1. The Solar Corona – contd.
We discussed:
Physical properties of the corona
We will now discuss:
Coronal scale height
Ionization equilibrium
Coronal heating

2. Scale heights
For some purposes, we can assume an atmosphere is in hydrostatic equilibrium (just like the solar interior). r = distance from Sun’s centre.
This means gas pressure is balanced by gravity. So
Acceleration due to gravity is g(r) given by
Perfect gas law is
where kB = Boltzmann’s constant, mH = mass of H atom, and μ = molecular weight, i.e. atomic mass per unit particle – for fully ionized hydrogen and helium, μ = 0.6 (Lecture 1).
(1)
(2)
(3)

3. Scale heights (contd.)
Substitute for g(r) in (1) and eliminate ρ(r) in (1) to give
Integrate Eq. (4) from r = 0 (P = P0) to r to get
If T(r) and g(r) are only slowly varying with r,
H = kBT0/[μ mH g0] = the scale height
and is a characteristic of the atmosphere.
[T0, g0 = (e.g.) the surface values.]
(4)
(5)

4. Summary of Ionization and Recombination Processes in Corona
Collisional Ionization plus
Auto-ionization
is balanced by:
Radiative Recombination plus
Dielectronic recombination

5. Coronal Equilibrium – ionization/recombination
Notation: e- stands for free electron, X an element (e.g. Si, Fe) and X+q an ion of element X with q electrons missing.
Ionization (rate coefft. Ce) is by collisions with free electrons:
e1- + X+q → X+q+1 + e1- + e (e1- loses energy, imparting it to ion)
Auto-ionization (rate coefft. CA) is break-up of a doubly excited ion:
(X+q)** → e1- + X+q+1
Recombination (rate coefft. Rr ) occurs by radiative processes:
e1- + X+q+1 → X+q + hν (hν = photon, removes excess energy)
and by dielectronic processes (rate coefficient RD):
e1- + X+q+1 → (X+q)** (** = doubly excited ion)

6. Collisional Ionization
... proceeds by the collision of the ion with free electrons.
The electrons are very energetic (hundreds of eV or a few keV) as coronal temperature is so large (1-2 MK or larger for active regions and much larger for flares).
Ions are therefore highly ionized – e.g. Fe in quiet corona is in the form Fe+9 to Fe+16. There is very little neutral H or He.
The process can be summarized as:
e1- + X+q → X+q+1 + e1- + e2-
There are too few photons with the required energy – there are plenty of photospheric photons with energy ~2 eV, so they pass through the corona without any effect – no photo-ionization.

7. Radiative Recombination
A free electron is captured by the ion – the electron generally has more energy than required, and the excess is released as a photon – hence “radiative” recombination.
Process is:
e1- + X+q+1 → X+q + hν
where hν is a photon.

8. Dielectronic Recombination
A free electron is captured causing double excitation of the ion. Process is:
e1- + X+q+1 → (X+q)**
where ** means that the ion is doubly excited, i.e. two electrons are excited (not just one as in normal excitation).
There is no photon to take away any excess energy, the incoming electron has just the right amount of energy for the recombination.
Although seemingly an exotic transition, dielectronic recombination is more important than radiative recombination at relatively high temperatures.
Its importance in the solar corona was pointed out by Alan Burgess (UCL) in 1964.

9. Alan Burgess’ original paper in Astrophysical Journal (1964)

10. Pictorial representations of coronal ionization and recombination processes

11. Coronal ionization equilibrium
In equilibrium (which occurs in most of corona), rate of ionizations from ion X+q per unit volume is balanced by rate of recombinations from ion X+q+1 per unit volume. Nq =N(X+q), Nq+1 = N(X+q+1).
That is, Nq Ne (Ce + CA) = Nq+1 Ne (Rr + RD).
The ratio of Nq to Nq+1 is therefore given by
(i)
So the ratio normally only depends on electron temperature Te, not density.
The ratio of any particular ion X+q to all other ionization stages,
Nq / NΣq , can be calculated from all ratios given in Eq. (i).

