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 4πr2v  CHAPTER 4. UNIFIED STELLAR WIND MODELS 114  Equation of State  CHAPTER 4. UNIFIEDIn 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 ji 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. From here the value gular diameter of a star can be measuredF precisely, a fundamental effective temperature can ∞ ∞ for the4 abundance is adjusted until the observed equivalent width is reproduced. Abundances = Fλ dλ = B(Teff )λ dλ = σTeff (14) computed from F 0 0 derived from linesNormalized Flux with (1) different excitation potentials, (2) different strengths, and fromof 1 where Fλ is monochromatic flux4 at 4each wavelengthconnected to λtheand fl (3)uxσ atis several the Stefan-Boltzmann different ionization con- stages of the same element, should yield the same abundance stant. The stellar continuumTeff = fromF a geometricallyearth via the thin angular photosphere diameter emanates from(17) layers σθ2  in a chemically homogeneousΔ atmosphere. Wavelength If they do not, the model parameters ( Teff , log g, near where τ = 16 in the continuum, consequently these layersabundance, have temperatures turbulent very line near width) are adjusted and the analysis proceeds in an iterative where θ is the angular diameter. CurrentlyStellar Atmospheric it is possible Structure to measureJasonθ accurately P. Aufdenberg, for10 July only 2003, Michelson Interferometry Summer School Teff . fashion until the abundances derived from different lines produce consistent results. bright stars, but the situation will improve soon with optical interferometers and space-based A qualitative estimate of T can be made by observing the color of a star. The mea- eff The use of equivalent widths discards all the shape information present in the line astrometric missionssurement set to of make the ratio precise of blue measurements to red light will of capture fainter this stars. brightness In thedistribution, absence the of mea- profile: only the integrated strength is used. This is appropriate for observations madewith accurate angular diametersurement measurements, being made either synthetic through filters stellar transmitting continua from broad model bands atmospheres of the blue andred moderate wavelength resolution because the equivalent width is not altered by the resolution, are employed to estimatespectralT regionseff . or with instruments designed to sample the light monochromatically from widely separated wavelengths. As observed in the laboratoryexpect and for described possible by systematic the Planck effects related to the estimation of the continuum reference The line spectrumfunction, is also incandescent used to objects estimate radiateTeff most. This brightly is possible at thelevel, short because whereaswavelengths, the the state line blue shape of or vio- is dominated by instrumental effects. Measurements made a gas, the excitationlet colors, and ionization whereas cool of stars the radiate constituent most strongly atoms at and longerwith molecules,wavelengths, high-resolution, is red governed colors. however, The minimize the instrumental contamination, permitting a direct by radiation, collisions,wavelength and the at the principle peak intensity of detailed of a black balance body. spectrum Detailedcomparison must balance satisfy of in the a computedthermal and observed line shapes. Both the equivalent width and the line profile approaches assume that the absorption line is isolated in the stellar spectrum. This plasma occurs when any reaction is balanced by an inversedB (T ) reaction. For example, the balance λ = 0 is generally not true, especially(15) for cool stars and at blue and ultraviolet wavelengths where between ionization (by radiation and collisions) anddλ recombination of an ion and electron. λmax severe crowding and overlapping of lines occurs. Important absorption lines are frequently  When collisions dominate,which yields this the principle relation can be used to derive the Boltzmannbadly blended distribution together, and, there is no clean spectrum of the measurement of an equivalent

λmaxT = 0.294 cm K width or a line profile. (16)In such cases it is necessary to compute the predicted spectrum n2 g2 E21/kT = e− including all the overlapping(18) and blended absorption lines, though this pushes our knowledge known as Wien’s displacementn1  g law.1 Low temperatures yield large values for λmax and visa- versa, consistent with observations. of atomic physics to the limit. If the synthetic spectrum does not match the observed for the relative population, n2/n1, of two energy levels in a atom separated by an energy E21. The ratio statistical weights, g2/g1, depends on the quantum mechanical properties of the two levels. In a similar way, the Saha equation,

+ + N 2u 3 χ /kT log n = λ− e− ion (19) N 0 e u e where, h λe = 1/2 (20) (2πmekT ) Spectroscopic Information Continued... Surface Gravity, Pressure Structure Rotational Velocity

β Canis Majoris α Cygni B2 III A2 Ia

Outfl ow Velocities Space, Orbital Motion H� double line

β Orionis 1 ζ Ursa Majoris aka Mizar A B8 Ia A2 V + A2 V C II

7 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Spectroscopic Information Continued... Surface Structure via Doppler Tomography Radial and non-Radial Pulsation; Convection

“wiggles”

“no wiggles”

Vogt & Penrod (1983) ApJ 275, 661

Solar Spectrum Gray: OASP

Donati et al. (1999) MNRAS 302, 437

8 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School 3 Intensity: The basic macroscopic quantity of RT

Start with...

dEλ = Iλ dω dσ dλ dt (1) 4 Radiative Energy Transport dEλ/dt: Energy per sec though area dσ through an opening angle dω at a wavelength λ. 4 Radiative Energy Transport

Intensity4 Radiative and the Radiative Energy Transport Transport 3 Intensity: The basic macroscopic quantity of RT

Start with... Intensity: The basic macroscopic quantity of absorptionTransport coefficient: of intensityκλ along a ray: Radiative Transfer. emission coefficient: λ absorption coefficient: κλ

dE��λ = Iλ dω dσ dλ dt ������� �� ����� ���������optical(1) �������� depth, τλ: dτλ = κλ ds emission coefficient: λ I� I� + dI� absorption coefficient: κ Along a ray:λ dEλ/dt: Energy per sec though�� area �������dσ�� ���through �������������� an �����optical opening ���� ��������� depth, angle ����� ����dωτ �����λat: dτ a ���λ �����= κ ���λ ds ���� � ���������� ��� ����������� ����������� �� ���������� ��� ������ ������� �� � ����� dIλ wavelength λ. � emission coefficient: λ Energy per�� time ��������� per ������� wavelength ����� � ������ ������ per �����area ������� ��� ��� ����� ������� �� �������� = κλ Iλ + λ (4) �������� ������������� ��������� �� ��� �����Along ����������� a ray: �������� �� ��� �� ����������� ds − Intensity (strictly the “specific intensity”):� per solid angle.��������� �������� � �� ����������� ��� ���������optical ��� ��������� depth, �������τλso, ����: dτ ���λ ���������= κλ dIdsλ s s + ds ����� � ��� �� ������ ���� ��� �������� ���� ������ �������� � ������� �� ��� ����� = κλ Iλ + λ (4) ��������������dEλ absorption dscoeffi −cient emission coeffi cient Iλ = Along a ray: (2) dIλ so, > 0 λ > κλ Iλ (5) dω dσ dλ dt z dIλ ds r sin θ ∆ϕ = κ I +  (4) dIλ λ λ and depends on: dIλ ds − λ so, > 0 λ >< κλ0Iλ λ < κλ Iλ (5) (6) ds pathds length Intensity depends on: r ∆θ 4 Radiative Energy Transport 1. location in space θ dIλ Divide bydIκλ<, 0 λ < κλ Iλ (6) y ds λ> 0 λ > κλ Iλ beam enhanced (5) 2. direction 1. location in space ϕ ds dIλ dI  93. The wavelength Flux2. and direction the Effective TemperatureDivide by κλ, < 0 λλ < κλ Iλ λ (6) x ds = Iλ + beam= extinguishedIλ + Sλ (7) 3. wavelength κ ds − κλ − Note: Intensity is independent������ of ���� the ����� distance ����� �� ����� ������������ from the ��� ���� source. �� ��� ������ ���� For ������ ex-� ������� �� ��� � � ��� 9.1 Flux: A Definition ��� ��� � � ��� �� �� ��� � �� �� �� ���� �� � ��� � �� ��� dIλ λ ample, the solar disk’s intensity is as bright from SaturnDivide as from bySource theκλ, Function:= ISλλ+= λ/κ= λ Iλ +SourceSλ Function (7) Note: intensity is independent of the distance κ ds − κλ − Earth.The The flux, flux thehowever net energy is not independent flow rate of distance.normal to a surface,We isamoment now have the basic radiative transfer equation (RTE): Intensity• (strictlyfrom the the “specific source.���� ���������� The intensity”): ���fl ����ux is ��������� not independent�� �������� ���� ��� ���������� ��� of the radiation field. This means: Source Function: Sλ = dIλ/κλ λ λ � � �� � = Iλ + absorption= Iλ + Sλ coefficient: κλ (7) of distance. ������ �� �� �� � �� ��� � �� ��� κ����� ds − κλ − dEλ � �� �� � � dI I = � We� � now have the(2) basic radiative transfer equationλ (RTE): λ � � � � = Iλ + Sλ (8) ������ ��� ��dω� � dσ� �� dλ� ���� dt� � ���� �� ��� ��� �� ����� �������� ���� ��� ������ emission coefficient:Opticalλ Depth π�� ��Fλ �����= ��� ����Iλ ��cos ��������θ dω ���������Source ������������� Function: ����� ������(37) ��������Sλ ��������= λ� /κλ dτ − � Radiative � ���������· ��� ����� �� � �� ��� � �� � �� �� ���� � ��� �� dIλ and depends on:  � We� nowNote: haveTransfer theWhen basic solving radiative= complicated transferI optical+ S equation problems depth, (RTE):τ evenλ: dτ forλ = complexκλ(8)ds geometries � � � �� λ λ where the solid angle dω is: �� ��� � ����� �� �� ��� � �� � �� ��� �� ��� ����� dτ − �� � � � theEquation RTE boils down to this simple one dimensional equation. 1. location in space � �� Along a ray: ���� �������� �� ��� ��� �� ��� ���� ����Note: ��� ������������When �� ������� solving �� �� ��������� complicated ��������� dIλ problems even for complex geometriesdI 9 ���� �� ��� ������ ������� ��� ������� ���� ���������� ��� ������ �� ��������� ���������� = Iλ + Sλ (8)λ Stellar Atmospheric Structurethe RTEJason boils P. Aufdenberg, down to this 10 simpleJuly 2003, one Michelson dimensional Interferometry equation. Summer School= κ λ Iλ + λ (4) 2. direction ��� �����������dω = sin θ dθ dφ dτ − ds −

