Stellar Atmospheres. Lecture on stellar atmospheres

Transcription

1 Wlcom to th Lctur on stllar atmosphrs Stfan Drzlr Unrstäts-Strnwart, Unrsty of Göttngn Grmany Stllar Atmosphrs Outln: Introducton Radaton fld Radaton transfr Emsson and absorpton Radat qulbrum Hydrostatc qulbrum Stllar atmosphr modls

2 Stllar Atmosphrs: Motaton Stllar Atmosphrs: Ltratur Dmtr Mhalas Stllar Atmosphrs, W.H. Frman, San Francsco Albrcht Unsöld Physk dr Strnatmosphärn, Sprngr Vrlag (n Grman) Rob Ruttn Lctur Nots Radat Transfr n Stllar Atmosphrs Stllar Atmosphrs: Motaton Why physcs of stllar atmosphrs? Physcs Stllar atmosphrs as laborators Astronomy Spctral analyss of stars Plasma-, atomc-, and molcular physcs, hydrodynamcs, thrmodynamcs Structur and oluton of stars Basc rsarch Galaxy oluton Tchncal applcaton Eoluton of th Unrs

3 Stllar Atmosphrs: Motaton Magntc flds n wht dwarfs and nutron stars Shft of spctral lns wth ncrasng fld strngth 3 Stllar Atmosphrs: Motaton Hrtzsprung Russll Dagram R 4 6 W L~R T 4 ff R 7Km 58K. R 4

4 Stllar Atmosphrs: Motaton Mass stars 5 Stllar Atmosphrs: Motaton Chmcal oluton of th Galaxy Carrtta t al., AJ 4,

5 Stllar Atmosphrs: Motaton SN mo 7 Stllar Atmosphrs: Motaton SN Ia 8 4

6 Stllar Atmosphrs: Motaton SN Ia cosmology Ω M, Ω Λ,.5,.5,.5, -.5,,, Rdshft z 9 Stllar Atmosphrs: Motaton SN Ia Kosmolog Ω Λ 5

7 Stllar Atmosphrs: Motaton Uranum-Thorum clock Stllar Atmosphrs: Motaton Stllar atmosphr dfnton From outsd sbl, obsrabl layrs of th star Layrs from whch radaton can scap nto spac Dmnson Not stllar ntror (optcally thck) No nbula, ISM, IGM, tc. (optcally thn) But: chromosphrs, corona, stllar wnds, accrton dsks and plantary atmosphrs ar closly rlatd topcs 6

8 Stllar Atmosphrs: Motaton Sonn 3 Stllar Atmosphrs: Motaton Fraunhofr lns 4 7

9 Stllar Atmosphrs: Motaton Spctrum - schmatcally Intnsty Walngth / nm 5 Stllar Atmosphrs: Motaton Spctrum formaton 6 8

10 Stllar Atmosphrs: Motaton Formaton of absorpton lns Intror outr boundary obsrr contnuum ln cntr contnuum stllar atmosphr ntnsty 7 Stllar Atmosphrs: Motaton Ln formaton / stllar spctral typs spctral ln tmpratur structur flux tmpratur dpth / km walngth ntror 8 9

11 Stllar Atmosphrs: Motaton Th spctral typs on th man squnc O B A F G K M A7 O5 O7 B4 B6 A A5 A8 A9 F3 F8 G G5 G8 9 Stllar Atmosphrs: Motaton D Spktraltypn dr Hauptrh O B A F G K M F6 A7 F8 F3 G F8 G6 G G5 G8 G9 K4 K5

12 Stllar Atmosphrs: Motaton Classfcaton schm M9 L3 L5 L8 Stllar Atmosphrs: Motaton Classfcaton schm T dwarfs

13 Stllar Atmosphrs: Motaton Stllar atmosphr dfnton From outsd sbl, obsrabl layrs of th star Layrs from whch radaton can scap nto spac Dmnson Not stllar ntror (optcally thck) No nbula, ISM, IGM, tc. (optcally thn) But: chromosphrs, corona, stllar wnds, accrton dsks and plantary atmosphrs ar closly rlatd topcs 3 Stllar Atmosphrs: Motaton Optcal tlscops Calar Alto (Span) 3.5m tlscop 4

14 Stllar Atmosphrs: Motaton Optcal tlscops ESO/VLT 5 Stllar Atmosphrs: Motaton UV / EUV obsratons Why s t mportant? flux walngth / Å flux walngth / Å 6 3

15 Stllar Atmosphrs: Motaton UV/optcal tlscops HST 7 Stllar Atmosphrs: Motaton X-ray tlscops XMM 8 4

16 Stllar Atmosphrs: Motaton Gamma-ray tlscops INTEGRAL 9 Stllar Atmosphrs: Motaton Infrard obsrators JWST ISO 3 5

17 Stllar Atmosphrs: Motaton Sub-mm tlscops 3 Stllar Atmosphrs: Motaton Rado tlscops m dsh at Efflsbrg 3 6

18 Stllar Atmosphrs: Motaton Stllar atmosphr dfnton From outsd sbl, obsrabl layrs of th star Layrs from whch radaton can scap nto spac Dmnson Not stllar ntror (optcally thck) No nbula, ISM, IGM, tc. (optcally thn) But: chromosphrs, corona, stllar wnds, accrton dsks and plantary atmosphrs ar closly rlatd topcs 33 Stllar Atmosphrs: Motaton PN NGC675 - HST 34 7

19 Stllar Atmosphrs: Motaton Plantary nbula spctrum 35 Stllar Atmosphrs: Motaton ISM spctrum 36 8

20 Stllar Atmosphrs: Motaton Quasar + IGM spctrum 37 Stllar Atmosphrs: Motaton Stllar atmosphr dfnton From outsd sbl, obsrabl layrs of th star Layrs from whch radaton can scap nto spac Dmnson Not stllar ntror (optcally thck) No nbula, ISM, IGM, tc. (optcally thn) But: chromosphrs, corona, stllar wnds, accrton dsks and plantary atmosphrs ar closly rlatd topcs 38 9

21 Stllar Atmosphrs: Motaton Eta Carna - HST 39 Stllar Atmosphrs: Motaton Stllar wnd spctrum 4

22 Stllar Atmosphrs: Motaton Formaton of wnd spctrum (P Cygn ln profls) 4 Stllar Atmosphrs: Motaton Stllar wnds P Cyg profls 4

23 Stllar Atmosphrs: Motaton Accrton dsks 43 Stllar Atmosphrs: Motaton AM CVn dsk spctrum modls 44

24 Stllar Atmosphrs: Motaton Tmpratur structur of an accrton dsk Hght [km] Dstanc from star [km] 45 Stllar Atmosphrs: Motaton Plantary atmosphrs 46 3

25 Stllar Atmosphrs: Motaton Quanttat spctral analyss what can w larn? Shap of ln profl: Tmpratur Flm Dnsty Flm Abundanc Flm Rotaton Turbulnc Magntc fld Ln poston: Chmcal composton Vlocts Rdshft Tmporal araton: Companon Surfac structur Spots Pulsaton 47 Stllar Atmosphrs: Motaton Zman ffct 48 4

26 Stllar Atmosphrs: Motaton L Magntc flds optcal spctrum l spctrum of a wht dwarf (PG ) wth fld strngth of about 5 MG crcular polarzaton poston of ln componnts 49 Stllar Atmosphrs: Motaton L Magntc flds optcal spctrum Wht dwarf Grw B3MG Crcular polarzaton poston of ln componnts 5 5

27 Stllar Atmosphrs: Motaton Extrasolar plants 5 Stllar Atmosphrs: Motaton Vlocty flds ~.Å Solar dsk Dstanc / km dstanc / km Walngth / Å tm / mn 5 6

28 Stllar Atmosphrs: Motaton Non-radal pulsaton mods 53 Stllar Atmosphrs: Motaton Tm rsold spctroscopy 54 7

29 Stllar Atmosphrs: Motaton Tm dpndnt ln profls Flux Tm 55 Stllar Atmosphrs: Motaton Dopplr tomography 56 8

30 Stllar Atmosphrs: Motaton Summary stllar atmosphrs thory Th atmosphr of a star contans lss than on bllonth of ts total mass, so, why do w car at all? Th atmosphr of a star s that what w can s, masur, and analyz. Th stllar atmosphr s thrfor th sourc of nformaton n ordr to put a star from th color-magntud dagram (.g. B-V,m ) of th obsrr nto th HRD (L,T ff ) of th thortcan and, hnc, to dr th thory of stllar oluton. Atmosphr analyss ral lmnt abundancs and show us rsults of cosmo-chmstry, startng from th arlst momnts of th formaton of th Unrs. Hnc, workng wth stllar atmosphrs nabls a tst for bg-bang thory. Stars ar th buldng blocks of galaxs. Our undrstandng of th most dstant (hnc most arly mrgd) galaxs, whch cannot b rsold n sngl stars, s not possbl wthout knowldg of procsss n atmosphrs of sngl stars. Work on stllar atmosphrs s a bg challng. Th atmosphr s that rgon, whr th transton btwn th thrmodynamc qulbrum of th stllar ntror nto th mpty blacknss of spac occurs. It s a rgon of xtrm non-qulbrum stats. 57 Stllar Atmosphrs: Motaton Summary stllar atmosphrs thory Important sourc of nformaton for many dscplns n astrophyscs rsarch for pur knowldg, contrbuton to our cultur ambalnt applcatons (.g. nuclar wapons) Applcaton of drs dscplns physcs numrcal mthods Stll a ry act fld of rsarch, many unsold problms.g. dynamcal procsss 58 9

31 Stllar Atmosphrs: Th Radaton Fld Th Radaton Fld Stllar Atmosphrs: Th Radaton Fld Dscrpton of th radaton fld Macroscopc dscrpton: Spcfc ntnsty I (, nrt r, r, ) as functon of frquncy, drcton, locaton, and tm; nrgy of radaton fld (no polarzaton) n frquncy ntral (,+d) n tm ntral (t,t+dt) n sold angl dω around n through ara lmnt dσ at locaton r n 4 r r I (, n,, t) : d E d dt dω dσ

32 Stllar Atmosphrs: Th Radaton Fld Th radaton fld ϕ n r dω ϑ da dσ 3 Stllar Atmosphrs: Th Radaton Fld Rlaton I I λ Enrgy n frquncy ntral (,+ ) Enrgy n walngth ntral (λ,λ+ λ) I I λ.. d 4 E I λ da cosϑ dt dλ dω thus wth I d I dλ λ d c c c λ c I I I I d λ λ λ λ Dmnson Unt I nrgy ara tm frq. sold angl rg cm s Hz strad I λ nrgy ara tm walngth sold angl rg o cm s A strad 4

33 Stllar Atmosphrs: Th Radaton Fld Inaranc of spcfc ntnsty Irradatd nrgy: de I ( ϑ, ) d cosϑda dω da as sn from da subtnds sold angl dω dω cos ϑ da / d cos ϑ da cos ϑ da de I ( ϑ, ) d d now, da as sn from da cos ϑ da cos ϑ da de I ( ϑ, ) d d f no sourcs or snks along d: de de I I Th spcfc ntnsty s dstanc ndpndnt f no sourcs or snks ar prsnt. 5 Stllar Atmosphrs: Th Radaton Fld Irradanc of two ara lmnts ϑ d ϑ da da 6 3

34 Stllar Atmosphrs: Th Radaton Fld Spcfc Intnsty Spcfc ntnsty can only b masurd from xtndd objcts,.g. Sun, nbula, plants Dtctor masurs nrgy pr tm and frquncy ntral de I cos ϑ dω A.g. A s th dtctor ara dω ~ ( ) s th sng dsk 7 Stllar Atmosphrs: Th Radaton Fld Spcal symmtrs Tm dpndnc unmportant for most problms In most cass th stllar atmosphr can b dscrbd n plan-paralll gomtry Sun: atmosphr km << radus 7 km 35 µ : cos ϑ I I (, µ, z) z ϑ 8 4

