Mon.Not.R.Astron.Soc.000,1–12(2004) Printed5February2008 (MNLaTEXstylefilev2.2) The distance to the Orion Nebula Cluster R.D. Jeffries Astrophysics Group, School of Physical and Geographical Sciences, Keele University,Keele, Staffordshire ST5 5BG SubmittedDecember52006 7 ABSTRACT 0 The distance to the Orion Nebula Cluster (ONC) is estimated using the rotational 0 propertiesofitslow-masspremain-sequence(PMS)stars.Rotationperiods,projected 2 equatorial velocities and distance-dependent radius estimates are used to form an observationalsini distribution (where i is the axial inclination), which is modelled to n a obtain the distance estimate. A distance of 440±34,pc is found from a sample of 74 J PMSstars with spectraltypes between G6 andM2, but this falls to 392±32pc when 8 PMSstarswithaccretiondiscs areexcludedonthe basisoftheir near-infraredexcess. Since the radii of accreting stars are more uncertain and probably systematically 1 underestimated,thenthiscloserdistanceispreferred.Thequoteduncertaintiesinclude v statistical errors and uncertainties due to a number of systematic effects including 6 binarity and inclination bias. This method is geometric and independent of stellar 8 evolutionmodels,thoughdoesrelyontheassumptionofrandomaxialorientationsand 1 theCohen&Kuhi(1979)effectivetemperaturescaleforPMSstars.Thenewdistance 1 isconsistentwith,althoughlowerandmoreprecise,thanmostpreviousONCdistance 0 estimates. A closer ONC distance implies smaller luminosities and an increased age 7 based on the positions of PMS stars in the Hertzsprung-Russell diagram. 0 / h Key words: stars:formation–stars:distances–methods:statistical–openclusters p and associations: M42 - o r t s a 1 INTRODUCTION portant parameter in determining absolute dimensions, ve- : locities and massloss rates intheclusteraswell asfor esti- v TheOrionNebulacluster(ONC)isamongthebeststudied i matingstellar luminosities andhencemasses andagesfrom X star forming regions. It is perhaps the premier cluster for evolutionary tracks. investigating star formation and early stellar evolution be- r a causeitisrelatively nearby,veryyoung(<2Myr)andcon- In this paper the distance to the ONC is determined tainsa large population of stars andbrown dwarfs covering usingtherotationalpropertiesofpremain-sequence(PMS) the entire (sub)stellar mass spectrum (0.01 < M/M⊙ < 30 stars(axialrotationperiodsP,andprojectedequatorialve- – see Hillenbrand 1997; Slesnick, Hillenbrand & Carpenter locitiesvsini,whereiistheinclinationangleofthespinaxis 2004).TheONCliesjustinfrontofthedenseOMC-1molec- tothelineofsight),togetherwith estimatesofstellar radii. ular cloud, so background contamination of cluster candi- From these, the observed distribution of sini can be com- dates is small and proper motion studies have successfully paredtoamodelthatassumesrandomspinaxisorientation ascribed membershiptoalmost 1000 stars(Jones & Walker andappropriate contributionsfrom uncertainties in theob- 1988).TheONCisafocusforunderstandingtheinitialstel- servational parameters. Because the estimated stellar radii larmassfunction,theevolutionofcircumstellar discs,early scalelinearlywithdistance,thedistancecanbetreatedasa stellar angular momentum loss, the influence of high mass freeparameterin themodeltooptimise thematchbetween starsonlowermasssiblingsandtheirdiscs,X-rayactivityin observed and predicted distributions. youngstars,thehistoryofstarformationandtheformation of star clusters in general (e.g. see Hillenbrand et al. 1998; This technique was developed initially by Hendry, O’Dell1998;Ladaetal.2000;Muenchetal.2002;Herbstet O’Dell & Collier Cameron (1993) and used to derive a dis- al.2002;Preibischetal.2005;Huff&Stahler2006;Shuping tance of 132 10pc to the Pleiades, which compares well ± et al. 2006 among many others). with the currently accepted Pleiades distance from numer- The distance to the ONC is quite poorly constrained ous other techniques (O’Dell, Hendry & Collier Cameron – anywhere between 350pc and 550pc, with a most likely 1994). Preibisch & Smith (1997) used a variant of the rangeof400–500pcdependingonwhichtechniquesarecon- methodtofindadistancetoT-TauristarsintheTaurusstar sidered most reliable (see section 2). Thedistance is an im- formingregion.Again,thedistancetheyfoundof152 10pc ± (cid:13)c 2004RAS 2 R.D. Jeffries nowcompareswellwithdistancesfromHipparcosparallaxes (Bertout, Robichon & Arenou 1999). 0.3 Myr 1.5 1 Myr In section 2 previous estimates of the distance to the 2.0 3 Myr 10 Myr ONCarereviewed.ThedatabaseofONCrotationmeasure- 1.0 mentsthatwereusedforthisworkisdiscussedinsection 3. 1 Themodellingtechniqueisdescribedinsection4,whichalso 0.5 discussessystematiceffectsthatmustbetakenintoaccount. Lsun 0.5 Taghaeinrsetsuvlatrsiaatrieonpsriensetnhteedmiondeslecptairoanm5etaenrdsatnhdeiarsrsoubmupsttnioensss g L/bol 0 o is tested. Conclusions and a brief discussion are presented l 0.2 in section 6. -0.5 -1 2 PREVIOUS DISTANCE ESTIMATES TO 5500 5000 4500 4000 3500 3000 THE ONC Teff (K) Most previous distance estimates were based on fits to the Figure1.AHertzsprung-Russelldiagramforthe95starsofthe uppermainsequenceoftheONCintheHertzsprung-Russell ONC rotation sample (see text) assuming an ONC distance of (HR) diagram and hampered by colour excesses, variable 470pc. Model isochrones and evolutionary tracks from Siess et extinction, the nebular background and uncertain binarity. al.