Precision Scaling Relations for Disk Galaxies in the Local Universe

We build up RC templates for galaxies of different luminosities by exploiting the universal rotation curve (URC) statistical approach (see Persic et al. 1996; Salucci & Burkert 2000; Salucci et al. 2007). This is based on the notion, implicit even in the seminal paper by Rubin et al. (1985), that the RCs of local spiral galaxies with given luminosity are remarkably similar in shape. Then the mass modeling is performed not on the individual RCs, but on the stacked, suitably normalized RCs of galaxies within a given luminosity bin.

Such a statistical approach has some relevant advantages over the brute-force fit to individual RCs: (i) it increases the signal-to-noise ratio, allowing for a precise description of the average RCs, smoothing out small-scale fluctuations induced by bad data points and/or by physical features such as spiral warps; and (ii) it allows the simultaneous mass decomposition of RCs for galaxies with similar luminosity but with different spatial sampling.

Typically, the stacked data are constructed as follows: the individual RCs V(R) are normalized in both the dependent and independent variables in terms of the optical radius R_opt and of the corresponding velocity ${V}_{\mathrm{opt}}\equiv V({R}_{\mathrm{opt}})$, to obtain a normalized curve $\tilde{V}(\tilde{R})\equiv V(R/{R}_{\mathrm{opt}})/{V}_{\mathrm{opt}};$ then all the normalized RCs of individual galaxies falling in the same magnitude bin are co-added (properly weighting measurement uncertainties), to derive a stacked curve $\tilde{V}(\tilde{R}| {M}_{I})$. Note that the choices of R_opt and V_opt as normalization points are meant to minimize the uncertainties in velocity estimates because there the RC is flattening and the disk surface brightness is quickly decreasing.

We base our analysis on three stacked RC compilations from the literature. The first one has been constructed by Persic et al. (1996) from about 900 individual RCs extending out to $R\lesssim {R}_{\mathrm{opt}}$, separated into 11 luminosity bins with average I-band magnitude from $\langle {M}_{I}\rangle \sim -18.5$ to −23.2. The second stacked curve compilation has been constructed by Catinella et al. (2006) from a sample of about 2200 individual RCs extending out to 1.5R_opt–2R_opt, separated into 10 I-band magnitude bins with average values from $\langle {M}_{I}\rangle \sim -19.4$ to −23.8. The third stacked curve compilation has been constructed by Yegorova et al. (2011) from a sample of 30 individual RCs extending out to $\lesssim 2\,{R}_{\mathrm{opt}}$ for high-luminosity spirals with average I-band magnitude $\langle {M}_{I}\rangle \sim -23.3$. Uncertainties around the mean for the stacked normalized curves are of order $\lesssim 5 \%$, typically increasing toward the outer regions.

We analyze these stacked RCs with the following physical template:

$$\begin{eqnarray}&&{\tilde{V}}_{\mathrm{TOT}}^{2}(\tilde{R}| {M}_{I})={\tilde{V}}_{\mathrm{DISK}}^{2}(\tilde{R}| {M}_{I})+{\tilde{V}}_{\mathrm{HALO}}^{2}(\tilde{R}| {M}_{I})\,,\end{eqnarray} \tag{ 1 }$$

which is an addition of two terms: (i) an exponential, infinitesimally thin disk profile (Freeman 1970)

$$\begin{eqnarray}&&{\tilde{V}}_{\mathrm{DISK}}^{2}(\tilde{R}| {M}_{I})=\kappa \,{\tilde{R}}^{2}\,\displaystyle \frac{B(1.6\tilde{R})}{B(1.6)},\end{eqnarray} \tag{ 2 }$$

where $B(x)\equiv {I}_{0}(x){K}_{0}(x)-{I}_{1}(x){K}_{1}(x)$ is a combination of modified Bessel functions (for finite thickness the rotation curve will have a very similar shape but for a 5% lower peak; see Casertano 1983); and (ii) a cored dark halo profile following the Burkert shape (see Burkert 1995; Katz et al. 2017)

$$\begin{eqnarray}&&{\tilde{V}}_{\mathrm{HALO}}^{2}(\tilde{R}| {M}_{I})=(1-\kappa )\displaystyle \frac{1}{\tilde{R}}\,\displaystyle \frac{G(\tilde{R}/{\tilde{R}}_{c})}{G(1/{\tilde{R}}_{c})}\,,\end{eqnarray} \tag{ 3 }$$

where $G(x)=\mathrm{ln}(1+x)-{\tan }^{-1}(x)+0.5\,\mathrm{ln}(1+{x}^{2})$. This mass decomposition involves two parameters, namely, the normalized core radius ${\tilde{R}}_{c}({M}_{I})$ and the relative contribution $\kappa ({M}_{I})$ of the disk to the overall template at the optical radius. The main assumption in the construction of a universal template is that these parameters assume a single value within a (small) magnitude bin; this implies (see next section) that the stellar ${M}_{\star }\propto \kappa \,{V}_{\mathrm{opt}}^{2}\,{R}_{\mathrm{opt}}$ and halo masses ${M}_{{\rm{H}}}\propto (1-\kappa ){V}_{\mathrm{opt}}^{2}\,{R}_{\mathrm{opt}}/G({R}_{\mathrm{opt}}/{R}_{c})$ of each galaxy within a magnitude bin are determined only by further specifying the observed optical radius R_opt and velocity V_opt.

