1. THE SOLAR PHOTOSPHERE (contd.)

2. The Sun’s effective and surface temperature
Limb
Temperature minimum region
Lines of sight reach down to optical depth 2/3
Sun centre
Deepest photospheric level
Observer
Sun’s effective temperature is a measure of the Sun’s radiation coming from the deepest photosphere (T = 6400K) visible at Sun centre to the “upper photosphere” or temperature minimum region (T = 4400K) visible at the limb. Thus, there is a limb darkening (decrease of solar intensity with angle θ).
Teff is a kind of average of the kinetic temperatures in the photosphere.

3. Limb darkening: white light image of Sun
Partial solar eclipse, 2011 January 4: APOD, T. Legault.

4. Observed limb darkening as a function of wavelength
→ cos θ = μ
Observations of limb darkening are used to infer the temperature vs. height
profile in the photosphere.

5. Solar granulation
Observational evidence of convection in upper part of solar interior. Appears in ‘quiet’ photosphere as polygonal cells, roughly 1000 km across (1.5 arcsecs in angular measure), lasting ~15 minutes (largest granules last longest). Appear in sunspots too, as “umbral dots”. In quiet photosphere, granules are 30% brighter than intergranular lanes – this translates to a temperature difference of 400K. Very bright dots appear in intergranular lanes – “filigree” – sometimes forming chains which trace out a network. This is governed by a large-scale supergranulation (next lecture).

6. Solar granulation: time sequence

7. Spot penumbral filaments
Swedish Solar
Telescope image
of sunspot group and surrounding granulation.
Spot umbra
Spot penumbral filaments

8. Source of photospheric opacity
Why is the Sun so opaque? In the Sun’s photosphere, where T=6400K, there are neutral atoms (of H, He, etc.) and some ions (once-ionized Na, Mg, Fe) and free electrons e-. The free electrons attach themselves to neutral H atoms to form a negative hydrogen ion, H- : H + e- → H- Then an H- ion absorbs photons hν with wavelength < 1600nm, i.e. from the visible to the infrared: H- + hν → H + e- In the photosphere, there are only 10-8 H- ions to every H atom, but this is still enough to be the main cause of solar opacity (i.e. the absorption of photons).

9. Sharpness of the Solar Limb
The reaction H + e- → H- occurs at a rate N(H) Ne R(T) [where R(T) originally calculated by Rupert Wildt (1938) with input from Sir Harrie Massey (UCL)]. Now the density falls off in the solar atmosphere quite fast going from the photosphere upwards. But the solar opacity falls off even more sharply, as N(H) Ne (approx. proportional to N(H)2) – this explains why the solar limb is so sharp.

10. Some collegiate pride...

11. TE, LTE, NLTE and coronal equilibria
Thermodynamic equilibrium (TE): matter in complete equilibrium with radiation – a black body. Intensity of radiation = source function = Bν(T) (where T = temperature). Applies in the deep solar interior.
Local thermodynamic equilibrium (LTE): thermodynamic equilibrium defined by the local value of T. Holds in the solar photosphere, where radiation escapes and T changes, but slowly enough that LTE can be assumed.
Non-LTE (NLTE): Material has a kinetic temperature T different from the temperature characterizing the radiation passing through it, and which interacts with it. Applies in the solar chromosphere.
Coronal equilibrium: Equilibrium controlled entirely by the interactions of ions and electrons of corona; photospheric radiation passes through coronal material without any effect.

12. Definition of Rate Coefficients
Rate coefficients define how fast various reactions – excitation, ionization, or recombination – proceed. In the corona, these processes occur by collisions with free electrons.
Rate coefficients are formed by an integral over v, the velocity of the colliding free electrons (mass = m), of the cross section of the process times the velocity distribution of the electrons – generally a Maxwell-Boltzmann distribution:
f(v)
Ne Ni qe(X+m) = rate of ionizations etc. per unit volume (m-3 s-1). v
Rate coefficient definition:
m3 s-1
where sc is the cross section for collisional ionization or excitation or recombination by electrons with velocity v. v0 = a threshold velocity.

