4 Velocity dispersion

The linewidth in a single synthesized beam carries information about the velocity dispersion of the ensemble of HI clouds contained within the beam. This information is, unfortunately, confused with the signatures of other effects such as warps or flares of the HI layer, the extent to which HI structures are resolved, and the rotation velocity gradient over the beam. Warps and flares bring HI from different galactocentric radii into the line of sight, but they only cause confusion in highly inclined galaxies.

Figure 2: The mean velocity dispersion in the low resolution maps is plotted as a function of the area of the synthesized beam. The velocity dispersions have been corrected for instrumental resolution and the velocity gradient over the beam according to Eq. (4). The beam area is defined as ${1 \over 4} \pi b_{\alpha } b_{\delta }$, where $b_{\alpha }$ and $b_{\delta }$ are the FWHM beamsize in right ascention and declination. Open symbols represent the 13'' resolution data, filled symbols are the 27''resolution data. The pair of points with log(Beam area) < -1.6 represents the Local Group dwarf DDO 216.

Figure 3: The mean velocity dispersion as a function of $\sin(i)$ for the low resolution maps. Closed symbols are objects for which the inclinations were determined with a tilted ring fit. The inclination of the remaining objects (open symbols) was calculated from the optical axial ratios listed in Melisse & Israel (1994) and an intrinsic axial ratio of 0.15. The upper panel shows the uncorrected mean velocity dispersions. The lower panel shows the corrected values. The dashed line in the upper panel is the relation expected for a galaxy with a solid body rotation curve with a slope $1 ~{\rm {km~s^{-1}}}$ arcsec-1 (if seen edge-on) and a velocity dispersion of $9.5 ~{\rm {km~s^{-1}}}$.

Excluding lines of sight with low signal-to-noise ratios, we have used Gaussian fitting to obtain the mean velocity dispersions tabulated in Table 5. To first order, we corrected the maps for finite velocity resolution and linear velocity gradients over the beam according to the relation

   $$\sigma^2 = \sigma^2_{\rm obs} - \sigma^2_{\rm inst} - {1 \over 2} b^2 \Bigl( \nabla v \Bigr)^2$$

where $\sigma_{\rm obs}$ is the dispersion of a Gaussian e $^{-{1\over2}v^2/\sigma^2_{\rm obs}}$ fitted to the line profile at each position, $\sigma_{\rm inst}= 2 \cdot 0.8493 \cdot \Delta v$ is the dispersion of a Gaussian corresponding to the velocity resolution of the Hanning-smoothed data, and $\nabla v$ is the local velocity gradient over the beam, assumed to be of the form e ^{-x2 / b2}. We calculated the velocity gradient at each position from model velocity fields constructed with the rotation curves presented in the previous section. No correction for galaxy inclination was applied. The procedure is described in more detail in Appendix A.

   

Table 5: Mean velocity dispersions from single Gaussian fits.

