Part I Stellar Interiors 1 1 Stellar Interiors Copyright 2003 George W Collins II 1 Introduction and Fundamental Principles The development of a relatively complete picture of the structure and evolution of the stars has been one of the great conceptual accomplishments of the twentieth century While questions still exist concerning the details of the birth and death of stars scientists now understand over 90 of a star s life Furthermore our understanding of stellar structure has progressed to the point where it can be studied within an axiomatic framework comparable to those of other branches of Physics It is within this axiomatic framework that we will study stellar structure stellar spectra the traditional source of virtually all information about stars 2 1 Introduction and Fundamental Principles This book is divided into two parts stellar interiors and stellar atmospheres While the division between the two is fairly arbitrary it is a traditional division separating regimes where different axioms apply A similar distinction exists between the continuum and lines of a stellar spectrum These distinctions represent a transition zone where one physical process dominates over another The transition in nature is never abrupt and represents a difference in degree rather than in kind We assume that the readers know what stars are that is have a working knowledge of the Hertzsprung Russell diagram and of how the vast wealth of knowledge contained in it has been acquired Readers should understand that most stars are basically spherical and should know something about the ranges of masses radii and luminosities appropriate for the majority of stars The relative size and accuracy of the stellar sample upon which this information is based must be understood before a theoretical description of stars can be believed However the more we learn about stars the more the fundamentals of our theoretical descriptions are confirmed The history of stellar astrophysics in the twentieth century can be likened to that of a photographer steadily sharpening the focus of the camera to capture the basic nature of stars In this book the basic problem of stellar structure under consideration is the determination of the run of physical variables that describe the local properties of stellar material with position in the star In general the position in the star is the independent variable s in the problem and other parameters such as the pressure P temperature T and density are the dependent variables Since these parameters describe the state of the material they are often referred to as state variables Part I of this book discusses these parameters alone In Part II when we arrive near the surface of the star we shall also be interested in the detailed distribution of the photons particularly as they leave the star Although there are some excursions into the study of nonspherical stars the main thrust of this book is to provide a basis for understanding the structure of spherical stars Although the proof is not a simple one it would be interesting to show that the equilibrium configuration of a gas cloud confined solely by gravity is that of a sphere However instead of beginning this book with a lengthy proof we simply take the result as an axiom that all stars dominated by gravity alone are spherical We describe these remarkably stable structures in terms of microphysics involving particles and photons which are largely in equilibrium Statistical mechanics is the general area of physics that deals with this subject and contains the axioms that form the basis for stellar astrophysics Our discussion of stellar structure centers on the interaction of light with matter We must first describe the properties 3 1 Stellar Interiors of the space in which the interaction will take place It is not the normal Euclidean three dimensional space of intuition but a higher dimension space This higherdimension space called phase space includes the momentum distribution of the particles which make up the star as well as their physical location 1 1 Stationary or Steady Properties of Matter a Phase Space and Phase Density Consider a volume of physical space that is small compared to the physical system in question but still large enough to contain a statistically significant number of particles The range of physical space in which this small volume is embedded may be infinite or finite as long as it is significantly larger than the small volume First let a set of three Cartesian coordinates x1 x2 and x3 represent the spatial part of the volume Then allow the additional three Cartesian coordinates v1 v2 and v3 represent the components of the velocity of the particles Coordinates v1 v2 and v3 are orthogonal to the spatial coordinates This simply indicates that the velocity and position are assumed to be uncorrelated It also provides for a sixdimensional space which we call phase space The volume of the space is dV dx1dx2dx3dv1dv2dv3 Figure 1 1 shows part of a small differential volume of phase space It must be remembered that the position and velocity coordinates are orthogonal to each other 4 1 1 1 1 Introduction and Fundamental Principles If the number of particles in the small volume dV is N then we can define a parameter f known as the phase density by f x1 x2 x3 v1 v2 v3 dV N 1 1 2 The manner in which a number of particles can be arranged in an ensemble of phase space volumes is described in Figure 1 1 b Macrostates and Microstates A macrostate of a system is said to be specified when the number of particles in each phase space volume dV is specified That is if the phase density is specified everywhere then the macrostate of the system has been specified Later we shall see how the phase density can be used to specify all the physical properties of the system To discuss the notion of a microstate it must be assumed that there is a perceptible difference between particles because in a microstate in addition to the number of particles in each volume it makes a difference which particles are in which volumes If the specification of individual particles can be accomplished then it can be said that a microstate has been specified Clearly one macrostate could consist of many microstates For example the number of balls on a pool table might be said to be a macrostate whereas the specification of which balls they are would denote a specific microstate In a similar manner the distribution of suits of playing cards in a bridge hand might