11. Spectral Lines II. Curve of Growth

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1 . Spetra Lines Curve of Growth Line formation Eington-Mine Curve of Growth Curve of Growth is a theoretia pot of ine equivaent with against ine optia epth, usuay potte as ogw og. The W reation is seen in the Shuster-Shwarshi moe atmosphere: ontinuous spetrum is emitte from a eep ayer, T R, seon reversing ayer at T L absorbs the spetra ines. Raiation we observe is from L: TR e TL e where is the optia epth of the reversing ayer given by: σ N i π e m e f a, v a, v where N i is the oumn ensity aong LOS m - of parties in ower eve i. N i

2 The reative ine epression is: TR TL e T R e Where T R - T L ] / T R the imum epression for very strong ines. The equivaent with is W showing how W reates to an hene aso N i an f. The urve of growth an be spit into three regimes epening on the optia epth ine strength: weak, saturate, an strong ines. e ine. Weak Lines: For <<, exp- ~, so ~. The Voigt profie is approximate by opper beause opaity is too sma to map amping wings into emergent spetrum. Repaing a,v with exp- / ] with area π ½ gives: W e / π π e f N i me W ~ f N i. Part of urve of growth is opper part an has sope :. Linear inrease ue to optia thinness of reversing ayer.. Saturate Lines: For >, ine annot be eeper than. The ine with inreases with an W Q Where Q ~ 4. This is part of urve of growth or the shouer.

3 . Strong Lines: For >>, the ore oesn t hange any more. Line entre is fixe at, but ine wings have < an may grow in optiay thin fashion to inrease W. For arge the wings ontribute a ot beause they map the amping part of A rop off with that is ess steep than the exponentia eay of the opper ore. n the amping part of a,v we write: whih gives: π v a / π / ~ / a, v a / W ~ ine a π v e a a / a π Thus part or the amping part saes as W ~ a ½ ~ f N i γ ½. π ine e / u u og W og T R T L

4 Cassia Curve of Growth Fitting The strategy is to anaye ines via a urve of growth to etermine abunanes, amping parameter a, exitation temperature T ex, an miroturbuene ξ miro. Pot measure W s as ogw / against og X ogc og g f χ 54 / T ex Where C ontains unknowns suh as, miroturbuene, Saha popuation fator, ontinuous extintion, eementa abunane. Changing parameters hanges the shape of theoretia urve of growth. y minimiing satter of ata an erive physia parameters. Resiua Fux in a Line Opaity ue to ine an ontinuum: Continuum opaity varies sowy with > onstant aross ine. Write: η an assume η inepenent of. / Now onsier tota / ontinuum optia epth: ρ ρ η

5 n ERT want energy reate an energy estroye aong beam: ρ ρ ρ ρ ρ ρ ρ ρ C ontinuum sattering ontinuum absorption ine absorption ine sattering ontinuum emission ine emission ine sattering ontinuum sattering ERT beomes: Whih simpifies to where ] ρ µ ] µ η η

6 Eington-Mine Soution Assumptions to sove ERT above:. inepenent of i.e., η inepenent of. T] inear funtion of, ontinuum optia epth a b a p p b / η Can now sove ERT above in simiar way to sattering in Eington soution. Form moment equations: ] K Use Eington approximations K, to get from seon moment equation above: Substitute this in first moment equation above an use inearity of to get: Soution: Appy bounary onitions as before: no inient raiation, at arge epth >, Eington approximation, or from exat soution / ] ] ] - e e C C

7 Using these approximations an bounary onitions gives a p a an the emergent fux: - ] p e / p ]/ ] / a R For the ontinuum η so we get the resiua fux as / p a b a Tutoria Exampe The above soution has been outine. erive it in etai, going through a the steps an agebra. Then erive equations for the resiua fux for the foowing situations. Sattering ines: no sattering in ontinuum, pure sattering in the ine. Absorption ines: Pure sattering in ontinuum, no sattering in ine int: what are the ine an ontinuum vaues for the above ases? What are the resiua fuxes for very strong ines in the above ases? Comment on the spetra appearane of the ine ores. For a very strong ine, onsier the imit of very arge η