Astronomy&Astrophysicsmanuscriptno.VanLoo February5,2008 (DOI:willbeinsertedbyhandlater) Non-thermal radio emission from single hot stars S.VanLoo,M.C.Runacres, andR.Blomme RoyalObservatoryofBelgium,Ringlaan3,B-1180Brussel,Belgium Received/Accepted 4 0 Abstract. Wepresentatheoreticalmodelforthenon-thermal radioemissionfromsinglehotstars,intermsofsynchrotron 0 radiationfromelectronsacceleratedinwind-embedded shocks.Themodelisdescribedbyfiveindependent parameterseach 2 withastraightforwardphysicalinterpretation. Applyingthemodel toahigh-quality observation ofCygOB2No.9(O5If), n weobtainmeaningfulconstraintsonmostparameters.Themostimportantresultisthattheouterboundaryofthesynchrotron a emissionregionmustliebetween500and2200stellarradii.Thismeansthatshocksmustpersistuptothatdistance.Wealsofind J thatrelativelyweakshocks(withacompressionratio< 3)areneededtoproducetheobservedradiospectrum.Theseresults 0 are compatible with current hydrodynamical predictions. Most of our models also show a relativistic electron fraction that 2 increasesoutwards.Thispointstoanincreasingefficiencyoftheaccelerationmechanism,perhapsduetomultipleacceleration, oranincreaseinthestrengthoftheshocks.Implicationsofourresultsfornon-thermalX-rayemissionarediscussed. 1 v Key words. stars: early-type – stars: mass-loss – stars: winds, outflows – radio continuum: stars – radiation mechanisms: 9 non-thermal 8 3 1 0 1. Introduction sil magnetic field, or a field created by a dynamo mechanism 4 (MacGregor&Cassinelli2003). 0 ManyhotstarsofspectraltypeOandBareobservableatradio h/ wavelengths, due to thermal emission from the circumstellar The model proposed by White (1985) was further devel- opedbyChen&White(1991,1994),whoshowedthatasyn- p ionized gas in the stellar wind. The thermal radiation is free- chrotronmodelcansuccessfullyreproducetheradiospectrum - freeemission(Bremsstrahlung)fromanelectronacceleratedin o of9SgrandCygOB2No.9. theCoulombfieldofanion.Theemergentradiospectrumhasa r t characteristicspectralindex1α 0.6(Wright&Barlow1975), Inthepresentpaper,weinvestigatewhatpropertiesasyn- s ≈ a whichfitstheobservationsformostOstarsquitewell. chrotronmodelshouldhaveinordertoreproduceanobserved : v However, 25% of the brightest O stars have a radio spec- non-thermalradio spectrum.More specifically, we look at all Xi trum with a spectral index which is very different from the modelsthatfit a givenset of observationsandthusderivethe thermal wind emission (Abbott et al. 1984). It is believed permittedrangeofanumberofparameters.Thestringencyof r a that this non-thermal radio emission is synchrotron radiation theconstraintontheparametersdependscriticallyonthequal- by relativistic electrons (White 1985). The preferred mech- ity of the available observations,as well as on the number of anism to accelerate electrons to relativistic energies is parti- observedwavelengths.WeselectedCygOB2No.9astheob- cle accelerationin shocks(Fermi1949), asshocksare known jectofthisstudybecauseasetofthreesimultaneousobserva- to exist in stellar winds. In single stars, shocks are gener- tions(at2,6and20cm)withsmallerrorbarsisavailable. ated by the instability of the driving mechanism of the wind Forthe sake of clarity itis desirable thatthe propertiesof (Owocki & Rybicki 1984). For binaries, shocks also arise in themodelbecontainedinalimitednumberofparameterswith colliding winds (Eichler & Usov 1993). Non-thermal radio aclearphysicalinterpretation.Ratherthancalculatingthemo- emission fromcollidingwind binarieshasbeen furtherinves- mentum distribution of relativistic electrons from a model of tigatedbyDoughertyetal.(2003).Inthispaperwe limitour- theshocksinthestellarwind,weuseapower-lawdependence selvestosinglestars. onthecompressionratio.Theuncertaintyabouttheexactshock Besides shocks, a magnetic field is needed to produce a structure is so large that an ab initio calculation of the mo- synchrotronspectrum.Magneticfieldsofnormalhotstarsare mentumdistributionwouldleadtoagreatercomplexityofthe below current detection limits ( 100 Gauss). It is plausible modelwithoutincreasingitsphysicalaccuracy. ∼ thatasmallmagneticfieldispresentinthestar,eitherthefos- The remainder of this paper is organised as follows. In Sect. 2 we describe the model for the synchrotron radiation Sendoffprintrequeststo:S.VanLoo,e-mail:Sven.VanLoo@oma.be indetail.Theeffectofcoolingmechanismsonthemomentum 1 TheradiospectralindexαisdefinedbyF να λ α distribution is investigatedin Sect. 3. In Sect. 4 we apply the ν ∝ ∝ − 2 S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars model to Cyg OB2 No. 9. Finally, we