Are pre-main-sequence stars older than we thought?
MMon. Not. R. Astron. Soc. , 1–13 (2002) Printed 30 October 2018 (MN L A TEX style file v2.2)
Are pre-main-sequence stars older than we thought?
Tim Naylor
School of Physics, University of Exeter, Stocker Road, Exeter, EX4 4QL ???
ABSTRACT
We fit the colour-magnitude diagrams of stars between the zero-age main-sequence andterminal-age main sequence in young clusters and associations. The ages we derive area factor 1.5 to 2 longer than the commonly used ages for these regions, which arederived from the positions of pre-main-sequence stars in colour-magnitude diagrams.From an examination of the uncertainties in the main-sequence and pre-main-sequencemodels, we conclude that the longer age scale is probably the correct one, which implieswe must revise upwards the commonly used ages for young clusters and associations.Such a revision would explain the discrepancy between the observational lifetimes ofproto-planetary discs and theoretical calculations of the time to form planets. It wouldalso explain the absence of clusters with ages between 5 and 30Myr.We use the τ Key words: stars: formation – stars: pre-main-sequence – methods: statistical – openclusters and associations: general – stars: fundamental parameters – stars: early-type.
The ability to determine the ages of pre-main-sequence(PMS) stars is crucial for advancing our understanding ofthe early phases of stellar evolution. There are two key ap-plications. Firstly, and perhaps most obviously, we need stel-lar ages if one is to carry our experiments such as tracingthe evolution of stellar angular momentum, or following thefraction of stars with proto-planetary discs as a functionof time. Secondly, for PMS stars, the conversion from ob-servables such as temperature and luminosity into mass ishighly age dependent, making accurate ages vital for deter-mining the mass function. The primary method of deter-mining the ages required for these studies is to compare theobserved properties of PMS stars with models. The mosteasily accessible observables are a star’s temperature andluminosity, since they can be measured from its colours andmagnitudes. The problem is that for the same colours andmagnitudes different models can predict ages which differby a factor two, and even the same models will predict dif-ferent ages depending on which colours and magnitudes areused. This makes meaningful comparisons between the agesquoted in the literature for clusters or associations at bestdifficult, and often impossible. It was these problems whichled us to devise a model-independent age ordering of youngclusters and associations based on their colour-magnitude diagrams (Mayne et al. 2007). For PMS stars the primaryage diagnostic is based on the fact that stars fade as theyget older and contract towards the main sequence (MS). Weused this movement of the sequence towards progressivelyfainter magnitudes to derive an age ordering, although todo so we also had to measure a consistent set of distances,which we derived from the more massive stars which havealready reached the MS (Mayne & Naylor 2008).Whilst an age ordering such as ours is useful, for ex-ample it has showed unambiguously that different clusterstake different times to reach the same disc fraction or an-gular momentum distribution, for quantitative work an ab-solute scale is required. For PMS clusters and associationsthere are several usable age indicators, each of which re-lies on comparing stellar properties with models. For thisreason it is best to group them according to the underly-ing physics. First is the contraction of PMS stars as theyapproach the MS. As pointed out above, and discussed atlength in Mayne et al. (2007) these “contraction” or “PMS”ages are highly model dependent, and given the current dis-agreements between the models cannot yield an absolute agescale. Although most stars in a young cluster or associationare in the PMS phase, the evolution of the most massivestars proceeds so fast that they may not only have reachedthe MS, but evolved beyond it. This gives us access to twomore age measures. First, having reached the MS, stars move c (cid:13) a r X i v : . [ a s t r o - ph . S R ] J u l T. Naylor redwards and to higher luminosities away from the zero-agemain sequence (ZAMS), due to the increasing helium con-tent of their cores. This movement continues until the pointof core hydrogen exhaustion, when the star has reached thereached the terminal age MS (TAMS or MS turn-off). Fi-nally, after the turn-off, the post-main-sequence evolutionis driven by the burning of heavier elements which leads tomuch more rapid movement in the CMD. This relativelyhigh velocity in the CMD means that post-main-sequenceevolution has the potential to give precise ages. However,for young galactic clusters the paucity of stars in this re-gion of the CMD means such an age can depend on justone star, and such ages are rightly treated with some scep-ticism. Conversely, the main-sequence evolution (from theZAMS to the turn-off) has a larger number of stars, but themovement is often subtle, and using the normal techniqueof simply plotting isochrones over the data leads to largeuncertainties in age, and to questions over objectivity. How-ever, we have been developing a method of making objectivefits to colour-magnitude data, which should allow us to un-lock the information in this stage of a star’s evolution. Thetechnique, called τ fitting, can be viewed as an extension of χ to data points with uncertainties in two or more observ-ables, and to models which are distributions (not just lines)in the data space.The aim of this paper is to apply the τ fitting tech-nique to the main-sequence evolution of young stars, anduse the resulting ages to create a revised age scale for PMSstars. Surprisingly, this leads to a significantly older agesthan the commonly used contraction ages, a result whichwe will discuss in Section 11. To derive this result we firsthave to update our statistical techniques originally describedin Naylor & Jeffries (2006), since, as we discuss in Section 4,the technique will not work for the isochrones we wish to fit.We therefore lay out the changes which need to be made byfollowing an example through fitting (Section 5), testing thegoodness of fit (Section 6) and determining the uncertain-ties in the derived parameters (Section 7). Before doing so,however, we discuss the data and models we use (Sections2 and Sections 3). We deal with the effects of interstellarextinction in Section 8, and the details of each cluster inSection 9. We draw all the results together in our discussionin Section 11. To compare a set of ages derived from MS evolution withcontraction ages we need a sample of clusters and associ-ations which have contraction ages, and for each of whichdata are available for MS fitting. Our sample is, therefore,based on the groups we placed in age order using the PMS inMayne & Naylor (2008). Clearly, for each of these groups werequire stars in the appropriate mass range to show signifi-cant MS evolution, but we also require extinctions and reli-able distance measurements.
