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Stellar Atmospheric Structure S S + Ds metry Su fero mme ter r S In ch n o o ol ����cos-1 � ls 2 z e 0 h 0 ic 3 r0 M r1 r2 rN r3 p I� I� + dI� Stellar Atmospheric Structure s s + ds ) J A . f P C . n A a u ni fd o en hs be mit rg (Harvard-S On Stellar Atmospheres "Stars are massive and they have no walls." Steve Shore: The Tapestry of Modern Astrophysics (2002) "...the transition from confinement in the stellar interior to open-ended interstellar emptiness... ...will keep you and me busy for years to come." Rob Rutten: Radiative Transfer and Stellar Atmospheres (2000) 2 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Whatʼs to Come... * Stellar Interiors vs. Stellar Atmospheres * Parameters and Equations from a Stellar Atmosphere Model * Spectroscopic Information on Stellar Atmospheric Structure * From Basic Radiative Transport to Limb-Darkening * with diversions for spherical atmospheres and the Sunʼs temperature structure * Concept of Radiative Equilibrium * Real Stars * Altair: rapid rotation * Deneb: Stellar Winds * β Peg: Extended M-giant atmospheres * Summary & References 3 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Stellar Atmospheres in Contrast Stellar Interiors versus Stellar Atmospheres Interiors Atmospheres Radiative flow of energy Diffusion Equation Radiative Transfer Equation Thermodynamic State: Radiation Field * Thermodynamic Equilibrium (TE) *Non-Local TE *Radiation enclosed by matter *Matter "sees" radiation of at approx. the same temperature different temperatures *Radiation field is Planckian *Radiation field is non-local *Radiation field is isotropic *Radiation field is anisotropic Thermodynamic State: Collisional Processes *Radiative processes dominate *Saha & Boltzmann Eqns. --> detailed balance describe ionization and and excitation *Saha & Boltzmann don't describe ionization and * Maxwellian velocity excitation distribution of ions and electrons *Maxwellian velocities (except chromospheres) 4 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School CHAPTER 4. UNIFIED STELLAR WIND MODELS 129 Example Parameters and Equations for Stellar Atmosphere Models* * In this case, hot stars with winds lEffective Temperature Input Parameters lReference Radius Teff , R, g(R), Z ˙ lOuter Radius lGravity @ Reference Radius Rmax, M, v , βwind ∞ lMass-Loss Rate lChemical Composition lMax. Stellar Wind Velocity lVelocity fi eld of the Wind Atmospheric Structure Outer Structure Hydrodynamic Hydrostatic Inner Structure β R wind dP g l lVelocity Field v = v 1 r dτ = κ Pressure gradient goes as gravity/opacity ∞˙ − l ρ = M Density structure via continuity equation 4πr2v CHAPTER 4. UNIFIED STELLAR WIND MODELS 114 Equation of State CHAPTER 4. UNIFIEDIn equation STELLAR (4.19), WINDµ is theMODELS direction-cosine, I is the specific114 intensity, the veloc- l LTE ionization and level lSpherical geometry populations, chemistry P = P (ρ, T ) Special Relativistic Radiative Transfer Equation In equation (4.19),ity isµ measuredis the direction-cosine, in units of theI speedis the of specific light c, βintensity,(r) = v(r )/cthe, andveloc- γ = 1/(1 β2)1/2. ∂I ∂ 2 (1+βµ) 2 ∂β l γ(µ + β) ∂r + ∂µ γ(1 µ ) r γ (µ + β) ∂r I − − − 2 2 1/2 ity is measured in units∂ of theβ speed(1 µ ) of light2 c, β(r)∂β = v(r)/c, and γ = 1/(1 β ) . The∂ν principalγ −r reason+ γ forµ(µ choosing+ β) ∂r νI to solve the radiative transfer problem in the co- − l − 2µ+β(3 µ2) 2 2 ∂β +γ r − + γ (1 + µ + 2βµ) ∂r I lµ is direction-cosine The principal reasonmoving, for choosing fluid torest solve frame the radiative is that transferalthough problem the frequency in the co- spectrum may be complex, Rate Equations = η χI lr is the radius − lNon-LTE level populations; b nj ∗ (R + C ) l j<i j ni ji ji I is the intensity radiative and collisional rates moving, fluid restwith frame many is overlapping that although lines, the the frequency opacity isl isotropic.spectrum This may leads be complex, to great simpli- bi j<i(Rij + Cij) η is the emissivity − κ Radiative Equilibrium +bi (Rij + Cij) lχ is the opacity j>i with many overlappingTemperaturefication lines, Correctionsin the the opacity radiation-matter is isotropic. Thisinteraction leads to greatterms relative simpli- to the observer’s frame κ nj ∗ = bj (Rji + Cji) − j>i ni ∞ fication in the equation.radiation-matter(ηλ χλJλ For) dλ example,= interaction 0 the flow terms velocity relative is tonot the anobserver’s argument frame of the line profile − 0 l Temperature structure adjusted equation. For example,function the in flow the comoving velocityto conserve frame.is not Frequenciesenergy; an argument in cool are measured of the line in profile the frame of the For Interferometry stars convective equilibrium can be Full radiation fi eld: intensities @ every angle function in the comoving frame. Frequenciesestablished are measured in the inner in structure the frame of the Synthetic Spectrummoving fluid, therefore we can use a static line profile. In this frame onecandi- moving fluid, thereforerectly integrate we can over use anglea static and