FINDING THE CENTER: AN ANALYSIS OF THE TILTED RING MODEL FITS TO THE INNER AND OUTER PARTS OF SIX DWARF GALAXIES

The rotation curves of galaxies can be used to compute their total mass distributions. The dark matter density profile can be inferred by computing the mass contribution of the visible components—stellar bulge, stellar disk, and gas disk—and subtracting it from the total mass. The functional form of the dark matter density profile has been the subject of much debate (van den Bosch & Swaters 2001; Rhee et al. 2004; de Blok 2010; Governato et al. 2010, 2012; Pontzen & Governato 2012; Teyssier et al. 2013). Observations tend to favor the pseudo-isothermal core (ISO) model (Gentile et al. 2004; Donato et al. 2009; de Blok 2010), whereas large-scale N-body simulations tend to favor the Navarro–Frenk–White (NFW, Navarro et al. 1997) halo model. In the center, the NFW density profile rises to a cusp and the ISO density profile is constant. The NFW model was derived by fitting the density profiles of the dark matter halos formed in large-scale dark matter only simulations (Navarro et al. 1997). More recently, the Einasto halo model has been used with some success (Navarro et al. 2004; Chemin et al. 2011). The model has one more parameter than the former two models, which adjusts the slope of the density profile in the center, allowing for both NFW-like and ISO-like profiles.

In light of the debates concerning the exact form of the dark matter density profile, we thought it would be fruitful to examine the accuracy of the data analysis by comparing the tilted ring model fits to both the inner ($r\lt 1$ kpc) and outer ($r\gt 1$ kpc) radii of a sample of six dwarf galaxies drawn from the H i nearby galaxy survey, THINGS (Walter et al. 2008). We focused on the simple question of how accurately one could determine the center of the galaxy using H i at different radii. The consistency of the center determination is a measure of how reliable the tilted ring model is at determining the rotation speed of the H i gas.

For H i observations of spiral galaxies, the rotation curve is determined using the tilted ring model (Begeman 1989). The model assumes that a galaxy is a rotating disk tilted at an angle of inclination to the observer's line of sight. A galaxy disk appears as a circle if the galaxy is not inclined, while at other inclinations the galaxy appears as an ellipse. To determine the rotation velocity at a given radius, the galaxy emission (ellipse) is divided into individual annuli of similar width and a velocity for the gas within each annulus is found from the radial velocity of positions within each annulus. This velocity is taken as the rotation velocity of the gas within the annulus and is a single point on the rotation curve. When the model is fit to the data, some parameters, such as the dynamical center (hereafter, center) location, are often derived from other observations. This simplifies the model fitting. For example, radio emission coming from the nuclear center of the galaxy is assumed to occur at the location of the potential well center (Trachternach et al. 2008). We can also determine the center of the galaxy from 3.6 μm observations that map old stellar populations (Oh et al. 2008; Trachternach et al. 2008) that have had time to relax into a stable configuration. Most of the data analysis to date has been done using the tilted ring method (Begeman 1989), although more recently improved methods have been developed (Gentile et al. 2004).

In this paper, we have selected a sample of dwarf galaxies for analysis. Using only H i observations, we fitted a tilted ring model to the data with all model parameters left free. This best determines the accuracy of the tilted ring model in the central regions of dwarf galaxies, since this is where the controversy has arisen. The validity of this procedure is tested by applying it to a simple N-body model of an exponential disk galaxy. These tools must be rigorously tested because new blind H i of all-sky surveys, such as WALLABY (Duffy et al. 2012), will start collecting data soon. WALLABY will observe 10⁵ galaxies and there initially will not be observations of those galaxies in other wavelengths to help constrain the tilted ring model.

The paper is organized as follows. We begin by describing the selection criteria for the six galaxies that were the subject of this study and we give a description of their properties. We then describe the methods we used to analyze the data. We give a brief review of the titled ring model and a more detailed description of our implementation of it. In Section 4 we present the results of our analysis and our conclusions are presented in Section 5.

This paper focuses on six nearby galaxies: NGC 6822, DDO 154, NGC 2366, IC 2574, DDO 53, and NGC 3627. NGC 6822 was observed with the Australian Telescope Compact Array (Weldrake et al. 2003), while the remaining five come from the THINGS Survey undertaken at the Jansky Very Large Array (Walter et al. 2008). Various properties of these galaxies are displayed in Table 1. These nearby dwarf galaxies have been selected for this study because they may be dominated by dark matter at all radii and the core is resolved at sub-kiloparsec scales.

