Chapter 2: The Photosphere

Transcription

1 3 Chaptr : Th Photosphr Outr Atmosphr Cor - Enrgy Gnraton Zon.3R.7R Radaton Zon Convcton Zon Photosphr 3 km Fg -: Th Sun - Ovrall Structur.: Th Larg Scal Structur of th Photosphr Th photosphr, th vsbl surfac of th sun, s th upprmost opaqu lvl n th sun. Lght from dpr rgons wll not scap and hghr matral, bng almost transparnt, wll mt rlatvly lttl lght. Thus, th photosphr s th transtonal rgon btwn dpr opaqu rgons of th sun and ovrlyng rlatvly transparnt matral. Ths lads to th mportant faturs of th photosphr; n th photosphr th opacty drops from hgh to low, and th tmpraturs fall (although not as fast as th opacty) wth ncrasng hght.

2 4 Solar Ln Asymmtrs Rlatvly transparnt layrs of th sun h 3 km, T 4957 K, P.3 4 dyn cm -, ρ gcm -3 Dnsty, tmpratur and opacty dcras wth ncrasng hght h km, T 6533 K, P.5 5 dyn cm -, ρ gcm -3 Opaqu layrs of th sun Fg -: Larg Scal Structur of th Photosphr Th tmpratur falls wth ncrasng hght untl th tmpratur mnmum n th lowr chromosphr s rachd, aftr whch th tmpratur rss wth ncrasng hght. Th structur of th photosphr s dscussd n mor dtal n sctons..3 and..4. Fgur -3 shows th varaton of varous proprts of th photosphr wth hght...: Th Solar Intror Thrmonuclar fuson ractons n th cor of th sun provd th nrgy that rachs us from th sun. Ths ractons only occur n th nnrmost rgon of th sun, whr th tmpratur and dnsty ar vry hgh (out to about.3r ) and ths nrgy must thn b transportd to th surfac of th sun bfor t can scap nto spac. Ths nrgy can b transportd by radaton or convcton; whthr or not convctv flow occurs dpnds on th convctv stablty of th mdum, whch can b dtrmnd by th Schwarzschld crtron for th occurrnc of convcton: dt dt < dr dr adabatc radatv (-) Aftr Karl Schwarzschld, who proposd t n 96.

3 Chaptr : Th Photosphr 5 Thus, f th adabatc tmpratur gradnt s lss than th gradnt n th absnc of convcton, convcton wll occur. In a dns opaqu mdum, convcton, f t occurs, s vry ffcnt and th actual tmpratur gradnt of th mdum wll b vry clos to th adabatc gradnt. In th sun, radatv transport domnats to a dstanc of about.7r from th cntr, and thn convctv nstablty sts n, and convctv transport domnats from.7r to th surfac of th sun. Th photosphr tslf, wth ts lowr opacts and supradabatc tmpratur gradnt, s stabl aganst convcton. Th solar granulaton (s chaptr 6) s blvd to ars as a rsult of convcton n ths zon, as convctv motons wll tnd to ovrshoot nto stabl rgons; th flow n th convcton zon can caus varatons n th lowr rgons of th photosphr. An upward flow wll not smply stop as soon as t coms to a stabl rgon; ts momntum wll caus t to procd nto th stabl rgon, and thn t wll fall back down aftr comng to a stop. Larg convctv vlocts can b mantand n ths way for som dstanc nto th stabl rgon, as, du to th rapd dcras of dnsty wth hght, only a fracton of th mass n th flow nds to contnu upwards n ordr to mantan th sam volum flow...: Th Outr Atmosphr of th Sun Th outr solar atmosphr, consstng of th chromosphr and th corona, s rlatvly transparnt and of vry low dnsty. It s rsponsbl for som faturs of th solar spctrum, such as ultravolt msson lns, but has vry lttl ffct on Fraunhofr lns.

