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Astronomy&Astrophysicsmanuscriptno.vlemmings˙2430 February2,2008 (DOI:willbeinsertedbyhandlater) Astrophysical water masers: Line profiles analysis W.H.T.Vlemmings1 andH.J.vanLangevelde2 1 DepartmentofAstronomy,CornellUniversity,Ithaca,NY14853-6801,U.S.A. 2 JointInstituteforVLBIinEurope,Postbus2,7990AADwingeloo,theNetherlands 5 Received;accepted 0 0 2 Abstract. Thechangesinthespectrallineprofileofthe22GHzH2Omaserarecalculatedasafunctionofemergingmaser flux.Weaddressnotonlythenarrowingandre-broadeningofthemaserlines,butalsothedeviationsofGaussiansymmetryas n aresultofthevarioushyperfinecomponentsofthe22GHzmaserline.Non-LTEmodelsoftheH Omasertransition,including a 2 J hyperfinestructureandcross-relaxation,arecomparedwithhighspectralresolutionVeryLongBaselineInterferometry(VLBI) observationsoftheH Omasersinthecircumstellarenvelopes(CSEs)ofasampleoflate-typestars.Thisyieldsestimateson 8 2 2 thethermalwidthinthemaserregionaswellastheemergingmaserfluxandthusthelevelofsaturation.Wealsodiscussthe effectofavelocitygradientalongthemaserpathonthelinewidthsandshapesofthelineprofileandtheeffectofthegeometry 1 ofthemaserregion.Wefindthatthetypicalvelocityshiftalongthemaserpathisoftheorderof1.0kms−1.Theeffectofthis v shiftontheshapeofthemaserspectrumisdifficulttodistinguishfromtheeffectofthehyperfinecomponents. 7 2 Key words. masers – stars: circumstellar matter – stars: individual (U Her, S Per,NML Cyg, VY CMa) – stars: AGB and 6 post-AGB–Line:profiles 1 0 5 0 1. Introduction rameter. In this paper we examine the feasibility of using the / skewnessoftheprofiletogetherwith thelinewidthstodisen- h The spectral line profiles of astrophysicalmasers contain im- tangle the thermal line width of the 22 GHz masing medium p portant information on the conditions of the masing region, - and the level of saturation and beaming angle. We examine a o such as the thermal line width of the particle distribution and sampleoflate-typestarsandcomparetheirhighresolutionto- r the optical depth. However, the degree of saturation of the t talintensityspectratoournon-LTEmodels. s maser has a strong influence on the maser profile and can- a The H O masers around late-type stars are thoughtto oc- not be determined directly. It has been shown before, that a 2 v: detailed analysis of the maser line widths can provide some curintheexpandingwindatafewhundredAUfromthecentral i star(e.g.Laneetal.1987).Thetemperatureofthemasinggas X information on the maser saturation (e.g. Goldreich & Kwan isexpectedtobelessthan1000K,accordingtotheanalysisof 1974; Litvak 1970). The line profiles first get narrower with r H O maser pumpingschemes by Neufeld & Melnick (1990). a increasing radiative flux and then re-broadenwhen the maser 2 Asaresulttheintrinsic,thermallinewidthofthemasershould becomes radiatively saturated. However, if there is an effec- be<∼1.5kms−1.Largerlinewidthscanbecausedbyturbulence tive redistributionof the populationsof the excited states due ofthemasinggas.Modelestimatesoftheintrinsicthermalline tocross-relaxation,there-broadeningwillonlyoccuruntilthe widths can thus give an upper limit to the temperature in the radiativefluxislargeenoughthattherateofstimulatedemis- maserregionandtotheeffectofturbulence.Atafewhundred sion exceedsthe rate forcross-relaxation.Additionally,in the AU from the centralstar the outflowingmaterialis still being caseofthe22GHzrotationalH Omasertransition(6 −5 ), 2 16 23 accelerated.Therefore,asshowninRosenetal.(1978),theve- the interaction between the different hyperfine components locitycoherentpathsinthetangentialandintheradialdirection (F = 7−6,6−5,5−4,5−6,6−6 and 5−5) complicates are of similar length. As a result, for the H O masers neither a straightforwardlineanalysis.Severalpapershaveaddressed 2 radially nor tangentially beaming necessarily dominates. The this issue (e.g. Nedoluha & Watson 1991). The manifestation H Omaserspotshavetypicalsizesof1013cm(Reid&Moran ofthehyperfinecomponentsintheobservedmaserspectraalso 2 1981),butasdeterminedfromVLBIobservationstheycanbe dependson the maser thermalline width, as well as the satu- assmallasafewtimes1012cm(Imaietal.1997).Duetobeam- ration and beaming angle. This is due to the fact that the hy- ing,theactualsizeoftheH Omasercloudisgenerallymuch perfinecomponentscausesignificantdeviationsfromGaussian 2 largerthantheobservedsize.Goldreich&Keeley(1972)have symmetry.Thedeviationscanbe classified bya skewnesspa- shown that for H O masersa factor of 50 differenceis likely. 