In the Appendix we discuss to what extent the results of our analysis are affected by adopting shapes of the DM halo different from Burkert's, such as the classic NFW profile extracted from N-body simulations (e.g., Navarro et al. 1997,2010; Ludlow et al. 2013; Peirani et al. 2017) and the profile indicated by hydrodynamical simulations including baryonic effects on the DM halo (e.g., Di Cintio et al. 2014). In a nutshell, cuspier halo density profiles in the inner region yield a lower disk contribution and hence a smaller stellar mass. The inferred halo mass is to some extent less sensitive to the halo profile, since for the majority of galaxies it depends on the outer behavior of the RC, which is similar. However, in faint galaxies where the disk contribution is marginal or negligible, the inferred halo mass is also sensitive to the inner shape of the RCs, and it is found to be larger for cuspier density profiles. All that adds to the well-known diversity in the fitting accuracy of the RCs for strictly dwarf galaxies (not included in our sample), which appreciably increases when cored profiles are exploited (e.g., Gentile et al. 2004; Karukes & Salucci 2017).

For each of the three samples described above and for any of the respective magnitude bins, we fit the stacked RCs with Equations (1)–(3) and determine the best-fit parameters κ and ${\tilde{R}}_{c};$ the results of the fitting to the stacked RCs with our templates are reported in Figures 1 and 2. The overall fits are reasonably good, although for some high-luminosity bins the shape of the outer RCs is not very well constrained, and this causes large uncertainties on the estimated ${\tilde{R}}_{c}$. In these respects, the sample by Yegorova et al. (2011) provides the best constraints for high-luminosity disks, suggesting rather flat shapes beyond R_opt. This is also confirmed from recent, high-resolution radio observations of a few RCs out to $\lesssim 3\,{R}_{\mathrm{opt}}$ by Martinsson et al. (2016). RC measurements of such a wide spatial extension in an ensemble of local bright spirals would be extremely relevant to improve the robustness of the stacked template at high luminosity.

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The values and uncertainties of the best-fit parameters for the three different samples are reported in Figure 3. We find that the luminosity dependence of the parameters is rather weak and can be rendered with the approximated expressions

$$\begin{eqnarray}\mathrm{log}{\tilde{R}}_{c}({M}_{I}) & \simeq & 0.182+0.421\,\mathrm{log}({\tilde{L}}_{I})+0.178\,{[\mathrm{log}({\tilde{L}}_{I})]}^{2}\\ \kappa ({M}_{I}) & \simeq & 0.656+0.369\,\mathrm{log}({\tilde{L}}_{I})-0.072\,{[\mathrm{log}({\tilde{L}}_{I})]}^{2},\end{eqnarray} \tag{ 4 }$$

where ${\tilde{L}}_{I}={10}^{-({M}_{I}+21.9)/2.5};$ these are also illustrated, together with the associated uncertainties, in Figure 3.

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The resulting templates as a function of radius and I-band magnitude are plotted in Figure 4, with the contribution of the disk and of the halo highlighted. As expected, it is seen that in moving toward brighter galaxies the contribution of the disk in the inner regions becomes predominant, while the rising portion of the RC enforced by the dark halo extends to outer radii. We stress that the above method does not require adopting a specific value for the mass-to-light ratio M_⋆/L, but only assuming its dependence on radius (see Portinari & Salucci 2010).

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We now exploit the templates to derive global properties for a sample of 546 local disk-dominated galaxies selected from the compilation by Persic & Salucci (1995; see also Mathewson et al. 1992). For the selection we require the following properties: (i) the galaxy is Hubble-type classified as from Sb to Sm (to minimize the impact of bulge contamination to the stellar mass estimate), with an I-band magnitude M_I in the range from −18.5 to −23.8 (for most of the sample M_I falls in the range from −19.5 to −22.5); (ii) the galaxy has a well-measured I-band photometry extended out to several disk scale lengths $\gtrsim 4\,{R}_{d}$, from which the optical radius R_opt ≡ 3.2R_d can be determined within 10% accuracy; and (iii) the rotation velocity V_opt at the optical radius is robustly measured at a few percent level. The ESO ID of the galaxies in the sample and the corresponding magnitude M_I, disk scale length R_d, and optical velocity V_opt are reported in Table 2. Note that we have updated the distance measurements for each galaxy of this sample by adopting the most recent determination from the NED database (see https://ned.ipac.caltech.edu/). As a further check on our criteria aimed at selecting disk-dominated galaxies, we computed the I-band light concentration C_I of the objects in the sample; this quantity is defined as the ratio of the radii containing 90% and 50% of the light. The related distribution is nearly Gaussian with peak $\langle {C}_{I}\rangle \approx 2.3$ and variance ${\sigma }_{C}\approx 0.1$. The average value is very close to the expectation ${C}_{I}\approx 2.32$ for a pure thin exponential disk; the range of C_I spanned by the galaxies in our sample is also similar to that measured for disk-dominated galaxies in the Sloan Digital Sky Survey (see Bernardi et al. 2010, their Figure 4, referring to light concentration in the r band).

For each galaxy in the sample, we compute the template with the values of the parameters $\kappa ({M}_{I})$ and ${\tilde{R}}_{c}({M}_{I})$ appropriate for the galaxy magnitude M_I. The RC in physical units is recovered by de-normalizing the template ${\tilde{V}}_{\mathrm{TOT}}(\tilde{R})$ with the observed values of R_opt and V_opt for each galaxy, to get the total ${V}_{\mathrm{TOT}}(R)$ and the associated disk ${V}_{\mathrm{DISK}}(R)$ and halo ${V}_{\mathrm{HALO}}(R)$ components. We have checked a posteriori that the denormalized template provides indeed a good description of the individual RCs; the distribution of reduced chi-squared residuals over the full sample has a mean around 1.1 and a dispersion around 0.4. Note that, on the other hand, our fitting procedure applied to the individual galaxy RCs (in place of the stacked data) gives, for most of the sample, very loose constraints on the parameters κ and ${\tilde{R}}_{c}$ and hence on the stellar and halo masses.