4 6 Solar Ln Asymmtrs..3: Th Photosphr Th photosphr xhbts many complx motons and othr nhomognts. Dspt ths, th larg scal structur s domnatd by th varaton of proprts such as prssur and dnsty wth hght. Thus, a rasonabl approxmaton of th photosphr can b obtand by consdrng th photosphr to b composd of rlatvly unform horzontal strata; ths s known as th plan-paralll approxmaton. Th dpndnc of th physcal proprts of th photosphr on hght must thn b consdrd. Th chang n prssur wth hght for an atmosphr n hydrostatc qulbrum wll b dp dr GM( r) ρ (-) r whr r s th dstanc from th cntr of th sun, M(r) s th mass nclosd by ths radus, and ρ s th dnsty at ths radus. Th assumpton of hydrostatc qulbrum s a strong assumpton, spcally consdrng that th photosphr s n moton. Th motons n th photosphr ar not rapd nough to caus a strong dpartur from hydrostatc qulbrum, so th assumpton of hydrostatc qulbrum wll gv a rasonabl approxmaton. In th photosphr, M(r) s ssntally constant as th contrbuton to th mass of th sun du to th low dnsty photosphr s small, so w can wrt ths n trms of a local gravtatonal acclraton, g, as dp dr gρ. (-3) Th gravtatonal acclraton g n th photosphr s about 74 ms -. Th prssur and dnsty ar also rlatd by th dal gas law P kt kt ρ µ (-4) whr k s Boltzmann s constant and µ s th man mass of th partcls contrbutng to th prssur, whr Th prssur varatons drvng th flow ar small compard to th prssur dffrncs nvolvd wth th stratfcaton. As th surfac gravty of th sun s hgh, th atmosphr s hghly stratfd, and th rsultant prssur dffrncs ar larg.

5 Chaptr : Th Photosphr 7 µ m all partcl typs. (-5) Th tmpratur gradnt s dtrmnd by th total nrgy flow and th rsstanc of th mdum to th nrgy flow. Thus, t wll dpnd on th nrgy flow mchansm. For radatv transport of nrgy, dt dr L 3 κρ 4ac T 4πr 3 (-6) whr κ s th flux man opacty, or mass absorpton coffcnt avragd ovr all wavlngths (and sutably wghtd to account for th wavlngth dstrbuton of th flux), and L s th lumnosty, or radatv nrgy flux, at th radus n quston. For convctv nrgy transport, th tmpratur gradnt wll b clos to th adabatc tmpratur gradnt dt dr γ T P dp dr, (-7) whr th rato of th spcfc hats, γ, for a monatomc dal gas s gvn by γ 5 3. Radatv transport s th domnant mchansm n th photosphr, wth about 6% of th nrgy bng transportd by convcton at th bas of th photosphr, and vrtually non at a hght of 6 km. 3 Ths dcras n th fracton of nrgy transportd by convcton s a ncssary rsult of th rapd dcras n dnsty wth ncrasng hght; th vlocts of a convctv flow would hav to rs normously n ordr to mantan th sam mass flow ndd to mantan th sam convctv nrgy flux. Th radatv transport s not hamprd at all, th dcras n opacty rsultng from th dcras n dnsty only maks t vn asr for th nrgy to radat outwards. As th tmpratur changs n th photosphr ar small compard to th changs n dnsty and prssur, th prssur and dnsty fall at an approxmatly xponntal rat as th hght ncrass: z H( T ) P P (-8) 3 S pg 44 n Durrant, C. J. Th Atmosphr of th Sun Hlgr (988). Ths valus ar dtrmnd from an analyss of th tmpratur and vlocty varatons n th granulaton.

6 8 Solar Ln Asymmtrs ρ ρ z ( ) H T (-9) whr H(T) s th scal hght for prssur and dnsty. Ths scal hght s gvn by ( ) H T kt, (-) g whr s Avogadro s numbr. For a tmpratur of 63 K, th scal hght s 5 km. 4 Th man fatur of th photosphr s th xtrmly rapd drop n prssur and dnsty wth hght. Th tmpratur also falls wth ncrasng hght n th photosphr, 5 although much mor slowly than th prssur and dnsty. As th opacty of th photosphr s proportonal to th numbr of absorbrs, th opacty must also drop rapdly...4: Th Modl Photosphr As condtons n th photosphr cannot b drctly masurd, a modl atmosphr can b constructd so that ths condtons ar satsfd, and th spctrum producd matchs th obsrvd spctrum. Modfcatons mad ncssary by nhomognts wll b consdrd latr. 6 Th photosphrc modl usd n ths work s shown n tabl - blow. 4 S pg n Durrant, C. J. Th Atmosphr of th Sun Hlgr (988). 5 Th uppr chromosphr has a tmpratur of about K, and corona s vn hottr, wth a tmpratur n th mllons of dgrs. Lns from F XVII and othr hghly onsd lmnts hav bn dntfd n th coronal spctrum. 6 S chaptr 7.