UBV photometry can provideall three of these. First the U − B / B − V diagram providesextinctions. Second, the upper part of V / B − V diagram isage sensitive, tracing the evolution of stars from the ZAMSto the turn-off. Finally, in the age range of interest the lowermass stars are still close to the ZAMS, and the sequenceturns redwards, making it ideal as a distance measure. Fur- thermore, the UBV photo-electric system is very consistentand well characterised. However, to ensure we maintain thehighest level of consistency we have restricted ourselves asfar as possible to the data of Johnson and collaborators, pri-marily taken in the 1950s and 1960s. As we shall show later,the quality of these data when combined with the transfor-mations of Bessell et al. (1998) is impressive, giving τ valueswhich mean the model is a good fit to the data. Clearly wewish to avoid PMS stars contaminating our sample at faintmagnitudes and red colours, and so for most objects we ap-ply a cut in observed B − V which roughly corresponds to( B − V ) < . V and B − V , and in previous work we have always been care-ful to include that correlation when modeling the uncertain-ties. The starting point for such an analysis is that V and B are measured independently, and so the uncertainties in V and B − V are δV and √ δV + δB respectively. Such ananalysis also leads to the conclusion that the uncertainty in B − V must be larger than that in V , in direct contradictionto the quoted uncertainties for most of the data presentedhere. This is because it is not photon statistics which arethe driver of the uncertainties, but changes in the trans-parency. In this work, we therefore model the uncertaintiesas uncorrelated. Although we will try other models later, we begin by using“Geneva-Bessell” isochrones. For the stellar interior we fol-low the suggestion of Lejeune & Schaerer (2001), and usethe “basic model set” (i.e. set “c”) of the Geneva isochrones(Schaller et al. 1992). Temporal interpolation is a muchmore significant issue for post-MS isochrones than the PMSisochrones we have fitted in the past, as there are sharp dis-continuities in the rate of change of magnitude and colourwith time, as exemplified by the MS turn-off. We there-fore use the code provided on the website to interpolate theisochrones to the appropriate age. We then convert fromluminosity and effective temperature to colours and magni-tudes using the tables of Bessell et al. (1998), assuming thecolours of Vega are zero (though V = 0 . σ Ori. In this case we have used the con-version given in Bessell (2000) to convert the Geneva-Bessellisochrones into the Tycho system. (Høg et al. 2000, statethat the Tycho-1 and Tycho-2 systems should be identical.) c (cid:13) , 1–13 he PMS age scale We used the reddening vector derived in Mayne & Naylor(2008).
In Naylor & Jeffries (2006) we introduced a solution to thelong-standing problem of how to fit photometric data toisochronal models in colour-magnitude diagrams (CMDs).Whilst fitting an isochronal model (a curve) to a set of datapoints may at first appear to be a simple χ problem, thefacts that the data points have uncertainties in two dimen-sions, and that the curve is smeared by binarity into a two-dimensional distribution, means a more sophisticated tech-nique is required. We have now used our solution to deriveages and distances for the young clusters NGC2547 (Nay-lor & Jeffries 2006) and NGC2169 (Jeffries et al. 2007), andconsistent distances to some of the best-studied star-formingregions (Mayne & Naylor 2008). In Jeffries et al. (2009) wederive distances to Vel OB2 and the association around γ Vel, and Joshi et al. (2008) use our technique for measuringthe distance to NGC7419.Naylor & Jeffries (2006) provide a rigorous developmentof τ , but for the purpose of understanding the improve-ments we have had to make to the method, a relativelysimple intuitive interpretation gives a better insight into theproblems. Figure 1 shows a typical fit of a dataset (shown ascircled error bars) to a model (the colour scale). The modelis a simulation of roughly a million stars (including binaries)using a specific age, metallicity, mass function and distance,which is then sampled onto a grid in colour-magnitude space.A fit in (for example) distance can be viewed as movingthe model in the y -direction until one obtains the strongestoverlap between the model and the data. This overlap canbe quantified for a single data point by taking the functionwhich represents its position and uncertainties (normally atwo-dimensional Gaussian), multiplying it on a gridpoint-by-gridpoint basis by the grid, and then summing the re-sulting values. If the grid is ρ ( c, m ) (where c and m are thecolour and magnitude co-ordinates respectively) and the i thdata point and its uncertainties U i ( c − c i , m − m i ) (where( c i , m i ) are its co-ordinates), then mathematically the over-lap is the integral of U i ρ over the entire space. The productof these integrals for all the data points will therefore reflectthe overall overlap between the data points and the model,and so we define a statistic τ = − (cid:88) i =1 ,N ln (cid:90) U i ( c − c i , m − m i ) ρ ( c, m )d c d m, (1)whose minimum value corresponds to the best fit.In Naylor & Jeffries (2006) we showed that this defini-tion will, for models which are curves in ( c, m ) space, andonly have uncertainties in the m -axis, reduce to that for χ .However, this is only the case if one chooses to multiply ρ bya normalisation factor which is dependent on the gradientof the isochrone. Unfortunately this normalisation factor be-comes infinite if the isochrone is vertical, and double-valuedat any magnitude at which the isochrone is double valued.This means our χ -like normalisation will fail for the CMDfitting required here, because as one moves up the sequencetowards bright magnitudes the isochrones become vertical,before finally switching to a negative gradient. Furthermore, if we wish to fit in U − B / B − V space, the isochrones aredouble-valued for certain values of B − V .In what follows, we therefore develop an alternative nor-malisation, which allows us to fit the data. In doing so weexpose the limitations of an approximation we made whencalculating the probability that the data are a good fit tothe model. We must first create a probability density function to fitto the data. As in Naylor & Jeffries (2006), we create thisby simulating stars over the appropriate range of masses.For each star we choose a mass randomly from the SalpeterIMF, and if the star is a binary, we assign it a companion ofa mass drawn from a uniform distribution between zero andthe mass of the primary. The stellar model then