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Lecture 7 Evolution of Massive Stars on the Main Sequence and During Helium Burning Basics Key Physics and Issues • Nuclear Physics • Equation of state • Opacity • Mass loss • Convection • Rotation (magnetic fields) • Binary membership • Explosion physics • Evolution in HR diagram • Nucleosynthesis • Surface abundances • Presupernova structure • Supernova properties • Remnant properties • Rotation and B-field of pulsars Massive Stars Generalities: Because of the general tendency of the interior temperature of stars to increase with mass, stars of just over one solar mass are chiefly powered by the CNO cycle(s) rather than the pp cycle(s). This, plus the increasing fraction of pressure due to radiation, makes their cores convective. The opacity is dominantly due to electron scattering. Despite their convective cores, the overall main sequence structure can be crudely represented as an n = 3 polytrope. Following eq. 2- 286 through 2-289 in Clayton it is easy to show that for Pgas Ptotal N A kT 1 4 aT = (1- )Ptotal 3 Prad = that 1/ 3 Ptotal N k 4 3 1 = A a 4/3 That is, a star that has constant throughout its mass will be an n = 3 polytrope It is further not difficult to show that a sufficient condition for = constant in a radiative region is that L(r ) (r ) (1 - ) = constant M (r ) That is, if the energy generation per unit mass interior to r times the local opacity is a constant, will be a constant and the polytrope will have index 3. This comes from combining the equation for radiative equilibrium with the equation for hydrostatic equilibrium and the definition of On the other hand, convective regions (dS = 0), will have n between 1.5 and 3 depending upon whether ideal gas dominates the entropy or radiation does. Ideal gas (constant composition): P = const T 2/3 = 5/3 since = T3/2 constant n+1 3 n= n 2 Radiation dominated gas: P= since = T3 1 4 aT 4/3 3 constant n+1 n=3 n Half way through hydrogen burning For “normal” mass stars, ideal gas entropy always dominates on the main sequence, but for very massive stars Srad and Sideal can become comparable. So massive stars are typically a hybrid polytrope with their convective cores having 3 > n > 1.5 and radiative envelopes with n approximately 3. Overall n = 3 is not bad. Then 1- 2 2 M 18 M e = (1+Z i ) Yi Clayton 2-307 1 0.84 for 35% H, 65% He For very large M, M -1/2 ; for 18 M e 0.80 1 fraction of the pressure from radiation inner ~5 Msun is convective Near the surface the density declines precipitously making radiation pressure more important. L (1 ) decreases as M(r) M because L is centrally concentrated, so increases with M(r) inner ~8 Msun convective Convection plus entropy from ideal gas implies n = 1.5 d ln 1 d ln P 5 / 3 for ideal gas at constant entropy 40% of the mass Most of the mass and volume. d ln 1 d ln P 4 / 3 for standard model (with = const) in radiative regions For the n=3 polytrope 2/3 M 6 Tc 4.6 10 K 1/c 3 Me For stars on the main sequence and half way through hydrogen burning, 0.84 and, unless the star is very massive, 0.8 - 0.9. Better values are given in Fig 2-19 of Clayton. declines slowly with mass The density is not predicted from first principles, but detailed models 10 M e --3 (following page) give c 10 gm cm ,So M M 7 Tc 3.9 10 10 M e 1/ 3 K M All evaluated in actual models at a core H mass fraction of 0.30 for stars of solar metallicity. 