Abstract We develop the theory of instabilities in a rotating gaseous disc and in shallow water for the case where there is a break in the surface density and sound velocities, as well as the rotation rate, at a particular radius. Different instabilities of sub-sonic and supersonic flows have been investigated. We also prove the identity of the linearised dynamical equations for the gaseous disc of the Galaxy and for our rotating shallow water experiments.

Introduction

The present paper pursues two aims: (1) to prove that gradient instabilities can lead to spiral structure in galaxies, and (2) to give the theory of gradient instabilities in rotating shallow water, when viscosity effects can be neglected. The behaviour in rotating shallow water has been investigated in an experiment known as “Spiral” at the Plasma Physics Department of the Institute of Atomic Energy.

It is natural to ask why such different subjects as galactic discs and shallow water are combined in this one paper. The reason is that the dynamical behaviour of a gaseous galactic disc and rotating shallow water are described by one and the same set of differential equations. Clearly, shallow water may be considered as a 2-D gaseous dynamical system (Landau & Lifshitz 1986) similar to the gaseous disc of our galaxy. However, viscosity effects near the bottom in the experimental set-up are absent in galaxies and the latter contain forces of self-gravitation that are absent in shallow water.

Seyferts with companions

Galaxy activity is correlated with companions (Keel et al. 1985, van der Hulst et al. 1986). Using matched samples of Seyferts and controls, Dahari (1984) searched for companions, measuring the galaxy-companion separations and their sizes. He measure the tidal perturbation strength by a parameter P = (companion mass)/(separation)3 in units of the galaxy mass and radius. Dahari found that more Seyferts (37%) have companions than do normal spirals (21%), and that Seyferts with companions are perturbed more strongly. Selection effects cause companions of higher redshift Seyferts to be missed and Byrd et al. (1987) estimate that 75% to 90% of Dahari's Seyferts have companions.

Byrd et al. (1986) tested the correlation using computer models of tidally perturbed spiral galaxies. Observations require a gas mass inflow rate of > 0.5 M⊙ yr1 for Seyfert activity. We used a self-gravitating 60 000 particle disc and inert “halo” perturbed by a companion on a parabolic orbit. Tidal perturbation of the disc throws gas clouds into nucleus-crossing orbits to fuel activity. The experiments demonstrated that the inflow rate exceeded the required value at perturbation levels matching those where Dahari finds many more Seyferts than normals. We therefore conclude that observed companions of Seyferts do have tidal fields sufficient to trigger activity.

Seyferts in rich clusters

If individual gravitational encounters are responsible for activity, the incidence of activity should correlate with the enounter rate. Gavazzi & Jaffe (1987) argue that individual encounters should be less important in rich clusters than in groups.

The model

We consider the linear stability of a plane shear layer, including the effects of compressibility and viscosity, as the simplest model for viscous supersonic shear flows occurring in accretion processes. Details of this investigation can be found in Glatzel (1989). Measuring lengths and velocities in units of half of the thickness of the shear layer and the flow velocity at its edge respectively, the flow may then be described by dimensionless numbers, the influence of compressibility is described by the Mach number, M, and viscosity by two Reynolds numbers, Reν and Reµ, corresponding to shear and volume viscosity respectively.

Instabilities and critical Reynolds numbers

We distinguish two types of modes, viscous modes and sonic modes, according to their physical origin: shear viscosity and compressibility. Shear-driven pairing of viscous modes, and distortion of the pattern speed of sonic modes, leads to mode crossings among the sonic, and between viscous and sonic modes, which unfold into bands of instability. The viscous-sonic resonances provide a new example of viscous instability; the role of viscosity is merely to provide an additional discrete spectrum, while shear is needed to produce mode crossings. The instability is ultimately caused by resonant exchange of energy between the crossing modes.

Critical Reynolds numbers for some resonances are plotted in Figure 1 as a function of the Mach number, M, for zero volume viscosity (Reµ = 3Reν).

Self-absorption curves

In previous work (Friedjung & Muratorio 1987, Muratorio & Friedjung 1988), we developed methods using self-absorption curves (SACs) to study stars having Fe II emission lines in their spectra. Such a curve is obtained by plotting log(Fλ3/gf) against log(gfλ), where F is the total flux, λ the wavelength, g the lower level statistical weight, f the oscillator strength. gfλ is proportional to the optical thickness. If no selective excitation mechanisms exist for particular levels, and the levels inside a term have populations proportional to their statistical weights, such a plot for emission lines of the same multiplet will have points lying on the same self-absorption curve. The shape of the curve is characteristic of the nature of the medium where the line is formed. Shifting the curves for different multiplets (which should have the same shape) relative to each other so as to superpose them, will give at the same time the relative populations of their upper and also their lower terms. Until now, we have calculated SACs for various simplified cases, and a comparison was made with observations of luminous stars whose spectra contained many Fe II emission lines. It was found that observations of certain Magellanic cloud stars could not be fitted by spherically symmetric wind models. Another line emitting medium seemed to be present (a slab or a thin disc with constant opening angle), which is also suggested by the continuum energy distributions.

