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Register a new userSecular diffusion in discrete self-gravitating tepid discs II: accounting for swing amplification via the matrix method
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Added by
Jean-Baptiste Fouvry
Publication date
2015
fields
Physics
and research's language is
English
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The secular evolution of an infinitely thin tepid isolated galactic disc made of a finite number of particles is investigated using the inhomogeneous Balescu-Lenard equation expressed in terms of angle-action variables. The matrix method is implemented numerically in order to model the induced gravitational polarization. Special care is taken to account for the amplification of potential fluctuations of mutually resonant orbits and the unwinding of the induced swing amplified transients. Quantitative comparisons with ${N-}$body simulations yield consistent scalings with the number of particles and with the self-gravity of the disc: the fewer particles and the colder the disc, the faster the secular evolution. Secular evolution is driven by resonances, but does not depend on the initial phases of the disc. For a Mestel disc with ${Q sim 1.5}$, the polarization cloud around each star boosts up its secular effect by a factor of the order of a thousand or more, promoting accordingly the dynamical relevance of self-induced collisional secular evolution. The position and shape of the induced resonant ridge are found to be in very good agreement with the prediction of the Balescu-Lenard equation, which scales with the square of the susceptibility of the disc. In astrophysics, the inhomogeneous Balescu-Lenard equation may describe the secular diffusion of giant molecular clouds in galactic discs, the secular migration and segregation of planetesimals in proto-planetary discs, or even the long-term evolution of population of stars within the Galactic centre. It could be used as a valuable check of the accuracy of ${N-}$body integrators over secular timescales.
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The long timescale evolution of a self-gravitating system is generically driven by two-body encounters. In many cases, the motion of the particles is primarily governed by the mean field potential. When this potential is integrable, particles move on nearly fixed orbits, which can be described in terms of angle-action variables. The mean field potential drives fast orbital motions (angles) whose associated orbits (actions) are adiabatically conserved on short dynamical timescales. The long-term stochastic evolution of the actions is driven by the potential fluctuations around the mean field and in particular by resonant two-body encounters, for which the angular frequencies of two particles are in resonance. We show that the stochastic gravitational fluctuations acting on the particles can generically be described by a correlated Gaussian noise. Using this approach, the so-called $eta$-formalism, we derive a diffusion equation for the actions in the test particle limit. We show that in the appropriate limits, this diffusion equation is equivalent to the inhomogeneous Balescu-Lenard and Landau equations. This approach provides a new view of the resonant diffusion processes associated with long-term orbital distortions. Finally, by investigating the example of the Hamiltonian Mean Field Model, we show how the present method generically allows for alternative calculations of the long-term diffusion coefficients in inhomogeneous systems.
The principal results of the classic analysis of the shearing sheet and swing amplification by Julian & Toomre (1966) are re-derived in a more accessible way and then used to gain a better quantitative understanding of the dynamics of stellar discs. The axisymmetric limit of the shearing sheet is derived and used to re-derive Kalnajs 1965 dispersion relation and Toomres 1964 stability criterion for axisymmetric disturbances. Using the shearing sheet to revisit Toomres important 1969 paper on the group velocity implied by Lin-Shu-Kalnajs dispersion relation, we discover that two rather than one wavepackets emerges inside corotation: one each side of the inner Lindblad resonance. Although LSK dispersion relation provides useful interpretations of both wavepackets, the shearing sheet highlights the limitations of the LSK approach to disc dynamics. Disturbances by no means avoid an annulus around corotation, as the LSK dispersion relation implies. While disturbances of the shearing sheet have a limited life in real space, they live on much longer in velocity space, which Gaia allows us to probe extensively. C++ code is provided to facilitate applications of winding spiral waves.