(2000)areshown.Massesarelabelledinsolarunits. Walker (1969) found a distance modulus of 8.37 0.05 ± (472pc) from UBV photometry of relatively unobscured theobject appears older bya few Myrthan thebulkof the starsintheoutskirtsoftheONC.However,Penston,Hunter ONCpopulation. &O’Neill(1975)obtainedopticalandnearinfraredphotom- etry in a 40 arcminute box around the ONC, determining distance moduli of 8.1 0.1 (417pc) from the V,B V ± − colour magnitude diagram (CMD), and only 7.7 (347pc) 3 CONSTRUCTION OF THE DATABASE from the V,V I CMD. More recent estimates used the − TheobservedsinidistributionintheONCwasinvestigated largeUBVubvyβ datasetofWarren&Hesser(1978).These using a database founded on the catalogue of ONC axial authorsfoundadistancemodulusof8.42 0.24(483pc)for ± rotation periods constructed by Herbst et al. (2002). This the Orion OB1d1 subregion, which is a 30 arcminute di- ∼ catalogue also includes rotation periods measured by Stas- ameterbox aroundtheONC,butexcludingthecentralfew sunetal.(1999)andHerbstetal.(2000).Fromthis,allob- arcminutes around the central Trapezium cluster of high- jectswereselectedthathadanentryintheJones&Walker massstars.Anthony-Twarog(1982)usedthesamedatawith (1988) catalogue of proper motions within 15 arcminutes somewhat different calibrations to obtain a distance modu- of the ONC centre and a spectral type, effective tempera- lusof8.19 0.10(435pc).Breger,Gehrz&Hackwell(1981) ± ture(T ),luminosity(L)andradius(R)estimatelistedby usedinfraredandbroadbandpolarisation measurementsto eff Hillenbrand (1997). The L and R estimates in Hillenbrand excludeobjectsthatshowedabnormalreddeningorevidence (1997) assume an ONC distance of 470pc. Almost all of forcircumstellarmaterial andfoundthattheremainingob- these stars also have a measurement of their I K excess jectshaveanaveragedistancemodulusof8.0 0.1(400pc) − ± – i.e. the excess colour over that expected from the photo- in theV,B V CMD. The significant discrepancy between − sphereofa“normal”starofsimilarspectraltype.Measure- for example Walker’s (1967) and Breger et al.’s (1981) dis- mentsoftheequivalentwidth(EW)oftheCaii8542˚Aline, tance estimates points to the fact that there may be sub- which is also diagnostic of accretion, are also available for stantialadditional systematicuncertaintiespresentinthese most of the sample from Hillenbrand et al. (1998). HR diagram-based results. From this subset, vsini measurements were found in Unfortunately, no light is shed on these varying dis- the catalogues of Rhode et al. (2001) and Sicilia-Aguilar tance estimates by other independent techniques. Only one et al. (2005). The latter work was performed at a higher star in the ONC has a direct parallax measurement from Hipparcos, yielding a distance of 361+168pc (Bertout et al. resolving power (34000 versus 21500) and so values from −87 Sicilia-Aguilaretal.werepreferredforvsinivalueslessthan 1999). A more indirect but frequently cited constraint was 20kms−1 . For higher velocities the measurement with the providedbyGenzeletal.(1981).Usinganexpandingcluster best quoted fractional precision was chosen. Objects with parallax method they analysed the proper motions and ra- onlyupperlimitstotheirvsiniwereexcluded.Threeobjects dialvelocities ofH2O masers intheKleinmann-Lownebula (JW 192, JW 381 and JW 710) where Herbst et al. (2002) (part of OMC-1), thought to be just behind the ONC, and state that the measured periods are highly uncertain were obtained a distance of 480 80pc. Finally, Stassun et al. ± also excluded. (2004) found an eclipsing binary just 20 arcminutes south oftheTrapeziumstars.Theydeterminedabsoluteradiiand effective temperatures for the components and hence esti- 3.1 Measurement uncertainties mated a distance of 419 21pc from the system photome- ± try.However,Stassunet al.suggest thatthisbinarysystem The periods measured for low-mass PMS stars are gener- might be foreground to the ONC because at that distance ally very precise and accurate, relying on the co-rotation (cid:13)c 2004RAS,MNRAS000,1–12 The distance to the ONC 3 Table1.Databaseofobjectsconsideredinthiswork.Thefulltableisavailableinelectronicformandcontains95rows.Asampleisgiven heretoillustrateitscontent.Column1givestheidentifyingnumberfromJones&Walker(1988);columns2and3listthespectraltype andTeff fromHillenbrand(1997);columns4,5and6listthevsini,itsuncertaintyandthesourceoftherotationalvelocitymembership (1–Rhodeetal.[2001],2–Sicilia-Aguilaretal.[2005]);column7liststherotational periodas givenbyHerbstetal.