Solar Interior (continued)

1
Solar Interior (continued)

• Role of convection energy transport in the solar
  interior

2
The onset of convection

Convection occurs in liquids and gases when the
temperature gradient gt some critical value, i.e.
it begins because the state of the fluid is
unstable. In a gas, convection occurs if a
rising element is less dense than its
surroundings. This depends on the rate at which
the element expands due to decreasing pressure,
and the rate at which the surrounding density
decreases with height.
In the Sun and other stars the temperature falls
going from the core outwards. Thus both dT/dr and
dP/dr are negative. For r lt 0.71 R?, dT/dr is
determined by the Rosseland absorption
coefficient (approximating free-free and
free-bound absorption). When r gt 0.71 R?, as T
falls still further, the free electrons start to
recombine with nuclei to form ions which are
effective absorbers of radiation. As a result the
opacity increases and so dT/dr becomes steeper
(i.e. T falls off with r more quickly).
3
ONSET OF CONVECTION
Illustrating an element of gas rising in the
solar interior and principle of convection.
Element of gas after rising
Values of pressure (P), density (?) outside gas
element

Element of gas before rising
Temperature T1, T1 initially, then T2, T2
after element has risen.
4

Onset of convection (contd.)

• Lets assume that the element rises adiabatically
  and does not exchange heat with its surroundings.
• So, in the figure, for ?2' lt ?2 and T2' gt T2, the
  element keeps rising towards the photosphere.
• Here it radiates, cools, becomes denser, and
  falls.
• Hence a convective cycle is established.
• Evidence for convection can be seen in the
  patterns of granulation and supergranulation in
  the surface layers of the Sun.

5
Figure shows the gas element starting out at P1'
P1 and ?1' ?1. It moves to a new position
with P2' P2 but ?2 ? ?2' and T2 ' ? T2. The
transfer is adiabatic if the element rises
quickly compared with the time to absorb or emit
radiation. Hence, or where ? is the ratio
of specific heats at constant pressure
volume (as we had before) for a fully ionized
gas.
6

In terms of density (?1/?2 V2/V1) Hence
the condition for instability is


(13a) i.e. density inside the
element lt density of surroundings. Before the
element starts rising, it is indistinguishable
from its surroundings, so Also, after the
element rises, the pressures inside outside the
element must be equal Now let ?1 ?, ?2 ?
d? and P1 P, P2 P dP, then inequality
(13a) is


(13b) using the binomial expansion, (1 a)n 1
na if a is small.


7
Now if we consider radial gradients (subscript r)
as in the Suns interior, the condition for
instability becomes Eliminate ?r using ideal
gas equation (µ molecular weight, mH H atom
mass) or
First recall differential calculus rule for
quotients
(13)

8
So inequality (13) becomes and
expanding or But note that both dT/dr and
dP/dr are negative and hence the criterion for
instability is



(14)
9
The RHS of inequality (14) is called the
adiabatic temperature gradient. So there is
convection when the magnitude of dTr /dr is
greater than the adiabatic value. Proceeding
outwards in the solar interior, the absorption
(opacity) increases because of the increased
recombination of free electrons with ions
(particularly Fe ions). This causes the radial
temperature gradient dTr /dr to steepen. Onset of
convection occurs at a level in the solar
interior where the temperature 106 K. The
condition for convective instability is called
the Schwarzschild Criterion. It implies that
when the actual T gradient, dTr /dr is
greater than the dTr /dr adiabatic convection
will begin. (See Schwarzschilds book pp. 44 et
seq. for further description of convective energy
transport.)

10

• We have now derived the basic stellar structure
  equations
• Equation of hydrostatic equilibrium (3)
• Equation of state (8)
• Mass as a function of radius (4)
• Radiation flux Lr as a function of radius (9)
• Radiative temperature gradient (12)
• Condition for convective transport (14)
• The boundary conditions for the solution of these
  equations are M(r0) 0 L(r0) 0
• M(rR?) M? radius of Sun at age 4.6 Gyr is
  R?.
• These can be integrated numerically to give T, P,
  M, and L as a function of radius r a model of
  the solar interior.
• A standard solar model has been developed by J.
  N. Bahcall (see his book Neutrino Astrophysics,
  chap. 4).

