THE DISKMASS SURVEY. IV. THE DARK-MATTER-DOMINATED GALAXY UGC 463

Given the basic image reduction and photometric calibration provided by SDSS and 2MASS, our surface photometry is primarily concerned with sky-background subtraction and masking sources other than UGC 463.

Source catalogs have been created for each band using Source Extractor.⁹ Each catalog has been visually inspected and pruned of erroneous source identifications, such as along meteor streaks or diffraction spikes; these features are masked from our final results by including pseudo sources in our catalog. We have created a master catalog for the region surrounding UGC 463 by merging the catalogs from all bands, identifying sources detected in multiple bands.

Using the Image Reduction and Analysis Facility (IRAF)¹⁰ task imsurfit, we determine the sky background of each image by fitting a Legendre polynomial surface (with cross terms) to each image where all sources and artifacts are replaced, initially, by a (±3) sigma-clipped mean of the image. After the lowest-order surface fit (2 × 2 order), masked regions are replaced by the fitted surface values as the fit order is increased. By inspection, we find there is little improvement in the sky flatness when using surfaces of more than 9 × 9 order (terms up to x⁸), and higher order fits begin to introduce artificial structure. The backgrounds of SDSS images are generally well behaved, whereas the 2MASS H- and K-band data exhibit significant background structure.

Surface photometry has been performed on each image after subtracting the sky background and masking all artifacts and sources, except for UGC 463. Source masking is forced to be identical in every band. From these masked images, we perform elliptical aperture photometry over a range of radii, each with an aspect ratio and orientation coinciding with the derived geometry of the disk discussed in Section 3.1. Given the shallow depth of the 2MASS data, we have also produced a JHK surface brightness profile using the unweighted sum of the J, H, and K surface brightness profiles. This extends the NIR surface brightness profile to larger radii. The "JHK bandpass" has an effective band width of δλ/λ = 0.62 and a Vega zero point of 1062 Jy.

Figure 2 provides all the surface photometry used in our present study of UGC 463. We apply Galactic extinction corrections¹¹ of A_g = 0.357, A_r = 0.242, A_i = 0.173, and A_K = 0.034; no extinction correction is applied to μ_JHK (see Section 2.2.2). We also plot the UGC 463 K-band photometry from de Jong & van der Kruit (1994), which is in very good agreement with our 2MASS photometry at R < 40''. Table 2 provides the result of fitting an exponential disk to all bands, including the JHK data, demonstrating marginal change in the best-fitting scale length and no evident trend with wavelength. The table provides the radii over which the exponential disk is fit; the minimum radius is always 15'' in order to avoid non-exponential features seen near the galaxy center.

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We apply two corrections to μ_JHK and use it as the primary NIR surface brightness measurement for our study of UGC 463: (1) a color correction to produce a more accurate K-band surface brightness profile at large radius and (2) an instrumental-smoothing correction. The color correction compares μ_K and μ_JHK from Figure 2, such that it includes the Galactic extinction correction to μ_JHK. We find μ_K − μ_JHK = −0.47 ± 0.03 at R < 40'' with a maximum deviation from the mean of 0.05 mag, which is at most ∼(μ_K). The NIR color gradients are small over this radius, with Δ(J − H) ≲ 0.2 and Δ(J − K) ≲ 0.1, and roughly consistent with no color gradient to within the photometric error.

Our instrumental-smoothing correction effectively performs a one-dimensional bulge–disk decomposition of μ_JHK. We assume that the central light concentration intrinsically follows a Sérsic (1963) profile. After first subtracting an exponential surface brightness profile fitted to μ_JHK data between 10'' < R < 33'', we model the central light concentration within R < 8'' by a Sérsic profile convolved with a Gaussian kernel, assuming the latter is a good approximation for all instrumental effects. We find a best-fitting Sérsic index of n = 1.5 and an effective (half-light) radius of R_e = 17. Random errors in this modeling are approximated by the root-mean-square (rms) difference between the measured and modeled central light profile. We assume that the "intrinsic disk surface brightness" is the remainder of the profile after subtracting the model of the central-light concentration, the "intrinsic central light concentration" is the Sérsic component of the model, and the "intrinsic μ_JHK profile" is the sum of these two components; our instrumental-smoothing correction then consists of the difference between the measured and "intrinsic" μ_JHK profile. We adopt a conservative 50% systematic error in this correction due to the inherent uncertainties in the true parameterization of the central light concentration and instrumental-smoothing kernel. Although relevant to our assessment of any central mass concentration in our calculations of Σ_dyn (Section 5.3), ϒ^disk_{*, K} (Section 5.6), and the mass budget (Section 6), our instrumental-smoothing correction is immaterial to the fundamental conclusions of our paper.

Hereafter, we refer to the corrected JHK measurement as μ'_K; h_R always refers to the measurement based on this profile unless otherwise stated. The fully corrected μ'_K profile is shown in, e.g., Figure 4.

In Section 3.1.2, we use M_K to calculate the inverse-TF inclination of UGC 463 based on the TF relations derived by V01. Therefore, we calculate the total apparent magnitude, m_K, using the 2MASS K-band data according to the procedure used by V01. We measure a roughly isophotal magnitude at μ_K = 21.5 mag arcsec⁻² (the rough surface brightness limit occurring at R ≲ 55'') and extrapolate to infinity based on the fitted exponential disk (h_R = 123 ± 03). We find m_K(R ⩽ 55'') = 9.45 ± 0.01 and a correction of −0.09 mag for the extrapolation of the disk. Accounting for Galactic-extinction (A_K = 0.034), we find m_K = 9.32 ± 0.02.

Left uncorrected, μ'_K may overestimate our dynamical mass-to-light ratios (Section 5.6) due to internal dust extinction in the K band, Aⁱ_K. Using the dust-slab model from Tully & Fouque (1985)¹² and i = 27° ± 2° (Section 3.1), we calculate the function A^{i = 27}_K(R) as shown in Figure 2, which has a maximum of A^{i = 27}_K(R = 0) = 0.02 ± 0.01 at the galaxy center. Assuming A_V/E(B − V) = 3.1 (Cardelli et al. 1989), we calculate A_K/E(g − i) = 0.18 ± 0.02, which we use to apply an internal reddening correction to our g − i color when comparing our dynamical ϒ_* measurements to those predicted by SPS modeling. Figure 2 plots the uncorrected g − r, r − i, and g − i colors, as well as our internal-reddening-corrected (g − i)₀ color. Compared to more realistic radiative-transfer modeling including a clumpy ISM and spiral structure, these simple dust-slab model predictions tend to overestimate both the level of dust extinction and reddening toward high inclination; the predictions are more reasonable at the low inclination appropriate for UGC 463 (Schechtman-Rook et al. 2011). Given the marginal extinction in K band and reddening of g − i, the simpler model is sufficient for our purposes.

We note here that the K band also contains emission from hot dust and polycyclic aromatic hydrocarbons; however, based on a preliminary modeling of the spectral energy distribution of our full suite of NIR and Spitzer imaging, we expect this to be no more than a 3% contribution, which is immaterial to the conclusions of this paper.

SparsePak integral-field spectroscopy (IFS) of UGC 463 was obtained on the nights of UT January 2 and UT 2002 October 20, following the setup provided for the Hα region as listed in Table 1 of Paper I. We obtained four pointings during the January run and an additional three pointings during the October run. The pointings nominally followed the three-pointing dither pattern designed to fully sample the 72'' × 71'' FOV of SparsePak (Bershady et al. 2004);¹⁵ the fourth pointing during the 2002 January run was a repeat of the center pointing. We obtained 2×15 minute exposures for each pointing. Each exposure pair is combined, before extraction of the spectra, while simultaneously removing cosmic rays. Spatially overlapping fibers among the seven pointings are not combined but treated individually throughout our analysis. Further details of the reduction of these data (basic image reduction, spectral extraction, wavelength calibration, and sky and continuum subtraction) are provided by R. A. Swaters et al. (2011, in preparation), largely following methods described in Andersen et al. (2006) with continuum-subtraction techniques described in Bershady et al. (2005). The rms difference between the cataloged and measured line centroids for the ThAr lines used in our wavelength calibration, i.e., the "wavelength calibration error," is typically ≲ 0.1 km s⁻¹.