2 Sendoffprintrequeststo:WV([email protected]) Themaserpathlengthcouldthusbeaslongas1014−1015cm. BasedonresearchcarriedoutwhileWVwasbasedinLeiden. Along this path the velocity can change by a significantfrac- 2 W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 2 Fig.1. Scaled intensity lines profiles for the non-LTE models Fig.2.LinewidthasafunctionofemergingmaserfluxT ∆Ω b withv = 0.6kms−1 (left)andv = 1.0kms−1 (right).From formodelswithdifferentthermalwidth.Thethicksolidlineis th th inside out, the dashed line has emerging maser flux T ∆Ω = the modelfor v = 1.0 km s−1 with [Γ+Γ ]= 5.0, the other b th ν 108,theshort-dashedlinehasT ∆Ω = 1010 andthesolidline linesarefor[Γ+Γ ]=1.0. b ν hasT ∆Ω=1011. b Heren(v)andn (v)arethepopulationnumbersforrespec- i j tionoftheintrinsicthermallinewidth,especiallyneartheedge tively the upper (616) and the lower (523) rotational levels. i of the envelopewhere the velocity gradientis steep. Here we denotestheupperlevelhyperfinestates(F =7,6or5), jisthe also examinethe effectof suchvelocitychangesonthe shape related lower hyperfinestate (F′ = 6,5 or 4). The population ofthe22GHzH Omaserprofiles. levels are solved as a function of molecular velocity v and at 2 differentpositionsalongthemaserpath.Thestatisticalweights aredesignatedbyg andg ,andarefromKukolich(1969).The i j 2. Background pumprateλ(v)isassumedtobethesameforthedifferenthy- i perfinecomponentsandhasaMaxwelliandistribution.There- The rateequationsforthe populationsof the upperand lower sultsdependonthetheratio∆λ/λ where∆λisthedifference energylevelsofthevarioushyperfinelinesofthe22GHzH O i 2 betweenthepumprateintotheupperandlowerlevels.Thisra- maserhavebeensolvedalongauni-directionalmaserbeamus- tioisoftheorderofafewpercent(Anderson&Watson1993). ingthemethoddescribedin Nedoluha&Watson(1992,here- TherateforstimulatedemissionR iscalculatedusingthelo- afterNWa)andusedinVlemmingsetal.(2002,hereafterV02). ij calmaserintensityandthehyperfineinteractioncoefficientsas We have included only the three strongest hyperfine compo- described in NWa. Γ is the decay rate for the molecularexci- nents(F =7−6,6−5and5−4),sinceithaspreviouslybeen tations.Thecross-relaxationrateΓ describesthereabsorbtion shown that the weakest of the hyperfine lines are negligible ν of previously emitted infrared pump photons trapped in opti- (Nedoluha&Watson1991,Deguchi&Watson1986).InV02, cally thick transitions of the maser system, which generate a therateequationsaresolvedincludingamagneticfield,thusin- newly excited molecule at random, within a Maxwellian ve- cludingallthedifferentmagneticsubstates.Themagneticfield locitydistribution(φ(v)).Thishastheeffectofpostponingthe doesnotnoticeablyinfluencethelineshapesandlinewidthsof rebroadening of the maser line due to maser saturation. The the total intensity maser lines. Here we limit ourselves to the rate for redistribution in velocity is not exactly the same for caseofnomagneticfield. the redistribution among the different hyperfine components. Forthepopulationlevelsoftheupperstateswesolve However,bothareexpectedtoberoughlyequaltotheinverse 0 = λ(v)−(Γ+Γ )n(v)+R (v)(n (v)−n(v)) lifetime of the state due to emission of infrared radiation. In i ν i ij j i Γ NWa it was shown that the results scale by [flux/(Γ + Γν)]. +φ(v)( νgi)Z dvXgini(v). (1) tMhooustgohfwtheeaclsaolciunlvaetisotingsaaterehpigehrfeorrvmaeludefsoorf[ΓΓ+aΓndν]Γ=.1A.0ssi−m1ialal-r P ν Asimilarbutreversedequationissolvedforthepopulationof effect as the cross-relaxation due to reabsorbtion is produced thelowerlevels. by elastic collisions between the maser molecules and inter- W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 3 2 netic temperatureT inKelvin,v ≈ 0.5×(T/100)1/2 (NWa). th Lineprofilesarepresentedfordifferentvaluesofemergingflux (T ∆Ω = 108,1010 and 1011). Due to the hyperfine interac- b tion the maser profiles show clear deviations from Gaussian symmetry with increasing maser brightness. The effects are largestfor