3. wavelength ����� ��� ������������� ��� � ��� Note: When solving complicated problems evenso, for complex geometries For azimuthal symmetry: � ��the� RTE boils down to this simple one dimensional equation. ����� ��� �� �� ��� � �� � �� ��� � ��� � �� ��� ����� dI Note: Intensity is independent of the� distance� from the� source.� For ex- λ � � � > 0 λ > κλ Iλ (5) � +1 � � � � ample, the solar disk’s intensity������ ��� �� is��� as�� bright� �� � � from� ��� � Saturn ���� �� ��� as ��� from ��� �� ������ the ��� ������ ds ������� �� ���� ������ �� �������� � ������������� �� � � �� �� ��� �������� �� ��� ��� ����� π Fλ = 2π Iλ cos θ d cos θ (38) dIλ Earth. The flux however· ������ is ��� not ���������� independent1 �� ��������� ��������� of distance. ������� ������� ���������� ������� ������������ < 0 λ < κλ Iλ (6) ������� ��� ������������− ����� ����� ����� ���� ������ ���������� ��������� �� ������� ��� �� ���� ���� ds 9.2 Isotropic Radiation Field Divide by κλ, In this case the intensity is independent of direction (angle) so: • dIλ λ = Iλ + = Iλ + Sλ (7) +1 κ ds − κλ − π Fλ = 2πIλ cos θ d cos θ = 0 (39) · 1 Source Function: Sλ = λ/κλ − The flux is zero within an isotropic source We now have the basic radiative transfer equation (RTE): dI λ = I + S (8) dτ − λ λ Note: When solving complicated problems even for complex geometries the RTE boils down to this simple one dimensional equation. Now, Now,

dIλ > 0 Sλ > Iλ (9) dτ dIλ > 0 Sλ > Iλ (9) dIλ dτ < 0 Sλ < Iλ (10) dτ dIλ < 0 S < I (10) dτ λ λ 4.1 The Source Function and a Perfect Blackbody 4.1 The Source Function and a Perfect Blackbody Transfer in a perfect blackbody: Transfer in a perfect blackbody: dIλ = κλ Iλ + κλ Sλ (11) ds − dIλ = κ I + κ S (11) ds − λ λ λ λ with a perfect BB, Iλ is isotropic and homogeneous with a perfect BB, Iλ is isotropic and homogeneous dI λ 0 ds −→ dIλ 0 therefore, ds −→ therefore, S I B 4 Radiative Energy Transport λ ≡ λ ≡ λ Sλ Iλ Bλ Planck function: ≡ ≡ 2hc2 1 PlanckB = function: (12) λ λ5 ehc/λkT 1 2hc2 1 − Bλ = (12) λ5 ehc/λkT 1 Numerically, − Numerically, 5 c1λ− π Bλ = (13)5 absorption coefficient: κ · ec2/λT 1 c1λ− λ − π Bλ = (13) 5 2 1 c2/λT c1 = 3.741 10− [erg cm s− ] · e 1 emission coefficient: λ × 5 2 1 − c = 1.438 [cm K] c = 3.741 10− [erg cm s− ] 2 · 1 × optical depth, τ : dτ with= κ λdsin [cm] & T in [K].c2 = 1.438 [cm K] λ λ λ · 4 Radiative Energy TransportPerfect Black Bodies; Kirchoffʼswith λ in [cm] (Radiation) & T in [K]. Law Along a ray: Note: Sλ = Bλ = λ/κλ which leads to Kirchoff’s Law Now, dIλ Note: Sλ = Bλ = λ/κλ which leads to Kirchoff’s Law Now, = κλ Iλ + λ (4) ds −  = κ B (14) so, λ λ · λ dIλ λ = κλ Bλ (14) dIλ > 0 Sλ > Iλ (9) “A good emitter is a· > 0 dτ Sλ > Iλ dIλ (9) dτ > 0 λ > κλ Iλ good absorber”(5) dIλ ds dIλ < 0 Sλ < Iλ (10) < 0 dτ Sλ < Iλ dIλ (10) dτ < 0 λ < κλ Iλ (6) This must be absorption coefficient: κλ ds Good Good 4.1 The SourceFor a Functionperfect blackbody and a Perfect, the intensity Blackbody is true in general 4.1 The Source Function and a Perfect Blackbody Emitter Absorber emission coefficient: λ isotropic andDivide homogeneous. by κλ, not to violate Transfer in a perfect blackbody: Transfer in a perfect(A blackbody: good approximation deep in a stellar energy optical depth, τλ: dτλ = κλ ds dIλ λ screen atmosphere.) dIλ = I + = I + S (7) conservation. dI = κλ Iλ + κλ Sλ λ λ (11)λ λ κ ds − κλ − Along a ray: = κdsλ Iλ +−κλ Sλ (11) dIλ ds − From detailed Source= κλ I Function:λ + λ Sλ = λ/κλ (4) with a perfectds BB,−Iλ is isotropic and homogeneous balance, this with a perfect BB, Iλ is isotropicWe now and have homogeneous the basic radiative transfer equation (RTE): so, prism must be true dI λ isotropic at each wave- dIλ dIλ 0 dI > 0 λ > κλ 0Idsλ −→ and λ =(5)I + S (8) length λ in ds ds −→ dτ − λ λ therefore,dI homogeneous order that the therefore, λ < 0  < κ I (6) ds Note: Whenλ λ solvingλ complicated problems even for complex geometries 2nd law of ther- the RTE boils down to this simple one dimensional equation. Sλ Iλ Bλ modynamics is Divide by κλ, S I B≡ ≡ λ ≡ λ ≡ λ violated! Planck function: Planck function: 2 heated dIλ λ 2 2hc 1 = Iλ + 2hcB=λ =Iλ +1 5Sλ hc/λkT (7) (12) sodium κ ds −Bλ =κλ − λ e 1 (12) λ5 ehc/λkT 1 − vapor Source Function:Numerically,S =  /κ − incandescent Numerically,λ λ λ Planck Function lamp We now have the basic radiative transfer equation (RTE):5 5 c1λ− π cB1λλ−= c /λT (13) π Bλ = · e 2 1 (13) 10 dIλ c2/λT Stellar 5=Atmospheric· Iλ +e2Sλ 1 1 Structure− (8)Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School c1 = 3.7415 10dτ− 2 [erg−1 cm s− ] − c1 = 3.741 10− [erg× cm s− ] c2 ×= 1.438 [cm K] Note: Whenc = solving 1.438 [cm complicatedK] · problems even for complex geometries 2 with λ· in [cm] & T in [K]. the RTEwith boilsλ in down [cm] & to T this in [K]. simple one dimensional equation. Note: Sλ = Bλ = λ/κλ which leads to Kirchoff’s Law Note: Sλ = Bλ = λ/κλ which leads to Kirchoff’s Law

