Academic literature on the topic 'Keplerian disk'

AbstractSeveral theories have been proposed to explain the origin of Be star disks. Among them are Wind-Compressed Disks, accretion disks, decretion disks, and “explosive” ejections. In reviewing these mechanisms, I first concentrate on the current status of the Wind-Compressed Disk model. In particular, I discuss how non-radial forces may prevent disk formation and then discuss various physical effects that may restore the disk. Second, I examine the observational evidence and what it tells us about the structure of the disk. Of particular interest is evidence in favor of Keplerian disks. Finally, I discuss theories for Keplerian disk formation and some of the constraints such theories must satisfy.

AbstractSeveral examples of thin, Keplerian, sub-parsec megamaser disks have been discovered in the nuclei of active galaxies and used to precisely determine the mass of their host black holes. We show that there is an empirical linear correlation between the disk radius and black hole mass and that such disks are naturally formed as molecular clouds pass through the galactic nucleus and temporarily engulf the central supermassive black hole. For initial cloud column densities below about 1023.5 cm−2 the disk is non-self gravitating, but for higher cloud columns the disk would fragment and produce a compact stellar disk similar to that observed around Sgr A* at the galactic centre.

We study the characteristics of global oscillation modes of masing disks in megamasers and the effect of the modes on the disk kinematics. We find that the eccentric mode is responsible for the observed sub-Keplerian velocity distribution of the maser source of NGC 1068, whereas in the masing disk of NGC 4258 the warping mode is dominant so that the angular rotation velocity remains near Keplerian.

AbstractThe nature of MHD and hydrodynamical turbulence in accretion disks is discussed. Comparison is made with planar Couette flow, a classical system prone to nonlinear shear instability resulting in enhanced turbulent transport. Both Keplerian and non-Keplerian hydrodynamical disks are studied, and it is found that only constant angular momentum disks are unstable to nonlinear disturbances and develop enhanced turbulent transport. Convective instabilities do not lead to enhanced turbulent transport. Hydrodynamical Keplerian disks are quite stable to nonlinear disturbances. Several lines of argument are presented which all lead to this conclusion, but the key to disk turbulence is the interaction between the stress tensor and the mean flow gradients. The nature of this coupling is found to determine completely the stability properties of disks (hydrodynamics and magnetic), and the nature of turbulent transport. The weak field MHD instability, which is of great astrophysical importance, displays the same type of stress tensor – mean flow coupling that all classical local shear instabilities exhibit. Hydrodynamical Keplerian disks, on the other hand, do not. Accretion disk turbulence is MHD turbulence.

Abstract Observations of protoplanetary disks have shown that forming exoplanets leave characteristic imprints on the gas and dust of the disk. In the gas, these forming exoplanets cause deviations from Keplerian motion, which can be detected through molecular line observations. Our previous work has shown that machine learning can correctly determine if a planet is present in these disks. Using our machine-learning models, we identify strong, localized non-Keplerian motion within the disk HD 142666. Subsequent hydrodynamics simulations of a system with a 5 M J planet at 75 au recreate the kinematic structure. By currently established standards in the field, we conclude that HD 142666 hosts a planet. This work represents a first step toward using machine learning to identify previously overlooked non-Keplerian features in protoplanetary disks.

AbstractWe test whether Be star disks rotate in a Keplerian or an Angular Momentum Conserving fashion. This is done by employing sub-milli arcsecond spectroastrometry around Hα. We spatially resolve the disks, and are the first to do so at such a high spectral resolution. We fit the emission line profiles with parametric models. The Keplerian models reproduce the spectro-astrometry, whereas the AMC models do not, thereby supporting the viscous disk model for Be stars.

Abstract In this paper, the mechanisms of star formation and the formation of the equatorial gas and dust disk of protostars are considered. The viscous dynamics of the interstellar matter of gas and dust disks is mainly determined by perturbations of the matter density during gas accretion onto the equilibrium core of the protostar. Using the model of pulsating perturbations of the density of the gas-dust envelope of the protostar and the Navier-Stokes equations, the formulas for the dynamic viscosity of Keplerian and almost Keplerian disks are obtained. It is shown that in the regime of unstable equilibrium of the envelope, accretion of gas onto the core of the protostar begins. In the regime of stable equilibrium, the fragmentation of the gas-dust envelope and the equatorial disk of the protostar occurs. In the ring-shaped fragments of the disk, the process of formation of “embryos” of planets begins and accretion on the “embryos” of the planet also begins.

