Transcript Die Entstehung der Sonne

Internal structure of
the Sun
Main characteristics of the Sun and
some constants
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Mass: 1.9891 1030 kg (333 000 x Earth)
Luminosity: 3.846 1026 W
Mean diameter: 1.3914 106 km (109 x Earth)
Flattening: 9 10-6
Surface area 6.09 1012 km2
Volume 1.41 10 18 km3
Average density: 1.408 gcm-3
Escape velocity 617.7 kms -1
Temperature in the centre: ~1.57 107 K,
Temperature of the photosphere: 5777 K
Distance: 149 597 870 700 m
Mass-loss due to wind: (2-3)10-14 Msunyr-1
G=6.67408 10-11 m3kg-1s-2
k=1.38064852 10 -23 J/K
h=6.626070040 10-34 Js
The 5 basic equations:
1. The Condition for Hydrostatic Equilibrium
2. Conservation of mass
3. Thermal equilibrium
The energy that is produced is (with , the “energyproduction-rate”):
Two equation for the transport of energy:
In the Sun, energy is transported in two
forms: convection (Fc) and radiation (FR)
The Transport equation for radiation (FR):
 is the absorption coefficient,
 the sources of radiation within that volume.
If we observe a tube in which radiation is absorbed and
produced under a certain angle  we get:
 and  are very complicated functions but we can take a simplified
approach here. Let us take the Kirchhoff-Planck function:
to define the Rosseland-Mean-Oacity:
Integrating over all frequencies gives
(Stephan-Boltzmannsches-constant ):
Energy transport via convection I
In principle the energy transport has to be calculated hydro-dynamically but this is
quite complicated. From 1915 to 1930 G.I. Taylor, W. Schmidt und L. Prandtl
(famous to have developed the theory of the boundary layer which is essential for
aeronautical engineering) developed a simplified theory of the convective energy
transport. This theory is called the mixing-length theory, and was introduced to
astronomy by Erika Böhm-Vitense (born 1923 in Curau, SH).
If the actual temperature gradient exceeds the adiabatic temperature gradient the
layer is unstable and convection sets in (Schwarzschild criterion):
Energy transport via convection II
An important aspect of the mixing-length theory is that only
average values are
used. If we take as the temperature gradient within a bubble,
and as the temperature gradient outside the bubble
we obtain the equation for the energy transport by convection as:
The 5 basic equation for the star are:
1838: First measurement of the solar
constant : 3.846 1026 W
• John Herschel
(1792-1871)
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• Claude Poullet
(1790-1868)
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Energy production inside the sun
Thermal equilibrium: what actually is ?
Proton-Proton Zyklus
• 4 Protons -> He + 2 Positrons +2 Neutrinos
• Energy production in the core of the sun only: 276 Wm-3!
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Energy production inside the sun II
Important reactions: PPI, PPII, PPIII (CNO only 1%).
Branching ratio: 87:13:0.015
Eddington first proposed that the Sun was powered by nuclear
fusion. It is now known:
4 H → 1 He
4 MH = 4.032 atomic units
MHe = 4.0026 atomic units
If we assume that the sun is 100% hydrogen and that all is converted to Helium:
E = 0.007 Msun c2 = 1.3 ×1045 J
Lifetime of sun = 1011 years
Energy Generation in different types of stars
A hint to the source of the Sun‘s energy comes from the fact that the
luminosity of a star depends on the central temperature:
Sp. T
Teff
M (solar)
Tc (106K)
L (solar)
O5
40000
40
48
500000
B0
28000
18
36
20000
B5
15500
6.5
26
800
A0
9900
3.2
21
80
A5
8500
2.1
18
42
F0
7400
1.7
17
6.3
F5
6580
1.3
15
2.5
G0
6030
1.1
14.4
1.3
G5
5520
0.9
13.7
0.8
K0
4900
0.8
12.9
0.4
K5
4130
0.7
12.4
0.16
M0
3480
0.5
10.9
0.06
L/Lsun = 4.4 ×10–13 T610.7
The Standard Solar Model
Results:
Core chemical composition has changed signficantly since the sun arrived
on the main sequence. There has been a 25% increase in the luminosity
since the sun arrived on the main sequence
Region
Extent
Composition
Core
0 – 0.2 R‫סּ‬
35% H, 63% He, 2% Z
Radiative Zone
0.2 – 0.7 R‫סּ‬
75% H, 23% He, 2% Z
Convection Zone
0.7 – 1.0 R‫סּ‬
75% H, 23% He, 2% Z
Oscillations of the Sun
Solar oscillations
Discovery
In 1975 solar oscillations were
discovered by Franz-Ludwig
Deubner and independently by
Robert B. Leighton using Dopplershifts of spectral lines.
The figure shows the Doppler motions
in the photosphere.
The oscillations have typical periods of
about 5 minutes,
or
=0.01-0.03 s -1.
Basics about oscillations
The length of the signal T (length of the time that the signal is observed)
allows resolve frequencies of =2/T.
The highest frequency that we can observe is given by the sampling rate
(Nyquist frequency) t: Ny=/t.
In frequency domain: 2/T≤  ≤ /t
In space domain: 2/Lx≤ kx ≤ /x
P and G-modes
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P-modes: (acoustic modes) restoring force pressure. They live in the convection zone.
