Mon. Not. R. Astron. Soc. 319, 457±466 (2000)

A comparison of the rotational properties of T Tauri stars in Orion and Taurus C. J. Clarkew² and J. Bouvier

Laboratoire d'Astrophysique, Observatoire de Grenoble, Universite Joseph Fourier, B.P. 53, F-38041, Grenoble Cedex 9, France Accepted 2000 July 10. Received 2000 May 22; in original form 2000 March 8

A B S T R AC T

We analyse the distribution of projected equatorial velocities (v sin i) for a magnitudelimited sample of stars in Taurus, in order to assess whether this sample can contain a population of fast rotators (missed in previous photometric monitoring campaigns) similar to those recently discovered in Orion by Stassun et al. We find strong evidence, in line with the results of photometric monitoring campaigns in Taurus, that there is no such population of stars in Taurus that rotate at a large fraction of breakup velocity. We thus demonstrate that the stellar rotation distributions in the two star-forming regions are intrinsically different (with a statistical significance of this discrepancy in excess of 3s ), and discuss possible origins for this difference. Key words: stars: magnetic fields ± stars: pre-main-sequence ± stars: rotation.

1

INTRODUCTION

The recent study by Stassun et al. (1999) of the photometric periods of pre-main sequence stars in Orion has seriously challenged current beliefs about the rotational properties of young stars. Periodic photometric variations in T Tauri stars are conventionally ascribed to rotationally modulated emission from spots on the stellar photosphere and have therefore been used as a direct probe of the stellar rotation-period distribution. Previous studies of this sort in Taurus and Orion (Bouvier et al. 1993; Herbst et al. 1994; Choi & Herbst 1996) had indicated that T Tauri stars rotate at a small fraction (typically less than 10 per cent) of breakup velocity. These studies also suggested that Classical and Weak Line T Tauri stars have rather different rotational properties: whereas Weak Line systems are rather broadly distributed in rotational period, Classical systems appeared to be more narrowly bunched towards longer periods (Edwards et al. 1993). The problem of maintaining a contracting star at less than breakup velocity, especially when it is accreting high angular momentum material from a disc, has given rise to a number of models for braking young stars. Motivated, in part, by the fact that the T Tauri stars which are associated with discs (i.e. the Classical T Tauri stars, henceforth CTTs) are rotating somewhat more slowly on average than their disc-less counterparts (the Weak Line T Tauri stars, henceforth WTTs), many of these models sited the braking mechanism in the disc/accretion flow (Konigl 1991; Cameron & Campbell 1993; Shu et al. 1994; Yi 1994; Ghosh 1995; Ostriker & Shu 1995; Armitage & Clarke 1996; Popham 1996; Tout & Pringle 1996; Ferreira, Pelletier & Appl 2000). In w

E-mail: [email protected] ² On leave from Institute of Astronomy, Madingley Road, Cambridge CB3 0HA. q 2000 RAS

particular, a number of models (which differ substantially in detail) invoked magnetic linkage between the star and slowly rotating disc material at several stellar radii; all these models require surface fields of about a kilo-Gauss in order that the star be strongly coupled to disc material rotating at an angular velocity similar to the observed angular velocity of T Tauri stars (see, however, Safier 1998 for a dissenting view on the efficacy of magnetic braking, and Miller & Stone 1998 for a discussion of early numerical results of magnetically threaded discs which do not reproduce many of the anticipated features of magnetic braking models). The study by Stassun et al. (henceforth SMMV) has upset this picture in two respects. First, it contains a substantial fraction of stars that are rotating at a high fraction of breakup velocity. Secondly, the rotational properties of the CTTs and WTTs are indistinguishable in this sample; in particular, there is no evidence that CTTs are prevented from acquiring angular velocities as high as those of the WTTs. Taken at face value, therefore, this study suggests that the braking is less efficient than previously believed (at least in a subset of stars) and that its efficiency is not obviously correlated with the current presence of circumstellar material. Although disc-braking models can be modified to some degree so as to obtain a higher fraction of faster rotators (e.g. by dispersing the disc more quickly relative to the stellar contraction timescale), they cannot explain the existence of the very rapidly rotating CTTs (which, moreover, show spectral evidence for active accretion onto the star) in SMMV's data. According to discbraking models, accretion onto the central star cannot occur unless the disc is strong enough (relative to the magnetic field strength of the central star) to extend down to radii inward of the corotation radius between disc and star (e.g. Armitage 1995). The field and disc properties invoked in standard disc-braking models allow the disc to extend to a radius of around 5 stellar radii, and hence

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Table 1. V sin i measurements for Taurus stars. Name

AA Tau BP Tau CI Tau CW Tau CX Tau CY Tau DE Tau DF Tau DG Tau DH Tau DI Tau DK Tau DL Tau DM Tau DN Tau DO Tau E DQ Tau DR Tau DS Tau FP Tau FX Tau GG Tau GH Tau GI Tau GK Tau GM Aur GO Tau HDE 283572 HK Tau HN Tau HP Tau HP Tau G2 Haro 6-37N Haro 6-37S Hubble 4 IP Tau IQ Tau IW Tau LkCa 1 LkCa 3 LkCa 4 LkCa 5 LkCa 7 LkCa 14 LkCa 15 LkCa 19 LkCa 21 LkHa 266S LkHa 332 G2 RW Aur RY Tau SU Aur T Tau UX Tau A UX Tau B UY Aur UZ Tau E UZ Tau W V410 Tau V710 Tau V773 Tau V819 Tau V826 Tau V827 Tau V830 Tau V836 Tau V927 Tau V928 Tau V955 Tau VY Tau

C/W TTs

v sin i (km s21)

C C C C C C C C C C C C C C C C C C C C C C C C C C C W C C C W C C W C C W W W W W W W C W W W W C C C C W C C C C W C W W W W W W W W C W