Table 1.  The Galaxy Sample

Galaxy	${{ \mathcal S }}_{{\rm{H}}{\rm{I}}}$	MH i	${V}_{20,\mathrm{sys}}$	Dist	σ	Rd	Ref
	(Jy km s−1)	$({10}^{8}{M}_{\odot })$	(km s−1)	(Mpc)	(mJy)	(kpc)
NGC 6822	1745.2 ± 174.5	0.99	−53.6 ± 1.6	0.49	4.26	0.67	1
DDO 154	79.3 ± 7.9	3.46	375.0 ± 2.6	4.3	0.4494	0.68	2
NGC 2366	229 ± 22.9	6.25	100.0 ± 2.6	3.4	0.5267	1.59	3
IC 2574	401 ± 40.1	15.2	48.3 ± 2.6	4.0	0.5235	2.23	2
DDO 53	18.2 ± 1.8	0.56	17.4 ± 2.6	3.6	0.4469	0.43	4
NGC 3627	39.1 ± 3.9	7.98	719.7 ± 5.2	9.3	0.3868	2.95	2

Note. S_{H i} is the total flux density, M_{H i} is the H i mass, ${V}_{20,\mathrm{sys}}$ is the velocity from the global H i profile taken at the 20% level, and σ is an estimation of the noise fluctuations. These numbers were computed after the removal of spurious pixels and using a noise level of $2.0\sigma$ for NGC 6822 and $2.5\sigma$ for the rest. The distances are from Walter et al. (2008), with the exception of NGC 6822,which is from Weldrake et al. (2003). R_d is the exponential disk scale-length measured from the stellar disk.

References. (1) Weldrake et al. (2003), (2) Leroy et al. (2008), (3) Hunter et al. (2001), (4) Hunter et al. (2011).

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NGC 6822 is classified as a IB(s)m galaxy, an irregular galaxy with a bar. The SE side (the receding side) has a prominent hole in the H i emission whose origin is still unknown (Cannon et al. 2012). This galaxy is less than a half a megaparsec from the Milky Way. DDO 154 and NGC 2366 are both classified as Im galaxies, irregular galaxies, although they do exhibit rotation and seem to be flattened systems (Walter et al. 2008). IC 2574 is classified a Sm galaxy, which means it has a spiral structure but is irregular in appearance. This galaxy also exhibits rotation and seems like a flattened system, but has strong random motions within the galaxy disk (Walter et al. 2008). DDO 53 is another Im galaxy; the velocity maps indicate some rotation, but the global H i maps indicate little rotation (see Figure 12). This is the smallest galaxy in the sample, in both angular and physical size. NGC 3627 is classified as a Sb galaxy. It has two prominent spiral arms clearly seen in the H i map (Walter et al. 2008). This galaxy is not a dwarf galaxy; it was selected because it is very close by. The intensity map, velocity field, and H i profiles are shown in Figure 4 through 14.

We checked the accuracy of the algorithm by applying it to an N-body simulation of an exponential disk galaxy. This model is introduced as a simple way to test our algorithm on an exponential disk whose center is known in advance. It is not intended as a realistic model of the very complex gas and stellar dynamics of spiral galaxies. The model is a thin disk embedded in an NFW halo potential with a high concentration (model IV from Rhee et al. 2004). Since the model's center, inclination angle, position angle, and rotation curve are known in advance, we can run the code to see how accurately these parameters can be recovered. The Adaptive Refinement Tree (ART) N-body code was used to run the simulation (Kravtsov et al. 1997; Kravtsov 1999). The relevant model parameters are shown in Table 2. The model consists of $1.6\times {10}^{6}$ particles in total. There are no stellar or gaseous components to the model, just a dark matter halo and disk component. The model was run for 1.6 Gyr to ensure the final state of the disk was in dynamic equilibrium. The model disk was "observed" at two different inclinations to mimic the actual sample galaxy inclinations (see Section 4.1).

