1. Atmospheres of Cool Stars
Radiative Equilibrium Models Extended Atmospheres Heating Theories

2. Radiative Equilibrium Models
Gustafson et al. (2005): MARCS code difficult because of UV “line haze” (millions of b-b transitions of Fe I in nm range, and Fe II in nm range)
Convection important at depth
Metallicity and line blanketing causes surface cooling and back warming

3. Solid line = Solar abundances [Fe/H]=0
Dwarfs Giants
Solid line = Solar abundances [Fe/H]=0
Dashed line = Metal poor [Fe/H]=-2
Dot-dashed = Kurucz LTE-RE model

4. Semi-empirical Models Based on Observations of Iλ(μ=1,τ=1)
Solar spectrum shows non-thermal components at very long and short wavelengths that indicate importance of other energy transport mechanisms

5. Major b-b and b-f transitions for solar opacity changes

6. Determine central specific intensity across spectrum
Get brightness temperature from Planck curve for Iλ
From opacity get optical depth on standard depth scale
T(h) for h=height

9. Reality: Structured and Heated
Optical: photosphere
EUV: higher
X-ray: higher yet

10. Extended Atmospheres Photosphere Chromosphere Transition region Corona
Wind

11. Corona
Observed during solar eclipses or by coronagraph (electron scattering in optical)
Nearly symmetric at sunspot maximum, equatorially elongated at sunspot minimum
Structure seen in X-rays (no X-ray emission from cooler, lower layers)
Coronal lines identified by Grotrian, Edlén (1939): Fe XIV 5303, Fe X 6374, Ca XV 5694
High ionization level and X-rays indicate T~106 K

12. X-ray image of Sun’s hot coronal gas

13. Chromosphere
Named for bright colors (“flash spectrum”) observed just before and after total eclipse
H Balmer, Fe II, Cr II, Si II lines present: indicates T = 6000 – K
Lines from chromosphere appear in UV (em. for λ<1700 Å; absorption for λ>1700 Å)
Large continuous opacity in UV, but lines have even higher opacity: appear in emission when temperature increases with height

14. Transition Region
Seen in high energy transitions which generally require large energies (usually in lines with λ<2000 Å)
Examples in solar spectrum: Si IV 1400, C IV (resonance or ground state transitions)

15. Stellar Observations
Chromospheric and transition region lines seen in UV spectra of many F, G, K-type stars (International Ultraviolet Explorer)
O I 1304, C I 1657, Mg II 2800
Ca II 3968, 3933 (H, K) lines observed as emission in center of broad absorption (related to sunspot number in Sun; useful for starspots and rotation in other stars)
Emission declines with age (~rotation)

18. Chromospheres in H-R Diagram
Emission lines appear in stars found cooler than Cepheid instability strip
Red edge of strip formed by onset of significant convection that dampens pulsations
Suggests heating is related to mechanical motions in convection

19. Coronae in H-R Diagram
Upper luminosity limit for stars with transition region lines and X-ray coronal emission
Heating not effective in supergiants (but mass loss seen)

20. Theory of Atmospheric Heating
Increase in temperature cannot be due to radiative or thermal processes
Need heating by mechanical or magneto-electrical processes
Nanoflares and magnetic recombination (Brosius et al. 2014, ApJ, 790, 112)

21. Acoustic Heating
Large turbulent velocities in solar granulation are sources of acoustic (sound) waves
Lightman (1951), Proudman (1952) show that energy flux associated with waves is where v = turbulent velocity and cs = speed of sound

22. Acoustic Heating
Acoustic waves travel upwards with energy flux = (energy density) x (propagation speed) = ½ ρ v2 cs
If they do not lose energy, then speed must increase as density decreases → form shock waves that transfer energy into the surrounding gas

23. Wave Heating
In presence of magnetic fields, sound and shock waves are modified into magneto-hydrodynamic (MHD) waves of different kinds
Damping (energy loss) of acoustic modes depends on wave period: ex. 5 minute oscillations of Sun in chromosphere with T = K yields a damping length of λ = 1500 km

24. Wave Heating Change in shock flux with height is
Energy deposited (dissipated into heat) at height h is where

25. Wave Heating
Energy also injected by Alfvén waves (through Joule heating caused by current through a resistive medium)
Observations show spatial correlation between sites of enhanced chromospheric emission and magnetic flux tube structures emerging from surface: magnetic processes cause much of energy dissipation

26. Balance Heating and Cooling
Energy loss by radiation through H b-f recombination in Lyman continuum (λ < 912 Å) and collisional excitation of H
In chromosphere, H mainly ionized, primary source of electrons
for H recombination for H collisional excitation
Similar relations exist for other ions

27. Radiative Loss Function
Below T = K, f(T) is a steep function of T because of increasing H ionization
Above T = K, H mostly ionized so it no longer contributes much to cooling
He ionization becomes a cooling source for T > K:
Above T=105 K, most abundant species are totally ionized → slow decline in f(T)

28. Radiative Loss Function

29. Energy Balance
In lower transition region (hot) Pg = 2 Pe Electron density: Radiation loss rate: (almost independent of T since f(T)~T 2.0)
Set T(h) by Einput = Erad
Suppose Einput = Fmech(h) / λ

30. (1) Einput = constant T(h) increases with h
Increasing height in outer atmosphere
Each line down corresponds to a 12% drop in Pg or a 26% drop in Pg2

31. (2) Einput declines slowly with h T(h) still increases with h

32. (3) Einput declines quickly with h T(h) may not increase with h
No T increase for damping length λ and pressure scale height H if λ < H/2 H is large in supergiants so heating in outer atmosphere does not occur.

33. Temperature Relation for Dwarfs
Suppose λ >> H in lower transition region so that Fmech(h) / λ ≈ constant
Constant temperature gradient

34. Heating in the Outer Layers
T>105 K, rad. losses cannot match heating
T increases until loss by conductive flux downwards takes over (+ wind, rad. loss)
Conductive flux (from hot to cool regions by faster speeds of hotter particles)
Find T(h) from (η ~ 10-6 c.g.s.)