Atmospheres of Cool Stars Radiative Equilibrium Models Extended Atmospheres Heating Theories

Radiative Equilibrium Models Gustafson et al. (2005): MARCS code difficult because of UV “line haze” (millions of b-b transitions of Fe I in 300-400 nm range, and Fe II in 200-300 nm range) Convection important at depth Metallicity and line blanketing causes surface cooling and back warming

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

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

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

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

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

Extended Atmospheres Photosphere Chromosphere Transition region Corona Wind

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

X-ray image of Sun’s hot coronal gas

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 – 10000 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

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 1550 (resonance or ground state transitions)

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)

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

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)

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)

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

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

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 = 10000 K yields a damping length of λ = 1500 km

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

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

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

Radiative Loss Function Below T = 15000 K, f(T) is a steep function of T because of increasing H ionization Above T = 15000 K, H mostly ionized so it no longer contributes much to cooling He ionization becomes a cooling source for T > 20000 K: Above T=105 K, most abundant species are totally ionized → slow decline in f(T)

Radiative Loss Function

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

(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

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

(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.

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

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.)