We also measure the instrumental dispersion, σ_inst, as a function of wavelength for all spectra, using the ThAr emission lines from our calibration lamp spectra (Paper II). The intrinsic widths of the ThAr features are negligible such that the second moment of these lines is equivalent to σ_inst to good approximation. After identifying a set of appropriate (unblended) lines from the calibration spectrum, we fit single Gaussian functions to each line using the same code described by Andersen et al. (2008) to fit the emission-line features in our galaxy spectra (see also R. A. Swaters et al. 2011, in preparation). For each fiber, we fit a quadratic Legendre polynomial to σ_inst(λ), which is used to interpolate σ_inst at any wavelength. The average instrumental broadening across the full spectral range for all fibers is 13 km s⁻¹.

Our optical continuum spectra in the Mg i region taken with both SparsePak and PPak—described in Sections 2.4.1 and 2.4.2, respectively—have sufficient spectral coverage to include the [O iii]λ5007 emission feature. Therefore, we also use these lines as tracers of the ionized-gas kinematics. No adjustment of the continuum-data reduction recipe was needed to accommodate the proper handling of the emission features. Instrumental dispersions are calculated as described in Sections 2.4.1 and 2.4.2 for the SparsePak and PPak data, respectively.

Ionized-gas kinematics are measured for all available emission lines. Following Andersen et al. (2008; see also Andersen et al. 2006), both single and double Gaussian line profiles are fitted in a 20 Å window centered around each line. All Gaussian fits have been visually inspected to ensure that each emission line was fitted properly. Velocities (cz) of each atomic species are calculated separately using the wavelength of the Gaussian centroid. Of all fitted line profiles, 27% are better fit by a double Gaussian profile (Andersen et al. 2008, D. R. Andersen et al. 2011, in preparation); in these cases, a single component is used to measure the line-of-sight (LOS) velocity. Ionized-gas velocity dispersions, σ_gas, also use a single component and are corrected for the measured instrumental line width.

For our Hα region spectroscopy, we combine all available velocity and velocity dispersion measurements (any combination of the [N ii]λ6548, Hα, [N ii]λ6583, [S ii]λ6716, and [S ii]λ6731 lines) into an error-weighted mean velocity for each fiber. Due to the large uncertainties in σ_gas for lines other than Hα, we only include measurements with (σ_gas) < 3 km s⁻¹ in the combined value. In this spectral region, both velocities and σ_gas are dominated by the Hα line measurements due to the higher S/N of these lines; hereafter, we refer to these kinematics as "Hα" kinematics, despite their inclusion of other ionized atomic species.

In addition to the consistency check among measurements made by SparsePak and PPak, the [O iii] kinematics provide a useful comparison with the Hα results (see Section 4). However, the Hα line generally provides higher quality kinematics and velocity fields: the line S/N is higher on average and the filling factor of the kinematic measurements in the disk of UGC 463 is more uniform. We eventually combine all ionized-gas kinematics into a single, axisymmetric set of measurements (Section 4.3); however, it is useful to keep in mind this distinction between the merit of the Hα and [O iii] kinematic data.

SparsePak IFS of UGC 463 was obtained using the Mg i-region setup as listed in Table 1 of Paper I—primarily targeting Fe i and Mg i stellar-atmospheric absorption lines. The observation and reduction of these data are described by Westfall (2009); we review the salient details here.

UGC 463 was observed on consecutive nights during a single run from UT 2006 September 23 to 25. Four, six, and four 45 minute exposures were taken during the three nights of observations, respectively. No dithering of the pointing was applied between exposures; the original pointing was repeated to the best of our ability for each night. Exposures taken within a given night have been combined into a single image, using an algorithm that simultaneously rejects cosmic rays, and reduced (basic image reduction, spectral extraction, wavelength calibration, and sky-subtraction) on a night-by-night basis. The basic reduction procedures are nearly the same as that used for the Hα data (Section 2.3.1). The wavelength calibration errors are typically ≲ 0.1 km s⁻¹. Error spectra have been calculated in a robust and parallel analysis. The repeatability of the pointing across the three nights of observation was good to less than 1 arcsec, determined by Westfall (2009) by forcing the kinematic centers of all the SparsePak Hα and [O iii] data to follow from the same on-sky kinematic geometry. Thus, the extracted spectra from each night have been combined on a fiber-by-fiber basis and weighted by the spectral (S/N)². The weighting is relevant due to changing conditions; moon illumination increased for each night (from 1% to 7%) and a number of exposures (∼30%) suffered from variable transparency losses due to passing clouds, particularly during the second night of observation. The combined spectra, from all 10.5 hr of integration, are analyzed in Section 2.4.3 to measure stellar kinematics for UGC 463.

Using our calculated error spectra, Figure 3 plots the mean S/N of our SparsePak IFS as a function of radius against the g-band surface brightness profile, demonstrating that we have measured stellar kinematics for fibers with μ_g ∼ 22.5 at S/N ∼ 3. The scale translation between mean S/N and μ_g assumes that the data are detector-limited (S/N∝ flux), which is only an approximation for our data.

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Finally, we estimate σ_inst for both the template and galaxy spectra for use in measuring instrumental-broadening corrections (Section 2.4.4). We use the same approach as described for the Hα-region spectroscopy in Section 2.3.1 for both the galaxy and template observations. The average instrumental broadening for these galaxy spectra is 11 km s⁻¹.

PPak IFS of UGC 463 was obtained using the Mg i-region setup as listed in Table 1 of Paper I. Compared to the SparsePak optical continuum spectra, individual PPak spectra have approximately 1.5 times the spectral range, 0.7 times the spectral resolution, and 0.6 times the on-sky aperture per fiber; however, PPak contains 4.4 times as many fibers as SparsePak in its main fiber bundle. These data are described by Martinsson (2011), which includes all Mg i-region data taken by PPak. Here, we briefly review the acquisition and reduction of the data specifically for UGC 463.

We obtained two consecutive 1 hr exposures targeting UGC 463 on each of three nights from UT 2004 November 13 to 15, yielding six hours of total on-target integration. Flexure corrections have been applied on an exposure-by-exposure basis, requiring spectral extraction, wavelength calibration, and sky subtraction to be performed on each exposure individually. The rms wavelength calibration error is typically of the same order as that found for the SparsePak observations (∼0.1 km s⁻¹). Error spectra have been calculated in a robust and parallel analysis. For UGC 463, no pointing offsets were detected; the guide-camera CCD of the Potsdam Multi-Aperture Spectrophotometer (PMAS) allows for accurate reacquisition of our target galaxies on subsequent nights. All six spectra for a given fiber have been combined using weights depending on the instrumental resolution and the spectral S/N as described by Martinsson (2011).

