Thermonuclear Reactions

Transcription

1 Thermonuclear Reactions

2 Eddington in 1920s hypothesized that fusion reactions between light elements were the energy source of the stars. Stellar evolution = (con) sequence of nuclear reactions E kinetic kt c T~ kev, but E Coulomb barrier = Z 1Z 2 e 2 r = 1.44 Z 1Z 2 r[fm] higher than the kinetic energy of the particles. ~ MeV, 3 orders Tunneling effect in QM proposed by Gamow (1928, Z. Physik, 52, 510); applied to energy source in stars by Atkinson & Houtermans (1929, Z. Physik, 54, 656) 2

3 George Gamow ( ) Russian-born physicist, stellar and big bang nucleosynthesis, CMB, DNA, Mr. Thompkins series 1929 U Copenhagen 1960s U Colorado 3

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5 Quantum mechanics tunneling effect 5

6 e πz 1Z 2 e 2 /ε 0 hν This as v e mv2 /2kT This as v 6

7 Clayton 7

8 Clayton 8

9 Resonance very sharp peak in the reaction rate ignition of a nuclear reaction So there exists a narrow range of temperature in which the reaction rate a power law an ignition (threshold) temperature For a thermonuclear reaction or a nucleosynthesis (fusion) process, the reaction rate is expressed as q energy released per mass ρ m T n Resonance reactions Energy of interacting particles Energy level of compound nucleus 9

10 σ v t n v N = n (σvt) N t = n σv t col = 1/ n σv l = vt col = 1/n σ 10

11 r 12 n 1 n 2 σv n 1 n 2 exp C z 1 2 z 2 2 n i X i q 12 Q ρ X 1 X 2 /m 1 m 2 [erg g 1 s 1 ] T 6 1/3 [cm 3 s 1 ] n i m i = ρx i 11

12 Nuclear Fission (e.g., power planets) Fusion (e.g., stars) 12

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14 Star Brown dwarf Planet Burrows 14

15 Stars Brown Dwarfs Planets M M > 0.08, core H fusion Spectral types O, B, A, F, G, K, M > M M > 0.013, core D fusion > M M > 0.065, core Li fusion Spectral types M6.5 9, L, T, Y Electron degenerate core 10 g cm 3 < ρ c < 10 3 g cm 3 T c < K M M < 0.013, no fusion ever 15 39

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18 Brown dwarfs and very lowmass stars partial P e deg White dwarfs completely degenerate, R as M Terrestrial planets R as M complicated EoSs Mass-radius relation M Jupiter ( ) M 18

19 M c 2 = ergs 1 amu = 931 Mev/c 2 19

20 n + p D + γ (production of D) D + D 4 He + γ (destruction) faster The lower the mass density, the more the D abundant D as a sensitive tracer of the density of the early Universe 4 He 4 He n/2 (n+p)/4 = 2n n+p n/p He 2/9 20

21 D H 156 ppm Terrestrial seawater ~26 ppm Jupiter 17 ppm Saturn 55 ppm Uranus 200 ppm Halley s Comet 21

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23 Clayton 23

24 Protostars are heavily embedded in clouds, so obscured, with no definition of T eff Birthline=beginning of PMS; star becomes optically visible deuterium main sequence Stahler (1983, 1988), Palla & Stahler (1990) 24

25 compared with observations 25

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27 Presence of Li I λ6707 absorption stellar youth Ca I λ6718 prominent in late-type stars Stahler & Palla 27

28 For protostars with T c K, the central lithium is readily destroyed. Stars 0.9 M become radiative at the core, so Li not fully depleted. Li abundance age clock 7,000 K 4,000 K Stahler & Palla 28

29 Stahler & Palla29

30 0.420 MeV to the positron and neutrino; position and electron (each MeV rest energy) annihilate MeV 30

31 p + p 2 He p + p 31

32 32

33 pp I important when T c > K Q total = = 27.7 MeV Q net = = 26.2 MeV The baryon number, lepton number, and charges are all conserved. All 3 branches operate simultaneously. pp I is responsible for > 90% stellar luminosity 33

34 Exercise Assuming that the solar luminosity if provided by 4 1 H 4 He, liberating MeV, and that the neutrinos carry off about 2% of the total energy. Estimate how many neutrinos are produced each second from the sun? What is the solar neutrino flux at the earth? 34

