different cold gas phases will be discussed in a sequel paper.

Having established the M–j–β relation in Section 3.1, we now compare this relation against published data. No significant sample of spiral galaxies with detailed measurements of angular momentum, based on summation of sub-kiloparsec maps, has yet been published. All approximate measurements are restricted to stellar angular momentum without including gas. Thus, this comparison is restricted to samples of approximate stellar angular momentum. The largest and broadest sample was recently published by Romanowsky & Fall (2012), who estimated the stellar angular momenta in a broad mass-range of spiral, lenticular, and elliptical galaxies. Here, we focus on the 67 spiral galaxies listed in Table 4 of Romanowsky & Fall. This table contains bulge mass fractions β, based on the r-band bulge-disk decomposition of Kent (1986, 1987, 1988), stellar masses M_* based on Two Micron All Sky Survey K-band photometry, and approximate stellar angular momenta j_* estimated from global size and velocity measurements (see Section 2.4). Given the irreducible 3D-correlation between M_*, j_*, and β, the average M_*–j_* relation is a poor and potentially misleading estimator for the comparison of two data sets with different β distributions. Therefore, the comparison of the 16 THINGS galaxies against the 67 Romanowsky galaxies must be performed in (M_*, j_*, β)-space or several projections thereof.

The top panels in Figure 8 show two projections of the M_*–j_*–β relation: the M_*–j_* relation and the $(j_{\ast }M_{\ast }^{-1})$–β relation. These are the same projections as those in the bottom panels of Figure 6 (but for stars instead of all baryons). Clearly, the Romanowsky data deviate significantly and systematically from the THINGS data in all three coordinates. This deviation is dominated by differences in measurement techniques, not by systematic differences between the two samples, as can be seen from the six galaxies that are in both samples, connected by lines in Figure 8. Upon careful inspection, the following features explain the offset of the Romanowsky points.

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M_* axis. Since Romanowsky & Fall use a K-band mass-to-light ratio of 1 M_☉/L_{☉, K}, while THINGS data (Leroy et al. 2008) assumed 0.5 M_☉/L_{☉, K}, the stellar masses of Romanowsky masses must be rescaled by a factor 0.5 for the purpose of this comparison.

j_* axis. As explained in Section 2.4, the approximate values $\tilde{j_{\ast }}$ of Romanowsky systematically differ from the fully measured j_*. This systematic offset can be corrected by rescaling the $\tilde{j_{\ast }}$ values using Equation (6).

β axis. The bulge mass fractions of the Romanowsky data, which were adopted from Kent (1986, 1987, 1988), differ significantly from those of the THINGS data, as revealed by the six overlapping objects in Figure 8(b). Their numerical values are as follows.

$$\begin{equation*} \begin{array}{lcccccc}{\rm NGC:} & 2403 & 2841 & 3198 & 4736 & 5055 & 7331\\ \beta _{\rm Kent}= & 0.00 & 0.36 & 0.00 & 0.35 & 0.05 & 0.26\\ \beta _{\rm THINGS}= & 0.02 & 0.10 & 0.03 & 0.32 & 0.17 & 0.16 \end{array} \end{equation*}$$

Kent decomposed the galaxies by performing 2D fits to r-band images. They only assumed that the disk and the bulge have elliptical isophotes in projection, without imposing a disk/bulge model. By contrast, most other studies, including this paper, fit a specific disk and bulge model. The comparison of these two methods is difficult, even more so when applied to different wavebands. However, the Kent decompositions can be verified against the more recent 2D bulge–disk decompositions in H band by Weinzirl et al. (2009). They assumed exponential disks and Sérsic bulges, analogous to our decomposition of the THINGS galaxies. Of their 143 galaxies, five overlap with those in the Romanowsky sample.⁶ The respective bulge mass fractions disagree considerably.

$$\begin{equation*} \begin{array}{lccccc}{\rm NGC:} & 1087 & 2775 & 4062 & 4698 & 7217\\ \beta _{\rm Kent}= & 0.00 & 0.20 & 0.03 & 0.55 & 0.25\\ \beta _{\rm Weinzirl}= & 0.00 & 0.61 & 0.02 & 0.22 & 0.54 \end{array} \end{equation*}$$