Figure 3 provides the mean S/N of the PPak spectra, alongside that of the SparsePak spectra. Despite the shorter integration time and smaller fiber aperture, the PPak data have slightly higher mean S/N, an effect of both the lower spectral resolution and the better efficiency of PMAS over the WIYN Bench Spectrograph.¹⁶

Instrumental broadening measurements are performed differently for PPak data than described above for SparsePak data. PPak observations provide simultaneous calibration spectra in 15 fibers, evenly distributed along the pseudo-slit among the galaxy spectra. These spectra have proven critical for applying the necessary flexure corrections (Martinsson 2011). Moreover, they provide simultaneous measurements of the instrumental dispersion, obtained by fitting Gaussian functions to the ThAr emission lines. Martinsson (2011) has fit a quadratic Legendre polynomial surface to these measurements of σ_inst for each object frame. We use this description of σ_inst to calculate the instrumental-dispersion corrections for both the ionized-gas and stellar kinematics measured from the PPak spectra. The mean instrumental dispersion across all fibers and all spectral channels is 17 km s⁻¹ for our PPak data of UGC 463.

We use the Detector-Censored Cross-Correlation (DC3) software presented by Westfall et al. (2011, hereafter Paper III) to determine spatially resolved stellar kinematics in UGC 463 from both our SparsePak and PPak spectra. Based on a preliminary analysis (Paper II), we find that a K1 III star provides a minimum template-mismatch error of ∼5% in the observed velocity dispersion (σ_obs) with no systematic trend in radius. In the future, we can improve upon this by using composite templates; however, a 5% template-mismatch error in σ_obs is satisfactory for the present study.

As per our survey protocol (Paper I), we have observed template stars in the Mg i region using both SparsePak and PPak. For the analysis here, we specifically use the K1 III stars HD 167042 and HD 162555 for our SparsePak and PPak stellar kinematics, respectively; Table 3 presents salient information regarding the template spectra. Template star observations are performed under nominally the same spectrograph/telescope configuration as for our galaxy data. For both SparsePak and PPak, template stars are observed by drifting the star through the full FOV, yielding many spectra that are combined to provide high-S/N templates; the final S/N of each template is provided in Table 3.

We fit any galaxy–template cross-correlation (CC) function that peaks within a few hundred km s⁻¹ of the systemic velocity of UGC 463 (as recorded by NED) regardless of the S/N. For each fiber, we adopt a Gaussian broadening function, and we use a cubic Legendre polynomial to minimize continuum differences between the broadened template and the fitted galaxy spectrum. Additionally, we mask the [O iii]λ5007 and $\rm [N\,\mathsc{i}]$ (λ5198 and λ5200) nebular emission regions from both the template and galaxy spectra. For the PPak data, the redshifted [O iii]λ4960 line is also visible; however, DC3 masks the CC to a rest-wavelength range common to both the galaxy and template spectrum, thereby automatically masking this line. Each CC fit has been visually inspected to insure that the proper peak was considered by the fitting algorithm and that any unexpected artifacts—poorly removed sky lines and/or cosmic-ray detections—were masked. Based on this inspection, spectra have been refit as necessary. Our stellar kinematic analysis follows the expectations derived for random errors in Paper III. As assessed via χ² and the velocity shift with respect to spatially neighboring fibers, we find reasonable fits to spectra with mean S/N approaching unity, albeit with large errors. Systematic errors should be negligible for velocity measurements at all S/N, and they should be ≲ 20% in σ_obs at S/N ≳ 2; systematic errors are always smaller than the calculated random error (Paper III).

We correct our observed stellar kinematics for the system response function by considering the following two separable components: (1) the broadening of the intrinsic absorption-line widths due to the spectrograph optics, accounted for using an "instrumental-broadening" correction, δσ_inst, and (2) the smearing of the intrinsic surface brightness, velocity, and velocity dispersion distributions by the response of the atmosphere+telescope system, accounted for using a "beam-smearing" correction, σ_beam. The final LOS dispersion is σ²_* = σ_obs² − δσ²_inst − σ_beam² (Paper II). Unlike σ_beam, δσ_inst is independent of the on-sky geometry and intrinsic kinematic structure of the observed galaxy; therefore, we calculate δσ_inst here. We calculate σ_beam before combining our SparsePak and PPak kinematics in Section 4 using the projection geometry derived in Section 3.

Each CC is used to compare template and galaxy absorption-line shapes such that δσ_inst is determined by the difference in σ_inst measured for the template and galaxy spectrum; we calculate δσ_inst following Appendix A of Paper III using our measurements of σ_inst for both the template and galaxy spectra. We adopt a 4% error in δσ_inst (Paper II), which is marginal when compared to (σ_obs). These corrections differ rather dramatically between SparsePak and PPak; however, in both cases, δσ²_inst is typically small. Corrections to σ_obs—i.e., the ratio (σ²_obs − δσ_inst²)^1/2/σ_obs—are ≲ 4% and ≲ 20% for, respectively, 90% and 99% of all measurements; a few measurements have rather large corrections due to dispersion measurements of σ_obs < 10 km s⁻¹, which are likely erroneously low (Paper III). We always find δσ_inst < (σ_inst).

The survey strategy for all our Spitzer observations are provided in Section 6.2.3 of Paper I. In general, 24 μm images collected for the DMS demonstrate significant background structure, due to both detector effects and intrinsic structure in the Galactic ISM, with fluctuations on angular scales close to that of our galaxies. To account for these fluctuations, we mask out all statistically significant sources, including a substantial radial region surrounding UGC 463, and create a 47'' × 47'' boxcar-smoothed background image. Masked regions are iteratively filled by the boxcar smoothing, effectively interpolating the sky background and its gross structure, across all detected sources. We simply subtract this smoothed image from our 24 μm image of UGC 463 and use the result to calculate the 24 μm surface brightness profile.

Our background-subtraction procedure has been carefully assessed to ensure that the low surface brightness extent of UGC 463 has not been systematically oversubtracted. Preliminary tests with UGC 463 and other galaxies in our survey demonstrate that our I_24 μm profiles become strongly affected by the sky-subtraction errors at a source intensity below I_24 μm < 0.05 MJy sr⁻¹ ($\Sigma _{\rm H_2}< 0.21$ $\mathcal {M}_\odot$ pc⁻² in Figure 4). UGC 463 is the third brightest 24 μm emitter in our entire sample, meaning that this surface brightness limit falls outside the radial region relevant to this paper. Our measured 24 μm surface brightness profile uses elliptical apertures following a geometry identical to that used for the optical and NIR photometry in Section 2.2.1. Figure 4 provides the 24 μm surface brightness profile and the result of its conversion to $\Sigma _{\rm H_2}$, according to the discussion in the next two sections. The random errors in our I_24 μm measurements incorporate a constant 4% calibration error (Engelbracht et al. 2007) and a sky-subtraction error estimated by the change in I_24 μm introduced by a factor of two change in the smoothing-box size; the latter results in 1% and 10% sky-subtraction errors at R ∼ 30'' and ∼47'', respectively.

We use measurements of both CO and 24 μm emission provided by L08 (see their Table 7) to measure the correlation between I_24 μm and I_COΔV. Twelve of the twenty-three galaxies studied by L08 include both CO and 24 μm observations; however, four of those galaxies (NGC 2841, NGC 3627, NGC 4736, and NGC 5194) are listed in NED as having either LINER or Seyfert activity, unlike UGC 463. Therefore, we calibrate I_24 μm/I_COΔV using only the remaining eight galaxies, hereafter the "I_24 μm/I_COΔV subsample."

The quantities provided by L08 are matched-resolution, azimuthally averaged radial profiles of $\Sigma _{\rm H_2}$ (based on CO emission and a value for X_CO) and the contribution of embedded star formation (determined from the 24 μm surface brightness) to the total star formation rate surface density. We revert these quantities to I_COΔV (in K km s⁻¹) and I_24 μm (in MJy sr⁻¹) using Equations (A2) and (D1) from L08. All eight galaxies in the I_24 μm/I_COΔV subsample were observed as part of the HERACLES Survey (Leroy et al. 2009), observing only the ¹²CO(J = 2 → 1) emission line, where L08 adopt a line ratio of ¹²CO(J = 2 → 1)/¹²CO(J = 1 → 0) = 0.8. The CO surface brightness has been determined by integrating the emission profile over the full line width and converting the flux units per beam to Kelvin using the Rayleigh–Jeans limit. The 24 μm fluxes are determined from surface photometry of Spitzer imaging data obtained by the SINGS Survey (Kennicutt et al. 2003).

