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Trieste 23-25 Sept. 2002

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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

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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

Ts

L

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The basic inputs

The physical structure of a star and its evolution are determined by these main inputs:

-initial chemical composition Xi

-the equation of state for stellar matter-the radiative opacity (, T, Xi)

-the energy production per unit mass (,T, Xi)

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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…...

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Comparison among EOS

•Perfect gas law accurate at 10-3 in the core • worsen in the outer regions, 2-4%

(for a fixed solar structure)

gas

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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)

ρκ1

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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 (1960 -1988)

• Other compilations now available: – for the sun: Adelberger et al. 1998– for a large class of reactions: NACRE 1999

* energy /unit mass/unit time

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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)

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The basic equations1)Hydrostatic equilibrium2)Continuity equation3)Transport equation4) Energy Production5)Equation of state

6)Time evolution

2r

3 r4L

Tρ

ac43

drdT

•First 1-5) is solved for a given Xi (r) [5 eqs and 5 unknowns:can be solved if we know (T Xi) and (T Xi)]•Next 6) is applied for a step t and the new values for Xi(r) is used to solve again 1-5)

2r

rρGM

drdP

ρr4drdM 2

) XT, ρ,(PP i

ε ρr4drdL 2

kik

jij svsv

ρ

m

dt

dX ii

spati

al sy

stem

See Kippenham and Weigert, “stellar structure and evolution”,Springer Verlag, 1990

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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”.

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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 Lo=3.844(1 0.4%) 1033 erg/s -solar radius Ro=6.9598(1 0.04%) 1010 cm

-photospheric (Z/X)photo=0.0245(1 6%) composition

Mo= 1.989 (1 0.15%) 1033

grto=4.57(1 0.4%) GyrX= hydrogenY= heliumZ=metals

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The 3 “free” parameters

For producing a SSM one can tune 3 parameters:

• the initial Helium abundance Yin

• the initial metal abundance(s) Zin

• “the mixing length parameter” (a parameter describing the

convection efficiency)

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The effect of the parameters• The luminosity of the sun is mainly sensitive

to Yin

(increasing Yin the sun is brighter and a given luminosity is reached in a shorter time )

• the mixing length affects only Ro (to reproduce Ro one adjusts the efficiency of external convection: if , convection is more efficient, dT/dr , Tsur since Lo is fixed , radius decreases)

• Zin essentially determines the present metal content in the photosphere, Zphoto

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Results of SSM calculations

Density [gr/cm3]

Temperature [107 K]

R/Ro

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Comparison among different calculations

BP2000 FRANEC GARSOM

Tc 15.696 15.69 15.7 [107K]

c 152.7 151.8 151[gr/cm3]

Yc 0.640 0.632 0.635

Zc 0.0198 0.0209 0.0211

• Good agreement: differences at % level or less

<1%

6%

1%

1%

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Comparison of calculated neutrino fluxes

• 1% BP2000 FRANEC GARSOM

pp 5.96 5.98 5.99 [1010/s/cm2]

Be 4.82 4.51 4.93[109/s/cm2]

B 5.15 5.20 5.30[106/s/cm2]

CNO 1.04 0.98 1.08[109/s/cm2]

..see Episode III

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Z/X Lo AgeSpp

q/q (1) 6% 3% 0.4% 0.4% 2%

dlogTc/dlogq 0.08 0.14 0.34

0.08 -0.14

• Tc is an important observable for calculation of neutrino fluxes.

• It is strongly sensitive to solar quantities:

(Tc/Tc)q =0.6% (1)

The accuracy of the central solar temperature

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Remarks• 3 input parameters to be tuned

(Yin,Zin,

• 3 observables to be reproduced by the evolutionary calculation (Lo,(Z/X)photo, Ro)

• 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.

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The impact of helioseismic data

• Helioseismology determines the present value of the photospheric helium abundance,

Y= 0.249 (1± 1.4%)

• and the transition between the radiative and convective regimes Rb =0.711 (1 ± 0.14%) Ro

• When this is taken into account, one has now 3 parameters and 5 data.

• Acutally there is much more….

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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/

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Relative differences of sound speed squared

• Agreement between model and data at less than 0.5%

U/U

= (

SSM

-su

n )

/SS

M

BP2000

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The accuracy of helioseismic determinations*

Systematic errors in the inversion procedure dominates (starting solar models, numerical …)

3

1U

/U

* Dziembowki et al. Astrop. Phys. 7 (1997) 77

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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?)

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Extraction of U*• Let us invert the

helioseismic data by starting from two (non standard) models.

(u/umod=1% at R=0)

• Inversion gives quite similar seismic models, even near the center

(u/usei=0,1% at R=0)

Z/X + 10% Z/X 10%

Starting models

Results of inversion

Nucl Phys B Suppl 81(2000)95

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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

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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)

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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

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Helioseismic determination of p+p cross section (Spp)

•Consistency with helioseismology requires:Spp=Spp (SSM)(1 ± 2%)

•This accuracy is comparable to the theoretical uncertainty:

Spp(SSM)=4(1 ± 2%)x 10-22KeVb

U/U

(m

od

-SS

M)/

SS

M

Remind: Spp is not measured

Degli Innocenti et al. PLB 416 (1998) 365

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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

(m

od-S

SM

)/S

SM

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Solar axion production• If Axions are

produced ( +Z A +Z ) one has an extra energy loss mechanism in the solar interior (LA)

• LA depends on A- coupling constant (gA)

• gA > 5 10-10 GeV-1 is excluded at 3 level

3

Schlattl et al. Astrop. Phys. 10 (1999) 353

gA=(5,10,15,20) /1010 GeV

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List of applicationsBy 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/0206211]

• Existence of extra-dimensions [PLB 481(2000)291]

• Possible deviation from standard Maxwell-Boltzmann distribution [PLB 441(1998)291]

S

UN

EX

OTIC

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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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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 Hp=1/(dlnP/dlnR) through L= Hp and is used as a free parameter

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Inversion method• Calculate frequencies i as a function of u

=> ii(uj) j=radial coordinate

• Assume SSM as linear deviation around the true sun:

ii, sun + Aij(uj-uj,sun)• Minimize the difference between the

measured i and the calculated i

• In this way determine uj=uj-uj, sun

2

i i

ii2