4 Matching HI linewidths to $\vec {V}_{\bf max}$ and $\vec {V}_{\bf flat}$

In this section, it will be investigated how the linewidth correction for turbulent motion can be used to match the finally corrected global HI linewidths to the actual rotational velocities measured from the rotation curves.

After the correction for instrumental resolution, the profile widths are generally corrected for broadening due to turbulent motions of the HI gas by applying TFq's formula

$$\begin{eqnarray*}W^2_{R,l} = W^2_l + W^2_{t,l} \left[ 1 - 2 {\rm e}^{-\left(\fra... ...ft[1 - {\rm e}^{-\left(\frac{W_l}{W_{c,l}}\right)^2} \right] \\ \end{eqnarray*}$$

where the subscript l refers to the widths at the $l=20\%$or the $l=50\%$ level of peak flux. This formula yields a linear subtraction of W_t,l if W_l>W_c,l and a quadratic subtraction if W_l<W_c,l. Values of W_t,l and W_c,l are different for line width corrections at the 20% and 50% levels. The values of W_c,l indicate the profile widths where the transition from a boxy to a Gaussian shape occurs. The amount by which a global profile is broadened due to random motions is given by $W_{t,l}=2k_l\sigma$ where, for a Gaussian velocity dispersion $\sigma$, k₂₀=1.80 and k₅₀=1.18.

The generally adopted values for W_c,l are $W_{c,20}=120~{\rm km}\, {\rm s}^{-1}$ and $W_{c,50}=100~{\rm km}\, {\rm s}^{-1}$. The more important values of W_t,l, however, have been subject of some debate among various authors. With our new HI synthesis data we can give a meaningful contribution to this debate.

Bottinelli et al. (1983) came up with an empirical approach, based on a minimization of the scatter in the TF-relation. They assumed an anisotropic velocity dispersion of the HI gas of $\sigma_x=\sigma_y=1.5\sigma_z$ and a velocity dispersion perpendicular to the plane of $\sigma_z=10~{\rm km}\, {\rm s}^{-1}$. They determined the values of k_l by minimizing the scatter in the TF-relation and found k₂₀=1.89 and k₅₀=0.71, indicating deviations from a Gaussian distribution (broader wings). Due to the assumed velocity anisotropy, W_t,l has become a function of inclination angle and varies in the range 45<W_t,20<57 and 17<W_t,50<21 for inclinations ranging between $45^\circ<i<90^\circ$.

The same value of k₂₀=1.89 was adopted by TFq but they assumed an isotropic velocity dispersion of $\sigma_x=\sigma_y=\sigma_z=10~{\rm km}\, {\rm s}^{-1}$ and consequently advocate $W_{t,20}=2\cdot 1.89 \cdot 10=38~{\rm km}\, {\rm s}^{-1}$, independent of inclination. They did not address the situation at the 50% level.

Fouqué et al. (1990) also assumed isotropy but adopted $\sigma=12~{\rm km}\, {\rm s}^{-1}$. They determined k_l in a more direct way by comparing the corrected line width to the observed maximum rotational velocity $V_{\rm max}$ as derived from HI velocity fields. They found k₂₀=1.96 and k₅₀=1.13, indicating a near-Gaussian distribution, contrary to the findings of Bottinelli et al. Consequently, Fouqué et al. advocate the much larger values of $W_{t,20}=47~{\rm km}\, {\rm s}^{-1}$ and $W_{t,50}=27~{\rm km}\, {\rm s}^{-1}$ respectively.

A similar procedure was followed by Broeils (1992) using a sample of 21 galaxies with well defined HI velocity fields. Broeils made no a priori assumptions about the intrinsic velocity dispersion and did not decouple k_l and $\sigma$. He did, however, recognize that $V_{\max}$ may exceed $V_{\rm flat}$ and he determined for each galaxy the values of $W^{\max}_{t,l}$ and $W^{\rm flat}_{t,l}$ for which the differences

$$\Delta W^{\max}_{R,l} = W_{R,l} - 2 V_{\max}\sin(i)$$

and

$$\Delta W^{\rm flat}_{R,l} = W_{R,l} - 2 V_{\rm flat}\sin(i)$$

become zero for each galaxy. He found mean values of

$$W^{\max}_{t,20} = 21\pm2, \quad W^{\max}_{t,50}= 7 \pm 1 \\$$

$$W^{\rm flat}_{t,20} = 37\pm5, \quad W^{\rm flat}_{t,50}= 25 \pm 4 .$$

(Note that he quoted the much larger scatters instead of the errors in the mean quoted above.) He rejected his results, probably discouraged by the large scatters, and adopted the values W_t,20=38 and $W_{t,50}=14~{\rm km}\, {\rm s}^{-1}$ which he erroneously identifies with Bottinelli et al.'s results.

Finally, Rhee (1996a) performed the same investigation using 28 galaxies, most of them in common with Broeils' (1992) sample. Not surprisingly, he found

$$W^{\max}_{t,20} = 20\pm2, \quad W^{\max}_{t,50}= 8\pm2$$

$$W^{\rm flat}_{t,20} = 30\pm3,\quad W^{\rm flat}_{t,50} = 18\pm3$$

similar to Broeils' result.