λ = κλ Bλ (14)  = κ B · (14) λ λ · λ ml then small htmgtohrieb oiae ya xenlbihe ore hsrlto hw that shows relation This at where source. greatest brighter external is an intensity by the dominated be otherwise might that pia depth, optical fe asn hog h slab, the through passing after lentvl,at Alternatively, I is,dvdn by dividing First, τ htnbfr titrcswt atr h ai fteeiso n bopincoefficients absorption and emission function, the of source ratio the The called matter. is with interacts it before photon Here hspoie h nest safnto fdphadi rd u sflapproximation useful but crude term a The is atmosphere. and planetary depth a of into function passing a light as for intensity the provides This since twvlntsweethe where wavelengths At oeotcldpho neet o xml,o h te ieo h slab. the of side other the on example, for interest, of depth optical some rmteba and beam the from safrtapoiain ti o a oasm hta atclrdepth ataparticular that toassume bad spectrum. solar not the is of it formation the understand approximation, to first used a be can As results basic These results. topeecnb hrceie yasnl eprtr.Ti supini nw slocal as known is assumption This temperature. single a by characterized be can atmosphere h nest is intensity the then, λ λ  = ntescn aealgtsuc srqie,so required, is source light a case second the In h ouino h rnfreuto o niohra,hmgnossa fgsis gas of slab homogeneous isothermal, an for equation transfer the of solution The eunn otefrtcs ihn ih source, no light with case first the to Returning S κ .Teslto ftetase qainnwbecomes now equation transfer the of solution The 1. e λ κ − λ λ τ = ds slre hsi utKrhf’ a rmeuto 4.I h pia depth isnot optical If the (4). equation from law Kirchoff’s just is This large. is eoe 1 becomes B e sdfnda h pia depth, the optical as defined is − λ . τ I hc nLEcnb written be can LTE in which λ hroyai qiiru LE n utlk h stemlcvt,tesuc function source the cavity, isothermal function, the Planck like the just equals and (LTE) equilibrium thermodynamic pia et fuiyi ece tasalwrpyia et,det h ogrgeometric longer the to an due limb, depth, the physical near shallower sun to a the at sees observe reached we one core. is when line Similarly, example, unity the of dark. for of are depth outside lines, cores optical sees line absorption one the than photospheric layers reason cooler this strong For fainter, of to thus cores depths, the physical shallower In depth observer optical the 1). intensity the The Figure when absorbed function. above, be any source out will the an pointed photon to sees simply As a reduces that intensity depth. even the that large than from becomes better exiting is before chance scattered the or means which unity, about of hneuto 1)sosta ic h nest nrae ihhih,a msinspectrum emission an height, with increases produced. intensity is the since that shows (11) equation Then ultraviolet at contrast, satisfy: In corona absorption and an is chromosphere “reversing”. spectrum of solar term temperature visible the the Thus, hence lines. at spectrum, spectral reduced in is example intensity for depth, the optical (11) equation to according Therefore, r rgtrta h ue layers, (see outer the atmosphere optical than outer At brighter the tenuous photosphere. are visible non-thermal structure the to Consider that due PHYSICS). atmo- likely fields SOLAR is solar magnetic rise the with temperature of This connected a temperature processes corona. the and as chromosphere however, the the treated detail, surface, into visual In was height the chemical below spectrum star. depth same with the the absorption was increases of with sphere the sunlight layer” rest “reversing the interior, of a as treatments dense called composition atmosphere earliest hot, gaseous the isothermal a in thin, from cool, which come spectrum, to continuous assumed the to contrast In ost eoadteitniyi ipytesuc function. source the simply is intensity the and zero to goes λ aeegh where wavelengths 0,seiistelgtsuc neigteslaband the entering source light the specifies (0), usrpsfrtemoment, the for subscripts κ λ ti motn opitottea any at the out point to important is It S , − λ (1 τ − o eysmall very for S λ e aeegh hr h pia et shget nohrwords other in highest, is depth optical the where wavelengths aeeghi setal h orefnto ta pia et fuiy(see unity of depth optical an at function source the essentially is wavelength I − ssalcmae ihtelgtsuc h nest sreduced is intensity the source light the with compared small is λ S τ I ( λ λ λ τ steitniyaddt h em h nest tzero at intensity The beam. the to added intensity the is ) ( λ I = τ = ) λ 11 = ) ( S τ  Basic SpectralLineFormation:IsothermalSlabs ml then small htmgtohrieb oiae ya xenlbihe ore hsrlto hw that shows relation This at where source. greatest brighter external is an intensity by the dominated be otherwise might that pia depth, optical fe asn hog h slab, the through passing after lentvl,at Alternatively, I is,dvdn by dividing First, τ htnbfr titrcswt atr h ai fteeiso n bopincoefficients absorption and emission function, the of source ratio the The called matter. is with interacts it before photon Here hspoie h nest safnto fdphadi rd u sflapproximation useful but crude term a The is atmosphere. and planetary depth a of into function passing a light as for intensity the provides This since twvlntsweethe where wavelengths At oeotcldpho neet o xml,o h te ieo h slab. the of side other the on example, for interest, of depth optical some rmteba and beam the from safrtapoiain ti o a oasm hta atclrdepth ataparticular that toassume bad spectrum. solar not the is of it formation the understand approximation, to first used a be can As results basic These results. topeecnb hrceie yasnl eprtr.Ti supini nw slocal as known is assumption This temperature. single a by characterized be can atmosphere h nest is intensity the then, λ λ κ λ a. Sourcefunction λ λ I ����� ��������� ����� ������� ������ ����������� � ���� ����� �������� ���� ������ �� � �������� �� ����� � ���� ���� �� ����� ��� ����������� ��������� �� �� ���� ������ ���� ����� � �������� ��������� ��� �� �� ������ ������ ��� ��� ���� ���� ������ ������ ����� �������� ����� �� �������������� �� ������ �������� ����� dI ) /κ dI dτ λ � I λ I I > I 1. τ Stellar Atmospheric Structure τ  = ��� λ I < (0) ds λ ν λ ( τ λ I (0)>1 I (0)1 λ I (0)>1 I (0)1 � τ − τ ν ν 0 ν 0 0 B ����������� ���� � ���� ���� ����� ���� ��������� �� ������ ��� ���� ���� ������ ����� ���������� � λ I − slre hsi utKrhf’ a rmeuto 4.I h pia depthisnot optical If the (4). equation from law Kirchoff’s just is This large. is = ) eoe 1 becomes I I I λ λ I λ λ I B + S � e eplayer deep ( ( ( ν ν λ hc esrstema rept fa of path free mean the measures which , 0 0 ν sdfnda h pia depth, the optical as defined is τ 0 ����������� ���� � ���� ���� ����� ���� ��������� �� ������ ��� ���� ���� ������ ����� ���������� � + τ T T − photospher λ λ λ + S = ) . τ τ ) ) S + ) S � λ eplayer deep layer deep λ I I � ost eoadteitniyi ipytesuc function. source the simply is intensity the and zero to goes ( . κ I Emission λ (1  λ λ τ � aeegh where wavelengths λ λ . I λ Height � ( 0,seiistelgtsuc neigteslaband the entering source light the specifies (0), I = ) S minimum temperatur (0) � λ τ κ − � 0 + (0) 0 ,adati lb so slab, thin a and 0, = (0) ν ν λ λ � ν + λ � ➪ ) ( S 0 0 0 , I e − τ � ������� ���������� ��������� ���� �� ����� ��������� �� ������ ��� ���� > not B ν ν λ I > λ τ e − ν – 6 – λ � aeegha bevrse oa pia depth optical an to sees observer an wavelength (0) (1 0 0 ( a B < B > τ ) 0 ( τ ������� ���������� ��������� ���� �� ����� ��������� �� ������ ��� ���� ,aogwt hnsa,again slab, thin a with along 0, λ B . τ τ hc nLEcnb written be can LTE in which λ hroyai qiiru LE n utlk h stemlcvt,tesuc function source the cavity, isothermal function, the Planck like the just equals and (LTE) equilibrium thermodynamic pia et fuiyi ece tasalwrpyia et,det h ogrgeometric longer the to an due limb, depth, the physical near shallower sun to a the at sees observe reached we one core. is when line Similarly, example, unity the of dark. for of are depth outside lines, cores optical sees line absorption one the than photospheric layers reason cooler this strong For fainter, of to thus cores depths, the physical shallower In depth observer optical the 1). intensity the The Figure when absorbed function. above, be any source out will the an pointed photon to sees simply As a reduces that intensity depth. even the that large than from becomes better exiting is before chance scattered the or means which unity, about of hneuto 1)sosta ic h nest nrae ihhih,a msinspectrum emission an height, with increases produced. intensity is the since that shows (11) equation Then ultraviolet at contrast, satisfy: In corona absorption and an is chromosphere “reversing”. spectrum of solar term temperature visible the the Thus, hence lines. at spectrum, spectral reduced in is example intensity for depth, the optical (11) equation to according Therefore, r rgtrta h ue layers, (see outer the atmosphere optical than outer At brighter the tenuous photosphere. are visible non-thermal structure the to Consider that due PHYSICS). atmo- likely fields SOLAR is solar magnetic rise the with temperature of This connected a temperature processes corona. the and as chromosphere however, the the treated detail, surface, into visual In was height the chemical below spectrum star. depth same with the the absorption was increases of with sphere the sunlight layer” rest “reversing the interior, of a as treatments dense called composition atmosphere earliest hot, gaseous the isothermal a in thin, from cool, which come spectrum, to continuous assumed the to contrast In ) ➪ afunctionofopticaldepth λ − .Euto 9 a ewitn rpigthe dropping written, be can (9) Equation ). ( ue layer outer o eysmall very for . e S 0 n neiso spectrum emission an and (0) S e usrpsfrtemoment, the for subscripts − λ λ λ e aeegh hr h pia et shget nohrwords other in highest, is depth optical the where wavelengths ν ν ( ν ( ( τ I − τ λ T T ssalcmae ihtelgtsuc h nest sreduced is intensity the source light the with compared small is chromospher ti motn opitottea any at the out point to important is It λ S τ ) S S I S I I ν ν ( ν ) ) λ steitniyremoved intensity the is ν λ λ ν − τ ν ν ν ue layer outer layer outer steitniyaddt h em h nest tzero at intensity The beam. the to added intensity the is ) (0) (0) ( λ I S I S S I = τ = ) λ ν I I ν − ν ν ν = ) ( (0) photosphere λ (0) (0)) S τ  ( ��� ����������������� ��������� ����� �� ������� B λ λ τ κ to corona λ aeeghi setal h orefnto ta pia et fuiy(see unity of depth optical an at function source the essentially is wavelength I dI ) /κ dI λ dτ eplayer deep λ I λ I I > sitniyat intensity is ) ��� ����������������� ��������� ����� �� ������� λ I < (0) ds λ . . λ e λ λ τ (0)(1 ( o ntecs fauiomtmeauecavity temperature uniform a of case the in So, . τ ic h eimi o,i mt t w light own its emits it hot, is medium the Since . = λ ml then small htmgtohrieb oiae ya xenlbihe ore hsrlto hw that shows relation This at where source. greatest brighter external is an intensity by the dominated be otherwise might that pia depth, optical fe asn hog h slab, the through passing after lentvl,at Alternatively, I is,dvdn by dividing First, τ htnbfr titrcswt atr h ai fteeiso n bopincoefficients absorption and emission function, the of source ratio the The called matter. is with interacts it before photon Here hspoie h nest safnto fdphadi rd u sflapproximation useful but crude term a The is atmosphere. and planetary depth a of into function passing a light as for intensity the provides This since twvlntsweethe where wavelengths At oeotcldpho neet o xml,o h te ieo h slab. the of side other the on example, for interest, of depth optical some rmteba and beam the from safrtapoiain ti o a oasm hta atclrdepth ataparticular that toassume bad spectrum. solar not the is of it formation the understand approximation, to first used a be can As results basic These results. topeecnb hrceie yasnl eprtr.Ti supini nw slocal as known is assumption This temperature. single a by characterized be can atmosphere h nest is intensity the then, λ = e 0 then (0) – 5 – λ λ λ aeegh fappreciable of wavelengths = ) � τ − aeegh,dee layers deeper wavelengths, − 0,ada bopinsetu sproduced. is spectrum absorption an and (0), � dτ −  = 1934ZA...... 8..124G τ � �������� ��� ����������� ����������� �� �� ���� �� ������� ��� �� � ��� ) λ h solar the I − � n loicesswith increases also and , λ λ I � ntescn aealgtsuc srqie,so required, is source light a case second the In h ouino h rnfreuto o niohra,hmgnossa fgsis gas of slab homogeneous isothermal, an for equation transfer the of solution The eunn otefrtcs ihn ih source, no light with case first the to Returning τ + S S aeegh,tehigh the wavelengths, � �������� ��� ����������� ����������� �� �� ���� �� ������� ��� �� � ��� λ hc esrstema rept fa of path free mean the measures which , κ .Teslto ftetase qainnwbecomes now equation transfer the of solution The 1. I e λ 2. τ + λ λ κ − λ ( λ + S  τ τ + ) λ τ = S ds λ � λ = ) τ I (6) (7) (8) (9) (5) τ . κ I � λ � (1  slre hsi utKrhf’ a rmeuto 4.I h pia depthisnot optical If the (4). equation from law Kirchoff’s just is This large. is eoe 1 becomes λ I I (0)=0 I (0)>S ν τ τ ν 1, B λ λ . I λ ν ν ν ( � e ν ( I (0)>S ν � (D) <1 S τ sdfnda h pia depth, the optical as defined is (0) λ (D) <1 B B τ 0 τ τ ν ν − � − � λ (D) <1 I (0)=0 I (0)>S ν τ (D) >1 0 ,adati lb so slab, thin a and 0, = (0) τ B λ λ ν = ) . τ I ν ν ν � λ λ ν I (0)>S ν � (D) <1 ) ( I Jason P. Aufdenberg, 10July 2003, Michelson Interferometry Summer School I (D) <1 I 0 I e eplayer deep ( � ν ν ( ( � τ I (D) <1 ost eoadteitniyi ipytesuc function. source the simply is intensity the and zero to goes > (D) >1 ν λ I > τ ν 0 T T − ν 0 λ λ 0 Photosphere � � aeegh where wavelengths (0) I = ) I τ ) 0,seiistelgtsuc neigteslaband the entering source light the specifies (0), I ) ) � S � ,aogwt hnsa,again slab, thin a with along 0, λ . ν κ ν 0 eplayer deep layer deep ν 0 0 b. Opticaldepthismuchlessthanone � ) ( λ � Absorption λ . τ � e 0 n neiso spectrum emission an and (0) S , I ➪ − − = ) � � ����� ���� �� ��������� ����������� λ � 0 + (0) τ � (1 ��� + λ τ (13) (12) (10) (11) � b ����� ���� �� ��������� ����������� � steitniyremoved intensity the is B τ − ��� – 6 – o eysmall very for ν � � ν ➪ aeegha bevrse oa pia depth optical an to sees observer an wavelength S ν ( B < B > hc nLEcnb written be can LTE in which λ hroyai qiiru LE n utlk h stemlcvt,tesuc function source the cavity, isothermal function, the Planck like the just equals and (LTE) equilibrium thermodynamic pia et fuiyi ece tasalwrpyia et,det h ogrgeometric longer the to an due limb, depth, the physical near shallower sun to a the at sees observe reached we one core. is when line Similarly, example, unity the of dark. for of are depth outside lines, cores optical sees line absorption one the than photospheric layers reason cooler this strong For fainter, of to thus cores depths, the physical shallower In depth observer optical the 1). intensity the The Figure when absorbed function. above, be any source out will the an pointed photon to sees simply As a reduces that intensity depth. even the that large than from becomes better exiting is before chance scattered the or means which unity, about of spectrum emission an height, with increases produced. intensity is the since that shows (11) equation Then ultraviolet at contrast, satisfy: In corona absorption and an is chromosphere “reversing”. spectrum of solar term temperature visible the the Thus, hence lines. at spectrum, spectral reduced in is example intensity for depth, the optical (11) equation to according Therefore, r rgtrta h ue layers, (see outer the atmosphere optical than outer At brighter the tenuous photosphere. are visible non-thermal structure the to Consider that due PHYSICS). atmo- likely fields SOLAR is solar magnetic rise the with temperature of This connected a temperature processes corona. the and as chromosphere however, the the treated detail, surface, into visual In was height the chemical below spectrum star. depth same with the the absorption was increases of with sphere the sunlight layer” rest “reversing the interior, of a as treatments dense called composition atmosphere earliest hot, gaseous the isothermal a in thin, from cool, which come spectrum, to continuous assumed the to contrast In 0 ( 0 B 0 λ e aeegh hr h pia et shget nohrwords other in highest, is depth optical the where wavelengths τ τ I ν � � ν I � ν − usrpsfrtemoment, the for subscripts λ .Euto 9 a ewitn rpigthe dropping written, be can (9) Equation ). ( ue layer outer ssalcmae ihtelgtsuc h nest sreduced is intensity the source light the with compared small is 0 � �������� ��� � λ 0 S 0 S ( τ I ( λ λ λ τ � λ ti motn opitottea any at the out point to important is It λ ( τ ( ( steitniyaddt h em h nest tzero at intensity The beam. the to added intensity the is ) λ � �������� ��� � τ ( λ I T T sitniyat intensity is ) = τ = ) ) λ ) ) Absorption ν Chromosphere = ) ( ν ν − ue layer outer layer outer S τ  λ λ S κ λ S I ν S I ν I I ν − dI ) τ /κ dI ν dτ λ ν ν photosphere ν I λ (0)) I I > ν hc nLEcnb written be can LTE in which λ hroyai qiiru LE n utlk h stemlcvt,tesuc function source the cavity, isothermal function, the Planck like the just equals and (LTE) equilibrium thermodynamic pia et fuiyi ece tasalwrpyia et,det h ogrgeometric longer the to an due limb, depth, the physical near shallower sun to a the at sees observe reached we one core. is when line Similarly, example, unity the of dark. for of are depth outside lines, cores optical sees line absorption one the than photospheric layers reason cooler this strong For fainter, of to thus cores depths, the physical shallower In depth observer optical the 1). intensity the The Figure when absorbed function. above, be any source out will the an pointed photon to sees simply As a reduces that intensity depth. even the that large than from becomes better exiting is before chance scattered the or means which unity, about of spectrum emission an height, with increases produced. intensity is the since that shows (11) equation Then ultraviolet at contrast, satisfy: In corona absorption and an is chromosphere “reversing”. spectrum of solar term temperature visible the the Thus, hence lines. at spectrum, spectral reduced in is example intensity for depth, the optical (11) equation to according Therefore, r rgtrta h ue layers, (see outer the atmosphere optical than outer At brighter the tenuous photosphere. are visible non-thermal structure the to Consider that due PHYSICS). atmo- likely fields SOLAR is solar magnetic rise the with temperature of This connected a temperature processes corona. the and as chromosphere however, the the treated detail, surface, into visual In was height the chemical below spectrum star. depth same with the the absorption was increases of with sphere the sunlight layer” rest “reversing the interior, of a as treatments dense called composition atmosphere earliest hot, gaseous the isothermal a in thin, from cool, which come spectrum, to continuous assumed the to contrast In (0) aeeghi setal h orefnto ta pia et fuiy(see unity of depth optical an at function source the essentially is wavelength (0) S λ S I I < (0) ds λ S I λ λ B h solar the λ τ ν (0)(1 ( ν ν ν usrpsfrtemoment, the for subscripts o ntecs fauiomtmeauecavity temperature uniform a of case the in So, . τ ν (0) τ ic h eimi o,i mt t w light own its emits it hot, is medium the Since . eplayer deep = (0) λ λ λ = e 0 then (0) – 5 – . . λ ti motn opitottea any at the out point to important is It = ) −  − 0,ada bopinsetu sproduced. is spectrum absorption an and (0), dτ − τ (6) (7) (8) (9) (5) λ I − λ λ I aeegh fappreciable of wavelengths 1, aeegh,dee layers deeper wavelengths, + S λ hc esrstema rept fa of path free mean the measures which , τ + λ ) λ + S n loicesswith increases also and , τ Spectrum -DiskCenter I + ) S λ λ aeegh,tehigh the wavelengths,

