Radiative Processes, Stellar Atmospheres and Winds

Radiative processes, stellar atmospheres and winds

Master of Science in Astrophysics ‒ P5.0.2 Master of Science in Physics with main focus on Astrophysics ‒ P4.0.5, P5.2.5, P6.0.5

The Tarantula Nebula (30 Dor) in the LMC

The wind-blown bubble N44F in the LMC

The bubble nebula NGC 7635 in Cassiopeia: A Spitzer view of R 136 in the a wind-blown bubble around heart of the Tarantula Nebula BD+602522 (O6.5IIIf)

Joachim Puls, University observatory Munich (LMU) 1 Content USM

Part I 1. Prelude: What are stars good for? A brief tour through present hot topics (not complete, personally biased) 2. Quantitative spectroscopy: the astrophysical tool to measure stellar and interstellar properties 3. The radiation field: specific and mean intensity, radiative flux and pressure, Planck function 4. Coupling with matter: opacity, emissivity and the equation of radiative transfer (incl. angular moments) 5. Radiative transfer: simple solutions, spectral lines and limb darkening 6. Stellar atmospheres: basic assumptions, hydrostatic, radiative and local thermodynamic equilibrium, temperature stratification and convection 7. Microscopic theory 1. Line transitions: Einstein-coefficients, line-broadening and curve of growth, continuous processes and scattering 2. Ionization and excitation in LTE: Saha- and Boltzmann-equation 3. Non-LTE: motivation and introduction

Part II Intermezzo: Stellar Atmospheres in practice A tour de modeling and analysis of stellar atmospheres throughout the HRD

8. Stellar winds – an overview 9. Line driven winds of hot stars – the standard model 1. Radiative line-driving and line-statistics 2. Theoretical predictions for line-driven winds (incl. wind-momentum luminosity relation) 10. Quantitative spectroscopy: stellar/atmospheric parameters and how to determine them, for the exemplary case of hot stars

2 Literature USM

. Carroll, B.W., Ostlie, D.A., “An Introduction to Modern Astrophysics”, 2nd edition, Pearson International Edition, San Francisco, 2007, Chap. 3,5,8,9 . Mihalas, D., "Stellar atmospheres", 2nd edition, Freeman & Co., San Francisco, 1978 . Hubeny, I., & Mihalas, D., “Theory of Stellar Atmospheres”, Princeton Univ. Press, 2014 . Unsöld, A., "Physik der Sternatmosphären", 2nd edition, Springer Verlag, Heidelberg, 1968 . Shu, F.H., "The physics of astrophysics, Volume I: radiation", University science books, Mill Valley, 1991 . Rybicki, G.B., Lightman, A., "Radiative Processes in Astrophysics", New York, Wiley, 1979 . Osterbrock, D.E., "Astrophysics of Gaseous Nebulae and Active Galactic Nuclei", University science books, Mill Valley, 1989 . Mihalas, D., Weibel Mihalas, B., “Foundations of Radiation Hydrodynamics”, Oxford University Press, New York, 1984 . Cercignani, C., "The Boltzmann Equation and Its Applications", Appl. Math. Sciences 67, Springer, 1987 . Kudritzki, R.-P., Hummer, D.G., "Quantitative spectroscopy of hot stars", Annual Review of Astronomy and Astrophysics, Vol. 28, p. 303, 1990 . Sobolev, V.V., “Moving envelopes of stars”, Cambridge: Harvard University Press, 1960 . Kudritzki, R.-P., Puls, J., "Winds from hot stars", Annual Review of Astronomy and Astrophysics, Vol. 38, p. 613, 2000 . Puls, J., Vink, J.S., Najarro, F., "Mass loss from hot massive stars", Astronomy & Astrophysics Review Vol. 16, ISSUE 3, p. 209, Springer, 2008 . Puls, J., “Radiative Transfer in the (Expanding) Atmospheres of Early-Type Stars, and Related Problems”, in: “Radiative Transfer in Stellar and Planetary Atmospheres”, Canary Islands Winter School of Astrophysics, Vol. XXIX, Cambridge University Press, 2019, in press 3 Chap.1 ‒ Prelude USM . cosmology, galaxies, dark energy, dark matter, …

What are stars good for?

. … and who cares for radiative transfer and stellar atmospheres?

. remember • galaxies consist of stars (and gas, dust) • most of the (visible) light originates from stars • astronomical experiments are (mostly) observations of light: have to understand how it is created and transported 4 The cosmic circuit of matter USM Life planet systems

What are stars good for?

. Us! star formation . (whether this is really proto- stars good, is another ISM: dust, molecules, gas

question...) stellar evol. dust + gas => red/blue (super)giants

planetary low stellar nebulae masses Joni Mitchell - Woodstock (1970!) “… We are stardust Billion year old carbon…” WD core collapse Supernovae NS, BH adapted from http://astro.physik.tu-berlin.de/~sonja/Materiekreislauf/index.html 5 First stars and reionization USM

WMAP = Wilkinson Microwave Anisotropy Probe color coding: ΔT range ± 200 μK, ΔT/T~ few 10-5 => “anisotropy” of last scattering surface (before recomb.) white bars: polarization vector  CMB photons scattered at electrons (reionzed gas) [NOTE: newer data from PLANCK]

credit: NASA/WMAP Science Team

6 The first stars … USM

• cosmic reionization:

• z=7.7 ± 0.8 (from PLANCK, assuming instantaneous reionization, state 2018)

• z ≈ 11 (begin) to 7 (from WMAP) • quasars alone not capable to reionize Universe at that high redshift, since rapid decline in space density for z > 3 (Madau et al.1999, ApJ 514, Fan et al. 2006, ARA&A 44) Bromm et al. (2001, ApJ 552) • (almost) metal free: Pop III But: theoretical models indicate more typical masses • very massive stars (VMS) with around 40 M (fragmentation!, Hosokawa et al. 2011), 1000 M > M > 100 M , L prop. to M, T ̴100 kK    eff though (much) more massive stars might have formed • large H/He ionizing fluxes: as well 1048 (1047) H (He) ionizing photons Present status: Massive stars important for per second and solar mass reionization, but not exclusive • assume that primordial IMF is heavy, i.e., favours formation of VMS see also: Abel et al. 2000, ApJ 540; Bromm et al. 2002, ApJ 564; Cen 2003, ApJ 591; Furnaletto & Loeb 2005, ApJ 634; Wise & • then VMS capable to reionize universe alone Abel 2008, ApJ 684; Johnson et al. 2008, Proc IAU Symp 250 (review); Maio et al. 2009, A&A 503; Maio et al. 2010, MNRAS 407; Weber et al. 2013, A&A 555 … and many more publications 7 … might be observable in the NIR USM

with a ≥ 30m telescope, e.g. via HeII λ1640 Å (strong ISM recomb. line)

Standard IMF Heavy IMF, zero metallicity 1 Mpc (comoving)

GSMT Science Working Group Report, 2003, Kudritzki et al. http://www.aura-nio.noao.edu/gsmt_swg/SWG_Report/SWG_Report_7.2.03.pdf As observed through 30-meter telescope (Hydro-simulations by R=3000, 105 seconds (favourable conditions, see Davé, Katz, & Weinberg) also Barton et al., 2004, ApJ 604, L1) 8 Massive vs. intermediate-/low-mass stars USM

. massive stars (MZAMS > 8 Msun) • short life-times (few to 20 million years) • end products: core-collapse SNe (sometimes as slow GRBs) → neutron stars, black holes (or even complete disruption in case of pair-instability SNe) • Grav. waves from BH mergers! . intermediate-/low-mass stars

