1 Trieste Sept. 2002

2 Standard and non-standard solar models Success of stellar evolutionary theory Basic inputs of the theory Standard solar model: inputs and outputs Relevance of helioseismic data What can be learnt more on solar models from helioseismology

3 Stellar structures and the standard model Stellar evolution theory can explain in good detail the different phases of stellar life. The iscochrone calculation of globular cluster (parameter is the cluster age) is a good summary of its successes. Metter efigura ammasso  T s LL

4 The basic inputs The physical structure of a star and its evolution are determined by these main inputs: -initial chemical composition X i -the equation of state for stellar matter -the radiative opacity  ( , T, X i ) -the energy production per unit mass  ( ,T, X i )

5 Equation of state Perfect gas law is the first approximation One has to evaluate the ionization degree for all nuclei Also plasma effects must be included (screening, degeneracy, Coulomb interactions) Over the years study of EOS has been improved and accurate tabulations are available Anyhow…...

6 Comparison among EOS Perfect gas law accurate at in the core worsen in the outer regions, 2-4% (for a fixed solar structure) gas T=  P/(  R gas )  =1/[X+1/2 Y+ 1/2 Z]

7 Radiative opacity  Opacity is connected with photon mean free path. In the radiative region,  governs the temperature gradient (…see next). The evaluation of  requires detailed knowledge of several processes involving photons (scattering, absortion, inverse bremsstrahlung…) and of knowledge of atomic levels in the solar interior Used: OPAL tables of Livermore group   /   3 % (assumed 1  ) ionization degree and the population of excited states

8 Nuclear energy production   The expression for the nuclear energy production  is obtained by using tables of nuclear reaction rates. Fowler’s group compiled and updated the tables for many years ( ) Other compilations now available: –for the sun: Adelberger et al –for a large class of reactions: NACRE 1999 *   energy /unit mass/unit time

9 Spherical symmetry The sun is described as a spherically symmetric system, so that one has an effectively one dimensional problem. Radial coordinate or Mass coordinate are used Rotation is neglected Magnetic field is neglected ( see Episode I )

10 The basic equations 1)Hydrostatic equilibrium 2)Continuity equation 3)Transport equation 4) Energy Production 5)Equation of state 6)Time evolution First 1-5) is solved for a given X i (r) [5 eqs and 5 unknowns:can be solved if we know  (  T X i ) and  (  T X i )] Next 6) is applied for a step  t and the new values for X i (r) is used to solve again 1-5) spatial system See Kippenham and Weigert, “stellar structure and evolution”,Springer Verlag, 1990

11 Standard Solar Model (SSM) Stix (1989): “the standard model of the sun could be defined as the model which is based on the most plausible assumptions” i.e inputs are chosen at their central values Bahcall (1995): “A SSM is one which reproduces, within uncertainties, the observed properties of the Sun, by adopting a set of physical and chemical inputs chosen within the range of their uncertainties”.

12 The 3 main properties to be reproduced : In order to produce a SSM one studies the evolution of an initially homogeneous solar mass model up to the sun age so as to reproduces the: -solar luminosity L o =3.844(1  0.4%) erg/s -solar radius R o =6.9598(1  0.04%) cm -photospheric ( Z/X) photo =0.0245(1  6%) composition M o = (1  0.15%) gr t o =4.57(1  0.4%) Gyr X= hydrogen Y= helium Z=metals

13 The 3 “free” parameters For producing a SSM one can tune 3 parameters: the initial Helium abundance Y in the initial metal abundance(s) Z in “the mixing length parameter”  (a parameter describing the convection efficiency)

14 The effect of the parameters The luminosity of the sun is mainly sensitive to Y in (increasing Y in the sun is brighter and a given luminosity is reached in a shorter time ) the mixing length  affects only R o ( to reproduce R o one adjusts the efficiency of external convection: if , convection is more efficient, dT/dr , T sur  since L o is fixed, radius decreases) Z in essentially determines the present metal content in the photosphere, Z photo T   P/  Y    T 

15 Results of SSM calculations Density [gr/cm 3 ] Temperature [10 7 K] R/R o

16 Comparison among different calculations BP2000 FRANEC GARSOM T c [10 7 K]  c [gr/cm 3 ] Y c Z c Good agreement: differences at % level or less <1% 6% 1%

17 Comparison of calculated neutrino fluxes 1% BP2000 FRANEC GARSOM pp [10 10 /s/cm 2 ]  Be [10 9 /s/cm 2 ] B [10 6 /s/cm 2 ] CNO [10 9 /s/cm 2 ]..see Episode III

18 Z/X  L o AgeS pp  q/q (1  ) 6% 3% 0.4% 0.4% 2% dlogT c /dlogq T c is an important observable for calculation of neutrino fluxes. It is strongly sensitive to solar quantities:  (  T c /T c ) q =0.6% (1  ) The accuracy of the central solar temperature Z/X 0.08  0.14 L 0.34 age 0.08 I

19 Remarks 3 input parameters to be tuned (Y in,Z in,  3 observables to be reproduced by the evolutionary calculation (L o,(Z/X) photo, R o ) Up to this point, the SSM is “no so big success”. Confidence in the SSM is gained from the successes of stellar evolution theory for describing more adavanced phases of stellar life.

