The outer atmospheres of luminous cool giant stars and early-type stars can be driven outward by the strong radiation fields from the stellar photospheres. In the case of the cool stars, radiative driving occurs because of absorption of photons by dust grains that can form in the outer atmospheres. The grains can absorb radiation over a broad range of wavelengths, so the outflows of the cool stars are said to be ‘continuum driven’ winds. In the case of hot early-type stars the winds are driven by the scattering of radiation by line opacity, so their outflows are called ‘line driven’ winds.

The essential difference between continuum driven and line driven winds is the role of the Doppler shift between a parcel of outflowing matter and the photosphere. In the acceleration of a stellar wind to terminal velocity, the stellar light incident on the parcel of the wind is increasingly redshifted up to the final value of Δλ = λv∞/c. For a cool star with a continuum driven wind, this redshift corresponds to a few Å, which is such a narrow band that within it neither the continuum opacity nor the incident radiation field changes significantly. So the Doppler shifting is not important in continuum driven winds. In the case of line driven winds both the line opacity and the radiation field in the lines change significantly over the Doppler shifts associated with the winds.

The purpose of this chapter is to describe and explain some of the fundamental properties of the stellar wind models. This is done by deriving the equations for idealized simple winds. For these simple models the equation of motion can be solved easily so that the velocity and density structures of the wind are known. The solutions show how the velocities and densities depend on the forces in the wind. They also show that the mass loss rate of a stationary wind model is uniquely determined by the solution of the equations, i.e., given the lower boundary conditions in the wind and the forces and energy gains and losses, a physically realistic solution exists for only one specific value of the mass loss rate. The simple solutions discussed in this chapter show how this so-called critical solution depends on the forces and the energy of the wind. Although only simplified models are considered in this chapter, the conclusions are qualitatively valid for the more complicated and detailed models which will be described in later chapters.

Section 3.1 describes the simplest possible model of an isothermal wind in which gas pressure provides the outward force. In §§ 3.2, 3.3 and 3.4 the effects of additional forces in isothermal wind models are considered; first as simple analytic expressions, such as a force which varies as r-2, or as v dv/dr, and later in more general terms.

Early-type stars often show rotationally broadened photospheric lines that indicate that they are rotating with equatorial speeds in the range 100 to 400 km s-1. These stars have radiatively driven winds owing to the strong line opacities in their outer atmospheres, as described in Chapter 8. The rotation of the stars leads to interesting effects, the most prominent of which is the tendency to concentrate the outflowing material toward regions near the equatorial plane. The equatorial material is moving outwards from a star whose surface is rotating at a speed below the critical speed. Therefore these disks are called outflowing disks or de-cretion disks, in contrast to the ‘accretion disks’ around pre-main sequence stars or around the gaining stars in binary systems with mass transfer.

In this chapter we consider only the formation of outflowing disks. For a star that has a stellar wind and also an outflowing disk, the contrast in density from equator to pole is typically about a factor of ten or so. We discuss two basic pictures for producing such a contrast. The first is a piece-wise spherical outflow in which the equatorial density is enhanced because the mass flux from the near-equatorial latitudes is larger or the wind velocity is lower than those in the polar regions. Such a wind could be the result of the ‘rotation induced bi-stability’ (RIB) model of Lamers and Pauldrach (1991). The second is the wind compression picture in which the streamlines of the gas from both hemispheres of a rotating line driven wind are bent towards the equatorial plane.

Mass loss has a profound effect on the evolution of stars. In the case of stars with initial masses greater than about 30 M⊙, mass loss occurs at a considerable rate throughout their whole life. So it affects their evolution from the beginning to the end. In the case of lower mass stars, mass loss is only important in the late stages of their evolution. For those stars only their late evolution is changed dramatically by mass loss. In this chapter we discuss some of the important effects of mass loss on the evolution of the stars. We first discuss the effects in general terms. Later we discuss the evolution of massive stars and of low mass stars under the influence of mass loss. We describe two characteristic examples in some detail: the evolution of a massive star of 60 M⊙ in § 13.2 and of a low mass star of 3 M⊙ in § 13.3. The effect of mass loss on stellar evolution has been described in several reviews: e.g. Iben and Renzini (1983), Chiosi and Maeder (1986) and at several conferences: e.g. Mennessier and Omont (1990) and Leitherer et al. (1996).