From the disk component ${V}_{\mathrm{DISK}}(R)$, the stellar mass is straightforwardly derived as ${M}_{\star }\equiv {V}_{\mathrm{DISK}}^{2}({R}_{\mathrm{opt}}){R}_{\mathrm{opt}}/1.1\,G$. In our analysis, we shall also use the central surface density in stars ${{\rm{\Sigma }}}_{0}\equiv {M}_{d}/2\pi \,{R}_{d}^{2}$ and the effective radius ${R}_{e}\approx 1.68\,{R}_{d}\,\approx 0.52\,{R}_{\mathrm{opt}}$, wherein half of the stellar mass is encompassed. Finally, the specific angular momentum (per unit mass) of the stellar component is simply given by

$$\begin{eqnarray}&&{j}_{\star }\equiv 2\,{f}_{R}\,{R}_{d}\,{V}_{\mathrm{opt}}\end{eqnarray} \tag{ 5 }$$

in terms of a shape factor ${f}_{R}\equiv \int {dx}\,{x}^{2}\,{e}^{-x}\,{V}_{\mathrm{TOT}}({{xR}}_{d})/2\,{V}_{\mathrm{opt}}$ of order unity, which mildly depends on the shape of the RC template; for most of the galaxies in our sample, ${f}_{R}\simeq 0.93$ holds to an extremely good approximation.

As for the DM mass, we proceed as follows. From the dark halo component ${V}_{\mathrm{HALO}}(R)$, we derive the radius ${R}_{{\rm{\Delta }}}$ where the average halo density $\langle {\rho }_{{\rm{\Delta }}}\rangle \equiv 3\,{M}_{{\rm{\Delta }}}(\lt {R}_{{\rm{\Delta }}})/4\pi \,{R}_{{\rm{\Delta }}}^{3}=3{V}_{\mathrm{HALO}}^{2}({R}_{{\rm{\Delta }}})/4\pi \,G\,{R}_{{\rm{\Delta }}}^{2}$ equals a given overdensity Δ of the critical one ${\rho }_{\mathrm{crit}}\equiv 3{H}^{2}/8\pi \,G$, with H the Hubble parameter. Setting $\langle {\rho }_{{\rm{\Delta }}}\rangle ={\rm{\Delta }}\,{\rho }_{\mathrm{crit}}$ yields the implicit equation ${V}_{\mathrm{HALO}}({R}_{{\rm{\Delta }}})/{R}_{{\rm{\Delta }}}=H\,\sqrt{{\rm{\Delta }}/2}$, which is easily solved for R_Δ; the associated halo mass is simply estimated from ${M}_{{\rm{\Delta }}}\equiv {\rm{\Delta }}\,{H}^{2}\,{R}_{{\rm{\Delta }}}^{3}/2\,G$. For the standard cosmology adopted here, the quantities R₁₀₀ and M₁₀₀ corresponding to Δ = 100 constitute a very good approximation (e.g., Eke et al. 1996; Bryan & Norman 1998) to the virial radius and mass of halos at redshift $z\approx 0;$ hereafter we use the notations ${R}_{{\rm{H}}}={R}_{100}$, ${M}_{{\rm{H}}}={M}_{100}$, and ${V}_{{\rm{H}}}^{2}=G\,{M}_{{\rm{H}}}/{R}_{{\rm{H}}};$ in the literature, the radius ${R}_{200}\approx 0.77\,{R}_{100}$, mass ${M}_{200}\approx 0.87\,{M}_{100}$, and velocity ${V}_{200}={(G{M}_{200}/{R}_{200})}^{1/2}\,\approx 1.07\,{V}_{{\rm{H}}}$ are also often exploited. Finally, the specific angular momentum of the halo is defined as (see Mo et al. 1998, 2010)

$$\begin{eqnarray}{j}_{{\rm{H}}} & = & \sqrt{2}\,\lambda \,{R}_{{\rm{H}}}\,{V}_{{\rm{H}}}\approx 4.8\times {10}^{4}\,\lambda \,E{(z)}^{-1/6}\\ & & \times {\left(\displaystyle \frac{{M}_{{\rm{H}}}}{{10}^{12}{M}_{\odot }}\right)}^{2/3}\,\,\mathrm{km}\,{{\rm{s}}}^{-1}\,\mathrm{kpc}\\ & \approx & 2.9\times {10}^{4}\,\lambda \,E{(z)}^{-1/2}{\left(\displaystyle \frac{{V}_{{\rm{H}}}}{100\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}\,\,\mathrm{km}\,{{\rm{s}}}^{-1}\,\mathrm{kpc}\,,\end{eqnarray} \tag{ 6 }$$

where $E{(z)=(H/{H}_{0})}^{2}\equiv {{\rm{\Omega }}}_{M}\,{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}$ is a redshift-dependent factor (close to unity for local spirals) and λ is the spin parameter of the host DM halo. Numerical simulations (see Barnes & Efstathiou 1987; Bullock et al. 2001; Macció et al. 2007; Zjupa & Springel 2017) have shown that λ exhibits a lognormal distribution with average value $\langle \lambda \rangle \approx 0.035$ and dispersion ${\sigma }_{\mathrm{log}\lambda }\approx 0.25$ dex, nearly independent of mass and redshift.