thesmallest thermalwidths. We also noticethe rebroadening of the profiles when the maser starts to get saturated.Thiscanalso be seen in Fig. 2, whichshowsthe maser line width as a function of emerging flux. In the unsaturated casethemasergetsincreasinglynarrow,whenthemasersstarts togetsaturatedthelinewidthincreases.Thefigureshowsthe relations for [Γ+Γ ] = 1 and 5 s−1. As expected, the maser ν is able to remain unsaturated longer, for higher values of the decayandcross-relaxationrate. Aside fromthe narrowingand re-broadeningof the maser lines,Fig.1showsthat duetothe contributions ofthehyper- fine components,theshapeofthe22GHzH Omaserprofile 2 is also a good indicator of the saturation level. The hyperfine componentswillcausetheprofiletobeskewedtowardtheneg- ative velocities and the skewness will depend on the level of saturation. The deviationfrom Gaussian shape can be quanti- fiedbytheskewnessparameterdefinedas: Fig.3. Skewness parameter as a function of emerging maser 1 (v−v ) Skewness= dv[ψ(v) 0 ]3. (2) brightness Tb∆Ω. The lines are labeled for different thermal ΨZ σ width.Theshadedareasindicatethe3σerrorswhenincluding Hereψ(v)istheprofileofthemaserline,σthestandardde- observationalerrorsasdescribedinthetext. viationandv thepositionofmaximumintensity.Theparame- 0 terisscaledbyΨ,whichistheareaundertheprofile.Apositive mixed H2 molecules, that change the velocity but not the en- valueoftheskewnessparametersignifiesadistributionwithan ergyofthecollidingparticles.Thiseffectisdescribedindetail asymmetric tail outtoward positive velocity,while a negative inElitzur(1990),whereitwasshownthatduetoelasticcolli- skewnessparameterindicatesanegativetail.Fig.3showsthe sions,thenarrowingofthemaserlinecanbeextendedsignifi- skewnessofthemaserprofileasafunctionofemergingbright- cantly. However, the analysis by Elitzur (1990) indicates that ness for different values of v , as determined from our mod- th for maser transitions that have large infrared optical depths, els.Forthehighlyunsaturatedmaserstheskewnessparameter suchastheH2Omaserdiscussedhere,theeffectofelasticcol- isstronglynegativeduetothemultiplehyperfinecomponents. lisionsissmallcomparedtothatofΓandΓν. The profiles become more symmetric when the F=7-6 hyper- Eq. 1 is iteratively solved along the maser beam, as the fine component starts to dominate. When the maser starts to maser intensity itself, determined using the radiative trans- saturate, at T ∆Ω ≈ 1010 for [Γ+Γ ] = 1 s−1, the skewness b ν fer equations in NWa, depends on the level populations. We parameterdecreasesagain,becausetheF=7-6hyperfinecom- assume nearly one-dimensional maser propagation, with the ponentisthe first to saturate. Asa result,the profilebecomes beaming of the maser radiation represented by the solid an- moreasymmetric.Thisis the mostpronouncedforthe lowest gle ∆Ω. The emerging maser fluxes will thus be represented valuesofv .Forv < 1.0kms−1 theskewnessparameterin- th th inTb∆Ω,withTbthebrightnesstemperature.Thebeamingan- creasesagainforTb∆Ω > 1011afterastrongdecrease.Forthe gle ∆Ω is nota constantpropertythroughoutthe entiremaser higher thermal velocity widths the skewness parameter keeps regime.Itdependsonthemaseropticaldepth|τ|asbeamingis decreasingslightlyinthesaturatedregime. morepronouncedforthe strongermasers.Thebeamingangle Becausetheskewnessparameteristhethirdmomentofthe intheunsaturatedcasesisroughlyproportionalto|τ|−1,while maserprofile,itisstronglyaffectedbyobservationalerrors.To in the saturated case it is proportionalto |τ|−2 (Elitzur 1992). includethe influenceoftheerrors,wehavesimulatedourob- AllcalculationswereperformedwithTb∆Ω=0.1Ksr,forthe servation by adding Gaussian distributed errors on top of the radiationincidentontothemaserregion.ItwasverifiedinNWa model profiles. We have used a signal-to-noise of 100, typi- andV02thattheresultsareinsensitivetothechosenvalue. cal for the weakest of the observedmaser sources. The result is indicated with the shaded areas in Fig. 3, these correspond to 3σ errors. The effect of the errors is strongest in the un- 2.1.Profileshapes saturatedregimeanddecreaseswhenthemasersaturates.The In Fig. 1, we show an example of the line profiles resulting spectralresolutionofthemaserprofilesalsoaffectstheerrorsin