(2002); columns 8,9and10listthelog bolometricluminosity(insolarunits), stellarradius(assumingaclusterdistance of470pc)and ∆(I−K)from Hillenbrand(1997); column 11 gives the equivalent width of the 8542˚A CaII line fromHillenbrand et al. (1998) [negative indicates an emissionline];column12liststhederivedsini. JW SpT Teff vsini ∆vsini Source P logL/L⊙ R/R⊙ ∆(I−K) EW[Ca] sini (K) (kms−1) (kms−1) (d) (mag) (˚A) 3 K8 3801.8 29.8 4.5 1 3.43 0.24 3.036 0.55 2.0 0.665 17 M4 3228.4 19.9 3.9 1 3.14 -0.20 2.537 0.10 0.8 0.486 20 M3.5 3296.0 70.3 11.6 1 0.67 -0.40 1.933 0.10 0.7 0.481 25 M4.5 3162.2 18.0 4.0 1 2.28 -0.88 1.209 0.52 0.0 0.671 ofmagneticinhomoheneities(starspots)inthephotosphere. random errorsin R areabout 0.11dex.Itis fair tosaythat Providingasufficient timebaselineandsampling frequency this uncertainty is itself uncertain, and so values between areobtained,periodprecisionsofbetterthan1percentare 0.07dex and 0.15dex are tested in the modelling. Hillen- usually achieved. Studies of objects with periods measured brand(1997)alsohadconcernsthatbecauseofdifficultiesin atmorethanoneepochsuggestthat 90percentofperiod assigninganextinctionvalue,theLandhenceRofaccreting measurementshaveatleastthislevel≃ofprecision(Herbstet classicalT-Tauristars(CTTS)wouldbeunderestimated.As al.2002).About10percentofperiodmeasurementscanbe the rotational database contains information on the I K − catastrophically incorrect. The periods could be too short excess and EW[Ca], both of which are diagnostic of strong byafactoroftwoifmorethanonespottedregion onastar accretion, such stars can be optionally excluded. leads to a “double-humped” light curve that is interpreted The final database contains 95 objects with both ade- incorrectly.Alternatively,themeasuredperiodcanbemuch quate P and vsini measurements and estimates of L, T eff longerthanthetrueperiodsbecauseofaliasingwiththe 1 andR.TheseobjectsarelistedinTable1(availableinelec- ≃ day sampling interval that is present in almost all datasets tronicformonly)andtheHRdiagramiscomparedwiththe (see discussion in Herbst et al. 2002). Many of these in- stellar evolutionary models of Siess, Dufour & Forestini in correct periods have already been weeded out by Herbst et Fig. 1 (2000 – the variant with a metallicity of 0.02 and no al. (2002) and others are suspected (e.g. the three stars ex- convectiveovershoot). cluded in the previous subsection), but it is possible that a small fraction remain. Fornow an uncertaintyof 1percent 3.2 Selection effects and biases on all the period measurements is assumed. The resolution of the rotational velocity studies limits Thedatabaseissubjecttoanumberofselection effectsand the vsini measurements to values of more than 11kms−1 biases (see also the discussion in Herbst & Mundt 2005). and about 5kms−1 from Rhode et al. (2001) and Sicilia- Thereisabiasagainst theinclusion ofCTTS andinfavour Aguilar et al. (2005) respectively. Above these limits the ofweak-linedT-Tauristars(WTTS).ThisisbecauseCTTS fractional precision of the vsini measurements are of order often show accretion-related, non-periodicvariability which 10percent,thoughvaryfromobject-to-object.Rhodeetal. masks the true rotation period of the star. There is also a (2001) discusshowproblemsincalibratingtheintrinsicres- bias towards objects with shorter rotation periods simply olution of fibre spectrographs can lead to problematic and because these are easier to measure from datasets with a possibly incorrect values for vsini when rotational broad- limitedtimespan.Neitheroftheseselectioneffectswillbias ening is close to the detection threshold. Rhode et al. sug- theobservedsinidistributionunlessCTTSandmoreslowly gestusingvsini=13.6kms−1asamoreplausibledetection spinningstars havenon-random spin-axis orientation. thresholdfortheirdata,andthatiswhatisusedhere.With Of more concern is the observational bias against slow similar reasoning only values of vsini>10kms−1 are used rotators and objects with small sini because of the limited fromtheworkofSicilia-Aguilaretal.(2005). Thesampleis sensitivity of the vsinimeasurements. In principlethis can alsofilteredofpoorqualitydatawithafractionalvsiniun- beaccountedforinourmodellingbyputtingavsinicut-off certainty greater than 25 per cent to prevent unnecessarily inthemodel.Thisthresholdimposesasmoothroll-offinthe broadening theobserved sini distribution. valuesofsinithatarecapableofdetection.Theexactshape Stellar radii are estimated from combining L and T oftheroll-off dependsonthedistributionoftrueequatorial eff (Hillenbrand 1997). L comes from extinction-corrected I- velocities (see section 4.2). bandmagnitudesandbolometriccorrections,andassumesa Afurtherconsideration isthebiasawayfrom lowincli- distancetotheONCof470pc.T valuesareestimatedfrom nationsystemsdueeithertothelackofvisibilityofstarspots eff the relationship between spectral type and T (with some at equatorial latitudes caused by limb darkening, or to the eff small modifications) proposed by Cohen & Kuhi (1979). reduced amplitude of spot modulation caused by starspots Hillenbrand (1997) estimates that L is uncertain by about athigherlatitudes(seediscussioninO’Dell&Hendry1994). 