3 Galactic mass derived from rotation curves

In this section, we compare the apparent rotation curves obtained from the numerical results (Sect. 3.1) and the true rotation curves, then we get probability to overestimate/underestimate the galactic mass (Sects. 3.2 and 3.3).

  
3.1 Observing rotation curves in models

We obtain a position-velocity (p-v) diagram by observing our calculated gas disks edge-on, and then determine a rotation curve from the p-v diagram. We assign gas particles in a position-velocity grid using the cloud-in-cell method (Hockney & Eastwood 1981); the spacing for the grid is set to a typical resolution in recent interferometry observations of the CO gas for Virgo galaxies, i.e. $100\pc$ ($\sim$1 $\hbox{$^{\prime\prime}$ }$) in space and $5{\rm ~ km ~ s^{-1}}$ in velocity (Sofue in private communication). Then we determine a rotation curve by tracing the gas at the highest velocity for each radius in the p-v diagram. Some examples for p-v diagrams and rotation curves are shown in Fig. 6.

Figure 6: Position-velocity diagrams and rotation curves in model E, observed from a variety of viewing angles, at $t=500\rm ~Myr$. Solid lines show the distorted rotation curves derived from the calculated p-v diagrams (gray scale), while dashed lines indicate the rotation curve derived from the model potential. The mark "Detected'' indicates that our criterion with $\alpha _{\rm crit}=2$ is satisfied.

There are several ways to derive a rotation curve from an observed p-v diagram. One traces the peak-intensity velocity or intensity-weighted mean velocity at each radius (Rubin et al. 1980, 1982, 1985; Mathewson et al. 1992; Mathewson & Ford 1996), while another traces the 20% envelope of the peak-intensity velocity at each radius (Sofue 1996). These intensity-based methods cannot be applied to our density-based p-v diagram, because intensity is not a simple function of density, especially in edge-on systems. Compared with these methods, our method provides generally higher velocity.

  
3.2 Error estimation in rotation curves and mass

We compare the observed rotation curves $V_{\rm obs}(R)$ derived from p-v diagrams in simulations (Sect. 3.1), with the true rotation curves $V_{\rm pot}(R)$ from the gravitational potential ($\Phi_0(R)$). We estimate the errors in rotation curves by defining a function, i.e.

$$\alpha(R) \equiv \frac{V_{\rm obs}(R)}{V_{\rm pot}(R)},$$ (5)

and examining whether $\alpha$ exceeds an arbitrary critical value  $\alpha _{\rm crit}$ as

$$\alpha(R) \geq \alpha_{\rm crit}.$$ (6)

The ratio of the mass $M_{\rm obs}$, derived from an observed rotation curve $V_{\rm obs}$, over the true mass $M_{\rm pot}$ from the potential is,

$$\gamma(R) \equiv \frac{M_{\rm obs}}{M_{\rm pot}} [= \alpha(R)^2];$$ (7)

the second equal comes from Eq. (1). Then Eq. (6) becomes equivalent to

$$\gamma(R) \geq \gamma_{\rm crit},$$ (8)

where $\gamma_{\rm crit}= \alpha_{\rm crit}^2$.

A rotation curve rises steeply from a galactic center, having a peak, or at least a shoulder, at an innermost region, then reaching the flat rotation. A central galactic mass is always estimated at the radius of the peak or shoulder in observations (see Sofue et al. 1999). We thus consider the case that the mass is overestimated/underestimated at the radius of the first peak or shoulder. The radius depends on, and changes with a viewing angle for the gas disk (see Fig. 6). Hence, we define a reference region of $400\pc < R < 2{\rm ~ kpc}$ to which the above criterion, i.e. Eq. (6), is applied; in our models the first peak or shoulder always fall in this region. If Eq. (6) or (8) is satisfied in this reference region, our observed rotation velocity differs from the true velocity at least by a factor of $\alpha _{\rm crit}$.

3.3 Probability of erroneous mass estimation

We define the probability that the central rotation velocity is larger/smaller by a factor of $\alpha _{\rm crit}$, as the fraction of viewing angles. We obtain rotation curves by observing the gas disk for viewing angles at ten-degree intervals, and calculate the probabilities for three-hundred snapshots in each simulation. We hereafter describe the probability as $P[\alpha>\alpha_{\rm crit}]$ or $P_{\alpha}(\alpha_{\rm crit})$. Figure 7 shows a time evolution of $P_{\alpha}$ in model E. Corresponding to the three evolutionary phases (Sect. 2.4), the probability rises steeply in the linear perturbation phase, approaching a constant value in the transient phase, then reaching the constant in the quasi-steady phase. We also calculate the averaged probability in the quasi-steady phase, i.e. $300{-}500\rm ~Myr$, and describe it as  $\tilde{P}_{\alpha}$. Similarly $P_{\gamma}(\gamma_{\rm crit})$, the probability that the estimated mass differs from the true galactic mass by a factor of $\gamma _{\rm crit}$, is defined. $\tilde{P}_{\gamma}$ is a time-average of $P_{\gamma}(\gamma_{\rm crit})$.

