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STELLAR ATMOSPHERES A VERY SHORT INTRODUCTION PART I Ewa Niemczura Astronomical Institute, UWr [email protected] Stellar spectra Stellar spectra One picture is worth 1000 words, but one spectrum is worth 1000 pictures! Ivan Hubeny What is a Stellar Atmosphere? Stellar atmosphere: • medium connected physically to a star; from this medium photons escape to the surrounding space • region where the radiation observable by a distant observer originates What is a Stellar Atmosphere? Stellar atmosphere: • usually, a very thin layer on the surface of the star • late stars: photosphere, chromosphere, corona What is a Stellar Atmosphere? Stellar atmosphere: • usually, a very thin layer on the surface of the star • late stars: photosphere, chromosphere, corona • early stars: photosphere, expanding regions Why study stellar atmospheres? „Why in the world would anyone want to study stellar atmospheres? They contain only 10-10 of the mass of a typical star! Surely such a negligible fraction of a star mass cannot possibly affect its overall structure and evolution!” Question to D. Mihalas, about 50 years ago From the lecture of Ivan Hubeny Why study stellar atmospheres? Atmospheres are all we see; we have to use this information in the fullest. Stars: Stellar atmospheres: Determination of atmospheric parameters. Stellar spectra What can we obtain? • Spectral classification • Atmospheric parameters • Effective temperature Teff • Surface gravity logg • Chemical abundances • Metallicity [m/H] • Microturbulence, macroturbulence • Chemical peculiarities • Stratification of elements • Rotation velocity Vsini • Stellar wind parameters • Magnetic field parameters Stellar spectra What can we obtain? • Stellar classification • Atmospheric parameters • Chemical peculiarities • Stratification of elements • Rotation velocity • Stellar wind parameters • Magnetic field parameters • Multiple systems • Variability in spectral lines • Radial velocities • Orbit determination • Cluster membership • Pulsations • … Why study stellar atmospheres? Stars: Stellar atmospheres: Determination of primary (Teff, logg, chemical composition) and secondary atmospheric parameters (rotation velocity, turbulence etc.) Stellar structure and evolution: Determination of basic stellar parameters (M, R, L) 4 → = 42 → 2 Determination of the detailed physical state; Boundry for the stellar structure/evolution models; Atmospheres do influence the stellar evolution after all (mass loss from the atmosphere). Why study stellar atmospheres? Global context: Galaxies are made of stars (special case: very bright stars in distant galaxies); Sources of chemical species; (…) Why study stellar atmospheres? Methodological importance: Radiation determines the physical structure of the atmosphere, and this structure is probed only by the radiation; Sophisticated modeling approach needed – stellar atmospheres are guides for modeling other astronomical objects (e.g. accretion discs, planetary nebulae, planetary atmospheres etc.). Models: typical assumptions Geometry • Plane-parallel symetry ∆/ ≪ 1, very small curvature (e.g. main-sequence stars); • Typically for stellar photospheres: ∆/ ≪ 1 • Sun: = 700,000 km • Photosphere: ∆ = 300 km; ∆ = 4 × 10−4 • Chromosphere: ∆ = 3000 km; • Corona: ∆ ≈3 ∆ = 4 × 10−3 Models: typical assumptions Geometry • Plane-parallel symetry ∆/ ≪ 1, very small curvature (~, e.g. main-sequence stars); • Spherical symetry ∆/ ≈ 1, significant curvature ( ≠ , e.g. giants, supergiants); • … Models: typical assumptions Homogeneity • We assume the atmosphere to be homogeneous. • But it’s not always the case, e.g. sunspots, granulations, non-radial pulsations, magnetic Ap-stars (stellar spots), clumps and shocks in hot star winds etc. Models: typical assumptions Stationarity • We assume the atmosphere to be stationary • In most cases this assumption can be accepted • Exceptions: pulsating stars, supernovae, mass transfer in close binaries etc. Models: typical assumptions Conservation of momentum and mass • We assume hydrostatic equilibrium; • plane-parallel geometry: • spherical geometry: = − = − ( − ) • Exceptions: effects of magnetic fields, interaction in binary systems etc. • no hydrostatic equilibrium: = − ( − + ) Models: typical assumptions Conservation of energy • Nuclear reactions and production of energy: stellar interiors • Stellar atmospheres: negligible production of energy • We assume that the energy flux is conserved at any radius: • = / ∙ • 4 2 = = Different stars – different atmospheres Temperature: • MS stars, T~2000 – 60,000K • Brown dwarfs, T < 2000K • Hot, degenerate objects, T~104 – 108 K • White dwarfs, T < 100,000K • Neutron stars, T~107K Density: • MS stars, N~1010 – 1015 cm-3 • WD, N~1021 – 1026 cm-3 Basic Structural Equations Stellar atmosphere: plasma composed of particles (atoms, ions, free electrons, molecules, dust grains) and photons. Conditions: temperatures: ~103 – ~105 K; densities: 106 – 1016 cm-3. Starting point for physical description: kinetic theory Distribution function (most general quantity which describes the system): , , - number of particles in a volume of the phase space at position , momentum , and time t. Basic Structural Equations Kinetic (Boltzmann) equation (describes a development of the distribution function): + ∙ + ∙ = , – nabla differential operators with respect to the position and momentum components – particle velocity – external force – collisional term (describes creations and destructions of particles of type with the position (, + ) and momentum (, + ). Kinetic equation – complete description of the system Problem – number of unknowns (e.g. different excitation states of atoms etc.) Simplification – moments of the distribution function – integrals over momentum weighted by various powers of Basic Structural Equations Moment equations: (moment equations of the kinetic equation, summed over all kinds of particles; hydrodynamic equations): Continuity equation (1): Momentum equation (2): + ∙ = 0 () + ∙ = − + Energy balance equation (3): 1 2 2 + + ∙ 1 2 2 + + = ∙ − ∙ + – macroscopic velocity – total mass density – pressure – external force – internal energy , – radiation and conductive flux Basic Structural Equations Additional equation (zeroth-order moment equation): conservation equation for particles of type : + ∙ = – number density (or occupation number, or population) of particles of type . Basic Structural Equations Significant simplification of the system: Stationary = 0 , static medium ( = 0), 1-D (all quantities depend on one coordinate): Statistical equilibrium equation (0): =0 Hydrostatic equilibrium equation (2; assumption): = ⟹ = − Radiative equilibrium equation (3; assumption): 4 = 0 ⟹ = ≡ Basic Structural Equations Convection: transport of energy by rising and falling bubbles of material with properties different from the local medium; non-stationary and nonhomogeneous process. In 1-D stationary atmosphere, simplification – mixing-length theory; Radiative equilibrium equation with convection: 4 + = – convective flux; specified function of basic state parameters TE and LTE • TE – thermodynamic equilibrium: • simplification; particle velocity distribution and the distribution of atoms over excitation and ionisation states are specified by two thermodynamic variables: absolute temperature and total particle number density (or the electron number density ). • stellar atmosphere is not in TE – we see a star – photons are escaping – there are gradients of state parameters. TE and LTE • LTE – local thermodynamical equilibrium: • simplification; standard thermodynamical relations are employed locally for local values of , or , despite of the gradients that exist in the atmosphere. • equilibrium values of distribution functions are assigned to massive particles, the radiation field can depart from equilibrium (Planckian) distribution function. LTE LTE is characterised by three distributions: 1. Maxwellian velocity distribution of particles: = 2 3 2 2 4 2 exp − 2 , – particle mass and velocity – Boltzmann constant LTE 2. Boltzmann excitation equation ( − ) = exp − , − statistical weight of levels , , – level energies (measured from the ground state) LTE 3. Saha ionisation equation −3 2 = exp +1 +1 – total number density of ionisation stage – ionisation potential of the ion – partition function defined by ∞ = = ℎ2 2 1 3 2 exp − = 2.07 × 10−16 (cgs) In LTE the same temperature applies to all kind of particles and to all kinds of distributions. LTE • Maxwell, Saha, Boltzmann equations – LTE from macroscopic point of view • Microscopically – LTE is hold if all atomic processes are in detailed balance • the number of processed → is exactly balanced by the number of inverse processes → , • , – any particle state between which there exists a physically reasonable transition. • e.g. – is an atom in an excited state , the same atom in another state , etc. LTE vs. non-LTE • non-LTE (NLTE) – any state that departs from LTE (usually it means that populations of some selected energy levels of some selected atoms/ions are allowed to depart from their LTE value, but velocity distributions of all particles are assumed to be