0.2dex due to variability and uncertainties in extinction. Theactuallatitudedistributionofspotsonveryyoungstars Random errors in spectral types lead to uncertainties of isstilldebatable.Therehavebeentheoreticalpredictionsof about 0.02dex in the T values. As R L−1/2T2 , the polarconcentrationsforspotsonrapidlyrotatingstarswith eff ∝ eff (cid:13)c 2004RAS,MNRAS000,1–12 4 R.D. Jeffries 470pc deep convection zones (e.g. Schu¨ssler et al. 1996). However, Robs =Rtrue b(q)10δlogRU, (5) Granzer et al. (2000) predict spots over a wide range of (cid:16) D (cid:17) latitude for fast-rotating T-Tauri stars, with an equatorial where δ is the uncertainty (in dex) of the logarithmic logR concentrationinslowerrotators.TheevidencefromDoppler radius estimate (see section 3.1). imagingofspotsonveryyoungT-Tauristarsismixed.Some Hence,combiningequations1–5themodelfor(sini) obs show spot activity at low latitudes, some at high latitudes can beexpressed as andotherhavespotsatalllatitudes(seeJoncour,Bertout& Bouvier 1994; Johns-Krull& Hatzes 1997; Neuhäuser et al. (sini) = D (1+δPU1)(1+δvU2) sini, (6) 1998).FornowImaketheassumptionthatthereissomein- obs (cid:20)470pc(cid:21) (cid:20) b(q)10δlogRU3 (cid:21) clinationi ,belowwhichstarspotmodulationisneverseen th whereU1,U2,U3 indicatethatthesearethreedifferentran- (see also O’Dell et al. 1994). This threshold is allowed to dom numberstaken from a unit Gaussian distribution. vary over some plausible range or can be tuned to give the ToestimatethedistancetotheclusterIadoptthesim- best fit to thelower end of theobserved sini distribution. plest,unbiasedapproachsuggestedbyHendryetal.(1993). Ifequation 6 is re-expressed as 4 MODELLING THE OBSERVED D (sini) = α, (7) obs INCLINATION DISTRIBUTION (cid:20)470pc(cid:21) 4.1 The general approach thentakingtheaverageoverallthestarsinthesampleand the average over all the trials in a model Monte-Carlo sim- The observational estimates of sini for each star are given ulation, the best estimate for the distance to thecluster is by (sini) (sini)obs = k Pobs(vsini)obs , (1) D=470 (cid:18)h αobsi(cid:19)pc. (8) (cid:16)2π(cid:17) Robs h i whereP istheobservedperiod,(vsini) istheobserved obs obs projectedequatorialvelocity,Robsistheestimateofthestel- 4.2 The true equatorial velocity distribution larradiusbasedonadistancetotheONCof470pc,andkis a constant which depends on the units used for the various The model distribution for vtrue has some influence on the quantities.Theaimistomodelthedistributionof(sini) derivedclusterdistance.Thereasonisthatthelowerthresh- obs with a Monte Carlo simulation. To that end I define old for vsini measurements leads to a higher threshold in k=2π Rtrue (2) t(hsiendi)isotbrsibfourtiloowneorfv(sailnuei)sofvretrquue.irTeosaobretaaisnonaagboleoddemscordipetlifoonr Ptruevtrue obs of the distribution of vtrue. An incorrect vtrue distribution where the subscript “true” indicates the actual values of leads to a systematic error in the distance estimate. these three parameters in the absence of measurement un- Withthesizeofsampleconsideredherethiserrorisnot certaintiesandothersystematiceffects(seebelow).Iassume entirely negligible compared with the statistical uncertain- that Pobs is related to Ptrue by ties in D (see section 5.2). Fortunately we can check that Pobs =Ptrue(1+δPU), (3) the assumed vtrue distribution is reasonable by comparing the modelled distribution of vsini from equation 4 (after whereδ isthefractionaluncertaintyintheperiodandU is P uncertainties and selection thresholds have been applied) arandomnumberdrawn from aGaussian distributionwith a mean of zero and unit standard deviation. Similarly, with that seen in the data. For instance a flat vtrue distri- butionwouldnotexplaintheobservedvsinidistributionin (vsini)obs =vtrue sini(1+δvU) (4) theONC(see section 5.2). where sini is drawn randomly assuming that cosi is dis- tributeduniformlybetween0and1(i.e.randomorientation 4.3 The binary correction factor of the spin axes) and δ is the fractional uncertainty in the v vsini measurements. The true velocity and sini are split Unresolved binary systems will have systematically overes- into separate factors because the effect of an observational timatedL,overestimatedRandhenceunderestimatedsini. lower limit to vsini can only be modelled properly if a dis- A neglect of this effect would lead to an underestimated tribution of vtrue is initially specified. For instance a star distance in equation 8. In older clusters where stars have with vtrue=50kms−1 cannotbeintheRhodeet al.(2001) reachedtheZAMS,unresolvedbinarieswithq&0.5areeas- observational sample unlesssini>13.6/50, butclearly this ily detectedas theylie significantly abovetheclustersingle threshold changes with vtrue (see section 4.2). stars locus in the HR diagram. This becomes impractical ThereisasimilarexpressionrelatingRobsandRtruebut in young clusters like the ONC where there may be an age this must also take into account that the true distance, D, spread that is a significant fraction of the mean age, and which is assumed common to all stars in the sample, may wheredifferentialextinctionandvariabilitycausesignificant differ from the 470pc assumed by Hillenbrand (1997). In spreads in any CMD. additionatermmustbeincludedthatadmitsthepossibility I have calculated a correction to R that should be ap- that a given star could be an unresolved binary system. In plied to an unresolved binary system of total luminosity L whichcase,ListhesumoftwocomponentsandRtrue could and mass ratio q. The calculation uses a model of L versus be smaller than R by a factor, 1 6 b 6 √2, for binary T calculated at a range of masses. For this paper I have obs eff mass ratios 06q61. In this case usedthemodelsofSiessetal.(2000)andBaraffeetal.