Table 2.  The Parameters of the Simulated Disk Galaxy

Model IV
Halo mass, Mhalo (M⊙)	$6.0\times {10}^{10}$
Disk mass, Mdisk (M⊙)	$5.7\times {10}^{8}$
Disk scale-length, Rd (kpc)	2.14
Disk stability parameter, Q	3.0
Number of halo particles	$3.45\times {10}^{6}$
Number of disk particles	$1.6\times {10}^{5}$
Halo Concentration, C	12.0
Duration of evolution (Gyr)	1.6
Time step (year)	$1.4\times {10}^{4}$

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Spurious pixels were eliminated from the data using a $2.5\sigma$ clipping. The noise level is determined using the imexam task from the iraf software package. Emission is required to be in at least three consecutive velocity pixels to be acceptable, and each pixel is required to have a signal-to-noise ratio (S/N) of at least 2.5σ. The S/N requirement adopted for NGC 6822 is 2σ. NGC 6822 is traveling toward us, and its H i spectrum is blueshifted near the Milky Way emission. Care was taken to eliminate any contamination due to intervening H i clouds in the Milky Way.

${V}_{\mathrm{sys}}$ is extracted from the global H i profiles by taking the velocities on each side of the spectrum at 20% of the peak and averaging them. The uncertainty is equal to the width of a velocity pixel. The velocities extracted are in excellent agreement with those reported by Walter et al. (2008) and Fisher & Tully (1981). The total flux density is calculated by taking the integral of the global H i profile. Errors in ${{ \mathcal S }}_{{\rm{H}}{\rm{I}}}$ are dominated by uncertainties in the flux calibration, taken to be 10%. The total flux densities found are in agreement with those published by Walter et al. (2008). H i emission is optically thin, so all of the atomic hydrogen in the line of sight is detected. The total flux density can be converted directly into a column density and then into H i mass. ${{ \mathcal S }}_{{\rm{H}}{\rm{I}}}$, H i mass, ${V}_{\mathrm{sys}}$, distance, and σ for these galaxies are reported on Table 1. Radial velocities are extracted from each sky position using the intensity-weighted method.

Spiral and rotationally dominated dwarf galaxies tend to have a flattened baryonic distribution that can be represented by a rotating disk. The tilted ring model is designed to describe rotating disks. The model assumes that the H i gas is rotating in a plane with near circular orbits, i.e., the circular motions dominate the non-circular (random) motions. The model breaks the galaxy emission into separate annuli of widths determined by the beamsize; generally a few annuli will fit within a single beam. The emission in each annulus is described independently by six parameters. They are the following:

• 1.  
  x_c: the dynamical center x coordinate;
• 2.  
  y_c: the dynamical center x coordinate;
• 3.  
  V_sys: the velocity of the galaxy with respect to the Sun;
• 4.  
  ϕ: the position angle of the major axis, defined as the angle between the northern direction of the sky to the major axis of receding emission;
• 5.  
  ${i};$ the inclination angle of the galaxy, defined as the angle between the line of sight and a line normal to the plane of the galaxy;
• 6.  
  V_c: the rotation velocity, at a distance R from the dynamical center.

The dynamical center center is not necessarily the same as the optical center of the galaxy because the dynamical center depends on the velocity field of the galaxy. Three of the model parameters, the center and the systemic velocity, should not vary throughout the galaxy disk.

The method determines the rotation curve by extracting a rotation velocity from each annulus. Every annulus contains many sky positions, each providing a radial velocity measurement. We require at least 35 sky positions within each annulus. These line of sight velocities are related to the tilted ring parameters by

$$\begin{eqnarray}&&{V}_{\mathrm{rad}}(x,y)={V}_{\mathrm{sys}}+{V}_{c}(R)\;\times \;\mathrm{sin}({i})\times \;\mathrm{cos}(\theta ).\end{eqnarray} \tag{ 1 }$$

This expression breaks the radial velocities into two terms, one representing the entire galaxy receding away from the Sun (at V_sys) and the other representing the internal rotation of the galaxy. R is taken as the mean radius of the annulus. θ represents the azimuthal angle within the galaxy. This angle can be computed as follows:

$$\begin{eqnarray}&&\mathrm{cos}(\theta )=\displaystyle \frac{-(x-{x}_{c})\;\times \;\mathrm{sin}(\phi )+(y-{y}_{c})\times \;\mathrm{cos}(\phi )}{R}.\end{eqnarray} \tag{ 2 }$$

Substituting Equation (2) into Equation (1) yields the full tilted ring model. ${V}_{\mathrm{rad}}(x,y)$ is determined from the velocity field, the line of sight spectra at each sky position within the annulus. The free parameters describe the geometry of the annulus: its inclination, position angle, center, systemic velocity, and circular velocity.