Due to this discrepancy, the values β_Kent adopted in the Romanowsky sample are not appropriate for the purpose of comparison with the present data. We therefore re-estimate the bulge mass fractions of the Romanowsky galaxies from their numerical Hubble types T (drawn from HyperLeda (Paturel et al. 2003) and listed in Table 4 of Romanowsky & Fall 2012). To compute β from T, we use the mean T–β relation of Weinzirl et al. (2009), shown as blue squares in their Figure 14.⁷

The three adjustments of M_*, j_*, and β in the Romanowsky data are justified and necessary for a fair comparison with the present study. Given these adjustments, the data become consistent with the THINGS data (Figure 8, lower panels). In fact, the trivariate fit of Equation (10) to the Romanowsky data with assumed statistical uncertainties of 0.1 dex in M_* and j_* and 20% for β, gives (k, α, g) = (0.99  ±  0.15, 0.92  ±  0.06, 7.63  ±  0.99) in full agreement with the respective parameters of the THINGS galaxies for (M, j) = (M_*, j_*). The reduced χ² of this fit is 1.7; thus the scatter of the Romanowsky data is roughly accounted for by observational uncertainties. In this sense the Romanowsky data fully supports the scaling relations of Section 3.1.

In the model of a singular isothermal spherical CDM halo (Mo et al. 1998) of truncation radius R_h and dynamical mass M_h, Newtonian gravity sets the circular velocity to

$$\begin{equation} V_{\rm h}=(GM_{\rm h}/R_{\rm h})^{1/2}, \end{equation} \tag{ 12 }$$

where G denotes the gravitational constant. For V_h to be constant (isothermicity), the mass density needs to vary as $\rho (r)=V_{\rm h}^2(4\pi G r^2)^{-1}$ ∀r ⩽ R_h. Thus the potential energy becomes $E_{\rm pot}=-M_{\rm h}V_{\rm h}^2$. Following the virial theorem (2E_kin = −E_pot), the total energy is

$$\begin{equation} E_{\rm h}= -0.5M_{\rm h}V_{\rm h}^2. \end{equation} \tag{ 13 }$$

Halos are embedded in the cosmic background field of mean density ρ_c = 3H²(8πG)⁻¹, where H is the Hubble "constant" at the considered epoche. The halo radius R_h can then be defined as the radius to which orbits are approximately virialized. In the spherical collapse model (Cole & Lacey 1996), the mean density enclosed by R_h is about 200ρ_c, thus ρ(R_h) = (200/3)ρ_c. It follows that

$$\begin{equation} R_{\rm h}^3=10^{-2}GH^{-2}M_{\rm h}. \end{equation} \tag{ 14 }$$

Equations (12)–(14) are the essential scaling relations of the isothermal CDM halo. This model is manifestly scale-free (at fixed H) in that all global quantities depend on a single scale-factor (e.g., on R_h) via

$$\begin{equation} V_{\rm h}\propto R_{\rm h}\propto M_{\rm h}^{1/3}\propto |E_{\rm h}|^{1/5}. \end{equation} \tag{ 15 }$$

When dealing with the halo angular momentum J_h, the spin parameter (Steinmetz & Bartelmann 1995)

$$\begin{equation} \lambda \equiv J_{\rm h}|E_{\rm h}|^{1/2}G^{-1}M_{\rm h}^{-5/2} \end{equation} \tag{ 16 }$$

has the advantage of being approximately invariant during the growth of a halo in the absence of major mergers (Figure 1 in Stewart et al. 2013). Combining Equation (16) with the scaling equations of the isothermal halo,

$$\begin{equation} j_{\rm h}\equiv \frac{J_{\rm h}}{M_{\rm h}} = \sqrt{2}\,\lambda \,R_{\rm h}V_{\rm h}= \frac{\sqrt{2}\,\lambda \,G^{2/3}}{(10H)^{1/3}}\,M_{\rm h}^{2/3}. \end{equation} \tag{ 17 }$$