(2002 (cid:13)c 2004RAS,MNRAS000,1–12 The distance to the ONC 5 1.45 is set by assuming that a fraction f of star-systems are bi- S00 logL=+1.0 age=1 Myr S00 logL=+1.0 age=3 Myr naries, with q drawn randomly from a defined distribution. 1.4 S00 logL=+0.5 age=3 Myr The appropriate value of f and the q distribution can be S00 logL=+0.0 age=1 Myr B02 logL=+0.0 age=1 Myr inferred from observational studies of the ONC. 1.35 B02 logL=+0.0 age=3 Myr B02 logL=-0.5 age=1 Myr Using different high spatial resolution surveys Prosser B02 logL=-0.5 age=3 Myr R(q) 1.3 et al. (1994), Padgett, Strom & Ghez (1997), Petr et al. ∆ (1998),Simon,Close&Beck(1999)andKöhleretal.(2006) Correction factor 11 ..112.552 h1tno0aaw0vry0earascdyuossnitlcseolmswuidmesreaidlrmaetrahystaesottetstthohianebtebaoipfsnuifiamberlylidilsafhrsretewadqru,asybe.anunDctydpetrpinaerilolitsmbhafeionbrralayrcnyldgoreeseces1lriu0nbl0eti–ss- suggestthattheclosebinaryfrequencyisalsosimilartothat 1.1 of field stars (Stassun & Mathieu 2006). 1.05 I make the assumption that binary properties in the ONC are similar to that of the field. Duquennoy & Mayor 1 0 0.2 0.4 0.6 0.8 1 (1991)showthattheoverallbinaryfrequencyforsolar-mass Mass ratio (q) stars is 57 per cent and at distances of 450pc, the vast ∼ majority of these would be unresolved binaries in theONC Figure 2.Themultiplicativebinarycorrectionfactorasafunc- tion of mass ratio for ages of 1Myr or 3Myr, for −0.5 < sample. The binary frequency declines towards 30–40 per log(L/L⊙)<1.0andfortwodifferentevolutionarymodels(Siess cent in M dwarfs with 0.3–0.5M⊙ (Fischer & Marcy 1992). etal.2000–S00andBaraffeetal.2002–B02). Themass ratio distribution appears reasonably uniform, at least over the range 0.50.3 befurtherrestrictedtoarangeoftemperatureandwhether Spline All they show signs of accretion or not (see section 5.1). Spline WTTS Spline CTTS (ii) A model is chosen for the true equatorial velocity 1.5 (ainvrterturheae)ndsdaoimsmtrpliylbeou)rtoiieofnnmt.aoTtedhdee,lnav,ssaeistns(uitmyvpainliucgaelstlhyaa1rte0t4rhaternidarolostmaptleyiornogbeanjxeeecrst- (sin i) obs 1 ated and perturbed according to equation 4 using the frac- tionaluncertaintiesin(vsini) astheδ values.Themodel obs v 0.5 vtrue distribution is adjusted until a satisfactory match to the (vsini) distribution is indicated by a Kolmogorov- obs Smirnov (K-S) test between the observed and modelled cu- 0 mulative distribution functions (CDFs). 5500 5000 4500 4000 3500 3000 (iii) The trial values of sini are used to calculate the α Teff (K) valuesinequation7usingδ ,fractionalerrorsintheperiod v Figure3.Theobservedvalueofsini(fromequation1)versusthe δ = 0.01, and assuming a suitable value for the fractional p effective temperature. CTTS/WTTS are indicated by different error in the log stellar radius (δ ). A fraction f of the logR symbols.ThelinesshowlowordersplinefitsasafunctionofTeff trials arerandomly assigned abinarystatusandtheappro- to(a)allthedata,(b)theCTTSonly(∆(I−K)>0.3),(c)the priate correction factor to the observed radius, b(q), is ap- WTTS(∆(I−K)<0.3). plied assuming that the binary mass ratio (q) is uniformly distributed. (iv) The results of the trials pass through a filter which onlyallowsthosetrialstoproceedwhichhavevsinigreater WTTS EW[Ca]>1A 2 CTTS EW[Ca]<1A than the observational thresholds and axial inclinations Spline All greater than a user-definedthreshold, i . Spline WTTS th Spline CTTS (v) Atthisstage theaveragevalueof(sini)obs isdivided 1.5 by the average value of α in order to find D (equation 8). Aα(tDth/4e7s0a)maendtim(seinai)Kob-sS. TtehstisiissmusaeddetboetdweceiednetwhheeCthDeFrsthoef (sin i) obs 1 modelisareasonable description ofthedata(thesampleis too small to consider χ2 tests). 0.5 (vi) Statistical uncertainties in the results are estimated by generating many fake datasets of the appropriate size. The(sini)obsvaluesaredrawnrandomlyusingequation7at 0 the estimated distance and these are modelled in the same 5500 5000 4500 4000 3500 3000 way as the data. The standard deviation of the distance Teff (K) estimates is used as a statistical uncertainty. Figure 4.SimilartoFig.3,butheretheWTTSandCTTSare (vii) The sensitivity of the results to the model assump- classified according to whether the equivalent width of the Caii tions and parameters are tested using different subsamples 8542˚Alineismoreorlessthan1˚A(seeHillenbrandetal.1998). of data (e.g. with or without signs of accretion), different valuesofi ,f,δ ordifferentmodelsfor∆R(q)andthe th logR vtrue distribution (see section 5.3). forstronglyaccretingobjects.WTTSaregenerallyfoundto haveEW[Ca]>1˚AandFig.4repeatsFig.3usingthiscrite- riatoseparateWTTSandCTTS.Althoughcrude,thesedi- agnosticsshouldbesufficienttoidentifycaseswherebroad- 5 RESULTS bandcolours, magnitudes and henceestimated luminosities and radii are likely to be affected, which is theissue here. 5.1 The observed sini values Two sample selection issues are highlighted by Figs. 3 InFig. 