0.8 9 12 15 20 25 40 60(57) 85(78) 120(99) Tc/107 3.27 3.45 3.58 3.74 3.85 4.07 4.24 4.35 4.45 9.16 6.84 5.58 4.40 3.73 2.72 2.17 1.85 1.61 C L/1037 2.8 7.0 13 29 50 140 290 510 810 L M2.5 c decreases with mass as a general consequence of the fact that Tc3 c M 2 3 3 and H burning happens at a relatively constant temperature. Until about 40 M e , the density decreases roughly as M -1 . After that it decreases more slowly. Recall M -1/2 for very large masses Homology n ~ 18 const if eleectron scattering From homology relations on the previous page and taking = constant (electron scattering), an ideal gas equation of state, and n = 18, one obtains: L Tc 4M 3 o M 1/ 3 M 4 / 21 4 / 21 M if composition constant 1/ 21 ( o o ) R where was defined in lecture 1. If radiation pressure dominates, as it begins to for very large values of mass, 1/ 42 L M o T M ( o o )1/ 21 hence M L Overall L ~ M2.5 M 2 (lower end) to a constant (high end) M 100M Competition between the p-p chain and the CNO Cycle The Primary CNO Cycle In a low mass star The slowest reaction is 14N(p,)15O. For temperatures near 2 x 107 K. nuc T n n= -2 3 1/ 3 22 7 1 14 1 4.248 0.020 14 1 60.0 n =18 (More on nucleosynthesis later) CNO tri-cycle CN cycle (99.9%) O Extension 1 (0.1%) Ne(10) F(9) O Extension 2 O Extension 3 O(8) N(7) C(6) 3 4 5 6 7 8 9 neutron number All initial abundances within a cycle serve as catalysts and accumulate at largest Extended cycles introduce outside material into CN cycle (Oxygen, …) Mainly of interest for nucleosynthesis In general, the rates for these reactions proceed through known resonances whose properties are all reasonably well known. There was a major revision of the rate for 14 N(p, )15O in 2001 by Bertone et al., Phys. Rev. Lettr., 87, 152501. The new rate is about half as large as the old one, so the main sequence lifetime of massive stars is longer (but definitely not linear in the reciprocal rate). Mainly affected globular cluster ages (0.7 to 1 Gy increase in lifetime due to the importance of the CNO cycle at the end of the MS life and during thick H shell burning). Equation of state Well defined if tedious to calculate up to the point of iron core collapse. Ions - ideal gas - P = NA k T 1 4 Radiation P = aT 3 Electrons - the hard part - can have arbitrary relativity and degeneracy (solve Fermi integrals). At high T must include electron-positron pairs. Beyond 1011 g cm -3 - neutrino trapping, nuclear force, nuclear excited states, complex composition, etc. Opacity In the interior on the main sequence and within the helium core for later burning stages, electron capture dominates. In its simplest form: e ne Th N A Ye Th Ye(N A Th ) 8 e2 Th 3 me c 2 2 e 0.40 Ye cm 2 gm -1 Recall that for 75% H, 25% He, Ye 0.875, so e 0.35 For He and heavier elements e 0.20. There are correction terms that must be applied to es especially at high temperature and density 1) The electron-scattering cross section and Thomson cross section differ at high energy. The actual cross section is smaller. Klein-Nishina KN 2h 26 h 2 Th 1 ... 2 2 me c 5 me c h 1020 Hz 2) Degeneracy – at high density the phase space for the scattered electron is less. This decreases the scattering cross section. 3) Incomplete ionization – especially as the star explodes as a supernova. Use the Saha equation. 4) Electron positron pairs may increase at high temperature. Effects 1) and 2) are discussed by Chin, ApJ, 142, 1481 (1965) Flowers & Itoh, ApJ, 206, 218, (1976) Buchler and Yueh, ApJ, 210, 440, (1976) Itoh et al, ApJ, 382, 636 (1991) and references therein Electron conduction is not very important in massive stars but is important in white dwarfs and therefore the precursors to Type Ia supernovae Itoh et al, ApJ, 285, 758, (1984) and references therein For radiative opacities other than es, in particular bf and bb, Iglesias and Rogers, ApJ, 464, 943 (1996) Rogers, Swenson, and Iglesias, ApJ, 456, 902 (1996) see Clayton p 186 for a definition of terms. “f” means a continuum state is involved Note centrally concentrated nuclear energy generation. convective During hydrogen burning Convection All stellar evolution calculations to date, except for brief snapshots, have been done in one-dimensional codes. In these convection is universally represented using some variation of mixing length theory. Caveats and concerns: • The treatment must be time dependent • Convective overshoot and undershoot • Semiconvection (next lecture) • Convection in parallel with other mixing processes, especially rotation • Convection in situations where evolutionary time scales are not very much longer than the convective turnover time. Kuhlen, Woosley, and Glatzmaier exploried the physics of stellar convection using 3D anelastic hydrodynamics. QuickTime™ and a YUV420 codec decompressor are needed to see this picture. QuickTime™ and a YUV420 codec decompressor are needed to see this picture. The model shown is a 15 solar mass star half way through hydrogen burning. For now the models are not rotating, but the code includes rotation and B-fields. (Previously used to simulate the Earth’s dynamo). QuickTime™ and a YUV420 codec decompressor are needed to see this picture. Convective structure Note growth of the convective core with M from Kippenhahn and Wiegert The adiabatic condition can be written in terms of the temperature as dT dP 2 0 P 1 2 T This defines 2 (see Clayton p 118) For an ideal gas 2 5 / 3, but if radiation is included the expression is more complicated Convective instability is favored by a large fraction of radiation pressure, i.e., a small value of 1 T dP dT 1 convection dr star 2 P dr 32 24 3 2 4 5 2 (Clayton 2-129) 2 2 24 18 3 3 3 1 1 For =1, 1- 0.4; for = 0, 1- 0.25 2 2 1 for = 0.8 1- = 0.294 2 So even a 20% decrease in causes a substantial decrease in the critical temperature gradient necessary for convection. decreases with increasing mass. Hence more massive stars are convective to a greater extent. (1 )1/ 2 M 18.0 Me 2 2 M M 9 12 15 20 25 40 60(57) 85(78) 120(99) All evaluated at a core H mass fraction of 0.30 for stars of solar metallicity. Tc/107 3.27 3.45 3.58 3.74 3.85 4.07 4.24 4.35 4.45 9.16 6.84 5.58 4.40 3.73 2.72 2.17 1.85 1.61 L/1037 C 2.8 7.0 13 29 50 140 290 510 810 Q conv core 0.26 0.30 0.34 0.39 0.43 0.53 0.60 0.66 0.75 Several things to note: a) Increasing size of convective core 12 - 15 M b) d ln Tc d ln M 0.17 20 - 25 M 0.13 60 - 85 M 0.10 4 1 0.19 ( =1) to 0.048 ( =0) 21 21 c) Polytropic index varies from >1.5 to 3.0 as M increases. n = 3 not bad overall Homology gives d) C M 2 3 TC 3 constant, though n does vary significantly; in fact, c M Tc 3 constant First 3 Burning Stages in the Life of a Massive Star 0 blue = energy generation purple = energy loss green = convection Surface convection zone H-burn He-burn The convective core shrinks during hydrogen burning During hydrogen burning the mean atomic weight is increasing from near 1 to about 4. The ideal gas entropy is thus decreasing. As the central entropy decreases compared with the outer layers of the star it is increasingly difficult to convect through most of the star’s mass. For an ideal gas plus radiation: (see Clayton p. 123) 1 4 for pure hydrogen; for pure helium 2 3 change in entropy during He burning is small The convective core grows during helium burning. During helium burning, the convective core grows largely because the mass of the helium core itself grows. This has two effects: a) As the mass of the core grows so does its luminosity, while the radius of the convective core stays nearly the same (density goes up). For a 15 solar mass star: He mass fraction Radius conv core Lum conv core Lum star 3.0 M 1 0.87 x 1010 cm 3.2 x 1037 erg s-1 2.16 x 1038 erg s-1 3.6 M 0.5 1.04 x 1010 cm 6.8 x 1037 erg s-1 2.44 x 1038 erg s-1 The rest of the luminosity is coming from the H shell.. blue = energy generation purple = energy loss green = convection Surface convection zone H-burn He-burn b) As the mass of the helium core rises its decreases. T3 M 2 3 1/ 3aT 4 1 N A kT M 2 3 This decrease in favors convection. The entropy during helium burning also continues to decrease, and this would have a tendency to diminish convection, but the and L effects dominate