Nonlocal density wave theory for gravitational instability of protoplanetary disks without sharp boundaries

Rüdiger, G.; Kitchatinov, L. L.

doi:10.1002/1521-3994(200008)321:3<181::aid-asna181>3.3.co;2-y

Cited by 4 publications

(4 citation statements)

References 23 publications

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“…Without this assumption, Fourier modes are no longer solutions of the linearized equations and we have to switch to much more complicated mathematics involving integral equations (see Rüdiger & Kitchatinov 2000). We prefer to present numerical simulations for the stability of (isothermal) density-stratified Keplerian disks under the influence of finite disturbances.…”

Section: Numerical Simulationsmentioning

confidence: 99%

Hydrodynamic stability in accretion disks under the combined influence of shear and density stratification

Rüdiger

Arlt

Shalybkov

2002

A&A

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Abstract. The hydrodynamic stability of accretion disks is considered. The particular question is whether the combined action of a (stable) vertical density stratification and a (stable) radial differential rotation gives rise to a new instability for nonaxisymmetric modes of disturbances. The existence of such an instability is not suggested by the well-known Solberg-Høiland criterion. It is also not suggested by a local analysis for disturbances in general stratifications of entropy and angular momentum which is presented in our Sect. 2. This confirms the results of the Solberg-Høiland criterion also for nonaxisymmetric modes within the frame of ideal hydrodynamics but only in the frame of a short-wave approximation for small m. As a necessary condition for stability we find that only conservative external forces are allowed to influence the stable disk. As magnetic forces are never conservative, linear disk instabilities should only exist in the magnetohydrodynamical regime which indeed contains the magnetorotational instability as a much-promising candidate. To overcome some of the used approximations in a numerical approach, the equations of the compressible adiabatic hydrodynamics are integrated, imposing initial nonaxisymmetric velocity perturbations with m = 1 to m = 200. Only solutions with decaying kinetic energy are found. The system always settles in a vertical equilibrium stratification according to pressure balance with the gravitational potential of the central object.

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Section: Numerical Simulationsmentioning

confidence: 99%

Hydrodynamic stability in accretion disks under the combined influence of shear and density stratification

Rüdiger

Arlt

Shalybkov

2002

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“…1 1 By numerically solving the perturbed Poisson equation using the Fourier–Bessel transform, Rüdiger & Kitchatinov (2000) approached the linear stability of the disc in terms of an eigenvalue problem and could also treat the density wave problem globally. Their formulation differs from ours in two major aspects.…”

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Structures in a class of magnetized scale-free discs

Shen¹,

Liu²,

Lou³

2005

Monthly Notices of the Royal Astronomical Society

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We construct analytically stationary global configurations for both aligned and logarithmic spiral coplanar magnetohydrodynamic (MHD) perturbations in an axisymmetric background MHD disc with a power-law surface mass density $\Sigma_0\propto r^{-\alpha}$, a coplanar azimuthal magnetic field $B_0\propto r^{-\gamma}$, a consistent self-gravity and a power-law rotation curve $v_0\propto r^{-\beta}$ where $v_0$ is the linear azimuthal gas rotation speed. The barotropic equation of state $\Pi\propto\Sigma^{n}$ is adopted for both MHD background equilibrium and coplanar MHD perturbations where $\Pi$ is the vertically integrated pressure and $n$ is the barotropic index. For a scale-free background MHD equilibrium, a relation exists among $\alpha$, $\beta$, $\gamma$ and $n$ such that only one parameter (e.g., $\beta$) is independent. For a linear axisymmetric stability analysis, we provide global criteria in various parameter regimes. For nonaxisymmetric aligned and logarithmic spiral cases, two branches of perturbation modes (i.e., fast and slow MHD density waves) can be derived once $\beta$ is specified. To complement the magnetized singular isothermal disc (MSID) analysis of Lou, we extend the analysis to a wider range of $-1/4<\beta<1/2$. As an example of illustration, we discuss specifically the $\beta=1/4$ case when the background magnetic field is force-free. Angular momentum conservation for coplanar MHD perturbations and other relevant aspects of our approach are discussed.Comment: 25 page

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“…Without this assumption, Fourier normal modes are no longer solutions of the linearized equations and one has to switch to far more complicated mathematics (e.g., Rüdiger & Kitchatinov 2000). We intend to address the issue in the following publications of the series.…”

Section: Discussionmentioning

confidence: 99%

Fine-scale density wave structure of Saturn's rings: A hydrodynamic theory

Griv

Gedalin

2010

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Aims. We examine the linear stability of the Saturnian ring disk of mutually gravitating and physically colliding particles with special emphasis on its fine-scale ∼100 m density wave structure, that is, almost regularly spaced, aligned cylindric density enhancements and optically-thin zones with the width and the spacing between them of roughly several tens particle diameters. Methods. We analyze the Jeans' instabilities of gravity perturbations (e.g., those produced by a spontaneous disturbance) analytically by using the Navier-Stokes dynamical equations of a compressible fluid. The theory is not restricted by any assumptions about the thickness of the system. We consider a simple model of the system consisting of a three-dimensional ring disk that is weakly inhomogeneous and whose structure is analyzed by making a horizontally local short-wave approximation. Results. We demonstrate that the disk is probably unstable and that gravity perturbations grow effectively within a few orbital periods. We find that self-gravitation plays a key role in the formation of the fine structure. The predictions of the theory are compared with observations of Saturn's rings by the Cassini spacecraft and are found to be in good agreement. In particular, it appears very likely that some of the quasi-periodic microstructures observed in Saturn's A and B rings -both axisymmetric and nonaxisymmetric onesare manifestations of these effects. We argue that the quasi-periodic density enhancements revealed in Cassini data are flattened structures, with a height to width ratio of about 0.3. One should analyze high-resolution of the order of 10 m data acquired for the A and B rings (and probably C ring as well) to confirm this prediction. We also show that the gravitational instability is a potential cluster-forming mechanism leading to the formation of porous 100-m-diameter moonlets of preferred mass ∼10 7 g each embedded in the outer A ring, although this has yet to be directly measured.

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Nonlocal density wave theory for gravitational instability of protoplanetary disks without sharp boundaries

Cited by 4 publications

References 23 publications

Hydrodynamic stability in accretion disks under the combined influence of shear and density stratification

Hydrodynamic stability in accretion disks under the combined influence of shear and density stratification

Structures in a class of magnetized scale-free discs

Fine-scale density wave structure of Saturn's rings: A hydrodynamic theory

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