Figure 5 presents the data for the I_24 μm/I_COΔV subsample regardless of the galaxy or radial region from which it has been measured. Table 7 from L08 is used to calculate (I_COΔV) directly, whereas we adopt a uniform (I_24 μm) = 0.15I_24 μm due to insufficient information; we expect this (I_24 μm) to be an upper limit. We fit a power-law relationship between I_24 μm and I_COΔV to all available data, incorporating errors in both coordinates (Section 15.3 of Press et al. 2007), finding a best fit of

$$\begin{equation} \log \left[ \frac{I_{\rm CO}\Delta V}{\rm K\ km\ s^{-1}}\right] = \left(1.08\ \log \left[ \frac{I_{24\,\mu {\rm m}}}{\rm MJy\ sr^{-1}}\right] + 0.15\right), \end{equation} \tag{ 1 }$$

with a weighted standard deviation of ±0.11 dex, in good agreement with the previous result from Paladino et al. (2006). Thus, given that UGC 463 has physical parameters that are comparable to the I_24 μm/I_COΔV subsample, our calibration is expected to estimate I_COΔV for this galaxy to within ∼30%.

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Using Equation (1), we convert our sky-subtracted 24 μm image of UGC 463 to a CO surface brightness map. Subsequently, we calculate the H₂ mass surface density using the traditional conversion factor, X_CO, following

$$\begin{eqnarray} \left[\frac{\Sigma _{\rm H_2}}{\mathcal {M}_\odot \ {\rm pc}^{-2}}\right] &=& 1.6 \left[\frac{I_{\rm CO}\Delta V}{\rm K\ km\ s^{-1}}\right]\nonumber\\ &&\times \left[\frac{X_{\rm CO}}{\rm 10^{20}\ cm^{-2}\ (K\ km\ s^{-1})^{-1}}\right] \cos i,\quad\ \end{eqnarray} \tag{ 2 }$$

where i is the galaxy inclination and X_CO is the ratio of the H₂ column density to the CO line strength. This use of X_CO to calculate $\Sigma _{\rm H_2}$ is a common procedure for calculating the molecular-gas content of external galaxies; however, it may suffer from substantial systematic error.

A large number of studies have been devoted to measuring X_CO both in our own Galaxy and within the Local Group. Empirical and theoretical studies suggest X_CO likely depends on multiple physical parameters, such as metallicity, radiation field, gas mass surface density, and density structure (Arimoto et al. 1996; Mihos et al. 1999; Boselli et al. 2002; Bell et al. 2006; Narayanan et al. 2011). Moreover, direct measurement of X_CO is observationally challenging. For example, the assumption of virial equilibrium and the finite spatial resolution of giant molecular clouds, even in Local Group galaxies, may both lead to inflated values of X_CO (Blitz et al. 2007; Bolatto et al. 2008); see Bolatto et al. (2008) for a more general review of X_CO measurements. Keeping these complications in mind, our analysis here adopts a simple approach: combining the Galactic measurement of X_CO = 1.8 ± 0.3 from Dame et al. (2001) with the measurements for M31 (X_CO = 3.6 ± 0.3) and M33 (X_CO = 2.6 ± 0.4) from Bolatto et al. (2008), we find a mean and range of X_CO = (2.7 ± 0.9) × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. We assume this value to be representative of UGC 463, given that the Milky Way, M31, and M33 are arguably the only spiral galaxies with well-resolved observations of giant molecular clouds or associations from which robust measurements of X_CO can be made.

Figure 1 provides the 24 μm image, converted to $\Sigma _{\rm H_2}$ in units of $\mathcal {M}_\odot \ {\rm pc}^{-2}$ using the I_24 μm/I_COΔV calibration from Section 2.6.2 and assuming X_CO = 2.7 × 10²⁰ cm⁻² (K km s⁻¹)⁻¹. Figure 4 provides the azimuthally averaged surface density profile $\Sigma _{\rm H_2}(R)$ using the 24 μm surface brightness profile from Section 2.6.1. Errors in $\Sigma _{\rm H_2}$ are plotted separately for random and systematic components; the former includes errors from the I_24 μm calibration and photometry and the inclination, whereas the latter includes the I_24 μm/I_COΔV calibration error and range in X_CO. This estimate of $\Sigma _{\rm H_2}(R)$ agrees with the expectation that H₂ is concentrated toward the "hole" in the $\rm H\,\mathsc{i}$ mass surface density, also seen in Figure 1 and studies of other galaxies (e.g., L08).

Out to 15 kpc (4.2 h_R), we find $\mathcal {M}_{\rm \rm H_2}/\mathcal {M}_{\rm \rm H\,\mathsc{i}}= 3.2$, a value that is reasonable with respect to direct CO and $\rm H\,\mathsc{i}$ studies in the literature. In particular, Young & Knezek (1989) find a range of 0.2 and 4.0 for, respectively, late- and early-type spiral galaxies, comparable to the range measured by the more recent COLD GASS survey (Saintonge et al. 2011). However, despite having a total $\mathcal {M}_{\rm \rm H_2}/\mathcal {M}_{\rm \rm H\,\mathsc{i}}$ that is decreased by ∼10% compared to the measurement within 15 kpc (Martinsson 2011), UGC 463 is more rich in molecular gas than the mean $\mathcal {M}_{\rm \rm H_2}/\mathcal {M}_{\rm \rm H\,\mathsc{i}}$ calculated by Saintonge et al. (2011) for the COLD GASS survey by approximately twice the standard deviation. Similarly, by combining our 24 μm imaging, the I_24 μm/I_COΔV and X_CO values derived herein, and the $\rm H\,\mathsc{i}$ data presented by Martinsson (2011) for 24 galaxies in the DMS, we find a mean of $\mathcal {M}_{\rm \rm H_2}/\mathcal {M}_{\rm \rm H\,\mathsc{i}}= 0.48$, consistent with the Saintonge et al. (2011) measurement after accounting for the difference in their adopted X_CO; UGC 463 has the maximum value of $\mathcal {M}_{\rm \rm H_2}/\mathcal {M}_{\rm \rm H\,\mathsc{i}}$ and 21 of 24 galaxies have $\mathcal {M}_{\rm \rm H_2}/\mathcal {M}_{\rm \rm H\,\mathsc{i}}< 1$. Therefore, consideration of the molecular mass component in UGC 463 is relatively more important to our dynamical ϒ_* measurements and the baryonic mass budget than the majority of galaxies in the DMS. In Section 5.6, we discuss both the total dynamical mass-to-light ratios as well as stellar-mass-only measurements to illustrate the effects of the gas-mass corrections.

We use the method described in Andersen et al. (2008) to measure kinematic inclinations; see also Andersen & Bershady (2003). The strength of this method is in its simultaneous use of the full two-dimensional information in our observed velocity fields. However, it assumes (1) a single set of geometric projection parameters for the entire disk (i.e., a "one-zone" model), (2) all motion is purely circular rotation in the disk plane, and (3) the rotation curve follows a parameterized model. We justify these assumptions below.

