Contents

STELLAREVOLUTION ANDNUCLEOSYNTHESIS

Introduction9

Chapter 1Main-sequence stars 11 Introduction11 1.1 TheHertzsprung–Russell diagram11 1.2 TheSun as atypical star 14 1.3 Theequations of stellar structure 18 1.4 Theproton–proton (p–p) chain 24 1.5 Themassdefect 25 1.6 Thecarbon–nitrogen–oxygen(CNO) cycle 28 Chapter 2Gravitational contraction31 Introduction31 2.1 Aself-gravitatinggas cloud31 2.1.1Self-gravity versus internalpressure 31 2.1.2 Freefall 34 2.1.3Hydrostatic equilibrium 36 2.2 Stabilityand thevirial theorem37 2.3 Thelower mass limit for stars39 2.4 Theuppermasslimit for stars42 2.5 TheKelvin–Helmholtz contractiontimescale 43 2.6 Whyare starshot?Puttingfusioninits place 46 Chapter 3Nuclear fusion 49 Introduction49 3.1 Quantum-mechanical properties of particles 49 3.1.1Nuclear dimensions 49 3.1.2Wavefunctions andSchrodi¨ nger’sequation50 3.1.3Barrier penetrationand quantum tunnelling52 3.1.4The Gamowenergy54 3.2 Fusion interactions 55 3.2.1Cross-sections56 3.2.2Mean freepath 58 3.2.3Energy-dependenceoffusioncross-sections59 3.3 Thermonuclear fusion reaction rates 60 3.3.1The speed-averagedcross-section 61 5 Contents

3.3.2 TheGamow peak 63 3.3.3The widthofthe Gamowwindow64 3.3.4The integrated fusion rate equation65 3.3.5The CN cycle revisited 68 3.4 Thetemperature-dependenceofthe fusionrate 72 3.4.1The temperature derivative of thefusionrate 72 3.4.2The dependenceofhydrogen burning on stellar mass 74 3.4.3The energy generationrate 75 Chapter 4Frommainsequence to red-giant branch 78 Introduction78 4.1 Hydrogen-burning timescales 78 4.2 Assigning ages from hydrogen-burning timescales 79 4.3 Rescaling the stellar structure equations 81 4.3.1Massdistribution82 4.3.2Ideal gaslaw 83 4.3.3Hydrostatic equilibrium 83 4.3.4 Radiative diffusion84 4.3.5Energygeneration 85 4.3.6Taking stock86 4.4 Relations between starsofdifferent mass alongthe main sequence86 4.5 Theeffect on astar of its changing composition88 4.6 Thesubgianttransitiontoshell-hydrogen burning 89 4.6.1 Expansionofthe envelope,contractionofthe core 89 4.6.2 TheHertzsprung gap93 4.7 Thered-giantphase:shell-hydrogen burning 93 4.7.1 Convective energy transport 94 4.7.2The increaseinluminosity on thered-giantbranch98 4.7.3First dredge-up 98 Chapter 5Helium-burning stars 101 Introduction101 5.1 Nucleosynthesiswithout heliumburning 101 5.2 Equilibrium andthe chemical potential 102 5.3 Heliumburning 104 5.3.1The triple-alpha process: Step 1105 5.3.2The triple-alpha process: Step 2106

6 Contents

5.3.3 Thetriple-alphaprocess: Step 3107 5.3.4The energy released by thetriple-alphaprocess108 5.3.5Furtherhelium-burningreactions108 5.4 Quantum states anddegeneracy109 5.5 Electron degeneracy111 5.5.1Non-relativistic degenerateelectrons 112 5.5.2Ultra-relativistic degenerateelectrons 113 5.6 Thehelium flash 114 5.7 Core-heliumburning stars116 5.8 Helium-burning in ashell 119 5.9 Otherfactorsinfluencingstellar evolution122 Chapter 6Latestagesofstellar evolution126 Introduction126 6.1 Planetarynebulae 126 6.2 White dwarfs 129 6.2.1 TheChandrasekharmass129 6.2.2The white-dwarfmass–radius relation131 6.2.3Different typesofwhite dwarf133 6.2.4Fading (and cooling) of white dwarfs 134 6.3 Advanced nuclear burning 135 6.3.1 Carbonburning 135 6.3.2Neonburning 136 6.3.3Oxygenburning 136 6.3.4Siliconburning 136 6.3.5The end-pointsofcarbon, neon, oxygenand silicon burning 137 6.4 Neutron-capturenucleosynthesis139 6.4.1The chartofnuclides 139 6.4.2Neutron captureand β−-decay 140 6.4.3Closedneutron shells andneutron magicnumbers 141 6.4.4Competitionbetween neutron captureand β−-decay 142 6.4.5Different products of s- andr-processnucleosynthesis144 Chapter 7Supernovae, neutronstars &black holes150 Introduction150 7.1 Supernovae150 7.1.1Nuclear photodisintegration150 7 Contents

7.1.2 Electron capture152 7.1.3Supernovaeexplosions 154 7.1.4Typesofsupernovae155 7.2 Neutron stars157 7.2.1Neutron star composition158 7.2.2The radius of aneutron star 160 7.2.3The maximummassofaneutron star 161 7.3 Pulsars162 7.3.1The discovery of pulsars163 7.3.2The rotationperiod of pulsars 164 7.3.3 Theenergylossfrom arotating pulsar166 7.3.4The magnetic field strength of apulsar166 7.3.5The ages of pulsars 168 7.3.6 Typesofpulsar169 7.4 Stellar-massblack holes 172 Chapter 8Starformation 176 Introduction176 8.1 TheJeanscriterionfor gravitationalcollapse 177 8.2 Collapse andcoolinginthe temperature–density diagram179 8.2.1The temperature–density diagram179 8.2.2Adiabatic collapse181 8.2.3Non-adiabatic collapseand theimportanceofcooling183 8.2.4Fragmentation 185 8.2.5The non-adiabatic free-fall collapse 186 8.3 Kelvin–Helmholtz contractionofthe protostar 187 8.3.1The Hayashitrack 188 8.3.2The Henyey contractiontothe main sequence188 8.4 Observationsofyoung pre-main-sequenceobjects 193 8.5 Therange of stellar births 196 8.5.1The stellar mass function 196 8.5.2The single-star fraction 197 8.6 Stellar lifecycles 198 Appendix201 Solutions203 Acknowledgements230 Index 231 8