THE MOSDEF SURVEY: DYNAMICAL AND BARYONIC MASSES AND KINEMATIC STRUCTURES OF STAR-FORMING GALAXIES AT 1.4 ≤ z ≤ 2.6

We make use of data from the MOSDEF survey (Kriek et al. 2015), conducted using the MOSFIRE spectrograph (McLean et al. 2012) on the 10 m Keck I telescope. In this work, we use the spectra obtained during semesters 2012B, 2013A, and 2014A. When complete, the MOSDEF survey will contain moderate resolution (R = 3000–3650) rest-frame optical spectra for ∼1500 H-band selected galaxies at $1.4\leqslant z\leqslant 3.8$ in several CANDELS (Grogin et al. 2011; Koekemoer et al. 2011) fields. A detailed overview of the survey, observations, data reduction, line measurements and sensitivities, success rate, redshift measurements, stellar population properties, and sample characteristics are given in Kriek et al. (2015).

For all galaxies observed with MOSFIRE, we measure structural parameters, including the Sérsic index, n (Sérsic 1968), the effective radius, ${R}_{E}$ (assumed to be the semimajor axis, unless explicitly stated otherwise), the axis ratio, b/a, and the major axis PA from the HST/F160W images (released by the CANDELS team) using Galfit (Peng et al. 2010). See L. de Groot et al. (2016, in preparation) for more details on the structural parameter measurements.

Stellar masses for all MOSDEF galaxies are derived by fitting the 0.3–8.0 μm photometry from the 3D-HST survey (Brammer et al. 2012; Skelton et al. 2014; Momcheva et al. 2015) with the flexible stellar population models (Conroy et al. 2009; Conroy & Gunn 2010) using FAST (Kriek et al. 2009), while adopting the MOSFIRE redshifts (${z}_{\mathrm{MOS}}$). We assume a Chabrier (2003) stellar IMF, along with a Calzetti et al. (2000) dust attenuation curve, a delayed exponentially declining star formation history, and solar metallicity. To account for template mismatch, we assume the default FAST template error function. Confidence intervals are calibrated using 500 Monte Carlo simulations. Hence, the stellar mass uncertainties do not include systematic uncertainties due to the choice of IMF, dust attenuation curve, or other assumptions. See Kriek et al. (2015) for more details on the stellar population modeling.

Following Taylor et al. (2010), we correct the stellar masses by the difference between the Galfit (${m}_{{\rm{G}}{\rm{ALFIT}}}$) and total photometric F160W magnitudes (${m}_{\mathrm{phot}}$), using

$$\begin{eqnarray}&&{\mathrm{log}}_{10}{M}_{*}\;=\;{\mathrm{log}}_{10}{M}_{*,\mathrm{FAST}}+0.4({m}_{\mathrm{phot}}-{m}_{{\rm{G}}{\rm{ALFIT}}}).\end{eqnarray} \tag{ 1 }$$

This correction makes the size and stellar mass measurements self-consistent.

Emission line fluxes are measured from the optimally extracted MOSFIRE 1D spectra by fitting adjacent lines simultaneously with Gaussians plus a linear fit to account for the underlying continuum. The ${\rm{H}}\alpha$ and ${\rm{H}}\beta$ lines are corrected for the underlying Balmer absorption, as estimated from the best-fit stellar population models. See Kriek et al. (2015) and Reddy et al. (2015) for more details on the emission line measurements.

We use the ${\rm{H}}\alpha$ emission lines to estimate SFRs and gas masses (${M}_{\mathrm{gas}}$) using the following method. For galaxies with detected ${\rm{H}}\beta$, the Balmer absorption-corrected ${\rm{H}}\alpha$ fluxes are corrected for dust using the Balmer decrement, assuming a Cardelli et al. (1989) extinction curve (Reddy et al. 2015). When ${\rm{H}}\beta$ is undetected, we assume the reddening of the nebular regions is related to that of the continuum by ${A}_{V,\mathrm{neb},\mathrm{Calzetti}}\;=\;1.86\;{A}_{V,\mathrm{cont},\mathrm{Calzetti}}$ (Price et al. 2014). Because this relation was derived by assuming the Calzetti et al. (2000) attenuation curve for both the continuum and line emission, we convert the inferred nebular attenuation to the Cardelli et al. (1989) curve, and correct the ${\rm{H}}\alpha$ fluxes accordingly. In order for ${\rm{H}}\beta$ to be used in the dust correction, it must be detected with a signal-to-noise ratio (S/N) ≥ 3, and the spectrum transmission at ${\rm{H}}\beta$ must be at least 50% of the maximum transmission.

The dust-corrected ${\rm{H}}\alpha$ fluxes are converted into ${\rm{H}}\alpha$ luminosities, that are then used to calculate the ${\rm{H}}\alpha$ SFRs following the relation of Kennicutt (1998) for a Chabrier (2003) IMF (Shivaei et al. 2015). Finally, the relation between ${{\rm{\Sigma }}}_{\mathrm{gas}}$ and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ by Kennicutt (1998) is used to estimate the gas masses, using ${{\rm{\Sigma }}}_{\mathrm{gas}}={M}_{\mathrm{gas}}/(\pi {R}_{E}^{2})$ and ${{\rm{\Sigma }}}_{\mathrm{SFR}}=\mathrm{SFR}/(\pi {R}_{E}^{2})$, where R_E is the best-fit Galfit major axis. In Section 5.7, we discuss the validity of this relation at high redshift. The gas mass uncertainties include uncertainties on the ${\rm{H}}\alpha$ and ${\rm{H}}\beta$ fluxes and on the slit-loss corrections. An uncertainty of $0.2\;\mathrm{dex}$ on ${A}_{{\rm{V}},\mathrm{cont},\mathrm{Calzetti}}$ is assumed when ${\rm{H}}\beta$ is undetected.

For this work, we select objects in the redshift ranges $1.34\leqslant z\leqslant 1.75$ and $2.075\leqslant z\leqslant 2.6$, to ensure coverage of the ${\rm{H}}\alpha$ emission line. We also require that ${\rm{H}}\alpha$ is detected (i.e., S/N $\geqslant \quad 3$) and that there is HST/F160W coverage, to make use of the Galfit structural parameter measurements.

We use additional selection criteria to ensure that we include only high-quality spectra. First, we consider only primary MOSDEF targets, excluding any serendipitously detected galaxies that happened to fall within the slit. Second, we exclude objects with non-negligible contamination to the ${\rm{H}}\alpha$ flux from neighboring skylines, to provide clean kinematic measurements. Third, we impose quality cuts for both the stellar population and structural parameters to ensure that the best fits adequately model the data. For the stellar population fits, we exclude objects for which the best-fit reduced chi-square ${\chi }_{\mathrm{red}}^{2}\gt 10.$ For the structural parameters, we flag and exclude objects for which (a) the Galfit runs did not converge or (b) the Galfit and HST/F160W total magnitudes differ by more than 0.5 mag. Fourth, we exclude any objects that fall within the quiescent region in the UVJ diagram (Wuyts et al. 2007; Williams et al. 2009). Fifth, we exclude AGNs with outflow signatures or with very broad emission lines (W. R. Freeman et al. 2016, in preparation). Sixth, we exclude objects that appear to have an interacting counterpart at a similar redshift because the velocity signatures of these systems may not reflect the internal kinematics. We consider AGNs, as identified by their X-ray luminosity, IRAC color, or rest-frame optical emission lines ratios (Coil et al. 2015; M. Azadi et al. 2016, in preparation) separately from our sample of star-forming galaxies.

Our final sample includes 178 unique galaxies, with ${\rm{H}}\beta$ detected in 138. One object has been observed twice. We also consider 21 unique AGNs (14 with ${\rm{H}}\beta$ detected) that meet all selection criteria, with 2 AGNs having been observed twice.

We show the effective radii versus stellar masses for the galaxies and AGNs in our sample in Figure 1. For comparison, we also show the best-fit size-stellar mass relations found by van der Wel et al. (2014a) for a complete sample of star-forming (late-type) galaxies at $z\;=\;2.25$ and $1.75$. The samples are complete down to ${M}_{*}\sim {10}^{9.5}\;{M}_{\odot }$ at z = 2.25 and down to ${M}_{*}\sim {10}^{9.1}\;{M}_{\odot }$ at z = 1.75, and are, therefore, a good representation of the star-forming galaxies at these redshifts. Our galaxies generally follow these best-fit size-stellar mass relations in both redshift ranges, though our galaxies at $z\sim 1.5$ may be somewhat smaller in size than the average as determined by van der Wel et al. (2014a).

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In this section, we constrain the rotational velocity and velocity dispersion simultaneously by modeling the 2D spectra in combination with the F160W structural parameters for each MOSDEF galaxy in our sample. Previous studies have presented methods for fitting 2D spectra, including Vogt et al. (1996), Simard & Pritchet (1999), and Weiner et al. (2006). However, the models of Simard & Pritchet (1999) and Weiner et al. (2006) do not account for misalignment between the slit and major axis, while Vogt et al. (1996) and Simard & Pritchet (1999) exclude velocity dispersion.

Instead, we define kinematic models that explicitly include the PA misalignment and inclination, and simultaneously fit the rotation velocity and velocity dispersion. The kinematic models are discussed in detail in Appendix A.1. In summary, the kinematic models include both rotation and a constant velocity dispersion over the galaxy, and have a total of three free parameters: the asymptotic velocity (V_a) and turnover radius (r_t) of the arctan rotation curve model, and the constant intrinsic velocity dispersion (${\sigma }_{V,0}$). We assume the best-fit Galfit parameters and the PAs from the F160W observations. The model is collapsed along the line of sight, and convolved to match the seeing conditions of each spectrum. Using the PA, inclination, brightness profile, and seeing information, we determine what portions of the model fall within the slit for each object. Finally, the model is collapsed in the spatial direction perpendicular to the slit and convolved by the instrumental resolution.

To fit the emission lines, we start by subtracting the continuum from each ${\rm{H}}\alpha$ 2D spectrum. We also trim the spectrum to exclude the $[{\rm{N}}\;{\rm{III}}]$ emission lines and to include only the positive emission line image. We then construct a mask for the emission line spectrum to exclude missing data and low signal-to-noise rows from our fitting procedure. A detailed description of this procedure can be found in Appendix A.2.

We then find the best-fit models to the trimmed 2D ${\rm{H}}\alpha$ spectra and the corresponding confidence intervals by performing parameter space exploration using the python Markov-Chain Monte Carlo (MCMC) package emcee (Foreman-Mackey et al. 2013), following the method detailed in Appendix A.2. As the rotation curve turnover is not well constrained in our data, there is a degeneracy in the values of V_a and r_t. Nonetheless, the values of $V({R}_{E})$ and ${V}_{2.2}\;=\;V(2.2{r}_{s})$ (assuming the arctan rotation curve model, see Equation (11)) are well constrained. We note that we explicitly include the structural parameters and projection effects in our model, so we directly constrain the intrinsic galaxy parameters, without projection or blending effects. Examples of the 2D ${\rm{H}}\alpha$ emission line fits are shown in Figure 2.