Based on edge-on galaxies, literature studies have repeatedly found that warps in non-interacting galaxies only influence disk morphology at large radii. Highlighting two recent studies, van der Kruit (2007) have shown that gas disks (as traced by H i) typically begin to warp at approximately the outer truncation radius of the stellar disk and Saha et al. (2009) have shown that stellar–disk warps occur at R > 3h_R. Thus, we expect no warping within the FOV of our IFS of UGC 463, and only a marginal warping of our H i data. Indeed, our $\rm H\,\mathsc{i}$ data only begin to show a position-angle warp for the last measured radial bin (R = 45''; Martinsson 2011). For our IFU data, post-analysis of the velocity-field residuals demonstrates little to no radial dependence of the geometric parameters, as determined by translating velocity residuals with respect to our nominal model (as developed in this and subsequent sections) into model-parameter residuals. This is done by holding all but one parameter fixed and adjusting the free parameter until the velocity residual is nearly or identically zero. We find no correlation between the parameter residuals and radius, with the possible exception of the position angle. There is some indication of a positive slope in position angle with radius; however, the magnitude of the position angle change is small (less than 5° over the full radial range) and the significance of the slope is marginal. This means that the use of a radially dependent position angle is only marginally justified and, more importantly, inconsequential to our measurement of the rotation curve. Therefore, a one-zone velocity-field model provides an adequate description of our optical kinematic data.

As briefly noted in Section 3.3, the isovelocity contours in our velocity fields appear to show slight non-circular motions. These motions are most prevalent for the Hα data where some coherent structure is seen in the velocity-field residual map, particularly along the spiral-arm to the southwest of the galaxy center (on the approaching side of the velocity field). These coherent residuals likely represent streaming motions along this spiral arm toward the galaxy center given their spatial correlation to the photometric feature and the sign of the residual. The magnitude of this streaming is less than 10 km s⁻¹ along the LOS (less than 25 km s⁻¹ in the disk plane) and the covering fraction of all non-circular motions is small. Therefore, we expect that our best-fitting velocity-field model should suffer only marginally from these motions, particularly given the benefits afforded the one-zone model in this respect.

The parameterization of V^proj_rot(R) = V_rot(R)sin i does not adversely bias the derived geometric parameters: Andersen & Bershady (2003) and Andersen et al. (2008) have chosen a hyperbolic tangent (tanh ) function—a simple two-parameter model that enforces an asymptotically flat rotation curve. Although inappropriate for the rare declining rotation curve in the DMS sample, kinematic inclinations derived for such galaxies using a more appropriate parameterization (e.g., Courteau 1997) are within the formal errors of those derived using a tanh  model.

Given our highly sampled and high-quality velocity fields of UGC 463, here we use a step function to define V^proj_rot(R), effectively fitting a set of coplanar "rings" with constant rotation speed. We fit up to 13 rings, each with a width of 3'' (approximately the diameter of a single PPak fiber) such that rotation-speed gradients within each ring are small, except for possibly the central ring. The final ring includes all data at R > 36'' and may be omitted for some tracers if no data exist at these radii. Despite our use of the term "ring" here, we emphasize that this fit is not a typical tilted-ring fit given that we are defining only a single set of geometric parameters.

We measure independent kinematic inclinations for the three kinematic data sets provided by the SparsePak Hα data and the PPak [O iii] and stellar data. All geometric and rotation-curve parameters are fit simultaneously, with one exception: Martinsson (2011) has used reconstructed continuum images to determine the morphological center of UGC 463 relative to the PPak fibers, to which we affix the dynamical center when modeling these data. Greater detail regarding our velocity-field fitting approach is provided in Appendix B, including a full description of which measurements are omitted from consideration during the fit. However, Appendix B is primarily focused toward an assessment of the optimal data-weighting scheme for modeling the velocity field of UGC 463. Therein, we use bootstrap simulations (see Section 15.6.2 of Press et al. 2007) to produce inclination probability distributions based on four different weighting schemes. We thereby demonstrate that we obtain the most correspondent inclinations among the different data sets by adopting weights defined by the derivative of the model LOS velocity, V_LOS, with respect to the inclination, i.e., ∂V_LOS/∂i. These weights approximately follow a sin ²(2θ) function in azimuth and a direct proportionality in radius; therefore, data with the most leverage on the fitted inclination (at approximately ±45° from the major axis; Andersen & Bershady 2003) have the highest weight. The best-fitting inclination, position angle, and systemic velocity for each tracer are given in Table 4; bootstrap simulations are used to calculate the 68% confidence intervals. The results provided for our $\rm H\,\mathsc{i}$ data from Martinsson (2011) are based on traditional tilted-ring fitting (Begeman 1989).

Table 4. Kinematic Geometry

Data Set	ikin	ϕ0	Vsys
	(deg)	(deg)	(km s−1)
Hα	24.1+4.5−2.1	68.8+0.3−0.3	4458.6+1.0−0.5
[O iii]	26.5+4.6−3.9	68.8+0.4−0.7	4460.4+0.6−0.8
Stars	25.5+4.7−10.8	68.4+0.6−0.7	4461.3+0.6−0.8
$\rm H\,\mathsc{i}$	⋅⋅⋅	68.8 ± 1.5	4459.5 ± 1.5
Mean	25.1 ± 2.5	68.8 ± 0.3	4460 ± 1

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The geometric parameters listed in Table 4 for each dynamical tracer are in general agreement; the systemic velocities exhibit the most statistically significant differences. Such differences are likely due to systematic errors in the heliocentric velocities of the template stars and/or shifts in the pointing center. In any case, these shifts are small and irrelevant to our analysis of the mass distribution in UGC 463. Using the half-width of the 68% confidence interval from Table 4 as the error, we calculate error-weighted means of i_kin = 251 ± 25 and ϕ₀ = 688 ± 03. The unweighted mean value V_sys = 4460 ± 1 km s⁻¹ has been used in Section 2.1 to calculate the distance to UGC 463.

Following the discussion in Paper II, inverse-TF inclinations are calculated according to

$$\begin{equation} i_{\rm TF}= \sin ^{-1}\left[ 2V_{\rm rot}^{\rm proj}\ {{\rm d}{\rm exp} }\left(\frac{c_{1,\lambda } - M_\lambda }{c_{2,\lambda }}\right)\right], \end{equation} \tag{ 3 }$$

where c_{1, λ} and c_{2, λ} are, respectively, the zero point and slope of the TF relation in wavelength band λ and M_λ is the total absolute magnitude. Combining D = 59.67 ± 4.15 Mpc (the error here is the quadrature sum of the random and systematic error from Section 2.1), m_K = 9.32 ± 0.02 (Section 2.2.3), and a ${\mathcal K}$-correction of 0.035 mag (Bershady 1995), we find M_K = −24.59 ± 0.15 for UGC 463. We use the K-band TF relations derived by V01 to calculate i_TF based on measurements of the projected rotation speed.

We measure the projected rotation curve for all gas tracers in UGC 463 for use in calculating i_TF; stellar measurements are not considered due to significant asymmetric drift (AD) (Section 4.3). Figure 6 presents V^proj_rot for the Hα and [O iii] data resulting from all four weighting schemes implemented in Appendix B; $\rm H\,\mathsc{i}$ measurements are directly from Martinsson (2011). It also provides the error-weighted mean measurements 〈V^proj_rot〉 for data at R > 24''. No beam-smearing (Section 4.1) or pressure (Section 4.3.1) corrections have been applied; these are negligible considerations for the measurement of 〈V^proj_rot〉. The Hα rotation curve exhibits less dependence on the applied weighting than does the [O iii] rotation curve; however, they both compare well with each other and with the $\rm H\,\mathsc{i}$ rotation curve, regardless of the weighting scheme. The Hα data, in particular, appear to asymptote at R > 24''; hence this radial region is chosen for measuring 〈V^proj_rot〉. Table 5 provides 〈V^proj_rot〉 for each tracer; measurements of 〈V^proj_rot〉 from Hα and [O iii] are the unweighted mean of the results from all weighting schemes. Using all tracers, we find a mean and standard deviation of 〈V^proj_rot〉 = 107 ± 2 km s⁻¹.