Figure 7: Change of the probability $P[\alpha >\alpha _{\rm crit}]=P_{\alpha }(\alpha _{\rm crit})$ that Eq. (6) will be satisfied. $\alpha _{\rm crit}= 1.6$ and 2.0 indicate the cases that an observed rotation velocity exceeds the true value by a factor of 1.6 and 2.0 times, respectively.

Figure 8: The probability that an observed rotation velocity would exceed $\alpha _{\rm crit}$ times that inferred from the galaxy potential. Probability $\tilde{P}_{\alpha}$ is averaged over the time $t=300{-}500\rm ~Myr$ when the systems are in quasi-steady states. Models with different pattern speeds $\Omega _{\rm b}$ and bar strengths  $\epsilon _0$ are arranged vertically and horizontally, respectively.

Figure 9: The probability that an observed mass would exceed $\gamma _{\rm crit}$ times that inferred from the galaxy potential by chance. Probability $\tilde{P}_{\gamma}$ is averaged in $t=300{-}500\rm ~Myr$ when the systems are in quasi-steady states. Models with different pattern speeds $\Omega _{\rm b}$ and bar strengths $\epsilon _0$ are arranged vertically and horizontally, respectively.

Figure 8 shows the averaged probability $\tilde{P}_{\alpha}$ as a function of  $\alpha _{\rm crit}$ for all nine models. Different pattern speeds $\Omega _{\rm b}$ and bar strengths $\epsilon _0$ are arranged vertically and horizontally, respectively, as in Fig. 4. If the gas follows pure circular rotation, i.e. $V_{\rm obs}= V_{\rm pot}$, then these $\tilde{P}_{\alpha}$-profiles must be a step function: $\tilde{P}_{\alpha}= 1$ for $\alpha_{\rm crit}\leq 1$ and $\tilde{P}_{\alpha}= 0$ for $\alpha_{\rm crit}> 1$. However, the non-circular motion changes the $\tilde{P}_{\alpha}$-profiles as seen in the plots. In model E for example, $\tilde{P}_{\alpha}= 0.8$ at $\alpha_{\rm crit}= 1.0$ means that 80% of the rotation curves $V_{\rm obs}$ observed from random angles would apparently show higher velocities than the true rotation curve  $V_{\rm pot}$ which traces the mass, and another 20% would show lower velocities than the true one. Thus the overestimation in mass occurs more frequently than the underestimation in model E. $\tilde{P}_{\alpha}= 0.0$ at $\alpha_{\rm crit}= 2.4$ indicates that the observed rotation curve cannot be overestimated by more than the factor of 2.4 in model E.

We discussed in Sect. 2.4 that the final structure depends strongly on the pattern speed $\Omega _{\rm b}$ and a little on the bar strength  $\epsilon _0$. This is also evident in Fig. 8; the global profiles in the same  $\Omega _{\rm b}$ are quite similar, but $\tilde{P}_{\alpha}$ increases slightly with increasing $\epsilon _0$. Models A, B and C have no ILRs, not showing non-axisymmetric structures in the central regions (Sect. 2.4), thus the $\tilde{P}_{\alpha}$-profiles are similar to the step functions. Models D, E and F show the most prominent streaming motions in their central regions, and therefore they have the largest $\tilde{P}_{\alpha}$. Although models G, H and I have the same $\Omega _{\rm b}$, the $\tilde{P}_{\alpha}$-profile for model G is different from those for models H and I, because gaseous x₂-orbits remain in models H and I, but not in model G (see Sect. 2.4).

Figure 9 shows the probability $\tilde{P}_{\gamma}$ vs. $\gamma _{\rm crit}$. All the plots show properties similar to those in Fig. 8. In model E for example, $\tilde{P}_{\gamma}=0.15$ at $\gamma_{\rm crit}=5.0$ means that the central galactic mass derived from an observed rotation curve is overestimated by a factor of five in the probability of 15%. $\tilde{P}_{\gamma}$s in all models become zero at $\gamma_{\rm crit}=6.0$, meaning that the central mass from an observed rotation curve can be overestimated by at most a factor of six in our models.