Massive Stars - Key Physics and Issues • Nuclear Physics • Evolution in HR diagram • Equation of state • Nucleosynthesis Lecture 7 • Opacity • Mass loss • Surface abundances Evolution of Massive • Convection • Presupernova structure • Rotation (magnetic fields) Stars on the Main Sequence • Binary membership • Supernova properties and During Helium Burning - Basics • Explosion physics • Remnant properties • Rotation and B-field of pulsars Massive Stars 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 ⎟ + Generalities: 2 ⎝ h ⎠ 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): Because of the general tendency of the interior temperature of main sequence stars to increase with mass, stars of over two P = const × ρ × T ∝ρ × ρ 2/3 = ρ 5/3 = ργ For non relativistic, but possibly partly degenerate 3/2 solar mass are chiefly powered by the CNO cycle(s) rather than T electrons, the electrons are since = constant the pp cycle(s). This, plus the increasing fraction of pressure due to ρ given as a separate term see Clayton 2-145. radiation, makes their cores convective. The opacity is dominantly n+1 3 γ = ⇒ n = due to electron scattering. Despite their convective cores, the n 2 overall main sequence structure can be crudely represented as Radiation dominated gas or a gas with constant : an n = 3 polytrope. This is especially true of the outer radiative 1 P = aT4 ∝ρ 4/3 part of the star that typically includes the majority of the mass. 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 Not surprisingly then, it turns out that massive stars main sequence that are more radiation - dominated 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. 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. continuing farther out 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 Most of the mass and volume. d ln ρ 1 = 40% of the mass 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 P P (though not necessarily dominant) and energy transport gas rad Define β = = 1- where P= Pgas +Prad , is by radiative diffusion P P dPrad d ⎛ 1 4 ⎞ 4 3 dT = ⎜ aT ⎟ = aT dr dr ⎝ 3 ⎠ 3 dr then Prad = P −Pgas =(1− β) P and dT 3κρ L(r) But for radiative diffusion, = so dPrad κL(r) L(r) dr 16πacT 3 r 2 = (1− β)= = dP 4πGmc L dP κρ L(r) Edd rad = − dr 4πc r 2 but hydrostatic equilibrium requires If, and it is a big IF, β (or 1-β) were a constant throughout dP Gmρ = − the star, then one could write everywhere, including the surface dr r 2 Divide the 2 eqns L(r) = 1 L dP κL(r) L(r) ( − β ) Ed (Main sequence only) rad = = dP 4πGmc LEdd 4πGMc where L = Ed κ is nearly constant 3/2 3/2 n +1 dθ ⎞ ⎛ K ⎞ 3−n Fom polytropes M ( ) 2 2n = − ξ1 ⎟ ⎜ ⎟ ρc 1− β = fraction of the pressure 4π dξ ⎠ ⎝ G⎠ (Clayton 155- 165) ξ1 from radiation 1/3 4 ⎡3(NAk) (1− β)⎤ K = ⎢ 4 ⎥ ⎣⎢ a(µβ) ⎦⎥ For n = 3 (β= constant), ρc drops out and this becomes 3/2 3/2 2.01824 Near the surface P 4 2 ⎛ dθ ⎞ ⎛ K ⎞ ⎛ K ⎞ gas M 4.56 