Disc evolution and the relationship between L acc and L ∗ in T Tauri stars
aa r X i v : . [ a s t r o - ph ] J a n Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 25 November 2018 (MN L A TEX style file v2.2)
Disc evolution and the relationship between L acc and L ∗ in T Tauristars I. Tilling, C. J. Clarke, J. E. Pringle and C. A. Tout
Institute of Astronomy, The Observatories, Madingley Road, Cambridge, CB3 0HA
25 November 2018
ABSTRACT
We investigate the evolution of accretion luminosity L acc and stellar luminosity L ∗ in pre-mainsequence stars. We make the assumption that when the star appears as a Class II object,the major phase of accretion is long past, and the accretion disc has entered its asymptoticphase. We use an approximate stellar evolution scheme for accreting pre-mainsequence starsbased on Hartmann, Cassen & Kenyon, 1997. We show that the observed range of values k = L acc / L ∗ between 0.01 and 1 can be reproduced if the values of the disc mass fraction M disc / M ∗ at the start of the T Tauri phase lie in the range 0.01 – 0.2, independent of stellarmass. We also show that the observed upper bound of L acc ∼ L ∗ is a generic feature of suchdisc accretion. We conclude that as long as the data uniformly fills the region between thisupper bound and observational detection thresholds, then the degeneracies between age, massand accretion history severely limit the use of this data for constraining possible scalingsbetween disc properties and stellar mass. Key words: accretion, accretion discs – stars: pre-main sequence – planetary systems: pro-toplanetary discs
The claimed relationship between accretion rate, ˙ M , and mass, M ∗ ,in pre-mainsequence stars (e.g. Natta, Testi & Randich 2006 andreferences therein) originates from the more direct observationalresult that in Class II pre-mainsequence stars the accretion lumi-nosity, L acc , is similarly found to correlate with stellar luminosity, L ∗ . The accretion luminosity is deduced from the luminosity in theemission lines of hydrogen (in Natta et al., 2006, these are Pa β andBr γ ), together with modeling of the emission process (Natta et al.2004; Calvet et al. 2004). The modeling appears to be relativelyrobust. In order to deduce the quantities ˙ M and M ∗ from the quanti-ties L acc and L ∗ , it is necessary to be able to deduce the stellar prop-erties. This is done by placing the object in a Hertzsprung-Russelldiagram and making use of theoretical pre-mainsequence tracks fornon-accreting stars (e.g. D’Antona & Mazzitelli 1997).The result of this exercise indicated a correlation between ˙ M and M ∗ of the form ˙ M ∝ M α ∗ , where α is close to 2. The simplesttheoretical expectation (in the case of disc accretion where neitherthe fraction of material in the disc nor the disc’s viscous timescalescale systematically with stellar mass) is instead that ˙ M ∝ M ∗ . Theclaimed steeper than linear relationship thus motivated a varietyof theoretical ideas about specific scalings of disc parameters withstellar mass (Alexander & Armitage 2006, Dullemond et al 2006).Clarke & Pringle (2006) however suggested that the claimedcorrelation between ˙ M and M ∗ could be understood as follows.They noted that the distribution of datapoints in the plane of L ∗ -6-5-4-3-2-1 0 1 2 -3 -2 -1 0 1 log L acc log L * L a c c = L * L a c c = L * / 1 0L a c c = L * / 1 0 0 Figure 1.
