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2. Rotation curves

Since the data on the galaxies in table 1 of chapter 2 stem from various sources, the technique used for the determination of the rotation curve varies somewhat from galaxy to galaxy. For 8 galaxies we have determined rotation curves ourselves with the method described below. Data on 4 edge-on galaxies, where the rotation curves have been derived from the 21-cm line data by taking the terminal velocity of the profiles along the major axis, and for NGC 5383 have been kindly provided by Dr. R. Sancisi. For the other 11 galaxies we have taken data from the literature as given by the authors, but we rejected the use of fitted Brandt curves, employed for instance by Rogstad and Shostak (1972) since these curves are biased towards a particular mass distribution. In practice this meant that we derived a rotation curve for NGC 2403 and NGC 4236 from Shostak (1973) and took Huchtmeier's version of the curve for IC 342, excluding the points obtained from single dish measurements (Huchtmeier, 1975). We did, however, include the model B rotation curve of M83 (Rogstad et al., 1974), to be able to compare this galaxy with other warped galaxies. We did not use the rotation curves based on 21-cm observations with single dish telescopes, because such curves are likely to be influenced by sidelobes, especially in the outer parts (cf. discussion by Sancisi, 1978). Moreover, these curves have been mainly derived from points along the optical major axis, which in view of the possible presence of warps may be incorrect.

For eight galaxies (the seven described in chapter 4, and M101) we have used the method outlined by Warner et al. (1973) and Rots (1974) to derive the rotation curve. This method works as follows. If the major axis of the disk of the galaxy has a position angle of the line of nodes psi and if the plane of the disk makes an angle i with the plane of the sky then, at a point with polar co-ordinates (R, theta) in the disk, the radial velocity will be:

Equation 1 (1)

where Vs is the radial velocity of the center of gravity of the galaxy, and Vc(R) is the rotation velocity at radius R. A minimization procedure is now set up whereby we specify Vs, psi and i and calculate Vc for a set of concentric annuli in the plane of the galaxy. We also calculate the dispersion D(R) around the value of Vc(R). We vary Vs, psi and i to achieve a minimum for D(R) over the entire radial range; the values for Vs, psi and i obtained in this way are used for the final determination of the rotation curve. The centre of each of the annuli is the centre of gravity of the galaxy; for the latter we take either the measured optical position of the nucleus or the position of the radio continuum source associated with the nucleus. The width of the annuli has been taken roughly equal to the halfpower beamwidth; this gives us enough datapoints per annulus to form a mean Vc(R); R is the mid-point. The datapoints are weighted with 1 / cos (theta - psi) to achieve zero weight at the minor axis, and with the HI distribution, since the accuracy in the individual datapoints is roughly proportional to the signal-to-noise ratio. We will refer to this method as the "standard method".

As discussed in chapter 5, in nearly all galaxies deviations from the simple model of a disk in circular motion have been found. The deviations are usually an order of magnitude smaller than the rotational velocities themselves. Therefore we can still derive an estimate of the rotation curve from the distorted velocity fields. For the five types of non-circular motions described in chapter 5 we made the following changes, if any, to the standard method:

  1. Motions associated with individual spiral arms: for M81 we have adopted Visser's (1978) corrected curve, otherwise we made no changes.

  2. Large scale symmetric deviations thought to be oval distortions: here we have adopted the standard method.

  3. Large scale symmetric deviations thought to be kinematical warps: we have modified the standard method by allowing psi and i to vary for each annulus, and we minimize D(R) for each annulus independently. Thus we assume that within each annulus, independent of its spatial orientation, the observed radial velocity is due to circular motion only.

  4. Large scale asymmetries: we use the standard method for the symmetric inner parts, and we ignore the outer parts (example: M101).

  5. Small scale asymmetries: these are ignored.

As discussed in chapter 3, the determination of large changes in the rotation curve in the inner parts is not possible of the beam if the telescope is larger than the scale over which the change takes place. If available, we have used optical data, mainly from spectroscopy of emission lines, to determine the rotation curve in the inner parts. For galaxies with substantial nuclear bulges no HI emission has been detected in the bulge. In some of these cases (like M81) the available optical data do not show unambiguously that the observed motions can be approximated by a simple rotation model. For edge-on galaxies the rotation in the inner parts cannot be measured optically. We think that the rotation curves for 14 of the 25 galaxies for which we calculated mass models are uncertain in the inner parts (see table 1). In a few other cases there is a lack of data between the last measured optical point and the first reliable 21-cm point; we have interpolated between these points.

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