The predicted velocity field can be calculated at each location using the model parameters from Equation (1). ${V}_{c}(R)$ for a particular annulus is determined using a linear least-squares fit of the observed radial velocity versus $\mathrm{cos}\theta$. The intercept is V_sys. The slope is a measure of ${V}_{c}(R)\;\times \;\mathrm{sin}({i})$, as these parameters are coupled. The reduced ${\chi }^{2}$ of the fit is calculated by

$$\begin{eqnarray}&&{\chi }_{\mathrm{red}}^{2}(R,\phi ,{i},{x}_{c},{y}_{c},{V}_{\mathrm{sys}})=\displaystyle \frac{1}{{n}_{\mathrm{anu}}-5}\displaystyle \sum _{i=1}^{{n}_{\mathrm{anu}}}\left(\displaystyle \frac{{V}_{\mathrm{extr},i}(x,y)-{V}_{\mathrm{fit}}}{{\sigma }_{\mathrm{extr},i}}\right).\end{eqnarray} \tag{ 3 }$$

where ${V}_{\mathrm{extr}}(x,y)$ is the velocity field from the data, V_fit is the velocity calculated by fitting Equation (1), and n_anu is the number of sky positions in the annulus. The minimization of Equation (3) provides an estimate of portions of the radial velocity field which are due to the rotational motion of the gas on an inclined disk. The ${\chi }^{2}$ minimized in other works are equivalent to Equation (3), except for a numerical factor in front; minimizing either equation is essentially equivalent (Corbelli et al. 2010).

3.1. Implementation of the Tilted Ring Model

In order to determine the best-fit parameters (ϕ, ${i}$, x_c, y_c) at each radius, we compute ${\chi }_{\mathrm{red}}^{2}$ from the fitting, Equation (3). For each galaxy, the fits were done for three annuli with respective radii of 1, 2, and 3 kpc. For NGC 3627, a 4 kpc annulus was used instead of 1 kpc because of the scarce emission in the central region. The outer regions of galaxies often exhibit a systematic trend in inclination and/or position angle with radius. In the inner regions, projection effects are known to complicate the analysis since H i emission is optically thin (Rhee et al. 2004).

The three radii were chosen in order to see at which radius the center of rotation could be most reliably determined. The annuli and their widths are shown in Table 3. The tilted ring model was applied to the annuli by simply varying the center and orientation parameters over a grid and searching for the best values of ${\chi }_{\mathrm{red}}^{2}$ (Equation (3)). The ϕ and ${i}$ parameters were varied in increments of one degree. Formal fitting errors are too small to be used for the uncertainty of the measured rotational velocity, so instead, the velocity dispersion within an annulus (represented by the rms) is taken as the uncertainty (de Blok et al. 2008).

Table 3.  Average Radius and Width of Each Annulus

Galaxy	Radius Sampled		dR		dR/Rd
	(kpc)	(pxl)	(pc)	(pxl)
NGC 6822	3.00	314	28.5	3	0.043
	2.00	209
	1.00	104
DDO 154	3.00	96	135.5	$4\tfrac{1}{3}$	0.199
	2.00	64
	1.00	32
NGC 2366	2.98	120	98.9	4	0.062
	2.00	81
	1.04	42
IC 2574	4.01	134	116.4	4	0.052
	3.00	103
	2.01	69
DDO 53	0.89	34	96.0	$3\tfrac{2}{3}$	0.223
NGC 3627	4.06	60	225.4	$3\tfrac{1}{3}$	0.076
	2.98	44
	1.89	28
N-body #1	2.97	100	118.8	4	0.056
	1.96	66
	0.95	32
N-body #2	2.97	100	118.8	4	0.056
	1.96	66
	0.95	32

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In several cases the minimization of the rms alone results in an implausible center location and a system velocity that is not consistent with that derived from the summed H i emission. Since V_sys can be determined independent of the model, it can be used to determine if the tilted ring model is providing physically meaningful results. An additional criterion for assessing the quality of the tilted model fit is the difference between the titled ring radial velocity value and the actual value ($| a-{V}_{\mathrm{sys}}|$). Each of the models must pass three criteria to be deemed acceptable: the ${\chi }_{\mathrm{red}}^{2}$ must be less than 1, each annulus is required to contain at least 35 sky positions, and the value of the quantity $| a-{V}_{\mathrm{sys}}|$ must be less than 4 velocity channels.