discuss the results in of0.1Gauss,radiationbetween2 20cm(15 1.4GHz)comes − − Sect.5andgivesomeconclusions. from electrons with momenta between 50 150 MeV/c, ≈ − wherewesetsinθ=1. From Eq. (3) we see that the Razin effect is important 2. Themodel when ν ν p/(m c). If on the other hand ν ν p/(m c), 0 e 0 e ≪ ≫ 2.1.Power the influence of the Razin effect is small. We can then use Eq. (2) to substitute p and find the frequenciesfor which the Inthepresenceofshocksandamagneticfield,electronsareac- influence of the Razin effect is small. In the literature (e.g. celeratedto highvelocities(Bell1978)throughthe first-order Ginzburg&Syrovatskii1965),thisisgivenby Fermimechanism(Fermi1949)andemitsynchrotronradiation n astheygyratearoundthemagneticfieldlinesatrelativisticve- ν 20 e (5) ≫ B locities. The synchrotron power radiated per unit volume per unitfrequencybyasingleelectronwithmomentump,moving where, in view of the approximatenature, the factor 0.29 has inamagneticfieldB(dependentondistancer)withpitchangle notbeentakenintoaccount.Inatypicalhot-starwind,thether- θ,canbeexpressedas(Westfold1959) malelectrondensityat100R isoforder107cm−3.Foramag- neticfieldof0.1Gaussthism∗eansthattheRazineffectbegins √3e3 to suppress radiation at wavelengths larger than 15 cm, well P (p,θ,r) = Bf(ν,p)sinθ ν m c2 within the observable range. For smaller magnetic fields, the e ν radiationissuppressedatevensmallerwavelengthse.g.1.5cm F , (1) × f3(ν,p)ν (p,r)sinθ! fora0.01Gaussfield. s wherem ande are the electronmassandcharge,c the speed e 2.3.Flux of light, ν the frequency, F(x) = x x∞dηK5/3(η) and K5/3 the modified Bessel function. For theRmeaning of f(ν,p), see ThesynchrotronemissivityofadistributionN(p,r)ofrelativis- Sect. 2.2.The synchrotronpower spectrumextendsup to fre- ticelectronscanbeexpressedas quenciesof orderν beforefalling away,with the critical fre- s 1 pc quencyνs definedas jν(r)= dpN(p,r)P¯ν(p,r), (6) 4πZ 3 eB p 2 p0 ν (p,r)= . (2) where P¯ (p,r) is the synchrotron power P (p,θ,r) averaged s 4πm c m c! ν ν e e over solid angle assuming that the electron velocity distribution is locally isotropic.The low (high)momentumcut-off of 2.2.Razineffect theelectrondistributionisdefinedby p (respectively p ).We 0 c willspecifythecut-offvaluesinSect.3. The presence of f(ν,p) in the synchrotron power is a conse- Theemergentfluxisgivenby quenceoftheTsytovitch-Razineffect(orRazineffectforshort; Tsytovitch 1951; Razin 1960). Relativistic beaming plays a 1 Rmax F = dr4πr2j (r), (7) dominant roˆle in the explanation of synchrotron emission. In ν D2 Z ν the presence of a cold plasma, the refraction index is smaller Rν whereDisthedistancetothestar.Theintegrationboundaries than unity. This increases the beaming angle, thereby reduc- ing the beaming effect. Thus the synchrotron emitting power reflect the fact that, at any given radio frequency,only a lim- itedpartofthewindcontributestotheflux.Thechoiceofthese is greatly reduced. Ginzburg & Syrovatskii (1965) expressed boundariesisrathersubtle.Thelowerboundarymimicstheef- f(ν,p)as fectofthelargefree-freeopticaldepthofthewind,whicheffec- ν2 p 2 −1/2 tivelyshieldstheinnerpartofthesynchrotronemittingregion f(ν,p)=1+ ν02 m c!  , (3) from observation.The lower boundarymust increase with in-  e  cdreepatshinogftwhaevwelienndg.tIht,istocruesfltoemctatrhye(ien.gc.reWashinitgef1r9e8e5-f)retoeuospetitchael withν = nee2 theplasmafrequencyandn thenumberden- 0 qπme e characteristicradiusfromWright&Barlow(1975)asthelower sityofthermalelectrons. boundary.However,thisradiuscorrespondsto a smalloptical IntheabsenceoftheRazineffect(f(ν,p)=1),theessential depth (τ 0.25) and was introduced to produce the correct ν partofthesynchrotronemissionofanindividualelectronwith ≈ thermalradioflux with a simple one-dimensionalintegral.Its momentumpisemittedbelowthecriticalfrequencyν ,witha s physicalmeaningis often misstated and is actuallyquite lim- maximumat ited. It is not true that all observed radio emission originates Bsinθ p 2 fromaboveRν.ItisquiteeasytoreplaceEq.(7)bythecorrect ν=0.29νssinθ 1.3 MHz. (4) expression which is, however, rather cumbersome. We there- ≈ 1Gauss! m c! e forerelegateitsderivationtoAppendixAandprefertousethe Away from the maximum, the synchrotron radiation falls off simple approximationEq. (7) in the discussion of ourresults. rapidly.We can thereforeuse Eq. (4) to estimate the momen- Thecalculationsweredoneusingtheexactexpressiongivenin tumrangerelevantfortheradiospectrum.Foramagneticfield Eq.