7 Chaptr : Th Photosphr 9 Tabl -: Th Holwgr-Müllr Modl Atmosphr 7 Hght 8 (km) Optcal Dpth (τ 5 ) Tmpratur ( K) Prssur (dyns cm - ) Elctron Prssur (dyns cm -) Dnsty (g cm -3 ) Opacty (κ 5 ) Holwgr, H. and Müllr, E. A. Th Photosphrc Barum Spctrum: Solar Abundanc and Collson Broadnng of Ba II Lns by Hydrogn, Solar Physcs 39, pg 9-3 (974). Extra ponts hav bn cubc spln ntrpolatd by J. E. Ross. Th optcal proprts (such as th optcal dpth and th opacty) of a modl atmosphr ar, obvously, vry mportant, and wll b consdrd latr. S tabl C-4 for complt dtals of th Holwgr-Müllr modl atmosphr ncludng all dpth ponts usd. 8 Th hght scal s not arbtrary. Th bas of th photosphr (hght km) s chosn to b at standard optcal dpth of on (.. τ 5Å ).

8 3 Solar Ln Asymmtrs 6 Hght vs Tmpratur 6 Hght vs Dnsty Hght (km) Hght (km) Tmpratur (K) Dnsty ( -7 g cm -3 ) 6 Hght vs Prssur 6 Hght vs Elctron Prssur Hght (km) Hght (km) Prssur ( 5 dyns cm - ) Elctron Prssur (dyns cm - ) Fgur -3: Th Holwgr-Müllr Modl Atmosphr Th varaton of physcal proprts wth hght can b radly sn for ths modl. Th tmpratur ncrass wth dcrasng hght and th prssur ncrass xponntally. Whn th tmpraturs bcom hgh nough, th lctron prssur ncrass rapdly as varous atomc spcs bgn to ons mor. Th lctronc contrbuton to th total prssur bcoms sgnfcant, and th dnsty dos not rs as rapdly at ths dpth, as th prssur ncras s provdd by th ncrasd onsaton. Othr modl atmosphrs dffr n dtal, but hav th sam gnral structur as th on shown hr. Th rgons rsponsbl for th producton of th solar spctrum ar vry smlar btwn modl atmosphrs; th rgons from whch vry lttl radaton mrgs ar whr most of th dffrncs ar.

9 Chaptr : Th Photosphr 3.: Chmcal Composton of th Photosphr Th photosphr, lk th rst of th sun, s mostly composd of hydrogn. Hlum s also common, and othr lmnts ar lss abundant. Th abundancs of many lmnts ar only poorly known, but th abundancs of th most mportant lmnts ar known to a rasonabl dgr of accuracy. 9 Th abundanc of lmnts s usually gvn as abundanc lmnt log. (-) H lmnt + Th factor of s addd to th logarthmc abundanc rato to mak th abundancs of most lmnts gratr than zro. Fgur -4 shows th chmcal composton of th photosphr. (S tabl C-3 for abundancs usd n ths work.) Atomc umbr Fgur -4: Solar Abundanc of Elmnts Th chmcal composton of th photosphr sms to b th sam as th avrag composton of th ntr solar systm, so mtorc abundanc masurmnts can b usd to mprov th accuracy of solar dtrmnatons, or solar masurmnts can b usd n cass outsd th sun. 9 S Ross, J. and Allr, L. Th Chmcal Composton of th Sun Scnc 9, pg 3-9 (976), Grvss,. Accurat Atomc Data and Solar Photosphrc Spctroscopy Physca Scrpta T8, pg (984), and Andrs, E. and Grvss,. Abundancs of th Elmnts: Mtorc and Solar Gochmca t Cosmochmca Acta 53, pg 97-4 (989).