Introduction

We have recently shown (Tagger et al. 1989a,b and references therein) that the linear theory of density waves in a flat self-gravitating disc contains, in addition to the usual tightly wound waves, another type of perturbation which is essentially bar-like and which dominates the mode structure in the vicinity of the co-rotation radius. We briefly discuss this analytical result, which can be illustrated by numerical calculations, and by its relationship to the disc response to external forcing.

Two different descriptions of density waves have been used in the past: steady waves and shearing perturbations. The difficulty of a unique description stems from the flat disc geometry where the solution of the Poisson equation in the vertical dimension involves an integral operator. The WKBJ approximation, in practice the assumption of tightly wound spirals, allows us to calculate waves with well defined physical properties and has the important advantage of incorporating a properly defined boundary condition at infinity, but it cannot be used to describe the efficient swing amplification mechanism.

Spirals and Bars

Swing amplification is most simply described in the shearing sheet model, where the relevant equations can be Fourier transformed very easily. The solution φ(k) can be easily computed for “large” radial wavenumber k, but we found that a problem arises when one transforms back to real space. It has already been noted that when one computes the inverse Fourier transform the integrand oscillates rapidly at large k, except at saddle points Kj. The contributions of these saddle points Ci exp(ikjx) to φ(x) can be identified with the usual short and long, leading and trailing spiral waves (Goldreich & Tremaine 1978).

Abstract Most Seyfert 1 nuclei and quasars show strong excess continuum flux in the blue and ultraviolet, relative to an extrapolation of their spectra at longer wavelengths. The arguments for identifying this “Big Blue Bump” as thermal emission from an optically thick accretion flow are outlined. Further (less secure) arguments are presented that the flow is flattened, possibly in a disc. The close agreement between simple accretion disc models and the observations is summarized. Several modifications needed to make disc models more realistic are discussed. Finally, the extent to which these models are constrained by observations, such as detctions of “Soft X-Ray Excesses”, and prospects for obtaining future observational evidence of AGN accretion discs is considered.

UV excess

Almost from the first multi-frequency observations of Seyfert 1 nuclei and quasars, it was realized that their optical and ultraviolet spectra were far flatter than their infrared spectra, which have typical slopes of −1.2 (fv ∼ V−1.2). The different variability properties of the infrared and optical/ultraviolet continuum further suggest that they are produced by physically separated components (Cutri et al. 1985). The blue component (also known as the “UV excess” or “Big Blue Bump”) has a flux density rising with frequency in the optical, and a broad maximum somewhere in the ultraviolet. It falls (probably rather steeply) in the far- or extreme-UV. A falling high-frequency tail may be observed in the soft X-rays. This characteristic shape strongly suggests thermal emission from optically thick gas (Shields 1978, Malkan & Sargent 1982).

Introduction

Since the discovery of bipolar molecular outflows, a significant observational effort has been made to study the role of the dense molecular cores (n(H2) ≤ 104 cm−3) in the collimation processes. Dense molecular gas is almost always found in association with the central regions of a bipolar outflow. As a matter of fact, there is practically a one-to-one correspondence. This association of dense gas with the central parts of bipolar outflows supports the notion that the energy source of the outflows is a very young star (Torrelles et al. 1986a). There is also evidence that molecular toroids or discs with interstellar dimensions are present in several regions and that they play, at least on the scale of tenths of pc, an important role in the collimation and channelling of the high-velocity gas. See Rodríguez (1988) and Snell (this volume) for reviews.

In the last few years, our group has obtained Very Large Array (VLA) NH3 observations toward regions of molecular outflows. These observations have revealed the morphology of the high-density molecular gas on scales of ∼ 3″. This program allowed: (1) the study of dense gas as a possible focusing mechanism of bipolar outflows, (2) the study of local heating effects produced by star formation, and (3) the analysis of the kinematics of the regions. Here we present VLA NH3(1,1) and NH3(2,2) observations toward four regions with molecular outflows. These observations were obtained with the VLA of the National Radio Astronomy Observatory (NRAO)5.