useline comoving-frame profile. In momentsthis frame of onecandi- the radiation field 5 Stellar AtmosphericFigure Structure 4.1: Flow chartrectly showingJason integrate the P. Aufdenberg, most overin important angle the computation and 10 use equationsJuly comoving-frame 2003, of the (and Michelson radiation their moments inter- pressureInterferometry of the and radiation radiative Summer field equilibrium School equations. relationships) which characterize our model for the expanding stellar atmosphere problem in the STANDARD-WINDin the computationcase. In theThe ofREAL-WIND the angular radiation dependencecase pressure the velocity of and the radiative opacity field is in equilibrium the observer’s equations. frame leads to computa- calculated from a solution to the hydrodynamical momentum equation (see Eq. 4.2). The angular dependencetional difficulties,of the opacity inand the in observer’s flows with frame small leads velocity to computa- gradients with many overlapping tional difficulties,spectral and in lines, flows escape with probability small velocity methods, gradients like the with Sobolev many approximation, overlapping are not spectral lines, escapevalid. probability methods, like the Sobolev approximation, are not valid. The conservation of energy, assuming radiative equilibrium, can be specified in The conservationthe of Lagrangian energy, assuming frame (Hauschildt,radiative equilibrium, 1992b) by can be specified in the Lagrangian frame (Hauschildt, 1992b) by ∞ (ηλ χλJλ) dλ = 0 (4.20) 0 − ∞ (ηλ χλJλ) dλ = 0 (4.20) 0 − where ηλ and Jλ describe the emissivity and mean intensity respectively. This where ηλ and Jλ describeassumes the that emissivity each element and of mean material intensity absorbs respectively. the same amount This of energy that it assumes that eachemits. element of material absorbs the same amount of energy that it emits. – 27 – Spectroscopicbut the lineInformation profiles also have extensive wings due to pressure broadening and assucha good estimate of the surface gravity is needed to correctly interpret these lines. Figure 12 – 7 – shows both a typical curveChemical of growth Abundance and the kinds of lines that are found on its three main Energy– 8 Distribution; – Color Temperature segments. path length, compared with the physical depth seen at the center of the solar disk. Thus, the limb of the sun appears less bright than the center. In the simple case of a homogeneous slab of uniform temperature the equivalent width To determine Teff quantitatively the spectral energy distributionof a ofweak the line star transition must be from level i to j is given by known relative to someThe reference shape of standard the overall stellar such as spectrum a calibrated and the character oven. Near and strength the end of of its the spectral Linear Curve-of-Growth lines are most sensitive to the temperature and pressure structure of atmosphere. How and 2 Equivalent Width Equivalent πe 19th century ultra-sensitive alcohol thermometers were being employed to study the sun’s W = λ2N f (35) what we know about stellar temperatures and pressures are the subjects of the next two λ mc2 i ij heat distribution betweensections. 3000 A˚ and 3 µm. For other stars visual comparisons were made between starlight and a lamp calibrated on an oven. Assuming thewhere shapeN thei is spectrum the number ob- of ions in level i of element Z and fij is the atomic oscillator strength of the transition. TheLog Boltzmann (Abundance) (18) and Saha (19) equations relate N to the tained in this way was a black body, a value of T could be determined. Stellar continua are i eff total abundance of the element Z at a given temperature and pressure. The first segment of black bodies only to first order and so these2.1. Stellar first Temperaturesmeasurements were subject to considerable the curve of growth is established by the linear relationship between Wλ and Ni. systematic error. ModernTo first spectrophotometric order, in between observations the spectral of lines, stars arethe radiative calibrated energy relative distribution to froma The equivalent width of a given line therefore is dependent on the abundance, the the bright star Vega,stellar whose atmosphere spectral follows energy the Planck distribution function has(3) of been a characteristic derived relative temperature to pre-known as oscillator strength, and the thermal excitation. For a given line the f value and excitation cisely calibrated freezing-pointthe effective temperature, copper blackTeff . bodies. Formally, If the both effective the absolute temperature flux does distribution, not correspond to potential are fixed and our model temperature and pressure structure (constant forthe corrected for extinctiona particular by the temperature interstellar in mediumthe stellar and atmosphere, the earth’s but to atmosphere, the total radiative and energy the an- or the bolometric flux, , surface fl ux; fl ux per unit area at the photospherehomogeneous slab) sets the ionization and excitation state of the gas.
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