Do all Sun-like stars have planets? Inferences from the disc mass reservoirs of Class 0 protostars

Abstract

Protostars ∼0.1 Myr old are heavily obscured, but their circumstellar dust discs can be studied by millimetre interferometry that resolves out the obscuring envelope. Consistent estimates are made for the disc masses of Class 0 protostars, and these range over 7–660 M_Jup. A simple grain coagulation model reproduces the mass distributions of more evolved discs around Class I protostars and Class II T Tauri stars (at ≲1, ≲5 Myr), implying that their observed dust is remnant material. The conversion of most dust grains into planetesimals thus occurs very early, at ∼0.1 Myr. As this is concurrent with the formation of the star itself, much of the disc is expected to accrete on to the central object, and a correlation of bolometric luminosity and disc mass is observed that agrees well with disc models. The observed Class 0 discs all contain upwards of 20 M of dust, allowing for the formation of ‘super-Earths’ around many Sun-like stars. Only 10 per cent of the dust mass needs to be converted into planetary cores to match the range of such core masses presently known.

protoplanetary discs, circumstellar matter

1 Introduction

Dust particles in discs around pre-main-sequence stars are expected to stick together to form larger rocky bodies, and ultimately planetesimals and planetary cores. The minimum mass solar nebula (MMSN) needed ≈50 M of such dust to sum to the present-day solid content of the planets (Davis 2005), so if analogue planetary systems are common, MMSN discs should frequently be seen around young stars. However, it has recently been discovered that such discs are uncommon – in particular, surveys of optically thin millimetre-wavelength dust emission have found that the discs of T Tauri stars are usually a few times less massive (Andrews & Williams 2005, 2007). It has thus been suggested that planet formation starts earlier; in particular, protostars in the Class 0, I stages have more circumstellar dust than pre-main-sequence stars (Andrews & Williams 2007; Greaves & Rice 2010). Then planetesimal growth could be occurring very early in the evolution of young solar analogues, even while the star is still accumulating its final mass. This would allow the low-mass dust discs of T Tauri stars to be explained as simply remnants of the initial solid reservoir available to form planetesimals.

Here the discs of Class 0 protostars only around 0.1 Myr old (Evans et al. 2009) are assessed as the true reservoirs of dust available to form planetary cores before the grains aggregate into larger bodies. These protostars are very heavily obscured by much of the system's mass remaining in a circumstellar envelope. However, millimetre wavelength interferometry resolves out the large-scale dust emission of the envelope, detecting compact or point-like emission associated with only the circumstellar disc. These wavelengths are ideal to probe through very large extinctions and to trace particles that are large compared to interstellar dust and of comparable size to the observing wavelength (Draine 2006).

Dullemond & Dominik (2005) have modelled dust growth and find that by a few ×0.1 Myr, a significant fraction of the mass is already expected to be in metre-sized bodies. The observational consequence is that such ‘boulders’ are inefficient millimetre emitters per unit mass, and so Class I and II discs should show a decline in millimetre emission if the solid mass has already been partly converted into unobservably large forms. In that case, the Class I, II discs with median ages of about 0.5 and 2 Myr (Evans et al. 2009) are evolved remnants, but at only ∼0.1 Myr the Class 0 discs should not have had time to convert much of their solid mass into boulders. We therefore make a systematic assessment of all the existing millimetre-interferometry data on Class 0 systems and consider the prospects for forming planetesimals and so future planetary systems.

2 DATA COMPILATION

As the Class 0 phase is short, few objects exist at any one time in nearby regions of star formation, and as the observations also require many hours per target, large surveys do not exist. The data included here are for the majority of objects with good detections in millimetre interferometry. Four sources were rejected where the signal-to-noise ratio is poor; only angular resolutions of 5 arcsec upwards are available (which are found to inflate masses by up to an order of magnitude, presumably by including part of the envelope); or the compact-source flux is formally zero, but no upper limit is given. The 15 objects in Table 1 have compact-source fluxes taken from Guilloteau et al. (1992), Rodríguez et al. (1998), Brown et al. (2000), Looney et al. (2003), Schöier et al. (2004), Rodríguez et al. (2005), Enoch et al. (2009), Jørgensen et al. (2009), Jørgensen & van Dishoeck (2010) and Ward-Thompson et al. (in preparation). The highest angular resolution in each data set is also listed, along with disc radii in au, derived from these authors' Gaussian fits to the source extent, and the assumed distances in the table.