Maxwellian, with the same kinetic temperature ). LTE vs. non-LTE When we have to take non-LTE into account? • LTE breaks down if the detailed balance of at least one transition → breaks down • Radiative transitions – interaction involves particles and photones • Collisional transitions – interactions between two or more massive particles • Collisions tend to maintain LTE (their velocities are Maxwellian) • Validity of LTE depends on whether the radiative transitions are in detailed balance or not. LTE vs. non-LTE Departures from LTE: 1. Radiative rates in an important atomic transition dominate over the collisional rates and 2. Radiation is not in equilibrium (intensity does not have Planckian distribution) Collisional rates are proportional to the particle density – in high densities departures from LTE will be small. Deep in the atmosphere photons do not escape and intensity is close to the equilibrium value – departures from LTE are small even if the radiative rates dominate over the collisional rates. LTE vs. non-LTE non-LTE if: rate of photon absorptions >> rate of electron collisions LTE if: low temperatures and high densities non-LTE if: high temperatures and low densities LTE vs. non-LTE LTE if: low temperatures and high densities non-LTE if: high temperatures and low densities R. Kudritzki lecture Transport of energy Mechanisms of energy transport: a. radiation: (most important in all stars) b. convection: (important especially in cool stars) c. conduction: e.g. in the transition between solar chromosphere and corona d. radial flow of matter: corona and stellar wind e. sound and MHD waves: chromosphere and corona Intensity of Radiation (, , , ) specific intensity of radiation at position , travelling in direction , with frequency at time – the amount of energy transported by radiation in the frequency range (, + ), across an elementary area into a solid angle in a time interval : = , , , cos dS – angle between and the normal to the surface . − specific intensity, proportionality factor; dimention: erg cm-2 sec-1 hz-1 sr-1 Intensity of Radiation Photon distribution function : , , , – number of photons per unit volume at location and time , with frequencies in the range (, + Intensity of Radiation From the comparison of energy of photons: ∙ cos ℎ with defined before: = , , , cos Relation between specific intensity and photon distribution function: = ℎ Using this relation we define moments of the distribution function (specific intensity): energy density, flux and stress tensor. Intensity of Radiation Energy density of the radiation ( is the number of photons in an elementary volume, ℎ is the energy of photon): 1 = ℎ = Energy flux of the radiation ( is the vector velocity); how much energy flows trough the surface element: = ℎ ∙ = Ratiation stress tensor: = (ℎ) = 1 Photon momentum density (momentum of an individual photon is ℎ 1 = = 2 ℎ ): Absorption and Emission Coefficient The radiative transfer equation describes the changes of the radiation field due to its interaction with the matter. Absorption coefficient – removal of energy from the radiation field by matter: = , , , , , , Element of matterial of cross-section and length remove from a beam of specific intensity an amount of energy . The dimention of is cm-1 Absorption and Emission Coefficient The radiative transfer equation describes the changes of the radiation field due to its interaction with the matter. Absorption coefficient – removal of energy from the radiation field by matter: = , , , , , , 1/ – dimention of length; measures a characteristic distance a photon can travel before it is absorbed – a photon mean free path. Absorption and Emission Coefficient Emission coefficient – the energy released by the material in the form of radiation. Elementary amount of material of cross-section and length releases into a solid angle in direction within a frequency band an amount of energy: = , , , The dimention is erg cm-3 hz-1 sec-1 sr-1 dS Absorption and Emission Coefficient Microscopic physics – all contributions from microscopic processes that give rise to an absorption or emission of photons with specified properties. True absorption and scattering: True (thermal) absorption – photon is removed from a beam and is destroyed. Scattering – photon is removed from a beam and immediately reemitted in a different direction with slightly different frequency. , , , = , , , + , , , Absorption and Emission Coefficient Quantum theory of radiation – processes that give rise to an absorption or emission of a photon: • Induced absorption – an absorption of a photon and transition of an atom/ion