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3. Motions associated with individual spiral arms

For a few galaxies the 21-cm maps have sufficient angular resolution to allow a study of the individual spiral arms. The data on M81 (Rots and Shane, 1975) and M101 (Allen, 1975) have shown beyond doubt that the HI closely coincides with the optically visible spiral arms. These data, in particular those of M81, have been considered as supporting evidence for the density wave theory of spiral structure formulated by Lin and collaborators (Lin, 1971, and references therein). Visser (1978) has analyzed the M81 data on the basis of these concepts. His results, in particular those outlining the effects of smoothing of the telescope beam, can be used to estimate the effects of the motions associated with spiral arms on the determination of the rotation curve.

Visser has calculated model velocity fields of M81 for a variety of resolutions: the unsmoothed response, and this response convolved to resolutions of 25", 50" and 120". In the unsmoothed case jumps in the equivelocity curves occur when an arm is crossed. These jumps can be large (30-50 km s-1) when the gas passes through a shock. When the resolution is poorer they diminish in amplitude, and at a resolution of 120" they disappear. Since the arms in M81 have a "grand design" the jumps create a typical pattern of "wiggles" superimposed on the axisymmetric velocity field. This pattern is observable 1) if the amplitude exceeds the typical uncertainty in the determination of the radial velocities, and 2) if the arm separation is at least twice as large as the beamsize (see also Allen, 1975).

In M81 a resolution of 120" means a ratio radius/beamsize (R/B) of about 8.5. M81 is one of those galaxies in which strong shocks may be expected to occur (Roberts, et al., 1975). Hence for the galaxies with R/B less than 10 we do not expect to see density wave patterns in the velocity fields. Still the presence of these patterns, whether they are smoothed or not, can influence the determination of the rotation curve. For M81 Visser has derived the "true" rotation curve, i.e. the one of the equilibrium disk, by an iteration process. This curve differs from the curves determined directly from the observed velocity fields. The differences are not large, but work through in the derived mass surface density distribution, sigmaM(r). The resulting uncertainty in sigmaM(r) is not likely to exceed about 20%.

We may make the following notes about those galaxies studied in sufficient detail to show the spiral structure in the 21-cm line maps:

  1. M81. There is an asymmetry between the western and the eastern arm, the latter being more pronounced. Visser's models agree best with the eastern arm.

  2. M51. Here the resolution is good enough to show the individual arms in the HI distribution (Shane, l975; Segalovitz, 1976), but no clear sign of a pattern is visible in the velocity field, probably because R/B is too low and the galaxy is seen too much face-on.

  3. M101. Again the spiral structure can be seen in the HI distribution (Allen, 1975), but not so clearly in the velocity field. There are some indications in the data that the deviations from circular motions in the regular inner parts are associated with the individual arms.

  4. M31. This galaxy is so highly inclined (i = 78°) that the individual arms cannot be easily identified. The HI distribution (Emerson, 1974, 1976) clearly coincides with optical spiral arms; in the velocity field large non-circular motions associated with the arms are found, but no clear pattern seems to be present.

  5. M33. In this galaxy the arms are not very pronounced. Rogstad et al. (1976) claim to have found a velocity pattern consistent with a model of circular motion perturbed by a small amplitude (linear) density wave.

  6. NGC 2403. Shostak (1973) argues for a systematic pattern of deviations from circular motion associated with the arms, but R/B is rather small (about 7)

From this list it becomes clear that only under favourable conditions (large R/B, open arms, strong shocks, inclination between 30° and 70°) systematic deviations from circular motions associated with spiral arms are clearly observed. In most galaxies the HI distribution is a better tracer of the arms than are the specific patterns in the velocity fields. The deviations from axial symmetry have, most likely, a small effect (20% at most) on the derived axi-symmetric parameters.

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