and the helium burning convective core grows until near the end when it shrinks both due to the decreasing central energy generation. This growth of the helium core can have two interesting consequences: • Addition of helium to the helium convection zone at late time increases the O/C ratio made by helium burning • In very massive stars with low metallicity the helium convective core can grow so much that it encroaches on the hydrogen shell with major consequences for stellar structure and nucleosynthesis. Convective Overshoot Mixing Initially the entropy is nearly flat in a zero age main sequence star so just where convection stops is a bit ambiguous. As burning proceeds though and the entropy decreases in the center, the convective extent becomes more precisely defined. Still one expects some overshoot mixing. A widely adopted prescription is to continue arbitrarily the convective mixing beyond its mathematical boundary by some fraction, a, of the pressure scale height. Maeder uses 20%. Stothers and Chin (ApJ, 381, L67), based on the width of the main sequence, argue that a is less than about 20%. Doom, Chiosi, and many European groups use larger values. Woosley and Weaver use much less. Nomoto uses none. Overshoot mixing: Overshoot mixing has many effects. Among them: • Larger helium cores • Higher luminosities after leaving the main sequence • Broader main sequence • Longer lifetimes • Decrease of critical mass for non-degenerate C-ignition. Values as low as 5 solar masses have been suggested but are considered unrealistic. Overshoot mixing: DeMarque et al, ApJ, 426, 165, (1994) – modeling main sequence widths in clusters suggests a = 0.23 Woo and Demarque, AJ, 122, 1602 (2001) – empirically for low mass stars, overshoot is < 15% of the core radius. Core radius a better discriminant than pressure scale height. Brumme, Clune, and Toomre, ApJ, 570, 825, (2002) – numerical 3D simulations. Overshoot may go a significant fraction of a pressure scale height, but does not quickly establish an adiabatic gradient in the region. Differential rotation complicates things and may have some of the same effects as overshoot. See also http:/www.lcse.umn.edu/MOVIES METALLICITY Metallicity affects the evolution in four distinct ways: • Mass loss • Energy generation (by CNO cycle) • Opacity • Initial H/He abundance lower main sequence: decreases if Z decreases L Tc o 16 /13 o1/13 o o 2 /15 For example, 1 M at half hydrogen depletion Z = 0.02 Z = 0.001 log Tc 7.202 7.238 L L 2.0 L Because of the higher luminosity, the lifetime of the lower metallicity star is shorter (it burns about the same fraction of its mass). Upper main sequence: The luminosities and ages are very nearly the same because the opacity is, to first order, independent of the metallicity. The central temperature is a little higher at low metallicity because of the decreased abundance of 14N to catalyze the CNO cycle. For example in a 20 solar mass star at XH = 0.3 log Tc Z 0.02 7.573 Z 0.001 7.647 log L / L 4.867 4.872 Qcc 0.390 0.373 M /M 19.60 19.92 (mass loss) Schaller et al. (1992) M d ln MS d ln M M d ln MS d ln M 1-2 -2.9 2-4 4-9 -2.6 -2.0 12 - 15 20 - 25 85 - 120 -1.4 -1.2 -0.3 There is a slight difference in the lifetime on the upper main sequence though because of the different initial helium abundances. Schaller et al. used Z = 0.001, Y = 0.243, X=0.756 and Z = 0.02, Y = 0.30, X = 0.68. So for the higher metallicity there is less hydrogen to burn. But there is also an opposing effect, namely mass loss. For higher metallicity the mass loss is greater and the star has a lower effective mass and lives longer. Both effects are small unless the mass is very large. For helium burning, there is no effect around 10 solar mases, but the higher masses have a longer lifetime with higher metallicity because mass loss decreases the mass. For lower masses, there is