Introduction

Although circumstellar discs play an important role in many of the phenomena associated with star formation there is little direct evidence for them at optical wavelengths. In polarization studies of reflection nebulae surrounding young stars and protostars we have noticed a deviation in the expected polarization pattern that appear to indicate the presence of circumstellar discs. We call this feature the ‘polarization disc’.

Examples and properties of polarization discs

Figure 1 shows a polarization map of the reflection nebulosity illuminated by the star HL Tau. At large distances from the star the polarization pattern has the expected centrosymmetric form but in inner regions the pattern deviates to form an anomalous band running across the illuminating star which itself is linearly polarized. We identify this inner pattern, the so called polarization disc, with a circumstellar disc of dusty material. Table 1 gives a comprehensive list of objects possessing such discs and indicates any additional peculiarities in the polarization data.

The properties of polarization discs are summarized below. Obviously not all of these properties are found in every object but they seem to represent various facets of the same phenomenon.

1. (1) The polarization disc consists of an anomalous band of polarization centred on the apical region of reflection nebulae.

2. (2) This band is normally present regardless of the visibility of the central source.

3. […]

Abstract We discuss the linear theory of non-axisymmetric normal modes in self-gravitating gaseous discs. These instabilities occur when the disc is stable to axisymmetric modes. They can have co-rotation situated either inside or outside the disc. The profile of the ratio of vorticity to surface density is found to be important in determining the properties of the normal modes. These modes may be important for redistributing the angular momentum in the disc.

Introduction

Discs and rings in which the internal self-gravity plays an important role are important in astronomy. Examples are the rings around Saturn and Uranus, (Goldreich & Tremaine 1982), and the Milky Way and other spiral galaxies (Toomre 1977, 1981). They may also exist around active galactic nuclei and T. Tauri stars. In both of these cases, instabilities may be important for driving mass accretion and angular momentum transport (Paczynski 1977, Lin & Pringle 1987). An understanding of non-axisymmetric instabilities is clearly important because they may play a significant role in determining the structure and evolution of all of these objects. In this paper we discuss the linear theory of stability as applied to self-gravitating gaseous discs. We find various kinds of instabilities, some of which are generalizations of those found in the non-self-gravitating case (Papaloizou & Pringle, 1984, 1985, 1987). These are essentially due to the unstable interaction of waves on either side of co-rotation. However, when self-gravity is included, there are other modes which have co-rotation outside the system.

Introduction

The subject of discs is central to most of astrophysics, from the formation and dynamics of planetary systems to the formation of protogalaxies in the early universe. Our meeting this week has been very successful, I feel, in bringing out the connections between disc phenomena of very different types and scale sizes. The planning of sessions has played an important part in bringing this about. I would like to thank the organisers for their careful planning, and for all their efforts in making the meeting a success. It was good that all speakers were given sufficient time explain their ideas.

The Compact Oxford Dictionary defines a disc as a “round flattened part in body, plant etc.” In this spirit, and in my capacity as an observer, I will concentrate in my summary on those discs which are actually observed, and which can be assigned a shape. I will start with the smallest discs and work up in size.

Planetary rings

Smallest but by no means the least interesting are the planetary disc and ring systems which were reviewed by Jack Lissauer and Nicole Borderies. Because these systems are relatively simple they are ideal to test theories of density waves, bending waves, edge phenomena and gaps. The purely dynamical phenomena can be studied without having to worry about such messy problems as changes of state (star formation), interactions of the disc with central jets or stellar winds, and interactions with unseen halos.

The hydrogen lines deserve special consideration, not only because they are the strongest lines, but also because they are a very important tool in analyzing stellar spectra. The reason is that they are broadened by the so-called ‘molecular’ Stark effect, that is by the electric fields due to the passing ions. When a radiating atom or ion finds itself in an electric field, the energy levels are shifted to slightly different values of the excitation energy n or χexc The emitted lines therefore occur at slightly different wavelengths. Even more important is that for the hydrogen atom the different orbitals contributing to the energy level with a main quantum number n, are normally degenerate, i.e., when there is no external force field they all have the same excitation energy (this is, of course, indicated by the statistical weight for each level). In the presence of an external force field this degeneracy is removed, so that in an external field the different orbitals contributing to a given level with main quantum number n now occur at slightly different values for the excitation energy. This in turn means that, instead of one line being emitted in the force free case, there are now several lines emitted or absorbed at slightly different wavelengths. Fig. 11.1 shows the different components which are observed for the different Balmer lines.

This shifting of the energy levels in the electric field of the neighboring ions has mainly two effects. First, it shortens the lifetime of the undisturbed level which, according to equation (10.15), results in a broadening of the undisturbed energy level.