We used two grids to search the parameter space, one over a broader area at lower resolution, and a smaller area at higher resolution. The (x_c, y_c) grid is initially chosen to cover an elongated region about 2 kpc in size, centered on the maximum in the H i emission. After searching these sky locations, a smaller grid roughly 1 kpc in diameter is drawn around acceptable centers of the initial search and searched at higher spatial resolution. To be acceptable, a model (i.e., tilted ring) must meet the three criteria listed in the previous paragraph. For each center on the grid we vary the position angle (ϕ) and inclination (${i}$) in one degree increments (∼8000 ellipses for each center). For each possible center we compute the number of acceptable models. We then compute the mean number of acceptable models for each center on the grid. We reject all centers that have a number of acceptable models smaller than the mean. This produces a list of centers that are close to the plausible center of the galaxy, determined by visual inspection. However, there are outliers that are not plausible. We then compute the mean absolute deviation of the center distribution (x_c, y_c). We reject centers that are more than two mean absolute deviations from the mean. The size of annulus, width of annulus, number of centers searched, and the spacing of the centers in both pixel and projected physical coordinates are shown in Table 4 and the search areas are shown in panel (d) of Figure 1 through 14. The locations of the surviving centers are used to define the next grid of centers to be searched.

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Table 4.  Centers in the Galaxy Sample

Galaxy	ncenters	Δ(x, y)c	$\tfrac{{\rm{\Delta }}{(x,y)}_{c}}{{R}_{d}}$	n'centers	Δ(x', y')c	$\tfrac{{\rm{\Delta }}{(x^{\prime} ,y^{\prime} )}_{c}}{{R}_{d}}$
		(pc) (pxl)			(pc) (pxl)
NGC 6822	99	95.0 (10)	0.141	456	28.5 (3)	0.042
DDO 154	91	250.0 (8)	0.368	317	62.5 (2)	0.092
NGC 2366	92	247.3 (10)	0.156	413	49.5 (2)	0.031
IC 2574	101	261.8 (9)	0.117	347	58.2 (2)	0.026
DDO 53	93	157.1 (6)	0.365	184	52.4 (2)	0.122
NGC 3627	99	270.5 (4)	0.092	72	135.3 (2)	0.046
N-body #1	49	237.6 (8)	0.111	145	59.4 (2)	0.028
N-body #2	49	237.6 (8)	0.111	145	59.4 (2)	0.028

Note. The number of centers for the coarse and finely sampled search areas and their separations are shown for each galaxy. Separations are shown in both parsecs and pixels.

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The dynamical center, position angle, and inclination angle are estimated using the second grid. The weighted average of the centers, weighted by ${\chi }_{\mathrm{red}}^{2}$, was taken as the center location and the σ of the centers was taken as the error.

The position and inclination angle were estimated by running the algorithm with the determined center. We then varied the position angle and inclination through all the combinations and kept models (i.e., tilted rings) that met the three criteria outlined above (consistent system velocity, 35 or more velocity values in the ellipse, and ${\chi }_{\mathrm{red}}^{2}$). For the surviving models we computed the mean position angle and inclination and rms of these values. We then did a 2σ clipping to remove outliers and recomputed the mean and sigma as our final position angle and inclination values for that radius. The results of running this algorithm on each galaxy (see Table 3) are shown in the next section.

The results of the our calculations, the dynamical center, position angle, and inclination angle are shown in Tables 5–12. The centers for each annulus are plotted on the velocity fields of the galaxy in Figure 2 and every other Figure from 3 to 15.

