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4.1 NGC 3198 Once More

Bottema (1993), from an analysis of velocity dispersions, claims that the maximum velocity of the disc component is 63% of the maximum observed velocity. The path to this result is strewn with assumptions, the most important of which are that discs are exponential with a velocity ellipsoid close to that in the solar neighbourhood, that Freeman's (1970) law holds, and that (B - V)old disc = 0.7 for all discs. As already discussed above, for NGC 3198 Bottema's result corresponds closely to the ``no m = 2'' solution proposed by Athanassoula et al. (1987).

A quick re-evaluation of Bottema's result can be formulated as follows. From the good agreement between the stellar and gas rotation data presented in Bottema (1988), it follows that the asymmetric drift in NGC 3198 is small, and thus the epicyclic approximation can be used. This states that kappa / 2Omega = sigmaphi/sigmaR, where Omega is the angular velocity, kappa is the epicyclic frequency, and sigmaR and sigmaphi are respectively the radial and azimuthal velocity dispersion. From the measurement of the rotation curve and the velocity dispersion along the major axis, ignoring the z-axis contribution for the moment, we can then calculate the radial velocity dispersion sigmaR as function of radius, and from the mass model we can calculate the critical radial velocity dispersion needed for axisymmetric stability. The ratio between these two is Toomre's (1964) parameter Q:

Equation 3 (3)

For NGC 3198 we then find for the ``maximum disc'' model a mean Q of 0.92, and for a ``no m = 2'' model a mean Q of 1.92. The real uncertainties in these values are about 20%. The important result is that indeed for NGC 3198 the measured velocity dispersion seems too low to have a stable ``maximum disc'' solution (see also Fuchs's contribution in this volume).

One can argue with this in several ways. First of all, the observed major axis velocity dispersion is a combination of sigmaphi and sigmaz, so the determination of sigmaR may not be entirely correct.

Second, the numerical factor 3.36, which corresponds to a Schwarzschild distribution of the random velocities in an infinitesimally thin disc, should perhaps be corrected for the effect of thickness, possibly also for a different shape of the distribution function, and certainly for the effect of the gas which is important for a late type spiral. The thickness correction alone (calculated assuming all the material in the disc, and the solar neighbourhood value for the ratio of vertical to radial velocity dispersions) lowers the numerical factor to 2.6 (cf. Shu 1968, Vandervoort 1970) and the effect of 10% of gas enhances then the factor to about 2.9 (cf. Toomre 1974). Other distribution functions may apply: Fuchs & Von Linden (1998) rediscuss and extend work by Graham (1967) and by Toomre (unpublished) on an exponential distribution function, for which the numerical factor is 3.944 instead of 3.36, so that with the thickness and 10% gas corrections we end up at 3.40. From Graham's (1967) Figure 1, where he shows results for several other distribution functions, it becomes clear that the uncertainty in the numerical factor could be easily 20-30%.

Finally, as argued by Kormendy (priv. comm., see also discussion after Fuchs's contribution), the influence of younger stellar populations in the spectra could result in lower measured velocity dispersions. In view of all these arguments, perhaps we should not be overly worried about the result that Q is just below 1.00 in NGC 3198.

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