Nuclear reactions
in the Sun
Gianni Fiorentini @Santa Tecla 2003
• A first look at the Sun
• A deeper look:
helioseismology
• Neutrino: the spy of the
innermost sun
• Nuclear reactions in the
Sun, after SNO and
KamLAND
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A first look at the Sun
M= 2 1030 Kg
R=7 108m
Np=M/mp=1057
L=4 1026W
• Observables:
-Mass
-Luminosity
- Radius,
- Metal content of the photosphere
- Age
• Inferences on the typical scales of the
solar interior (r, P, T)
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Who measured the solar mass?
Galileo
Kelvin
Cavendish
Rolfs
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Solar mass
M= 2 1030 Kg
• Astronomy only deals with the
extremely well determined Gaussian constant:
GNMo=(132 712 438  5) 1012 m3/s2
• Astrophysics needs Mo, since:
- opacity is determined by
Ne  Np  Mo/mp  1057
- energy content/production of the star
depends on Np.
• Cavendish by determining GN provided a
measurement of Mo
• The (poor) accuracy on GN (0.15%)reflects on Mo
Mo= 1.989 (1  0.15%)
1033 gr

1057mp
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Solar Luminosity
L=4 1026W
• Derived from measurements of the solar
constant Ko: the amount of energy per unit time
and unit area, impinging from the Sun onto Earth.
• Not precisely a constant, it varies with time
(0.1% in a solar cycle). The average over 12
years of solar irradiance (and over different
satellite radiometers) multiplied by Sun Earth
distance, gives the solar luminosity:
Lo=4pd2Ko =3.844(1  0.4%) 1033 erg/s
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(Lo= 10 17 nuclear power plants)
Solar Radius
R=7 108m
• The distance from the center of the
sun to its visible surface (the
photosphere)
• Difficult to define the edge of the sun
• Different methods and different
experiments:
Ro=6.9598(1 0.04%) 1010 cm
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Solar Age
• Method: radioactive dating of oldest
objects in the solar system
(chondritic meteorites)
• Problems:
– relationship between the
age of the meteorite and the
age of the sun
– what is the the zero time
for the sun?
• The age of the sun referred to
Zero Age Main Sequence
t=4.57(1  0.4%) Gyr
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Solar Metal abundance
• Spectroscopic measurements of the solar
photosphere yield the relative abundances
(in mass) of “metals“ to H
(Z/X)photo=0.0233(1  6%)
• Most abundant metals: O, C, N, Fe
• Results are generally consistent with
the meteoritic abundances
• A remarkable exception: the solar Li content
is depleted by 100 with respect to
meteorites
Note: Hydrogen abundance X~ 0.75
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A remark: Helium abundance
• Helium was discovered in the Sun (1895),
but its abundance cannot be accurately
measured there
• Until a few years ago, estimate of the
photospheric He abundance was taken
the result of solar models
• Helioseismology provides now an indirect
measurement…..
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Typical scales
• density:
r =3Mo/(4pRo3) 1.5 g/cm3
• pressure
P  GMo2/Ro4 1016 dine/cm2
• Sound speed vs  (P/r) 1/2  800 Km/s
• photon mean free path*:
l=1/(ne sTh)
 1/(1024 cm-3 10-24 cm2)  1 cm
(photon escape time 2 104 yr)
• Cumulated energy production
• E =Lot  10-4 Moc2
typical nuclear energy scale
Enuc 1MeV
nuc 

2
1GeV
mc
E chem 1eV
 chem 

2
1GeV
mc
*astrophyscists use “opacity” k: l=1/(r k)
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Sun energy inventory
• The present heat flow L can be sustained by an energy source U
for an age t provided that U>Lt :
a)chemistry:
b)gravitation
c)nuclear
U (1eV)Np
U  GM 2/R
U (1MeV)Np
[yr]
-> tch=2 104
-> tgr=3 107
-> tnu=2 1010
Only nuclear energy can sustain the Solar luminosity over the sun
age, t=4.5 109 y
Can we prove that actually the Sun shines due to
nuclear energy?