35 Stllar Atmosphrs: Th Radaton Fld For xtndd objcts,.g. gant stars (xpandng atmosphrs) sphrcal symmtry can b assumd sphrcal coordnats: Cartsan coordnats: I (, µ, r) I (, p, z) rr outr boundary z ϑ p r ϑ 9 Stllar Atmosphrs: Th Radaton Fld Intgrals or angl, momnts of ntnsty Th -th momnt, man ntnsty r J I ( n) dω wth sphrcal coordnats 4π 4π π π / I sn ϑ d ϑ dφ wth µ : cosϑ 4π π / π I dµ dφ 4π In cas of plan-paralll or sphrcal gomtry J I d µ nrgy rg ara tm frquncy / / cm s Hz 5

36 Stllar Atmosphrs: Th Radaton Fld J s rlatd to th nrgy dnsty u radatd nrgy through ara lmnt da durng tm dt : de I d dt dω da l c dt dv l da c dt da hnc, th nrgy contand n olum lmnt dv pr frquncy ntral s udvd I dω d dtda 4πJ u 4π c J 4π 4π c nrgy rg olum frquncy 3 cm Hz total radaton nrgy n olum lmnt: u u d J d dv / c d nrgy rg olum 3 cm dv l da dω Stllar Atmosphrs: Th Radaton Fld, x Th st momnt: radaton flux ( ) ω z r r r F I n n d 4π propagaton ctor n sphrcal coordnats: snϑcosφ r n snϑsnφ cosϑ F I( ϑφ, )snϑcosφsn ϑdϑdφ n plan-paralll or sphrcal gomtry: F F, F F π I ( µ ) µ dµ, x, y, z nrgy rg ara tm frquncy / / cm s Hz ϑ y ϕ x 6

37 Stllar Atmosphrs: Th Radaton Fld Manng of flux: Radaton flux ntto nrgy gong through ara z-axs Dcomposton nto two half-spacs: F π I( µ ) µ dµ π I( µµ ) dµ + π I( µµ ) dµ π I( µµ ) dµ π I( µµ ) dµ F F + ntto outwards - nwards Spcal cas: sotropc radaton fld: F Othr dfntons: F astrophyscal flux H Eddngton flux F πf 4 πh 3 Stllar Atmosphrs: Th Radaton Fld Ida bhnd dfnton of Eddngton flux In -dmnsonal gomtry th n-th momnts of ntnsty ar -th momnt: st momnt: nd momnt: J ( ) I µ dµ H ( ) I µ µ dµ K I( ) d µ µ µ n-th momnt: I( ) d n µ µ µ 4 7

38 Stllar Atmosphrs: Th Radaton Fld Ida bhnd dfnton of astrophyscal flux Intnsty aragd or stllar dsk astrophyscal flux ϑ n r prsnϑ R p p Rsnϑ R ( µ ) dp p R µ dµ pdp R d µ µ 5 Stllar Atmosphrs: Th Radaton Fld Ida bhnd dfnton of astrophyscal flux Intnsty aragd or stllar dsk astrophyscal flux R I I ( ) p π pdp π R I ( ) R d R µ π µ µ π + + F / π F F no nward flux at stllar surfac I F p dp 6 8

39 Stllar Atmosphrs: Th Radaton Fld Flux at locaton of obsrr I R d dω f r r r F I ( n) n dω 4π Flux at dstant obsrr's dtctor normal to th ln of sght: R f I d I R d F d R d * * ω π / πf 7 Stllar Atmosphrs: Th Radaton Fld Total nrgy radatd away by th star, lumnosty Intgral or frquncy at outr boundary: + F F d F d Multpld by stllar surfac ara ylds th lumnosty L 4πR F 4π R F 6π R H * * * nrgy tm rg s 8 9

40 Stllar Atmosphrs: Th Radaton Fld Th photon gas prssur Photon momntum: Forc: p E / c dp dt F de c dt cosϑ Prssur: dp P( ) c F de cosϑ da c dt da I cos ϑ dω d c I 4π π cos ϑ dω c 4π I µ dµ K c Isotropc radaton fld: I ( µ ) I J 4π I 4π P( ) u I P( ) u J 3K c 3 c 3 9 Stllar Atmosphrs: Th Radaton Fld Spcal cas: black body radaton (Hohlraumstrahlung) Radaton fld n Thrmodynamc Equlbrum wth mattr of tmpratur T r I (, n, r, t) I ( ) I B(, T) bzw. Iλ Bλ(, T) r n caty: F J I B 3 h h B (, T) xp c kt hc hc B ( λ, T) xp 3 λ λkt hc hc Bλ ( λ, T) xp 5 λ λkt 5 h h Bλ (, T) xp 3 c kt

41 Stllar Atmosphrs: Th Radaton Fld Hohlraumstrahlung Stllar Atmosphrs: Th Radaton Fld Asymptotc bhaour In th rd Raylgh- Jans doman In th blu Wn doman h << kt k T B (, T ) c ckt B ( λ, T ) 4 λ h >> kt h h xp + kt kt h xp kt xp 3 h h B (, T ) xp c kt hc hc B ( λ, T ) xp 3 λ λkt h kt

42 Stllar Atmosphrs: Th Radaton Fld max Wn s law 3 d d h h B (, T) xp d d c kt 3 x x B + x d B 3 x max / d xmax xmax ( ) xmax ( ) x 3 h max numrcal soluton: xmax.8 λmaxt.5 cm dg kt d B xmax λ x max 5( ) dλ hc numrcal soluton: xmax λmaxt.897 cm dg λ kt max x:h/kt 3 Stllar Atmosphrs: Th Radaton Fld Stfan-Boltzmann law Intgraton or frquncs 3 h h BT ( ) B( T) d xp d c kt k 4 x π k 4 T 3 dx T x 3 5 ch ch 4 π 5 σ π k T π 5 ch wth rg cm s dg 3 Enrgy dnsty of blackbody radaton: 4π u J c 4π ( ) d B( T ) c 4σ T c 4 4

43 Stllar Atmosphrs: Th Radaton Fld Stars as black bods ffct tmpratur Surfac as opn caty (... physcally nonsns) + I B, I B for µ > I for µ σ 4 wth F B and F B( T) T π lumnosty: L 4π R F 4σπR T 4 * * ( σπ ) /4 / 4 / ff * hnc, ff. tmpratur: T 4 L R Attnton: dfnton dpndnt on stllar radus! 5 Stllar Atmosphrs: Th Radaton Fld Stars as black bods ffct tmpratur 6 3

44 Stllar Atmosphrs: Th Radaton Fld Exampls and applcatons Solar constant, ffct tmpratur of th Sun f ( ) d f.36 kw/m.36 rg s cm Sun s cntr F f wth d.5 cm R 6.69 cm π R* F T 4 ff d - -. rg s cm flux at solar surfac π F Tff 578 K σ max max 7.4 K Planck maxmum at λ 3.4Å ( B ) or λ.å ( B ) wth Å.4 kv maxmum 4 kv T c λ 7 Stllar Atmosphrs: Th Radaton Fld Exampls and applcatons Man squnc star, spctral typ O R R, T 6 K * * ff * 4 * T ff R * ff L 6 L* L L T R λ 88Å ( B ) or λ 5Å ( B ) max max λ Intrstllar dust T K, λmax.3mm ( B ) 3K background radaton T.7 K, λmax.9mm ( B ) 8 4

45 Stllar Atmosphrs: Th Radaton Fld Radaton tmpratur... s th tmpratur, at whch th corrspondng blackbody would ha qual ntnsty hc I ( λ) λ hc hc hc 3 xp rad ln + 3 ktrad kλ λ I T Comfortabl quantty wth Kln as unt Oftnusdn radoastronomy 9 Stllar Atmosphrs: Th Radaton Fld Th Radaton Fld - Summary - 3 5

46 Stllar Atmosphrs: Th Radaton Fld Summary: Dfnton of spcfc ntnsty 4 de I(,n,r,t): rr d dt d ω dσ ϕ n r dω ϑ da dσ 3 Stllar Atmosphrs: Th Radaton Fld Summary: Momnts of radaton fld In -dm gomtry (plan-paralll or sphrcally symmtrc): -th momnt: J ( ) I µ dµ Man ntnsty st momnt: H ( ) I µ µ dµ Eddngton flux nd momnt: K ( ) I µ µ dµ K-ntgral F astrophyscal flux H Eddngton flux F flux F πf 4 πh nrgy dnsty total flux at stllar surfac 4π c u u d J d F Fd Fd + stllar lumnosty L 4πR F 4π R F 6π R H * * * 3 6

47 Stllar Atmosphrs: Th Radaton Fld Summary: Momnts of radaton fld 4π prssur of photon gas P( ) K c 3 h h blackbody radaton B (, T) xp c kt Wn s law λ T max constant σ Stfan-Boltzmann law BT ( ) B ( T) d T π 4σ nrgy dnsty of blackbody radaton u T c ffct tmpratur L 4π R F 4π R B 4σπR T 4 * * * ff

48 Stllar Atmosphrs: Radaton Transfr Radaton Transfr Stllar Atmosphrs: Radaton Transfr Intracton radaton mattr Enrgy can b rmod from, or dlrd to, th radaton fld Classfcaton by physcal procsss: Tru absorpton: Tru msson: Scattrng: photon s dstroyd, nrgy s transfrrd nto kntc nrgy of gas; photon s thrmalzd photon s gnratd, xtracts kntc nrgy from th gas photon ntracts wth scattrr drcton changd, nrgy slghtly changd no nrgy xchang wth gas

49 Stllar Atmosphrs: Radaton Transfr Exampls: tru absorpton and msson photoonzaton (bound-fr) xcss nrgy s transfrrd nto kntc nrgy of th rlasd lctron ffct on local tmpratur photoxctaton (bound-bound) followd by lctron collsonal d-xctaton; xctaton nrgy s transfrrd to th lctron ffct on local tmpratur photoxctaton (bound-bound) followd by collsonal onzaton rrs procsss ar xampls for tru msson 3 Stllar Atmosphrs: Radaton Transfr Exampls: scattrng procsss -ll atom absorbs photon b wth frquncy, r-mts photon wth frquncy ; frquncs not a xactly qual, bcaus lls a and b ha non-anshng nrgy wdth Dopplr ffct bcaus atom mos Scattrng of photons by fr lctrons: Compton- or Thomson scattrng, (anlastc or lastc) collson of a photon wth a fr lctron 4

50 Stllar Atmosphrs: Radaton Transfr Fluorscnc Nthr scattrng nor tru absorpton procss c b c-b: collsonal d-xctaton b-a: radat a 5 Stllar Atmosphrs: Radaton Transfr Chang of ntnsty along path lmnt gnrally: di ds plan-paralll gomtry: sphrcal gomtry: di I dr I dµ + ds r ds µ ds di ds di ds I I µ µ + r µ r di µ wth dt µds dt dr dr ds cosϑ µ ds rdϑ sn( ϑ + dϑ) snϑ ds dµ dµ dϑ snϑ ( snϑ) ds dϑ ds r ( µ ) / r 6 3