provides aluminosity, gravity and effective temperature for each star,which we then convert into colour and magnitude using theappropriate bolometric corrections. If a binary companion isso low mass, or so cool, that it does not appear in the mod-els it is assigned a flux of zero. (Note that this assignment isa change from Naylor & Jeffries (2006), but has been usedin all our subsequent work.) The value of each pixel in theimage is then simply the number of stars whose colours andmagnitudes lie within the pixel. We typically simulate 10 stars, and for this work have used pixels of size 0.0025 mag-nitudes in each axis. This is half the value we have used inprevious work, but is necessitated by the small uncertaintiesof the current data. We find the residual effects of the place-ment of pixel boundaries are much smaller ( ∼ ρ Before proceeding further we must address the normalisationof the model image, ρ . In Naylor & Jeffries (2006) we usedour χ -like normalisation which was a function of magni-tude. Here, we instead explore the results of a much simplernormalisation, setting the integral of ρ over the entire imageto one. This raises the question of how faint magnitudes wemust integrate down to. In fact, the strictly correct way toproceed would be to first multiply the image by the pho-tometric completeness function, such that below a certainmagnitude ρ was zero, and then set the integral of whatremains to one. Such a normalisation has an interesting,though subtle implication. When fitting for distance mod-ulus as the distance modulus increases, there is a decreasein the non-zero area of ρ , the region between the faintestobservable absolute magnitude (as defined by the complete-ness function) and the brightest model star. Given that theintegral over the model remains one, this means the valueof any non-zero pixel will increase, implying that τ will de-crease, and hence the fit improve. This is actually the correctbehaviour since it means that a model which fully populatesthe upper part of the sequence is better than one which doesnot. Practically, for our data, we can use a simpler normali-sation, where we make the integral between the faintest and c (cid:13) , 1–13 T. Naylor the brightest data points one. This means we have thrownaway one possible source of information, but in practice thisdoes not significantly affect the fits.Comparing the results obtained using this normalisa-tion with that used in Naylor & Jeffries (2006) simplychanges the values of τ in a given τ parameter grid byan additive factor; it does not change the best-fitting pa-rameters. This is at first a surprising result since we arechanging the value of ρ in one part of the isochrone com-pared with another, which may appear as a weighting of thepoints. However, it should be remembered that adding thelogarithms of the integrals in Equation 1 is equivalent tomultiplying them together, so changing the relative valuesof ρ as a function of magnitude is a normalisation, not aweighting process. The only possibility for altering the bestfit is if the length scale for changes in ρ is small comparedto the size of an error bar. Then data points will drag the fitso that they lie in the higher-valued regions of ρ . Since thedata point is more likely to originate in the higher-densitypart of the model this would again be the correct behaviour.Finally, it is important to note that we have no longer“normalised out” the mass function as we did in Naylor &Jeffries (2006). Changing the mass function will change thevalue of τ . In practice we have chosen to fix it such thatdN / dM ∝ − .
35, which results in good fits to the models. U At the same time as considering the normalisation of themodel, we should also consider that of the uncertainty func-tion ( U in Equation 1). In χ fitting this is set such thatthe maximum value of U is always the same, so the high-est probability attainable is always the same, correspond-ing to a perfect fit, i.e. χ = 0. This is the normalisationwe adopted in Naylor & Jeffries (2006). However, there isanother obvious possibility, setting the integral of U to beone. This would have a very significant advantage in caseswhere the error bars seem to have been significantly under-estimated, and to obtain a good fit (i.e. a value of τ whichcorresponds to a Pr( τ ) of approximately 0.5) one has toadd an extra uncertainty to U , in addition to those fromthe observations. This could well be due to mis-matches be-tween photometric systems. In such cases the procedure wehave previously adopted has been to calculate τ , and thenPr( τ ) for increasing values of the added uncertainty, un-til Pr( τ ) exceeds 0.5. However, if one normalises U suchthat its integral is one, then conceptually one is comparinga model which includes the uncertainties with data pointswhich are δ -function. One can, therefore, simply adjust thevalues of the uncertainties until one obtains the lowest valueof τ .This normalisation has an additional conceptual advan-tage. In the case where the uncertainties are very small onecan now approximate U as a 2-dimensional δ -functions. Thiseffectively removes the integral in Equation 1, and meansone can evaluate τ by simply multiplying together the val-ues of ρ at the positions of the data points.We will refer to a normalisation where the integral of themodels and the integrals of the uncertainty functions are allone as the natural normalisation. This clearly distinguishesit from the χ -like normalisation used in Naylor & Jeffries(2006). -0.2 -0.1 04.555.566.577.588.59 Intrinsic B-V E x t i n c t i on F r ee V Figure 1.
The data and best-fitting model for Cep OB3b. Thecolour scale is the model ( ρ in Equation 1) and the encircled errorbars are the data. Given we now have the correct normalisations we can nowfit our example data, which is a sample from Cep OB3b de-scribed in detail in Section 9.6. We calculated the extinctionon a star-by-star basis as described in Section 8, and aftercorrecting for it, searched in both age and distance modu-lus, evaluating Equation 1 at values of our fitting parameterswhich cover the range of interest. The resulting τ space isshown in Figure 2. The best fit, which lies at 10 Myr anda true distance modulus of 8.7 mags, is shown overlayed onthe data in Figure 1.In some fits we find that there are data points whichclearly do not lie on the sequence, and are presumably non-members. To deal with these objects we first fit the datawith a variant of the “soft clipping” first described in Sec-tion 7.1 of Naylor & Jeffries (2006). We adapt this to the newnormalisation by imposing a maximum τ for any one data c (cid:13) , 1–13 he PMS age scale T RU E D I S T ANC E M O DU L U S Figure 2.