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Table 5.  Tilted-ring Analysis of the N-body Galaxy Model Viewed at Angles ϕ = 48 ${i}$ = 65

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
3	510.8 ± 4.0, 512.3 ± 4.1	48.6 ± 3.5	64.2 ± 7.1	0.079
2	511.0 ± 4.2, 512.5 ± 4.4	49.0 ± 4.9	60.7 ± 11.6	0.084
1	510.8 ± 4.7, 512.1 ± 4.9	48.6 ± 10.1	57.7 ± 10.9	0.094
Real	512 ± 1.0, 511 ± 1.0	48 ± 1.0	65 ± 1.0	0.020

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Table 6.  Tilted-ring Analysis of the N-body Model Viewed at Angles ϕ = 300 ${i}$ = 42

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
3	512.9 ± 4.7, 512.0 ± 4.7	300.9 ± 4.3	51.6 ± 10.2	0.094
2	513.2 ± 4.6, 511.8 ± 4.8	300.8 ± 6.9	51.8 ± 10.2	0.092
1	512.7 ± 4.8, 511.8 ± 5.0	301.4 ± 11.4	53.3 ± 10.1	0.096
Real	512 ± 1.0, 511 ± 1.0	300 ± 1.0	42 ± 1.0	0.020

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Table 7.  Tilted-ring Analysis of NGC 6822

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
3	401.1 ± 6.1, 409.3 ± 6.0	116.1 ± 8.9	44.7 ± 10.8	0.121
2	397.7 ± 5.9, 407.5 ± 6.1	115.1 ± 8.7	54.3 ± 10.6	0.120
1	397.8 ± 5.7, 408.1 ± 5.7	108.2 ± 12.7	58.1 ± 10.1	0.114
Weldrake 2003	395, 406	106	57	...

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4.1. N-body Galaxies

4.2. NGC 6822

NGC 6822 is the closest dwarf galaxy in the sample; it is observed with the highest resolution of all the galaxies in this work. The velocity field shows signs of warping in both inclination and position angle. The signature of warping is a distortion in the velocity contours (spider diagram) and also in the H i emission image. The most striking feature is the absence of emission (large hole) in the receding side of the galaxy. The algorithm found dynamical center positions that did not vary significantly with radius and were consistent with Weldrake et al. (2003). The position and inclination angles for each annulus follow a slight warp in the velocity field. The 3 kpc annulus has a lower inclination than that published for the whole galaxy. It should be noted that the tilted ring is not well-sampled by the H i emission.

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Table 8.  Tilted-ring Analysis of DDO 154

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
3	503.8 ± 5.5, 510.6 ± 5.6	225.9 ± 6.2	66.1 ± 8.6	0.361
2	501.4 ± 5.7, 509.7 ± 5.6	224.1 ± 10.8	64.1 ± 10.9	0.367
1	503.1 ± 6.1, 508.7 ± 5.8	227.5 ± 14.0	51.4 ± 10.2	0.387
Trach. 2008	504 ± 1.8, 510 ± 2.3	230	66	0.134
Oh 2011	506, 510	229	66	...

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Table 9.  Tilted Ring Analysis of NGC 2366

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
3	528.2 ± 5.8, 508.1 ± 6.8	47.8 ± 8.5	73.0 ± 3.6	0.139
2	531.1 ± 5.9, 508.8 ± 6.7	51.5 ± 12.6	60.1 ± 9.1	0.139
1	532.2 ± 6.3, 511.7 ± 6.5	39.8 ± 15.5	53.9 ± 10.3	0.141
Trach. 2008	515 ± 2.5, 500 ± 5.2	40	64	0.090
Oh 2011	517, 509	39	63	...

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4.3. DDO 154

DDO 154 is a dwarf galaxy with a well-behaved velocity field. There is a well-defined minor axis with a slight warp in the outer radii of the galaxy. The dynamical center positions and position angles are consistent with each other, as well as with Trachternach et al. (2008) and Oh et al. (2011). The inclination angles were found to be consistent with Trachternach et al. (2008) and Oh et al. (2011) for the 3 and 2 kpc annuli. However, the 1 kpc annulus has an inclination that is lower than that of the outer radii and also lower than that quoted for the whole galaxy. This difference is likely due to projection effects.

