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Standard Solar Models I
Aldo Serenelli
Institute for Advanced Study, Princeton
SUSSP61: Neutrino Physics - St. Andrews, Scotland – 8th to 23rd, August, 2006
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
1D Euler equation – Eulerian description (fixed point in space)

v
v
P
 v  
f
t r
r
r
Numerically, Lagrangian description (fixed mass point) is easier (1D)

t

m

r

t r t
m

r
here m denotes a concentric mass shell
Gm
and using f   2 and m  4r 2 r Euler equation becomes
r
P
Gm
1  2r


4
m
4r 4r 2 t 2 m
Stellar structure – Hydrostatic equilibrium 2/2
P
Gm
1  2r


4
m
4r 4r 2 t 2
Hydrodynamic time-scale hydr:
1
Gm
r
1
P  0  2  2   hydr  G  2  27 min
r
 hydr
2
hydr << any other time-scale in the solar interior:
hydrostatic equilibrium is an excellent approximation
P
Gm

m
4r 4
(1)
Stellar structure – Mass conservation
We already used the relation
m  4r 2 r
leading to
r
1

m 4r 2 
(2)
Stellar structure – Energy equation 1/2
Lm is the energy flux through a sphere of mass m; in the absence
of energy sources
Lm
dt  dq
m
where dq  du  PdV .
Additional energy contributions (sources or sinks) can be
represented by a total specific rate e (erg g-1 s-1)
Lm
dq
e 
m
dt
Possible contributions to e: 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):
e  e n – en
(taking en > 0)
Lm
ds
 e n  en  T  e n  en  e g
m
dt
(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: L  
Lm
dm   e n  en dm
m
Stellar structure – Energy transport
Radiative transport 1/2
Mean free path of photons lph=1/k k opacity,  density)
Typical values k0.4cm2g-1, 1.4 g cm-3  lph2cm
lph /R31011  transport as a diffusion process
If D is the diffusion coefficient, then the diffusive flux is given by
F  DU
1
and in the case of radiation D  cl ph and U  aT 4 where
3
c is the speed of light and a is the radiation-density constant
and U is the radiation energy density.
In 1-D we get
4ac T 3 T
F 
3 k r
Stellar structure – Energy transport
Radiative transport 2/2
Lm
F
4r 2
The flux F and the luminosity Lm are related by
and the transport equation can be written as
T
3 k Lm

r
16ac r 2T 3
kLm
T
3


or, in lagrangian coordinates
m
64 2 ac r 4T 3
Using the hydrostatic equilibrium equation, we define the
radiative temperature gradient as
d ln T
3 kLm P
 rad 

d ln P rad 16acG mT 4
and finally
T
GmT

 rad
4
m
4r P
(4)
Stellar structure – Energy transport
Convective transport 1/3
r+Dr: P+DP, T+DT,
D
b
Adiabactic
displacement
s
s
r: P, T, 
Stability condition:
b
d ln 
d ln 
Db  D s 
Dr 
Dr
dr b
dr s
Using hydrostatic equilibrium, and
d ln T
d ln T
d ln T


dr b
dr S
dr
ad
d ln T

dr
d ln   d ln P  d ln T
rad
Stellar structure – Energy transport
Convective transport 2/3
d ln P
Divide by
and get
dr
d ln T
d ln T

  ad   rad
d ln P ad d ln P rad
Schwarzschild criterion for
dynamical stability
When does convection occur?
 rad
3 kLm P  large Lm (e.g. cores of stars M*>1.3M)


4
16acG mT
 regions of large k (e.g. solar envelope)
Stellar structure – Energy transport
Convective transport 3/3
Lm

F

Using definition of rad and
we can write
2
4r
4
4acG T m
F
 rad and, if there is convection: F=Frad+Fconv
2
3 k Pr
4acG T 4 m
where Frad 

2
3 k Pr
 is the actual temperature gradient and satisfies  ad     rad
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
T
GmT


4
m
4r P
(4b)
Stellar structure – Composition changes 1/4
Relative element mass fraction:
Xi 
ni mi
; Xi 1
 i
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)
Stellar structure – Composition changes 2/4
Convection (very fast) tends to homogenize composition
2
ni
ni 
 
