Transcript Stellar structures and the standard solar model

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
Metter efigura ammasso
L
• 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.
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Ts
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 k(r, T, Xi)
-the energy production per unit mass
e(r,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
(for a fixed solar structure)
gas
•Perfect gas law accurate at 10-3 in the core
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• worsen in the outer regions, 2-4%
Radiative opacity k
• Opacity is connected with photon
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
mean free path.
ρκ
• In the radiative region, kgoverns the
temperature gradient (…see next).
• The evaluation of k 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 group7
Dk/k 3 %
(assumed 1s)
Nuclear energy production
*
e
• The expression for the nuclear energy
production e 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
* e  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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spatial system
The basic equations
1)Hydrostatic equilibrium
2)Continuity equation
3)Transport equation
4) Energy Production
5)Equation of state
6)Time evolution
dP
GMr ρ

dr
r2
dM
 4 r 2ρ
dr
dT
3 ρk L r

dr
4ac T 3 4 r 2
dL
 4 r 2ρ ε
dr
P  P(ρ, T, X i )

dX i mi 
   sv ij    sv ik 


dt
ρ  j
k

•First 1-5) is solved for a given Xi (r) [5 eqs and 5
unknowns:can be solved if we know k(rT Xi) and e(rT Xi)]
•Next 6) is applied for a step Dt and the new values for
Xi(r) is used to solve again 1-5)
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See Kippenham and Weigert, “stellar structure and evolution”,Springer Verlag, 1990
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 gr
to=4.57(1  0.4%) Gyr
X= hydrogen
Y= helium
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Z=metals
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
(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 a affects only Ro
(to reproduce Ro one adjusts the efficiency of external
convection: if a, 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
<1%
rc
152.7
151.8
151
1%
Yc
0.640
0.632
0.635
1%
Zc
0.0198
0.0209
0.0211
6%
[107K]
[gr/cm3]
• Good agreement: differences at %
level or less
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Comparison of calculated
neutrino fluxes
• 1%
pp
BP2000
FRANEC GARSOM
5.96
5.98
5.99
Be
4.82
4.51
4.93
B
5.15
5.20
5.30
CNO
1.04
0.98
1.08
[1010/s/cm2]
[109/s/cm2]
[106/s/cm2]
[109/s/cm2]
..see Episode III
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The accuracy of the central
solar temperature
• Tc is an important observable
for calculation of neutrino fluxes.
• It is strongly sensitive to solar
quantities:
Dq/q (1s)
Z/X
6%
dlogTc/dlogq 0.08
k
3%
Lo
0.4%
Age
0.4%
Spp
2%
0.14
0.34
0.08
-0.14
(DTc/Tc)q =0.6% (1s)
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Remarks
• 3 input parameters to be tuned (Yin,Zin,a)
• 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/r)
profile of the solar interior (inversion method):
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DU/U= (SSM-sun )/SSM
Relative differences of
sound speed squared
BP2000
• Agreement between model and
data at less than 0.5%
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The accuracy of helioseismic determinations*
3s
DU/U
1s
Systematic errors in the inversion procedure
dominates (starting solar models, numerical …)
* 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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Nucl Phys B Suppl 81(2000)95
Extraction of U*
• Let us invert the
helioseismic data by
starting from two
(non standard)
models.
(Du/umod=1% at R=0)
• Inversion gives quite
similar seismic
models, even near
the center
(Du/usei=0,1% at R=0)
Z/X + 10%
Z/X 10%
Starting models
Results of inversion
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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:
m =1/[2x+3/4 Y+1/2 Z]
mean molecular weight
u=P/r= T/m
• The abundances of elements (and
EOS) is needed to translate sound
speed in temperature.
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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++ne
-plasma physics: screening effects
-new physics: solar axion emission
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Helioseismic determination of p+p
cross section (Spp)
Degli Innocenti et al.
•Consistency with
helioseismology
requires:
Spp=Spp (SSM)(1 ± 2%)
PLB 416 (1998) 365
•This accuracy is
comparable to the
theoretical uncertainty:
Spp(SSM)=4(1 ± 2%)
x 10-22KeVb
Remind: Spp is not measured
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Screening of nuclear charges
in the plasma
Fiorentini et al.
•Screening modifies
nuclear reactions rates
PLB 503(2001) 121
•Thus it can be tested by
means of helioseismology
•TSYtovitch anti-screening
is excluded at more than 3s
•NO Screening is also
excluded.
•Agreement of SSM with
helioseismology shows that
(weak) screening does exist.
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Solar axion production
Schlattl et al. Astrop. Phys.
• If Axions are produced
(g +Z A +Z ) one has
an extra energy loss
mechanism in the solar
interior (LA)
10 (1999) 353
3s
• LA depends on A-g
coupling constant (gA)
10-10
GeV-1
• gA > 5
is
excluded at 3s level
gA=(5,10,15,20) /1010 GeV
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SUN
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]
EXOTIC
List of applications
•
•
•
•
Axion production in the sun
WIMPs-matter interaction
Existence of extra-dimensions
Possible deviation from standard
Maxwell-Boltzmann distribution
[hep-ph/0206211]
[PLB 481(2000)291]
[PLB 441(1998)291]
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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=a Hp and a is used as a
free parameter
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Inversion method
• Calculate frequencies wi as a function of
u =>
wi wi(uj)
j=radial coordinate
• Assume SSM as linear deviation around
the true sun:
wiwi, sun + Aij(uj-uj,sun)
• Minimize the difference between the
measured Wi and the calculated wi

2
 W i  wi
 
 DW
i
i





2
• In this way determine Duj =uj -uj, sun
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