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

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7. Global modes in a multi-component disk

To compare the behavior of a multi-component disk with the dynamics of a corresponding one-component system, the surface densities of all phases were initially distributed in accordance with Eqs. (16) or (18). In both cases we choose the initial mass of the gaseous component equal to the mass of the one-component system. Masses of the "admixture", i.e. stars and remnants, were initially set to 0.01 each which is about two percent of the initial mass of the gaseous component. All components were set at the beginning into centrifugal equilibrium, with the circular rotation supported by the pressure gradients of the components, the total gravitational field of the three components and the gravity of the external halo. Fig. 7 shows the rotation curve and the epicycle frequency of the gaseous component for the initial exponential surface density distributions. The rotation curve of the gaseous component is very similar to the equilibrium profiles of the stellar disk discussed in the previous section (Fig. 1). However, the gaseous component is less stable compared to the purely stellar disk, and the broad trough of Toomre's Q-parameter lies below the corresponding Q-distribution of the stellar disk.

[FIGURE] Fig. 7. Radial dependence of the angular velocity [FORMULA], density distribution and the Toomre-parameter Q for the gaseous component in a multi-component disk with the exponential surface density distribution.

Despite the centrifugal balance, the components are not in equilibrium. Mass transformations given by the right-hand-sides of the continuity Eqs. (6)-(8) change the densities and total masses of the components, and the system evolves even without initial perturbations. Fig. 8 illustrates such mass transformation. At the beginning, the mass of the system is contained in the gaseous phase, and by the end of the computation about 90% of the gas has been converted into stellar remnants or long-lived low mass stars. The fraction of mass contained in massive stars drops from the initial value to 0.1% at the end of the simulation.

[FIGURE] Fig. 8. Temporal evolution of the masses of gaseous and the stellar component in a multi-component disk with an exponential density distribution.

With the [FORMULA] perturbation given by Eq. (20), all three phases develop a two-armed spiral pattern. Fig. 9 shows the time dependence of the global amplitudes for the [FORMULA],2,3 and 4-armed global modes growing in the stellar component. Again, the [FORMULA] global mode prevails over its competitors, and compared to the purely stellar disk (Fig. 2) it grows about two orders of magnitude faster developing a nonlinear spiral pattern.

[FIGURE] Fig. 9. Temporal evolution of the global amplitudes [FORMULA] ([FORMULA]) for stellar component of the multi-phase disk with an exponential density distribution.

Figs. 10 and 11 show the time sequence of the contour plots of the density distributions of the stellar and gaseous components of a disk with an exponential density profile. Initially, the density distribution evolves similarly to the dynamics of the one-component disk, but the subsequent behavior is different. A comparison of Fig. 10 with the simulations in a one-component disk clearly shows that spirals are better developed in a multi-component disk.

[FIGURE] Fig. 10. Contour maps of the logarithmic surface density ([FORMULA]) of the stars for the multi-phase exponential disk at different times: [FORMULA] (upper left ), [FORMULA] (upper right ), [FORMULA]. The contours are logarithmic-equally spaced. The equilibrium model is perturbed by an [FORMULA]-mode.

[FIGURE] Fig. 11. Contour maps of the logarithmic surface density ([FORMULA]) of the gas for the multi-phase exponential disk at different times: [FORMULA] (upper left ), [FORMULA] (upper right ), [FORMULA]. The contours are logarithmic-equally spaced. The equilibrium model is perturbed by an [FORMULA]-mode.

A similar behavior was observed for the disk with the Gaussian surface density profile which was seeded by the [FORMULA] perturbation. Fig. 12 shows the growth of the global amplitude in the stellar component of the multi-phase disk accompanied by the mass transformations illustrated on Fig. 13. Again, a comparison with Fig. 5 demonstrates, that the growth rate of the [FORMULA] spiral perturbation of the multi-phase disk is an order of magnitude larger than that of the one-component system.

[FIGURE] Fig. 12. Temporal evolution of the global amplitudes [FORMULA] of the [FORMULA] and [FORMULA] modes for the stellar component of the multi-phase disk with the Gaussian density profile.

[FIGURE] Fig. 13. Mass transformations between the gaseous and stellar components in the multi-phase disk with the Gaussian density distribution.

The exponential growth of the [FORMULA] perturbation is changed by the lingering saturation phase occurring at the amplitude level [FORMULA]. This nonlinear saturation of global modes is known for one-component simulations (Laughlin & Rózyczka 1996, Laughlin et al. 1997, 1998). Our result shows that the nonlinear saturation of exponentially growing global modes is a common phenomenon which occurs as well in the multi-phase gravitating disks experiencing phase transitions.

The three-armed nature of the perturbations in the multi-component disk is perfectly illustrated in Figs. 14 and 15 which show the contour map of the surface density and the radial velocity of the gaseous phase in a multi-component disk. We note that at late phases of the evolution the gas contains about 8% of the total disk mass, but nevertheless it is still a good tracer of the spiral structure. Moreover, our simulations allow to conclude, that the gas "helps" to develop spiral arms. A comparison of the density distribution of the collisionless remnants phase in a multi-component disk (Fig. 16) with perturbations in a purely stellar disk (Fig. 6) supports this conclusion: The density perturbation in remnants is more organized, and clearly depicts a three-armed spiral.

[FIGURE] Fig. 14. Contour maps of the logarithmic surface density ([FORMULA]) of the gas for the multi-phase Gaussian disk at different times: [FORMULA] (upper left ), [FORMULA] (upper right ), [FORMULA]. The contours are logarithmic-equally spaced. The equilibrium model is perturbed by an [FORMULA]-mode.

[FIGURE] Fig. 15. Contour maps of the radial velocity (in k ms-1) of the gaseous component in the Gaussian disk 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. The equilibrium model is perturbed by an [FORMULA]-mode.

[FIGURE] Fig. 16. Contour maps of the logarithmic surface density ([FORMULA]) of the stellar remnant component for the multi-phase Gaussian disk at different times: [FORMULA] (upper left ), [FORMULA] (upper right ), [FORMULA]. The contours are logarithmic-equally spaced. The equilibrium model is perturbed by an [FORMULA]-mode.

The destabilizing role of the cold gaseous component was studied for the linear regime by various authors. Local analysis performed by Lin and Shu (1966), Lynden-Bell (1967), Miller et al. (1970), Quirk (1971), Jog & Solomon (1984), Sellwood & Carlberg (1984), and semi-analytical global modal analysis of Bertin & Romeo (1988) demonstrated that a small admixture of gas in a stellar self-gravitating disk may considerably destabilize the system. Our simulations are in agreement with this conclusion, and illustrate, how the spiral structure behaves on a nonlinear stage. Spiral structure remains well developed and is self-sustained after a rapid star-formation process in a gaseous disk when most of the gas is transformed into a "remnants" phase with higher velocity dispersion, and with a more rigid equation of state.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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