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Stellar Structure and Evolution Stellar structure and evolution Warrick Ball November 3 & 4, 2014 Contents 1 Stellar modelling 4 1.1 Basic assumptions . 4 1.1.1 Mass conservation . 6 1.1.2 Hydrostatic equilibrium . 6 1.1.3 Energy generation (and conservation) . 7 1.1.4 Energy transport . 7 1.2 Composition equations . 10 1.3 Matter equations . 11 1.3.1 Opacity . 12 1.3.2 Nuclear reactions . 13 1.3.3 Neutrino loss rates . 15 1.3.4 Equation of state . 16 1.4 Boundary conditions . 18 1.4.1 Centre . 18 1.4.2 Surface . 18 1.4.3 Composition . 18 1.4.4 Initial models . 19 2 Stellar evolution 20 2.1 Characterizing stellar evolution . 20 2.1.1 The Hertzsprung{Russell diagram . 20 2.1.2 Colour{magnitude diagrams . 22 2.1.3 The ρ-T diagram . 23 2.1.4 Kippenhahn diagrams . 23 2.2 Evolution of a Sun-like star . 25 2.2.1 The main sequence . 25 2.2.2 The red giant branch . 25 2.2.3 The helium flash and core helium burning . 27 2.2.4 Thermal pulses and the asymptotic giant branch . 28 2.2.5 Envelope expulsion and the white dwarf cooling track . 29 2.3 More massive stars . 30 2.3.1 Convective cores on the main sequence . 30 2.3.2 Non-degenerate helium ignition . 30 1 2.3.3 Mass loss on the main sequence . 31 2.3.4 Carbon burning and beyond . 34 2.4 Parting thoughts . 34 2 Prologue About the course These lecture notes are from the two lectures I gave at the DWIH Winter School in 2014. They were forked from the notes I had so far prepared for the course on stellar modelling taught at G¨ottingenin August 2014. There are two lectures, distinct in content and character. The first lecture aims to introduce the equations of stellar structure and evolution, with a keen focus on the principles and concepts behind the models, rather than detailed derivations to study the theory in depth. The second lecture aims to simply outline how we study stars and what the models say about the evolutionary states of stars in different phases. I tend to use the Sun as a fiducial case, both since it's a special target and because it happens to undergo many distinct phases of evolution. I then identify things that change in stars of increasing mass. 3 Chapter 1 Stellar modelling Over the course of the 20th century, a basic formulation of stellar structure and evolution emerged that is able to explain most observational results (or, rather, the observed properties of most stars). This is sometimes called classical, standard or even vanilla stellar structure and evolution. It is well-studied, and serves as the starting point for studying the stars. If you read a paper and the writer doesn't specify precisely how their models are calculated, they probably use the standard picture. Conversely, authors should be specific about non-standard properties (or properties for which there is no standard!). We will now consider how this standard model is constructed, and only allude to where \non-standard" modelling can enter. There is a long line of fine textbooks on stellar structure and evolution, arguably starting with the theoretical works of Eddington (1926) and Chandrasekhar (1939, 1957). The post-war era saw the rise of computers, and textbooks shifted in flavour towards formulating the equations, solving them numerically, and interpreting the results. Notable subsequent works include Schwarzschild (1958) and Cox & Giuli (1968), among many others. But since it's publication, the work of Kippenhahn & Weigert (1990) has been definitive, and a second edition was recently released (Kippenhahn et al. 2012). There are also many lecture notes available online, but two notable entries are those of Jørgen Christensen-Dalsgaard1 and Onno Pols 2. I'm a particular fan of Onno Pols' notes, which are near textbook quality and have excellent figures, many of which I have borrowed. If you do use these notes at any point, also glance at some of the exercises to test yourself. 1.1 Basic assumptions To formulate equations that represent the structure and evolution of a star, we make several assumptions. We assume that a star is 1. a fluid; 2. spherically symmetric; 1http://astro.phys.au.dk/~jcd/evolnotes/LN_stellar_structure.pdf 2http://www.astro.ru.nl/~onnop/education/stev_utrecht_notes/ 4 3. self-gravitating; 4. stationary on dynamical timescales; and 5. in local thermal equilibrium We can correspondingly write down the consequences of these assumptions. 1. Our formulation will start essentially from the equations of fluid dynamics. This is done explicitly in the textbook by Collins (1989)3. 2. The structure will depend only on the radial co-ordinate in the star, or any function that varies monotonically with radius (e.g. the mass or pressure). This assumption also requires that the star is not rotating. 