Table 1

Disc masses (in increasing order) for Class 0 protostars, as derived from interferometric compact-source fluxes and a unified set of assumptions described in the text. Each mass estimate is listed along with highest angular resolution in the data set in brackets; the highest mass estimates are preferred assuming these are the most optically thin. The 3 and 1.4mm bands encompass observations at 2.6-3.4mm and 1.3-1.5 mm. Also listed are disc radii from Gaussian fits and bolometric luminosities (Hatchell et al. 2007; Jørgensen et al. 2009) plus for VLA 1623 (André, Ward-Thompson & Barsony 1993), L1551 (van Kempen et al. 2009), Ser FIRS1 (Hurt & Barsony 1996) and IRAS16293 B (Evans et al. 2009). For most binaries, these are only known for the whole system and so values marked by superscript ‘b’ are estimated as system-L_bol/2.

For systematic purposes, the disc masses were recalculated, rather than taken directly from these papers, to unify the assumptions used for dust temperature and opacity and source distances. This implicitly assumes that the dust evolves similarly in all these young systems. The dust properties adopted here are mid-range values of T= 30 K and κ= 0.015 cm² g⁻¹ at 1 mm wavelength with a spectral index of 1 (κ_ν∝ν). This opacity is from Draine (2006) for a population of grains up to 1 cm in size and also matches the widely used value from Ossenkopf & Henning (1994) from their comparison of a range of grain models. It folds in the assumed gas component of the discs via a gas-to-dust mass ratio of 100 inherited from the interstellar medium. The total disc mass is then

(1)

(Hildebrand 1983) if the observed flux F_ν is optically thin. The distances to each region are recent literature estimates, and all protostars in a region are taken to be at the same distance, which is true to about 10 per cent (Loinard et al. 2008).

Table 1 lists the derived disc masses, including from multiple wavelengths where suitable data exist. Notably, the mass estimates are then in fair agreement, with discrepancies of only factors of ≈2 even at the high-mass end. In part, this error is due to the difficulty of fitting the data for the point-source flux (e.g. Brown et al. 2000). We adopt the largest mass estimate for each disc from here on, and this is generally also the value from the longest wavelength observed, where the emission should be least optically thick. At 0.8 mm, disc models suggest opacity is a problem above ≈10 M_Jup (Andrews & Williams 2005), but there are no mass estimates relying solely on this wavelength. At the longest wavelength of 7 mm, there can be contamination from free–free emission, but the disc studies estimated this to be at a low level. More important biases could arise from the range of angular resolutions used, i.e. large-scale disc flux could be resolved out, or conversely extended envelope emission included. However, disc radii that were inferred from Gaussian fits (Table 1) are not correlated with the disc masses – small massive discs and large insubstantial ones both exist – suggesting this bias is small. There is a tendency for the smaller discs to be in binary systems observed at high resolution, but this could be a genuine tidal truncation effect. A remaining bias may arise in target selection for millimetre interferometry, with only brighter systems chosen for these time-intensive observations. In the Class 0 catalogue compiled by Froebrich (2005), single-dish millimetre data are used to derive envelope masses, and the Table 1 objects have M_env≥ 0.7 M_⊙, in the upper two-thirds of the population. There is a good correlation between log M_env and our log M_disc (coefficient of 0.9), and by downward extrapolation the disc masses may be incomplete below 20 M_Jup.

