The Sun As a Typical Star Stellar Spectra

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Stellar Spectroscopy The Sun as a Typical Star Resolved observaons of the sun allow us to look at variaons across the surface, but we can only look at • Solar Interior cannot be probed directly (almost all) other stars in integrated light. (except neutrinos & helioseismology) In the visible, we see the photosphere as a disk (projected • Emerging radiaon from solar atmosphere hemisphere). The surface temperature is ~5700K tells us compared to 15,000,000K in the core. Photons journey outwards being mulply scaered, absorbed and re- – Total Flux (Luminosity) emied before they emerge. Random walk process R~ λN1/2 and with mean free path, λ ~ 1cm in the solar core, – Photospheric Temperature, Density 20 N~10 encounters before emerging from the – Surface Abundances photosphere The visible connuum spectrum and emission and/or – Dynamics absorpon lines inform us about the surface layers. • Photosphere well deﬁned layer Energy ﬂow and stellar models allow us to infer the interior -4 -3 structure (checked via astroseismology) – T~ 5800K, ρ~ 10 kg m What you see depends on how you look: – ~300 km thick, (0.0005 R) see EUV, Visible images

Above the photosphere, the major regions are the – Fraunhofer Spectrum Chromosphere and Corona and the solar wind – Limb Darkening, Granulaon, The temperature in the tenuous Corona is ~ 2 106 K with Sunspots emission lines from highly ionized species. The heang mechanism is not fully understood, but involves magnec reconnecon.

Stellar Spectra Note the appearance and • Stars classiﬁed via spectral lines then disappearance of the • HD Spectral Sequence OBAFGKMLT 40,000 -> 1500 K break in the spectra near • Sun is a G2V main sequence star (V= dwarf, high surface gravity) 360nm (the Balmer • Fraunhofer absorpon lines - element abundances seen against jump) , the weakening of connuous (approx black body) spectrum hydrogen absorpon lines in G-type stars and the • Note that the density is high and we do not see forbidden lines onset of molecular band from the photosphere (but we do from the Corona) emission in the cool M • Variability (sun spot cycle, rotaon, pulsaon, ﬂares) stars

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

H Balmer series ⊕ Atmospheric absorpon

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Emission and Absorpon

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

r To Observer θ

Consider radiaon of Speciﬁc Intensity Iν and solid angle dΩ normally incident on a slab of stellar atmosphere with cross secon dA, thickness ds and density ρ. 2 Annulus on stellar surface has an area 2πr dr = 2πR sinθcosθdθ normal As it propagates through, the beam loses energy through absorpon to line of sight dEν = kν Iν ρdsdAdΩdνdt and subtends solid angle dΩ= 2π(R/D)2sinθcosθdθ with distance D where kν is the exncon coeﬃcient per unit mass, or opacity, 2 and consists of scaering and and absorpon terms k = + Flux measured by observer is Fν = (R/D) Fν(0) . σ α

Radiave Transfer Opcal Depth Consider a path through the slab ds = dz/cos = dz/ Energy emied in the direcon of propagaon θ µ dz

dEν = jν ρdsdAdΩdνdt µdIν /ρdz = Jν - kν Iν where is the emission coefficient per unit mass, containing contribuons jν In general k will vary as a funcon of x, and the integral of the from scaering and thermal emission exncon coefficient w.r.t. distance is the Opcal Depth, τ. The rao j / k - emissivity/opacity - is known as the Source Funcon, ν ν τ = ∫ k(x) ρ dz and Specific Intensity falls off with τ as: denoted by Sν -τ Note that in a purely absorbing atmosphere, Sν = Bν (T) I = I0 e and opcal depth τ = ln (I0/I) (note: τ is measured inwards) - liming case for thermodynamic equilibrium and is a direct measure of the absorpvity of the medium

where jν = kνBν (T) (Kirchoff’s Law) Purely eming medium: µdIν /dz = ρ Jν in a pure scaering atmosphere with no absorpon, S = J . ν ν and no emission : µ dIν / dz = - kν ρ Iν Dividing the top equaon by k gives the standard form of the The difference between energy emied and absorbed in the element is ν related to the change in Specific Intensity of the beam: Radiave Transfer Equaon : dI µ ν = I − S dI dAdΩdνdt = (J ρds - k ρ I ds)dAdΩdνdt ν ν ν ν ν ν dτν

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Consider radiaon of Speciﬁc Intensity I ν Emergent Flux and solid angle dω emerging from a The Grey Atmosphere Mulplying the equaon of Radiave Transfer by integraon factor e-τνsecθ surface dσ at an angle θ to the normal. dI The ﬂux: gives ν −τν secθ −τν secθ e = (Iν − Sν )e d(τν secθ) 4π πF (τ ) = I (τ ,θ)cosθdω ν ν ∫0 ν ν dI or ν −τν secθ −τν secθ (−Iν + )e = −Sν e d(τν secθ) 2π π € −τ secθ πF (τ ) = I (τ ,θ)cosθ sinθdθdφ ν ν ν ∫ ∫ ν ν dI e −τ secθ € 0 0 ν = −S e ν d( sec ) ν And with d(cosθ)/dθ = -sinθ, τν θ € And ∞ I e−τν secθ = I (0,θ) = S e−τν secθ d(τ secθ) € [ ν ]0 ν ∫ ν ν 1 πF (τ ) = 2π I (τ ,θ)cosθ d(cosθ ) We approximate the depth dependence of the source funcon S€ ν by Sν = aν + bν τν ν ν ∫−1 ν ν along the normal direcon

So with a plane-parallel atmosphere € −τν secθ −τν secθ So Iν (τν ,θ) = ∫ aν e d(τν secθ) + bν ∫ τν e d(τν secθ)

