Emission and Absorption

Transcription

1 Stllar Atmosphrs: Emissio ad Absorptio Emissio ad Absorptio Stllar Atmosphrs: Emissio ad Absorptio Chmical compositio Stllar atmosphr mixtur, composd of may chmical lmts, prst as atoms, ios, or molculs Abudacs,.., i as mass fractios β k Solar abudacs β.7 β H H.8 β.4 C β. β.9 β O F M. M Uirsal abudac for Populatio I stars

2 Stllar Atmosphrs: Emissio ad Absorptio Chmical compositio Populatio II stars Chmically pculiar stars,.., hlium stars βh. << βh βh.964 >> βh βc.9 >> βc β. β β. < β O O βh βh βh βh β.l. β Z Z Chmically pculiar stars,.., PG59 stars βh.5 << βh β H.5 >> βh βc.55 >> βc β <. β.5 >> β O O Stllar Atmosphrs: Emissio ad Absorptio Othr dfiitios Particl umbr dsity k umbr of atoms/ios of lmt k pr uit olum rlatio to mass dsity: β ρ A k k m H k with A k ma mass of lmt k i atomic mass uits (AMU) m H mass of hydro atom Particl umbr fractio k k loarithmic k k umbr of atoms pr 6 Si atoms (mtorits) ε H lo( / ) +. 4

3 Stllar Atmosphrs: Emissio ad Absorptio Th modl atom Th populatio umbrs (occatio umbrs) i umbr dsity of atoms/ios of a lmt, which ar i th ll i E io Ery fr stats ioizatio limit boud stats, lls E i ry lls, quatizd E E(roud stat) E io ioizatio ry 5 Stllar Atmosphrs: Emissio ad Absorptio Photo absorptio cross-sctios Trasitios i atoms/ios E io Eri. boud-boud trasitios lis. boud-fr trasitios ioizatio ad rcombiatio procsss. fr-fr trasitios Brmsstrahlu W look for a rlatio btw macroscopic quatitis κ ( ), η ( ) ad microscopic (quatum mchaical) quatitis, which dscrib th stat trasitios withi a atom 6

4 Stllar Atmosphrs: Emissio ad Absorptio Photo absorptio cross-sctios ( ) Li trasitios: Ebb ± E Elow Boud-fr trasitios: thrmal ara of lctro locitis (Maxwll distributio, i.., lctros i thrmodyamic quilibrium) ( ) uboud stat io + fr lctro / m Ebf > Eth Eio Elow + Fr-fr trasitio: fr lctro i Coulomb fild of a io, Brmsstrahlu, classically: ump ito othr hyprbolic orbit, E ff arbitrary For all procsss holds: E ca oly b splid or rmod by: Ilastic collisios with othr particls (mostly lctros), collisioal procsss By absorptio/missio of a photo, radiati procsss I additio: scattri procsss (i)lastic collisios of photos with lctros or atoms - scattri off fr lctros: Thomso or Compto scattri - scattri off boud lctros: Raylih scattri 7 Stllar Atmosphrs: Emissio ad Absorptio Th li absorptio cross-sctio Classical dscriptio (H.A. Lortz) Harmoic oscillator i lctromatic fild Dampd oscillatios (-dim), i-frqucy ω Dampi costat γ Priodic xcitatio with frqucy ω by E-fild Equatio of motio: iωt mx &&+ γmx& + mω x E irtia + dampi + rstori forc xcitatio iωt Usual Asatz for solutio: x( t) x E iωt ( ω + iωγ + ω ) x m 8 4

5 Stllar Atmosphrs: Emissio ad Absorptio Th li absorptio cross-sctio E iωt ( ω + iωγ + ω ) x(t) x(t) m E m iωt ( ω ω + iωγ) ( ) E ω ω iωγ iωt xpad x(t) m ( ω ω ) + ω γ E ω ω γω ral part R(x(t)) cosωt + siωt m ( ω ω ) + ω γ ( ω ω ) + ω γ Elctrodyamics: radiatd powr p(t) (&& x) c E ω ω γω && x(t) ( ω )cos ωt + ( ω )siωt m ( ω ω ) + ω γ ( ω ω ) + ω γ E ( ω ω ) ω γ ( ω ω ) ω γω (x(t)) && cos ωt + cosωt siωt + si ωt m 9 Stllar Atmosphrs: Emissio ad Absorptio ara or o priod cos si /, ( x) & Th li absorptio coss-sctio ωt ωt cosωtsiωt E ω ( ω ω ) + γ ω m ( ( ω ) ) ω + γ ω 4 E ( ω x&& ) m ( ) + ω ω γ ω powr, arad or o priod p ( x& ) c ( ) E mc 4 4 ω ( ) + ω ω γ ω 4 E p ϕ ( )/ C Cormalizatio costat ( ω/π) mc 4 C ϕ ( ) profil fuctio + ( γ / π) 5