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Table 5. Projected Rotation Speed

Data Set	〈Vprojrot〉
	(km s−1)
Hα	107 ± 1
[O iii]	108 ± 2
$\rm H\,\mathsc{i}$	105 ± 2
Mean	107 ± 2

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The calculation of i_TF and its uncertainty relies on two additional factors: (1) the choice of the TF relation and (2) the estimation of its intrinsic scatter. V01 created multiple samples based on a rotation-curve- and an asymmetry-based taxonomy, each sample yielding different TF coefficients (c_{1, λ} and c_{2, λ}) and intrinsic-scatter estimates. UGC 463 exhibits $\rm H\,\mathsc{i}$ and ionized-gas properties that are most consistent with the "RC/FD" sample from V01 (see his Sections 4 and 5 for a detailed definition of this sample); UGC 463 fits within the definition of the "RC/FD" sample because its rotation curve asymptotes to a nearly constant rotation speed and exhibits neither strong rotation asymmetries (Section 4.2) nor signs of ongoing interaction. Additionally, V01 produced TF coefficients based on two fiducial velocities, one measured at the rotation-curve peak (V_max) and the other where it "flattened" to a constant value (V_flat). UGC 463 exhibits a well-defined V_flat (Figure 6); however, in most cases V_flat = V_max for galaxies studied by V01. The smallest observed scatter in the K-band TF relations derived by V01 (0.26 mag) was found by excluding the outlying NGC 3992 measurements from the "RC/FD" sample (leaving measurements for 21 galaxies) and using V_flat for the fiducial rotation measurement; this TF relation is consistent with having zero intrinsic scatter.

Given the range in c_{1, K} and c_{2, K} for the "RC/FD" sample (with and without NGC 3992 and based on either V_max or V_flat) from Table 4 of V01, we find 28° ⩽ i_TF ⩽ 30°; and we find (i_TF) = 1° and 2° assuming, respectively, 0.0 and 0.2 mag of intrinsic TF scatter, regardless of the assumed c_{1, K} and c_{2, K}. Taking a mean across the four relevant TF relations and assuming 0.2 mag for the intrinsic TF scatter, we measure i_TF = 29° ± 2° for UGC 463; this measurement of i_TF is dominated by systematic error with roughly equal contributions from the uncertainties in H₀ and the TF relation. Our conservative approach to measuring i_TF is justified given that, at M_K = −24.59 ± 0.15, UGC 463 is more luminous than any galaxy considered by V01, far away from the "pivot" point of the fitted K-band TF relations (M_K ∼ −22).

Dalcanton & Stilp (2010) derive

$$\begin{eqnarray} V_{\rm c}^2 & = & V_{\rm rot}^2 - \sigma _{\rm gas}^2 \left(\frac{d\ln \sigma _{\rm gas}^2}{d\ln R} + \frac{d\ln \Sigma _{\rm gas}}{d\ln R}\right) \nonumber \\ & = & V_{\rm rot}^2 + \delta _P \sigma _{\rm gas}^2, \end{eqnarray} \tag{ 4 }$$

where σ_gas is assumed to be dominated by turbulence and produced by an isotropic gas velocity ellipsoid (cf. Agertz et al. 2009). We, thereby, correct our gas rotation curve to the circular speed using our measurements of the $\overline{\sigma _{\rm gas}}$ in Figure 8 and Σ_gas in Figure 4.

One expects σ_gas and Σ_gas to decrease with radius such that δ_P > 0.¹⁸ Dalcanton & Stilp (2010) propose exponential functions for use in calculating δ_P, which is appropriate for our Σ_gas measurements; however, we adopt the linear function plotted in Figure 8 for σ_gas. We calculate 0.2 ⩽ δ_P ⩽ 3.3 such that the circular-speed corrections range from 0.4 ⩽ (V_c − V_rot) ⩽ 1.6 km s⁻¹, always below the measurement error in the gas rotation speed. If we treat the circular-speed corrections independently for [O iii] and Hα, we find the Hα-based and [O iii]-based circular speeds to be more consistent than if one correction is applied to both. However, the combined circular speeds are roughly independent of whether or not the Hα and [O iii] data are treated separately.

We create an approximate probability distribution (combining random and systematic components) for our measurements of ϒ^disk_{*, K}(R) and 〈ϒ^disk_{*, K}〉 by MC sampling individual probability distributions for each component of our ϒ^disk_{*, K} calculation. In contrast to the random error contributors, systematic errors—particularly in q, k, and X_CO—may not be normally distributed, providing the primary motivation for this test. Our simulation is limited by the exclusion of any parameter covariance, such as might be manifest in a refitting of α and β after adjusting i; however, we do not expect parameter covariance to dramatically change the fundamental conclusions drawn from this MC simulation.

Random-error contributors—all quantities with measurement errors contributing to the random error in ϒ^disk_{*, K}—are assigned Gaussian probability distributions according to their derived . We also assign a Gaussian distribution for D (combining the random and systematic error in quadrature) and I_24 μm/I_COΔV (using the ±0.11 dex error in Equation (1)). Information on the probability distribution for X_CO, particularly for spiral galaxies like UGC 463, is limited; therefore, we simply assume a uniform distribution with the range X_CO = (2.7 ± 0.9) × 10²⁰ cm⁻² (K km s⁻¹)⁻¹ (Section 2.6.3). We have derived an empirical probability distribution for q as presented in Figure 1 from Paper II (and references therein). For our MC simulation, we smooth the growth curve of this empirical distribution by a low-order polynomial to avoid the discrete-measurement quantization noise.

A robust probability distribution for k is elusive, lacking an empirical measurement. For UGC 463, the measurements of a relatively massive gas disk (Section 2.7) and DM halo (Section 6) are particularly relevant to the value of k. In an extreme scenario, the stellar disk is vertically exponential and the gas disk is razor thin, yielding an effective 30% decrease in k for our measured values of Σ_gas and Σ_*. Despite the resulting increase in Σ_dyn, we would still infer a massive DM halo that increases k by 20%–30% (Section 6.3.2) and, therefore, roughly offsets the effect of the massive gas disk. In view of further complications introduced by finite-thickness gas disks, multi-component stellar disks, and triaxial halos, we have decided to take a simple approach and assume that k is quantized and equally distributed among the exponential (k = 1.5), sech (k = 1.71), and sech² (k = 2.0) cases.

The results of our MC simulation are shown in Figure 13 based on 10⁶ recalculations of ϒ^disk_{*, K} for each measurement of $\overline{\sigma _{\ast }}$ at R ⩾ 2 kpc. Each recalculation is binned in the two-dimensional (R, ϒ^disk_{*, K}) plane; R is binned in physical units, incorporating the MC sampling of D. For each radial bin, we create a growth curve for ϒ^disk_{*, K} such that higher intensity (darker) cells in the left panel of Figure 13 represent more probable measurement of ϒ^disk_{*, K}. We overplot 68%, 95%, and 99% confidence contours for ϒ^disk_{*, K}(R), as well as a contour following the median value. We also overplot the nominal measurements of ϒ^disk_{*, K} from Figure 12 for reference, again differentiating between random and systematic errors. Figure 13 also provides the growth curve of 〈ϒ^disk_{*, K}〉, calculated for each of the 10⁶ recalculations of ϒ^disk_{*, K}(R).