11.4 7.8 10.4 27.4 18.2 10.0 10.0 16.1 21.7 10.0 10.5 11.4 16 10.0 8.1 11.1 30.8 #10.0 11.2 26.6 10.0 10.2 22.1 11.2 18.7 12.4 17.5 $75.0 10.0 52.8 15.4 100 10.1 10.2 12.8 #11.0 11.5 6.9 23.3 21.8 26.1 37.0 12.9 21.9 12.5 18.6 60.0 12.9 22.3 17.2 52.2 65.0 20.1 25.4 10.0 19.3 15.9 13.7 71.0 15.9 55.0 #15.0 4.2 18.5 29.1 #15.0 13.4 18.8 10.1 10.0

s (km s21) 2.0 9.8 0.8 5.3 6.3

2.0 2.9 1.6 6.0 3.5 1.8 9.7 2.4 1.6 20.0 1.7 1.0 1.8 3.2 1.9 2.4 6.0 2.9 1.0 1.3 1.9 11.0 3.0 3.1 1.5 1.8 3.5 1.5 1.6 1.9 4.0 2.3 9.5 1.4 3.4 0.7 4.2 2.4 4.3 3.3

Teff

L/L(

M/M(

R/R(

v sin i ref.a

4060. 4060. 4060. 4730. 3580. 3720. 3580. 3470. 3955. 3720. 3850. 4060. 4060. 3720. 3850. 3720. 3850. 4060. 4350. 3370. 3720. 4060. 3580. 4205. 4060. 4730. 3850. 5770. 3785. 4350. 4730. 6030. 4060. 4205. 4060. 3850. 3785. 4060. 3370. 3720. 4060. 3580. 4060. 3850. 4350. 5250. 3470. 3470. 4060. 4730. 5080. 5860. 5250. 4900. 3720. 4060. 3720. 3470. 4730. 3720. 4730. 4060. 4060. 4060. 4060. 4060. 3145. 3785. 4060. 3850.

0.74 0.95 0.87 1.35 0.41 0.47 0.81 1.60 0.90 0.68 0.62 1.32 0.70 0.25 0.91 0.62 0.72 0.85 0.65 0.33 1.02 1.50 0.79 0.85 1.17 0.83 0.28 6.50 0.47 0.22 1.30 5.60 0.30 1.29 1.70 0.43 0.65 0.87 0.37 1.66 0.85 0.29 0.89 0.63 0.74 1.50 0.62 0.32 1.20 2.20 7.60 10.70 8.91 1.35 0.37 2.00 1.60 0.52 2.14 0.54 5.50 0.81 0.89 0.89 0.68 0.47 0.33 1.02 0.93 0.47

0.76 0.75 0.76 1.41 0.41 0.47 0.41 0.34 0.65 0.47 0.57 0.74 0.77 0.47 0.56 0.48 0.57 0.76 1.09 0.30 0.47 0.73 0.41 0.93 0.75 1.17 0.59 1.70 0.52 0.79 1.40 1.52 0.81 0.91 0.73 0.58 0.52 0.76 0.30 0.47 0.76 0.41 0.76 0.57 1.11 1.25 0.35 0.34 0.75 1.58 2.37 1.97 2.41 1.36 0.48 0.74 0.47 0.35 1.59 0.48 1.98 0.76 0.76 0.76 0.77 0.80 0.23 0.51 0.75 0.57

1.66 1.89 1.77 1.63 1.57 1.53 2.22 3.29 1.91 1.86 1.66 2.21 1.56 1.10 2.05 1.78 1.78 1.77 1.35 1.58 2.28 2.38 2.21 1.62 2.07 1.27 1.08 2.42 1.47 0.80 1.65 2.06 1.03 1.99 2.42 1.39 1.78 1.77 1.67 2.92 1.77 1.33 1.77 1.66 1.42 1.40 2.06 1.50 2.07 2.14 3.33 3.02 3.47 1.52 1.37 2.63 2.88 1.88 2.07 1.69 3.25 1.66 1.77 1.77 1.56 1.30 1.82 2.20 1.83 1.41

hs89 bou hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 bo86 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 w88 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hss87 hs89 bou hss87 hss87 hss87 hss87 bou hss87 hss87 hss87 hss87 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 hs89 fei w88 bo90 bo86 bo86 w88 hs89 hs89 hs89 hs89

a Ref: bou ˆ Bouvier, unpublished, bo86 ˆ Bouvier et al. 1986, bo90 ˆ Bouvier 1990, fei ˆ Feigelson, private communication, hs89 ˆ Hartmann & Stauffer 1989, hss87 ˆ Hartmann, Soderblom & Stauffer 1987, w88 ˆ Walter et al. 1988.

q 2000 RAS, MNRAS 319, 457±466

Rotational properties of T Tauri stars couple the star to disc material whose orbital speed is much less than the breakup velocity of the star. If the star is rotating at nearly breakup, however, the radius of corotation between disc and star is nearly at the stellar surface, so that if accretion onto the star is to occur, the disc must be able to extend almost down to the stellar surface. This means that the disc must be much stronger relative to the magnetic field of the star than in standard models. A much stronger disc, however, can be readily ruled out. If the stellar field is kept constant, standard braking models would require a disc accretion rate of around 1024 M( yr21 for the disc to penetrate close to the stellar surface, a value that would correspond to an accretion luminosity far greater than the observed luminosity of the system. Thus, if magnetic-braking models are not to be abandoned in anything like their current form, one has to postulate that Orion contains stars with magnetic properties ± either in terms of the magnitude of the field or else its topology ± that are very different from those usually invoked to explain the rotational data in star formation regions. If the SMMV study appears to imply serious adjustment, if not downright rejection, of magnetic-braking models, it is important to understand why these properties were not discovered in previous investigations. Does this discrepancy result from observational selection effects or from genuine physical differences in the samples studied ± for example for stars in different mass ranges or environments? (Herbst et al. 2000 have pointed out, for example, that the SMMV sample contains a much higher proportion of low-mass stars than previous studies in either Orion or Taurus). Existing photometric data cannot be used to answer this question, however, because other monitoring campaigns have not been sensitive to the shortest periods …P , 2 d† and could, therefore, have simply missed a substantial minority of rapid rotators. Note that monitoring campaigns are generally able to assign photometric periods to a fraction in the range 30±50 per cent of the CTT stars monitored, presumably due to intermittent spot activity on the stellar surface, while the success rate is usually higher for WTT stars. There are, therefore, plenty of stars with unassigned periods in existing studies that could in principle conceal a population of undiscovered fast rotators. Another constraint on the rotational properties of young stars is provided by rotational broadening of photospheric absorption features (v sin i) data (Vogel & Kuhi 1981; Bouvier et al. 1986; Hartmann et al. 1986; Hartmann & Stauffer 1989). This has not been the preferred tool for investigating the rotation of individual T Tauri stars because of the ambiguity introduced by the unknown angle between the stellar rotation axis and the line of sight (i.e. in the value of sin i). This method has the great advantage, however, for studying the properties of an entire population in that it yields data for every star. It therefore by-passes the concern that one might have about photometric periods, i.e. that they are derivable only in a subset of stars whose rotational properties may not necessarily reflect that of the overall population. Accordingly, in this paper, we analyse the v sin i distribution for a sample of stars in Taurus, in order to answer the simple question: is the distribution of rotational velocities in these stars compatible with those measured in the SMMV study in Orion? We find overwhelming evidence that the two samples are different, and that the Taurus sample is extremely unlikely to contain the high fraction of fast rotators found in the SMMV study. Section 2 describes the samples used and Section 3 the methods by which projected velocities in Taurus are compared with the rotationperiod distribution measured by SMMV in Orion. Section 4 q 2000 RAS, MNRAS 319, 457±466