3(sini) versusT isplotted forthesample. The and 4. The first is that stars with a near infrared excess or obs eff data have been distinguished on the basis of whether their EW[Ca]< 1˚A have a larger (sini) on average than those obs nearinfrared excesses,∆(I K),arelarger orsmaller than without. To illustrate this, smooth spline fits as a function − 0.3. This division approximately separates stars with and ofT areplottedforallthedataandfortheCTTS/WTTS eff without active accretion and thus the two subsets are la- subsets separately. The effect is small at cool temperatures belled as CTTS and WTTS respectively. Hillenbrand et al. butincreasesto 30percentinhotterstars.Forstarswith ≃ (1998) show that because of physical effects (disc inclina- T > 3499K (see below) the average sini is 0.78 0.04, eff ± tion, inner holes etc) and also observational uncertainties, but the averages for theWTTS samples are 0.69 0.04 for ± this threshold is fuzzy and not capable of establishing or 34 stars with ∆(I K) < 0.3 and 0.67 0.04 for 44 stars − ± excludingaccretioninallcases.Unfortunately,mid-IRmea- withEW[Ca]>1˚A.Astheestimateddistanceislinearlyde- surementswhich aremuchmoresensitiveareonlyavailable pendent on (sini) , this bias is of concern. There are no obs for a small fraction of the sample (see Rebull et al. 2006). physical reasons to suppose that CTTS have larger inclina- An alternative is to use EW[Ca], which goes into emission tion angles, but thereare good reasons to suppose that the (cid:13)c 2004RAS,MNRAS000,1–12 The distance to the ONC 7 radii of CTTS havebeen underestimated (leading to a sini following an exponentially decaying distribution between overestimate), either because of a systematic underestima- 10 3500K) this relationship is relatively propertiesdrawnfromthedistributionsusedineachmodel. uncontroversial in the sense that alternative scales (for in- Observational errors for the fake datasets matched those in stanceKenyon&Hartmann1995;Leggettetal.1996)differ the real dataset considered here. The fake datasets passed by less than 2 per cent and not in a systematic way. For through the same analysis procedure and the standard de- cooler stars the relationship is more uncertain and espe- viations of the distance estimates (from 300 fake datasets) cially so for very cool PMS stars which have significantly were used as estimates of thestatistical error. lower gravities than main-sequence stars of similar spectral Theresults arecollected inTable 2, andFig. 6 demon- type. The luminosity estimates are also reliant on an ac- strates the sensitivity of the model distribution to changes curate assessment of the extinction which requires intrin- in the various parameters in the form of both the differ- sic broadband colours for very cool PMS stars. Hillenbrand ential and cumulative distributions of α(D/470) compared (1997)usedtheintrinsiccoloursofmain-sequencestarstab- with those of thedata. ulated by Bessell & Brett (1988), but very cool PMS stars Analysis number 1 from Table 5 is used as a baseline mayhavemore giant-likecolours. Dwarfsare bluerthangi- for thediscussion as thisuses theparameters that could be antsby 0.3maginV I forspectraltypesM3-M5,which considered tomost likelyrepresent themodelled ONCpop- ∼ − would lead to a larger extinction estimate by 0.5 mag ulation. The average value of (sini) is significantly lower ∼ obs (at I). This gives a larger luminosity estimate by 0.2dex, thanwouldbeexpectedforrandomlyinclinedaxesinagree- a larger radius estimate by 0.1dex and hence a decreased ment with the analysis of Rhode et al. (2001) based on a siniestimate byafactor of 1.26 that would almost entirely smallersample.ThesimplestexplanationisthattheONCis explain thetrend seen. muchcloserthanthe470pcassumedbyHillenbrand(1997). For the reasons above the analysis here is initially re- IndeedD=470pcisrejected with97percentconfidence.I stricted to the sini distribution of those 34 objects in Ta- findadistanceof 392pcgivesa muchbetterrepresentation ble 1 with ∆(I K) <0.3 and Teff >3499K. This sample of the data as may be judged from Figs. 6a and 6b. The − is labelled WTTS1 in what follows. Two alternate samples statistical uncertainty is 23pc which compares well with are also considered, namely 44 objects with Teff > 3499K simulations and uncertain±ty estimates presented by O’Dell andEW[Ca]>1˚A(labelledWTTS2)andall74objectswith et al. (1993) for similar sized samples. Teff > 3499K, regardless of their accretion status (labelled In addition to the statistical uncertainties I have tried ALL). other models to test the sensitivity of the derived distance to variations in model parameters that are only partially constrained. Fig. 6 shows examples of the model distribu- 5.2 The vtrue distribution tionsatafixeddistanceof392pcinordertohighlightthese Forthereasonsexplainedinsection4.2,areasonablemodel differences.Thereadershouldnotethatinallcasesthevari- of the vtrue distribution is required and can be constrained ationsintheintrinsicαdistributioncanbecompensatedfor by comparing thesimulated vsini distribution (from many bychangesintheclusterdistancetoyieldareasonablevalue trials) with that observed. forP(K-S)(thesearetheresultsquotedinTable2).There- In Figure 5 two of these comparisons are shown. A foretheobserved(sini) distributioniscurrentlyincapable obs formal statistical comparison has been made using a 1- ofconstrainingsomeofthemoreinterestingparameterslike dimensional K-S test of the CDFs. The models shown have i , dueto thesmall sample size and experimental errors. th i =30◦,butthemodelvsinidistributiondoesnotdepend Figs. 6c and 