5 A comparison of inclinations

Present day instrumentation allows accurate measurements of the luminosities and global HI profiles of galaxies. In general, the observed scatter in the TF-relation is larger than can be explained by the observational uncertainties in these measured parameters alone. However, the uncertainty in corrections sensitive to inclination contribute significantly to the observed scatter. For a sample of randomly oriented galaxies more inclined than 45 degrees, an uncertainty of 1, 3 or 5 degrees in the inclination angle contributes respectively 0.04, 0.12 or 0.19 magnitudes to the scatter due to the uncertainty in line widths alone, assuming a slope in the TF-relation of -10. Therefore, it is important to determine the inclination angle of a galaxy as accurate as possible and this issue deserves some special attention.

From the photometric and HI synthesis data available, three independent measurements of the inclination angle of a galaxy can in principle be obtained; $i_{\rm opt}$ from the optical axis ratio, $i_{\rm HI}$ from the apparent ellipticity of the HI disk, and $i_{\rm VF}$ from fitting tilted rings to the HI velocity field. Each of these methods has its own systematic limitations which are important to recognize when estimating the actual inclination of a galaxy. In the following discussion we will briefly address those limitations and make an intercomparison of $i_{\rm opt}$, $i_{\rm HI}$, and $i_{\rm VF}$.

5.1 i_opt from optical axis ratios

The most widely used formula to infer the inclination angle from the observed optical axis ratio $(b/a)\equiv q$ was provided by Hubble (1926):

$$\cos^2({i}_{\rm opt}) = \frac{{q}^2 - {q}_0^2}{1 - {q}_0^2}$$

where q₀ is the intrinsic thickness of an oblate stellar disk. Holmberg (1946) determined an average value of q₀=0.20 which is still commonly used although it is obvious from images of edge-on systems that large variations in q₀ exist. For instance, Fouqué et al. (1990) found q₀ to vary from 0.30 to 0.16 for spirals of morphological types Sa to Sd respectively and q₀=0.42 for galaxies of type Sdm-Im. Apart from the debate on the intrinsic thickness, the observed axis ratio q itself has limited meaning since it is often defined at a certain isophote around which q may still vary as a function of radius. From images of edge-on disks in the Ursa major cluster (see Paper I) it can often be observed that the axis ratio keeps increasing outward until the faintest isophotes. An extreme example is NGC 4389, dominated by a narrow bar and surrounded by an extended faint halo. The axis ratios presented in Table 1 were not determined at a fixed isophote but were chosen to represent the stellar disk instead of a bulge, lopsided structures or a faint halo.

Figure 3: Intercomparison of the three independently determined inclination angles $i_{\rm opt}$, $i_{\rm HI}$ and $i_{\rm VF}$. Only the filled symbols ( $i_{\rm HI}<80^\circ$ when $i_{\rm VF}$ is involved) are considered for the unweighted quantitative assessment

5.2 i_HI from the inclined HI disk

Apart from the oblate stellar disk, the HI disk can also be used to determine the inclination. In general, the HI disk is much thinner than the stellar disk and its intrinsic thickness is of no concern. However, its patchiness, lopsidedness and the existence of warps and tidal tails may complicate the interpretation of the results from fitting ellipses to a certain HI isophote. Here, no correction for the intrinsic thickness of the HI layer was applied. However, the relatively large synthesized beams of imaging arrays at 21 cm may smear the observed HI disks to a rounder appearance. Therefore, a simple correction for beam smearing was applied to our measurements and the inclination of the HI disk was determined according to

$$\cos^2({i}_{\rm HI})=\frac{d^2_{\rm HI}-\Theta^2_d}{D^2_{\rm HI} -\Theta^2_D}$$

where $D_{\rm HI}$ and $d_{\rm HI}$ are the observed major and minor axis diameters of the inclined HI disk obtained by fitting an ellipse to the outer column density levels. $\Theta_D$ and $\Theta_d$ are the sizes of the synthesized beam in the direction of the major and minor axis of the HI disk.