The main effects of mass loss

Changes in the surface composition

The outer layers of stars are peeled off by mass loss. Nuclear fusion occurs in the interior of stars. This nuclear fusion changes the chemical composition and the abundance ratios of the elements in the layers where the fusion occurs.

The winds of luminous hot stars are driven by absorption in spectral lines and they are called line driven winds.

Hot stars emit the bulk of their radiation in the ultraviolet where the outer atmospheres of these stars have many absorption lines. The opacity in absorption lines is much larger than the opacity in the continuum. The opacity of one strong line, say the C IV resonance line at 1550 Å, can easily be a factor of 106 larger than the opacity for electron scattering.

The large radiation force on ions due to their spectral lines would not be efficient in driving a stellar wind if it were not for the Doppler effect. In a static atmosphere with strong line-absorption, the radiation from the photosphere of the star will be absorbed or scattered in the lower layers of the atmosphere. The outer layers will not receive direct radiation from the photosphere at the wavelength of the line, and so the radiative acceleration in the outer layers of the atmosphere due to the spectral lines is strongly diminished. However, if the outer atmosphere is moving outward, there is a velocity gradient in the atmosphere allowing the atoms in the atmosphere to see the radiation from the photosphere as redshifted. This is because in the frame comoving with the gas the photosphere is receding. As a result the atoms in the outer atmosphere can absorb radiation from the photosphere which is not attenuated by the layers in between the photosphere and the outer atmosphere.

As we move away from an isolated body, its gravitational field decreases and tends to zero as we approach infinity. In a spacetime picture, where gravity is described by curvature, it would therefore seem entirely reasonable to model an isolated body on a spacetime that is, in some sense, asymptotically flat. If the body possesses some sort of symmetry – it might, for example, be an axisymmetric or spherically symmetric star – then it would also seem reasonable to model it on spacetime with the same type of symmetry. But what exactly do we mean by asymptotic flatness, and what do we mean by a spacetime symmetry? In this chapter we shall attempt to answer these two questions.

Asymptotically Flat Spacetimes

In order to use general relativity to study the gravitational field of an isolated body, such as a star, it is necessary to have some well-defined notion of asymptotic flatness. An asymptotically flat spacetime represents the idealized situation of a gravitating body that, to all intents and purposes, is totally isolated from the rest of the universe by virtue of its great distance from all other bodies. As we move away from such an object, its gravitational field should decrease, and thus we expect that spacetime should become flat at asymptotic distances. Of course, no physical system truly can be isolated from the rest of the universe, but for a system such as a star the gravitational influence of all other matter is so slight that it is entirely reasonable to consider it as being totally isolated, essentially a single system in an otherwise empty universe.

Introduction

Numerical experiments are making important contributions to the study of turbulence in the interstellar medium (ISM). Since any numerical simulation is restricted in the range of spatial and temporal scales which it can describe, it is important to develop a prescription for treating the effects of turbulence at the smallest scales, which are generally omitted from this range. Although very little energy resides at the smallest scales, the small scale motions dramatically increase momentum and magnetic flux transport in the ISM, and can also produce rapid thermal and chemical mixing. The most common way to account for these subgridscale effects is to simply assume that the viscosity, electrical resistivity, and other transport coefficients are much larger than their molecular values. The difficult problem of justifying this approach and calculating the so-called eddy diffusivities has received more attention in the atmospheric and stellar turbulence communities than it has, so far, among interstellar turbulence theorists.

I describe the multivariate technique of Principal Component Analysis and its application to spectroscopic imaging data of the molecular interstellar medium. The technique identifies differences in line profiles with respect to the noise level at various scales. It is assumed that such differences arise from fluctuations within turbulent flows. From the resultant eigenvectors and eigenimages, a size line width relationship, (δv ∼ τα), can be constructed which describes the relationship between the magnitude of velocity fluctuations and the angular scale over which these occur for a given region. From a sample of selected molecular regions in the outer Galaxy, I find the power law exponent varies from 0.4 to 0.7. Thus, the turbulent flows within molecular regions of the Galaxy do not follow the Kolmogorov-Obukhov relation for incompressible turbulence. Implications of these results are discussed with respect to the injection and dissipation of kinetic energy in molecular regions.