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Based on our nominal calculation (Figure 12), we find 〈ϒ^disk_{dyn, K}〉 = 0.34 ± 0.09 ± 0.15 and 〈ϒ^disk_{*, K}〉 = 0.22 ± 0.09^+0.16_−0.15 at R > 2 kpc, in units of $\mathcal {M}_\odot /\mathcal {L}_\odot ^K$. In contrast, B03 and Z09 predict 〈ϒ^SPS_{*, K}〉 = 3.6〈ϒ_{*, K}^disk〉 and 〈ϒ^SPS_{*, K}〉 = 0.56〈ϒ_{*, K}^disk〉, respectively. From our derived probability distribution we find a median of 〈ϒ^disk_{*, K}〉 = 0.17^+0.12_−0.09 such that the Z09 prediction is within our 68% confidence interval; however, B03 predict a measurement that occurs for less than 1 in 10⁵ recalculations of 〈ϒ^disk_{*, K}〉. The median value of our probability distribution is below our nominal measurement because the median value for k is above our nominal value of k = 1.5. Within the parameter space probed by our MC simulation, we find a maximum measurement of 〈ϒ^disk_{*, K}〉 = 0.82.

The prescriptions adopted for the many ingredients of SPS modeling—such as the initial mass function (IMF), star formation and chemical-enrichment history, dust content, and the treatment of specific phases of stellar evolution—play an important role in setting ϒ^SPS_{*, K} and its trend with color (Portinari et al. 2004; Conroy et al. 2009, 2010; Conroy & Gunn 2010). Indeed, in their discussion of the differences between their ϒ_*–color relations and those from B03, Z09 isolate their treatment of the star formation histories and thermally pulsating asymptotic giant branch (TP-AGB) phases of stellar evolution from intermediate-age populations as the primary culprits. The latter particularly affects differences in SPS predictions of ϒ_* in the NIR bands (Maraston 2005). Aside from the factor of ∼7 difference in the mean B03 and Z09 predictions for UGC 463, it is also important to keep in mind that there is an additional factor of two or more internal variation in the ϒ^SPS_{*, K} calibration associated with each study as determined by their search of the SPS modeling parameter space. It is encouraging that the advancement in SPS modeling, as represented by the Z09 study, are more consistent with our dynamical measurements. However, Figure 12 shows that neither the zero point nor the trend of ϒ_* with (g − i)₀ color from these models are a good match to our measurements; therefore, we cannot conclude that this specific SPS treatment—either for the TP-AGB phase or star formation histories—is correct or applicable to our entire sample. An absolute calibration of ϒ_* using a multi-color approach for the full DMS Phase-B sample will be presented in forthcoming papers.

As expected from the discussion in Section 5.1, Figure 12 shows that ϒ^disk_{dyn, K} and ϒ^disk_{*, K} generally increase with radius. Considering only the data at R = 5.6 and 10.8 kpc, ϒ^disk_{dyn, K} and ϒ^disk_{*, K} increase by a factor of 2.2 ± 1.0^+1.2_−1.1 and 3.4 ± 2.3^+2.8_−2.5, respectively. Figure 13 shows that the radial trend holds for the median of the probability distribution in ϒ^disk_{*, K}(R). Variation in ϒ^SPS_{*, K} is expected in galaxy disks given the observed arm/inter-arm and radially averaged color gradients. Yet, in terms of the latter, Figure 12 shows that the variation in ϒ^disk_{*, K} for UGC 463 is consistent with neither of the plotted ϒ^SPS_{*, K} predictions. Given both the errors in our measurement and the errors in the ϒ^SPS_{*, K} calibration, it is difficult for us to conclude that our measurements are inconsistent with a radially invariant ϒ^disk_{*, K}. Indeed, a radially independent ϒ^disk_{*, K} is consistent with our 68% confidence limits derived in Figure 13.

Moreover, the more shallow decline of σ²_z with respect to the surface brightness profile, which is the primary driver for our measurement of a radially varying ϒ^disk_{*, K}, may also be interpreted as a flaring of the stellar disk (Section 5.1). A flared stellar disk has been measured for the Galaxy (López-Corredoira et al. 2002; Robin et al. 2003; Momany et al. 2006) and one might expect disks to be flared due to, e.g., interactions with dark and/or luminous satellites (Hayashi & Chiba 2006; Dubinski et al. 2008; Read et al. 2008; Kazantzidis et al. 2009). Indeed, Herrmann et al. (2009) suggest that their measurements of a nearly constant σ_z at large radii in M83 and M94 provide evidence for such interactions. However, the onset radius for disk flaring is expected to be beyond the region relevant to our dynamical measurements for UGC 463. Robust photometric evidence for stellar-disk flares in edge-on galaxies remains elusive. The empirical and theoretical foundation for the vertical structure of galaxy disks with radially independent scale heights developed by van der Kruit & Searle (1981a, 1981b, 1982a, 1982b) remains the current paradigm due to repeated confirmations of little to no variation in scale height measurements from surface photometry, particularly for galaxies of similar Hubble type to UGC 463 (SABc; e.g., de Grijs & Peletier 1997; Bizyaev & Mitronova 2002). However, claims of factors of two or more increase in scale height within the optical extent of some stellar disks exist in the literature (e.g., Narayan & Jog 2002; Saha et al. 2009). For our UGC 463 data, the measured increase in ϒ^disk_{*, K} with radius is consistent with these claims; however, we cannot claim that a stellar-disk flare exists in UGC 463 based solely on our data. Therefore, barring more detailed information on the disk structure of UGC 463, we simplify our mass decomposition in Section 6 by largely focusing on results that assume ϒ^disk_{*, K} and h_z are constant for the entire disk.

Finally, we note that our data provide a few, limited assessments of the presence of a "DM disk" in UGC 463. Such a structure has been predicted by recent simulations (e.g., Read et al. 2009) and modeling of the Galaxy by Kalberla (2003) (cf., Moni Bidin et al. 2010). DM disks are expected to be more extended both radially and vertically than stellar thin disks; Kalberla (2003) fit a DM disk that has a scale length and scale height that are, respectively, 3 and 10 times larger than for the stellar thin disk. Our stellar kinematic data are expected to trace the thin disk mass distribution only such that we can place an upper limit on any DM distributed identically to this structure as follows. Assuming the ϒ^SPS_{*, K} prediction from Z09 is exactly correct, our ϒ^disk_{*, K} measurements suggest a thin DM disk that has ∼80% of the stellar mass surface density (or ∼30% of the baryonic mass surface density). One can increase the mass surface density of a DM disk in UGC 463 by proportionally increasing its scale height with relatively moderate effects on our calculation of Σ_dyn. Assuming no influence on Σ_dyn, the scale height of the DM disk would need to be roughly the same as the thin-disk scale length to reach the mass ratio that Kalberla (2003) measure for the Galaxy. Although one may accommodate such a disk within the current understanding of DM disks, such a structure is mostly conjectural with respect to our data. Given the uncertainty in both the expected vertical distribution of a DM disk and the SPS modeling results for ϒ^SPS_{*, K}, our mass decomposition assumes that no DM disk exists in UGC 463, which is consistent with our observations.

Figure 16 provides $\mathcal {F}_{\ast }^{\rm disk}$ and $\mathcal {F}_{\rm b}$ as a function of radius assuming a constant ϒ_{*, K}(R). Individual data points use our direct measurements of V_c, whereas the gray lines calculate V²_c = V_b² + V²_DM using V_b from Figure 14 and V_DM from the NFW and pseudo-isothermal halos fitted in Figure 15. Both $\mathcal {F}_{\ast }^{\rm disk}(R)$ and $\mathcal {F}_{\rm b}(R)$ are effectively constant between 1.0 ≲ R/h_R ≲ 3.5; the roughly constant $\mathcal {F}_{\rm b}-\mathcal {F}_{\ast }^{\rm disk}$ at these radii reflects the similarity between the radial distribution of the gas and stars. Figure 16 also provides the expected $\mathcal {F}_{\rm disk}$ for the fiducial maximum disk, which is the same for the NFW and pseudo-isothermal halos at R ≳ 0.5h_R. The shape of $\mathcal {F}_{\ast }^{\rm disk}(R)$ is very similar to that calculated for a maximal disk, which is essentially a statement that μ'_K is very close to an exponential after subtracting the central light concentration; however, the normalization is very different. At small radius, $\mathcal {F}_{\rm b}$ and $\mathcal {F}_{\ast }^{\rm disk}$ diverge largely due to the exclusion of the central mass concentration in the calculation of the latter.

