A&A 384, 872-878 (2002)
DOI: 10.1051/0004-6361:20020098
N. Orlova^{1} - V. Korchagin^{1} - Ch. Theis^{2}
1 - Institute of Physics, Stachki 194, Rostov-on-Don, Russia
2 -
Institut für Theoretische Physik und Astrophysik d. Univ. Kiel,
24098 Kiel, Germany
Received 22 June 2001 / Accepted 19 October 2001
Abstract
We performed two-dimensional non-linear
hydrodynamical simulations of two-component gravitating
disks aimed at studying stability properties of these systems.
In agreement with previous analytical and numerical simulations,
we find that the cold gas component
strongly affects the growth rates of the unstable global
spiral modes in the disk.
Already a five percent admixture of cold gas increases
approximately two-fold the growth rate of the most
unstable global mode while a twenty percent gas admixture
enhances the modal growth rate by a factor of four.
The local stability properties of a two-component disk
coupled by self-gravity are governed by a stability
criterion similar to Toomre's Q-parameter derived for one-component
systems. Using numerical simulations, we analyse
the applicability of a two-component local stability criterion
for the analysis of the stability properties
of global modes.
The comparison of non-linear simulations with the
analytical stability criterion shows that
the two-component disks can be globally unstable while
being stable to the local perturbations. The minimum value of the
local stability criterion provides, however,
a rough estimate of the global stability properties of
two-component systems.
We find that two-component systems with a content of cold gas
of ten percent or less are globally stable
if the minimum value of the stability parameter exceeds 2.5.
Key words: galaxies: kinematics and dynamics - galaxies: spiral
Global spiral arms observed in disk galaxies are the result of density waves propagating in the galactic disks (e.g. Binney & Tremaine 1987). Starting from pioneering works by B. Lindblad (1941, 1942), a number of authors extensively investigated this idea, and a variety of techniques have been developed to study the stability properties of gravitating disks. Most of these studies concentrated on one-component gaseous or collisionless stellar disks. Galactic disks, however, are multi-component systems whose properties depend on all the constituents. Lin & Shu (1966) first addressed the question showing the relative importance of gas in providing the spiral gravitational field. Jog & Solomon (1984) (hereafter JS) considered the influence of a cold component on the stability properties of gravitating disks. They derived a local dispersion relation for a two-fluid instability of a star-gas disk and discussed a relative contribution of two components in the stability properties of the disk. In further investigations the results of JS were generalized by Bertin & Romeo (1988) who studied the stability properties of the discrete global spiral modes by applying asymptotic WKB-methods. They also introduced the concept of the effective Q-parameter of the two-component disks. Elmegreen (1995) re-considered the original equations derived by JS and reduced them to a single effective Q-parameter. Romeo (1992, 1994) included the effects of the finite thickness of stellar and gaseous disks. Recently, Rafikov (2001) gave the analysis of the axisymmetric gravitational stability of the thin rotating disk consisting of several components.
The non-asymptotic global modal stability properties of multi-component disks were studied to a lesser extent. Noh et al. (1991) studied the global stability properties of a thin two-fluid protoplanetary disk composed of gas and dust coupled by gravity and friction and found that the stability properties of these disks are strongly affected by a small amount of dust. Kikuchi et al. (1997) studied global stability properties of a three-phase disk with phases coupled by self-gravity and interchange processes between the phases. Korchagin & Theis (1999) numerically studied the behavior of global modes in multi-component disks at the non-linear stage of global instability.
In this paper, we return to the problem of the global stability properties of multi-component disks. We consider the simplest possible system of that kind - a two-fluid thin disk coupled by the self-gravity of the components. We compare the results of our non-linear two-dimensional numerical simulations with the local stability criteria derived by JS and study the applicability of the Jog-Solomon criterion for the prediction of the global stability properties of multi-phase gravitating disks. We also study the non-linear behaviour of the unstable global modes in multi-phase disks determining the dependence of the saturation levels of the unstable global modes on the mass fraction of a cold component.
