The KMOS^3D Survey: Data Release and Final Survey Paper^*

A number of processing steps were performed on the individual detector images before subsequent processing by SPARK. These include corrections for the readout channel-dependent bias level, alternating column noise (ACN), and picture-frame noise effects described by Rauscher (2015). For each science exposure, we first removed a channel-dependent bias level using reference pixels around the perimeter of each of the three KMOS HAWAII-2RG detectors. This bias removal included a correction for ACN, which was also estimated from the reference pixel arrays. In a subset of KMOS exposures—particularly those with large negative or positive median reference pixel values—we found that the bias- and ACN-corrected frames showed significant spatial nonuniformity around their perimeters. This appears to be a manifestation of the so-called "picture-frame" noise discussed by Rauscher et al. (2013) and Rauscher (2015) and, in KMOS, appears to be due to drifts in the bias voltage (E. George 2019, private communication). Offsets of up to ±2 counts are present in 10%–15% of all exposures taken with detectors 1 and 2 and in nearly 30% of exposures taken with detector 3; in the most extreme cases, offsets of up to ±10 counts are observed. In order to correct for these effects, we used a set of ∼7500 dark frames (per detector) to estimate the correlation between the median reference pixel value and each pixel in the science array. We then estimated and removed the residual picture-frame noise based on the reference pixel values in each science exposure.

The corrected cubes were reconstructed using standard SPARK routines, including a frame-by-frame correction for the wavelength solution based on cross-correlation with a reference OH spectrum. Sky subtraction was then performed external to SPARK in two steps: first, a simple O–S subtraction based on the adjacent sky cubes, followed by a removal of sky-line residuals using a modified version of the ZAP principal component analysis (PCA) sky-subtraction code (Soto et al. 2016). Due to the relatively small size of individual KMOS IFUs (14 × 14 spaxels), principal components (PCs) for ZAP were estimated from a subsample of sky exposures taken over the lifetime of KMOS^3D  observations. The final set of reference spectra consisted of ∼5 × 10⁵ individual spectra in each band with approximately uniform distributions in elevation angle and time of year. We found that PCs measured in this way were better able to account for variations in the relative strength of different rovibrational OH transitions and spectrograph line-spread function compared to PCs constructed from individual observing runs or semesters (J. T. Mendel et al. 2019, in preparation).¹⁹

The application of PCA sky subtraction to the KMOS data provides a noticeable reduction to both the residual OH lines and molecular features such as the O2 feature at 1.26−1.28 μm in the YJ band. Figure 4 shows the ratio of the standard deviation for extracted galaxy spectra with the standard sky-subtraction routine implemented in SPARK to the PCA sky-subtraction routine implemented in the KMOS^3D  data. The standard deviation is calculated over ±7500 km s⁻¹ around the detected or expected location of Hα from a spectrum extracted by summing spaxels in a 2'' × 2'' window centered on the cube. A reduction in the standard deviation of the galaxy spectra is measured in all bands. The improvement is most dramatic in the H band (orange histogram), where sky emission lines are both stronger and more closely packed than in the YJ (blue) and K (red) bands.

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A heliocentric correction is applied to all data frames before they are combined. The corrections range from −30 to +30 km s⁻¹. The correction is especially important for the observations of the same objects in different semesters. Uncorrected data can lead to inaccurate redshifts and inflated integrated velocity dispersions, particularly for narrow emission lines that are near the instrument resolution limits.

A rotator angle–dependent illumination correction is done per O–S pair using the internal flat with the closest rotator angle. The matching angle, of the six available (30, 90, 150, 210, 270, 330), provides the best illumination correction with residual nonuniformity of ∼±3% in IFUs 1 and 23, which show the strongest gradients, and smaller elsewhere.

Observations of A0, B, and G stars were taken before and after observing a KMOS pointing when the conditions allowed. The observed "standard" stars were selected from the Hipparcos catalog (Perryman et al. 1997) with known IR magnitudes (Cutri et al. 2003). The star observations are used to apply both a telluric transmission correction and flux calibration to all individual science frames. The standard KMOS observing procedure was followed such that a single standard star is observed in three IFUs (one per detector). Observed stars were chosen to be at a similar airmass as the science data.

Photometric zero-points are calculated in the AB system using custom IDL routines. The observations of standard stars are collapsed to a 1D spectrum. The mean counts within a predefined wavelength range, matched to the central wavelengths of the 2MASS J, H, and K filters, are used to derive the zero-point. A model Moffat function is fit to the stars to correct for the small fraction of flux lost outside of the IFU (typically 1%–3%). The zero-point in each band is stable, with standard deviations over the full survey equal to 0.19, 0.39, and 0.26 mag for the YJ, H, and K bands, respectively. In the cases where the zero-point is more than 2σ deviant from the mean, the standard star cubes are visually inspected. The few deviant zero-points can be attributed to pointing errors or conditions with >60% humidity. Therefore, deviant zero-points are replaced by the mean of all of the zero-points measured in the same band and detector over the duration of the survey. The zero-points do not correlate with other recorded observing conditions such as airmass or seeing.

Many pointings were observed continuously for multiple hours, during which conditions changed. Before applying a telluric and flux calibration, we account for variations in the observing conditions of each frame over time. To make this correction, the total flux in the PSF stars of each frame are compared to the median flux of the same stars in the three science frames observed closest in time to the standard star observations.

A telluric transmission spectrum is created by dividing the standard star spectrum by a blackbody function of the effective temperature of the standard star and removing the intrinsic stellar absorption features. Each spaxel is then divided by the telluric spectrum observed in the same detector.

An airmass correction is applied to account for the difference in elevation between observations of the standard star and each science frame. The zero-point is then applied to derive the absolute flux scale for each science frame. Figure 5 shows the comparison of magnitudes derived from individual KMOS exposures of the PSF stars and known magnitudes from HST (F125W, F140W) and ULTRAVISTA (Ks) in similar wave bands to the KMOS YJ, H, and K filters. No color correction has been applied to match the different filters. We compare the observed and known magnitudes for each of the three PSF stars in individual frames, as well as the final combined PSF star images discussed in Section 4.7. To derive the magnitudes, the total flux in the flux-calibrated star cube is summed in the wavelength range defined for calibration of KMOS data²⁰ and corrected for flux loss outside of the IFU using a Moffat model. The fluxes measured from the combined star data agree within σ ≲ 19% with the fluxes derived from known magnitudes, with minor offsets with mean and standard deviations of $-0.09\pm 0.13$, $-0.07\pm 0.18$, and $-0.04\pm 0.19$ in the YJ, H, and K bands, respectively.

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The steps described in Section 4.1 (detector-level corrections) provide an initial correction for the detector-level background in individual exposures. In many cases, an additional correction is required to remove residual spatial nonuniformity driven by channel-to-channel variation, as well as correct nonzero background levels, which otherwise limit our ability to push to low (continuum) surface brightness levels.

We model the background in each reconstructed data cube as the combination of individual (detector) output channels and a spatially and spectrally uniform background component. Because individual dispersed spectra are tilted with respect to the KMOS detectors, the relative contribution of output channels at a given spatial position in the reconstructed cubes varies as a function of wavelength. We model this variation using "channel cubes" that provide a mapping between pixels in the reconstructed IFU data (in x, y, λ) and their corresponding detector output channels. On average, four distinct output channels contribute to any given IFU. The median background correction derived per detector is small, ∼a few ×10⁻²¹ erg s⁻¹ cm⁻² Å⁻¹, with interquartile values in the range $\pm 5\times {10}^{-20}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ Å⁻¹. Channel-to-channel variations are typically an order of magnitude smaller. For comparison, the median surface brightness at r_e of KMOS^3D sources implies flux densities of order ${10}^{-20}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ Å⁻¹ such that, if left uncorrected, this background variability severely limits the detection of faint sources.