AbstractMany accretion disks surrounding supermassive black holes in nearby AGN are observed to host 22 GHz water maser activity. We have analyzed single-dish 22 GHz spectra taken with the GBT to identify 32 such “Keplerian disk systems,” which we used to investigate maser excitation and explore the possibility of disk reverberation. Our results do not support a spiral shock model for population inversion in these disks, and we find that any reverberating signal propagating radially outwards from the AGN must constitute <10% of the total observed maser variability. Additionally, we have used ALMA to begin exploring the variety of sub-mm water megamasers that are also predicted, and in the case of the 321 GHz transition found, to be present in these accretion disks. By observing multiple masing transitions within a single system, we can better constrain the physical conditions (e.g., gas temperature and density) in the accretion disk.

This paper focuses on the polarized profiles of resonance scattering lines that form in magnetized disks. Optically thin lines from Keplerian planar disks are considered. Model line profiles are calculated for simple field topologies of axial fields (i.e., vertical to the disk plane) and toroidal fields (i.e., purely azimuthal). A scheme for discerning field strengths and geometries in disks is developed based on Stokes Q − U diagrams for the run of polarization across line profiles that are Doppler-broadened by the disk rotation. A discussion of the Hanle effect for magnetized disks in which the magnetorotational instability (MRI) is operating is also presented. Given that the MRI has a tendency to mix the vector field orientation, it may be difficult to detect the disk fields with the longitudinal Zeeman effect, since the amplitude of the circularly polarized signal scales with the net magnetic flux in the direction of the observer. The Hanle effect does not suffer from this impediment, and so a multi-line analysis could be used to constrain field strengths in disks dominated by the MRI.

博士
國立臺灣大學
天文物理研究所
101
Circumstellar disks around young stellar objects are sites of planet formation. It is intriguing to understand the formation process of such circumstellar disks. In the formation process of a low-mass star, a circumstellar disk is expected to form in the innermost (<100 AU) region of a protostellar envelope around a protostar at an early evolutionary stage, and envelope material infalls and feeds the central protostar and circumstellar disk. Formation and evolution of circumstellar disks are closely related to the mechanism of angular momentum transportation in protostellar envelopes from the envelope scale (thousands of AU) down to the disk scale (~100 AU). To study this mechanism, it is essential to reveal the kinematics of protostellar envelopes around representative protostellar sources and to compare the kinematics among sources at different evolutionary stages. Therefore, I have conducted SMT, ASTE, SMA and ALMA observations toward a sample of protostellar sources, and studied (1) the rotational motion on the scales from 10,000 AU to 100 AU around a Class 0 protostar B335, (2) evolution of the rotational motions on the scales of 1000-100 AU of a sample of Class 0 and I protostars, and (3) the connection between the disk and the protostellar envelope around a Class I protostar L1489 IRS. To study rotational motion from large to small scales in protostellar sources, I have conducted observations in the millimeter C18O (2-1) and submillimeter CS (7-6) lines with the SMT, ASTE and SMA toward a prototypical Class 0 protostar, B335. In B335, the C18O (2-1) emission traces the protostellar envelope on the scales from ~10,000 AU to a few hundred AU, while the CS (7-6) emission shows a compact envelope component with a size of ~800 AU surrounded by an east-west elongated outflow component with a size of ~3000 AU. On the scale of 10,000 AU, the C18O envelope exhibits rotational motion with a specific angular momentum of ~2 x 10^{-3} km/s pc (V ~ 0.04 km/s at a radius of 9000 AU), comparable to those of other NH3 dense cores. On the scale of a few hundred AU, the C18O envelope exhibits infalling motion but no signature of rotational motion (V < 0.04 km/s at a radius of 370 AU). The CS (7-6) line, having a higher upper energy level and a higher critical density than the C18O (2-1) line, can trace an inner dense and warm region around protostars, where rotational velocity is likely higher than that in an outer region. On the scale of ~100 AU, the CS envelope shows rotational motion (V = 0.11 km/s at a radius of 110 AU) but no signature of infalling motion. These results show that the specific angular momenta of the rotational motion in B335 decrease from radii of 10,000 AU to a few hundred AU, and the specific angular momenta on the scale of a few hundred AU are one to two orders of magnitude lower than those in other Class I and II sources. To study evolution of rotational motions of protostellar sources, I have conducted observations in the C18O (2-1) line with the SMA toward three Class 0, one Class 0/I, and two Class I protostars. My observational results show that two Class 0 sources, B335 and NGC 1333 IRAS 4B, do not exhibit detectable rotational motion on hundreds of AU scale, while L1527 IRS (Class 0/I) and L1448-mm (Class 0) exhibit rotational motions with radial profiles of V ~ r^{-1.0+/-0.2} and ~ r^{-1.0+/-0.1}, respectively. The other Class I sources, TMC-1A and L1489 IRS, exhibit the fastest rotational motions among the sample, and their rotational motions have flatter radial profiles of V ~ r^{-0.6+/-0.1} and ~ r^{-0.5+/-0.1}$, respectively. The rotational motions with the radial dependence of ~ r^{-1} can be interpreted as rotation with a conserved angular momentum in a dynamically infalling envelope, while those with the radial dependence of ~ r^{-0.5} can be interpreted as Keplerian rotation. To study the connection between circumstellar disks and their surrounding protostellar envelopes, I have conducted observations in the 1.3 mm continuum and the 12CO (2-1), C18O (2-1), and SO (5_6-4_5) lines with the ALMA toward a Class I protostar L1489 IRS. A circumstellar disk in Keplerian rotation around L1489 IRS is clearly identified in the 12CO and C18O emission, and the central protostellar mass is estimated to be 2.0 Msun. In addition, there are arm-like structures attached to the circumstellar disk, and their kinematics cannot be explained by the Keplerian rotation. These non-Keplerian structures could trace accretion flow following parabolic trajectories toward the disk. The SO emission primarily traces the transitional regions between the accretion flow and the disk, which could be due to the enhancement in the SO abundance in the regions of accretion shocks. From my observational results as well as those from literatures, I have found the kinematics of protostellar envelopes on 100-1000 AU scales around Class 0 and I protostars can be categorized into three groups, (1) infalling motion with little rotational motion around Class 0 protostars, (2) both infalling and rotational motions around Class 0 and I protostars, and (3) Keplerian rotation around Class I protostars. I propose that the three categories reflect the evolution sequence from infalling envelopes to formation of Keplerian disks. In an early stage of collapse of a dense core, the envelope material with a small angular momentum in the vicinity of the protostar collapses first, and the protostellar envelope on 100-1000 AU scales shows infalling motion but little rotational motion. As the expansion wave propagates outwardly, the envelope material with a larger angular momentum in an outer region start to collapse. As more angular momenta travel toward the center with the infalling motion, rotational velocities of the protostellar envelope on 100-1000 AU scales and the size of the central disk increase. With time, the protostellar envelope dissipates due to the mass ejection from outflows and the mass accretion onto the central protostar and disk. At a later evolutionary stage, the envelope material is infalling along few parabolic flows (not isotropically) due to the protostellar envelope is partially dissipated, and a Keplerian disk with an outer radius of hundreds of AU appears. Based on the inside-out collapse theory of protostellar envelopes, I have constructed an analytical model and computed evolution of radial profiles of rotational velocities, to interpret the observed results in the context of formation of large-scale (>100 AU) disk.