20 The impact of helioseismic data Helioseismology determines the present value of the photospheric helium abundance, Y= (1± 1.4%) and the transition between the radiative and convective regimes R b =0.711 (1 ± 0.14%) R o When this is taken into account, one has now 3 parameters and 5 data. Acutally there is much more….

21 Sound speed squared profiles From the thousands measured oscillation modes one reconstruts the sound speed squared (u=P  ) profile of the solar interior (inversion method): U= P/ 

22 Relative differences of sound speed squared Agreement between model and data at less than 0.5%  U/U= (SSM-sun )/SSM BP2000

23 The accuracy of helioseismic determinations* Systematic errors in the inversion procedure dominates (starting solar models, numerical …) 33 11 BP2000  U/U * Dziembowki et al. Astrop. Phys. 7 (1997) 77

24 The sound speed near the solar center The observed p-modes do not reach the solar center. Can we believe in the helioseismic determination near the solar center? Maybe we are just getting out what we put in?, (i.e. the output is just the value of the model used as a starting point of the inversion method?) Note: The observed frequencies are compared with the predicted ones -> inversion method

25 Extraction of U* Let us invert the helioseismic data by starting from two (non standard) models. (  u/u mod =1% at R=0) Inversion gives quite similar seismic models, even near the center (  u/u sei =0,1% at R=0) Z/X + 10% Z/X  10% Starting models Results of inversion Nucl Phys B Suppl 81(2000)95

26 Can helioseismogy measure the solar temperature? NO : the sound speed depends on temperature and chemical composition, e.g, for a perfect gas: u=P/  =  T/  The abundances of elements (and EOS) is needed to translate sound speed in temperature.  =1/[2x+3/4 Y+1/2 Z] mean molecular weight

27 Helioseimic tests of SSM Helioseismology has provided severe tests and constraints on solar models building. Recent SSM calculations (including element diffusion) are in excellent agreement with helioseismic data. (see previous slides)

28 Helioseismic constraints of solar models Helioseismology can be used to test the basic ingredients of the solar models and to study possible new effects: 3 examples: -nuclear physics: the pp-> d+e + + e -plasma physics: screening effects -new physics: solar axion emission

29 Helioseismic determination of p+p cross section (S pp ) Consistency with helioseismology requires: S pp =S pp (SSM)(1 ± 2%) This accuracy is comparable to the theoretical uncertainty: S pp (SSM)=4(1 ± 2%) x KeVb  U/U (mod-SSM)/SSM Remind: S pp is not measured Degli Innocenti et al. PLB 416 (1998) 365

30 Screening of nuclear charges in the plasma Screening modifies nuclear reactions rates Thus it can be tested by means of helioseismology TSYtovitch anti-screening is excluded at more than 3  NO Screening is also excluded. Agreement of SSM with helioseismology shows that (weak) screening does exist. Fiorentini et al. PLB 503(2001) 121  U/U (mod-SSM)/SSM

31 Solar axion production If Axions are produced (  +Z  A +Z ) one has an extra energy loss mechanism in the solar interior (L A ) L A depends on A -  coupling constant (g A ) g A > GeV -1 is excluded at 3  level 33 Schlattl et al. Astrop. Phys. 10 (1999) 353 g A =(5,10,15,20) /10 10 GeV It is of interest to know the maximum axion-photon coupling strength which is compatible with well established solar properties

32 List of applications By means of helioseismology one can constrain: p+p cross section screening effect solar age [A&A 343 (1999) 990] diffusion efficiency [A&A 342 (1999) 492] existence of a mixed core [Astr. Phys. 8(1998) 293] Axion production in the sun WIMPs-matter interaction [hep-ph/ ] Existence of extra-dimensions [PLB 481(2000)291] Possible deviation from standard Maxwell-Boltzmann distribution [PLB 441(1998)291] SUN EXOTIC

33 Summary SSM (and stellar evolutionary theory) is in good shape: agreement between observations and predictions Helioseismology added new constraints to SSM builders Moreover helioseismic data can be used to confirm (exclude) standard (non standard) solar models

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35 Mixing length As matter becomes too opaque, convection dominates the energy transport. The precise description of the convection is an essentially unsolved problem. The process is described in terms of a phenomenological model, the so called mixing length theory The mixing legnth L is the distance over which a moving unit of gas can be identified before it mixes appreciably. L is relatedd to the pressure scale height H p =1/(dlnP/dlnR) through L=  H p and  is used as a free parameter

36 Inversion method Calculate frequencies  i as a function of u =>  i  i (u j ) j=radial coordinate Assume SSM as linear deviation around the true sun:  i  i, sun + A ij (u j -u j,sun ) Minimize the difference between the measured  i and the calculated  i In this way determine  u j  =u j  -u j, sun