Coherent Stellar Motion in Galactic Spiral Arms by Swing Amplification

First, we examine the typical evolution of structures. Initially the surface density is almost uniform but includes the small density fluctuation due to the particle noise. Thus, the density fluctuation can grow by the self-gravity. In any model, the density structures appear readily.

In the fiducial model (model k4), at $\Omega t/2\pi=0.5$, the trailing structures are generated spontaneously, which correspond to the spiral arms. Figure 1 presents the surface density at $\Omega t/2\pi=1.5$. We find the clear trailing structures with pitch angle of about $20^{\circ}$. The separations between spiral arms in the $x$ and $y$ directions is roughly $\sim\lambda_{\mathrm{cr}}$ and $\sim 2\lambda_{\mathrm{cr}}$, respectively. These results are consistent with the swing amplification model as shown in Papers I and II.

These spiral structures are not steady but transient and recurrent, which are generated and destroyed continuously. This activity continues throughout the simulation time. This behaviour has been observed in the local simulations (Toomre & Kalnajs, 1991, Paper I) and the global simulations (Sellwood & Carlberg, 1984; Baba et al., 2009; Sellwood, 2000, 2010; Fujii et al., 2011). Since the properties of these structures do not change during $\Omega t/2\pi=1.0\mbox{--}5.0$, in the following we analyze the spiral arms during this period.

In order to analyze the averaged time-evolution of various quantities, at first we introduce the time average over $T$ of the space-time cross-correlation of quantity $f$ with $\Sigma$ as

where $s$ is the lag for space-time cross-correlation. This function means the correlation between $\Sigma$ and $f$ at two different times and points, which traces the typical time evolution of $f$ around the overdense region. It should be noted that in this formulation the shear motion is taken into account. Because the averaged velocity of the focused region in the $y$ direction is given by the shear velocity $-2Ax^{\prime}$, the typical displacement of the region in the $y$ direction during time $s$ is expected to be $-2Ax^{\prime}s$, where $A$ is Oort’s $A$ constant. Thus we introduce the offset $-2Ax^{\prime}s$ into the $y$ component.

Choosing $f=\Sigma/\Sigma_{0}-1$, we define the space-time autocorrelation as

which shows the typical evolution of the surface density fluctuation around the overdense region. Setting $x=y=0$, we obtain the time autocorrelation function as

which means the typical time evolution at the center of the overdense region.

The time autocorrelation functions for models k0, k4, and k8 are shown in Figure 2. The time autocorrelation decreases with increasing $s$. This means that the overdense region declines with time. We find the local minimum and the local maximum. On average the density at an overdense region tends to increase again after its first decay, which seems to be a damped oscillation (Julian & Toomre, 1966).

The damping time of the time autocorrelation function is the typical timescale of the activity of spiral arms. We define $s_{\mathrm{min}}$ as the time when $\bar{\Psi}$ takes the first local minimum. Similarly we define $s_{\mathrm{max}}$ as the time when $\bar{\Psi}$ reaches the local maximum after $s_{\mathrm{min}}$. The dependencies of $s_{\mathrm{min}}$ and $s_{\mathrm{max}}$ on $\kappa/\Omega$ are summarized in Figure 3. Both $s_{\mathrm{min}}$ and $s_{\mathrm{max}}$ decrease with increasing $\kappa/\Omega$. We compare them with two dynamical timescales, the epicycle period $t_{\mathrm{e}}=2\pi/\kappa$ and the shear timescale $t_{\mathrm{s}}=1/(2A)$. If the spiral arms are destroyed by the shear or the tidal force, it is expected that the damping time is characterized by the shear timescale. However, the shear timescale increases with $\kappa/\Omega$, and its dependence on $\kappa/\Omega$ is completely different from those of $s_{\mathrm{min}}$ and $s_{\mathrm{max}}$. On the other hand, the epicycle period decreases with increasing $\kappa/\Omega$, which has a similar dependence to $s_{\mathrm{min}}$ and $s_{\mathrm{max}}$. Thus the epicycle motion relates to generation and destruction processes.

The time autocorrelation function evolves like a damped oscillation. This is consistent with the swing amplification model discussed in Paper III (Julian & Toomre, 1966). The elementary process of the swing amplification is the phase synchronization of the epicycle motion. Therefore, the timescale of the spiral activity is also described by the epicycle period.

If we neglect the self-gravity of stars, the motion of a star is separated into two components: a guiding center and an epicycle, which are given as (e.g., Binney & Tremaine, 2008)

where $(x_{\mathrm{g}},y_{\mathrm{g}})$ is the position of the guiding center, $x_{\mathrm{a}}$ is the amplitude of the epicycle oscillation, and $\phi$ is its phase. The $x$ component of the guiding center $x_{\mathrm{g}}$ remains constant while its $y$ component $y_{\mathrm{g}}$ is given as

where $y_{\mathrm{g0}}$ is the initial $y$ component of the guiding center. The phase $\phi$ varies with time as

where $\phi_{0}$ is the initial phase. Using Equations (12), (13) and (14) we can calculate $x_{\mathrm{g}},y_{\mathrm{g}},\phi$ and $x_{\mathrm{a}}$ from the position $(x,y)$ and velocity $(\mathrm{d}x/\mathrm{d}t,\mathrm{d}y/\mathrm{d}t)$.