If the baryon angular momentum remains conserved during galaxy formation, then the initial equality j_b = j_h for a uniform mixing of baryons and dark matter applies at all times. More generally, we can define the ratio f_j ≡ j_b/j_h, which is unity in the conserved case. Further introducing the baryon mass fraction f_M ≡ M_b/M_h,

$$\begin{equation} j_{\rm b}= \frac{\sqrt{2}\,\lambda \,f_{\rm j}\,f_{\rm M}^{-2/3}\,G^{2/3}}{(10H)^{1/3}}\,M_{\rm b}^{2/3}. \end{equation} \tag{ 18 }$$

Adopting the local H = 70 km s⁻¹ Mpc⁻¹ and conventional units, Equation (18) becomes

$$\begin{equation} \frac{j_{\rm b}}{10^3\,\rm kpc\,{\,{\rm km\,s}^{-1}}} = 1.96\,\lambda f_{\rm j}f_{\rm M}^{-2/3}\left[\frac{M_{\rm b}}{10^{10}\,{M}_{\odot }}\right]^{2/3}. \end{equation} \tag{ 19 }$$

This equation is equivalent to Equation (15) of Romanowsky & Fall (2012) upon adopting the same H and substituting f_M = f_bf_⋆, where f_b = 0.17 is the universal baryon fraction (Komatsu et al. 2011).

To compare Equation (19) against the THINGS data, the dimensionless parameters need to be given sensible values. The spin parameter λ can be determined from cosmological simulations that tackle the formation of halos, including the tidal build-up of angular momentum. N-body simulations find present-day values around λ ≈ 0.04 with an intrinsic scatter of about 0.02 and no significant correlation to M_h (Macciò et al. 2008; Knebe & Power 2008). The baryon fraction f_M depends on the galaxy mass and is maximal for intermediate, Milky Way mass galaxies (McGaugh et al. 2010; Behroozi et al. 2013). The mean of the stellar mass considered here being approximately equal to that of the Milky Way, we adopt the constant⁸ value of the Milky Way, estimated⁹ to f_M ≈ 0.05. Regarding the spin fraction j_b, high-resolution simulations of four Milky Way-type galaxies (Stewart et al. 2013) find present day values of f_j ≈ 1 within about 50%. Given those choices, $1.96\lambda f_{\rm j}f_{\rm M}^{-2/3}$ can vary between 0.14 and 1.3, spanning the gray-shaded zone of Figure 9.

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In summary, isolated spiral galaxies evolved without major mergers, abnormal feedback, or otherwise exotic histories, are predicted to lie in the shaded zone of Figure 9. This prediction is consistent with the data. Coupling this prediction of a mean relation $j_{\rm b}\propto M_{\rm b}^\alpha$, where α = 2/3, with the empirical finding of α ≈ 1 for fixed β's (solid lines in Figure 9), implies that more massive spiral galaxies tend to have higher bulge fractions than less massive ones. This trend qualitatively agrees with observations of the stellar mass function split into Sa, Sb, Sc, and Sd types (Read & Trentham 2005).

The (M, j) plane is linked to the fundamental plane (FP) for spiral galaxies (Koda et al. 2000; Han et al. 2001; Courteau et al. 2007), a 3D relation between total luminosity L, disk scale radius R, and asymptotic velocity¹⁰ V, forming a plane in log-space. Projected onto 2D (Figure 10), the FP reduces to the L–R relation, the R–V relation, and the V–L relation. The last, known as the TF relation, appears to be a nearly edge-on projection of the FP (Shen et al. 2002).

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In the scale-free approximation of Section 4.1, a direct link between the FP and the (M, j) plane appears, because the three quantities L, R, and V scale with M and j (here, M ≈ M_b ≈ M_*, j ≈ j_b ≈ j_*, and R ≈ R_*). First, the luminosity is a linear proxy of mass, L∝M. Second, in an exponential disk at constant circular velocity, the scale radius becomes R = j/(2V) (Equation (4)). Third, the velocity V can be approximated by the halo velocity $V_{\rm h}\propto M_{\rm h}^{1/3}$ (Equation (15)); thus, V∝M^1/3 when assuming a constant disk mass fraction M/M_h. Hence the transformation (M, j)↦(L, R, V) writes

$$\begin{equation} L\propto M,\quad R\propto j M^{-1/3},\quad V\propto M^{1/3}. \end{equation} \tag{ 20 }$$

