Stars of all ages and evolutionary stages are seen to be surrounded by discs of matieral. during the formation of a stellar system the stars are orbited by a massive protoplanetary disc composed of interstellar gas and dust, in which planet formation occurs. Betewwen 1 and 10 Myr the protoplanetary disc disperses, leaving behind the newly formed system of planets and smaller bodies. The remaining material which has not formed into planets is referred to as a debris disc. Even though the interstellar dust grains from the protoplanetary disc have long been removed from the system, debris discs can contain large quantities of dust due to collisions between larger bodies and cometary activity. such dust can be detected by its thermal emission. This thesis focuses on observational studies at far-infrared and sub-millimetre wavelengths of debris discs and the late stages of protoplanetary disc evolution. An overview of surveys for debris discs performed to date is presented, highlighting the limitations and statistical biases. the motivation, design and sample selection for two large surveys for debris discs around nearby stars, with the Hershel space observatory and the SCUBA-2 sub-millimetre camera on the James Clerk Maxwell Telescope, are described. The combination of a uniform obstevational strategy, longer wavelengths than previous surveys, and a large, clearly chosen sample - unbiased by stellar properties - will allow robust statistical conclusions of how the incidence and properties of debris discs depend on system parameters such as stellar mass, age, metallicity, binarity and the presence of planets. As a precursor to the Hershel and SCUBA-2 surveys, a volume-limited ample of 130 A type star systems was surveyed using observations at 24 and 70 μm, which were required to determine the presence of emission from dust, were predicted by fitting model flux distributions to optical and near-infrared photometry. Debris discs were detected around 46 systems, 12 of which including the system with the largest dust mass - are new discoveries. This survey adds to the results of previous studies which show that debris disc incidence is not correleated with host star metallicity despite the wll known giant planet - metallicity correlation, This is in accordance with what is predicted from the core accretion theory of planet formation. The most signigicant result from this survey is that, contrary to results reported in a previous work, debris discs are oberall less common around binary stars. Further investigation shows that systems with separations of ~3-150 AU are especially deficient of debris, while closer binaries and the primaries of wider binaries show debris detection rates consisten with those for single stars. A sample of circumstellar discs around 29 young stellar systems with ages of 5-30 Myr were observed with the LABOCA sub-millimetre instrument on the APEX telescope at 870μm, to provide disc masses or mass upper limits in support of a large Hershel programme. These targets included the η Chamaeleontis cluster and four bright Herbig Ae/Be stars which have not previously been observed at this wavelength. All but the Herbig Ae/Be stars were not detected, and 3σ dust mass upper limits of ~ 0.1-3 M are determined, with corresponding total disc masses of ~0.03-1Mjup. These mass limits indicate that there is insufficient remaining material in these discs to form gas giant planets, and add to the prevailing view that protoplanetary discs typically disperse within 10 Myr and that gas giant planet formation must be completed before this time. A search for cold dust emimission from two of the Solar System's nearest neighbours - α Centauri AB and ε Indi - was also performed with LABOCA. In both cases no debris disc emission was detected. A bright resolved feature was detected near α Centauri AB, nowever, follow-up observations at a second epoch, two years after the initial observations, showed that the feature is not co-moving with the stars. It is argued that the feature is most likely a pre-stellar core. The stars α Centauri A and B are detected, which is one of only very few detections of main sequence stellar photospheres at sub-millimetre wavelengths.

The last twenty-five years have seen our understanding of the formation and abundance of planets revolutionised, thanks to the first detections of debris discs, and, a decade later, of the first extrasolar planets. Hardly a week now goes by without a planet discovery, and the range of methods used to search for planets has expanded to include techniques that are efficient at detecting different types of planets. By combining the discoveries of the various methods, we therefore have the opportunity to build a picture of planet populations across the Galaxy. In this thesis, I am presenting work done as a basis towards such an effort: first I present work carried out to improve modelling methods for gravitational microlensing events. Since the first microlensing observing campaigns, the amount of data of anomalous events has been increasing ever faster, meaning that the time required to model all observed anomalous events is putting a strain on available human and computational resources. I present work to develop a method to fit anomalous microlensing events automatically and show that it is possible to conduct a thorough and unbiased search of the parameter space, illustrating this by analysing an event from the 2007 observing season. I then discuss the possible models found with this method for this event, and their implication (Kains et al. 2009), and find that this algorithm locates good-fit models in regions of parameters that would have been very unlikely to be found using standard modelling methods. Results indicate that it is necessary to use a full Bayesian approach, in order to include prior information on the parameters. I discuss the analytical priors calculated by Cassan et al. (2009) and suggest a possible form of an automatic fitting algorithm by incorporating these priors in the algorithm used by Kains et al. (2009). Another topic with which this thesis is concerned is the evolution of debris discs around solar-type stars. Late-type stars are expected to be the most numerous host stars of planets detected with the microlensing technique. Understanding how their debris discs evolve equates to understanding the earliest stages of planet formation around these stars, allowing us to truly put our Solar System in perspective. Using the analytical model of Wyatt et al. (2007a), I modelled the evolution of infrared excess flux at 24 and 70 microns using published data of debris discs around solar-type (spectral types F, G and K) stars from the Spitzer Space Telescope. By comparing the results of this study to an analogous study carried out by for A stars by Wyatt et al. (2007b), I find that although best-fit parameters are significantly different for solar-type stars, this may be due to the varying number of inefficient emitters around stars of different spectral types. I suggest that although effective properties are different by an order of magnitude or more, intrinsic properties, while still different, are so by a much smaller factor. These differences may be due to the longer timescales over which solar-type stars evolve, which allow for the formation of larger and stronger planetesimals.

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.