I have carried out three-dimensional numerical simulations of self-gravitating discs to determine under what circumstances they fragment to form bound clumps that may grow into giant planets. Through radiation hydrodynamical simulations using a Smoothed Particle Hydrodynamics code, I find that the disc opacity plays a vital role in determining whether a disc fragments. Specifically, opacities that are smaller than interstellar Rosseland mean values promote fragmentation (even at small radii, R < 25AU) since low opacities allow a disc to cool quickly. This may occur if a disc has a low metallicity or if grain growth has occurred. Given that the standard core accretion model is less likely to form planets in a low metallicity environment, I predict that gravitational instability is the dominant planet formation mechanism in a low metallicity environment. In addition, I find that the presence of stellar irradiation generally acts to inhibit fragmentation (since the discs can only cool to the temperature defined by stellar irradiation). However, fragmentation may occur if the irradiation is sufficiently weak that it allows the disc to attain a low Toomre stability parameter. With specific reference to the HR 8799 planetary system, I find that it is only possible for fragments to form in the radial range where the HR 8799 planets are located (approximately 24-68 AU) if the disc is massive. In such a high mass regime, mass transport occurs in the disc causing the surface mass density to alter. Therefore, fragmentation is not only affected by the disc temperature and cooling, but also by any restructuring due to the gravitational torques. The high mass discs also pose a problem for the formation of this system because the protoplanets accrete from the disc and end up with masses greater than those inferred from observation and thus, the growth of planets would need to be inhibited. In addition, I find that further subsequent fragmentation at small radii also takes place. By way of analytical arguments in combination with hydrodynamical simulations using a parameterised cooling method, I explore the fragmentation criteria which in the past, has placed emphasis on the cooling timescale in units of the orbital timescale, beta. I find that at a given radius the surface mass density (i.e. disc mass and profile) and star mass also play a crucial role in determining whether a disc fragments or not as well as where in the disc fragments form. I find that for shallow surface mass density profiles (p<2, where the surface mass density is proportional to R^{-p}), fragments form in the outer regions of the disc. However for steep surface mass density profiles (p is greater than or similar to 2), fragments form in the inner regions of a disc. In addition, I find that the critical value of the cooling timescale in units of the orbital timescale, beta_crit, found in previous simulations is only applicable to certain disc surface mass density profiles and for particular disc radii and is not a general rule for all discs. I obtain an empirical fragmentation criteria between the cooling timescale in units of the orbital timescale, beta, the surface mass density, the star mass and the radius. Finally, I carry out crucial resolution testing by performing the highest resolution disc simulations to date. My results cast some serious doubts on previous conclusions concerning fragmentation of self-gravitating discs.

Photometry of eclipsing dwarf-nova EX Draconis was performed at the observatory of Astronomical Institute of the Academy of Sciences of the Czech Republic in Ondřejov and at the Astronomical Observatory on Kolonické sedlo during 63 nights. I calculated times of minimum light by two methods - the mirror method and the derivative method. The mirror method is more precise for these measurements with longer exposure time and smaller coverage of the light curve of eclipse. 53 new times of minima calculated by the mirror method and times of minima obtained from the older articles about EX Dra were included in the O-C diagram and fitted by the sine function and theoretical curve of LITE caused by an unseen third body. Period of cyclic changes for the sine function is 25 years, instead of 4 or 5 years period given in older papers.Period of the third body orbit is aproximately 17 years and its minimum mass is 53.5 Jupiter's mass. The sum of squared deviations is 5 times smaller for the LITE, which means that the LITE ilustrates the O-C diagram better. I determined the average outburst period for three observational seasons and I drew the phase curve of outburst. Finally, I calculated the short period of the light changes outside eclipse.

