1 Part I. Basics of Stellar Evolution1. Introduction 2. Basic Equations of Stellar Evolution 3. Evolution 3.1. Evolution of Low and Intermediate-mass Stars Mounib El Eid AUB campus looking north to the Mediterranean You think you are in Italy june 11, 2017
2 Quoting Maurizio Paradise: Hell: Sun Spanish Sun German Clocks SwissWine Italian Organization German Hell: Sun German Clocks Spanish Wine Swiss Organization Italian Well, we like the hell to be organized by the Italian june 11, 2017
3 1. Introduction 1a) Some remarksStars are physical systems in which the four natural forces work together. Their formation and evolution are complex, but they are based on fundamental physical laws. This is one of the manifestation that the physical laws are universal. Stars reveal how human being are connected to the cosmos, simply because the bulk of the elements are synthesized in stars of many generations. Think about the DNA based on carbon and nitrogen made in stars, and hydrogen produced in the early universe. Stars are always in interaction with surroundings thru their radiation, mass loss and ejection of enriched matter by supernova explosion or though planetary nebulae formation. The understanding of star formation and evolution is only possible by the combined effort of observational astronomer and astrophysicists and this was possible after the development of digital computers, which became faster in the course of time. The understanding of the internal structure of stars is accomplished by usinmg basic stellar structure equation which are discussed in the following. june 11, 2017
4 1b) Some motivations Lifetimes and sizes of stars: the bigger, the shorter lifetime june 11, 2017
5 Nobody will see this ever 1AU=150x106 kmWhat about the Sun? impressive Nobody will see this ever 1AU=150x106 km The question is: what is a red giant? We will see june 11, 2017
6 The Central Evolution of StarsIron Disintegration, supernovae II Electron-Positron –Creation. Pair creation supernova Rapid Electron Capture. supernova Origin of white dwarfs Up to 8 MSun june 11, 2017
7 2. Summary of the basic equations of stellar structure& evolutionDependent variables: u= velocity r=radius T=Temperature =density L=Luminosity P=pressure Independent variables : M r =interior mass (Lagrangian coordinate) t=time Spherical Symmetry is assumed, 1.Momentum equation: (gravitational interaction) (Derivation in A1) 2. Mass conservation 3. Definition of velocity: 4. Energy equation: (Strong, weak & EM interaction) Derivation In A2 5. Energy Transport (see A3 for details) Radiative – Convective-Conductive = mean opacity june 11, 2017
8 2a. What is the consequence of hydrostatic equilibrium ?Hydrostatic equilibrium is a clear manifestation of the fact that a pressure gradient balances gravity. It can be shown (derivation in appendix A3 that the free fall time scale (pressure gradient is neglected in the above equation) is given by: Sun: red giant (M=MSun , R=100 Rsun)) White Dwarf ( M=Msun , R/Rsun=1/50) Conclusion: the Sun and stars cannot live so long without achieving hydrostatic equilibrium june 11, 2017
9 2b. Virial Theorem We can derive this important theorem by using the equations above (equations Of hydrostatic equilibrium and mass conservation): Partial integration from center to surface yields: june 11, 2017
10 Potential energy= 2 * thermal energyOr: Virial Theorem Where : For a non-relativistic non-degenerate gas: Potential energy= 2 * thermal energy Hydrogen atom: Physics is unified It is the world of stars: they contract, heats up and radiate at the same time following the virial theorem june 11, 2017
11 2c. Energy Equation and Energy Conservation(strong, weak and coulomb interaction) L r is the luminosity (or power) passing through a sphere of radius r . At r=0: L r =0 At r=R: L r =Lstar In between, L is a complicated function resulting from energy transport by radiation, convection and conduction Electromagnetic interaction, strong interaction and weak interaction are involved in this equation. We have discovered the role of gravity previously God gave the universe the four forces that the stars can be created to create the elements in our DNA Thus, a star exists, because of the coordinated effort of the four natural forces. june 11, 2017