Figure 2: Comparison of the global profile widths WR,l, corrected for instrumental broadening and random motions, with $2V_{\max}\sin(i)$ (upper panels) and with $2V_{\rm flat}\sin(i)$ (lower panels). The left panels consider WR,20 and the right panels WR,50. Different values of the random motion parameters Wt,l are used. Open symbol indicate galaxies with declining rotation curves ( $V_{\max}>V_{\rm flat}$) and filled symbols indicated galaxies without a declining part ( $V_{\max}=V_{\rm flat}$). See Sect. 4 for further details

Here, with our new and independent dataset, we follow the same strategy as Broeils and Rhee by investigating which values of W_t,l allow an accurate retrieval of $V_{\max}$ and $V_{\rm flat}$ from the broadened global profile. For this purpose we will only consider those 22 galaxies in our Ursa Major sample that show a flat part in their rotation curves (with a significant amount of HI gas) and that are free from a major change in inclination angle. Of these 22, there are 6 galaxies with $V_{\max}>V_{\rm flat}$. Note that both Broeils and Rhee used Bottinelli et al.'s prescription to correct for instrumental broadening which we are forced to adopt here to ensure a valid comparison between their and our results. We calculated the values of $W^{\max}_{t,l}$ and $W^{\rm flat}_{t,l}$ for which the average values

$$\begin{eqnarray*}\overline{\Delta W^{\max}_{R,l}} &=& \frac{1}{N}\sum\left({W}_{... ...& \frac{1}{N}\sum\left({W}_{R,l} - 2V_{\rm flat}\sin(i) \right) \end{eqnarray*}$$

become zero. This is done for both the entire sample of N=22 galaxies and for the subsample of N=16 galaxies with $V_{\max}=V_{\rm flat}$. For the entire sample we find

$$W^{\max}_{t,20} = 22, \quad W^{\max}_{t,50}= 5$$

$$W^{\rm flat}_{t,20} = 32, \quad W^{\rm flat}_{t,50} = 15.$$

These values are in good agreement with the (rejected) results of Broeils and in excellent agreement with the results of Rhee. The values of $W^{\rm flat}_{t,l}$ are larger than the values of $W^{\max}_{t,l}$ because the galaxies with $V_{\max}>V_{\rm flat}$ in our sample have considerable amounts of HI gas at their peak velocity in the rotation curve. This gas, rotating at $V_{\max}$broadens the global profile somewhat further. If we consider only the 16 galaxies for which $V_{\max}=V_{\rm flat}$ we find

$$W^{\max}_{t,20} = W^{\rm flat}_{t,20}= 23, \quad W^{\max}_{t,50}= W^{\rm flat}_{t,50} = 6$$

in agreement with the values of $W^{\max}_{t,l}$ we found when using all 22 galaxies.

Our results are illustrated in Fig. 2 where we show, for each of the 22 galaxies, the deviations $\Delta W^{\max}_{R,l}$(upper panels) and $\Delta W^{\rm flat}_{R,l}$ (lower panels) as a function of $V_{\rm flat}$. Galaxies with $V_{\max}=V_{\rm flat}$ are indicated by filled symbols, galaxies with $V_{\max}>V_{\rm flat}$ are indicated by open symbols. The upper two panels in each block show the results one obtains when using Broeil's adopted values of W_t,20=38 and $W_{t,50}=14~{\rm km}\, {\rm s}^{-1}$.

From the upper panels in the upper block it is clear that the maximum rotational velocity as derived from the corrected global profiles is severely underestimated when using the values of W_t,l derived by TFq and adopted by Broeils. This systematic underestimation disappears when W_t,20 is decreased from 38 to 22 km s^-1 and W_t,50 is decreased from 14 to 5 km s^-1. The upper two panels in the lower block show that if one is interested in the amplitude of the flat part, which is smaller than the maximum rotational velocity for galaxies with a declining rotation curve (open symbols), the average offset becomes less significant simply because the open symbols scatter upward. In this case, to obtain an average zero offset, we find similar values for W_t,l as those adopted by Broeils. However, we find the curious situation that the corrected width of the global profile systematically overestimates $V_{\rm flat}$ for galaxies with a declining rotation curve (open symbols) and systematically underestimates $V_{\rm flat}$ for galaxies with a purely flat rotation curve (filled symbols).

From this we can conclude that, in a statistical sense, the maximum rotational velocity of a galaxy can be reasonably well retrieved from the width of the global profile when using W_t,20=22 or $W_{t,50}=5~{\rm km}\, {\rm s}^{-1}$. The amplitude of the flat part can not be retrieved consistently for a mixed sample containing galaxies with declining rotation curves. Note that we have explored only a restricted range of rotational velocities: $8{-}200~{\rm km}\, {\rm s}^{-1}$.

Our results also indicate a non-Gaussian distribution of random velocities in the sense that $W_{t,20}/W_{t,50}\neq 1.80/1.18$. Interpreting W_t,20 and W_t,50 in terms of velocity dispersions it follows that

$$\begin{eqnarray*}\sigma_{20} & = W_{t,20}/2k_{20} &= 6.1~{\rm km}\, {\rm s}^{-1} \\ \sigma_{50} & = W_{t,50}/2k_{50} &= 2.1~{\rm km}\, {\rm s} \end{eqnarray*}$$

where k₂₀=1.80 and k₅₀=1.18 for a Gaussian distribution. Recall, however, that we advocate a different correction for instrumental broadening than Bottinelli et al.'s scheme used by Broeils and Rhee. With our correction method for instrumental broadening we find the somewhat smaller values of:

$$W_{t,20}= 22, \quad W_{t,50}= 2.$$

These smaller values of W_t,l allow to retrieve $V_{\rm flat}$ from the global profiles of galaxies with purely flat rotation curves and $V_{\max}$ for galaxies with declining rotation curves. Applying our correction method for instrumental resolution and the above-mentioned value of $W_{t,20}=22~{\rm km}\, {\rm s}^{-1}$ we find an rms scatter in $\Delta W_{20}=0.5W^i_R -V_{\max}$ of 6.8 km s^-1.