(0.1…0.8 Msun < MZAMS < 8 Msun) • long life-times (0.1 to 100 billion years) • end products: White dwarfs, SNIa

. brown dwarfs (13 MJupiter< M < 0.08 Msun) • ‘failed stars’, core temperature not sufficient to ignite H-fusion Credit: ESO • instead, Deuterium and, for higher masses, Lithium fusion

ZAMS: Zero Age Main Sequence MS: Main sequence, core hydrogen burning 9 low-mass vs. massive star during the MS USM low-mass star (e.g., the sun) massive star factor 10…20 larger than sun factor 104…106 more luminous than sun

convective envelope subsurface convect. zone radiative envelope

radiative core convective core chromosphere dense wind corona/thin wind X-ray, UV, (X-ray radiation) radio radiation

photosphere photosphere (most of observed radiation) (most of optical radiation)

. radiative core . convective core . convective envelope . radiative envelope with subsurface convection zone . geometrically thin photosphere . geometrically thin photosphere + dense wind . level populations collisionally dominated . level populations radiatively dominated →Local thermodynamic equilibrium (LTE): →non-LTE level population Saha-Boltzmann population (all transition-probabilities need to be . chromosphere (temp. begins to increase outwards) calculated explicitly) . hot corona/thin wind (very low mass-loss rate)

NOTE: evolved objects (red giants and supergiants) and brown dwarfs are fully convective 10 Examples for current research:

USM Observations … . … in all frequency bands . both earthbound and via satellites . Gamma-rays (Integral), X-rays (Chandra, XMM-Newton), (E)UV (IUE, HST), optical (VLT), IR (VLT, →JWST, →ELT) , (sub-) mm (ALMA) , radio (VLA, VLBI, →SKMA) … . photometry, spectroscopy, polarimetry, interferometry, gravitational waves (aLIGO!) . current telescopes allow for high S/N and high spatial resolution . because of their high luminosity, massive stars can be spectroscopically observed not only in the Milky Way, but also in many Local Group (and beyond) galaxies (‘record-holder’: blue supergiants in NGC 4258 at a distance of ≈ 7.8 Mpc, Kudritzki+ 2013)

Abbreviations: IUE ‒ International Ultraviolet Explorer HST ‒ Hubbble Space Telescope VLT ‒ Very Large Telescope (Cerro Paranal, Chile) JWST ‒ James Webb SpaceTelescope ELT ‒ Extremely Large Telescope (Cerro Armazones, Chile, 20 km away from VLT)) ALMA ‒ Atacama Large Millimeter/Submillimeter Array (Chajnantor-Plateau, Chile, 5000 m altitude) VLA ‒ Very Large Array (Socorro, New Mexico, USA) VLBI ‒ Very Large Baseline Interferometer SKMA ‒ Square Kilometer Array (South Africa and Australia) 11 Examples for current research: USM Star formation . Star formation ‒ formation of massive stars

• until 2010, it was not possible to ‘make’ stars with M > 40 Msun

From Kuiper+ (2013)

12 Examples for current research: USM Star formation . Star formation ‒ formation of massive stars

• until 2010, it was not possible to ‘make’ stars with M > 40 Msun

From Kuiper+ (2010)

• Radiation pressure barrier for spherical infall: when core becomes massive, high luminosity heats ‘first absorption region’, radiation pressure due to re-processed IR radiation stops and reverts accretion flow. • If accretion via disk, re-processes radiation-field becomes highly anisotropic, the radial component of the radiative acceleration becomes diminished, and

further accretion becomes possible. Stars with M > 40 Msun (… 140 Msun) can be formed. (see work by R. Kuiper and collaborators) 13 Examples for current research:

USM Stellar structure and evolution

. Stellar structure and evolution • implementation/improved description of various processes, e.g., – impact of mass-loss and rotation (mixing!) in massive stars – generation and impact of B-fields – convection, mixing processes, core-overshoot etc. still described by simplified approximations in 1-D (e.g., diffusive processes), needs to be studied in 3-D (work in progress)

14 Examples for current research: USM Stellar structure and evolution . vrot vs. Teff , for rotating Galactic massive-star models from Ekström+(2012, ‘GENEC’) and Brott+ (2011, ‘STERN’), with vrot(initial) ≈ 300km/s . The main difference on the MS is due to the lack (Ekström) and presence (Brott) of assumed internal magnetic fields and the treatment of angular momentum transport.

. NOTE: Even at main sequence, stellar evolution of massive stars unclear in many details!!!! . Do not believe in statements such as ‘stellar evolution is understood’ 15 Examples for current research: USM Stellar structure and evolution

. Stellar structure and evolution • NOTE: binarity fraction of Galactic stars M-stars: 25%, solar-type: 45%, A-stars: 55% (Duchene & Kraus 2013, review) O-stars in Galactic clusters: – 70% of all stars will interact with a companion during their lifetime (Sana+ 2012)

• THUS: needs to be included in evolutionary calculations – even more approximations regarding tidal effects, mass-transfer, merging … (e.g., ‘binary_c’ by Izzard+ 2004/06/09)

• predictions on pulsations – frequency spectrum of excited oscillations – period-luminosity relations as a function of metallicity

16 Asteroseismology: Revealing the internal structure USM

. non-radial pulsations: examples for different models following slides adapted from C. Aerts (Leuven) Blue: Moving towards Observer Red: Moving away from Observer (l,m)=(3,0) (l,m) = (3,2) (l,m)=(3,3) axisymmetric tesseral sectoral : azimuthal order order azimuthal : m : nonradial degree, degree, nonradial : l

17 Internal behaviour of the oscillations USM The oscillation pattern at the surface propagates in a continuous way towards the stellar centre.

Study of the surface patterns hence allows to characterize the oscillation throughout the star.

18 Probing the interior USM

The oscillations are standing sound waves that are reflected within a cavity

Different oscillations penetrate to different depths and hence probe different layers

19 Doppler map of the Sun USM

The Sun oscillates in thousands of non-radial modes with periods of ~5 minutes

The Dopplermap shows velocities of the order of some cm/s

20 Solar frequency spectrum from ESA/NASA satellite USM SoHO: systematics !

21 Frequency separations in the Sun USM

Result: internal sound speed and internal rotation could be determined very accurately by means of helioseismic data from SoHO

22 Internal rotation of the Sun USM

X 10¯ Hz

Ω/2π [nHz]

Beginning of outer convection zone Solar interior has rigid rotation

23 … towards massive star seismology USM

  Cep: low order p- and g-modes

 SPB slowly pulsating B-stars high order g-modes

 Hipparcos: 29 periodically variable B-supergiants (Waelkens et al. 1998)

 no instability region predicted at that time

 nowdays: additional region for high order g-mode instability

asteroseismology of evolved massive stars becomes possible

p-modes: pressure (radial) order: number of nodes between center and surface g-modes: gravity as restoring force 24 Space Asteroseismology USM

COROT: COnvection ROtation and planetary Transits French-European mission (27 cm mirror) launched December 2006

Kepler: NASA mission (1.2m mirror), launched March 2009

MOST: Canadian mission (65 x 65 x 30 cm, 70 kg) launched in June 2003

BRITE-Constellation: Canadian-Austrian-Polish mission (six 203 cm nano-satellites, 7kg) first one launched 2013 asteroseismology of bright (= massive) stars 25 Examples for current research:

USM End phases of evolution

. End phases • evolutionary tracks towards ‘the end’ • models for SNe and Gamma-ray bursters

26 Long Gamma Ray Bursts USM

. long: >2s . Collapsar: death of a massive star Collapsar Scenario for Long GRB (Woosley 1993) . massive core (enough to produce a BH) . removal of hydrogen envelope . rapidly rotating core (enough to produce an accretion disk) credit: NASA requires chemically homogeneous evolution

meridional of rapidly rotating massive star circulation . pole hotter than equator (von Zeipel) convective . rotational mixing due to meridional core circulation (Eddington-Sweet)

27 Chemically Homogeneous Evolution … USM

 …if rotational mixing during main W/Wk: rotational frequency in units of critical one sequence faster than built-up of chemical gradients due to nuclear fusion (Maeder 1987)

 bluewards evolution directly towards Wolf- Rayet phase (no RSG phase). Due to meridional circulation, envelope and core are mixed -> no hydrogen envelope

 since no RSG phase, higher angular momentum in the core (Yoon & Langer 2005) massive stars as progenitors of high redshift GRBs: early work: Bromm & Loeb 2002, Ciardi & Loeb 2001, Kulkarni et al. 2000, Djorgovski et al. 2001, Lamb & Reichart 2000 At low metallicity stars are expected to be rotating faster because of weaker stellar winds

28 Examples for current research:

USM End phases of evolution

. End phases • evolutionary tracks towards ‘the end’ • models for SNe and Gamma-ray bursters • models for neutron stars and white dwarfs • accretion onto black holes • X-ray binaries (‘normal’ star + white dwarf/neutron star/black hole) • synthetic spectra of SN-remnants in various phases • observations (now including gravitational waves) and comparison with theory – first detection of aLIGO was the merger of two black holes with masses

around 30 Msun (Abbott et al. 2016) – Corresponding theoretical scenario published just before announcement of detection (Marchant+ 2016), predicting one BH merger for 1000 cc-SNe, and a high detection rate with aLIGO

29 Examples for current research: USM Impact on environment

. Impact on environment • cosmic re-ionization and chemical enrichment • chemical yields (due to SNe and winds) • ionizing fluxes (for HII regions) • Planetary nebulae (excited by hot central stars) • impact of winds on ISM (energy/momentum transfer, triggering of star formation) • stars and their (exo)planets

30 Feedback

• massive stars determine energy (kinetic and radiation) and momentum budget of Bubble Nebula surrounding ISM (NGC 7635) in Cassiopeia • kinetic energy and momentum budget via winds (of different strengths, in dependence of evolutionary status) wind-blown bubble around • massive stars enrich environment with metals, via winds and SNe, determine chemo- BD+602522 dynamical evolution of Galaxies (exclusively (O6.5IIIf) before onset of SNe Ia) • in particular: first chemical enrichment of Universe by First (VMS) Stars

→“FEEDBACK”

31 Chap. 2 ‒ Quantitative spectroscopy USM theory Astrophysics experiment Experiment in astrophysics = Collecting photons from cosmic objects 0.1 nm = 1 Å= 12.4 keV

hydrogen Lyman edge 1 Å = 10-8 cm = 10-4 μm (micron); 1 nm = 10 Å Collecting: earthbound and via satellites! Note: Most of these photons originate from the atmospheres of stellar(-like) objects. Even galaxies consist of stars! 32 Astrophys. monographs,A Univ. Chicago Press (1943) USM AN ATLAS OF STELLAR SPECTRA

WITH AN OUTLINE OF SPECTRAL CLASSIFICATION

Morgan, Keenan, Kellman

33 USM

Empirical system => Physical system

34 Digitized spectra USM

from Silva & Cornell, 1992 35 Spectral lines formed in USM (quasi-)hydrostatic atmospheres

ESO 3.6m CASPEC SMC

  0.5Å S/N 30...70 LMC

(Walborn et al.,1995 LMC

SMC

H H

36 P-Cygni lines formed in

USM hydrodynamic atmospheres

HST-FOS SMC

LMC

SMC

37 UV spectrum of the O4I(f) supergiant ζ Pup USM

montage of Copernicus ( 1500 Å, high res. mode, Å, Morton & Underhill 1977) and IUE (Å) observations 38 Supernova Type II in different phases USM

photospheric phase

transition to nebular phase

figure prepared by Mark M. Phillips, reproduced from McCray & Li (1988) 39 Spectrum of Planetary Nebula USM

pure emission line spectrum with forbidden [OIII] lines of O III 5007, 4959Å ratio 3:1

H

From Meatheringham & Dopita, 1991, ApJS 75

40 Quasar spectrum in rest frame of quasar USM

41 “UV”-spectra of starburst galaxies USM

obs= o(1+z) 1225->4560 Å galaxy at z = 2.72

local starburst galaxy, wavelengths shifted

From Steidel et al. (1997)

42 Atmospheres and nebulae - an overview USM Sun hot, massive star Supernova Ia Gaseous Nebula r PN

 R  R  R Photosphere Envelope Photosphere 5 v 00 km/s v H II sun v 100 km/s star Photosphere Core solar Corona Chromosphere wind v 20 000 km/s v  2000 km/s wind Quasar v v  v vmax Core: 33 M =2 10 g Core: R =7 1010 cm Core:  Core : MPN  1 M L =4 1033 erg s-1 M 50 M M 1.44 M 5    MHII  50  10 M R 20 R Photosphere:   exploded 9 11 MQ  10  10 M L 105 106 L 11 R  200 km    Lmax  10 L 4 LPN  10 L T  6000 K Photosphere: L 106 L 15 -3 HII   n  10 cm R/ R  R/R  L 1012 L Chromosphere: T 20000  50000 K Q   14 12 -3 R  1000 km n 10  10 cm Envelope: Envelope: T  20000 K Wind: 5 12 -3 R10 R R 0.1pc (PN) n  10 cm R  100 R*     T  12000 K R 10 pc (H II) Corona: T  0.9...0.3 Teff   6 -3 3 R R 12 8 -3 n  10 cm R 10 pc (Quasar)   n  10 ...10 cm   6 T  10 K 6 -3 n  2 10 cm 43 Stellar atmospheres - an overview USM Sun hot, massive star Supernova Ia Gaseous Nebula r PN

…gives insight into and understanding of our cosmos

Quantitative spectroscopy = quantitative diagnostics of spectra . provides • stellar properties, mass, radius, luminosity, energy production, chemical composition,properties of outflows • properties of (inter) stellar plasmas, temperature, density, excitation, chemical comp., magnetic fields

. INPUT for stellar, galactic and cosmologic evolution and for stellar and galactic structure

. requires • plasma physics, plasma is "normal" state of atmospheres and interstellar matter (plasma diagnostics, line broadening, influence of magnetic fields,...) • atomic physics/quantum mechanics, interaction light/matter (micro quantities) • radiative transfer, interaction light/matter (macroscopic description) • thermodynamics, thermodynamic equilibria: TE, LTE (local), NLTE (non-local) • hydrodynamics, atmospheric structure, velocity fields, shockwaves,...