G-modes: restoring force gravity: they live in the core but have not yet been detected
on the sun yet. We thus discuss only P-modes here (A is the acoustic cutoff
frequency) .
Propagation of the waves
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Let us assume a wave that travels downwards into the sun under a small angle.
The speed with which the waves propagate is the thermal velocity of the atoms. Thus, the speed is
proportional to sqrt(T)
Because the temperature increases inside the sun, speed of that part of the wave that is closer to the
surface is slower.
The wave bends, and is finally reflected back to the surface.
How are the oscillations described
Because the sun is a sphere, we should use
spherical coordinates. We thus use the same
description which is familiar to us from
quantum-mechanics (Kugelfächenfunktionen
mit Legendre-Funktionen
)
The meaning of the ”quantum-numbers” for a star:
• n : number of radial knots.
• l : total number of knots on the stellar surface.
• m : Number of knots that go over the poles.
The solar oscillations are studied with a network of telescopes (GONG)
Results
• Chemical composition inside the sun
• Depth of the convection zone
• Ration inside the sun
Solar-like
oscillations
SONG network of 1-m-telescopes to
observe stellar oscillations:
8 telescopes around the word planed. Spectrograph
with R=60000, and R=12000; Two knots are already
operational:
Izaña Tenerife
Delinghai, Qinghai plateau, west China (3200m).
Proposed additional sites:
Virgin islands, US continental site, Hawaii.
The Granulation
The different types of Granulation
1.) Granulation: cells with the size of 1000 km
2.) Mesogranulation: cells with a diameter of
5000-10000 km
3.) Supergranulation: cells of 20000 to 50000
km diameter
4.) Giant Cells: cells of 100000 km diameter
The granulation is caused by the convection in the outer layers
of the sun but it is not the convection itself.
The granulation is caused by convective cells the are
„overshooting“ into a none-convective layer.
The granulation is nevertheless an indication that convection
plays a role in the outer layers of the sun.
The thickness of this overshoot layer is only 200 km.
The granulation consists of the granules and intergranular
layers.
The Granulation
• There are typically one million Granules on the solar
surface.
• The life-time of a Granule is about 10 minutes (35%
fragment, 60% dive down, 4% merge with other
cells).
• The vertical velocities are about 2 km/s.
• The intensity contrast in the optical is about 30%.
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Spectra of the Granulation
• If we take a spectrum of the granulation, we can really
see that the material is moving upwards in the granules,
and downwards in the in the intergranular lanes.
• Under good seeing conditions, the spectra-lines are
wiggly.
• Under very good conditions we can even see that the
down-streaming material in the intergranular lanes is
faster than the up-streaming in the granules.
„Line Wiggles“
bisectors
Bisectors of Granules, intergranular lanes in
active and quite regions )
The Mesogranulation
The number of Mesogranules on
the solar surface is about
100000.
The life-time of a Mesogranule is
about 3 hours.
The vertical velocities are about
60 m/s.
Mesogranules were discovered
in so-called „Cork-Movies“ of
the granulation.
The Super-Granulation
• The number of Super-granules on the solar surface is
about 1000.
• The life-time of a Super-granule is about one day.
• The vertical velocities are about 40 m/s.
• The Super-Granulation can detected by studying the
velocity-pattern on the sun.
• The Super-Granulation nicely shows up in images taken
with a CaII Filter as „chromosphereic emission network“.
Giant Cells
• The giant cells were only found after a long, and
careful analysis of the very large -scale velocitypattern on the sun.
• The diameter of the cells is with 100000 really large.
• Detecting the giant cells was really difficult, because
the differential rotation of the Sun had to be
subtracted carefully.
Giant Cells
W hat is the origin of the
structures?
• At a depth of 2000 km hydrogen is ionized by 50% -> Granulation
• At a depth of 7000 km Helium I is ionized by 50%
(50% of the Helium atoms have lost one electron)-->
Mesogranulation
• St a depth of 30000 km Helium II is ionized by 50%
(50% of the Helium atoms have lost two electrons).
• Giant Cells: Depth of the convection zone in the sun.
Magnetic fields,
and the Zeeman effect
The Zeeman effect
In the case of LS-coupling (weak-field) we define a magnetic quantum number MJ
which describes the total angular momentum in the specific direction.
Possible values are -J to +J. J can have values from
|L-S| to L+S.
2S1
Terme:
LJ (L  S,P,D,F..)
In the case of a magnetic field we thus get 2J+1 energy
Levels (MJ).

