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Solar Energy Solar Energy Solar energy production Solar luminosity 33 Solar radiation spectrum M =1.989·10 g 10 R=6.96·10 cm 33 L=3.847·10 erg/sec Tsurface 6000K Math and Units kg m2 T :Temperature in units K 1 J 1 2 s E :Energy in units J 1 erg 107 J S :Power in units W 1 eV 1.6022 1019 J 23 W A g 6.022 10 particles F :Flux in units 2 c 2.998108 m / s m MeV J 1 amu 1.661027kg 931.49 L :Luminosity in units c2 s http://en.wikipedia.org/wiki/Electronvolt Example 33 26 Solar luminosity: Lsun 3.8510 erg / s 3.8510 J / s kg m2 1J 1 s2 E According to Einstein: E m c2 m c2 kg m2 3.851026 / s 2 1 s 9 kg tons ms 2 4.2810 4,280,000 2 m s s 3108 2 s That amount of energy can only be generated by nuclear fusion processes! Nuclear masses and energy Isotopes: nuclei with Z=constant, N varies! The mass M of the nucleus is smaller than the mass of its proton and neutron constituents! 2 2 M c Z mp N mn c The mass difference is the binding energy B 2 2 B M c Z mp N mn c http://ie.lbl.gov/toimass.html http://nucleardata.nuclear.lu.se/database/masses/ Nuclear Binding Energy Yield from nuclear fission 56Fe, 56Ni, maximum binding energy Yield from nuclear fusion Example: What is the binding energy of the oxygen isotope 18O? -24 mp= 1.007825 · 1.66 · 10 g -24 mn= 1.008665 · 1.66 · 10 g M(18O) = 17.99916 · 1.66 · 10-24 g Z=8, N=10 B = (Z · m + N · m - M) · c2 p n B(18O) = 1.398 · 105 keV = 2.24·10-11 J 1g 18O contains 7.5·1011 J (W·s) Atomic Mass Unit: 1 amu=1/12(M12C)=1.66 · 10-24 g Nuclear Reactions and Q-values 2 A(a,b)B (mA+ma-mB-mb)·c =Q • Mass difference between initial particles and final particles is called: Q-value (energy) • If a reaction needs energy to take place it is called: Endothermic (Q<0) • If a reaction releases energy when taking place it is called: Exothermic (Q>0) Product b http://ie.lbl.gov/toi2003/MassSearch.asp(p, n, ) Projectile a (p, n, ) http://nucleardata.nuclear.lu.se/database/masses/ Reaction occurs with a certain probability (cross section), which depends on the energy dependent Coulomb interaction and the probability for forming a Target A (17N, 17O, 14C) new quantum system ! Recoil B Hydrogen & helium burning Calculate the energy release of solar hydrogen fusion: 4 1H1 4He -24 mp = 1.007825 · 1.66 · 10 g -24 m4He = 4.002603 · 1.66 · 10 g 27 29 M 4H 1He 41.007825 4.0026031.6610 kg 4.7610 kg 2 12 7 E4H 1He Q4H 1He M 4H 1He c 4.2810 J 2.67 10 eV E4H 1He Q4H 1He 41.007825 4.002603931.48MeV 26.7MeV This is the energy release per fusion reaction MeV 2.41039 J MeV reactions L 3.851026 2.41039 R s 91037 sun s s 26.7MeV s reactions tons M 41.0078251.661027kg91037 6.02 108 H s s Stellar helium burning Helium burning process in red giant stars 3 4He 1 12C Betelgeuse In Orion -24 m4He = 4.002603 · 1.66 · 10 g -24 m12C = 12.000000 · 1.66 · 10 g 27 29 M 4 12 34.00260312.00 1.6610 kg 1.3010 kg 3 He1 C 2 12 6 E 4 12 Q 12 M c 1.1710 J 7.2710 eV 3 He1 C 3He1 C 4H 1He E 4 12 Q 12 34.00260312.00 931.48MeV 7.27MeV 3 He1 C 3He1 C Solar Energy Source 1H 1H 1H 1H 4 1H 1 4He Q Q 26.7MeV 4 2 2.410 X 1/ 3 H e3.380 T erg s1 g 1 1H 1H pp T 2/3 2H=D 2H=D The energy generation rate in the pp chain, where XH is hydrogen mass fraction. pp-I 3He 3He A N 1 N 4 N X i i X H X He i H He N A N A N A 1H 1H Particle density: Ni 23 Avogadro’s number: NA=6.022·10 part/A g Hydrogen mass fraction: X 0. 5, 4 H He Solar core density (g/cm3): 160 g/cm3 Burning temperature (GK): T 0.015 GK 4 2 2.4 10 X 1/ 3 H e3.380 T erg s1 g 1 pp T 2 / 3 4 2 3 2.4 10 0.5 160g / cm 1/ 3 e3.380 0.015 erg s1 g 1 pp 0.0152 / 3 1 1 pp 17.6 erg s g 33 1 Lsun 3.8510 erg s 3.851033erg s1 M 2.19 1032g core 17.6 erg s1 g 1 33 M sun 1.989 10 g M core 0.11 M sun About 10% of the solar material is undergoing hydrogen burning reactions in the solar core ! Additional pp-chain sources 3He 4He 18% e- 1H 7Be pp-III pp-II 8B 1 7Li H e- 8Be 4He 4He 4He 4He Additional energy sources 25 4.410 X Z 1/ 3 H e15.228 T erg s1 g 1 CNO T 2/3 The energy generation rate in the pp chain, where XH is hydrogen mass fraction and Z the average CNO mass fraction (metallicity). Xi X H X He Z 1 Z 0.02 i This requires CNO seed abundance as catalyst! More massive stars in CNO mode Vega in Lyra Sirius in Canis Majoris Distance 25 Ly Distance 8.6 Ly Mvega=2.5Msun Mvega=2.0Msun Lvega=50Lsun Lvega=25Lsun Homework 1 The solar hydrogen fuel will eventually get exhausted. The stellar core material contracts to increase the internal pressure to balance the gravitational forces. Assuming the stellar core material follows the ideal gas law, at what temperature will the CNO energy production be equal to the energy production by the pp-chains?. pp CNO T 0.017GK.
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