The presence of an accretion disk around a neutron star (NS) or a black hole (BH) is mainly responsible for the observed X-rays from the night sky. NS or BH is a part of the binary systems, and matter flows from a companion star to them. That is why they are categorized as X-ray binaries (XRBs). However, accretion flow around an astrophysical object is not only restricted to XRBs. They are ubiquitous in different astrophysical sites e.g., around a white dwarf (WD), supermassive black hole, young stellar object etc. Nevertheless, the hard X-rays present in the observed spectral energy distribution of XRBs cannot be explained by the so-called standard, geometrically thin, optically thick, Keplerian Shakura-Sunyaev disks (SSDs), which is responsible for the softer spectrum. The hard X-rays are responsible for optically thin, geometrically thick, hot advective accretion disks around the central accretor. Till now, the outer region of XRB and active galactic nucleus (AGN) disks is understood to be colder SSD and inner region to be advective accretion disk, together forming a disk-wind system. Although we try to explore different scientific objectives, throughout this thesis work, our study is centered around hot advective accretion flows around WD, NS and BH.

A rich variety of discs are found orbiting massive bodies in the universe. These could be accretion discs composed of gas around stellar mass compact objects fueling micro-quasar activity; protoplanetary discs, mainly composed of dust and gas, are the progenitors for planet formation; accretion discs composed of stars and gas around super-massive black holes at the centers of galaxies fueling the active galactic nuclei activity; discs in spiral galaxies; and many more. Structural and kinematic properties of these discs in several astrophysical systems are correlated to the global properties; for example, over a sample of thousands of galaxies, a correlation has been found between lopsidedness, black hole growth, and the presence of young stellar populations in the centers of galaxies. Galaxy formation and evolution of the central BH are some of the contexts in which such correlations become important. Studying the dynamics of these discs helps to explain their structural properties and is thus of paramount importance. In most astrophysical discs(a notable exception being the stellar discs in spiral galaxies),the dynamics are usually dominated by the gravity of the central object, and is thus nearly Keplerian. However, there is a small contribution to the total force experienced by the disc due to the disc material. Discs mentioned above differ from each other due to different underlying force that dominates the non-Keplerian dynamics of these discs. Two important numbers which are useful in describing physical properties of any disc structure in astrophysics are: (1) Mach number M , and(2) Toomre Q parameter. If thermal pressure gradient and/or random motion dominate the non-Keplerian forces, then M « Q, and in the case when the self-gravity of the disc is more important then Particles constituting the disc orbit under Keplerian potential due to central object, plus the small contribution from the non-Keplerian potential due to disc self-gravity, or the thermal pressure gradient. For a Keplerian potential, the radial and azimuthal frequencies are in 1 : 1 ratio w.r.t. each other and hence there is no precession in the orbits. In case of nearly Keplerian potential(when non-Keplerian contributions are small), the orbits precess at a rate proportional to the non-Keplerian forces. It is this non-zero but small precession that allows the existence of modes whose frequencies are proportional to the precession rate. These modes are referred to as slow modes (Tremaine 2001). Such modes are likely to be the only large-scale or long-wavelength modes. The damping they suffer due to viscosity, collisions, Landau damping, or other dissipative processes is also relatively less. Hence, these modes can dominate the overall appearance of discs. In this thesis we intend to study slow modes for nearly Keplerian discs. Slow modes innear-Kepleriandiscscantobethereasonforvariousnon-axisymmetricfeatures observed in many systems: 1 Galactic discs: Of the few galaxies for which the observations of galactic nuclei exist, two galaxies: NGC4486B(an elliptical galaxy) andM31(spiral galaxy), show an unusual double-peak distribution of stars at their centers. In order to explain such distributions, Tremaine in 1995 proposed an eccentric disc model for M31; this model was then further explored by many authors. In addition, lopsidedness is observed in many galaxies on larger scales, and such asymmetries need to be explained via robust modeling of galactic discs. 2 Debris disc: Many of the observed discs show non-axisymmetric structures, such as lopsided distribution in brightness of scattered light, warp, and clumps in the disc around β Pictoris; spiral structure inHD141569A,etc. Most of these features have been attributed to the presence of planets, and in some cases planets have also been detected. However, Jalali & Tremaine(2012)proposed that most of these structures can be formed also due to slow (m =1 or 2) modes. 3 Accretion Discs around stellar mass binaries have also been found to be asymmetric. One plausible reason for this asymmetry can be m =1slowmodes in these systems. Slow modes are studied in detail in this thesis. The main approaches that we have used, and the major conclusions from this work are as follows: Slow pressure modes in thin accretion disc Earlier work on slow modes assumed that the self-gravity of the disc dominates the pressure gradient in the discs. However, this assumption is not valid for thin and hot accretion discs around stellar mass compact objects. We begin our study of slow modes with the analysis of modes in thin accretion discs around stellar mass compact objects. First, the WKB analysis is used to prove the existence of these modes. Next, we formulate the eigenvalue equation for the slow modes, which turns out to be in the Sturm-Liouville form; thus all the eigenvalues are real. Real eigenvalues imply that the disc is stable to these perturbations. We also discuss the possible excitation mechanisms for these modes; for instance, excitation due to the stream of matter from the secondary star that feeds the accretion disc, or through the action of viscous forces. Slow modes in self-gravitating, zero-pressure fluid disc We next generalize the study of slow m = 1 modes for a single self-gravitating disc of Tremaine(2001) to a system of two self-gravitating