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Appendix A: The tilted ring model

To better constrain the disk orientation using the VLA+ GBT 21 cm datacube we first spatially smooth the data described in the previous section at 130 arcsec resolution to gain sensitivity. At this spatial resolution, we reach a brightness sensitivity of 0.25 K. Considering a typical signal width of 20 km s^-1 our sensitivity should be appropriate for detecting HI gas at column densities as low as 10¹⁹ cm^-2. To determine the disk orientation, we compare the tilted ring models directly against the full spectral database considering channels 12.88 km s^-1 wide (rather than fitting the moments of the flux distribution). The cube consists of 2475 positions (i.e., pixels 40 × 40 arcsec² wide) in which 21 cm emission has been detected, and for each position we have 25 velocity channels covering from –20 to –342 km s^-1 heliocentric velocities. We summarize below the main features of our method.

We use 110 tilted concentric rings in circular rotation around the center to represent the overall distribution of HI. Each ring is characterized by its radius R and by seven additional parameters: the H I surface density S_HI, the circular velocity V_c, the inclination i and the position angle PA with respect to the line of sight, the systemic velocity V_sys and the position shifts of the orbital centers with respect to the galaxy center (Δx_c,Δy_c). These last three parameters allow the rings to be nonconcentric and to have velocity shifts with respect to the systemic due to local perturbations (such as gas outflowing or infalling into the ring or M 31 tidal pull). Of these large set of rings, we allow only the parameters of eleven equally spaced rings, called the “free” rings, to vary independently. We set the properties of the innermost free ring to be the same as those of the next adiacent free ring (because of its small size it turns out to be highly unconstrained) and we keep the free rings to be the 11,22,33 ... 110th ring. The properties of rings between each of the free ring radii were then linearly interpolated. Each of the seven parameters of the ten free rings were allowed to vary. We assume that the emission is characterized by a Gaussian line of width w, which is an additional free parameter of the model, centered at V_c. We compute the 21 cm emission along each ring as viewed from our line of sight, and the synthetic spectrum at each pixel by convolving the emission from various ring pieces with the beam pattern. We then test how well the synthetic and observed spectra match by comparing the flux densities in 25 velocity channels. With this method, we naturally account for the possibility that the line flux in a pixel might result from the superposition along the line of sight of emission from different rings. As an initial guess for the free parameters of the tilted ring model we follow the results of Corbelli & Schneider (1997).

The assignment of a measure of goodness of the fit is done following two methods: the “shape” and the “v-mean” method. In the shape method, we minimize a χ² given by the sum of two terms: the flux and the shape term. The flux term is set by the difference between the observed and modeled fluxes in each pixel. The shape term retains information about the line shape only, which are lost when just the first few moments are examined. This is essential in the regions of M 33 where the emission is non-Gaussian, for example when the velocity distribution of the gas is bimodal (this is indeed the case for some regions in the outer disk of M 33, see CS). The shape term is given by the difference between the observed and the normalized modeled fluxes in each pixel and for every spectral channel. The normalized model spectrum is the flux predicted by the model in a given channel multiplied by the ratio of the observed to model integrated emission. In doing so the shape term is no longer dependent on the total flux. The shape term is computed only for pixels with flux larger than 0.2 Jy km s^-1/beam i.e. N_HI ≥ 1.3 × 10¹⁹ cm^-2. The error for the shape term is the rms in the baselined spectra, σ_n,i, which is the experimental uncertainty in the flux in the i-channel at the n-pixel. The noise in each channel of the datacube, σ_ch, is uniform and estimated to be 0.0117 Jy/beam/channel. As suggested by Corbelli & Schneider (1997) we estimate σ_n,i as: (A.1)where w_i = 12.88 km s^-1 is the width of the channel used to compare the data with the model and 2.57 km s^-1 is the database channel width. The flux term is affected by the uncertainty in the integrated flux and by calibration uncertainties, proportional to the flux. The calibration error forces the minimization to be sensitive to weak-line regions and is 5%. The resultant reduced χ² formula is the following: (A.2)where N is the number of pixels (2475) and N_f is the number of free parameters (71 for our basic model), leaving a large number of degrees of freedom. Given the difficulty in finding a unique minimum, we use a two-step method to converge toward the minimal solution. Since some of the parameters might be correlated, we begin by searching for minima over a grid of the parameters surrounding our first guess. We carried out several optimization attempts under a variety of initial conditions and with different orderings for adjusting the parameters. After iterating to smaller ranges of variation, we choose the parameter values that yield the minimum χ². In the second step, we check the minimal solution by applying a technique of partial minima. We evaluate the χ² by varying each parameter separately. We checked our solution by surrounding the galaxy with zero-flux observations for stabilizing the outermost ring. It is important to notice that the flux and the shape term yield a similar contribution to the minimum χ² value.

