In stars with a convection zone just below the photosphere, the convective motions might create acoustic waves which propagate outwards through the photosphere. These sound-waves produce an extra pressure, i.e. ‘wave pressure’ in the atmosphere. This pressure will depend on the density and on the amplitudes of the waves. The gradient of the wave pressure results in a force that can drive a stellar wind. If a stellar wind is driven by acoustic wave pressure it is called a ‘sound wave driven wind’.

In this chapter we will first explain the concept of wave pressure by studying the motion of a particle in the presence of an oscillating force. This simple case, first developed by Landau and Lifshitz (1959) shows that oscillations may result in a net force in the direction of the oscillations. In § 6.1 we discuss the motions of particles in an oscillatory field, such as in a sound wave, and we show that this produces a ‘wave pressure’. In § 6.2 we introduce the concepts of the ‘wave action density’ and the ‘acoustic wave luminosity’. These are useful concepts for describing sound wave driven winds. The pressure due to acoustic waves is described in § 6.3. Section (6.4) descibes sound wave driven wind assuming no dissipation of acoustic energy. This results in estimates of the both the mass loss rate and the wind velocity. In § 6.5 we discuss sound wave driven winds with dissipation of the acoustic energy.

During the last thirty years astronomers have discovered that nearly all stars are losing mass in the form of stellar winds through a major fraction of their lives. This mass loss affects their evolution from their origin to their death. It also leads to spectacular interactions between the supersonic stellar winds and the interstellar medium in the form of planetary nebulae and ring nebulae and in the form of interstellar bubbles and superbubbles. The return of matter from stars into the interstellar medium and the formation of bubbles and superbubbles changes the chemical composition of the galaxies and affects their kinematical properties.

Literature in this field has grown tremendously over the past three decades. On the one hand this is due to the advance of spectroscopic observations over the full range of the spectrum and to the enormous improvements in image resolution from ground based telescopes and the Hubble Space Telescope which results in spectacular images of the nebulae formed by stellar winds. On the other hand it is the result of many theoretical studies to explain the basic mechanisms for stellar winds and the interactions with their surroundings. Many reviews have been published that give an overview of specific aspects of stellar winds or mass loss from stars.

Stellar winds are the continuous outflow of material from stars. The ejection of material plays a major role in the life cycle of stars. In the case of massive stars, the winds remove more than half of the star's original mass before the star explodes as a supernova. In this book we will explore the many mechanisms that can lead a star to eject matter in the form of a steady stellar wind. We will also discuss the interaction of winds with the interstellar medium of our galaxy, and the effects of mass loss on the evolution of a star. We start by giving in this chapter a brief overview of the historical development of the subject, especially focusing on the early observations and theoretical advances that led us to our current level of understanding.

Historical introduction

The early developments

The names ‘solar wind’ and ‘stellar winds’ were both coined by Eugene Parker (1958, 1960). However, the origins of the basic ideas regarding mass loss from stars arose long before that.

The earliest phase in the development of the subject concerns the realization that a few stars are like ‘novae’, in having spectra with very broad emission lines. Novae are sudden outbursts of light from certain types of stars, and the outbursts are also associated with the high speed ejection of material. Tycho Brahe's observation of a ‘new star’ or nova in 1572 marks the birth of stellar astronomy as a study of objects that are not perfect celestial objects, but rather ones that can change in interesting ways.

Stars interact with the surrounding interstellar medium (ISM), both through their ionizing radiation and through the mass, momentum, and energy that is transferred by way of their winds. The extreme ultraviolet radiation from hot stars leads to ionized nebulae or H II regions around young stars. In the case of low mass stars about to become white dwarfs, the radiation leads to the ionization of planetary nebulae.

