Outline I. Overview. Equations of Stellar structure and Evolution. Pre Main sequence evolution. Main sequence evolution. Post-Main sequence evolution

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Morris Atkins
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1 1 Outline I Overview Equations of Stellar structure and Evolution Pre Main sequence evolution Main sequence evolution Post-Main sequence evolution AGB evolution White dwarfs Super-AGB stars Massive stars

2 2 Stellar evolution The evolution of stars is primarily determined by its mass and then by its composition and by other processes such as extra mixing, mass loss,..

3 3 Conservation of mass In star dr dt dm dt so the Lagrangian description is more appropriate. Quantities such as radius or density are then function of (m,t) : r(m,t), ρ(m,t),... In the Lagrangian description, the mass conservation equation simply writes m r We also define the Lagrangian velocity v L as = 4πr 2 ρ (2.1) t Dr = v L (2.2) Dt m

4 Conservation of momentum D 2 r Dt 2 m = 4πr 2 P m Gm r 2 (2.3) Free fall or dynamical time scale : τ eff If P = 0, no opposition to gravitational confinement collapse D 2 r Dt 2 Gm R GM r 2 τ 2 R ff 2 R τ ff 3 ( R ) 3/2( M ) 1/2 GM 1600 s R M

5 5 Explosion timescale τ exp if gravity becomes negligible, D 2 r Dt 2 1 P ρ r R τ 2 exp P ρr τ exp R ρ P R c s with c s the sound speed When the configuration is in hydrostatic equilibrium P r = Gm τ r 2 exp τ ff = τ hydro For a white dwarf R R /50 τ exp 4 5 sec

6 6 Energy conservation δq = (L m L m+dm ) t } {{ } + energy in out (ε nuc ε ν ) m t } {{ } local nuclear energy production where δq (erg) = amount of heat absorbed (> 0) or emitted (< 0) by the shell of mass m during the time interval t L r (L r+dr ) = energy entering (escaping) the shell per unit time ( luminosity in erg s 1 ) ε nuc local rate of nuclear energy production (erg g 1 s 1 ) ε ν local rate of neutrino energy loss (erg g 1 s 1 )

7 7 From the first law of thermodynamics, we have δq = (du int + PdV) m (V = 1 ρ : specific volume) Now defining the gravothermal energy production rate ε grav ( ) ε grav = δq t m = Dq = Du 1 D int ρ P = T Ds Dt Dt Dt Dt the equation for the conservation of energy finally writes (2.4) L m = ε nuc ε ν + ε grav (2.5) Integration of Eq 2.5 leads to L = L nuc L ν U int K Ω (2.6)

8 8 Equation of transport There are 3 ways of transporting energy within stars 1. Radiative transport In stars the photon mean free path is much smaller than the stellar dimension (l ν R) so the transport of energy by photons can be considered as a diffusive process. The energy flux (per unit time and area) is given by : F rad = D rad U rad where D rad is the diffusion coefficient which, from kinetic theory, writes D rad = 1 3 v l = 1 3 c l ν U rad = at 4 is the energy density of radiation, l ν = 1/κρ the mean free path and κ the opacity F rad = 4ac T 3 dt dt = k rad (2.7) 3 κρ dr dr

9 9 where k rad = 4ac 3 T 3 κρ Using, the luminosity L instead of F rad, 2.7 writes L 4πr = 4acT 3 dt 2 3κρ dr = c dp rad κρ dr (2.8) (2.9) and in the Lagrangian form dt dm = 3κL (2.10) 64π 2 acr 4 T 3 which leads to the definition of the radiative gradient rad Note that rad κ and L rad = ( dlnt ) 3κLP = (2.11) dlnp rad 16πacGmT 4

10 10 2. Transport by conduction Conduction results from collisions between electrons and atomic nuclei. This process works exactly as radiative transport through Fick s equation : F cond = D cond dt dr = 4acT 3 dt 3ρκ cond dr (2.12) where κ cond is the conductive opacity which contribution is included in the total opacity via the relation 1 κ = (2.13) κ cond κ rad

11 3. Transport by convection At r + r, the buoyancy force per unit of volume acting on a perturbed convective element is F buo = g(ρ e ρ s) If F buo < 0, i.e. if ρ e > ρ s the blob sinks back to its original position radiatively stable If F buo > 0, i.e. if ρ e < ρ s, the blob continues to rise convective instability The condition for convective instability then writes ( dρ ) ( dρ ) < dr e dr s (2.14)