from our calculations. The example profiles have been cal- the skewness determination.Higher spectral resolutionobser- culated for two different values of the FWHM thermal width vations have smaller errorsbecause of a more accurate deter- (v ) of the Maxwellian particle distribution. Assuming a ki- minationofthescalingparameterΨandvelocityv ofthepeak th 0 4 W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 2 skewed than the model with ∆V = 0.8 km s−1, while this is m notseeninthemodelswithv = 1.0.Mostmodelsshowless th deviationsofGaussiansymmetryinthesaturatedcase.Allthe modelsinFig.4werecalculatedwithanegativevelocitygra- dientalongthemaserpath.Agradientinthepositivevelocity directionincreasesthenegativeskewnessoftheprofilesforall cases. Sincethe frontside of the envelopehasthe mostnega- tivevelocity,thevelocitygradientthroughtheenvelopetoward theobserverispredominantlynegative,forfeaturesoriginating frombothsidesofthecircumstellarenvelope. We find that a velocity gradient along the maser path increases the maser line width with an approximately equal amount in both the unsaturated as well as the saturated case. As soon as ∆V approaches the intrinsic thermal line width, m themeasuredlinewidthsincreaserapidly.Nedoluha&Watson (1988)haveshownthatfor∆V afewtimesv ,themaserline m th splits intoseveraldistinct, narrowfeatures,each becomingan independentmaser.ThishasbeenobservedforH Omasersin 2 star-forming regions (e.g. Genzel et al. 1979, Walker 1984). OurobservationsofthecircumstellarH Omasersdonotshow 2 thislinesplitting,indicatingthatthelargestvelocityshiftalong themaserpathisnotmorethan≈2.0kms−1. Fig.4.Skewnessandline widthsforfromv = 0.8 (left)and th 1.0km s−1 (right)whenincludinga velocityshift ∆V of 0.4 m (thin solid), 0.6 (short-dashed), 0.8 (dashed-dotted) and 1.2 3. Observations kms−1 (long-dashed)asafunctionofemergingmaserbright- The observations were performed at the NRAO1 Very Long ness T ∆Ω. The thick solid line indicatesthe result with con- b BaselineArray(VLBA)atDecember13th1998aspartofthe stantvelocityalongthemaser path.Due to the computational projectdesignedtomeasurethecircularpolarizationofcircum- requirementswehavenotcalculatedthefullbrightnesstemper- stellar H O masers. The results of the polarization measure- aturerangeforthemodelswhichhaveavelocitygradient. 2 mentsarepresentedinV02.Atthefrequencyofthe6 −5 16 23 rotational transition of H O, 22.235 GHz, the average beam 2 width is ≈ 0.5 × 0.5 mas. This allows us to resolve the dif- intensity.Ourmodelswere determinedwith a spectralresolu- tionof0.027kms−1,similartotheobservations. ferent H2O maser features in the CSE. As explained in V02, the data were correlated twice, once with modest (7.8 kHz = 0.1 km s−1) spectral resolution, and once with high spec- 2.2.Velocitygradients tralresolution(1.95kHz=0.027kms−1).Inthispaperweuse the totalintensity profilesobtainedwith the highspectralres- Obviouslythe line profile can also be determined by velocity olution data. We have performed 6 hours of observations per gradients along the radiation path, giving rise to observable source-calibrator pair. The calibrator was observed for 1 1/2 skewness. Therefore, the calculations above have also been hoursinanumberofscansequallydistributedoverthe6hours. performed including a shift of velocity along the maser path We used 2 filters of 1 MHz width, which were overlappedto (∆V ).Thishasbeendonebyreplacingthepumprateλ(v)in m getavelocitycoverageof≈22kms−1.Thiscoversmostofthe Eq.1forallhyperfinecomponentsbyλ(v+δv),whereδvin- velocityrangeoftheH Omaser. creases for every step along the maser path. The effects have 2 beencalculatedforthemodelswithv = 0.8and1.0 kms−1. th Resultsforthelinewidthsandskewnessparameterareshown 3.1.Sample inFig.4for∆V ofupto1.2kms−1. Inourcalculations,the m Weobserved4latetypestars,thesupergiantsSPer,VYCMa linear velocity gradient along the maser path is characterized andNMLCygandtheMiravariablestarUHer.Theyarelisted by∆V ,thevelocityshiftbetweenthestartandtheendofthe m inTable.1withtype,position,distanceandperiod.Thesources path. The small variationson the lines are due to the spectral wereselectedonthebasisof2criteria;strongH Omasersand resolutionofourmodels,whichisequaltotheresolutionofthe 2 theavailabilityofSiO and/orOH maserpolarizationobserva- observations. tions.Thepeakfluxesmeasuredintheobservationsareshown The skewness is very sensitive to ∆V , especially in the m in Table. 