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Astron. Astrophys. 347, 442-454 (1999)

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6. Global instability of a one-component disk

In order to understand the influence of mass and momentum exchange processes on the evolution of global modes in a self-gravitating disk we performed simulations of the dynamics of a one-component stellar disk. The equilibrium stellar disk was chosen to have the surface density distribution (16). The parameter [FORMULA] for the stellar disk was selected so that the disk's total mass is equal to 0.5.

Fig. 1 shows the equilibrium properties of the stellar disk used in the analysis. The Toomre Q-parameter, which is defined for the stellar disk as [FORMULA], has a profile typical for the density distribution given by Eq. (16). The Q-profile rises towards the boundaries of the disk, with a minimum value of 1.64 at radius 0.73 indicating a globally stable disk with respect to Toomre's stability criterion [FORMULA].

[FIGURE] Fig. 1. Radial dependence of the angular velocity [FORMULA], the Toomre-parameter Q, and the equilibrium density [FORMULA] for a purely stellar disk ([FORMULA]) with exponential surface density distribution.

The one-component stellar disk was perturbed with an [FORMULA]armed perturbation of the form

[EQUATION]

We chose [FORMULA] and [FORMULA] perturbations in studying the stability properties of our models.

Fig. 2 plots the development of the global amplitudes for [FORMULA], and [FORMULA] spiral modes in a stellar disk seeded by an [FORMULA] perturbation of the form (20). The global amplitudes defined by the expression

[EQUATION]

illustrate the overall dynamics of the particular global mode.

[FIGURE] Fig. 2. Temporal evolution of the global amplitudes [FORMULA] ([FORMULA]) for a purely stellar disk ([FORMULA]) with exponential surface density distribution.

Fig. 2 shows a slow exponential growth of the [FORMULA] mode. The [FORMULA], [FORMULA] and [FORMULA] armed spirals grow, too, but due to the initial conditions, the [FORMULA] global mode outstrips the other competitor modes during the whole computation. However, even at the late stages of evolution the amplitude of spiral perturbations is less then half percent, and the spiral pattern does not emerge from the background. The slow development of the perturbation is best seen on the sequence of the snapshots shown in Fig. 3 illustrating the contour plots of the perturbed radial velocity. (The orbital periods at the inner and outer boundary are 0.59 and 4.16, respectively.)

[FIGURE] Fig. 3. Contour maps of the radial velocity (in k ms-1) of a purely stellar disk ([FORMULA]) with an exponential surface density distribution at different times: [FORMULA] (upper left ), [FORMULA] (upper right ), [FORMULA]. The contours give 30%, 50%, 70% and 90% of the maximum velocity in each diagram. The dotted lines correspond to negative velocities, whereas the solid lines give positive velocities. The zero-velocity contour is shown with a dashed line.

Similar behavior was observed in the stellar disk with the Gaussian surface density distribution (18) and the rotation curve (19). Fig. 4 shows the equilibrium properties of this disk which is rather stable with a minimum value of Toomre's Q-parameter equal to 1.78 at a disk radius of [FORMULA].

[FIGURE] Fig. 4. Radial dependence of the angular velocity [FORMULA], the Toomre-parameter Q, and the equilibrium density [FORMULA] for a purely stellar disk ([FORMULA]) with the Gaussian surface density distribution.

This disk was seeded with a three-armed perturbation of the form (20). As it is seen from Fig. 5, the disk develops a set of slowly growing modes. The behavior of the global amplitudes is somewhat similar to the previous case. (The orbital periods at the inner and outer boundary amount here to 0.97 and 3.65, respectively.) The [FORMULA] spiral mode has higher amplitude compared to the other competitors, and has a tendency to be saturated at the level [FORMULA]. The perturbation, however, does not have any properties of a regular spiral pattern (Fig. 6). Even at the late stages of the evolution the contours remain quite patchy, and do not resemble the regular spiral pattern.

[FIGURE] Fig. 5. Temporal evolution of the global amplitudes [FORMULA] ([FORMULA]) for a purely stellar disk ([FORMULA]) with the Gaussian surface density profile.

[FIGURE] Fig. 6. Contour maps of the logarithmic surface density ([FORMULA]) for a purely stellar Gaussian disk ([FORMULA]) at different times: [FORMULA] (upper left ), [FORMULA] (upper right ),

In the next section we will discuss how the morphological properties of the global modes are affected by the [FORMULA] of the equation of state which is a necessary consequence of a phase transformation in a star-forming disk.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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