counter–rotating, zero-pressure fluid discs, where the disc particles interact via softened-gravity. Counter– rotating streams of matter are susceptible to various instabilities. In particular, Touma(2002)found unstable modes in counter–rotating ,nearly Keplerian systems. These modes were calculated analytically for a two-ring system, and numerically for discs modeled assuming a multiple–ring system. Motivated by this, the corresponding problem for continuous discs was studied by Sridhar & Saini(2010),who proposed a simple model, with dynamics that could be studied largely analytically in the local WKB approximation. Their work, however, had certain limitations; they could construct eigenmodes only for η =0&12, where η is the mass fraction in the retrograde disc. They could only calculate eigenvalues but not the eigen functions. To overcome the above mentioned limitations, we formulate and analyze the full eigenvalue problem to understand the systematic behaviour of such systems. Our general conclusions are as follows 1 The system is stable for m = 1 perturbations in the case of no-counter rotation. 2. For other values of mass fraction , the eigenvalues are generally complex, and the discs are unstable. For η =12,theeigenvalues are imaginary, giving purely growing modes. 2 The pattern speed appears to be non-negative for all values of , with the growth(or damping) rate being larger for larger values of pattern speed. 3 Perturbed surface density profile is generally lopsided, with an overall rotation of the patterns as they evolve in time, with the pattern speed given by the real part of the eigenvalue. Local WKB analysis for Keplerian stellar disc We next urn to stellar discs, whose dynamics is richer than softened gravity discs. Jalali & Tremaine(2012)derived the dispersion relation for short wavelength slow modes for a single disc with Schwarzschild distribution function. In contrast to the softened gravity discs(which have slow modes only for m = 1), stellar discs permit slow modes for m 1. The dispersion relation derived by Jalali & Tremaine makes it evident that all m 1 slow modes are neutrally stable. We study slow modes for the case of two counter–rotating discs, each described by Schwarzschild distribution function, and derive the dispersion relation for slow m 1 modes in the local WKB limit and study the nature of the instabilities. One of the important applications of the dispersion relation derived in this chapter is the stability analysis of the modes. For fluid discs, it is well known that the stability of m = 0 modes guarantees the stability of higher m modes; and the stability criterion for such discs is the well known Toomre stability criterion. However, this is not the case for collisionless discs. Even if the discs are stable for axisymmetric modes, they can still be unstable for non-axisymmetric modes. The stability of axisymmetric modes is governed by the Toomre stability criterion The non-axisymmetric perturbations were found to be unstable if the mass in the retrograde component of the disc is non-zero. We next solve the dispersion relation using the Bohr-Sommerfeld quantization condition to obtain the eigen-spectrum for a given unperturbed surface density profile and velocity dispersion. We could obtain only the eigenvalues for no counter– rotation η = 0, where η is the mass fraction in the retrograde disc and equal counter–rotation(η =12). All the eigenvalues obtained were real for no counter– rotation, and purely growing/damping for equal counter–rotation. The eigenvalue trends that we get favour detection of high ω and low m modes observationally. We also make a detailed comparison between the eigenvalues for m = 1 modes that we obtain with those obtained after solving the integral eigenvalue problem for the softened gravity discs for no counter–rotation and equal counter–rotation. The match between the eigenvalues are quite good, confirming the assertion that softened gravity discs can be reasonable surrogates for collisionless disc for m =1 modes. Non-local WKB theory for eigenmodes One major limitation of the above method is that eigenfunctions cannot be obtained as directly as in quantum mechanics because the dispersion relation is transcendental in radial wave number . We overcome this difficulty by dropping the assumption of locality of the relationship between perturbed self-gravity and surface density. Using the standard WKB analysis and epicyclic theory, together with the logarithmic-spiral decomposition of surface density and gravitational potential, we formulate an integral equation for determining both WKB eigenvalues and eigenfunctions. The application of integral equation derived is not only restricted to Keplerian disc; it could be used to study eigenmodes in galactic discs where the motion of stars is not dominated by the potential due to a central black hole (however we have not pursued the potential application in this thesis). We first verify that the integral equation derived reduces to the well known WKB dispersion relation under the local approximation. We next specialize to slow modes in Keplerian discs. The following are some of the general conclusions of this work 1 We find that the integral equation for slow modes reduces to a symmetric eigenvalue problem, implying that the eigenvalues are all real, and hence the disc is stable. 2 All the non-singular eigenmodes we obtain are prograde, which implies that the density waves generated will have the same sense of rotation as the disc, albeit with a speed which is compared to the the rotation speed of the disc. 3 Eigenvalue ω decreases as we go from m =1 to 2. In addition, for a given , the number of nodes for m =1 are larger than those for m =2. 4 The fastest pattern speed is a decreasing function of the heat in the disc. Asymmetric features in various types of discs could be due to the presence of slow m =1 or 2 modes. In the case of debris discs, these asymmetric features could also be due to the presence of planets. Features due to the presence of slow modes or due to planets can be distinguished from each other if the observations are made for a long enough time. The double peak nucleus observed in galaxies like M31 and NGC4486B differ from each other: stellar distribution in NGC4486B is symmetric w.r.t. its photocenter in contrast to a lopsided distribution seen in M31. It is more likely that the double peak nucleus in NGC4486B is due to m = 2 mode, rather than m = 1 mode as is the case for M31. NGC4486B being an elliptical galaxy, it is possible that the excitation probability for m =2 mode is higher.