In the case of bright sources, where the object continuum is detected in an individual 300 s exposure, the fitting procedure outlined above can systematically overestimate the true background level. The magnitude of this overestimation is small, ≲10% of the variation in background level between frames, but it is systematic and can otherwise bias flux measurements for bright objects. Correcting for this effect requires a comparison with the surface brightness profile derived from HST imaging and is described in the next section.

Finally, the reduced science frames are combined for each galaxy with the standard KMOS pipeline using 3σ clipping and then taking the average to create a single data cube for each observed object ("ksigma" combine method). To produce the final combined data cubes, the following steps are taken. A small fraction, 6%, of the frames were observed at a nonzero rotator angle. As a result, 39 galaxies have data observed at multiple rotator angles. To resolve the angle mismatch, all individual frames are first derotated to 0°.

Individual exposures are inspected for any bad or failed data or reductions. This can include failed sky subtraction or telluric correction, spectral fringing, bad seeing, clouds, bright background from twilight, odd continuum shapes, and or a failed reference star fit. Possible bad frames and nights with poor conditions are automatically flagged and then manually checked. For some exceptional cases (e.g., humidity >60%), individual IFUs are flagged as bad within a science frame. Flagged frames or IFUs are not included in the final combination. In total, 554 frames, or 6.6% of the frames, were thus rejected.

Astrometric shifts between exposures are computed using the average measured offsets from the three stars included in the same pointing. The measured astrometric shifts are the combined effect of dithering and gradual drift. For data taken over the same night or series of nights, this method provides an improvement with respect to using the shifts recorded in the header keywords. However, instrument interventions and longer timescale variations in the instrument and telescope over the 5 years of the survey can result in larger spatial shifts that are not well accounted for by the standard correction applied from the PSF stars. These offsets of >1−2 pixels can lead to spurious "double images" of the observed galaxy. To correct for these spatial shifts, partial combined data cubes are created for each galaxy that are the sum of all useful data taken for a given galaxy within a given KARMA setup (based on jumps in the arm telemetry). Each galaxy may have one to six partial combined frames, depending on the number of observations throughout the survey.

In brief, model KMOS images are generated by convolving HST postage stamp images in the closest available band to the KMOS PSF, cropping to the KMOS field of view, and binning in KMOS 02 pixels. This is compared to the KMOS data cube, optimally collapsed to form a continuum image. The centroid is allowed to vary, and the best fit is the one that minimizes χ². This provides a visually good solution in every case where the source is clearly visible in the continuum, which includes  98.6% (730/739) accepted after visual inspection. Shifts in four of the remaining 10 cubes are confirmed via the Hα image: the remaining six objects are all fainter than Ks = 22.3. The resulting shifts are applied to the cube headers, with a median shift of ∼1 KMOS pixel and ∼10% of cubes exceeding shifts of 2 pixels.

The partial combined frames are each registered to the HST imaging following a procedure that is described in detail by Wilman et al. (2019). In brief, model KMOS continuum images are generated by convolving HST images in the closest available band to the KMOS PSF, cropping to the KMOS field of view, and binning to KMOS 02 pixels. This is compared to a KMOS continuum image obtained by taking a weighted average of the data cube in wavelength (masking contamination by sky emission). The centroid is allowed to vary, and the best fit is determined through a nonlinear least-squares fitting algorithm. The resulting astrometric shifts are visually inspected. Corrections for relative shifts between partial combined frames are then applied to all of the individual exposures to generate the final total combined data cube. Of 355 objects with multiple setups, the median residual shift is ∼1.33 KMOS pixels (0267), with ∼27% of shifts above 2 KMOS pixels (04), ranging as high as 4.35 pixels (087). Not accounting for such shifts can artificially blur the combined cube by an average of ∼2 kpc and up to ∼7 kpc.

The final data cubes are astrometrically registered to the HST imaging by repeating the same procedure now using the fully combined cubes for each galaxy. This provides a visually good solution in every case where the source is clearly visible in continuum, which includes 98.6% (730/739) of the targeted sources. Shifts in four of the other 10 cubes are confirmed via the Hα image. The remaining six objects are all fainter than Ks = 22.3 mag. The resulting shifts are applied to the cube headers, with a median shift of ∼1 KMOS pixel and ∼10% of cubes exceeding shifts of 2 pixels.

For each science frame, the PSF stars are collapsed in the wavelength range used for the flux calibration to produce PSF images. The PSF images are fit with a Moffat profile to extract the centroid and the total flux of the star. Then each image is normalized to a total flux value equal to unity, allowing different stars to be combined, as the same PSF stars may not have been observed across the multiple pointings. Different stars are combined to produce a PSF image representative of the observing conditions for galaxies observed in multiple pointings. In this case, the stars for the final PSF image are preferentially selected to be from the same detector as the galaxy observations. Once all of the relevant individual PSF images are selected, they are shifted and combined using Swarp (Bertin et al. 2002) on a 21 × 21 pixel grid. This process resamples and coadds the input frames onto a final common grid. This process also generates a noise frame from the standard deviation of the input frames. We then obtain a PSF image that reflects the specific observing conditions for each object. The natural sampled images (02 pixel⁻¹) are refit with a Moffat and Gaussian function separately using custom IDL routines to characterize the PSF and photometric conditions, as shown in Figure 2. The units are normalized flux units such that the total flux in the PSF model is unity.

The spectral resolution of KMOS varies from IFU to IFU, as well as in both the wavelength direction and across spaxels (Davies et al. 2013). It is sensitive to the focus of the optical elements in the KMOS instrument and, as a result, can change after an instrument intervention (when KMOS is warmed up for maintenance). We account for all but the spatial variations by measuring the resolution as a function of wavelength for each galaxy data cube. Combined arc lamp cubes are assembled from the arc frames for each IFU. Sky cubes for each target are created from the same science frames as the science cubes prior to the sky subtraction and heliocentric correction being applied. Therefore, the combined sky and galaxy cubes were created from the same raw data. They are combined following the same procedure as the final science cubes and thus take into account changes in resolution for targets observed in multiple IFUs. Then we fit Gaussian profiles to the arc lines in each spaxel and average the spectral resolution values obtained for a given line. Finally, we fit a fourth-order polynomial to the spectral resolution values as a function of wavelength. However, the spectral resolution obtained with this method is likely to be inaccurate, since the science data have been passed through different steps of the reduction procedure (e.g., shifting of the wavelength solution to match the OH line spectrum). To overcome this issue, we fit ∼10 bright and isolated night-sky lines in the non-sky-subtracted cubes and adjust the zeroth-order term of the polynomial to fit these values' instrumental resolution while preserving the overall shape of the polynomial. The resolution at a given wavelength can then be recovered such that

$$\begin{eqnarray}\begin{array}{rcl}R & = & \mathrm{RES}\ \mathrm{COEFF}0+\mathrm{RES}\,\mathrm{COEFF}1\times {\lambda }_{\mathrm{obs}}\\ & & +\mathrm{RES}\ \mathrm{COEFF}2\times {\lambda }_{\mathrm{obs}}^{2}\\ & & +\mathrm{RES}\ \mathrm{COEFF}3\times {\lambda }_{\mathrm{obs}}^{3}\\ & & +\mathrm{RES}\ \mathrm{COEFF}4\times {\lambda }_{\mathrm{obs}}^{4},\end{array}\end{eqnarray} \tag{ 1 }$$

where λ_obs is the wavelength of the observed line, and RES COEFF1 to RES COEFF4 are the coefficients of the polynomial fit. There are a few cases where the polynomial solutions extend toward very low or high values at the edges of the spectrum. A minimum and maximum spectral resolution are given for these cases: RES MIN, RES MAX. The minimum and maximum spectral resolutions are derived separately in each band from the upper and lower 3σ limits of the resolution distribution (excluding data within ∼200 Å of the ends of the wavelength range).