11
Solar evolution

• Progressive gravitational contraction of an
  interstellar gas cloud some gravitational
  energy is converted to thermal energy
• Temperature (T ) and particle density (n)
  increase at the centre of the collapsing cloud
  when T and n become large enough, nuclear fusion
  of H begins (nucleosynthesis) and contraction
  stops. Sun is on main sequence, where it spends
  nearly all its lifetime (next slide).
• After 1010 years 10 Gyr, the H is exhausted,
  contraction will resume. Fusion continues in a
  thin H shell whose radius increases with time.
  The layers outside the H shell are heated so that
  the Sun expands to become a red giant (see next
  but 1 slide).
• As core contraction continues the core
  temperature will increase until He fusion begins
  (T 108 K) and contraction halts again.
• When He is exhausted, contraction continues until
  electron degeneracy provides sufficient pressure
  to halt the collapse the white dwarf phase. A
  tenuous envelope of gas is shed to form a
  planetary nebula.

12
Hertzsprung-Russell Diagram
4M?
1M?

0.5 M?
Stellar masses indicated in units of 1 solar mass
1 M?
0.25 M?
13

• Evolution of the Sun
• Contraction
• Main Sequence
• Red giant
• He-burning stage
• White dwarf

14
NGC 3132 nebula
Examples of Planetary nebulae
Cats Eye nebula
15
Solar evolution core temperature and density
MAIN SEQUENCE
Core density
Shaded area convectively unstable
Core temperature
PRESENT
16
Nuclear fusion reactions

• For nuclear reactions to occur, the K.E. of the
  interacting particles must be large enough to
  overcome the Coulomb (charge) barrier and bring
  them within the range of the (residual) strong
  nuclear force 1.7 10-15 m ( 1.7 femtometers
  1.7 fm).
• Four factors determine the most likely nuclear
  reactions
• Abundance of nuclear species
• Reaction probability at the core temperature
• Nuclear charge, Z
• Production of stable nuclei as reaction products
• Since the Coulomb force scales with Z 2,
  reactions occur preferentially between low-Z
  nuclei. The most abundant are
• H, He, C, N, O, Ne, Mg Si.
• At the temperature of the Suns core, some
  reaction rates are very slow only one reaction
  per particle occurs in 1.41010 years.

17
Nuclear reaction sequences in Sun
Particle energies in the core (temperature 15.6
MK) are at the few keV level, but typical Coulomb
barriers for light elements are MeV. Fusion in
main sequence stars only begins as the result of
quantum mechanical tunnelling (i.e. wave
functions of interacting particles penetrate
each other).

• Two nuclear reaction sequences occur in the Sun
• Proton-proton (p-p) chain gives 99 of the
  energy
• Carbon-Nitrogen (CN) cycle gives 1 of the
  energy.

18
Protonproton chain

• There is a series of reactions. First, deuterium
  (2H) is formed
• from the collision of two protons.
• 1a) p p ? 2H e ?e with Q 1.422 MeV,
  E? lt 0.420 MeV
• Alternatively, the pep reaction can start the
  chain
• 1b) p e- p ? 2H ?e with Q 0.4 MeV,
  E? 1.442 MeV
• where Q is the amount of energy produced (e.g. in
  ?-ray photons, positron annihilation, and
  particle kinetic energy) which is available for
  heating the Sun (i.e. to produce its luminosity)
  
• E? is the neutrino energy (neutrinos escape from
  the Sun and so carry away this energy).
• Both reactions proceed very slowly first, 2
  protons must penetrate each other to within 1.7 x
  10-15 m (only 10-8 of them do), secondly, one
  must undergo an inverse ß decay during the
  collision process
• p ? n e ?e

19
Protonproton chain (contd.)

• The next step happens within 10s
• 2H p ? 3He ? Q 5.493 MeV
• followed by
• 3He 3He ? 4He 2p Q 12.859 MeV
• Note that (1a) and (2) or (1b) and (2) must occur
  twice for each time (3) occurs.
• The net result is 4p ? 4He 2e 2?e
  plus energy available for heating the Sun.

.
20
Protonproton chain (contd.)

• Important side reactions (producing neutrinos
  observable if in blue)
• 4) 3He 4He ? 7Be ? with Q 1.587 MeV
• followed by either of these two reactions
• 5) 7Be e- ? 7Li ?e with Q 0.862 MeV and
  E? 0.862 MeV (89.7)
• or
• 6) 7Li p ? 2 4He with Q 17.347 MeV
  and E? 0.384 MeV (10.3)
• Or these reactions
• 7Be p ? 8B ? Q 0.135 MeV
• 8B ? 8Be e ?e Q 15.079 MeV E? lt
  14.02 MeV
• 8Be ? 2 4He Q 2.995 MeV
• In addition there is a very rare reaction (hep
  reaction)
• 10) 3He p ? 4He e ?e E? lt 18.773 MeV

21
Carbon-Nitrogen cycle

• This set of reactions contributes 1 of the
  solar luminosity and is only important in the
  interiors of higher temperature stars. The
  summary reaction is
• 4p ? 4He 2 e 4? 2 ?e Q
  25.01 MeV
• with C and N nuclei involved, acting as
  catalysts.