12. Fractions of Fe ionization stages in coronal equilibrium
Fractions of Fe ions as a function of T (plotted logarithmically)
Ion notation:
9 = Fe+9
19 = Fe+19 etc.
Note: for corona, Fe+8 to Fe+14 are abundant ionization stages
Mazzotta et al. (1998)

13. Spectral line excitation in the corona, spectroscopy and spectral diagnostics

14. Coronal excitation equilibrium: Excitation of ion with 2 levels
Collisional excitation (rate coefft. C12): X1 + e- → X2 + e-
is balanced by radiative de-excitation (rate A21): X2 → X1 + hν
m-3 s-1
where A21 is the radiative transition probability (in s-1), I21 is the photon emission rate (m-3 s-1). Ne = electron density, N1, N2 = number densities of ions in levels 1 and 2. Unlike LTE, there is in general no collisional de-excitation C21.
So photon emission rate = collisional excitation rate in a simple level ion.
This ratio is always < 1 in coronal equilibrium, usually much less.

15. Level diagram for 2-level atom
Collisional excitation
Rate = N1 Ne C12 m-3 s-1
Radiative de-excitation
Rate = N2 A21 m-3 s-1
Level 1 *

16. Optically thin emission lines in the coronal spectrum
More generally, for any upper level, j, and lower level, i, the probability of spontaneous emission of a photon is Aji (s-1).
Let number density of atoms (m-3) in excited level is Nj (also called the population of level j), so the total emissivity is:
J m-3 s-1 (or W m-3)
So from a volume on the Sun, ΔV (m3), the flux of radiation at Earth from a spectral line emitted by transitions from level j to level i is:
J m-2 s-1 (or W m-2)
where R is Sun-Earth distance = 1 astronomical unit (AU).

17. Optically thin emission lines in the coronal spectrum (contd.)
Number density of ions in excited level j = Nj (m-3) can be expressed in terms of other known solar parameters:
where:
Nj /Nion is the relative population of the excited level;
Nion /Nel is the ionization fraction (from ionization equilibrium calculation, is a function of T);
Nel/NH is the abundance of the element relative to H;
NH/Ne is approximately 0.8 (electrons supplied by H and He atoms).

18. Optically thin emission lines in the coronal spectrum (contd.)
So the total flux is given by:
J m-2 s-1 or W m-2
In the coronal approximation, the populations of excited levels are so small that practically all ions are in the ground state.
So level j is populated exclusively by collisions from the ground state i and de-populated exclusively by radiative de-excitation: collisional de-excitation and radiative excitation are negligible.
For “resonance” (very strong) lines only, the ground level and excited level are important for calculating the line flux – this is the 2-level approximation.

19. Flux of spectral line in a two-level approximation
This is the case we saw before, applying to resonance lines. Collisional excitation balances spontaneous radiative de-excitation. So:
m-3 s-1
If N1 is assumed to be Nion, the flux in the line becomes:
W m-2
where
m3 s-1
C12 is the collisional excitation rate coefficient: W12 is the “collision strength” of the transition, g1 is the statistical weight of level 1, ΔE12 is the excitation energy (difference in energy between levels 1 and 2).

20. Flux of spectral line in a two-level approximation (contd.)
The 2-level approximation simplifies the calculation of emission
line fluxes (in reality, a complete set of level population equations is normally required ).
where NH/Ne ~ 0.8. Often the temperature-dependent terms are grouped together as the line’s contribution function G(T):
Sometimes this is called the emissivity of the line.

21. Contribution functions for some high-temperature coronal X-ray resonance lines
Temperature (in MK)

22. Spectroscopic diagnostics for studying the Sun’s atmosphere
We can study or “diagnose” laboratory high-temperature plasmas using probes inserted into the plasma.
Thus, the Joint European Torus (JET) tokamak at Culham Laboratory has a large array of such probes.
To determine the temperature and number density of particles forming the ~20 MK plasma during “shots”, monochromatic (i.e. narrow spectral line) laser beams are shone into the plasma.
The scattered radiation is detected by spectrometers. The width of the spectral line determines the ion temperature and the intensity of the scattered radiation determines the particle density.
We can’t do this for diagnosing the solar atmosphere but we can use spectroscopic diagnostic techniques, using spectral lines in the extreme ultraviolet or X-ray region – “remote sensing”.