The average effective spectral resolutions at the location of Hα detections in KMOS^3D  are 3515, 3975, and 3860 in the YJ, H, and K bands, respectively. The variation of spectral resolution across the wave band and IFU is shown in Figure 6. The measured spectral resolution obtained is close to the nominal KMOS resolution in each band, with the exception of the K band, which yielded a lower resolution by ΔR ∼ 200. In each wave band, there is variation between IFUs up to ΔR = 1000. We therefore stress the importance of using the wavelength- and IFU-dependent effective spectral resolution (provided with the data release) for scientific analysis, in particular to determine accurate velocity dispersions.

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We generated 100 bootstrap realizations of each final combined data cube by selecting random exposures to combine with replacement. The KMOS pipeline produces a noise cube corresponding to the rms of all pixels contributing to an x, y, λ position in the final combined cube. The bootstrap cubes complement this default noise cube and provide more realistic noise estimates, which are typically ∼2−3× larger.

In this section, we characterize the KMOS^3D  detection fractions on and off the MS and as a function of color. We adopt the separation between star-forming and passive galaxies in terms of rest-frame colors following the UVJ criteria of Williams et al. (2009) and in terms of SFR using a threshold of ΔMS = −0.85 dex. A high spectroscopic redshift success rate was a key factor in the design and, ultimately, success of the survey. Hα and [N ii] emission was searched for manually in each reduced cube around the 2D continuum center, using the spectroscopic or grism redshift as a prior. The global detection fraction of 79% splits between subsets is as follows. Among the 36% of targeted galaxies that had a previous spectroscopic redshift (from Mignoli et al. 2005; Vanzella et al. 2008; Popesso et al. 2009; Cooper et al. 2012; Kurk et al. 2013; Tadaki et al. 2013; Kriek et al. 2015; see also Skelton et al. 2014), 84% are detected in Hα. Among the other 64% with only a grism redshift at the time of observation, the detection fraction is 76%.

Figure 8 shows the ${z}_{\mathrm{best},\mathrm{orig}}$ redshift distribution of targeted galaxies, where z_best,orig is defined as the most accurate redshift available at the time of targeting from either grism or higher-resolution spectroscopy. We detect 245, 159, and 177 galaxies in the redshift ranges of $0.602\lt {z}_{\mathrm{KMOS}}\lt 1.039$, $1.275\,\lt {z}_{\mathrm{KMOS}}\lt 1.924$, and $1.996\lt {z}_{\mathrm{KMOS}}\lt 2.675$, respectively. We achieve a comparable detection fraction within each redshift slice of ∼79%, as shown in Table 1.

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Table 1.  Target Detection Fractions

	All	${\rm{\Delta }}\mathrm{SFR}\gt -0.85$	${\rm{\Delta }}\mathrm{SFR}\lt -0.85$
All	79% (581/739)	91% (541/592)	27% (40/147)
$0.602\lt z\lt 1.039$	77% (245/319)	90% (219/243)	34% (26/76)
$1.275\lt z\lt 1.924$	79% (159/201)	93% (148/160)	27% (11/41)
$1.996\lt z\lt 2.675$	81% (177/219)	92% (174/189)	10% (3/30)

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To compare z_KMOS and z_best,orig, we calculate the prior redshift accuracy for detected galaxies as

$$\begin{eqnarray}&&{\sigma }_{\mathrm{NMAD}}=1.48c\times \mathrm{median}\left|\displaystyle \frac{{\rm{\Delta }}z-\mathrm{median}({\rm{\Delta }}z)}{(1+{z}_{\mathrm{kmos}})}\right|,\end{eqnarray} \tag{ 2 }$$

where σ_NMAD is equal to the standard deviation for a Gaussian distribution following Brammer et al. (2008), Δz = z_best−z_kmos, and c is the speed of light. The overall 3D-HST redshift accuracy for the detected KMOS^3D  galaxies corresponds to a velocity offset of 463 km s⁻¹ from the previously known redshift. For galaxies selected on a prior spectroscopic redshift and detected with KMOS, σ_NMAD is 155 km s⁻¹. For galaxies selected on a grism redshift and detected with KMOS, ${\sigma }_{{\rm{NMAD}}}=1020$ km s⁻¹. Below the MS, where grism redshifts are more often constrained by the continuum rather than emission lines, the standard deviation is higher, with ${\sigma }_{{\rm{NMAD}}}=1546$ km s⁻¹. Possible nondetections due to larger redshift uncertainties would decrease the quoted accuracies, discussed further in Section 5.1.1. The grism redshift accuracy also decreases with increasing observed K-band magnitude, as demonstrated in Momcheva et al. (2016). Within the detected sample, 41 galaxies have redshifts from KMOS^3D  deviant from the expected redshift from 3D-HST by >10,000 km s⁻¹ (seven galaxies at z ∼ 1, 12 galaxies at z ∼ 1.5, and 22 galaxies at z ∼ 2).

There are 60 galaxies with a possible Hα detection, but the S/N is low or sky features put the validity of the detection in question. These galaxies are considered detected but are plotted separately as the blue histogram in Figure 8. They are found primarily in the lowest-redshift bin, in which a higher number of UVJ passive galaxies were observed. The remaining detected galaxies have a high-quality detection with detection flag ${z}_{{\rm{q}}}=0$.

Detection fractions are a strong function of MS offset and galaxy colors, as shown in Figure 9. Table 1 gives the detection fraction and number of galaxies observed above and below ${\rm{\Delta }}\mathrm{MS}=-0.85\,\mathrm{dex}$ in each redshift slice. The detection fraction is as high as 91% (541/592) when considering only galaxies near and above the MS at ${\rm{\Delta }}\mathrm{MS}\gt -0.85\,\mathrm{dex}$. It is fairly constant across redshifts, with 90%, 93%, and 92% detected in the redshift slices z ∼ 1, 1.5, and 2, respectively. In contrast, the detection fraction below the MS (${\rm{\Delta }}\mathrm{MS}\lt -0.85\,\mathrm{dex}$) is 34%, 27%, and 10%, respectively, in the same redshift ranges. At blue colors, ${(U-V)}_{\mathrm{rest}}\lt 1.3$, and on/above the canonical MS (ΔMS > −0.25), we detect >90% of all galaxies. We detect all galaxies at ΔMS > 0.6. The detection fractions fall rapidly when moving to redder (U − V)_rest colors or below the MS. Galaxies with no Hα detection below the MS do not correlate with magnitude or exposure time. They are typically small and have SFRs derived from SEDs (because they are undetected in the far-IR).