Section 2 describes the basic equations and the numerical code used in simulations. In Sect. 3 we present the equilibrium model used in our simulations. In Sect. 4 we briefly describe the two-fluid local stability criterion. Section 5 presents the results of our simulations and gives a comparison of the results of the numerical simulations with the two-fluid analytical stability criterion.
To model the behaviour of the global perturbations in
a gravitationally coupled multi-component disk
we use the hydrodynamic approximation, modelling both a stellar and a
gaseous component as polytropic fluids:
The behaviour of a two-fluid gravitationally interacting system
is described by the continuity and momentum equations for
each component:
To solve the two-component hydrodynamical Eqs. (2)-(4) we use the multi-component fluid-dynamical code developed by Korchagin & Theis (1999). The code is a multi-phase realization of the Eulerian ZEUS-2D code (Stone & Norman 1992) with the Van-Leer advection scheme. The two-fluid hydrodynamical equations are solved using equally spaced azimuthal and logarithmically spaced radial zones. In the numerical simulations discussed here we employed a grid of cells. The Poisson equation is solved by the 2D Fourier convolution theorem in polar coordinates. We look for the global modes starting from initial random perturbations at the level 10^{-6}.
The equilibrium disk taken in our simulations is based
on the equilibrium properties of real galaxies.
Specifically, we applied the model of the spiral galaxy NGC 1566
by Korchagin et al. (2000). The
disk of NGC 1566 is observationally well studied.
Its rotation curve
taken from Bottema (1992) can be approximated by
The radial velocity dispersion of the
stellar and gaseous components can be derived from the equation of state by
The equilibrium rotation of the disk given by the Eq. (6)
is balanced by the self-gravity, the pressure gradient of the disk and
the external potential of a rigid halo:
Inner disk radius , kpc | 0.2 |
Outer disk radius , kpc | 10.0 |
Disk thickness (z_{0}), kpc | 0.7 |
Scale length of velocity dispersion (h_{cz}), kpc | 2.6 |
Scale length of surface density , kpc | 1.3 |
Central velocity dispersion (c_{z0}), kms^{-1} | 168 |
Total mass of the disk , | |
Rotation curve parameters: | |
V_{1}, V_{2}, kms^{-1} | 399, 82 |
R_{1}, R_{2}, kpc | 4.4, 0.6 |
The local stability of a thin gravitating disk can be
expressed in terms of Toomre's Q parameter which for the
gaseous disk has the form (Safronov 1960;
Toomre 1964):
The stability criterion (12) was derived for tightly wound perturbations. However, it can be successfully used as an indicator for the stability of global modes in gravitating disks. Numerical simulations show that a locally stable stellar or gaseous disk can be globally unstable if the minimum value of Q does not exceed 2. For example, a one-component disk with the rotation and surface density distribution given by Eqs. (6) and (9) can be globally unstable, if the minimum value of Qdoes not exceed 1.6 (Korchagin et al. 2000).
Disks of real galaxies consist of stellar and gaseous components.
The stellar component comprises the largest fraction
of the total mass of the disk. However, colder gaseous components
can play a significant role for the stability properties
of multi-component disks.
JS derived a local stability criterion for a two-component fluid
disk which can be written in the form
where
We will use the criterion given by Eq. (13) for the analysis of instability of the global modes in two-component gravitating disks.
Figure 1: Equilibrium curves for the one-component "stellar'' disk. Angular velocity and epicyclic frequency are given in units of . The surface density is given in units , radius is in units of length L=2 kpc. | |
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To study the influence of a cold component on the overall dynamics of a two-fluid gravitating disk, we start our analysis with a one-component "stellar'' disk which we choose to be close to marginal stability. Figure 1 shows the equilibrium properties of this disk. The local stability parameter Q is well above unity with a minimum value .
