Discrete spiral modes, spiral waves, and the local dispersion relationship

Lau, Y. Y.; Bertin, G.

doi:10.1086/156635

Cited by 105 publications

(154 citation statements)

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“…If the disk mass grows above 0.1−0.2M * , however, transport becomes inherently global, and thus an increase in mass ratio corresponds to an increase in the maximum rate of transport. This is consistent with the Lau & Bertin (1978) dispersion relation. According to the ξ − Γ boundary derived above, a more massive disk corresponds to a larger value of Γ for fixed ξ, which will thus be stable at the same infall rate.…”

Section: When Saturation Fails: Fragmentationsupporting

confidence: 90%

“…This is due to the fact that such global modes are not captured by the WKB tightly wound approximation. However, a WKB description of such modes can still be obtained under less restrictive conditions than the tightly wound approximation (Lau & Bertin 1978). The resulting dispersion relation is more complicated (it is a cubic rather than quadratic expression in k) and depends on a new dimensionless parameter J:…”

Section: Dispersion Relationsmentioning

confidence: 99%

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Gravitational Instabilities in Circumstellar Disks

Kratter

Lodato

2016

Annu. Rev. Astron. Astrophys.

453

380

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Star and planet formation are the complex outcomes of gravitational collapse and angular momentum transport mediated by protostellar and protoplanetary disks. In this review we focus on the role of gravitational instability in this process. We begin with a brief overview of the observational evidence for massive disks that might be subject to gravitational instability, and then highlight the diverse ways in which the instability manifests itself in protostellar and protoplanetary disks: the generation of spiral arms, small scale turbulence-like density fluctuations, and fragmentation of the disk itself. We present the analytic theory that describes the linear growth phase of the instability, supplemented with a survey of numerical simulations that aim to capture the non-linear evolution. We emphasize the role of thermodynamics and large scale infall in controlling the outcome of the instability. Despite apparent controversies in the literature, we show a remarkable level of agreement between analytic predictions and numerical results. In the next part of our review, we focus on the astrophysical consequences of the instability. We show that the disks most likely to be gravitationally unstable are young and relatively massive compared to their host star, M d /M * ≥ 0.1. They will develop quasi-stable spiral arms that process infall from the background cloud. While instability is less likely at later times, once infall becomes less important, the manifestations of the instability are more varied. In this regime, the disk thermodynamics, often regulated by stellar irradiation, dictates the development and evolution of the instability. In some cases the instability may lead to fragmentation into bound companions. These companions are more likely to be brown dwarfs or stars than planetary mass objects. Finally, we highlight open questions related to (1) the development of a turbulent cascade in thin disks, and (2) the role of mode-mode coupling in setting the maximum angular momentum transport rate in thick disks. We dedicate this article to the memory of Francesco Palla, who has been a source of inspiration for star formation and disk studies throughout the years. Through his support and encouragement of young scientists, his contributions live on.

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Section: When Saturation Fails: Fragmentationsupporting

confidence: 90%

Section: Dispersion Relationsmentioning

confidence: 99%

Gravitational Instabilities in Circumstellar Disks

Kratter

Lodato

2016

Annu. Rev. Astron. Astrophys.

453

380

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“…J controls the shape and growth rate of the unstable mode. The spiral mode appears for smaller J , and the bar mode for larger J (Lau & Bertin 1978). The number of spiral arms and their pattern speed cannot be determined in the framework of the tight-winding densitywave theory of Lin-Shu-Kalnajs (Section 2.1.1).…”

Section: Global Mode Theorymentioning

confidence: 99%

“…Lau & Bertin (1978) derived the asymptotic dispersion relation of open spiral density waves in the fluid disc (BertinLau-Lin dispersion relation; BLL dispersion relation):…”

Section: Global Mode Theorymentioning

confidence: 99%

Dawes Review 4: Spiral Structures in Disc Galaxies

Dobbs

Baba

2014

Publ. Astron. Soc. Aust.

255

272

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The majority of astrophysics involves the study of spiral galaxies, and stars and planets within them, but how spiral arms in galaxies form and evolve is still a fundamental problem. Major progress in this field was made primarily in the 1960s, and early 1970s, but since then there has been no comprehensive update on the state of the field. In this review, we discuss the progress in theory, and in particular numerical calculations, which unlike in the 1960s and 1970s, are now commonplace, as well as recent observational developments. We set out the current status for different scenarios for spiral arm formation, the nature of the spiral arms they induce, and the consequences for gas dynamics and star formation in different types of spiral galaxies. We argue that, with the possible exception of barred galaxies, spiral arms are transient, recurrent and initiated by swing amplified instabilities in the disc. We suppose that unbarred m = 2 spiral patterns are induced by tidal interactions, and slowly wind up over time. However the mechanism for generating spiral structure does not appear to have significant consequences for star formation in galaxies.

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“…Here κ is the epicyclic frequency; since we are interested in Keplerian disks, we take κ ≈ Ω, the disk orbital frequency. Note that on large scales, angular momentum (κ 2 k −2 ) enters in a similar way to c 2 s and suppresses fluctuations, which follows directly from the form of the dispersion relation for density perturbations (e.g., Lin et al 1969;Toomre 1977;Lau & Bertin 1978); accounting for this is necessary to ensure mass conservation.…”

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confidence: 99%

Turbulent Disks Are Never Stable: Fragmentation and Turbulence-Promoted Planet Formation

Hopkins

Christiansen

2013

ApJ

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A fundamental assumption in our understanding of disks is that when the Toomre Q 1, the disk is stable against fragmentation into self-gravitating objects (and so cannot form planets via direct collapse). But if disks are turbulent, this neglects a spectrum of stochastic density fluctuations that can produce rare, high-density mass concentrations. Here, we use a recently developed analytic framework to predict the statistics of these fluctuations, i.e., the rate of fragmentation and mass spectrum of fragments formed in a turbulent Keplerian disk. Turbulent disks are never completely stable: we calculate the (always finite) probability of forming self-gravitating structures via stochastic turbulent density fluctuations in such disks. Modest sub-sonic turbulence above Mach number M ∼ 0.1 can produce a few stochastic fragmentation or "direct collapse" events over ∼Myr timescales, even if Q 1 and cooling is slow (t cool t orbit ). In transsonic turbulence this extends to Q ∼ 100. We derive the true Q-criterion needed to suppress such events, which scales exponentially with Mach number. We specify to turbulence driven by magneto-rotational instability, convection, or spiral waves and derive equivalent criteria in terms of Q and the cooling time. Cooling times 50 t dyn may be required to completely suppress fragmentation. These gravo-turbulent events produce mass spectra peaked near ∼(Q M disk /M * ) 2 M disk (rocky-to-giant planet masses, increasing with distance from the star). We apply this to protoplanetary disk models and show that even minimum-mass solar nebulae could experience stochastic collapse events, provided a source of turbulence.

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Discrete spiral modes, spiral waves, and the local dispersion relationship

Cited by 105 publications

References 0 publications

Gravitational Instabilities in Circumstellar Disks

Gravitational Instabilities in Circumstellar Disks

Dawes Review 4: Spiral Structures in Disc Galaxies

Turbulent Disks Are Never Stable: Fragmentation and Turbulence-Promoted Planet Formation

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