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Figure 10 shows the composite rest-frame SEDs of detected and undetected galaxies for each of the subsets above and below ΔMS = −0.85 dex. These SEDs were constructed from the optical–to–8 μm broad- and medium-band photometry, normalizing the individual SEDs at rest frame 5000 Å and computing the running median and inner 68% range of the distributions. For comparison, we also constructed composite SEDs in a similar manner for the 3D-HST parent population (at 0.7 < z < 2.7, $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 9$, F160W < 25.1 mag, and K < 23 mag), split in the same ΔMS bins. Among the star-forming subset, the median SED of the detected KMOS targets and the 68% range around it are nearly identical to those for the 3D-HST parent SFG population, while the undetected targets tend to be significantly redder. This difference likely reflects higher levels of dust extinction among undetected targets at ${\rm{\Delta }}\mathrm{MS}\gt -0.85\,\mathrm{dex}$, which would lead to fainter emergent Hα fluxes. In support of this explanation, we find that the median IR/UV ratio (as a measure of dust obscuration; see Section 5.3) of undetected targets is systematically higher by 3.5× than that of Hα-detected targets matched in M_⋆ (within ±0.2 dex), z (same band), and observing time (±1 hr) at any SFR level (UV+IR or SED SFRs). In contrast, there is very little difference in SEDs between detected and undetected targets at ΔMS < −0.85 dex, with both subsets having substantially redder SEDs than the overall galaxy population, as expected, and consistent with their broad- and medium-band SEDs being dominated by older stellar populations (e.g., Kriek et al. 2008; Fumagalli et al. 2016). The Hα+[N ii] spectral properties of the detected objects in the quiescent regime indicate that half of them may be undergoing low-level rejuvenation events, and the other half exhibits dominant signatures of gas outflows and shocks (Belli et al. 2017).

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5.1.1. Nondetections

5.1.2. Serendipitous Galaxies

5.1.3. Final Sample Distributions

Figure 11 compares the 1D distributions of KMOS^3D  sample galaxies in log(M_⋆), ΔMS, ${(U-V)}_{\mathrm{rest}}$, and ΔMR_e to those of the parent 3D-HST source catalog, subjected to the same cuts in redshift, mass, and magnitude applied to select our targets. Here ΔMR_e is defined as the logarithmic offset in R_e from the mass–size relation for the SFGs of van der Wel et al. (2014b; and shown in Figure 7) at the same M_⋆ and z as each target. In stellar mass, the full KMOS^3D  sample selected for observations has an excess above $\mathrm{log}({M}_{\star }/{M}_{\odot })\sim 10.5$ and a deficit below that mass relative to the parent 3D-HST population. This difference reflects our strategy of emphasizing a more homogeneous coverage in mass and redshift compared to the underlying galaxy population. The KMOS^3D  target distribution closely follows the parent population in ΔMS and ΔMR_e, as well as in ${(U-V)}_{\mathrm{rest}}$ colors, although with a slight deficit in the bluer half that is tied to the more uniform mass and redshift distribution of our selection. In turn, the distributions for the Hα-detected subset reflect the lower success rate in the redder, higher-mass regime well below the MS discussed above in Section 5.1. Unsurprisingly, the distributions for the serendipitously detected line-emitting sources show that they are all SFGs with SFRs within a factor of ∼10 of the MS and half-light radii within a factor of ∼3 of the mass–size relations. They have a broad mass distribution covering the same range as the primary targets and are typically in the bluer part of the ${(U-V)}_{\mathrm{rest}}$ range. Similar trends as just discussed are seen when considering the distributions in the three redshift slices corresponding to the objects observed in the YJ, H, and K bands.

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A total galaxy spectrum is extracted for each galaxy. The spectrum is extracted from the data cube within a $1\buildrel{\prime\prime}\over{.} 5$ radius aperture centered on the continuum center. Serendipitous galaxies, as discussed in the previous section, are masked when extracting an aperture spectrum.

Prior to extracting a galaxy spectrum, the continuum is subtracted from each pixel of the data cube. The continuum in the KMOS cubes is a combination of real galaxy continuum and possible residual background and is not well captured by a simple polynomial fit. To more robustly estimate the shape of the continuum, we mask channels within 1000 km s⁻¹ (2200 km s⁻¹ for seven galaxies with particularly broad emission lines) of strong emission lines (O i λ6300, [N ii] λ6548, Hα, [N ii] λ6584, [S ii] λ6716, or [S ii] λ6731), calculate the moving median across each spectrum in 30 pixel wide windows, and then perform linear interpolation across the line channels. After the continuum is subtracted, spikes (defined as channels more than 1000 km s⁻¹ from strong emission lines with values exceeding twice the rms or three times the median value of all nonline channels in a given pixel spectrum) are masked. The galaxy spectrum is then extracted, and the continuum subtraction is performed again to remove any residual continuum.

We fit the combined Hα, [N ii] λλ6548,6563 emission complex with Gaussians for all aperture spectra. For the multiline fit, the positions of the [N ii] lines are tied to the Hα position, and the widths of the lines are equal. In most cases, having constraints from the [N ii] emission improves the fit to the Hα emission, particularly when atmospheric contamination is present near the emission lines. In contrast, when the detection of [N ii] is weak or contaminated, a single Gaussian component can provide a better fit to the data. In these cases, the resulting Hα fluxes from single and multiple Gaussian fits are compared. The spectral fits for galaxies with large differences between measurements are visually inspected, and the χ² of the two fits are considered. The flux from the better fit is adopted. For the majority of galaxies, the Hα flux measurement from the joint Hα+[N ii] fit is adopted. The resulting Hα and [N ii] fluxes and a flag indicating which fitting method was used are given in Table 6. Errors on the line fluxes are calculated by repeating the spectrum extraction and line fitting for each of the bootstrap cubes and taking the standard deviation across all fit values for each of the parameters.

The choice of a single aperture for all galaxies despite size differences provides a simple, robust, and repeatable measurement to characterize the full sample. We note that, while optimal for the majority of the sample, for many sub-MS galaxy detections, the relatively large 15 radius aperture reduces the S/N of detections and is not optimal for the study of compact galaxies near our detection limits. However, when the aperture size is reduced to 12 radius, for example, the median Hα flux decreases by 23% for the z ∼ 1 sample, 16% for the z ∼ 1.5 sample, and 15% for the z ∼ 2 sample. Thus, we adopt the 15 radius aperture. Given the continuum half-light sizes of massive galaxies, especially at z ∼ 1; the effects of beam smearing; and the size of the KMOS IFUs, it is likely that a fraction of the total Hα flux is outside of the presented IFU observations. Wilman et al. (2019) present Hα fluxes, addressing these issues, derived from exponential Hα profile fitting for a subset of the KMOS^3D  sample. By comparing our aperture flux measurements to total fluxes derived from exponential profile fitting for this subset, we find that the estimated flux losses outside the aperture show the expected dependence on the ratio 15/r_e, such that, for instance, ∼50% flux is lost where r_e = 15. For the purpose of discussing Hα SFRs in the next section, we estimated simple aperture corrections for all detected galaxies based on the results from the more detailed work by Wilman et al. (2019) and using the structural parameters fitted on HST F160W images (van der Wel et al. 2014b). We created mock 2D Hα exponential profiles with the same axis ratio and a half-light radius equal to 1.19 × r_e(F160W), convolved with the associated KMOS PSF (Section 4.7). The aperture correction was then taken as the ratio between the total flux and that within r = 15 of the model center.