12 Comments on the Energy Equation(a) It turns out that the term plays a decisive role in the evolution of massive stars starting carbon burning. Later more. (b) One can derive from the gravitational term (containing the time derivatives), the so called Kelvin Helmholtz time scale. One obtains (see appendix 2): For the Sun: If this would be the life time of the Sun, we would not be here on Earth The Sun and other stars need a much longer time scale and this will be the nuclear time scale (see below) june 11, 2017
13 Conclusion: Nuclear time scale Derivation: see appendix 3If a star evolves on a time scale nuc >> KH, then the energy equation is well described by: The star will be in hydrostatic and thermal equilibrium. The nuclear luminosity Appears at the star’s surface. Such conditions are fulfilled in particular during the main sequence. june 11, 2017
14 2d. Radiative Energy TransportRadiation escapes the star’s surface. This is accomplished by: Effective energy transport A temperature gradient must exist throughout the star The temperature gradient existing in the star means that there is a gradient of the radiation pressure as well. This leads to net movement of the photons toward the surface which we call radiative flux (Frad) The whole process is described by: Where <> is a mean absorption coefficient or opacity interpretation The net radiative flux is driven by the differences in radiation pressure with a photon wind blowing from high to low radiation pressure. We talk about a beam of photons traveling to the surface, although photons “suffer” a lot (unless they like it) until they reach the surface – they diffuse out or randomly walk out. june 11, 2017
15 In thermal equilibrium, we have:Equating these last two equations , we get: With: Transport Equation in case of Diffusion of radiation june 11, 2017
16 2e. Convective Energy TransferConvection is a matter of instability of a radiative stellar layer e r+r T+T, + P+P s e Convection may be described as dynamical instability: Meaning: r T,P, With the assumption that a moving mass element has not enough time to exchange an appreciate amount of heat with its surroundings, that is it moves Adiabatically, and we speak of dynamical instability. e=element s= surroundings For a physical variable A: (1) Take A=T (temperature): Then DT> 0 means a hotter element than surroundings. If DP>0 element will expand immediately until the pressure balances with surrounding is restored and this expansion occurs with the speed of sound the fastest the element can move. june 11, 2017
17 The assumed DT>0 requires D<0 for an ideal gasThen, 0ne can assume that the element remains in a pressure balance with surrounding, or The assumed DT>0 requires D<0 for an ideal gas Proof : (2) Learn Effect Element lighter than surrounding. It will be lifted upward by buoyancy forces. Conclusion Temperature fluctuation leads to radial motion of the convective element. Let us test the stability of a convective layer june 11, 2017
18 Element lighter f r >o upward unstable situationConsider a radial shift r>0, and write: The buoyant force per unit volume exerted on the element by surrounding is: Archimedes principle The downward gravitational force per unit volume is: The net force is then: if Element lighter f r >o upward unstable situation if Element heavier f r
19 Criterion for stability against convectionThis relation impractical since the density gradients do not appear in the basic equations. In principle, evaluation of (d/dr)e requires inclusion of heat exchange of the element with surroundings. For the adiabatic motion: Ideal gas Equation (3) can be rewritten (see appendix 3) to get a suitable expression: Criterion for stability against convection The adiabatic motion means: june 11, 2017
20 Convective Energy TransferLedoux criterion for stability against convection If a layer is homogenous, then; Schwarzschild criterion for stability has a stabilizing effect, because element of heavier material is pulled back by gravity. The term with is positive. Convective Energy Transfer It is an exchange of energy between hotter and cooler layers in dynamically unstable layers. The simplest description is based on the mixing length theory (MLT), which is a local phenomenological theory We had: How to get for convection? june 11, 2017
21 In summary, five equations are obtained for 5 variables:Using the MLT, the convective flux can be written as (see ref [5] for all details): In summary, five equations are obtained for 5 variables: They can be solved, if : Are known., they are known in the stellar model, except the mixing length parameter , a free parameter on the MLT. It reflects the local description of convection. june 11, 2017
