16/05/2016

Stellar Resolved observaons of the allow us to look at variaons across the , but we can only look at (almost all) other in integrated . In the visible, we see the as a disk (projected hemisphere). The surface is ~5700K compared to 15,000,000K in the core. journey outwards being mulply scaered, absorbed and re- emied before they emerge. Random walk process R~ λN1/2 and with mean free path, λ ~ 1cm in the core, N~1020 encounters before emerging from the photosphere The visible connuum spectrum and emission and/or absorpon lines inform us about the surface layers. flow and stellar models allow us to infer the interior structure (checked via astroseismology) What you see depends on how you look: see EUV, Visible images Above the photosphere, the major regions are the and Corona and the solar The temperature in the tenuous Corona is ~ 2 106 K with emission lines from highly ionized species. The heang mechanism is not fully understood, but involves magnec reconnecon.

The Sun as a Typical

• Solar Interior cannot be probed directly (except & ) • Emerging radiaon from solar tells us – Total Flux () – Photospheric Temperature, – Surface Abundances – Dynamics • Photosphere well defined layer – T~ 5800K, ρ~ 10-4 kg m-3

– ~300 km thick, (0.0005 R) – Fraunhofer Spectrum – , Granulaon,

1 16/05/2016

Stellar Spectra • Stars classified via spectral lines • HD Spectral Sequence OBAFGKMLT 40,000 -> 1500 K • Sun is a G2V star (V= dwarf, high surface ) • Fraunhofer absorpon lines - element abundances seen against connuous (approx ) spectrum • Note that the density is high and we do not see forbidden lines from the photosphere (but we do from the Corona) • Variability (sun spot cycle, rotaon, pulsaon, flares)

H ⊕ Atmospheric absorpon

Note the appearance and then disappearance of the break in the spectra near 360nm (the Balmer jump) , the weakening of absorpon lines in G-type stars and the onset of molecular band emission in the cool M stars

These gross change reflect changes in the dominant source of in the

2 16/05/2016

Measurement of Stellar Flux

R r To Observer θ

Annulus on stellar surface has an area 2πr dr = 2πR2sinθcosθdθ normal to line of sight and subtends dΩ= 2π(R/D)2sinθcosθdθ with distance D 2 Flux measured by observer is Fν = (R/D) Fν(0) .

Emission and Absorpon

In stars, photons are absorbed and scaered (absorbed energy may be thermalised before being re-emied).

Consider radiaon of Specific Intensity Iν and solid angle dΩ normally incident on a slab of with cross secon dA, thickness ds and density ρ. As it propagates through, the beam loses energy through absorpon

dEν = kν Iν ρdsdAdΩdνdt where kν is the exncon coefficient per unit , or opacity, and consists of scaering and and absorpon terms k = σ + α

3 16/05/2016

Radiave Transfer Energy emied in the direcon of propagaon

dEν = jν ρdsdAdΩdνdt where jν is the emission coefficient per unit mass, containing contribuons from scaering and emission The o jν / kν - emissivity/opacity - is known as the Source Funcon, denoted by Sν Note that in a purely absorbing atmosphere, Sν = Bν (T) - liming case for thermodynamic equilibrium

where jν = kνBν (T) (Kirchoff’s Law)

in a pure scaering atmosphere with no absorpon, Sν = Jν .

The difference between energy emied and absorbed in the element is related to the change in Specific Intensity of the beam:

dIνdAdΩdνdt = (Jν ρds - kνρ Iν ds)dAdΩdνdt

Opcal Depth Consider a path through the slab ds = dz/cos = dz/ θ µ dz µdIν /ρdz = Jν - kν Iν In general k will vary as a funcon of x, and the integral of the exncon coefficient w.r.t. distance is the Opcal Depth, τ. τ = ∫ k(x) ρ dz and Specific Intensity falls off with τ as: -τ I = I0 e and opcal depth τ = ln (I0/I) (note: τ is measured inwards) and is a direct measure of the absorpvity of the medium

Purely eming medium: µdIν /dz = ρ Jν and no emission : µ dIν / dz = - kν ρ Iν Dividing the top equaon by kν gives the standard form of the Radiave Transfer Equaon : dI µ ν = I − S ν ν dτν

€ 4 16/05/2016

Consider radiaon of Specific Intensity I ν Emergent Flux and solid angle dω emerging from a surface dσ at an angle θ to the normal. The flux:

4π πF (τ ) = I (τ ,θ)cosθdω ν ν ∫0 ν ν

2π π πFν (τν ) = ∫ ∫ Iν (τν ,θ)cosθ sinθdθdφ € 0 0 And with d(cosθ)/dθ = -sinθ,

€ 1 πF (τ ) = 2π I (τ ,θ)cosθ d(cosθ ) ν ν ∫−1 ν ν

So with a plane-parallel atmosphere

€

The Grey Atmosphere Mulplying the equaon of Radiave Transfer by integraon factor e-τνsecθ

dI gives ν −τν secθ −τν secθ e = (Iν − Sν )e d(τν secθ)

dI or (−I + ν )e−τν secθ = −S e−τν secθ ν d(τ secθ) ν € ν −τν secθ dI e −τ secθ ν = −S e ν d( sec ) ν τν θ € And ∞ I e−τν secθ = I (0,θ) = S e−τν secθ d(τ secθ) [ ν ]0 ν ∫ ν ν We approximate the depth dependence of the source funcon S€ ν by Sν = aν + bν τν along the normal direcon

€ −τν secθ −τν secθ So Iν (τν ,θ) = ∫ aν e d(τν secθ) + bν ∫ τν e d(τν secθ)

€ 5 16/05/2016

The Grey Atmosphere

−τν secθ −τν secθ Iν (τν ,θ) = ∫ aν e d(τν secθ) + bν ∫ τν e d(τν secθ)

sec ∞ b sec ∞ b sec ∞ = −a "e−τν θ $ − ν "e−τν θ $ − ν "τ secθe−τν θ $ ν # %0 # %0 # ν %0 secθ secθ € Iν (τν ,θ) = aν + bν cosθ At the stellar 1 surface, need only πF (τ ) = 2π (a + b cosθ)cosθ d(cosϑ ) ν ν ∫0 ν ν consider 0 < cosθ < 1 € % 2 ( So πFν (τν ) = π aν + bν &' 3 )* € and with Sν = aν + bν τν , we see that Fν (0) = Sν (τν = 2/3) This is the Eddington-Barber relaon which shows that the flux that emerges from a stellar surface is equal to the Source Funcon at a depth of τ = ⅔ € or that the effecve temperature of a star is equal to the temperature at € τ = ⅔

Limb Darkening

The

To Observer θ

6 16/05/2016

Limb Darkening

As the line of sight moves from the centre to the edge of the stellar disk, it passes through an increasing path length of atmosphere. of limb darkening depends on the opcal depth and temperature gradient. As τ increases, it approximates an opaque surface, with a hard edge, so see into very similar physical depths at centre and edge. Opacity is higher at wavelengths and the effect of temperature gradient in outer layers is lower.

Solar limb scans at different wavelengths.

Limb Darkening The emergent intensity at a posion on the stellar disk is given by ∞ sec I (θ ) = S e−τν θ secθdτ ν ∫0 ν ν and for a grey atmosphere 3 2 S(τ ) = (τ + )F(0) 4π 3 Note that with B(τ)

when τ=2/3, T = Teff ; so that the effecve depth at which the connuum is emied is τ=2/3

and we have S(τ) = B(τ) in the form S(ν ) = a + bτν

7 16/05/2016

Limb Darkening

S(ν ) = a + bτν

I ( ) I (0)(a bcos ) or with τ α cosθ ν θ = ν + θ

measurements in the visible I(θ ) approximate with good = (0.4 + 0 .6cosθ ) agreement down to cosθ ~ 0.1 I(0)

as we observe closer to the limb of the sun, we see into progressively shallower and cooler regions.

Limb darkening in HD 209458

Transing shows that the limb of HD 209458 is more darkened at short wavelengths, as in the sun.

8 16/05/2016

Opacity in Stars

Hydrogen is the most abundant species but in solar-type stars, both the ionizaon fracon and the populaons in excited states are low. H is predominantly neutral and in the ground state Bound-free or bound-bound transions will dominate

Contribuons to H opacity

Bound-free absorpon: Ionizaons from the ground state (Lyman series) lie in the UV, while the Balmer edge (ionizaon limit from n=2) occurs at 365nm so cannot contribute to visible opacity Paschen edge (ionisaon limit from n=3) occurs at 820nm so will contribute opacity in the visible

9 16/05/2016

Paschen edge (ionisaon limit from n=3) occurs at 820nm so can contribute opacity Opacity in the visible, but we need the n=3 level to be populated for this to contribute significantly. With T<6000K, the populaon in the n=3 level (12 eV) is very small (esmated from the Boltzmann distribuon) However, in hoer stars the higher n levels are populated and H bound-free opacity becomes important at T > 10000 K. In even hoer stars, scaering (wavelength independent Thompson scaering) becomes dominant