35 Solution 3% is carried away by neutrinos, so the actual energy produced for radiation E = ( MeV) erg/ev Each alpha particle produced 2 neutrinos, so with L = ergs/s, the neutrino production rate is ν/s, and the flux at earth is /4π (1 AU) ν cm 2 s 1 35

36 The thermonuclear reaction rate, r pp = n 2 2 p T exp T 6 1 ( T T T 6 ) [cm 3 s 1 ], where the factor n 2 p = ρ 2 2 X H q pp = ρ X H 2 T exp T ( T T T 6 ) [erg g 1 s 1 ] 36

37 6.54 MeV per proton n ~ 6 for T K n ~ 3.8 for T K (Sun) n ~ 3.5 for T K 37

38 Among all fusion processes, the p-p chain has the lower temperature threshold, and the weakest temperature dependence. Q pp = (M 4H M He ) c 2 = MeV But some energy (up to a few MeV) is carried away by neutrinos. 38

39 CN cycle more significant NO cycle efficient only when T > K Recognized by Bethe and independently by von Weizsäcker CN cycle + NO cycle Cycle can start from any reaction as long as the involved isotope is present. after that carried away by the neutrinos 39

40 Schwarzschild At the center of the Sun, q CNO /q pp 0.1 CNO dominates in stars > 1.2 M, i.e., of a spectral type F7 or earlier large energy outflux a convective core This separates the lower and upper MS. CN cycle takes over from the PP chains near T 6 =18. He burning starts ~10 8 K. 41

41 The Solar Standard Model (SSM) Best structural and evolutionary model to reproduce the observational properties of the Sun L = [ergs/s] R = [cm] M = [gm] Spectroscopic observations Z/X = (latest value seems to indicate Z = 0.013) Neglecting rotation, magnetic fields, and mass loss (dm/dt ~ M /yr) Sun Fact Sheet 42

42 bottleneck A succession of α, γ processes 16 O, 20 Ne, 24 Mg (the α-process) 43

43 C-buring ignites when Tc ~ ( ) 10 9 K, i.e., for stars M O-buring ignites when Tc ~ ( ) 10 9 K, i.e., for stars > M The p and α particles produced are captured immediately (because of the low Coulomb barriers) by heavy elements isotopes O burning Si 44

44 q PP = ρ X 2 T 6 2/3 exp 33.8 T 6 1/3 erg g 1 s 1 q ρ X H 2 T 4 q CN = ρ X X CN T 6 2/3 exp T 6 1/3 erg g 1 s 1 q ρ X H X CN T 16 X CN X H = 0.02 ok for Pop I q 3α = ρ 2 X 3 α T 3 8 exp 42.9 T 8 erg g 1 s ρ 2 X 3 40 α T 8 erg g 1 s 1 (if T 8 1) Clayton 45

45 Photoionization Photodisintegration 46

46 For example, 16 O + α 20 Ne + γ If T < 10 9 K but if T K in radiation field So 28 Si disintegrates at K to lighter elements (then recaptured ) Until a nuclear statistical equilibrium is reached But the equilibrium is not exact pileup of the iron group nuclei (Fe, Co, Ni) which can resist photodisintegration until K 47

47 Nuclear Fuel Process T threshold (10 6 K) Products Energy per nucleon (MeV) H p-p ~4 He 6.55 H CNO 15 He 6.25 He 3α 100 C, O 0.61 C C + C 600 O, Ne, Na, Mg 0.54 O O + O 1,000 Mg, S, P, Si ~0.3 Si Nuc. Equil. 3,000 Co, Fe, Ni <0.18 From Prialnik Table

48 56 Fe MeV 13 4 He + 4 n If T, even 4 He p + + n 0 So stellar interior has to be between a few T 6 and a few T 9. Lesson: Nuclear reaction that absorb energy from ambient radiation field (in stellar interior) can lead to catastrophic consequences. 49

49 Time Scales Different physical processes inside a star, e.g., nuclear reactions (changing chemical composition) are slow (longer time scales); structural adjustments ( dp dt) take places on relatively shorter time scales. Dynamical timescale Thermal timescale Nuclear timescale Diffusion timescale 52