The τ grid for Cep OB3b. The contour is at the 68percent confidence level. point. The value used is the minimum value of τ amongstall the data points, plus a fixed value, normally 20. We im-plement this by calculating the probability corresponding tothe imposed maximum τ and adding this to the calculatedprobabilities for each data point before calculating their τ values. We then performed a second fit removing the datapoints which had τ values close to the clipping limit, withno clipping limit applied. To test whether the model is a good fit, one must evaluatethe chance of obtaining a given τ or below. One does thisby calculating Pr( τ ), the cumulative distribution of the ex-pected value of τ . In Naylor & Jeffries (2006) we showedhow to calculate this, for no free parameters, in such a waythat it was insensitive to an incorrect choice of mass func-tion. We then suggested that one allow for free parametersby multiplying the τ axis of the distribution by ( N − n ) /N .Although our numerical simulations in Naylor & Jeffries(2006) showed that the above approach to the free param- Figure 3.
The probability of obtaining a given τ for a fit of30 data points to a main-sequence. This is the distribution onewould obtain if one created a large number of datasets at a givendistance modulus and extinction, and then “fitted” the data withthe distance modulus and extinction fixed at their original values.The right-hand solid curve is for the χ -like normalisation, theleft-hand solid curve for a natural normalisation where the sumof the probability over all colours is independent of magnitude.For comparison the dashed curves show the χ distribution for 30degrees of freedom and the Gaussian distribution for σ = 60, withtheir expectation values shifted to match those for the χ -like andnatural distributions respectively. eter problem may be approximately correct for the χ -likenormalisation, it is straightforward to show that it cannotbe correct in an arbitrary normalisation, such as the onedescribed in Section 5.2. Consider a plot of the cumulativedistribution of Pr( τ ) as a function of τ (Figure 3). Chang-ing the normalisation of the model means multiplying ρ inEquation 1 by a constant. This has the effect of adding aconstant to the values of τ as shown by the solid curvesin Figure 3. Allowing for free parameters by scaling the τ axis of the distribution by ( N − n ) /N would yield differentvalues for the decrease in τ when adding extra parameters,depending on the normalisation. This cannot be correct, thedecrease must be additive for the shape of the distributionto be invariant for a change in normalisation.There is an approximate solution to this problem,though, based on the fact that for CMD fitting, the distri-bution of Pr ( τ ) is similar to Pr ( χ ), save a additive factor.The reason for this is that both distributions derive fromthe distribution of probability, and therefore τ , in the CMDplane. For a χ problem this distribution is a line, smearedby a one-dimensional Gaussian. For the τ CMD problemthe distribution approximates to two sequences (those ofsingle-stars and of equal-mass binaries) smeared by a two-dimensional Gaussian. It is, therefore, unsurprising that theresulting distributions of Pr ( τ ) and Pr ( χ ) are similar. Sowe could approximate Pr ( τ ) by simply using the χ dis-tribution directly. However, for large values of N − n thedifferential form of the χ distribution tends to a normaldistribution whose mean is N − n and whose σ is 2( N − n ). It is interesting that the differential form of the τ distributionc (cid:13) , 1–13 T. Naylor
This means we can allow for n free parameters by subtract-ing the expectation value from the distribution of τ , mul-tiplying the τ axis by ( N − n ) /N ), and adding back theexpectation value less n . We have implemented the latterapproach, as it retains any asymmetry in the distribution.Applying this to the Cep Ob3b data results in a value ofPr( τ ) of 0.05. This is on the margins of acceptability, butno one datapoint is clearly discrepant. We have found a faster method for calculating the uncertain-ties than that presented in Naylor & Jeffries (2006). The aimof the calculation is to place a contour in the τ grid of Fig-ure 2 which represents a region within which the parameterslie with a given confidence. We can derive the uncertaintiesby first converting the values of τ in the grid into probabil-ity, and then integrating over the entire grid. We then dividethis into the integral of the probabilities below progressivelyhigher values of τ to obtain the cumulative τ distribution.We can then pick off values of τ at given confidence limits,and draw contours on the τ space.There are four practical issues which have to be solvedwhen using this method. The first is that to carry out theintegral one must multiply each pixel by its area. If the axesare linear then the the area of the pixels is the same, and thesum of the pixels will suffice, as we normalise by the integralover the whole area. However, if the age axis is logarithmicthe simplest method is to multiply the probability by theage for that pixel, before performing the sum.The second problem is the underlying assumption thatthe model is correct. This means that the fitting to create thegrid must be carried out using only those data points whichare consistent with the model. So practically this means asecond fit must be carried out excluding any points whichthe first fit clipped, without any further clipping (see Section5.4). Even so, this means one fit as opposed to fitting typ-ically 100 Monte-Carlo datasets for the previous technique,giving a speed improvement of a factor of 100.The third issue is that one must sum the grid out to in-finity. This is less demanding than it might at first appear.For example, if fitting a single data point in one dimensionwith Gaussian uncertainties, one only has to move ± σ fromthe best fit to include 99.7 percent of the total probability,which is accurate enough for calculating a 95 percent confi-dence interval. Note, however, that the probability enclosedfor a given σ declines as the power of the number of observ-ables measured for each data point, and so if generalising τ to many dimensions one would have to act with caution.The final issue is that the machine precision may be ex-hausted for some data points towards the edge of the τ grid,where the corresponding values of ρ are very close to zero.For example, if the smallest representable number greaterthan zero is 1 × − , the highest τ which can be obtained (like the χ distribution) tends towards a Gaussian. This is a nat-ural consequence of the central limit theorem, since multiplyingfunctions together and then taking the logarithm is equivalentto averaging their logarithms. Whilst the mean and width of thedistribution are problem dependent, this may still provide a keyto the solution in more general cases. is approximately 168. Once the τ for any data point lies be-low this probability, the computer will calculate τ for thewhole data set to be infinite. To flag such points in the gridwe set them to a high value of τ (the number of data pointstimes the τ resulting from a probability of twice the small-est representable number). Since such a τ is guaranteed toreturn a probability of zero, this works transparently in codewhich implements the new method of calculating confidencelimits. However, we also note this number in the header ofthe grid file, so its meaning is clear if plots are made from thefile. Applying this technique to the Cep OB3b data resultsin the contour shown in Figure 2.For the work here, the distance is a nuisance parameter,and we need to be able to quote an uncertainty in age alone.We therefore integrate the probability in Figure 2 over alldistance modulii at each value of the age to create a run ofprobability with age. We then define a confidence limit asthat region in age which integrates to give 68 percent of theprobability, and which excludes equal integrals of probabilityabove and below it. For Cep OB3b this gives an age rangeof 8.6 – 10.9 Myr. Now having shown how we can fit for age, we must returnto the question of the extinction. We follow an improvedversion of the two-strand approach developed in Mayne &Naylor (2008). We first attempt to fit the the U − B / B − V data with just the reddening as a free parameter. We cannow (in contrast to Mayne & Naylor 2008), test whether themodel is a good description of the data. If it is, we assumethe extinction is uniform, and apply the derived extinctionto all the data points. If Pr( τ ) is too high, we conclude theextinction is non-uniform, and resort to deriving individualextinctions for each star by moving them along the (colourdependent) reddening vector until they reach the single-star U − B / B − V isochrone. This is essentially a modern versionof the Q method of Johnson & Morgan (1953). As explainedin Mayne & Naylor (2008) the disadvantage of this methodis that it cannot allow for the fact the star may be a binary.This has the effect of narrowing the dereddened sequencein V / B − V space, hence our preference for the τ methodwhere the extinction can be shown to be uniform. We can now apply our technique to the rest of our sampleof clusters and associations to derive MS ages. Each datasetwe fit is given as an (electronic only) table as summarised inTable 1, though we show the data for λ Ori as an examplein Table 2.
