T2i11T 451080861on3TheopacityduetofreefreeboundfreeandboundboundelectronictransitionscanbeapproximatedwiththesocalledKramersformulaK410251XZ0001 T35on4Thisformul ID: 129099
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OPACITYAgooddescriptionofopacityisprovidedbySchwarzschildin"StructureandEvolutionofStars"(ChapterII).InpracticalapplicationsopacitiescalculatedandtabulatedbytheLosAlamosgroupareused(e.g.Cox,A.N.,andTabor,J.E.1976,Ap.J.Suppl.,31,271).Heresomeapproximateanalyticalformulaearegiven.Thesearefairlyaccurateforelectronscattering(Paczynski,B.,1983,Ap.J.,267,315)andforelectronconductivity(Yakovlev,D.G.,andUrpin,V.A.1980,SovietAstron.,24,303).Allotherareverycrudeapproximationstothetabulatedopacities.Alowdensity,relativelylowtemperaturefullyionizedgashasopacitydominatedbyThompsonelectronscattering.ThisisisgivenasTh=neTh;(on:1)wherene=(1+X)=2Histhenumberoffreeelectronspercubiccentemeter,andTh=8r2e=3=0:66510 24cm2isThompsonscatteringcross-sectionperelectron.PuttingnumericalvaluesforallconstantsweobtainTh=0:2(1+X);cm2g 1:(on:2)Whentemperatureishigh,sothatenergyofphotonsbecomesanappreciablefractionoftheelectronrestmass,thecross-sectionforscatteringisreducedaccordingtotheKlein-Nishinaformula,andtheelectronscatteringopacityisreduced.WhenelectrongasbecomesdegenerateandmostenergylevelsbelowEFermiare lledin,photonsmaybescatteredonlyonsomeelectrons,andtheelectronscatteringopacityisreducedtoo.ThenumericalresultspresentedbyBuchler,J.R.,andYueh,W.R.,1976,Ap.J.,210,440)canbe ttedwiththefollowingfairlyaccurateformulae=0:2(1+X)h1+2:71011 T2i 1"1+T 4:51080:86# 1:(on:3)Theopacityduetofree-free,bound-free,andbound-boundelectronictransitionscanbeapprox-imatedwiththesocalled\Kramersformula":K41025(1+X)(Z+0:001) T3:5:(on:4)Thisformulaisimportantwhenhydrogenandheliumandotherelementsarepartlyionized,roughlyspeakingforT2104K.Theopacityduetothenegativehydrogenion,H ,isthedominantopacitysourceinthesolaratmosphere,ingeneralfor4103T8103K:H 1:110 25Z0:50:5T7:7:(on:5)Notice,theextremelysteeptemperaturedependenceoftheH opacity.Atverylowtemperaturemoleculesdominatetheopacity,withH2OandCObeingthemostimportantfor1:5103T3103K:m0:1Z:(on:6)Thetotalradiativeopacitymaybecrudelyapproximatedwiththefollowingformula,whichmaybeusedoverahugerangeoftemperatures,1:5103T109K:radm+ 1H +(e+K) 1 1:(on:7)on|1 Atveryhighdensityandlowtemperaturetheradiativeopacitybecomesveryhigh,aslongasthegasremainssubstantiallyionizedandnon-degenerate[eq.(on.4)],andtheelectronthermalconductivitybecomesamuchmoreecientheatcarrier.Areasonableapproximationtotheelectronconductivityexpressedinaformof\opacity"is:cond2:610 7hZiT2 2 1+ 21062=3!;(on:8)wherehZiistheaverageelectricchargeperion.Finally,thetotale ectiveopacity,whichincludesbothheatcarriers:photonsandelectrons,canbecalculatedas:= 1rad+ 1cond 1:(on:9)Thislastformulaisexact,asthetwoheatcarrierso ertwoindependentmodesofenergytransport,i.e.thecorrespondingcoecientsofheatconductivityareadditive,whichisequivalenttotheinverseofthecorresponding\opacities"tobeadditive.Throughoutthischapterallopacitiesaregiveninc.g.s.units,i.e.in[cm2g 1].Also,inallformulaeXandZindicatethehydrogenandheavyelementfractionbymass.on|2