51 Stllar Atmosphrs: Radaton Transfr di ds outr boundary Plan-paralll gomtry di µ wth dt µds dt gomtrcal dpth t dt ϑ ds 7 Stllar Atmosphrs: Radaton Transfr Sphrcal gomtry ϑ+dϑ dr ϑ ds -rdϑ di ds I r di ds dr I dµ + ds µ ds I I µ µ + r µ r -dϑ dr dr ds cosϑ µ ds rdϑ sn( ϑ + dϑ) snϑ ds dµ dµ dϑ snϑ( snϑ ) ds dϑ ds r ( µ ) / r 8 4

52 Stllar Atmosphrs: Radaton Transfr Chang of ntnsty along path lmnt gnrally: di ds plan-paralll gomtry: sphrcal gomtry: di ds I r di ds dr I dµ + ds µ ds I I µ µ + r µ r di di µ wth dt µds ds dt dr dr ds cosϑ µ ds rdϑ sn( ϑ + dϑ) snϑ ds dµ dµ dϑ snϑ ( snϑ) ds dϑ ds r ( µ ) / r 9 Stllar Atmosphrs: Radaton Transfr Rght-hand sd of transfr quaton No absorpton (acuum) di I const. naranc of ntnsty ds Absorpton only, no msson I ds I + di nrgy rmod from ray: s proportonal to nrgy contnt n ray: and to th path lmnt: ds de di I d dt dω dσ d dt dω dσ 5

53 Stllar Atmosphrs: Radaton Transfr Absorpton coffcnt thus: di κi ds κ absorpton coffcnt, opacty dmnson: /lngth unt: cm - but also oftn usd: mass absorpton coffcnt,.g., pr gram mattr κ n gnral complcatd functon of physcal quantts T, P, and frquncy, drcton, tm... r r κ κ, n,, t ( ) oftn thr s a coordnat systm n whch κ sotropc,.g. co-mong fram n mong atmosphrs κ κ r, ( ) countr-xampl: magntc flds (Zman ffct) Stllar Atmosphrs: Radaton Transfr only absorpton, plan-paralll gomtry gomtrcal dpth t outr boundary dt ϑ ds di( µ, t) di κ (,) ti( µ,) t µ ( µ,) t κ (,) ti( µ,) t ds dt wth optcal dpth d : κdt (, t) κ (, t ) dt wth at t di ( µ, ) I ( µ, ) d µ t t / µ (, ) ntgraton constant, fxd by I µ c c boundary alus 6

54 Stllar Atmosphrs: Radaton Transfr Schustr boundary-alu problm I outr boundary max nnr boundary + I µ < : I I µ > : I I + + ( µ, ) c ( µ, ) I ( µ, ( µ, ) I max + / µ ( µ, ) ) c ( µ, c max max / µ / µ ) ( max ) / µ 3 Stllar Atmosphrs: Radaton Transfr Exampl: homognous mdum.g. glass fltr κ ( t, µ ) κ κ t κ d d thcknss of fltr max I ( µ, ) I ( µ, ) + + ( κ d )/ µ max I ( µ, ) I ( µ, ) max κ d / µ I + I - d/µ s d/ µ s 4 7

55 Stllar Atmosphrs: Radaton Transfr s κ s/ / : Half-wdth thcknss / Matral Rr watr Wndow glass Cty ar Glas fbr S / / mtr Solar atmosphr 5 Stllar Atmosphrs: Radaton Transfr Physcal ntrprtaton of optcal dpth What s th man pntraton dpth of photons nto mdum? p( ) d I ( )/ I ( ) (mathmatcally: xpctaton alu of probablty functon p( ) ) [ d ] p( ) d : probablty for absorpton n ntral, + / µ / µ d x { d not normalzaton: p( ) d µ µ / µ x d x dx µ µ µ µ man pntraton dpth t f s f man fr path µ µ κ κ 6 8

56 Stllar Atmosphrs: Radaton Transfr Th rght-hand sd of th transfr quaton transfr quaton ncludng msson I ds I + di Enrgy addd to th ray: s proportonal to path lmnt: msson coffcnt η de + di ds di η ds d dt dω dσ dmnson: ntnsty / lngth unt: rg cm -3 strad - 7 Stllar Atmosphrs: Radaton Transfr Th rght-hand sd of th transfr quaton Transfr quaton ncludng msson η n gnral a complcatd functon of physcal quantts r r T,P,..., and frquncy η η, n,, t ( ) η s not sotropc n n statc atmosphrs, but s usually assumd to sotropc (complt rdstrbuton) f constant wth tm: ( ) η η r, 8 9

57 Stllar Atmosphrs: Radaton Transfr Th complt transfr quaton di ds η κ ( ) I Dfnton of sourc functon: di ds κ ( ) ( S I ) S η κ ( ) Plan-paralll gomtry di (, µ, t) κ (, t) dt Sphrcal gomtry ( S (, µ, t) I (, µ, t) ) µ I (, µ, r) µ I + r r (, µ, r) κ(, r) µ ( S (, µ, r) I (, µ, r) ) µ 9 Stllar Atmosphrs: Radaton Transfr Soluton wth gn sourc functon: Formal soluton Plan-paralll cas di di ( I S ) or: + I S d µ d µ µ lnar st-ordr dffrntal quaton of form y + f ( x) y g( x) x has th ntgratng factor M ( x) xp f ( x) dx x x und thus th soluton y(x) g(x)m(x)dx + C Cy(x ) M(x) x (proof by nsrton) n our cas: x f ( x) / µ g ( x) / µ S y( x) I ( ) ( )

58 Stllar Atmosphrs: Radaton Transfr Formal soluton for I + Rfrnc pont x : max for I + (µ> ) outgong radaton µ µ µ µ µ µ µ µ max max max max max max )xp ( )xp ( ) ( ) ( )xp ( - xp ) ( xp - xp ) ( max max max I d S I I d S I d M wghtd man or sourc functon xponntally absorbd ngong radaton from nnr boundary I + + +µ pn pont /µ Hnc, as rough approxmaton: ) ( ) ( µ + + S I S Stllar Atmosphrs: Radaton Transfr Formal soluton for I - Rfrnc pont x : for I - (µ< ) ngong radaton + + µ µ µ µ µ µ µ µ () xp )xp ( ) ( () )xp ( - xp ) ( xp - xp ) ( I d S I I d S I d M wghtd man or sourc functon xponntally absorbd ngong radaton from outr boundary

59 Stllar Atmosphrs: Radaton Transfr Emrgnt ntnsty max + d + max () ( )xp + ( max )xp µ µ µ I S I for sm-nfnt atmosphrs: : max + d I () S( )xp µ µ max + hnc, approxmatly: I () S ( µ ) Eddngton-Barbr-Rlaton Rlaton s xactly ald f sourc functon s lnar n :.. wth S( ) S + S and x : / µ w ha: + x x I () S dx+ S µ x dx S + S µ S ( µ ) 3 Stllar Atmosphrs: Radaton Transfr Th sourc functon In thrmodynamc qulbrum (TE): for any olum lmnt t s: absorbd nrgy mttd nrgy pr scond pr scond κi dsdσdωd η dsdσdωd κb η Krchhoff s law η S B κ Th local thrmodynamc qulbrum (LTE): w assum that max r r S (, ) B (, T ( )) z.b. I + () B ( T ( ))xp d µ µ Local tmpratur, unfortunatly unknown at th outst In stllar atmosphrs TE s not fulflld, bcaus Systm s opn for radaton T(r) const (tmpratur gradnt) 4

60 Stllar Atmosphrs: Radaton Transfr Sourc functon wth scattrng Exampl: thrmal absorpton + contnuum scattrng (Thomson scattrng of fr lctrons) dω r r r κ ( ) χ( ) + σ( ) η χb + σ R, n ;, n I, n d tru absorpton η χb + σj scattrng ( ) ( ) 4π rdstrbuton functon r r sotropc, cohrnt: R n n χb + σj S ρj + ( ρ) B wth ρ σ /( σ + χ) χ + σ Insrtng nto formal soluton: (, ;, ) δ (, ) + d d I () ( ρ) B xp + ρj xp µ µ µ µ dω I (, µ ) ntgral quaton for I 4π 5 Stllar Atmosphrs: Radaton Transfr Th Schwarzschld-Mln quatons Exprssons for momnts of radaton fld obtand by ntgraton of formal soluton or angls µ -th momnt J( ) (, ) I µ dµ d d J ( ) ( )xp ( + )xp dµ S dµ S µ µ µ µ (wrttn for sm-nfnt atmosphr wthout rradaton from outsd) dw dw xchang ntgrals ( S, ndpndnt of µ ) w, m dµ m µ dµ µ w dw dw J ( ) ( ) xp ( ( )) ( ) xp ( ( + )) d S w w d S w w w w dw dw J ( ) ( ) xp ( ( )) + ( ) xp( ( )) d S w d S w w w 6 3

61 Stllar Atmosphrs: Radaton Transfr Th Schwarzschld-Mln quatons -th momnt dw dw J( ) ( ) xp ( ( )) ( ) xp( ( )) d S w + d S w w w J( ) S( ) E( ) d + S( ) E( ) d - xt wth E (x): t dt xponntal ntgral of st ordr J ( ) ( ) ( ) S E d Karl Schwarzschld (94) 7 Stllar Atmosphrs: Radaton Transfr Th Lambda oprator Dfnton Λ [ f () t ] f () t E ( t ) dt J ( ) Λ ( S ) In analogy, w obtan th Mln quatons for th st momnt H ( ) S( t) E( t ) dt S( t) E( t) dt ( S) Φ 4 nd momnt K ( ) S ( t) E t dt Χ S ( ) ( ) 3 4 xt wth En( x) dt n t 8 4

62 Stllar Atmosphrs: Radaton Transfr LTE Strct LTE ( T ( )) J ( ) ΛB Includng scattrng S ρj J ( ) ΛρJ + ( ρ) B + Λ( ( T ( )) ρ) B ( T ( )) Intgral quaton for ( ) J Sol J ( ) S ( ) I ( ) H ( ) 4ΦS K ( ) 4 ΧS ( ) ( ) 9 Stllar Atmosphrs: Radaton Transfr Excurson: xponntal ntgral functon s Chandraskhar: Radat Transfr III.8 For classcal LTE atmosphr modls, >5% of computaton tm s ndd to calculat E n (x) In non-lte modls, E n (x) s ndd to calculat lctron collsonal rats Rcurson formula n xt ntgraton by parts E ( x) t dt ( n) xt ( n) xt wth product rul n( ) ( ) n n n E x t t x dt ( n) x ( n) xt + x t dt n n x En( x) xen ( x) for n> n E ( x) E ( x) n 3 5

63 Stllar Atmosphrs: Radaton Transfr Excurson: xponntal ntgral functon dffrntaton d n d xt n xt ( n) xt En( x) t ( ) dt t ( t) dt t dt dx dx d E n( x ) E n( x ) n > dx x xt xt xt ( ) ( ) ( ) d d E x t dt t t dt dx dx x x d dx E ( x) x x 3 Stllar Atmosphrs: Radaton Transfr ntgrals Excurson: xponntal ntgral functon s l x E n(x)dx s l+ s s l+ l x x n n n l+ l+ s l+ s l+ x rpatd ntgraton by parts x E (x)dx E (x) E (x)dx for s x x d E(x) n + E n(x)dx tc. untl E( x) l+ l+ dx x l+ l+ l+ n s s x E n(s) + E n(s) + L+ E() s l+ ( l+ )( l+ ) ( l+ )( l+ ) L( l+ n) + ( l + )( l + ) L( l + n) l+ n x l l+ n x n L s x dx ( l+ n )! ( l+ n)! l! l! xe(x)dx x dx ( l+ )( l+ ) ( l+ n) ( l+ )( l+ ) L( l+ n) ( l+ n)! l+ n 3 6