13. LTE - ionization/recombination equilibrium
Ionization (rate coefft. q) occurs by electron impact collisions:
e1- + X+q → X+q+1 + e1- + e2-
Recombination (rate coefft. R3) dominated by a 3-body process:
e1- + e2- + X+q+1 → e1- + X+q (e1- removes some energy in the collision)
where Nq = N(X+q), Nq+1 = N(X+q+1)
From statistical mechanics, using principle of detailed balance:
where Ei is the ionization energy from q to q+1 stage, g’s are the “statistical weights” of the atomic states of the ion in its q and q+1 states.
In LTE, ratio of populations of two adjacent ions = Nq+1 / Nq depends on Ne.
Generally, in photosphere, atoms are either neutral or once-ionized (q = 0).

14. LTE – Excitation of an ion with two levels (1 and 2)
Both excitation (rate coefft. C12) and de-excitation (rate coefft. C21) of an ion are 2-body collision processes:
e- + X1 → e- + X (excitation from 1 to 2)
e- + X2 → e- + X (de-excitation from 2 to 1)
and in LTE they are balanced:
(N1 = N(X1), N2 = N(X2)
Using detailed balance leads to the Boltzmann distribution:
In LTE, the ratio of level populations within one ion = N2/N1 is independent of Ne

15. Non – LTE (NLTE) calculations
The details of the chromospheric spectrum are calculated in NLTE by taking into account all the various excitation and de-excitation mechanisms for a particular ion which may have many levels (not just 2).
This requires detailed knowledge of the rate coefficients for excitation and de-excitation.
This can be done (e.g.) for the chromospheric Hα and Ca II H and K lines.
The cores of these lines are formed high in the chromosphere – the wings are formed nearer the photosphere.

16. THE SOLAR CHROMOSPHERE

17. Solar Chromosphere: Temperature Profile

18. The chromosphere and supergranules
mag. field lines
isothermals
downflow
downflow
upflow
supergranule
From Gabriel (1976)
Cell boundary
Cell centre
Cell boundary

19. Chromosphere: supergranular flow
Supergranules: polygonal convective cells in photosphere, approx. 30,000 km across, lasting ~ 30 hours.
The horizontal flow from the cell centre sweeps out material and magnetic fields.
The magnetic fields form a polygonal network across the quiet Sun – the chromospheric network. (This mirrors the photospheric network.)
The network is evident in magnetograms (maps of the line-of-sight mag. field) and spectroheliograms in Hα and Ca II H and K absorption lines.

20. Chromospheric network in magnetogram and Hα spectroheliogram
Magnetogram (line of sight field): white = +ve field, black = -ve field.
spicules
Hα line (image in line wings)
Big Bear Observatory image

21. Hα spicules: schematic

22. Ca II K line (392 nm) spectroheliogram
Network marked by bright and dark mottles.
Active regions are bright areas.
U.S. Nat. Solar Observatory image

23. Chromosphere in profile: an eclipse flash spectrum
Fe XIV nm “green line”
Hα 656.3nm Hβ
He I nm
Photograph during 1970 eclipse at 2nd contact of eclipse

24. The Extreme Ultraviolet Spectrum of the Chromosphere and Corona
log Flux
121.6nm
Spectrum is dominated by the Lyman lines of neutral H and some He lines – other lines in this region ( nm) are due to the solar transition region (ionized C, N, O, Ne etc.)

25. The network in the extreme ultraviolet from SOHO
Sun in the He II 30.4nm emission line, 1997 September 14. Image from EIT instrument on SOHO.