50 Dynamical Timescale hydrostatic equilibrium perturbation motion adjustment hydrostatic equilibrium Free-fall collapse Equation of motion r = GM r r 2 1 ρ Near the star s surface r = R, M r = M, so R = GM dp dr R 2 1 ρ Free-fall means pressure gravity, so R GM R 2 Assuming a constant acceleration R = R 2 τ 2 ff, so τ ff = 2R 3 GM 1 2 = 1 ρ ρ πg ρ [d] dp dr 53

51 Stellar Pulsation The star pulsates about the equilibrium configuration same as dynamical timescale τ pul 1/ ρ Propagation of Sound Speed (pressure wave) Pressure induced perturbation, R τ 2 ff = R = GM + 1 dp 1 dp 2 R 2 ρ dr ρ dr 1 ρ P R so R τ ff P ρ c s (sound speed) T (for ideal gas) τ s R c s In general, τ dyn 1 G ρ n s = 1000 R R 3 M M [S] 54

52 Thermal Timescale Kelvin-Helmholtz timescale (radiation by gravitational contraction) E total = E grav + E thermal = 1 E 2 grav = 1 α GM 2 R 2 This amount of energy is radiated away at a rate L, so timescale τ KH = E total = 1 α GM 2 RL L 2 = M 2 RL [yr] in solar units τ KH M M 2 R R L L [yr] 55

53 M = 1 M, R = 1 pc τ dyn yr τ ther 1 yr M = 1 M, R = 1 R τ dyn s 30 min τ ther yr 56

54 Nuclear Timescale Time taken to radiate at a rate of L on nuclear energy 4 1 H 4 He (Q = erg/g) τ nuc = E nuc L = M L τ nuc M M L L [yr] From the discussion above, τ nuc τ KH τ dyn 57

55 Main-Sequence Lifetime of the Sun M g L [ergs/s] Nuclear physics τ MS M c 2 L Stellar physics = s = [yr] Given L MS L M M 4 τ MS M M 3 [yr] 58

56 Diffusion Timescale Time taken for photons to randomly walk out from the stellar interior to eventual radiation from the surface r e = 1 e 2 4πε 0 m e c2 ( classical radius of the electron) σ Thomson = 8π 3 r e 2 = [m 2 ] for interactions with photon energy hν m e c 2 (electron rest energy) Thus, mean free path l = 1/ σ T n e, where for complete ionization of a hydrogen gas, n e = M/(m p R 3 ). So, l m p R 3 σ T M = 4 [mm] for the mean density. At the core, it is 100 times shorter. τ dif 10 4 [yr] (Exercise: Show this.) 59

57 For an isotropic gas P = p vp n p dp - p and v p : relativistic case - n p : particle type & quantum statistics For a photon gas, p = hν/c, so P rad = hν n ν dν = u = at4, a = ergs cm 3 K 4 60

58 Radiation Pressure P total = P radiation + P gas Since P rad ~ T 4 ~ But P tot ~ M 2 R 4 P rad P tot ~ M 2 M 4 R 4 So the more massive of stellar mass, the higher relative contribution by radiation pressure (and γ decreases to 4/3.) 61

59 P rad F = d P rad dr κρ dp rad dr = 4ac 3 ~ κρ c T3 dt L 4πr 2 dr = L 4πr 2 dp tot dr dp rad dp tot = Gm = ρ r 2 κl 4πcGm 62

60 P gas P rad P tot dp tot > dp rad L < 4πcGM κ m = M, P = 0, κl 4πcGm This is the Eddington luminosity limit = Maximum luminosity of a celestial object in balance between the radiation and gravitational force. L Edd L = μ e M M L X < erg sec 1 63

61 L 4πGm p σ T M L Edd M M [erg s 1 ] M L Edd L, M bol = 7.0 M M bol = 11.0 L L M bol = 11.6 M 120 M 64

62 NGC 3372 α = 10: 45.1, b = 59: 52 (J2000) l = 287.7, b = 0.8 D = 2.3 kpc HST 3.0 x2.5. DSS image. AAO/ROE 65

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64 Evolution of the Sun in the HRD 67

65 Comparison of 1, 5, and 25 M stars 68