64 Stllar Atmosphrs: Radaton Transfr Excurson: xponntal ntgral functon asymptotc bhaour x xt x xt 6 L 3 t x x t x x x x x : E ( x) dt + dt + + L u u u xt du du du x : E + ( x) dt t u u u x x u u u du du du du ( ) + + ( ) u u u u x E ( x) γ ln x+ ( γ L Eulr s constant srs xpanson for th ntgral: n n x E ( x) γ ln x+ ( ) n nn! x E > E n E E n n Valus at : n() n() (), 3() x x u ) du u 33 Stllar Atmosphrs: Radaton Transfr Exampl: lnar sourc functon S( ) a + b J( ) Λ S (a + b )E ( ) d a E ( ) d b E ( ) d +. J( ) a + b+ [ be 3( ) ae ( ) ] H() b+ [ ae() 3 be() 4 ] 3... on can show ths x Conclusons: >> : En / x J a + b S Th man ntnsty approachs th local sourc functon H b /3 Th flux only dpnds on th gradnt of th sourc functon 34 7

65 8 Stllar Atmosphrs: Radaton Transfr 35 Momnts of transfr quaton Plan-paralll gomtry -th momnt st momnt µ S I d di Ldµ (I) ) ( ) ( µ µ µ µ µ µ S J H d d d S d I d I d d Lµdµ (II) ) ( ) ( µ µ µ µ µ µ µ µ H K d d d S d I d I d d Stllar Atmosphrs: Radaton Transfr 36 ( ) ( ) ( ) (I) κ κ κ κ µ µ µ µ κ µ κ µ µ µ µ µ J S H r r r J S H r H r J S d I r I r H r d I d S d I r d I r Momnts of transfr quaton Sphrcal gomtry -th momnt Ldµ ( ) ),, ( ),, ( ), ( ),, ( ),, ( r I r S r r I r r r I µ µ κ µ µ µ µ µ +

66 9 Stllar Atmosphrs: Radaton Transfr 37 ( ) (II) κ κ κ µ µ µ µ µ µ κ µ µ κ µ µ µ µ µ µ H r J r K K r H K J r K r H d I r I r K r d I d S d I r d I r Momnts of transfr quaton st momnt Lµdµ Stllar Atmosphrs: Radaton Transfr 38 Soluton of momnt quatons Problm: n-th momntum quaton contans (n+)-st momnt always on mor unknowns than dffrntal quatons to clos th systm, anothr quaton has to b found Closur by ntroducton of arabl Eddngton factors Eddngton factor, s found by traton J f K f nw startng stmat for ( ) ( ), sol / f I II K f K J +

67 Stllar Atmosphrs: Radaton Transfr ( I) ( II) Soluton of momnt quatons dh J S d d( f J ) H d dffrntal qs. for f J, H Start: approxmaton for, assumpton: ansotropy small,.. substtut I by J (Eddngton approxmaton) 3 K ( ) I µ dµ J µ dµ J µ J 3 3 K f J Stllar Atmosphrs: Radaton Transfr Eddngton approxmaton Is xact, f I lnar n µ (on can show by Taylor xpanson of S n trms of B that ths lnar rlaton s ry good at larg optcal dpths) I() µ I +µ I J I ( µ )dµ I 3 µ H I ( µ ) µ dµ I I µ µ µ µ 3 3 K I ( ) d I I K J 3 4

68 Stllar Atmosphrs: Radaton Transfr Summary: Radaton Transfr 4 Stllar Atmosphrs: Radaton Transfr Transfr quaton di η κ ( ) I ds Emsson and absorpton coffcnts η, κ ( ) Dfntons: sourc functon optcal dpth S η / κ ( ) d κ ds Formal soluton of transfr quaton max + d + max ( ) ( )xp + ( max )xp µ µ µ I S I + Eddngton-Barbr rlaton I () S( µ ) LTE Local Thrmodynamc Equlbrum r r S (, ) B (, T( )) T( r ) local tmpratur Includng scattrng: S ρj + ( ρ) B wth ρ σ /( σ + χ) 4

69 Stllar Atmosphrs: Radaton Transfr Schwarzschld-Mln quatons Momnt quatons of formal soluton J ( )Λ( S ) H ( ) Φ( S ) Λ, Φ ntgral oprators 4 dh d( K) Momnts of transfr quaton (plan-paralll) J S H d d Dffrntal quaton systm (for J,H,K), closd by arabl Eddngton factor f : K / J 43 Summary: How to calculat I and th momnts J,H,K (wth gn sourc functon S)? Stllar Atmosphrs: Radaton Transfr Sol transfr quaton di d µ ( I S) (no rradaton from outsd, sm-nfnt atmosphr, drop frquncy ndx) + d Formal soluton: I ( ) S( )xp ( µ >), I analogous µ µ How to calculat th hghr momnts? Two possblts: n. Insrt formal soluton nto dfntons of J,H,K: Iµ dµ J( )Λ( S) H( ) Φ( S) K ( ) Χ( S ) 4 4 Schwarzschld-Mln quatons. Angular ntgraton of transfr quaton,.. -th & st momnt dh d( K ) J S H momnt quatons for 3 quantts J,H,K d d Elmnat K by Eddngton factor f: K f J n... µ dµ dh d( f J ) J S H sol: J,H,K nw f (K/J) traton d d 44

70 Stllar Atmosphrs: Emsson and Absorpton Emsson and Absorpton Stllar Atmosphrs: Emsson and Absorpton Chmcal composton Stllar atmosphr mxtur, composd of many chmcal lmnts, prsnt as atoms, ons, or molculs Abundancs,.g., gn as mass fractons β k Solar abundancs β.7 β H H.8 β.4 C β. N β.9 β O F M. M Unrsal abundanc for Populaton I stars

71 Stllar Atmosphrs: Emsson and Absorpton Chmcal composton Populaton II stars β β Chmcally pcular stars,.g., hlum stars β. << β β H H.964 >> β β.9 >> β C β.3 β N β. < β O N O H H H H H β.l. β H C Z β β Z Chmcally pcular stars,.g., PG59 stars β.5 << β β H H.5 >> β β.55 >> β C β <. N β.5 >> β O H H C O 3 Stllar Atmosphrs: Emsson and Absorpton Othr dfntons Partcl numbr dnsty N k numbr of atoms/ons of lmnt k pr unt olum rlaton to mass dnsty: β ρ A k k m H N k wth A k man mass of lmnt k n atomc mass unts (AMU) m H mass of hydrogn atom Partcl numbr fracton k N k logarthmc k k Numbr of atoms pr 6 S atoms (mtorts) ε N H log( N / N ) +. 4

72 Stllar Atmosphrs: Emsson and Absorpton Th modl atom Th populaton numbrs (occupaton numbrs) n numbr dnsty of atoms/ons of an lmnt, whch ar n th ll E on Enrgy fr stats onzaton lmt bound stats, lls E nrgy lls, quantzd E E(ground stat) E on onzaton nrgy 5 Stllar Atmosphrs: Emsson and Absorpton Photon absorpton cross-sctons Transtons n atoms/ons E on Enrg 3. bound-bound transtons lns. bound-fr transtons onzaton and rcombnaton procsss 3. fr-fr transtons Brmsstrahlung W look for a rlaton btwn macroscopc quantts κ ( ), η ( ) and mcroscopc (quantum mchancal) quantts, whch dscrb th stat transtons wthn an atom 6 3

73 Stllar Atmosphrs: Emsson and Absorpton Photon absorpton cross-sctons ( ) Ln transtons: Ebb ± Eup Elow Bound-fr transtons: thrmal arag of lctron locts (Maxwll dstrbuton,.., lctrons n thrmodynamc qulbrum) ( ) unbound stat on + fr lctron / m Ebf > Eth Eon Elow + Fr-fr transton: fr lctron n Coulomb fld of an on, Brmsstrahlung, classcally: jump nto othr hyprbolc orbt, E ff arbtrary For all procsss holds: E can only b suppld or rmod by: Inlastc collsons wth othr partcls (mostly lctrons), collsonal procsss By absorpton/msson of a photon, radat procsss In addton: scattrng procsss (n)lastc collsons of photons wth lctrons or atoms - scattrng off fr lctrons: Thomson or Compton scattrng - scattrng off bound lctrons: Raylgh scattrng 7 Stllar Atmosphrs: Emsson and Absorpton Th ln absorpton cross-scton Classcal dscrpton (H.A. Lorntz) Harmonc oscllator n lctromagntc fld Dampd oscllatons (-dm), gn-frquncy ω Dampng constant γ Prodc xctaton wth frquncy ω by E-fld Equaton of moton: ωt mx &&+ γmx& + mω x E nrta + dampng + rstorng forc xctaton ωt Usual Ansatz for soluton: x( t) x E ωt ( ω + ωγ + ω ) x m 8 4

74 Stllar Atmosphrs: Emsson and Absorpton Th ln absorpton cross-scton E ωt ( ω + ωγ + ω ) x(t) xpand x(t) x(t) m E m ωt E m ωt Elctrodynamcs: radatd powr ( ω ω + ωγ) ( ω ω ωγ) p(t) (&& x) 3 3 c E ω ω γω && x(t) ( ω )cos ωt + ( ω )snωt m ( ω ω ) + ω γ ( ω ω ) + ω γ ( ) ( ) ( ω ω ) + ω γ E ω ω γω ral part R(x(t)) cosωt + snωt m ( ω ω ) + ω γ ( ω ω ) + ω γ E ω ω ω γ ω ω ω γω (x(t)) && cos ωt + cosωt snωt + sn ωt m N N N 9 Stllar Atmosphrs: Emsson and Absorpton arag or on prod cos sn /, ( x) & Th ln absorpton coss-scton ωt ωt cosωtsn ωt E ω ( ω ω ) + γ ω m ( ( ω ) ) ω + γ ω ( x&& ) m ( ω ω ) + γ ω powr, aragd or on prod 4 E ω p ( x& ) 3 3 c ( ) E 3mc ω ( ) + ω ω γ ω 4 E p ϕ ( )/ C Cnormalzaton constant ( ω/π) 3 3mc 4 C ϕ ( ) profl functon + ( γ / π) 5

75 Stllar Atmosphrs: Emsson and Absorpton Th ln absorpton cross-scton snc - <<, : + ( ) (( + )( )) 4 ( ) C C ϕ ( ) 4( ) ( γ / π) 4 ( ) ( γ / 4 π) + + now: calculatng th normalzaton constant ϕ ( ) d 4π substtuton: x : ( ) γ + + C 4π dx π γ ϕ( ) d 4 C C γ + x γ π π Stllar Atmosphrs: Emsson and Absorpton Th ln absorpton cross-scton ϕ ( ) Profl functon, Lorntz profl Max 4 / γ γ / 4π ϕ( ) ( ) + ( γ / 4π ) proprts: Symmtry: ϕ + ( )) ϕ( ( ( Asymptotcally: ) FWHM FWHM: γ / 4π γ γ FWHM γ ( ) + ( γ / 4π ) 4π π )) ϕ( ) ( FWHM Max 6

76 Stllar Atmosphrs: Emsson and Absorpton Th dampng constant Radaton dampng, classcally (othr dampng mchansms latr) Dampng forc ( frcton ) F γmx(t) & powrforc locty p ( t) γm( x& ( t) ) lctrodynamcs p ( t) (& x ( t) ) 3 3 c Hnc, Ansatz for frctonal forc s not corrct Hlp: dfn γ such, that th powr s corrct, whn tmaragd or on prod: 4 ωt γmω ω (whr w usd x( t) x 3 ) 3 c ω γ 3 3 classcal radaton dampng constant mc 3 Stllar Atmosphrs: Emsson and Absorpton Half-wdth Insrt nto xprsson for FWHM: γ 4π FWHM 3 π 3mc λ c 4π λ 3mc FWHM FWHM 4 λfwhm FWHM.8 Å 4 7