22
Reaction Rates
To get energy generation rates per unit mass per
second, we evaluate the reaction rates (units
m-3 s-1)
where N1 and N2 are the number densities of
nuclei with atomic number Z1 and Z2 and atomic
mass A1 and A2. The factor lt?vgt12 is a rate
coefficient or temperature-averaged cross
section, and is an integral of the product of
the cross section ?(v), the particle velocity v,
and the Maxwell-Boltzmann velocity distribution
(velocities v are in the centre-of-mass system
of the reacting nuclei)
23
where
is the reduced mass, E is the centre-of-mass
energy,
T is the temperature, S(E) is a function which
defines the cross sections and f0 is a factor
that describes the screening effect of
electrons. The additional exp (-2??) term
expresses the probability of penetrating the
Coulomb barrier and is called the Gamow
penetration term
where A is related to the atomic masses A1, A2 by
S(E) can be measured, but only at E ? 200 keV.
S(E) for lower energies are obtained by
extrapolation to lower energies.
24
The energy generation rate, ?, is set by the
reaction rate per unit volume, the energy per
reaction, and the density. Hence, e f0 X1
X2 ? T a
where X1 and X2 are reactant fractions (X1 X2
for p-p chain) and a 4.5 for the p-p chain (it
is a 20 for the C-N cycle). So e is
extremely dependent on the core temperature.
Fusion suddenly turns on during the solar
contraction phase when T reaches a particular
value.
25
Standard solar model

• The points discussed so far allow us to specify
  the standard solar
• model describing the physical conditions in the
  solar interior as follows.
• Abundances
• The model begins with a homogeneous chemical
  composition (H or
• X 0.71, He or Y 0.27, metals or Z 0.02)
  changes that occur
• during the 4.6 109 years to the present day
  are assumed only to occur
• due to fusion reactions.
• Hydrostatic equilibrium
• The Sun is assumed to be in hydrostatic
  equilibrium.
• Energy transport
• In the deep interior (out to 0.71 R?), energy
  transport is by radiation.
• Outside this region, energy transport is by
  convection.
• Energy generation
• Nuclear fusion reactions (mostly p-p chain) are
  the primary source of
• energy.

26
Conditions inside Sun Bahcalls models
Luminosity
Hydrogen fraction
Temperature
Density
r/R0
27
Solar neutrinos
There are uncertainties in the solar standard
model because of the Suns internal composition
and age, and from errors in opacities, nuclear
reaction rates and the equation of state
(relating pressure and density) in the dense
central regions.
In the p-p chain, neutrinos of the following
energies are produced lt 0.420 MeV 1.442 MeV
0.862 MeV 0.384 MeV lt 14.02 MeV lt 18.773
MeV and for the C-N cycle lt 1.199 MeV lt
1.732 MeV lt 1.740 MeV
28
Calculated Neutrino Fluxes
Spectra of solar neutrinos p-e-p and 7Be
reactions produce neutrinos with single
energies. p-p chain reactions produce neutrinos
with continuous energies.
Energy range of Neutrino experiments
hep neutrinos are those produced by 3He p
reactions.
29
Predicted neutrino fluxes at Earth (standard
solar model)
Source Reaction no. Neutrino energy (MeV) Flux (1014 m-2 s-1)
Proton-proton
p-p 1a lt 0.420 6.0
p-e-p 1b lt 1.442 0.014
hep 10 lt 18.773 8x10-7
7Be 5 0.862,0.384 0.47
8B 8 lt 14.02 5.8x10-4
CN cycle
13N lt 1.199 0.06
15O lt 1.732 0.05
17F lt 1.740 5.2x10-4
30
Neutrino production as a function of
solar radius
31
The Homestake Neutrino Detector
The first solar neutrino experiment was set up in
1968 by Raymond Davis and collaborators. It
operated continuously from 1970 to 1994. The
experiment was 1.5 km deep in the Homestake gold
mine in South Dakota (to shield it from cosmic
rays). Its operation was based on the
reaction 37Cl ?e ? 37Ar e- The neutrino
energy threshold is 0.814 MeV. From the
standard model most (77) detectable neutrinos
for this reaction are from reaction (8) in the
pp chain, i.e. ? decay of 8B to 8Be. 13 are
from reaction (5). The detector consisted of a
large (400 m3 ) tank of the cleaning fluid
perchloroethylene (C2Cl4) containing 2.2 1030
Cl atoms. Just under 2 solar neutrino-induced
reactions were expected per day.
32
The Homestake Mine Neutrino Detector in South
Dakota. The tank contained 400 m3 of
perchloroethylene. Raymond Davis Jr
(1914-2006), Nobel Prize winner in 2002.
33