We divide the computational domain into $150\times 150$ cells. Selecting the stars whose guiding centers are in each cell, we calculate the average of the relative position of stars to their guiding centers $(x-x_{\mathrm{g}},y-y_{\mathrm{g}})$ and the corresponding phase $\bar{\phi}$ from

where angle brackets denote the average in each cell.

Figure 4 shows the surface densities of stars $\Sigma$ and their guiding centers $\Sigma_{\mathrm{g}}$ and $\bar{\phi}$. At $t=0$, because the stars are distributed uniformly, in other words, their guiding centers and epicycle phases are given randomly, $\Sigma$, $\Sigma_{\mathrm{g}}$ and $\bar{\phi}$ have no structure completely. After that, the gravitational instability takes place and the spatial structure appears. At $\Omega t/2\pi=2.0$, $\Sigma$ and $\bar{\phi}$ show trailing structures, while $\Sigma_{\mathrm{g}}$ is nearly uniform. At $\Omega t/2\pi=4.0$, basically the structure remains the same. These results clearly show that the phase synchronization of the epicycle motion enhances the surface density of stars in the spiral arms. During the phase synchronization the spatial distribution of the guiding centers is kept almost uniform since their change is not significant.

The stellar motion in spiral arms is important to understand generation and destruction processes of spiral arms (Baba et al., 2013). We extract a typical spiral arm and investigate dynamics of stars in it.

We search the highest surface density cell at $\Omega t/2\pi=4.0$ for model k4, which is located at $(x,y)=(1.25\lambda_{\mathrm{cr}},-0.75\lambda_{\mathrm{cr}})$. Next we extract the group of the high density cells with $\Sigma/\Sigma_{0}>1.4$ that include the highest surface density cell and connect to each other. These cells consist of the amplified spiral arms. We investigate the motion of stars in this region. Figure 5C shows these stars at $\Omega t/2\pi=4.0$. The stars are separated into 7 groups by their $x$ position and we distinguish them by color.

Figure 5 shows the spatial distribution of the selected stars at $\Omega t/2\pi=3.6\mbox{--}4.4$. At $\Omega t/2\pi=3.6$ (Fig. 5A), the stars are scattered in the leading pattern. Although they diffuse in some degree, we can see the coherency of stars. At $\Omega t/2\pi=3.8$ (Fig. 5B), they come close to the center. The width in the $x$ direction becomes small. At $\Omega t/2\pi=4.0$ (Fig. 5C), they concentrates on the spiral arm, which show the clear trailing pattern. At $\Omega t/2\pi=4.4$ (Fig. 5D), the width of the pattern widens and the density decreases finally. During the rotation of the pattern from leading to trailing, the density is enhanced. The rotation of the pattern and the density enhancement coincide, which is consistent with the swing amplification mechanism.

Figure 6 shows the evolution of the averaged $x$ position of the stars and their guiding centers of each group. The variations of the guiding centers are smaller than those of their positions. This is consistent with the fact that the guiding center distribution remains uniform although the spiral arms are generated as discussed in Section 3.2. To clarify the amplification, we examine the evolution of the amplitudes of the epicycle and vertical motion. The amplitude of the epicycle motion $x_{\mathrm{a}}$ is defined in Equation (13), and that of the vertical motion $z_{\mathrm{a}}$ is defined by

where $\psi_{0}$ is the phase of the vertical motion at $t=0$. We introduce $x_{\mathrm{a},\mathrm{rms}}$ and $z_{\mathrm{a},\mathrm{rms}}$, that are the root mean squares of $x_{\mathrm{a}}$ and $z_{\mathrm{a}}$, respectively. Figure 7 displays the time evolution of $x_{\mathrm{a},\mathrm{rms}}$ and $z_{\mathrm{a},\mathrm{rms}}$. We find that is the largest $x_{\mathrm{a},\mathrm{rms}}$ around $\Omega t/2\pi=4$. Thus the density enhancement and the increase of the epicycle amplitude coincide. This is consistent with the swing amplification model. On the other hand, the $z_{\mathrm{a},\mathrm{rms}}$ barely changes during the amplification. In the swing amplification model, the motion in the $z$ direction is not considered. The numerical simulation shows that this treatment is valid. The swing amplification is essentially two-dimensional phenomenon.