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discusses possible origins for the strikingly different results found in the two samples and Section 5 summarizes the conclusions.

2

T H E D ATA

We compare a sample of stars in Taurus which have measured values of v sin i with a subset of the stars in Orion for which SMMV measured photometric periods. The Taurus data (see Table 1) is essentially a magnitude-limited (approximately V # 14) sample of T Tauri stars known prior to the ROSAT era. We excluded X-ray sources discovered by ROSAT projected onto the Taurus cloud since most of them appear not to be related to star formation in Taurus (Briceno et al. 1999). For the same reason, we also excluded Einstein-discovered X-ray sources except for a few whose membership is not in doubt. The Orion data comprises the stars measured by SMMV for which spectral types had been assigned by Hillenbrand (1997). These 108 stars are located within 15 arcmin of the centre of the ONC and therefore represent around 40 per cent of the stars for which SMMV obtained photometric periods. As stressed by SMMV, however, the rotational properties of this subset are indistinguishable from the entire sample. In Fig. 1 we plot the locations of the stars in Taurus (open circles) and those measured by SMMV (filled circles) in the HR diagram. It can be seen that there is substantial overlap in the properties of the two populations, but that the SMMV stars contain a population that is somewhat fainter and significantly cooler than the Taurus stars. Although deep charge-coupled device imaging in Taurus (Briceno et al. 1998) has revealed a small number of late-type T Tauri stars not included in our Taurus sample, it would nevertheless appear that relatively luminous latetype T Tauri stars are intrinsically less numerous in Taurus than in Orion. The above remarks are based on the empirical properties of the two populations. In our analysis of the rotation rates of these stars relative to breakup velocity, we also have to assign masses and

Figure 1. HR diagram for Taurus stars (open circles) with measured v sin i and Orion stars (filled circles) for which both spectral types and photometric periods are available. The pre-main sequence tracks of Siess, Dufour & Forestini 2000 are superimposed.

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radii to these stars and hence, in the case of the mass, interpret these properties in terms of theoretical pre-main sequence tracks (see, however Tout, Livio & Bonnell 1999 for a cautionary note on estimating masses of accreting T Tauri stars from pre-main sequence tracks). It is evidently desirable that we compare the properties of the two samples using consistent mass estimates; we employ the tracks of Siess et al. (2000), which are overlaid in Fig. 1. Note that Hillenbrand (1997) used the tracks of D'Antona & Mazzitelli (1994) which yield masses that are around a factor of 2 lower than those obtained with the tracks of Siess et al. This means that p the resultant velocities normalized to breakup are around 2 lower than those quoted by SMMV (note that this adjustment removes most of the systems that are apparently rotating at more than breakup in SMMV's data). We employ stellar radii determinations from Kenyon & Hartmann (1995) for Taurus stars (except for DG Tau, DL Tau, DR Tau and RW Aur from Basri & Bertout 1989) and from Hillenbrand (1997) for SMMV's stars. Finally, since we wish to compare the properties of low-mass pre-main sequence stars in the two regions, without the complicating contamination of small numbers of Herbig Ae Be stars, we impose an upper mass cut-off of 1 M( in both samples, corresponding (for the Siess et al. tracks) to an upper limit on the effective temperature of around 4200 K. This cut leaves the Taurus sample with 55 stars and the SMMV sample with 102. 3 C O M PA R I S O N O F S T E L L A R R O TAT I O N I N TA U R U S A N D O R I O N We wish to examine whether some form of braking prevents premain sequence stars from rotating at or near breakup velocity. Therefore the quantity of interest for each star, which we term its normalized velocity is its equatorial velocity expressed as a fraction of its breakup velocity vb. For a star of mass M, radius R, p vb ˆ GM=R; so that the calculation of the normalised velocity involves an estimation of both its mass and radius. Mass and radius estimates are obtained as discussed in Section 2 above. In the SMMV sample, the observed quantity is the photometric period P (assumed to be the rotation period), so that the equatorial velocity is simply 2pR/P, where R is obtained as above. In the Taurus sample, however, the observed quantity is v sin i and thus comparison of the values of normalized velocity in the two samples involves some correction for the unknown sin i value for each star. We defer a discussion of how these two distributions can be most meaningfully compared to Section 3.3, and first discuss the results for the projected velocity distribution in Taurus. 3.1

The projected velocity distribution in Taurus

Fig. 2 depicts the values of v sin i normalized to the breakup velocity of each star for the 55 stars in the Taurus sample with a mass less than 1 M(. Masses are assigned using the pre-main sequence tracks of Siess et al. and the crosses and dots denote respectively, WTTs (total 19) and CTTs (total 36), classified according to both Ha emission strength and near-IR excess. It is immediately evident that most T Tauri stars in Taurus have projected velocities that are a small fraction of breakup (,0.1), in line with the conclusions reached by Bouvier et al. (1993) and Vrba et al. (1993) from photometric monitoring campaigns in Taurus. This result does not of course prove the absence of a minority population of fast rotators in Taurus, since unfavourable