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We use the values of $V({R}_{E})$ to determine which objects have spatially resolved rotation. We take objects with $V({R}_{E})\ne 0$ within the 95% one-sided distribution to be our sample with resolved rotation, and the objects that fail this cut to be our dispersion-only sample. The position-velocity diagrams of the 35 galaxies with detected rotation are shown in Appendix A.3.

For all objects without resolved rotation (see Section 3.1), we measure the kinematics from the 1D spectra. Because our sample consists of star-forming galaxies, we may expect that their intrinsic velocity support is at least partially rotational. This assumption is reinforced by the work of Newman et al. (2013), who find that galaxies that were initially classified as dispersion-dominated in fact do show evidence for rotation in observations with higher spatial resolution. Thus, we model the composite unresolved kinematics by assuming a value for ${(V/{\sigma }_{V,0})}_{{R}_{E}}=V({R}_{E})/{\sigma }_{V,0}$, while taking into account the Galfit parameters and seeing conditions. We then use this model to convert the measured velocity dispersion to intrinsic quantities.

We measure the velocity dispersion from the optimally extracted 1D spectra by fitting ${\rm{H}}\alpha$, the $[{\rm{N}}\;{\rm{III}}]$ doublet, and the continuum simultaneously with a triple Gaussian and a linear component. We fit the spectrum between $6480\;\mathring{\rm A} \ \leqslant \lambda /(1+{z}_{\mathrm{MOS}})\leqslant 6650\;\mathring{\rm A}$, and mask pixels with no coverage. We vary the coupled line centers, while constraining ${\lambda }_{{\rm{H}}\alpha ,\mathrm{obs}}$ to within $\pm 20\;\mathring{\rm A}$ of ${\lambda }_{{\rm{H}}\alpha }(1+{z}_{\mathrm{MOS}})$. The widths of the emission lines (in Å) are coupled in velocity space, with ${\sigma }_{\lambda ,N{\rm{II}}\lambda \lambda \mathrm{6584,48}}\ ={\sigma }_{\lambda ,{\rm{H}}\alpha }({\lambda }_{N{\rm{II}}\lambda \lambda \mathrm{6584,48}}/{\lambda }_{{\rm{H}}\alpha })$. We also assume $F([{\rm{N}}\;{\rm{III}}]\;\lambda 6548)\ =1/3\;F([{\rm{N}}\;{\rm{III}}]\;\lambda 6584)$ (Osterbrock & Ferland 2006). Finally, we enforce ${\sigma }_{\lambda ,{\rm{H}}\alpha }\geqslant {\sigma }_{\lambda ,\mathrm{sky}}$, with the instrumental resolution measured from the median skyline width.

The ${\rm{H}}\alpha$ line widths are corrected for the line broadening due to the instrumental resolution by subtracting ${\sigma }_{\lambda ,\mathrm{sky}}$ in quadrature from ${\sigma }_{\lambda ,{\rm{H}}\alpha }$. Each corrected ${\rm{H}}\alpha$ line width ${\sigma }_{\lambda ,{\rm{H}}\alpha ,\mathrm{corr}}$ is converted to an observed velocity dispersion, ${\sigma }_{V,\mathrm{obs}}$, using the best-fit redshift.

The errors on the observed velocity dispersions are estimated by creating 500 realizations where the spectra are perturbed according to the corresponding error spectra. We then perform the same fitting and correction procedure on the perturbed spectra, and convert the corrected line widths to velocity dispersions using the best-fit redshift of each realization.

For each object, we convert the observed velocity dispersion into an intrinsic root mean square (rms) velocity, ${V}_{\mathrm{rms}}\ =\sqrt{{V}^{2}+{\sigma }_{V}{}^{2}}$, which explicitly includes both intrinsic rotation and dispersion velocities. This method is discussed in detail in Appendix B. In summary, we model each object as an inclined disk (using the Galfit structural parameters ${R}_{E}$, n, b/a), with the major axis offset from the slit by ${\rm{\Delta }}\mathrm{PA}$. The rotation and velocity dispersion kinematics are included by assuming a fixed ratio of ${(V/{\sigma }_{V,0})}_{{R}_{E}}$, and then the model is convolved to match the MOSFIRE seeing resolution. We determine which portions of the model fall within the slit width and the extracted width in the spatial direction, and then apply the optimal-extraction weighting. For this model, we calculate the ratio of the luminosity-weighted second velocity moment (${\sigma }_{V,\mathrm{model}}$) to the rms velocity at ${R}_{E}$ (${V}_{\mathrm{rms}}{({R}_{E})}_{\mathrm{model}}$), and use this ratio to convert the observed, integrated velocity dispersion to the composite rms velocity at ${R}_{E}$ following

$$\begin{eqnarray}&&{V}_{\mathrm{rms}}\;{({R}_{E})}_{1{\rm{D}},\mathrm{corr}}\;=\;{\sigma }_{V,\mathrm{obs}}{\left(\displaystyle \frac{{\sigma }_{V,\mathrm{model}}}{{V}_{\mathrm{rms}}{({R}_{E})}_{\mathrm{model}}}\right)}^{-1}.\end{eqnarray} \tag{ 2 }$$

van Dokkum et al. (2015) use an α parameterization to infer a rotational velocity from an observed velocity dispersion. This α value is empirically derived and combined with an inclination correction, with $\alpha \;=\;{\sigma }_{V,\mathrm{obs}}/(V\mathrm{sin}i)$. Hence, this correction does not take into account the exact portion of the galaxy observed through the slit or partial support from random motions. However, the galaxies by van Dokkum et al., in general, are smaller than those in our sample and will suffer less from slit losses.

In Figure 3, we show ${\sigma }_{V,\mathrm{obs}}$ and ${V}_{\mathrm{rms}}\;{({R}_{E})}_{\mathrm{corr}}$ versus ${R}_{E}$ for galaxies and AGNs without detected rotation. For now, we assume ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;2.1$, the median of the values measured for galaxies with detected rotation. For reference, we also show the velocity dispersions measured from the integrated 1D spectra of the galaxies with detected rotation, with the corrections for these objects calculated using the exact ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ measured for each object. The median observed 1D velocity dispersion for our sample with $M\geqslant {10}^{9.5}\;{M}_{\odot }$ (the approximate completeness limit for star-forming galaxies in the MOSDEF survey, see Shivaei et al. 2015) is ${({\sigma }_{V,\mathrm{obs}})}_{\mathrm{median}}=78\;\mathrm{km}\;{{\rm{s}}}^{-1}$. The median observed velocity dispersion ${({\sigma }_{V,\mathrm{obs}})}_{\mathrm{median}}=70\;\mathrm{km}\;{{\rm{s}}}^{-1}$ of the galaxies at $z\sim 1.5$ is slightly lower than the value of ${({\sigma }_{V,\mathrm{obs}})}_{\mathrm{median}}\;=\;80\;\mathrm{km}\;{{\rm{s}}}^{-1}$ for the galaxies at $z\sim 2.3$.

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Because our primary sample consists of star-forming galaxies, in Section 3.2, we have treated the objects for which we only observe velocity dispersion as being intrinsically disks, with some amount of thickening. Here we consider whether this is a reasonable assumption by considering the necessary conditions to observe rotation in a galaxy.

One reason why intrinsic rotation may not be observable is that the galaxy is small with respect to the seeing size. If there is only one resolution element across the galaxy disk, then any rotation signature will be washed out and we would only observe velocity dispersion. Additionally, the galaxy major axes may be misaligned with the slit axis. A PA offset reduces the ability to detect rotation, as for more misaligned objects, the rotational information is projected into fewer resolution elements along the slit. At the most extreme, if a galaxy is completely misaligned with the slit (i.e., ${\rm{\Delta }}\mathrm{PA}\;\;=\;90^\circ$), the rotational signature is collapsed into the same resolution element, and again we would only observe a velocity dispersion.

We consider the dual effects of object size and ${\rm{\Delta }}\mathrm{PA}$ , by calculating how much of the stellar light major axis falls within the slit, projected along the slit direction. The projected size of the object falling within the slit ($2\;{R}_{E,\mathrm{proj}}$) should be larger than the seeing for the object to be spatially resolved, or ${R}_{E,\mathrm{proj}}\geqslant {\mathrm{FWHM}}_{\mathrm{seeing}}/2$.

We divide our sample into four categories, based on the combination of the projected spatial resolution criterion given above (resolved versus unresolved) and whether we detect rotation or not (rotation versus dispersion, see Section 3.1). This classification scheme gives 15 spatially resolved galaxies with observed rotation, 20 spatially unresolved galaxies with observed rotation, 20 spatially resolved galaxies with only dispersion observed, and 123 spatially unresolved galaxies with only dispersion observed. In Figures 4 and 5, we show example HST images and spectra, respectively, for objects in each category.

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We show PA offset versus object size (${R}_{E}$) as a function of seeing in Figure 6. At fixed ${R}_{E}$, we tend to have more "dispersion-only" objects and fewer objects with detected rotation as the PA offset increases from aligned (0°) to completely misaligned (90°). We also note that, on average, galaxies for which we observe rotation tend to be larger than those without observed rotation. This finding supports the possibility that we may not observe rotation for some objects simply because they are physically too small to resolve, even if there is no PA offset between the semimajor axis and the slit. We can also see these trends in Figure 4 because the objects with observed rotation (left panels) tend to have better alignment between their major axes and the slit, and tend to have larger angular sizes relative to the objects for which we only observe dispersion (right panels).

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However, not all objects follow the expected classifications. For instance, we observe rotation for some spatially unresolved objects. These galaxies are shown as yellow diamonds in Figure 6, with example images and spectra shown in the lower left panels of Figures 4 and 5. Also, if all galaxies in our sample are intrinsically disk galaxies and have at least partial rotational support, we would expect to see rotation in all objects that are spatially resolved. However, we do not observe rotation in some of the galaxies that meet the projected spatial resolution criterion, which are shown as green triangles in Figure 6, with examples in the upper-right panels of Figures 4 and 5.

Other effects may influence the classification of our sample into these four sub-samples, which could explain why we see objects in the unexpected classification categories. First, the PA offset between the kinematic major axis and the slit could be incorrect. This error may be due to uncertainties in the photometric major axis PA estimation or to a misalignment between the kinematic and photometric major axes. The latter effect has been observed by Wisnioski et al. (2015) in galaxies at $z\sim 2$, and possibly indicates disturbed kinematics due to mergers (Epinat et al. 2012). If the PA offset is incorrect, our projected size along the major axis may not match the true intrinsic projected size of the region that we probe with the kinematics. Thus objects may scatter from the "spatially resolved" category into the "spatially unresolved" category or vice versa.

Second, we use the rest-frame optical ${R}_{E}$ in this classification, but we measure the kinematics from ${\rm{H}}\alpha$ emission. We show in Section 5.6 that the rest-frame optical and ${\rm{H}}\alpha$ sizes of the galaxies with detected rotation are very similar, so using ${R}_{E}$ to determine spatial resolution is a reasonable approximation. Still, if an object is close to the detection limit, small differences between ${R}_{E}$ and ${\rm{H}}\alpha$ size could change the spatial resolution classification.

Third, we do not incorporate the inclination angle in our classification procedure. For face-on galaxies, we do not expect to detect rotation. If we consider galaxies with $b/a\geqslant 0.9$ (i.e., $i\lesssim 26^\circ$, assuming ${(b/a)}_{0}\;=\;0.19$) to be face-on, only five of our galaxies satisfy this criterion, three of which have detected rotation, and two of which had been classified as "spatially resolved."

