## 3. Density-wave theory argumentsOnce the photometric parameters have been determined, a model rotation curve of the form can be constructed, where and denote the contributions due to the bulge and disk, respectively. The bulge contribution is given according to Eq. (2) by where where denotes the central face-on
surface density of the disk.
NGC 1288 is not a grand-design spiral. The spiral structure of such galaxies is almost certainly due to swing amplification of perturbations in the disks (Toomre 1981, 1990). If the spiral arms of NGC 1288 were rigidly rotating spiral modes of the disk, they would have to be closely connected to the bulge, because it acts as a reflector of density waves in modal theory (Bertin et al. 1998a,b, Fuchs 2000). We conclude from the models presented below that the bulge dominates the dynamics of the disk at galactocentric radii up to 0.5 kpc. In the regions adjacent to the bulge the epicyclic frequency is relatively high and the critical wave length is rather small (cf. Eqs. (7) and (8)); 6 kpc in the models presented below. The appearance of the shapes of modes is dominated by the short wave-length solutions of the modal equations. These have typical wave lengths less than (Athanassoula 1984). Thus modal theory can account only for the innermost structures in the disk, such as the small bar with an outer boundary radius of 2.6 kpc, but not for the spiral arms reaching out to galactocentric distances of 16 kpc. Swing amplified shearing density waves can form over the disk
coherent global spiral patterns with low multiplicity, which shear and
eventually wrap up (Toomre
1981) where denotes the epicyclic frequency, The value of the and is shown in the lower panel of Fig. 4. The theoretical expectation according to a pure bulge/disk model is a two-armed spiral, which is clearly contradicted by the observation (cf. Fig. 2). Since it is generally expected that galaxies are imbedded in dark matter halos, we have also considered an additional dark halo component in the construction of the model rotation curve. The dark halo is modelled by a quasi-isothermal sphere, which leads to a further term, in Eq. (4). Since the rotation curve of NGC 1288 has not been
measured in detail, the parameters in Eqs. (10) and (11) are not
known. However, plausible models can be constructed, where the
expected multiplicity of spiral arms is in accordance with the
observations. In Fig. 5 we show such a model with parameters
= 2000
pc
Furthermore, after the introduction of a dark matter halo in the model the mass-to-light ratio is lowered to = 14 , which is more consistent with determinations in other nearby spiral galaxies. © European Southern Observatory (ESO) 1999 Online publication: November 23, 1999 |