To examine the phase synchronization, we consider the displacement from the guiding center $\delta x=x-x_{\mathrm{g}}$ and $\delta y=y-y_{\mathrm{g}}$. If the epicyclic oscillation is uniform, the average of $\delta x$ and $\delta y$ should be zero. We define $\overline{\delta x}$ and $\overline{\delta y}$ as the average of $\delta x$ and $\delta y$ in each group, respectively. The absolute values of these quantities show the degree of the phase synchronization. If the epicycle phases of stars are not synchronized, these are close to $0$. In Figure 8, some groups show the upward trend while the other groups shows the downward trend. Before the density is amplified ($\Omega t/2\pi<3.8$), $|\overline{\delta x}|\lesssim 0.05$ and $|\overline{\delta y}|\lesssim 0.07$. Thus, the phase in each group is not well synchronized. After the density amplification ($\Omega t/2\pi>4$), $|\overline{\delta x}|$ and $|\overline{\delta y}|$ in each group increase up to 0.13. Thus, the phase in each group is synchronized after the amplification.

As discussed above the stars gather in the $x$ direction and the width of the spiral arm shrinks in Figure 5 when the density gets enhanced ($\Omega t/2\pi=3.8\mbox{--}4.2$). This behavior cannot be understood by the previous analytical works based on the linear theory (Toomre, 1981, Paper IV). In their analytical works, the density enhancement is caused by the displacement normal to a spiral arm. Thus, the averaged displacement of all stars parallel to a pattern is zero. This discrepancy between the simulations and the analytical analyses suggests the importance of the finite length of spiral arms.

In order to analyze the typical spiral evolution, we use the space-time autocorrelation given by Equation (10). Figure 9 shows the space-time autocorrelation function at $\Omega s/2\pi=-0.90,-0.60,-0.30,0.00,0.30,$ and $0.60$. At $\Omega s/2\pi=-0.90$, we can observe the faint leading structure. The overdense region has the leading structure before the density enhancement. Because of the shear, the pitch angle increases gradually. At $\Omega s/2\pi=-0.60$, there are two overdense regions at the both ends of the leading structure. Each overdense region has the dim trailing tails. At $\Omega s/2\pi=-0.30$, although the pitch angle at the center is quite large, it has already trailing tails. At $\Omega s/2\pi=0.0$, the spiral arms are amplified to the maximum, and the structure is almost along the line. After the amplification, the amplitude begins to decrease. At $\Omega s/2\pi=0.3$, the spiral structure bends at the center, that is, the pitch angle at the center is larger than those in the tails. At $\Omega s/2\pi=0.6$, the spiral structure splits into two halves and the narrow leading structure develops.

The amplification processes in the numerical simulation and the analytical analyses share the basic fact that the amplification occurs while the pattern rotates from leading to trailing. However, the overall process depicted here is somewhat different from the behavior considered in the analytical analyses. In the analytical analyses, we consider only the rotating single wave and do not consider the structure parallel to the wave. It seems that the leading pattern forms from the interaction between two trailing spiral arms.

In order to elucidate the physical process in more detail, we investigate the evolution of the displacement from the guiding center and the relative velocity to the guiding center using Equation (9). We calculate the displacement vector from the guiding center by $\langle x-x_{\mathrm{g}}\rangle(x,y,s)$ and $\langle y-y_{\mathrm{g}}\rangle(x,y,s)$ and the relative velocity to the guiding center by $\langle v_{x}\rangle(x,y,s)$ and $\langle v_{y}+2Ax_{\mathrm{g}}\rangle(x,y,s)$, and show them in Figure 10. At $\Omega s/2\pi=-0.4$, we can see the leading structure. In this case, the rotation of the pattern cancels out that of the coordinate system. In the comoving frame of the leading pattern, the stabilizing effect due to rotation weakens. Thus, the stars are pulled towards the center of the density pattern, and the relative velocity is almost parallel to the leading pattern. The stars move along the density pattern and the two density peaks come close to each other. Note that, this behavior is unpredictable by the analytical theory of the swing amplification. Since, in the analytical theory, we postulate the infinite plane wave that has no structure in the direction parallel to the wave, the resulting perturbed flow towards the density peak is perpendicular to the wave. At $\Omega s/2\pi=-0.2$, the significant density enhancement takes place due to the flow towards the density peak. Due to the Coriolis force, the velocity field rotates in the anticlockwise direction around the density peak. At the same time, the long trailing tail from the density peak appears. At $\Omega s/2\pi=0.0$, the clear trailing pattern appears. The displacement from the guiding center has a convergent field towards the density peak. This indicates that the phase synchronization causes the density enhancement. On the other hand, the corresponding velocity field rotates in the anticlockwise direction around the density peak. Hence, around the density peak the region where $y>0$ moves to the left and the region where $y<0$ moves to the right. At $\Omega s/2\pi=0.4$, this anti-parallel motion splits the trailing pattern into two halves.