(A.6). S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars 3 As for the outer integration boundary, it is well known 10 thatinstability-generatedshocksdecayas they moveoutwith the wind (e.g. Runacres & Owocki 2002). It can thereforebe expected that they eventually become too weak to accelerate 8 electronstorelativisticenergies.Becauserelativisticelectrons rapidly lose momentum as they move away from the shock 5 GHz (Sect.3.4),theouterboundaryofthesynchrotronemissionre- 6 gshioonck(Rstmraoxn)gceannoaulgsohtboeavcicseulaelriasteedealesctthroenpsotsoitiroenlatoivfisthtiecleans-t F (mJy)ν 15 GHz 1.4 GHz ergies. It does not depend on frequencyand is a fundamental 4 parameterofourmodels. model parameters: The total emergent flux has a minor contribution from free-free emission by thermal electrons. This contribu- 2 Bn==21.8050; Gδ=a1u;s Rs;m fa=x=17.0126× R10*;−7; tion is calculated using the Wright & Barlow formalism p*=1 MeV/c; p=*15 GeV/c 0 c (Wright&Barlow1975)andaddedtothenon-thermalflux. 0 5 10 15 20 25 λ (cm) 2.4.Numberdistributionandmagneticfield Fig.1.ThefulllinerepresentsamodelthatfitstheVLA-observations Inordertoderiveanexplicitexpressionfortheemissivity,we (21 Dec 1984) for Cyg OB2 No. 9 (Bieging et al. 1989), while the need to specify the number distribution N(p,r) and the mag- dashedlineneglectstheRazineffect.Thediamondsaretheobserva- tionsandtheerrorbarisgivenbythesizeofthesymbol.Thestellar neticfieldB(r). parametersarelistedinTable1. It was shownby Bell (1978) that the accelerationof elec- tronsthroughtheFermimechanismresultsinapower-lawmo- mentumdistributionwithexponentn,wherenisafunctionof the velocityhasnotbeentakenintoaccount),butmerelypro- thecompressionratioχoftheshock:n = (χ+2)/(χ 1).For videsthecorrectscalingfactorfortheelectrondensityatlarge − reasons of simplicity, we also assume that the distribution is distances. a continuousfunction of distance, given by a power law with In Eq. (1) the magnetic field B must be specified. exponentδ.N(p,r)isthenexpressedas Unfortunately,the magnetic fields of hot stars are below current detection limits. Therefore, we can say little about their N(p,r)= N r −δp n, (8) radialdependence.For wantof anythingbetter,we willadopt 0 R ! − theexpression(Weber&Davis1967) ∗ where N0 isa normalisationconstantrelatedtothenumberof B(r)= B vrotR∗, (11) relativisticelectrons.Thetotalnumberofrelativisticelectrons ∗v r atthestellarsurfaceisgivenby pcdpN p n.Thisgives ∞ p0 0 − where B is the surface magnetic field and vrot the rotational R velocity∗of the star. This expressionis only valid at largedis- N = f n (n 1)pn 1, (9) 0 ∗ ∗e − 0− tancesfromthestar(r≥10R∗),whichcoverstheregionwhere the synchrotron radiation is emitted. For the surface mag- where we introduced f and n as the fraction of relativistic ∗ ∗e netic field,the detectionlimit, whichis oforder 100Gauss electrons and the total electron number density at the stellar ∼ (Mathys1999),servesasanupperlimit.Atentativelowerlimit surface. We also assumed p p (see Sect. 3). The radial c ≫ 0 can be derived from the synchrotron emission itself. At local variation of the fraction of relativistic electrons in the wind magnetic fields smaller than 0.005Gauss at 100 R , all ra- is described by f∗(r/R∗)−δ+2. It must be emphasised that f∗ is dio emission would be suppr∼essed by the Razin effec∗t. Using merelyaparameterusedtosettheoveralllevelofthisfraction. Eq. (11) and v /v = 250/2900,this translates to a surface Itshouldnotbe too strictly interpretedas a physicalquantity. rot ∞ magneticfieldoforder5Gauss. Indeed,closetothestellarsurface,therelativisticelectronpop- OnceachoicefortheparametersB ,n,δ,R , f , p and ulationmaybestronglysuppressedbyinverseComptoncool- max 0 ∗ ∗ p hasbeenmade,Eq.(A.6)canbeintegratednumerically.The ingintheintenseUV-radiation.Therefore f neednotrepresent c ∗ total emergent flux (i.e. non-thermal+ thermal) for one such theactualrelativisticelectronfractionatthestellarsurface. modelisshowninFig.1. Forn weuse ∗e γM˙ n = , (10) 2.5.Tests ∗e 4πR2v µm H ∗ ∞ IfweneglecttheRazineffectandapproximatethecut-offval- where M˙ isthemasslossrate,v theterminalvelocityofthe ues p and p by 0 and , the emissivity can be calculated 0 c ∞ ∞ stellarwind,m themassofahydrogenatom,γthemeannum- analytically(e.g.Rybicki&Lightman1979)fromEq.(6).The H berofelectronsperionandµthemeanionicweight.Forsim- expressionfortheemissivitythenbecomesapowerlaw: j ν ∝ plicity we assume γ = 1 and µ = 1.4. Note that n is notthe ν (n 1)/2. Introducing this into Eq. (7), an analytic expression ∗e − − actualelectrondensityatthestellarsurface(asthevariationof isfoundforthesynchrotronflux(Ginzburg&Ozernoy1966), 4 S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars which can then be compared to the results of the numerical 3.2.Lowmomentumcut-off code. Synchrotron emission at wavelengths shorter than 20 cm Inthecode,themomentumintegraloftheemissivity,given is produced by relativistic electrons with momenta above inEq.