77 Stllar Atmosphrs: Emsson and Absorpton Th absorpton cross-scton Dfnton absorpton coffcnt κ di κ ( ) I ds wth n low numbr dnsty of absorbrs: κ ( ) σ ( ) nlow σ ( ) absorpton cross-scton (dfnton), dmnson: ara Sparatng off frquncy dpndnc: σ ( ) σ ϕ( ) Dmnson : ara frquncy σ Now: calculat absorpton cross-scton of classcal harmonc oscllator for plan lctromagntc wa: E I x E ωt c (, µ ) E 8π δ ( ) δ ( µ ) 5 Stllar Atmosphrs: Emsson and Absorpton Powr, aragd or on prod, xtractd from th radaton fld: 4 E π ω p ϕ ( ) wth γ γ 3 class. 3 3mc γ 3 mc 4 3 E π3mc E p ϕ ( ) ϕ ( ) 3 3mc 4π 8m c On th othr hand: p σ ( ) I(, µ ) d dµ σ ( ) E 8 π µ c E Equatng: σ ( ) E ϕ ( ) 8π 8m π σ ( ) ϕ ( ) σ.6537 cm Hz mc Classcally: ndpndnt of partcular transton Quantum mchancally: corrcton factor, oscllator strngth π π σ lu flu κ ( ) nlow f luϕ( ) mc mc ndx lu stands for transton lowr uppr ll 6 8

78 Stllar Atmosphrs: Emsson and Absorpton Oscllator strngths Oscllator strngths f lu ar obtand by: Laboratory masurmnts Solar spctrum Quantum mchancal computatons (Opacty Projct tc.) λ/å Ln Ly α Ly β Ly γ H α H β H γ f lu g low g up Allowd lns: f lu, Forbddn: <<.g. H I s S ss 3 S f lu -4 7 Stllar Atmosphrs: Emsson and Absorpton Opacty status rport Connctng th (macroscopc) opacty wth (mcroscopc) atomc physcs Classcal crosscton π κ ( ) nlow flow,upϕ ( ) mc σlow, up Populaton numbr of lowr ll Profl functon QM corrcton factor Vw atoms as harmonc oscllator Egnfrquncy: transton nrgy Profl functon: racton of an oscllator to xtrnal drng (EM wa) Classcal crosscton: radatd powr dampng 8 9

79 Stllar Atmosphrs: Emsson and Absorpton Extnson to msson coffcnt Altrnat formulaton by dfnng Enstn coffcnts: κ h 4π ϕ h π 4π mc ( ) nlow B lu ( ).. Blu f lu Smlar dfnton for msson procsss: η n nducd up η n spontanous up h BulIψ( ) 4π h Aulψ( ) 4π ψ ( ) profl functon, complt rdstrbuton: ϕ ( ) ψ ( ) 9 Stllar Atmosphrs: Emsson and Absorpton Rlatons btwn Enstn coffcnts Draton n TE; snc thy ar atomc constants, ths rlatons ar ald ndpndnt of thrmodynamc stat In TE, ach procss s n qulbrum wth ts nrs,.., wthn on ln thr s no ntto dstructon or craton of photons (dtald balanc) mttd ntnsty absorbd ntnsty h h h Bul Inup + Aul nup Blu Inlow TE: I B( T) 4π 4π 4π B B ( T) + A n B B ( T) n ( ) ul ul up lu low ( ) B ( T) n B n B n A low lu up ul up ul B T A n B ul low lu ( ) B ul nupbul

80 Stllar Atmosphrs: Emsson and Absorpton Rlatons btwn Enstn coffcnts A n B h kt ( ) ul low lu up up ( ) wth Boltzmann quaton: B ul nupb ul nlow glow B T B T A g B B ul gupb ul h kt ul low lu h kt ( ) comparson wth Planck blackbody radaton: 3 h B ( T) c 3 Aul h B c g ul B g B g B low lu low lu up ul gupbul n g Stllar Atmosphrs: Emsson and Absorpton Rlaton to oscllator strngth B lu 4 π f mch lu g g 4 π B B f up up ul lu lu glow glow mch 3 h g up 8 π gup Aul B ul f 3 lu 3γ ul flu dmnson A ul tm c g mc g low low Intrprtaton of A ul as lftm of th xctd stat ordr of magntud: at 5 Å: lftm: A ul γ ul 8 s 8 s

81 Stllar Atmosphrs: Emsson and Absorpton Comparson nducd/spontanous msson Whn s spontanous or nducd msson strongr? wth η I spontanous 3 ul * up ul * nducd * * 3 η B( T ) Bul h* nupψ( ) 4 π Bul B ( T ) c h* * h * kt * : h * kt ln * * At walngths shortr than λ spontanous msson s domnant * h * kt ( ) Ah n ψ( ) 4π A h c *.g. T K : λ A T * B 5K : λ 46 A o o 3 Stllar Atmosphrs: Emsson and Absorpton Inducd msson as ngat absorpton Radaton transfr quaton: di spontanous nducd η κi wth η η + η ds di spontanous nducd η + η κi ds h nducd h κlu Blu nlowϕ ( ), ηlu Bul nupiϕ( ) 4π 4π Usful dfnton: κ corrctd for nducd msson: η transton low up di spontanous h η + ( Bn ul up Bn lu low ) ϕ( ) I ds 4π π g low κlu f lu n low n up ϕ( ) mc g up h π g f n 3 spontanous low lu lu up c mc gup ϕ( ) So w gt (formulatd wth oscllator strngth nstad of Enstn coffcnts): 4

82 Stllar Atmosphrs: Emsson and Absorpton Th ln sourc functon Gnral sourc functon: S η κ Spcal cas: msson and absorpton by on ln transton: h lu Aulnup ϕ( ) 3 lu η 4 n h up S π lu κ h g up ( Blunlow Bulnup ) ϕ( ) c nlow - nup 4π g S lu h c 3 g g up low n n low up low Not dpndnt on frquncy Only a functon of populaton numbrs In LTE: 3 h [ ] lu h kt S B (, T ) c 5 Stllar Atmosphrs: Emsson and Absorpton Ln broadnng: Radaton dampng Ery nrgy ll has a fnt lftm aganst radat dcay (xcpt ground ll) A ul Smpl cas: rsonanc lns (transtons to ground stat) xampl Lyα (transton ): γ A 3γcl g g f 3γcl 8.4.3γ cl xampl Hα (3 ): l< u Hsnbrg uncrtanty prncpl: E h Enrgy ll not nfntly sharp q.m. profl functon Lorntz profl γ + Auk + Al j u l k< u j< l γ 3γ g f g + f g + f 3γ γ cl 3 3 cl cl g g3 g

83 Stllar Atmosphrs: Emsson and Absorpton Ln broadnng: Prssur broadnng Rason: collson of radatng atom wth othr partcls Phas changs, dsturbd oscllaton E()~ t ω t t tm btwn two collsons 7 Stllar Atmosphrs: Emsson and Absorpton Ln broadnng: Prssur broadnng Rason: collson of radatng atom wth othr partcls Phas changs, dsturbd oscllaton E()~ t t ω t tm btwn two collsons Intnsty spctrum (powr spctrum) of th cut wa tran: I ~ Fourr transform t / ωt ω t I( ω)~ dt t / ωω sn t ωω 8 4

84 Stllar Atmosphrs: Emsson and Absorpton Ln broadnng: Prssur broadnng Probablty dstrbuton for t Aragng or all t gs ( ) Wt ( ) dt dt arag tm btwn two collsons t / ω ω t / t dt / ω ω I ( ω) const sn Prformng ntgraton and normalzaton gs profl functon of ntnsty spctrum: π ϕ( ω) ( ω ω ) + ( ).. profl functon for collsonal broadnng s a Lorntz profl wth - γ, ~ N N partcl dnsty of colldrs γ N approxmatly constant (to calculat γ : calculaton of ncssary; for that: assumpton about phas shft ndd,.g., gn by sm-classcal thory) 9 Stllar Atmosphrs: Emsson and Absorpton Ln broadnng: Prssur broadnng Sm-classcal thory (Wsskopf, Lndholm), Impact Thory Phas shfts ω: p Ansatz: ω C r, p,3,4,6, r(t) dstanc to colldng partcl p fnd constants C p by laboratory masurmnts, or calculat p nam lnar Stark ffct rsonanc broadnng quadratc Stark ffct an dr Waals broadnng domnant at hydrogn-lk ons nutral atoms wth ach othr, H+H ons mtals + H Good rsults for p (H, H II): Unfd Thory H Vdal, Coopr, Smth 973 H II Schönng, Butlr 989 For p4 (H I) Flm logg Barnard, Coopr, Shamy; Barnard, Coopr, Smth; Bauchamp t al. 3 5

85 Stllar Atmosphrs: Emsson and Absorpton Thrmal broadnng Thrmal moton of atoms (Dopplr ffct) Vlocts dstrbutd accordng to Maxwll,.. for on spatal drcton x (ln-of-sght) Thrmal (most probabl) locty th : th x x A 4 ( T ) x th x x x x w x ( x ) ~ x th x d x th d C π th C π th / kt m.85 A km/s xampl: T 6K, A 56 (ron):.33 km/s.. w ( ) C, wth w ( ) d w obtan: C C x w ( ) x π th x th th m A x kt 3 Stllar Atmosphrs: Emsson and Absorpton Dopplr ffct: profl functon:, c Ln profl Gauss cur Symmtrc about Maxmum: th Half wdth: Tmpratur dpndncy: ( ) th Ln profl th c + C th x( x) ϕ ( ), wth ϕ w obtan: π c th w ( )d ϕ ( ) π th ϕ ( ) th FWHM π ln. FWHM th 67 th ~ T th Max Max th π 3 6

86 Stllar Atmosphrs: Emsson and Absorpton Exampls At λ 5Å: T6K, A56 (F): λ th.å T5K, A (H): λ th.5å Compar wth radaton dampng: λ FWHM.8-4 Å But: dcln of Gauss profl n wngs s much stpr than for Lorntz profl: 43 Gauss ( λ ) : In th ln wngs th Lorntz profl s domnant th Lorntz ( λ ) : rad 6 33 Stllar Atmosphrs: Emsson and Absorpton Ln broadnng: Mcroturbulnc Rason: chaotc moton (turbulnt flows) wth lngth scals smallr than photon man fr path Phnomnologcal dscrpton: Vlocty dstrbuton: w x π x ( ) x mcro mcro.., n analogy to thrmal broadnng mcro s a fr paramtr, to b dtrmnd mprcally Solar photosphr: mcro.3 km/s 34 7

87 Stllar Atmosphrs: Emsson and Absorpton Jont ffct of dffrnt broadnng mchansms y y x x profl A + profl B jont ffct Mathmatcally: conoluton commutat: multplcaton of aras: Fourr transformaton: 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 ~ ~ ~..: n Fourr spac th f A f B π f A f B conoluton s a 35 multplcaton dy x Stllar Atmosphrs: Emsson and Absorpton Applcaton to profl functons Conoluton of two Gauss profls (thrmal broadnng + mcroturbulnc) 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 wth C A + B A B Rsult: Gauss profl wth quadratc summaton of half-wdths; proof by Fourr transformaton, multplcaton, and backtransformaton Conoluton of two Lorntz profls (radaton + collsonal dampng) A/ π B/ π LA( x) L ( ) B x x + A x + B C / π LC( x) LA( x) LB( x) x + C wth C A+ B Rsult: Lorntz profl wth sum of half-wdths; proof as abo 36 8

