Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Standard Solar Models I Aldo Serenelli Institute for Advanced Study, Princeton SUSSP61: Neutrino Physics - St. Andrews, Scotland – 8th to 23rd, August, 2006 John N. Bahcall (1934-2005) Plan Lecture 1 • Motivation: Solar models – Solar neutrinos connection • Stellar structure equations • Standard Solar Models (SSM) - setting up the problem • Overview of helioseismology • History of the SSM in 3 steps Lecture 2 • SSM 2005/2006 • New Solar Abundances: troubles in paradise? • Theoretical uncertainties: power-law dependences and Monte Carlo simulations • Summary Motivation • The Sun as a paradigm of a low-mass star. Standard test case for stellar evolution. Sun is used to callibrate stellar models • Neutrinos from the Sun: only direct evidence of solar energy sources (original proposal for the Homestake experiment that led to the Solar Neutrino Problem) • Neutrino oscillations: onstraints in the determination of LMA solution. However, SNO and SK data dominate importance of SSM minor Motivation • Transition between MSW effect and vacuum oscillations at ~5 MeV. 99.99% of solar neutrinos below 2 MeV: additional neutrino physics at very low energies? • Direct measurements of 7Be (pep, pp?) (Borexino, KamLAND, SNO+) key to astrophysics. Check the luminosity constraint • Future measurement of CNO fluxes? Answer to Solar Abundance Problem? What is inside a Standard Solar Models? Stellar structure – Basic assumptions • • • • The Sun is a self-gravitating object Spherical symmetry No rotation No magnetic field Stellar structure – Hydrostatic equilibrium 1/2 1D Euler equation – Eulerian description (fixed point in space) v v P v f t r r r Numerically, Lagrangian description (fixed mass point) is easier (1D) t m r t r t m r here m denotes a concentric mass shell Gm and using f 2 and m 4r 2 r Euler equation becomes r P Gm 1 2r 4 m 4r 4r 2 t 2 m Stellar structure – Hydrostatic equilibrium 2/2 P Gm 1 2r 4 m 4r 4r 2 t 2 Hydrodynamic time-scale hydr: 1 Gm r 1 P 0 2 2 hydr G 2 27 min r hydr 2 hydr << any other time-scale in the solar interior: hydrostatic equilibrium is an excellent approximation P Gm m 4r 4 (1) Stellar structure – Mass conservation We already used the relation m 4r 2 r leading to r 1 m 4r 2 (2) Stellar structure – Energy equation 1/2 Lm is the energy flux through a sphere of mass m; in the absence of energy sources Lm dt dq m where dq du PdV . Additional energy contributions (sources or sinks) can be represented by a total specific rate e (erg g-1 s-1) Lm dq e m dt Possible contributions to e: nuclear reactions, neutrinos (nuclear and thermal), axions, etc. Stellar structure – Energy equation 2/2 In a standard solar model we include nuclear and neutrino contributions (thermal neutrinos are negligible): e e n – en (taking en > 0) Lm ds e n en T e n en e g m dt (3) In the present Sun the integrated contribution of eg to the solar luminosity is only ~ 0.02% (theoretical statement) Solar luminosity is almost entirely of nuclear origin Luminosity constrain: L Lm dm e n en dm m Stellar structure – Energy transport Radiative transport 1/2 Mean free path of photons lph=1/k k opacity, density) Typical values k0.4cm2g-1, 1.4 g cm-3 lph2cm lph /R31011 transport as a diffusion process If D is the diffusion coefficient, then the diffusive flux is given by F DU 1 and in the case of radiation D cl ph and U aT 4 where 3 c is the speed of light and a is the radiation-density constant and U is the radiation energy density. In 1-D we get 4ac T 3 T F 3 k r Stellar structure – Energy transport Radiative transport 2/2 Lm F 4r 2 The flux F and the luminosity Lm are related by and the transport equation can be written as T 3 k Lm r 16ac r 2T 3 kLm T 3 or, in lagrangian coordinates m 64 2 ac r 4T 3 Using the hydrostatic