 have to fulfil the requirement
Transitions
(selection rule):
MJ  0,1
The Landé-factor is given by:
J(J  1)  S(S  1)  L(L  1)
g  1
2J(J  1)
We define the effective Landé-factor of the transition geff as:
geff  gM  g' M'
the Landé-factors of the transition are g and g‘ , and M und M‘ the magnetic
 using
quantum number.


n2S1LJ ;L  S,P,D,F

Zeeman splitting:
= 4.668598 10-13 Å-1 G-1* geff * 2 [Å] * B [G]
The Stokes parameters
Lets us assume monochromatic, electromagnetic wave
wave the is moving z-direction:
E x   x cos 
E y   y cos(  )
with the phase-difference between Ex and Ey
  t  kz


The amplitudes
are
x und  y

where
describes the (elliptical) polarisation.



I   2x   2y
Q   2x   2y
U  2  x y cos
V  2  x y sin 
Because for none-polarized
radiation we Q=U=V=0, and

for fully polarized radiation I2=Q2+U2+V2, we can define the
degree ofpolarization by:
Q2  U 2  V 2
P
I2
π-compnents: ∆MJ=0, linear polarized,
produced by radiation that is emitted
perpendicular to the magnetic field
lines. In this case we have: Q≠0, U≠0.
-components ∆MJ=±1, circular
polarised. Produced by radiation the is
emitted along the field-lines V≠0.



An example
Let us take the FeI line at 5250.2 Å, and let us
furthermore assume that the magnetic field is weak.
This line has geff=3. With a magnetic field the width of
this line would be 42mÅ, and the central depth r=0.7.
For a magnetic field of 10 Gauss we would have a
degree of polariazation of only 1%.

V
 9.6 104 B(Gauss)cos 
I
In order to determine the orientation of the magnetic field-line, we also
need Q and U. If we assume again a magnetic field of 10, the degree in
polarization in Q and U would be only of the order of 0.1%. The is quite
difficult to measure.
Q
6
2
2
10 B (Gauss)sin 
I