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λ λ ( c Emission B ➪ B ) ( τ I e B τ > = ) I λ I > λ λ − λ (0) eplayer deep ( ( ( τ ) τ T T ,aogwt hnsa,again slab, thin a with along 0, λ . ) = ) λ ) ) S . I e 0 n neiso spectrum emission an and (0) eplayer deep layer deep ( − ( c. LTE () τ τ τ I = ) λ = ) 0 + (0) I + steitniyremoved intensity the is ( B B τ B (13) (12) (10) (11) τ B Spectrum -Limb – 6 – = ) I I λ λ aeegha bevrse oa pia depth optical an to sees observer an wavelength ( B < B > ( λ eplayer deep ( ( ( B τ τ τ T T ( τ .Euto 9 a ewitn rpigthe dropping written, be can (9) Equation ). ( ue layer outer = ) ) ) S λ S eplayer deep layer deep λ λ sitniyat intensity is ) ( ( ( ( τ τ T T I = ) ) ) ) 0 + (0) − + ue layer outer layer outer τ B I τ − – 6 – h solar the photosphere aeegha bevrse oa pia depth optical an to sees observer an wavelength (0)) ( B < B > ( τ B B τ τ λ .Euto 9 a ewitn rpigthe dropping written, be can (9) Equation ). ( ue layer outer eplayer deep  S λ λ . . ( ( ( (6) (7) (8) (9) (5) τ T T 1, ) ) ) aeegh fappreciable of wavelengths − ue layer outer layer outer aeegh,dee layers deeper wavelengths, ) I − n loicesswith increases also and , photosphere (0))

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aeegh,dee layers deeper wavelengths, ) n loicesswith increases also and , aeegh,tehigh the wavelengths, (13) (12) (10) (11) (13) (12) (10) (11) 8.2 Surface Intensities We want to find the intensities that we can actually measure, thatis • the intensity emerging at the surface, τ = 0, into all angles between –π/2 and π/2. 8.2 Surface Intensities Solve the RTE with an integrating factor: • We want to find the intensities that we can actually measure, thatis • the intensity emerging at the surface, τ = 0, into all angles between dI(θ) –π/2 and π/2. = I(θ) S (24) sec θdτ − Solve the RTE with an integratingτ sec factor:θ • factor = e−

