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

Standard Solar Models IAldo Serenelli

Institute for Advanced Study, Princeton

John N. Bahcall (1934-2005)

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

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

Motivation

• 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?

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

Stellar structure – Hydrostatic equilibrium 1/2

fr

P

r

vv

t

v

r

1D Euler equation – Eulerian description (fixed point in space)

Numerically, Lagrangian description (fixed mass point) is easier (1D)

rt

r

tt mrm

here m denotes a concentric mass shell

mt

r

rr

Gm

m

P2

2

24 4

1

4

and using

and

2r

Gmf rrm 24 Euler equation becomes

Stellar structure – Hydrostatic equilibrium 2/2

min272

10 2

1

22

G

r

r

GmP hydr

hydr

Hydrodynamic time-scale hydr:

2

2

24 4

1

4 t

r

rr

Gm

m

P

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

hydrostatic equilibrium is an excellent approximation

44 r

Gm

m

P

(1)

Stellar structure – Mass conservation

We already used the relation

rrm 24

(2) 24

1

rm

r

leading to

Stellar structure – Energy equation 1/2

Lm is the energy flux through a sphere of mass m; in the absence of energy sources

dqdtm

Lm

.PdVdudq where Additional energy contributions (sources or sinks) can be represented by a total specific rate (erg g-1 s-1)

dt

dq

m

Lm

Possible contributions to nuclear reactions, neutrinos (nuclear and thermal), axions, etc.

Stellar structure – Energy equation 2/2

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

n– (taking > 0)

(3)gnn

m

dt

dsT

m

L

In the present Sun the integrated contribution of g to the solar luminosity is only ~ 0.02% (theoretical statement)

Solar luminosity is almost entirely of nuclear origin

Luminosity constrain:

dmdmm

LL n

m

Stellar structure – Energy transportRadiative transport 1/2

Mean free path of photons lph=/ opacity, density)

Typical values 0.4cm2g-11.4 g cm-3 lph2cm

lph /R transport as a diffusion process

If D is the diffusion coefficient, then the diffusive flux is given byUDF

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 phclD3

1 4aTU and where

In 1-D we get

r

TTacF

3

3

4

Stellar structure – Energy transportRadiative transport 2/2

The flux F and the luminosity Lm are related by

3216

3

Tr

L

acr

T m

24 r

LF m

and the transport equation can be written as

or, in lagrangian coordinates

34264

3

Tr

L

acm

T m

Using the hydrostatic equilibrium equation, we define the radiative temperature gradient as

416

3

ln

ln

mT

PL

acGPd

Td m

radrad

and finally radPr

GmT

m

T

44 (4)

Stellar structure – Energy transportConvective transport 1/3

Stability condition:

b

b

s s

r+r: P+P, T+T,

r: P, T,

rdr

dr

dr

d

sbsb

lnln

Adiabactic displacement

TdPdd lnlnln Using hydrostatic equilibrium, and

radadSb dr

Td

dr

Td

dr

Td

dr

Td lnlnlnln

Stellar structure – Energy transportConvective transport 2/3

Divide by dr

Pd lnand get

radadradad Pd

Td

Pd

Td

ln

ln

ln

ln Schwarzschild criterion for dynamical stability

416

3

mT

PL

acGm

rad

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

regions of large (e.g. solar envelope)

When does convection occur?

Stellar structure – Energy transportConvective transport 3/3

Pr

GmT

m

T44

(4b)Energy transport equation

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

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

rad

mTacGF

2

4

Pr3

4

Using definition of

rad and

24 r

LF m

we can write

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

is the actual temperature gradient and satisfiesradad

2

4

Pr3

4

mTacG

Fradwhere

Stellar structure – Composition changes 1/4

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)

Relative element mass fraction:

1; i

iii

i Xmn

X

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

Stellar structure – Composition changes 2/4

Microscopic diffusion (origin in pressure, temperature and concentration gradients). Very slow process: diff>>1010yrs

iidiff

i wnrmt

n 24

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

m

nDr

mt

n iconv

conv

i 224

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

Stellar structure – Composition changes 3/4

Sun: main sequence star hydrogen burning

low mass pp chains (~99%), CNO (~1%)

Basic scheme: 4p 4He + 2+ + 2e+ ~25/26 MeV

Nuclear reactions (2 particle reactions, decays, etc.)