The distribution of Classical T Tauri stars in Ophiuchus in the L ∗ , L acc plane. Detections (filled squares) and upper limits (crosses) deducedfrom emission line data. Adapted from Natta et al 2006.c (cid:13) I. Tilling, C. J. Clarke, J. E. Pringle & C. A. Tout versus L acc (Figure 1) more or less fills a region that is bounded,at high L acc by the condition k = L acc / L ∗ ∼ L acc , byobservational detection thresholds, which roughly follow a relationof the form L acc ∝ L . ∗ . They noted that when this relation between L acc and L ∗ is combined with specific assumptions about the rela-tionship between L ∗ and M ∗ , and between L ∗ and stellar radius R ∗ (based on placing the stars on pre-mainsequence tracks at a partic-ular age, or narrow spread of ages) then the resulting relationshipbetween ˙ M and M ∗ is indeed ˙ M ∝ M ∗ . They thus claimed thatit is currently impossible to reject the possibility that the claimed,steeper than linear, relationship is an artefact of detection biases.Clarke & Pringle (2006) also pointed out that the distributionof detections in the L acc – L ∗ plane may nevertheless be used totell us something about conventional accretion disc evolutionarymodels. In this paper we explore the extent to which this might beachieved. The first question which needs to be addressed is whythere is a spread of values of L acc at a given value of L ∗ . Secondlywe need to understand why the upper locus of detections corre-sponds to L acc ∼ L ∗ , since there seems no obvious reason why thisshould correspond to a detection threshold or selection e ff ect.We begin here (Section 2) by setting out a simple argumentwhy – as a generic property of accretion from a disc with massmuch less than the stellar mass – one would expect the distributionto be bounded by the condition L acc ∼ L ∗ . In Section 3 we use theideas of Hartmann, Cassen & Kenyon (1997) to develop an approx-imate evolutionary code for accreting pre-mainsequence stars anddemonstrate its applicability by comparison with the tracks of pre-mainsequence stars computed by Tout, Livio & Bonnell (1999). InSection 4 we apply the code to pre-mainsequence stars accretingfrom an accretion disc with a declining accretion rate and investi-gate the range of disc parameters required to obtain the observedrange of L acc / L ∗ . We draw our conclusions in Section 5. From a theoretical point of view there are three relevant timescalesin the case of star gaining mass by accretion from a disc. The firstis t M = M ∗ / ˙ M , (1)which is the current timescale on which the stellar mass is increas-ing. The second is t disc = M disc / ˙ M , (2)which is the current timescale on which the disc mass is decreasing.The third is the Kelvin-Helmholtz timescale t KH = GM ∗ R ∗ L ∗ , (3)which is the timescale on which the star can radiate its thermal en-ergy. It is also the timescale on which a star can come into thermalequilibrium. On the pre-main sequence, when the luminosity of thestar is mainly due to its contraction under gravity, t KH is also theevolutionary timescale.We see immediately (since L acc ∼ GM ∗ ˙ M / R ) that L acc L ∗ = t KH t M . (4)The pre-mainsequence stars in which we are interested,Class II (the Classical T Tauri stars), all have the following proper-ties: (i) They have passed through the major phase of mass accre-tion which occurs during the embedded states Class 0 and Class I.This implies typically that they have been evolving for some timewith their own gravitational energy as the major energy source.Deuterium burning interrupts this briefly but only delays contrac-tion by a factor of 2 or so. Thus for these stars we may expect thatthey have an age T ≈ t KH roughly.