We used the data and uncertainties of Walker (1957), whichare for a sample which is unbiased in colour and taken froman specific area of the cluster. To ensure we excluded thePMS, we selected only those stars blueward of B − V =0.28 c (cid:13) , 1–13 he PMS age scale -0.4 -0.3 -0.2 -0.1 0 0.167891011 Intrinsic B-V E x t i n c t i on F r ee V -0.3 -0.2 -0.1 0 0.16789101112 Intrinsic B-V E x t i n c t i on F r ee V Figure 4.
The best-fitting model for NGC6530 with the agefixed at 2Myr (left) and the best fitting model with both age anddistance as free parameters (right). Note how the five brightestdatapoints are better fitted in the right-hand plot, and how thegroup of datapoints below them would have the be interpreted asbinaries in the model on the left. The very faintest single starsalso lie marginally closer to the single-star sequence in the better-fitting model. and brighter than V = 13 and derived individual extinc-tions. The resulting fit is shown on the right-hand side ofFigure 4. σ Ori
Our sample consisted of the members listed by Sherry et al.(2008) that are blueward of B − V = 0 .
1. We omitted thetwo stars noted by Sherry et al. (2008) as variable, and HD37333 which is above the MS, and probably a PMS star.We could not find a consistent Johnson
UBV dataset forthis cluster, and so used the Tycho-2 catalogue and its firstsupplement (Høg et al. 2000), although it does not containa magnitude for σ Ori C. In this dataset a combined mag-nitude is given for σ Ori A and B. As σ Ori A is itself abinary, we removed the effect of σ Ori B on the combinedmagnitude by assuming the magnitude difference betweenthe components is the mean of the values for the differencefound from speckle (Horch et al. 2001) and adaptive optics(ten Brummelaar et al. 2000) work. σ Ori B will have littleeffect on the combined colour.The disadvantage of using the Tycho-2 data is that wecannot determine the extinction, as there are no U -banddata. We therefore simply adopted E ( B − V ) = 0 .
06 from Brown et al. (1994). The resulting τ contour does not closeat low ages, and so we have only an upper limit on the age.We therefore quote (in Table 1) the upper limit below which68 percent of the probability lies, but exclude this clusterfrom further analysis. We used the photo-electric data of Walker (1956), as pre-sented in his Table 1. Fitting all stars blueward of B − V = 0for extinction in U − B / B − V space gives Pr( τ )=0.37, im-plying uniform extinction over the field. We then fitted in V / B − V and obtained a Pr( τ ) of 0.06. This is on themargins of acceptability, and there is a case that the twodata points furthest redward from the sequence should beremoved. However, in not doing so we simply enlarge ouruncertainty estimate, and so are being conservative. λ Ori (Collinder 69)
We used the data from Murdin & Penston (1977) takingonly those stars within half a degree of λ Ori. We excludedobjects with B − V > .
2, which results in a sample whichis almost complete blueward of this colour, and has no starsredward of ( B − V ) = − .
04. After applying reddeningsdetermined on a star-by-star basis, we obtained a value ofPr( τ ) of 0.52, provided we assumed the uncertainties were0.01 mags in V and 0.008 mags in B − V (Murdin & Penston1977, do not provide error bars) and removed two objects(HD36881 and HD36913) which appear to be non-membersbased on their position in the V /( B − V ) diagram. Theresulting fit is shown on the right-hand side of Figure 5. We used the data and associated uncertainties for NGC2362from Johnson & Morgan (1953), which were taken as part ofa programme to define what became the
UBV system. Weused only those stars blueward of B − V = 0 .
04 and excludedstars noted as non-members by Johnson & Morgan (1953).We also excluded the brightest star ( τ CMa) as it is clearlybeyond the turnoff. Finally we found that star 36 gave ahigh τ in both U − B / B − V and V / B − V , and star 50in V / B − V , and so removed them from the fit as well. Wethen measured a global extinction from the U − B / B − V diagram, before fitting in V / B − V . Blaauw et al. (1959) carried out a photometric survey ofstars of spectral type A0 and earlier identified from objec-tive prism plates. We take the membership list from Pozzo(2001), but exclude BHJ11 for which the measurement is acombined light measurement for a rather wide ∆ m = 2 . τ )=0.05. c (cid:13)000
04 and excludedstars noted as non-members by Johnson & Morgan (1953).We also excluded the brightest star ( τ CMa) as it is clearlybeyond the turnoff. Finally we found that star 36 gave ahigh τ in both U − B / B − V and V / B − V , and star 50in V / B − V , and so removed them from the fit as well. Wethen measured a global extinction from the U − B / B − V diagram, before fitting in V / B − V . Blaauw et al. (1959) carried out a photometric survey ofstars of spectral type A0 and earlier identified from objec-tive prism plates. We take the membership list from Pozzo(2001), but exclude BHJ11 for which the measurement is acombined light measurement for a rather wide ∆ m = 2 . τ )=0.05. c (cid:13)000 , 1–13 T. Naylor -0.2 -0.1 0345678910 Intrinsic B-V E x t i n c t i on F r ee V -0.2 -0.1 045678910 Intrinsic B-V E x t i n c t i on F r ee V Figure 5.