In the case of cold accretion disk, coupling between charge neutral gas and magnetic field is too weak such that the magneto-rotational instability will be less effective or even stop working. In such a situation it is of prime interest to investigate the pure hydrodynamic turbulence and transport phenomenon. As the Reynolds number increases, the relative importance of the non-linear term in the hydrodynamic equation increases and in the case of accretion disk where molecular viscosity is too small the Reynolds number is large enough for the non-linear term to bring new effects. We investigate a scenario, the ‘weakly non-linear’ evolution of amplitude of linear mode when the flow is bounded by two parallel walls. The unperturbed flow is similar to plane Couette flow but with Coriolis force included in the hydrodynamic equation. Although there is no exponentially growing eigenmode, due to self-interaction the least stable eigenmode will grow in an intermediate phase. Later on this will lead to higher order non-linearity and plausible turbulence. Although the non-linear term in the hydrodynamic equation is energy conserving, within the weakly non-linear analysis it is possible to define a lower bound of the energy needed for flow to transform to turbulent phase. Such an unstable phase is possible only if the Reynolds number ≥ 103−4. In Chapter-2 we set up equation of amplitude for the hydrodynamic perturbation and study the effect of weak non-linear evolution of linear mode for general angular momentum distribution, where Keplerian disk is obtained as a special case. As we know that to explain observed hard X-rays the choice of Keplerian angular momentum profile is not adequate, we consider the sub-Keplerian regime of the disk. In Chapter-3 we assume that the cooling mechanism is dominated by bremsstrahlung process (without any strict knowledge of the magnetic field structure).We show that in a range of Shakura-Sunyaev viscosity 0.2 ≥ α ≥ 0.0005, flow behavior varies widely, particularly by means of the size of disk, efficiency of cooling and corresponding temperatures of ions and electrons. We also show that the disk around a rotating black hole is hotter compared to that around a Schwarzschild black hole, rendering a larger difference between ion and electron temperatures in the former case. We finally reproduce the observed luminosities(L) of two extreme cases—the under-fed AGNs and quasars and ultra-luminous X-ray sources at different combinations of mass accretion rate, ratio of specific heats, Shakura-Sunyaev viscosity parameter and Kerr parameter. In Chapter-4 we investigate the viscous two temperature accretion disk flows around rotating blackholes. We describe the global solution of accretion flows, unlike that in Chapter-3, with a sub-Keplerian angular momentum profile, by solving the underlying conservation equations including explicit cooling processes self-consistently. Bremsstrahlung, synchrotron and inverse comptonization of soft photons are considered as possible cooling mechanisms. We focus on the set of solutions for sub-Eddington, Eddington and super-Eddington mass accretion rates around Schwarzschild and Kerr black holes with a Kerr parameter 0.998. We analyse various phases of advection–general advective paradigm to radiatively inefficient paradigm. The solution may potentially explain the hard X-rays and γ-rays emitted from AGNs and X-ray binaries. We also compare the solutions for two different regimes of viscosity. We finally reproduce the observed luminosities of the under-fed AGNs and quasars, ultra-luminous X-ray sources at different combinations of input parameters such as mass accretion rate and ratio of specific heats.

In a magnetized plasma, naturally occurring pressure anisotropies facilitate instabilities that are expected to modify the transport properties of the system. In this thesis we examine two such instabilities and, where appropriate, their effects on transport. First we consider the collisional (fluid) magnetized magnetorotational instability (MRI) in the presence of the Braginskii viscosity. We conduct a global linear analysis of the instability in a galactic rotation profile for three magnetic field configurations: purely azimuthal, purely vertical and slightly pitched. Our analysis, numerical and asymptotic, shows that the first two represent singular configurations where the Braginskii viscosity's primary role is dissipative and the maximum growth rate is proportional to the Reynolds number when this is small. For a weak pitched field, the Braginskii viscosity is destabilising and when its effects dominate over the Lorentz force, the growth rate of the MRI can be up to 2√2 times faster than the inviscid limit. If the field is strong, an over-stability develops and both the real and imaginary parts of the frequency increase with the coefficient of the viscosity. Second, in the context of the ICM of galaxy clusters, we consider the pressure-anisotropy-driven firehose instability. The linear instability is fast (~ ion cyclotron period) and small-scale (ion Larmor radius ρi) and so fluid theory is inapplicable. We determine its nonlinear evolution in an ab initio kinetic calculation (for parallel gradients only). We use a particular physical asymptotic ordering to derive a closed nonlinear equation for the firehose turbulence, which we solve. We find secular (α t) growth of magnetic fluctuations and a k-||3 spectrum, starting at scales >~ ρi. When a parallel ion heat flux is present, the parallel firehose instability mutates into the new gyrothermal instability. Its nonlinear evolution also involves secular magnetic energy growth, but its spectrum is eventually dominated by modes with a maximal scale ~ρilT/λmfp,(lT is the parallel temperature gradient scale). Throughout we discuss implications for modelling, transport and other areas of magnetized plasma physics.

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.