dI(θ) τ sec θ τ sec θ τ sec θ dI(θ) e− = I(θ) e− S e− (25) = secI(θ)θdτ S· · (24)− · sec θdτ 8.2 Surface− Intensities Let x = τ sec θ: τ sec θ factor = e− • We want to find the intensities that we can actually measure, thatis • the intensity emerging at the surface, τ = 0, into all angles between d x x Ie− = Se− (26) dI(θ) τ sec θ –τπ/sec2θ and π/2.τ sec θ e− = I(θ) e− Sdxe− − (25) sec θdτ · · − ·   8.2 Surface Intensities Now we integrateSolve from theτ = RTE 0 with anwhich integrating is x = 0 factor: : Let x = τ sec θ: • • −→ ∞ −→ ∞ We want• to find the intensities that we can actually measure, thatis • ∞ d ∞ the intensity emerging at the surface, τ = 0, into all angles betweenx x AST 598: Atmospheres – Session #2 d x x Ie− dx = Se− dx (27) More Radiative Transfer: NowIe− with= Se Angles!− dx dI(26)−(θ) –π/2 and π/2. dx − 0   0 = I(θ) S (24) AST 598: Atmospheres1. Setup Equation – Session #2 2. Solve for the Intensity at the Surface!sec θdτ − Solve the RTENow with we integrate an integrating from τ factor:= 0 whichx ∞ is x =τ 0sec θ : 0 ∞ x 8 More Radiative Transfer: Now with angles! (use integratingIefactor− factor = Iee − −∞ )I(0)e = Se− dx (28) • • −→ ∞ −→− ∞ − 8 More Radiative Transfer: Now with angles!  0 0 8.1 Plane Parallel Approximation ∞ d ∞ x x dI(θ) τ sec θ τ sec θ τ sec θ Ie− dx = Se− dx ∞ e (27)= I(θ) e S e (25) 8.1 Plane Parallel Approximationθ dI(θ) dx I−(0, θ) = Se τ sec− θ d(τ sec θ) − (29)− Valid for the solar photosphere. Not valid for the extended atmo- 0 = I(θ) S 0 sec(24)θdτ−· · − · • sec θdτ − 0 spheresValid offor giants, the solar supergiants, photosphere. stellar Not winds, valid novae for the or supernovae. extended atmo- Let x = τ secθ: • τ sec θ 8.3 Linearx Source∞ Function• 0 and Limb∞ Darkeningx spheres of giants, supergiants, stellar winds,factor novae = e− or supernovae.Ie− =ThisIe−∞ in knownI(0)e as= the formalSe− solutiondx to the(28) equation of transfer Write down the RTE along an arbitraryPlane Parallel path s along an angle θThis 0is the formal− solution to0 the− equation of transfer. • Let’s let the source function be a linear function of thed opticalx depth: x measuredWrite down from the the RTEz axis. along (We an dispense arbitraryGeometry with path the λssubscriptsalong an• here): angle θ Ie− = Se− (26) • dI(θ) τ sec θ Whereτ secI(0θ , θ) is theτ sec surfaceθ intensity (τ =dx 0) leaving− the atmosphere measured from the z axis. (We dispense with the λ subscriptse− here):= I(θ) e− ∞ Sτ sece−θ (25)   sec θdτ · I(0,at θ)· =angle is theθ surfaceSe.− − · (τd=0)(τ sec viewedθ) from angle(29)θ. dI(θRadiative) Transfer Equation S = a + bτ0 Nowreally we integrateSλ from= aλ τ+=bλ 0τλ (30)which is x = 0 : = κI(θ) +  = κILet(θ)x +=κSτ sec θ: 8.3(18) Linear Source Function• and Limb Darkening −→ ∞ −→ ∞ dsdI(θ) − − Intensity varies with angle because = κI(θ) +  =• κI(θ) + κSThis in knownLet’s(18) stick as the thisformal form solution of S intoto the the formal equation solution: of transfer That optical depth alongds the− path s is defined:− • Let’s let8.3 thethe Linear source source function Source function be Function avaries linear with function and depth.∞ Limbd of the Darkening opticalx depth:∞ x • • d x x Ie− dx = Se− dx (27) Ie− = Se− (26) dx − That optical depth along the path s is defined: Where I(0, θ)dx is the surfaceLet’s− intensity let the source (τ = 0) function leaving be the a0 atmospherelinear function of the0 optical depth: • dτs = κds (19) • ∞ ∞  S = a +3.τbτ secTrialθ Sourcereally FunctionSτ secλ =θ aλ + bλτλ (30) at angle θ. I(0, θ) = a e− d(τ sec θ) + b τe− ∞ d(τ sec θ) (31) dτs = κds (19) x 0 ∞ x Now we integrate from τ = 0 0 which is x = 0 : 0 Ie− = Ie−∞ I(0)e = Se− dx (28) We make a sign change here. Path s and• optical depth τ convention-Let’s stick−→ ∞ thisplug form in: S of=S ainto+−→bτ the ∞ formal really solution: S−λ = aλ + bλτλ − (30) • z •  0 0 8.2 SurfaceallyWe increase Intensities make a in sign opposite change directions, here. Path so:s and optical depth τ convention-∞Thisd reduces8.4 to... The∞ Source Function as a Power Series • Ie x dx =Let’s stickSe thisx dx form of S into(27) the formal solution: ally increase in opposite directions,θ so: − • − ∞ τ sec θ We want to find the intensities that we can actually measure,0 thatis dx In0 general− S can be written asI a(0 power, θ) = seriesSe in− τ. d(τ sec θ) (29) dz    ∞ τ sec θ∞ ∞ τ sec θ ∞ • dτ = κds (20) I(0, θ) =• a e−τ sec θ d(τ sec θ) +sec θτ b τe1− τ sec θd(τ0 sec θ) (31) s I(0, θ) = a e− + b − − e−  (32) the intensity emerging at the surface, τ −= 0, into all angles between 0 0 x ∞ −0 · ∞ 0∞ x ·τ sec θ sec θ ∞ 0 τ sec θ –π/2 andTheπ/ optical2. depth along the z dτaxiss = is then:κds Ie− = (20)Ie−∞ I(0)e I=(0, θ) = Se This−adx e− in knownd(τ (28)sec asθ the) +formalb τe solutioni − dto(τ thesec θ equation) (31) of transfer − Sλ(τλ) = aλ iτλ (34) 0 − 0 − 0 0 The optical depth along the z axis is then:8.4 The Source FunctionThis reduces as a to...Power Series i Solve the RTE with an integrating factor: I(0, θ) = a + b cos θ  (33) • Where I(0, θ) is the surface intensity (τ = 0) leaving the atmosphere dτ = dτz = dτs sec θ In general S can(21) be written∞ asThisτ secThe a powerθ reduces formal series to... solution in τ. is then evaluated to be I(0, θ) = Se d(τ sec θ) ∞ (29) ∞ • So normal to surface−• of theτ sec atmosphere:atθ angle θ. sec θτ 1 τ sec θ dI(θ) dτ = dτz = dτs sec θ (21) 0I(0, θ) = a e− + b − − e− (32) The term can be written in terms of τelmz: •  − · · sec θ • ds dI(θ) I, S and τ are  i 0 τ sec θ ∞ sec θτi0 1 τ sec θ ∞ dI(θ) = I(θ) S This in known as(24) the formal solutionSλ(τλ)to =II the(0(0,,0) θa equation)λ i= =τλa +a b ofecos(0)− transferIλ(0 =,a θ+)+ =bb (34)−Aλ i cos−θ e− (32)(35) The term ds can be written in termsall of functionsτelmz: of − · · sec θ • sec θdτ − i  0 i 0 dI(θ) dI(θ) wavelengthdI(θ)  I(0, θ) = a + b cos θ  (33) τ sec θ = = κ (22) factor = e− dsdI(θ) dτdI/κ(θ) − secdIθdτ(θThe) formal solutionand parallel is thento evaluated surfaceA power of to theseries be atmosphere: in cos θ, where the coefficients A are given by: s Where• I(0, θ) is the• surface intensity (τ = 0) leaving the atmosphereI(0, θ) = a + b cos θ i (33) 12 =− = κ JasonSo(22) normal P. Aufdenberg,to surface 10 July of the 2003, atmosphere: Michelson Interferometry Summer School Stellards Atmosphericdτs/κ −atsec angle θdτStructureθ. • Now the RTE is: − I(0, π/2) = a + b cos(π/2) = a • dI(θ) τ sec θ τ sec θ τ sec θ So normal to surfacei of the atmosphere: e = I(θ) e S e (25) Iλ(0, θ) = AI(0λ i,cos0) =θ a + b cos(0)A =λa i =+(35)abλ i i! (36) Now the RTE is:− − − • · • sec θdτ · · − · i dI(θ) This means for positive values of a and b the intensity is a maximum = I(θ) S • (23) Typo in BV PageI(0 233:, 0)A =λa i += baλcos(0) i! = a + b Let x = τ sec θ: forand cos θ parallel= 1, normalto surface to the of atmosphere. the atmosphere: That’s a good thing because • secdIθdτ(θ) − A power series in• cos θ, where the coefficients Ai are given by: = I(θ) S it means(23) the intensity is greatest from the center of the solar disk and sec θdτ − and parallelI(0, π/to2) surface = a + ofb thecos( atmosphere:π/2) = a d x x smallest when• observed at angles near cos θ = 0, near the limb. Ie− = Se− (26) A = a i! (36) dx − λ i λ i · I(0, π/2) = a + b cos(π/2) = a   TheThis Sun means and a few for positive other stars values show of limba and darkening.b the intensity HST and is a ground maximum Typo in BV Page• 233: A = a ! Now we integrate from τ = 0 which is x = 0 : • for cos θ λ= i 1, normalλ i to the atmosphere. That’s a good thing because • −→ ∞ −→ ∞ based interferometric This means observations for positive have values resolved of a aand handb the full intensity of bright is a maximum • giantsit means and supergiants. thefor intensity cos θ = is1, greatest normal to from the the atmosphere. center of the That’s solar a disk good and thing because ∞ d x ∞ x smallest when observed at angles near cos θ = 0, near the limb. Ie− dx = Se− dx (27) it means the intensity is greatest from the center of the solar disk and 0 dx 0 −     The Sun andsmallest a few other when stars observed show at limb angles darkening. near cos HSTθ = and 0, near ground the limb. The intensity• varies with angle because the source function changes with depth based interferometric observations have resolved a hand full of bright x ∞ 0 ∞ x The Sun and a few other stars show limb darkening. HST and ground Ie− = Ie−∞ I(0)e = Se− dx (28) − − giants and• supergiants.  0 0 based interferometric observations have resolved a hand full of bright giants and supergiants. ∞ τ sec θ I(0, θ) = Se− d(τ sec θ) (29)The intensity varies with angle because the source function changes with depth 0 The intensity varies with angle because the source function changes with depth This in known as the formal solution to the equation of transfer

Where I(0, θ) is the surface intensity (τ = 0) leaving the atmosphere at angle θ. ���� ���� ����������� ��

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Determining the Temperature Structure of the Sun: θ Limb-Darkening

S DiskASTAST Center 598: 598: Atmospheres Atmospheresν a – Session –I Sessionν a #3 #3 GreyGrey Temperature Temperature Distribution Distribution under under Radiative Radiative ���� ����AST ����������� 598: AtmospheresEquilibriumb – Session #3 b �� GreyNear Temperature Limb Equilibrium Distribution under Radiative AST 598: Atmospheres0Equilibrium – Session0 #3 h 0 sin θ 1 1010 EmpiricalGrey Empirical Temperature Depth Depth Dependence DependenceDistribution of the of under the Source Source Radiative Func- Func- ������ ���� ����� ���� ���������� ��� ������� ����� � ��������� ���� ��� ���������� ������ ��� � ��� � tion Equilibrium � Rayleigh-Jeans 10��tion ��� Empirical �������� ����� ����� Depth ��� �������� Dependence ���������b ������� of ��������� the Source ������ ������� Func- ��� ����� ���� ������� ������ ��������� ��� ���� ���� �� ��� � � ����� ��� ������� ����� �� ����� ��� ��� ������� ��������� Approximation � �����tionRecall ���� ���� for ����������� an observed �������� intensity �� � � ��� distributiona� �� ������� with���� ����� angle �� ��� (limb ����� dark- ��� ��� � ��� � � 4 Recall� �� for� an observed intensity distribution with angle (limb dark- � Bλ = 2ckT λ− 10��• � Empirical� ����� �� ����� Depth ������ ��� �� Dependence ���� ����� ��� ������� of ���������the Source �� ���� ������ Func- �� ����� �� ������ ��� • �ening) expressed as a power series in cos θ: ening)��Recall ���� expressed ���� for an observed as a power intensity series in distribution cos θ: with angle (limb dark- d Bλ 2 4 tion θ Relative Intensity = kλ− • d T c ening) expressed as a power series in cos θ: i Recall for anIλ(0 observed, θ) =S intensityAλ iicos distributionθ A withλ i = a angleλ i i! (limb dark- (47) Iλ(0, θ) = Aν λa i cos θ AλI ν i = a λ i i!· (47) • � ����������� ���������Solve �������������i Inverse Problem: · ening) expressed asi a power seriesi in cos θ: � ������� ���������Iλ(0, θ) = �������������Aλ i cos θ Aλ i = aλ i i! (47) We can solve1. the Measure inverse problemintensitiesb and find the source· functionb S (τ ): We can solve the inverse problemi and find the source function Sλ(τλλ): λ � ����������� ����������� i Wavelength Iλ(0, θ) = 0 Aλ i cos θ0 Aλ i = aλ i i! (47) We can solve the inverse problem and find the source· function Sλ(τλ): � ����������� �������������i h i 0 sin θ 1 Sλ(τλ) = aiλ iτ (48) Sλ(τλ) = aλ iτλ λ (48) ������ ���� ����� ���� ���������� ��� ������� �����i � ��������� ���� ��� ���������� ������ ��� � ��� � We can solve the inverse problemi and findi the source function Sλ(τλ): � �� ��� �������� ����� ����� ��� ��������Sλ(τλ) ��������� = a �������λ iτλ ��������� ������ �������(48) ��� ����� ���� ������� �����������So ��������� an �������� intensity ��� ���� ���������� distribution ���� �� ��� such� � ����� as ��� ������� ����� �� ����� ��� ��� ������� ��������� So an intensity distribution such� as i • �����• ���� ���� ����������� �������� �� � � ��� � �� �������i ���� ����� �� ��� ����� ��� ��� � ��� � � ����������� �������������2. DetermineS fi���( tτ coeffi) �������� = cients,a ���������τ A0, A1, ��� A2 ����������� ������������(48) � ��� ��So� an����� intensity����� �� ����� distribution ������ ��� ��λ such ����λ ����� as ���λ ������� i λ ��������� �� ���� ������ �� ����� �� ������ ��� � •�� ���� ���� i 2 Iλ(0, θ) = Ao(λ) + A1(λ) cosθ + A2(λ)2 cos θ Iλ(0, 0) (49) Iλ(0�,�� θ) =����������Ao(λ) + �����������A1(λ) cos θ ���+ A �����������2(λ) cos θ Iλ ������������(0, 0) (49) ���� ������ So an intensity� distribution such as ����� � �� ����� � ��� ��� ��� �������� ��2 ����� �� • Iλ(0, θ) = Ao(λ) + A1(λ) cos θ + A2(λ) cos θ Iλ(0, 0) (49) impliesimplies a source a source function function like: like: Central disk ��� �������������� ������� ��3. ���������� ��������� Find source ��� ������������� �� function ��� ����� ����������� ���� ��� �� ���� ��� ���� �������� 2 intensity ��implies ���������� �������I aλ(0 �� source,����� ��������� θ) =� function��Ao � �������������(λ� )� + like:���A1(λ��) cos ���θ �������������+ A2(λ) cos θ ������Iλ(0, ��0) ������ (49) �� ������������� A �� A A Relative Intensity Ao o��� A1 1 A���2 2 2 in absolute Cos (θ) ���������������������S�� �����������λ( ��τλ) � =� � � (λ�)�� +�� ����(λ)τ�λ + �(2λ��)τ� ���Iλ(0 ���������, 0) ������� (50) ��������� Sλ(τλ) = (λ0!) + (λ1!)τλ + (λ�2!)τλ Iλλ(0, 0)units (50) ����implies ��������� a source ���������� function0! ������� like:1! ������ ��2! �������� ����������� ������������ ��� ��� � ����������� ������������� A o A1 A2 2  � ������ ��� ����Sλ( ������τλ) = �� � �(λ�)� +����( ��λ) �����τλ + ��( ���λ)τ ���������������λ Iλ(0, 0) ������� (50) ������� In theIn the non-grey non-grey case case we’ll0! we’ll have have1! a different a different2! run run of I (0 of I,λ θ(0), for θ) foreach each • •  A A A  λ 13 Stellar Atmospheric Structure wavelengthJasonwavelength P. Aufdenberg, and and therefore therefore 10 July ao different 2003, a different Michelson1 run ofrunS Interferometry���( ofτS2λ)( atτ�λ) each2 at each wavelength.Summer wavelength. School In the non-greySλ(τλ case) = we’ll(λ have) + a(λ different)τλ +�λ λ�( �λ�run)τλ of IIλ(0(0,,0) θ) for each (50) ����� �������� ����������0!��� ���1! � 2! λ � ������ • � �� � �� � ���������� wavelength and therefore a different run� of Sλ(τλ) at each wavelength. ����������� ������������� ��� ��������� ��������� ��� ����������� ������������ ��� In the non-grey case we’ll have a different run of Iλ(0, θ) for each ���• ��� ��������� �� � ��������� ���� ��� ������� ���������� ��� ���������� ����������� ������ ���� ��������� ��������� ����� ������ �� ��� �������� ����� ���� ���������� ��� �����wavelength ����� ����� and therefore a different run of Sλ(τλ) at each wavelength. ��� ��� �������� ��������������� ��� ����������� ����������� ��� ��� ���� ����������� �� ��������� ������������ �� ����� �� ���������� ��������� ����� �������� ����� �������� ����� �������� �� ��� �������� ������ �� � � ��� �� ����� ������ ���� �� ����� � � ��� ��� �� ������������� � ����� ������������ ��� ��� ��� ����������� ��� �������� ������������ �� ����� �� ���� ��� �� �� ����� �� ��� ���������� �� � �� ��� �� �� ������ ��� ��������� ���� ��� ���������� �� ���� ���� ��������������� ���������� ��������� ������ ����� �� ��� �������� ����� ����� ��� �������� ��������������� ������� ����� ��������� � � ������ ��� �� ��������� �������� �������� ��� ��������� ����� ����� ����� ������������ ��� ������ ������� �������� �� ���������� ��� �� ��� ����� ����������� ���� ��� �� ���� ��� ���� �������� ���� ��� ����� ����� ���� ��� ������� �� ������ ����� ������������ ����� ��� �������� �������������� ��������������� ��������� ����� �������� �� ��������� �� ���� ������ �� ������� ��� �� �� ��������� �� ����� � �� �� � ����� �� ��� ������������� ������ �� ������ �� ������������� ����� �� ��� ��� ���� ������ �������� ���� �� ��� ���� � ��� �� ���������� ��������������� �� ��� � ��������� �������� ���� ����� ���� ���������� ���������� ��� ������ ��� ������� ���� ��� ��� ������ ���������� ���� �� ��� �������� ���� ��� � ���� �� � ���� �� ��� ��������������� � �� ���� ����� ������ �� �� � � ����� ���� � ���� ��� ��������� ������� ��������� � � ���� ��������� ���� ��������� �������� ��� � ������� ���������� ��� �������� ���� ���� ��� ����������� �� ��� �������� ������� ��� �������� ����� ���� ���������� �� �������� ��������� ��������� ��� ���� �� ���������� ������� ������ �� �������� ����������� ������������ ��� ��� ��������� �� ����� ������������ �� ������� �������������� ���� ������ ��� ������� ������� � ������ ��� ���� ������ �� � � �� ���� �� ����� �� ��� ��������������� ������� ������� 1 �������� ��� � � � � ������ � � �� � �� � ���������� � � ��� ��� ��������� �� � ��������� ���� ��� ������� ���������� ��� ���������� ����������� ��� ��� ����� ����� ��� ����������� ��� ���� �� ��������� �� ����� �� � ��� � � ����� �� � � ��� � � �� �������� ��� ������ ��� ����������� ������������ ���� ��� ��� �� ����� �� � � ��� ������� ������ ��� ��������������� ����� �������� �� ��������� �� ���� ������ �� ������� ��� �� ���� �� � �������� ���� ����� ���� ���������� ���������� ��� ������ ��� ������� ���� ���� ��������� ���� ��������� �������� ��� � ������ ���������� ��� �������� ���� ���� AST 598: Atmospheres – Session #3 Grey Temperature Distribution under Radiative Equilibrium