23. The JET tokamak
Coils carrying current that forms containing mag. field
Andrei Sakharov, tokamak designer (with I. Tamm)
Torus
High-temperature plasma formed in short “shots”: T~20MK
Port holes in torus allow probes to view the hot plasma

24. Note about ion and spectrum notation
Atom or Ion produces spectrum:
Neutral H H I
Ionized H No spectrum
Neutral He He I
Ionized He (He+1) He II
Neutral C C I
Once-ionized C (C+1) C II
C+4 (He-like) C V
Fe+16 (Ne-like) Fe XVII
Fe+24 (He-like) Fe XXV
Fe+25 (H-like) Fe XXVI
Fully stripped Fe (Fe+26) No spectrum

25. Nature of the solar spectrum
In infra-red and visible wavelengths, the solar spectrum is a continuous background crossed by absorption (Fraunhofer) lines: cooler gas (temperature minimum) in front of hotter gas.
At wavelengths < 160 nm (extreme ultraviolet), the solar spectrum is in the form of emission lines with a relatively weak continuum: hotter gas (chromosphere/transition region/corona in front of photosphere.
These follow Kirchhoff’s (and Bunsen’s) laws: see Böhm-Vitense’s book (vol. 2).

26. Sun’s extreme ultraviolet spectrum
H Lyman-α
log spectral irradiance
Emission lines
Recombination continuum

27. Solar temperature diagnostics
To determine the temperature of the corona (approx. 1—2 MK) or active region within the corona (approx. 2—5 MK), we could use spectral lines from adjacent ionization stages.
E.g. for corona, Fe+8 to Fe+14 are abundant ionization stages. So we could use the ratio of these extreme ultraviolet lines to determine the corona’s temperature:
Fe IX (17.1nm) / Fe X (17.48nm)
Fe X (17.48nm) / Fe XI (18.04nm)
Fe XI (18.04nm) / Fe XII (19.5nm)

28. Hinode EIS spectrum of solar coronal spectrum, 17.5—21 nm

29. Another temperature diagnostic: two lines from same ionization stage
Consider two lines from same ionization stage, excited from ground level (g) to upper levels 1 and 2. If the lines 1→g and 2→g are strong (resonance) lines, then approximately:
NgNeCg1 = I1 and NgNeCg2 = I2 or:
and
Hence:
If energy difference (E2 – E1) >> kT, I2/I1 sensitively depends on T.

30. Dielectronic Satellite Lines in the X-ray Spectrum of the Sun
In the solar X-ray spectrum, we observe lines that are excited by hot active regions or flares – temperatures range from 2—5MK (non-flaring active regions) to > 20 MK (flares).
We can use lines formed in the dielectronic recombination process – satellite lines – to get information about temperature in such plasmas.
Satellite lines are so called because they occur just to the long-wavelength side of associated “resonance” (strong) lines.
The ratio satellite/resonance line has a fairly strong dependence on electron temperature Te (= constant × 1/Te).
The satellite line intensity depends strongly on the ion atomic number Z (= constant × Z 4).