Fourth, we rely on single-component Galfit fits to determine the photometric PA, axis ratio b/a, and stellar effective radius ${R}_{E}$. Our galaxies often exhibit clumps, so they are not perfectly fit by a smooth Sérsic profile. Furthermore, Galfit is unable to recover extreme inclination angles (i.e., close to edge-on or face-on; Epinat et al. 2012). These limitationscould further influence the accuracy of the PA, effective radius, and axis ratio, and could influence whether an individual galaxy is categorized as "spatially resolved" or not.

Fifth, the S/N of the observed spectra will influence the object kinematic categorization. Our 95% one-sided $V({R}_{E})$ detection requirement may result in classifying objects with intrinsic rotation, but low S/N spectra, as "dispersion-only." The example spectra in Figure 5 in the spatially resolved, dispersion-only quadrant (green upper-right panel) do appear to have either similar or lower S/Ns relative to the spectra of the objects with detected rotation (shown in the left panels).

Sixth, neighboring skylines may overlap portions of a rotation curve, which may also cause an object to fail the $V({R}_{E})$ detection criterion. In Figure 5, we see some objects in the green quadrant with signifiant overlap with skyline contaminated columns (i.e., the fourth object, top row, and second object, third row, of the green, upper-right quadrant of Figure 5).

Between the four categories, 89% of the objects are consistent with having rotation. This includes the 69% of the galaxies that are unresolved and have no detected rotation, for which the kinematic structures of the individual galaxies are unknown. In Section 4.2, we find that galaxies without detected rotation are consistent with having kinematic support from both rotation and random motions. The only galaxies inconsistent with the assumption of intrinsic rotational support are those that are spatially resolved without observed rotation (green objects, Figure 6). However, as mentioned previously, there are sources of uncertainty in our categorizations that may imply that these objects may still have intrinsic rotation.

We combine the kinematic and structural information to calculate the dynamical masses of our galaxies. The rotation and dispersion velocities are combined by taking the rms velocity, ${V}_{\mathrm{rms}}\;({R}_{E})=\sqrt{{\sigma }_{V,0}^{2}+V{({R}_{E})}^{2}}$, and we calculate the total dynamical mass as

$$\begin{eqnarray}&&{M}_{\mathrm{dyn}}\;=\;{k}_{\mathrm{eff}}\displaystyle \frac{{V}_{\mathrm{rms}}\;{({R}_{E})}^{2}{R}_{E}}{G},\end{eqnarray} \tag{ 3 }$$

where G is the gravitational constant. Here we define an "effective" virial coefficient to account for the relative contribution to the rms velocity from the rotation and dispersion velocities (i.e., including an "asymmetric drift" correction from the velocity dispersion, as in Meurer et al. 1996; Epinat et al. 2009; Daddi et al. 2010; Newman et al. 2013):

$$\begin{eqnarray}&&{k}_{\mathrm{eff}}\;=\;\displaystyle \frac{{k}_{\mathrm{disp}}+{k}_{\mathrm{rot}}{({(V/{\sigma }_{V,0})}_{{R}_{E}})}^{2}}{1+{({(V/{\sigma }_{V,0})}_{{R}_{E}})}^{2}},\end{eqnarray} \tag{ 4 }$$

where ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;V({R}_{E})/{\sigma }_{V,0}$.⁶ We assume ${k}_{\mathrm{disp}}\;=\;5$ as the virial coefficient corresponding to the dispersion kinematic component, from the simple case of a sphere of uniform density (Pettini et al. 2001). We estimate the virial coefficient for the rotational kinematic component following Miller et al. (2011), who find k = 1.33 for the dynamical mass within $r=2.2{r}_{s}=1.3{R}_{E}$. To convert to the total dynamical mass, we approximate ${k}_{\mathrm{rot}}\approx 2k=2.66$.

To calculate ${k}_{\mathrm{eff}}$ and the dynamical masses, we use the best-fit values of $V({R}_{E})$ and ${\sigma }_{V,0}$ measured in Section 3.1 for the galaxies with detected rotation. For the galaxies without observed rotation, we have to assume a value of ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ to calculate ${k}_{\mathrm{eff}}$, ${V}_{\mathrm{rms}}\;({R}_{E})\;=\;{V}_{\mathrm{rms}}\;{({R}_{E})}_{1{\rm{D}},\mathrm{corr}}$, and ${M}_{\mathrm{dyn}}$. We will discuss the assumption of ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ in Section 4.

We test the method for correcting the kinematics of disk galaxies without detected rotation using the spatially resolved rotation sample of galaxies, for which we have more detailed kinematic information.

First, we measure the dynamical masses ${M}_{\mathrm{dyn},2{\rm{D}}}$ using the exact values of $V({R}_{E})$ and ${\sigma }_{V,0}$ from the 2D spectral fitting method in Section 3.1. We then measure the velocity dispersions from the optimally extracted 1D spectra for the same sample of galaxies. We assume the exact ${(V/{\sigma }_{V,0})}_{{R}_{E}}\ =V({R}_{E})/{\sigma }_{V,0}$ measured from the 2D spectral fitting for each object to calculate the corrected 1D velocity dispersions (Equation (2)), ${k}_{\mathrm{eff}}$ (Equation (4)), and the resulting dynamical masses ${M}_{\mathrm{dyn},1{\rm{D}}}$.

We compare ${M}_{\mathrm{dyn},1{\rm{D}}}$ with ${M}_{\mathrm{dyn},2{\rm{D}}}$ in Figure 7. The corrected ${M}_{\mathrm{dyn},1{\rm{D}}}$ values are in excellent agreement with the ${M}_{\mathrm{dyn},2{\rm{D}}}$ values, with little scatter between them (${\sigma }_{\mathrm{rms}}\;=\;0.11\;\mathrm{dex}$). We find a median offset of ${\rm{\Delta }}{\mathrm{log}}_{10}{M}_{\mathrm{dyn},\mathrm{offset}}=0.03\;\mathrm{dex}$ between the two measurements, such that the 2D-kinematic derived values of ${M}_{\mathrm{dyn}}$ are slightly larger than the values derived from the aperture-corrected 1D spectra.

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However, if these galaxies had not been resolved, we would not have known their intrinsic ${(V/{\sigma }_{V,0})}_{{R}_{E}}$, which was used in the dynamical mass estimate. Thus, we also calculate ${M}_{\mathrm{dyn},1{\rm{D}}}$ using the median ${[{(V/{\sigma }_{V,0})}_{{R}_{E}}]}_{2{\rm{D}},\mathrm{median}}=2.1$ for each object (shown as the gray points in Figure 7). We find a slightly larger offset (${\rm{\Delta }}{\mathrm{log}}_{10}{M}_{\mathrm{dyn},\mathrm{offset}}=0.04\;\mathrm{dex}$) and scatter (${\sigma }_{\mathrm{rms}}=0.18\;\mathrm{dex}$). Hence, for the galaxies with detected rotation, assuming the average value of ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ for each object yields dynamical masses that are nearly as accurate as the dynamical masses derived from the rotation curves. Based on this test, we conclude that the 1D velocity dispersion correction method works well and should produce reasonable dynamical masses for the remainder of the galaxies without observed rotation if the average ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ is known.

In order to measure the dynamical masses for all galaxies in our sample, we need a ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ ratio for the galaxies without resolved kinematics. We assume that the kinematically resolved objects have a similar structure as the unresolved objects, and adopt the median ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;2.1$ as measured from the rotation objects (see Section 3.1 and Appendix A).

In Figure 8, we compare the dynamical masses to the baryonic masses, ${M}_{\mathrm{baryon}}\;=\;{M}_{*}+{M}_{\mathrm{gas}}$ (as given in Section 2.1). They show a remarkable agreement, with a median offset of ${\rm{\Delta }}{\mathrm{log}}_{10}M\;=\;0.04\;\mathrm{dex}$, where ${M}_{\mathrm{dyn}}$ is slightly larger than ${M}_{\mathrm{baryon}}$ at a given ${M}_{\mathrm{baryon}}$. The scatter about the median ${\rm{\Delta }}{\mathrm{log}}_{10}M$ is low, with ${\sigma }_{\mathrm{rms}}\;=\;0.34\;\mathrm{dex}$. Additionally, objects with and without detected rotation follow the same ${M}_{\mathrm{baryon}}$–${M}_{\mathrm{dyn}}$ relation, which may support the assumption that the galaxies are all intrinsically rotating disks. We will investigate this in more detail in the next section.

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The offset between the masses implies a dark matter fraction within an ${R}_{E}$ of 8%. This fraction is lower than the ∼30%–50% dark matter fractions within 2.2 r_s for disk galaxies at $z\sim 0$ (which increase with decreasing stellar mass; Pizagno et al. 2005; Dutton et al. 2011a), and the ∼20%–30% within $r\lt 10\;\mathrm{kpc}$ at $z\sim 2$ (Förster Schreiber et al. 2009). This is expected because these studies consider larger radii, and the dark matter fraction increases with increasing radius. Additionally, our measurement is dependent on several systematic uncertainties that we discuss in Section 5.7.

In the previous section, we simply assumed that all kinematically resolved and unresolved objects have a similar median ${(V/{\sigma }_{V,0})}_{{R}_{E}}$. However, from Figure 6, we know that, on average, the unresolved objects are smaller, and thus they may be structurally different. In this section, we use the average properties of the sample without resolved rotation to independently estimate the average ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ for these objects.

The effects of varying the ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ for all objects without observed rotation are demonstrated in Figure 9. In the left panel, we show the assumption of ${(V/{\sigma }_{V,0})}_{{R}_{E}}\to \infty$, or no intrinsic velocity dispersion, for the objects without detected rotation. For comparison, we show the galaxies with detected rotation, using the the dynamical masses calculated from the rotation velocities and velocity dispersions measured from the 2D fitting procedure (see Section 3.1). There is a positive correlation between the mass offset ${\rm{\Delta }}{\mathrm{log}}_{10}M$ and b/a with Spearman correlation coefficient $\rho \;=\;0.18$ at $2.2\sigma$. We quantify this trend by fitting a line to median binned ${\rm{\Delta }}{\mathrm{log}}_{10}M$ and b/a in bins of b/a. This trend of increasing ${M}_{\mathrm{dyn}}$ relative to ${M}_{\mathrm{baryon}}$ as b/a increases indicates that we have over corrected for inclination and that our assumption of ${(V/{\sigma }_{V,0})}_{{R}_{E}}\to \infty$ is too extreme.

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If we instead assume lower values of ${(V/{\sigma }_{V,0})}_{{R}_{E}}$, the inclination correction will be reduced at higher b/a, resulting in a reduced mass offset. Hence, we can constrain ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ for the galaxies without detected rotation by examining the offset ${\rm{\Delta }}{\mathrm{log}}_{10}M\;=\;{\mathrm{log}}_{10}({M}_{\mathrm{dyn}}/{M}_{\odot })-{\mathrm{log}}_{10}({M}_{\mathrm{baryon}}/{M}_{\odot })$ versus the axis ratio, b/a over a range of assumed ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ values. For each assumed ${(V/{\sigma }_{V,0})}_{{R}_{E}}$, we calculate the dynamical masses for the galaxies without detected rotation. We then measure the ${\chi }^{2}$ between the data and the value if there were no trend with b/a, or a constant ${\rm{\Delta }}{\mathrm{log}}_{10}M$ equal to the median of the ${\rm{\Delta }}{\mathrm{log}}_{10}M$ values. We determine the best-fit ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ by minimizing the ${\chi }^{2}$ statistic.