This mapping is sketched schematically in Figure 10. It implies that one can reconstruct the FP of spiral galaxies from their distribution in the (M, j) plane. This distribution is described by j = kM^2/3 (Equation (18)), where k is a λ-dependent, scattered parameter (gray shading in Figure 9). Combining j = kM^2/3 with Equations (20), the projected relations of the FP become

$$\begin{equation} R\propto k L^{1/3},\quad V\propto k^{-1}R,\quad L\propto V^3. \end{equation} \tag{ 21 }$$

These scalings are remarkably similar to those found by Courteau et al. (2007) in I band for a sample of 1300 spiral galaxies of all Hubble types (S0a-Sm). Their fits are R∝L^{0.32  ±  0.02} (scatter σ_ln R = 0.33), R∝V^{1.10  ±  0.12} (σ_ln R = 0.38), V∝L^{0.29  ±  0.01} (σ_ln V = 0.13). The scatter of the third scaling—the TF relation—is significantly smaller than that of the other two, relative to the range spanned by the data (Figure 3 in Courteau et al. 2007). This difference in scatter is elegantly explained by the fact that the first two relations in Equations (21) depend on k, while the TF relation does not, as it is an exactly edge-on projection of the FP in our simplistic model.

The three relations of Equation (21) are consistent with the present sample, where L = L_K and R = R_*, as shown in Figure 11 (first three panels). Solid lines are power-laws with zero-points fitted to the data and exponents fixed according to Equation (21). Shaded regions denote standard deviations. The location of a galaxy in these planes depends on its position in the (M, j) plane, which systematically depends on β (Section 3); thus the visible offset from between open and filled points in Figure 11. The TF relation exhibits the smallest scatter relative to the range of the data. This had to be expected from the TF being an edge-on projection of the FP in the model discussed so far. In reality, the TF relation is not exactly an edge-on projection of the FP, as explained by the more detailed theory of Shen et al. (2002). Therefore, the offset of galaxies from the mean TF relation correlates with their location in the (M, j) plane, thus with β (hence departing from Equation (21), right). This explains the slight morphology dependence of the TF relation (Kannappan et al. 2002), also visible in the present sample. In fact, we can minimize the scatter of the TF relation by heuristically substituting  lg L_K for  lg L_K − uβ with u ≈ 2 (last panel in Figure 11).

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In summary, within the model of an exponential disk inside a CDM halo, the FP results from mapping the 2D (M, j) plane into 3D (L, R, V) space via Equations (20). This mapping approximately explains the three classical scaling relations that are the 2D projections of the FP, such as the TF relation. The morphology dependence of these three relations can then be traced back to the M–j–β relation established empirically in Section 3.

When discussing the Hubble type-dependence of the M–j relation, Fall (1983) and Romanowsky & Fall (2012) invoked the idea that this dependence might result from different, fixed M–j relations for pure disks and pure bulges. While this idea might be valid for classical bulges in bulge-dominated systems, the data of this paper dispels the hope for such an elegant explanation in the case of spiral galaxies with smaller (pseudo-)bulges.

Let us assume—ad absurdum—that disks and bulges do indeed obey independent M–j relations. This assumption can be understood in two ways, formalized via the following models. In "model 1," disk and bulge are strictly independent in the sense that they obey different relations $j_{\rm disk}=kM_{\rm disk}^\alpha$ and $j_{\rm bulge}=fkM_{\rm bulge}^\alpha$ with constants k > 0 and f > 0. In "model 2," the angular momenta of disk and bulge both depend on the same total mass M = M_disk + M_bulge (i.e., j_disk = kM^α and j_bulge = fkM^α). In both models, the total specific angular momentum j = (1 − β)j_disk + βj_bulge becomes