(ii) The masses of their accretion discs are now small, M disc < M ∗ . Further, accretion disc models at such late stages – i.e. the socalled asymptotic stage when most of the mass that was originallycontained in them has been accreted on to the central object – ex-hibit a power law decline in ˙ M with time and thus have the genericproperty that the age of the disc is about t disc . If most of the star’slifetime has been spent accreting in this asymptotic regime, we mayroughly equate the age of the disc and the age of the star and write T ≈ t disc .Given this, we see immediately that, for these stars, we expect t M = M ∗ / ˙ M > M disc / ˙ M = t disc ≈ t KH and therefore that L acc < L ∗ .In the following section we test this argument by evolving asuite of model star-disc systems in the L ∗ − L acc plane. Rather than attempting to carry out full stellar evolution computa-tions of accreting and evolving pre-mainsequence stars (e.g. Toutet al. 1999), we here adopt the simplified approach of Hartmann,Cassen & Kenyon (1997).We begin with a stellar core of mass M i = . ⊙ and radius R i = ⊙ . We prescribe the accretion rate ˙ M ( t ) and thus the stellarmass as a function of time M ∗ ( t ) = M i + Z t ˙ M ( t ) dt . (5)Hartmann et al. (1997) write the energy equation as L ∗ = − GM ∗ R
13 ˙ MM ∗ + ˙ RR + L D . (6)In deriving this equation they assume that the star is always in hy-drostatic balance and, being fully convective, can be treated as an n = / pdV work as well energyinput from deuterium burning L D and energy loss via radiative cool-ing at the photosphere L ∗ . In the form written above, it is assumedthat the material accreting on to the star arrives with zero thermalenergy. This is probably a good approximation to the case wherematerial enters the star via a radiative magnetospheric shock orfrom a disc boundary layer. The rate of energy input to the stardelivered by deuterium burning is parameterised (following the ex-pression given by Stahler 1988 for an n = / L D / L ⊙ = . × X f X D M ∗ M ⊙ ! . RR ⊙ ! − . , (7)where X is the mass fraction of hydrogen and f is the fraction ofdeuterium remaining in the star relative to the accreted material.We assume that the accreted material has the same abundance asthe star before deuterium begins burning and we take this initialmass fraction to be X D = . × − as did Tout et al. (1999). Wefix the hydrogen mass fraction at X = . The variation of f is given by (equivalent to Stahler 1988). Our notation di ff ers from that of Stahler (1988) and Hartmann et al.c (cid:13)000
13 ˙ MM ∗ + ˙ RR + L D . (6)In deriving this equation they assume that the star is always in hy-drostatic balance and, being fully convective, can be treated as an n = / pdV work as well energyinput from deuterium burning L D and energy loss via radiative cool-ing at the photosphere L ∗ . In the form written above, it is assumedthat the material accreting on to the star arrives with zero thermalenergy. This is probably a good approximation to the case wherematerial enters the star via a radiative magnetospheric shock orfrom a disc boundary layer. The rate of energy input to the stardelivered by deuterium burning is parameterised (following the ex-pression given by Stahler 1988 for an n = / L D / L ⊙ = . × X f X D M ∗ M ⊙ ! . RR ⊙ ! − . , (7)where X is the mass fraction of hydrogen and f is the fraction ofdeuterium remaining in the star relative to the accreted material.We assume that the accreted material has the same abundance asthe star before deuterium begins burning and we take this initialmass fraction to be X D = . × − as did Tout et al. (1999). Wefix the hydrogen mass fraction at X = . The variation of f is given by (equivalent to Stahler 1988). Our notation di ff ers from that of Stahler (1988) and Hartmann et al.c (cid:13)000 , 000–000 he ratio of L acc to L ∗ in T Tauri Stars -2-1.5-1-0.5 0 0.5 3.4 3.45 3.5 3.55 3.6 3.65 3.7 log L * log T eff Figure 2.