We find a best-fit ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;{2.1}_{-0.3}^{+0.2}$ for the objects without detected rotation. We show the effects of assuming this ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ value in the right panel of Figure 9. When we adopt ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;2.1$, we notice very little offset in ${\rm{\Delta }}{\mathrm{log}}_{10}M$ as a function of b/a, and the total scatter in ${M}_{\mathrm{dyn}}$–${M}_{\mathrm{baryon}}$ is also slightly lower. We note that this measurement reflects an estimate of the typical ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ of this sample; the scatter in the ${M}_{\mathrm{dyn}}$–${M}_{\mathrm{baryon}}$ relation also includes variations introduced by a range of intrinsic ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ values for the sample without observed rotation. The typical ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ of the galaxies with and without observed rotation are identical, suggesting that, on average, all galaxies have similar support from random motions.

The ratio $V/\sigma$ is a measure of the amount of rotational support versus support from random motions, and thus provides information about the structure of a galaxy. Higher $V/\sigma$ values are indicative of thin disks, while lower $V/\sigma$ values may indicate thicker disks or high gas turbulence. To understand the physical processes setting the internal structure of star-forming galaxies at $z\sim 2$, we investigate the relation between $V/\sigma$ and other galaxy properties.

We show the measured ${(V/{\sigma }_{V,0})}_{2.2}\;=\;V(2.2{r}_{s})/{\sigma }_{V,0}$ values for our sample of galaxies as a function of ${\rm{H}}\alpha$ specific star formation rate (SSFR) and stellar mass in Figure 10. For the galaxies without observed rotation, we show the median ${(V/{\sigma }_{V,0})}_{2.2}$ values in bins of ${\rm{H}}\alpha$ SSFR and stellar mass, calculated using the method described in Section 4.2 and assuming ${r}_{t}\;=\;0.4{r}_{s}$. For comparison, we also show the KMOS${}^{3{\rm{D}}}$results of Wisnioski et al. (2015) at z = 1 and z = 2 as the gray open triangles and diamonds, respectively. We note that Wisnioski et al. measure the rotation velocity from the difference between the observed minimum and maximum velocities along the major kinematic axis. However, we do not directly constrain the flat portions of the rotation curves, so instead we consider ${V}_{2.2}=V(r=2.2{r}_{s})$—the radius at which an exponential rotation curve peaks—to attempt to provide a reasonable comparison with existing measurements. We also note that we assume a constant ${\sigma }_{V,0}$, while Wisnioski et al. measure ${\sigma }_{0}$ in the outer portions of the galaxies.

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We see a trend of decreasing ${(V/{\sigma }_{V,0})}_{2.2}$ with increasing ${\rm{H}}\alpha$ SSFR, especially when considering the binned measurement for galaxies without observed rotation and when adding the results by Wisnioski et al. (2015). As suggested by Wisnioski et al. (2015), this trend may reflect disk settling, where galaxies with lower gas fractions (and lower SSFRs) have lower velocity dispersions. For the galaxies without observed rotation, this trend may also reflect a higher fraction of dispersion-dominated galaxies at high SSFRs. Our measured ${(V/{\sigma }_{V,0})}_{2.2}$ values do not show a trend with stellar mass, but when considering our data together with the results of Wisnioski et al. (2015), we may see a weak trend of increasing $V/\sigma$ with increasing stellar mass.

When measuring the kinematics, we assume that the velocity dispersion is constant. If the true velocity dispersion profile rises toward the center of a galaxy (as discussed in Section 5.7), our measured ${\sigma }_{V,0}$ may be systematically higher than what would be measured in the outer portions of our galaxies. This would systematically shift our measured ${(V/{\sigma }_{V,0})}_{2.2}$ to lower values than those found by Wisnioski et al. (2015). Thus the trends of $V/\sigma$ with ${\rm{H}}\alpha$ SSFR and stellar mass found from the combined samples may be partially due to a systematic difference between the adopted $V/\sigma$ values.

Previous studies have compared stellar masses to dynamical masses, especially using the stellar kinematics of quiescent galaxies (e.g., Newman et al. 2010; Taylor et al. 2010; Bezanson et al. 2013; van de Sande et al. 2013; Belli et al. 2014). For quiescent galaxies, this may be a fair comparison, but for high-redshift star-forming galaxies the gas mass is found to be a non-negligible fraction of the total baryonic mass (e.g., Tacconi et al. 2013).

To assess this finding, we consider the difference between the dynamical masses and the stellar masses alone versus ${\rm{H}}\alpha$ SSFR in the left panel of Figure 11. There is an offset between the masses, with the dynamical masses generally exceeding the stellar masses. When we split the sample into bins of ${\rm{H}}\alpha$ SSFR (${\mathrm{log}}_{10}({\mathrm{SSFR}}_{{\rm{H}}\alpha })\lt -9.25$, $-9.25\ \leqslant {\mathrm{log}}_{10}({\mathrm{SSFR}}_{{\rm{H}}\alpha })\lt -8.75$, ${\mathrm{log}}_{10}({\mathrm{SSFR}}_{{\rm{H}}\alpha })\geqslant -8.75$), we find that the higher SSFR bins have larger offsets between the dynamical and stellar masses than the lower SSFR bins. This result is not surprising because the median inferred ratios of ${f}_{\mathrm{gas}}={M}_{\mathrm{gas}}/({M}_{*}+{M}_{\mathrm{gas}})$ are ${f}_{\mathrm{gas}}=0.22$, ${f}_{\mathrm{gas}}=0.44$, and ${f}_{\mathrm{gas}}\;=\;0.58$ for the lowest to highest SSFR bins, respectively.

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In contrast, the baryonic masses (right panel of Figure 11) show a much weaker SSFR-dependent offset with respect to the dynamical masses. The ${M}_{\mathrm{dyn}}$/${M}_{\mathrm{baryon}}$–SSFR relation also show a smaller observed scatter (${\sigma }_{\mathrm{rms}}\;=\;0.34\;\mathrm{dex}$) than the ${M}_{\mathrm{dyn}}/{M}_{*}$–SSFR relation (${\sigma }_{\mathrm{rms}}\;=\;0.37\;\mathrm{dex}$). The agreement between the baryonic and dynamical masses, as well as the larger ${M}_{\mathrm{dyn}}/{M}_{*}$ offset for higher SSFR galaxies, suggests that our ${\rm{H}}\alpha$ SFR-based gas masses are reasonable estimates of the true values.

Thus, at least for star-forming galaxies, it is important to include the gas masses in the total baryonic mass when comparing it against the dynamical masses.

Because our sample consists of star-forming galaxies, we have assumed that there is some amount of rotational support for all galaxies, even for those that are not kinematically resolved. However, it might be the case that some, if not all, of these objects without observed rotation are early-type galaxies. To assess this possibility, we derive dynamical masses assuming instead that all of the kinematically unresolved galaxies are dispersion-dominated ellipticals or lenticulars (S0s).

We calculate the dynamical masses as

$$\begin{eqnarray}&&{M}_{\mathrm{dyn}}\;=\;\beta (n)\displaystyle \frac{{\sigma }_{e,\mathrm{corr}}^{2}{R}_{E,\mathrm{circ}}}{G}.\end{eqnarray} \tag{ 5 }$$

Here we use the Sérsic index dependent virial coefficient, $\beta (n)$, given in Cappellari et al. (2006), where $\beta (n)$ is a quadratic function of n (e.g., $\beta (n\;=\;1)\sim 8$ and $\beta (n\;=\;4)\sim 6$). The observed integrated velocity dispersions (${\sigma }_{V}\;$) are corrected (to ${\sigma }_{e,\mathrm{corr}}$) using an aperture correction similar to that presented in van de Sande et al. (2013), modified to include the axis ratio and PA offset relative to the slit. When calculating the dynamical masses, we use the circularized effective radii, ${R}_{E,\mathrm{circ}}={R}_{E}\sqrt{b/a}$, with ${R}_{E}$ as the Galfit effective radius (which is equal to the semimajor axis) and b/a as the axis ratio. It is necessary to use ${R}_{E,\mathrm{circ}}$ in this case because $\beta (n)$ is derived using circularized radii. The resulting mass difference ${\rm{\Delta }}{\mathrm{log}}_{10}M$ as a function of axis ratio, b/a, is shown in Figure 12.

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Under these assumptions, the dynamical masses are also in reasonable agreement with the baryonic masses (${\rm{\Delta }}{\mathrm{log}}_{10}M\ =-0.07\;\mathrm{dex}$, ${\sigma }_{\mathrm{rms}}=0.37\;\mathrm{dex}$), though the mass offset is somewhat lower than measured for the galaxies with detected rotation. Remarkably, there is no trend of ${\rm{\Delta }}{\mathrm{log}}_{10}M$ with b/a, which is due to the circularization of the effective radii. For objects with smaller axis ratios b/a, the circularized radii will be smaller, and thus the dynamical masses will be lower than if they were calculated using non-circularized radii. Thus, the Sérsic dependent virial coefficient, combined with the circularized radius has a similar effect as the inclination correction applied when assuming that the galaxies are primarily supported by rotation.

The agreement between the dynamical masses for the two separate methods may explain why both star-forming and quiescent galaxies, when treated as early-type galaxies, share "one mass fundamental plane," as found by Bezanson et al. (2015). In this work, both star-forming and quiescent galaxies fall on the same surface in the 3D parameter space defined by effective radius ${R}_{E}$, velocity dispersion ${\sigma }_{V}$, and stellar mass surface density ${{\rm{\Sigma }}}_{*}$. We note that when assuming that all unresolved galaxies are early-type galaxies, the median offset ${\rm{\Delta }}{\mathrm{log}}_{10}M$ is negative, with the typical galaxy lying in the unphysical regime where ${M}_{\mathrm{dyn}}\lt {M}_{\mathrm{baryon}}$. This result could suggest that more accurate dynamical masses are obtained when assuming the galaxies are rotationally supported. However, both the baryonic and dynamical masses are subject to systematic uncertainties, and thus we cannot definitively state whether the unresolved galaxies are rotationally supported or not.

In this section, we consider the implications of the measured ${M}_{\mathrm{baryon}}$ and ${M}_{\mathrm{dyn}}$ values of our galaxy sample for the stellar IMF. In estimating the baryonic masses, we have assumed a Chabrier (2003) IMF. An alternative choice is a Salpeter (1955) IMF, which would result in higher stellar masses, by a factor of 1.8 (e.g., Erb et al. 2006). The relations between $L({\rm{H}}\alpha )$ and SFR, and ${{\rm{\Sigma }}}_{\mathrm{SFR}}$ and ${{\rm{\Sigma }}}_{\mathrm{gas}}$ given by Kennicutt (1998) both contain an IMF dependence, but the relation between the intrinsic $L({\rm{H}}\alpha )$ and ${{\rm{\Sigma }}}_{\mathrm{gas}}$ does not depend on the IMF, so the gas masses do not change whether one is using a Salpeter or a Chabrier IMF.