A rich variety of discs are found orbiting massive bodies in the universe. These could be accretion discs composed of gas around stellar mass compact objects fueling micro-quasar activity; protoplanetary discs, mainly composed of dust and gas, are the progenitors for planet formation; accretion discs composed of stars and gas around super-massive black holes at the centers of galaxies fueling the active galactic nuclei activity; discs in spiral galaxies; and many more. Structural and kinematic properties of these discs in several astrophysical systems are correlated to the global properties; for example, over a sample of thousands of galaxies, a correlation has been found between lopsidedness, black hole growth, and the presence of young stellar populations in the centers of galaxies. Galaxy formation and evolution of the central BH are some of the contexts in which such correlations become important. Studying the dynamics of these discs helps to explain their structural properties and is thus of paramount importance. In most astrophysical discs(a notable exception being the stellar discs in spiral galaxies),the dynamics are usually dominated by the gravity of the central object, and is thus nearly Keplerian. However, there is a small contribution to the total force experienced by the disc due to the disc material. Discs mentioned above differ from each other due to different underlying force that dominates the non-Keplerian dynamics of these discs. Two important numbers which are useful in describing physical properties of any disc structure in astrophysics are: (1) Mach number M , and(2) Toomre Q parameter. If thermal pressure gradient and/or random motion dominate the non-Keplerian forces, then M « Q, and in the case when the self-gravity of the disc is more important then Particles constituting the disc orbit under Keplerian potential due to central object, plus the small contribution from the non-Keplerian potential due to disc self-gravity, or the thermal pressure gradient. For a Keplerian potential, the radial and azimuthal frequencies are in 1 : 1 ratio w.r.t. each other and hence there is no precession in the orbits. In case of nearly Keplerian potential(when non-Keplerian contributions are small), the orbits precess at a rate proportional to the non-Keplerian forces. It is this non-zero but small precession that allows the existence of modes whose frequencies are proportional to the precession rate. These modes are referred to as slow modes (Tremaine 2001). Such modes are likely to be the only large-scale or long-wavelength modes. The damping they suffer due to viscosity, collisions, Landau damping, or other dissipative processes is also relatively less. Hence, these modes can dominate the overall appearance of discs. In this thesis we intend to study slow modes for nearly Keplerian discs. Slow modes innear-Kepleriandiscscantobethereasonforvariousnon-axisymmetricfeatures observed in many systems: 1 Galactic discs: Of the few galaxies for which the observations of galactic nuclei exist, two galaxies: NGC4486B(an elliptical galaxy) andM31(spiral galaxy), show an unusual double-peak distribution of stars at their centers. In order to explain such distributions, Tremaine in 1995 proposed an eccentric disc model for M31; this model was then further explored by many authors. In addition, lopsidedness is observed in many galaxies on larger scales, and such asymmetries need to be explained via robust modeling of galactic discs. 2 Debris disc: Many of the observed discs show non-axisymmetric structures, such as lopsided distribution in brightness of scattered light, warp, and clumps in the disc around β Pictoris; spiral structure inHD141569A,etc. Most of these features have been attributed to the presence of planets, and in some cases planets have also been detected. However, Jalali & Tremaine(2012)proposed that most of these structures can be formed also due to slow (m =1 or 2) modes. 3 Accretion Discs around stellar mass binaries have also been found to be asymmetric. One plausible reason for this asymmetry can be m =1slowmodes in these systems. Slow modes are studied in detail in this thesis. The main approaches that we have used, and the major conclusions from this work are as follows: Slow pressure modes in thin accretion disc Earlier work on slow modes assumed that the self-gravity of the disc dominates the pressure gradient in the discs. However, this assumption is not valid for thin and hot accretion discs around stellar mass compact objects. We begin our study of slow modes with the analysis of modes in thin accretion discs around stellar mass compact objects. First, the WKB analysis is used to prove the existence of these modes. Next, we formulate the eigenvalue equation for the slow modes, which turns out to be in the Sturm-Liouville form; thus all the eigenvalues are real. Real eigenvalues imply that the disc is stable to these perturbations. We also discuss the possible excitation mechanisms for these modes; for instance, excitation due to the stream of matter from the secondary star that feeds the accretion disc, or through the action of viscous forces. Slow modes in self-gravitating, zero-pressure fluid disc We next generalize the study of slow m = 1 modes for a single self-gravitating disc of Tremaine(2001) to a system of two self-gravitating counter–rotating, zero-pressure fluid discs, where the disc particles interact via softened-gravity. Counter– rotating streams of matter are susceptible to various instabilities. In particular, Touma(2002)found unstable modes in counter–rotating ,nearly Keplerian systems. These modes were calculated analytically for a two-ring system, and numerically for discs modeled assuming a multiple–ring system. Motivated by this, the corresponding problem for continuous discs was studied by Sridhar & Saini(2010),who proposed a simple model, with dynamics that could be studied largely analytically in the local WKB approximation. Their work, however, had certain limitations; they could construct eigenmodes only for η =0&12, where η is the mass fraction in the retrograde disc. They could only calculate eigenvalues but not the eigen functions. To overcome the above mentioned limitations, we formulate and analyze the full eigenvalue problem to understand the systematic behaviour of such systems. Our general conclusions are as follows 1 The system is stable for m = 1 perturbations in the case of no-counter rotation. 