Accreting pre-mainsequence tracks computed using the simpli-fied evolutionary scheme, with f min = . − , 10 − and 10 − M ⊙ yr − . Ini-tially the mass is 0 . ⊙ and the radius is 3 R ⊙ . The tracks are terminatedwhen the approximations cease to be valid (because a radiative core has de-veloped or hydrogen burning begins in the core) or at a maximum age of20 Myr. Points are plotted along the tracks at masses of 0.1, 0.2, 0.5 and1 . ⊙ up to the maximum mass allowed by each track. The correspondingpoints from the Tout et al. (1999) tracks are joined to these by dotted lines. d fdt = ˙ MM ∗ − f − L D Q D X D ˙ M ! , (8)where Q D = . × erg g − is the energy available from fusionof deuterium. Because of timestep problems we set f = f falls below some small value f min . Except whereexplicitly mentioned, we take f min = − .For a fully convective star, the stellar entropy is controlled bythe outer (photospheric) boundary condition which takes accountof the strong temperature sensitivity of the surface opacity for starson the Hayashi track. Hartmann et al. (1997) adopt a fit to the(non-accreting) stellar evolutionary tracks of D’Antona & Mazz-itelli (1994) in the form L ∗ / L ⊙ = M ∗ . M ⊙ . R R ⊙ . . (9)They stress that this fit is only approximate and is adequate only inthe mass range 0 . M ∗ / M ⊙ M = − , − and 10 − M ⊙ yr − . On each track (where possible) we plot the pointat which the stellar mass is M ∗ / M ⊙ = . , . . (1997) because we prefer not to use their non-standard definition of [D / H].We recover their β D ≡ Q D X D in equation (8) with X D = . × − . with the corresponding points on the tracks. Both sets of compu-tations had the same initial values of M ∗ = . ⊙ and R = ⊙ .The stars first descend a Hayashi track, while being driven acrossto higher e ff ective temperature by the accreting matter. When thecore temperature reaches about 10 K they evolve up the deuterium-burning main sequence. Once deuterium fuel is exhausted theyresume their descent towards the zero-age main sequence, leav-ing their Hayashi tracks when the stellar luminosity drops to thepoint that the Kelvin timescale becomes greater than the accretiontimescale. At this point, accretion is again important and the tracksare then driven to higher e ff ective temperatures. Up to these pointsour simplified model appears to be in reasonably good agreementwith Tout et al (1999)’s models, even at early times and low masses,when the equations are formally incorrect. All the points are accu-rate to within a factor of two in luminosity and to within 10 per centin e ff ective temperature. We conclude that this simplified model isadequate for the task in hand. In view of our ignorance about formation history of pre-mainsequence stars in terms of their accretion history, we need tomake some assumptions about the form of ˙ M ( t ). In doing so, weare guided by some simple physical ideas. It needs to be borne inmind, however, that although our simple assumptions give rise toresults which provide a reasonable description of the data, this doesnot imply that the assumptions are necessarily correct nor that theyare the only ones which might work.We assume that most of the mass which has accreted on to aT Tauri star has been processed through an accretion disc. This as-sumption, coupled with the observational fact that protostellar discsare at most 10 – 20 per cent of the stellar mass and usually muchless, tells us that the evolution of the accretion disc has progressedinto its late, asymptotic phase. Lynden-Bell & Pringle (1974; seealso Hartmann et al. 1988) present similarity solutions for accre-tion disc evolution using the simple assumption that the kinematicviscosity varies as a power law of radius. They find that for suchdiscs, the disc accretion rate varies at late times as a power law intime with ˙ M ∝ t − η , for some η >
1. This is true more generallyfor discs in which the viscosity also varies as powers of local discproperties, such as surface density (Pringle 1991). Here we take η = .
5, corresponding to kinematic viscosity being proportionalto radius (Hartmann et al. 1988).We assume that T Tauri stars only become visible at a time t = t after the onset of formation and take t = × yr. Weconsider four sets of models in terms of the disc mass fraction f disc = M disc / M f at time t , where M f is the final stellar mass. Weconsider f disc = . , . , .
037 and 0 . . M f / M ⊙ . t =
0. At t = M ∗ = M i = . ⊙ . In principle,the evolution of the star for t > t can depend critically on the ac-cretion rate prior to this point. This is simply because the evolutiontimescale for a pre-mainsequence star is always comparable to itsage, so that it has never had time to forget its initial conditions. Inorder to minimise this e ff ect, we assume that the accretion rate doesnot increase with time. This is probably reasonable for stars whichform in isolation (e.g. Shu, Adam & Lizano 1987), but may not bevalid for more dynamical modes of star formation, when late inter-action and merging may be normal (e.g. Bate, Bonnell & Bromm c (cid:13) , 000–000 I. Tilling, C. J. Clarke, J. E. Pringle & C. A. Tout -1.5 -1 -0.5 0 0.5-4-3-2-101
Figure 3.