26. The network in the extreme ultraviolet from SOHO (contd
The network in the extreme ultraviolet from SOHO (contd.) – region at Sun centre
chromosphere (~30,000K)
transition region (~105 K)
corona (~106 K)

27. Interpretation spectroheliograms: equation of radiative transfer
Light beam Iν(θ)
Consider light beam with intensity Iν(θ) passing through solar photosphere/chromosphere. The atmosphere both absorbs and emits radiation, so along its path, the change of intensity is where κν = absorption coefft., jν = emission coefft., and Sν = the source function. Optical depth along the beam defined by so equation of radiative transfer is: (1)
solar surface
See Stellar Atmospheres, vol. 2 (E. Böhm-Vitense, CUP), p.40

28. Interpretation of Hα and Ca II H and K spectroheliograms
Solving the radiative transfer equation: Put μ = cos θ and multiply both sides of Eq. (1) by exp(- τν / μ). Then: (2) or (3) Integrate with respect to (τν / μ) from 0 to ∞ to give (4) - the emergent intensity at optical depth τν and angle θ from the Sun’s centre.

29. Interpretation of spectroheliograms (contd.)
Approximate Sν by Sν = Aν + Bν τν. Then:
(5)
Now, definition of outward flux is:
(6)
Rewrite this in terms of
(7)

30. Interpretation of spectroheliograms (contd.)
This leads to
(8)
This is the Eddington—Barbier relation.
It means that the emergent flux is equal to π × the source function at optical depth τν = 2/3.
Suppose we look at a region of the visible-light continuum at Sun centre. The emergent flux is from a point where the source function is τν = 2/3, which is low down, at the bottom of the photosphere.
But for an optically thick line like Hα or the Ca II H and K lines, τν = 2/3 corresponds to high up in the chromosphere.
So in these lines, we view the chromosphere.

31. THE SOLAR CORONA

32. White-light solar corona during total solar eclipses
White-light corona during total solar eclipse, 2006 March 29

33. STEREO images of corona (and chromosphere) – 2006 Dec. 22
19.5nm
(Fe XII)
17.1nm
(Fe X/Fe XI)
30.4nm (He II: chrom.)
28.4nm
(Fe XV)

34. High Altitude Observatory (WL) and Yohkoh (X-rays)
White-light and X-ray Corona images superimposed:
High Altitude Observatory (WL) and Yohkoh (X-rays)

35. Corona: Physical Properties
Corona has much higher temperature than chromosphere or photosphere: 106 K – 2×106 K (1—2 MK).
Densities are lower than those of chromosphere or photosphere: n(particles m-3) ~ 1013 – 3×1014 m-3, out to 2 solar radii from Sun’s centre.
Pervaded by a magnetic field, strength ~1mT, mostly in form of loop structures.
Temperature, density, and magnetic field strength much enhanced over sunspot regions (“active regions”)
Reduced density and temperature and open magnetic fields at poles, particularly at solar minimum: coronal holes.
Corona undergoes an 11-year activity cycle like photosphere and chromosphere: brighter and more irregular at solar maximum.

36. Corona is a fully ionized gas or plasma
Because of the high temperature, corona is practically a fully ionized gas (or plasma).
Nearly all H and He atoms are fully stripped.
So particles are almost entirely protons, α particles (He nuclei), and free electrons.
These charged particles perform helices along magnetic field lines, radii of gyration depending on particle speed, charge, and mass, and field strength.

37. Paths of charged particles in magnetized plasma
Radius of gyration = m v / (Z e B)
where m = particle mass,
v = component. of velocity perpendicular to the magnetic field B,
Ze = particle charge. (e = 1.6 x Coulomb.)

38. White-light emission from corona
White-light emission is photospheric radiation scattered by free electrons – Thomson scattering.
Electron cross section for Thomson scattering is
6.65×10-29 m2.
If there are N free electrons along a line of sight, the coronal surface brightness in terms of the photospheric brightness is N × 6.65×10-29.
In fact, surface brightness is ~ 10-6 × photospheric brightness (about the brightness of the full Moon).
So N × 6.65×10-29 = 10-6, or N = 1.5×1022 electrons.
A typical coronal streamer extent ~100,000 km = 108 m.
So very roughly the density of electrons is ~1014 m-3.

