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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