AST 598: Atmospheres – Session #310 Empirical Depth Dependence of the Source Func- Grey Temperature Distribution under Radiativetion Equilibrium Recall for an observed intensity distribution with angle (limb dark- • 10 Empirical Depth Dependence of the Sourceening) Func- expressed as a power series in cos θ: tion I (0, θ) = A cosi θ A = a i! (47) λ λ i λ i λ i · Recall for an observed intensity distribution with angle (limb dark- i •  ening) expressed as a power series in cos θ: We can solve the inverse problem and find the source function Sλ(τλ):

i Iλ(0, θ) = Aλ i cos θ Aλ i = aλ i i! (47) i · Sλ(τλ) = aλ iτλ (48) i i   WeNow, can solve the inverse problem and find the source functionSoSλ an(τλ intensity): distribution such as • i Sλ(τλ) = aλ iτλ (48) 2 DeterminingdIλ the Temperature StructureIλ(0, θ) =of Atheo(λ) +Sun:A1(λ) cosLimb-Darkeningθ + A2(λ) cos θ Iλ(0, 0) (49) > 0 i S > I (9) dτ  λ λ   So an intensity distributiondI such as Continued... • λ < 0 S < I implies(10) a source function like: dτ λ λ I (0, θ) = A (λ) + A (λ) cos θ + A (λ) cos2 θ I (0, 0) (49) Ao A1 A2 2 4.1 Theλ Source Functiono 1 and a Perfect2 Blackbodyλ Sλ(τλ) = (λ) + (λ)τλ + (λ)τ Iλ(0, 0) (50) 0! 1! 2! λ     Transfer in a perfect blackbody: implies a source function like: In the non-grey case we’ll have a different run of I (0, θ) for each • λ dIλ wavelength and therefore a different run of Sλ(τλ) at each wavelength. A A= κλ Iλ +Aκλ Sλ (11) S (τ ) = o (λ)ds + 1 (−λ)τ + 2 (λ)τ 2 I (0, 0) (50) Leadsλ toλ the solar0! temperature1! λ 2! structureλ λ assuming:   with a perfect BB, Iλ is isotropic and homogeneous In the non-grey case we’ll have a different run of I (0, θ) for each • 1. Plane-parallel geometry is valid. λ wavelengthThe sunʼs and photosphere therefore a differentis roughlydI 1000run of Skmλ( τthickλ) at versus each wavelength.a solar 5 λ 0 radius of 7 x 10 km, or an extenstionds −→ of 0.1%. therefore, 2. Local thermodynamic equilibrium is a good approximation.

Sλ Iλ Bλ The Planck function connects ≡S(τ) to≡ T(τ), the temperature structure. 9.4 Effective TemperaturePlanck function: for a Grey Atmosphere 2hc2 1 If we assume LTE and take S B , theB surface= flux at a given (12) • λ ≡ λ λ λ5 ehc/λkT 1 wavelength will depend only on the temperature at an optical− depth τλ = 2/3. Further moreNumerically, we must have a grey opacity such that τλ is independent of wavelength. This way we see to the same depth at all 5 frequencies and this depth has a single temperature. So thec1λ radiation− π Bλ = (13) field is completely discribed by the temperature.· ec2/λT 1 −

5 2 1 Log Gravity c = 3.741 10− [erg cm s− ] 1 ×

c = 1.438 [cm K] Temperature 2 · κλ κwithand λτλin [cm]τ for & T grey in [K].atmosphere −→ −→ Depth Log Effective Temperature Note:14SλStellar= Bλ = Atmosphericλ/κλ which leads toStructureKirchoff’s LawJason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School π F (0) = π B (τ = 2/3) (43) · λ · λ λ λ = κλ Bλ (14) The bolometric flux is, ·

∞ ∞ 4 π Fλ(0) = π Bλ(τλ = 2/3) = σT (τ = 2/3) (44) 0 0 where σ is the Stefan-Boltzman constant. The effective temperature T of a star is defined as • eff

4 πF (0) = σTeff (45) Equating these last two equations gives

σT 4(τ = 2/3) = σT 4 T = T (τ = 2/3) (46) eff −→ eff No such thing as an effective temperature in a spherical atmosphere. • Models for Limb-Darkening: Plane-Parallel vs. Spherical Geometry

a) plane-parallel case

The semi-infi nite nature of plane-parallel models means that the atmosphere is optically ����cos-1 � thick at all angles.

z

Drop off characteristic of spherical models.

b) spherical case

The rays of a spherical model impact nested shells, of which the outer most are optically thin.

����cos-1 � z

r0 r1

r2 rN r3

p 15 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Spherical Models and the 3rd Parameter: Mass (or Radius) Gravity, Mass, Radius: Choose Two! The common use of plane-parallel model atmospheres GM g(r) = (e.g. Kuruczʼs ATLAS programs) gets us accustomed r2 to thinking that for a given chemical composition, a model hydrostatic atmosphere is primarily In spherical models the gravity is characterized by 1) the effective temperature a function of depth. The gravity and 2) the surface gravity. Spherical models require a parameter must refer to a reference 3rd parameter, mass or radius, to established the radius consistent with the stellar luminosity of the star (remember, fl ux is mass. not conserved in the spherical case).

Atmospheric Extension is a function of Mass The ratios of angular diameter measurements at several wavelengths, some of which probe strong molecular bands, will depend of the extension of the atmosphere and therefore the mass.

Mass is an additional free parameter for fi tting visibility functions, however for same Teff near by stars the mass may be well and gravity constrained and provide a strong test for the models.