Accretion luminosity L acc against stellar luminosity L ∗ for thevarious runs with parameters given in Table 1. The filled symbols lead tofinal mass M f = . ⊙ and the open symbols to M f = . ⊙ . Each line ofsymbols represents one evolutionary sequence, starting at top right at time t = × yr and finishing bottom left at time t = yr, with one symbolplotted every 5 × yr. For each mass, the disc mass fractions at time t are f disc = . = . , = .
037 and = .
014 (lowermosttrack). The dashed diagonal lines represent L acc / L ⊙ = . , . . M = ˙ M i = const ., t t s (10)and that˙ M = ˙ M i tt s ! − η , t s t , (11)where t s and ˙ M i are chosen to ensure that the integral of ˙ M betweenzero and infinity is M f − M i and that the integral of ˙ M from t toinfinity is f disc M ∗ (see Table 1). We note that, since t ≫ t s , the discis always in the asymptotic regime at the stage that it is classified asa T Tauri star. We stress that the form of the evolutionary prescrip-tion that we have adopted for ˙ M ( t ) prior to t is not the only one thatproduces a given disc and stellar mass at t and that - depending onthe temporal history and location of matter infall onto the disc - avariety of prior histories, with the same integrated mass, are possi-ble. Our intention here is however not to explore all possibilities butto adopt a plausible and easily adjustable prescription with whichone can investigate the system’s evolution in the L acc − L ∗ plane.In Fig. 3 we show the results of these computations showingthe accretion luminosity ( L acc ) as a function of stellar luminosity( L ∗ ). We plot results for two final masses M f / M ⊙ = . . f disc = . , . , . .
2. We plot the positions starting at time t = × yr and thenat intervals of 5 × yr until time t = yr. In Table 1 we givethe values of t s and of ˙ M i corresponding to these parameters. Wenote that in all our models t s ≪ t ; thus, by the time that the staris deemed to be a Classical T Tauri star, the accretion rate has beendeclining as a power law over the majority of the system’s lifetime. From Fig. 3 we see that our data points more or less fill therange 10 − < L acc / L ∗ < L ∗ which areaccessible to our computations. It is reasonable to expect that witha more detailed evolutionary scheme, valid at lower masses, thistrend would continue to lower values of L ∗ . Thus these simple as-sumptions, coupled with this simplified evolutionary scheme arecapable of producing the typical observed spread of points in the L acc − L ∗ diagram (see Fig. 1).The stars in Fig. 3 are evolving at essentially constant massbecause, by assumption, M disc ≪ M ∗ and have already con-sumed all of their initial deuterium. At constant mass, Equation 9gives L ∗ ∝ R . . For a star evolving on the Kelvin-Helmholtztimescale this implies R ∗ ∝ t − / . and thus that L ∗ ∝ t − . .Since L acc = GM ∗ ˙ M / R ∗ and we have assumed ˙ M ∝ t − η we have L acc ∝ t − ( η − / . . From this we deduce that as time varies, we haveapproximately L acc ∝ L ( η − . / . ∗ . (12)For η = . L acc ∝ L . ∗ . Thus the slope of the tracks in Fig. 3is a result of the ratio of the rates at which the star and the accretiondisc evolve. Owing to the photospheric boundary condition this ra-tio just depends on η .It might be thought suggestive that the relation L acc ∝ L ! . ∗ (which holds in the case η = .
