Stellar Evolution, Supernovae, White Dwarfs, Neutron Stars, and Black Holes Feryal Ozel University of Arizona As we start… • I will assume little to no background in fluid dynamics, general relativity, statistical mechanics, or radiative processes. (If you’ve seen them, some of this will be easy for you). • Because I’m charged with covering a wide range of topics, I made some choices based on personal preferences. (Neutron stars and black holes really ARE very interesting). • Still, I am leaving out a lot. You can find more background material in e.g., “Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects” by Shapiro & Teukolsky. For current research on individual subjects, I’ll try to give references as we go, and you’re welcome to ask for more after the lectures. • I’m going to focus on their structure, their interiors, and their appearance as it relates to determining the properties of their interiors. • Please ask questions, interrupt, ask for more explanation, etc. Lives of Stars End Stages of Stellar Evolution Main Sequence stars: H burning in the core, synthesizing light elements Heavier elements form in the later stages, after H in the core is exhausted and core contracts, central T rises to ignite “triple-α” reaction 3 He4 --> C12 Which stars can ignite He? If they cannot, what happens during the contraction phase? The stellar mass determines if there is sufficient contraction (and thus heating) to ignite further nuclear reactions or if matter becomes degenerate (at very high densities) before nuclear reactions set in. Energy (Temperature) Dependence of Nuclear Reactions Reaction rates are determined by two competing effects: Two charged nuclei have to overcome a large Coulomb barrier Z Z e2 Z Z V = 1 2 "1.44 1 2 MeV R (R /1 fm) However, their thermal energies are much lower # T & kT " 9% ( keV ! $ 108 K' Nuclear reactions occur only because of quantum tunneling, with a probability given by the Gamow factor ! 2#Z Z e 2 " 1 2 u P ~ e h ! Energy (Temperature) Dependence of Nuclear reactions On the other hand, particles should approach within ~ a de Broglie wavelength for efficient tunneling, so that the reaction cross section goes as: 1 ! ~ E The combination of the Gamow factor and 1/E leads to well determined optimal temperatures for nuclear reactions, with higher temperatures required for higher Z’s. End Stages of Stellar Evolution Main Sequence stars: H burning in the core, synthesizing light elements Heavier elements form in the later stages, after H in the core is exhausted and core contracts, central T rises to ignite “triple-α” reaction 3 He4 --> C12 Which stars can ignite He? If they cannot, what happens during the contraction phase? The stellar mass determines if there is sufficient contraction (and thus heating) to ignite further nuclear reactions or if matter becomes degenerate (at very high densities) before nuclear reactions set in. Let’s first look at equation of state of degenerate Fermions. Kinetic Theory Preliminaries Let’s start with the distribution function and define number density: df n = d 3 p " d 3 x d 3 p All averages, such as energy density are given by df 3 2 2 2 4 1/ 2 " = E d p ; E = (p c + m c ) includes particle rest mass # d 3 x d 3 p ! For an ideal fermion/boson gas in equilibrium, !1 Fermion (half-integer spin particles) f (E) = ! exp[(E " µ)/kT]±1 Boson (integer-spin particles) ! Some limits of f(E): High temperature, low density: µ # E f (E) " exp( ) kT ! For fermions, chemical potential (energy cost of adding one particle) is the Fermi energy 1 f (E) = exp[(E " E F )/kT]+1 1 where Fermi energy EF is defined such that f (E ) = F 2 ! ! Fermions at zero temperature (complete degeneracy): 1 (E ≤ E ) f(E) ~ F { 0 (E > EF) For comparison, let’s look at bosons: Statistical distributions of photons detected at different times following the startup of the laser oscillation. At short times the source is chaotic and the distribution is of Bose-Einstein type. At longer times the source is a laser and the distribution becomes Poissonian. Unlike Fermions, as T--> 0, an unlimited number of bosons condense to the ground state. • We can write the available number of cells in terms momentum: V N(p)dp = 2 " 4# p2dp h 3 or in terms of energy by using E=p2/2m 8"V ! N(E)dE = (2m3)1/ 2 E1/ 2dE h 3 Thus, at a given E and for fixed V, the phase space available to the system of particles decreases with the particle mass, and electrons can fill the phase space much more easily than protons. ! • The pressure associated with the degenerate electron gas is given by 1 " 1 pF Pe = # Ne (p) f (p)v pdp Pe = " Ne (p)v pdp 3 0 3 0 Which can be evaluated for the nonrelativistic case v<<c (v=p/me) and for extreme relativistic case (v~c) to give 2 2 / 3 2 (3" ) h 5!