45 one example ... USM atomic levels and allowed transitions ("Grotrian-diagram") in OIV

gf oscillator strength, measures "strength" of transition (cf. Chap. 7)

46 Concept of spectral analysis USM

comparison observed spectrum synthetic spectrum

detector theory of atmospheres model calculations (simulation) physical concept spectrograph hydro-/thermodynamics numerical solution of atomic physics theoretical equations radiative transfer telescope

approximations for INPUT of L, M, R, specific objects chemical composition geometry, symmetry, homogeneity, stationarity

48 The VLT-FLAMES survey of massive stars (‘FLAMES I’ )

USM The VLT-FLAMES Tarantula survey (‘FLAMES II’)

FLAMES = Fibre Large Array Multi Element Spectrograph

image credit: ESO image credit: ESO . FLAMES I: high resolution spectroscopy of massive Major objectives stars in 3 Galactic, 2 LMC and 2 SMC clusters . rotation and abundances (test rotational mixing) (young and old) . stellar mass-loss as a function of metallicity - total of 86 O- and 615 B-stars . binarity/multiplicity (fraction, impact) . FLAMES II: high resolution spectroscopy of more . detailed investigation of the closest ‘proto-starburst’ than 1000 massive stars in Tarantula Nebula (incl. 300 O-type stars) summary of FLAMES I results: Evans et al. (2008) summary of FLAMES II results: Evans et al. (2019, in prep.)49 Optical spectrum of a very hot O-star USM

BI237 O2V (f*) (LMC) ‒ vsini = 140 km/s

. Synthetic spectra from Rivero-Gonzalez et al. (2012) NIV

red: HI blue: HeI green: HeII orange: NIV Teff ≈ 53,000 K, log g ≈ 4.1, low mass loss, magenta: NV Nitrogen moderately enriched (≈ 0.4 dex)

NIV

50 . Tarantula Nebula (30 Dor) in the LMC

. Largest starburst region in Local Group

. Target of VLT-FLAMES Tarantula survey (‘FLAMES II’, PI: Chris Evans)

. Cluster R136 contains some of the most massive, hottest, and brightest stars known

. Crowther et al. (2010): 4 stars with initial masses

from 165-320 (!!!) Mʘ

. problems with IR-photometry (background-correction), lead to overestimated luminosities → initial masses become reduced:

140 - 195 Mʘ (Rubio-Diez et al., IAUS 329, 2016, and in prep. for A&A) Spectral energy distribution of the most massive stars USM in our “neighbourhood” – theory vs. observations initial current mass mass (Msun) (Msun) 320 265 → <194 240 195

→ <140

165 135

→ 141

220 175 from Crowther et al. 2010 etal. Crowther from al2016 et Rubio-Diez masses from lower typical uncertainty ± 40 Msun 52 Chap. 3 ‒ The radiation field USM

f fr  f  p Number of particles in (r , r dr ) with momenta ( ppp ,  dt ) at time  i   i  t  rti  pt i

 N(,,)rp t f (,,) rp t d3 rp d 3 For a detailed derivation and 0 Vlasov discussion, see, e.g., f  (v  ) f ( F p ) f   f  Cercignani, C., "The t { Boltzmann distribution function f    Boltzmann Equation and Its  t coll if collisions Applications", Appl. Math. i) f(,,r p t) is Lorentz-invariant Sciences 67, Springer, 1987 D/Dt f, Lagrangian derivative 3 3 total derivative of f measured in fluid frame, ii)  N0 f(,rp 0 0 ,) t 0 d r 0d p 0 at times t, t+Δt and positions r, r + v Δt evolution  implications for photon gas

3 3  N f (rrpp0 d , 0  d , t 0  dt ) d rp d h p n 3 3 c ( p  F) f(rvpF0  dt ,0  dt ,) t 0  dt d rpd

d3p  p 2 dpd   solid angle with respect to n Theoretical mechanics: If no collisions, conservation of phase space volume: absolute value d3rp d 3 d 3 rp d 3 0 0 2 3

h  h h 2 =   d d3  d d  and c  c c

3 N0   N (particles do not "vanish", again no collisions supposed) h  f(,,)rp t d3 rp d 3 2 f ( rn, ,  ,) t d 3 r d  d  c3  f(r , p , t ) const, if no collisions = (rn , ,,t) d3 r d  d 

53 USM

d3p J( pp , ) d 3 p  , p   ( p , ,  ) cartesian Jacobi-det. spherical

px  p sin cos  z py  p sin sin  pz  p cos

p = |p|n  px  p x  p x   p       sin cos pcos  cos -psin  sin   py  p y  p y    J  det   = det sin sin pcos  sin psin  cos p        cos -psin  0      pz  p z  p z    p     = (exercise) p2 sin y x d3p dp dp dp  p 2 dpsin d  d  x y z    d

54 The specific intensity USM

Number of photons with  , +d  which propagate through summarized surface element dS into direction n and solid angle d, h4 3 Ich  f function of r , n , ,t at time t and with velocity c: c2 h3 2 N f (r ,n ,  , t ) d 3 r d d  c3 specific intensity is radiation energy, which is transported into direction n through surface d S , per frequency, time and solid angle. specific intensity is a distribution function, and the basic quantity in theory of radiative transfer

invariance of specific intensity h3 2 =3 f (r , n , , t ) cos  cdt dS d  d c Df (n ,dS ) since  0 without collisions (Vlasov equation) and  Dt without GR (i.e., F 0 ), we have  corresponding energy transport h4 3 I  f  E = h  N =f (r ,n , , t ) cos  dS d  dt d c2  I(,r n , ,) t specific intensity  I  const in fluid frame , as long as no interaction with matter! [erg cm-2 Hz-1 s-1sr-1] If stationary process, i.e. tthen nI = d/ds I = 0, where ds is path element, i.e. I = const also spatially! (this is the major reason for working with specific intensities) 55 s USM

s' s n

n n'

d' emitter receiver Invariance of specific intensity

Consider pencil of light rays which passes through both area elements (emitter) and ' (receiver).

If no energy sinks and sources in between, the amount of energy which passes through both areas is given by

E I cos  d dtd  E' I ' cos  ' d ' dtd ', and, cf. figure, specific intensity is radiation energy with  frequencies (,  dwhich is transported projected area cos 'd  ' through projected area element dcosθ into d   directionn per time interval dt and solid distance2r 2 angle d cosd  d '  r 2 E=Irn( ,  ,  ,)cos t  dddtd I  I ' , independent of distance ... but energy/unit area in pencil decreases with r 2 ! 56 Plane-parallel and spherical symmetries USM

stars = gaseous spheres => spherical symmetry

BUT: rapidly rotating stars (e.g., Be-stars, vrot ≈ 300 ... 400 km/s) rotationally flattened, only axis-symmetry can be used

AND: atmospheres usually very thin, i.e. Δr / R << 1

example: the sun

Rsun 700,000 km r (photo) 300 km

R => r / R  4 10-4 BUT corona r / R (corona) 

r

57 USM as long as r / R << 1 => plane-parallel symmetry

light ray through atmosphere

lines of constant temperature and density (isocontours) curvature of atmosphere insigni- significant curvature : ab ficant for photons' path : ab spherical symmetry

examples solar photosphere / cromosphere solar corona atmospheres of atmospheres of main sequence stars supergiants white dwarfs expanding envelopes (stellar winds) giants (partly) of OBA stars, M-giants and supergiants 58 Co-ordinate systems/symmetries USM

intensity has direction n into d Cartesian spherical n requires additional angles ,  with respect to

eeex, y , z eee ,  ,r and

 =  (ez , n )  = (er , n )

Ixyzt( , , , , , ) I  ( ,  , rt ,  , , ) p-p symmetry spherical symmetry independent of azimuthal direction,  reeexyzxyz reee   r r Izt( , , )  Irt  (,  ,)

e , e eeexyz , , right-handed, orthonorma l eee ,  , r z r specific intensity: Ixyzt( , , ,n , , ) Irt ( ,  , , n ,  , ) n important symmetries plane-parallel spherically symmetric physical quantities depend .... depend P only on z , e.g. only on r, e.g. I(rn , , , tIzt ) ( , n , , ) I ( rn , ,  , tIr )  ( , n , ,t ) P at ey, e r ex, e 59 0 Moments of the specific intensity USM

dA = d (r=1!)

ratio between area element (on sphere) and r2

In plane-parallel or spherical symmetry:

60 The Planck function USM

61 Planck function B()

max(T1)

Wien

 (T ) max 2 Rayleigh-Jeans

 

62 1st moment: radiative flux USM

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63 USM

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64 Effective temperature USM

• total radiative energy loss is flux (outwards directed) times surface area of star =

luminosity L = F + 4π R2

33 dim[L] = erg/s (units of power), Lsun=3.83 10 erg/s

• definition: “effective temperature” is temperature of a star with luminosity L at radius R*, if it were a black body (semi-open cavity?)