2. For other values of mass fraction , the eigenvalues are generally complex, and the discs are unstable. For η =12,theeigenvalues are imaginary, giving purely growing modes. 2 The pattern speed appears to be non-negative for all values of , with the growth(or damping) rate being larger for larger values of pattern speed. 3 Perturbed surface density profile is generally lopsided, with an overall rotation of the patterns as they evolve in time, with the pattern speed given by the real part of the eigenvalue. Local WKB analysis for Keplerian stellar disc We next urn to stellar discs, whose dynamics is richer than softened gravity discs. Jalali & Tremaine(2012)derived the dispersion relation for short wavelength slow modes for a single disc with Schwarzschild distribution function. In contrast to the softened gravity discs(which have slow modes only for m = 1), stellar discs permit slow modes for m 1. The dispersion relation derived by Jalali & Tremaine makes it evident that all m 1 slow modes are neutrally stable. We study slow modes for the case of two counter–rotating discs, each described by Schwarzschild distribution function, and derive the dispersion relation for slow m 1 modes in the local WKB limit and study the nature of the instabilities. One of the important applications of the dispersion relation derived in this chapter is the stability analysis of the modes. For fluid discs, it is well known that the stability of m = 0 modes guarantees the stability of higher m modes; and the stability criterion for such discs is the well known Toomre stability criterion. However, this is not the case for collisionless discs. Even if the discs are stable for axisymmetric modes, they can still be unstable for non-axisymmetric modes. The stability of axisymmetric modes is governed by the Toomre stability criterion The non-axisymmetric perturbations were found to be unstable if the mass in the retrograde component of the disc is non-zero. We next solve the dispersion relation using the Bohr-Sommerfeld quantization condition to obtain the eigen-spectrum for a given unperturbed surface density profile and velocity dispersion. We could obtain only the eigenvalues for no counter– rotation η = 0, where η is the mass fraction in the retrograde disc and equal counter–rotation(η =12). All the eigenvalues obtained were real for no counter– rotation, and purely growing/damping for equal counter–rotation. The eigenvalue trends that we get favour detection of high ω and low m modes observationally. We also make a detailed comparison between the eigenvalues for m = 1 modes that we obtain with those obtained after solving the integral eigenvalue problem for the softened gravity discs for no counter–rotation and equal counter–rotation. The match between the eigenvalues are quite good, confirming the assertion that softened gravity discs can be reasonable surrogates for collisionless disc for m =1 modes. Non-local WKB theory for eigenmodes One major limitation of the above method is that eigenfunctions cannot be obtained as directly as in quantum mechanics because the dispersion relation is transcendental in radial wave number . We overcome this difficulty by dropping the assumption of locality of the relationship between perturbed self-gravity and surface density. Using the standard WKB analysis and epicyclic theory, together with the logarithmic-spiral decomposition of surface density and gravitational potential, we formulate an integral equation for determining both WKB eigenvalues and eigenfunctions. The application of integral equation derived is not only restricted to Keplerian disc; it could be used to study eigenmodes in galactic discs where the motion of stars is not dominated by the potential due to a central black hole (however we have not pursued the potential application in this thesis). We first verify that the integral equation derived reduces to the well known WKB dispersion relation under the local approximation. We next specialize to slow modes in Keplerian discs. The following are some of the general conclusions of this work 1 We find that the integral equation for slow modes reduces to a symmetric eigenvalue problem, implying that the eigenvalues are all real, and hence the disc is stable. 2 All the non-singular eigenmodes we obtain are prograde, which implies that the density waves generated will have the same sense of rotation as the disc, albeit with a speed which is compared to the the rotation speed of the disc. 3 Eigenvalue ω decreases as we go from m =1 to 2. In addition, for a given , the number of nodes for m =1 are larger than those for m =2. 4 The fastest pattern speed is a decreasing function of the heat in the disc. Asymmetric features in various types of discs could be due to the presence of slow m =1 or 2 modes. In the case of debris discs, these asymmetric features could also be due to the presence of planets. Features due to the presence of slow modes or due to planets can be distinguished from each other if the observations are made for a long enough time. The double peak nucleus observed in galaxies like M31 and NGC4486B differ from each other: stellar distribution in NGC4486B is symmetric w.r.t. its photocenter in contrast to a lopsided distribution seen in M31. It is more likely that the double peak nucleus in NGC4486B is due to m = 2 mode, rather than m = 1 mode as is the case for M31. NGC4486B being an elliptical galaxy, it is possible that the excitation probability for m =2 mode is higher.

