1968ApJ. . .151. . 991C + + continuum foreachobjectunderconsideration. numerator anddenominatorofequation(2)by of thestellareffectivetemperatureandisgivenby representative oftheatmospherescentralstarsplanetarynebulae. Harman andSeatonsequencesofstarswithmassesbetween0.51solarmassareassumedtobe Lyman continuumineachcaseThemodelatmospheresofBöhmandDeinzerthatcorrespondtothe where 5gisthefilterfunctionforphotographic region.F(Tg)istheemergentflux distribution forastarofeffectivetemperatureT andgravityg.Bymultiplyingthe Lyman limitintheH°continuumforeachofthirty-fiveplanetarynebulae. the assumptionthatcentralstarsarenon-grayandthereiscompleteabsorptioninHe The AsteophysicalJouenal,Vol151,March1968 by thecentralstarinH°Lymancontinuumthatisconvertedtolineradiation in eachcasethetemperatureofcentralstarandthatfractionradiationemitted where F(4686)istheenergyfluxatEarthinHe (4686)line.W{Tg)isafunction and thecentralstartemperaturegravitybyequation each ofthirty-fiveplanetarynebulae,HarmonandSeaton(1966)wereabletodetermine by BöhmandDeinzer(1966).NewestimatesoftheopticalthicknessesinH°con- and Seatoninordertoestimateanupperlowerlimittheopticalthicknessat thirty-five planetarynebulaearemadeusingnon-graystellaratmospheresconstructed the nebula.Thecentralstarswereassumedtoradiatelikeblackbodies. tinua aremadeusingthesenewtemperatures. P vSl s Sj + + © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem On theassumptionthatthereiscompleteabsorptionincontinuumofHefor The fluxintheHePaschen-alinecanberelatedtophotographicmagnitude,Wp, Upper andlowerlimitsaresettothecharacteristicopticalthicknessatLymanlimitinH° The effectivetemperaturesofthecentralstarsthirty-fiveplanetarynebulaearedeterminedon In thefollowingsections,newestimatesoftemperaturescentralstars Capriotti (1967;hereinafterreferredtoas“PaperI”)usedtheresultsofHarmon g EFFECTIVE TEMPERATURESOFTHECENTRAL II. DETERMINATIONOETHECENTRALSTARTEMPERATURES Perkins Observatory,OhioStateandWesleyanUniversities w (T,g) =fjFAT^g)^/fJsF,(Td v, (2) sim Eugene R.CapriottiandWilliamS.Kovach TU(r,g) =8.2 s STARS OFPLANETARYNEBULAE Received July6,1967 I. INTRODUCTION ABSTRACT M4686) E(4686) 991 1Q(0 4m+470) » pg} (1) 1968ApJ. . .151. . 991C + Lyman limitisdefinedtobe numerically byfittingpolynomials betweensuccessiveabsorptionedgesand theninte- black bodywiththesametemperature. of thefactthataBöhmandDeinzermodelparticulareffectivetemperatureshows model islargerineachcasethantheassignedblack-bodytemperature.Thisaresult where B(T)isthePlanckfunction. grating theresultingpolynomial. TheintegralsinvolvingB(T)weredone inthesame manner soastoavoidsystematic errors.Thevalueofthesecondratioin equation (5), is alreadyknownforvariouscombinationsofT andr.Theintegralswereevaluated column andtheeffectivetemperaturesfornon-graymodelsaregiveninthird tion. sorption byNe,He,N,C,andOionsinthestellaratmosphere)whencomparedtoa a depletioninthecontinuumbeyondHeLymanlimit(dueprimarilytoab- column. Onecanseethatthetemperatureofcentralstarfoundfromnon-gray F(T) andtabulatedbyO’Dell(1963).Equations(1)(3)wereusedtodetermine 0.5 and1solarmass.Therefore,thegravitygdependsontemperatureTinaknown star radiatesasablackbody,by way. Thefirstratioinequation(3)isnearlyequalto Deinzer (1966)fortheHarmon-Seatonsequencesofstarswithmassesbetween 992 EUGENER.CAPRIOTTIANDWILLIAMS.KOVACHVol.151 the effectivetemperaturesofcentralstarsthirty-fiveobjectsunderconsidera- when theBöhmandDeinzerfluxdistributionsareused.Thelatterratioisdenotedby This canbeexpressedintermsofGi,whichisdefinedPaperIastheprobability we canwrite absorption ofthecentralstarradiationbeyond Lymanlimit,assumingthecentral va VS s a 8 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The probabilityofabsorptiondirectradiationfromthecentralstarbeyond The determinationofGithenreducestotheevaluation oftheaboveintegralssinceGi In Table1theblack-bodytemperatureslistedinPaperIaregivensecond Here weusetheemergentfluxdistributionsthatweredeterminedbyBöhmand 01=01_exp /.|[ W{Tg)=:sT '’[f, /fo^‘>s)^] III. CALCULATIONSOFTHEOPTICALTHICKNESSES f„ //, 0 z* ñv/r IN THEH°LYMANCONTINUA T I— Po V.ir.,«)*. (4) (3) 1968ApJ. . .151. . 991C s where ÿ(T«)iheredefinedtobe and correspondingly onrbytherelations NGC 7662. NGC 7354 NGC 7139 NGC 7009 IC 5217 NGC 7026 NGC 7008 NGC 6905 NGC 6881 NGC 6894 NGC 6891 NGC 6853 NGC 6818 BD 30°3639 NGC 6826 NGC 6804 NGC 6778 NGC 6772 NGC 6445 NGC 3242 NGC 6781 NGC 6751 NGC 6572 NGC 6543 NGC 6439 NGC 6309 NGC 6058 NGC 3587 NGC 2392 NGC 2371-2 NGC 2022. NGC 1535 NGC 1501 IC 351 NGC 650-1 largest valueofTis150000°K,wemustextrapolatewhenastar’seffectivetempera- ture appearsgreaterthan150000°K. tion ofrandT.ThevalueswerethoseusedbyBöhmDeinzer.Sincethe the areaunderaplotofB(T¿)versusv.ThevaluesGiaregiveninTable2asfunc- No. 3,1968CENTRALSTARSOFPLANETARYNEBULAE993 done bythismethod,differedlessthan1percentfromthevaluefoundconsidering s s8 V © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Following theargumentsinPaperI,upperandlower limitscanbeplacedonG\and T (PaperI) a 100000 102000 102000 132000 112000 182000 195000 105000 100000 182000 98000 98000 98000 81000 98000 69000 46000 74000 69000 91000 95000 93000 56000 61000 68000 85000 66000 91000 72000 76000 72000 72000 91000 74000 72000 , œ ÿ(r,)= f"F(T,g)^ -/ fF„{T,g)- re s Effective TemperaturesofCentralStars 222000? 118000 120000 116000 116000 116000 206000? 120000 148000 116000 110000 100000 130000 123000 109000 112000 118000 108000 108000 195000? 90000 98000 85000 60000 92000 85000 88000 71500 88000 88000 90000 82000 84000 88000 a
1968Apj. . .151. . 991C the Asteophysical Jouenal, Vol 151, March 1968 EFFECTIVE TEMPERATURES of the CENTRAL STARS of PLANETARY