(6),iscomputedusingthetrapezoidalruleandthestep 30 MeV/c. Any value p 30 MeV/c can therefore be 1 size isrefineduntilitreachesa desiredfractionalprecisionof u≈sedasapracticallowerlimitt≪othemomentumrange.Acon- 10−3.Thisemissivityhastobecalculatedatdifferentpositions venientchoiceof p1 is p1 = 1MeV/c.Althoughthechoiceof in the wind. Then, the spatial integral of the flux, Eq. (7), is p doesnotinfluencetheflux,itdoesinfluencethenumberof 1 treated in the same way as the momentum integral, but up to electrons(asseeninEq.(9)). a precision of order 1%. The number distribution is given by Westillneedtocheckwhetherelectronsofmomentum p 1 Eq.(8),whilethemeansynchrotronpowerP¯νiscalculatedus- exist,i.e.whether p1 > p0 (where p0 isthemomentumcut-off ing a linear interpolationbetween tabulated values. This way, set by Coulomb cooling). We therefore compare the acceler- the mean synchrotron power P¯ν needs to be calculated only ation time of electrons with momentum p1 = 1 MeV/c with once (as a function ν/f3(ν,p)νs(p,r)) which reduces the cal- thetime-scaleforCoulombcollisionsofelectronswiththermal culationtimesignificantly.Thenextstepistoapproximatethe ions.Collisionshappenontime-scalesgivenby(Spitzer1956) momentumcut-offvalues,p andp by0and .Westartwith 0 c ∞ m2v3 p0 = 1 MeV/c and pc = 10 GeV/c and then we expand the tD = 8πe4ne lnΛ, (12) intervalbydoublingthevaluefor p andhalvingthe valueof g c p until the solution for the integral has converged. We can wherevisthespeedoftherelativisticelectrons,n thenumber 0 g thencomparethenumericalresultstotheanalyticsolution.We densityoftheionsandlnΛtheCoulomblogarithm.Inhot-star calculated 5000 models with B in the range 10–100 Gauss, windsthe Coulomblogarithmis 10.The time to accelerate ∗ ∼ n between 1.5–7.5, δ between 0–5 and R between 100– anelectronuptoamomentumpisgivenby(Chen1992) max 10000 R . The relative errors of the numerical models com- 3 pc paredtot∗heanalyticmodelsdidnotexceedthe1%level. tacc = 2∆u(r)eB, (13) where ∆u(r) is the shock velocity difference at radius r. For anelectronof1MeV/cataradialdistanceof100R wherethe numberdensityoftheionsisn =107cm 3andB=∗0.1Gauss, g − 3. Coolingmechanisms wefindt 105 s,whilet 5 10 3 s(whereweassume D acc − ≈ ≈ × ∆u = 100 kms 1). Further out in the wind the ratio t /t − D acc 3.1.Practicalmomentumlimits increases.Thismeansthatelectronswithmomentum1MeV/c willnotbeCoulomb-cooledsignificantly. Inthissectionwedeterminepracticalintegrationlimitsforthe momentum distribution, which we need to calculate the synchrotronemissivity,givenbyEq.(6). 3.3.Highmomentumcut-off Relativistic electrons can lose their energy through a va- Synchrotron emission at wavelengths greater than 2 cm riety of cooling mechanisms. Coulomb collisions, inverse is produced by relativistic electrons with momenta below Compton scattering, adiabatic cooling, synchrotron cooling 500 MeV/c (for B = 100 Gauss). Any value p 2 and Bremsstrahlung cooling can all play a roˆle. The effect 5≈00MeV/ccantherefor∗ebeusedasapracticalupperlimit≫to of these cooling mechanisms has been discussed in detail by themomentumrange,providingp < p (where p isthehigh 2 c c Chen (1992). At the high energy end of the spectrum, elec- momentumcut-off). trons are mainly cooled by inverse Compton scattering. This Thehighestmomentumattainableforrelativisticelectrons setsanupperlimitpctothemomentumrange.Atthelow(non- isdeterminedbythebalancebetweenaccelerationandinverse relativistic) end of the spectrum, Coulomb collisions are the Compton scattering at the shock. This results in a maximum dominant cooling mechanism. This sets a lower boundary p0 attainablemomentumpc,thatcanbeexpressedas(Chen1992) to the momentum range. As momenta below p or above p 0 c eB(r) 1/2 caafnrancottioonccoufr,ththisermanogmeeancttuumallryacnognetirsibluimteistetodotobs[epr0v,apbcl]e.rOandliyo pc(r)=mec 4πσTL ∆u(r)r2! , (14) ∗ emission,ascanbeseenfromEq.(4).ForB =100Gaussand whereσ istheThomsoncrosssectionandL thestellarbolo- T assumingthatthe synchrotronradiationis f∗ormedsomewhere metricluminosity.Usingtypicalhotstarvalu∗es(L = 106L , between30and1000R (seeSect.4.2.1),therangeofmomenta v = 2500 kms 1, v = 200 kms 1, R = ∗20 R an⊙d − rot − thatcontributetotheob∗servableradiofluxis30 500MeV/c. ∆∞u(r) = 100 kms 1), we can estimate a num∗ erical val⊙ue for − − Itisthereforeconvenienttointroduceanad-hocintegrationin- thehighmomentumcut-off, terval[p ,p ] thatis containedin [p ,p ] butis largeenough 1 2 0 c GeV r 1/2 toincludeallmomentathatproduceobservableradioemission. p (r) 0.97 B . (15) c Withinthese constraintswe choose[p ,p ] assmallaspossi- ≈ c ∗R ! 