• Teff corresponds roughly to stellar surface temperature (more precise → later)

4 2 2 1/4 L =: σBTeff 4π R or Teff = (L/ σB 4π R )

65 USM

see exercise

66 2nd moment: radiation pressure (stress) tensor USM

67 USM

Das Bild kann nicht angezeigt werden. For symmetric tensors Tijij ( , ,  , r ) one can prove the following relations divergence of radiation pressure tensor (e.g., Mihalas & Weibel Mihalas, "Foundations of Radiation Hydrodynamics", Appendix ) 1(r2 T rr ) 1 ()T  fTfT ()()(r  r   T  T  ) gas pressure → r r 2 r r  here: radiative acceleration = volume forces exerted by 1r 1(sinT )  1 r  ()TfT  ()   fTTT ()(cot)  radiation field rrsin  r  ith component of divergence  1r 1 T   (Cartesian) ()T  fTfT ()()   fT (cot)  rsin r sin   where f are (different) functions of the tensor-elements which are not relevant here.

Since in spherical symmetry the radiation pressure tensor P is diagonal (i.e., symmetric),

and since puR and are functions of r alone, we have rr rr 1rr 2 P  1    P 1 rr   ()2Pr 2  rPr    ( PP  ) (2P  P P ) r r  r r r Prr p (which in the isotropic case would yield (P )   R ) r r  r  1T  1  (P ) 2 cos P  sin   2 cot  P  0 (in spherical symmetry) rsin   r

(P )  0 (in spherical symmetry).

Finally, we obtain

pR 1 1    ()()P Prer   22 p RRR  p  (3) p  u     r r 2   

pR 1  =er   (3pR  u )  , q.e.d. r r 

68 Summarizing comparison: USM from p-p to spherical symmetry specific intensity and moments similarly defined if z→r

I(z , ) I (r , ) with =cos and = (er , n ) [in the following,  - and t -dependence suppressed] from symmetry about azimuthal direction: 1 1 nth moment =  Ir(, ) nd  , as in p-p case when z  r ; n=0,1,2  J (),r Hr (), Kr () 2 1 0    flux(-density) F = 0 : only r-component different from zero, prop. to Eddington-flux   4 H  radiation stress tensor P : only diagonal elements different from zero only difference refers to divergence of radiation stress tensor,   P in pp-symmetry, only z-component different from zero, and p 4 P   R with p (radiation pressure scalar) = Kz ( ) z z R c in spherical symm e try, only r-component different from zero, and p3 p  u 4 P  R  R with u (radiation energy density) =J ( r ) r r r c

69 Chap. 4 ‒ Coupling with matter USM

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70 USM

71 The equation of transfer for specific geometries USM

so-called p-z geometry

General (without proof) for

72 Source function and Kirchhoff-Planck law USM

73 True absorption and scattering USM

"true" absorption radiation energy => thermal pool processes: if not TE, temperature T(r) is changed examples: photo-ionization bound-bound absorption with subsequent collisional de-excitation scattering: no interaction with thermal pool absorbed photon energy is directly reemitted (as photon) no influence on T(r) But direction n -> n' is changed (change in frequency mostly small)

examples: Thomson scattering at free electrons Rayleigh scattering at atoms and molecules resonance line scattering

ESSENTIAL POINT true processes: localized interaction with thermal pool, drive physical conditions into local equilibrium often (e.g., in LTE – page 127/130): ην(true) = κνBν(T) scattering processes: (almost) no influence on local thermodynamic properties of plasma propagate information of radiation field (sometimes over large distances) ην (Thomson) = σTH Jν (-> next page)

74 Thomson scattering USM

Motivation - energy conservation of scattering,

with    , and assuming isotropic scattering

absorbed energy/area element  Id  4 J      emitted energy/area element d   4   

   J 

75 Moments of the transfer equation USM

0th moment: frequency-dependent, stationary and static

F 4    J 

static: v=0 (or v << vsound) stationary: time-independent, ∂/∂t=0 76 USM

1st moment: frequency-dependent, stationary and static 1 P   F c 

The change in radiative pressure drives the flux!

static: v=0 (or v << vsound) stationary: time-independent, ∂/∂t=0 77 Summary: moments of the RTE USM

general case, 0th moment general case, 1st moment

4  1 1 JF    I d  F P    I  nd  c t    c2 t c  plane-parallel, stationary ( / t 0) and static (v 0)

dH dK     J   H dz   dz   spherically symmetric, stationary and (quasi-)static [no/negligible Dopplershifts  no winds or continuum problems(except for edges) Otherwise, opacities become angle-dependent (Doppler-shifts), and cannot be put in front of the integrals]

2 K3 KJ  1 (r H )       J  H  r2  r   r r

when frequency integrated, = -f when frequency integrated, = 0, if ONLY rad radiation energy transported: radiative equilibrium → (for stationary conditions) flux conservation

78 Chap. 5 – Radiative transfer: simple solutions USM

79 USM

80 USM

dt μ

81 Eddington-Barbier-relation USM em I () ≈ S(=

We “see” source function at location =(remember: radial quantity) (corresponds to optical depth along path / 

Generalization of principle that we can see only until = 1

i) spectral lines (as before)

for fixed  / is reached further out in lines (compared to continuum) line line cont cont => S ( /  S ( / => "dip" is created

ii) limb darkening for (central ray), we reach maximum in depth (geometrical) temperature / source function rises with  => central ray: largest source function, limb darkening

iii) "observable" information only from layers with  deepest atmospheric layers can be analyzed only indirectly 82 Solar limb-darkening USM Empirical temperature stratification

Das Bild kann nicht angezeigt werden. empirical temperature structure of solar photosphere by Holweger & Müller (1974)

(one absolute measurement

required, e.g., Bν(0))

83 Lambda operator USM

84 Diffusion approximation USM

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85 The Milne-Eddington model USM

(1929, ‘The formation of absorption lines’)

( Chap. 7)

(see problem sheet 6)

86 USM

• assume isothermal atmosphere, bν = 0 (possible, if gradient not too strong)

→ Jν(0) < Bν(0) !!!

87 Chap. 6 ‒ Stellar atmospheres USM

Basic assumptions

1. Geometry plane-parallel or spherically symmetric (-> Chap. 3)

2. Homogeneity atmospheres assumed to be homogenous (both vertical and horizontal)

BUT: sun with spots, granulation, non-radial pulsations ... white dwarfs with depth dependent abundances (diffusion) stellar winds of hot stars (partly) with clumping (<2> 2)

HOPE: "mean" = homogenous model describes non-resolvable phenomena in a reasonable way [attention for (magnetic) Ap-stars: very strong inhomogeneities!]