Figure 2. Projected velocity normalized to breakup velocity is plotted as a function of mass for the 55 stars in Taurus with masses ,1 M(. Filled circles are CTTs, crosses WTTs.

orientation to the line of sight can conceal such systems. We investigate this possibility further in Section 3.3 below. In line with previous analyses of the v sin i distribution in Taurus (e.g. Hartmann & Stauffer 1989; Bouvier 1990), we find no significant difference between the CTTs and WTTs in the sample (KS probability ± i.e. the probability that two datasets would be at least this discrepant if drawn from the same parent distribution ± being 0.12). This of course does not demonstrate that the underlying period distributions of CTTs and WTTs are necessarily the same, since the effect of random viewing angles may smear out any such differences in the v sin i distribution. In fact, the photometric period distributions in Taurus are apparently very different for the WTTs and the CTTs. Of the 55 stars in our Taurus sample, photometric periods have been obtained for 19 CTTs and 11 WTTs. The WTTs have considerably shorter periods, with an associated KS probability that they sample the same distribution of 2  1023 : This marked difference between the WTTs and CTTs in Taurus contrasts strongly with the situation in the SMMV sample. Inspection of Fig. 2 suggests that the projected velocities (normalized to breakup velocity) increase somewhat towards lower masses. If the sample is divided into two subsamples (comprising, respectively, 19 and 36 stars) in the mass ranges 0±0.5 and 0.5±1 M(, the KS probability is 0.057. Although this lends some support (at the 2s level) to the notion that lower-mass stars rotate somewhat closer to breakup velocity, we note that this result relies heavily on the unusually fast rotator (LkCa 21) in the low-mass bin. The anomalous radial velocity for this star (Hartmann et al. 1987) however suggests that this may be a spectroscopic binary. If this star is removed from the analysis, there is no significant difference between the v sin i distributions of stars with masses, respectively, greater than or less than 0.5 M(. 3.2

The equatorial velocity distribution in Orion

Fig. 3 shows a corresponding plot for the equatorial velocity distribution for the stars in SMMV's sample, again normalized to q 2000 RAS, MNRAS 319, 457±466

Rotational properties of T Tauri stars

461

Figure 3. Equatorial velocity normalized to breakup velocity is plotted as a function of mass for the 102 stars in the Orion sample with masses ,1 M(. Filled circles are CTTs, crosses WTTs.

Figure 4. The cumulative distribution of equatorial velocities (normalized to breakup) for the Orion sample (solid line) is compared with the result of deprojecting the Taurus v sin i data (dashed line).

breakup velocity. The sample comprises 45 CTTs and 57 WTTs in the mass range 0±1 M(. Fig. 3 clearly suggests a different rotational velocity distribution for the SMMV sample compared with that in Taurus. Whereas in Taurus only 1/55 stars has a projected velocity greater than 0:2 breakup, the corresponding figure in the SMMV sample is 31/102. Dots and crosses in Fig. 3 again denote, respectively, CTTs and WTTs, differentiated by values of D…I 2 K† . or , 0:3: As noted by SMMV, there is no evidence that the rotational properties of the CTTs and WTTs are different in this sample (KS probability of 0.24), a result that contrasts strongly with the results of photometric monitoring in Taurus. (Perhaps more persuasively, given the uncertainties in what value of D…I 2 K† should be used to separate CTTs from WTTs, SMMV find essentially no correlation between rotation rate and accretion diagnostics, whether measured through infrared excess or the strength of Ha or Ca ii emission). We also find that when the sample is divided into those with masses less than and greater than 0.4 M( according to the premain sequence tracks of Siess et al. (2000) (Fig. 1) (i.e. subsamples comprising 61 and 41 stars, respectively), the two subsamples are statistically indistinguishable (KS probability of 0.28).

range y to y 1 dy is f (y) dy], then for an isotropic distribution of viewing angles, the distribution of intrinsic velocities f …v† [such that f(v) dv is the fraction of stars with normalized velocity in the range v to v 1 dv† is given by (Gaige 1993):    …z … …z 2 1 d 21 z f …v† dv ˆ f…y† dy 1 f…y† y sin dy …1† p z dy y 0 0

3.3 Deprojection of the projected velocity distribution in Taurus Comparison of Figs 2 and 3 leads us to suspect that the underlying rotational distributions of the two samples are very different. Nevertheless, it has to be borne in mind that Fig. 2 refers to projected velocities and it is therefore not completely clear a priori how statistically significant this apparent discrepancy is when one takes into account the convolution of the intrinsic distribution with a finitely sampled distribution of viewing angles. In an attempt to address this, we have numerically inverted the distribution of observed v sin is (normalized to breakup velocity). If the distribution of observed velocities is represented by f (y) [such that the probability that a star has normalized v sin i in the q 2000 RAS, MNRAS 319, 457±466

The advantage of obtaining the cumulative distribution is that it bypasses the problems inherent in differentiating a function [in this case f (y)] obtained from finitely sampled data. In the above expression, the first term is just the fraction of systems with projected velocity less than a given value, z, and can therefore be obtained directly without any smoothing of the data. The second term is the fraction of systems subtracted from the above because they in reality correspond to systems with v . z but which have been projected into the domain y , z: Evaluation of this second term involves some smoothing of the data. We have simply used a histogram to evaluate f (y) in the second term, so that each bin ( j) in this histogram (with f ˆ fj † contributes a quantity 1 fj ‰y sin21 …z=y†Š2 ; where 1 and 2 denote the evaluation of the function in square brackets at the upper and lower boundaries of the jth bin. In practise we find that (for our sample of 55 stars) the resulting deprojected cumulative distribution is not unduly sensitive to our choice of bin width. Fig. 4 shows the resulting cumulative distribution of deprojected normalized velocities for the Taurus sample (dashed curve). We have verified that this represents a reasonable inversion of the Taurus data by running Monte Carlo simulations in which this deprojected distribution is viewed at random angles and can confirm that in the majority of cases the resulting projected distribution is indeed compatible with Taurus (but see below). The cumulative distribution of normalized velocities in the SMMV sample (solid curve) is strikingly different. The KS probability that these data were drawn from the same distribution is 1026! However, before concluding that there is a definite incompatibility between the datasets, one needs to examine the nature of the inversion process. There are two points to be borne in