The best-fitting model for λ Ori with the age fixed at3Myr (left) and the best fitting model with both age and distanceas free parameters (right). The brightest three stars are clearlybetter fitted by the model on the right.
Our data and uncertainties are taken from Walker (1969),who aimed to obtain photometry for as many stars as possi-ble within the outline of the dark cloud, since this area willbe the least contaminated by background stars. We removedstars redward of B − V =0.0, those marked as variables orvisual doubles, and three stars which lie away from the se-quence in the U − B / B − V diagram. We fitted the data fora single extinction in U − B / B − V space, and after remov-ing two outliers in τ obtained a good fit with Pr( τ )=0.46.At first this may sound counter-intuitive, since it impliesuniform extinction, yet it is well known that the extinctionof ONC members is highly variable. In fact it seems thisonly applies to stars in the central cluster. We then fitted in V / B − V to obtain the results in Table 1. The resulting fitis shown on the right-hand side of Figure 6. -0.3 -0.2 -0.1 0 0.13456789 B-V V -0.3 -0.2 -0.1 0 0.13456789 B-V V Figure 6.
The best-fitting model for the ONC with the agefixed at PMS age of 3Myr (left) and the best fitting model withboth age and distance as free parameters (right). The 68 percentconfidence interval for the best-fitting age just encompasses 3Myr,and so we expect the improvement in the fit from left to right tobe only marginal. We can see the improvement in the fit is dueentirely to the improvement in the fit for the brightest star, andeven that is at the expense of a worse fit for the second brighteststar. Thus the conclusion that the statistics drive us to, that theimprovement is marginal, seems reasonable.
We took the photometry from Claria (1982), and used anuncertainty of 0.02 mags in both magnitude and colours.This is an estimate for U − B and corresponds to the devi-ations for single observations derived by Claria (1982) fromcomparison with the data of Fernie (1959). Although Claria(1982) often has four observations per star we prefer to takethe view that the uncertainty represents the difference be-tween the photometric systems. We use the membership listof Claria (1982), which is based on proper motions, photom-etry and spectroscopy, and select only stars with B − V < . τ ). Furthermorethis star is right on the edge of the proper motion distribu-tion of the bulk of the members, so we excluded it. c (cid:13) , 1–13 he PMS age scale We used the data of Eggen (1972), excluding stars with B − V > .
0, and HD93163, which lies away from the sequence.Unfortunately Eggen (1972) does not provide uncertainties,but we found a single extinction would yield Pr( τ )=0.79 foruncertainties of 0.014 and 0.014 mags in B − V and U − B respectively, which suggests the extinction is uniform. Usingthat extinction, and an uncertainty in V of 0.025 mags givesPr( τ )=0.27 when fitted in V vs B − V . We again used the data and memberships from Johnson &Morgan (1953). The U − B / B − V diagram, especially theregion where the gradient is reversed, shows that there isvariable extinction to this cluster. We therefore dereddenedthe data on a star-by-star basis, which limits us to B − V < .
0. Before fitting we also excluded Hertzsprung 371 (whichappears to be reddened).
10 PRE-MAIN-SEQUENCE AGES
For the PMS ages we require a set of consistent ages, andwe therefore adopt the ages of Mayne & Naylor (2008), withthe following exceptions.
We take the PMS age for NGC2547 from Naylor & Jef-fries (2006) as 38.5Myr, which is derived from isochrone fit-ting, though also agrees with the Lithium depletion age. Theage for IC2602 (25Myr) is taken from Stauffer et al. (1997),which again is based on isochrone fitting to the PMS stars.We use a PMS age of 4.5Myr for Cep OB3b, which is fromLittlefair at al (in prep), but is based on the system of Mayne& Naylor (2008).
The position on the sky of our MS sample is shown in Fig-ure 7, along with the positions of the sample of Hillenbrand(1997), which represents stars in the ONC itself, and theflanking fields of Ram´ırez et al. (2004). Given the distribu-tion of stars, it is clear that the PMS age we should use isthat of the flanking fields. Although Ram´ırez et al. (2004)calculate this, they do so on the assumption that the ONC is470pc away, whilst a more modern estimate is 400pc (Mayne& Naylor 2008, and references therein). In the V /( V − I ) diagram Ram´ırez et al. (2004) place the flanking fields 0.3mags above NGC2264. Correcting the distance to 400pc willbring the flanking fields PMS to the same magnitude as thatof NGC2264, and therefore to an age of 3Myr on the scaleof Mayne & Naylor (2008).
11 DISCUSSION
We collect together our measurements of the ages of thegroups and clusters in Table 1, along with the other pa-rameters from our fits. For completeness we include the dis-tances, though as these are derived from two-parameter fits
Figure 7.
The positions of stars in the vicinity of the OrionNebula Cluster. The dots in the central region are one in five of thestars from Hillendbrand (1997). The dots around the peripheryare X-ray sources from Ram´ırez et al. (2004). The filled circlesare the main-sequence sample.
Table 2.