12 The instability criterion can also be rewritten as ( ) ( ) dlnt dlnt > dlnp s dlnp e } {{ }} {{ } rad ad and is referred to as the Schwarzschild criterion if rad > ad convectively unstable The Mixing Length Theory (MLT) estimates the temperature gradient in the convective zone and there : = conv

13 13 Equation of nucleosynthesis Nuclear reactions release energy and change the composition of the stellar matter. For each nucleus, there is an equation governing the evolution of its abundance and is the difference between the production and its destruction DY i Dt = 1 ρn Y n Y m < σv > nm Y i Y j < σv > ij (2.15) A nm ij } {{ }} {{ } m+n i+j destruction of i where < σv > nm is the nuclear reaction rate and Y i the molar mass fraction (abundance per mole unit)

14 14 The equations of stellar structure Independent variables are t et m m r Dr Dt = u (2.16) = 4πr 2 ρ (2.17) D 2 r Dt 2 = 4πr 2 P m m Gm r 2 (2.18) L m = ε nuc ε ν + ε grav (2.19) = lnt 3κL r P lnp = rad = 16πac GT 4 m if rad < ad (2.20) conv if rad > ad DY i = 1 ( ) r nm r ij (2.21) Dt ρn A nm ij

15 15 Virial theorem Assume that the star is in hydrostatic equilibrium (i.e. τ hydro τ evol ) then M 0 4πr 2 P m = Gm r 2 4πr 3 P m dm = [4πr 3 P] M 0 3 M 0 4πr 2 r m P dm from which we obtain the Virial theorem (P is the surface pressure 0) : M 0 Gm M P dm = 3 r 0 ρ dm 4πR3 P (2.22) The Virial theorem (2.22) connects the different energy sources of the star, namely the gravitational energy on the left hand side M Ω = 0 Gm dm = ζ GM2 r R (2.23) where the parameter ζ 1 (ζ = 3/5 for ρ(r) = cste) and another term (right hand side) which meaning will become obvious in the following examples.

16 16 Virial theorem : Applications In these illustrations, we assume that 1. the star is in hydrostatic equilibrium : K = 0 2. that the pressure vanishes at the stellar surface : P = 0 Stellar stability Equation of state : ideal γ law : P ρ = (γ 1)u int where γ = c P c v Total energy : E = Ω + The star is gravitationally bound if E < 0 γ > 4/3 when γ < 4/3 : star is hydrodynamically unstable Ω = 3(γ 1)U int (2.24) }{{} K +U int = 3γ 4 3(γ 1) Ω (2.25) =0

17 17 Energy partition If we neglect the nuclear energy sources, the total luminosity is L = Ω U int = 3γ 4 3γ 3 Ω In case of a perfect monoatomic gas (γ = 5/3) L = 1 2 Ω = U int half of the energy released by gravitational contraction ( Ω < 0) is radiated away and the other half goes into heating the star ( U int > 0) For a degenerate gas : γ = 4/3 E = 0 This means that such a configuration is unstable. A slight perturbation may lead to collapse (E < 0) or explosion (E > 0) Note: a star has a negative specific heat : it heats up ( U int > 0) while it looses energy (L > 0)!

18 18 Kelvin Helmholtz timescale Kelvin Helmholtz timescale If the star is in hydrostatic equilibrium and L nuc L whence L = Ω U int = Ω 2 GM2 2R τ KH τ KH = GM2 2RL M M 2 R R L yr (2.26) L τ KH represents the time scale during which a star can maintain its luminosity by gravitational contraction. τ KH also corresponds to the thermal adjustment timescale of the star.