We show the implications of assuming a Salpeter rather than a Chabrier IMF in Figure 13. The median ${\mathrm{log}}_{10}({M}_{\mathrm{dyn}}/{M}_{\mathrm{baryon}})$ for the whole sample is shown as the solid black line. If we had instead adopted a Salpeter IMF (right panel), the zero-point of ${\mathrm{log}}_{10}({M}_{\mathrm{dyn}}/{M}_{\mathrm{baryon}})$ is lower.

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Physically, we expect that the dynamical mass, which traces all mass in a galaxy, must at least be as large as the observed baryonic mass, depending on the dark matter fraction within an effective radius. The median ${\mathrm{log}}_{10}({M}_{\mathrm{dyn}}/{M}_{\mathrm{baryon}})$ we measure assuming a Chabrier IMF is consistent with this expectation, with 46% of the galaxies fall within the unphysical regime. However, for a Salpeter IMF, 63% of the galaxies fall within the unphysical regime where ${M}_{\mathrm{dyn}}\lt {M}_{\mathrm{baryon}}$. Thus a Salpeter IMF is inconsistent with our measured values of ${M}_{\mathrm{baryon}}$ and ${M}_{\mathrm{dyn}}$. This inconsistency with a Salpeter IMF is in agreement with the findings of other studies of star-forming, disk galaxies (Bell & de Jong 2001; Tacconi et al. 2008; Dutton et al. 2011a; Brewer et al. 2012). Nonetheless, our result is subject to potential systematic uncertainties that might decrease the measured dynamical masses, such that our measurements would be inconsistent with both a Salpeter (1955) and a Chabrier (2003) IMF. We discuss these systematic uncertainties in detail in Sections 5.6 and 5.7.

Finally, there is a suggestive trend between ${M}_{\mathrm{dyn}}$/${M}_{\mathrm{baryon}}$ and ${M}_{\mathrm{dyn}}$, such that more massive galaxies may have a steeper IMF (or a larger dark matter fraction). However, this trend primarily reflects the cutoff in observed baryonic masses (upper envelope). Thus, at fixed dynamical mass, we miss galaxies with the lower baryonic masses. These missed galaxies could increase the median baryonic-dynamical mass offset, leaving room for more dark matter, or bringing the Salpeter IMF into agreement with our data.

The TFR (Tully & Fisher 1977)—which relates the luminosity of disk galaxies to their rotation velocity—captures information about the interplay between the build-up of disk galaxies and their dark matter halos. Because the luminosity-based TFR is subject to luminosity evolution (due to aging populations) and a possible evolution in the gas mass fraction, more recent works have focused on measuring the stellar mass or baryonic mass TFRs because mass allows for more straight-forward comparisons between redshifts (e.g., Dutton et al. 2011b; Gnerucci et al. 2011; Miller et al. 2012; Vergani et al. 2012).

Furthermore, Kassin et al. (2007) argue that the stellar and baryonic TFRs may evolve due to the increase of non-rotational support in higher redshift galaxies. To account for the increased non-rotational support, they argue for the adoption of the ${S}_{0.5}$ kinematic indicator, with ${S}_{0.5}\;=\;\sqrt{0.5{V}_{\mathrm{rot}}^{2}+{\sigma }_{V}{}^{2}}$. This study shows a reduction of scatter in the ${S}_{0.5}$–${M}_{*}$ TFR relative to the standard ${M}_{*}$-TFR at $z\sim 0.2-1$. Furthermore, they find that there is barely any evolution in the ${S}_{0.5}$–${M}_{*}$ TFR out to $z\sim 1$.

We use our kinematic measurements to examine the ${S}_{0.5}$–${M}_{*}$ and ${S}_{0.5}$–${M}_{\mathrm{baryon}}$ TFRs for our sample of star-forming galaxies at z ∼ 1.5–2.6, shown in Figure 14. We perform a weighted linear fit of ${S}_{0.5}$ versus ${M}_{*}$ (left panel, Figure 14) to our sample of star-forming galaxies by fixing the slope to the average value Kassin et al. (2007) find at z ∼ 0.1–1.2 (i.e., $\mathrm{slope}\;=\;0.34$). We find a moderate correlation between ${S}_{0.5}$ and ${M}_{*}$, with scatter in ${S}_{0.5}$ of ${\sigma }_{\mathrm{rms}}\;=\;0.17\;\mathrm{dex}$ for our star-forming galaxies. The scatter is similar to the average scatter Kassin et al. (2007) find (0.16 dex). Our best-fit relation is offset to higher ${S}_{0.5}$ compared to the average relation found by Kassin et al. (2007; black dashed line, left panel of Figure 14), which may be explained by lower average gas fractions of star-forming galaxies at lower redshifts. We find a similar intercept if we fit the ${S}_{0.5}\mbox{--}{M}_{*}$ relation using only the galaxies with detected rotation.

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We follow the same general method to fit ${S}_{0.5}$ versus ${M}_{\mathrm{baryon}}$ (right panel, Figure 14), adopting the slope of the ${S}_{0.5}$–${M}_{\mathrm{baryon}}$ TFR at $z\sim 0.2$ measured by Kassin et al. (2007) (i.e., $\mathrm{slope}\;=\;0.39$), and find a correlation between ${S}_{0.5}$ and ${M}_{\mathrm{baryon}}$ with a scatter in ${S}_{0.5}$ of 0.16 dex. Our ${S}_{0.5}$ intercept is somewhat lower than found by Kassin et al. (2007) at $z\sim 0.2$ (dashed black line, right panel), suggesting an increase in ${S}_{0.5}$ at fixed ${M}_{\mathrm{baryon}}$ over time. We find the same result if we exclude the galaxies without detected rotation. The offset between ${S}_{0.5}$ intercepts may be explained by a higher dark matter fraction at later times. This trend may reflect the increasing radii of similar-mass star-forming galaxies with decreasing redshift (e.g., Williams et al. 2010; van der Wel et al. 2014a); because dark matter halo profiles are less concentrated than stellar mass profiles, a larger radius results in a higher dark matter fraction.

Nonetheless, systematic differences may affect the comparison of the ${S}_{0.5}-M$ TFRs. In particular, the rotation velocities are not measured uniformly, which could introduce systematic offsets. Kassin et al. (2007) use the maximum rotation velocity ${V}_{\mathrm{max}}$ (i.e., the velocity at the flat portion of an arctan rotation curve or at $2.2{r}_{s}$ for an exponential disk). Our data do not sample the flat portion of the arctan rotation curve, so we instead adopt ${V}_{2.2}$ as our velocity measurement, as we have reasonable velocity constraints at this radius. Future work is necessary to quantify the differences between the literature measurements, in order to study the M-TFR evolution over time in a consistent manner.

In this section, we consider the kinematics of the AGNs that fall within our galaxy sample. Interpreting the kinematics of AGNs can be challenging, as the line profiles may have contributions from nuclear emission tracing non-virial motions. Therefore, we did not include the AGNs in our analysis so far. Here we assess whether the kinematics may still provide a probe of the host-galaxy structure.

We calculate the stellar, gas, and baryonic masses following the procedure of Section 2.1. We measure the ${\rm{H}}\alpha$ kinematics from the resolved 2D or integrated 1D spectra, and derive dynamical masses following the procedures of Sections 3.1, 3.2, and 3.4. The resulting baryonic and dynamical masses for the AGNs, along with those of the galaxy sample, are shown in Figure 15.

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The AGNs that meet our sample selection criteria generally follow the same relation of ${M}_{\mathrm{dyn}}$–${M}_{\mathrm{baryon}}$ as the primary galaxy sample. For the AGNs, we find a median offset ${\rm{\Delta }}{\mathrm{log}}_{10}M\;=\;0.01\;\mathrm{dex}$, which is slightly lower than the median offset for the star-forming galaxies (${\rm{\Delta }}{\mathrm{log}}_{10}M\;=\;0.04\;\mathrm{dex}$), and a scatter of ${\sigma }_{{\rm{s}}}\;=0.35\mathrm{dex}$. We note that the effective radii may be underestimated for some objects, due to the influence of a nuclear source.

The good agreement between the AGNs and galaxies in the ${M}_{\mathrm{dyn}}$–${M}_{\mathrm{baryon}}$ plane suggests that the rest-frame optical lines of most AGNs in our sample are not dominated by line emission from the nuclear regions, and that we are probing the kinematics of the host galaxies. Our findings support the results of Gabor & Bournaud (2014), who show that in high-resolution hydrodynamic simulations of disk galaxies at $z\sim 2$, AGNs have relatively little impact on the gas in galaxy disks.

When constructing kinematic models, we rely on structural parameters and radii measured from the stellar light distribution, but measure the kinematics from ${\rm{H}}\alpha$ emission. Ideally, we would measure the kinematics from stellar absorption features, but our galaxies are too faint for these measurements. Instead, we assume that the ionized gas and stellar mass have the same distribution and that the gas follows the gravitational potential of the galaxy.

When using kinematic models to measure rotation and interpret the velocity dispersion of unresolved objects, we weight the model velocity field by a luminosity distribution. For the galaxies with detected rotation, this weighting determines the composite velocity profile within a spatial slice, as there is a mix of line-of-sight velocity components, as well as components that fall within the slit, parallel to the spatial direction. The weighting also predicts a light profile along the spatial direction, but our method of fitting the 2D spectra with the models removes this variation in the spatial direction through the scaling factor S(y). For the kinematically unresolved galaxies, the luminosity weighting affects all directions.

To be fully consistent, we should weight the model velocities by the ${\rm{H}}\alpha$ light profile, so there are no mis-matches between the model and observed luminosity profiles. However, the ${\rm{H}}\alpha$ profiles are not well constrained by the MOSDEF data, and most galaxies in our sample lie at redshifts where ${\rm{H}}\alpha$ falls outside of the wavelength coverage of the WFC3 grism (Brammer et al. 2012). We therefore assume that the stellar light profiles are a reasonable approximation for the luminosity weighting. We assess this assumption by comparing the stellar light profiles with the ${\rm{H}}\alpha$ profiles for the galaxies with detected rotation.

To measure the stellar light profile, we convolve the HST/F160W image to match the seeing resolution of the corresponding MOSFIRE spectrum. We then collapse all light falling within the slit along the slit direction. We approximate the ${\rm{H}}\alpha$ profile by collapsing the scaled emission line model, $S(y){f}_{\mathrm{model}}(y,\lambda )$, over the wavelength direction. We show example F160W and ${\rm{H}}\alpha$ profiles in Figure 16 for the same galaxies shown in Figure 2. For three of the four examples, the stellar and ${\rm{H}}\alpha$ profiles are very similar. Only the first object, AEGIS-17437, has a noticeable deviation between the ${\rm{H}}\alpha$ and stellar profiles, where the ${\rm{H}}\alpha$ profile is wider than the stellar light profile.