the density declines β = = − ξ1 ⎜ ⎟ ⎜ ⎟ = ⎜ ⎟ P dξ G G precipitously making total π ⎝ ⎠ ξ ⎝ ⎠ ⎝ ⎠ 1 radiation pressure 1/2 ⎡3(N k)4(1− β)⎤ inner ~5 Msun is more important. M = 4.56 A lim M →0 convective ⎢ 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 ⎠⎟ −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 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) from Clayton p. 163 inner ~8 Msun convective 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⎟ ⎝ ⎠ ⎝ ⎠ For M not too far from M β is close to 1 and L ∝M3. L(r) = (1- β ) LEdd ⎛ aG3 ⎞ π 4πGc L 4 4M 2 M =⎜ 4 ⎟ 2 µ β At higher masses however the mass dependence of β 3(N k) 4 ⎝ A ⎠ 16 d / d κ 4 −2 ξ1 ( θ ξ) becomes important. Eventually M so that L M. ξ1 β ∝ ∝ 2 4 4 4 In fact, the luminosity of very massive stars approaches π ⎛ acG ⎞ ⎛ µ β ⎞ 3 = M the Eddington limit as 0 ( L(r) = 1 L ) 4 2 ⎜ 4 ⎟ ⎜ ⎟ β → ( − β ) Edd 12 d / d ⎝ (NAk) ⎠ ⎝ κ ⎠ ξ1 ( θ ξ) ξ1 4 3 2 −1 ⎛ ⎞ Mass luminosty 4πGMc 38 -1 ⎛ M ⎞ ⎛ 0.34⎞ 4 ⎛ µ ⎞ ⎛ 1 cm g ⎞ M L 1.47 10 erg s 5.5 L Edd = = × ⎜ ⎟ ⎜ ⎟ = β ⎜ ⎟ ⎜ ⎟ Relation κ M ⎝ κ ⎠ ⎜ 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 n = 3 one can also derive useful equations for the central conditions based upon the original polytropic equation for mass For the n=3 polytrope 2/3 3 2 dθ ⎞ 3 ⎛ M ⎞ M = − 4πα ρ ξ = 2.01824 (4πα ρ ) T = 4.6 × 106 K µβ ρ1/3 (in general for n = 3) c 1 d ⎟ c c ⎜ ⎟ c ξ ⎠ ξ ⎝ M ⎠ 1 1/2 1/2 ⎡P (n +1)⎤ ⎡ P ⎤ For stars on the main sequence and half way through and the definitions α = ⎢ c ⎥ = ⎢ c ⎥ 4 G 2 G 2 hydrogen burning, µ ≈ 0.84 and, unless the star is very massive, ⎣⎢ π ρc ⎦⎥ ⎣⎢π ρc ⎦⎥ β ≈ 0.8 - 0.9. Better values are given in Fig 2-19 of Clayton P ρ N kT ρ 4πR3 ρ and P = ideal = c A c and c = c = 54.18 replicated on the next page. c β µβ ρ 3M The density is not predicted from first principles since the actual 2 ⎛ M / M ⎞ 17 ( ) radius depends upon nuclear burning, but detailed main Pc =1.242×10 ⎜ 4 ⎟ ⎜ R / R ⎟ ⎛ 10 M ⎞ --3 ⎝ ( ) ⎠ sequence models (following page) give ρ ≈ 10 gm cm ,So c ⎝⎜ M ⎠⎟ ⎛ M / M ⎞ 6 ( ) 1/3 T =19.57×10 βµ ⎜ ⎟ K 7 ⎛ M ⎞ c R / R T ≈ 3.9 ×10 β K (main sequence only) ⎝ ( ) ⎠ c ⎜ ⎟ ⎝ 10M ⎠ 2/3 T 4.62 106 M / M 1/3 K c = × βµ ρc ( ) 7 37 M Tc/10 C L/10 Competition between the p-p 9 3.27 9.16 2.8 chain and the CNO Cycle 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 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 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) In a low mass star C(6) The slowest reaction is 14N(p,)15O. For temperatures near 2 x 107 K. 1/3 3 4 5 6 7 8 9 ⎛ 2 2 14 ⋅1 ⎞ 7 1 neutron number n τ -2 ⎜ 14 + 1 ⎟ ε nuc ∝T n = τ = 4.248 ⎜ ⎟ =60.0 3 ⎜ 0.020 ⎟ ⎝ ⎠ All initial abundances within a cycle serve as catalysts and accumulate at largest ! n =18 Extended cycles introduce outside material into CN cycle (Oxygen, …) (More on nucleosynthesis later) 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).