22 2f. Equation of state of stellar materialIdeal Gas: For a gas of n particles per unit volume (number density) that are non-interacting and Non-degenerate (Ideal gas), the pressure is well known: k is the Boltzmann constant, is the mean molecular weight and is the gas constant per unit mass (8.31 x10 7 erg K-1 g-1 ). So we have introduced the mean molecular weight as the average mass of a free particle in the gas measured in atomic mass unit m u= x10-24 g=1/NA where NA=Avogadro number. With The EOS above is recovered. The mean molecular weight depends on the composition of the gas as well as the state of ionization of each atoms. In the stellar interior, the gas becomes fully ionized. one deals with a mixture of nuclei and electrons. To describe such a state, one introduce the mass fraction xi of a nuclear species. june 11, 2017
23 (mass number: number of grams of the ith Particle per gram of the mixture) The total pressure is then: If the gas is completely ionized, then Is the number of free particles (Z i electrons + the ions) Then: Introducing the mean molecular weight: Recovers the usual form of the EOS So, the meaning of is that a mixture of gases can be treated as a uniform gas A neutral gas has a mean molecular weight june 11, 2017
24 which is similar to the law of mass action in Chemistry.But in general the gas may be partially ionized and the mean molecular weight is calculated on line using ionization equilibrium leading to the so called Saha Equation which is similar to the law of mass action in Chemistry. Now, how to include the pressure of degenerate electrons? The Fermi-Integrals are evaluated (tables exist) Electron Gas degeneracy parameter Center of an evolving 20 Msun star june 11, 2017
25 2g. OPAL Opacities C,O,Fe He H The heavy-element peaks in the OPALOpacities (Livermore) was missing in the older calculation in Los Alamos (LAOL) june 11, 2017
26 I usually chose the beautiful.3.EVOLUTION We still need to talk about the nuclear transmutation as agent of evolution. This will be done in the next lecture. Let’s see some example of evolution. A message My work always tried to unite the true with the beautiful, but when I had to choose between one or the other, I usually chose the beautiful. Hermann Weyl There is no terror in any religion, but terrorists in each religion june 11, 2017
27 Luminosity behavior from the main sequence very sensitive to the mean molecular weight (). See illustration in Appendix 5 You also see that the radius is uniquely determined by the nuclear energy source in thermal equilibrium (see appendix 5) june 11, 2017
28 3.1 Low and Intermediate Mass StarsLet us follow first the evolution of our Sun or solar-like stars june 11, 2017
29 Contracting CO-core +H and He shells SUMMARY: AGB Star: Contracting CO-core +H and He shells SUMMARY: Evolution of Solar-like stars planetary Nebula About half of the mass lost Red Giant: contacting He-core+ H-shell White Dwarf Main-Sequence: Radiative core, convective envelope The end stage
30 5 M Intermediate-mass Star Blue loop AGB 2nd Dredge up end coreH-shell burning 2nd Dredge up end core He-burning 1st Dredge up Or mix up to surface End core H-burning 4 Begin core He-burning Main sequence Recent work on blue loop: Halabi, El Eid and Champagne (2012) 1st dredge up: of H-burning (He, 12C and 14N) to the surface
31 How are the loops initiated?N energy generation rate LH XH 1 T 2 3 It is the enforcement of the H-burning shell which initiate the loop
32 Cepheid variables Loops explain the Cepheid variablesSee Halabi et al (2012) Loops extension influences by: Overshooting, that is mixing beyond the edge as determined by the Schwarzschild criterion for convection, and by the rate of Essential rate in the CNO cycle Cepheid variables june 11, 2017