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A digression:
The Earth energy inventory
It is not at all easy to understand the
dominant contribution to the Earth energy
production.
M= 6 1024 Kg
R=6 106m
Np=M/mp=1051
HE=4 1013W
• The present heat flow HE=40 TW can be
sustained by an energy source U for a time t=U/HE :
•a)”chemistry”*):
U=(0.1 eV)Np=6 1031J
->tch=5 1010 y
•b)gravitation
U=GM 2/R=4 1032J
->tgr=3 1011y
•c)nuclear **)
U=(1MeV)Np,rad =6 1030J
-> tgr=5 109 y
•-> Thus all energy sources seem capable to sustain HE on
geological times.
*)actually it means the solidification (latent) heat
**)the amount of radioactive material is taken as Mrad  10-8 MEarth
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What is the source of
terrestrial heat?
J Verhoogen, in “Energetics of Earth”
(1980) N.A.S.:
•“What emerges from this morass of fragmentary and
uncertain data is that radioactivity itself could possibly
account for at least 60 per cent if not 100 per cent of the
Earth’s heat output”.
•“If one adds the greater rate of radiogenic heat production in
the past, possible release of gravitational energy (original
heat, separation of the core…) tidal friction … and possible
meteoritic impact … the total supply of energy may seem
embarassingly large…”
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•Determination of the radiogenic component is thus important.
The birth of Nuclear
Astrophysics
Eddington: Nature (1920)
“Certain physical investigations in the
past year make it probable to my mind
that some portion of sub-atomic energy
is actually being set free in a star. … If
five per cent of a star's mass consists
initially of hydrogen atoms, which are
gradually being combined to form more
complex elements, the total heat
liberated will more than suffice for our
demands, and we need look no further
for the source of a star's energy…”
E =Lot 10-4 Moc2
•In the same paper: “If indeed the sub-atomic energy in the stars is being freely
used to maintain their great furnaces, it seems to bring a little nearer to
fulfillment our dream of controlling this latent power for the well-being of the
human race - or for its suicide”
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Nuclear reactions in the sun?
The temperature scale
• We have found scales for P and r
• Need equation of state for deriving T.
• Use perfect gas equation assuming the
Sun consists of fully ionized hydrogen
kT= P/(2np)=P mp/(2r) 1 keV
(T  1.2 107 K)
• [Remark: kT>> e2/r  e2n1/3 implies
perfect gas reasonable]
• However: kT<< e2/rnuc  1MeV
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The gross solar structure
• Core
R < 0.1 Ro
M  1/3 Mo
(the site of nuclear reactions)
• Radiative zone
(0.1  0.7) Ro
M  2/3 Mo
• Convective zone
(0.7  1) Ro
M  2% Mo
(As temperature drops, opacity
increases and radiation is not
efficient for energy transport)
• Photosphere
the very tiny layer of the Sun that
we can observe directly
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How can we look at the solar
interior?
• “If there are more things in
heaven and Earth than are
dreamt of in our natural
philosphy, it is partly because
electromagnetic detection alone
is inadequate”
• We have two probes:
Sound waves and Neutrinos
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Helioseismology
•Birth: in 1960 it was found that the
solar surface vibrates with a period T
 5 min,
•Plan: reconstruct the properties of
the solar interior by studying how the
solar surface vibrates
(as one studies the deep Earth ’s
structure through hearthquakes or
just like you can tell something about
a material by listening to the sounds
that it makes when something hits it)
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Method
• By means of Doppler effect on
the emitted radiation, one can
measure oscillations of the solar
surface with a very high
accuracy
l v v o
 
sint
l
c
c
• Most recent measurements
performed with SOHO satellite
(SOlar and Heliospheric
Observatory)
http://sohowww.nascom.nasa.gov/
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The modes
• The observed oscillations are
decomposed into discrete
modes.