1. On the total intensity maps the noise is between unsaturated case. The effect of ∆V becomes less for models m ≈0.08−0.3Jy. witharelativelylargethermalwidth.When∆V becomessig- m nificantlylargerthanthethermalwidth,theskewnessnolonger 1 The National Radio Astronomy Observatory is a facility of the dependsontheintrinsicwidth.Thisexplainsthat,inFig.4,the National Science Foundation operated under cooperative agreement modelforv = 0.8anda∆V = 1.2kms−1 islesspositively byAssociatedUniversities,Inc. th m W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 5 2 Table1.StarSample Name Type RA Dec Distance Period V Peakflux rad (hm s) (◦′”) (pc) (days) (kms−1) (Jy) SPer Supergiant 022251.72 +583511.4 1610 822 -38.1 76.1 UHer Mira 162547.4713 +185332.867 277 406 -14.5 12.5 NMLCyg Supergiant 204625.7 +400656 1220 940 0.0 48.2 VYCMa Supergiant 072258.3315 -254603.174 1500 2000 22.0 244.1 Table2.Results Name V Flux(I) ∆v Skewness 3σ+ 3σ− rad L skew skew (kms−1) (Jy) (kms−1) SPer -23.2 10.0 0.44 -0.29 +0.11 −0.24 -27.2 76.1 0.88 -2.21 +0.31 −0.43 -27.2 14.2 0.51 0.04 +0.06 −0.10 -28.9 29.7 0.61 4.35 +0.84 −0.66 -30.4 15.4 0.92 -0.10 +0.04 −0.09 -30.9 57.7 0.77 -0.73∗ +0.11 −0.16 -33.9 11.4 1.21 -0.59 +0.10 −0.15 -34.2 14.9 0.48 -0.20 +0.06 −0.14 -37.1 34.7 0.54 -0.25 +0.05 −0.09 -39.2 26.6 0.64 1.31 +0.27 −0.22 -39.3 12.9 0.69 -0.17 +0.06 −0.11 -40.1 35.6 0.53 -0.19 +0.04 −0.06 VYCMa 10.6 155.5 0.67 -0.35∗ +0.05 −0.09 10.8 244.1 0.66 -0.53∗ +0.08 −0.11 12.9 115.2 0.74 -1.19 +0.17 −0.25 13.7 47.9 0.83 -0.11 +0.02 −0.04 15.0 17.8 0.94 0.01 +0.02 −0.05 18.3 58.5 0.80 -0.36 +0.06 −0.08 24.6 126.2 0.60 0.07 +0.02 −0.03 25.2 65.6 0.79 0.34 +0.07 −0.06 27.1 25.3 0.88 0.22 +0.06 −0.06 28.8 43.1 0.61 -0.32 +0.05 −0.06 NMLCyg -20.0 3.66 0.54 0.64 +0.32 −0.36 -20.2 5.42 0.54 -0.14 +0.11 −0.30 -20.4 10.5 0.40 -0.09 +0.08 −0.19 -21.2 48.2 0.66 -0.03 +0.02 −0.03 UHer -17.6 2.91 0.50 5.47 +1.65 −1.60 -17.6 1.16 0.73 0.69 +0.55 −0.63 -17.8 12.20 0.75 0.69 +0.18 −0.16 -19.2 2.46 0.47 -0.63 +0.38 −0.93 -19.2 2.07 0.51 -0.14 +0.33 −0.76 -19.3 1.36 0.51 -0.27 +0.46 −1.14 ∗Badspectralchannelsaffectskewnessdetermination 4. Results noiseratioofthemaserfeature,andonthespectralresolution oftheobservations. We will plot the observed line width and skewness as a TheresultsoftheobservationsareshowninTable2.Thetable function of brightness temperature in order to make a com- givestheLSRvelocityofthemaserfeature,themaserstrength parison with our models. One should realize that there is not inJyandthelinewidthandskewnessparameter.Thefeatures labeled∗ haveafewbadspectralchannelsinthewingsofthe necessarily a single value for width and velocity shift that is valid for all maser features in any star. It is natural to expect profile due to interference. This can have a significant effect theresultstobequitescattered. on the skewness of the profile; we have verified that the val- ueslistedaretheupperlimitsforthesefeaturesandtheactual Figs. 5 and 6 show the observed line widths ∆v and L skewnesscouldbeupto≈1.0lower.Thequoted3σerrorval- Skewnessparametersplottedagainstthelogarithmofthemaser uesoftheskewness,are basedontheerrorsinthedetermina- brightness temperature T . This is calculated using the fluxes b tionofv andσforuseinEq.2.Thesedependonthesignal-to- listedinTable2usingthebeamsizeoftheobservationsasthe 0 6 W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 2 the determined brightness temperatures to be 0 with the real values. 4.1.UHer For the Mira variable star U Her most of the maser features showlinewidthsof≈ 0.5kms−1 with2ofthefeaturesshow- ing widths of ≈ 0.75 km s−1. There is no clear trend with increasing maser brightness. The H O masers around U Her 2 havealsobeenobservedwiththeVeryLargeArray(VLA)by Colomeretal.