5, at least over the range of L ∗ for which (9) is valid), is very close to the mean relationship inthe observational data. It is then tempting to see this as observa-tional support for a viscosity law with this value of the power lawindex. However, we stress that when we experimented with a dif-ferent viscosity law, which changed the slope of the individual stel-lar trajectories according to equation (12), the distribution of starsin the L ∗ − L acc plane was indistinguishable, filling, as it did, theavailable parameter space between L acc ∼ L ∗ and observational de-tection thresholds.We note that at fixed L ∗ there is a trend for lower values of L acc to correspond to lower values of M disc , the disc mass fractionat the time when the star is deemed to appear as a classical T Tauristar. Thus in this picture the spread of values of L acc / L ∗ at fixed L ∗ comes about because of the spread of disc masses at late times.Such a spread could be due to a number of reasons such as a rangeof initial disc sizes (i.e. initial angular momenta) or a variety of disctruncations caused by dynamical interactions.There is also a trend for lower values of stellar mass M ∗ to cor-respond to lower values of L ∗ . It is evident, however, that the tracksin this diagram intermingle to some extent. At the times plotted M disc ≪ M ∗ , so each track corresponds essentially to a fixed stel-lar mass. Thus it is clear that at a given value of L ∗ there can be aspread of stellar masses depending on the assumed stellar age. We have shown, using simplified stellar evolution calculations forstars subject to a time dependent accretion history, that a plausiblemorphology of the L ∗ − L acc plane for T Tauri stars has the followingfeatures,(i) an upper locus corresponding to L ∗ ≈ L acc and(ii) diagonal tracks for falling accretion rates as stars descendHayashi tracks.We have argued that (i) is a generic feature of stars which areaccreting from a disc reservoir with mass less than the central starcoupled with the assumption that the mass depletion timescale of c (cid:13) , 000–000 he ratio of L acc to L ∗ in T Tauri Stars Table 1.
The parameters for the runs shown in Fig. 3. M f is the final stellarmass. f disc is the fraction of M f which is still in the disc at time t = × yr. The accretion rate takes on a constant value of ˙ M i until time t s andthereafter declines as ∝ t − η with η = . M f / M ⊙ f disc t s / yr ˙ M i / M ⊙ yr − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − . × . × − the disc is comparable to the age of the star. . The first is necessaryfor stars to be classified as Classical T Tauri stars while the secondresults from the gradual decay of the accretion rate.We show that the slope of of the diagonal tracks (ii) is relatedto the index of the power law decline of accretion rate with timeat late times. This itself depends on the scaling of viscosity withradius in the disc. For reference, if ν ∝ R , then L acc ∝ L . ∗ alongsuch tracks, whereas if ν ∝ R . then L acc ∝ L . ∗ .We however argue that, as long as observational datapoints ap-pear to fill the whole region of the L ∗ − L acc plane that is boundedby the upper limit (i) and luminosity dependent sensitivity thresh-olds, it will remain impossible to deduce what viscosity law or whatdependence of disc parameters on stellar mass will be required inorder to populate this region. This is because of the severe degen-eracies that exist between stellar mass, age and accretion history(our results show that in order to populate the desired region, it isnecessary that there is, at the least, a spread in both stellar massand initial disc properties). It is not, however, necessary to positany particular scaling of disc properties with stellar mass in orderto populate the plane, in contrast to the hypotheses of Alexander& Armitage 2006 and Dullemond et al 2006. Likewise, although,as noted above, the trajectories followed by individual stars in thisplane are a sensitive diagnostic of the disc viscosity law, this in-formation is washed out in the case of an ensemble of stars whichsimply fill this plane.We therefore conclude that such data could only yield usefulinsights if regions of the plane turned out to be unpopulated (or verysparsely populated). There is a shadow of a suggestion in Figure 