88 Stllar Atmosphrs: Emsson and Absorpton Applcaton to profl functons Conolng Gauss and Lorntz profl (thrmal broadnng + dampng) ( ) γ /4π D G( ) L( ) π + γ π D ( ) ( ) /4 V G L dpnds on,, γ, : V( ) G( ) L( )d Transformaton: : ( ) a : γ/( 4 ) y : ( ) D D D y y a / Dπ a G( y) L(y) V π y + a π π ( y) + a D Vogt functon, no analytcal rprsntaton possbl. (approxmat formula or numrcal aluaton) D D a Df: V H( a, ) wth H( a, ) dy π π ( y) + a D Normalzaton: H ( a,) d π y dy 37 Stllar Atmosphrs: Emsson and Absorpton Vogt profl, ln wngs 38 9

89 Stllar Atmosphrs: Emsson and Absorpton Tratmnt of ry larg numbr of lns Exampl: bound-bound opacty for 5Å ntral n th UV: Möllr Dploma thss Kl Unrsty 99 Drct computaton would rqur ry much frquncy ponts Opacty Samplng Opacty Dstrbuton Functons ODF (Kurucz 979) 39 Stllar Atmosphrs: Emsson and Absorpton Bound-fr absorpton and msson Enstn-Mln rlatons, Mln 94: Gnralzaton of Enstn rlatons to contnuum procsss: photoonzaton and rcombnaton Rcombnaton spontanous + nducd Transton probablts: [, + d ] P : probablty for photoonzaton n F(): spontanous rcaptur probablty of lctron n, G() : corrspondng nducd probablty [ + d] lctron locty I) numbr of photoonzatons nlow P I ddt II) numbr of rcombnatons nup n ()[ F() + G() I ] ddt Photon nrgy h Eon + m d m h d In TE, dtald balancng: I) II) 4

90 Stllar Atmosphrs: Emsson and Absorpton Enstn-Mln rlatons ()[ () () ] wth [ ] n P I ddt n n F + G I h m ddt I B low up n PB n n () F() + G() B h m low up 3 F() nlow Pm h h kt G() nupn () hg() c B 3 F() h G() c n Pm n n () hg() low up h kt 3/ nlow π mkt up Eon kt nlow nup from Saha quaton: nup n h glow m m kt n () : Maxwll dstrbuton: n () d n 4π d π kt 3/ g 4 Stllar Atmosphrs: Emsson and Absorpton Enstn-Mln rlatons P h G() m P h m up low 3/ h m up 3/ m m h glow up 3 G() h glo w 3/ h kt π mkt up n h glow g 4π 8π m h kt g n n n () g Eon kt m n π kt 3/ m kt 4π Enstn-Mln rlatons, contnuum analogs to A j, B j, B j 4

91 Stllar Atmosphrs: Emsson and Absorpton Absorpton and msson coffcnts absorpton coffcnt (opacty) msson coffcnt (mssty) κ ( ) n P h n low low σ [ F() G() I ] h m η ( ) n n () / up + dfnton. of cross-scton σ And agan: nducd msson as ngat absorpton and LTE: κ n Ph n n G h m ( ) low up () () / η ( )... κ ( ) n η ( ) n * n G() h n n Ph n n n M η ( ) κ ( ) B up up h / kt low ()... σ low up nlow P m nlow up h / kt σ n 3 up c nlow low up h P h kt [ h ] n () F() h n * / m (usng Enstn-Mln rlatons) 43 Stllar Atmosphrs: Emsson and Absorpton Contnuum absorpton cross-sctons H-lk ons: sm-classcal Kramrs formula σ ( ) 3 th th for > th σ ( ) ls 3 8h n 8 n th thrshold frquncy, σth 7.96 cm 3 3π mc Z n prncpal quantum numbr, Z nuclar charg Quantum mchancal calculatons yld corrcton factors ( ) 3 σ ( ) σth th gbf ( n, ), gbf ( n, ) Gaunt factor Addng up of bound-fr absorptons from all atomc lls: xampl hydrogn Z κ n tot max n bf ( ) σ bf ( ) nn n 44

92 Stllar Atmosphrs: Emsson and Absorpton Contnuum absorpton cross-sctons Optcal contnuum domnatd by Paschn contnuum 45 Stllar Atmosphrs: Emsson and Absorpton Th solar contnuum spctrum and th H - on H - on has on bound stat, onzaton nrgy.75 V Absorpton dg nar 7Å, hnc, can potntally contrbut to opacty n optcal band n + n H 4 H 7.5 Sun: T 6K, log n 3.6 Saha quaton:, n n H H H almost xclusly nutral, but n th optcal Paschn-contnuum,.. populaton of H(n3) dcs: n H n n H H ( n 3) g3 ( n ) g n H ( n 3) n H.V / kt n H n ( n ) n 8 H H ( n ) 3 ( n 3) 6 5 Bound-fr cross-sctons for H - and H ar of smlar ordr H - bound-fr opacty thrfor domnats th sual contnuum spctrum of th Sun

93 Stllar Atmosphrs: Emsson and Absorpton Th solar contnuum spctrum and th H - on Ionzd mtals dlr fr lctrons to buld H - 47 Stllar Atmosphrs: Emsson and Absorpton Th solar contnuum spctrum and th H - on 48 4

94 Stllar Atmosphrs: Emsson and Absorpton Th solar contnuum spctrum and th H - on 49 Stllar Atmosphrs: Emsson and Absorpton Scattrng procsss Thomson scattrng at fr lctrons Absorpton coffcnt κ n σ follows from powr of harmonc oscllator ( Thomson cross-scton) σ 4 4 E p 3 3 mc ( ) ( γ π) + fr lctrons: no rsonanc frquncy, no frcton: ; γ 4 E c p, on th othr hand w had p σ E 3 3mc 8 π 4 8π 5 σ 6.65 cm 4 3 mc Thomson cross-scton s walngth-ndpndnt 5 5

95 Stllar Atmosphrs: Emsson and Absorpton Scattrng procsss Raylgh scattrng of photons on lctrons bound n atoms or molculs 4 4 E p 3 3mc ( ) ( γ π) + sm-classcal: << lu R 4 4 lu lu 4 3 mc lu lu R l lu 4 l lu E c p on th othr hand w had p σ E 3mc 8π π σ f σ f κ ( ) nσ f 4 lu R (hr w ha ncludd th oscllator strngth as th quantum mchancal corrcton) Raylgh scattrng on Lyα mportant for stllar spctral typs G and K 5 Stllar Atmosphrs: Emsson and Absorpton Raman scattrng Dscord n symbotc noa RR Tl Raman scattrng of O VI rsonanc ln (Schmd 987) rtual n3 ll n 5Å n 6Å 3/38Å Raman-scattrd ln 685/78Å λ λ OVI λ Ly Schmd 989, Espy t al

96 Stllar Atmosphrs: Emsson and Absorpton Two-photon procsss 53 Stllar Atmosphrs: Emsson and Absorpton Fr-fr absorpton and msson Assumpton (also ald n non-lte cas): Elctron locty dstrbuton n TE,.. Maxwll dstrbuton ff ff ff S ( ) η ( ) / κ ( ) B (, T) Fr-fr procsss always n TE Smlar to bound-fr procss w gt: ff κ σff k ( ) ( )n n h / kt ( ) 6 6π Z σff ( ) g 3/ 3 ff (n,,t) 3 3 hc(πm) T gnralzd Kramrs formula, wth Gauntfaktor from q.m. Fr-fr opacty mportant at hghr nrgs, bcaus lss and lss bound-fr procsss prsnt Fr-fr opacty mportant at hgh tmpraturs σ ~ T, but σ ~ T (Saha), thrfor: κ / κ T / 3/ ff bf ff bf 54 7

97 Stllar Atmosphrs: Emsson and Absorpton Computaton of populaton numbrs Gnral cas, non-lte: In LTE, just n n ( ρ, T, I ) n n ( ρ, T) In LTE compltly gn by: Boltzmann quaton (xctaton wthn an on) Saha quaton (onzaton) 55 Stllar Atmosphrs: Emsson and Absorpton Draton n txtbooks Boltzmann quaton n g ( E )/ statstcal wght Ej kt g n g E xctaton nrgy j j Othr formulatons: Rlatd to ground stat (E ) n g E / kt n g Rlatd to total numbr dnsty N of rspct on n n n n n Ej / kt nj n nj n n j n g j n n g, wth ( ): N n UT ( ) partton functon U T n g g Ej / kt j 56 8

98 Stllar Atmosphrs: Emsson and Absorpton Drgnc of partton functon.g. hydrogn: g n g, E E Ion.. n N lm lm E/kT Normalzaton can b rachd only f numbr of lls s fnt. Vry hghly xctd lls cannot xst bcaus of ntracton wth nghbourng partcls, radus H atom: r ( n) an At dnsty 5 atoms/cm 3 man dstanc about -5 cm r(n max ) -5 cm n max ~43 Lls ar dssold ; dscrpton by concpt of occupaton probablts p (Mhalas, Hummr, Däppn 99) g g p wth p whn g lm 57 Stllar Atmosphrs: Emsson and Absorpton Hummr-Mhalas occupaton probablts 58 9

99 Stllar Atmosphrs: Emsson and Absorpton Saha quaton Draton wth Boltzmann formula, but uppr stat s now a -partcl stat (on plus fr lctron) Enrgy: E Eon + p m plctron momntum) Statstcal wght: g gup G( p) wght of on * wght of fr lctron Insrt nto Boltzmann formula Statstcal wght of fr lctron numbr of aalabl stats n ntral [p,p+dp] (Paul prncpl): G p dp Summarz or all fnal stats By ntgraton or p dω( p) h n ( p) g G( p) n g ( ) spns 3 phas spac cll up up ( Eon + p m Elow )/ kt low low n up gup ( Eup Elow )/ kt p mkt G( p) dp n g phas spac olum low low dω ( p) dxdydz dp dp dp dv 4πp dp n 4 πp dp G( p) 8πp h n x y z 3 59 Stllar Atmosphrs: Emsson and Absorpton Saha quaton Insrton nto Boltzmann formula gs: n g p dp wth x p/ mkt n g h n up up ( Eup Elow )/ kt 8π p mkt 3 low low n g g g g up ( E 3/ up Elow )/ kt 8π x 3 ( ) low h n up low m kt x dx ( Eup Elow )/ kt 8π 3 h n 3/ up π mkt up 3 low low n n h g g ( m kt) ( Eup Elow )/ kt Saha quaton for two lls n adjacnt onzaton stags 3/ π 4 Altrnat: n up n n low 3/ T gup ( Eup Elow ) / kt f ( T ) C.7 C g low 6 K 3/ cm 3 6 3

100 Stllar Atmosphrs: Emsson and Absorpton Exampl: hydrogn Modl atom wth only on bound stat: n n n(h I ground stat) g low I I n n n(h II ) g up II II nn n I II 3/ T 5.58 K/ T f ( T) C pur hydrogn: n n, N n + n n nii onzaton dgr: x N N ft ( ) f( T) f( T) x x x N N xn + f ( T) f( T) f( T) x + + N N N x x( T, N) II I II 6 Stllar Atmosphrs: Emsson and Absorpton Hydrogn onzaton Ionzaton dgr x Tmpratur / K 6 3