(2000).Theyobserveline widthsseveraltimes widerthanfoundinourobservations,buttheVLAisunableto resolvetheindividualspotsandtheyonlyusedaspectralreso- lutionof0.33kms−1.Itislikelythatthelargerlinewidthsare duetotheblendingofseveralsmallercomponents. Because U Her has the weakest masers, the errors on the skewness parameter are relatively large. The weighted average of the skewness is 0.55, a possible indication of large ∆V . The skewness is positive for the maser features with m V ≈ −17.5kms−1,whileitisnegativeforthefeatureswith LSR V ≈ −19.0km s−1. Allthe maser featuresare blue-shifted LSR with respect to the stellar velocity of V = −14.4 km s−1. LSR Fig.5.Linewidthsoftheobservedmaserfeaturesasafunction As discussed above, ∆V will be negative because the maser m ofemergingflux.Theerrorbarsarethe3σerrors. exists in a radially accelerating outflow. This results in a fre- quencyshiftalongthemaserfeatureinthedirectionofthehy- perfinecomponents.Ifthefeatureswithavelocityclosertothe stellar velocity are located closer to the star, they will experi- encemoreaccelerationalongthemaserpathandthusalarger velocity gradient.This can explainthat the skewness of these featuresismorepositivethanthethatoftheotherfeatures.The maser feature with the skewness of 5.47 however, is likely a blendoftwooverlappingmaserfeatures. 4.2.NMLCyg The line widths of the maser features around NML Cyg are foundtoliebetween0.4and0.7kms−1.Againnocleartrend canbedetermined,especiallysincewehaveonlybeenableto detect4maserspots.MERLINobservationsbyRichardsetal. (1996) show a much more complex structure with many features.Thelinewidthstheyfindcoveralargerangebetween0.4 and2.5kms−1. The weighted average of the skewness is −0.01. The line widths and skewness, indicate an intrinsic thermal width of between 0.8 and 1.0 km s−1 and ∆V of ≈ 0.75 km s−1. A m larger∆V would resultin a morepositive skewness, while a m larger thermal width does not allow for the observed narrow linewidths. Fig.6. Skewness parameter of the observedmaser features as a functionofemergingflux.Theerrorbarsindicate3σerrors andthedashed-dottedlineistheweightedaverage. 4.3.SPer The line widths of the maser features around S Per show someindicationofanincreasewithmaserbrightness.Theline maser spots are unresolved, thus T is formally only a lower width increases from 0.4 to 0.9 and varies approximately as b limit.However,atthedistanceofthestarthebeamhasadiam- ∆v∝(T )0.3.However,2featuresshowawidthwhichishigher b eterofapproximately1–2×1013 cmandiscomparabletothe than expected. But, as seen in Fig. 4, this can be due to a typicalmaserspotsize (Reid&Moran1981).We thusexpect velocity shift along the path which is more than 1.2 km s−1. W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 7 2 Alternatively, the actual spot size of these features could be Thelargespreadintheobservedskewnessparameterindi- less than 1012 cm leading to an underestimation of T . The catesthatvelocitygradientsalongthemaserpathplayalarge b skewness of the other features indicate a ∆V of not more role in determining the shape of the H O maser line profiles. m 2 than0.8kms−1.Similarvelocityshiftshavebeenobservedby 35%ofthemaserfeatureshaveapositiveskewness,whichonly Richardsetal.(1999).Theobservedlinewidthscorrespondto occursformasersthatarestartingtosaturateandwhichhavea athermallinewidthbetween0.8and1.0kms−1. relativelylargenegative∆V . m Acombinationofboththeskewnessandlinewidthanaly- Theobserved∆v ∝ (Tb)−α dependence,withdifferentval- sisallowsustoestimate,fromourmodels,theactualemerging uesofα(α≈ −0.3forSPerand0.3forVYCMa),canbeex- maser brightness temperature and beaming angle. For the es- plainedbytherelationbetweenbeamingangle∆Ωandmaser timated velocity shifts along the maser path and thermal line opticaldepth|τ|.Asnotedbefore,inthecompletelyunsaturated widths, our models give observed line widths of ≈ 0.4−0.9 case∆Ω∝ |τ|−1,whilethisincreasesto∆Ω∝ |τ|−2inthesatu- km s−1 for T ∆Ω between 1010 and 1011 K. With observed ratedcase.Additionally,intheunsaturatedregimetheamplifi- b brightnesstemperatureTb = 1012–1013, thisindicatesthatthe cationisexponential,soTb ∝e|τ|,whileinthesaturatedregime beamingsolidangle∆Ω≈10−2sr. the amplification becomes linear and Tb ∝ |τ|. This means that, while unsaturated, the beaming angle ∆Ω ∝ (ln T )−1, b and when saturated ∆Ω ∝ T−2. Meanwhile, our models in 4.4.VYCMa b Fig. 2 show, that before the maser starts re-broadening the line width is almost constant with increasing T ∆Ω. after re- The line widths of the H O masers around VY CMa show a b 2 broadening, when