1. Standard Solar Models I Aldo Serenelli Institute for Advanced Study, Princeton
SUSSP61: Neutrino Physics - St. Andrews, Scotland – 8th to 23rd, August, 2006

2. John N. Bahcall ( )

3. 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

4. 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

5. Motivation
Transition between MSW effect and vacuum oscillations at ~5 MeV % 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?

6. What is inside a Standard Solar Models
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

7. Stellar structure – Hydrostatic equilibrium 1/2
1D Euler equation – Eulerian description (fixed point in space)
Numerically, Lagrangian description (fixed mass point) is easier (1D)
here m denotes a concentric mass shell
and using
and
Euler equation becomes

8. Stellar structure – Hydrostatic equilibrium 2/2
Hydrodynamic time-scale thydr:
thydr << any other time-scale in the solar interior:
hydrostatic equilibrium is an excellent approximation
(1)

9. Stellar structure – Mass conservation
We already used the relation
leading to
(2)

10. Stellar structure – Energy equation 1/2
Lm is the energy flux through a sphere of mass m; in the absence of energy sources
where
Additional energy contributions (sources or sinks) can be represented by a total specific rate e (erg g-1 s-1)
Possible contributions to e: nuclear reactions, neutrinos (nuclear and thermal), axions, etc.

11. Stellar structure – Energy equation 2/2
In a standard solar model we include nuclear and neutrino contributions (thermal neutrinos are negligible):
e = en – en (taking en > 0)
(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:

12. Stellar structure – Energy transport Radiative transport 1/2
Mean free path of photons lph=1/kr (k opacity, r density)
Typical values k=0.4cm2g-1, r=1.4 g cm-3  lph2cm
lph /R8310-11  transport as a diffusion process
If D is the diffusion coefficient, then the diffusive flux is given by
c is the speed of light and a is the radiation-density constant and U is the radiation energy density.
and in the case of radiation
and
where
In 1-D we get

13. Stellar structure – Energy transport Radiative transport 2/2
The flux F and the luminosity Lm are related by
and the transport equation can be written as
or, in lagrangian coordinates
Using the hydrostatic equilibrium equation, we define the radiative temperature gradient as
and finally
(4)

14. Stellar structure – Energy transport Convective transport 1/3
Stability condition: b s
r+Dr: P+DP, T+DT,
r+Dr
r: P, T, r
Adiabactic displacement
Using hydrostatic equilibrium, and

15. Stellar structure – Energy transport Convective transport 2/3
Divide by
and get
Schwarzschild criterion for dynamical stability
large Lm (e.g. cores of stars M*>1.3M8)
regions of large k (e.g. solar envelope)
When does convection occur?

16. Stellar structure – Energy transport Convective transport 3/3
Using definition of
and
we can write
and, if there is convection: F=Frad+Fconv
is the actual temperature gradient and satisfies
where
Fconv and  must be determined from convection theory (solution to full hydrodynamic equations)
Easiest approach: Mixing Length Theory (involves 1 free param.)
(4b)
Energy transport equation

17. Stellar structure – Composition changes 1/4
Relative element mass fraction:
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)

18. Stellar structure – Composition changes 2/4
Convection (very fast) tends to homogenize composition
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: tdiff>>1010yrs
here wi are the diffusion velocities (from Burgers equations for multicomponent gases, Burgers 1969)
Dominant effect in stars: sedimentation H  Y & Z 

19. Stellar structure – Composition changes 3/4
Nuclear reactions (2 particle reactions, decays, etc.)
here
(v) is the relative velocity distrib. and s(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

20. Interlude on hydrogen burning – pp chains
ppI
88-89%
Q=1.44 MeV, <Qn>= pp neutrinos
Q=Qn= pep neutrinos
Q=5.49
Q=12.86
ppII
10%
Q=1.59
Q=Qn=0.86 (90%)-0.38 (10%)
Q=17.35
7Be neutrinos
ppIII 1%
Q=0.137
8B neutrinos Q=17.98, <Qn>=6.71
Marginal reaction:
Q=19.795, <Qn>= hep neutrinos

21. Interlude on hydrogen burning – CNO cycle
CN-cycle
Q=2.22, <Qn>= N neutrinos
Q=1.94
Q=7.55
Q=2.75, <Qn>= N neutrinos
Q=7.30
Q=4.97
NO-cycle
Q=1.19
Q=0.600
Q=12.13
Q=2.76, <Qn>= F neut.
CNO cycle is regulated by 14N+p reation (slowest)

22. Stellar structure – Composition changes 4/4
i=1,…..,N
(5)

23. Stellar structure – Complete set of equations
(1)
(2)
(3)
(4)
(5)
Microscopic physics: equation of state, radiative opacities, nuclear cross sections

24. 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
M8=1.9891033g
0.1%
Kepler’s 3rd law
Solar age
t8=4.57 109yrs
0.5%
Meteorites
Quantities to match
Solar luminosity
L8=3.842 1033erg s-1
0.4%
Solar constant
Solar radius
R8= 1010cm
0.1%
Angular diameter
Solar metals/hydrogen ratio
(Z/X)8=
Photosphere and meteorites

25. Standard Solar Model – What we do 2/2
3 free parameters:
Convection theory has 1 free parameter: aMLT 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 1M8 initial model with Xini, Zini, (Yini=1- Xini-Zini) and aMLT, evolve it during t8 and match (Z/X)8, L8 and R8 to better than one part in 10-5

26. 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 (r), sound speed (c)
Surface helium Ysurf (Z/X and 1=X+Y+Z leave 1 degree of freedom)
Depth of the convective envelope, RCZ

27. 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

28. 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

29. 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

30. The Sun as a pulsating star - Overview of Helioseismology 4/4
Oscillation frequencies depend on r, P, g, c
Inversion problem: using measured frequencies and from a reference solar model determine solar structure
Output of inversion procedure: dc2(r), dr(r), RCZ, YSURF
Relative difference of c between Sun and BP00

31. 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)T 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 ( ).
SSM correct in solar interior to better than 1%
Bahcall et al. 1996

32. History of the SSM in 3 steps
Step 3. The BP00 model and Sudbury Neutrino Observatory
BP00: Bahcall, Pinsonneualt & Basu (2001)
RCZ=0.714 / ± 0.001
YSUP=0.244 / ± 0.003
F(8B)= (5.05 ± 0.91) x 106 cm-2 s-1
FSK(8B)= (2.32 ± 0.09) x 106 cm-2 s-1 (only sensitive to ne)

33. History of the SSM in 3 steps
Step 3. SNO: direct measurement of the (8B) flux.
SNO collaboration (2002)
Solution to the Solar Neutrino Problem !!!!
FBP00(8B)/ FSNO(8B)= 0.99