19 19 Nuclear timescale When the nuclear energy production dominates : L L nuc L nuc = M 0 ε nuc dm M i Q i τ nuc where Q i : energy released by nuclear burning of species i ([erg g 1 ]) M i mass of fuel made of species i ([g]) In the sun, MH = 0.15M, Q H = erg g 1, X H = 0.7 and L = L = erg s 1 leading to τ nuc yr

20 20 The pace of evolution of stars if L L nuc > 0 the star is in complete equilibrium, evolutionary timescale τ evol τ nuc Central burning phases if L nuc 0 or if L nuc < L in a small or large part (in mass) of the star then L grav 0 so the thermal equilibrium is not reached : τ evol τ KH Core contraction phases

21 Pre main sequence evolution radiative core development central Li burning contraction T > H(p,γ) 3 He T > Li(p,α)α radiative core H ignition CNO equilibrium (ZAMS) PP equilibrium D burning envelope Li burning radiative track beginning of nuclear burning Hayashi phase ZAMS

22 22 Evolution of the internal structure M = 1M M = 3M

23 PMS structural evolution of a 1 M star 23

24 24 Overview of stellar calculations Structure of 5 M Kippenhahn diagram of the 5 M star. Kippenhahn diagrams provide temporal information on the overall structure: position of convective regions position of burning regions mass loss main chemical species: H in the whole star (begin) He built in the MS core C+O built in the HeB core in a 5M, H burning in a convective core of decreasing extent

25 25 Overview of stellar calculations Structure of 1.2 M Kippenhahn diagram of the 1.2 M star. First half of main sequence: radiative core Second half of main sequence: convective core End MS + RGB: H burn in shell Mass loss on the RGB

26 pp chains 26

27 27 CNO cycles F F 19 F O 16 O 17 O 18 O N 14 N N C 13 C C (p, γ) (p, α) (β + )

28 28 Burning regimes Stars with masses above 1.2M generate most of their energy via the CNO cycle beause they develop higher temperatures (> K)

29 29 Importance of convection in main sequence stars Solid lines give the mass values at 1/4 and 1/2 the stellar radius Dashed lines show the mass within which 50% and 90% of the stellar luminosity is produced

30 30 Post Main sequence stars When X c < 0.05 (point 2), the MS has ended First the core contracts L and T eff increase. For higher mass stars, the convective core recedes. The star moves from 2 3. At point 3, X When H burning stops, because of T, energy flows outward (T ) but at the same time the core contracts (T ) core becomes isothermal and contraction slows down ( T 0 L 0)

31 31 Hertzsprung gap Past point 4 (Schronberg-Chandrasekhar limit), the core contracts rapidly the star crosses the HRD on a Kelvin-Helmholtz timescale (τ KH ), T ε HBS nuc the luminosity remains constant because the increase in ε HBS nuc is compensated for by the narrowing of the H-burning shell (HBS)

32 32 The first dredge-up As the stars expands at practically constant luminosity, the envelope temperature decreases, κ(kramers) rad and convection sets in Chemical signature He 12 C 14 N 7 Li 12 C/ 13 C

33 33 The Bump Observationally, there is an accumulation of stars along the red giant branch: it is called the bump more pronounced in low metallicity stars

34 34 The passage of the Bump It occurs when the HBS meets the chemical discontinuity left by the convective envelope (for M < 2 2.5M ) When the HBS crosses the µ-discontinuity (M r 0.285M in the figure) composition changes nuclear energy production readjusts the luminosity temporarily decreases due to change in µ and κ and goes back up again in the HR diagram the star goes down and up : it spends more time in this luminosity bin

35 35 Structure of the giant stars The luminosity produced by the HBS depends almost exclusively on the core mass. At the core edge dp dm = GM 4πr 0 (r R ) 4 The pressure drops by many orders of magnitude in a small region and the envelope has almost no influence on the shell properties

36 36 Core helium flash For 0.5 < M < 2.25M partial degeneracy + neutrino loss off-center helium ignition HBS switched off R, L as M zams ignition closer to the center flash less powerful series of flashes of decreasing strength increasing duration For stars M > 2.25M, 4 He ignites at the center under non-degenerate conditions.

37 37 Central helium burning M = 2M, L > L H > L He M = 5M, L L H L He

38 38 Core helium burning stars with degenerate cores expand much more in low mass stars, after the flash, the temperature in the HBS has decreased radius and luminosity smaller. in more massive stars, He ignites gently. Core expansion causes temperature HBS L, R as the envelope contracts, κ convection retreats the star moves blueward in the HRD : blue loop the convective core grows during central He burning (different from MS) convective cores are smaller than during central H burning

39 39 He-burning Network 25 Mg 26 Mg need high T (> 10 8 K) to overcome Coulomb barrier. Mains reactions 18 F 16 O 18 O 22 Ne 3α 12 C (7.28 MeV) 12 C + α 16 O (7.16 MeV) 14 N 12 C 13 C 4 He