33 Why second dredge up? The way to the AGB stageThis happens in stars more massive than 4 M sun and it is the phase of early AGB. As seen in the previous Figure the convective envelope penetrates into the H-depleted zone. This happens when the helium-burning shell becomes effective causing the star to expand, so that shell H-burning is switched off allowing the envelope convection to penetrate deeper. In stars of lower mass, H-burning shell remains active and prevent the second dredge up. The way to the AGB stage Main sequence phase: every star formed in the universe must go through the phase of hydrogen burning either by the proton-proton chain, or by the CNO cycle. Relatively simple evolutionary phase, but it has deep meaning. It is characterized by the low efficiency of nuclear burning, namely a rate of transformation H to He of 0.71%. This is the ratio of the mass of a helium atom divided by the mass of four hydrogen atoms. This number is one of six numbers shaping the universe (see just six numbers by Martin Rees, and his challenging author M. Rowan-Robinson with his 9 numbers)
34 Asymptotic Giant Branch Stars AGBjune 11, 2017
35 3.0 solar mass MH MCO (He-exhausted) Stancliffe et al (2004) MNRAS LHe
36 Notice that the Lhe the stronger the smaller LHLstar not influenced by the extremely large Lhe (why? Homework) He-shell Thermally pulsating 3 solar mass star Basic physics of pulsation, see Appendix A6 H-shell
37 Two pulses for illustration.Research problems: How to control the diffusion of protons into the carbon-rich region in order t get the an amount of 13C of the order of several solar masses, so that C (,n) 16O can deliver the required neutron density. How to control the diffusion of protons into the carbon-rich region in order to get the an amount of 13C of the order of several solar masses, so that 13C (,n) 16O can deliver the required neutron density. The thin helium shell in AGB star cannot deal with this effective nuclear energy source and a helium flash is initiated See Appendix A6) How to describe correctly the mixing process during the AGB pulsations.
38 STOP june 11, 2017
39 3.2 Massive stars Why the difference? See nextDarker areas: CONVECTIVE see El Eid, Meyer, The APJ 611, 452 (2004) The, El Eid, Meyer: APJ, 655, 1058, (2007) Note that the energy production does not comprise the whole convective core NACRE NACRE Why the difference? See next
40 Nuclear physics essential for understanding the structure of starsIt is the rate Nuclear physics essential for understanding the structure of stars He burning;15-30 MSun CF85: Caughlan et al ; Atom.. Data Nucl. Data Tables, 32, 197 (1985) Kunz: Kunz etal. APJ, 567, 643 (2002) NACRE: Angulo et al , Nucl. Phys. A, 656, 3A, 1999 Buchmann: 1996, APJ, 468, L127 (1996) june 11, 2017
41 Limongi, Straniero , Chieffi APJS, 129, 625 (2000)25 Msun star Imbriani et al (2001), ApJ 558, 903 Rate of CF85 > CF88 No convective carbon-burning core CF85 X 12=0.18 X12 lower Remaining car bon mass fraction No convective core But here CF88 X12=0.42 See also : Limongi, Straniero , Chieffi APJS, 129, 625 (2000) june 11, 2017
42 Learn Effect The carbon burning is an important evolutionary phase in massive stars: It is a branching point between the formation of white dwarfs and the evolution toward core collapse supernovae It influences strongly the subsequent evolution, especially through carbon-shell burning, where the s-process nucleosynthesis operates in part It depends on several factors: Mass of star Carbon mass fraction left over from Helium burning, thus on the rate of Neutrino losses june 11, 2017
43 It is a count down No fusion No fusion What the star did in more than 7 million years, it is undone in few seconds
44 Why does a massive star collapse?Plotted the mass difference/nucleon Remarkable: iron is a dividing point between the two sides. It marks the minimum of the curve. Iron nuclei are so compact that energy cannot be extracted either by combining them or by splitting them. Actually splitting them costs energy fusion Iron at minimum fission The iron core collapses and can lead to supernovae explosion. This property of iron initiate an inferno in the core of a massive stars june 11, 2017
45 Core collapse supernova in schematic pictures1. Prior to collapse, the star has developed onion shell structure 2. Iron cannot undergo fusion, iron core collapses 3. Within seconds, the core collapses to nuclear density. The high density squeezes the neutrons and they stop promptly the collapse. Shock wave is created which expelled the stellar material More details by Dr. Chieffi 4. The out streaming neutrinos complete the job and lead to a supernova Bang
46 Appendices A1: Derivation of the momentum equationA1: Fee-Fall time scale A2: Kelvin Helmholtz time scale A3: Nuclear time scale A4: Derivation of convective criteria A5: Meaning of Thermal equilibrium A6: Basics of Thermal pulsations of AGB stars june 11, 2017