• A typical run lasted 80 days, after which He
  gas was bubbled through the tank
• to pick up 37Ar atoms.
• A measured volume of 37Ar gas was then placed in
  a proportional counter (similar to a Geiger
  counter but with pulse height proportional to
  photon energy).
• 37Ar decays by electron capture and emits 2.82
  keV electrons that
• are detected by the proportional counter.
• Counting was carried out over 8 months to
  estimate the background -
• 37Ar decays in 3 months.
• New unit introduced by Bahcall - the solar
  neutrino unit or SNU
• 1 SNU ? 1 neutrino capture per sec in 1036 target
  atoms
• 5.35 SNU ? 1 37Ar atom per day
• Observed 37Ar production rate 2.55 0.25 SNU
• Prediction rate from Bahcalls standard solar
  model 9.3 1.3 SNU

34
Other experiments include

Chemical detection in large Ga targets, either
aqueous (GALLEX) or solid (SAGE). Ge decays by
electron capture with a half-life of 11 days
the resulting electrons are detected as for the
37Cl experiment. Neutrino scattering by
electrons orbiting atoms in water molecules in
highly purified water (KAMIOKANDE and SUPER
KAMIOKANDE) or heavy water (Sudbury Neutrino
Observatory, SNO). The electron generates
Cerenkov radiation which is detected by arrays
of photomultipliers. SUPER KAMIOKANDE and SNO
also detect muon neutrinos. Cerenkov radiation
is generated by electrons with v gt c / n where n
is the refractive index. Radiation is emitted in
a cone with half-angle cos ? 1 / n ?, where ?
v / c. KAMIOKANDE thus determines the
direction of neutrinos.


35

SUPER KAMIOKANDE neutrino detector (Japan) The
detector, which measures 40m (tall) 40m (wide),
is filled with purified water, and has 13,000
photomultiplier tubes that detect Cerenkov
radiation from recoiling electrons struck by
incoming neutrinos, in the form of a cone of
light. The detector is 1km underground. Unlike
other neutrino detectors, it can determine the
neutrino direction.
36
Super Kamiokande under construction filling up
with water
37
Suns image in neutrinos 500 days of Super
Kamiokande data
90
38
Measured and predicted neutrino fluxes
Target Experiment Threshold energy (MeV) Measured neutrino flux (SNU) Predicted neutrino flux (SNU) Ratio measured /predicted
Chlorine 37 Homestake 0.814 2.56 9.5 0.27?0.02
Water Kamiokande 7.5 2.80 6.62 0.42?0.06
Gallium 71 Gallex 0.2 69.7 136.8 0.51?0.06
Gallium 71 SAGE 0.2 72 136.8 0.53?0.10

39
Observed neutrino rate is 1.93.6 times less than
predicted by theory.
Possible explanations for the discrepancy

1. The standard solar model is wrong
2. The experiments are wrong
3. Standard model of particle physics is wrong

The neutrinos detected in the 37Cl ? ? 37Ar
e- reaction (in the Homestake expt.) are mostly
those due to 8B decays. The 8B neutrino
production rate ? T 15, so only a slightly wrong
T in the standard solar model could account for
the discrepancy.
40

1. Low Z model (dT/dr)rad ?

hence a lower central temperature
can be achieved by decreasing
. This can be done with a
lower abundance of heavy elements in the core.
Need to assume Z/X 0.1 (Z/X)surface to obtain
1/3 of standard model neutrino flux. But! This
is ruled out by helioseismology (p-mode
oscillations).

• Rapid core rotation If the core were to rotate
  1000 times faster
• than the surface, the thermal pressure
  required to support the Sun
• against gravity would be reduced and the
  neutrino flux would
• decrease. But! analysis of p-mode
  oscillations shows no
• increase of rotation rate between r 0.2
  R? and r R?. This is
• outside the region of the core where 8B
  neutrinos are produced.