3. Stationarity vast majority of spectra time-independent => t = 0

BUT: explosive phenomena (supernovae) pulsations close binaries with mass transfer ...

88 Density stratification USM

89 Hydrodynamic description USM

Hydrodynamic description: inclusion of velocity fields Equation of continuity:  2 M (v )  0 r v const  (I) t 4  with (v)  0 Equation of momentum  stationarity, i.e.,   0 v p ("Euler equation" ) t v    g ext (II) and spherical symmetry , r  r r v ext i.e.,u  1  (r2 u )  (v  v) p  g 2 r r "advection term", t  r (from inertia) v( v )    v   v

I: Conservation of mass-flux II: "Equation of motion" Exercise: with gravity and radiative acceleration Show, by using the cont. eq.,

v p GM *  that the Euler eq. can   (rr)v()   () r 2 grRad ( )  rr   r  be alternatively written as v p or, to be compared with hydrostatic equilibrium (v  ) v   gext t  p GM*  v ()r 2  grRad ()   (r )v( r ) r r  r

hydrostatic equilibrium p GM*   ()z 2  gzRad ( )  in p-p symmetry:  z R* 

90 When is (quasi-)hydrostatic approach justified? USM

kB T 2 By using p   = vsound (equation of state, with  mean molecular weight, and v sound the isothermal sound speed ) , mH and M 4 r 2  vc  onst (for the hydrodynamic case) the equations of motion and of hydrostatic equilibrium can be rewritten:

2 2 2 2  dvsound 2v (r)  vsound v()r   (rg ) grav ( r )  g Rad ( r )    [hydrodynamic] rd r r  2 2  dvsound  vsound (z )g grav ( R * ) g Rad (z )   [hydrostatic, p-p] z dz 

Conclusion:

 for v << vsound, hydrodynamic density stratification becomes (“quasi”-) hydrostatic  this is reached in deeper photospheric layers, well below the sonic point, defined by v(rs)=vsound example: vsound (sun)  6 km/s, vsound (O-star)  20 km/s

Thus: p-p atmospheres using hydrostatic equilibrium give reasonable results even in the presence of winds as long as investigated features (continua, lines) are formed below the sonic point.

91 Barometric formula USM

( → Chap. 4)

92 Hydrostatic equilibrium USM

(Teff = 40,000 K, log g = 3.6)

93 Unified atmospheres ‒

USM density/velocity stratification for stars with winds photosphere + wind = unified atmosphere (Gabler et al. 1989)

Two possibilities: a) stratification from theoretical wind models [Castor et al. 1975, Pauldrach et al. 1986, WM-Basic (Pauldrach et al. 2001), see lecture part 2] Disadvantage: difficult to manipulate if theory not applicable or too simplified b) combine quasi-hydrostatic photosphere and empirical wind structure [PHOENIX (Hauschildt 1992), CMFGEN (Hillier & Miller 1998), PoWR (Gräfener et al. 2002), FASTWIND (Puls et al. 2005), see lecture part 2] Disadvantage: transition regime ill-defined

deep layers: at first  (r) calculated (quasi-hydrostatic, with ggrav (r ) and g rad (r)) M  v(r )  for v v (roughly: v < 0.1 v ) 4r2  () r  sound sound bR outer layers: at first v(r) = v (1 * )b , "beta-velocity-law", from observations/theory (b from transition velocity)  r M  (r )  4 r2 v() r transition zone: smooth transition from deeper to outer stratification

Input/fit parameters: M , v , b , location of transition zone

94 Unified atmospheres ‒

USM density/velocity stratification for stars with winds

abscissa: τRoss Rosseland optical depth (frequency averaged opacity, see below)

corresponds to log ρ = log m – log H

cores of strong lines

dotted: hydrostatic solid: unified model with thin wind dotted: hydrodynamic, from wind-theory

From Puls et al. (2008) al. et Puls From dashed: unified model with dense wind solid: unified model, with similar v∞ and β=0.8

NOTE: at same τ or m, wind-density (for v ≥ vsound ) lower than if in hydrostatic equilibrium 95 Plane-parallel or unified model atmospheres? USM

-2  Unified models required if τRoss ≥ 10 at transition between photosphere and wind (roughly at 0.1*vsound)

 rule of thumb using a typical velocity law (=1) R v M M  (  102 at 0.1 v ) 6108M yr  1    max Ross sound  10R 1000 kms1 

 if M (actual) < M  max for considered object, then (most) diagnostic features formed in quasi-hydrostatic part of atmosphere

→ plane-parallel, hydrostatic models possible for optical spectroscopy of late O-dwarfs and B-stars up to luminosity classes II (early subtypes) or Ib (mid/late subtypes)

 check required!

96 Eddington limit USM

97 Summary: stellar atmospheres – the solution principle

Solution of differential equations A and B by discretization Eq. of radiative transfer (B) differential operators => finite differences usually solved by the so-called all quantities have to be evaluated on suitable grid Feautrier and/or Rybicki scheme 98 Ray-by-ray solution ‒ USM p-z geometry for spherically symmetric problems

NOTE: the following method (based on Hummer & Rybicki 1971 ) works ONLY for spherically symmetric problems and no Doppler-shifts! a) define p-rays (impact-parameter) tangential to each discrete radial shell b) augment those by a bunch of (equidistant) p-rays resolving the core c) use only the forward hemisphere, i.e., 2 2  1 zdi rp d i and z di  0 41

all points zidi ,  1,NP, are located on the same r d -shell, i.e., have the same physical parameters such as emissivities, opacities, velocities, ... r4 (due to spherical symmetry, and neglect of Doppler-shifts) z4i -1 4i -1  Now one solves the RTE along each p-ray : from first principles, r4 dI (, z p )  i (r ) ( r ) I ( z , p ) (with '  ' for  0 and ' ' for  0) dz   i using appropriate boundary conditions (core vs. non-core rays), and standard methods (finite differences etc.) 4i  0  After being calculated, Izrpi (di ( d ), i ),  1,NP, samples the specific intensity at the same radius, r d , but at different angles,

zdi di, starting at  di  1 for i 1 and d 1,NZ (central ray, pi =0) to di  0 (tangent ray, where prid  and thus z di 0). rd

  In other words, along individual rd -shells, the specific intensities IrIz ( d , )  ( d , ) are sampled for all relevant  , and corresponding moments can be calculated by integration. 99 Feautrier-variables USM

 + - In fact, the RTE is not solved for I seperately, but for a linear combination of II  and , using the so-called Feautrier-variables u and v  ,which allows to construct a 2nd order scheme as in the plane-parallel case: higher accuracy, diffusion limit can be easily represented

1 u( z , p ) ( I+ ( z , p )  I - ( z , p )) mean intensity like 2   1 v (z , p ) ( I+ ( z , p )  I - ( z , p )) flux like 2  

v u   (S  u ),   v z   dz   2  u 2 u  S (2nd order, with d    dz ) 

... and corresponding boundary conditions inner boundary: for core rays, first order, using the diffusion approximation; for non-core rays, 2nd order, using symmetry arguments - outer boundary: either I (zmax , p ) 0, or higher order for optically thick conditions (e.g., shortward of HeII Lyman edge)