I review recent progresses in the dynamics and the evolution of self-gravitating accretion discs. Accretion discs are a fundamental component of several astrophysical systems on very diverse scales, and can be found around supermassive black holes in Active Galactic Nuclei (AGN), and also in our Galaxy around stellar mass compact objects and around young stars. Notwithstanding the specific differences arising from such diversity in physical extent, all these systems share a common feature where a central object is fed from the accretion disc, due to the effect of turbulence and disc instabilities, which are able to remove the angular momentum from the gas and allow its accretion. In recent years, it has become increasingly apparent that the gravitational field produced by the disc itself (the discs self-gravity) is an important ingredient in the models, especially in the context of protostellar discs and of AGN discs. Indeed, it appears that in many cases (and especially in the colder outer parts of the disc) the development of gravitational instabilities can be one of the main agents in the redistribution of angular momentum. In some cases, the instability can be strong enough to lead to the formation of gravitationally bound clumps within the disc, and thus to determine the disc fragmentation. As a result, progress in our understanding of the dynamics of self-gravitating discs is essential to understand the processes that lead to the feeding of both young stars and of supermassive black holes in AGN. At the same time, understanding the fragmentation conditions is important to determine under which conditions AGN discs would fragment and form stars and whether protostellar discs might form giant gaseous planets through disc fragmentation.
The formation of astrophysical structures, such as stars, compact objects but also galaxies, entail an,enhancement of densities by many orders of magnitude which occurs through gravitational collapse. The role played by turbulence during this process is important. Turbulence generates density fluctuations, exerts a support against gravity and possibly delivers angular momentum. How turbulence exactly behave during the collapse and get amplified remains a matter of investigation. Spherical averaging of the fluid equations is carried out, leading to 1D fluid equations that describe the evolution of mean quantities in particular the mean radial velocity as well as the mean radial and transverse turbulent velocities. These equations differ from the ones usually employed in the literature. We then perform a series of 3D numerical simulations of collapsing clouds for a wide range of thermal and turbulent supports with two polytropic equation of state, $P propto rho^Gamma$, with $Gamma=1$ and 1.25. For each 3D simulations we perform a series of 1D simulations using the spherically averaged equations and with the same initial conditions. By performing a detailed comparison between 3D and 1D simulations, we can analyse in great details the observed behaviours. Altogether we find that the two approaches agree remarkably well demonstrating the validity of the inferred equations although when turbulence is initially strong, major deviations from spherical geometry certainly preclude quantitative comparisons. The detailed comparisons lead us to an estimate of the turbulent dissipation parameter that when the turbulence is initially low, is found to be in good agreement with previous estimate of non self-gravitating supersonic turbulence. abridged.
Horseshoe-shaped brightness asymmetries of several transitional discs are thought to be caused by large-scale vortices. Anticyclonic vortices are efficiently collect dust particles, therefore they can play a major role in planet formation. Former studies suggest that the disc self-gravity weakens vortices formed at the edge of the gap opened by a massive planet in discs whose masses are in the range of 0.01<=M_disc/M_*<=0.1. Here we present an investigation on the long-term evolution of the large-scale vortices formed at the viscosity transition of the discs dead zone outer edge by means of two-dimensional hydrodynamic simulations taking disc self-gravity into account. We perform a numerical study of low mass, 0.001<=M_disc/M_*<=0.01, discs, for which cases disc self-gravity was previously neglected. The large-scale vortices are found to be stretched due to disc self-gravity even for low-mass discs with M_disc/M_*>=0.005 where initially the Toomre Q-parameter was <=50 at the vortex distance. As a result of stretching, the vortex aspect ratio increases and a weaker azimuthal density contrast develops. The strength of the vortex stretching is proportional to the disc mass. The vortex stretching can be explained by a combined action of a non-vanishing gravitational torque caused by the vortex, and the Keplerian shear of the disc. Self-gravitating vortices are subject to significantly faster decay than non-self-gravitating ones. We found that vortices developed at sharp viscosity transitions of self-gravitating discs can be described by a GNG model as long as the disc viscosity is low, i.e. alpha_dz<=10^-5.
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