40 39 He-burning Network 25 Mg 26 Mg need high T (> 10 8 K) to overcome Coulomb barrier. Mains reactions 18 F 16 O 18 O 22 Ne 3α 12 C (7.28 MeV) 12 C + α 16 O (7.16 MeV) 12 C 14 N 13 C Nucleosynthesis 4 He Destruction of 14 N and 13 C Creation of 22 Ne and 25/26 Mg α captures on 13 C and 22 Ne = provide neutrons s-process elements

41 40 Second dredge-up and early-agb After central He exhaustion, core contracts envelope deepens 2DUP H burning shell narrows and gets more powerful HeBS extinguishes

42 41 AGB stars Structure degenerate CO core He burning shell H burning shell convective envelope circumstellar envelope Characteristics pulsating stars formation of grains and dust site of thermal pulses experience heavy mass loss

43 AGB star 42

43 Structural Evolution Development of recurrent convective instabilities in the HeBS : thermal pulse H and HeBS advance at different rates intershell mass core contracts HeBS temperature convection

44 43 Structural Evolution Development of recurrent convective instabilities in the HeBS : thermal pulse H and HeBS advance at different rates intershell mass core contracts HeBS temperature convection develops Pulse acts as a piston extinguishes HBS lifts envelope possible descent of the envelope in the pulse region : 3rd dredge up some numbers : pulse duration : yr, interpulse : yr, between 5 and 500 pulses, numbers strongly depend on M and Z

45 Evolutionary properties of AGB stars 44

46 45 Making sodium 14 N produced by the CNO in the HBS engulfed in the pulse

47 45 Making sodium In the pulse 14 N converted in 22 NE : 14 N(α, γ) 18 O(α, γ) 22 Ne

48 45 Making sodium the 3DUP pollutes the envelope with the 22 Ne produced in the pulse

49 45 Making sodium Below the convective envelope, at higher temperature 22 Ne(p,α) 23 Na

50 45 Making sodium 23 Na produced in the intershell is engulfed in the pulse

51 45 Making sodium The 3DUP brings 23 Na in the envelope

fast stellar wind ( 2000 km s 1 ) fast winds runs into previously

52 46 Post AGB evolution As the HBS approaches the surface T eff : the star moves to the left in the HRD As the hot core is exposed, very fast stellar wind ( 2000 km s 1 ) fast winds runs into previously ejected matter and stellar UV ionize the ejected material planetary nebula shines

53 47 White dwarfs Remnant cores of stars with M < 8M. supported against gravity by electron degeneracy pressure M ZAMS < 6 8M : C-O White Dwarfs 6 8 < M ZAMS < 9 12M : O-Ne White Dwarfs Properties: Mass < 1.4M Radius R earth (< 0.02R ) Density g cm 3 Escape Speed: 0.02 c (2% speed of light) Very thin atmosphere M No nuclear fusion or gravitational contraction. It shines by residual heat.

54 48

55 48 the WD mass distribution is strongly peaked around 0.6M Stars manage to loose most of their mass during the AGB phase. (Liebert et al 2005)

56 49 H and He burning phase Standard evolution up to C ignition convective H core burning 1DUP chemical signatures similar to that of intermediate mass stars 4 He, 13 C, 14 N H, 12 C, 16 O convective He core burning

50 C ignition When T max 6 10 8 K carbon ignites off center where the degeneracy η 2 3 Burning proceeds in 2 steps carbon flash : short lived high energy

57 50 C ignition When T max K carbon ignites off center where the degeneracy η 2 3 Burning proceeds in 2 steps carbon flash : short lived high energy release 10 6 < L C /L < used to lift the degeneracy and expand the structure quenches the instability deflagration : laminar flame that propagates to the center

58 51 Thermally pulsing super-agb phase L He < 10 6 L short interpulse periods without extra-mixing : NO 3DUP high temperatures at the base of the convective envelope (10 8 K)