An sample of Tables 2-11, the fitted dataset for λ Ori.Table 1 gives the number of the electronic table for each dataset,along with the reference for the star numbering system. As shownhere, for each cluster we give, along with the uncertainties, thefitted V , B − V and U − B . In the case of groups where extinctionswere derived on a star-by-star basis these are the reddening andextinction free values, and the E ( B − V ) used is given in the lastcolumn.Star V B − V U − B E(B-V)number mag σ mag σ mag σ we emphasise that those of Mayne & Naylor (2008) are to bepreferred. We plot PMS against MS age in Figure 8. Whilstthe PMS and MS ages for individual clusters may agree towithin the uncertainties, the average of the MS ages is sig-nificantly older than the average of the PMS ages. If wetake only those clusters and associations less than 10Myrold, the MS ages are, on average, a factor two larger. Theissue is clearly which of these age scales is correct. Explanations as to why the MS ages may be incorrect fallinto two groups, those associated with the statistical tech-niques and those associated with the models. We can rule c (cid:13)000
An sample of Tables 2-11, the fitted dataset for λ Ori.Table 1 gives the number of the electronic table for each dataset,along with the reference for the star numbering system. As shownhere, for each cluster we give, along with the uncertainties, thefitted V , B − V and U − B . In the case of groups where extinctionswere derived on a star-by-star basis these are the reddening andextinction free values, and the E ( B − V ) used is given in the lastcolumn.Star V B − V U − B E(B-V)number mag σ mag σ mag σ we emphasise that those of Mayne & Naylor (2008) are to bepreferred. We plot PMS against MS age in Figure 8. Whilstthe PMS and MS ages for individual clusters may agree towithin the uncertainties, the average of the MS ages is sig-nificantly older than the average of the PMS ages. If wetake only those clusters and associations less than 10Myrold, the MS ages are, on average, a factor two larger. Theissue is clearly which of these age scales is correct. Explanations as to why the MS ages may be incorrect fallinto two groups, those associated with the statistical tech-niques and those associated with the models. We can rule c (cid:13)000 , 1–13 T. Naylor
Table 1.
Main-sequence and pre-main-sequence ages.Cluster or PMS Age MS Age (Myr) Pr( τ ) Distance E(B-V) Data NumberingGroup (Myr) Best fit 68% confidence Modulus table system reference λ Ori 3 6.6 5.8–7.5 0.52 8.05 0.12 σ Ori 3 0.4 <
11 Hertzsprung (1947) Median from individual extinctions. From Brown et al. (1994).
Figure 8.
The main-sequence and pre-main-sequence ages forour sample. The groups at PMS ages of 3 and 4.5Myr have beenseparated slightly in age to aid visibility. out problems with the fitting procedure by comparing ourages with those obtained by Meynet et al. (1993). They usesimilar isochrones to the ones presented here and measurethe age of the Pleiades as 100 Myr and the environs of theONC as 4Myr. Both these ages are compatible with those wemeasure, suggesting our technique gives similar ages to “byeye” fitting. Equally importantly, we match the lithium de-pletion age for the NGC2547 (34–36Myr Jeffries & Oliveira2005) and are very close to the depletion age for the Pleiades(125–130Myr Stauffer et al. 1998).To check that our uncertainties are at least reasonablewe tested how the result changes if one star is removed fromeach fit. As one might expect, the brightest star in the fitprovides the tightest limits on the age. We therefore removedthe brightest star from each dataset and replotted Figure 8.As Figure 9 shows, the result remains clear, though as onemight expect the error bars are larger, and one more dataset(the ONC) returns an upper limit for the age. This stronglysuggests that our uncertainty estimates are reasonable, andthe result is robust. This experiment also shows what theeffect might be of a non-member being included in the fit.Were a non-member very far from the fitted sequence itwould have been clipped out by the procedure described at
Figure 9.
As figure 8 but with pre-main-sequence ages calculatedwithout the brightest star for each dataset. the end of Section 5.4. Were it close to the sequence, thenit could deviate the fit sufficiently to have a reasonable τ ,but then would only change the best fit by a small amount,similar to the effect of removing a data point.As a final check of the uncertainties, in Figures 4 to 6 weplot the data over the best fitting models if the age is fixedat the PMS age (left) or left as a free parameter (right). Thefirst two examples (NGC6530 and λ Ori) are ones where thePMS age lies far outside the 68 percent confidence region forthe upper-main-sequence age. As one expects, we see thatthe brightest stars lie to the right of the model when the ageis fixed at the PMS age. Our final example, the ONC, is onewhere the PMS age lies almost exactly on the edge of the 68percent confidence limit. Here the improvement in the fit is,as it should be, marginal. Although such comparisons withour expectations are at best subjective, that they fit withour expectations adds to our confidence in the result. Whencombined with the experiment of missing out the brightestdatapoint, we have a strong case that our uncertainties arecorrect, and the result is robust.The obvious problems with the models are the absenceof rotation, uncertainties as to the mass-loss rates, and thetreatment of convective core overshoot. Figure 9 of Meynet& Maeder (2000) shows that if the stars were rotating, and c (cid:13) , 1–13 he PMS age scale we fitted them with isochrones for stationary stars, the re-sulting ages would be too young by about 10 percent. Thistherefore exacerbates the discrepancy between the PMS andMS ages.All modern models include a degree of core overshoot,which has the effect of mixing more hydrogen into the core,and hence lengthening the MS lifetime. Naively, models withno overshoot will have shorter MS lifetimes than those usedhere, by roughly the decrease in available hydrogen (per-haps 20-40 percent), which is of the right order to bring theMS and PMS ages back into agreement. However, our CMDfitting does not measure lifetime on the MS, but how farfrom the ZAMS a star at a given luminosity (not mass) hasmoved. A close comparison of Figures 4 and 5 of Maeder &Mermilliod (1981) shows that for the youngest ages they cal-culate (25Myr) the difference in the position of the isochronecorresponds to an age difference of around 5 percent.Mass-loss rates for early-type stars are uncertain, andso in addition to using the Geneva models with the standardmass-loss rates (set “c”of Maeder & Meynet 1994) we alsotried the higher mass-loss rate, set “e”. Comparison of theresulting isochrones for the masses and ages we are inter-ested in shows differences in colour which are too small toaffect our