e.t.11

klkl

kl

lkij

j ij

ji

nuc

i vnn

vnn

t

n

0

)()( dvvvvv here (v) is the relative velocity distrib. and (v) is cross section

Interlude on hydrogen burning – pp chains

2pHeHeHe

HepH

Hepp

Hpp

433

32

2-

2

e

ee

ppI88-89%

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

Q=Q=1.44 pep neutrinos

Q=5.49

Q=12.86

HeHeBe

eBeB

BpBe

448

88

87

e

ppIII1%

Q=0.137

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

HeHepLi

LieBe

BeHeHe

447

7-7

743

e

ppII10%

Q=1.59

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

Q=17.357Be neutrinos

Marginal reaction:

eeHepHe 43

Q=19.795, <Q>=9.625 hep neutrinos

Interlude on hydrogen burning – CNO cycle

HeCpN

eNO

OpN

NpC

eCN

NpC

41215

1515

1514

1413

1313

1312

e

e

CN-cycle

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

Q=1.94

Q=7.55

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

Q=7.30

Q=4.97

Q=1.19

Q=0.600

Q=12.13

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

HeNpO

eOF

FpO

OpN

41417

1717

1716

1615

e

NO-cycle

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

Stellar structure – Composition changes 4/4

diff

i

conv

i

nuc

ii

t

n

t

n

t

n

dt

dn

(5)

Composition changes

i=1,…..,N

Stellar structure – Complete set of equations

Nit

n

t

n

t

n

dt

dn

diff

i

conv

i

nuc

ii ,....,1;

(5)

rad

Pr

GmT

m

T44

(4)

(3)

dt

dsT

m

Ln

m

44 r

Gm

m

P

(1)

(2) 24

1

rm

r

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

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 M=1.9891033g

0.1%

Kepler’s 3rd law

Solar age t=4.57 109yrs

0.5%

Meteorites

Quantities to match

Solar luminosity L=3.842 1033erg s-1

0.4%

Solar constant

Solar radius R=6.9598 1010cm

0.1%

Angular diameter

Solar metals/hydrogen ratio

(Z/X)= 0.0229 Photosphere and meteorites

Standard Solar Model – What we do 2/2

3 free parameters:

• Convection theory has 1 free parameter: MLT 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 1Minitial model with Xini, Zini, (Yini=1- Xini-Zini) and MLT, evolve it during t and match (Z/X), L and R to better than one part in 10-5

Standard Solar Model – Predictions

• Surface helium Ysurf (Z/X and 1=X+Y+Z leave 1 degree of freedom)

• Thermodynamic quantities as a function of radius: T, P, density, sound speed (c)

• 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)

• Depth of the convective envelope, RCZ

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

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 Hz: very long observations are needed. BiSON network (low-l modes) has data collected for 5000 days• Relative accuracy in frequencies 10-5

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

The Sun as a pulsating star - Overview of Helioseismology 4/4

• Oscillation frequencies depend on , P, g, c• Inversion problem: using measured frequencies and from a reference solar model determine solar structure

)()()()()( 22 ,2

2

, isurfi

c

i

ci

i FdrrrKdrrc

crK

Output of inversion procedure: c2(r), (r), RCZ, YSURF

Relative difference of c between Sun and BP00

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)T20-25. 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 (1995-1997).SSM correct in solar interior to better than 1%

Bahcall et al. 1996

History of the SSM in 3 steps

RCZ=0.714 / 0.713 ± 0.001

YSUP=0.244 / 0.249 ± 0.003

005.0/

001.0/

22

22

ccc

8B)= (5.05 ± 0.91) x 106 cm-2 s-1

SK8B)= (2.32 ± 0.09) x 106 cm-2 s-1 (only sensitive to e)

• Step 3. The BP00 model and Sudbury Neutrino Observatory BP00: Bahcall, Pinsonneualt & Basu (2001)

History of the SSM in 3 steps

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

1-2-6 scm1066.009.5

(ES)

MeV222 (NC)

MeV441(CC)

SNO

xx

xx

e

evev

.- vnpdv

. - eppdv

SNO collaboration (2002)

Solution to the Solar Neutrino Problem !!!!

BP008B)/ SNO8B)= 0.99