to a higher energy state; • Spontaneous emission – an emission of a photon and a spontaneous transition of an atom/ion to a lower energy state; • Induced emission – an interaction of an atom/ion with a photon and an emission of another photon with identical properties (negative absorption). Radiative Transfer Equation Basic Radiative transfer equation: We express conservation of the total photon energy when a radiation beam passes through an elementary volume of matter of crosssection and length (measuread along the direction of propagation): + Δ, , , + Δ − , , , = , , , − , , , , , , The difference between spectific intensity before and after passing through the elementary volume of path length is equal to the difference of the energy emitted and absorbed in the volume. Iν + ΔIν Radiative Transfer Equation Basic + Δ, , , + Δ − , , , = , , , − , , , , , , The differences of intensities: 1 diff = + = + General form of the radiative transfer equation: 1 + ∙ , , , = , , , − , , , , , , Radiative Transfer Equation Basic Special case: one dimentional planar atmosphere: = = ≡ – angle between direction of propagation of radiation and the normal to the surface dz Time independent RTE: ,, = , , , − , , , , , , z Radiative Transfer Equation Basic Special case: spherical coordinates 1 − 2 = + Time independent: , , 1 − 2 (, , ) + = , , − , , , , Radiative Transfer Equation Optical Depth, Source function 1-D transfer equation: = − Elementary optical depth: ≡ − Source function: ≡ The emission and absorption coefficients are local quantities, so the definition of source function is good for all geometries. = − z Radiative Transfer Equation Optical Depth, Source function Physical meaning of optical depth: RTE in the absence of emission: = With the solution: = ( + Δ)exp−Δ ⇒ the optical depth is the e-folding distance for attenuation of the specific intensity due to absorption; ⇒ the probability that a photon will travel an optical distance is: = exp(−) Radiative Transfer Equation Optical Depth, Source function Physical meaning of source function: ≡ = 4 ℎ number of photons emitted in an elementary volume to all direction (4 comes from integration over the all solid angles, and ℎ transforms energy to the number of photons). 4 = ℎ ⟹ he source function is proportional to the number of photons emitted per unit optical depth interval. Radiative Transfer Equation Elementary Solutions 1. No absorption, no emission, = = = ⟹ Radiation intensity remains constant. 2. No absorption, only emission, > , = , = 0, + ′ ′ / 0 Outcoming radiation from an optically thin radiative slab (e.g. forbidden line radiation from planetary nebulae, radiation from solar transition region or/and corona). 3. No emission, only absorption = , > 0, = , exp − Radiative Transfer Equation Elementary Solutions 4. Absorption and emission, > , > General formal solution of the transfer equation – formal because it is assumed that both and are specified functions of position and frequency: 2 − 1 1 , = 2 , exp − + 2 1 − 1 exp − / 5. Semi-infinite atmosphere – emergent radiation (1 = 0, 2 = ∞) ∞ 0, = 0 exp − / observed intensity is a weighted average of the source function along the line of sight. Radiadive Transfer Equation Moments Moments of specific intensity (of the photon distribution function): photon energy density, radiation flux, radiation stress tensor. = 1 Integration of transfer equation (kinetic equation) – relations between these moments: + ∙ = − 1 + ∙ = − Structure of the moment equations of the kinetic equation: + = − Radiadive Transfer Equation Moments In astrophysics: moments are angle averaged, not angle-integrated quantities: 1 = 4 1 = 4 1 In the plane-parallel approximation moments of specific intensity are scalar quantities: 1 1 = 2 −1 1 1 = 2 −1 1 1 2 = 2 −1 Radiative transfer equation Moments The moment equations of RTE: = − = Eddington factor: ≡ Combination of two moment equation: 2 = − 2 There is no dependence on the angle; useful in numerical solution. Model atmosphere Definition and terminology Model atmosphere – • Specification of all atmospheric state parameters as function of depth. • Table of values of the state parameters in the discrete depth points. • State parameters: depend on the type of the model (on the basic assumption under which the model is constructed) • Massive particle state parameters – from this we can determine the radiation field by a formal solution of the transfer equation. Basic equations Classic stellar atmosphere problem Radiative transfer equation First order form: = − Second order form: 2 ( ) = − 2 NA x NF vs. NF NA – number of angle points NF – number of frequency points Basic equations Classic stellar atmosphere problem Hydrostatic equilibrium equation: = − = − ⟹ = – mass in the column of a cross-section of 1 cm2 above a given point in the atmosphere Basic equations Classic stellar atmosphere problem Total pressure: = + + 4 = + ∞ 0 1 2 + 2 The hydrostatic equilibrium equation – effective gravity acceleration: 4 =− ∞ 0 4 =− ∞ 0 true gravity acceleration (acting downward) – the radiative acceleration (acting upward). Basic equations Classic stellar atmosphere problem Radiative equlibrium equation: • the total radiation flux is conserved ∞ 0 4 = = 4 • using the radiative transfer equation: ∞ 0 ∞ − = 0 − = 0 Basic equations Classic stellar atmosphere problem Statistical equilibrium eqations = rate equations The conservation equation for a particle : =0 Explicitly: + = ≠ ( + ) ≠ and – radiative and collisional rate; Total number of transitions out of level = total number of transitions into level ; Radiative rates depend on radiation intensity; Collisional rates depend on temperature and electron density. Basic equations Classic stellar atmosphere problem Total number conservation equation (or abundance definition equation): = Only a limited number of levels of an atom/ion is treated explicitly (the rate equation is written and solved, low-lying levels); remaining levels – approximations, and the abundance definition: + = Basic equations Classic stellar atmosphere problem Charge conservation equation: • global electric neutrality of the medium − = 0 is the charge associated with level (0 for levels of neutral atoms, 1 for once ionised atoms etc.); summation extends over all levels of all ions of all species. Additional equations Classical stellar atmosphere problem Definition equations of the absorption and emission coefficients: = − > + = • • • • 2 , 1 − −ℎ + 2ℎ 3 ( − ∗ −ℎ/ ) + [ > + ∗ −ℎ/ ℎ − (, ) ] + Bound-bound transitions (i.e. spectral lines) Bound-free transitions (continua) Free-free absorption (inverse brehmstrahlung) Electron scattering Another equations: relevant cross-sections, definition of LTE populations, etc. Basic equations: summary Equation State parameter Radiative transfer Mean intensities, Radiative equilibrium Temperature, Hydrostatic equilibrium Total particle density, Statistical equilibrium Populations, Charge conservation Electron density, Types of model atmosphere Static models: • assumption of hydrodynamical equilibrium • applies only to atmospheric layers that are close to hydrodynamic equilibrium, i.e. the macroscopic velocity is small compared to the thermal velocity of atoms – stellar photospheres • basic input parameters: effective temperature, surface gravity and chemical composition, • additional parameters: microturbulence, and in case of convective models: mixing length. Types of model atmosphere Static LTE models: • LTE assumption, two state parameters, temperature and density (or electron density) describe the physical state of the atmosphere at any given depth. • standard models • Example: Kurucz ATLAS9 code Types of model atmosphere Static non-LTE models: model which takes into account some kind of departure from LTE: • models solving for the full structure (TLUSTY) • restricted non-LTE problem: • the atmospheric structure is assumed to be known from previous calculations (LTE or simplified non-LTE), • radiative transfer and statistical equilibrium for a chosen atom/ion is solved simultaneously (DETAIL/SURFACE). • non-LTE line-blanketed models: • non-LTE is considered in most energy levels and transitions between them – lines and continua – that influence the atmospheric structure. • number of such lines may go to millions. non-LTE models When departures from LTE may be important in stellar atmospheres? non-LTE models For a star of any spectral type, there is always a wavelength range, and of course a layer in the atmosphere, where non-LTE effects are important. „important non-LTE effects” – • arbitrary – it depends what precision do we want; • e.g. for B stars, visual part 10% - LTE will be OK, 2-5% non-LTE is necessary; • EUV, the same star – 10-20% requires non-LTE effects. Types of model atmosphere Unified models: • no assumption of hydrostatic equilibrium in the whole atmosphere ranging from a static photosphere to a dynamical outher parts. Literature • D. Mihalas: „Stellar Atmospheres” • I. Hubeny: „Stellar atmospheres theory: an introduction” in: Stellar atmospheres: Theory and Observations, Lecture note in physics, J.P. De Greeve, R.Blomme, H. Hensberg (Eds.), Springer • D. Gray: „The observations and analysis of stellar photospheres”