results.Finally, we have tested the effect of using different MSmodels. As an alternative to the Geneva models with theBessell et al. (1998) conversions to colour and magnitude weused the conversions presented with the isochrones in Leje-une & Schaerer (2001). We obtained ages somewhat olderthan those from the Geneva-Bessell models, exacerbatingthe age difference problem. More importantly, the values ofPr( τ ) are much worse than those for the Geneva-Bessellmodels, typically around 0.01 or 0.001, showing that thesemodels can be ruled out as good descriptions of the data.To test whether this is the interior models or the atmo-spheres, we fitted the data to the Padova models (Girardiet al. 2002) but with the same model atmospheres (Bessellet al. 1998) as we used for the Geneva-Bessell models. Wefind this gives slightly younger ages (a factor 1.5 older thanthe PMS ages in the range 1-10Myr), but very similar valuesof Pr( τ ) to the Geneva-Bessell models. In summary, there-fore, our fitting gives strong support for the Bessell et al.(1998) conversions, and there is only a weak effect from theinterior models, which can explain some, but not all of, theage discrepancy. The PMS ages are much less robust than the MS ones. Wehave adopted the PMS age scale of Mayne & Naylor (2008).However, as Mayne & Naylor (2008) and Mayne et al. (2007)make clear, the primary aim of this scale is an age ordering.The age scale itself is rather arbitrary, though was chosento match as closely as possible the commonly quoted agesfor the young groups. The problem is that there is no singlePMS age scale, a point nicely illustrated in Jeffries et al.(2009). They show that the γ Vel association could have aPMS age between 5 and 15Myr depending on which PMSmodels are used, and which part of the sequence is consid-ered. They estimate that the association is about 7Myr oldon the Mayne & Naylor (2008) scale, so doubling the ages ofthese young associations is consistent with some PMS mod- els. Our conclusion, therefore, is that the MS age scale isprobably the correct one.
Before discussing the implications of a longer timescale, weshould be wary of over interpreting Figure 8. Whilst itclearly shows a discrepancy between mean PMS and MSages, the error bars for individual data points are large. Allwe can say with any certainty is that there is a differenceof approximately a factor two at PMS ages of 3Myr. By30Myr our data are consistent with the age scales matching,though a difference of a factor 1.5 is still, in the statisti-cal sense, likely. We therefore limit ourselves to discussingthe implications of a lengthening of the timescales in the1-10Myr PMS age range. Even here, however, we find thereare problems it might solve.There is a long-standing issue that the observedtimescale for the dissipation of proto-stellar discs (3Myr;Haisch et al. 2001) may be shorter than the time required bythe models for planet formation (10Myr; Pollack et al. 1996).In recent years there has been significant effort to find mech-anisms which will shorten the planet forming timescales.Whilst a case can be made that this problem has been solved(Mordasini et al. 2008), there is a view that significant prob-lems remain (see, for example, the introductory sectionsof Ayliffe & Bate 2009; Dodson-Robinson et al. 2008). Afair summary is probably that whilst there are mechanismswhich could shorten the timescale, such as dust settling (Hu-bickyj et al. 2005) and planetary migration (Alibert et al.2005), the uncertainties in the physics remain such that it isnot clear they do. Our result offers an interesting alternativesolution. If the clusters used to measure the disc dissipationtimescale are 50-100 percent older than previously thought,there may be no contradiction with the Pollack et al. (1996)timescale.Jeffries et al. (2007) point out that there is a lack ofclusters in the age range 5–30Myr. Revising the age scalein the way suggested by the MS fitting would move clustersfrom the youngest ages into this age range. Furthermore ifthe age scales come back into register at around 30Myr, asFigure 8 suggests they might, there would not be a compen-sating movement out of the 5-30Myr range, leading to anincreased number of clusters at these ages.
12 CONCLUSIONS
We have shown that there is a systematic difference betweenthe ages of clusters and associations measured from the MSand ages commonly used which are based on the PMS. Thedifference is in the sense that the MS ages are a factor 1.5-2.0greater than the PMS ages in the age range 2-5Myr (on thePMS scale). The most straightforward solution is to adoptthe MS age scale, as there are PMS models which fit withthe longer timescale. Adopting the longer timescale offers asolution to the problem that the lifetimes of discs aroundstars (3-5Myr on the PMS age scale) are shorter than thetime taken to form planets, and to the apparent absence ofclusters in the 5-30Myr age range.Finally we should be clear that although we favour the c (cid:13) , 1–13 T. Naylor age-scale given by MS fitting, we are not recommending itas a method for deriving ages for individual clusters andassociations. As Figure 8 and Table 1 make clear, the uncer-tainties for individual groups are large. Nor can we at thispoint make any clear recommendation as how one shouldreflect this result when quoting PMS ages. Whilst it is clearthat the youngest ages need to be increased, how far downthe age scale that should be propagated is unclear. We there-fore continue to commend the Mayne et al/Mayne & Nay-lor age ordering, though recommend that if these ages arequoted one states clearly that they are on the Mayne etal/Mayne & Naylor scale. If absolute ages are required forclusters younger than 10Myr for comparison with other datawe recommend multiplying the Mayne et al/Mayne & Nay-lor values by 1.5 and quoting the age scale as originatingfrom this paper.
ACKNOWLEDGMENTS
I am grateful to three people for provoking significant partsof this work. First, Herbie (H.D.) Deas for conversationsmany years ago demonstrating that one has choices in math-ematics; a view which led me to the re-examination of thenormalisation presented in Section 5.2. Second, the (anony-mous) referee of Naylor & Jeffries (2006), whose efforts em-phasised to me the inelegance of the method presented inthat paper for calculating the uncertainties in the parame-ters, and hence led to Section 7. Third, the referee of a veryearly version of this paper forcefully made the point that itwould be improved by the inclusion of data, which led meto the main result presented here.This research has made use of the WEBDA database,operated at the Institute for Astronomy of the University ofVienna, from where much of the electronic form of the dataused here orginated. Finally, the referee of this version ofthe paper, John Stauffer prompted several improvements.
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