10 3 Solar Ln Asymmtrs.3: Mcroscopc Proprts and Bhavour.3.: Thrmodynamc Equlbrum If th nrgy n a systm s qually dstrbutd among th avalabl stats, th rato of th occupaton numbrs of any two stats s dtrmnd by th tmpratur T and s gvn by g g E kt E kt (-) whr k s Boltzmann s constant, g and g ar th statstcal wghts of th two stats (th ffctv numbr of sub-stats makng up ach stat), and E and E ar th nrgs of th two stats. Ths dpndnc of occupaton of stats upon th tmpratur only s charactrstc of systms n thrmodynamc qulbrum. A systm s n tru thrmodynamc qulbrum only f t dos not xchang nrgy wth ts surroundngs, but f th nrgy flows n and out of th systm ar balancd, and th occupaton of stats dpnds only on th tmpratur, th systm can b rgardd as bng n thrmodynamc qulbrum. Th stats of th systm nclud th moton stats of th partcls, th xctaton and onsaton stats of atoms, and th nrgy of photons n th radaton fld. Th qu-partton of nrgy among th photon nrgy stats gvs th radaton fld n thrmodynamc qulbrum: hc I B ( T) λ λ 5 (-3) λ hc λkt whch s th wll-known Planck radaton functon for th black-body radaton fld. Ths qulbrum radaton fld s obtand through ntracton wth th partcls n th systm (du to th absnc of photon-photon ntracton). Th statstcal wght s gvn n trms of th atomc quantum numbr j by g j +.

11 Chaptr : Th Photosphr 33.3.: Local Thrmodynamc Equlbrum - th LTE Approxmaton Th photosphr, howvr, cannot b rgardd as bng n tru thrmodynamc qulbrum. Although thrmodynamc qulbrum prvals n th solar ntror, n th photosphr, du to th lowr opacts and th hghr tmpratur gradnt, th radaton fld at any pont contans contrbutons from rgons of dffrnt tmpraturs, and wll not b qual to th black-body fld. Also, f any atomc stats strongly ntract wth th radaton fld, thr populatons wll b affctd by th radaton fld and wll not b solly dtrmnd by th local tmpratur. If th partcls ntract wth ach othr much mor strongly than wth th radaton fld, thr stat populatons wll stll b gvn by th Boltzmann quaton, (qn -) vn f th radaton fld s not gvn by th Planck functon. A systm wth ths charactrstcs s sad to b n LTE, or local thrmodynamc qulbrum. Th tmpratur of a systm n LTE can b dfnd as th tmpratur of th partcls. Th radaton fld can b qut dffrnt from th Planck functon, bng n gnral ansotropc and non-planckan..3.3: Th LTE Equaton of Stat Th populaton of dffrnt nrgy lvls or stats for a systm n LTE (or n thrmodynamc qulbrum) can b found from th Boltzmann quaton (qn -). Snc th occupaton of a stat s proportonal to g E kt, th sum of th occupatons of all stats can b usd to normals ths to fnd th probablty of occupaton of a stat for on partcl, gvng: all j g g E kt j E j kt (-4) whr s th total numbr of partcls. Th normalsaton factor, g all j calld th partton functon. E j kt U( T) g j (-5) all j E j kt j s

12 34 Solar Ln Asymmtrs Th numbr of partcls n a partcular stat s thn gvn n trms of th total numbr of partcls by E g U( T) kt (-6) If w consdr only th populaton of atoms n a gvn onsaton stat, thn ths gvs th populaton of atoms n th nrgy lvl f s th total populaton of atoms n th onsaton stat, and th partton functon s calculatd ovr all nrgy stats avalabl to atoms n ths onsaton stat. If w consdr th rato btwn th populatons of atoms n th ground stat n two succssv onsaton stats, quaton (-) gvs g g, + + +,, ( χ KE ) kt lctron (-7) f w consdr th nrgy of th ground stat of th lowr onsaton stat to b zro. Th multplcty of th + stat s gvn by g g g, + lctron. (-8) Th numbr of stats avalabl to th lctron s g lctron m KE 8 3 π 3 h lctron dke lctron (-9) whr s th lctron numbr dnsty, m s th mass of an lctron, and h s Planck s constant, so w can thn ntgrat quaton (-7) ovr all lctron kntc nrgs to gv m kt g, π h g, + + χ + kt, 3, (-) or n trms of th total populatons of atoms n th two onsaton stats, m kt U π h U ( ) T ( T), + + χ + kt, 3 (-) whch s known as Saha s quaton. Ths can thn b usd rpatdly to fnd th fracton of th total populaton of any atom n any onsaton stat:

13 Chaptr : Th Photosphr (-) Thus, th fracton of th populaton n any onsaton stat can b found from ratos btwn populatons of succssv stats, whch can b calculatd usng Saha s quaton (quaton (-) ). Usually, t wll b suffcnt to only consdr th mor lkly onsaton stats whn usng quaton (-), calculatng only th frst fw trms n th sum n th dnomnator. Ths s a strong tchnqu, as w nd only know th approprat partton functons and th local tmpratur. Ths as of calculatng populatons s what maks th LTE approxmaton so attractv. Usng ths tchnqus to calculat onsaton fractons and populatons for Iron, t can b sn that th F I populaton s strongly dpndnt on hght n th photosphr. Th populaton s affctd by both th tmpratur and th lctron concntraton, whch n turn dpnds on th onsaton lvls of othr lmnts. (S fgur -5 blow.)

14 36 Solar Ln Asymmtrs 6 F onsaton fractons vs Hght 6 F I umbr Dnsty vs Hght Hght (km) 3 Hght (km) F I F II Fracton of total Dnsty (x atoms cm - ) Fgur -5: Varaton of Iron onsaton stats and populatons wth hght.3.4: on-lte Condtons If th LTE approxmaton s not vald, mattrs bcom mor complcatd. In th xtrm cas w would not b abl to mak us of any of th thrmodynamc qulbrum rsults, and w could not, n fact, vn snsbly dfn a tmpratur for th systm. As collsons domnat th transfr of nrgy btwn stats, th photosphr s almost n LTE, and only thos stats whch ntract strongly wth th radaton fld wll hav populatons dffrng sgnfcantly from th LTE valus. Th populatons of any wakly ntractng atomc xctaton stats wll stll b at thr LTE valus, th onsaton qulbrum wll stll b gvn by quaton (-) (at last for most atoms) and th vlocty dstrbuton of partcls wll stll b Maxwllan. Th populaton of a non-lte stat can b dtrmnd from th rat of all transtons to or from ths stat: j j j λ j j j> j A + B I + R collsons j Aj + Bj I λ + R j collsons j j< j j (-3) S scton 3.5 for th ffct of a Maxwllan vlocty dstrbuton on spctral ln profls.

15 Chaptr : Th Photosphr 37 whr A j s th Enstn spontanous msson coffcnt and B j s th Enstn nducd transton rat for th -j transton. If ths radatv transton rats ar small, th stat wll b n LTE (as th collson xctaton and d-xctaton rats act to produc a Boltzmann dstrbuton of stats f th vlocts ar Maxwllan), but f thy ar larg, th populaton wll dffr from th LTE populaton. (If all th radaton trms balanc, an LTE populaton can rsult by accdnt.) To calculat th populaton, w nd to know th varous racton rats, and th radaton fld as wll as th tmpratur. Th dvaton from th LTE cas can b xprssd n trms of a non-lte dpartur coffcnt b, whr b ( ) n LTE. (-4) Dpartur coffcnts dpnd on th hght n th photosphr and on th nrgy lvl nvolvd. Thy can b dtrmnd by fttng calculatd spctral lns to obsrvd spctral lns and thn ncorporatd nto th modl atmosphr, but n gnral, t s dsrabl to assum LTE whnvr possbl, and to rstrct ourslvs to transtons btwn lvls n LTE. LTE s most lkly to b a good approxmaton f th absorpton cross-sctons and msson rats for transtons to and from th lvl ar low, and collson xctaton and d-xctaton rats ar hgh. Ths condtons ar lkly to b satsfd for a gvn spctral ln f transtons nvolvng th uppr and lowr lvls ar not xcssvly strong, and f th ln s formd dp n th photosphr (whr th dnsty and collson rats ar hghr). In th outr atmosphr of th sun, whr xtrmly low dnsts rsult n much lowr collson rats, populatons can b far rmovd from LTE populatons. Cass whr dparturs from LTE wr sgnfcant wr avodd n ths work. In practc, non-lte cass usually nvolv vry strong lns, whch ar mor lkly to b blndd than wakr lns (du to th gratr wdths). Of th unblndd lns usd n ths work, only th potassum rsonanc ln at 7699Å shows srous dpartur from LTE. on-lte calculatons can b prformd, but ar mor nvolvd than LTE calculatons. S scton 5.. for a brf dscusson of non-lte mthods.

16 38 Solar Ln Asymmtrs