1that at low L ∗ the data falls further below the locus L acc = L ∗ than athigher luminosities although, given the sample size, this di ff erenceis not statistically significant (Clarke & Pringle 2006). Any suchedge in the distribution could in principle re-open the possibilityof constraining disc models. We however emphasise that this lowluminosity regime can in any case not be explored by the simpleevolutionary models that we employ here, and that – if it becomesnecessary to investigate this regime – it will be essential to use fullstellar evolution calculations.Finally, we note that we have chosen to relate our study to ob-servational data in the L acc − L ∗ plane because the ratio of thesequantities is straightforwardly derivable from emission line equiv- We may in fact argue that the existence of such an upper limit in the datais a good signature of temporally declining accretion through a disc, since ifClassical T Tauri stars instead accreted external gas at a constant rate (e.g.Padoan et al 2005), no such feature would be apparent in the data alent widths. Thus this comparison is more direct than the alter-native, i.e. the comparison with observational data that has beentransformed into the plane of ˙ M versus M ∗ via the use of isochronefitting and the application of theoretical mass-radius relations. It isnevertheless worth noting that the upper locus of L acc ∼ L ∗ corre-sponds to a line with ˙ M ∝ M ∗ . Although it is often taken as self-evident that this is the expected ratio (in the case that the maximuminitial ratio of disc mass to stellar mass is independent of stellarmass), this actually only corresponds to ˙ M ∝ M ∗ in the case that thedisc’s initial viscous time is independent of stellar mass. Given thepossible variety of viscosity values and radii of protostellar discs itis not at all evident that these should conspire together to give thesame viscous timescale – until, that is, that one realises that all discsin the asymptotic regime have viscous timescale which is equal tothe disc age. The conclusion that the upper locus should follow therelation ˙ M ∝ M ∗ is thus a very general one.In summary, we conclude, as in the study of Clarke & Pringle(2006), that the observed population of the L acc − L ∗ plane, andthe consequent relationship between ˙ M and M ∗ , is strongly drivenby luminosity dependent sensitivity thresholds and an upper locusat L acc ≈ L ∗ . The main new ingredient injected by our numericalcalculations and plausibility arguments is that we now understandthat this observed upper locus can be understood as a consequenceof plausible accretion histories, given the current properties of therelevant pre-mainsequence stars. In addition, our computations in-dicate the range of initial disc properties that are required in order topopulate the region of parameter space occupied by observationaldatapoints: we are able to reproduce the observed range of values of L acc / L ∗ at a given value of L ∗ by assuming once the T Tauri phasehas been reached, the disc mass fractions lie roughly in the range0 . M disc / M ∗ .
2, independent of stellar mass.
ACKNOWLEDGMENTS
CAT thanks Churchill College for his Fellowship. We acknowledgeuseful discussions with Lee Hartmann and Kees Dullemond.
REFERENCES
Alexander, R.D., Armitage, P.J., 2006. ApJL 639, 83Bate M. R., Bonnell I. A., Bromm V., 2002, MNRAS, 336, 705Calvet N., Muzerolle J., Brice˜no C., Hern´andez J., Hartmann L.,Suacedo J. L., Gordon K. D., 2004, AJ, 128, 1294Clarke C. J., Pringle J. E., 2006, MNRAS, 370, L10 c (cid:13) , 000–000 I. Tilling, C. J. Clarke, J. E. Pringle & C. A. Tout
D’Antona F., Mazzitelli I., 1994, ApJS, 90, 467D’Antona F., Mazzitelli I., 1997, Mem. Soc. Astron. Ital., 68, 807Dullemond, C.P., Natta, A., Testi, L., 2006. ApJL 645,69Hartmann L., Cassen P., Kenyon S. J., 1997, ApJ, 475, 770Hartmann L., Calvet N., Gullbring E., D’Alessio P., 1998, ApJ,495, 385Lynden-Bell D., Pringle J. E., 1974, MNRAS, 168, 603Natta A., Testi L., Muzerolle J., Randich S., Comer´on F., Persi P.,2004, A&A, 424, 603Natta A., Testi L., Randich S., 2006, A&A, 452, 245Padoan, P., Kritsuk, A., Norman, M., Nordlund, A., ApJ 622,L61Pringle J. E., 1991, MNRAS, 248, 754Stahler S., 1988, ApJ, 332, 804Tout C. A., Livio M., Bonnell I. A., 1999, MNRAS, 310, 360 c (cid:13)000