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mind here. First, (as noted above) when we ran Monte Carlo experiments in viewing the deprojected data at random angles, we mostly obtained distributions that were compatible with the original data (as measured by the KS test). However, we found that occasionally (about 1 in every 10 random realizations), we obtained distributions that were incompatible with the original data (KS probability of 0.02 or less). This shows that even though the deprojection procedure provides the correct intrinsic distribution corresponding to the observed distribution if it were perfectly sampled, the finite sampling of viewing angles breaks this correspondence in a significant minority of random realizations, even for a sample size as large as 55. The corollary of this, of course, is that there are other intrinsic distributions which will project into the observed distribution a significant fraction of the time. Thus though deprojection provides information on the most likely intrinsic distribution, it does not readily serve our purpose here, which is, instead, to explore the region of parameter space that is compatible with the observed distribution. There is a second reason why deprojection may be misleading in this context. As may be seen from equation (1), the deprojected distribution f …v† always goes to zero for values of v greater than the maximum value of v sin i in the sample. The reason for this is that if there is an upper cut-off in the intrinsic distribution, then a perfectly sampled projected distribution would always contain datapoints with projected velocities arbitrarily close to this maximum (i.e. viewed close to the equatorial plane). In a finitely sampled system, however, one would expect the intrinsic distribution to extend beyond the maximum value of v sin i in the data. Thus the deprojection procedure is going to exaggerate the lack of systems at high intrinsic velocity and therefore overestimate the discrepancy with a dataset like that of SMMV. 3.4

Projection of the SMMV data

Bearing the above arguments in mind, we compare here the two datasets by instead projecting the SMMV data on to the plane of the sky and then comparing with the Taurus data. Specifically, we perform Monte Carlo experiments which assign random viewing angles to each of SMMV's stars and hence construct the distribution of (normalized) v sin i that a hypothetical observer would measure if viewing the SMMV sample from an arbitrary direction. A KS test is used to compare this distribution (of 102 datapoints) with the observed Taurus data (55 datapoints), the resulting KS value providing a `figure of merit' for this particular random realization. Repeated random realization is then used to assess how often the projected SMMV data would resemble the Taurus data. Again, we find overwhelming incompatibility between the datasets. In 100 random realizations we found a maximum KS probability of 0.013 and an average KS probability of 1023. We thus conclude that there is a very small probability that the stars measured by SMMV have the same intrinsic rotation distribution as those in Taurus. 3.5

Dependence on stellar mass

Inspection of Fig. 1 reveals that the two samples occupy rather different regions of the HR diagram. The bulk of the stars in Taurus (48/55) lie at effective temperatures greater than around 3600 K (corresponding to masses greater than ,0.4 M( according to the models of Siess et al.). In SMMV's sample, while 41 of their stars lie in a similar region of the HR diagram, 61 are cooler

(presumably less massive) than this. Herbst et al. 2000 have also drawn attention to the high proportion of low-mass stars in SMMV's data. (Note that 0.4 M( according to the models of Siess et al. corresponds very roughly to 0.25 M( for the tracks of D'Antona & Mazzitelli 1994 used by SMMV and Herbst et al. in their analyses.) In order to test whether the differences in rotational properties between SMMV's data and that in Taurus is a result of the stars' different distributions in the HR diagram (and thus presumably in their masses), we construct synthetic subsamples from both datasets which are well matched in mass. We do this by binning both datasets in intervals of 0.1 M( in the range 0.1±1 M( and by randomly rejecting stars from whichever dataset contains more stars in that bin, until the number of stars in that bin is equal in both datasets. We then analyse the subsamples as in Section 3.4 above. Unfortunately, owing to the rather poor overlap in stellar properties in the two samples, this procedure produces subsamples containing only 35 stars. Comparison of the rotational properties of these mass-matched subsamples shows that they are not significantly different (KS probabilities typically in the range 0.1±0.3). However, before concluding that the different rotational properties of the Taurus and Orion samples are indeed a product of the different mass distributions of stars in the two samples, it is worth noting that the reason that the mass-matched subsamples are not significantly different from each other is almost entirely due to the reduced sample size (in other words, the distributions for Taurus and Orion are almost as different for the mass-matched samples as they are in the whole dataset, but such a difference is not statistically significant when it relates to samples containing only 35 stars). We therefore conclude that there is no strong evidence to suggest that the difference in rotational properties between Orion and Taurus is a mass-dependent effect. On the other hand we caution that the poor overlap in stellar properties between the two regions means that we are not well placed to address this issue. 3.6

Dependence on accretion characteristics

Another striking difference between the Taurus sample and that of SMMV is in the relative mix of CTTs and WTTs. In SMMV's sample they are roughly equally mixed (for stars less massive than 1 M( the CTT:WTT ratio is 45:57) whereas in the Taurus sample CTTs outnumber WTTs by almost 2:1 (36 vs 19). We have therefore tested the Taurus CTTs against random projections of the CTTs from the SMMV sample, and likewise for the WTTs. We find that the discrepancy between the samples persists for both CTTs and WTTs, albeit with somewhat lower statistical significance due to the reduction in sample sizes. We note in passing that this conclusion is not unduly sensitive to the value of D…I 2 K† used to discriminate between CTTs and WTTs in the SMMV sample ± essentially the SMMV stars in all ranges of IR colours rotate faster than either the CTTs or the WTTs in Taurus. We therefore conclude that the differences between Taurus and Orion cannot be attributed to the different mix of CTTs to WTTs in the two samples. 3.7

Incompleteness of SMMV data at long periods

We finally examine the possibility that the apparent difference in the rotational period distribution between Taurus and Orion derives simply from the decline in SMMV's detection efficiency at longer periods. The 10-d observing window employed by SMMV q 2000 RAS, MNRAS 319, 457±466