6 Stllar Atmosphrs: Emissio ad Absorptio Th li absorptio cross-sctio sic - <<, : + ( ) (( + )( )) 4 ( ) C C ϕ ( ) 4( ) ( γ / π) 4 ( ) ( γ / 4 π) + + ow: calculati th ormalizatio costat ϕ ( ) d 4π substitutio: x : ( ) γ + + C 4π dx π γ ϕ( ) d 4 C C γ + x γ π π Stllar Atmosphrs: Emissio ad Absorptio Th li absorptio cross-sctio ϕ ( ) Profil fuctio, Lortz profil Max 4 / γ γ / 4π ϕ( ) ( ) + ( γ / 4π ) FWHM proprtis: Symmtry: ϕ( + ( )) ϕ( ( )) Asymptotically: ϕ( ) ( ) FWHM: γ / 4π γ γ FWHM γ ( ) + ( γ / 4π ) 4π π FWHM Max 6

7 Stllar Atmosphrs: Emissio ad Absorptio Th dampi costat Radiatio dampi, classically (othr dampi mchaisms latr) Dampi forc ( frictio ) F γmx(t) & powrforc locity p ( t) γm( x& ( t) ) lctrodyamics p ( t) (& x ( t) ) c Hc, Asatz for frictioal forc is ot corrct Hlp: dfi γ such, that th powr is corrct, wh timarad or o priod: 4 iωt γmω ω (whr w usd x( t) x ) c ω ω ω γ classical radiatio dampi costat mc Stllar Atmosphrs: Emissio ad Absorptio Half-width Isrt ito xprssio for FWHM: γ 4π FWHM π mc λ c 4π λ mc FWHM FWHM 4 λfwhm FWHM.8 Å 4 7

8 Stllar Atmosphrs: Emissio ad Absorptio Th absorptio cross-sctio Dfiitio absorptio cofficit κ di κ ( ) I ds with low umbr dsity of absorbrs: κ ( ) σ ( ) low σ ( ) absorptio cross-sctio (dfiitio), dimsio: ara Sparati off frqucy dpdc: σ ( ) σ ϕ( ) Dimsio : ara frqucy σ ow: calculat absorptio cross-sctio of classical harmoic oscillator for pla lctromatic wa: E I x E iωt c (, µ ) E 8π δ ( ) δ ( µ ) 5 Stllar Atmosphrs: Emissio ad Absorptio Powr, arad or o priod, xtractd from th radiatio fild: 4 E π ω p ϕ ( ) with γ γclass. mc γ mc 4 E πmc E p ϕ ( ) ϕ ( ) mc 4π 8m c O th othr had: p σ ( ) I(, µ ) d dµ σ ( ) E 8 π µ c E Equati: σ ( ) E ϕ ( ) 8π 8m π σ ( ) ϕ ( ) σ.657 cm Hz mc Classically: idpdt of particular trasitio Quatum mchaically: corrctio factor, oscillator strth π σ lu f mc lu π κ ( ) low f luϕ( ) mc idx lu stads for trasitio lowr pr ll 6 8

9 Stllar Atmosphrs: Emissio ad Absorptio Oscillator strths Oscillator strths f lu ar obtaid by: Laboratory masurmts Solar spctrum Quatum mchaical computatios (Opacity Proct tc.) λ/å Li Ly α Ly β Ly γ H α H β H γ f lu low Allowd lis: f lu, Forbidd: <<.. H I s S ss S f lu -4 7 Stllar Atmosphrs: Emissio ad Absorptio Opacity status rport Cocti th (macroscopic) opacity with (microscopic) atomic physics Classical crossctio π κ ( ) low flow,ϕ ( ) 4444 mc σlow, Populatio umbr of lowr ll Profil fuctio QM corrctio factor Viw atoms as harmoic oscillator Eifrqucy: trasitio ry Profil fuctio: ractio of a oscillator to xtral drii (EM wa) Classical crossctio: radiatd powr dampi 8 9