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We quantify the profile differences for all galaxies with detected rotation by fitting the ${\rm{H}}\alpha$ and F160W profiles with Gaussians. We note that the profiles of a number of objects are not well described by a Gaussian profile, but the FWHM measurements should provide a reasonable, albeit rough, comparison. We compare the widths of the seeing-matched stellar light and ${\rm{H}}\alpha$ profiles in Figure 17. Generally, the FWHMs are in reasonable agreement, and only a few objects have FWHMs that differ by more than 10% (objects lying outside the gray shaded region). Additionally, we show the relation ${R}_{E,{\rm{H}}\alpha }\;=\;1.3{R}_{E}$ found by Nelson et al. (2012) as the dotted gray line, converted to FWHMs and convolved with a median seeing FWHM of $\sim 0\_\_AMP\_\_farcs;7$. All but a few of the galaxies lie below this line, suggesting that the ${\rm{H}}\alpha$ sizes for our sample are closer to the stellar light sizes than for the Nelson et al. sample. Thus, for most of our objects the stellar light profile is a reasonable substitute for the ${\rm{H}}\alpha$ profile, and hence the measured kinematics will not be biased.

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However, for galaxies with more extended ${\rm{H}}\alpha$, such as AEGIS-17437, we may overestimate the velocity dispersion and possibly the rotation velocity, when assuming the stellar light profile in the model construction. This velocity difference can be explained by the fact that the high velocities at large radii have been down-weighted when using the less extended stellar light instead of the ${\rm{H}}\alpha$ profile. The exact changes in the measured velocity and dispersion velocity from the 2D models depend on the misalignment between the major axis and the slit. If ${\rm{\Delta }}\mathrm{PA}=0$, there is symmetric mixing of velocities at different radii within a spatial slice, and the narrower stellar light profile, therefore, results in less broadening in the wavelength direction. Additionally, by weighting with the narrower stellar light profile, the composite of the velocity components along the line-of-sight direction also results in less broadening in the wavelength direction. Thus, when ${\rm{\Delta }}\mathrm{PA}=0$ and we weight the velocity fields with the stellar light profile, the measured $V({R}_{E})$ is the same, and ${\sigma }_{V,0}$ is larger than we would measure when using the ${\rm{H}}\alpha$ light profile. If a galaxy is misaligned with the slit, the measured $V({R}_{E})$ using the stellar light profile may also be larger than if we weighted with the ${\rm{H}}\alpha$ profile.

The 1D model for AEGIS-17437 would suffer a greater discrepancy if we would have weighted the velocities of the model with the ${\rm{H}}\alpha$ instead of the stellar light profile. The increased weight at large radii would increase the weighted integrated velocity dispersion of the model within the aperture. Thus the ratio ${\sigma }_{V,\mathrm{model}}/{V}_{\mathrm{rms}}\;{({R}_{E})}_{\mathrm{model}}$ is higher for the ${\rm{H}}\alpha$ profile than for the stellar light profile, when using the same underlying rotation curve and assumed ${(V/{\sigma }_{V,0})}_{{R}_{E}}$. Therefore, the corrected rms velocity values measured using the stellar light profile overestimate the true values.

An additional question is whether the stellar light sizes correctly probe the characteristic size of the underlying matter density profile. In particular, our current calculations have assumed that half of the total mass is enclosed within the half-light radius. However, the half-mass sizes are $\sim 25\%$ smaller than the half-light sizes on average (Szomoru et al. 2013). If we assume the same intrinsic rotation velocity curve, $V({R}_{E})\gt V({r}_{1/2,\mathrm{mass}})$, and thus the measured $V({R}_{E})$ (for the resolved models) and ${V}_{\mathrm{rms}}\;({R}_{E})$ (for the unresolved models) are larger than the velocities at ${r}_{1/2,\mathrm{mass}}$. If we assume a constant ${\sigma }_{V,0}$, this difference implies a lower $V/\sigma$. In combination with a smaller R_e, this results in a considerably lower dynamical mass. For example, if ${r}_{1/2,\mathrm{mass}}$ is 25% lower than ${R}_{E}$ (as in Szomoru et al. 2013), then a galaxy with ${(V/{\sigma }_{V,0})}_{{R}_{E}}=2.1$ at ${R}_{E}$ and ${r}_{t}=0.4{r}_{s}=0.4{R}_{E}/1.676$ (following Miller et al. 2011, see Appendix B) has ${M}_{\mathrm{dyn}}({r}_{1/2,\mathrm{mass}})$ which is 31% lower than ${M}_{\mathrm{dyn}}({R}_{E})$. When applying the correction found by Szomoru et al. (2013) the dynamical masses decline by 0.16 dex (to ${\rm{\Delta }}{\mathrm{log}}_{10}M=-0.12\;\mathrm{dex}$) and are inconsistent with both a Chabrier (2003) and a Salpeter (1955) IMF.

We illustrate a lower limit of this systematic variation in the left panel of Figure 18, by approximating the changes of ${\rm{\Delta }}{\mathrm{log}}_{10}M$ caused by varying ${r}_{1/2,\mathrm{mass}}/{R}_{E}$ and using ${M}_{\mathrm{dyn}}\;=\;{M}_{\mathrm{dyn}}({r}_{1/2,\mathrm{mass}})$ (i.e., including variations to the rms velocity and virial coefficient with the assumption of an arctan curve with ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;2.1$ and ${r}_{t}\;=\;0.4{r}_{s}\ =0.4{R}_{E}/1.676$). Even $\sim 10\%$ changes in ${r}_{1/2,\mathrm{mass}}/{R}_{E}$ result in a $\sim 10\%$ change to the inferred dark matter fraction. However, if we instead assume a decreasing velocity dispersion with increasing radius, as we discuss in the next section, the ${V}_{\mathrm{rms}}\;({r}_{1/2,\mathrm{mass}})$ values would be larger. This moves in the opposite direction as the above trend and may change the masses so they remain consistent with a Chabrier (2003) IMF.

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In this section, we consider caveats to assumptions made in the preceding analysis. Specifically, we focus on possible variations due to assumptions about the accuracy of the structural parameters, misalignment of the photometric and kinematic major axes, the intrinsic thickness of galaxy disks, the accuracy of our derived gas masses, the shape of the rotation curve, and the velocity dispersion profile being constant.

First, we have not fully accounted for the accuracy of the Galfit-derived structural parameters. We depend on the structural parameters to model the kinematics of the detected rotation curves, to correct the kinematics from integrated 1D spectra and to calculate the dynamical masses. We include estimated errors on the effective radii when calculating the dynamical masses, but do not include any errors when fitting the kinematic models. Thus, uncertainties in the structural parameters introduce scatter in our derived dynamical masses.

Second, for objects where the photometric and kinematic major axes are misaligned, the inferred velocities and dispersion velocities will be incorrect. If the true ${\rm{\Delta }}\mathrm{PA}\;$ is closer to aligned, the corrected rms velocities will be overestimated, while if it is more misaligned, the velocities will be underestimated (as seen in the first panel in Figure 23). We expect similar over- and underestimates in the measured velocities and dispersions for the galaxies with detected rotation. Additional misalignment uncertainties are introduced by slit alignment issues, which introduce the same trends as stated above.

Third, we assume an intrinsic disk thickness of ${(b/a)}_{0}\ =0.19$ to estimate inclination angles. If a galaxy is intrinsically thicker than the assumed value, the inferred inclination angle underestimates the true value. In this case, the inferred intrinsic rotation velocity and ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ would be overestimated. If a galaxy is thinner, the inclination angle will be overestimated, producing an underestimate of both the rotation and ${(V/{\sigma }_{V,0})}_{{R}_{E}}$. Thus, variations in disk thickness within our sample will add scatter and a possible systematic offset in our dynamical masses.

Fourth, we assume that the galaxies in our sample follow the SFR–gas mass relation of Kennicutt (1998) for star-forming galaxies in the local universe. Based on a local sample of normal and starburst galaxies, Kennicutt (1998) measure N = 1.4 for ${{\rm{\Sigma }}}_{\mathrm{SFR}}\propto {{\rm{\Sigma }}}_{\mathrm{gas}}^{N}$, where ${{\rm{\Sigma }}}_{\mathrm{gas}}$ includes both atomic and molecular gas. Analyses of galaxies at z∼1–3 find slopes that vary between N = 1.28 (Genzel et al. 2010) and N = 1.7 (Bouché et al. 2007). These values bracket the local slope, so our gas masses may be reasonable. However, if the true slope is lower than the local relation, our gas masses underestimate the true value, while a higher slope implies that our gas masses overestimate the true value. An alternate method would be to adopt the gas mass scaling relations presented in Genzel et al. (2015), which relate the gas mass to the stellar mass, the offset from the star-forming main sequence, and the redshift. If we adopt this scaling relation, we see an offset of ${\rm{\Delta }}{\mathrm{log}}_{10}M\;=\;-0.12\;\mathrm{dex}$ and a scatter of ${\sigma }_{\mathrm{rms}}\;=\;0.368\;\mathrm{dex}$ between the dynamical and baryonic masses, with the median mass difference lying in the unphysical region where ${M}_{\mathrm{baryon}}\gt {M}_{\mathrm{dyn}}$. However, the gas mass scaling relations of Genzel et al. (2015) were calibrated for UV+IR SFRs, while we use ${\rm{H}}\alpha$ SFRs in this work. Mismatches between these SFR indicators could be causing the large ($\sim 0.2\;\mathrm{dex}$) difference between the inferred baryonic masses when using the scaling relation method and the inverted Kennicutt–Schmidt relation. Because Kennicutt (1998) calibrated the ${{\rm{\Sigma }}}_{\mathrm{SFR}}\mbox{--}{{\rm{\Sigma }}}_{\mathrm{gas}}$ relation using ${\rm{H}}\alpha$ SFRs, we opt to estimate the gas masses following this prescription.

Fifth, we have assumed that the rotation velocity profiles of our disk galaxies are well described by arctan models, as shown by Courteau (1997), Weiner et al. (2006), and Miller et al. (2011). Some distant star-forming galaxies exhibit rotation following a Freeman exponential disk model (Freeman 1970), as found by Wisnioski et al. (2015), while van Dokkum et al. (2015) find indications of Keplerian rotation in compact star-forming galaxies. However, a preliminary analysis suggests that using an exponential disk rotation curve with our modeling produces poor fits to some of our galaxies. More detailed modeling is required to determine, in detail, whether an alternative rotation profile provides better agreement with our data and to assess the uncertainties introduced by this assumption.

Sixth, we have assumed that the intrinsic velocity dispersion is constant over all radii. However, the true velocity distribution may decrease with increasing radius, as seen in Genzel et al. (2014) and Wisnioski et al. (2015). For the unresolved objects, a decreasing velocity dispersion distribution would produce higher model rms velocities for a given ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ measured at ${R}_{E}$ ⁷ because matching the same ${\sigma }_{V}\;({R}_{E})\approx {\sigma }_{V,0}$ implies a higher central velocity dispersion, ${\sigma }_{V}\;(r\;=\;0)$. A velocity dispersion profile that rises toward the center would increase the integrated model velocity dispersions, but not the model rms velocity at ${R}_{E}$, leading to lower corrected rms velocities and lower dynamical masses. The trend of decreasing integrated rms velocity with increasing inclination will also be less strong for a fixed ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ than with a constant ${\sigma }_{V,0}$. The median ${(V/{\sigma }_{V,0})}_{{R}_{E}}$ required to remove the ${\rm{\Delta }}{\mathrm{log}}_{10}M$ trend for the kinematically unresolved galaxies will, therefore, be higher, increasing the implied amount of rotational support relative to the random motions, which would indicate more settled or thinner disks. Preliminary calculations assuming an additional dispersion term that rises toward the center of a galaxy confirms these general trends, but more careful analysis is necessary to determine the correct form of a rising velocity dispersion profile.

