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Lecture 13
Review: Static Stellar structure equations
Hydrostatic
equilibrium:
Mass conservation:
dT
3 Lr

dr
64 r 2T 3
Radiation
dP
GM r 

dr
r2
dM r
 4r 2 
dr
or
Equation of state: P 
Energy generation:
Polytrope
kT
mH
dLr
 4r 2 
dr
P  
dT  1  mH GM r
 1  
dr    k
r2
Convection
The Solar model
• In this way we can build up a model
of the interior structure of the
Sun
• In general the differential
equations are solved numerically
• Instead of assuming a polytrope,
choose the temperature gradient
depending on the mode of energy
transport
• Boundary conditions:
in the simplest case, , P and T =0
at r=R
M,L=0 at r=0
Convection zones in the Sun
For the solar model we can plot dlnP/dlnT as a function
of radius. Where this is >2.5, radiation is the most
effective form of energy transport.
The Solar interior
The interior can be
divided into three
regions:
1.
Core: site of nuclear
reactions
2. The radiative zone
3. The convective zone
Abundance distribution
• H is depleted in the core, where He is produced
• 23 He is an intermediate species in the pp chain. It is most
abundant at the top of the H-burning region, where the
temperature is lower.
• Abundances are homogeneous within the convective zone, since
the plasma is effectively mixed
The solar model: evolution
• As the abundances in the core change, the nuclear reaction rates change
accordingly, and the luminosity, temperature and radius of the star are
affected.
Energy production
Although nuclear reaction rates are higher where the
temperature is higher, most of the energy is not
produced at the centre of the Sun, because:
2
 The amount of mass in a shell at radius r is dM  4r dr
 i.e. there is more mass per unit volume at large radius (assuming
constant density)
 The mass fraction of hydrogen (X) at the centre has been
depleted due to fusion, and the rate equations depend on X2.
Recall: Proton-proton chain
The net reactions are:
PPI
411H 24He  2e   2e  2
PPII
PPIII
7
4
3
2
Be 11H 224He  e   e  
He  24He 11H  e  224He  e  
Direct observations of the core: neutrinos
• One type of neutrino detector on Earth uses an isotope of chlorine, which
will (rarely) interact with a neutrino to produce a radioactive isotope of
argon.
37
Cl  e 18
Ar  e 
37
17
• This reaction requires the neutrino to have an
energy of 0.814 MeV or more, and can only
detect neutrinos from the “side-reactions” in
the PP chain:
PPII
PPIII
3
4
7
7
1
8
2 He  2 He  4 Be  
4 Be 1 H 5 B  
7

7
8
8

4 Be  e  3 Li   e
5 B  4 Be  e   e
7
3
Li 11H 2 24He
8
4
Be 2 24He
• The Homestake detector contains ~400,000 L of
cleaning fluid
• 2x1030 atoms of Cl isotope
• Detect one Argon atom every 2-3 days.
Direct observations of the core: neutrinos
• More recently, the GALLEX (also SAGE) experiments uses 30 tons of
natural gallium in a 100 ton aqueous gallium chloride solution to detect
neutrinos via:
Ga  e  3271Ge  e 
71
31
• This is sensitive to lower neutrino energies (0.233 MeV) and can detect
neutrinos from the main branch of the PP chain
1
1
H 11H 12H  e   e
2
1
H 11H  23He  
3
2
He  23He  24He  211H
The Solar neutrino problem
Both the Homestake and GALLEX experiments detected fewer
neutrinos (by a factor 2-3) than were expected from the PP-chain
reactions. This problem existed for about 30 years.
The solution to the problem was
suggested by results from the
Super-Kamiokande detector in
Japan
 Results showed that electron
neutrinos produced in the upper
atmosphere can change into
tau- or muon-neutrinos
 This means neutrinos must have
some mass and can therefore
oscillate between flavours.
The Solar neutrino problem… solved
• The Sudbury Neutrino
Observatory uses heavy
water, and was able to
directly detect the flux of
all types of neutrinos from
the Sun.
• The results are now
completely consistent with
the standard solar model.
Break
The main sequence
• The atmospheres of most stars are mostly hydrogen, X=0.7.
• The fraction of metals varies from Z~0 to Z~0.03
• Because of the relative slow burning of hydrogen, the structure of
the star changes only slowly with time.
In general, the central
temperature is higher for
more massive stars
Tc  M 0.23
 Thus, low mass stars will be
dominated by the pp-chain
 Higher mass stars undergo
the CNO cycle
Central density is actually lower
for more massive stars.
Increasing mass
The main sequence
Assuming hydrogen-burning reactions in the core, we can
construct a theoretical relation between L, T and M
• Stars undergoing hydrogen
burning lie along the main
sequence
• For low-mass stars,
<0.08MSun, central
temperatures are not high
enough to allow nuclear
fusion
• At very high masses, M>90
MSun, the stars become
unstable: thermal
oscillations in the core
coupled with extreme
temperature sensitivity of
the nuclear reactions means
an equilibrium is never
attained.
Main sequence lifetimes
At the lower end of the main sequence,
M  0.085M Sun
Te  2.74 103 K
L  5.05 10 4 LSun
Such low-mass stars are entirely convective, so all the hydrogen
(70% by mass) is available for fusion. What is the lifetime of
such a star?
At the upper end of the main sequence,
M  90M Sun
Te  5.27 10 4 K
L  1.1106 LSun
Only the central ~10% of the mass is available for hydrogen fusion,
because the star is not fully convective. What is the lifetime of
such a star?
Stellar lifetimes
• From observations of the cosmic microwave background, we know
the Big Bang occurred about 13.7 billion years ago
• Galaxies have been observed at a time
when the Universe was less than 1
billion years old. Thus stars have been
forming for at least ~13 billion years
Main sequence lifetimes
• Bluer (hotter) stars must be
very young, formed within the
last 0.01% of the age of the
Universe
t star
• On the other hand, some of the
reddest (coolest) stars may
have been formed shortly after
the Big Bang, and would still be
around.
• The stars lying off the main
sequence are not explained by
the hydrogen-burning model:
something else must be going
on…
tUniverse
~ 0.00006
t star
tUniverse
~ 900
The Solar Atmosphere
T~106 K
T~25000 K
T~5770 K
Core
T~107 K
• The solar atmosphere
extends thousands of km
above the photosphere
(from which the optical
radiation is emitted)
• It is of much lower
density and higher
temperature than the
photosphere
The extended solar spectrum
While the solar radiation is similar to a blackbody
prediction at optical wavelengths, there is excess
radiation at very short wavelengths.
 This light is also highly variable.
The chromosphere
HeI emission
• UV (30.4 nm) images reveal the chromosphere
• Can sometimes see large prominences rising high above
the surface of the Sun.
•At the north and south poles of the Sun, less EUV light
is emitted - these regions often end up looking dark in
the pictures, giving rise to the term coronal holes.
 These are low density regions extending above the
surface where the solar magnetic field opens up
The X-ray sun
• The X-rays we see
all come from the
corona.
• The corona is a very
stormy place,
constantly changing
and erupting.
Movie from http://www.lmsal.com/SXT/sxt_movie.html
Sunspots
• Dark (cool) regions of the photosphere
• Number of spots changes on a 11 year cycle
• Concentrations
of magnetic
field lines
The Sun’s magnetic field
• By studying sunspots we can learn
about the nature of the Sun’s magnetic
field
• Switches polarity every 11 years