5.3 i_VF from HI velocity fields

The inclination angle of an HI disk can also be measured by fitting tilted-rings to its velocity field (Begeman 1989). However, the inclination angle and the rotational velocity are strongly coupled and reasonable results can only be obtained for inclination angles between roughly 50 and 75 degrees. This procedure requires accurate velocity fields with high signal-to-noise ratios as well as many independent points along a ring. The advantage that velocity fields offer is the possibility to identify warps and to check the kinematic regularity of the HI disk. For instance, the optical appearance of a galaxy may look very regular while the outer regions of the HI disk may be strongly warped toward edge-on (e.g. N3726). Such a warp would broaden the global profile and an inclination correction based on the optical axis ratio would lead to an overestimate of the rotational velocity when dividing the "warp-broadened'' line width by $\sin(i_{\rm opt})$. Note that the inclination measurement of a tilted ring may be affected by non-circular motions due to spiral arms, bars and lopsidedness.

5.4 The comparison

For the comparison between the three differently inferred inclination angles we considered only those 27 galaxies with fully reduced HI data for which the velocity fields and integrated HI maps are available. We excluded the interacting galaxies (N3769, N3893, U6973) because their outer isophotes (optical and HI) are affected by tidal tails. We also excluded galaxies with perturbed or inadequately sampled velocity fields (N4088, U6969, N4389), galaxies with excessively patchy HI maps (N4102) and obviously lopsided galaxies (N4051). These eliminations leave us with 19 galaxies that have smooth outer isophotes, well filled HI disks and regular HI velocity fields.

Figure 3 presents the comparison between the three differently inferred inclination angles using two different values for q₀. When calculating mean differences and scatters using $i_{\rm VF}$, only galaxies with $i_{\rm HI}<80^\circ$ are considered because kinematic inclinations of highly inclined galaxies are systematically underestimated. The error bars on $i_{\rm VF}$ are based on the variations in $i_{\rm VF}$ between the various fitted rings but are not considered any further here.

The upper most panel compares $i_{\rm VF}$ with $i_{\rm HI}$. No significant offset is found for the 14 galaxies that meet the above-mentioned criteria. Assuming that $i_{\rm VF}$ and $i_{\rm HI}$ contribute equally to the scatter of 3.1 degrees implies that the inclination angle can be determined with an accuracy of 2.2 degrees from either the velocity fields or from the inclined HI disk. Note that the correlation turns up for $i_{\rm HI}>80^\circ$ due to the systematic underestimation of $i_{\rm VF}$ for highly inclined disks.

Comparing $i_{\rm opt}$ with $i_{\rm VF}$and $i_{\rm HI}$ does show a significant offset of roughly 3 degrees when assuming q₀=0.20 (middle panels). This offset is biggest toward edge-on as would be expected in case of an overestimate of the intrinsic thickness. Note that there are several galaxies with an observed optical axis ratio less than 0.20 which have been assigned an inclination angle of $90^\circ$.

This 3$^\circ$ offset disappears when q₀=0.09 is used (lower panels) and the rms scatter is reduced to only 1.9 degrees for $i_{\rm opt}$ versus $i_{\rm VF}$ but is still 4.0 degrees in case of $i_{\rm opt}$ versus $i_{\rm HI}$. In the latter case, the scatter is caused by a few nearly edge-on systems for which the higher uncertainties have no influence on the deprojection of the rotational velocities.

The adopted inclinations and their errors, listed in Col. 11 of Table 1 are best estimates based on all the information available for a particular galaxy, including the morphology of dust lanes if present. For galaxies which lack fully reduced HI synthesis data, the inclination angles were inferred from the optical axis ratios using q₀=0.09 for galaxies of type Sc and later and q₀=0.24 for galaxies of type Sbc and earlier. The latter value of q₀ seemed justified by the observed axis ratios of the (nearly) edge-on systems N4013, N4026 and N4111 of types Sb, S0 and S0 respectively. Unfortunately, there are not enough suitable galaxies available to determine q₀ as a function of morphology.

Figure 4: Layout of the HI atlas pages for the 30 galaxies with fully reduced data. All the data for these galaxies are presented on two facing pages. Results for the 13 galaxies with partially reduced data are presented on a single page per galaxy and include only the channel maps, the global profile and the XV-diagram. The linear scale is 5.4 kpc per arcminute