16/05/2016 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 mul7ply scaered, absorbed and re- – Total Flux (Luminosity) emiLed 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 con7nuum spectrum and emission and/or – Dynamics absorp7on 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 magne7c reconnec7on. 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 absorp7on lines - element abundances seen against jump) , the weakening of con7nuous (approx black body) spectrum hydrogen absorp7on 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 absorp7on 1 16/05/2016 Emission and Absorp7on Measurement of Stellar Flux In stars, photons are absorbed and sca)ered (absorbed energy may be thermalised before being re-emied). R r To Observer θ Consider radiaon of Specific Intensity Iν and solid angle dΩ normally incident on a slab of stellar atmosphere with cross sec7on 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 absorp7on to line of sight dEν = kν Iν ρdsdAdΩdνdt and subtends solid angle dΩ= 2π(R/D)2sinθcosθdθ with distance D where kν is the ex7nc7on coefficient per unit mass, or opacity, 2 and consists of scaering and and absorp7on terms k = + Flux measured by observer is Fν = (R/D) Fν(0) . σ α Radiave Transfer Op7cal Depth Consider a path through the slab ds = dz/cos = dz/ Energy emiLed in the direc7on of propagaon θ µ dz dEν = jν ρdsdAdΩdνdt µdIν /ρdz = Jν - kν Iν where is the emission coefficient per unit mass, containing contribu7ons jν In general k will vary as a func7on of x, and the integral of the from scaering and thermal emission ex7nc7on coefficient w.r.t. distance is the Op6cal Depth, τ. The rao j / k - emissivity/opacity - is known as the Source Func7on, 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 op7cal depth τ = ln (I0/I) (note: τ is measured inwards) - limi7ng case for thermodynamic equilibrium and is a direct measure of the absorp7vity of the medium where jν = kνBν (T) (Kirchoff’s Law) Purely eming medium: µdIν /dz = ρ Jν in a pure scaering atmosphere with no absorp7on, 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 emiLed 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 € 2 16/05/2016 Consider radiaon of Specific Intensity I ν Emergent Flux and solid angle dω emerging from a The Grey Atmosphere Mul7plying 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 func7on S€ ν by Sν = aν + bν τν ν ν ∫−1 ν ν along the normal direc7on 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 Func7on at a depth of τ = ⅔ € or that the effec7ve temperature of a star is equal to the temperature at € τ = ⅔ 3 16/05/2016 Limb Darkening Limb Darkening The emergent intensity at a posi7on 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 op7cal 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 effec7ve 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 Transi7ng 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. 4 16/05/2016 Opacity in Stars Contribu7ons to H opacity Hydrogen is the most Bound-free absorp7on: abundant species but in Ionizaons from the solar-type stars, both the ground state (Lyman ionizaon frac7on 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 transi7ons 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 absorp7on will also significantly. give a con7nuum opacity, but With T<6000K, the populaon in the n=3 at a low level level (12 eV) is very small (es7mated from Hydrogen can form a negave the Boltzmann distribu7on) ion with a proton + 2 electrons. The dissociaon However, in hoLer 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 con7nuum opacity in the T > 10000 K. visible and near-infrared. In even hoLer 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 transi7ons may be populated and atomic H important.