31. Dielectronic recombination satellite lines
In dielectronic recombination, an electron recombines with an ion to form a doubly excited state (double asterix):
e1- + X+q+1 → (X+q)**
Take a real example – recombination of a free electron on to a He-like ion, ground level 1s2 : suppose the free electron goes into an n=2 shell and simultaneously excites one of the 1s2 electrons also to the n=2 shell (this is most common example):
1s2 + e- → 1s 2p2
The doubly excited state 1s 2p2 could auto-ionize (i.e. → He-like ion [1s2] + e-) or it could stabilize by de-exciting sequentially:
1s 2p2 → 1s2 2p + hνDS - hνDS is the diel. satellite line photon
1s2 2p → 1s2 2s - an EUV photon is emitted

32. Dielectronic recombination satellite lines (contd.)
So the excitation of the satellite line occurs from the He-like ionization stage, even though the satellite line is emitted by the Li-like stage (3-electron ion). Let Ni = He-like ion number density, Ne=electron density, and Cs = rate coefft. for dielectronic capture of the free electron.
The satellite photon emission rate (photons m-3 s-1) is given by:
where Ar / (Aa + Ar) is the branching ratio expressing the relative probability that the doubly excited state 1s 2p2 will de-excite (Ar) rather than autoionize (Aa). Coefficients Cs and Aa are related by the Boltzmann-Saha equation (gs = statistical weight):

33. Dielectronic recombination satellite lines (contd.)
Therefore the satellite line intensity is:
The associated He-like ion resonance line is 1s2 – 1s2p. Using the two-level approximation:
So the ratio satellite/resonance line photon emission rate is:

34. Dielectronic recombination satellite lines (contd.)
Now for many important satellites, Aa>>Ar (the doubly excited level is much more likely to auto-ionize than de-excite). Also, Ar = const. × Z4 (Z = ion’s atomic number) and (Er – Es) / kT is small.
Thus Is / Ir = constant × Z4 /Te approximately.
So X-ray satellite lines are useful “temperature diagnostics” of active regions and flares, especially for lines emitted by Mg, Si, S, Ca, and Fe ions. (They are not so useful for light elements such as Ne, O, or C.)
For flares especially, S, Ca, Fe lines are used to find the flare temperature.
The satellite lines are just to the long-wavelength side of the resonance lines, so X-ray spectrometers observing the spectra do not have to be extremely accurately intensity-calibrated.

35. Temperature dependence of Fe sat. j / Fe XXV w
T dependence of Ca sat. k / Ca XIX w

36. Solar flare spectra of Fe XXV lines and Fe XXIV satellites: observations by the SMM spacecraft
Te=15.0MK:
Satellites intense compared with Fe XXV resonance line w.
Te=17.5 MK:
Satellites weaker compared with Fe XXV line w.

37. Level diagram for a 3-level ion or atom
Line 3→1 is allowed, line m→1 is forbidden. 3
NmNeC* (collisional excitation)
m (metastable)
N3A31
N1NeC13
(collis. excitation)
N1NeC1m
NmAm1 – very small 1
Ground level
Rates per unit volume are indicated in m-3 s-1

38. Electron density diagnostics
Consider a 3-level diagram for an ion, with ground level 1, metastable m, and another excited level 3.
The transition 3→1 is an allowed transition (one that has a high transition probability). The transition m→1 is forbidden, i.e. would not normally appear in a laboratory plasma, but does appear in solar coronal plasmas because of their low densities.
Collisional excitation from level m to 3 can compete with radiative de-excitation m→1 if the density is high enough, i.e. if
Nm Am1 ~ Nm Ne C*
Collisional de-excitation of level m (m→1) can also occur; this is usually negligible.
The line intensity ratio I(m→1)/I(3→1)= Im / I3 thus depends on Ne.

39. For allowed level (transition 3→1), I3 ~ N1NeC13 m-3 s-1
This assumes that the populating of level 3 is mostly by collisional excitation from level 1 and that radiative transition 3→m is small.
For line m→1, Im = N1 Ne C1m × a branching ratio:
Ratio of line m→1 (forbidden) to line 3→1 (allowed) is:
and so depends on Ne if Ne  Am1 / C* (units of m-3).
C* scales approximately as Z -3 and Am1 (for a forbidden transition) approximately scales as Z n (where n = 6 to10), so the region of density sensitivity varies steeply with Z (Z 9 – Z 13).