Frozen in field-lines
The electric conductivity can be calculated for the case of a low
degree in ionization (ne<nn) by using Nagasawa (1955)
equation.
The degree of ionization of hydrogen in the Photosphere and
Chromosphere is in fact small, most of the Electrons in these
regions come from heavy elements.
For this case Nagasawa found for the electric conductivity:
S is the cross-sectionfor the collision between charged and neutral
particles.
Typical values for the conductivity in the photosphere are
100 to 1000 Ohm-1m.
The conductivity within the sun-spot is a bit lower, because the
temperature is lower.
If we think of the typical sizes of structures in the photosphere, the
ohmic resistance is almost zero.
Magnetic field lines are „frozen in“ if the conductivity is zero.
Magnetic flux-tubes
Many people think that the magnetic fields on the sun are located in
dark sun-spots but that is a misconcept.
The dark sunspot have a magnetic field that is correct, but there
are also bright regions on the sun which have strong magnetic
fields.
These are the so-called magnetic flux-tubes.
Already in 1963 Parker speculated that there should be regions
with strong magnetic field in interganular lanes, because
magnetic flux is washed in these regions, because the magnetic
field is sticks to matter.
How do these flux-tubes look like?
Flux-tubes
Up to now, we have only found bright, and very small regions in
the intergranular lanes but are these the flux-tubes?
Until recently we had no possibility to measure the magnetic
field strength in regions as small as 100x100 km2 on the Sun
but we can prove the idea indirectly: There is a very high
correlation between the number of facular points observed in
region on the Sun, and the total magnetic flux in that region.
Although until recently it was not possible to measure the flux of
a single flux-tubes, such studies show that the magnetic flux
in such a tube must be about 4.4 109 Wb. This number can
be compared with the flux of a large spot region which is
about ≈1014 Wb. However, the flux-density is about 10002000 Gauss, which means not much smaller than in a sunspot.
Why are flux-tubes bright?
• Flux-tubes are cooler than the surrounding region.
• The gas pressure is in the tube is smaller because in the tube
we have gas+magnetic pressure but outside only gas-pressure.
• Because the gas pressure is lower in the tube, the atmosphere
within the tube is more transparent than the surrounding
material. If we look into a tube, we see quite deep into the solar
atmosphere, at least deeper than in the surrounding
atmosphere.
• Because the diameter of the tubes are small, radiation from the
hot, surrounding material is scattered into the tube. In other
words: The walls of the fluxes tubes are hot and radiation from
there is scattered into the tube.
--> Flux-tubes are at the same geometrical depth cooler than the
surrounding material but they are hotter at the same optical
depth. That is why the tubes look „bright“
Discovery of Sunspots
Sunspots were already discovered by Galilei (1564-1642) and
Christorph Scheiner (1575-1650).
In 1843 H. Schwabe discovered that the sunspot number varies
with the cycle of 11 years. Schwabe did these observations
to study sunspots but he wanted to discover an planet the
orbits inside Mercury.
We define as a sunspot number:
R= K(10g+f)
f the number of individual spots,
g the number of groups
K an individual correction factor
Sizes of magnetic
elements on the sun
• Flux-tubes usually have diameters of only 100-200 km,
and are bright.
• Active regions larger than that are called pores.
• Once the size reaches 2400 km, a penumbra develops
and we call the region a sunspot. A sunspot thus consists
of a dark Umbra and a Penumbra.
The magnetic flux
We have spoken about sunspots, flux-tubes, and pores. The spots
have huge fluxes but there are only few of them. In contrast to this,
there are many flux-tubes but each carries only a relatively small
amount of flux. This raises the interesting question in which kind of
structures is most of the magnetic flux of the Sun?
The answer is that it is surprisingly neither in the spots, nor in the fluxtubes but in the so-called micro-pores which have typical diameters
of 350-650 km.
Sunspots
• Are sunspots really dark?
• The temperature in the Umbra is typically 3500K, and in
the photosphere typically 5780 K.
• Let us take as an example a spot that has a typical
diameter of 2400 km. Such a spot would be -11 mag when
viewed from earth. This means that a sunspot would look
like a point-source (like a star) that is only 1.7 mag fainter
than the full moon.
Umbral dots (UDs) and
penumbral grains (PGs)
In a sunspot the convective energy transport is not completely inhibited.
There is still a rest of convection present.
The signature of this “rest of convection” can be seen in the sunspots as
bright dots, and in the penumbra as bright penumbral grains.
Umbral grains have a typical size of 200 to 700 km.
Temperature differences between 500 and 1500 K have been reported.
Magnetic field measurements of the grains are somewhat controversial.
Values smaller than in the Umbra have been reported.
Penumbral grains have typical sizes of 300x1500 km.
Penumbra
Large spots have a penumbra.
Magnetic field lines in the Umbrae are
basically vertical, and in the Penumbra
basically horizontal.
Magnetoconvection on
different types of
main sequence
stars. Note the
different box sizes
(1400-30000 km)!