Seventh, in our derivation of the dynamical masses, we have not included the systematic uncertainties arising from the choice of virial coefficients, ${k}_{\mathrm{disp}}$ and ${k}_{\mathrm{rot}}$. The matter distributions assumed when deriving the virial coefficients may not match the underlying profiles of the star-forming galaxies in our sample, but more detailed analysis is required to quantify the uncertainties introduced by the adopted virial coefficients. We note that the systematic shifts from a different choice of virial coefficient can be non-negligible, and have implications especially for the IMF constraints. For instance, the combination of ${k}_{\mathrm{disp}}\;=\;\beta (n)$ (from Cappellari et al. 2006), n = 1, and ${(V/{\sigma }_{V,0})}_{{R}_{E}}\;=\;2.1$ would have resulted in dynamical masses that are larger by $\sim 0.07\;\mathrm{dex}$ and an inferred dark matter fraction of ${f}_{\mathrm{DM}}\sim 22\%$. We approximate the systematic changes due to changing only the combined ${k}_{\mathrm{eff},\mathrm{median}}$ in the right panel of Figure 18. In this plot, we see that if we assume ${k}_{\mathrm{eff},\mathrm{median}}\sim {k}_{\mathrm{rot}}\;=\;2.66$ for all galaxies (without changing the conversion from the integrated velocity dispersions to the rms velocities for the unresolved galaxies), then the median ${\rm{\Delta }}{\mathrm{log}}_{10}M\lt 0$, which lies in the unphysical region where ${M}_{\mathrm{dyn}}\lt {M}_{\mathrm{baryon}}$. If we instead assume ${k}_{\mathrm{eff},\mathrm{median}}\sim {k}_{\mathrm{disp}}\;=\;5$ (again with no other changes), we would instead infer a much higher dark matter fraction of ${f}_{\mathrm{DM}}\sim 45\%$ rather than the 8% measured from the default assumptions.

The multiplexing capabilities of MOSFIRE, which allow us to study many galaxies simultaneously, come at the price of not observing the galaxies along the kinematic major axis. Misalignment of the kinematic major and slit axes poses problems for the interpretation of kinematic measurements even for resolved disk galaxies. Issues to address include: How much kinematic information is lost because portions of the galaxy fall outside the slit? How much of the line broadening in a spatial row is caused by intrinsic velocity dispersion, and how much is caused by the inclusion of multiple line-of-sight velocities in that slice of the galaxy?

In this appendix, we describe how we model the internal kinematics of a disk galaxy, apply the appropriate inclination and PA offset to the model, and then collapse the model along the line of sight and slit direction to calculate the observed kinematic signature of the object as a function of position along the slit.

To model an ideal disk galaxy with an arbitrary PA offset with respect to the slit, we define coordinates as shown in Figure 19. First, we consider a point on the galaxy at $({x}_{\mathrm{int}},{y}_{\mathrm{int}},{z}_{\mathrm{int}})$, with a distance in the plane of the disk of

$$\begin{eqnarray}&&{r}_{\mathrm{int}}\;=\;\sqrt{{x}_{\mathrm{int}}^{2}+{y}_{\mathrm{int}}^{2}}\end{eqnarray} \tag{ 6 }$$

from the axis of rotation, and define the angle ψ as

$$\begin{eqnarray}&&\mathrm{cos}\psi \;=\;{y}_{\mathrm{int}}/{r}_{\mathrm{int}},\end{eqnarray} \tag{ 7 }$$

which is the counterclockwise angle between the major axis ${\hat{y}}_{\mathrm{int}}$ and $({x}_{\mathrm{int}},{y}_{\mathrm{int}},{z}_{\mathrm{int}})$ with respect to the rotation axis (see the left panel of Figure 19). We incline our galaxy model by rotating around the major axis ${\hat{y}}_{\mathrm{int}}$. The inclination angle i is estimated as

$$\begin{eqnarray}&&\mathrm{sin}i\;=\;\sqrt{\displaystyle \frac{1-{(b/a)}^{2}}{1-{(b/a\;)}_{0}^{2}}},\end{eqnarray} \tag{ 8 }$$

where a and b are semimajor and semiminor axes lengths, respectively, from the Galfit parameterization. We assume an intrinsic disk axis ratio of ${(b/a)}_{0}\;=\;0.19$ (Miller et al. 2011). By inclining the model (see the middle panel of Figure 19), the intrinsic coordinates are mapped to projected coordinates by

$$\begin{eqnarray}{x}_{p} & = & {x}_{\mathrm{int}}\mathrm{cos}i+{z}_{\mathrm{int}}\mathrm{sin}i\\ {y}_{p} & = & {y}_{\mathrm{int}}\\ {z}_{p} & = & -{x}_{\mathrm{int}}\mathrm{sin}i+{z}_{\mathrm{int}}\mathrm{cos}i.\end{eqnarray} \tag{ 9 }$$

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Finally, we apply the PA offset, ${\rm{\Delta }}\mathrm{PA}$ , between the galaxy major axis and the slit. We rotate the projected model by ${\rm{\Delta }}\mathrm{PA}$ in the ${\hat{x}}_{p}\mbox{--}{\hat{y}}_{p}$ plane, mapping the projected coordinates into observed coordinates relative to the slit layout (see the right panel of Figure 19) by

$$\begin{eqnarray}x & = & {x}_{p}\mathrm{cos}{\rm{\Delta }}\mathrm{PA}\;+{y}_{p}\mathrm{sin}{\rm{\Delta }}\mathrm{PA}\;\\ y & = & -{x}_{p}\mathrm{sin}{\rm{\Delta }}\mathrm{PA}\;+{y}_{p}\mathrm{cos}{\rm{\Delta }}\mathrm{PA}\;+{y}_{0}\\ z & = & {z}_{p},\end{eqnarray} \tag{ 10 }$$

where we also allow for an offset of the object center relative to the slit center in the $\hat{y}$ direction, through the parameter y₀. If we invert this set of coordinate transformations, we may calculate the intrinsic position $({x}_{\mathrm{int}},{y}_{\mathrm{int}},{z}_{\mathrm{int}})$ and intrinsic radius r_int within the galaxy for any given point $(x,y,z)$ in the slit coordinate system (see Figure 19, right panel).

To model the kinematics of a disk galaxy, we adopt the arctan model for rotation in exponential disks (Courteau 1997; Miller et al. 2011),

$$\begin{eqnarray}&&V(r,{r}_{t},{V}_{a})\;=\;\displaystyle \frac{2}{\pi }\;{V}_{a}\mathrm{arctan}\left(\displaystyle \frac{r}{{r}_{t}}\right),\end{eqnarray} \tag{ 11 }$$

where V_a is the asymptotic velocity and r_t is the turnover radius, which encodes a transition between the rising and flat parts of the rotation curve (Courteau 1997).

We must account for LOS velocity reductions due to projection effects. First, the inclination reduces the LOS velocity by a factor of $\mathrm{sin}i$. The LOS velocity is also reduced by $\mathrm{cos}\psi$, which accounts for the position of every point around the rotational axis of the galaxy. Together, the LOS velocity at each point $(x,y,z)$ is

$$\begin{eqnarray}&&{V}_{\mathrm{los}}(x,y,z)\;=\;V(r,{r}_{t},{V}_{a})\cdot \mathrm{sin}i\cdot \mathrm{cos}\psi ,\end{eqnarray} \tag{ 12 }$$

where $r\;=\;{r}_{\mathrm{int}}$ and $\mathrm{cos}\psi$ are evaluated given the slit coordinates $(x,y,z)$. We assume the rotation is independent of z_int, so the model consists of nested cylindrical shells of varying radii, with each shell rotating at the appropriate velocity. We show an example LOS velocity field of a galaxy, integrated along the line of sight, in the left panel of Figure 20.

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However, our galaxies may not be ideal disks. Thus the galaxies may also have an intrinsic velocity dispersion component, as is the case with thickened disks. We assume that the intrinsic dispersion component ${\sigma }_{V,0}$ is constant over the whole disk, or $\sigma (r)\;=\;{\sigma }_{V,0}$.

The composite kinematic profile of our model, as would be observed with slit spectroscopy, consists of the combination of all the kinematic information of the portions of the galaxy lying within each slit pixel. The relative weights of the individual kinematic components are determined by the associated intensities. Thus we must consider the light distribution of our galaxy model. We assume that the light follows a modified Sérsic intensity profile,

$$\begin{eqnarray}I(r,n,{R}_{E},{z}_{\mathrm{int}}) & = & {I}_{e}\mathrm{exp}\left\{-{b}_{n}\left[{\left(\displaystyle \frac{r}{{R}_{E}}\right)}^{1/n}-1\right]\right\}\\ & & \times \mathrm{exp}\left[-\displaystyle \frac{{z}_{\mathrm{int}}}{{q}_{0}{R}_{E}}\right],\end{eqnarray} \tag{ 13 }$$

where n and ${R}_{E}$ are set to the Galfit best-fit parameters, ${b}_{n}\approx 2n-0.324$ (Ciotti & Bertin 1999), and approximating the vertical scale height as ${z}_{0}\;=\;{q}_{0}{R}_{E}$. We show a simple example intensity profile, integrated over the line of sight, in the center panel of Figure 20.

The composite 2D line model is constructed by combining the line-of-sight velocity information and the assumed intensity profile. We begin by calculating the intensity $I(x,y,z)$ and observed velocity ${V}_{\mathrm{los}}(x,y,z)$ at every point in our slit coordinates. To include the velocity dispersion, ${\sigma }_{v}(x,y,z)$, we assume that at every point $(x,y,z)$ the intensity has a Gaussian distribution with wavelength, with center ${\lambda }_{\mathrm{los}}(x,y,z)\;=\;(1+{V}_{\mathrm{los}}(x,y,z)/c){\lambda }_{0}$ and standard deviation ${\sigma }_{\lambda }\;=\;({\sigma }_{V,0}/c){\lambda }_{0}$. We thus expand our intensity cube into wavelength space as

$$\begin{eqnarray}&&I(x,y,z,\lambda )\;=\;\displaystyle \frac{I(x,y,z)}{{\sigma }_{\lambda }\sqrt{2\pi }}\mathrm{exp}\left[-\displaystyle \frac{{(\lambda -{\lambda }_{\mathrm{los}}(x,y,z))}^{2}}{2{\sigma }_{\lambda }^{2}}\right],\end{eqnarray} \tag{ 14 }$$

where we normalize the intensity distribution to ensure ${\int }_{\lambda }I(x,y,z,\lambda )d\lambda \;=\;I(x,y,z)$.

We collapse the intensity over the z (line of sight) direction,

$$\begin{eqnarray}&&I(x,y,\lambda )\;=\;{\displaystyle \int }_{-\infty }^{+\infty }I(x,y,z,\lambda ){dz},\end{eqnarray} \tag{ 15 }$$

to estimate the observable spectral cube, which contains the combined line-of-sight velocity and velocity dispersion at every point (x, y).