Introduction

In the early, pioneering days of millimeter wave astronomy, the presence of turbulent flows within molecular regions of the Galaxy was inferred from the supersonic line widths of CO spectra. Since that time, telescope and detector technology has advanced such that one can now routinely construct detailed images of molecular emission from which the properties of interstellar turbulence can, in principle, be derived. In practice, statistical descriptions of the observations are required to fully exploit the available information.

As we have seen, a (charged) matter distribution on flat spacetime is described by the fields Ja, Fab, and Tab, where Ja determines the charge density, Fab describes the electromagnetic field, and Tab determines the four-momentum density. The background geometry of M determines ∇a and the constant tensor fields gab and εabcd. Nature does not, of course, allow Ja, Fab, and Tab to vary in an arbitrary manner, but imposes constraints in the form of conservation laws and equations of motion, which may be expressed in the form of field equations. In this chapter we shall discover the field equations satisfied by Ja, Fab, and Tab.

Conservation Laws

Consider a congruence whose curves are the particle world lines of a uniform dust distribution with particle-number current ja. Let us select one particular curve, l0, and three neighboring curves, l1, l2, and l3, which are joined to l0 by three space-like connecting vectors aa, ba, and ca (Fig. 8.1).

According to an observer with four-velocity va orthogonal to aa, ba, and ba, the volume V = εabcd vaabbccd will always contain the same number of particles. Note that va is determined uniquely by the condition that it is orthogonal to aa, ba, and ba, but this does not mean that it is Liepropagated along the congruence.

Introduction

The density and velocity structure within a molecular cloud is a remnant of its formation environment and the starting point for the creation of stars. It determines how deeply radiation can propagate through the cloud, and is a critical parameter for understanding the evolution of the ISM. How is it best described?

Beginning with Larson (1981), correlations between cloud properties such as linewidth and size have been fit by power laws. Since a power law does not have a characteristic scale, the implication is that clouds are scale-free and self-similar. This has led to statements in the literature that clouds are best described as fractals (e.g. Falgarone, Phillips, & Walker 1991; Elmegreen 1997). On the other hand, other recent studies (Larson 1995; Simon 1997; Goodman et al. 1998; Blitz & Williams 1997) suggest that there are characteristic size and velocity scales in star-forming regions.

Interstellar Turbulence, the second conference organized by the Guillermo Haro International Program on Advanced A strophysical Research, was an excellent forum to review and discuss one of the most intriguing features of cosmic and terrestrial fluids. Turbulence is universal and mysterious, and remains one of the major unsolved problems in physics and astrophysics. It is present in all terrestrial and astrophysical environments: close to our telescopes, it blurs and distorts our view of the skies, and in the interstellar and intergalactic media, somehow, it creates fluctuations and redistributes angular momentum, leading to star formation and large scale structure.

The Guillermo Haro Program was created in 1995 at the Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE), and is named in honor of its founder, the remarkable astronomer-lawyer Guillermo Haro. This second conference was aimed at revising our conceptions on the properties of turbulence, and at summarizing the present status in observational, theoretical, and computational research in interstellar turbulence. It was held in Puebla, México, at the Benemérita Universidad Autónoma de Puebla, during the week of January 12th to 16th, 1998. There were 130 participants, from four continents, and a large fraction of them were very young scientists. The program covered a wide variety of topics, ranging from atmospheric and interstellar turbulent flows, to magnetic fields and cosmic ray transportation, and energy dissipation, fragmentation and star formation.

Introduction

Over the last several decades it has become clear that the dynamics and evolution of star-forming interstellar clouds are difficult to explain without magnetic effects. A principal problem involves support of dense clouds against their own gravity. In general, such clouds are observed to be in approximate virial equilibrium between gravity and internal motions. Seemingly, therefore, they should be stable against collapse. However, observed line widths are almost invariably much greater than the sound speed. Therefore the internal motions that support the clouds are highly supersonic, and simple estimates indicate that shock-induced dissipation of mechanical energy should occur on about the free-fall time. In such a case, non-magnetic turbulence offers no effective support for the clouds (unless, of course, it can somehow be continuously regenerated).

Introduction

One of the main features of turbulence is its multi-scale nature (e.g., Scalo 1987; Lesieur 1990).