6d show how changes of 15◦ < i < 45◦ th th strongly on thisparameter. affectthemodelleddistributionatafixeddistanceof392pc. After experimenting with a variety of simple analyt- The changes are small and easily compensated for by mod- ical forms, I find that a two component vtrue model (la- est changes in the cluster distance of only 4.0 per cent. ± belled v1 in Fig. 5) provides a good (though not neces- Thisrangeseems areasonable estimate oftheuncertainties sarily unique) fit to the data, with 80 per cent of stars engendered by the inclination bias – if i <15◦ this would th (cid:13)c 2004RAS,MNRAS000,1–12 8 R.D. Jeffries 16 Exponential+Flat (v1) Exponential+Flat (v1) 14 Flat (v2) 1 Flat (v2) 12 0.8 10 F 0.6 N 8 D C 6 0.4 4 0.2 2 0 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 v sin i (km/s) v sin i (km/s) Figure 5. A comparison of the observed vsini distribution with Monte-Carlo simulations based on two different models of the vtrue distributions,v1andv2(describedinthetext).Theleftpanelshowsthe(binned)distributionsandtherightpanelshowsthenormalised cumulativedistributionsusedtoperformaKolmogorov-Smirnovtest.Thediscontinuityinthemodeldistributionsatvsini=13.6kms−1 isduetothemodelledsamplebeingdividedbetweenobjects withdetectable vsinithresholdsat13.6kms−1 and10kms−1. Table 2. The results of the distance analysis. The columns are: (1) An analysis identification number; (2) the analysed sample and number ofobjects inthesample–WTTS1indicates objects ∆(I−K)<0.3,WTTS2indicates objects EW[Ca]>1˚A(see section5.1), ALL indicates no ∆(I −K) restriction; (3) identifies which ∆R(q) model was used, r1 or r2 (see section 4.3); (4) identifies which vtrue distribution was assumed, v1 or v2 (see section 5.2); (5) the assumed value of ith (see section 3.2); (6) the assumed value of the binary fraction f (see section 4.3); (7) the assumed value of uncertainty in the observed (log) radius (see section 3.1); (8) the average value of (sini)obs; (9) the average value of α; (10) the distance implied by equation 8 and its 1-sigma statistical uncertainty; (11) the Kolmogorov-Smirnovnull-hypothesisprobability(theprobabilitythattheobserveddistributionisnotdrawnfromthemodeldistribution is1−P(K-S)). AnalysisNo. Sample(n) ∆R(q) vtrue ith bfrac δlogR h(sini)obsi hαi D P(K-S) model model degrees f (dex) (pc) 1 WTTS1(34) r1 v1 30 0.5 0.11 0.691 0.828 392±23 0.66 2 WTTS1(34) r1 v1 15 0.5 0.11 0.691 0.802 405±24 0.75 3 WTTS1(34) r1 v1 45 0.5 0.11 0.691 0.870 373±22 0.61 4 WTTS1(34) r1 v1 30 0.3 0.11 0.691 0.858 378±22 0.67 5 WTTS1(34) r1 v1 30 0.7 0.11 0.691 0.798 407±24 0.63 6 WTTS1(34) r1 v1 30 0.5 0.07 0.691 0.812 400±18 0.64 7 WTTS1(34) r1 v1 30 0.5 0.15 0.691 0.851 382±28 0.41 8 WTTS1(34) r1 v2 30 0.5 0.11 0.691 0.815 398±24 0.64 9 WTTS1(34) r2 v1 30 0.5 0.11 0.691 0.849 383±22 0.68 10 WTTS2(44) r1 v1 30 0.5 0.11 0.671 0.829 381±20 0.66 11 ALL(74) r1 v1 30 0.5 0.11 0.780 0.834 440±19 0.48 make little further difference to the distance estimate and inthemodelthatcanbecompensatedforbychangesinthe if i > 45◦ then there would be difficulties in measuring cluster distance of only 1,9 per cent. th ± rotation periods in a large fraction of WTTS. For completeness, I show the effects of considering Figs. 6e and 6f show the effects of changing the bi- the sample of 74 objects (ALL) including both CTTS and nary frequency between 0.3 < f < 0.7 at a fixed distance. WTTS(butallwithspectraltypesofM2orhotter)andalso Againthesmalldifferencescanbecompensatedforbysmall asampleofWTTSdefinedonthebasisofEW[Ca](WTTS2 changes( 3.7percent)indistance–largerf leadstolarger –seesection5.1).AsexpectedfromFigs.3and4andthedis- ± distances. cussioninsection5.1thedistributionfortheentiresampleis Figs. 6g and 6h show that changing the assumed value skewedtohigher(sini)obs values,resultinginasignificantly of δ in the plausible range 0.07 < δ < 0.15 sig- larger distance estimate with slightly better statistical pre- logR logR nificantly changes the shape of the distribution at a fixed cision(440 19pc).ThesampleofWTTSbasedonEW[Ca] ± distance. This can be partially compensated for by chang- is larger than sample WTTS1 but filters out a few more of ingthedistanceby 2.3 percent.Thebroaderdistribution the objects with large (sini)obs values. Hence the deduced ± is a poorer fit to the cumulative distribution function but distancebased on sampleWTTS2 ismarginally closer than cannot berejected. that based on sample WTTS1. Figs. 6iand6jshowthateventakingimplausibly large The “best” final ONC distance estimate is 392pc. The variations in the vtrue and ∆R(q) distributions (see sec- total uncertainty is 32pc, estimated from quadratic sum of tions 5.2 and 4.3 respectively) results in verysmall changes thestatisticaluncertaintyandthedistanceuncertaintiesdue (cid:13)c 2004RAS,MNRAS000,1–12 The distance to the ONC 9 12 D=392 pc 1 D=392 pc D=415 pc D=415 pc 10 D=470 pc D=470 pc 0.8 8 F 0.6 N 6 D C 0.4 4 2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (sin i) (sin i) obs obs 12 i =15 degrees i =15 degrees th 1 th i =30 degrees i =30 degrees 10 th th i =45 degrees i =45 degrees th th 0.8 8 F 0.6 N 6 D C 0.4 4 2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (sin i) (sin i) obs obs 12 f=0.3 1 f=0.3 f=0.5 f=0.5 10 f=0.7 f=0.7 0.8 8 F 0.6 N 6 D C 0.4 4 2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (sin i) (sin i) obs obs 12 δ =0.07 dex δ =0.07 dex δlog R=0.11 dex 1 δlog R=0.11 dex 10 δlog R=0.15 dex δlog R=0.15 dex log R log R 0.8 8 F 0.6 N 6 D C 0.4 4 2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (sin i) (sin i) obs obs Figure6.Theobserved(sini)obsdistributioncomparedwithvariousmodeldistributionsoftheformα(D/470pc).Thelefthandpanels showthedifferentialdistributionswhiletherighthandpanels showthenormalisedcumulativedistributions.