Lecture 7 Evolution of Massive Stars on the Main Sequence and During
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Lecture 7 Evolution of Massive Stars on the Main Sequence and During Helium Burning - Basics Massive Stars - Key Physics and Issues • Nuclear Physics • Evolution in HR diagram • Equation of state • Nucleosynthesis • Opacity • Mass loss • Surface abundances • Convection • Presupernova structure • Rotation (magnetic fields) • Binary membership • Supernova properties • Explosion physics • Remnant properties • Rotation and B-field of pulsars Massive Stars Generalities: Because of the general tendency of the interior temperature of main sequence stars to increase with mass, stars of over two 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. This is especially true of the outer radiative part of the star that typically includes the majority of the mass. For a non-degenerate gas, the entropy is given by (Clayton 2-136) For an ideal gas 3 ⎛ 2πmk ⎞ 5 st S = ln + integrate the 1 0 2 ⎝⎜ h2 ⎠⎟ 2 law of thermo- ( Reif - Statistical Physics - 7.3.6) dynamics T dS = dU +P dV The electrons are included in µ and in S0 Ideal gas (convective with negligible radiation entropy): P = const × ρ × T ∝ρ × ρ 2/3 = ρ 5/3 = ργ For non relativistic, but possibly partly degenerate 3/2 T electrons, the electrons are since = constant ρ given as a separate term see Clayton 2-145. n+1 3 γ = ⇒ n = n 2 Radiation dominated gas or a gas with constant : 1 P = aT4 ∝ρ 4/3 3 3 T P 1− β Pgas Pgas if ∝ rad = = constant β = = P P +P ρ Pideal β tot gas rad n+1 γ = ⇒ n = 3 n (S/NAk) half way through hydrogen burning 15 and 100 solar masses More massive stars have larger entropies on the main sequence that are more radiation - dominated For normal massive stars, the ionic entropy always dominates on the main sequence, but for very massive stars Selec, Srad and Sionic can become comparable. Not surprisingly then, it turns out that massive stars are typically hybrid polytropes with their convective cores having 3 > n > 1.5 and radiative envelopes with n approximately 3. Overall n = 3 is not bad. Convection plus entropy from ideal gas implies n = 1.5 d ln ρ 1 = d ln P γ γ ≈ 5 / 3 for ideal gas at constant entropy 40% of the mass continuing farther out Most of the mass and volume. d ln ρ 1 = d ln P γ γ ≈ 4 / 3 for standard model (with β = const) in radiative regions Eddington’s standard model (n=3) Consider a star in which radiation pressure is important (though not necessarily dominant) and energy transport is by radiative diffusion dP d ⎛ 1 ⎞ 4 dT rad = aT 4 = aT 3 dr dr ⎝⎜ 3 ⎠⎟ 3 dr dT 3κρ L(r) But for radiative diffusion, = so dr 16πacT 3 r 2 dP κρ L(r) rad = − dr 4πc r 2 but hydrostatic equilibrium requires dP Gmρ = − dr r 2 Divide the 2 eqns dP κL(r) L(r) rad = = dP 4πGmc LEdd 4πGMc where L = Ed κ P P Define β = gas = 1- rad where P= P +P , P P gas rad then Prad = P −Pgas =(1− β) P and dP κL(r) L(r) rad = (1− β)= = dP 4πGmc LEdd If, and it is a big IF, β (or 1-β) were a constant throughout the star, then one could write everywhere, including the surface L(r) = 1− β L (Main sequence only) ( ) Ed 3/2 3/2 n +1 dθ ⎞ ⎛ K ⎞ 3−n Fom polytropes M ( ) 2 2n = − ξ1 ⎟ ⎜ ⎟ ρc 4 dξ ⎠ ⎝ G⎠ (Clayton 155- 165) π ξ1 1/3 4 ⎡3(NAk) (1− β)⎤ K = ⎢ 4 ⎥ ⎣⎢ a(µβ) ⎦⎥ For n = 3 (β= constant), ρc drops out and this becomes 2.01824 3/2 3/2 P 4 ⎛ dθ ⎞ ⎛ K ⎞ ⎛ K ⎞ gas M 2 4.56 β = = − ξ1 ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ Ptotal ⎝ dξ ⎠ ⎝ G⎠ ⎝ G⎠ π ξ1 1/2 ⎡3(N k)4(1− β)⎤ M = 4.56 A lim M →0 ⎢ 4 3 ⎥ 1 ⎣⎢ a(µβ) G ⎦⎥ β→ 1/2 lim M →∞ Eddington’s quartic 18.1M ⎛ 1− β ⎞ β→0 M equation = 2 4 µ ⎝⎜ β ⎠⎟ is nearly constant 1− β = fraction of the pressure from radiation Near the surface the