40. Density sensitivity of lines of He-like O (O VII lines)
Density-sensitive ratio R = ratio of a forbidden line to “intercombination” line.
Note: densities are in cm-3 (multiply by 106 to get m-3).

41. Observations of O VII lines during a solar flare observed by the P78-1 spacecraft (1980)
Max. density spectrum
Doschek et al. (1981)
Low-density spectra

42. Density measurements from O VII X-ray lines during solar flares
Repeated measurements of O VII X-ray lines for solar flare on 1980 April 8 (P78-1 X-ray spectrometer) gave:
Maximum density Ne ~ 2 × 1018 m-3
Late in flare Ne ~ 5 × 1017 m-3.
Temperature (Te) of O VII lines ~ 2 MK.
So gas pressure at maximum density was
Ne kBTe ~ 55 Pa.
This is based on the paper by Doschek et al. (1981)

43. Other diagnostics
Spectral lines (not necessarily coronal emission lines) can give information about mass motions, e.g. flows of gas or plasma.
This is done through the Doppler shifts of spectral lines.
In solar flares, X-ray lines indicate huge upflowing plasma flows at the flare onset phase, velocities of several hundred km/s.
Spectral line profiles are often broadened beyond the thermal Doppler width, suggesting plasma turbulence. This occurs in the quiet Sun, active regions, and particularly flares.

44. CORONAL HEATING

45. Coronal Heating
The solar corona has a temperature of 1—2 MK (in active regions up to 2—5 MK). The temperature of the photosphere is only 6400 K.
There must therefore be a heating mechanism that raises the coronal temperature to these high values.
Radiative transfer of energy (as in the solar interior) does not play a part – photospheric radiation passes straight through the corona without being absorbed.
There is a correlation of temperature with magnetic complexity – complex active regions are hotter than “quiet” coronal regions where the field is relatively simple. So the heating mechanism is magnetic in origin.

46. Two main possibilities for the magnetic heating of the corona
Either the corona is heated by the dissipation of waves – magnetohydrodynamic or MHD waves –that are associated with the magnetic field B;
Or the corona is heated by lots of tiny flares that are beyond the limits of observation: flares are sudden releases of energy due to the conversion of magnetic energy (B2/2μ) to heat energy and the acceleration of particles. A large flare releases ~1025 J, so many “nanoflares” (energy/nanoflare ~1016 J or less) might heat the corona. (First proposed by E. N. Parker in 1988.)

47. MHD wave heating
A simple type of MHD wave is an Alfvén wave – a wave travelling along the magnetic field lines.
They do not easily dissipate their energy.
More likely, magneto-sonic waves (hybrid of Alfvén and sound wave) are involved in coronal heating – either slow-mode or fast-mode waves.
Observationally, intensity oscillations in the corona can be expected with very short periods (~1 second or less).
Observations during eclipses indicate that this may happen, but still no clear signals as yet.
Wave heating may or may not be a significant coronal heating mechanism.

48. Nanoflare heating of the corona
Flares with smaller energy releases E are more numerous than larger flares: roughly :
f(E) = dN/dE = A × E-n
with n = 1.8 ± 0.2.
Total energy released by all flares in a range E1 to E2 is
∫ f(E) E dE = A ∫ E-n+1 dE
= constant × (E2-n+2 – E1-n+2).
So only if n > 2 are small flares capable of supplying the corona with energy.
Observationally n is almost exactly equal to 2, so it is unclear whether the nanoflare heating mechanism is viable.

49. More recent developments
Searches for short-period waves in the corona haven’t been successful, but Tomczyk et al. (2007: Science 317, 1192) found striking evidence for long-period (few minutes) waves that might be Alfven waves emerging round an active region. Unfortunately, they are probably not significant for heating.

50. Spicules and coronal heating
De Pontieu et al. (2011: Science 331, 55) using the Hinode SOT observed large spicules in the chromospheric Hα (656.3nm) line which shoot outwards, disappear, then a short distance away a coronal brightening (0.1 – 1 MK) occurs. So spicules may supply the mass of the corona and its heating.