For Class I and II objects, the deep surveys of Taurus and Ophiuchus by André & Montmerle (1994) and Andrews & Williams (2005, 2007) were adopted, but the disc masses were recomputed to use the same assumptions as for the Class 0 protostars. The survey data are from single dishes rather than interferometers, but envelope contamination by these later phases should be minimal. At the high-flux end, the more optically thin 1.3 mm data are adopted, but at the low-flux end 0.8 mm detections or limits are taken preferentially where deeper. The 31 Class I data points are all detections, but of the 139 Class II measurements, 37 (27 per cent) are upper limits. The underlying population adopted was found by survival analysis, using the Kaplan–Meier estimator in the package asurv (Lavalley, Isobe & Feigelson 1992). Class III (remnant disc, weak-line T Tauri systems) are not considered in detail here, but were also surveyed. These objects have similar ages to Class II classical T Tauris (Winston et al. 2009) but have only remnant discs, with 6/99 of the Class III's having millimetre dust detections. The survival analysis of the combined (presumed similar-aged) II/III sample is hence only approximate at the low-mass end. The proportions of Class II and III objects searched for dust emission are roughly in line with real population fractions at ≲3 Myr (Winston et al. 2009; Luhman et al. 2010).

3 RESULTS

Fig. 1 shows the cumulative distributions of disc mass for the evolutionary Classes 0, I, II and II+III. Logarithmic bins covering a mass range of 3 are used to smooth out the factor of ∼2 variations between wavelengths (Table 1) and similar-level uncertainties in absolute masses from properties of different grain materials (Draine 2006). In this binned form, the global reduction in detected dust mass at later stages is clearly seen. The median disc mass declines by a factor of 10 from Class 0 to I and by a further factor of 3 going from I to II. For Classes II and III combined, the median disc comprises only 2 M_Jup of gas and dust, confirming that this material at a few Myr cannot be the reservoir for forming gas giant planets, but only a remnant. In contrast, discs are nearly all massive at Class 0, with 13/15 examples hosting more than an MMSN (20 M_Jup upwards). Also now apparent is a developing asymmetry towards more low-mass discs at later times, with, for example, nearly half the Class II+III discs lying in the lowest mass bin plotted.

Figure 1

Cumulative distributions of disc masses by evolutionary class. The three rightmost bins correspond to an MMSN or more in gas and dust. The Class II and II/III distributions below 20 M_Jup are estimated (see text).

3 .1 Dust evolution model

Fig. 2 shows the fractions of stars per mass bin in Classes 0, I and II (nearly all the Class III's pile-up in the leftmost bin and are not shown). Overlaid are the results of a simple parametrized model for dust coagulation. The hypothesis is that successive collisions of dust particles cause sticking until the aggregate is large enough so that it is an inefficient emitter at millimetre wavelengths and so is removed from the ‘detectable’ population. In effect, the observed dust declines because it is converted into boulders and planetesimals (metre to kilometre size scales). This can be formulated according to equation (7) of Dominik & Decin (2003), which considers removing two colliding particles from a population. Their case was collisional break-up of comets, but the same mathematical approach is used here for collisional growth, removing particles to a larger rather than a smaller regime. In terms of detectable mass, the expression becomes

(2)

where t is time and M_* is the stellar mass. The constant K folds in unknown quantities such as number of collisions to make a large aggregate, sticking efficiency and collision speeds (taken to be the same in all discs), and the square root term refers to the orbital period for any characteristic disc radius. The time factor enters linearly as a particle sweeps through the disc volume (Dominik & Decin 2003).

Figure 2

Disc mass distributions. The Class 0 distribution (red line plus shaded region indicating Poisson errors) is an input to the evolutionary model results (see text), shown with connecting lines and Poisson error bars for Classes I and II.