3. Mixing in the Sun the core progressively
depletes H by fusion (note diffusion
of nuclear species is negligible). Now the
energy generation rate ? ? X 2 ? T 4.5, so
somehow returning H to the core would allow a
reduced value of T. To obtain a factor 3
reduction in neutrino flux requires 60 of the
Suns mass to have been mixed for the Suns
lifetime. But! There is no evidence of mixing at
this level.
41
Neutrino mixing the Neutrino Problem solved
There are 3 different neutrino states or
flavours ?e, ?? , ?? associated with the
electron, muon, and tau particle in weak
interaction decays. Most solar neutrino
experiments so far have only detected electron
neutrinos. Mikheyev and Smirnov (1985) and
Wolfenstein (1978) showed that neutrinos can
change state in the presence of other matter if
they have a tiny rest mass. This change of state
is called the MSW effect. Masses are very small
the ?e has a mass that is lt 2.2 eV/c2. Hence the
emitted electron neutrinos could be mixed into
muon or tau particle neutrinos modified as they
leave the Sun. So electron neutrinos would only
be 1/3 of the total number of neutrinos. The
solar neutrino problem has now been resolved by
mixing of the neutrino flavours as they emerge
from the solar core many detectors only see
electron neutrinos which are 1/3 of the total
neutrinos arriving at Earth.
42
THE SOLAR PHOTOSPHERE

43
Solar rotation

• All stars must rotate - random eddy motion in
  protostellar clouds transfers angular momentum.
• The Sun rotates rather slowly compared with many
  similar stars.
• Pressure differences drive flows between the
  poles and the equator.
• The Sun does not rotate rigidly (rotates faster
  at equator).
• In the convection zone convective motions are
  much faster than circulation currents - this
  leads to a distribution of angular velocity that
  varies both with depth and latitude ?
  differential rotation.

44
Measured rates of solar rotation
Solar rotation rates observed by Doppler shifts
at the solar limb and by sunspots.
Latitude N or S (o) Period from Doppler shift (days) Spot period (days)
0 25.6 24.7
10 25.7 24.9
20 26.0 25.3
30 26.6 26.0
40 27.7 26.9
50 29.3
60 31.4
70 33.6
80 33.5
45
Flows in the solar interior

46
Flows in the solar interior (contd.)
Faster equatorial rotation ascribed to
large-scale motions meridional flows from
equator to pole near surface, return flows from
pole to equator well beneath surface. Flow
velocities measured by the Michelson Doppler
Imager instrument on SOHO are 10 m
s-1 Observations from sunspots are slightly
ambiguous small sunspots rotate slightly
faster than large spots.
47
Solar spectral irradiance ( f? or f?)
Theory (HRSA Harvard Reference Standard
Atmosphere) and observed distributions Black
body
48
Photospheric radiation
f? as a function of wavelength (or frequency) is
measured with bolometers, either Earth-based or
on spacecraft. The values of f? are corrected
to the mean Earth-Sun distance (149,600,000 km).
Its integral over frequency
is the total solar irradiance (formerly known as
the solar constant) total amount of radiation
per unit energy per unit time per unit area
reaching the top of the Earths atmosphere.
49
Variability of Total Solar Irradiance
The total solar irradiance is slightly variable
and depends on the 11-year sunspot cycle. Total
range of variations is 0.3, and is at a maximum
at sunspot maximum. But when there are large
sunspots the total solar irradiance may dip by
0.25. Its mean value is about 1.368 103 W
m-2 (1.368 kW m-2).
50
Irradiance (W m-2)
51
More recent information...
D. Pesnell (AGU 2008)
52
The Suns Effective Temperature

• The Suns mean total irradiance is 1.368 103 W
  m-2
• This is incident on a sphere, radius 1 A.U.
  1.496 1011 m.
• So total energy/second received by sphere is
• 1.368 103 4 p (1.496 1011)2 W.
• Radiation is uniformly emitted from the entire
  surface of the Sun. Radius of Sun is R? 6.96
  108 m.
• If the Sun were a perfect black body, the
  radiation emitted would be
• 4 p R?2 s Teff4 W
• where Teff is the Suns effective temperature and
  s Stefan-Boltzmann constant 5.67 10-8 W m-2
  K-4. This is the Suns luminosity.
• Therefore
• i.e.
• Teff 5778 K.