10 Stllar Atmosphrs: Emissio ad Absorptio Extsio to missio cofficit Altrati formulatio by dfii Eisti cofficits: κ h 4π ϕ h π 4π mc ( ) low B lu ( ) i.. Blu f lu Similar dfiitio for missio procsss: η iducd η spotaous h BulIψ( ) 4π h Aulψ( ) 4π ψ ( ) profil fuctio, complt rdistributio: ϕ ( ) ψ ( ) 9 Stllar Atmosphrs: Emissio ad Absorptio Rlatios btw Eisti cofficits Driatio i TE; sic thy ar atomic costats, ths rlatios ar alid idpdt of thrmodyamic stat I TE, ach procss is i quilibrium with its irs, i.., withi o li thr is o tto dstructio or cratio of photos (dtaild balac) mittd itsity absorbd itsity h h h Bul I + Aul Blu Ilow TE: I B( T) 4π 4π 4π B B ( T) + A B B ( T) ( ) ul ul lu low ( ) B ( T) B B A low lu ul ul B T A B ul low lu ( ) B ul Bul

11 Stllar Atmosphrs: Emissio ad Absorptio Rlatios btw Eisti cofficits A B h kt ( ) ul low lu ( ) with Boltzma quatio: B ul B ul low low B T B T A B B ul B ul h kt ul low lu h kt ( ) compariso with Plack blackbody radiatio: h B ( T) c Aul h B c ul B B B low lu low lu ul Bul Stllar Atmosphrs: Emissio ad Absorptio Rlatio to oscillator strth B lu 4 π f mch lu 4 π B B f ul lu lu low low mch h 8 π Aul B ul f lu γ ul flu dimsio A ul tim c mc low low Itrprtatio of A ul as liftim of th xcitd stat ordr of maitud: at 5 Å: liftim: A ul γ ul 8 s 8 s

12 Stllar Atmosphrs: Emissio ad Absorptio Compariso iducd/spotaous missio Wh is spotaous or iducd missio stror? with I B spotaous η Ah ul * ψ( ) 4π * Aul h* c h * kt iducd * * η B( T ) Bul h* ψ( ) 4 π Bul B ( T ) c h* * h * kt * : h kt l *.. T K : λ A T * 5K : λ 46 A * * * o o At walths shortr tha λ spotaous missio is domiat ( ) Stllar Atmosphrs: Emissio ad Absorptio Iducd missio as ati absorptio Radiatio trasfr quatio: di spotaous iducd η κi with η η + η ds di spotaous iducd η + η κi ds h iducd h trasitio low κlu Blu lowϕ( ), ηlu Bul Iϕ( ) 4π 4π Usful dfiitio: κ corrctd for iducd missio: η di ds spotaous h η + ( B ul B lu low ) ϕ( ) I 4π π ϕ( ) low κlu f lu low mc spotaous h π low lu f lu c mc ϕ( ) So w t (formulatd with oscillator strth istad of Eisti cofficits): 4

13 Stllar Atmosphrs: Emissio ad Absorptio Th li sourc fuctio Gral sourc fuctio: S η κ Spcial cas: missio ad absorptio by o li trasitio: h lu Aul ϕ( ) lu η 4 h S π lu κ h ( Blulow Bul ) ϕ( ) c low - 4π S lu h c low low low ot dpdt o frqucy Oly a fuctio of populatio umbrs I LTE: h [ ] lu h kt S B (, T ) c 5 Stllar Atmosphrs: Emissio ad Absorptio Li broadi: Radiatio dampi Ery ry ll has a fiit liftim τ aaist radiati dcay (xcpt roud ll) τ A ul Simpl cas: rsoac lis (trasitios to roud stat) xampl Lyα (trasitio ): γ A γcl f γcl 8.4.γ cl xampl Hα ( ): l< u Hisbr ucrtaity pricipl: E τ h Ery ll ot ifiitly sharp q.m. profil fuctio Lortz profil γ + Auk + Al τ τ u l k< u < l γ γ f + f + f γ γ cl cl cl

14 Stllar Atmosphrs: Emissio ad Absorptio Li broadi: Prssur broadi Raso: collisio of radiati atom with othr particls Phas chas, disturbd oscillatio E()~ t iω t t tim btw two collisios 7 Stllar Atmosphrs: Emissio ad Absorptio Li broadi: Prssur broadi Raso: collisio of radiati atom with othr particls Phas chas, disturbd oscillatio E()~ t i t ω t tim btw two collisios Itsity spctrum (powr spctrum) of th cut wa trai: I ~ Fourir trasform t / iωt iω t I( ω)~ dt t / ω ω si t ω ω 8 4