1 2 ∗ ble, for reasonsof computationalefficiency.The exactchoice For B = 100 Gauss and r = 10 R , Eq. (15) gives p = c of p and p isunimportant.Inthefollowingsectionweshall 30GeV∗ /c.Atlargerdistancesfromthe∗starp islarger.Aprac- 1 2 c derivevaluesof p and p thatcanbeusedinthemodels. ticalchoicefor p thatsatisfies p < p is p =15GeV/c. 1 2 2 2 c 2 S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars 5 Table1. Relevant stellar parameters for Cyg OB2No. 9 adopted in for a thermal wind source (see Fig. 1). Also, the radio emis- thispaper. Thenumbers aretaken fromBieginget al.(1989,BAC), sion of Cyg OB2 No. 9 is highly variable. On one occasion Leitherer(1988,L)andHerreroetal.(1999,HCVM). the radio spectrum even showed a typical free-free spectrum (with α = 0.6). If we determine the mass loss rate from this Parameter Ref. radioobservation,wefindthesamevalueasderivedfromHα v sini(kms 1)a 145 BAC rot − (Leitherer 1988). So we can assume that at that time we saw T (K)b 44500 HCVM eff theunderlyingfree-freeemissionfromthestellarwind.Ofthe R (R ) 22 HCVM d∗(kpc⊙) 1.82 BAC observationalradiodatathatareavailableforCygOB2No.9, we will use the observationthathasdetectionsof emissionin v (kms 1) 2900 L − M˙∞(M yr 1) 2.0 10 5 L three radiowavelengthbands.We chose the 21Dec 1984ob- − − ⊙ × servation with the VLA (Bieging et al. 1989) because of its a WefollowWhite(1985)inassumingthatv = 250kms 1,which smallerrorbarsonthedetections.Byfittingtheseobservations rot − isnotincontradictionwiththevalueforv sini. wecanderivethesolutionspaceofthemodelparameters.The rot bThewindtemperatureisassumedtobe0.3timesthestellareffective stellarparametersadoptedinourmodelarelistedinTable1. temperature(Drew1989). 4.2.Results Table 2. With the stellar parameters for Cyg OB2 No. 9 given in Table 1, we have the following values for the characteristic radius We recall that the parameter space is spanned by n, δ, f , ∗ Rν (see Eq. (A.7)) and for the free-free emission from the wind Rmax and B . The parameters have a straightforwardinterpre- (Wright & Barlow 1975). For simplicity we assume γ = 1,Z2 = 1 tation:ncor∗respondstoatypicalcompressionratiointhesyn- andµ=1.4.Asareferencewealsogivethevaluesofthenon-thermal chrotronformationregion,δ describesthe spatial dependence componenttothefluxandtheobservedfluxwitherrorbars. ofthenumberdistribution, f describesthefractionofrelativis- ∗ ticelectronsatthesurfaceandR istheouterboundaryofthe max λ ν Rν(Twind) Fνff Fνnt Fνobs synchrotronemission region.We take B = 100 Gauss in the (cm) (GHz) (R ) (mJy) (mJy) (mJy) ∗ fitting procedure,which correspondsto a local magneticfield ∗ 2 15 124 1.1 4.6 5.7 0.1 6 5 266 0.6 6.8 7.4±0.1 ofB=1.7×10−2Gaussatr=500R∗. ± Wecalculatethesynchrotronfluxinthethreewavelengths 20 1.4 614 0.3 4.6 4.9 0.1 ± foragridinn,δ,R and f andselectthosecombinationsof max ∗ the parameters that fit the VLA-observationswithin the error bars. This procedureproducesa strong constrainton n, δ and Now thata choicehasbeenmadefor p and p , theinde- 1 2 R , but not on f . The parameter f is well constrained lo- max pendentparametersarereducedtoB ,n,δ,Rmaxand f . cally(i.e.foreachc∗ombinationofn,δ∗andR ),butnotglob- ∗ ∗ max ally(i.e.fortheentirerangeofn,δandR ).Thisparameter max 3.4.Emittingregion doesplay a minorroˆle in limitingthe otherparametersas the numberofrelativisticelectronscannotexceedthetotalnumber Wehaveestimatedpracticalupperandlowerlimitstothemo- of electrons.In principlethislimitation dependson p , butin 1 mentumrangethatare compatiblewith the highandlow mo- practicetheeffectisminimal.Allpossiblecombinationsofn, mentum cut-offs due to Compton and Coulomb cooling, re- δandR liewithintheboomerang-shapedregioninFig.2. max spectively.Ofcourse,coolingcanalsochangethedistribution between these limits. Specifically, as the relativistic electrons moveawayfromtheshockinwhichtheywereaccelerated,in- 4.2.1. Rmax verseComptonscatteringwillrapidlycoolthemostenergetic ThemostimportantresultfromFig.2isthatR iswellcon- electrons.InverseComptonscatteringdoesnotchangethetotal max strainedandalmostnotinfluencedbynandδ.Theouterbound- number of relativistic particles, but quickly reduces their en- aryofthesynchrotronemittingregionmustliebetween520R ergybelowtheenergyofobservableradioemission.Thus,the ∗ and750R .Tounderstandthisresult,itisimportanttorealise synchrotron emitting gas is concentrated in more or less nar- thatthediff∗erentwavelengthsplayadifferentroˆleinthefitting row regions behind the shocks. For the sake of simplicity we procedure.Thefluxesat2and6cmconstrainn,δand f .The neglect this layered structure of the synchrotron emitting re- parametersnandδdeterminetheslopebetween2and6c∗mand gionandassumea continuousvolumeofrelativistic particles. f setstheglobalfluxlevel.Notethatdifferentcombinationsof TheadequacyofthisassumptionisdiscussedinSect.5. n∗andδcangivethesameslope.TheeffectofR ontheslope max is negligible. The flux at 20 cm is fitted by the only remain- 4. Applicationandresults ingparameterR . Thestrongdependenceofthe 20cm-flux max on R is due to the larger free-free opacity at larger wave- 4.1.CygOB2No.9:21Dec1984data max lengths.Thismeansthatmoreofthe synchrotronemittingre- WecannowapplythemodeltoCygOB2No.9(O5If).Itwas gionisshieldedat20cmthanat2and6cm.Thisalsoexplains firstdiscoveredbyAbbottetal.(1984)thatthespectralindex whyR hasonlyasmallinfluenceonthe2and6cm-fluxes. max ofitsradioemissiondoesnotfollowtheα = 0.6lawexpected Note thatthe rangein R (520– 750R ) extendsbelowthe max ∗ 6 S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars Cyg OB2 No. 9 Cyg OB2 No. 9 1000 3000 900 800 2500 700 max 2000 R 600 500 max R 1500 400 300 3 1000 2.5 2.5 2 2 1.5 1 5 6 1.5 1 3.5 4 δ 0.5 0 1 2 3 4 δ 0.5 0 2 2.5 3 n n Fig.3.SimilarfigureasFig.2,butnowcalculatedforasurfacemag- Fig.2. All the combinations of n, δ and Rmax with B = 100 Gauss netic field B = 10 Gauss (Note the different scales compared to thatfittheVLA-observations(21Dec1984) forCyg∗OB2No.9lie Fig.2). ∗ withintheboomerang-shaped region.Projectionsondifferentplanes areplottedtosituatethesolutionsintheparameterspace. radiationisemittedatradiowavelengths.Thusthesynchrotron spectrumisdominatedbythestrongershocks.Thusthevalue characteristicradioradiusat20cm(Rν(20cm)=614R ).This ofχwe obtainshouldbeinterpretedasamaximumcompres- ∗ isnota contradictionasradioemissionfrombelowRν canbe sionratiointhewind,ratherthanameancompressionratio. detected(seeSect.2.3). Note that most solutionshave δ-valueslower than 2. This While R is insensitive to n and δ, it depends strongly meansthatthefractionofrelativisticelectronsincreasesfurther max on the magnetic field. Assuming a smaller surface magnetic inthewind.Thisissurprisingasonewouldexpectaconstant field of 10 Gauss rather than the 100 Gauss used previously, efficiencyoftheaccelerationmechanismorevenadecreasing we find a larger value of R (see Fig. 3). R is still well efficiency (due to the decay of the shocks in the wind). The max max constrained, but now the range extends from 1500 R up to increasingrelativisticfractionpointstoanincreasingaccelera- ∗ 2200 R . The larger value of R is caused by the greater tionefficiency.Apossibleexplanationismultipleacceleration max importan∗ce of the Razin effect at lower magnetic fields. The of the electrons by subsequent shocks. Thermal electrons are 20 cm-flux is already greatly reduced by the Razin effect. To injected in the shock front and accelerated, but electrons, al- replenishthephotonstakenawaybytheRazineffect,thesyn- readyacceleratedbyapreviousshock,arealsore-accelerated. chrotronemittingregionmustbelargerthanthevalueofR Thiswouldincreasethenumberofrelativisticelectronstoward max foundforamagneticfieldof100Gauss.Itisnotpossibletofit theouterregionofthestellarwind. theobservationswithasurfacemagneticfieldof5Gauss,be- Another possible explanation is that, as hydrodynamical causeessentiallyallradioemissionissuppressedbytheRazin simulations show a variety of shocks, some stronger shocks effect(Sect.2.4).InthissensethevalueofB = 10Gausscan mightexistintheouterpartsofthewind.Thiswouldincrease ∗ beconsideredalowerlimitforthesurfacemagneticfield,and the synchrotronemissionwhich wouldshowupas a δ < 2 in likewisethevaluefoundforRmax isanupperlimittotheouter theparametrisationofthecurrentmodel. boundaryofthesynchrotronemissionregion. 