In Figure 5 we show the relationship between the halo mass M_H and the stellar mass M_⋆ from our analysis, both for the individual galaxies of the sample and for the mean within bins of given M_⋆. We also show a polynomial fit to our mean result for binned data, whose parameters are reported in Table 1.

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Table 1.  Fits to Global Relationships for Local Spiral Galaxies

Relation	Fitting Parameters
y–x	y0	y1	y2	y3
$\mathrm{log}{M}_{{\rm{H}}}\mbox{--}\mathrm{log}{M}_{\star }$	11.8562375	0.9284897	0.2066120	0.0005020
$\mathrm{log}{R}_{e}\mbox{--}\mathrm{log}{M}_{\star }$	0.7519445	0.2332562	0.0494138	0.0267494
$\mathrm{log}{{\rm{\Sigma }}}_{0}\mbox{--}\mathrm{log}{M}_{\star }$	8.6510640	0.4887056	−0.1162257	−0.0515896
$\mathrm{log}{M}_{\star }\mbox{--}\mathrm{log}{V}_{\mathrm{opt}}$	10.5269449	3.4242617	−1.2325658	1.3322386
$\mathrm{log}{M}_{\star }/{L}_{I}\mbox{--}\mathrm{log}{V}_{\mathrm{opt}}$	0.2377460	0.8276025	−0.3334509	4.9630465
${V}_{\mathrm{opt}}/{V}_{200}\mbox{--}\mathrm{log}{M}_{\star }$	1.2525990	−0.1582360	−0.1246982	−0.0099017
${V}_{\mathrm{opt}}/{V}_{200}\mbox{--}\mathrm{log}{V}_{\mathrm{opt}}$	1.2620718	−0.3582905	−1.8746358	−1.0999992
${V}_{\mathrm{opt}}/{V}_{200}\mbox{--}\mathrm{log}{R}_{200}$	1.2845604	−0.2741447	−1.5439286	1.0877142
$\mathrm{log}{j}_{\star }\mbox{--}\mathrm{log}{M}_{\star }$	2.9833952	0.4951152	0.0572170	0.0140116
$\mathrm{log}{j}_{\star }\mbox{--}\mathrm{log}{V}_{\mathrm{opt}}$	2.9945072	1.7240553	0.2421562	0.3331636
$\mathrm{log}{j}_{\star }\mbox{--}\mathrm{log}{M}_{{\rm{H}}}$	3.0521633	0.4920955	−0.0685457	0.0460152
${f}_{j}\mbox{--}\mathrm{log}{M}_{\star }$	0.7212290	−0.1962629	−0.0883662	0.0491686
${f}_{j}\mbox{--}\mathrm{log}{V}_{\mathrm{opt}}$	0.6918093	−0.8996696	−0.8156081	2.8074829
${f}_{j}\mbox{--}\mathrm{log}{M}_{{\rm{H}}}$	0.6864526	−0.2666190	−0.0725810	0.1276653
$\mathrm{log}{R}_{e}\mbox{--}\mathrm{log}{R}_{200}$	0.7066246	0.7334537	−0.4857772	1.2129571
$\mathrm{log}{R}_{d}\mbox{--}\mathrm{log}{V}_{\mathrm{opt}}$	0.5286874	0.6689526	0.0000000	0.0000000
$\mathrm{log}{V}_{\mathrm{opt}}\mbox{--}\mathrm{log}{R}_{d}$	2.1440626	0.6154330	0.0000000	0.0000000

Note. Relation y–x is fitted with the polynomial shape $y={y}_{0}+{y}_{1}\,{(x\mbox{--}\tilde{x})+{y}_{2}(x\mbox{--}\tilde{x})}^{2}+{y}_{3}\,{(x\mbox{--}\tilde{x})}^{3}$. Masses are in solar units, sizes in kpc, velocities in km s⁻¹, and specific angular momenta in km s⁻¹ kpc. Normalization points are chosen as $\mathrm{log}{\tilde{M}}_{\star }=10.5$, $\mathrm{log}{\tilde{V}}_{\mathrm{opt}}=2.2$, $\mathrm{log}{\tilde{R}}_{200}=2.2$, $\mathrm{log}{\tilde{R}}_{d}=0.5$, and $\mathrm{log}{\tilde{M}}_{{\rm{H}}}=12.0$. Fits hold to within 5% within the ranges: $\mathrm{log}{M}_{\star }\sim [9.0,11.5]$, $\mathrm{log}{V}_{\mathrm{opt}}\sim [1.8,2.5]$, logR₂₀₀ and $\mathrm{log}{R}_{d}\sim [1.9,2.5]$, and $\mathrm{log}{M}_{{\rm{H}}}\sim [11.0,12.5]$.

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We compare the relationship from our analysis to the independent determinations via weak gravitational lensing by Mandelbaum et al. (2016) and Reyes et al. (2012) and via abundance matching techniques (in terms of the average $\langle {M}_{{\rm{H}}}\rangle$ at given M_⋆) by Shankar et al. (2006), Aversa et al. (2015), and Rodriguez-Puebla et al. (2015; see their Figure 10), finding a remarkable agreement. Note that we derive dynamical estimates of the stellar mass for a well-defined sample of pure disks, while the above weak-lensing studies refer to stellar masses inferred from photometry of blue color-selected galaxies. We stress that the samples and methodologies are completely different (and suffer from different biases); therefore, the apparent agreement is of paramount relevance in order to check for the respective systematics and assess the relationship between stellar mass and halo mass in disk-dominated galaxies. In particular, our basic assumptions on the halo component (see Section 2.1) are validated and complemented by weak-lensing measurements, which are mostly sensitive to the DM distribution in the outskirts.

