3. Density-wave theory arguments
Once 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 G denotes the constant of gravity. The rotation curve of an infinitesimally thin exponential disk is given by
where denotes the central face-on surface density of the disk. x is an abbreviation for and I and K are Bessel functions (cf. Binney & Tremaine 1987). The same mass-to-light ratio is assumed for bulge and disk, respectively. Unfortunately, the actual rotation curve of NGC 1288 is not known, but Bottinelli et al. (1986) have measured with HI observations a maximal rotation velocity of = 468 km s-1. In Fig. 4 the model rotation curve according to Eq. (4) is shown. The central surface density of the disk according to this determination is = 3500 pc-2, implying an unrealistically high mass-to-light ratio of = 24 .
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) 1. Swing amplification is most effective if the circumferential wave length of the density wave is
where denotes the epicyclic frequency,
The value of the X parameter is about 2 in the case of a flat rotation curve, but less in the rising parts of the rotation curve (Athanassoula et al. 1987). We apply in Eq. (7) a relation for found by analyzing the stellardynamical equivalent of the Goldreich & Lynden-Bell sheet (Fuchs 1991). The expected number of spiral arms is given by
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-2, = 12 kpc, and = 0.035 pc-3. The expected multiplicity of spiral arms is very similar to the observed as measured by the Fourier coefficients (cf. Fig. 3). Close to the center at 3 kpc (10") we predict and observe a two-armed spiral. In the region around 10 kpc (35") the geometry changes to a three-armed spiral, again in accordance with the theoretical prediction. In the domain R = 15 kpc (50") to 20 kpc (70") most power goes into a four-armed structure, both observationally and theoretically. This behaviour is very similar in all three colours B,V,I. However. the blue Fourier coefficients show more fine-structure due to the more pronounced effects of star formation regions and dust. Near 70" part of the power is shifted to even higher order Fourier coefficients. This is still consistent with swing amplification theory, because the expected number of spiral arms according to Eq. (9) has been calculated from the peak amplification factor of the swing-amplification mechanism. Actually there is a broad distribution of the X parameter (cf. Fig. 1 in Fuchs 1991). The same coexistence of spiral patterns with various m can be seen also in the inner parts of the disk of NGC 1288, although to a lesser degree because the amplitudes of the residuals are smaller there (see also Fig. 2).
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