59 52 Massive stars What is a massive star? star that goes through all the hydrostatic burnings from H to Si and explodes as a core collapse supernova 812 < M < 1000M, represent 14% of the mass of all stars Why are massive stars important in the global evolution of our Universe? strongly responsible for the chemical enrichment of the interstellar medium short lifetimes 3-30My. Inject radiation (2/3 of the visible light of galaxies), mechanical energy and new products of nucleosynthesis in universe massive stars plays a key role in many cosmic evolution processes (star formation,...) main producers of γ ray emitters ( 26 Al, 60 Fe, 44 Ti, 56 Ni) can be seen far away in the universe parents of black holes and of a large fraction of neutron stars

60 53 Overview Massive stars are very luminous and are constrained to have L < L Edd Main sequence O-B stars post MS stars BSG : Blue Super Giants (transitory phase) LBV : Luminous Blue Variables (M > 85M ) WR : Wolf Rayet (M WR < M LBV ) RSG : Red Super Giants

61 At the end of He burning, core mass > M Ch contraction ignition of subsequent nuclear burnings 54

62 55 Main sequence From dimensional considerations, ρ MR 3 and dp dr dt dr P R Mρ R 2 T R P M2 R 4 ρl T 3 R 2 T 4 MLR 4 P ρt L M 3 P T 4 L M The main sequence lifetime τ MS M/L Radiation pressure dominant in massive stars

63 56 H burning : overshooting Observational problem : Main sequence (H-burning) stars spread over a much larger T eff than predicted Indicates presence of core overshooting during central H burning. Overshooting = additional mixing at the convective edge Overshooting increases the core size more fuel is available MS lifetime longer larger MS in the HRD

64 56 H burning : overshooting Observational problem : Main sequence (H-burning) stars spread over a much larger T eff than predicted Indicates presence of core overshooting during central H burning. Overshooting = additional mixing at the convective edge Overshooting increases the core size more fuel is available MS lifetime longer larger MS in the HRD

65 57 Mass loss rate In massive stars, mass loss is a consequence of radiation pressure on grains and atoms. Shocks and turbulence may be very important in more massive stars.

66 58 Note that He burning stars are not anymore an homogeneous group M < 60 M < M ignite He as RSG - He burnt as WR stars M < 25M < M partially as RSG always RGS - partially as WR stars

67 59 Advanced burnings During advanced burning stages, neutrino luminosity dominates reduction of stellar lifetime C burning occurs between K 20 Ne, 23 Na, 24 Mg, 27 Al main reactions 12 C+ 12 C Ne burning occurs between K 16 O, 24 Mg, 28 Si main reactions 20 Ne(γ, α) and 20 Ne(α, γ) O burning occurs between K 28 Si ( 0.55), 32 S ( 0.24) main reactions 16 O+ 16 O

68 60 Si burning With increasing temperature, photodisintegration reactions (γ,n),(γ,p) and (γ, α) become important. Equilibrium reactions : 28 Si + γ 24 Mg + α Lighter nuclei are also photodissociated : 24 Mg(γ, α) 20 Ne(γ, α) 16 O(γ, α) 12 C(γ, 2α)α The released α particles react with the abundant 28 Si : 28 Si + α 32 S + γ; 32 S + α 36 Ar + γ Cr + α 56 Ni + γ; With increasing temperatures reactions approach the nuclear statistical equilibrium

61 Structure at the pre-sn stage Above 5 10 9 K, statistical equilibrium is reached and the abundances of all elements become independent of the

69 61 Structure at the pre-sn stage Above K, statistical equilibrium is reached and the abundances of all elements become independent of the reaction rates, they only depend on T, ρ and Y e = Z i X i i (equivalent of the Saha equation) so X A i i = f(ρ, Y, Y e (t)) Onion-like structure :

70 62 Basic core collapse scenario Result Fe core contracts as no nuclear fusion is occurring, and e become degenerate. When core mass > M Ch the e degeneracy pressure is less than self-gravity and core contracts rapidly (for Fe M Ch 1.26M ) Gas is highly degenerate, hence as core collapses T rises unconstrained, and reaches threshold for Fe photodisintegration 56 Fe 13 4 He + 4 n (-100 Mev) Reaction is highly endothermic collapse turns into almost free fall. Infall continues, T, and photons energetic enough to photodissociate He: 4 He 2n + 2p (-25 Mev) Core contracts further, density becomes high enough for e capture e + p n + ν e The neutron gas becomes degenerate at densities of g cm 3 neutron star created.

71 Core collapse scenario 63

Stellar Structure and Evolution

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