$$\begin{equation} j = k \xi (\beta)\,M^\alpha, \end{equation} \tag{ 22 }$$

where ξ(β) = (1 − β)^{1 + x} + fβ^{1 + x} with x = α for model 1 and x = 0 for model 2. Intermediate models can then be obtained by choosing 0 < x < α. Considering this range for x, choosing f between f = 0 (zero-rotation bulge model of Mo et al. 1998) and f = 0.2 (empirical value of Fall & Romanowsky 2013), and adopting the empirical α ≈ 1 (Equation (10)), implies that ξ(β) falls within the shaded region of Figure 12. By contrast, ξ(β) = exp [ − gβ] determined empirically (see below Equation (10)) varies as the solid line in Figure 12. This measurement is clearly inconsistent with any plausible model of independent disk and bulge relations—an unrealistic α ≈ 6 or f < 0 would be required to match up the model with the data. Therefore, the initial assumption of independent M–j relations for disks and bulges cannot be true.

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This conclusion can be confirmed explicitly by measuring the M_*–j_*–β relation of the disk component only. The stellar mass of the disk M_disk = βM_* is drawn directly from the stellar bulge-disk decompositions (Appendix A). The specific stellar angular momentum of the disk j_disk is computed via Equation (2), substituting Σ(r) for the disk stellar mass surface density, again drawn from our bulge–disk decompositions. Figure 13 shows the resulting relation projected onto the $(j_{\rm disk}M_{\rm disk}^{-1},\beta)$ plane. It turns out that $j_{\rm disk}M_{\rm disk}^{-1}$ correlates strongly with β, hence explicitly rejecting the model of a fixed M–j relation for the disk component. Disks with more massive bulges in their centers have lower specific angular momentum, thus smaller radii for a given mass.

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In summary, disks "know" about the bulges via their angular momentum—an interesting feature that must be accounted for by any model of the M–j–β relation.

Late-type galaxies grow their (pseudo-)bulges in situ (Elmegreen et al. 2008; Weinzirl et al. 2009), rather than via major mergers (mass ratios >0.3) thought to produce the classical bulges of early-type galaxies (Koda et al. 2009). Yet, the β dependence of the M_disk–j_disk relation in late-type systems (Section 5.1) rules out the tempting idea that low-j material simply migrates toward the bulge until the surrounding disk satisfies a certain bulge-independent criterion, such as a universal stability threshold. A more dynamic explanation is needed to account for the β dependence of j_disk.

To uncover the origin of the M–j–β relation, let us note that this relation is approximately a monotonic relation between β and jM⁻¹, similarly for baryons and stars, as $j_{\rm b}M_{\rm b}^{-1}\approx j_{\ast }M_{\ast }^{-1}$ according to Section 3.2. Therefore, understanding the M–j–β relation reduces to understanding the quantity jM⁻¹ and its effect on bulge formation. As for the first step, it is easily shown that jM⁻¹ is a measure of the surface density. In fact, using Equation (4), the surface density scale Σ₀∝MR⁻² can be rewritten as Σ₀∝Mj⁻¹R⁻¹V. Assuming a constant velocity V = V_h and using V_h∝HR_h (from Equations (12) and (14)), gives Σ₀∝HMj⁻¹R_hR⁻¹, where R_h is the halo radius. If R∝R_h (corresponding to constant λ and f_j), then

$$\begin{equation} \Sigma _0\propto H M j^{-1}. \end{equation} \tag{ 23 }$$

Thus, jM⁻¹ scales inversely with the surface density (or the "concentration") of the galaxy baryons. In this way, our finding that the bulge mass fraction β scales inversely with jM⁻¹, confirms earlier evidence (Prieto et al. 1989) for a relation between the morphology of spiral galaxies and their mean surface density.

Less obvious is the physics behind the connection between the surface density and β. Assuming that the bulge forms from instabilities in the gas-rich protogalaxies, the characteristic bulge growth rate $\dot{M}_{\rm bulge}/M$ and the final bulge mass fraction β are expected to decrease monotonically with the stability of the protogalaxy. Locally, the stability of a flat disk against Jeans instabilities is quantified by the parameter Q = σ κ (3GΣ)⁻¹ (Toomre 1964), where σ is the local velocity dispersion, κ is the orbital frequency, and Σ is the local surface density. By extension, the mean stability of the disk is then characterized by a global parameter $\overline{Q}\propto \sigma _0\kappa _0\,\Sigma _0^{-1}$, where σ₀, κ₀, and Σ₀ are normalization factors of the dispersion, orbital frequency, and surface density, respectively. For circular orbits, κ₀∝VR⁻¹∝HR_h R⁻¹. Assuming again that R∝R_h, yields κ₀∝H and

$$\begin{equation} \overline{Q}\propto H\sigma _0\Sigma _0^{-1}. \end{equation} \tag{ 24 }$$