Rotational properties of T Tauri stars obviously limits their derived periods to timescales less than this; they estimate that their detection efficiency starts to fall off significantly at periods of around 8 d. They also argue that they would expect only a small contribution from rotators with periods longer than 10 d, basing this assertion on the results of previous monitoring campaigns in Orion by Herbst and collaborators. Subsequent to SMMV's paper, further data contained in Herbst et al. (2000) gives strong support to this claim (see in particular their fig. 8b, which shows that for stars in the mass range probed by SMMV, the fraction of stars with periods greater than 10 d is very low. Since Herbst et al. employed an observing window of around 150 d, one may be confident that this represents a genuine dearth of stars with long rotation periods.) We approach this problem by Monte Carlo simulation. As before, we view SMMV's sample at random angles and compare with the projected data in Taurus. We now, however, modify SMMV's sample by generating fake datapoints in the period bins in which SMMV is incomplete. Having specified the incompleteness level of each period bin (see below) we randomly generate periods in each bin, and then also randomly assign these to mass and radius values of stars in the sample with measured periods. We are thus assuming that the `missed' systems populate the same region of the HR diagram as the systems with measured periods. From the mass, radius and period of the fake datapoint we compute the normalized velocity of this fake datapoint, and from then on, treat it exactly like the genuine datapoints in the sample ± that is, assign a randomly chosen viewing angle and include it in the cumulative distribution of projected velocities. As a minimum level of correction, we should adopt the detection efficiencies given by SMMV in the range of periods 8± 10 d, and use the number of detected systems in this range to calculate the number of fake datapoints we need to introduce. This exercise results in three fake datapoints in the range 8±9 d and 12 in the range 9±10 d. If we then accept SMMV's assertion that there should be very few missed systems in the domain P . 10 d; then a typical random realization of this corrected data yields a KS probability (that this corrected data is drawn from the same distribution as the Taurus stars) that is still very low (typically a few times 1023). We now examine the possibility that there is a significant tail of slow rotators beyond P ˆ 10 d; that were not in principle detectable by SMMV. As noted above, photometric monitoring with longer observing windows by Herbst and collaborators demonstrates a marked decline in the number of stars with periods longer than 10 d. We nevertheless experiment with applying a highly excessive incompleteness correction to SMMV's data at long periods, by assuming that the intrinsic period distribution is roughly flat out to a period of 12 d. (This involves adding an extra 20 fake datapoints to the sample, uniformly distributed in the range 10± 12 d.) Even with this extreme correction, we find that the KS probability that it is drawn from the same population as the Taurus data is typically a couple of per cent, and that the difference is thus still statistically significant at more than the 2s level. We thus conclude that the discrepancy between SMMV's data and that in Taurus cannot be explained by any plausible population of long-period stars that were not detectable by SMMV. 4

DISCUSSION

The above analysis has proven at a very high level of significance that the rotational properties of the stars measured by SMMV are different from those of the Taurus sample. This difference cannot q 2000 RAS, MNRAS 319, 457±466

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be attributable to the different mix of CTTs and WTTs in the two samples, since both types of T Tauri star are highly discrepant between the two samples. We cannot rule out the possibility that the difference derives from the rather different mass ranges covered by the two samples, though the evidence that mass is the determining factor is at best weak. Before discussing this possibility further we first consider possible selection biases and sources of sample contamination. 4.1 Selection biases and sample contamination If the apparently overwhelming difference between the two samples is to be explained in terms of selection biases, we have to either argue that the Taurus sample has missed a population of fast rotators or that the SMMV sample is contaminated with an unrepresentative excess of fast rotators. If the rotation axes of stars in Taurus are randomly aligned with respect to the sky (as we have assumed throughout) then this first possibility is easily disposed of, since values of v sin i were obtained for all the stars measured in what is essentially a magnitude-limited sample. In order for the v sin i data to conceal a population of fast rotators, it would be necessary to argue that the rotational axes in Taurus are preferentially aligned close to the line of sight. Calculation of sin i for the Taurus stars with both v sin i data and measured photometric periods, however, gives no hint that this is the case (the mean value of sin i for these stars is close to the value of …p=4† expected for random viewing angles; see also Bouvier et al. 1993). The latter possibility (i.e. that fast rotators are over-represented in SMMV's data) is more problematic, since SMMV were able to obtain photometric periods only in the case of 10 per cent of the stars surveyed. It is usually assumed that this low efficiency of period extraction is due to intermittency in the mechanism producing the variability (e.g. the existence of starspots). It is also implicitly assumed that this level of intermittency is the same for all periods so that the measured period distribution is then a good measure of the period distribution for the whole population. In the hypothetical case, however, in which phenomena on short periods were more stable, these rapid rotators would preferentially find their way into the measured sample. One can think of several classes of objects that might in principle contaminate the sample. Foreground stars of age ,108 yr would exhibit similar photometric periods (Krishnamurthi et al. 1998), though of course such periods would only represent a modest fraction of breakup for main-sequence stars. In the overwhelming majority of stars in SMMV's sample, the equivalent width of lithium is sufficiently large that any contaminating object must itself be rather young (less than 10±20 Myr) so that such contamination would not unduly change our conclusions. Another class of star that complicates the interpretation of photometric variability is the close-binary systems. Close binaries can generate, or modify, photometric variability in several ways. The closest systems, which are virtually in contact and hence with orbital periods close to breakup, would produce light curves modulated at half the orbital period owing to geometric effects (Mazur, Krzeminski & Kaluzny 1995) The stellar rotation in premain sequence binaries is tidally synchronized out to periods of several days or more (for example, V826 Tau is a binary with a 4-d period which exhibits photometric variability on the same timescale; Mundt et al. 1983). In CTTs, another effect that can cause photometric variations on the orbital period is that of accretion onto an eccentric binary from a circumbinary disc