101 Stllar Atmosphrs: Emsson and Absorpton Mor complx modl atoms j,...,j onzaton stags,...,i(j) lls pr onzaton stag j Saha quaton for ground stats of onzaton stags j and j+: n j n 3 / j+ 3 h n πm kt g g j j+ E j Ion / kt Wth Boltzmann formula w gt occupaton numbr of -th ll: n g j E kt g j j j j / 3/ EIon / kt nj nj nj+ nct n g g n j j g g j j+ n j j+ n C T 3/ ( E j Ion j E ) / kt j+ 63 Stllar Atmosphrs: Emsson and Absorpton Mor complx modl atoms Rlatd to total numbr of partcls n onzaton stag j+ n N j+ j+ n j n n j+ j+ g g j j+ g U j+ j+ g U j+ j+ N n N j+ j+ j+ n C T g U j+ 3/ ( E j+ j Ion n j j+ E ) / kt g U n j+ j+ j N j+ g U j j+ N j+ n C T 3/ ( E j Ion j E ) / kt N j /N j+ N N N j j j+ N U U U n j+ j+ j j+ j n C T n C T g U j j+ N 3/ E 3/ E j Ion j+ j Ion n C T / kt / kt g n Φ 3/ ( E E / kt j ( T ) j j j Ion j E ) / kt N U j+ j+ n C T 3 / E j Ion / kt U j 64 3

102 Stllar Atmosphrs: Emsson and Absorpton Ionzaton fracton N N N N K N N N N j j j+ J- J j+ j+ J J J N j NJ- NJ- NJ- NJ- N N Nj NJ NJ K+ L j j NJ NJ NJ NJ- NJ N Nj Nj + NJ- K Nj Nj N N J j+ Nj + NJ N N N J N J- NJ- NJ- NJ- N K+ L N N N N N J- nφ k( T) N j k j J J- N + n Φ ( T) m k m k J J J- J 65 Stllar Atmosphrs: Emsson and Absorpton Ionzaton fractons 66 33

103 Stllar Atmosphrs: Emsson and Absorpton Summary: Emsson and Absorpton 67 Stllar Atmosphrs: Emsson and Absorpton Ln absorpton and msson coffcnts (bound-bound) 3 π g low h π glow κlu ( ) flu nlow nup ϕ( ) ηlu ( ) f lu nupϕ( ) mc gup c mc gup ϕ( ) profl functon,.g., Vogtprofl V( a,) π Contnuum (bound-fr) n κ σ up h /kt () nlow nup nlow * D h n η σ y ( y) + a * up h /kt ( ) n 3 up c nlow dy Contnuum (fr-fr), always n LTE ff κ σff k -h / kt ( ) ( ) ( )n n - ff ff η κ k ( ) ( )n n B (,T) Scattrng (Compton, on fr lctrons) κ n σ η () n σ J Total opacty and mssty add up all contrbutons, thn sourc functon S η /κ( ) 68 34

104 Stllar Atmosphrs: Emsson and Absorpton Exctaton and onzaton n LTE n n low up g g low ( Elow Eup )/ kt up Boltzmann n 3/ up π mkt up ( Eup Elow )/ kt 3 low low n n h g g Saha 69 35

105 Stllar Atmosphrs: Hydrostatc Equlbrum Hydrostatc Equlbrum Partcl consraton Stllar Atmosphrs: Hydrostatc Equlbrum Idal gas da r dr P+dP P P prssur k P ρ T ρ mass dnsty AmH A atomc wght forcs actng on olum lmnt: dv dadr dm ρdv GMrdm GMrρ dfg dadr r r buoyancy: df dpda (prssur dffrnc * ara) P

106 Stllar Atmosphrs: Hydrostatc Equlbrum Idal gas In stllar atmosphrs: MM r mass of atmosphr nglgbl rr thcknss of atmosphr << stllar radus GM ρ dfg dadr gρdadr R GM wth g : R surfac graty usually wrttn as log( g / cm s log g s bsdst ff th nd fundamntal paramtr of statc stllar atmosphrs - ) Typ Man squnc star Sun Suprgants Wht dwarfs Nutron stars Earth log g ~8 ~ Stllar Atmosphrs: Hydrostatc Equlbrum Hydrostatc qulbrum, dal gas buoyancy gratatonal forc: ρ() r H lmnat wth dal gas quaton : ( ) xampl: df + df P dpda gρdadr g dp gρ() r dr dp Arm () g Pr dr kt ( r) Tr ( ) T const, Ar ( ) A const (.., no onzaton or dssocaton) dp dr soluton: Am dp Am g Pr () g kt Pdr kt H P(r) P(r ) P(r) P(r ) ( rrgam ) H / kt ( rr )/ H kt H : prssur scal hght ga m H H 4

107 Stllar Atmosphrs: Hydrostatc Equlbrum Atmosphrc prssur scal hghts Earth: Sun: A 8 (N ) T 3 K H 9 km logg 3 A (H) T 6 K H 8 km logg 4.44 H kt gam H Wht dwarf: Nutron star: + A.5 (H + n ) T 5 K H.5 km logg 8 + A.5 (H + n ) 6 T K H.6 mm! logg 5 5 Stllar Atmosphrs: Hydrostatc Equlbrum nd momnt of ntnsty Effct of radaton prssur 4π P ( ) c R K st momnt of transfr quaton (plan-paralll cas) dk H d () dpr 4π H wth d( ) κ( dr ) d () c dpr 4π κ () H dr c ntgraton or frquncs: dpr 4π κ () Hd dr c 6 3

108 Stllar Atmosphrs: Hydrostatc Equlbrum Effct of radaton prssur Extndd hydrostatc quaton ff grr () ρ() ff dfnton: ffct graty g 4π ( r) : g ( ) rad (dpth dpndnt!) ( ) κ Hd g g c ρ r In th outr layrs of many stars: 4 dp dpr π gr ρ() gr ρ() dr dr c κ () Hd 4π gff <.. grad ( ) > () κ Hd g c ρ r Atmosphr s no longr statc, hydrodynamcal quaton Expandng stllar atmosphrs, radaton-drn wnds 7 Stllar Atmosphrs: Hydrostatc Equlbrum Th Eddngton lmt Estmat radat acclraton Consdr only (Thomson) lctron scattrng as opacty σ ( ) σ (Thomson cross-scton) q numbr of fr lctrons pr atomc mass unt Pur hydrogn atmosphr, compltly onzd q Pur hlum atmosphr, compltly onzd q / 4.5 4π 4π q 4π qσ g rad nhd Hd H c nmh / q σ σ c m H c m H σ 4 H Tff 4π 4 grad 4πqσ σ 4 M qσ 4πσRTff Γ Tff G g c m 4π R c m 4πG M Flux consraton: qσ L H LL / 4.5 Γ q 4πcGM m H MM / H 8 4

109 Stllar Atmosphrs: Hydrostatc Equlbrum Th Eddngton lmt Consqunc: for gn stllar mass thr xsts a maxmum lumnosty. No stabl stars xst abo ths lumnosty lmt. L L qmm 4.5 max Sun: Γ << Man squnc stars (cntral H-burnng) Mass lumnosty rlaton: ( ) 3 max Gs a mass lmt for man squnc stars Eddngton lmt wrttn wth ffct tmpratur 5. 4 and graty Γ qtff / g 5. + logq + 4logT logg Straght ln n (log T ff,log g)-dagram ff LL / MM / M 8M 9 Stllar Atmosphrs: Hydrostatc Equlbrum Th Eddngton lmt Postons of analyzd cntral stars of plantary nbula and thortcal stllar olutonary tracks (mass labld n solar masss) 5

110 Stllar Atmosphrs: Hydrostatc Equlbrum Computaton of lctron dnsty At a gn tmpratur, th hydrostatc quaton gs th gas prssur at any dpth, or th total partcl dnsty N: Pgas NkT NN atoms + Nons + nnn N + N N mass partcl dnsty Th Saha quaton ylds for gn (n,t) th on- and atomc dnsts N N. Th Boltzmann quaton thn ylds for gn (N N,T) th populaton dnsts of all atomc lls: n. Now, how to gt n? W ha k dffrnt spcs wth abundancs α k Partcl dnsty of spcs k: ( ) N αn α N n, and t s N N k k N k k N k K Stllar Atmosphrs: Hydrostatc Equlbrum Charg consraton Stllar atmosphr s lctrcally nutral Charg consraton lctron dnstyon dnsty * charg K jk n jn, N dnsty of j-th onzaton stag of spcs k k j jk Combn wth Saha quaton (LTE) by th us of onzaton fractons: W wrt th charg consraton as n K jk k j j N n ( N n ) K k f α jk k k j ( n, T ) jk j f jk jk K α ( N n ) k k j ( n, T ) F( n ) Non-lnar quaton, trat soluton,.., dtrmn zros of F n ) n ( jk lk Njk l j jk jk jk-m Nk + n m l jk- n f j f jk ( n, T) (T) ylds n and f j, and wth Boltzmann all ll populatons lk (T) us Nwton-Raphson, conrgs aftr -4 tratons; 6

111 Stllar Atmosphrs: Hydrostatc Equlbrum Summary: Hydrostatc Equlbrum 3 Stllar Atmosphrs: Hydrostatc Equlbrum Summary: Hydrostatc Equlbrum Hydrostatc quaton ncludng radaton prssur dp dpr 4π gr ρ() gr ρ() κ() Hd dr dr c Photon prssur: Eddngton Lmt Hydrostatc quaton N Combnd charg quaton + onzaton fracton n Populaton numbrs n jk (LTE) wth Saha and Boltzmann quatons 4 7

112 Stllar Atmosphrs: Hydrostatc Equlbrum 3 hours Stllar Atmosphrs pr day s too much!!! 5 8

113 Stllar Atmosphrs: Radat Equlbrum Radat Equlbrum Enrgy consraton Stllar Atmosphrs: Radat Equlbrum Radat Equlbrum Assumpton: Enrgy consraton,.., no nuclar nrgy sourcs Countr-xampl: radoact dcay of N 56 Co 56 F 56 n suprnoa atmosphrs Enrgy transfr prdomnantly by radaton Othr possblts: Concton.g., H concton zon n outr solar layr Hat conducton.g., solar corona or ntror of wht dwarfs Radat qulbrum mans, that w ha at ach locaton: Radaton nrgy absorbd / sc Radaton nrgy mttd / sc ntgratd or all frquncs and angls

114 Stllar Atmosphrs: Radat Equlbrum Radat Equlbrum Absorpton pr cm and scond: Emsson pr cm and scond: dω 4π dω 4π Assumpton: sotropc opacts and mssts Intgraton or dω thn ylds dκ ( ) J dη( ) κ ( )( J S ) d dκ ) I Constrant quaton n addton to th radat transfr quaton; fxs tmpratur stratfcaton T(r) ( dη() 3 Stllar Atmosphrs: Radat Equlbrum Consraton of flux Altrnat formulaton of nrgy quaton In plan-paralll gomtry: -th momnt of transfr quaton dh dt ( J S ) Intgraton or frquncy, xchang ntgraton and dffrntaton: d dt ( ) κ Hd κ J S d bcaus of radat qulbrum σ 4 H Hd const Tff for all dpths. Altrnatly wrttn: 4π dk σ 4 Hd Tff 4 d π d d( fj ) σ d T d 4π 4 ff (st momnt of transfr quaton) (dfnton of Eddngton factor) 4

115 3 Stllar Atmosphrs: Radat Equlbrum 5 Whch formulaton s good or bttr? I Radat qulbrum: local, ntgral form of nrgy quaton II Consraton of flux: non-local (gradnt), dffrntal form of radat qulbrum I / II numrcally bttr bhaour n small / larg dpths Vry usful s a lnar combnaton of both formulatons: A,B ar coffcnts, prodng a smooth transton btwn formulatons I and II. ( ) ) ( + H d d J f d B d S J A κ Stllar Atmosphrs: Radat Equlbrum 6 Flux consraton n sphrcally symmtrc gomtry -th momnt of transfr quaton: ( ) ( ) ( ) H R L L d H r d J S r d H r r J S H r r r 6 bcaus 6 const π π κ κ