The observed intensity is convolved with the atmospheric seeing. We model the PSF as a 2D Gaussian and take the blurred intensity cube to be $I(x,y,\lambda )\otimes \mathrm{PSF}(x,y)$. The MOSFIRE spectra are taken through a slit, so there is only one dimension of spatial information. Thus we collapse the intensity model in the slit width axis, x, over the width of the slit, x_ap by taking

$$\begin{eqnarray}&&I(y,\lambda )\;=\;{\displaystyle \int }_{-{x}_{\mathrm{ap}}/2}^{+{x}_{\mathrm{ap}}/2}[I(x,y,\lambda )\otimes \mathrm{PSF}(x,y)]{dx}.\end{eqnarray} \tag{ 16 }$$

Finally, we include the effects of instrumental resolution by convolving the model with a Gaussian with width ${\sigma }_{\lambda ,\mathrm{inst}}$ (measured from the width of sky lines). An example ${\rm{H}}\alpha$ emission line model is shown in the right panel of Figure 20.

In practice, we generate a model by performing these calculation over a finite grid of values in $x,y,z$, and λ. We set ${x}_{\mathrm{ap}}\;=\;0\_\_AMP\_\_farcs;7$ (MOSFIRE slit width), and y_ap equal to the spatial extent of the trimmed 2D spectrum to which we will compare the model. We set ${z}_{\mathrm{ap}}\;=\;{y}_{\mathrm{ap}}$, to probe the same spatial extent both along the line of sight and along the slit. We allow for sub-pixel sampling, and set the number of sub-pixels in $x,y,z$ so that the sub-pixel width in each dimension is nearly equal, while preserving an integer number of whole pixels in the spectrum spatial direction, y. Additionally, we pad the grid by $0.5\;{\mathrm{FWHM}}_{\mathrm{seeing}}\;\mathrm{arcsec}$ on both sides in the x and y directions, to accurately calculate the seeing-convolved intensity over the full ${x}_{\mathrm{ap}},{y}_{\mathrm{ap}}$. We sample the model at the wavelengths of the associated 2D spectrum in the range around ${\rm{H}}\alpha$. The array $I(y,\lambda )$ is oversampled in the y direction. Finally, we rebin the data to match the observed spatial pixel size by adding the sub-pixels in y. The final model ${f}_{\mathrm{model}}(y,\lambda )$ now samples the intensity at the exact spatial positions y and wavelengths λ covered by the data.

The resulting model of the observed kinematic signature of a disk galaxy depends on fixed parameters ${\rm{\Delta }}\mathrm{PA}$, n, ${R}_{E}$, and b/a, all derived from the Sérsic fits performed using Galfit. The seeing FWHM and instrument resolution are additional fixed parameters. Because we do not probe the flat part of the rotation curves for our galaxy sample, we fix y₀ and ${\lambda }_{0}$. We mask missing pixels and skylines for the 2D spectrum, collapse over λ and fix y₀ to the peak of a Gaussian fit. We similarly collapse over y to fit ${\lambda }_{0}$. The free parameters are V_a, r_t, and ${\sigma }_{V,0}$. V_a and r_t describe the arctan disk rotation model, while ${\sigma }_{V,0}$ introduces additional broadening in the wavelength direction.

In this appendix we describe how we measure rotation and dispersion velocities from ${\rm{H}}\alpha$ emission lines that exhibit resolved rotation. For each object, we start by subtracting out the continuum from the ${\rm{H}}\alpha$ 2D spectrum. First, we measure the continuum slope of the optimally extracted 1D spectrum using a noise-weighted linear fit in the wavelength range of $6454.6\;\mathring{\rm A} \leqslant \lambda /(1+{z}_{\mathrm{MOS}})\leqslant 6674.6\;\mathring{\rm A}$, where we mask the ${\rm{H}}\alpha$ and $[{\rm{N}}\;{\rm{III}}]$ lines from $6533.6\;\mathring{\rm A} \leqslant \lambda /(1+{z}_{\mathrm{MOS}})\ \leqslant 6599.6\;\mathring{\rm A}$. We then assume that the slope of the continuum in each spatial slice of the 2D spectrum is equal to the 1D continuum slope value, and perform a weighted linear fit in each spatial slice where only the intercept is allowed to vary. We subtract the best-fit continuum from each spatial slice to leave only the ${\rm{H}}\alpha$ line emission. Next, we trim the continuum-subtracted emission line 2D spectrum to the wavelength range of $6555.6\;\mathring{\rm A} \leqslant \lambda /(1+{z}_{\mathrm{MOS}})\leqslant 6573.6\;\mathring{\rm A}$, to exclude the $[{\rm{N}}\;{\rm{III}}]$ emission lines from our resolved line fitting. As we exclude objects with outflows or AGNs with very broad emission lines, $[{\rm{N}}\;{\rm{III}}]$ contamination within this trimmed range should be minimal. We also trim the spectrum in the spatial direction so that only the positive emission line is retained.

To ensure that the model comparisons include only high S/N portions of each spectrum, we construct a mask $m(x,y)$, where x is the wavelength dispersion direction of the spectrum and y is the spatial direction. First, we mask pixels with missing data. Second, we mask rows where ${\rm{S}}/{\rm{N}}(y)\lt 2$, leaving only contiguous rows with high S/N unmasked. The row S/N is estimated to be the total row flux over the total row error, with the pixel errors added in quadrature. We mask pixels with missing data or columns affected by skyline contamination in this S/N measurement. The columns affected by skyline contamination are identified as those where the total column error, added in quadrature, is two and three times greater than the median error of all columns in the spectrum, for the K and H bands, respectively.⁸ We do not mask pixels affected by skyline contamination when fitting the models to the data, as these pixels are down-weighted in the fitting procedure because of their large errors relative to the non-contaminated pixels.

To find the best-fit model to the data, we first match the model intensity to the data intensity profile. We perform a weighted least squares fit of the model ${f}_{\mathrm{model}}(x,y)$ to the data $f(x,y)$ (with errors ${\sigma }_{f}(x,y)$) at each y and measure the appropriate scaling S(y) between the model and data spatial rows

$$\begin{eqnarray}&&S(y)=\displaystyle \frac{{\displaystyle \sum }_{x}m(x,y)[f(x,y){f}_{\mathrm{model}}(x,y)/{\sigma }_{f}{(x,y)}^{2}]}{{\displaystyle \sum }_{x}m(x,y)[{f}_{\mathrm{model}}{(x,y)}^{2}/{\sigma }_{f}{(x,y)}^{2}]},\end{eqnarray} \tag{ 17 }$$

where we mask missing data and columns contaminated by skyline emission with $m(x,y)$ (discussed in Section 3.1). We do not fix the scaling to the convolved and integrated (in the slit direction) Galfit profile because the line emission may be distributed differently (see Nelson et al. 2012). Nonetheless, in modeling the kinematics, we adopt a Sérsic stellar light profile to determine the intensity-weighted velocities. In Section 5.6, we show that most objects have similar stellar light and ${\rm{H}}\alpha$ profiles, and also discuss the implications for the modeling results when the ${\rm{H}}\alpha$ profile differs from the stellar light profile.

The goodness-of-fit of the model is determined using a weighted ${\chi }^{2}$ value. We choose the following weighting scheme to up-weight lower S/N rows, so the information in the fainter parts of the rotation curve is not lost:

$$\begin{eqnarray}&&{w}_{y}\;=\;{[{\rm{S}}/{\rm{N}}(y)]}^{-1}.\end{eqnarray} \tag{ 18 }$$

The weighted goodness-of-fit criterion is then

$$\begin{eqnarray}{\chi }_{\mathrm{weighted}}^{2} & = & \displaystyle \sum _{x,y}{w}_{y}\\ & & \times {\left[m(x,y)\displaystyle \frac{f(x,y)-S(y){f}_{\mathrm{model}}(x,y)}{{\sigma }_{f}(x,y)}\right]}^{2}\end{eqnarray} \tag{ 19 }$$

where we mask missing data and low S/N rows with $m(x,y)$, and the spectrum is dispersed in the x direction.

We use the python MCMC package, emcee (Foreman-Mackey et al. 2013) to find the best-fit models and confidence intervals. We define flat priors $\mathrm{log}p(X)$ for each parameter X (V_a, r_t, ${\sigma }_{V,0}$), with bounds calculated based on the spatial and wavelength coverage of the trimmed 2D spectrum, yielding composite prior $\mathrm{log}p({V}_{a},{r}_{t},{\sigma }_{V,0})\;=\;{{\rm{\Sigma }}}_{\{X={V}_{a},{r}_{t},{\sigma }_{V,0}\}}\mathrm{log}p(X)$. The log posterior probability is taken to be

$$\begin{eqnarray}\mathrm{log}P({V}_{a},{r}_{t},{\sigma }_{V,0}| \lambda ,f) & = & \mathrm{log}{ \mathcal L }(f| \lambda ,{V}_{a},{r}_{t},{\sigma }_{V,0})\\ & & +\mathrm{log}p({V}_{a},{r}_{t},{\sigma }_{V,0})+\mathrm{const},\end{eqnarray} \tag{ 20 }$$

with log likelihood probability $\mathrm{log}{ \mathcal L }\;=\;-0.5{\chi }_{\mathrm{weighted}}^{2}$.

The rotation curve turnover is not well constrained in our data, so there is a degeneracy in the values of V_a and r_t. However, the values of $V({R}_{E})$ and ${V}_{2.2}\;=\;V(2.2{r}_{s})$ are much better constrained. Thus we calculate $V({R}_{E})$ and ${V}_{2.2}$ for each pair of $({V}_{a},{r}_{t})$ values in the posterior sampling, to determine the posterior distributions on $V({R}_{E})$ and ${V}_{2.2}$. We take the best-fit values of $V({R}_{E})$, ${V}_{2.2}$, and ${\sigma }_{V,0}$ to be the peaks of the respective marginalized posterior distributions, and calculate the confidence intervals using the lower and upper 68-percentile bounds of the posterior distributions (e.g., see Figure 21).

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To demonstrate the agreement between the observed and modeled kinematics, we measure velocity as a function of position from both the observed and modeled 2D spectra for each of the 35 galaxies with detected rotation. For each unmasked, high S/N row (see Appendix A.2), we fit the flux $f(x,y)$ with a Gaussian and determine the central wavelength, $\lambda (y)$, constraining λ to fall within $\pm 1.25\;{\mathrm{FWHM}}_{\lambda ,1{\rm{D}},\mathrm{obs}}$ of the fixed central wavelength ${\lambda }_{0}$ (see Appendix A.1). We then calculate ${V}_{\mathrm{obs}}(y)$ from $\lambda (y)$ and ${\lambda }_{0}$. We estimate the errors in ${V}_{\mathrm{obs}}(y)$ by creating 500 realizations in which we perturb the flux $f(x,y)$ by the errors ${\sigma }_{f}(x,y)$, and repeat the fitting procedure on each realization. Finally, we correct the observed velocities for inclination, yielding ${V}_{\mathrm{obs}}(y)/(\mathrm{sin}i)$. The velocity profiles are shown in Figure 22. The observed and model velocity profiles are in good agreement, suggesting that our modeling approach works well.

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