saturation occurs, the line widths becomes cleardependenceonthebrightnesstemperature.Thelinewidth varies with ∆v ∝ (T )−α, where α ≈ 0.3. A similar decrease approximatelyproportionalto(Tb∆Ω)0.5.Usingtheresultsfor b the variation of ∆Ω with T we then find that for unsaturated oflinewidthwithintensitywasobservedbeforeonthemaser b masers∆v ∝ (T )0.7, while the line width of saturatedmasers flareinCepheusA(Rowland&Cohen1986),thoughthereα≈ b 0.5.Weobservelinewidthsfrom0.6to0.94kms−1.MERLIN will be proportional to (Tb)−0.5. The value of α can thus in- crease from ≈ −0.7 in the mostly unsaturated regime to 0.5 observations(Richardsetal.1999)havefoundmanyfeaturesto haveavelocityextentofupto3kms−1,indicatingthepresence inthesaturatedregime.Thisseemstoindicatethatthemasers around S Per are still mostly unsaturated, while those around oflargevelocityshiftsalongthemaser. VY CMa areclose tobecomingsaturated.However,thepres- ForVYCMa,theskewnessandlinewidthsindicateather- ence of velocity shifts along the maser path complicates this mal width in the H O maser region of ≈ 1.0 − 1.2 km s−1, 2 straightforward explanation. For example, if the high bright- with∆V upto1.25kms−1.Usingthis,accordingtoourmod- m ness maser features trace the lower velocity shifts because a els,T ∆Ωisbetween1010to1011K,whiletheobservedlower b strong gradient disrupts the maser growth, we can also ob- limitforT ≈1012.4K.Thisindicatesabeamingsolidangleof b serve a decrease of the line width with increasing brightness. ≈10−3sr. Additionally,velocityshifts willintroducea bigspreadin the relationbetweenlinewidthandmaserbrightness. An additionaluncertainty in the analysis is introduced by 5. Discussion assuming a linear maser model. While astrophysical masers, The non-LTE method used here to model the shape of the andespeciallyH Omasers,arethoughttobehighlyelongated, 2 22 GHz H O maser spectrum was previously used to deter- theactualthree-dimensionalshapeofthemaserwillaffectthe 2 mine the circular polarizationof the maser in the presence of maser line width and shape as described by Elitzur (1994). amagneticfield(V02).Therethenon-LTEmodelingmethod, There it was shown that the linear maser model is sufficient is compared with a standard LTE method which is described to describe the maser profile out to ∼ 2v for spherical and th inVlemmingsetal.(2001).Thenon-LTEapproachwasdevel- ∼ 3–5v forelongatedcylindricalmasers.However,crossing th oped to analyze circular polarizationspectra and successfully maser beamswillsuppresstheemission in theouterwingsof reproducedthedistributionovermagneticcomponents.Inprin- the maser profile. This can cause narrowingof the maser line ciple,similarphysicalprocessesareimportantforunderstand- tobeobservedevenforsaturatedmasers.Inthecaseoftheob- ing the line shapes of the H O maser blend and we apply the jectsstudiedinthispaper,theeffectisexpectedtobesmallor 2 sameanalysishere.Recently,Watsonetal.(2002)haveuseda evennegligible,asthe suppressionofthe maser profilewings similar methodto examinethe line shapesof interstellar H O only becomes significant when the maser becomes saturated. 2 masers,althoughtheiranalysishasbeenbasedononlyatwo- Then,asshowninElitzur(1994),thesuppressioneffectsonthe levelmaser transition,thusonly1 hyperfinecomponentis in- profileareonlynoticeablewhenthedynamicrangeofthespec- cluded.Insteadofusingtheskewnessofthemaserprofile,they trumis large.Fora filamentarymaser,the dynamicrangehas examine the second order deviations of Gaussian symmetry, tobehigherthan≈7×103,whilethelargestdynamicrangefor the ’kurtosis’. They find that the interstellar H O masers are ourmasersourcesis≈ 7×102 forthestrongestmaserfeature 2 foundingaswithtemperaturesgreaterthan1200K,forwhich ofVYCMa.Ifthemaserswerepurelysphericalhowever,the the influence of multiple hyperfine components is thought to measured linewidth decrease for increasing maser brightness besmall.SincethegasinCSEsisatalowertemperature(e.g. forthe VY CMa H O maserscouldbepartlydueto thisgeo- 2 Goldreich & Scoville 1976), the effect of the other hyperfine metriceffect.Asaresulttheestimatedintrinsicthermalwidth componentsislarger,asisindeedseenourresults. andthebeamingsolidanglemightbeslightlyunderestimated. 