a significant metallicity dependence for the helium burning lifetime. The reason is not clear. Perhaps the more active H-burning shell in the solar metallicity case reduces the pressure on the helium core. For 2 solar masses half way through helium burning log Tc Z = 0.001 Z = 0.02 8.089 8.074 a small difference but the helium burning rate goes as as a high power of T at these temperatures. The above numbers are more than enough to explain the difference in lifetime. Zero and low metallicity stars may end their lives as compact blue giants – depending upon semiconvection and rotationally induced mixing For example, Z = 0, presupernova, full semiconvection a) 20 solar masses R = 7.8 x 1011 cm Teff = 41,000 K b) 25 solar masses R=1.07 x 1012cm Teff = 35,000 K Z = 0.0001 ZO a) 25 solar masses, little semiconvection R = 2.9 x 1012 cm Teff = 20,000 K b) 25 solar masses, full semiconvection R = 5.2 x 1013 cm Teff= 4800 K Caveat: Primary 14N production Caution - rotationally induced mixing changes these results Very massive stars As radiation pressure becomes an increasingly dominant part of the pressure, decreases in very massive stars. This implies that the luminosity approaches Eddington. E. g. in a 100 solar mass star L = 8.1 x 1039 erg s-1, or LEdd/1.8. But still is 0.55. In regions where the star is supported entirely by radiation pressure and the radiation is diffusing the luminosity is the Eddington luminosity. The stellar structure equation for the radiative temperature gradient is dT 3 L(r) dr 4ac T 3 4 r 2 Massive stars are always radiative near their surfaces, implies even though they can become c 2 d 1 4 L(r) 4 r aT convective throughout most dr 3 of their mass. if the pressure is given by Prad = (1- ) Ptot where = Pideal /Ptot , then 4 r 2 c d L(r) (1 ) Ptot dr which if is a constant gives (with hydrostatic equilibrium) GM (r) 4 GMc (1 ) . 2 r If =0, this is the Eddington luminosity L(r) 4 r 2 c (1 ) This suggests that massive stars, as approaches 0, will approach the Eddington luminosity with L proportional to M. In fact, except for a thin region near their surfaces, such stars will be entirely convective and will have a total binding energy that approaches zero as approaches zero. But the calculation applies to those surface layers which must stay bound. Completely convective stars with a luminosity proportional to mass have a constant lifetime, which is in fact the shortest lifetime a (main sequence) star can have. LEdd M 0.34 1.47 10 erg s M qnuc 4.8 1018 erg/g 4 GMc 38 -1 MS qnuc M / LEdd 2.1 million years (exception supermassive stars over 105 solar masses – post-Newtonian gravity renders unstable on the main sequence) Similarly there is a lower bound for helium burning. The argument is the same except one uses the q-value for helium burning to carbon and oxygen. One gets 7.3 x 1017 erg g-1 from burning 100% He to 50% each C and O. Thus the minimum (Eddington) lifetime for helium burning is about 300,000 years. Limit Limit Since ~ 4/3, such stars are loosely bound (total energy much less than gravitational or internal energy) and are subject to large amplitude pulsations. These can be driven by either opacity instabilities (the mechanism) or nuclear burning instabilities (the mechanism). is less than 0.5 for such stars on the main sequence, but ideal gas entropy still dominates. For solar metallicity it has long been recognized that such stars (say over 100 solar masses) would pulse violently on the main sequence and probably lose most of their mass before dying. Ledoux, ApJ, 94, 537, (1941) Schwarzschild & Harm, ApJ, 129, 637, (1959) Appenzeller, A&A, 5, 355, (1970) Appenzeller, A&A, 9, 216, (1970) Talbot, ApJ, 163, 17, (1971) Talbot, ApJ, 165, 121, (1971) Papaloizou, MNRAS, 162, 143, (1973) Papaloizou, MNRAS, 162, 169, (1973) Mass lost ~35 solar masses 100 M e (main sequence mass) star when the central