The model results were calculated from a grid of 10⁴ outcomes from the possible combinations of 10 values of each of the four parameters M_disc(0), M_*, R_disc and t. 10 equally probable values were assigned for each parameter, lying at the 5th, 15th … 95th percentile points of its distribution. For M_disc(0), 10 evenly spaced disc masses were taken from Table 1, and these form the base distribution, implicitly with no error and no evolution of dust during the short-lived Class 0 phase. The R_disc are based on Table 1 values, but since the distribution is not very well defined, the simple form of a rectangular log-radius function is assumed. This is taken to start at 10 au and go up to steps of 0.15 dex to the largest observed disc radius of ≈200 au. For the stellar mass, an initial mass function of dn/dm∝m^−2.35 from 0.3 to 2 M_⊙ is adopted, neglecting that the protostars will be only part-way to their final masses as this can be folded into the constant K in an averaged sense. Finally, the time t terms take into account that there is a dispersion in the ages at which Class I and II objects are seen if stars form at a steady rate, but they last a range of times in each evolutionary stage. Evans et al. (2009) find a median lifetime of 0.44 Myr for Class I after adopting a median duration of 2 Myr for Class II. Here we assume that the fraction of stars still in the class drops linearly with time, up to t_end defined as twice the median lifetime. Hence the oldest age for Class I is 0.9 Myr and for Class II is the sum of t_end for both phases, i.e. ≈5 Myr. The percentiles are then determined according to probabilities P(≤t/t_end) = 2t/t_end(1 −t/2t_end).

The model fits are optimized by adjusting only the constant in equation (1), and Fig. 2 shows the result with minimized residuals (for K= 90). The simple parametrized model fits the observed distributions well, with a reduced χ² value of 2.9. The shift towards low-end masses noted above occurs because the smaller discs evolve faster, with grains sticking more efficiently when they collide more frequently in shorter period orbits. The Class II fit is very close, while the Class I points lie further from the model lines. This could be a mixture of smaller number statistics and the neglect in the model of continued infall into the disc of small dust particles from the residual envelope.

A test was made adding in a few discs of 1–20 M_Jup in the region that is possibly incomplete as discussed above, but this significantly degraded the fit by overproducing low-mass systems. Similarly, a test was made combining Class II and III discs, but the fit was again degraded, as a large fraction of low-mass discs cannot be fitted in a smoothly evolving scenario. The dust masses of the Class III objects are on average less than a few tenths of an Earth mass (Andrews & Williams 2007), which is as low as limits for debris discs in the Pleiades at 100 Myr (Greaves et al. 2009). This suggests that the Class III systems are reduced to a low-dust state early on and perhaps lose their discs via some mechanism such as stripping by external radiation or perturbations. Luhman et al. (2010) found higher proportions of Class III systems in higher density regions of young stars, supporting a disruptive role played by the environment.

3 .2 Accretion by the star

These results suggest that most of the dust has already evolved into boulders and possibly planetesimals, even by the Class I stage at ∼0.5 Myr. The observed dust reservoir has then typically already dropped by an order of magnitude (Fig. 1), so the initial solid-mass reservoir for planet formation can only be assessed from the Class 0 objects. However, at this stage the star itself is still forming and much of the disc material should be accreted.

Fig. 3 plots the bolometric luminosities versus the disc masses for the Class 0 systems. The luminosity of an accreting protostar should be supplied mainly from the gravitational energy of infalling mass, given by

(3)

and if higher M_disc increases dM_*/dt and hence M_*, more luminous systems could result. There is in fact such a rising trend in Fig. 3. We also compute bolometric luminosities and disc masses from one-dimensional collapsing core calculations (Rice, Mayo & Armitage 2010) which assume that disc evolution is controlled primarily by disc self-gravity. A range of initial core masses and initial rotation rates is considered, and each model produces self-consistent mass accretion rates, dM_*/dt, surface density profiles Σ(r) and temperature profiles T(r) from the start of core collapse up to two free-fall times (∼180 000 yr in this case). The model bolometric luminosities can be calculated directly using equation (3). The millimetre flux, F_ν, can be calculated by summing over annuli using (Draine 2006)

(4)

where Σ(r) 2πdr is the mass in each annulus, ν is the frequency (corresponding here to 7 mm wavelength) and κ(ν) is 0.002 cm² g⁻¹. However, we also assume that the optical depth is approximated by τ=κ (ν) Σ(r) and modify equation (4) when τ≥ 1 by setting κ(ν) Σ(r) = 1 and replacing T(r) with T(r)/τ^1/4. Once the millimetre flux has been calculated, the disc mass can be estimated using equation (1).