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Table 10.  Tilted-ring Analysis of IC 2574

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
3	510.8 ± 5.7, 503.7 ± 5.5	67.1 ± 7.9	71.6 ± 4.6	0.103
2	509.8 ± 5.8, 505.3 ± 6.0	62.3 ± 12.0	63.5 ± 10.7	0.109
1	512.2 ± 5.7, 508.7 ± 5.8	45.3 ± 14.7	53.2 ± 10.7	0.106
Trach. 2008	517 ± 9.9, 526 ± 6.9	56	53	0.157
Oh 2011	518, 525	53	55	...

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Table 11.  Tilted-ring Analysis of DDO 53

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/ Rd
1	521.6 ± 4.4, 476.1 ± 4.3	125.4 ± 16.6	56.3 ± 10.6	0.375
Oh 2011	522, 484	131	27	⋯

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4.4. NGC 2366

Distortions from uniform circular motion are present in the velocity field. There is a counterclockwise distortion in the emission. The dynamical center positions are consistent with each other and with the centers of Trachternach et al. (2008) and Oh et al. (2011). The position angle tracks the warping of the disk. The inclination angles for the 2 and 1 kpc annuli are consistent with Trachternach et al. (2008) and Oh et al. (2011). The inclination for the 3 kpc annulus is about 10° larger ($2.5\sigma$ difference), but more importantly, the error is much smaller for this annulus ($\sim 4^\circ$) than for the others.

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Table 12.  Tilted-ring Analysis of NGC 3627

Radius (kpc)	Center (pxl)	PA (deg)	inc (deg)	${\sigma }_{r}$/Rd
4	508.5 ± 2.5, 503.7 ± 2.0	171.2 ± 3.5	52.9 ± 7.6	0.073
3	507.6 ± 2.0, 509.0 ± 2.0	168.5 ± 2.3	41.7 ± 3.2	0.065
2	515.3 ± 2.0, 515.9 ± 2.0	185.5 ± 1.8	76.6 ± 1.6	0.065
Trach. 2008	512, 513	173	62	⋯

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4.5. IC 2574

4.6. DDO 53

4.7. NGC 3627

With the advent of large-scale surveys such as the ASKAP H i all-sky survey (Koribalski 2012), we expect rotation curve data to become available for thousands of galaxies. It is necessary to develop automated methods for producing rotation curve data (Kamphuis et al. 2015). We find that without prior information the unguided tilted ring algorithm described in this paper can produce reliable estimates of the dynamical center, position angle, and inclination angle at annuli greater than 1–2 kpc. There are several reasons for this. In general, the central regions of galaxies have more non-circular motions and the projection effects are pronounced, which make the tilted ring model difficult to apply.

It is instructive to quantify the uncertainties in the center location, ${\sigma }_{r}$, in terms of the exponential disk scale, R_d. For the unconstrained center determinations, the uncertainties in the central kpc are typically 1–2 R_d. When we include the requirement that the titled ring model have a radial velocity consistent with that found for the galaxy using the total H i emission, the uncertainties are much smaller, mostly lying in the range 0.1–0.4 R_d (see Tables 7–12). These latter error estimates are probably reasonable, as they show our measurements to be consistent with estimates of the center location made using other methods (Trachternach et al. 2008; Oh et al. 2011).

In the central kiloparsecs, these algorithms have trouble arriving at a plausible center without outside guidance. It is natural to ask how these center uncertainties would affect the inferred dark matter halo properties. Rotation curves are very sensitive to the dynamical center location. Incorrect determinations of the dynamical center affect the overall shape of the rotation curve, including the slope in the inner kiloparsec, which would then result in an incorrect determination of the dark matter density in the central region. The scatter of center locations from published results and this work show that the formal errors in the center are underestimated and need to be reevaluated in order to draw reliable conclusions on the dark matter halo properties in the inner parts of disk galaxies.

Given our finding that one has to make prior assumptions in order to get physically plausible tilted ring solutions one has to find a way to include these prior assumptions in automated methods. One method of quantifying the assumptions used in deriving results is the Bayesian method (Sivia & Skilling 2006). The Bayesian methodology has the advantage that one has to clearly state assumptions in the form of priors, and it standardizes the computation of estimated parameters and associated errors. This will make it much easier to compare results from different algorithms. The nested sampling method (Sivia & Skilling 2006) should also speed up the computation of the best-fit titled ring solutions. We propose this as the next step in anticipation of the new, very large H i rotation curve databases that are about to arrive.

This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We thank the referee for useful comments and suggestions that improved the manuscript.