density declines precipitously making radiation pressure inner ~5 Msun is more important. convective κ L ∝(1− β) decreases as M(r) ↑ M because L is centrally concentrated, so β within a given star increases with M(r) (for radiative regions) inner ~8 Msun convective −1 ⎡ ⎤ µ =⎣∑(1+Zi ) Yi ⎦ = 0.73 for 50% H, 50% He 0.64 for 75% H, 25% He For 20 M β ≈0.80− 0.85 µ2 M ≈11 from Clayton p. 163 1/2 18.1M ⎛ 1− β ⎞ M = µ2 ⎝⎜ β 4 ⎠⎟ 2 4 ⎛ M ⎞ ⎛ µ ⎞ 1− β = 4.13×10−4 β 4 and since ⎜ M ⎟ ⎜ 0.61⎟ ⎝ ⎠ ⎝ ⎠ L(r) = (1- β ) LEdd ⎛ aG3 ⎞ π 4πGc L = µ4 β 4M 2 M ⎜ 4 ⎟ 4 2 ⎝ 3(NAk) ⎠ 16 d / d κ ξ1 ( θ ξ) ξ1 π 2 ⎛ acG4 ⎞ ⎛ µ4β 4 ⎞ = M 3 4 2 ⎜ 4 ⎟ ⎜ ⎟ 12 d / d ⎝ (NAk) ⎠ ⎝ κ ⎠ ξ1 ( θ ξ) ξ1 4 3 2 −1 Mass luminosty 4 ⎛ µ ⎞ ⎛ 1 cm g ⎞ ⎛ M ⎞ = 5.5β L Relation ⎜ 0.61⎟ ⎜ ⎟ ⎜ M ⎟ ⎝ ⎠ ⎝ κ surf ⎠ ⎝ ⎠ where κ surf is the value of the opacity near the surface. This was obtained with no mention of nuclear reactions. For M not too far from M is close to 1 and L M3. β ∝ At higher masses however the mass dependence of β becomes important. Eventually β 4 ∝ M −2 so that L∝M. In fact, the luminosity of very massive stars approaches the Eddington limit as β → 0 ( L(r) = 1− β L ) ( ) Edd 4πGMc ⎛ M ⎞ ⎛ 0.34⎞ L 1.47 1038 erg s-1 Edd = = × ⎜ ⎟ ⎜ ⎟ κ ⎝ M ⎠ ⎝ κ ⎠ For n = 3 one can also derive useful equations for the central conditions based upon the original polytropic equation for mass dθ ⎞ M = − 4πα 3 ρ ξ 2 = 2.01824 (4πα 3 ρ ) c 1 dξ ⎠⎟ c ξ1 1/2 1/2 ⎡P (n +1)⎤ ⎡ P ⎤ and the definitions α = ⎢ c ⎥ = ⎢ c ⎥ 4 G 2 G 2 ⎣⎢ π ρc ⎦⎥ ⎣⎢π ρc ⎦⎥ P ρ N kT ρ 4πR3 ρ and P = ideal = c A c and c = c = 54.18 c β µβ ρ 3M 2 ⎛ M / M ⎞ 17 ( ) Pc =1.242×10 ⎜ 4 ⎟ ⎜ R / R ⎟ ⎝ ( ) ⎠ ⎛ M / M ⎞ 6 ( ) T =19.57×10 βµ ⎜ ⎟ K c R / R ⎝ ( ) ⎠ 2/3 T = 4.62×106 βµ M / M ρ1/3 K c ( ) c For the n=3 polytrope 2/3 6 ⎛ M ⎞ 1/3 Tc = 4.6 × 10 K µβ ρc (in general for n = 3) ⎝⎜ M ⎠⎟ For stars on the main sequence and half way through hydrogen burning, µ ≈ 0.84 and, unless the star is very massive, β ≈ 0.8 - 0.9. Better values are given in Fig 2-19 of Clayton replicated on the next page. The density is not predicted from first principles since the actual radius depends upon nuclear burning, but detailed main ⎛ 10 M ⎞ --3 sequence models (following page) give ρ ≈ 10 gm cm ,So c ⎝⎜ M ⎠⎟ 1/3 7 ⎛ M ⎞ Tc ≈ 3.9 ×10 β K (main sequence only) ⎝⎜ 10M ⎠⎟ 7 37 M Tc/10 C L/10 9 3.27 9.16 2.8 12 3.45 6.84 7.0 All evaluated in 15 3.58 5.58 13 L M2.5 actual models at 20 3.74 4.40 29 ∝ a core H mass 25 3.85 3.73 50 fraction of 0.30 40 4.07 2.72 140 for stars of solar 60(57) 4.24 2.17 290 metallicity. 85(78) 4.35 1.85 510 L ~ (1− β) L µ ≈0.8 120(99) 4.45 1.61 810 Ed ρc decreases with mass as a general consequence of the fact that T3 c ∝ M 2β 3µ 3 and H burning happens at a relatively constant ρc temperature. Until about 40 M, the density decreases roughly as M-1. After that it decreases more slowly. Recall β ∝ M-1/2 for very large masses Competition between the p-p chain and the CNO Cycle The temperature dependence of the CNO cycle is given by the sensitivity of the proton capture rate of 14N. See previous lectures The Primary CNO Cycle In a low mass star The slowest reaction is 14N(p,)15O. For temperatures near 2 x 107 K. 1/3 14 1 ⎛ 7212 ⋅ ⎞ n τ -2 ⎜ 14 + 1 ⎟ ε nuc ∝T n = τ = 4.248 ⎜ ⎟ =60.0 3 ⎜ 0.020 ⎟ ⎝ ⎠ n =18 (More on nucleosynthesis later) CNO tri-cycle CN cycle (99.9%) O Extension 1 (0.1%) Ne(10) O Extension 2 F(9) 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 t Extended cycles introduce outside material into CN cycle (Oxygen, …) The extra loops are mainly of interest for nucleosynthesis and for bringing 16O into the cycle 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 14N(p,γ )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).