15 Stllar Atmosphrs: Emissio ad Absorptio Li broadi: Prssur broadi Probability distributio for t t ( ) / τ Wt dt dt τ τ ara tim btw two collisios Arai or all t is ( ) ω ω t / t τ dt / ω ω I ( ω) cost si τ Prformi itratio ad ormalizatio is profil fuctio of itsity spctrum: πτ ϕ( ω) ( ω ω ) ( ) + τ i.. profil fuctio for collisioal broadi is a Lortz profil with - γ τ, τ~ particl dsity of collidrs γ γ γ approximatly costat (to calculat γ : calculatio of τ cssary; for that: assumptio about phas shift dd,.., i by smi-classical thory) 9 Stllar Atmosphrs: Emissio ad Absorptio Li broadi: Prssur broadi Smi-classical thory (Wisskopf, Lidholm), Impact Thory Phas shifts ω: p Asatz: ω Cp r, p,,4,6, r(t) distac to collidi particl fid costats C p by laboratory masurmts, or calculat p 4 6 am liar Stark ffct rsoac broadi quadratic Stark ffct a dr Waals broadi domiat at hydro-lik ios utral atoms with ach othr, H+H ios mtals + H Good rsults for p (H, H II): Uifid Thory H Vidal, Coopr, Smith 97 H II Schöi, Butlr 989 For p4 (H I) Film lo Barard, Coopr, Shamy; Barard, Coopr, Smith; Bauchamp t al. 5

16 Stllar Atmosphrs: Emissio ad Absorptio Thrmal broadi Thrmal motio of atoms (Dopplr ffct) Vlocitis distributd accordi to Maxwll, i.. for o spatial dirctio x (li-of-siht) Thrmal (most probabl) locity th : x x 4 ( T ) th kt ma.85 A km/s xampl: T 6K, A 56 (iro):. km/s x th x x x x w x ( x ) ~ i.. w ( ) C, with w ( ) d w obtai: x th x d x th d C π th C π th C C x w ( ) x π th x th / th m A x kt Stllar Atmosphrs: Emissio ad Absorptio Li profil th th Dopplr ffct:, c c profil fuctio: + C th wx( x) ϕ ( ), with ( )d w obtai: c ϕ π th Max ϕ ( ) ( ) th ϕ ( ) π th Li profil Gauss cur Symmtric about Maximum: th Half width: Tmpratur dpdcy: FWHM π l. FWHM th 67 th ~ T th Max th π 6

17 Stllar Atmosphrs: Emissio ad Absorptio Exampls At λ 5Å: T6K, A56 (F): λ th.å T5K, A (H): λ th.5å Compar with radiatio dampi: λ FWHM.8-4 Å But: dcli of Gauss profil i wis is much stpr tha for Lortz profil: 4 Gauss ( λ ) : I th li wis th Lortz profil is domiat th Lortz ( λ ) : rad 6 Stllar Atmosphrs: Emissio ad Absorptio Li broadi: Microturbulc Raso: chaotic motio (turbult flows) with lth scals smallr tha photo ma fr path Phomoloical dscriptio: Vlocity distributio: w x π x ( ) x micro micro i.., i aaloy to thrmal broadi micro is a fr paramtr, to b dtrmid mpirically Solar photosphr: micro. km/s 4 7

18 Stllar Atmosphrs: Emissio ad Absorptio Joit ffct of diffrt broadi mchaisms y y x x profil A + profil B oit ffct Mathmatically: coolutio commutati: multiplicatio of aras: Fourir trasformatio: f A ( f f A f B )( x) f A( y) f B ( x y) B f B f A ( f A f B )( x) dx f A( x) dx f B ( x) dx ~ ~ ~ i..: i Fourir spac th f A f B π f A f B coolutio is a 5 multiplicatio dy x Stllar Atmosphrs: Emissio ad Absorptio Applicatio to profil fuctios Coolutio of two Gauss profils (thrmal broadi + microturbulc) C x A x B B π G ( x) A π G ( x) B A x C G ( x) G ( x) G ( x) C π with C A + B A B Rsult: Gauss profil with quadratic summatio of half-widths; proof by Fourir trasformatio, multiplicatio, ad backtrasformatio Coolutio of two Lortz profils (radiatio + collisioal dampi) A/ π B/ π LA( x) L ( ) B x x + A x + B C / π LC( x) LA( x) LB( x) x + C with C A+ B Rsult: Lortz profil with sum of half-widths; proof as abo 6 8