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Table 8 gives $\mathcal {F}_{\rm b}(R_e)$, $\mathcal {F}_{\rm b}(2.2h_R)$, and $\mathcal {F}_{\ast }^{\rm disk}(2.2h_R)$ for both the constant and variable ϒ_{*, K}(R) assumptions. Our measurement of $\mathcal {F}_{\rm b}(R_e)$ has significant uncertainty due to the error in measurement of V_c at this radius (largely due to centering errors), the uncertainty in the baryonic rotation speed, and the difference in the extrapolation when based on either the NFW or pseudo-isothermal halo. To the contrary, both $\mathcal {F}_{\rm b}$ and $\mathcal {F}_{\ast }^{\rm disk}$ at 2.2h_R are relatively well constrained. Following the definition proposed by Sackett (1997) and assuming a constant ϒ_{*, K}(R), these quantities demonstrate that UGC 463 has a substantially submaximal disk by a factor of ∼(0.85/0.46)² ≈ 3.4 in mass; this reduces to 1.9 if one instead defines $\mathcal {F}_{\rm b}(2.2h_R) = 0.85$ as a maximal disk. These factors are consistent with our previous discussion in Sections 5.6.2 and 6.1 regarding the comparison of our dynamical measurements with the SPS predictions from B03. Adopting a variable ϒ_{*, K}(R) results in a mass profile of the disk that yields a peak rotation speed at or beyond the limit of our calculation (R = 15 kpc), whereas the constant ϒ_{*, K}(R) disk has a peak rotation at 2.7h_R. However, in both cases the stellar (and baryonic) disk remains submaximal due to the relatively constant value of $\mathcal {F}_{\ast }^{\rm disk}$ (and $\mathcal {F}_{\rm b}$) at R > h_R. Although it is possible that the baryonic mass is close to satisfying the $\mathcal {F}_{\rm b}(R_e)\sim 1$ given the uncertainty, our data suggest that the circular speed may have substantial contributions from DM even within the bulge region.

Our mass-surface-density and ρ_DM measurements from Figures 14 and 15, respectively, provide the mid-plane volume-density ratio between the dark and baryonic matter, (ρ_DM/ρ_b)_{z = 0}. We assume that the stratification of the gaseous and stellar disks combine to produce an exponential vertical density distribution with a single scale height h_z and that the bulge is spherical. Thus, the baryonic volume density at the disk mid-plane is (ρ_b)_{z = 0} = (Σ_* + Σ_gas)/2h_z + ρ_bulge. If the gas is more confined to the plane, this calculation results in upper limits. Figure 17 provides the mid-plane volume–density ratio when assuming a constant ϒ_{*, K}(R), where the discrete measurements use the non-parametric calculation of ρ_DM and the gray lines use the NFW and pseudo-isothermal sphere parameterizations. We also plot the expectation for the fiducial maximal disk discussed above. For reference, Bienaymé et al. (2006) find (ρ_DM/ρ_b)_{z = 0} = 0.14 in the solar neighborhood assuming a spherical halo.

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At R ≫ z, the change in ρ_DM with z is much smaller than the change in ρ_b such that (ρ_DM/ρ_b)∝exp (|z|/h_z). Averaging the data at 1 < R/h_R < 3 in Figure 17, we find that (ρ_DM/ρ_b)_{z = 0} ∼ 0.2, which is a factor of five larger than the expectation for a maximal disk (only 40% larger than the Milky Way value). The mass volume density of UGC 463 is, therefore, dominated by DM at |z| ≳ 1.6h_z at R ≳ h_R. This result is particularly important for our understanding of out-of-plane motions in the disk of UGC 463. The derivation of Σ_dyn = σ²_z/πkGh_z assumes an isolated, plane-parallel, infinite disk. Deviations from these assumptions, such as embedding the disk in a very massive DM halo, introduce systematic errors in the calculation, as briefly discussed in Paper II. The result for (ρ_DM/ρ_b)_{z = 0} in UGC 463 suggests that such effects may be significant for this galaxy.

Bottema (1993) has discussed the influence of a massive DM halo on σ_z in disk stars, continuing the work of Bahcall (1984). These authors find that σ_z should be inflated relative to an isolated disk when embedded in a massive, spherical halo; the degree of the inflation is proportional to (ρ_DM/ρ_b)_{z = 0}, as shown in Figure 15 from Bottema (1993). For the fiducial maximal disk shown in our Figure 17, Bottema (1993) would predict a less than 5% increase in the σ_z over an isolated disk, whereas our measurements for UGC 463 from Figure 17 suggest σ_z could be increased by 10%–15%. This means that our calculation of Σ_dyn∝σ²_z could overestimate the mass of the disk by 20%–30%.

Ideally, one would calculate Σ_dyn by first assuming an isolated disk and then converging to a solution that incorporated the effects of the DM halo. However, we have not done so here for UGC 463 because (1) the random error in our isolated-disk measurements of Σ_dyn are of the same order as this systematic correction; (2) there is substantial error in our measurement of (ρ_DM/ρ_b)_{z = 0} ∼ 0.2, even if we adopt the parameterized solutions for ρ_DM; and (3) there are equally unknown competing systematics that work in the opposite direction, such as the inclusion of a massive, razor-thin gas disk. The continued study of the effects of a massive halo on the velocity dispersion, as opposed to just the rotation curve (Hayashi & Navarro 2006; Widrow 2008), is worthy of a dedicated effort. However, given ambiguities regarding the three-dimensional structure of galaxies and the vertical stratification of disks (see discussion in Section 5.6.1), a detailed understanding of the influence of the DM halo on the disk is complicated. Here, we simply note that the impact of a relatively massive DM halo in UGC 463 works to further lower the maximality of an already submaximal disk.

As discussed in Paper II, the total baryon fraction ${\mathcal F}_{\rm bar} = \mathcal {M}^{\rm tot}_{\rm b}/\mathcal {M}^{\rm tot}_{\rm dyn}$ is ill defined; however, the surface density and V_DM measurements from Figure 14 allow for a robust calculation of the enclosed-mass budget to a finite radius, assuming the halo is spherical. The resulting mass budget is presented in Table 7 and the mass growth curves are shown in Figure 18. We end the calculation at R = 15 kpc (R = 4.2h_R; ∼R₂₀₀/10 for our fitted NFW halo), well within the limiting radius of our μ'_K measurements (Figure 2) but extrapolating beyond our dynamical data. We calculate the DM mass at 15 kpc using the fitted halo parameterizations and the percentage errors from the measured data. The mass growth curve of the fiducial maximal disk discussed above is overplotted in Figure 18 for reference. Although Table 7 only provides the results when assuming a constant ϒ_{*, K}(R), the results assuming a variable ϒ_{*, K}(R) are insignificantly different.

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Figure 18 shows that the integrated baryonic mass of UGC 463 is, at most, equal to the integrated DM mass at R ∼ h_R. Beyond this radius, the integrated DM mass quickly begins to dominate, such that $\mathcal {M}_{\rm b}/\mathcal {M}_{\rm DM}\sim 0.3$ at 15 kpc. To the contrary, the integrated mass of the fiducial maximal disk always dominates over the DM mass; the minimum mass ratio $\mathcal {M}_{\rm disk}/\mathcal {M}_{\rm DM}\sim 1.4$ is at R = 15 kpc.