(a,b)Showstheeffects of distanceonthedata-modelcomparison.Thethreemodelsshowncorresponstomodel1inTable5atD=392pc,model1atD=415pc (corresponding to a 1-sigma error) and model 1 at D = 470pc. (c, d) Shows the effects of varying ith by displaying models 1, 2 and 3 (ith = 30◦, 15◦ and 45◦) at a common D = 392pc. (e,f) Shows the effects of varying the binary fraction f by displaying models 1, 4 and 5 (f = 0.5, 0.3 and 0.7) at a common D = 392pc. (g, h) Shows the effects of varing δlogR by displaying models 1, 6 and 7 (δlogR=0.11dex,0.07dexand0.15dex)atacommonD=392pc. (cid:13)c 2004RAS,MNRAS000,1–12 10 R.D. Jeffries 12 v2, r1, D=392 pc 1 v2, r1, D=392 pc v1, r1, D=392 pc v1, r1, D=392 pc 10 v1, r2, D=392 pc v1, r2, D=392 pc 0.8 8 F 0.6 N 6 D C 0.4 4 2 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (sin i) (sin i) obs obs 20 WTTS2, D=381 pc 1 WTTS2, D=381 pc ALL, D=440 pc ALL, D=440 pc 15 0.8 F 0.6 N 10 D C 0.4 5 0.2 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 (sin i) (sin i) obs obs Figure 5. continued: (i,j)Shows theeffects ofchoosingdifferent vtrue (v1, v2)and∆R(q)(r1,r2)distributionsbydisplayingmodels 1, 8 and 9 at a common D = 392pc. (k, l) Shows a model which best fits a sample of WTTS defined using EW[Ca] (see section 5.1) (model10atD=381pc)andamodelwhichbestfitsasamplecontainingbothCTTSandWTTS(model11atD=440pc). to theparameter variations discussed above. This of course cretion activity such as the I K excess and EW of the assumes these uncertainties are independent, ignores any Caii 8542˚A line. A more carefu−l analysis could use mid-IR systematicproblemwiththespectraltype-effectivetemper- measurements to exclude (accretion) discs, but for now it aturerelationship, andassumesthatitiscorrecttoexclude seems wisest to discard thesuspected CTTS and adopt the CTTSfromtheanalysis.Theequivalentfigureincludingthe closer of thetwo distance estimates above. CTTS would be 440 34pc, where the uncertainty in this A further issue is the validity of the spectral- ± case is dominated by systematic effects. type/effective temperature relationship. As stated in section 5.1,therelationship ofCohen &Kuhi(1979) isuncon- troversial in that it is consistent with several later studies. However, these are also fundamentally based on the atmo- 6 DISCUSSION AND CONCLUSIONS spheres, angular diameters and bolometric luminosities of The observed sini distribution of a group of WTTS (and coolmainsequencestars.YoungPMSstarsarelikelytohave CTTS) in theONChasbeen modelled. Thesinivaluesare different atmospheres and the fact that rotational modula- linearly dependent on the cluster distance and from this a tionhasbeenmeasuredinoursamplestarsmeansthatthey distancetotheONCof392 32pcisderivedor440 34pc must becovered bycool spotted regions. Assigning a single ± ± if CTTS are included in the sample. The uncertainties in- temperature to such magnetically active stars may be diffi- clude1-sigmastatisticaluncertaintiesandthecontributions cult. These difficulties could be worse in CTTS with their fromuncertaintiesinanumberofotherpartiallyconstrained attendantdiscs, veilingandaccretion hotspotsandthisisa modelparameterssuchasthebinaryfrequency,biasagainst furtherreasonforexcludingthemfromtheanalysis.Asyet, theinclusion oflowinclinationobjects,theformofthetrue thereisnocompellingevidenceforachangeinthespectral- equatorialvelocitydistributionanduncertantiesinthestel- type/temperaturerelationshipfornon-accretingPMSstars, larradiusestimates.Withthepresentanalysisandobserva- but as deduced distances will scale as Te2ff this issue should tionalsample(34WTTS),statistical andsystematicuncer- beborne in mind. tainties are of equal importance. ThededucedONCdistanceissmallerthanmostprevi- The observed sini values depend linearly on the esti- ousresults, although generally agrees withintheerror bars. matedradii,which aredifficulttodetermineforPMSstars. Ifproperaccountofsystematicuncertaintiesisincluded,the Hillenbrand(1997)discussesthesedifficultiesandconcluded distance derived here is also probably more accurate than thatextinctions,luminositiesandhenceradiimaybeunder- most previous determinations. It is worth noting however estimated foraccreting CTTS, resulting inoverestimates of theexcellent agreementwith the419 21pcdeterminedby ± siniandhenceoverestimatesofthedistanceusingtheanal- Stassunetal(2004)fromaneclipsingbinarythattheycon- ysis presented in this paper. This effect is readily apparent sidered tobe a possible ONCmember. inthedata,evenwhenusingimperfectdiscriminatorsofac- To recover themore usually used longer ONC distance (cid:13)c 2004RAS,MNRAS000,1–12

The distance to the Orion Nebula Cluster

Similar The distance to the Orion Nebula Cluster