hydrogen mass fraction hits 0.01. The luminosity of the star is 6.18 10 39 erg s-1 (about 0.1% as bright as a Type II supernova) L 100Me at hydrogen depletion Eddington luminosity for 65 solar masses LEdd 4 GMc 0.34 9.6 1039 erg s-1 So about model is ~2/3 Eddington luminosity for electron scattering opacity near the surface. As the star contracts and ignites helium burning its luminosity rises to 8 x 1039 erg s-1 and the mass continues to decrease by mass loss. Super-Eddington mass ejection? L excess R -1 M: : 0.001M e y 1 L 38 R13 M100 GM L excess depends on which is at least as large as electron scattering. Luminous blue variable stars? Structural adiabatic exponent is nearly 4/3 Radial pulsations and an upper limit 1941, ApJ, 94, 537 (1941) Also see Eddington (1927, MNRAS, 87, 539) Upper mass limit: theoretical predictions Stothers & Simon (1970) Upper mass limit: theoretical predictions Ledoux (1941) radial pulsation, e- opacity, H 100 M Schwarzchild & Härm (1959) radial pulsation, e- opacity, H and He, evolution 65-95 M Stothers & Simon (1970) radial pulsation, e- and atomic Larson & Starrfield (1971) pressure in HII region 50-60 M Cox & Tabor (1976) e- and atomic opacity Los Alamos 80-100 M Klapp et al. (1987) e- and atomic opacity Los Alamos 440 M Stothers (1992) e- and atomic opacity Rogers-Iglesias 120-150 M 80-120 M Upper mass limit: observation R136 Feitzinger et al. (1980) 250-1000 M Eta Car various 120-150 M R136a1 Massey & Hunter (1998) 136-155 M Pistol Star Figer et al. (1998) 140-180 M Damineli et al. (2000) ~70+? M LBV 1806-20 Eikenberry et al. (2004) 150-1000 M LBV 1806-20 Figer et al. (2004) 130 (binary?) M HDE 269810 Walborn et al. (2004) 150 M WR20a Bonanos et al. (2004) Rauw et al. (2004) 82+83 M Eta Car R122 in LMC Calculations suggested that strong non-linear pulsations would grow, steepening into shock in the outer layers and driving copious mass loss until the star became low enough in mass that the instability would be relieved. But what about at low metallicity? Ezer and Cameron, Ap&SS, 14, 399 (1971) pointed out that Z = 0 stars would not burn by the pp-cycle but by a high temperature CNO cycle using catalysts produced in the star itself, Z ~ 10-9 to 10-7. Maeder, A&A, 92, 101, (1980) suggested that low metallicity might raise Mupper to 200 solar masses. Surprisingly though the first stability analysis was not performed for massive Pop III stars until Baraffe, Heger, and Woosley, ApJ, 550, 890, (2001). Baraffe et al found that a) above Z ~ 10-4 solar, the instability dominated and below that it was negligible and b) that the -instability was also suppressed for metallicities as low as 10-7 solar. Reasons No heavy elements to make lines and grains High temperature of H-burning made the reactions less temperature sensitive. They concluded that stars up to about 500 solar masses would live their (main sequence) lives without much mass loss. In the same time frame, several studies suggested that the IMF for Pop III may have been significantly skewed to heavier masses. Abel, Bryan, and Norman, ApJ, 540, 39, (2000) Larson, astroph 9912539 Nakamura and Umemura, ApJ, 569, 549,(2002) So - very massive stars might have formed quite early in the universe, might have lived their lives in only a few million years and might have died while still in possession of nearly their initial mass. Or given the discussion of the Eddington limit, they might not As we shall see later the supernovae resulting from such large stars and their nucleosynthesis is special. These stars may also play an important role in reionizing the universe (even if only 0.01% of the matter forms into such stars). 8 Mev/nucleon 13eV/nucleon 106 There are really three important distinct issues: What masses were the early stars born with, what were there typical masses on the main sequence, and with what mass did they die? CAVEAT!