In addition, it is striking that the 1σ dispersion around the mean relation from our analysis amounts to about 0.08 dex. This is appreciably smaller than in any other previous determination and sets a new standard for the ${M}_{{\rm{H}}}\mbox{--}{M}_{\star }$ relationship of local spiral galaxies. Hereafter the quoted dispersions around the mean are meant to include the intrinsic scatter of the data for individual galaxies within the bin, the measurement errors in the determination of the optical radii R_opt and velocities V_opt, and, as a major contribution, the uncertainty in the reconstruction of the templates (associated with the uncertainties in the parameters κ and ${\tilde{R}}_{c}$; see Figure 3).

The inset illustrates the same outcome in terms of the star formation efficiency ${f}_{\star }\equiv {M}_{\star }/{f}_{b}\,{M}_{{\rm{H}}}\approx {M}_{\star }/0.16\,{M}_{{\rm{H}}}$, i.e., the efficiency at which the original baryon content of a halo is converted into stars. This is often considered a fundamental quantity, since it bears the imprint of the processes (gas cooling and condensation, star formation, energy/momentum feedback from supernovae and stellar winds, etc.) ruling galaxy formation and evolution. We find that it amounts to around 30% for stellar masses ${M}_{\star }\,\gtrsim$ a few × 10¹⁰ M_⊙, which is quite a high value if compared to the maximum of $\lesssim 20 \%$ for early-type galaxies (see Shi et al. 2017, and references therein).

The physical interpretation is that the stellar masses of late-type galaxies are being accumulated at low rates over several gigayears, regulated by continuous energy feedback from supernova explosions and stellar winds; this is to be contrasted with the much larger star formation rates and short duration timescales ≲1 Gyr (likely related to feedback from the central accreting supermassive black hole) for the progenitor of massive ellipticals (e.g., Lapi et al. 2011, 2017; Kriek et al. 2016; Glazebrook et al. 2017). Note that the low metallicity of late-type galaxies relative to ellipticals calls for processes able to continuously dilute the star-forming gas; viable possibilities are galactic fountains and/or continuous inflow of pristine gas from the outer regions of the dark halo (see Fraternali 2017; Shi et al. 2017).

Although our analysis here concerns only local spirals, Hudson et al. (2015) showed that the relationship between stellar and halo masses does not appreciably evolve out to $z\approx 0.7$. This is consistent with the aforementioned notion that, in late-type galaxies, the star formation and DM accretion go in parallel along cosmic times. It would be extremely important to further test this finding by independent measurements, and in particular by applying our analysis to a statistically significant high-redshift sample of disk-dominated galaxies. If the shape and tightness of the M_H versus M_⋆ relation stay put or evolve in a controlled way with respect to the local universe, then the halo masses of high-redshift spiral galaxies out to $z\sim 1$ could be robustly determined by evaluating the stellar mass content, hence dispensing with the systematic uncertainties related to clustering or to weak gravitational lensing measurements. In perspective, a reliable determination of halo masses in the redshift range z ≲ 1, an epoch where the dark energy starts driving the accelerated expansion of the universe, could potentially have profound implications for cosmological studies.

In Figure 6 we show the relationship between the effective radius R_e ≈ 1.68R_d of the stellar component and the stellar mass M_⋆ from our analysis, both for the individual galaxies of the sample and for the mean within bins of given M_⋆. We also show a polynomial fit to our mean results for binned data, whose parameters are reported in Table 1; this is in good agreement with the relationships for late-type galaxies by Reyes et al. (2011), by van der Wel et al. (2014), and by Shen et al. (2003; their $\mathrm{log}{R}_{e}$ have been corrected by +0.15 dex to transform from circularized to major-axis sizes).

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We compare the relationship from our analysis to the direct determinations via photometry and SED modeling by Dutton et al. (2011) and Huang et al. (2017), finding a remarkable agreement. We also confront our relation with the independent measurements from individual RC modeling by Romanowsky & Fall (2012) and Lelli et al. (2016), finding a good accord. The 1σ dispersion around the mean relation from our analysis amounts to about 0.05 dex. Note that the small scatter in our R_e versus M_⋆ relationship is partly due to the selection of pure disk-dominated galaxies in the sample analyzed here.

The same data can be recast in terms of the relationship between the central surface density ${{\rm{\Sigma }}}_{0}\equiv {M}_{\star }/2\pi \,{R}_{d}^{2}$ and the stellar mass M_⋆, which is plotted in Figure 7. We stress that the mean relationship flattens toward values Σ₀ ∼ 10⁹ M_⊙ kpc² for high stellar masses ${M}_{\star }\gtrsim$ a few × 10¹⁰ M_⊙. The physical interpretation is that to reach higher stellar surface densities, large masses must accumulate (rapidly) within quite small sizes, a condition regularly met in the progenitors of compact spheroids (see Lapi et al. 2011, 2017; Kriek et al. 2016; Glazebrook et al. 2017) but difficult to achieve in extended disks with quiet star formation histories.

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In Figure 8 we show the Tully–Fisher relationship between the stellar mass M_⋆ and the optical velocity V_opt. We plot the outcome from our analysis, both for the individual galaxies of the sample and for the mean within bins of given V_opt. We also show a polynomial fit to our mean results for binned data, whose parameters are reported in Table 1. This is in good agreement, especially for ${V}_{\mathrm{opt}}\gtrsim 100$ km s⁻¹, with the classic power-law determinations ${M}_{\star }\propto {V}_{\mathrm{opt}}^{3.7}$ by Dutton et al. (2010) and Reyes et al. (2011). The 1σ dispersion around the mean relation from our analysis amounts to about 0.08 dex, mainly contributed by the uncertainty in template reconstruction. We also confront the outcome of our analysis with independent results from individual RC modeling by Romanowsky & Fall (2012) and Lelli et al. (2016), finding a pleasing consistency.