In the second method we determine a solution using the deviations of the integrated flux and of the intensity-weighted mean velocity along the line of sight at each pixel. We carried this out with another two-step procedure, allowing all 71 parameters to be varied. We start by keeping the ring centers and systemic velocity fixed; then the rings centers and velocities are considered as free parameters in the minimization as well. In order to keep the model sensitive to variations of parameters of the outermost rings, each pixel in the map is assigned equal weight. Pixels with higher or lower 21 cm surface brightness contribute equally to determine the global goodness of the model fit. Since the original data has a velocity resolution of 1.25 km s^-1, we arbitrarily set σ_m, the uncertainty in the mean velocity, equal to the width of about five channels (~6 km s^-1). This is simply a scaling factor that gives similar weight to the two terms in the χ² formula. The equation below defines the reduced χ² of the v-mean method: (A.3)where V_mod is the mean velocity predicted by the tilted ring model at the pixel n.

Given the large number of degrees of freedom, the increase of χ² corresponding to 1σ probability interval for Poisson statistics would be very small. The χ² standard deviation is of order 0.03 for the flux term and of order 0.006 for the shape term. Since the presence of local perturbations does not allow the model χ² to approach unity, we consider χ² fractional variations corresponding to the mean value of the two terms (i.e., 2%). By testing the χ² variations for each variable independently, we should have an indication on which ring and parameter is well constrained by the fitting procedure. Hence, we first arbitrarily collect all possible sets of tilted ring models which give local minima in the χ² distribution with values within 2% of the lowest χ² (which is 7.3 and 6.8 for the v-mean and shape method respectively). In Fig. A.1, we show V_c, Δx_c, Δy_c and ΔV_sys corresponding to an assortment of models whose χ² is within 2% of the absolute minimum value found. The adopted systemic velocity is V_sys = −179.2 km s^-1. The displacements of the ring centers and systemic velocities are not very large and increase going radially outward, as does the scatter between solutions corresponding to partial minima. The value of the velocity dispersion we find from the minimization is of order 10 km s^-1.

Fig. A.1 Parameters Vc, Δxc, Δyc, and ΔVsys corresponding to an assortment of tilted ring models whose χ2 is within 2% of the absolute minimum value found. In the panels to the left the parameters for the v-mean minimization are shown, and in the panels to the right those for the shape minimization. The continuous line connects the parameters of the best fit tilted ring models used for deriving the rotation curve.

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For each minimization method we then select a tilted ring model between those with acceptable χ² using the maximum

north-south symmetry criterion for rotation curves relative to the two galaxy halves. The corresponding values of i and PA are shown in Fig. 3 of Sect. 4 with the relative uncertainties. The uncertainties are computed by varying one parameter of each free ring at a time, around the minimal solution until the χ² increases by 2%. Simultaneous parameter variations within the given uncertainties gives χ² variations larger than 2%. In deriving the rotation curve, we take into account the uncertainties considering deconvolution models in which the inclination or PA of all the rings vary simultaneously. In this case we consider only 35% of the uncertainties displayed in Fig. 3, in either PA or i, in order to have a χ² within 2% of the minimal solution.

Appendix B: Finite disk thickness corrections

The “rotation curve” is the azimuthal component of the velocity in the equatorial plane of the disk at a given galactocentric distance. However, the 21 cm spectrum observed at a certain position in an inclined disk depends not only on the azimuthal velocity in the plane but also on two additional effects: the smearing due to the extent of the telescope beam and the vertical extension of the disk; both become more severe with increasing inclination. The tilted ring model fit runs over a smooothed database, spatially and in frequency, whose final geometrical parameters are then used to derive the rotation curve from a higher resolution dataset. Therefore, instead of including disk finite thickness effects in the tilted ring models we prefer to account for this and for the beam smearing in the 21 cm spectral cube at the original resolution using a a set of numerical simulations.

We assume the gas to be in an azimuthally symmetric disk inclined with respect to the line of sight according to the tilted ring model fit. The gas radial distribution is set equal to that given by the integrated spectral profile, while the vertical profile is modeled by an exponential with a folding length of 0.3 kpc. Only a disk component is considered with no allowance for a halo component (Oosterloo et al. 2007). For the beam we used a Gaussian with FWHM = 20 arcsec truncated at a radius of 24 arcsec; the channel spacing in the spectrum is 1.25 km s^-1. The input rotation curve is that obtained by the observed peak and mean velocities, and the random velocity is assumed to be isotropic and spatially constant with FWHM = 8.0 km s^-1, as observed in most of the disk. We ran simulations with and without a vertical rotation lag according to Oosterloo et al. (2007).

Using the above parameters, we simulate a synthetic HI data cube. The cube is used to derive the corresponding velocity for each position either the peak or the flux-averaged mean, and then build a simulated rotation curve. The differences between the simulated and the input rotation give the average corrections to the rotation curve as function of radius. We shall refer to these corrections as finite disk thickness corrections. They are of order 2–3 km s^-1 and reach values of 5 km s^-1 only within the innermost 200 pc. We apply these corrections to the rotational velocities.

© ESO, 2014