47 A1. Momentum equation. (gravitational interaction)dm, dA or Net pressure force on the mass element Now apply Newton’s second law and using : Pressure gradient negative, because encountering gravity If acceleration very small: june 11, 2017
48 2c. Energy Equation and Energy Conservation(strong, weak and coulomb interaction) L r is the luminosity (or power) passing through a sphere of radius r . At r=0: L r =0 At r=R: L r =Lstar In between, L is a complicated function resulting from energy transport by radiation, convection and conduction n is the energy released per unit mass per Second or energy generation rate [erg/(g. s)] due to thermonuclear nuclear Reaction. Then: Using june 11, 2017
49 = internal energy Steady state.(thermal equilibrium: nuclear luminosity appears at the star’s surface ) Now consider a slowly contracting layer in a star: The amount of heat added (dQ) (per gram and second) to the stellar material is according to the first law of thermodynamics: = internal energy using Quasi-static change june 11, 2017
50 Consider momentum equation with pressure term zero (free fall)A1. Free-fall time scale Consider momentum equation with pressure term zero (free fall) Or: Integrate to get: When r=0, t=ff june 11, 2017
51 A2. Kelvin-Helmholtz time scale= relaxation time needed for a star to return to thermal equilibrium. The star loses this state if it is not producing energy by thermonuclear reaction. Roughly: june 11, 2017
52 Nuclear time scale It characterizes the time for changes due to nuclear reaction. Hydrogen Fusion (or burning): 4 1H 1 4He (see later details) (one fundamental number in our Universe) With Lsun=3.83x1033 erg/s This is what the sun is doing june 11, 2017
53 A5. The physical meaning of thermal equilibrium:For a given temperature (T), the luminosity (L) is chiefly determined by the opacity (). The lower the opacity, the brighter the star. Using With the equation of state: june 11, 2017
54 What is really influencing the radius of a star? See nextIf the opacity is given by the so called “Kramer opacity” As in low mass stars: Proportional 7.5 Weak dependence on R If the opacity is dominated by Thomson electron scattering, such as in massive stars): then: Proportional 4 no dependence on R What is really influencing the radius of a star? See next june 11, 2017
55 Answer: It is the nuclear energy source in the star:Write the nuclear luminosity: Very sensitive to temperature For the CNO cycle, we have: Was used Very clear dependence on the radius R In thermal equilibrium, the star seeks to make This determines R uniquely given the mass and initial chemical composition The main sequence evolution is a clear demonstration of this analysis See Figure (9.1) on next slide june 11, 2017
56 Basic mechanism of thermal pulsationsThis is a thermal runaway, or a secular instability which may occur when nuclear burning becomes unstable and is governed by thermal relaxation Secular instability in degenerate regions leads to core helium flash. Secular instability in thin non-degenerate regions leads to quasiperiodic thermal pulsation. Analyzing this instability is accomplished by considering the gravothermal specific heat . Stars are surprising since they have negative gravothermal specific heat reflecting the fact that they heats up while losing energy by radiation. The gravothermal specific heat (c*) is introduced by the following equation (see Kippenhahn & Weigert 1990 for derivation): Where: (adiabatic temperature gradient) obtained from the equation of state:
57 Conclusion: since cooling, overproduction of nuclear energy reduced Discussion: a) Ideal gas: Using : Conclusion: since cooling, overproduction of nuclear energy reduced stabilization of stellar layers c* < 0 acts as stabilizer b) Degenerate non-relativistic gas: With adding heat (dq >0), then dT >0 : heating leads eventually to thermal runaway. Indeed this is the case of helium flash One can show that : Therefore, in thermal runaway: Core helium flash is an example: let’s see in case of a 2 M
58 C) Thermal Pulsations (AGB stage)Thin shell D r0 r Shell source unstable, since: and for: Ideal Gas: It depends on D whether the shell source is stable or not. The point is that the temperature sensitivity of nuclear burning has to exceed a certain limit to have instability and this is the case for helium burning.