Figure 3

Bolometric luminosities against disc masses for the Class 0 protostars. In the binary systems shown by unfilled symbols, L_bol is taken to be the system total divided by 2, as individual values are not known. The black data points are tracks for model discs (see text) in systems with stellar mass up to 1.2 M_⊙.

Fig. 3 shows that model results compare well with the observed luminosities and disc masses, with both showing a broad trend of rising L_bol with M_disc. It should be noted that the Fig. 3 disc masses are from fluxes calculated via equation (4) and can differ by factors of a few from the actual M_disc in the models. At the low-mass end, M_disc can be underestimated as the actual surface temperature of the disc can be lower than the assumed 30 K, while at the high-mass end the opposite is true, so flux-based masses may be too high. Where ‘actual’ disc masses in the model are larger, it was found that there are inner regions that are optically thick even at millimetre wavelengths (Clarke 2009; Rice & Armitage 2009; Zhu, Hartmann & Gammie 2009), as a consequence of self-gravitating discs settling into a quasi-steady state with substantial centralized material.

The agreement between the models and observations suggests that there is indeed sufficient planet-building material at these early times, possibly even more than inferred from the millimetre fluxes. Model (and observed) accretion rates do support the expectation that a large fraction of the gas in the disc is accreted on to the star during the Class 0 and early Class I phases. However, initiating planet formation only requires that a modest amount of the available solid material be converted to large boulders or planetesimals – which decouple from the disc gas – at this early time.

4 IMPLICATIONS

Reservoirs of disc material have been compared at the Class 0, I, II and III stages of young low-mass stellar evolution. The sharp decline in mass implies that the observed dust represents the real mass budget for forming planetary cores only in the earliest protostellar stage. Growth into larger non-observable particles must be occurring as early as 0.1 Myr, and a simple model of grain sticking reproduces the disc mass distribution observed at later stages. That growth occurs at such early times is also consistent with the suggestion that particles could collect and grow in self-gravitating spiral structures (Rice et al. 2004, 2006) that would be present during these early stages. Such a process may, however, only be efficient at relatively large radii where the spirals are strongest (Clarke & Lodato 2009).

The Class 0 discs are substantial, allowing enough material to accrete on to the protostar to produce its luminosity by infall, while reserving a small fraction of the mass would be sufficient to build planets. For example, all of the discs in Table 1 contain upwards of 20 M of dust, which could be sufficient to form the ‘super-Earths’ now being discovered in radial velocity surveys. Many of these comprise only a few M, and they are thought to occur around as many as 40–60 per cent of stars (Mayor et al. 2011).

This raises the possibility that all Sun-like stars could host a planetary system. The efficiency achieved in converting the available dust into planetary cores can be estimated by comparing these two mass distributions. The lowest mass rocky body around a main-sequence star presently known is GJ 581e at 2–3 M (Mayor et al. 2009), while the most massive core inferred from transit measurements is that of CoRoT-10b at ∼120–240 M (Bonomo et al. 2010). This is a range of 2 orders of magnitude, which is very similar to the span covered by the Class 0 disc masses (Table 1; a slightly wider range is possible if low-mass discs are incomplete). The solid content of the Table 1 discs is 20–2000 M for a gas-to-dust mass ratio of 100, so if they went on to form the least and most massive planetary cores so far discovered, an efficiency of 10 per cent in usage of the available solid material would be required. All stars could then host a modest rocky body – for example a 0.3 M planet that can support plate tectonics, and an atmosphere (Raymond, Scalo & Meadows 2007) could be formed by a 1M_Jup disc at the lowest end estimated here for Class 0 discs. The only systems lacking planets might be those that lose their discs and/or planetesimals to destabilizing forces such as perturbations.

Acknowledgments

We thank STFC and SUPA for support of this work.