Nonlinear equations describing dynamics of 2D vortices are very important in the physics of the ocean and the atmosphere and in plasma physics and Astrophysics. Here linear and nonlinear 2D vortex perturbations of gravitating and light gaseous disks are examined in the geostrophic and post-geostrophic approximations. In the frame of geostrophic approximation, it is shown that the vortex with positive velocity circulation is characterized by low pressure with negative excess mass density of substance. Vortex with negative circulation has higher pressure and is a relatively tight formation with the positive excess mass density. In the post-geostrophic approximation, structures of the isolated monopole and dipole vortex (modons) solutions of these equations are studied. Two types of mass distributions in dipole vortices are found. The first type of modon is characterized by an asymmetrically positioned single circular densification and one rarefaction. The second type is characterized by two asymmetrically positioned densifications and two rarefactions, where the second densification-rarefaction pair is crescent shaped. The constant density contours of a dipole vortex in a light gas disk coincide with the streamlines of the vortex; in a self-gravitating disk, the constant density contours in the vortex do not coincide with streamlines. Possible manifestations of monopole and dipole vortices in astrophysical objects are discussed. Vortices play decisive role in the process of planet formation. Gas in a protoplanetary disk practically moves on sub-Keplerian speeds. Rigid particles, under the action of a head wind drags, lose the angular momentum and energy. As a result, the ~10 cm to meter-sized particles drift to the central star for hundreds of years. Long-lived vortical structures in gas disk are a possible way to concentrate the ~10 cm to meter sized particles and to grow up them in planetesimal. Here the effect of anticyclonic Burgers vortex on formation of planetesimals in a protoplanetary dusty disc in local approach is also considered. It is shown that the Burgers vortex with homogeneously rotating kernel and a converging radial stream of substance can effectively accumulate in its nuclear area the meter-sized rigid particles of total mass ∼1028 g for characteristic time ∼106 year.