19 Stllar Atmosphrs: Emissio ad Absorptio Applicatio to profil fuctios Cooli Gauss ad Lortz profil (thrmal broadi + dampi) ( ) γ /4π D G( ) L( ) π + γ π D ( ) ( ) /4 V G L dpds o,, γ, : V( ) G( ) L( )d Trasformatio: : ( ) a : γ/( 4 π ) y : ( ) D D D y y a / Dπ a G( y) L(y) V π y + a π π ( y) + a D Voit fuctio, o aalytical rprstatio possibl. (approximat formula or umrical aluatio) D D a Df: V H( a, ) with H( a, ) dy π π ( y) + a D ormalizatio: H ( a,) d π y dy 7 Stllar Atmosphrs: Emissio ad Absorptio Voit profil, li wis 8 9

20 Stllar Atmosphrs: Emissio ad Absorptio Tratmt of ry lar umbr of lis Exampl: boud-boud opacity for 5Å itral i th UV: Möllr Diploma thsis Kil Uirsity 99 Dirct computatio would rquir ry much frqucy poits Opacity Sampli Opacity Distributio Fuctios ODF (Kurucz 979) 9 Stllar Atmosphrs: Emissio ad Absorptio Boud-fr absorptio ad missio Eisti-Mil rlatios, Mil 94: Gralizatio of Eisti rlatios to cotiuum procsss: photoioizatio ad rcombiatio Rcombiatio spotaous + iducd Trasitio probabilitis: P : probability for photoioizatio i [, + d] F(): spotaous rcaptur probability of lctro i [, + d] G() : corrspodi iducd probability lctro locity I) umbr of photoioizatios low P I ddt II) umbr of rcombiatios ()[ F() + G() I ] ddt Photo ry h Eio + m d m h d I TE, dtaild balaci: I) II) 4

21 Stllar Atmosphrs: Emissio ad Absorptio Eisti-Mil rlatios ()[ () () ] with [ ] P I ddt F + G I h m ddt I B low PB () F() + G() B h m low F() lowpm h h kt G() () hg() c B F() h G() c Pm () hg() low h kt / low π mkt Eio kt low from Saha quatio: h low m m kt () : Maxwll distributio: () d 4π d π kt / 4 Stllar Atmosphrs: Emissio ad Absorptio Eisti-Mil rlatios P h G() m P h m low / h m / m m h low G() h lo w / h kt π mkt h low 4π 8π m h kt () Eio kt m π kt / m kt 4π Eisti-Mil rlatios, cotiuum aalos to A i, B i, B i 4

22 Stllar Atmosphrs: Emissio ad Absorptio Absorptio ad missio cofficits absorptio cofficit (opacity) missio cofficit (missiity) κ ( ) P h low low σ [ F() G() I ] h m η ( ) () / + dfiitio. of cross-sctio σ Ad aai: iducd missio as ati absorptio ad LTE: κ Ph G h m ( ) low () () / η ( )... κ ( ) η ( ) * G() h Ph M η ( ) κ ( ) B h / kt low ()... σ low low P m low h / kt σ c low low h P h kt [ h ] () F() h * / m (usi Eisti-Mil rlatios) 4 Stllar Atmosphrs: Emissio ad Absorptio Cotiuum absorptio cross-sctios H-lik ios: smi-classical Kramrs formula σ ( ) th th for > th σ ( ) ls 8h 8 thrshold frqucy, σ 7.96 cm π mc Z Z pricipal quatum umbr, Z uclar char th th Quatum mchaical calculatios yild corrctio factors σ ( ) σ ( ) th th bf (, ), bf (, ) Gaut factor Addi of boud-fr absorptios from all atomic lls: xampl hydro κ tot max bf ( ) σ bf ( ) 44

