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7.2.3 Line-Width Definitions and Corrections

As mentioned in Sec. 7.2 the parameter used in the TF relations for predicting the luminosity for a particular galaxy is Vmax, the maximum amplitude of the rotation curve. If the dark matter distribution is isothermal (implied by flat rotation curves), then Vmax is a measure of mass, provided that the distribution of mass within galaxies is similar (see Sec. 7.3). Since galaxies have flat rotation curves over much of their extent, Vmax should be insensitive to differing measurement techniques, and has the additional advantage of being well defined. In the original formulation, Tully and Fisher used the doppler width of the 21-cm line profile measured at 20% of peak (W20), corrected for receiver broadening and galaxy inclination (i.e., Wi = W20/sin i ~ 2Vmax). This was also the procedure used by Aaronson and collaborators. It has the advantage of being conceptually simple and it is easy to compare between different researchers when these minimally corrected line-widths are used. The 21-cm line is particularly well suited to sampling the outer rotation curves of galaxies since the extent of detectable HI is often several times that of the visible galaxy (e.g. Roberts 1962; Bosma 1981). It is also relatively easy to detect HI out to considerable distances (e.g. the Hercules Cluster at V0 ~ 11000 km s-1 with Arecibo). It is worth emphasizing that errors in the corrected line-widths are likely to be the dominant source of observational error, so a signal-to-noise level of gtapprox 10 is essential.

Other definitions of rotational broadening have been used (e.g. W50), particularly if the signal-to-noise is low. While alternative definitions are acceptable, in principle, the fact that estimates of Vmax using W20 closely correspond to those values obtained from classical rotation curves via Halpha implies that W20 is a good choice. The use of different definitions for line-width become important when trying to compare results from the Halpha and HI techniques, or even from different investigators.

Bottinelli et al. (1983, 1984) correctly pointed out that the simple inclination correction given above does not allow for the effects of the turbulent motion internal to the gas; these are expected to be a significant source of doppler broadening for low luminosity systems. Observations of significant line-widths for face-on systems (e.g. Lewis 1984) attests to the importance of this effect. Without a correction for turbulence, Wi, and hence Vmax, would be progressively over estimated with decreasing luminosity (mass), resulting in a curved TF relationship (e.g. Aaronson et al. 1986; Mould et al. 1989). Using the observed velocity ellipsoid for the flattest population I components in the Galaxy, Bottinelli et al. estimated the turbulence at ~ 15 km s-1. The amount of the correction was slightly inclination dependent due to the assumption of an anisotropic velocity ellipsoid. They then linearly subtracted this correction from the observed profile width and corrected for inclination to obtain 2Vmax.

Tully and Fouqué (1985) suggested a slight modification to the Bottinelli et al. procedure. They advocated an isotropic velocity dispersion for the HI and a quadrature subtraction of the turbulence term for dwarf galaxies on the grounds that the velocity profiles are more like Gaussians than the ``box car'' profiles of the higher luminosity galaxies. Their prescription for correcting line-widths is consequently more complicated as it also includes a smoothing function to ease the transition between the two correction regimes:

Equation 13 (13)

where W20 is the measured width at 20% of peak intensity, Wt is the expected 20% width due to turbulence (38 km s-1) for an isotropic velocity dispersion of 10 km s-1, Wt is the characteristic transition width from ``horned'' to Gaussian shaped profiles (120 km s-1), and finally, WRi = WR / sini is the corrected rotational parameter (~ 2Vmax). Note, that this procedure results in the transition width (Wt) being a function of inclination.

These two correction schemes are essentially indistinguishable for WRi geq 100 km s-1, but differ substantially from the case of no turbulence correction. Obviously, care must be taken when comparing corrected line-widths from different researchers. Naturally, the calibrators and the sample galaxies should span a similar range in luminosity and WR to the program galaxies in order to minimize any systematic errors introduced by these correction procedures.

An optical alternative to HI velocity widths has been developed by Rubin et al. (1985) based on rotation curves measured at Halpha. This technique is particularly promising because detection efficiencies are high for telescopes in the 2-3 m class out to velocities of ~ 10,000 km s-1; thus, they are competitive with 21-cm line-widths obtained with the Arecibo telescope, while offering full sky coverage. In addition, optical spectroscopy can be applied in the crowded cores of more distant clusters, where the large beam of even the Arecibo dish suffers from confusion. While the Halpha data can easily be corrected for inclination, the correction for internal turbulence in low luminosity systems is complicated by outflow near star-forming regions. Recently, Faber and Corteau (1989) have re-examined the application of Halpha rotation curves for estimating Vmax, and find good agreement between optical and radio estimates. Dressler and Faber (1990b) have used an optical version of the TF relation to examine the velocity field in the vicinity of the ``Great Attractor''.

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