Substituting Σ₀ in Equation (24) for Equation (23), the explicit H dependence disappears and

$$\begin{equation} \overline{Q}\propto \sigma _0j M^{-1}. \end{equation} \tag{ 25 }$$

This derivation shows that, up to variations in σ₀, a basic CDM-based galaxy model coupled with an instability-driven bulge can qualitatively account for the monotonic relation between jM⁻¹ and β.

The detailed processes governing the in situ formation of bulges as a function of the Q parameter, including the physics of the velocity dispersion σ, remain subject to numerical modeling. Recent high-resolution hydrodynamic simulations with radiative feedback (Elmegreen et al. 2008; Bournaud et al. 2014) suggest that the semi-stable, gas-rich progenitors of modern spiral galaxies partially collapsed into giant star-forming clumps, which survived the strong radiative feedback over timescales required to spiral to the galaxy center by dynamical friction. According to Bournaud et al., this clump-feeding of the bulge can approximately account for the bulge mass of typical spiral galaxies in the local universe and explain the observed structure and outflows of clumps in galaxies at redshift z ≈ 2 (Genzel et al. 2011). However, the question whether giant clumps survive long enough to migrate to the galaxy center remains debated as summarized by Glazebrook (2013): simulations still allow for both short (Genel et al. 2012) and long lifetimes (Ceverino et al. 2012), depending on the model assumptions, and the observations remain non-conclusive (Genzel et al. 2011; Wuyts et al. 2012; Guo et al. 2012). Details on clumps aside, the success of high-resolution hydrodynamic simulations with radiative feedback in explaining the structure of spiral galaxies is encouraging and suggests that such simulations might hold the key to explaining the M–j–β relation. However, to date, such simulations still represent a major computational challenge (Section 1).

In essence, the M–j–β scaling can be explained from similarity considerations summarizing Sections 4.1 and 5.2. Assuming self-similarity in 3D, the mass M_h of a halo is proportional to its characteristic volume $R_{\rm h}^3$. Newtonian gravity then implies a circular velocity V_h∝(M_h/R_h)^1/2∝R_h. Thus,

$$\begin{equation} V_{\rm h}\propto R_{\rm h}\propto M_{\rm h}^{1/3}. \end{equation} \tag{ 26 }$$

Given these relations and a fixed λ, the specific angular momentum is $j_{\rm h}=J_{\rm h}/M_{\rm h}\propto (\lambda R_{\rm h}M_{\rm h}V_{\rm h})/M_{\rm h}\propto M_{\rm h}^{2/3}$. This scaling extends to M and j in baryons/stars, up to variations in the ratios M/M_h and j/j_h. Thus,

$$\begin{equation} j\propto M^{2/3}. \end{equation} \tag{ 27 }$$

The scatter of this relation due numerically predicted variations in λ, M/M_h, and j/j_h, approximately covers the shaded region in Figure 9 for local spiral galaxies.

If the bulge grows from disk instabilities set by the 2D surface density MR⁻², then β scales monotonically with $MR^{-2}\propto MR_{\rm h}^{-2}\propto M j^{-1}$ (use Equations (26) and (27)). Hence, spiral galaxies of fixed β satisfy

$$\begin{equation} j \propto M \end{equation} \tag{ 28 }$$

with a proportionality factor that decreases monotonically with increasing β.

In brief, late-type galaxies scatter around a mean relation j∝M^2/3, representing 3D self-similarity (fixed volume density profile), while any subsample of fixed β follows a relation j∝M, representing 2D self-similarity (fixed surface density profile). Together, these scalings naturally explain why the bulge fraction of spiral galaxies tends to increase with their mass (see Figure 9 gray shading versus solid lines).