THE SOLAR PHOTOSPHERE(contd.)

1

The Sun’s effective and surface temperature

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.

Temperature minimum region

Deepest photospheric level Observer

Lines of sight reach down to optical depth 2/3

Limb

Sun centre

2

3

Limb darkening: white light image of Sun

Partial solar eclipse, 2011 January 4: APOD, T. Legault.

Observations of limb darkening are used to infer the temperature vs. heightprofile in the photosphere.

Observed limb darkening as a function of wavelength

→ cos θ = μ

4

Solar granulationObservational 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).

5

Solar granulation: time sequence

6

Swedish SolarTelescope image of sunspot group and surrounding granulation.

Spot umbra

Spot penumbral filaments

7

Source of photospheric opacityWhy 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).

8

Sharpness of the Solar LimbThe 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.

9

10

Some collegiate pride...

TE, LTE, NLTE and coronal equilibriaThermodynamic 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.

11

Definition of Rate CoefficientsRate 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:

dvvfvvXqv

cm

e )( )()(0

where cis the cross section for collisional ionization or excitation or recombination by electrons with velocity v. v0 = a threshold velocity.

kT

mvv

kT

mvf

2 exp

2 4)(

22

23

Ne Ni qe(X+m) = rate of ionizations etc. per unit volume (m-3 s-1).

Rate coefficient definition: m3 s-1

12

f(v)

v

LTE - ionization/recombination equilibriumIonization (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).

32

1 RNNqNN eqeq

kTE

q

q

q

eq i

eg

g

h

mkT

N

NN 1

3

23

1 ) 2(2

13

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- + X2 (excitation from 1 to 2)

e- + X2 → e- + X1 (de-excitation from 2 to 1)

and in LTE they are balanced:

(N1 = N(X1), N2 = N(X2)212121 CNNCNN ee

Using detailed balance leads to the Boltzmann distribution:

kTEeg

g

N

N12

1

2

1

2

In LTE, the ratio of level populations within one ion = N2/N1 is independent of Ne

14

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.

15

THE SOLAR CHROMOSPHERE

16

Solar Chromosphere: Temperature Profile

17

The chromosphere and supergranules

supergranule

upflow downflowdownflow

mag. field lines

isothermals

Cell boundary Cell centre Cell boundaryFrom Gabriel (1976)

18

Chromosphere: supergranular flowSupergranules: 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. 19

Chromospheric network in magnetogram and Hα spectroheliogram

Hα line (image in line wings)

Magnetogram (line of sight field): white = +ve field, black = -ve field.

spicules

Big Bear Observatory image

20

Hα spicules: schematic

21

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

22

Chromosphere in profile: an eclipse flash spectrum

Photograph during 1970 eclipse at 2nd contact of eclipse

Hα 656.3nmHβ Fe XIV 530.3 nm “green line” He I 587.5nm

23

The Extreme Ultraviolet Spectrum of the Chromosphere and Corona

Spectrum is dominated by the Lyman lines of neutral H and some He lines – other lines in this region (45-140 nm) are due to the solar transition region (ionized C, N, O, Ne etc.)

log Flux 121.6nm

24

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.

25

The network in the extreme ultraviolet from SOHO (contd.) – region at Sun centre

chromosphere (~30,000K) transition region (~105 K)

corona (~106 K) 26

Interpretation spectroheliograms: equation of radiative transfer

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)

27

SIjIds

dI )()(

)(

sec dds

)()( cos

SId

dI

See Stellar Atmospheres, vol. 2 (E. Böhm-Vitense, CUP), p.40

solar surface

Light beam Iν(θ)

Interpretation of Hα and Ca II H and K spectroheliograms

28

///

)/( eSeIe

d

dI

/

/

)/(

)(

eSd

eId

0

/ )/( )( )0,(

deSI

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.

Interpretation of spectroheliograms (contd.) Approximate Sν by Sν = Aν + Bν τν. Then:

(5)

Rewrite this in terms of

Now, definition of outward flux is:

(6)

(7)

29

0

//

0

)()/()/()0,(

SBAdeBdeAI

Interpretation of spectroheliograms (contd.)This leads to

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.

(8)

30

THE SOLAR CORONA

31

White-light solar corona during total solar eclipses

•

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

STEREO images of corona (and chromosphere) – 2006 Dec. 22

•

17.1nm

(Fe X/Fe XI)

28.4nm

(Fe XV)

19.5nm

(Fe XII)

30.4nm (He II: chrom.)

33

White-light and X-ray Corona images superimposed:

High Altitude Observatory (WL) and Yohkoh (X-rays)

34

Corona: Physical PropertiesCorona 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.

35

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.

36

Paths of charged particles in magnetized plasma

37

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 10-19 Coulomb.)

White-light emission from coronaWhite-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.

38

Density (particles m-

3) and surface brightness of the

solar corona

Figure shows the surface brightness of K (electron) and F (dust) corona separately.

Electron density near Sun is ~1014 m-3.

39

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.

40

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.

41

2/1

ion

ion

2/1

ion

ion 2665.1

22ln 2FWHM

m

Tk

cm

Tk

cBB

obs

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.

42

The Zodiacal Light from Hawaii (spring 2010)

43

Brian May’s contribution to F coronal physics

Courtesy: amazon.co.uk and Brainmay.com 44

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.

45

Polarization of the corona’s white-light emission (contd.)

Arrows indicated electric field vibration directions.

46

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˚

47

Solar corona’s 11-year cycle

Yohkoh X-rays

Sunspot number

Spacecraft white-light corona

48

Solar corona during the recent deep solar minimum: 2009 solar eclipse

M. Druckmuller, Brno Observatory, Czech Rep. 2009 July 22 49

Coronal “butterfly” diagram

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

50

Solar maximum Solar minimum

51

Yohkoh SXT data

Appearance of the Solar CoronaWhite-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.

52

Gas and magnetic pressuresCoronal 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 × 1.38.10-23 × 4.106 = 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. 53