The mass loss in stellar winds leads to a recycling of matter back to the interstellar medium, and because of the nuclear processing that occurs in the interiors of stars, the matter which is returned is often chemically enriched. In the cases of late type giants and carbon rich Wolf-Rayet stars, dust grains are produced in the winds, so the outflows may carry grain enriched material into the interstellar medium. These grains could play a role in the next generation of star formation. There are also dynamical effects associated with wind-interstellar medium interactions. The collisions of the winds with their surroundings produce ‘wind bubbles’, and the momentum transfer helps to maintain the random velocities of interstellar clouds that otherwise would be damped out by the dissipative effects of cloud collisions.

The winds of ‘massive stars’ tend to have the greatest effect on the ISM, because their mass loss rates are large, and the massive stars that are hot also have winds that are very fast and carry large momentum fluxes.

In the last chapter we have seen that if a star has an open magnetic field in the equatorial region and is also rapidly rotating, a very strong stellar wind can be produced. In this chapter we consider the effects of the magnetic field in absence of rotation. If oscillations are induced in the field at the base of the wind, transverse ‘Alfvén’ waves will be generated. The dissipation of energy and momentum associated with the wave propagation can lead to the acceleration of the outer atmosphere in the form of an ‘Alfvén wave driven wind’. Open field regions can arise in a variety of configurations, depending on the circulation currents or dynamo properties of the interior of the star. Furthermore, the strength and geometry of the magnetic field can vary significantly from one location on the star to another, and the wind flow tubes will vary accordingly.

In the absence of a magnetic field, a star that has a spherically symmetric hot corona will produce a steady, radial, structureless wind, driven by the thermal gas pressure gradients in the corona (Parker, 1958), as discussed in Chapter 5. Within a few years after the solar wind was predicted by Parker, interplanetary space probes proved that indeed there is a wind from the sun that occurs at all times. However, the wind was found to be far from steady and structureless. To understand the spatial and temporal variability of the wind, Parker (1965) considered outflow in open magnetic field structures.

Stars emit not only radiation but also particles. The emission of particles is called the stellar wind.

The two most important parameters regarding a stellar wind that can be derived from the observations are the mass loss rate Ṁ, which is the amount of mass lost by the star per unit time, and the terminal velocity v∞, which is the velocity of the stellar wind at a large distance from the star. By convention, the mass loss rate Ṁ is always positive and it is expressed in units of solar masses per year, with 1 M⊙ yr-1 = 6.303 × 1025 g s-1. A star with Ṁ = 1--6M⊙ yr-1, which is not an unusual value, loses an amount of mass equal to the total mass of the earth in three years. The terminal velocity v∞ of a stellar wind ranges typically from about 10 km s-1 for a cool supergiant star to 3000 km s-1 for a luminous hot star.

The values of Ṁ and v∞ are important because

(1) Ṁ describes how much material is lost by the star per unit of time. This is important for the evolution of the stars, because stars with high mass loss rates will evolve differently from those with low mass loss rates.

(2) Different stellar wind theories predict different mass loss rates and different terminal velocities for a star. So by comparing the observed values with the predictions we can learn which mechanism is responsible for the mass loss from a star.

Coronal winds are stellar winds driven by gas pressure due to a high temperature of the gas. In the case of the sun a coronal temperature of about 2 × 106 K is reached in the outer layers of the solar atmosphere. The solar photosphere, where the visual radiation from the sun is emitted, has a temperature of about 6000 K. Above the photosphere the temperature rises with height to a few times 106 K. The temperature rise beyond the photosphere is due to the dissipation of mechanical energy or the reconnection of magnetic fields that originate in the convection zone below the photosphere. Other forces, such as those produced by Alfvén waves, may play a role in the coronal holes which are regions of lower temperatures and higher mass flux. However in this chapter on coronal winds, we will only consider the effects of gas pressure and heat conduction in the production of a stellar wind.

All non-degenerate stars with effective temperatures less than about 6500 K are expected to have a convection zone below their surface, so in principle chromospheres and coronae could exist around all cool stars. However, very luminous cool stars can also have winds driven by other mechanisms such as wave pressure or radiation pressure on dust grains. If these stars have a high mass loss rate, then the heating cannot compete with the cooling of the outflowing gas.

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.