4.2.3. ComparisonwiththeChen&Whitemodel 4.2.2. nandδ We can compare our results with the results found in Fig. 2 shows solutionswith n extendingfrom 2.5 to 5. Using Chen & White (1991, 1994). We looked at all the models the relation χ = (n + 2)/(n 1), this translates to compres- that fit the observations of Cyg OB2 No. 9 and derived the − sion ratios from χ = 3 (moderately strong shocks) to χ = permitted range of the model parameters. We used a power 1.75 (weak shocks). Note that strong shocks (χ = 4) can- law to describe the number distribution of relativistic elec- notexplaintheobservations.Thisisconsistentwithwhatwas trons.Chen&Whitecalculatedthenumberdistributionfroma found by Rauw et al. (2002), who applied a similar model to model of shocks in the stellar wind. The parameters of their 9Sgr.Sincetime-dependenthydrodynamicalsimulations(e.g. synchrotron model which reproduce the radio spectrum of Runacres&Owocki2002)showavarietyofshocks(andcom- CygOB2No.9weren = 2.6,δ < 2andR = 500R .Even max pressionratios)inthestellarwind,χissomekindofcompres- though we used differentmethods to describe the mom∗entum sionratioaverage.Strongershockshoweverproducemorera- distributionofrelativisticelectronsinthewind,theparameters dioemissionthanweakershocks,becausetheyacceleratemore of their synchrotronmodel are very similar and can be found of the electrons into the momentumrange where synchrotron nearthesolutionspaceofourmodel. S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars 7 5. Discussionandconclusions chrotronemission,andmustmimicthequenchingofelectrons by inverse Compton scattering. Because f is not well con- In this paper we have presented a theoretical model for ∗ strained by the adopted fitting procedure,this is of no conse- non-thermal radio emission from single hot stars, in terms quenceforourresults.Asasecondaryeffect,inverse-Compton of synchrotron emission from electrons accelerated in wind- cooling can also change the momentum distribution, as it af- embedded shocks. Applied to a specific observation of fectsthehighmomentumelectronsmorethanthelowmomen- CygOB2No.9,themodelproducesconstraintsonalmostall tumelectrons.Theimportanceofthiseffectwillbeinvestigated ofitsparameters(f beingtheonlyexception).Amajorresult inasubsequentpaper. ∗ is that the outer boundary of the synchrotron emitting region Hydrodynamicalmodels predict a variety of compression (R )iswellconstrained(520–750R for B = 100Gauss). max ratios. Due to the strong dependence of the emission on the AstheefficiencyofComptoncoolingpr∗events∗relativisticelec- compression ratio, the strongest shock will dominate. In that tronsfromcarryingtheirkineticenergyfarfromtheshock,this case,theradiationwouldnotcomefromthewholevolume,but means that our models predict that shocks persist to at least onlyfrombehindthe strongestshocks.Thismay also explain 500R .Thisisconsistentwithrecenthydrodynamicalcalcula- thevariabilitythatisacharacteristicfeatureofnon-thermalra- ∗ tions(Runacres&Owocki2003). dioemission(Biegingetal.1989).Asthestrongershocksmove Another interesting result is the absence of very strong inandoutofthesynchrotronemittingregiontheyleadtovari- shocks (those with compression ratio χ near 4) in the outer ability. wind. We find shocks with χ = 3 at most. This is consis- Thepresenceofrelativistic electronsinthe windwillalso tent with the hydrodynamical prediction that shocks become leadtonon-thermalX-rays,duetotheinverseComptonprocess weakerastheymoveoutwiththestellarwind. (Chen & White 1991). The non-thermal X-rays should show Withintherangeofsolutionswefindbothδ 2andδ<2. upasa hardX-raytailin aspectrum.Clearevidenceforsuch ≈ Thephysicalpictureassociatedwiththesedifferentvaluesofδ a non-thermal tail has remained elusive. Rauw et al. (2002) isratherdifferenttoo.Theδ 2 solutionspointtoa constant show that the hard X-ray tail of the non-thermal radio emit- ≈ fractionofrelativisticelectrons.ItcanbeseenfromFig.2that ter 9 Sgr does not have the expected power-law dependence they also correspond to weak shocks (χ < 2). This offers a fornon-thermalX-rayemission.Thetailcanalsobemodelled naturalexplanationforRmax:theshockswe∼akenastheymove withahigh-temperature( 2 107K)thermalplasma.Itshould outwiththeflowandbeyondRmaxtheyaretoweaktoproduce benotedthatthenon-the≥rma×lX-rayemissionisformedmuch significantsynchrotronemission. closer to the stellar surface (where there is an abundant sup- However, most of the solutions have δ < 2, which points plyofUVphotons)thanthenon-thermalradioemission.One to the relativistic electron fraction increasing outwards. This could therefore surmise that the shocks closer to the star are could indicate an increasing compression ratio of the shocks, weaker and therefore result in an insufficient number of rela- or at least the presence of a stronger shock in the outer part tivisticelectronstocreatedetectablenon-thermalX-rays.This of the synchrotron emitting region. While this could inciden- suggestion,however,doesnotappeartobeconsistentwithcur- tallyhavebeenthecaseatthetimeofobservation,itdoesnot renthydrodynamicalmodels,astheshocksarestrongestinthe