In this paper, the mechanisms of star formation and the formation of the equatorial gas and dust disk of protostars areconsidered. The viscous dynamics of the interstellar matter of gas and dust disks is mainly determined by perturbations of thematter density during gas accretion onto the equilibrium core of the protostar. Using the model of pulsating perturbationsof the density of the gas-dust envelope of the protostar and the Navier-Stokes equations, the formulas for the dynamicviscosity of Keplerian and almost Keplerian disks are obtained. It is shown that in the regime of unstable equilibrium of theenvelope, accretion of gas onto the core of the protostar begins. In the regime of stable equilibrium, the fragmentation ofthe gas-dust envelope and the equatorial disk of the protostar occurs. In the ring-shaped fragments of the disk, the processof formation of “embryos” of planets begins and accretion on the “embryos” of the planet also begins.

Spectral and photometric monitoring of CTTS RY Tau in 2013–2021 revealed variations in the H α and Na i D lines atradial velocities of infall (accretion) and outflow (wind) with a period of about 22 days. The Na i D absorption in theinfalling and outflowing gas changes in anti-phase: an increase in the density of infalling gas is accompanied by a decreasein the wind density, and vice versa. The fluctuations remain coherent for several years of observation. We assume that theobserved effect may be due to processes at the disk-magnetosphere boundary in MHD propeller mode. The stability of theoscillation phase indicates the possible presence of a massive planet, causing azimuthal asymmetry of the accretion andwind flows. If the observed period (22 days) is Keplerian, then the planet is located at a distance of 0.2 AU and moves withan orbital velocity of 100 km s −1 .

The article is based on the plenary report at the VAK 2021. It describes the development of ideas and reviews somerecent achievements in modern celestial mechanics. We proceed from the fact that the classical definition of this sciencegiven in textbooks and encyclopedias does not fully reflect the content of modern celestial mechanics. A broader and morecapacious term is “dynamic astronomy”. This science is complex and includes not only classical but also relativistic celestialmechanics, as well as the theory of equilibrium figures and computer simulations; besides, we should not forget aboutqualitative methods, the peak of which is the creation of KAM theory. Here we trace the development of a chain of ideasfrom Keplerian orbits to osculating Lagrange ellipses, which leads eventually to Gaussian rings and two models (R-disk andR-toroid) based on precessing Gaussian rings. The review also includes a presentation of results obtained in recent years byresearch groups in the Russian Federation.