Non-thermal radio emission from single hot stars

0.66 MB·English

by  S. Van Loo

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:[email protected] 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 distribu-
tion 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 cur-
rent 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 syn-
chrotronemissivity,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. (A.1), we in fact replaced the star by stellar wind cludedin theoriginalintegrationtocompensateforthis).The
material.Thismathematicalsimplificationhasnoconsequence functionG(r/Rν)isplottedonFig.A.1.NotethatG(r/Rν)can
asthefree-freeopticaldepthissolargethatanycontributionof be approximated by a Heaviside function: H = 0 if r < Rν
thestarcanbeneglectedanyway. and H = 1 if r > Rν. This brings us back to the approxima-
The equation for τν(q,l) can be integrated to (using tionEq.(7)fortheflux.ThefunctionG(r/Rν)needstobecal-
Abramowitz&Stegun1965,3.3.24) culated only once, so that evaluating Eq. (A.6) does not take
significantlymoretimethanEq.(7).
K γA2 l/q l π
τ (q,l)= ff +arctan + . (A.3)
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