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We stress that the latter authors adopt a definite value of M_⋆/L in their analysis; specifically, Romanowsky & Fall (2012) assume ${M}_{\star }/{L}_{K}\approx 1$, while Lelli et al. (2016) assume ${M}_{\star }/{L}_{3.6\mu {\rm{m}}}\approx 0.5$. On the other hand, the ratio M_⋆/L_I constitutes an outcome of our approach; the resulting values are plotted in the inset of Figure 8 as a function of V_opt and range from ${M}_{\star }/{L}_{I}\approx 1$ for ${V}_{\mathrm{opt}}\sim 100$ km s⁻¹ to ${M}_{\star }/{L}_{I}\approx 3$ for ${V}_{\mathrm{opt}}\gtrsim 200$ km s⁻¹ to ${M}_{\star }/{L}_{I}\lesssim 0.5$ for ${V}_{\mathrm{opt}}\lesssim 70$ km s⁻¹. Note that the 1σ dispersion of the M_⋆/L_I values for individual galaxies and the dispersion around the mean are consistent with those of the ${M}_{\star }\mbox{--}{V}_{\mathrm{opt}}$ relationship. Despite the difficulty in soundly comparing the M_⋆/L values computed by us to those adopted by Romanowsky & Fall (2012) and Lelli et al. (2016), it is worth stressing that the corresponding stellar masses are consistent. However, the assumption of an M_⋆/L ratio constant with V_opt is likely at the origin of the slightly larger scatter in their stellar masses (mirrored in all relationships involving M_⋆) with respect to those resulting from our analysis. In addition, we notice that our dynamical determinations of M/L_I ratios are quite close to those adopted by Bell et al. (2003) and Reyes et al. (2011) based on (g–r) colors for disk galaxies, converted to a Chabrier initial mass function.

We also check the dependence of the derived Tully–Fisher relationship on the disk scale length (or equivalently on the effective radius). To this purpose, in Figure 8 we illustrate the average relationship when grouping the data in bins of $\mathrm{log}{R}_{d};$ the outcome is, within uncertainties, indistinguishable from the overall M_⋆ versus V_opt relation. Such an independence of the size (and consequently of the disk surface brightness) has been originally pointed out by Courteau et al. (2007).

In Figure 9 we present the fundamental space of spiral galaxies, involving three observable quantities: the effective radius R_e, the central stellar surface density ${{\rm{\Sigma }}}_{0}$, and the optical velocity V_opt, as derived from our analysis; we also show the projected relationships in the ${R}_{e}\,\mbox{--}\,{{\rm{\Sigma }}}_{0}$, ${R}_{e}\,\mbox{--}\,{V}_{\mathrm{opt}}$, and ${{\rm{\Sigma }}}_{0}-{V}_{\mathrm{opt}}$ planes. The shaded surface illustrates the best-fit plane to the data points for individual galaxies from a principal component analysis, which is given by the expression

$$\begin{eqnarray}\mathrm{log}{R}_{e}[\mathrm{kpc}] & \approx & 1.31-0.57\,\mathrm{log}{{\rm{\Sigma }}}_{0}[{M}_{\odot }\,{\mathrm{kpc}}^{-2}]\\ & & +1.99\,\mathrm{log}{V}_{\mathrm{opt}}[\mathrm{km}\,{{\rm{s}}}^{-1}];\end{eqnarray} \tag{ 7 }$$

this is analogous to the fundamental plane of ellipticals (see Djorgovski & Davis 1987; Dressler et al. 1987). At variance with the latter, the plane of spirals is substantially tilted with respect to all the axes, and so its projections are characterized by an appreciable dispersion; moreover, it does not provide a comparably good rendition of the data distribution in the full three-dimensional space.

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In Figure 10 we show the relationship between the stellar mass M_⋆ and the ratio ${V}_{\mathrm{opt}}/{V}_{200}$ of the optical velocity to the halo circular velocity at the radius where the overdensity is 200 times the critical one. We plot the outcome from our analysis, both for the individual galaxies of the sample and for the mean within bins of given M_⋆. We also show a polynomial fit to our mean results for binned data, whose parameters are reported in Table 1; this is consistent with the relationship by Dutton et al. (2010) for late-type galaxies. We also compare our relationship to the independent determinations via weak galaxy–galaxy lensing by Reyes et al. (2012), finding a remarkable agreement. The 1σ dispersion around the mean relation from our analysis amounts to about 0.06 dex, making our determination one of the most precise to date.

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The weak dependence of the ratio ${V}_{\mathrm{opt}}/{V}_{200}$ on M_⋆ implies that the optical velocity V_opt is a good proxy of the halo circular velocity V₂₀₀ in the outskirts. The inset of Figure 10 shows the same velocity ratio ${V}_{\mathrm{opt}}/{V}_{200}$ as a function of the optical velocity ${V}_{\mathrm{opt}};$ such a relation could be used to infer halo masses with high accuracy. In Section 3.7 we will exploit these results to theoretically interpret the ${R}_{d}\mbox{--}{V}_{\mathrm{opt}}$ relationship.