Name	$<\sigma>_{13''}$	$<\sigma>_{27''}$	Name	$<\sigma>_{13''}$	$<\sigma>_{27''}$	Name	$<\sigma>_{13''}$	$<\sigma>_{27''}$
[1]	[2]	[3]	[1]	[2]	[3]	[1]	[2]	[3]
	$~{\rm {km~s^{-1}}}$	$~{\rm {km~s^{-1}}}$		$~{\rm {km~s^{-1}}}$	$~{\rm {km~s^{-1}}}$		$~{\rm {km~s^{-1}}}$	$~{\rm {km~s^{-1}}}$
DDO 22	12.0	11.2	DDO 64	10.8 $\pm$ 3.3	10.9 $\pm$ 2.5	DDO 125	6.2	7.3 $\pm$ 1.5
DDO 43	8.4 $\pm$ 2.2	9.0 $\pm$ 1.7	DDO 68	10.9 $\pm$ 3.8	12.6 $\pm$ 4.4	DDO 133	-	8.2 $\pm$ 2.0
DDO 46	9.3 $\pm$ 2.3	10.0 $\pm$ 2.2	DDO 73	5.8	8.5 $\pm$ 2.8	DDO 165	9.3 $\pm$ 2.9	12.0 $\pm$ 3.8
DDO 47	8.2 $\pm$ 1.9	8.7 $\pm$ 3.2	DDO 83	9.9 $\pm$ 2.4	10.0 $\pm$ 2.5	DDO 166	9.1	11.6 $\pm$ 3.4
DDO 48	9.3 $\pm$ 3.1	10.1	DDO 87	-	6.0 $\pm$ 2.0	DDO 168	9.9 $\pm$ 2.9	10.6 $\pm$ 3.3
NGC 2537	7.2	11.5 $\pm$ 5.2	Mkn 178	-	7.6	DDO 185	8.1 $\pm$ 1.6	8.8 $\pm$ 1.7
DDO 52	6.8 $\pm$ 2.8	7.4 $\pm$ 2.3	NGC 3738	(12.2)	(18.2)	DDO 190	9.2 $\pm$ 2.4	10.0 $\pm$ 2.4
DDO 63	7.1 $\pm$ 2.1	8.9 $\pm$ 2.3	DDO 123	7.4 $\pm$ 2.5	9.0 $\pm$ 1.9	DDO 216	6.3 $\pm$ 1.7	5.4 $\pm$ 2.6
NGC 2976	11.1 $\pm$ 3.5	11.8 $\pm$ 3.2	Mkn 209	-	11.5	DDO 217	7.5 $\pm$ 2.7	8.6 $\pm$ 2.7

Notes: results are corrected for finite velocity resolution and velocity gradients over the beam as discussed in the text. Velocity dispersion scatter values are given only if the area considered was at at least ten times the synthesized beam area. Column [2] gives mean velocity dispersion and rms scatter of the velocity dispersion over the galaxy from the full-resolution data, and Col. [3] the corresponding mean from the low-resolution data.

The high velocity dispersion in NGC 3738 is probably an artifact resulting from too small a beam/velocity gradient correction caused by the marginally resolved steep velocity gradient of the galaxy. Excluding NGC 3738, we find a mean velocity dispersion of $8.6\pm0.34 ~{\rm {km~s^{-1}}}$ at 13'' resolution and $9.5\pm0.38 ~{\rm {km~s^{-1}}}$ at 27'' resolution. Although the difference between the two results is significant, its magnitude is only 10$\%$. Moreover, these values are consistent with those in the literature (Shostak & van der Kruit 1984; Skillman et al. 1988). As the physical area contributing to the measured velocity dispersion increases with galaxy distance squared, we show in Fig. 2 mean velocity dispersions as a function of beam surface area for both the 13'' and 27'' resolution maps. The systematic increase in mean velocity dispersion with physical beam area increasing by two orders of magnitude is no more than about $2 ~{\rm {km~s^{-1}}}$. As our results are thus effectively insensitive to linear resolution on scales of $\sim$0.1 kpc and larger, we may compare the velocity dispersions of galaxies at various distances without fear of introducing large systematic effects.

Finally, we show mean velocity dispersions as a function of inclination in Fig. 3. We used tilted-ring inclinations and, lacking these, inclinations estimated from optical axial ratios assuming an intrinsic axial ratio of 0.15. The result is not sensitive to the exact value of this intrinsic axial ratio. For instance, use of the higher values suggested by the work of Staveley-Smith et al. (1992), increases $\sin~i$ values by at most 0.08. The upper and lower panels in Fig. 3 show velocity dispersions before and after the correction for inclination. The upper panel shows velocity dispersions increasing at the highest inclinations ( $\sin~i>0.9$; $i > 65^\circ$). This increase has disappeared completely in the corrected set in the lower panel. The widths of local line profiles depend on inclination only through the observed velocity gradient over the beam, which is steeper on average for high inclination angles. Thus, all observed galaxies, irrespective their absolute luminosity (-12.8 mag > M_B > -17.6 mag) are have mean velocity dispersions of about 10 $~{\rm {km~s^{-1}}}$, very similar to that of spiral galaxy disks. We will return to this result in a forthcoming paper.

The velocity dispersion maps shown in Fig. 4 are corrected for the local velocity gradient over the beam. Note that the steep inner rotation curves of NGC 2537 and NGC 3738 are not completely resolved, resulting in artificially large linewidths.