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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 Massive Stars Generalities: Because of the general tendency of the interior temperature of main sequence stars to increase with mass*, stars of over two Lecture 7 solar mass are chiefly powered by the CNO cycle(s) rather than the pp cycle(s). The high temperature sensitivity of the CNO cycle (n = 17 instead of 4 for pp-cycle) makes the energy Evolution of Massive generation very centrally concentrated. This, plus the increasing Stars on the Main Sequence fraction of pressure due to radiation, makes their cores convective. Because of the greater temperature, the opacity in and During Helium Burning - their interiors is dominantly due to electron scattering. Basics 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. * To provide a luminosity that increases as M3 s15 3233 2.19951015502790E+14 c12( 1)= 7.5000E+10 R = 4.3561E+11 Teff = 2.9729E+04 L = 1.0560E+38 Iter = 37 Zb = 61 inv = 66 Dc = 5.9847E+00 Tc = 3.5466E+07 Ln = 6.9735E+36 Jm = 1048 Etot = -9.741E+49 B star 10,000 – 30,000 K 1 HHHHH 15 Solar mass He He HeH Convective history HH He He He He He .1 15 M half way ⊙ through hydrogen burning Elemental Mass Fraction .01 N N N O OOOO N C C N C Fe Fe Fe Fe Fe Fe Fe Fe IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIINeIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII NeIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII!,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Ne,,I,,,,,,,,,,Ii,,,,,,,,,,,,,,,,,,,,,,,,,I,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Ne,,,,,,,,,,,,............................................................................................................ NeC Ne..................................... Ne.......................................................... Ne.............................................. .001 0 2 4 6 8 10 12 14 Interior Mass (solar masses) The convective core (30% of the mass) resembles more an n=1.5 polytrope as expected for constant entropy for an ideal gas with P proportional to ρ5/3 (see appendix). Overall though γ = 4/3 is not bad blue 15 M H-burning 4/3 ⊙ P =const × ρ 15 M H-burning X = 0.50 ⊙ X = 0.50 Constant entropy and ideal gas T 3/2 Edge of implies is a constant hence ρ convective P ∝ρ5/3 core red 4.5 solar 5/3 P =const × ρ masses d ln ρ 1 = d ln P γ γ ≈ 4 / 3 for standard model γ (with β = const) in radiative P ∝ ρ n +1 regions γ = n = polytropic index n β is nearly constant outside the convective core If µ and β were constant throughout the star, this would imply that the star was an n=3 polytrope 1− β = fraction of the pressure from radiation Pgas Pgas β ≡ = ⇒ β P = P-Prad P Prad + Pgas 4 1/4 Prad aT ⎡3P(1− β)⎤ P= = ⇒T = ⎢ ⎥ (1− β) 3(1− β) ⎣ a ⎦ | 1/4 Pgas N k N k ⎡3P(1− β)⎤ Near the surface P = = A ρT ⇒ P = A ρ ⎢ a ⎥ the density declines β µβ µβ ⎣ ⎦ precipitously making 1/4 N k 3(1 ) radiation pressure 3/4 A ⎡ − β ⎤ inner ~4.5 Msun is hence P = ρ ⎢ ⎥ and more important. µβ ⎣ a ⎦ convective If and are constants throughout 1/3 µ β 4 the star this is an n = 3 polytrope ⎡3(N k) (1− β)⎤ 4/3 P = ⎢ A ⎥ ρ a(µβ)4 See also Clayton eq. 6-7 ⎣⎢ ⎦⎥ β will be constant if κ L(r) /M(r) is everywhere constant in radiative equilibrium. 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 inner ~8 Msun dr 16πacT 3 r 2 convective 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 EDDINGTON’S QUARTIC EQUATION 3/2 3/2 n +1 dθ ⎞ ⎛ K ⎞ 3−n Fom polytropes M ( ) 2 2n = − ξ1 ⎟ ⎜ ⎟ ρc 4π dξ ⎠ ⎝ G⎠ (Clayton 155- 165) ξ1 1/3 Since P = P −P =(1− β) P and 4 rad gas ⎡3(N k) (1 )⎤ A − β Pgas −1 K β = ⎡ (Z 1) X / A ⎤ = ⎢ 4 ⎥ µ = ⎣∑ i + i i ⎦ dP κL(r) L(r) Ptotal rad = (1− β)= = ⎣⎢ a(µβ) ⎦⎥ dP 4πGmc LEdd For n = 3 (β= constant), ρc drops out and this becomes 2.01824 3/2 3/2 4 ⎛ dθ ⎞ ⎛ K ⎞ ⎛ K ⎞ If, and it is a big IF, β (or 1-β) were a constant throughout M = − ξ 2 = 4.56 1 ⎝⎜ dξ ⎠⎟ ⎝⎜ G⎠⎟ ⎝⎜ G⎠⎟ the star, then one could write everywhere, including the surface π ξ1 1/2 ⎡3(N k)4(1− β)⎤ M 4.56 A L(r) = 1− β L = ⎢ 4 3 ⎥ ( ) Ed ⎣⎢ a(µβ) G ⎦⎥ 1/2 lim M →0 18.1M ⎛ 1− β ⎞ β→1 M = ⊙ µ2 ⎝⎜ β 4 ⎠⎟ lim M →∞ Eddington’s quartic β→0 equation 18.1M 1/2 ⎛ 1− β ⎞ −1 M = µ =⎡ (1+Z ) Y ⎤ = 0.73 for 50% H, 50% He µ2 ⎝⎜ β 4 ⎠⎟ ⎣∑ i i ⎦ 0.64 for 75% H, 25% He 2 4 ⎛ M ⎞ ⎛ µ ⎞ 1− β = 5.12×10−4 β 4 and since 2 ⎜ M ⎟ ⎜ 0.64⎟ For 20 M⊙ β ≈0.85 µ M ≈11 ⎝ ⊙ ⎠ ⎝ ⎠ 4π GM c L(r) = (1- β ) L = 1− β Edd ( ) κ For low M, (1− β )∝ M2, and L ∝ M3 from Clayton p. 163 For high M, β ≪1 and L → LEd ∝M This was obtained with no mention of nuclear reactions. 1/2 18.1M ⎛ 1− β ⎞ For n = 3 one can also derive useful equations for the central M = 2 ⎜ 4 ⎟ temperature based upon the original polytropic equation for µ ⎝ β ⎠ mass dθ ⎞ 3 M 4 3 2 2.01824 (4 3 ) For M not too far from M β is close to 1 and L ∝M . = − πα ρc ξ1 = πα ρc dξ ⎠⎟ ξ1 1/2 1/2 ⎡P (n +1)⎤ ⎡ P ⎤ At higher masses however the mass dependence of β and the definitions α = ⎢ c ⎥ = ⎢ c ⎥ 4πGρ 2 πGρ 2 4 2 ⎣⎢ c ⎦⎥ ⎣⎢ c ⎦⎥ becomes important. Eventually β ∝ M − so that L∝M. P ρ N kT ρ 4πR3 ρ In fact, the luminosity of very massive stars approaches and P = ideal = c A c and c = c = 54.18 c β µβ ρ 3M the Eddington limit as 0 ( L(r) = 1 L ) β → ( − β ) Edd 2 ⎛ M / M ⎞ 17 ( ⊙ ) Pc =1.242×10 ⎜ 4 ⎟ 4πGMc 38 -1 ⎛ M ⎞ ⎛ 0.34⎞ ⎜ R / R ⎟ L = =1.47 ×10 erg s ⎝ ( ⊙ ) ⎠ Edd κ ⎜ M ⎟ ⎝⎜ κ ⎠⎟ ⎝ ⎠ ⎛ 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 7 r 37 M Tc/10 C L/10 L/LEd All evaluated in 9 3.27 9.16 2.8 .02 For the n=3 polytrope actual models at 12 3.45 6.84 7.0 .04 2/3 a core H mass 15 3.58 5.58 13 .06 ⎛ ⎞ 2.5 6 M 1/3 fraction of 0.30 L ∝ M Tc = 4.6 × 10 K µβ ρc (in general for n = 3) 20 3.74 4.40 29 .10 ⎝⎜ M ⎠⎟ for stars of solar 25 3.85 3.73 50 .14 For massive stars on the main sequence and half way through metallicity (but 40 4.07 2.72 140 .24 outer layers still 60(57) 4.24 2.17 290 .35 hydrogen burning, µ ≈ 0.84 and, unless the star is very massive, unburned). 85(78) 4.35 1.85 510 .44 β ≈ 0.8 - 0.9. Better values are given in Fig 2-19 of Clayton 120(99) 4.45 1.61 810 .56 replicated on the next page. L ~ (1− β) L Ed decreases with mass as a general consequence of the fact that The density is not predicted from first principles since the actual ρc T3 radius depends upon nuclear burning, but detailed main c ∝ M 2β 3µ 3 and H burning happens at a relatively constant ρc ⎛ 10 M⊙ ⎞ --3 sequence models (following page) give ρc ≈ 10⎜ ⎟ gm cm ,So ⎝ M ⎠ temperature. Until about 40 M, the density decreases roughly 1/3 -1 -1/2 ⎛ M ⎞ as M . After that it decreases more slowly. Recall β ∝ M for T ≈ 3.9 ×107 β K (main sequence only) c ⎜ 10M ⎟ very large masses ⎝ ⊙ ⎠ β~ 0.8 – 0.9 Competition between the p-p Complications chain and the CNO Cycle • Beta not really constant, star not really an n=3 polytrope to begin with • Star evolves and develops variations in β and µ from center to surface • L varies almost a factor of two from beginning to end of main sequence • Opacity not constant and not all due to electron scattering. • L good to about a factor of two for conditions given (i.e., current µ, current M). Temperature to better than 10%. 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,γ)15O. For temperatures near 2.5 x 107 K. 1/3 3 4 5 6 7 8 9 ⎛ 14 ⋅1 ⎞ 7212 τ -2 ⎜ ⎟ neutron number ε ∝Tn n = τ = 4.248 14 + 1 =51.9 nuc 3 ⎜ 0.025 ⎟ ⎜ ⎟ ⎝ ⎠ All initial abundances within a cycle serve as catalysts and accumulate at largest t, i.e., where the (p,γ) reaction rate is smallest n=17 (less at higher T) (More on nucleosynthesis later) In general, the rates for these reactions proceed through known resonances whose properties are all reasonably well known.
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