€

The Grey Atmosphere Limb Darkening −τν secθ −τν secθ Iν (τν ,θ) = ∫ aν e d(τν secθ) + bν ∫ τν e d(τν secθ) The 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 I (τ ,θ) = a + b cosθ To Observer € ν ν ν ν θ 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

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Limb Darkening Limb Darkening The emergent intensity at a posion on the stellar disk is given As the line of sight moves from the by ∞ sec centre to the edge of the stellar disk, it I (θ ) = S e−τν θ secθdτ passes through an increasing path ν ∫0 ν ν and for a grey atmosphere length of atmosphere. 3 2 Degree of limb darkening depends on S(τ ) = (τ + )F(0) the opcal depth and temperature 4π 3 gradient. As τ increases, it Note that with B(τ) approximates an opaque surface, with a hard edge, so see into very similar physical depths at centre and edge.

Opacity is higher at infrared when τ=2/3, T = Teff ; so that the effecve depth at which the wavelengths and the effect of connuum is emied is τ=2/3 temperature gradient in outer layers is lower. and we have S(τ) = B(τ) in the form S(ν ) = a + bτν Solar limb scans at different wavelengths.

Limb Darkening Limb darkening in HD 209458

S(ν ) = a + bτν Transing planet shows that the limb of HD I ( ) I (0)(a bcos ) 209458 is more darkened or with τ α cosθ ν θ = ν + θ at short wavelengths, as in the sun. 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.

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Opacity in Stars Contribuons to H opacity

Hydrogen is the most Bound-free absorpon: abundant species but in Ionizaons from the solar-type stars, both the ground state (Lyman ionizaon fracon and the series) lie in the UV, populaons in excited while the Balmer edge states are low. (ionizaon limit from H is predominantly neutral n=2) occurs at 365nm and in the ground state so cannot contribute to Bound-free or bound-bound visible opacity transions will dominate Paschen edge (ionisaon limit from n=3) occurs at 820nm so will contribute opacity in the visible

Paschen edge (ionisaon limit from n=3) - occurs at 820nm so can contribute opacity Opacity H Opacity in the visible, but we need the n=3 level to be populated for this to contribute Free-free absorpon will also signiﬁcantly. give a connuum opacity, but With T<6000K, the populaon in the n=3 at a low level level (12 eV) is very small (esmated from Hydrogen can form a negave the Boltzmann distribuon) ion with a proton + 2 electrons. The dissociaon However, in hoer stars the higher n energy of H- is 0.75eV (1.65 levels are populated and H bound-free µm) and so it can provide opacity becomes important at connuum opacity in the T > 10000 K. visible and near-infrared. In even hoer stars, electron scaering H- will dominate in cool stars, (wavelength independent Thompson but with increasing scaering) becomes dominant photospheric temperature, higher n levels in H will be Bound-bound transions may be populated and atomic H important. H has few transions, but bound-free dominates in A- some metals (e.g. Fe) have many. type stars. In cool stars, molecules dominate the In the sun, the Saha equaon spectrum e.g. TiO,VO,CO,C gives the ionizaon balance 2 - Fracon of H atoms in the n=2 level as a funcon of temperature between H and H . N(H-)/N(H) ~ 3 10-8.

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LTE Level Populaons Saha Equaon Ionizaon equilibrium is described by the Saha equaon • Derived from the Boltzmann formula Atomic/ionic levels are populated thermally (Boltzmann • Gives the balance between successive stages of ionizaon as distribuon) a funcon of temperature: Ion and e- velocity distribuons are Maxwellian N N 2 g The source funcon is given by the Planck funcon. e i+1 i+1 −ΔE / kBT = 3 e Ni Λ gi Boltzmann Equaon N g i = i e−χ i / kT N1 g1 where Λ is the thermal deBroglie wavelength of the electron 2 (Λ=√h /2πm€ ekBT ), Ni+1 is the number in the state i+1 and gi+1 Where Ni is the populaon of level i with stascal weight gi and is the stascal weight of that state) excitaon energy χi of ionizaon stage j -8 • You have seen this in the cosmology lectures, describing the In the solar photosphere, the fracon of H excited to n=2 is ~10 ionizaon state of the Universe, and the same expression € -10 and fracon excited to n=3 is ~ 6 . 10 determines the ionizaon state of H in stars.

Ionizaon state in the solar photosphere Absorpon Lines • The Saha equaon gives Connuum opacity, resulng from the energy levels in Hydrogen (and • N(H+)/N(H) = 10-4 at T=6,000K and = 0.07 at 10,000K other species) dictates the overall shape of the observed spectra. With increasing temperature, the increasing numbers of H atoms in the n=2 level Absorpon lines arise in regions of enhanced opacity, corresponding increase the opacity shortwards of 365nm above that from the H- ion. The to transions from populated levels 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. Heavy elements with low lying levels above the ground state can In cool stars, the break can be characterised as : 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) k(365+) k(H )N(H ) = − − Ca is predominantly singly ionized : the Saha equaon gives k(365−) k(H )N(H ) + k(H)N H (n = 2) N(Ca II) / N(Ca I) ~ 900 for T= 5800 K - I.P. = 6.11 eV 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 and almost all Ca II ions are in the ground state : N2/N1 ~ 1/270. the Balmer jump is a good diagnosc of temperature. -5 So although the abundance is only 7 10 NH, almost all Ca is in the € k(365+) k(H)N (n = 3) + = H ground state of the Ca ion and the total number is much greater than k(365−) k(H)N (n = 2) the number of H atoms in n=2 (Balmer series) at T~ 6000K H

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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

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

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