Comparison of Bar Formation Mechanisms. IIIA. The role of classical bulges in spontaneous bar formation

Abstract

We run a suite of $N$-body simulations to investigate how classical bulges affect bar formation and properties under the internal formation mechanism. We incorporate bulges of varying mass and compactness into disk galaxy models and evolve them in isolation to examine the resulting bar pattern speeds and growth timescales. A more massive/compact bulge increases the Toomre $Q$ stability parameter and the circular velocity in the central region, while decreasing the disk mass fraction. It therefore delays the onset of bar formation and increases the bar growth timescale; sufficiently strong bulges can suppress bar formation entirely. During the formation stage, bars exhibit higher initial pattern speeds and faster deceleration rates when the bulges become more massive or compact. This faster deceleration persists after the bar buckling phase, leading to slower-rotating bars in the secular growth stage. However, when the bulge's "diluting" effect on the measured bar strength is removed or reduced, all bars within the same disk share similar distributions in the pattern speed-bar strength ($Ω_p$-$A_2$) space during the secular growth stage. They also show comparable ratios of the co-rotation radius to the bar length ($\mathcal{R}=R_{\mathrm{CR}}/R_{\mathrm {bar}}$) in this stage. These results suggest that the bulge's influence on the pattern speed is more significant during the bar formation stage, while in the secular growth stage, the bulge's effect may be less important, and the disk component dominates the pattern speed evolution.

Figures (7)

• Figure 1: Toomre $Q$ profiles (top row), circular speeds (middle row), and disk mass fraction $f_{\text{disk}}$ (bottom row) for the cold series models. Columns show increasing bulge compactness (left to right), with the scale radius (length) ratio $r_b/R_d$$=0.6$, 0.4, and 0.2. Solid lines trace models of varying bulge-to-disk mass ratio ($B/D$), with the minimum $Q$ value and $f_{\text{disk}} (2.2\;R_d)$ noted in the legend. The middle row decomposes $v_c(R)$ into contributions from bulge (dash-dotted), stellar disk (dashed), and DM halo (dotted) components. Vertical gray lines indicate 1$R_d$ and 2.2$R_d$ (where $R_d=2$ kpc).
• Figure 2: Bar strength $A_2$ (odd rows) and pattern speed $\Omega_p$ (even rows) for different models. Models are arranged by increasing bulge compactness from left to right, with $r_b/R_d$$=0.6$, 0.4, and 0.2. Rows show results of different dynamical hotness: the cold series ($\sigma_{R,0}=73 \; {\rm km/s}$, top two rows) and the warm series ($\sigma_{R,0}=124 \; {\rm km/s}$, bottom two rows). The results for the hot series ($\sigma_{R,0}=226 \; {\rm km/s}$) are not shown as no bar forms in isolation for any of the models in this series.
• Figure 3: The $\Omega_p-A_2$ space of all simulations. Panel layout and color scheme follow \ref{['fig:bar_pro']}. Three definitions of bar strength $A_2$ are employed: (1) $A_2$ from all stars within $4\;R_d$ (rows 1 & 4); (2) $A_{2,\text{disk}}$ from disk particles only within $4\;R_d$, removing the bulge (rows 2 & 5); (3) $A_{2,\text{excl}}$ from all stellar particles within $4\;R_d$ but excluding the central $1\;R_d$, mitigating bulge influence (rows 3 & 6). In the bar formation phase, higher bulge mass leads to higher $\Omega_p$ at fixed bar strength. In the secular growth stage, models converge to similar $\Omega_p$ at fixed $A_2$ when the bulge contribution is excluded or reduced.
• Figure 4: Similar to \ref{['fig:PS_A2']}, but in the $\Delta L_z-A_2$ space. The $y$-axis shows the change in angular momentum ($\Delta L_z$) of the stellar disk within 4 $R_d$. During bar formation, stronger bulges lead to greater angular momentum loss in the disk at a given bar strength. In the secular growth stage, however, models exhibit comparable $\Delta L_z$ at fixed $A_2$ once the bulge's contribution is removed or reduced.
• Figure 5: Time evolution of the co-rotation to bar-length ratio (denoted $\cal R$) since bar formation ($t_0$, defined at $A_2$$=0.1$). Layout and coloring match \ref{['fig:bar_pro']}. The vertical axis displays $R_{\mathrm{CR}}$, $R_{\mathrm{bar}}$, and their ratio $\mathcal{R}$. The horizontal dash-dotted line indicates $\mathcal{R}=1.4$, separating fast and slow bars. Reliable measurement of $R_{\mathrm{CR}}$ becomes impossible beyond ${\sim}10\;R_d$ due to low particle counts; these epochs are omitted. Bars in models with a more massive bulge show a lower $\cal R$ ratio at their birth but hold a similar $\cal R$ ratio after buckling.