For a small fraction of galaxies, we find that a single spectral component fit to the Hα emission provides an inadequate description of the observed line profiles. In most of these cases, there is excess emission in high-velocity wings that are associated with strong gas outflows (Genzel et al. 2014; Förster Schreiber et al. 2019). For these objects, two Gaussian components are used to fit the emission lines, with the narrower of the two emission lines assumed to be associated with star formation. The emission from the narrow component is the one used in the analysis in the next sections. More detail on the fitting and interpretation of the broad emission component is given by Förster Schreiber et al. (2014, 2019) and Genzel et al. (2014).

If Hα is not detected, we calculate an upper limit using a fixed line width corresponding to 120 km s⁻¹ by summing the errors on the Hα channels (defined as channels separated from the Hα line center by less than the FWHM of the Hα line) in quadrature and then multiplying by 3 to obtain the 3σ upper limit. For galaxies with a prior spectroscopic redshift, the upper limit is measured assuming Hα is centered at the known redshift. For nondetections with only a grism redshift, a weighted average upper limit is estimated using the 3D-HST redshift probability distribution.

We detect [N ii] λ6563 at S/N > 3 in 70% of galaxies. The detection fraction of [N ii] is a strong function of stellar mass, as expected from the mass–metallicity relation (e.g., Lequeux et al. 1979; Tremonti et al. 2004; Erb et al. 2006; Wuyts et al. 2016a). In the mass ranges $9.5\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 10.5$, $10.0\,\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 11.0$, and $10.5\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 11.5$, the detection fraction of [N ii] is 66%, 74%, and 80%. If the resulting [N ii] λ6563 flux is zero, we estimate an upper limit following the same procedure described above but assuming a line width fixed to the Hα line width.

In Figure 12, we show a stacked spectrum of all galaxies with a secure Hα detection. In the stack, we detect the [S ii] λλ6716,6731 doublet at S/N = 20 and [O i] λ6302 at S/N = 8. These additional lines are detected in a subsample of KMOS^3D galaxies. The interpretation of the [N ii]/Hα ratios is discussed further in Wuyts et al. (2014, 2016a), and that of the [S ii] ratios is discussed in Förster Schreiber et al. (2019).

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From the flux measurements described above, Hα-based SFRs, SFR_Hα, are calculated using Kennicutt (1998) adjusted to a Chabrier (2003) IMF such that

$$\begin{eqnarray}&&{\mathrm{SFR}}_{{\rm{H}}\alpha }=4.65\times {10}^{-42}{L}_{{\rm{H}}\alpha }{10}^{-0.4{A}_{\mathrm{extra}}}{10}^{-0.4{A}_{\mathrm{cont}}},\end{eqnarray} \tag{ 3 }$$

where ${A}_{\mathrm{extra}}=(0.9{A}_{\mathrm{cont}}-0.15{A}_{\mathrm{cont}}^{2})$ and ${A}_{\mathrm{cont}}=0.82{A}_{{\rm{v}},\mathrm{SED}}$ following Wuyts et al. (2013). A factor of 1.7 is used to convert from a Salpeter (1955) to Chabrier (2003) IMF. Here A_cont is the attenuation of the continuum light at the wavelength of Hα following Calzetti et al. (2000) and ${A}_{{\rm{v}},\mathrm{SED}}$ is the attenuation at the V band derived from the SED fitting described in Section 2. The derived aperture- and dust-corrected Hα SFRs range between 0.2 and 319 M_⊙ yr⁻¹, with an average SFR_Hα = 37 M_⊙ yr⁻¹, not including upper limits.

Figure 13 shows the comparison of SFR_Hα with the UV+IR or SED SFRs (hereafter SFR_phot) described in Section 2 as a function of ΔMS, A_V, and IR/UV, the IR-to-UV flux ratio for IR-detected sources. Each SFR indicator, ${\mathrm{SFR}}_{\mathrm{UV}}+160\,\mu$m, ${\mathrm{SFR}}_{\mathrm{UV}}+100\,\mu$m, ${\mathrm{SFR}}_{\mathrm{UV}}+24\,\mu$m, and SFR_SED, is shown by a different color. In general, we find good agreement between Hα and UV+IR SFR indicators for the majority of galaxies on the MS. Below the MS, where SED-based SFRs dominate (due to detection limits in the IR), SFR_SED is ∼0.5 dex lower than the derived SFR_Hα, shown in the left panel. This is discussed further in Belli et al. (2017) and may result from the assumptions in the SED models (e.g., SFHs). In contrast, above the MS, where UV+IR SFRs dominate (in particular ${\mathrm{SFR}}_{\mathrm{UV}}+160\,\mu$m and ${\mathrm{SFR}}_{\mathrm{UV}}+100\,\mu$m), SFR_phot is typically greater than the SFR_Hα.

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One possible explanation for the galaxies with ${\mathrm{SFR}}_{\mathrm{phot}}\,\gg$SFR_Hα is that the dust correction is underestimated at higher masses and/or in "starbursts" above the MS. In the middle and right panels of Figure 13, we look at the ratio of SFR indicators as a function of dust properties A_V and IR/UV. Galaxies with high SFR_phot relative to SFR_Hα have high IR/UV ratios, indicating more dust-attenuated star formation. This is most common among galaxies with far-IR SFR indicators, ${\mathrm{SFR}}_{\mathrm{UV}}+160\,\mu {\rm{m}}$ and ${\mathrm{SFR}}_{\mathrm{UV}}+100\,\mu$m. The same subset of galaxies has relatively low A_V < 1, suggesting that the A_V derived from the SED fitting does not capture the global dust attenuation. For these highly star-forming galaxies, reddening as a tracer of extinction may saturate regardless of dust geometry. In these cases, the extinction may be underestimated toward both the stellar and nebular regions (e.g., Wuyts et al. 2011a).

The Hα emission is fit in every spatial pixel in each galaxy using the IDL emission line fitting code LINEFIT (Davies et al. 2011; see further descriptions in Förster Schreiber et al. 2009, 2018; W15). In short, the code fits an intrinsic Gaussian convolved with a line profile representing the spectral resolution, thereby implicitly taking into account instrumental broadening. The uncertainties of the fit are determined by 100 Monte Carlo simulations, where the spectrum is perturbed according to a Gaussian distribution from the associated noise spectrum.

The peak, position, and width of the fitted Gaussians are used to create the Hα flux, velocity, and velocity dispersion maps, respectively. A mask is created automatically including pixels with S/N > 2 and velocities and velocity dispersions with errors <100 km s⁻¹. Single spaxels detached from the central source that are erroneously included due to fitted sky lines or noise spikes are removed. Visual inspection of the Gaussian fits in spaxels around the edges of each galaxy is performed to include or remove spaxels as appropriate. For example, in the low-S/N regime, velocity dispersions can be artificially inflated due to the surrounding noise. A simple S/N cut cannot unequivocally determine the robustness of a fit, especially when the emission line falls on or near a sky residual. Example spatially resolved maps are shown in Figure 14.

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We define a kinematic axis by identifying the highest and lowest 5% of spaxels in the velocity map for larger Hα detections (≥50 high-S/N spaxels) or the highest and lowest 5% of spaxels for smaller galaxies (<50 high-S/N spaxels). The positive and negative nodes of the velocity map are determined by taking the weighted average of the selected spaxels. The kinematic axis is defined as the angle between the north–south vertical axis and the line created by the positive and negative nodes. The kinematic center is defined as the halfway position between the two nodes. All kinematic axes and centroids are then inspected by team members to assess the success of this method. The kinematic axes and centroids are well defined by this process. However, for a small number of galaxies, real (e.g., differential dust extinction) or data-driven (e.g., sky-line contamination) artifacts can bias the automated method. In these cases, the kinematic axes and centroids are adjusted by hand.