equilibrium equation, we define the radiative temperature gradient as d ln T 3 kLm P rad d ln P rad 16acG mT 4 and finally T GmT rad 4 m 4r P (4) Stellar structure – Energy transport Convective transport 1/3 r+Dr: P+DP, T+DT, D b Adiabactic displacement s s r: P, T, Stability condition: b d ln d ln Db D s Dr Dr dr b dr s Using hydrostatic equilibrium, and d ln T d ln T d ln T dr b dr S dr ad d ln T dr d ln d ln P d ln T rad Stellar structure – Energy transport Convective transport 2/3 d ln P Divide by and get dr d ln T d ln T ad rad d ln P ad d ln P rad Schwarzschild criterion for dynamical stability When does convection occur? rad 3 kLm P large Lm (e.g. cores of stars M*>1.3M) 4 16acG mT regions of large k (e.g. solar envelope) Stellar structure – Energy transport Convective transport 3/3 Lm F Using definition of rad and we can write 2 4r 4 4acG T m F rad and, if there is convection: F=Frad+Fconv 2 3 k Pr 4acG T 4 m where Frad 2 3 k Pr is the actual temperature gradient and satisfies ad rad Fconv and must be determined from convection theory (solution to full hydrodynamic equations) Easiest approach: Mixing Length Theory (involves 1 free param.) Energy transport equation T GmT 4 m 4r P (4b) Stellar structure – Composition changes 1/4 Relative element mass fraction: Xi ni mi ; Xi 1 i X hydrogen mass fraction, Y helium and “metals” Z= 1-X-Y The chemical composition of a star changes due to •Convection •Microscopic diffusion •Nuclear burning •Additional processes: meridional circulation, gravity waves, etc. (not considered in SSM) Stellar structure – Composition changes 2/4 Convection (very fast) tends to homogenize composition 2 ni ni 2 4r Dconv t conv m m where Dconv is the same for all elements and is determined from convection treatment (MLT or other) Microscopic diffusion (origin in pressure, temperature and concentration gradients). Very slow process: diff>>1010yrs ni t diff 4r 2 ni wi m here wi are the diffusion velocities (from Burgers equations for multicomponent gases, Burgers 1969) Dominant effect in stars: sedimentation H Y & Z Stellar structure – Composition changes 3/4 Nuclear reactions (2 particle reactions, decays, etc.) ni t nuc j ni n j 1 ij v ij kl nk nl v kl e.t. 1 kl here v v (v)(v)dv 0 (v) is the relative velocity distrib. and (v) is cross section Sun: main sequence star hydrogen burning low mass pp chains (~99%), CNO (~1%) Basic scheme: 4p 4He + 2b+ + 2ne+ ~25/26 MeV Interlude on hydrogen burning – pp chains 2 p p H e n e 2 p p e H n e 2 H p3 He Q=1.44 MeV, <Qn>=0.265 Q=Qn=1.44 pp neutrinos pep neutrinos Q=5.49 He 3 He 4 He 2p 3 Q=12.86 ppI 88-89% Q=1.59 Q=Qn=0.86 (90%)-0.38 (10%) He 4 He 7 Be 7 Be e - 7 Li n e Li p 4 He 4 He 7 Q=17.35 7Be 3 neutrinos ppII 10% 8B neutrinos Marginal reaction: 3 He p4 He e n e Q=19.795, <Qn>=9.625 hep neutrinos Q=0.137 Q=17.98, <Qn>=6.71 7 Be p8 B 8 B8 Be e n e Be 4 He 4 He ppIII 1% 8 Interlude on hydrogen burning – CNO cycle CN-cycle 12 C p13 N 13 13 N C e n e 13 C p14 N 14 15 N p O 15 O15 N e n e 15 N p12 C 4 He NO-cycle Q=1.94 Q=2.22, <Qn>=0.707 13N neutrinos Q=7.55 Q=7.30 Q=2.75, <Qn>=0.996 13N neutrinos Q=4.97 15 N p16 O 16 17 O p F 17 17 F O e n e 17 O p14 N 4 He Q=12.13 Q=0.600 Q=2.76, <Qn>=0.999 Q=1.19 CNO cycle is regulated by 14N+p reation (slowest) 17F neut. Stellar structure – Composition changes 4/4 Composition changes dni ni dt t nuc ni t conv ni t i=1,…..,N diff (5) Stellar structure – Complete set of equations P Gm m 4r 4 (1) r 1 m 4r 2 (2) Lm ds e n en T m dt (3) T GmT rad 4 m 4r P dni ni dt t nuc ni t (4) conv ni t ; i 1,...., N (5) diff Microscopic physics: equation of state, radiative opacities, nuclear cross sections Standard Solar Model – What we do 1/2 Solve eqs. 1 to 5 with good microphysics, starting from a Zero Age Main Sequence (chemically homogeneous star) to present