4 Discussion and summary

Based on gas dynamical calculations in a fixed galactic potential with a weak bar-like distortion, we estimated errors in mass estimation from the rotation curves, and calculated the probability that observations would suffer from such errors. We found that, as well as the final morphologies of gas disks, the probability strongly depends on the pattern speed of a bar $\Omega _{\rm b}$, and weakly on the bar strength $\epsilon _0$. Among our nine models, the probability for the errors becomes maximal for the models with $\Omega_{\rm b}=0.8 \times (\Omega-\kappa/2)_{\rm max}$; the 15% of them have an error of a factor of five in mass estimation, if we observe the disks from an arbitrary viewing angle, and if we define rotation curves as the highest-velocity envelope of the p-v diagrams. Even in those erroneous cases, the galactic mass is not overestimated by more than a factor of six. In all of our models, the overestimation in mass is more probable than the underestimation. We consider only some particular cases for a weak bar, thus cannot obtain general conclusions. The above estimation however must be a guideline to consider the central galactic mass derived from an observed rotation curve.

Figure 10: Same as Fig. 9, but rotation curves are derived by taking the density-weighted mean velocity rather than the most rapidly rotating envelope of the p-v diagram.

Conventionally, rotation curves have been often defined as the peak-intensity velocity or intensity-weighted mean velocity of p-v diagrams. However, Sofue (1996) pointed out that these methods underestimate the rotation velocity, particularly in the central region, because the finite beam size causes the confusion with the gas with lower line-of-sight velocities on the p-v diagram; this effect is also demonstrated in Koda et al. (2002, in their Fig. 15). For rotation curves in highly inclined galaxies, this confusion can not be avoidable. Alternatively, the envelope-velocity of the p-v diagram is better suited to trace the central mass distribution (Sofue 1996; Sofue & Rubin 2001). Therefore we defined the highest-envelope velocity as our rotation curves in the above study. Here we repeat the same analysis for a comparison, using rotation curves derived from the density-weighted mean velocity, and shows the results in Fig. 10. $\tilde{P}_{\gamma}$s are always less than those in Fig. 9, and are almost zero at $\gamma_{\rm crit}=1.0$. This means that the mass derived from the mean-velocity rotation curves are almost always underestimated in the central regions of galaxies. These results suggest that the conventional method for deriving rotation curves from p-v diagrams is not also relevant to estimate the mass in galaxies with bar-like distortions.

Sofue et al. (1999) showed that most of the rotation curves rise steeply from the centers, reaching high velocities of about $100{-}300{\rm ~ km ~ s^{-1}}$ in the innermost regions. Owing to the large fraction of the rotation curves with these high central velocities, they discussed the idea that these velocities should be attributed to massive cores rather than to bars. We may have a chance to statistically clarify whether or not the massive cores exist by comparing a probability such as ours with the observed fraction of rotation curves with high central velocities. When we define the probability P averaged in all types of barred and non-barred galaxies by

$$P = \int \tilde{P}_{\alpha}(\Omega_{\rm b},\epsilon_0) f(\Omega_{\rm b}, \epsilon_0) {\rm d}\Omega_{\rm b}{\rm d}\epsilon_0,$$ (9)

where $f(\Omega_{\rm b}, \epsilon_0)$ is a distribution function of galaxies with a pattern speed $\Omega _{\rm b}$ and bar strength $\epsilon _0$, the existence of massive cores is confirmed if the fraction of galaxies with the central high velocities is more than P. For example, using our maximum calculated probability $\tilde{P}_{\alpha}\vert _{\rm max}=0.4$ and an observed fraction of barred galaxies $f_{\rm bar}\sim0.6$ (Knapen et al. 2000), P could be very roughly calculated as $P < \tilde{P}_{\alpha}\vert _{\rm max} \times f_{\rm bar} = 0.24$. Of course, we need more intensive studies for a number of barred potentials and parameters, and more precise knowledge of the distribution function of parameters for bars.

Acknowledgements

We are grateful to Y. Sofue for fruitful discussions. We also thank an anonymous referee and H. J. Habing, the editor, for useful comments. J.K. was financially supported by the Japan Society for the Promotion of Science (JSPS) for Young Scientists.