23 Stllar Atmosphrs: Emissio ad Absorptio Cotiuum absorptio cross-sctios Optical cotiuum domiatd by Pasch cotiuum 45 Stllar Atmosphrs: Emissio ad Absorptio Th solar cotiuum spctrum ad th H - io H - io has o boud stat, ioizatio ry.75 V Absorptio d ar 7Å, hc, ca pottially cotribut to opacity i optical bad + H 4 H 7.5 Su: T 6K, lo.6 Saha quatio:, H H H almost xclusily utral, but i th optical Pasch-cotiuum, i.. populatio of H() dcisi: H H H ( ) ( ) H ( ) H.V / kt H ( ) 8 H H.4 6 ( ) ( ) 6 5 Boud-fr cross-sctios for H - ad H ar of similar ordr H - boud-fr opacity thrfor domiats th isual cotiuum spctrum of th Su 8 46

24 Stllar Atmosphrs: Emissio ad Absorptio Th solar cotiuum spctrum ad th H - io Ioizd mtals dlir fr lctros to build H - 47 Stllar Atmosphrs: Emissio ad Absorptio Th solar cotiuum spctrum ad th H - io 48 4

25 Stllar Atmosphrs: Emissio ad Absorptio Th solar cotiuum spctrum ad th H - io 49 Stllar Atmosphrs: Emissio ad Absorptio Scattri procsss Thomso scattri at fr lctros Absorptio cofficit κ σ follows from powr of harmoic oscillator ( Thomso cross-sctio) σ 4 4 E p mc ( ) ( γ π) + fr lctros: o rsoac frqucy, o frictio: ; γ 4 E c p, o th othr had w had p σ E mc 8 π 4 8π 5 σ 6.65 cm 4 mc Thomso cross-sctio is walth-idpdt 5 5

26 Stllar Atmosphrs: Emissio ad Absorptio Scattri procsss Raylih scattri of photos o lctros boud i atoms or molculs 4 4 E p mc ( ) ( γ π) + smi-classical: << lu 4 4 E c p o th othr had w had p σ 4 R E mc lu 8π π σr f 4 4 lu σflu 4 mc lu lu 4 κr( ) lσflu 4 l lu (hr w ha icludd th oscillator strth as th quatum mchaical corrctio) Raylih scattri o Lyα importat for stllar spctral typs G ad K 5 Stllar Atmosphrs: Emissio ad Absorptio Rama scattri Discord i symbiotic oa RR Tl Rama scattri of O VI rsoac li (Schmid 987) irtual ll 5Å 6Å /8Å Rama-scattrd li 685/78Å λ λ OVI λ Ly α Schmid 989, Espy t al

27 Stllar Atmosphrs: Emissio ad Absorptio Two-photo procsss 5 Stllar Atmosphrs: Emissio ad Absorptio Fr-fr absorptio ad missio Assumptio (also alid i o-lte cas): Elctro locity distributio i TE, i.. Maxwll distributio ff ff ff S ( ) η ( ) / κ ( ) B (, T) Fr-fr procsss always i TE Similar to boud-fr procss w t: ff κ σff k 6 ( ) ( ) h / kt ( ) 6π Z σff ( ) / ff (,,T) hc(πm) T ralizd Kramrs formula, with Gautfaktor from q.m. Fr-fr opacity importat at hihr ris, bcaus lss ad lss boud-fr procsss prst Fr-fr opacity importat at hih tmpraturs σ ~ T, but σ ~ T (Saha), thrfor: κ / κ T / / ff bf ff bf 54 7

28 Stllar Atmosphrs: Emissio ad Absorptio Computatio of populatio umbrs Gral cas, o-lte: I LTE, ust ( ρ, T, I ) i i ( ρ, T) i i I LTE compltly i by: Boltzma quatio (xcitatio withi a io) Saha quatio (ioizatio) 55 Stllar Atmosphrs: Emissio ad Absorptio Driatio i txtbooks Boltzma quatio ( E )/ statistical wiht i E kt i i i E xcitatio ry Othr formulatios: Rlatd to roud stat (E ) i i Ei / kt i Rlatd to total umbr dsity of rspcti io i i i i E / kt i, with partitio fuctio U( T): UT ( ) i E / kt 56 8

29 Stllar Atmosphrs: Emissio ad Absorptio Dirc of partitio fuctio.. hydro:, E E i i i Io i.. lim i lim i E/kT i i ormalizatio ca b rachd oly if umbr of i lls is fiit. Vry hihly xcitd lls caot xist bcaus of itractio with ihbouri particls, radius H atom: r ( ) a At dsity 5 atoms/cm ma distac about -5 cm r( max ) -5 cm max ~4 Lls ar dissold ; dscriptio by cocpt of occatio probabilitis p i (Mihalas, Hummr, Däpp 99) p with i p i i i i wh i lim i 57 Stllar Atmosphrs: Emissio ad Absorptio Hummr-Mihalas occatio probabilitis 58 9