solar age Fixed quantities Solar mass M=1.9891033g 0.1% Kepler’s 3rd law Solar age t=4.57 109yrs 0.5% Meteorites Quantities to match Solar luminosity Solar radius Solar metals/hydrogen ratio L=3.842 1033erg s-1 0.4% Solar constant R=6.9598 1010cm 0.1% Angular diameter (Z/X)= 0.0229 Photosphere and meteorites Standard Solar Model – What we do 2/2 3 free parameters: • Convection theory has 1 free parameter: MLT determines the temperature stratification where convection is not adiabatic (upper layers of solar envelope) • 2 of the 3 quantities determining the initial composition: Xini, Yini, Zini (linked by Xini+Yini+Zini=1). Individual elements grouped in Zini have relative abundances given by solar abundance measurements (e.g. GS98, AGS05) Construct a 1M initial model with Xini, Zini, (Yini=1- Xini-Zini) and MLT, evolve it during t and match (Z/X), L and R to better than one part in 10-5 Standard Solar Model – Predictions • Eight neutrino fluxes: production profiles and integrated values. Only 8B flux directly measured (SNO) so far • Chemical profiles X(r), Y(r), Zi(r) electron and neutron density profiles (needed for matter effects in neutrino studies) • Thermodynamic quantities as a function of radius: T, P, density , sound speed (c) • Surface helium Ysurf (Z/X and 1=X+Y+Z leave 1 degree of freedom) • Depth of the convective envelope, RCZ The Sun as a pulsating star - Overview of Helioseismology 1/4 • Discovery of oscillations: Leighton et al. (1962) • Sun oscillates in > 105 eigenmodes • Frequencies of order mHz (5-min oscillations) • Individual modes characterized by radial n, angular l and longitudinal m numbers The Sun as a pulsating star - Overview of Helioseismology 2/4 • Doppler observations of spectral lines: velocities of a few cm/s are measured • Differences in the frequencies of order mHz: very long observations are needed. BiSON network (low-l modes) has data collected for 5000 days • Relative accuracy in frequencies 10-5 The Sun as a pulsating star - Overview of Helioseismology 3/4 • Solar oscillations are acoustic waves (p-modes, pressure is the restoring force) stochastically excited by convective motions • Outer turning-point located close to temperature inversion layer. Inner turning-point varies, strongly depends on l (centrifugal barrier) Credit: Jørgen Christensen-Dalsgaard The Sun as a pulsating star - Overview of Helioseismology 4/4 • Oscillation frequencies depend on , P, g, c • Inversion problem: using measured frequencies and from a reference solar model determine solar structure 2 i c i i K c , (r ) 2 (r )dr K ,c (r ) (r )dr Fsurf (i ) i c 2 2 Output of inversion procedure: c2(r), (r), RCZ, YSURF Relative difference of c between Sun and BP00 History of the SSM in 3 steps • Step 1. Predictions of neutrino fluxes by the SSM to high (factor 2.5/3) w.r.t. to radiochemichal experiments: solar neutrino problem. 8B flux too sensitive to central temperature 8B)T20-25. Problem with SSM? Specultive solutions of all kinds. This lasted about 30 years. • Step 2. Precise calculations of radiative opacities (OPAL group). Helioseismology: results from low and mid-l sample well the solar interior (1995-1997). SSM correct in solar interior to better than 1% Bahcall et al. 1996 History of the SSM in 3 steps • Step 3. The BP00 model and Sudbury Neutrino Observatory BP00: Bahcall, Pinsonneualt & Basu (2001) RCZ=0.714 / 0.713 ± 0.001 YSUP=0.244 / 0.249 ± 0.003 c c 2 / c 2 0.001 2 / 2 0.005 8B)= (5.05 ± 0.91) x 106 cm-2 s-1 SK8B)= (2.32 ± 0.09) x 106 cm-2 s-1 (only sensitive to ne) History of the SSM in 3 steps • Step 3. SNO: direct measurement of the 8B) flux. ve d p p e (CC) - 1.44MeV v x d p n v x (NC) - 2.22MeV SNO 5.09 0.66106 cm -2s -1 SNO collaboration (2002) v x e vx e (ES) BP008B)/ SNO8B)= 0.99 Solution to the Solar Neutrino Problem !!!!