The radial behavior of the Q-parameter is one of the key assumptions in the modal analyses of spiral density waves. In previous studies it was usually assumed that the profile of the stability parameter Q has a barrier at the central regions of the disk and is equal to unity in the outer regions of the disk (e.g. Bertin et al. 1989). Such a behavior of an "effective'' Q-parameter is supposed to model the absence of cold gas as well as the transition in geometry from the disk to the nuclear bulge in the central regions of a galaxy, and to describe the increasing role of a cold gaseous component in the outer regions of a galactic disk. In this paper we do not introduce an effective Q-parameter, but we try to base our studies on observational properties of disk galaxies. We choose the basic state in our fiducial model based on the properties of the nearby late type spiral galaxy NGC 1566. The "stellar'' Q-parameter shown in Fig. 1 has an outer barrier which is typical for disks with exponentially decreasing radial surface density distributions and exponentially decreasing velocity dispersions.
Linear global modal analysis of this model reveals a few slowly
growing modes with
the fastest growing spiral m=2.
Nonlinear 2D simulations
confirm this result giving an exponential growth rate of the m = 2 mode
of approximately 0.11 in dimensionless units. Linear theory predicts a
value of 0.12 which is in good agreement with non-linear simulations.
Figure 2 illustrates the
evolution of the
m = 1 - 4 global modes in
the one-component "stellar'' disk given in terms of the
global Fourier amplitudes
Figure 2: Temporal evolution of the m=1-4 global Fourier amplitudes computed for the one-component "stellar'' disk of Fig. 1. Time is in units of yr. | |
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In the next step we split the one-component disk into a "stellar'' and a "gaseous'' subsystem keeping the total mass of the disk constant. We consider a set of models with a cold component consisting of 5%, 10% and 20% of the total mass of the disk. In the first series of simulations we keep the "stellar'' and gaseous equations of state the same for all the models. As a result, the gas velocity dispersion, which depends on the gas fraction, is not constant. Thus, the velocity dispersion of gas is about 20% of the velocity dispersion of the stars in the model with 5% gas admixture. This ratio increases up to 36% for the model with 20% gas admixture.
The strong influence
of the gas fraction is shown in Fig. 3 which displays the growth rate of
the most unstable m=2 mode in a one-component "stellar''
disk compared to the modal growth rate of the two-component models
with 5%, 10%, and 20%
gas admixtures. Already a five percent gas
fraction enhances the modal growth rate by a factor of 1.7 compared
to the one-component disk. A 20% gas admixture results in an increase of
the growth rate by a factor of 3.7.
Table 2 gives the masses and the
dispersion ratio of the components in the computed models
together with the growth rates of the most unstable modes,
saturation levels and the minimum values of the
two-component stability criterion
calculated by
Eq. (13).
The growth rate of the dominant mode increases in the
models with higher gas content,
being the same for the gaseous and the stellar components.
However, the saturation level of the most unstable mode is systematically
higher for the gaseous disk.
In general, the saturation level
slightly decreases in the models with higher gas content.
A possible explanation of this behavior is that
spiral shocks play a more important role in the saturation
of exponentially growing modes when more cold gas is present.
Figure 3: Time dependence of the global Fourier amplitudes for the most unstable m=2 spiral mode computed for models with different gas fractions (I: 5%, II: 10%, III: 20%). Time is in units of yr. | |
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Saturation level | ||||||
Model | gas | stars | ||||
I | 0.05 | 0.2 | 0.18 | 0.35 | 0.27 | 2.2 |
II | 0.1 | 0.25 | 0.24 | 0.32 | 0.25 | 1.75 |
III | 0.2 | 0.36 | 0.4 | 0.29 | 0.22 | 1.36 |
Note. Dependence of the growth rates and the saturation
levels on the content of a cold component for the
governing m=2 mode.
The total mass of the disk is fixed to
.
The growth rates are given
in units of
.
Disks with gaseous admixture show qualitatively the same evolution as the pure stellar disk. However, they evolve on a shorter time scale. The appearance of the global modes in all models is quite similar despite the considerable difference in their growth rates. Figure 5 shows the logarithmically spaced surface density contour plots taken at the moments when the global amplitude of the m=2 mode in all four models is . All the models develop an open two-armed spiral of comparable winding. Note, however, that the spiral arms are more narrow, more tightly wound and less extended when more gas is present. Analysis of the global modes with help of the local dispersion relation (Bertin & Romeo 1988) also shows a shift towards tighter spirals with a smaller corotation radius in models with higher gas content.
