1. Internal structure of the Sun

2. Main characteristics of the Sun and some constants
Mass: kg ( x Earth)
Luminosity: W
Mean diameter: km (109 x Earth)
Flattening:
Surface area km2
Volume km3
Average density: gcm-3
Escape velocity kms -1
Temperature in the centre: ~ K,
Temperature of the photosphere: 5777 K
Distance: m
Mass-loss due to wind: (2-3)10-14 Msunyr-1
G= m3kg-1s-2
k= J/K
h= Js

3. The 5 basic equations:
1. The Condition for Hydrostatic Equilibrium

4. 2. Conservation of mass

5. 3. Thermal equilibrium
The energy that is produced is (with , the “energy-production-rate”):

6. Two equation for the transport of energy: In the Sun, energy is transported in two forms: convection (Fc) and radiation (FR)

7. 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:

8.  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 ):

9. 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):

10. 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:

11. The 5 basic equation for the star are:

12. 1838: First measurement of the solar constant : 3.846 1026 W
John Herschel ( )
Claude Poullet ( )

13. Energy production inside the sun
Thermal equilibrium: what actually is ?

14. Proton-Proton Zyklus 4 Protons -> He + 2 Positrons +2 Neutrinos
Energy production in the core of the sun only: 276 Wm-3!

15. Energy production inside the sun II
Important reactions: PPI, PPII, PPIII (CNO only 1%). Branching ratio: 87:13:0.015

16. 4 H → 1 He 4 MH = 4.032 atomic units MHe = 4.0026 atomic units
Eddington first proposed that the Sun was powered by nuclear fusion. It is now known:
4 H → 1 He
4 MH = atomic units
MHe = atomic units
If we assume that the sun is 100% hydrogen and that all is converted to Helium:
E = Msun c2 = 1.3 ×1045 J
Lifetime of sun = 1011 years

17. 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 42 F0
7400
1.7 17
6.3 F5
6580
1.3 15
2.5 G0
6030
1.1
14.4 G5
5520
0.9
13.7
0.8 K0
4900
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

18. 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

20. Oscillations of the Sun

21. Solar oscillations Discovery
In 1975 solar oscillations were discovered by Franz-Ludwig Deubner and independently by Robert B. Leighton using Doppler-shifts of spectral lines.
The figure shows the Doppler motions in the photosphere.
The oscillations have typical periods of about 5 minutes,
or = s -1.

22. 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

23. P and G-modes
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) .

24. Propagation of the waves
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.

25. 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 )

26. 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.

28. The solar oscillations are studied with a network of telescopes (GONG)

29. Results Chemical composition inside the sun
Depth of the convection zone
Ration inside the sun

30. Solar-like oscillations

31. 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.

32. The Granulation

33. The different types of Granulation
1.) Granulation: cells with the size of 1000 km
2.) Mesogranulation: cells with a diameter of km
3.) Supergranulation: cells of to km diameter
4.) Giant Cells: cells of km diameter

34. 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.

35. 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%.

38. 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.

39. „Line Wiggles“

40. bisectors

41. Bisectors of Granules, intergranular lanes in active and quite regions )

42. The Mesogranulation
The number of Mesogranules on the solar surface is about
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.

43. 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“.

44. Giant Cells
The giant cells were only found after a long, and careful analysis of the very large -scale velocity-pattern on the sun.
The diameter of the cells is with really large.
Detecting the giant cells was really difficult, because the differential rotation of the Sun had to be subtracted carefully.

45. Giant Cells

46. What 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 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.

47. Magnetic fields, and the Zeeman effect

48. 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.
In the case of a magnetic field we thus get 2J+1 energy
Levels (MJ).
Transitions have to fulfil the requirement (selection rule):

49. The Landé-factor is given by:
We define the effective Landé-factor of the transition geff as:
using the Landé-factors of the transition are g and g‘ , and M und M‘ the magnetic quantum number.

51. Zeeman splitting: = 4.668598 10-13 Å-1 G-1* geff * 2 [Å] * B [G]

52. The Stokes parameters
Lets us assume monochromatic, electromagnetic wave wave the is moving z-direction:
with the phase-difference between Ex and Ey
The amplitudes are
where describes the (elliptical) polarisation.

53. 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:

54. 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.

55. An example
Let us take the FeI line at 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%.

56. 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.

58. 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:

59. 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.

60. 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?

62. 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 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 Gauss, which means not much smaller than in a sun-spot.

63. 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“

68. Discovery of Sunspots
Sunspots were already discovered by Galilei ( ) and Christorph Scheiner ( ).
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

70. Sizes of magnetic elements on the sun
Flux-tubes usually have diameters of only 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.

71. 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 flux-tubes but in the so-called micro-pores which have typical diameters of km.

73. 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.

76. 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.

81. Penumbra Large spots have a penumbra.
Magnetic field lines in the Umbrae are basically vertical, and in the Penumbra basically horizontal.

83. Magneto-convection on different types of main sequence stars
Magneto-convection on different types of main sequence stars. Note the different box sizes ( km)!