116 Stllar Atmosphrs: Radat Equlbrum Anothr altrnat, f T d-coupls from radaton fld Thrmal balanc of lctrons Q Q Q Q Q Q Q H H ff C ff H bf C bf H c C c Q 4πn 4πn 4π 4π n n C nl nk l, m l, m l j j m l, k l, k n nq q N N lm lm j j α α α α bf, lk bf, lk ff, j ff, j ( T ) h ( T ) h lm (, T) J d (, T ) J lm h + c lk ( ) J d lk ( ) J J 3 h kt h + c 3 d h kt d 7 Stllar Atmosphrs: Radat Equlbrum Th gray atmosphr Smpl but nsghtful problm to sol th transfr quaton togthr wth th constrant quaton for radat qulbrum Gray atmosphr: Momnts of transfr quaton ( ) ( ) ( I ) JS ( II) ( I ) κ κ dh dk I J SII H wth κdt d d Intgraton or frquncy dh dk H d d Radat qulbrum κ( JSd ) κ ( JSdJ ) S J S dh and bcaus of consraton of flux d d K dk ( II ) K c + c from ( II ) follows c H, c s blow d d 8 4

117 Stllar Atmosphrs: Radat Equlbrum Th gray atmosphr Rlatons (I) und (II) rprsnt two quatons for thr quantts S,J,K wth pr-chosn H (rsp. T ff ) Closur quaton: Eddngton approxmaton Sourc functon s lnar n Tmpratur stratfcaton? In LTE: ( ) K 3J S J 3K 3H+ 3c III σ 4 S( ) BT ( ( )) T π σ 4 nsrt nto ( III ): T 3H + 3c π σ 4 wth H Tff w gt: 4π σ T ( ) σtff + 3 c ( IV) c s now dtrmnd from boundary condton ( ) π 4π 9 Stllar Atmosphrs: Radat Equlbrum Gray atmosphr: Outr boundary condton Emrgnt flux: wth H() 3 H S( ) E ( ) d ( 3H + 3c ) E ( ) d E ( ) d + c l l! ten ( t) dt ande ( t) l + n wth S from E( ) d t [ te ( t) ] ( III ) 3 H() H + c 3 c H from (IV): ff, 3 T T + S H + (from III)

118 Stllar Atmosphrs: Radat Equlbrum Aodng Eddngton approxmaton Ansatz: J ( ) 3H ( + q( )) 3 σ J ( ) T 4 π 4 ff q( ) Hopf functon ( + q( )) gnralzaton of ( III ) Insrt nto Schwarzschld quaton: J ( ) ΛS ΛJ ntgral quaton for J + q( ) ( + q( ) ) E ( ) d Approxmat soluton for J by traton ( Lambda traton ) J J () () 3H ( + 3) ΛJ () Λ.., start wth Eddngton approxmaton (*) ntgral quaton for q, s blow ( 3H ( + 3) ) 3H + E ( ) + E ( ) 3 3 (was rsult for lnar S) 3 Stllar Atmosphrs: Radat Equlbrum At th surfac At nnr boundary, E (), E3() () J 3H + + 3H 3 3 4, E ( ), E ( ) J () 3 3H + 3 ( +.583) xact: q().577. Basc problm of Lambda Itraton: Good n outr layrs, but dos not work at larg optcal dpths, bcaus xponntal ntgral functon approachs zro xponntally. Exact soluton of (*) for Hopf functon,.g., by Laplac transformaton (Kourganoff, Basc Mthods n Transfr Problms) Analytcal approxmaton (Unsöld, Strnatmosphärn, p. 38) q( )

119 Stllar Atmosphrs: Radat Equlbrum Gray atmosphr: Intrprtaton of rsults Tmpratur gradnt d 4 3dT 3 4 T 4T Tff d d 4 dt 4 Th hghr th ffct tmpratur, th stpr th ~ Tff d tmpratur gradnt. dt dt κ Th largr th opacty, th stpr th (gomtrc) tmpratur dt d gradnt. Flux of gray atmosphr ff ff [ ] LTE: SBT ( ( )) H ( ) BT ( ( )) Et ( ) dt BT ( ( )) E ( tdt ) wth α h kt, T T 3 4( + q( )) p( ) h kt αp( ) σ / 4 4 Hd α α Hd and H T 4π ff 4 H d 4π ktff 4πk 3 Et ( ) E( t) Hα ( )/ H H 4 α 3 dt dt Hdα σtff h hc σ xp( αp( )) xp( αp( )) h 4πk αk 3 h 3 3 c σ h Stllar Atmosphrs: Radat Equlbrum Gray atmosphr: Intrprtaton of rsults Lmb darknng of total radaton σ 4 σ 4 3 I(, µ ) S( µ ) B(T( µ )) T ( µ ) Tff µ+ π π 4 3 I(, µ ) µ+ / 3 3 ( + cos ϑ) I(,) + / 3 5.., ntnsty at lmb of stllar dsk smallr than at cntr by 4%, good agrmnt wth solar obsratons Emprcal dtrmnaton of tmpratur stratfcaton masur I(, µ ) S( µ ) S( ) B( T ( )) T Obsratons at dffrnt walngths yld dffrnt T- structurs, hnc, th opacty must b a functon of walngth 4 7

120 Stllar Atmosphrs: Radat Equlbrum Th Rossland opacty Gray approxmaton (κconst) ry coars, st thr a good man alu κ? What choc to mak for a man alu? transfr quaton -th momnt st momnt gray non-gray di di µ κ ( S I) µ κ ( )( S I ) dz dz dh κ ( S J ) dz dk κh dz dh dz κ ( )( S J ) For ach of ths 3 quatons on can fnd a man κ, wth whch th quatons for th gray cas ar qual to th frquncy-ntgratd non-gray quatons. Bcaus w dmand flux consraton, th st momnt quaton s dcs for our choc: Rossland man of opacty dk dz κ ( ) H 5 Stllar Atmosphrs: Radat Equlbrum H d const R R R κ κ κ dkd κ ( ) dz dk dz dbd κ ( ) dz db dz dbd κ ( ) dt 4σ 3 T π Th Rossland opacty dk κ ( ) dz wth Eddngton approxmaton wth d κ dk dz db db dz dt dt dz and Dfnton of Rossland man of opacty R db dz K / 3J d dz σ T π and LTE J B : 4 4σ T π 3 dt dz 6 8

121 Stllar Atmosphrs: Radat Equlbrum Th Rossland man of opacty κ ( ) Th Rossland opacty κ R wth wght functon s a wghtd man db dt Partcularly, strong wght s gn to thos frquncs, whr th radaton flux s larg. Th corrspondng optcal dpth s calld Rossland dpth Ross For >> th gray approxmaton wth κ R s ry good, Ross z ( z) κ d R ( z ) z T ( Ross ) Tff ( Ross + q( 4 Ross )) 7 Stllar Atmosphrs: Radat Equlbrum Concton Comput modl atmosphr assumng Radat qulbrum (Sct. VI) tmpratur stratfcaton Hydrostatc qulbrum prssur stratfcaton Is ths structur stabl aganst concton,.. small prturbatons? Thought xprmnt Dsplac a blob of gas by r upwards, fast nough that no hat xchang wth surroundng occurs (.., adabatc), but slow nough that prssur balanc wth surroundng s rtand (.. << sound locty) 8 9

122 Stllar Atmosphrs: Radat Equlbrum Insd of blob T + T T ( r+ r) ad ad ad ρ + ρ ρ ( r+ r) ad r Tr (), ρ() r outsd T + T T ( r+ r) rad rad rad ρ + ρ ρ ( r+ r) rad Tr (), ρ() r ρ ( r+ r) < ρ ( r+ r) furthr buoyancy, unstabl ad ad rad ρ ( r+ r) > ρ ( r+ r) gas blob falls back, stabl rad dρad > dρrad unstabl.. dr < dr stabl k wth dal gas quaton p ρt and prssur balanc ρa d T ad ρ rad T rad Am dt unstabl ad < dtrad dr > dr stabl H Stratfcaton bcoms unstabl, f tmpratur gradnt rss abo crtcal alu. dt dr ad 9 Stllar Atmosphrs: Radat Equlbrum Altrnat notaton Prssur as ndpndnt dpth arabl: AmH p hydrostatc quaton: dp ρgffdr gff dr ( dal gas) k T kt dr dp Am g p H ff dt AmH dt T AmH d(ln T ) gff gff dr k dpp k d(ln p) dt (ln ad) < dt (ln rad ) unstabl d(ln p) > d( lnp) stabl Schwarzschld crtron Abbratd notaton dt (ln ad ) dt (ln rad ) ad ; rad dp (ln ) dp (ln ) > ad rad stabl

123 Stllar Atmosphrs: Radat Equlbrum dq (no hat xchang) dq de + pdv Th adabatc gradnt (st law of thrmodynamcs) de cvdt ntrnal nrgy cd V TpdV + (*) Intrnal nrgy of a on-atomc gas xcludng ffcts of onsaton and xctaton 3 3 E NkT cv Nk But f nrgy can b absorbd by onzaton: 3 cv >> Nk Spcfc hat at constant prssur Q de dv dnktp ( ) Nk c + p c + p c + p p p V V T p const dt dt p const dt cc p V Nk Stllar Atmosphrs: Radat Equlbrum p V Th adabatc gradnt Nk ( cp cv) dt Idal gas: pv NkT Vdp + pdv dt Vdp + pdv dt cc p V + + p + V (**) Vdp + pdv from(*) wth (**) cv + pdv cc dpdv p V dp dv c pvc V p dv cc V c V / pv cc p c V V cp dv (ln ) dp (l c V n ) cp dv (ln ) dfnton: γ : c d(ln p) γ V

124 Stllar Atmosphrs: Radat Equlbrum dt (ln ) ndd: dp (ln ) Th adabatc gradnt ad TpVNk / lnt lnp+ lnv ln( Nk) dt (ln ) dv (ln ) + dp (ln ) dp (ln ) dt (ln ) γ dp (ln ) γ γ γ ad γ γ rad < st ab l Schwarzschld crtron γ 3 Stllar Atmosphrs: Radat Equlbrum Th adabatc gradnt -atomc gas cv 3 Nk cp cv + Nk 5 Nk γ 5 3 ad 5.4 wth onzaton γ ad concton starts γ ffct Most mportant xampl: Hydrogn (Unsöld p.8) + ( xx )( 5 + EIon kt) ad 5 + ( xx )( 5 + EIon kt) wth onzaton dgr f ( T) ft ( ) ft ( ) x + + N N N 4

125 Stllar Atmosphrs: Radat Equlbrum Th adabatc gradnt ( xx )( EIon kt) ( xx )( EIon kt) ad ft ( ) ft ( ) ft ( ) x + + N N N 5 Stllar Atmosphrs: Radat Equlbrum Exampl: Gry approxmaton Tff ( 4 3) ( 3 4 T ) ( 4 ) ff 3 d ( ln ( + 3) ) 4d 4( + ) T ( ) + 4lnT ln + ln + dt (ln ) d 3 dp g b hydrostatc quaton: Ansatz: κ Ap ( κ hr a mass absorpton coffcnt) d κ dp g g g p ntgrat p b+ d A b+ A Ap ( b + ) b b+ dp (ln ) dp g g b b+ d pd pap Ap ( b+ ) rad rad dt ln d ( b + ) dp ln d 4( + 3) bcoms larg, f opacty strongly ncrass wth dpth (.. xponnt b larg). Th absolut alu of κ s not ssntal but th chang of κ wth dpth (gradnt) rad larg (> ): concton starts, κ-effkt ad 6 3