3. Specifically, this assumption is intended to mean that gravity is the only external body force that we consider in the fluid equations. We exclude electric or magnetic fields. 4. We neglect net velocities and accelerations in the star. As we will see, there are regions of the star where large scale flows exist, but there are always equal upward and downward flows, so the net flow (averaged over a shell) is zero. 5. For each shell of material, we can define a local temperature that is constant in an infinitesimal neighbourhood of that shell. In addition, we assume that particle velocities are distributed like Maxwell distributions and that the radiation field is given by a blackbody at the same local temperature. The assumptions may have seemed quite mild at first but keep in mind precisely what they imply. Let us also give a moment's thought to when these assumptions will fail. 1. Generally speaking, the fluid approximation is excellent until different particle species decouple, above the photosphere. 2. Stars generally rotate, but the amount of rotation is often small enough that the structure of the star is not significantly altered. However, rotation induces fluid instabilities that lead to additional mixing of material, and this can be significant even when the structural effect of rotation is small. 3. Again, in truth we know that the Sun has a magnetic field and we have strong evidence that many stars with convective envelopes do too. In addition, many stellar remnants (e.g. white dwarfs, pulsars) have strong magnetic fields. So they are present but, as with rotation, probably not at a structurally-important level. 3Collins' book is out of print and available freely from http://ads.harvard.edu/books/1989fsa..book/ 5 4. There are two places where this breaks down. The more benign occurrence is at the end of a massive star's life, when iron has been produced in the core, which ultimately begins to collapse. The other widespread case is the occurrence of convection. Material simultaneously flows upwards and downwards, with zero net mass flux but effective heat (and composition) transport. Stellar models circumvent this problem by employing simplified steady-state approximations of the structure in convective zones, but in principle there can be structurally significant velocities. 5. Local thermodynamic equilibrium applies well throughout most of most stellar models but breaks down near the surface, where the surface boundary conditions of the stellar model must be applied. The radiation field diverges from the Planck function, owing to the development of spectral lines in the lower density atmospheres. 1.1.1 Mass conservation The mass dm contained in a shell at radius r, with width dr and local density ρ(r) is simply dm = 4πρ(r)r2dr (1.1) Written differentially, dm = 4πρ(r)r2 (1.2) dr Since the density is always positive, the function m(r) is monotonic. We can make m the dependent variable (instead of r) and instead write the equation as dr 1 = (1.3) dm 4πρ(r)r2 The formulation with r or m as the dependent variables are known as using Eulerian or Lagrangian co-ordinates. Early codes tended to use m as the dependent co-ordinate because the surface boundary is then fixed at the total mass of the star, M∗. Were we to use r as the dependent variable, then the surface boundary is the total stellar radius R∗, which moves over the star's life. In reality, modern codes use neither of these formulations, usually instead using some function of monotonic variables (pressure P , temperature T , mass m and radius r). 1.1.2 Hydrostatic equilibrium The general equation of motion for material in the star is @~v ρ + ~v · r~ ~v = −r~ p + f:~ (1.4) @t Normally, the pressure p is a tensor but the pressure is taken to be isotropic and reduces to a scalar. First, we suppose that the only body force is the gravity, so that f~ = −ρ~g. Second, we 6 suppose that the system is dynamically stationary and not rotating, so that ~v = 0. In this case, we write r~ p = −ρ~g (1.5) which is known as the equation of hydrostatic equilibrium. Finally, we suppose that the system is spherically symmetric, so that we can write dp Gmρ = −ρg = − (1.6) dr r2 This equation|hydrostatic equilibrium under spherical symmetry|is just a statement of the balance of forces, which is itself a statement of the conservation of linear momentum. Note that these two equations|hydrostatic equilibrium and mass conservation|are said to describe the mechanical (rather than thermal) structure of the star. If the pressure P is purely a function of ρ (as is the case for e.g.
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