2

 4r   Dconv

t conv m 
m 
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: diff>>1010yrs
ni
t
diff



4r 2 ni wi 
m
here wi are the diffusion velocities (from Burgers equations for
multicomponent gases, Burgers 1969)
Dominant effect in stars: sedimentation H  Y & Z 
Stellar structure – Composition changes 3/4
Nuclear reactions (2 particle reactions, decays, etc.)
ni
t
 
nuc
j
ni n j
1   ij
 v ij  
kl
nk nl
 v  kl  e.t.
1   kl

here  v   v (v)(v)dv
0
(v) is the relative velocity distrib. and (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
Interlude on hydrogen burning – pp chains
2


p  p  H  e  n e  

2

p  p  e  H  n e
2
H  p3 He  
Q=1.44 MeV, <Qn>=0.265
Q=Qn=1.44
pp neutrinos
pep neutrinos
Q=5.49
He  3 He  4 He  2p  


3
Q=12.86
ppI
88-89%
Q=1.59
Q=Qn=0.86 (90%)-0.38 (10%)
He  4 He 7 Be  
7
Be  e - 7 Li n e
Li  p 4 He  4 He  



7
Q=17.35
7Be
3
neutrinos
ppII
10%
8B
neutrinos
Marginal reaction:
3
He  p4 He  e  n e  
Q=19.795, <Qn>=9.625
hep neutrinos
Q=0.137
Q=17.98, <Qn>=6.71
7
Be  p8 B  
8
B8 Be  e  n e  
Be 4 He  4 He

ppIII
1%
8
Interlude on hydrogen burning – CNO cycle
CN-cycle
 12 C  p13 N  
 13
13

N

C

e
n e  

 13 C  p14 N  

 14
15
N

p

O

 15 O15 N  e  n  
e

 15 N  p12 C 4 He  
NO-cycle
Q=1.94
Q=2.22, <Qn>=0.707
13N
neutrinos
Q=7.55
Q=7.30
Q=2.75, <Qn>=0.996
13N
neutrinos
Q=4.97
 15 N  p16 O  
 16
17
 O  p F  
 17
17

F

O

e
n e  

 17 O  p14 N  4 He  

Q=12.13
Q=0.600
Q=2.76, <Qn>=0.999
Q=1.19
CNO cycle is regulated by 14N+p reation (slowest)
17F
neut.
Stellar structure – Composition changes 4/4
Composition changes
dni ni

dt
t

nuc
ni
t

conv
ni
t
i=1,…..,N
diff
(5)
Stellar structure – Complete set of equations
P
Gm

m
4r 4
(1)
r
1

m 4r 2 
(2)
Lm
ds
 e n  en  T
m
dt
(3)
T
GmT  rad


4
m
4r P 
dni ni

dt
t

nuc
ni
t
(4)

conv
ni
t
; i  1,...., N (5)
diff
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=1.9891033g
0.1%
Kepler’s 3rd law
Solar age
t=4.57 109yrs
0.5%
Meteorites
Quantities to match
Solar luminosity
Solar radius
Solar metals/hydrogen
ratio
L=3.842 1033erg s-1
0.4%
Solar constant
R=6.9598 1010cm
0.1%
Angular diameter
(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 1M initial 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
• 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 , sound speed (c)
• Surface helium Ysurf (Z/X and 1=X+Y+Z leave 1 degree of
freedom)
• 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 mHz: 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
2
i

c

i
i
  K c ,  (r ) 2 (r )dr   K  ,c (r ) (r )dr  Fsurf (i )
i
c

2
2
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
• 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
c  c 
2
/ c 2  0.001
   2 /  2
 0.005
8B)= (5.05 ± 0.91) x 106 cm-2 s-1
SK8B)= (2.32 ± 0.09) x 106 cm-2 s-1 (only sensitive to ne)
History of the SSM in 3 steps
• Step 3. SNO: direct measurement of the 8B) flux.
ve  d  p  p  e  (CC) - 1.44MeV 

v x  d  p  n  v x (NC) - 2.22MeV    SNO      5.09  0.66106 cm -2s -1

SNO collaboration (2002)
v x  e   vx  e  (ES)

BP008B)/ SNO8B)= 0.99
Solution to the Solar Neutrino Problem !!!!