In Figure 11 we illustrate the relationship between the specific angular momentum j_⋆ of the stellar component (see Equation (5)) and the stellar mass M_⋆. We plot the outcome from our analysis, both for the individual galaxies of the sample and for the mean within bins of given M_⋆. We also show a polynomial fit to our mean result for binned data, whose parameters are reported in Table 1. The dashed line is the relation with fixed slope ${j}_{\star }\propto {M}_{\star }^{2/3}$ for pure disks by Romanowsky & Fall (2012). It is seen that our relation features such a 2/3 slope at high masses but then flattens out for ${M}_{{\rm{H}}}\lesssim$ a few × ${10}^{10}\,{M}_{\odot };$ this is a consequence of the trivial scaling ${j}_{\star }\propto {j}_{{\rm{H}}}\propto {M}_{{\rm{H}}}^{2/3}$ from Equation (6) and angular momentum conservation, coupled with the behavior of the ${M}_{{\rm{H}}}\,\mbox{--}\,{M}_{\star }$ relation (see Figure 5), which is log-linear at high stellar masses and flattens appreciably toward smaller ones (see also Posti et al. 2018a).

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We confront our result with the independent measurements from individual RC modeling of disk-dominated galaxies by Romanowsky & Fall (2012) and Lelli et al. (2016), finding good agreement, though the scatter of their data points is appreciably larger, especially for massive spirals (likely due to their assumption of an M_⋆/L ratio constant with luminosity). In fact, the 1σ dispersion around the mean j_⋆ – M_⋆ relation from our analysis amounts to about 0.05 dex. The inset of Figure 11 shows the corresponding relation between j_⋆ and the optical velocity V_opt, which will be used in Section 3.7 to theoretically interpret the ${R}_{d}\,\mbox{--}\,{V}_{\mathrm{opt}}$ relationship.

Figure 12 presents j_⋆ as a function of the halo mass M_H. The 1σ dispersion around the mean relation from our analysis amounts to about 0.05 dex. The tightness of the ${j}_{\star }\mbox{--}{M}_{{\rm{H}}}$ relation suggests the possibility of inferring halo masses via the specific angular momentum ${j}_{\star }\sim 2\,{V}_{\mathrm{opt}}\,{R}_{d}$, which in turn can be accurately estimated by measurements of the optical velocity and of the disk scale length (see also Romanowsky & Fall 2012). The inset shows the fraction ${f}_{j}\equiv {j}_{\star }/{j}_{{\rm{H}}}$ representing the amount of the halo angular momentum (see Equation (6)) still retained or sampled by the stellar component (see Shi et al. 2017; Posti et al. 2018b). The standard and simplest theory of disk formation envisages sharing and conservation of the specific angular momentum between baryons (in particular, stars) and DM, to imply ${f}_{j}\approx 1$. Remarkably, our analysis shows that such a picture is closely supported by the data, since the values of f_j range from 0.6 to 1, with a weak dependence on M_H (see also Romanowsky & Fall 2012). Note that, from a theoretical perspective, a value f_j below 1 can be ascribed to inhibited collapse of the high angular momentum gas located in the outermost regions by stellar feedback processes (e.g., Fall 1983; Shi et al. 2017).

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Figure 13 shows the relationship between the stellar effective radius R_e and the halo radius R₂₀₀ where the overdensity is 200 times the critical value (see Section 2.2). We illustrate the outcome from our analysis, both for the individual galaxies of the sample and for the mean within bins of given R₂₀₀. The polynomial fit to our mean results for binned data is also reported (see Table 1). Our mean relation is in remarkable agreement with the determination via abundance matching by Huang et al. (2017); on the other hand, it is substantially higher than the relation proposed by Kravtsov (2013), whose sample is actually dominated by early-type galaxies.

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It is also interesting to compare our relation to that expected from the classic theory of disk formation, envisaging sharing and conservation of specific angular momentum between DM and stars (see Mo et al. 1998, 2010). To this purpose, we equalize ${j}_{\star }={f}_{j}\,{j}_{{\rm{H}}}$ with ${j}_{{\rm{H}}}\equiv \sqrt{2}\,\lambda \,{R}_{{\rm{H}}}\,{V}_{{\rm{H}}}$ from Equation (6) to ${j}_{\star }\,=2\,{f}_{R}\,{R}_{d}\,{V}_{\mathrm{opt}}$ from Equation (5). Introducing ${f}_{V}\equiv {V}_{\mathrm{opt}}/{V}_{{\rm{H}}}$, we obtain the relation

$$\begin{eqnarray}&&{R}_{d}\simeq \displaystyle \frac{\lambda }{\sqrt{2}}\,\displaystyle \frac{{f}_{j}}{{f}_{R}\,{f}_{V}}\,{R}_{{\rm{H}}}\,,\end{eqnarray} \tag{ 8 }$$

where fits to the dependencies of f_j and ${f}_{V}\approx 1.07\,{V}_{\mathrm{opt}}/{V}_{200}$ on ${R}_{200}\approx 0.77\,{R}_{{\rm{H}}}$ are given in Table 1; note that the factor ${f}_{j}/{f}_{R}\,{f}_{V}$, often neglected in the literature, bends downward the relation for ${R}_{200}\gtrsim 150\,\mathrm{kpc}$. For an average value of the spin parameter $\langle \lambda \rangle \approx 0.035$ as indicated by numerical simulations, the expectation of Equation (8) is in excellent agreement with our mean determination from data. We stress that the 1σ dispersion around the mean relation from our analysis amounts to about 0.05 dex.