Figure 4: Time evolution of the global Fourier amplitude for the most unstable mode (m=2) computed for models with gas admixture and a ratio of the velocity dispersions equal to 1.95, 3.9 and 7.8. Time is in units of yr. | |
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Figure 5: Surface density perturbations of the stellar component. Displayed are the spiral perturbations in the one-component model (upper left), and in the two-component models with different gas fractions. The snapshots are taken at the moment when the m=2global amplitude reaches . The contour levels are logarithmically spaced between the maximum value of the density perturbation and one-hundredth of the maximum value. Circles show the positions of the outer Lindblad resonance (dashed line) and corotation (solid line). | |
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To use the stability criterion (13), one needs to know the wavenumber
of the perturbations. Figure 6 shows the radial dependence of
the radial wavenumber for m = 2 perturbations computed
for the models I-III. The wavenumbers were determined numerically
by calculating the pitch angles of the spiral arms shown in Fig. 5
at different radii, i.e. we used the relation
Figure 6: The radial dependence of the radial wavenumber of two-armed spiral for the models I-III. Unit of length L=2 kpc. | |
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Figure 7: The radial dependence of the local two-fluid stability parameter for disk models with different gas fraction. Unit of length L=2 kpc. | |
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Figure 7 shows the radial profile of the dimensionless Jog-Solomon stability parameter computed with the wavenumbers taken from Fig. 6. Figure 7 further illustrates the destabilizing role of a cold gas component: the minimum value of the Jog-Solomon parameter decreases with increasing content of cold gas. The minimum value of the Jog-Solomon parameter is about 1.36 for the two-component system with 20% gas admixture versus a minimum value of 2.2 found for the two-component model with 5% gas fraction. In all three models, the Jog-Solomon stability criterion exceeds unity, and all the models are locally stable. Such a behaviour is quite similar to the behaviour of one-component disks which are known can be globally unstable even if the minimum value of Toomre's Q parameter is above unity.
Decreasing the total mass of the disk results in its stabilization which consequently leads to an overall growth of the Jog-Solomon parameter . Figure 8 illustrates this by displaying the growth rate of the dominant mode m=2 as a function of the minimum value of the Jog-Solomon parameter . The latter was obtained from numerical simulations of two-component disks with 10% gas admixture. These disks are stabilized when the minimum value of the Jog-Solomon parameter exceeds 2.5.
Figure 8: The growth rate as a function of the minimum value of local stability parameter . | |
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In this paper we studied the applicability of the two-fluid local stability criterion derived by Jog & Solomon (1984) for the onset and growth of non-axisymmetric perturbations in two-component gravitating disks with an open spiral structure. We performed three sets of numerical experiments: one set with fixed total mass of the disk and fixed polytropic equation of state, but variable gas fraction. Another set of simulations was performed with a fixed total mass of the disk and a fixed ratio of the stellar and gaseous velocity dispersions. A final set of calculations deals with a variable disk mass. Our numerical experiments have shown that the mass contribution of the cold component strongly influences the stability properties of the disk. Even a low gas fraction of destabilizes the disk substantially, e.g. the growth rate of m=2 perturbations increases by a factor 1.7. All our disk models are unstable to m=2 spiral modes. More unstable disks develop narrower and more tightly wound spiral arms. The saturation levels are almost constant for all simulations, with a slightly enhanced level for the gaseous component. A comparison of the results of the direct two-dimensional numerical simulations with the JS stability criterion allows us to conclude that similarly to Toomre's stability parameter Q derived for one-component disks, the Jog-Solomon stability parameter can be used for the prediction of the global stability properties of two-component gravitating disks which greatly simplifies the analysis of disk stability.
Acknowledgements
We thank the referee Dr. A. B. Romeo for his valuable comments on this work. NO and VK gratefully acknowledge support by the Deutsche Forschungsgemeinschaft under grant RUS 17/75/00(S).