sitwellwiththehydrodynamicalpredictionthatshocksdecay inner wind and decay as they move outward. The hard X-ray as they move outward. It is also hard to imagine why these tail of 9 Sgr could also be explainedby colliding wind emis- shockswouldrathersuddenlydisappearwhentheyreachRmax. sionfromalong-periodbinarycompanion(Rauwetal.2002). A more plausible explanation of δ < 2 is multiple accelera- To obtain non-thermal radio emission, all that is required tion:particlesintheouterwindhavehadgreateropportunityto isa magneticfieldandshocksin thewind.Asbothshouldbe passthroughmultipleshocksthanparticlesinthe innerwind. present in any early-type star, the question arises why not all The δ < 2 solutions are also those with larger compression these stars are non-thermal emitters. One possible answer is ratios.Inthecaseofmultipleaccelerationthemomentumdis- thatthemagneticfieldinthermalstarsislower.Alowermag- tribution is no longer a pure power law p−n, but a flatter netic field means that less synchrotron radiation is generated ∼ function (White 1985). Simply put, this means that the mo- andthattheRazineffectismoreefficientatsuppressingit.This mentumdistributionfor multiple accelerationby a numberof explanationsuggeststhatsearchestodetectmagneticfieldsin weakershocksresemblesadistributioncorrespondingtoasin- O-type stars might have a better chance in non-thermalradio glestrongershock.Thisisthenagainconsistentwiththeidea emitters. Another possible explanationis that the synchrotron ofanouterboundaryofthesynchrotronemittingregion,asall emitting region (essentially determined by the characteristic theshocksinvolvedintheaccelerationareweak. free-freeradiusandR )istoosmall,sothatthesynchrotron max Inthispaperwehaveassumedapower-lawmomentumdis- radiationemittedfallsbelowdetectionlevels. tributionwithasinglecompressionratio,whereallofthevol- Finally, one needs to ask the uncomfortable question umecancontributetotheemission.Theefficiencyofinverse- whether there is any such thing as synchrotron emission Comptoncooling,however,willlimitthesynchrotronemission from single O stars. For Wolf-Rayet stars, the connectionbe- regiontomoreorlessnarrowlayersbehindtheshocks.Thepri- tweennon-thermalemissionandbinarityisfirmlyestablished maryeffectofthis isto reducethe numberofelectronsin the (Dougherty&Williams2000).ForOstars,thesituationisless momentumrangerelevantforradioemission.Thismeansthat clear.Roughlyhalfofthenon-thermalOstarsarenotknownto thevaluesof f wefindinourfittingprocedureareunderesti- be membersof a binary system. However,the truism that ab- ∗ matedas itis thisparameterthatsetsthe overalllevelof syn- senceofevidenceisnotevidenceofabsenceobviouslyholds. 8 S.VanLooetal.:Non-thermalradioemissionfromsinglehotstars Also, there is an indication that one of the archetypal sin- 1 gle non-thermal O star, 9 Sgr, might actually be in a binary (Rauw et al. 2002). The possibility that all non-thermalemit- tingOstarsarebinariescannotbeexcluded. Acknowledgements. We thank S. Dougherty and J. Pittard for care- ful reading of the manuscript and S. Owocki for interesting discus- sions.SVLgratefullyacknowledgesadoctoralresearchgrantbythe Belgian State, Federal Office for Scientific, Technical and Cultural 0.5 Affairs(OSTC).Partofthisresearchwascarriedoutintheframework oftheprojectIUAPP5/36financedbytheOSTC. AppendixA: Flux:exactandapproximatedintegral We calculate the flux following Wright & Barlow (1975) for thefree-freeemission,butwereplaceχB ,whereχisthefree- ν freeabsorptioncoefficientandB thesourcefunction(e.g.the ν 0 Planck function),by the synchrotronemissivity j . The exact 0 1 5 10 ν r/R integralforthefluxis ν Fν = D12 Z Rmaxdq2πqZ ∞dljν[r(l)]e−τν(q,l), (A.1) dFaigsh.Aed.1l.inTehreepfruelslelnintsethisetHheeagveisoimdeetfruinccftuionnc.tion G(r/Rν), while the 0 −∞ whereDisthedistancetothestar,qtheimpactparameter,lthe and distancealongthelineofsightwithanobserveratl= and R the outer boundaryfor the synchrotronemission−r∞egion. 1 π max G(x) = dθsinθ (A.8) Theopticaldepthτν(q,l)isdefinedas 2Z0 mwτνie(tqahn,Ali)ng==,ZaMn˙−dl∞/(Kd4lπ′µ(tqmhK2eHff+vfγr∞lAe′)2e2,)-2wfr,eheeraebtshoerpstyiomnbcoolseffihacvieentthgeiirve(uAnsu.b2ay)l The θ×inteexgprat−io(cid:16)nΓi(cid:16)xs313o(cid:17)sni/n8ly3(cid:17)3θfrπo2m(si0nθtocoπs,θbe−caθu+seπt)he. originalin- ff Allen(1973).Notethatbyusing0asthelowerintegralbound- tegralonlyconsideredpositiveq values(a factorof 2 was in- ary in Eq. 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Non-thermal radio emission from single hot stars

Similar Non-thermal radio emission from single hot stars