• Some 104 modes are available
• Their frequencies are
accurately determined
(/  10-3 - 10-4)
• So far, only p-modes have
been observed, i.e. pressure
driven oscillations.
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Helioseismic inferences
By comparing the measured frequencies with the
calculated ones (inversion method) one can
determine:
• The sound speed profile
(with accuracy of order
0.5%…see later)
• Locate the transition
between radiative
transport and convection:
Rb =0.711 (1 ± 0.14%) R
• The photospheric He
abundance: Yphoto= 0.249
(1± 1.4%)
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Standard Solar Model (SSM)
• 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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SSM before Helioseismology:
3 data and 3 parameters
• In order to produce a SSM one had to study the
evolution of an initially homogeneous solar mass Mo
up to the sun age t so as to reproduce Lo, Ro and
(Z/X)photo
• One can tune 3 parameters:
-the initial Helium abundance Yin
-the initial metal abundance(s) Zin
-“the mixing length” parameter a
• Up to here, the SSM is not such a big success….
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The impact of helioseismology on SSM
U/U= (SSM-sun )/SSM
3s
1s
Agreement of SSM with helioseismic data within
very tiny errors (for u as well as Yph Rb..)
* Dziembowki et al. Astrop. Phys. 7 (1997) 77
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EXOTIC
SUN
List of applications of helioseismology
By means of helioseismology one can constrain:
• rotation in the solar interior
• 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]
•
•
•
•
Axion production in the sun
WIMPs-matter interaction
[hep-ph/0206211]
Existence of extra-dimensions
[PLB 481(2000)291]
Possible deviations from standard Maxwell- 26
Boltzmann distribution
[PLB 441(1998)291]
Solar rotation
• Solar surface does not rotate uniformely:
T=24 days (30 days) at equator (poles). And
the solar interior?
• Helioseismology
(after 6 years of data
taking) shows that
below the convective
region the sun rotates
in a uniform way
• Note: Erot =1/2 m rotR2  0.02 eV
Erot << KT
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An example: the helioseismic determination
of p+p cross section (Spp)
Degli Innocenti et al.
•Remind: Spp is not measured.
•The value used in SSM
Spp(SSM)=4(1 ± 2%)
x 10-22KeVb
is derived from theory.
•Build solar models with
different Spp
•Consistency with
helioseismology requires:
Spp=Spp (SSM)(1 ± 2%)
•This accuracy is comparable
to the theoretical uncertainty:
PLB 416 (1998) 365
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Can helioseismogy probe the
core of the Sun?
• Accuracy degrades
from 0.1 % to 1% as
moving towards the
solar center.
• This follows form the
fact that p modes do
not propagate in the
innermost part of the
Sun
m =mp/[2x+3/4 Y+1/2 Z]
mean molecular weight
energy
gener.
c
radiative
transport
convective
transport
o
r
e
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Can helioseismogy measure
the solar temperature?
• NO : to go from sound speed to
temperature one needs the chemical
composition
• e.g, for a perfect gas:
m =mp/[2x+3/4 Y+1/2 Z]
mean molecular weight
kT=P/n=(P/r)m
• The abundances of elements (and EOS) is
needed for translating sound speed into
temperature.
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Can helioseismogy prove that
the Sun is shining by nuclear
reactions?
• Just indirectly : Standard Solar Models,
which assume that solar energy is
generated by means of H-fusion into
Helium are consistent with
helioseismology.
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A first summary
• The knowledge of M, R, L and age immediately
determine the physical scales (T, P, r) of the solar
interior and show that nuclear reactions are the
only sufficient source of solar power.
• Helioseismology provides an accurate picture of
the solar interior, in excellent agreement with
SSM calculations
• The accuracy of helioseismology degrades in the
solar core.
• Helioseismology does not measure the
temperature profile.
• We need a proof that nuclear energy sustains the
Sun
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