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Standard Solar Models I Aldo Serenelli Institute for Advanced Study, Princeton

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Standard Solar Models IAldo SerenelliInstitute for Advanced Study, Princeton

SUSSP61: Neutrino Physics - St. Andrews, Scotland – 8th to 23rd, August, 2006

John N. Bahcall (1934-2005)

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

- 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

- Transition between MSW effect and vacuum oscillations at ~5 MeV. 99.99% 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?

- The Sun is a self-gravitating object
- Spherical symmetry
- No rotation
- No magnetic field

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

Hydrodynamic time-scale thydr:

thydr << any other time-scale in the solar interior:

hydrostatic equilibrium is an excellent approximation

(1)

(2)

We already used the relation

leading to

Lmis 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.

(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:

In a standard solar model we include nuclear and neutrino contributions (thermal neutrinos are negligible):

e = en– en (taking en > 0)

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

Mean free path of photons lph=1/kr (k opacity, r density)

Typical values k=0.4cm2g-1, r=1.4 g cm-3 lph2cm

lph /R8310-11 transport as a diffusion process

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)

r+Dr: P+DP, T+DT,

r+Dr

b

Adiabactic displacement

s

s

r: P, T, r

b

Stability condition:

Using hydrostatic equilibrium, and

Divide by

and get

Schwarzschild criterion for dynamical stability

When does convection occur?

large Lm (e.g. cores of stars M*>1.3M8)

regions of large k (e.g. solar envelope)

where

is the actual temperature gradient and satisfies

Fconv and must be determined from convection theory (solution to full hydrodynamic equations)

Easiest approach: Mixing Length Theory (involves 1 free param.)

Energy transport equation

(4b)

Using definition of

and

we can write

and, if there is convection: F=Frad+Fconv

The chemical composition of a star changes due to

Relative element mass fraction:

- Convection
- Microscopic diffusion
- Nuclear burning
- Additional processes: meridional circulation, gravity waves, etc. (not considered in SSM)

X hydrogen mass fraction, Y helium and “metals” Z= 1-X-Y

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

Convection (very fast) tends to homogenize composition

where Dconv is the same for all elements and is determined from convection treatment (MLT or other)

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

Q=1.44 MeV, <Qn>=0.265 pp neutrinos

Q=Qn=1.44 pep neutrinos

Q=5.49

Q=12.86

ppI

88-89%

Q=1.59

Q=Qn=0.86 (90%)-0.38 (10%)

Q=17.35

7Be neutrinos

ppII

10%

Q=0.137

8B neutrinos Q=17.98, <Qn>=6.71

Marginal reaction:

ppIII

1%

Q=19.795, <Qn>=9.625 hep neutrinos

Q=1.94

Q=2.22, <Qn>=0.707 13N neutrinos

CN-cycle

Q=7.55

Q=7.30

Q=2.75, <Qn>=0.996 13N neutrinos

Q=4.97

Q=12.13

Q=0.600

NO-cycle

Q=2.76, <Qn>=0.999 17F neut.

Q=1.19

CNO cycle is regulated by 14N+p reation (slowest)

Composition changes

i=1,…..,N

(5)

(1)

(2)

(3)

(4)

(5)

Microscopic physics: equation of state, radiative opacities, nuclear cross sections

Solve eqs. 1 to 5 with good microphysics, starting from a Zero Age Main Sequence (chemically homogeneous star) to present solar age

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

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)

- 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 heliumYsurf (Z/X and 1=X+Y+Z leave 1 degree of freedom)

- Depth of the convective envelope, RCZ

- 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

- 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 10-5

- 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

- 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

- Step 2. Precise calculations of radiative opacities (OPAL group). Helioseismology: results from low and mid-l sample well the solar interior (1995-1997).
- SSM correct in solar interior to better than 1%

Bahcall et al. 1996

- 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)T20-25. Problem with SSM? Specultive solutions of all kinds. This lasted about 30 years.

- Step 3. The BP00 model and Sudbury Neutrino Observatory

BP00: Bahcall, Pinsonneualt & Basu (2001)

RCZ=0.714 / 0.713 ± 0.001

YSUP=0.244 / 0.249 ± 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)

SNO collaboration (2002)

FBP00(8B)/ FSNO(8B)= 0.99

Solution to the Solar Neutrino Problem !!!!

- Step 3. SNO: direct measurement of the (8B) flux.