30 Stllar Atmosphrs: Emissio ad Absorptio Saha quatio Driatio with Boltzma formula, but pr stat is ow a -particl stat (io plus fr lctro) Ery: E Eio + p m plctro momtum) Statistical wiht: G( p) wiht of io * wiht of fr lctro Isrt ito Boltzma formula Statistical wiht of fr lctro umbr of aailabl stats i itral [p,p+dp] (Pauli pricipl): dω( p) ( p) G( p) ( Eio + p m Elow )/ kt low low ( E Elow )/ kt p mkt G( p) dp phas spac olum low low ( ) spis h phas spac cll Ω ( ) x y z 4π 4 π ( ) 8π G p dp Summariz or all fial stats By itratio or p d p dxdydz dp dp dp dv p dp p dp G p p h 59 Stllar Atmosphrs: Emissio ad Absorptio Saha quatio Isrtio ito Boltzma formula is: p dp with x p/ mkt h ( E Elow )/ kt 8π p mkt low low ( E / Elow )/ kt 8π x ( ) low h low m kt x dx ( E Elow )/ kt 8π h / π mkt low low h ( m kt) ( E Elow )/ kt Saha quatio for two lls i adact ioizatio stas / π 4 Altrati: low / T ( E Elow ) / kt f ( T ) C.7 C low 6 K / cm 6

31 Stllar Atmosphrs: Emissio ad Absorptio Exampl: hydro Modl atom with oly o boud stat: (H I roud stat) low I I (H II ) II II I II / T 5.58 K/ T f ( T) C pur hydro:, + II ioizatio dr: x ft ( ) f( T) f T x x x x ( ) + f ( T) f( T) f( T) x + + x x( T, ) II I II 6 Stllar Atmosphrs: Emissio ad Absorptio Hydro ioizatio Ioizatio dr x Tmpratur / K 6

32 Stllar Atmosphrs: Emissio ad Absorptio Mor complx modl atoms,...,j ioizatio stas i,...,i() lls pr ioizatio sta Saha quatio for roud stats of ioizatio stas ad +: / + h πm kt + E Io / kt With Boltzma formula w t occatio umbr of i-th ll: E kt i i i / / EIo / kt i + CT i i + + C T / ( E Io i E ) / kt + 6 Stllar Atmosphrs: Emissio ad Absorptio Mor complx modl atoms Rlatd to total umbr of particls i ioizatio sta + i+ + i i+ + i + U + i + U C T U + / ( E + Io i + E ) / kt U + + i + U i + + C T / ( E Io i E ) / kt / + + i U U U i i C T C T U i + / E / E Io + Io C T / kt / kt i Φ / ( E Ei / kt i ( T ) Io i E ) / kt U + + C T / E Io / kt U 64

33 Stllar Atmosphrs: Emissio ad Absorptio Ioizatio fractio K + J- J + + J J J J- J- J- J- J J K+ L J J J J- J + J- K J + + J J J- J- J- J K+ L J- Φ k( T) k J J- + Φ ( T) m k m k J J J- J 65 Stllar Atmosphrs: Emissio ad Absorptio Ioizatio fractios 66

34 Stllar Atmosphrs: Emissio ad Absorptio Summary: Emissio ad Absorptio 67 Stllar Atmosphrs: Emissio ad Absorptio Li absorptio ad missio cofficits (boud-boud) π low h π low κlu ( ) flu low ϕ( ) ηlu ( ) f lu ϕ( ) mc c mc ϕ( ) profil fuctio,.., Voitprofil V( a,) π Cotiuum (boud-fr) * κ () σ low low h /kt D y ( y) + a * h h /kt ( ) c low η σ dy Cotiuum (fr-fr), always i LTE ff κ σff k -h / kt ( ) ( ) ( ) - ff ff η κ k ( ) ( ) B (,T) Scattri (Compto, o fr lctros) κ σ η () σ J Total opacity ad missiity add all cotributios, th sourc fuctio S η /κ( ) 68 4

35 Stllar Atmosphrs: Emissio ad Absorptio Excitatio ad ioizatio i LTE low low ( Elow E )/ kt Boltzma / π mkt ( E Elow )/ kt low low h Saha 69 5