The Dependency of Bar Formation Timescale on Disk Mass Fraction, Toomre $Q$, and Scale Height

TL;DR

This work extends the Fujii relation for bar formation timescales from a single disk-mass parameter to a three-parameter space including disk mass fraction $f_ ext{disk}$, Toomre $Q$, and disk thickness $h_z$, using a large suite of N-body models. It confirms that the exponential dependence on $f_ ext{disk}$ persists and discovers an approximate linear dependence of the timescale on $Q$ and $h_z$, leading to a robust empirical formula: $rac{ au_ ext{bar}}{[T]} = A Q rac{h_z}{R_d} ext{exp}ig(rac{f_ ext{disk}}{B}ig)$ with best-fit $A$ and $B$. This relation enables quantitative predictions of bar formation in pure stellar disks, yielding timescales from $ oughly 0.2$ Gyr to beyond the Hubble time depending on the disk's dynamical state, while acknowledging that gas can delay bar formation further. The results illuminate how disk mass, stability, and thickness jointly regulate the emergence of galactic bars and offer a framework for interpreting the observed diversity of barred galaxies. All math is presented with proper $...$ delimiters.

Abstract

Bars are one of the most prominent galactic structures. The classical swing-amplification theory can qualitatively describe the spontaneous bar instability of stellar disks. Still, it cannot quantify the bar formation process or explain why some disk galaxies do not have a bar. Recent studies found that the bar formation timescale depends exponentially on the disk mass fraction of the host galaxy (dubbed as "Fujii relation"), but they only explored a limited parameter space, where the physical effects of Toomre $Q$ (local disk stability parameter) and disk scale height of the host galaxies are not fully explored. In this work, we check the robustness of the Fujii relation in a higher-dimensional parameter space of disk mass fraction, Toomre $Q$, and scale height. We find that the Fujii relation holds for disk galaxies with physically reasonable Toomre $Q$ and scale height. Furthermore, the bar formation timescale also approximately linearly depends on both Toomre $Q$ and scale height, with a more prolonged bar formation in a hotter or thicker disk. We propose an empirical relation to combine the dependency of the bar formation timescale on the three parameters. Based on the empirical relation and recent observations, we estimate that the bar formation timescale in pure stellar disks ranges from $0.20_{-0.06}^{+0.09}~\mathrm{Gyr}$ to $12.20_{-2.80}^{+3.37}~\mathrm{Gyr}$ or even significantly beyond the Hubble timescale in some extreme cases.

Figures (5)

• Figure 1: Left column: the "Fujii diagram", namely $\tau_\mathrm{bar}$ against $f_\mathrm{disk}$, for the sample with controlled $Q$ (upper panel) or $h_z$ (lower panel) indicated by the color of the data points. The solid lines are the linear fits of the data points in the corresponding color. For comparison, we also show the Fujii relations reported in fujii_etal_2018 (black dashed curve, Equation \ref{['eq: Fujii formula']}) and bland_etal_2023 (black dotted-dashed curve, Equation \ref{['eq: BH formula']}). Right columns: similar Fujii diagrams but for models with several fixed $(Q,\ h_z/\rm{kpc})$; the unshown models just show similar results. Consistent with bland_etal_2023, $\ln\tau_\mathrm{bar}$ in our sample also linearly anticorrelates with $f_\mathrm{disk}$, shown by the data points' diagonal distribution from top left to bottom right in each panel. Besides the Fujii relation, the $\tau_\mathrm{bar}$ secondarily depends on $Q$ and $h_z$: a hotter/thicker disk corresponds to a slower bar formation.
• Figure 2: The distribution of $\tau_\mathrm{bar}$ against $Q$ (left panel), $h_z$ (middle panel), and $Q\times h_z$ (right panel). The colors represent the models' $f_\mathrm{disk}$, and the solid lines are the linear fits of the data points in the corresponding color. Approximately, $\tau_\mathrm{bar}$ linearly depends on $Q$ and $h_z$ in a similar fashion, and a more prominent linear correlation relative to $Q\times h_z$ is visible in the right panel.
• Figure 3: To better visualize how the trend between $\tau_\mathrm{bar}$ and $f_\mathrm{disk}$ varies with $Q$ and $h_z$, we further show some results extracted from Figures \ref{['fig: Fujii relation']} and \ref{['fig: linear relation']}. Left panel: results with $(Q, h_z/\mathrm{kpc})=$$(1.0, 0.2)$ in blue, $(1.0, 0.8)$ in black, and $(1.8, 0.8)$ in red. Right panel: similar but for $\tau_\mathrm{bar}$ against $f_\mathrm{disk}$ with $Q\times h_z/\mathrm{kpc}=$ 0.2 in blue and 1.0 in red.
• Figure 4: Left two panels: the comparison between the measured $\tau_\mathrm{bar}$ (data points) and the predicted values of Equation \ref{['eq: fit']} (solid lines). Globally, the measured $\tau_\mathrm{bar}$ aligns with Equation \ref{['eq: fit']}. In both panels, we use black dashed curves to represent the locus of the Hubble timescale and gray shadows to represent the range of the bar formation timescale for pure stellar disks in the Universe (see more details in Section \ref{['sec: results: constrain']}). Right panel: the distribution of $\Delta\tau_\mathrm{bar}\equiv\tau_\mathrm{bar, predicted} - \tau_\mathrm{bar, measured}$, where $\tau_\mathrm{bar, predicted}$ is calculated with Equation \ref{['eq: fit']} based on the models' $f_\mathrm{disk}$, $Q$, and $h_z$. Note that in all runs $R_\mathrm{d}=3.0\ \mathrm{kpc}$.
• Figure 5: Fujii diagram for models with $(Q,\ h/\mathrm{kpc}) = (1.2,\ 0.6)$. Red points are taken from the main text, while green points correspond to models with twice the particle number in both the halo and disk. Colored solid lines show linear fits to the respective data points. For comparison, we also plot the relation reported by bland_etal_2023 (black dotted-dashed line, Equation \ref{['eq: BH formula']}). The higher-resolution models are consistent with the lower-resolution ones.