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Lecture 1 - the Solar Interior O Topics to Be Covered Lecture 1 - The Solar Interior o Topics to be covered: o Solar interior o Core o Radiative zone o Convection zone January 19, 2006 Lecture 1 - The Solar Interior The Solar Interior - “The Standard Model” o Core o Energy generated by nuclear fusion (the proton-proton chain). o Radiative Zone o Energy transport by radiation. o Convective Zone o Energy transport by convection. January 19, 2006 Lecture 1 - The Solar Interior The Solar Interior o Christensen-Dalsgaard, J. et al., Science, 272, 1286 - 1292, (1996). January 19, 2006 Lecture 1 - The Solar Interior The Solar Core o R: 0.0 - 0.25 Rsun o T(r): 15 - 8 MK o !(r): 150 - 10 g cm-3 o Temperatures and densities sufficiently high to drive hydrogen burning (H->He). o Ultimate source of energy in the Sun and Sun-like stars. January 19, 2006 Lecture 1 - The Solar Interior The Solar Core o What is the temperature and pressure in the core? dP GM# o Assume hydrostatic equilibrium: = " dr r2 dM and mass conservation: = "4#r2$ dr ! dP dM dP GM o Divide to cancel !’s => / = = " dr dr dM 4#r4 ! M dP o Therefore, LHS => " dM = P " P PC = pressure at core #0 dM C S ! PS = pressure at surface 2 M GM GM and RHS => dM = #0 4"r4 8"r4 ! 2 GM " P = P + C S 8#r4 ! January 19, 2006 Lecture 1 - The Solar Interior ! The Solar Core GM 2 o Assuming P << P and setting r = R, P ~ S C C 8"R4 "kT o Using the Ideal Gas Law PC = nkT = mH ! k = Boltzmann’s const n = number density atoms/cm3 ! = density = M/4"!R 3 GMmH o The core temperature is therefore TC ~ kR 7 7 o Which gives Tc ~ 2.7 x 10 K (actual value is ~1.5 x 10 K). ! January 19, 2006 Lecture 1 - The Solar Interior The Solar Core o Coulomb barrier between protons must be overcome for fusion to occur. o To overcome Coulomb barrier, particles must have sufficient thermal kinetic energy to exceed Coulomb repulsion: 3 e2 kT > 2 rnuc 2e2 => T > 3krnuc ! =1010K! o Particles have Maxwell-Boltzmann distribution: E ! # kT !P (E)dE " Ee dE o There is a high-energy tail, but not sufficient … need quantum mechanics. ! January 19, 2006 Lecture 1 - The Solar Interior The Solar Core o From Heisenberg Uncertainty Principle ( " x " p # / 2 ) a proton of a given h (insufficient) energy may be located within nucleus of neighbouring proton. o Combined with high-energy M-B! tail, we get the Gamow Peak. o So protons in 3-10 keV energy range can overcome the Coulomb barrier (i.e., T>15MK). o Fusion can therefore occur. January 19, 2006 Lecture 1 - The Solar Interior Proton-proton cycle o The p-p cycle occurs in three main steps. Step 1: 1H + 1H ! 2H + e+ + " (Q = 1.44 MeV) o Might then expect a 2H + 2H reaction, but because of the large numbers of 1H, the following is more probable: Step 2: 2H + 1H ! 3He + # (Q = 5.49 MeV) o 3He can then react with 1H, but the resultant 4Li is unstable (i.e. 3He + 1H ! 4Li ! 3He + 1H). o The final step is then: Step 3: 3He + 3He ! 4He + 21H + # (Q = 12.86 MeV) o The net result is: 4 1H ! 4He + 2e+ + 2 " (Q = 26.7 MeV) January 19, 2006 Lecture 1 - The Solar Interior Proton-proton cycle (cont.) o ~99% of the Sun’s energy is produced via the p-p cycle. o The remaining ~1% is produced by the Carbon-Nitrogen-Oxygen (CNO) cycle. o CNO cycle is more important in more massive stars. January 19, 2006 Lecture 1 - The Solar Interior Proton-proton vs. CNO January 19, 2006 Lecture 1 - The Solar Interior The Radiative Zone o R: 0.25 - 0.8 Rsun o T(r): 8 - 0.5 MK o !(r): 10 - 0.01 g cm-3 o Hydrogen burning cuts off abruptly . at r ~ 0.25 Rsun o Interior becomes optically thin or transparent as density decreases. o Energy transported radiatively. o Photons cannot be absorbed in the radiative zone as the temperature are too high to allow atoms to form. Therefore no mechanism for the absorption of photons. January 19, 2006 Lecture 1 - The Solar Interior The Radiative Zone o For T = 15MK Wien’s displacement law implies #max = 0.19 nm i.e., the center of the Sun is full of X-rays. o Photons do 3D random walk out of Sun. o Assume photon moves l between interactions (mean free path) and takes a total number of steps N. o On average it will have moved a distance d = l N o As t = N l / c and 2 difusion R = l N => tdiffusion = R /lc ! 4 => tdiffusion >10 yrs! January 19, 2006 ! Lecture 1 - The Solar Interior Solar Interior # 4 & o Total radiative energy inside Sun is: E = aT 4% "R3 ( J $ 3 ' where a = 4$/c is the radiation constant. ! E 16" o Can thus estimate solar luminosity from, L = = #T 4 Rl W tdiffusion 3 o Which gives, L ~ 3 x 1026 W. ! o Actual value is actually 4 x 1026 W. January 19, 2006 Lecture 1 - The Solar Interior The Convective Zone o R: 0.8 - 1 Rsun o T(r): 0.5 MK - 6000 K. o ! <0.01 g cm-3 o Photons now absorbed as temperature is sufficiently low to allow atoms to form. Gas is optically thick or opaque. TC o Continuous absorption of photons by lower layers causes a temperature gradient to build up between the lower and upper layers. r o Plasma become convectively unstable, and large convective motions become the dominant transport mechanism. TH TH > TC January 19, 2006 Lecture 1 - The Solar Interior The Convective Zone January 19, 2006 Lecture 1 - The Solar Interior Advanced Stellar Physics o Email: [email protected] / [email protected] o Office: 3.17A. o 80%: Final exam. o 20%: 2000 word essay and 10-min presentation. o Deadline and presentation: Last lecture (March 9) o Claire: “The Coronal Heating Debate” o Brian: “The Solar Activity Cycle” January 19, 2006 Lecture 1 - The Solar Interior.
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