16 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School

1 Temperature and Opacity of the Solar Atmosphere: Limb-Darkening ���� ������� �������� ��� Continued... 1. Limb darkening measurements 2. Source function vs. optical depth

3. Assuming LTE, temperature 4. For a one-to-one temperature-depth

vs. optical depth relationship, the optical���� vs. ������� wavelength �������� ��� is determined

� ������ ����� ���������� ���������� ���������� �� ���� �������� ��� ������� ��� ������� �������� ���� � 17 Jason P. Aufdenberg, 10��� July ��� ����2003, �������� Michelson ��������� ��� ��� Interferometry ����� �� � � ���������� Summer ������� ����� School �� � � ���� �� �� ��� Stellar Atmospheric Structure ������������ �� ����� ����� ������ ��� ���������� � ��� ���� �������� ��� ������� �������� ���� �� �������� ����� �� ����������� �������� ��������� ���������� ������� ��� ���������� �� ��� �������� ����� �� � � ���� ������� ��� �� �� ������� ����� ������ ���� ��� ��� �������������� ��� �� ����� �� ��� �� � ��� �� �� �������� �� ��� ������ ������� ����� ��� �� ��� ��� ���� ����� ��� ��� � ���� � ������ ����� ��� ��� � ���� � ������ ��� ������ �� ��� ������ ������ ��� ������ ���� �� ���� ����� ��� ��������� ����� ����� ������ ���������� ���� ���� �������

� ������ ����� ���������� ���������� ���������� �� ���� �������� ��� ������� ��� ������� �������� ���� ��� ��� ���� �������� ��������� ��� ��� ����� �� � � ���������� ������� ����� �� � � ���� ��� �� ��� ������������ �� ����� ����� ������ ��� ���������� � ��� ���� �������� ��� ������� �������� ���� �� �������� ����� �� ����������� �������� ��������� ���������� ������� ��� ���������� �� ��� �������� ����� �� � � ���� ������� ��� �� �� ������� ����� ������ ���� ��� ��� �������������� ��� �� ����� �� ��� �� � ��� �� �� �������� �� ��� ������ ������� ����� ��� �� ��� ��� ���� ����� ��� ��� � ���� � ������ ����� ��� ��� � ���� � ������ ��� ������ �� ��� ������ ������ ��� ������ ���� �� ���� ����� ��� ��������� ����� ����� ������ ���������� ���� ���� ������� 1976ApJS...30....1V

Determining the Temperature Structure of the Sun: Disk-Center Absolute Intensities 1976ApJS...30....1V 1. Compute the brightness temperature from the measured central intensities, I. 1976ApJS...30....1V 5

1976ApJS...30....1V 1976ApJS...30....1V

Assume I=B and solve for T at each wavelength:

2. Construct a model atmosphere which reproduces the brightness temperature as a function of wavelength.

Models are fi ne tuned to match the inten- sity as a function of wavelength.

The best fi t model yields a temperature-depth relationship.

18 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School 1 dF 1 dF = J S (68) = J S (68) 4 dτ − 4 dτ − in radiative equilibrium1 dF the gradient of the flux is zero so, in radiative equilibrium the gradient of the flux is zero so, = J S (68) 4 dτ − J = S (69) J = S (69) in radiative equilibrium the gradient of the flux is zero so, Otherwise the condition of R.E. holds overall wavelengths Otherwise the condition of R.E. holds overall wavelengths• • J = S (69)

∞ 1 dF ∞ ∞ 1 dF ∞ λ Otherwise the condition of R.E. holdsλ overall wavelengths dλ = (Jλ Sλ) dλ (70) • dλ = (Jλ Sλ) dλ (70) 4 dτ − 0 4 dτλ 0 − 0 λ 0   ∞ 1 dF ∞ can be rewritten as can be rewrittenλ asdλ = (J S ) dλ (70) 4 dτ λ − λ 0 λ 0 ∞ 1 dF ∞ ∞ 1 dF ∞ λ can be rewritten as λ dλ = 0 = (κλJλ κλSλ) dλ (71) dλ = 0 = (κλJλ κλSλ) dλ (71) 4 dz − 0 4 dz 0 − 0 0   ∞ 1 dF ∞ or finally, or finally, λ dλ = 0 = (κ J κ S ) dλ Radiative(71) Equilibrium 4 dz λ λ − λ λ 0 0 Radiative Equilibrium applies in the ∞ ∞ ∞ ∞ or finally, κλJλ dλ = κλSλ dλ (72) tenuous outer layersκλJ ofλ dλstars= whereκ radiationλSλ dλ (72) 0 0 is the dominant0 energy transport0 mechanism.   ∞ ∞ Absorbed Radiation = Emitted Radiation AbsorbedκλJλ dλ = RadiationκλSλ =dλ Emitted Radiation (72) R.E. is a special case of thermal equilibrium 0 0 We knew this already. It must be true in thermal equilibrium. We knewwhich this says already. that the It temperature must be true structure in thermal is equilibrium. Absorbednot changing Radiation with time = Emitted so the radiative Radiation fl ux Compare Blackbody vs. Pure Hydrogen vs. ATLAS model.Compare For a Blackbody vs. Pure Hydrogen vs. ATLAS model. For a • must be constant. • We knewgiven this effective already. It temperature must be true inthe thermal area under equilibrium. each flux givendistribution effective must temperature the area under each flux distribution must be equal. The bolometric flux must be the same, however the Fλ does Comparebe Blackbody equal.This The means vs. bolometric Pure the total Hydrogen flux energy must absorption vs. be ATLAS the must same, model. however For a the Fλ does • vary. given effectivevary. equaltemperature the total energy the areaemitted under in each each layer; flux distribution must be equal. The bolometricthere are no flux sources must or besinks the of radiativesame,however the Fλ does vary. energy. (F ) = (F ) (73) (Fλ)blackbody = (Fλ)pure hydrogen (73) λ blackbody  λ pure hydrogen Lyman Balmer Paschen but, but, (F ) = (F ) (73) λ blackbody  λ pure hydrogen but, σ σ 4 (F ) dλ = (F ) dλ = T 4eff (74) (Fλ)blackbody dλ = (Fλ)pure hydrogen dλ = T eff (74) λ blackbody λ pure hydrogen π π     This is the convergence criterion for iterative model atmospheres. This is the convergence criterion for iterativeσ model4 atmospheres. (Fλ)blackbody dλ = (Fλ)pure hydrogen dλ = T eff •(74) Model Spectra: 10,000 K • π   This is the convergence criterion for iterative model atmospheres. • Extended Atmospheres Black Body/Planck Function For non-plane parallel atmospheres the fl ux is not constant, but falls off like 1/r2. In this case, only the luminosity is constant with depth.

19 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School 9

Quirrenbach, A., Mozurkewich, D., Buscher, D. F., Hummel, C. A., Smith, M. A. 1985, ApJ, 297, 206 & Armstrong, J. T. 1996, A&A, 312, 160 van Belle, G. T., Ciardi, D. R., Thompson, R. R., Akeson, R. L., & Reynolds, R. J. 1985, AJ, 90, 92 Lada, E. A. 2001, ApJ, 559, 1155 —. 1997, Science, 277, 1446 van Belle, G. T., Lane, B. F., Thompson, R. R., Boden, A. F., Riddle, R. L., Bagnuolo, W. G., & Gies, D. R. 2001, American Colavita, M. M., Dumont, P. J., Mobley, D. W., Palmer, D., Shao, Astronomical Society Meeting, 199, 0 M., Vasisht, G. X., Wallace, J. K., Creech-Eakman, M. J., Koresko, Ridgway, S. T., Jacoby, G. H., Joyce, R. R., Siegel, M. J., & Wells, C. D., Kulkarni, S. R., Pan, X. P., & Gubler, J. 1999, AJ, 117, 521 D. C. 1982, AJ, 87, 680 von Zeipel, H. 1924a, MNRAS, 84, 665 Royer, F., Grenier, S., Baylac, M.-O., G´omez, A. E., & Zorec, J. —. 1924b, MNRAS, 84, 684 2002, A&A, 393, 897 Waldron, W. L. & Cassinelli, J. P. 2001, ApJ, 548, L45 Ryans, R. S. I., Dufton, P. L., Rolleston, W. R. J., Lennon, D. J., Walker, H. J. & Wolstencroft, R. D. 1988, PASP, 100, 1509 Keenan, F. P., Smoker, J. V., & Lambert, D. L. 2002, MNRAS, Wittkowski, M., Hummel, C. A., Johnston, K. J., Mozurkewich, D., 336, 577 Hajian, A. R., & White, N. M. 2001, A&A, 377, 981 S´egransan, D., Kervella, P., Forveille, T., & Queloz, D. 2003, A&A, Wolff, S. C. 1983, The A-Stars: Problems and Perspectives, 397, L5 Monograph Series on Nonthermal Phenomenona in Stellar Schaller, G., Schaerer, D., Meynet, G., & Maeder, A. 1992, A&AS, Atmospheres (NASA SP-463), 113 96, 269 Worley, C. E. & Douglass, G. G. 1997, A&AS, 125, 523 Shemansky, D. E. 2002 Wyatt, M. C. & Dent, W. R. F. 2002, MNRAS, 334, 589

APPENDIX A. spatial frequencies and position angles The spatial frequency and position angle coverage for each star is derived from the array’s sampling of the u, v-plane. The spatial frequency, x, is defined as x = u2 + v2 (A1) and the position angle, β, is defined as  1 β = tan− (u/v) (A2) which is the angle measured from the north through east. In this regard we follow Quirrenbach et al. (1997) who state: “...the baseline position angle β, i.e., the angle of the corresponding point in the (u, v)-plane, measured clockwise from the v-axis. For a north-south baseline, β=0 at meridian transit.” In Quirrenbach et al. (1997) position angles derived from interferometric data and polarimetric data are compared and thus the standard convention for position angle is adopted. The spatial frequencies are computed for the baselines, B, and bearings, ψ, listed in Table A4. The east and north components of the baseline vector are

Beast = B sin(ψ) (A3)

Bnorth = B cos(ψ) (A4) and the u, v values are u = (B cos(h) B sin(φ) sin(h))/(206265 λ ) (A5) east − north 0 v = (Beast sin(δ) sin(h) + Bnorth(sin(φ) sin(δ) cos(h) + cos(φ) cos(δ)))/(206265 λ0) (A6)