Along the kinematic axis, a velocity and velocity dispersion profile are extracted within apertures with diameters equivalent to the FWHM of the PSF. Emission line fitting for each aperture spectrum is made with LINEFIT, as described in Section 6.1 with associated errors from a Monte Carlo analysis. Each spectral fit along the kinematic axis is inspected. Resulting fits are included based on satisfying the following criteria: S/N > 2, the difference between two successive velocity points is less than 150 km s⁻¹, the error on the velocity is $\delta {V}_{{xy}}[\mathrm{km}\,{{\rm{s}}}^{-1}]\lt 25$, and the error on the velocity dispersion is $\delta {\sigma }_{{xy}}[\mathrm{km}\,{{\rm{s}}}^{-1}]\lt 100$, where δV_xy and δσ_xy are the error on the velocity offset and dispersion from an aperture spectrum at point x, y in the galaxy. The maximum radius of kinematic extraction, r_kin, is defined as the largest distance from the kinematic center that an emission line can be fit satisfying these criteria. Kinematic extractions are shown in Figure 14 for a few example galaxies. More 1D kinematic extractions from the z ∼ 1 and 2 data sets are given in the Appendix of W15 and Figure 3 of Wuyts et al. (2016b).

We resolve Hα emission at and beyond three resolution elements, ${r}_{\mathrm{kin}}\gt 3{r}_{\mathrm{PSF}}$, in 81% of the detected galaxies. Figure 15 shows the ratio of the maximum radius of kinematic extraction to the model Moffat PSF, ${r}_{\mathrm{kin}}/{r}_{\mathrm{PSF}}$, where ${r}_{\mathrm{PSF}}=\mathrm{FWHM}/2$. As seen in Figure 15, the resolved fraction is comparable in each of the observing bands, YJ:79%, H:82%, K:83%, as is expected given the increased observing time enabling detection of extended low surface brightness out to the highest-redshift bin. The marginally lower fraction of resolved galaxies at z ∼ 1 is due to the larger number of below-MS sources. Of the resolved galaxies, 50% have >4.25 resolution elements across the galaxies. We measure out to ∼2r_e in 60% of detected galaxies and ∼3r_e in 30% of detected galaxies, as shown in the right panel of Figure 15, beyond where a change in slope, or flattening, of the velocity curve is expected for a self-gravitating exponential disk.

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For resolved galaxies, we measure an observed velocity, v_obs, and velocity dispersion, σ₀. The observed velocity is calculated as the average of the absolute value of the minimum and maximum velocity measured along the kinematic axis. Inclination-corrected rotational velocity is defined as v_rot = v_obs/sini, where i is the inclination measured from F160W HST axis ratios assuming an intrinsic disk thickness, q₀ = 0.25. The measured velocity dispersion is calculated by taking the weighted mean of all data points from the 1D velocity dispersion profile at $\gt 0.75\times | {r}_{\mathrm{kin}}|$. This methodology is designed to measure the line-of-sight velocity dispersion in disk galaxies where beam smearing from large velocity gradients does not inflate the dispersion by spreading galaxy-wide rotational motions across multiple wavelength channels. We apply this method to all galaxies regardless of kinematic type. We stress that since one of our goals is to quantify the disk fraction among the full sample, the approach followed here is appropriate, as it can be applied to nondisk systems as well. All resulting v_obs and σ₀ measurements from these automated methods are checked against the corresponding kinematic maps and 1D profiles. In a small number of cases, adjustments are made when the automated method fails to capture the dispersion in the outer regions or is biased by an individual data point. The derived v_obs, σ₀, and associated errors are represented by the horizontal lines and gray bands, respectively, in the 1D kinematic profiles (last column) for the example galaxies shown in Figure 14.

In rotating galaxies where a flattening of velocity is not detected (i.e., the velocity curve is a simple linear gradient), the effects of beam smearing are large, inflating the measured σ₀ and reducing the measured v_obs. Even galaxies mapped with the most resolution elements may be mildly affected by beam smearing (Davies et al. 2011). In Appendix A.2.4 of Burkert et al. (2016), we derived corrections for beam smearing based on the intrinsic size of the galaxy, stellar mass, inclination, and observed PSF size. This approach has the advantages that it is easy to apply for all galaxies consistently (regardless of S/N), not time-intensive, and based on galaxy models. On the other hand, it is based on relatively simple models and the interpolation between a fixed set of galaxy parameters. The alternative is time-intensive 2D kinematic models simultaneously taking into account the measured velocities and dispersions (e.g., Bouché et al. 2015; Di Teodoro & Fraternali 2015; Übler et al. 2018) that may only converge for a subset of the highest-S/N data. For the substantial subset of disk galaxies that are sufficiently well resolved and have sufficiently high S/N to be modeled, the simpler approach described here yields v_rot (and σ₀) measurements that agree with the results of full forward modeling (e.g., Wuyts et al. 2016b; Übler et al. 2019) within 8 km s⁻¹ for velocities (and 10 km s⁻¹ for dispersions), on average. For the present paper, we adopt model-independent measured values corrected for beam smearing using the methods presented in Burkert et al. (2016).

The beam-smearing corrections are valid only for galaxies that are well described by a disk model. As shown in W15 and Section 7 the majority of resolved galaxies in KMOS^3D  fulfill this criteria. The median-velocity beam-smearing correction factor for the rotation-dominated galaxies identified in Section 7 is 1.36, with a range from 1.08 to 1.97. However, inferred intrinsic Hα sizes can be 1×−4× greater than the H-band sizes used to derive the corrections (Nelson et al. 2013; Wisnioski et al. 2018; Wilman et al. 2019). In these cases, the beam-smearing corrections may be overestimated when using the H-band size, as has been done for this work. In Figure 14, the intrinsic non-beam-smeared rotation curve assuming the exponential disk scale length is equal to ${r}_{{\rm{e}}}[{\rm{F}}160{\rm{W}}]/1.68$, as shown for the rotation-dominated galaxies by the dashed line. The gray band surrounding the dashed line reflects the errors on the observed velocity, inclination correction, and beam-smearing corrections.

Errors on the beam-smearing corrections are estimated from Monte Carlo simulations of the galaxy parameters that enter into the beam-smearing calculations. For the velocity beam smearing, only r_e is varied, as the correction depends very little on other parameters. The resulting 16th and 84th percentile errors on the velocity beam-smearing correction are small, typically a few percent. Beam-smearing corrections to σ₀ are dependent on M_*, i, and r_e, as detailed in Appendix A.2.4 of Burkert et al. (2016). Multiplicative corrections range from 0.13 to 0.98 for the full sample, with a median of 0.53. The 16th and 84th percentile errors on the dispersion beam-smearing correction are larger, typically 25%.