Formal solution for I ( ) (or u  (  ) and v  (  ) ) and corresponding angle-averaged quantities (momens t ) affected by inaccuracies, due to specific way of discretization, but ratios of moments much more precise (errors cancel to a large part)

100 Continuum transfer in extended atmospheres USM

Thus: variable Eddington-factor method solve the moments equations (only radius-dependent), and use Eddington-factors from formal solution to close the relations. Ensures high accuracy (since direct solution for angle-averaged quantities, and 2nd order scheme), whilst Eddington-factors (from the formal solution) quickly stablilize in the course of global iterations.

th st Using the 0 and 1 moment of the RTE and f KJ  / , we obtain 2 (r H ) 2 r( J  S  ) 

(fJ ) (3 f   1) J    H    r

r Introducing a "sphericality factor" q via ln( rq2 ) (3 f 1) /( rfrr ' ) d '  ln( 2 ), the 2nd equation becomes      core rcore 2 (fqrJ  ) 2 2  qr H  , and can be combined with the first one to yield a 2nd order scheme for r J 

2 2 2  (fqrJ  ) 1 2  (f J  ) 2 2 2  r( J S  ) with dX q  d [for comp.: in p-p , 2  (J S  ), limit for q   1 and r R* ] X q  101 Grey temperature stratification USM

102 USM

103 Rosseland opacities USM  thus, use additionally diffusion approximation

1 1 dB KB and H   3 3 d 1 1BdT  1  B  1  B d   d    d  1 H 03 T dz 0    T 0    T       dK / dz 1 BdT  B 4 B 3 R d  d T    03 T dz 0  T

  4   since B d BT4 B T 3        T  

 Rosseland opacity

4 B T 3    R  1 B   d 0  T

 can be calculated without radiative transfer

 harmonic weighting: maximum flux transport where  is small!

104 USM Now we define the stellar radius via  alternatively, from construction (for   1) 1 HH H H       R R(  2/3)  dK/ dz 1B 1 dT  B 1 4 B 3 dT * Ross R d  d  T   3  dz 03z 3 dz 0  T  as the average layer ('"stellar surface") where 16 dT the observed UV/optical radiation is created. i) F4 H  B T 3 3 d R ii) in spherical geometry Furthermore, if we approximate const  2/3 as in the (approx.) grey case, i.e.,

L( r) 16 B 3 dT 2  T (used for stellar structure) 4 4 3 4r3 R d r T(Ross ) T eff (  Ross  2/3) 4 4 iii) integrate i), + F  B Teff 3  T4 T 4 (  const ), as in grey case, but now with  then we obtain T(  2/3)  TRT ( )  eff4 Ross Ross Ross * eff 2 4 and the definition L 4 RT*  B eff has also a THUS possibility to obtain initial (or approx.) values for physical meaning (at least for LTE conditions): temperature stratification ( exact for large optical depths) "the effective temperature is the atmospheric temperature of a star at its surface".

calculate (LTE) opacities    Note: in reality, T ( Ross  2/3) deviates (slightly) calculate R ,  R  again, iteration required

 from Teff , since const  2/3, and because of calculate T ( R )  deviations from LTE 105 USM

(page 86)

106 H/He continuum of a hot star around 1000 Å USM

Predictions

Lyman cont: JB at 500Å) R)

Balmer cont: JB at 912Å)R)

Balmer cont. ( )  Lyman edge: JB Å)  ) Lyman cont.  R ( )  note: large opacity leads to very small effective T-gradient,

minimum value JB cf. page 7 H-Ly edge ()

107 Convection (simplified) USM

Why??? later

108 The Schwarzschild criterion USM

109 USM

≤ 1 in photosphere

3, and 3/16 ≈ 0.19 mono-atomic gas : 0.4, and  0.19 ad 16

ad as function of T and p 110 Note: • mixing length theory only 0th order approach Mixing length theory • modern approach: calculate consistent USM hydrodynamic solution (e.g., solar convective layer+photosphere, Asplund+, see ‘Intermezzo’)

radiative vs. adiabatic T-stratification

Model for solar photosphere radiative

radiative

convective, l = H convective

111 Mixing length theory – some details USM

T l 1  FCv    CvT a    , with conv p a iH 2 2 p a i E   CTp  is excess energy density delivered to ambient medium l mixing length parameter a = (from fits to observations, a =O(1)) when bubble merges with surroundings. H

Cp is specific heat per mass. The average velocity is calculated by assuming that the work done by the buoyant force is (partly) converted to kinetic energy, where the average of this  Fconv EvC  pT  v is convective flux (transported energy) work might be calculated via with v average velocity of rising bubble l / 2 over distance r ( v mass flux). w F(  rdr )( ),  b 0 T is temperature difference between bubble and ambient medium. and the upper limit results from averaging over elements passing the point under consideration. The buoyant force is given by (see page 108) dT  dT   T   r > 0 when convective instable,     Fb g  g(  i a )  0 dra   dr i   since then T  T  > 0  a  i  Using the equation of state, and accounting for pressure equilibrium (pi  pa ),  T  ln   From the definiton of  , we find  Q with Q  1  , to account for ionization effects.  T   lnT  dT T p    , with pressure scale height H (see problem set 8), dr H assuming hydrostatic equilibrium and neglecting radiation pressure;   dT  dT    Fb  g gQ  T  gQ       r T T dr dr (inclusion of prad possible, of course) a  i    gQ   r :  A  r . Thus, F is linear in  r , an d Defining l as the mixing length after which element dissolves, and averaging H a i b 2 l l / 2 l2 H l  w A rd( r) A  gQ  over all elements (distributed randomly over their paths), we may write r  .  a i   2 0 8 8 H  112 Mixing length theory – some details USM

Let's assume now that 50% of the work is lost to friction (pushing aside the turbulent elements), and 50% is converted into kinetic energy of the bubbles, i.e.,

1/ 2 1/ 2 1 1 2 w   gQH  1/ 2 w v  v      a  i  a, 22    8  and the convective flux is finally given by

1/ 2 gQH  3 / 2 2 Fconv   CT p   ai  a . 32 

NOTE : different averaging factors possible and actually found in different versions!

Remember that still ad  i  a  rad .

The gradients i and  a are calculated from the efficiency  and the condition that the total flux remains conserved (outside the nuclear energy creating core), i.e., L rF2( F )  rF 2  R2FR( )  R 2 T 4  conv rad tot ***rad B eff 4 or from the condition that L (F F )  r with L the luminosity at r . covn r ad 4 r 2 r Usually, a tricky iteration cycle is necessary. An example for a simple case will be discussed in problem set 8. 113 Convective vs. radiative energy transport USM

. major difference in internal structure at MS ‒ convective vs. radiative energy transport: • if T-stratification shallow (compared to adiabatic gradient) → radiative energy transport; • else convective energy transport . cool (low-mass stars) during MS: • interior: p-p chain, shallow dT/dr → radiative core • outer layers: H/He recombines → large opacities → steep dT/dr, low adiabatic gradient → convective envelope . hot (massive) stars during MS: • interior: CNO cycle, steep dT/dr → convective core • outer layers: H/He ionized → low opacities → shallow dT/dr, large adiabatic gradient → radiative envelope

Note: (i) transition from p-p chain to CNO cycle around 1.3 to 1.4 Msun at ZAMS (ii) most massive stars have a sub-surface convection zone due to iron opacity peak (iii) evolved objects (red giants and supergiants) and brown dwarfs are fully convective

convective envelope radiative envelope radiative convective core core

114 Chap. 7 Microscopic theory USM Absorption- and emission coefficients can calculate now a lot, if absorption- and emission-coefficients given, e.g.