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(DQ Tau is a binary with a 15-d orbital period which appears to show this effect; Basri, Johns-Krull & Mathieu 1997). Of these mechanisms, the second is spot related and would be expected to manifest itself in the data with the same frequency as the spot activity in wider binaries and single stars. The other two mechanisms would be expected whenever there was a binary with suitable separation (and/or accretion characteristics) and thus might be over-represented in the sample with measured photometric periods. In order to investigate whether these contaminants are likely to change our over-all conclusion concerning the incompatibility of the rotational properties in Taurus and in SMMV's sample, we have performed the following exercise. We have simply asked how many of the fastest rotators would have to be removed from the SMMV sample before one obtains reasonable agreement with the Taurus data. As before, we view the (now modified) SMMV sample at random viewing angles and check the compatibility with the Taurus v sin i distribution with a KS test. We find that we have to remove about 20 per cent of SMMV's sample (i.e. about 20 stars) before we obtain a reasonable level of compatibility with the Taurus data (KS probability of 0.1 or more). This means eliminating all the objects with normalized velocities of greater than around 0.25. If all these rapid rotators are to be ascribed to binary-related phenomena, then the required orbital period of the binaries is less than a couple of days. If we take the most conservative line (i.e. that which brings the two datasets into closest agreement) we would assume that the systems we have removed from SMMV's sample in order to bring about this agreement are detected with 100 per cent probability in the parent population. Since SMMV detected rotation periods in 10 per cent of the stars monitored, the 20 per cent of the sample that we have removed represents 2 per cent of the total population. Duquennoy & Mayor's (1991) survey of 164 mainsequence G-type primaries yielded two or three systems in the required period range. Therefore, given the small number statistics of the Duquennoy and Mayor sample in this period range, it is not impossible that there are sufficient close binaries in the sample examined by SMMV to account for these 20 fast rotators. We nevertheless stress that this is only a tenable hypothesis if these systems are indeed detected with close to 100 per cent efficiency. We also remark that in the case of unequal mass binaries, the membership of such a tight binary pair would imply a radial velocity of the dominant member of the system that might well cause such stars to have been rejected as cluster members. 4.2 The origin of the different rotational properties in Taurus and Orion There appear to be two principal differences between the rotational properties of the stars in Taurus and Orion. Most obviously, the Orion stars rotate much faster. Whatever processes are invoked to explain the slow rotation of stars in Taurus must therefore be modified in Orion. Secondly, while the photometric period distribution in Taurus is significantly different for the CTTs and WTTs, in Orion there is apparently no dependence of rotation period on accretion characteristics. In principle, differences in rotation rates may derive either from initial conditions or from subsequent evolution. Initial conditions (i.e. the angular momentum content of the proto stellar core) are probably not relevant however, since all hydrodynamic collapse calculations (i.e. those not including any form of braking, e.g. Bonnell 1994) show that such cores are maintained at a high

fraction of breakup throughout the phase during which they acquire the bulk of their mass. Subsequent evolution is controlled by several factors: the timescale on which some mechanism can brake this rapid rotation (tV), the equilibrium spin period (Peq) for this mechanism and the factor (fspin) by which stars spin up when subsequently released from this brake. The last of the three is determined purely by the factor by which the star contracts after it is released from its brake, and therefore depends both on the contraction history of the star (determined mainly by its mass) and by the time (tbrake) over which the brake stays on. Canonical magnetic-braking models (designed with the Taurus observations in mind) have Peq , 10 d and therefore invoke torques for which tV , 105 yr (Armitage & Clarke 1996). Because stars of around a solar mass do not experience large changes in their moments of inertia between an age of a few times 105 and 107 yr (when they develop a radiative core), fspin for such stars is not much greater than unity unless they are released from their brakes at early times. One obvious way of generating more rapidly rotating stars is to invoke a weaker brake. In the context of magnetic-braking models, this produces a shorter Peq as the star is brought into corotation with relatively rapidly rotating material close to the star. If some of the rapid rotators have been brought into a state of spin equilibrium by a weaker brake, then (since the equilibrium spin period scales almost linearly with the star's surface magnetic field) one would have to invoke surface fields perhaps an order of magnitude lower than those invoked in Taurus. If the origin of such fields is through dragging in of some primordial field (e.g. Tayler 1987), one might expect there to be environmental differences in stellar magnetic fields. It is widely assumed, however, that the magnetic fields in T Tauri stars are maintained through dynamo activity, owing to the generic similarity of magnetic activity in pre-main sequence and main-sequence stars (see for example the discussion in Feigelson & Montmerle 1999). For a dynamo field, the field strength is expected to scale only (inversely) with the Rossby number (the ratio of rotation period to convective turnover time). Since the convective turnover time during the pre-main sequence is neither a strong function of mass or age (Gilliland 1986), it would therefore be puzzling if stars in Orion would spin faster and yet have weaker fields. Although we cannot rule out the possibility that the surface fields in Orion are substantially weaker, it does not fit well with the expectations of dynamo theory. Some empirical evidence for a weakly magnetized population in Orion might be provided by the relatively low rates of detection by ROSAT of stars in Orion compared with other starforming regions (GagneÂ, Caillault & Stauffer 1995), although more recent (Chandra) measurements suggests that this low detection efficiency may merely reflect the sensitivity and confusion limits of the ROSAT data (Garmire et al. 2000). Another way of producing more rapidly rotating stars is to release the brake more quickly. It is tempting here to invoke environmental differences between Taurus and Orion: the higher density in Orion would enhance the role of close encounters leading to the dispersal of the disc and loss of its associated braking power (Clarke & Pringle 1991; Sterzik & Durisen 1995; Armitage & Clarke 1997). Photoevaporation of discs by the massive stars in the core of the Orion Nebula Cluster is another environment-dependent disc-destruction mechanism (Johnstone, Hollenbach & Bally 1998). Two points have to be borne in mind here however. First, the disc mass at which braking becomes ineffective is very low and (because it depends on the relative moments of inertia of the star and disc) it depends on the age of the system. Models (for solar mass stars) suggest that at an age of q 2000 RAS, MNRAS 319, 457±466