/ 3 ! Pe = ne , v << c 5 me 2 1/ 3 (3" ) 4 / 3 Pe = hcne , v ~ c 4 Notice! there is no dependence on me in the relativistic limit (except through n). ! Note: P-ρ relations of the type P=K ρΓ are called polytropic equations of state. This is exact for degenerate matter (e.g., inside a white dwarf ) and a good approximation for some normal stars. Now back to the fate of the evolving stars: During the contraction of a star, nuclear reactions must start when P ≥ Pe. (otherwise, pressure due to degenerate electrons stops the contraction before necessary T for the reactions is reached) Because stellar temperatures scale with mass, this condition equates to a minimum mass for the onset of reactions: (i) For H-burning at 106 K: M ≥0.031 M Smaller masses do not become MS (ii) For He-burning at 2x108 K: Just the mass of the He core; smaller mass stars M ≥0.439 M become degenerate before triple-α kicks in. The evolu)on of a low-mass star Core already degenerate Iben 1966 Low-mass stars show similar evolu)onary tracks Mayer 1994 Heavier stars show dis)nct evolu)onary tracks Mayer 1994 The more massive stars live shorter H- and He-burning stages Nuclear Reactions • We saw in stellar evolution the basic fusion processes: H fusion through the pp and CNO cycles Triple alpha (He fusion) process Binding Energy Curve FISSION FUSION The Segre chart of isotopes Z=N Why determines this valley of stability? The Weizsacker semi-empirical mass formula is Z(Z "1) (A " 2Z)2 E = a A " a A2 / 3 " a " a B V S c A1/ 3 a A volume term av~ 16 MeV surface term Coulomb E. Symmetry energy a ~ a aa ~ 23 MeV ! s V ac ~ 0.7MeV By maximizing the binding energy with respect to Z, keeping A=N+Z constant, i.e., "E B = 0 "Z A we get 1 A Z = For small A, Z=A/2 2 2 / 3 ac ! 1+ A Z/A < 1/2 for large A 4aa ! Nuclear Reactions • How are elements beyond Fe produced? • A large number of different nuclear reactions take place during the late stages of stellar evolution • Fusion of large nuclei become rare and unimportant Nuclear reac)ons proceed towards heavy elements mostly via neutron-, proton- and alpha-captures. Why? Two charged nuclei have to overcome a large Coulomb barrier Z Z e2 Z Z V = 1 2 "1.44 1 2 MeV R (R /1 fm) However, their thermal energies are much lower # T & kT " 9% ( keV ! $ 108 K' Nuclear reac)ons occur only because of quantum tunneling, with a probability given by the Gamow factor ! 2#Z Z e 2 " 1 2 P ~ e hu Reac)ons between nuclei with high atomic numbers (Z) are not favored. ! Nuclear Reactions • A large number of different nuclear reactions take place during the late stages of stellar evolution • Fusion of large nuclei become rare and unimportant • Instead, alpha capture, beta decay and inverse beta decay (equivalently electron capture/decay), neutron capture and emission (drip), proton capture and emission become the dominant processes Beta Decay and Inverse β-decay One other reaction we should briefly talk about in the evolution of stars into compact objects is the inverse β-decay p + e → n + νe In “ordinary” environments, β-decay _ n → p + e + νe also proceeds efficiently and enables an equilibrium between electrons, protons, and neutrons. But at high densities, when electron Fermi energy is high and the electron produced by β-decay does not have sufficient energy, the inverse decay proceeds to primarily create more neutrons. Neutron/Proton Capture The neutron capture leads to A A+1 Z X + n ! Z X And proton capture leads to A A+1 Z X + n ! Z+1 X Nuclear reac)ons lead to a steady climb along the beta-stability valley of the isotope chart A]er carbon and oxygen burning, an equilibrium sets in between alpha-captures and photodisintegraon of various elements which can, eventually, lead to 56Ni -> 56Co -> 56Fe (Note: these reac)ons proceed in both direc)ons) Beyond this point, it is not energe)cally favored for alpha-par)cles to be captured and heavier elements to be generated. The only available path to the produc)on of heavier elements in this environment is via neutron-captures. Neutron captures are typically slow (compared to photodisintegraon) and their products are neutron-heavy nuclei that are unstable to beta decays.