Bound-bound transions may be important. H has few transions, but some metals (e.g. Fe) have many. In cool stars, molecules dominate the spectrum e.g. TiO,VO,CO,C2 Fracon of H atoms in the n=2 level as a funcon of temperature

H- Opacity Free-free absorpon will also give a connuum opacity, but at a low level Hydrogen can form a negave with a + 2 . The dissociaon energy of H- is 0.75eV (1.65 µm) and so it can provide connuum opacity in the visible and near-infrared. H- will dominate in cool stars, but with increasing photospheric temperature, higher n levels in H will be populated and atomic H bound-free dominates in A- type stars. In the sun, the Saha equaon gives the ionizaon balance between H and H-. N(H-)/N(H) ~ 3 10-8.

10 16/05/2016

LTE Level Populaons Ionizaon equilibrium is described by the Saha equaon Atomic/ionic levels are populated thermally (Boltzmann distribuon) Ion and e- distribuons are Maxwellian The source funcon is given by the Planck funcon.

Boltzmann Equaon N g i = i e−χ i / kT N1 g1

Where Ni is the populaon of level i with stascal gi and excitaon energy χi of ionizaon stage j In the solar photosphere, the fracon of H excited to n=2 is ~10-8 and € fracon excited to n=3 is ~ 6 . 10-10

Saha Equaon • Derived from the Boltzmann formula • Gives the balance between successive stages of ionizaon as a funcon of temperature:

NeNi+1 2 gi+1 −ΔE / kBT = 3 e Ni Λ gi

where Λ is the thermal deBroglie wavelength of the electron 2 (Λ=√h /2πmekBT ), Ni+1 is the number in the state i+1 and gi+1 is the stascal weight of that state) € • You have seen this in the lectures, describing the ionizaon state of the , and the same expression determines the ionizaon state of H in stars.

11 16/05/2016

Ionizaon state in the solar photosphere • The Saha equaon gives • N(H+)/N(H) = 10-4 at T=6,000K and = 0.07 at 10,000K With increasing temperature, the increasing numbers of H atoms in the n=2 level increase the opacity shortwards of 365nm above that from the H- ion. The increased opacity leads to emission at these wavelengths arising from higher layers, where the temperature is lower, and appears as a break in the spectrum. In cool stars, the break can be characterised as :

− − k(365+) k(H )N(H ) = − − k(365−) k(H )N(H ) + k(H)N H (n = 2) While in hot stars, H- becomes insignificant and the opacies arise from bound-free absorpon from n=2 shortwards of 365nm and n=3 longwards of 365nm, and so the Balmer jump is a good diagnosc of temperature.

€ k(365+) k(H)N (n = 3) = H k(365−) k(H)N (n = 2) H

€ Absorpon Lines Connuum opacity, resulng from the energy levels in Hydrogen (and other species) dictates the overall shape of the observed spectra. Absorpon lines arise in regions of enhanced opacity, corresponding to transions from populated levels Heavy elements with low lying levels above the ground state can produce absorpon lines in the visible, even in relavely cool stars. e.g. the Alkali metals (the Na D lines 2p-2s have ΔE ~2eV and lie in the visible) Ca is predominantly singly ionized : the Saha equaon gives N(Ca II) / N(Ca I) ~ 900 for T= 5800 K - I.P. = 6.11 eV

and almost all Ca II are in the ground state : N2/N1 ~ 1/270.

-5 So although the abundance is only 7 10 NH, almost all Ca is in the ground state of the Ca+ ion and the total number is much greater than the number of H atoms in n=2 (Balmer series) at T~ 6000K

12 16/05/2016

Absorpon line formaon

Prominent Stellar Absorpon Lines

Line opacies depend on the ionizaon state and level populaons, and so depend on the temperature of the region where the opcal depth ~1

13 16/05/2016

Castelli & Kurucz Stars 1994 Dense, opcally thick objects. At a given wavelength, we see in to a layer corresponding to τ ~ 1 In a region of enhanced opacity (line transion) we do not probe as deeply as at an adjacent connuum wavelength, and so the temperature will be lower, producing an absorpon line. In a real star, there may be complex regions of high and low temperature that will complicate the interpretaon (Corona, Chromosphere etc) and/or stellar and oulows. Detailed modeling is needed for a complete picture

Comparison with models permits refinement of photospheric temperature, , rotaonal and turbulent moons and element abundances

Self-consistent Spectral energy distribuons

Mulple components stellar populaons, gas, , exncon, ionizaon, abundances, geometries….

14