In Figure 14 we illustrate the relationship between two directly observable quantities, namely, the disk scale length R_d and the optical velocity V_opt. We plot the data points for the individual galaxies of our sample and for the mean within bins of given R_d or of given V_opt. We also show the linear fits to the mean relationships ${R}_{d}\,\mbox{--}\,{V}_{\mathrm{opt}}$ and ${V}_{\mathrm{opt}}\,\mbox{--}\,{R}_{d}$ for binned data and the associated bisector fit (see Table 1). We confront the outcome with the independent measurements by Romanowsky & Fall (2012) and Lelli et al. (2016), finding a very good agreement. Remarkably, the 1σ dispersion around the mean relation amounts to about 0.04 dex.

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We now turn to physically interpreting this fundamental relation within the classic picture of disk formation, which envisages sharing of specific angular momentum between baryons and DM at halo formation, and then a retention/sampling of almost all momentum into the stellar component. We again equalize ${j}_{\star }={f}_{j}\,{j}_{{\rm{H}}}$ with j_H expressed as a function of V_H from (Equation (6), last term) to ${j}_{\star }=2\,{f}_{R}\,{R}_{d}\,{V}_{\mathrm{opt}}$ from Equation (5). Introducing ${f}_{V}\equiv {V}_{\mathrm{opt}}/{V}_{{\rm{H}}}$, we get the straight relation

$$\begin{eqnarray}&&{R}_{d}\,[\mathrm{kpc}]\simeq 1.5\,\displaystyle \frac{\lambda \,{f}_{j}}{{f}_{R}\,{f}_{V}^{2}}\,E{(z)}^{-1/2}\,{V}_{\mathrm{opt}}\,[\mathrm{km}\,{{\rm{s}}}^{-1}],\end{eqnarray} \tag{ 9 }$$

where fits to the dependencies of f_j and ${f}_{V}\approx 1.07\,{V}_{\mathrm{opt}}/{V}_{200}$ on V_opt are given in Table 1. When adopting the average value of the spin parameter $\langle \lambda \rangle \approx 0.035$ indicated by numerical simulations, the relation ${R}_{d}\mbox{--}{V}_{\mathrm{opt}}$ after Equation (9) is plotted in Figure 15; it remarkably agrees in normalization and shape with the observed one. We note that using the simplified expression ${R}_{d}\,[\mathrm{kpc}]\simeq 0.64\,\lambda \,{V}_{\mathrm{opt}}\,[\mathrm{km}\,{{\rm{s}}}^{-1}]$, which involves the mean value $\langle {f}_{j}/{f}_{R}\,{f}_{V}^{2}\rangle \approx 0.43$ over the different V_opt bins, yields in turn a very good representation of the bisector fit relation presented in Figure 14.

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There is an issue, however, that needs to be discussed with some care. The 1σ variance expected from the theoretical relation of Equation (9), which is mainly determined by that in the halo spin parameter λ, amounts to about 0.25 dex; this 1σ dispersion is seen to embrace almost all the data, and indeed it is appreciably larger (by a factor 2) than the observed 1σ scatter (cf. Equation (8) and Figure 12; see also Saintonge et al. 2008; Saintonge & Spekkens 2011; Hall et al. 2012). This may be partly due to the fact that stable disks must be hosted by halos with quiet recent merging histories (see Wechsler et al. 2002; Dutton et al. 2007). In addition, specific regions (highlighted by hatched areas) of the ${R}_{d}\mbox{--}{V}_{\mathrm{opt}}$ diagram that correspond to very high or very low values of λ are actually prohibited by simple physical arguments. Specifically, too high values of λ, which would imply large values of R_d at given V_opt, are not permitted because otherwise the gravitational support would not be sufficient to sustain the rotational motions of the stars; this condition can be naively expressed as $G\,{M}_{\mathrm{tot}}/{R}_{d}^{2}\sim {j}_{\star }^{2}/{R}_{d}^{3}$, implying $\lambda \lesssim {f}_{b}/2\sqrt{2}\,{f}_{V}\,{f}_{R}$. Contrariwise, too low values of λ, which would imply small values of the disk scale length R_d at given V_opt, are not permitted since otherwise the specific angular momentum ${j}_{{\rm{H}}}\propto \lambda \,{V}_{{\rm{H}}}^{2}\propto \lambda \,{V}_{\mathrm{opt}}^{2}$ of the halo (cf. Equation (6) and Figure 10) would be smaller than the measured stellar one ${j}_{\star }\propto {f}_{j}\,\langle \lambda \rangle ;$ this implies $\lambda \gtrsim {f}_{j}\,\langle \lambda \rangle$ with $\langle \lambda \rangle \approx 0.035$. A similar criterion is obtained when basing on theoretical disk instability arguments (see Mo et al. 1998).

The tightness and the theoretical understanding of the mean ${R}_{d}\,\mbox{--}\,{V}_{\mathrm{opt}}$ relation opens up the possibility of exploiting it in the next future for cosmological studies. First of all, such a relation involves two directly observable quantities, of which R_d is determined from photometry and as such depends on the angular diameter distance, while V_opt can be estimated from kinematic measurements and is independent of it (see also Saintonge et al. 2008; Saintonge & Spekkens 2011). Second, the redshift evolution of the relationship after Equation (9) is expected to be mild and controlled, since λ is known from simulations to be almost independent of mass and redshift; the other shape factors are weakly varying functions of V_opt and should be related only to the internal structure of pure disks, which is known to be weakly dependent on redshift at least out to z ≲ 0.7 (see Hudson et al. 2015; Huang et al. 2017). Therefore, observations at increasing redshift can in principle be exploited to constrain the Hubble scale and the cosmological parameters, with particular focus on the dark energy properties.