where h is the hour angle, δ is the declination, φ is the latitude, and λ0 is the central wavelength of the observation, for the K-band λ 2.2µm. 0  A.1. Position Angles and the Rapid Rotators It must be noted that the position angle convention adopted above differs from that used by (van Belle et al. 2001, hereafter vB01) in their calculation of the position angle for Altair (α Aql). Their position angle is rotated 90◦ relative to the Quirrenbach et al. convention: β = 270◦ β. Rapid Rotation: Structural Distortion & von Zeipel Gravity Darkening− This has the effect that the position angle ofthe α Aql’s equatorial bulge, β = 245◦ as reported by vB01, corresponds to β =Stars 25◦. The must locations satisfy of the both stellar mechanical poles are therefore and thermal at angles 65◦ and 115◦. The ellipsoid fit from vB01 gives an equatorial diameter of a = 3.46 mas and a polar diameter of b = 3.04− mas. The diameter as a function of position angle is givenrequirements by: for stability. 2 2 θ(β) = (x + y ) (A7) Mechanical: rotation distorts a starʼsx fi= agurecos( β 25◦) (A8)  − as it adjusts its structure to maintainy hydrostatic= b sin(β 25 ◦) (A9) − We areequilibrium. able to predict the ratio of a/b for the other rapid rotators (ζ Oph, α Leo, α Oph) however we have no constraints on the orientation of their rotation axes. This will be determined from the observations. For11 the purposes of the computing the spatial frequency sampling only, we have assumed that the stars have the same11 orientation on the sky as α Aql with the predicted axial ratios appropriate for each star. where G is the universal gravitational constant,InterferometricM is the stellar measurement mass, and Rof Altairis the (A7 polar V)! radius. The ratio of the where G is the universal gravitationalThe constant, axial ratiosM is arethe calculated stellar mass, employing and Rp anispIDL the polarprogram radius. written The by ratio Steve of Cranmer the based Cranmer & Owocki (1995). The critical angular rotation velocity, ω (the angular velocity below which the star is mechanically stable) is given by equatorialequatorial radius to radius the to polar the polar radius radius as a as function a functionRatio of of ofequatorial the the dimensionless dimensionless and polar radii rotational rotationalcrit speedCritical speedu, ωangular=u,uω ωcrit= rotation,u is ωcrit velocity, is van Belle et al 2001 33 ππ++ arccos( arccos(u)u) 8 GM 11 R (u)/R = cos ω = (A11) (A10) Re(ue )/Rpp= u cos 3 crit 27 R3(ω) (A11) u  3   p where G is the universal gravitational constant, M is the stellar mass, and R is the polar radius. The ratio of the The equatorialThe equatorial rotation rotation speed speed is is Equatorial velocity p equatorial radius to the polar radius as a function of theVeq dimensionless= u ωcrit Re(u) rotational speed u, ω = u ωcrit, is (A12) Veq = u ωcrit Re(u) (A12) Taking the observed v sin i value for each star as a lower3 limitπ on+ the arccos( equatorialu) rotation speed, V we obtain a lower limit R (u)/R = cos eq (A11) Taking theon observedu which yieldsv sin thei value equatorial-to-polar for each stare radius as aplower via equationu limit on A11. the3 The equatorial value for ω rotationcrit is constrained speed, V usingeq we the obtain published a lower limit on u whichangular yields diameter, the equatorial-to-polar parallax, and mass fromradius comparison viaThermal equation with: At evolutionary A11. the Theequator, value tracks. lower for Theωcrit expectedgravityis constrained equator-to-polarreduces using radius the published ratios are listed in Table A5. Theangular equatorial diameter, rotation parallax, speed is and mass from comparisonboth the with pressure evolutionary and temperature tracks. The gradients. expected equator-to-polar radius ratios are listed in Table A5. Veq = u ωcrit Re(u) (A12) B.Localsynthetic fl ux visibilities~ local gravity Taking the observed v sin i value for each star as a lower limit on the equatorial rotation speed, V we1/4 obtain a lower limit We compute the monochromatic syntheticB. visibilitiesLocalsynthetic effective with a visibilities Hankel temperature transform using ~ (local the model gravity) specificeq intensities and on u whichcorresponding yields the equatorial-to-polar angles: radius via equation A11. The value for ωcrit is constrained using the published angularWe diameter, compute the parallax, monochromatic and mass synthetic from comparison visibilitiesEffective with∞ with evolutionaryTemperature a Hankel transform tracks. varies The with using expected stellar the model latitude: equator-to-polar specific intensities radius and J xρ(λ/λ ) I (ρ)ρ dρ ratioscorresponding are listed in angles: Table A5. 0 0 λ V (λ)cooler = 0 equator, hotter pole. (B1) ∞  ∞  I (ρ)ρ dρ B. syntheticJ0 xρ visibilities(λ/λλ 0) Iλ(ρ)ρ dρ 0 0 Cooler, wider V (λ) =Effect: Limb-darkening varies as a function of (B1) We computewhere theV is monochromatic the visibility, J0 is synthetic the zeroth visibilities order Bessel with function, a∞ Hankelx is the transform spatial frequency, using theρ is model the angular specific radius, intensities and λ and equator Hotter, narrow I (ρ)ρ dρ correspondingis the angles: wavelength. Thelatitude model intensities as a functionstellar of latitudewavelengthλ and angular position are Iλ(ρ). The mean wavelength λ for a passband with a sensitivity0 function, S is 20 Stellar Atmospheric Structure0 Jason P.∞ Aufdenberg, 10 July 2003,λ Michelson Interferometry Summer School where V is the visibility, J0 is the zeroth order BesselJ0 function,xρ∞(λ/λ0x) Iisλ( theρ)ρ spatial dρ frequency, ρ is the angular radius, and λ 0 SλFλλ dλ is the wavelength. The model intensitiesV as(λ a) = function of wavelength0 and angular position are Iλ(ρ). (B1) λ0 = ∞  (B2) The mean wavelength λ0 for a passband with a sensitivity∞ function, Sλ is IλS(ρF)ρdλ dρ 0 λ λ  0∞ where V isThe the broad-band visibility, J averageis the visibility zeroth is order computed Bessel from function,SxλFisλλ the dλ spatial frequency, ρ is the angular radius, and λ 0 0 is the wavelength. The model intensities as a functionλ0 of=wavelength and angular position are I (ρ). (B2) ∞ ∞ 2 λ V (λ)SλFλ dλ The mean wavelength λ0 for a passband with a sensitivity2 0 function,SλFλ dλSλ is V =  0 (B3) | | ∞∞ 2  The broad-band average visibility is computed from S FSλλFλ dλdλ 0 λ λ 0  λ0 =  ∞   2 (B2) ∞ V (λ)S F dλ TableS F A4dλλ λ 2 0 λ λ V =CHARA 0 Baselines (B3) | |  ∞ 2  The broad-band average visibility is computed from SλFλ dλ Name Baseline Bearing 0 ∞ (meters) (degrees)2 V (λ)SλFλ dλ V 2 S1-S2= 0 Table 34.09 A4 350.1 (B3) | |E1-E2∞ 65.882  236.5 W1-W2CHARA 110.88S BaselinesλFλ dλ 97.5 W2-E20 152.22 63.3 NameS2-W2 Baseline 178.39  Bearing 340.2 S1-W2Table(meters) 212.04 A4 (degrees) 341.8 E1-W2 217.76 241.2 S1-S2S2-E2CHARA Baselines 248.11 34.09 17.7 350.1 E1-E2S2-W1 249.38 65.88 317.0 236.5 NameW1-W2W1-E2 Baseline 110.88 251.31 Bearing 77.6 97.5 W2-E2S1-W1(meters) 152.22 278.50 (degrees) 320.9 63.3 S1-E2 278.76 14.5 S1-S2S2-W2S2-E1 34.09 178.39 302.30 350.1 25.5 340.2 E1-E2S1-W2E1-W1 65.88 212.04 313.50 236.5 253.2 341.8 W1-W2E1-W2S1-E1 110.88 217.76 330.65 97.5 22.1 241.2 W2-E2S2-E2 152.22 248.11 63.3 17.7 S2-W2S2-W1aFrom 178.39 Table 249.38 4 of CHARA 340.2 317.0 S1-W2W1-E2Technical 212.04Report 251.31 No. 48. 341.8 77.6 E1-W2S1-W1 217.76 278.50 241.2 320.9 S2-E2S1-E2 248.11 278.76 17.7 14.5 S2-W1S2-E1 249.38 302.30 317.0 25.5 W1-E2E1-W1 251.31 313.50 77.6 253.2 S1-W1S1-E1 278.50 330.65 320.9 22.1 S1-E2 278.76 14.5 S2-E1aFrom Table 302.30 4 of CHARA25.5 E1-W1Technical Report 313.50 No. 48. 253.2 S1-E1 330.65 22.1

aFrom Table 4 of CHARA Technical Report No. 48. CHAPTER 4. UNIFIED STELLAR WIND MODELS 129

Input Parameters Teff , R, g(R), Z Rmax, M˙ , v , βwind ∞ StellarMa Winds,ss Los Ans a Examplend Limb Profiles Limb-darkening Effects & Mass Loss Rates for Deneb (A2 Ia)  Atmospheric Structure Hydrodynamic Hydrostatic R βwind dP g v = v 1 r dτ = κ ∞˙ − ρ = M 4πr2v 

 Equation of State  The limb brightness is sensitive to the density structure, which is P = P (ρ, T )  Special Relativistic Radiative Transfer Equation a function of the mass-loss rate. ∂I ∂ 2 (1+βµ) 2 ∂β  γ(µ + β) ∂r + ∂µ γ(1 µ ) r γ (µ + β) ∂r I  Predicted N−POI Vis−ibilities  ∂ β(1 µ2) 2 ∂β  γ − + γ µ(µ + β) νI   Data in the 1st lobe −do∂ν not r ∂r  for Alph2 a Cygni   2µ+β(3 µ ) 2 2  ∂β constrain the mass-loss+γ rate.r − + γ (1 + µ + 2βµ) ∂r I Rate Equations  NPOI Configuration  = η χI  nj ∗ − ji Temperature Corrections Darkening 23.3 m κ nj ∗ to break the angular diameter/mass-loss = bj (Rji + Cji) − j>i ni 94.9 m   ∞(η χ J ) dλ = 0 rate degeneracy.  λ − λ λ 104.2 m 0

 Simulations indicate that measurements Synthetic Spectrum of the 2nd lobe could provide a mass- loss diagnostic for hot supergiant, like Deneb and Rigel (B8 Ia). Figure 4.1: Flow chart showing the most important equations (and their inter- relationships) which characterize our model for the expanding stellar atmosphere problem in the STANDARD-WIND case. In the REAL-WIND case the velocity field is calculated from a solution to the hydrodynamical momentum equationH (seeigh Eq.Mass-Loss 4.2). 21 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Extended Cool Star Atmospheres, An Example: β Pegasi (M 2.5 II) Titanium Oxide Band Head; Narrow Band Filters Model Spectra versus Observed Spectra

Spherical wind model fi ts interferometry, but not spectrum. Data: Quirrenbach et al (1993) Spherical ApJ, 406, 215 hydrostatic model fi ts spectrum, but not interferometry.

22 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Closing Thoughts, Summary

Spectroscopy probes stellar fl uxes, however interferometry probes stellar intensities, the basic quantity of radiative transfer in stellar atmospheres. Thatʼs very cool!

Spectroscopy and interferometry are complementary. How well does that best fi t stellar atmo- sphere model fi t both the visibility dataand the stellar spectrum.

Stellar atmospheres are not black bodies. Published spectrophotometry exists for thousands of bright stars.

Most bright stars are variable. Contemporaneous spectrophotometry/spectroscopy and interferometry should be the goal.

You always see to an optical depth of unity. In spherical atmospheres the limb is very “fuzzy” and optically thin.

Spherical models are parameterized by Teff, log(g) and Mass.

Outer boundaries of real stellar atmospheres are complicated by winds, shells, chromospheres, convection, magnetic fi elds, pulsation, etc. Realistic physical models are beyond challenging. Interferometry will help to further constrain these fascinating problems.

23 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Synthetic Visibilities

24 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School Some References for Stellar Atmospheres

* Lecture Notes: “Radiative Transfer in Stellar Atmospheres” -- R. J. Rutten http://www.astro.uu.nl/~rutten/node20.html (and references there in)

* Introduction to Stellar Astrophysics: Volume 2: Stellar Atmospheres E. Böhm-Vitense (Cambridge UP)

* The Observation and Analysis of Stellar Photospheres D. Gray (Cambridge UP)

* Introduction to Stellar Atmospheres and Interiors E. Novotny (Oxford)

* Kinetic Theory of Particles and Photons: Theoretical Foundations of Non- LTE Plasma Spectroscopy -- J. Oxenius (Springer)

* The Analysis of Star Light: One hundred and fi fty years of astronomical spectroscopy -- J. B. Hearnshaw

* Mapping the Spectrum - Techniques of visual representation in research and teaching -- K. Hentschel (Oxford)

25 Stellar Atmospheric Structure Jason P. Aufdenberg, 10 July 2003, Michelson Interferometry Summer School