In W15, we presented fractions of "rotation-dominated" and "disklike galaxies" of 83% and 71%, respectively, in our combined z ∼ 1 and 2 samples using five morpho-kinematic criteria. The high incidence of rotationally dominated kinematics in SFGs was more than ∼2× what had been previously reported (e.g., Förster Schreiber et al. 2009; Law et al. 2009). In the largest deep, high-resolution AO-assisted IFU survey at $1.5\lesssim z\lesssim 2.5$, including 35 galaxies probing massive MS SFGs, as many as ∼70% of the objects are kinematically classified as rotation-dominated disks (Förster Schreiber et al. 2018). It has become increasingly clear from the literature that large samples and high-S/N data are crucial to accurately characterize disk fractions at any redshift. Recently, it has been proposed that the rotation-dominated fraction among SFGs is monotonically increasing over cosmic time, with fractions as low as 35% at z ∼ 3.5 (Kassin et al. 2012; Stott et al. 2016; Simons et al. 2017; Turner et al. 2017). Although there was a hint of evolution between the z ∼ 2 and 1 galaxy samples with our first year of data, at that time it was not clear whether this was a result of the evolving galaxy sizes and morphologies, making it increasingly difficult to resolve higher-redshift samples and fairly apply the same criteria.

Here we present disk fractions for galaxies on the MS, $-0.85\lt {\rm{\Delta }}\mathrm{MS}\lt 0.85$, across three redshift slices for the full KMOS^3D  sample at a greater depth and larger stellar mass range than our previous results. Following W15, we use the same five disk criteria outlined below, with minor adjustments. However, we note the known caveats relevant to this selection as outlined in Förster Schreiber et al. (2018) and discuss the validity of this kinematic selection in Section 7.2.

The criteria applied are as follows.

• (1)  
  The Hα velocity map exhibits a continuous velocity gradient along a single axis. In larger systems, this is synonymous with the detection of a "spider" diagram (van der Kruit & Allen 1978; third column in Figure 14).
• (2)  
  Here v_rot/${\sigma }_{0}\gt \sqrt{3.36}$, where v_rot is the rotational velocity corrected for inclination, i, by v_rot = v_obs/sini, and both kinematic parameters are corrected for beam smearing (Förster Schreiber et al. 2018).
• (3)  
  The position of the steepest velocity gradient, as defined by the midpoint between the velocity extrema along the kinematic axis, is coincident within the uncertainties (∼1.6 pixels) with the peak of the velocity dispersion map.
• (4)  
  For inclined galaxies (q < 0.6), the photometric and kinematic axes are in agreement (<30°).
• (5)  
  The position of the steepest velocity gradient is coincident within the uncertainties with the centroid of the KMOS continuum center (a proxy for the center of the potential; i.e., in the higher-mass galaxies, this is usually a bulge).

For the first criterion, careful visual inspection is performed on the 2D maps and 3D cubes, as well as from the extracted major axis profiles, and compared to simple disk models (e.g., as plotted in Figure 14). To satisfy this criterion, the galaxy must exhibit a unique kinematic axis and be monotonically increasing/decreasing, as expected for disk velocity curves.

For a subset of observations, discussed in Section 5.1.2, multiple objects are detected within a single IFU. This includes cases where the multiple galaxies detected and segmented in 3D-HST and the CANDELS catalogs are within ∼500 km s⁻¹ and likely merging. If the galaxies in these cases are clearly spatially segmented, the kinematic parameters discussed in the previous section and used for the disk criteria are derived for the primary galaxy, while the serendipitous galaxy is masked. Some examples include GS4_29773, COS4_21030, and GS4_19676. When there is a smaller physical separation, e.g., <5 kpc, between the bright knots in the imaging, it is ambiguous whether the imaging reflects multiple galaxies in a merger or multiple clumps of star formation within a galaxy (e.g., COS4_19648, GS4_16776, U4_25808, U4_36568), and the image segmentation is variable. In edge-on systems, it can be especially difficult to identify clumps within a galaxy, in comparison to face-on systems, as shown in Figure 16. In these cases, the KMOS data can help disentangle the nature of the system. Similar conclusions are reached using simulations of galaxies and the kinemetry analysis of Shapiro et al. (2008).

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We first update the fraction of rotation-dominated systems (criteria 1 and 2) for the most massive galaxies, $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 10$, for comparison with W15, as shown in Table 2. We find excellent agreement with the results from our first year of data, presented in Table 1 of W15. With the larger sample presented here, we find that the main evolution between z ∼ 2.3 and 0.9 is a result of the evolving v_rot/σ₀, or criterion 2, primarily driven by the evolution of σ₀ (Übler et al. 2019).

Table 2.  Percentage of Galaxies Satisfying Disk Criteria

Criteria	1, 2	1, 2, 3	1, 2, 3, 4	1, 2, 3, 4, 5
$10.0\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 11.75$
Full sample	79%	65%	64%	59%
$z\sim 1.0$	91%	73%	70%	66%
z ∼ 1.5	79%	68%	68%	65%
z ∼ 2.0	70%	56%	56%	49%

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With the complete KMOS^3D  survey, we have a large enough sample to investigate rotation-dominated (criteria 1 and 2) and disk (criteria 1–5) fractions as a function of stellar mass, as shown in Table 3 and Figure 17. For the two highest-redshift bins, we split the sample into two mass bins, $9.5\,\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 10.5$ and $10.5\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 11.5$. For the lowest-redshift bin, we are able to include an additional low-mass bin of $9.0\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 9.5$. The percentage of galaxies satisfying each criterion is given in Table 3. The fraction of rotation-dominated galaxies (criteria 1 and 2) depends on both mass and redshift, with galaxies in the mass bin $9.5\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 10.5$ and lowest-redshift bin, z ∼ 1, having the highest fraction of rotation-dominated galaxies, 93%. The largest evolution is seen in the $9.5\,\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 10.5$ mass range, evolving from 58% at z ∼ 2 to 93% at z ∼ 1. In contrast, the highest-mass bin shows a significantly shallower evolution from 82% to 89%, respectively.

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Table 3.  Percentage of Galaxies Satisfying Disk Criteria

Criteria	1, 2	1, 2, 3	1, 2, 3, 4	1, 2, 3, 4, 5
$9.0\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 11.75$
Full sample	77%	61%	60%	55%
z ∼ 1.0	87%	67%	65%	61%
z ∼ 1.5	72%	57%	57%	54%
z ∼ 2.0	69%	56%	55%	48%
$10.5\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 11.75$
Full sample	85%	72%	71%	66%
z ∼ 1.0	88%	75%	74%	70%
z ∼ 1.5	86%	75%	75%	72%
z ∼ 2.0	81%	66%	66%	58%
$9.5\lt \mathrm{log}({M}_{\star }/{M}_{\odot })\lt 10.5$
Full sample	73%	54%	52%	49%
z ∼ 1.0	91%	66%	62%	60%
z ∼ 1.5	66%	49%	49%	46%
z ∼ 2.0	58%	46%	45%	39%
9.0 < log(M⋆/M⊙) < 9.5
Full sample	...	...	...	...
z ∼ 1.0	63%	47%	47%	42%
z ∼ 1.5	...	...	...	...
z ∼ 2.0	...	...	...	...