8 W.H.T.VlemmingsandH.J.vanLangevelde:H Omaserlineprofilesanalysis 2 The presence of large velocity gradients along the maser Colomer,F.,Reid,M.J.,Menten,K.M.,Bujarrabal,V.,2000, path,andtheuncertaintyintheexactmasergeometry,increases A&A,355,979 the difficulty of determining underlyingthermal widths, level Deguchi,S.,Watson,W.D.,1986,ApJ,302,750 of saturation and beaming angle. However, from comparison Elitzur,M.,1990,ApJ,350,L17 withthe modelswe havebeenabletogiveestimatesformost Elitzur, M., Astronomical Masers, Kluwer Academic ofthesourcespresentedhere.Ingeneralthethermallinewidth Publishers (Astrophysics and Space Science of the maser medium in our sources is between 0.8 and 1.0 Library.Vol.170),365p. kms−1.Thislinewidthcanbecausedbytheactualtemperature Elitzur,M.,1994,ApJ,422,751 of the maser gas, as well as by turbulent motions. The H O Genzel,R.,Downes,D.,Moran,J.M.,etal.,1979,A&A,78, 2 maser region should not have temperatures higher than T ≈ 239 1000K(Neufeld&Melnick1990),whichimpliesthattheline Goldreich,P.,Keeley,D.A.,1972,ApJ,174,517 widths due to the thermal motions should not be higher than Goldreich,P.,Kwan,J.Y.,1974,ApJ,190,27 ≈ 1.5kms−1. Thisisconsistentwith ourobservations,which Goldreich,P.,Scoville,N.,1976,ApJ,205,144 indicate a temperature of T ≈ 400 − 600 K. The line width Imai,H.,Sasao,T.,Kameya,O.,etal.1997,A&A,317,L67 due to turbulentmotions should then be of the same order of Kukolich,S.G.,1969,J.Chem.Phys.,50,3751 magnitudeor less. Thisis significantly less than the turbulent Lane,A.P.,Johnston,K.J.,Bowers,P.F.,etal.1987,ApJ,323, widthdeterminedbyColomeretal.(2000),whogiveawidthof 756 the orderof 2−4 km s−1. But as discussed abovethey found Litvak,M.M.,1970,Phys.Rev.A,2,2107 larger line widths, possibly due to the fact that the VLA was Nedoluha,G.E.,Watson,W.D.,1988,ApJ,335,L19 notabletoresolveindividualmaserfeatures. Nedoluha,G.E.,Watson,W.D.,1991,ApJ,367,L63 Similar to earlier observations (e.g. Spencer et al. 1979), Nedoluha,G.E.,Watson,W.D.,1992,ApJ,384,185(NWa) our observations suggest that the circumstellar H O masers Neufeld,D.A.,Melnick,G.J.,1990,ApJ,352,L9 2 are either unsaturated or only slightly saturated, with intrin- Richards, A.M.S., Yates, J.A., Cohen, R.J., 1996, MNRAS, sicbrightnesstemperaturesbetweenT =1010−1011K.From 282,665 b themodelswecanestimatethetypicalbeamingangletobebe- Richards, A.M.S., Yates, J.A., Cohen, R.J., 1999, MNRAS, tween∆Ω=10−2−10−3sr,similartovaluesfoundbefore(e.g. 306,954 Nedoluha&Watson1991). Reid,M.J.,Moran,J.M.,1981,ARA&A,19,231 Rosen,B.R., Moran,J.M.,Reid, M.J., etal. 1978,ApJ, 222, 132 6. Conclusions Rowland,P.R.,Cohen,R.J.,1986,MNRAS,220,233 Adetailedanalysisofthelineprofilesof22GHzH Omasers Spencer,J.H.,Johnston,K.J.,Moran,J.M.,etal.1979,ApJ, 2 can be a valuable tool to estimate saturation, thermal veloc- 230,449 ity widthandvelocityshiftalongthe maserpathin themaser Vlemmings, W., Diamond, P.J., van Langevelde,H.J., 2001, region.Becausethe22GHztransitionconsistsofmultiplehy- A&A,375,L1 perfine components, the maser profile shows clear deviations Vlemmings, W., Diamond, P.J., van Langevelde,H.J., 2002, fromsymmetryat variousstages of saturation. Unfortunately, A&A,394,589(V02) theeffectofvelocityshiftsalongthemaser path,cannoteas- Walker,R.C.,1984,ApJ,280,618 ily be detached from the effects of hyperfine contributions. Watson,W.D.,Sarma,A.P.,Singleton,M.S.,2002,ApJ,570, Additionally, for VY CMa, the actual unknownmaser geom- L37 etryintroducesanotherlevelofcomplexity. However, we have been able to obtain some estimates on thethermallinewidthsandbeaminganglesontheH Omasers 2 around a sample of 4 late-type stars. It is clear that velocity shifts along the maser path of up to 1.25 km s−1 play an im- portantrole,andwillhavetobetakenintoaccountforfurther comparisons between models and observations. The thermal linewidthsinthemaserregionsarefoundtobemostlyaround 1.0kms−1,indicatingtemperaturesofapproximately400Kin themasingregion.Mostmasersarethoughttobeonlyslightly saturatedatthemost,withTb∆Ω<∼ 1011K.Beamingsolidan- glesforthecircumstellarH Omasersarefoundtobetypically 2 between10−2and10−3sr. acknowledgments: WV thanks the Niels Stensen FoundationforpartlysupportinghisstayatCornellUniversity. References Anderson,N.&Watson,W.D.,1993,ApJ,407,620