39. Density (particles m-3) and surface brightness of the solar corona
Electron density near Sun is ~1014 m-3.
Figure shows the surface brightness of K (electron) and F (dust) corona separately.

40. The K (electron) corona
K corona is the electron-scattered component: Kontinuerlich.
It dominates out to 2 R (measuring from Sun centre).
It has a white-light spectrum consisting of a featureless continuum – no Fraunhofer lines.
This is due to high speeds of coronal electrons (0.03c). Fraunhofer lines are broadened so much that they are no longer visible.

41. Note on spectral line profiles
Spectral lines can be broadened by several mechanisms.
In a gas with Maxwell-Boltzmann distribution of atoms or ions, the observed line width (full width at half maximum flux, FWHMobs) due to thermal Doppler broadening is:
where the line wavelength is λ, c = vely. of light, Tion is the ion (or atom) temperature, and mion is the mass of the ion (or atom).
So a very hot gas like the corona produces very broad line profiles.

42. The F (dust) corona
F corona is the dust-scattered component: Fraunhofer corona.
It dominates beyond 2.5 R, becoming the zodiacal light at large distances.
The white-light spectrum is photospheric, with Fraunhofer lines.
The Fraunhofer lines are still visible as the dust grains doing the scattering move slowly in an orbital motion round the Sun.

43. The Zodiacal Light from Hawaii (spring 2010)

44. Brian May’s contribution to F coronal physics
Courtesy: amazon.co.uk and Brainmay.com

45. Polarization of the corona’s white-light emission
Thomson scattering of photospheric light gives rise to polarization.
The (unpolarized) photospheric light comes from a particular direction as seen from a point P in the corona.
At sufficiently large distances, white-light radiation from P scatters only radiation towards the Earth with the electric field vibrations parallel to the Sun’s limb.

46. Polarization of the corona’s white-light emission (contd.)
Arrows indicated electric field vibration directions.

47. Observed polarization during an eclipse
Images of white-light corona during total eclipse, 1970 March 7: successive images taken with polaroid filter rotated by 45˚

48. Solar corona’s 11-year cycle
Yohkoh X-rays
Sunspot number
Spacecraft white-light corona

49. Solar corona during the recent deep solar minimum: 2009 solar eclipse
M. Druckmuller, Brno Observatory, Czech Rep July 22

50. Coronal “butterfly” diagram
Images constructed from green-line limb scans (1944 to 2002) from coronagraph at Tatranska Lomnica Observatory, Slovakia (courtesy J. Rybak)

51. Solar maximum
Solar minimum
Yohkoh SXT data

52. Appearance of the Solar Corona
White-light corona has loops (closed structures) and streamers (open structures), changing with the solar cycle.
Streamers are mostly concentrated around the solar equator at sunspot minimum. Loop-like structures dominate at sunspot maximum, occurring over all latitudes during maximum.
X-ray coronal structures related to white-light structures – practically identical at low altitudes above photosphere.
Smaller but brighter loops associated with sunspots: active regions -- are seen in extreme ultraviolet and X-ray emission.
Numerous X-ray bright points – actually very small loops – occur over all phases of sunspot cycle and at all latitudes.
Coronal holes – reduced X-ray emission with well-defined boundaries – occur at sunspot minimum at the N and S poles. Lower-latitude holes also occur during the declining phase of the cycle that may grow and coalesce with polar holes of the same polarity.

53. Gas and magnetic pressures
Coronal structures are loops and streamers, closely following magnetic field. This must mean that the plasma (ionized gas) pressure N kB T << magnetic pressure B2/2μ0 (μ0 = 1.26 x 10-6 H m-1). For an active region, T ~ 4 MK, N ~ 1017 m-3, so plasma pressure is 1017 × × = 5.5 Pa. B cannot (easily) be directly measured in the corona but can be inferred from models: above an active region, B ~ .01 T. Magnetic pressure is about 40 Pa.