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In Figure 17, we show the dependence of the fraction of rotation-dominated galaxies on stellar mass and cosmic time. To facilitate comparison to literature results, we adopt a v/σ > 1 threshold to define the rotation-dominated fraction of galaxies, rather than a threshold of $\sqrt{(}3.36)$, as adopted in criterion 2. The galaxy samples are split into overlapping stellar mass bins of 1.0 dex. Our results are in general agreement with Simons et al. (2017) and Stott et al. (2016). With the KMOS^3D  survey, we are able to add an additional high-mass bin with $\mathrm{log}({M}_{\star }/{M}_{\odot })=10.5\mbox{--}11.5$ that shows higher fractions of $v/\sigma \gt 1$ than at $\mathrm{log}({M}_{\star }/{M}_{\odot })=10.0\mbox{--}11.0$ in the z ∼ 1.5 and 2.0 samples. In contrast, at z ∼ 1.0, the rotation-dominated fraction as a function of mass from the KMOS^3D  survey is higher than the literature data. The apparent disagreement between surveys at z ∼ 1 in the mass bin $\mathrm{log}({M}_{\star }/{M}_{\odot })\,=9.0\mbox{--}10.0$ may be due to the large bin size and distribution of masses in each bin. When a smaller bin size of 0.5 dex is used, we have a large enough sample to explore the KMOS^3D  data at z ∼ 1 in the range $\mathrm{log}({M}_{\star }/{M}_{\odot })=9\mbox{--}9.5$, shown by the dark blue point. This subset of the low-mass data is in better agreement with the literature for low-mass galaxies.

In addition to the caveats already discussed, we also note additional uncertainties in the comparison between samples. First, the depth of the KMOS^3D  data may help identify more galaxies with v/σ > 1, as the Hα emission is probed to larger radii. For example, at z ∼ 1, the KMOS^3D  data are typically twice the depth of the KROSS data. Methods to extract rotational velocity, velocity dispersion, and beam-smearing corrections also differ across publications. Finally, slit-based analyses can lead to higher velocity dispersions than IFU-based analyses due to slit misalignment (for discussion, see Price et al. 2016).

The right panel of Figure 17 compares the kinematic ratio, ${v}_{\mathrm{rot}}/{\sigma }_{0}$, measured for SFGs in the KMOS^3D  sample and simulated SFGs in the Illustris-TNG50 sample (Pillepich et al. 2019). While the kinematics of the simulations are extracted with comparable methods as the KMOS data, the effects of noise, beam smearing, and inclination are not included in the Illustris simulation data. Each data point shows the median value of ${v}_{\mathrm{rot}}/{\sigma }_{0}$ from KMOS^3D  at a given redshift and mass; however, we note that a wide range in values is present in each bin. This variation is reflected in the error bars, which represent the interquartile range. Interestingly, the data and simulated galaxies show qualitative agreement in both trends with mass and cosmic time despite the idealized extraction of kinematic values from the simulations. We note that due to both resolution limits of the data and simulations and completeness limits with the target selection, the comparison of results below 10¹⁰ M_⊙ is more uncertain.

Ordered rotation yields clear observational signatures that lead to the criteria defined in this section (see also W15; Förster Schreiber et al. 2018). A small fraction of galaxies dominated by rotation, as captured by our disk fractions, may be in some stage of an instability or interaction sequence, as it is not always a binary distinction (e.g., Rodrigues et al. 2017). In contrast, some accretion events, internal processes, and interactions are capable of perturbing rotational motions, producing a variety of kinematic signatures ranging from subtle to extreme (dependent on, e.g., timescale, halo mass, orientation, and gas fraction; Naab & Burkert 2003; Robertson & Bullock 2008; Shapiro et al. 2009; Font et al. 2017). Here we report high rotation fractions providing constraints on the duty cycle of such processes for the gas-rich SFGs. The high fraction of SFGs on/near the MS characterized by rotational motions, along with the typically disklike distributions of Hα and stellar light and mass (Wuyts et al. 2011b; Nelson et al. 2013, 2016; Lang et al. 2014; van der Wel et al. 2014a), point to major mergers playing a minor role in setting the galactic structure observed at z ∼ 1−3 or disks being largely preserved or regrown on short timescales, as indicated by numerical simulations of gas-rich systems (e.g., Robertson & Bullock 2008; Hopkins et al. 2009; Font et al. 2017; Sparre & Springel 2017; Martin et al. 2018). Taking advantage of the large KMOS^3D  sample size and extensive characterization of galaxies in the 3D-HST/CANDELS fields, we further explore the role of interactions at various stages of the kinematics, as well as metallicity and star formation, through a statistical analysis of neighboring galaxies (J. T. Mendel et al. 2019, in preparation).

In this and previous works, the alignment of the kinematic and photometric axis is used to assess whether a galaxy is perturbed from a recent or ongoing interaction (Flores et al. 2006; Epinat et al. 2012; Wisnioski et al. 2015; Rodrigues et al. 2017; Förster Schreiber et al. 2018), under the assumption that interactions can cause changes in the distribution of angular momentum (van de Voort et al. 2015). We test whether it is possible to recover these differences using our observables (Hα kinematics representing the gaseous component and i- or H-band imaging representing the stellar component) with the recently published environment catalog (Fossati et al. 2017), utilizing high-quality grism redshifts from 3D-HST and all available spectroscopic redshifts in COSMOS, GOODS-S, UDS, GOODS-N, and AEGIS. Position angles from both the i and H band are used to assess misalignment, as they may be sensitive to different mass components of the galaxy (e.g., Rodrigues et al. 2017; Förster Schreiber et al. 2018).

In Figure 18, we show the distribution of photometric to kinematic axis misalignment, Ψ, where ${\rm{\Psi }}=\arcsin (| \sin ({\mathrm{PA}}_{\mathrm{phot}}-{\mathrm{PA}}_{\mathrm{kin}})| )$. For this analysis, only resolved galaxies with i-band axis ratios (A. van der Wel 2019, private communication), $0.2\lt {q}_{{\rm{F}}814{\rm{W}}}\lt 0.8$, are included, as the ability to measure accurate photometric position angles is reduced at high axis ratios when galaxies appear mostly spherically symmetric or face-on, and the ability to recover axis misalignments in edge-on systems is reduced due to projection effects. A quarter of galaxies, 24%, have a kinematic axis misaligned from the i- or H-band photometric axes by >30°. The majority of galaxies with large axis misalignments have axis ratios >0.5, where determination of the photometric position angle becomes more uncertain (W15).

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In Figure 18, the KMOS^3D  sample is separated by the relative overdensity of galaxies in an aperture of 0.75 Mpc radius, δ_0.75 (Fossati et al. 2017). Galaxies in a more overdense environment (δ_0.75 ≥ 3) are shown in red (N = 43), compared to galaxies in a less overdense environment (δ_0.75 < 3) shown in blue (N = 214). The possible signal of misalignments on order ∼10°−20°, shown in Figure 18 to more commonly be found in dense environments, is comparable to the errors on the photometric and kinematic position angle. A two-sided K-S test using the i-band position angles gives a 10% probability of the two populations being drawn from the same distribution. In contrast, using the H-band position angles, there is a 60% chance of the populations being drawn from the same distribution. While mitigated by having a cut on the axis ratio, issues remain in determining photometric parameters such as position angle and axis ratio in face-on galaxies and galaxies with strongly light-weighted features, such as clumpy star formation, spiral arms, bulges, etc. Furthermore, due to the typically high stellar masses of the resolved sample, KMOS^3D  galaxies are also more likely to be classified as "central" galaxies in massive halos, as defined by the Fossati et al. (2017) environment catalog. This test would therefore miss any signatures of misalignment present in "satellite" galaxies. A more robust measurement of kinematic misalignment would result from comparing stellar and gas kinematics (Barrera-Ballesteros et al. 2014, 2015). Furthermore, SFGs at this epoch have high gas fractions (e.g., Daddi et al. 2010; Tacconi et al. 2010, 2013), making gas and star misalignments either unlikely or very short-lived (van de Voort et al. 2015).