Rotational properties of T Tauri stars 5

around 10 yr, stars are freed from discs that fall below a threshold of around 1023 M(; after several million years (when the stars have contracted), the threshold mass is considerably less than this (Armitage & Clarke 1996). Secondly, releasing the brake only has a large effect on the stellar rotation if it occurs at a stage when the star has still to contract significantly on the pre-main sequence, and can thereafter spin up at constant angular momentum. Both these arguments work in the same direction: i.e. if disc stripping is to have any effect on stellar rotation it must occur early on and must deplete the disc mass on the smallest scales (probably at the sub au level). Obviously suitable star±disc encounters are negligible at the average density even of the central parts of the ONC and can only be justified if the primordial state of Orion was very tightly sub-clustered (note that although there is no evidence for such sub-structure in the current state of Orion, one cannot rule out that such clustering has already been erased by dynamical effects; Bate, Clarke & McCaughrean 1998). Likewise photoionization (which strips off weakly bound material at disc radii .10 au), is unlikely to deplete discs on the smallest scales at sufficiently early times, especially as the bulk of stars measured by SMMV are unlikely to have been exposed to the intense UV radiation field from the massive stars residing in the core of the Orion Nebula Cluster. It is worth noting that if discs are indeed dispersed by interactions, then the lower-mass stars may well be more susceptible to this process. Obviously, a disc around a low-mass star is more readily dispersed by interaction with a more massive neighbour than vice versa. In addition, lower-mass stars contract more slowly; in particular, at an age of around a million years, the contraction of lower mass T Tauri stars is temporarily slowed by the onset of deuterium burning (Herbst et al. 2000). While a 0.5 M( star would spin up by only around 50 per cent over the following million years if released from its brake at an age of 106 yr, a 0.2 M( star would spin up by nearly a factor of 3 over the same period as it exhausts its deuterium and recommences relatively rapid contraction. Since the stars in Orion are systematically lower mass than in Taurus, these considerations work in the right direction. We finally consider how one might understand the different correlations between spin and possession of a disc in Taurus and Orion. If more stars in Orion have weak magnetic fields, then this is readily explained, since rapid rotators would include both discless stars and those that were magnetically braked (at a rather short Peq) and still accreting. If however the rotational properties of the Orion stars are determined largely through disc dispersal by interactions, then the problem remains: we would expect the production of fast rotators but we would not expect these to be spectroscopically classified as CTTs. 5

CONCLUSIONS

We have shown that the rotational properties of stars in Taurus are very different from those in Orion. In Orion, a significant fraction of stars are rotating at a large fraction of breakup velocity, whereas in Taurus the projected (v sin i) velocity distribution is heavily weighted towards values that are a small fraction (less than 10 per cent) of breakup velocity. We have used a variety of deprojection and Monte Carlo techniques to demonstrate that there is a negligible probability that the two samples are drawn from the same distribution of intrinsic rotation properties. The statistical significance of this result depends somewhat on the method of analysis used, but is always well in excess of 3s . Since v sin i q 2000 RAS, MNRAS 319, 457±466

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values are available for all the stars in what is essentially a magnitude-limited sample in Taurus, we can rule out the possibility that the fast rotators have simply been missed in Taurus. There are several possible interpretations of these results. Firstly, it is marginally possible that the Orion sample contains a population of close binaries that is over-represented in the sample of periodic variables measured by SMMV. Analysis of highresolution spectra of the faster rotators in SMMV's sample should settle this issue. Alternatively, and more likely, there is a genuine difference in the rotational evolution of stars in the two regions, which could result from differences in intrinsic properties or different interactions with their environment or a combination of the two. (We note, for example that owing to the poor overlap in stellar mass of the two samples, we cannot rule out that there is a mass-dependent effect, although the evidence that this is the case is weak.) One possible intrinsic difference would be if Orion possessed a population of weakly magnetized stars for which the magnetic-braking mechanisms invoked in Taurus are less effective. Alternatively, it is in principle possible that stars in Orion are released from their disc braking at an earlier stage, due to the more frequent incidence of close interactions in the denser environment of Orion. The very close encounters required to free a star from its brake however makes this unlikely unless the initial conditions of the Orion Nebula Cluster were highly sub-clustered. We also note that while in Taurus there is a good correlation between stellar rotation and a star's possession of a disc (in the sense that CTTs rotate more slowly on average), in Orion there is no such correlation. This result is most readily understood if Orion contains a significant population of weakly magnetized stars which are thus able to accrete material even when rotating at a high fraction of breakup velocity. If, on the other hand, the faster rotation of stars in Orion is attributed to the higher incidence of close encounters, there is no still no obvious way, within our current understanding of magnetic-braking models, to generate the rapidly rotating CTTs found in SMMV's sample. AC K N O W L E D G M E N T S CJC gratefully acknowledges the hospitality of the Laboratoire d'Astrophysique de Grenoble and financial support through an Associate Professorship granted by the Universite Joseph Fourier. REFERENCES Armitage P. J., 1995, MNRAS, 274, 124 Armitage P. J., Clarke C. J., 1996, MNRAS, 280, 458 Armitage P. J., Clarke C. J., 1997, MNRAS, 285, 540 Basri G., Bertout C., 1989, ApJ, 341, 340 Basri G., Johns-Krull C. M., Mathieu R., 1997, AJ, 114, 781 Bate M. R., Clarke C. J., McCaughrean M. J., 1998, MNRAS, 297, 1163 Bonnell I. A., 1994, MNRAS, 269, 837 Bouvier J., 1990, in Goupil M.-J., Zahn J.-P., eds, Rotation and mixing in stellar interiors, Lect. Not. Phys. 366, 165 Bouvier J., Cabrit S., Fernandez M., MartõÂn E. L., Matthews J. M., 1993, A&A, 272 Briceno C., Calvet N., Kenyon S., Hartmann L., 1999, AJ, 118 Briceno C., Hartmann L., Stauffer J., MartõÂn E., 1998, AJ, 115 Cameron A. C., Campbell C. G., 1993, A&A, 274 Choi P. I., Herbst W., 1996, AJ, 111 Clarke C. J., Pringle J. E., 1991, MNRAS, 249 D'Antona F., Mazzitelli I., 1994, ApJS, 90 Duquennoy A., Mayor M., 1991, A&A, 248

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