lunl (r)

cross-section profile function occupation (quantum mech.) (normalized) numbers

LTE NLTE if necessary, "free-free" Saha- detailed next higher ion (Bremsstrahlung) Boltzmann calculation

absorption ionization edge h+ El -> Eu

u excited level emission spontaneous isotropic E -> E + h l excited level u l stimulated

ground level h+ Eu -> El + 2 h "bound-free" "bound-bound" (ionization/ (line transition) same angular distribution recombination) 115 Line transitions USM

116 USM

117 Line broadening USM

118 USM

atomic

119 USM

‘ ‘

fully drawn: Voigt profile H(a,v) dotted : exp(-v2 ), Doppler profile (core) dashed: a / (√π v2), dispersion profile (wings)

120 Curve of growth method USM

(result of advanced reading)

121 √π dv

 Voigt profile with AO = 0.5, b=:b

122 USM

* logb log bo /(   D )

123 USM

measure W() for different lines (with different strengths) of one ionization stage

5040E1l plot as function of log( gf l lu  )   log C , with "C" fit-quantity Te * n1 shift horizontally until theoretical curve of growth Wb  is matched => log C => 0 => n1 c

5040E1l log(gl f lu  )  Te

Ti I lines shifted horizontally to define a unique relation 124 Continous processes USM

125 USM

for ν << ν0

example: Rayleigh wings of Ly-alpha in metal-poor, cool stars (G/K-type, few electrons, thus few H‒, see next paragraph) become important opacity source, even in the optical 126 Ionization and Excitation USM

127 USM

Ionization

128 USM

LTE bf and ff opacities for hydrogen

Lyman edge

Balmer edge

1 1 N n    e a()T a2 () T a () T

/neutral atom

129 LTE and NLTE USM

(L)TE: for each process, there exists an inverse process with identical transition rate

LTE = detailed balance for all processes!

processes = radiative + collisional

collisional processes (and those which are essentially collisional in character, e.g., radiative recombination, ff-emission) in detailed balance, if velocity distribution of colliding particles is Maxwellian (valid in stellar atm., see below)

radiative processes: photoionization, photoexcitation (= bb absorption) in detailed balance only if radiation field Planckian and isotropic (approx. valid only in innermost atmosphere )

J(r) = W(r) B

J(r)

radiative processes couple regions with different temperatures, anisotropy as a function of frequency: 1 130 USM

Question: is f(v) dv Maxwellian?

elastic collisions -> establish equilibrium inelastic collisions/recombinations disturb equilibrium inelastic collisions: involve electrons only in certain velocity ranges, tend to shift them to lower velocities recombinations : remove electrons from the pool, prevent further elastic collisions

-5 -7 can be shown: in typical stellar plasmas, tel / trec ≈ 10 ... 10 ≈ tel / tinel => Maxwellian distribution

under certain conditions (solar chromosphere, corona), certain deviations in high- energy tail of distribution possible

Question: is T(electron) = T(atom/ion)?

equality can be proven for stellar atmospheres with 5,000 K < Te < 100,000 K

When is LTE valid???

roughly: electron collisions >> photoabsorption rates ∝ n T ½ ∝ I (T) ∝ Tx, x  1 however: e  NLTE- effects also in cooler LTE: T low, ne high dwarfs (giants), late B and cooler NLTE: T high, n low all supergiants + rest stars, e.g.. e iron in sun 131 TE – LTE – NLTE : a summary USM

TE LTE NLTE

velocity distribution of particles Maxwellian    (Te=Ti)

excitation Boltzmann   no

ionization Saha   no

B(T), except ff source function B(T) scattering only S = B(T) component

J  B (T) J  B(T),   equality only for radiation field J = B(T) dito 1/ 2 1       

132 Statistical equilibrium USM

133 Solution of the rate equations – a simple example USM

HAD: for each atomic level, the sum of all populations must be equal to the sum of all depopulations (for stationary situations) example: 3-niveau atom with continuum assume: all rate coefficients are known (i.e., also the radiation field) => rate equations (equations of statistical equilibrium)

nRCRCRC1 1kk  1 12 12 13 13 nRC 2( 21  21 ) nRC 3 ( 31  31 ) nRC kkk ( 1  1 ) 0

nRC1 ( 12 12 ) nRC 2 2kk  2 RCRC 21 21 23 23  nRC 3 ( 32  32 ) nRC kkk ( 2  2 ) 0

nRC1 ( 13 13 ) nRC 2 ( 23  23 ) nRC 3 3kk  3 RCRC 31 31 32 32  nRC kkk ( 3  3 ) 0

n1 ( R11221331kkC) nRC ( kk  ) nRC ( kkkkkkkkk  ) nRCRCRC 112233  0

with

Rij, radiative bound-bound transitions (lines!) C ij collisional bound-bound transitions

Rik radiative bound-free transitions (ionizations) C ik collisional bound-free transitions

Rki radiative free-bound transitions (recombinations) C ki collisonal free-bound transitions

in matrix representation =>

134 USM

(RCRCRC1kk 1 12 12 13 13 )() RC 21  21 () RC 31  31 ( RC kk 1  1 )  P  ()(RC12 12  RCRCRC 2kk 2 21 21 23 23 )() RC 32  32 () RC kk 2  2  ()RC13 13 ()( RC 23  23  RCRCRC 3kk 3 31 31 32 32 )( RC kk 3  3 )  ()RC1kk 1 () RC 2 kk  2 ()( RC 3 kk  3 RCR kkk 1 1 2Ck 2  R k 3  C k 3 )   rate matrix, diagonal elements sum of all depopulations

Rate matrix is singular, since, e.g., last row linear combination of other rows (negative sum n 0 1    of all previous rows) n  0  P * 2     THUS: LEAVE OUT arbitrary line (mostly the last one, corresponding to ionization  n3  0      equilibrium) and REPLACE by inhomogeneous, linearly independent equation for all ni, n4 ( nk )  0  to obtain unique solution particle number conservation for considered atom: N ni a k N H, with a k the abundance of element k i1

NOTE 1: numerically stable equation solver required, NOTE 2: occupation numbers ni depend on since typically hundreds of levels present, and (rate-) radiation field (via radiative rates), coefficients of highly different orders of magnitude and radiation field depends (non-linearly) on ni (via opacities and emissivities) => Clever iteration scheme required!!!! 135 Example for extreme NLTE condition Nebulium (= [OIII] 5007, 4959) in Planetary Nebulae mechanism suggested by I. Bowen (1927): low-lying meta-stable levels of OIII(2.5 eV) collisionally excited by free electrons (resulting from photoionization of hydrogen via "hot", diluted radiation field from central star) Meta-stable levels become strongly populated radiative decay results in very strong [OIII] emission lines impossible to observe suggested process in laboratory, since collisional deexitation (no photon emitted)) much stronger than radiative decay under terrestrial conditions. Thus, after detection new element proposed , "nebulium"

Condition for radiative decay

2 7 [OIII] NOTE: Aml  10 (typical values are 10 ) ratio 3:1

nAnnqTmml meml( e ), with metastable level m

 ne n e (crit), H a Aml 6 (, lm ) ne (crit) , qml  8.63  10 qml( T e ) gm T e

Hb (l , m ) collisional strength, order unity

H for typical temperatures Te  10,000K and [OIII] 5007, 5 -3 we have ne (crit) 4.9  10 cm , much larger than typical nebula densities 136