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Abstract

Ices, including water ice, prefer to recondense on to preexisting nuclei rather than spontaneously forming grains from a cloud of vapour. Interestingly, different potential recondensation nuclei have very different propensities to actually nucleate water ice at the temperatures associated with freeze-out in protoplanetary discs. Therefore, if a region in a disc is warmed and then recooled, water vapour should not be expected to refreeze evenly on to all available grains. Instead, it will preferentially recondense on to the most favorable grains. When the recooling is slow enough, only the most favorable grains will nucleate ice, allowing them to recondense thick ice mantles. We quantify the conditions for preferential recondensation to rapidly create pebble-sized grains in protoplanetary discs and show that FU Orionis type outbursts have the appropriate cooling rates to drive pebble creation in a band about 5 au wide outside of the quiescent frost line from approximately Jupiter's orbit to Saturn's (about –10 au). Those pebbles could be of the appropriate size to proceed to planetesimal formation via the Streaming Instability, or to contribute to the growth of planetesimals through pebble accretion. We suggest that this phenomenon contributed to the formation of the gas giants in our own Solar system.

planets and satellites: formation, protoplanetary discs

1 INTRODUCTION

Ices are a major player in planet formation. In decreasing order of condensation temperature, rocky material, water ice, and all other ices each makes up about 0.5 per cent of a protoplanetary disc's total mass, and one-third of the potential solid mass (Lodders 2003). Accordingly, just outside the water frost line where water is in solid form, the disc has about twice the material available to participate in the growth of solids as inside the frost line. Ice-rimmed dust is also stickier and more collision resilient than bare silicates (Dominik & Tielens 1997), further promoting the collisional growth of solids. The temperature at which the water ice saturation vapour pressure equals the water vapour partial pressure depends on the local number density of water molecules.

Nonetheless, cooling from T = 170 to 166 K more than halves the saturation vapour pressure (Murphy & Koop 2005). Thus, it is reasonable to approximate that the temperature window in which water is not effectively entirely in solid or gaseous phases is extremely narrow, justifying the traditional assumption that the window occurs at T = 170 K (Sasselov & Lecar 2000), although this is complicated by disc vertical structure (Podolak & Zucker 2004). Disc regions at larger orbital separations have less ambient material, and therefore must be colder to condense water ice, but the difference in condensation temperatures is small enough that the radial temperature gradient dominates, and the fraction of water condensed rapidly approaches unity outside of the frost line.

We can assume that water vapour is in equilibrium with ice-mantled dust grains as long as the system evolves slowly enough and, crucially, as long as a significant portion of the water ice remains condensed and exposed. If a parcel of gas and ice-mantled dust experiences a temperature fluctuation sufficient to evaporate the ice before cooling, equilibrium cannot be assumed. Homogeneous freezing, spontaneous freezing in the absence of preexisting nuclei, is more difficult than inhomogeneous freezing on to existing potential ice nuclei (henceforth INs), and does not occur under the same temperature and pressure conditions as evaporation (Koop et al. 2000). Interestingly however, not all INs are created equal (Cziczo et al. 2013a): At temperatures associated with water freezing in protoplanetary discs, i.e. T < 170 K, INs of differing qualities can require saturation ratios (SRs) of factors of several to begin nucleating ice (Cziczo et al. 2013b). Ice is the best surface at condensing more ice, so once mantles are accreted, the differences between ice-mantled INs vanish.

If, then, a parcel of gas and ice-mantled dust is heated sufficiently to evaporate all the ice, and then recooled slowly, we can expect the most favourable potential INs to accrete ice mantles first. With those mantles in place, the favoured few grains will maintain equilibrium between their water ice surfaces and the water vapour, i.e. a saturation ratio SR = 1, preventing water ice from recondensing on the other grains even if they had originally possessed ice mantles. We refer to this process as preferential recondensation. If, on the other hand, the parcel is cooled sufficiently rapidly, the rise in SR due to cooling would outpace the drop in S due to condensation enough for the next tier of INs to also nucleate ice.

One scenario where we would expect preferential recondensation to occur with important consequences is the aftermath of a major accretion event such as an FU Orionis type outburst (Hartmann & Kenyon 1996). FU Orionis events occur early in a protostar's existence and lead to significant disc heating (Cieza et al. 2016). As we will show, during an FU Orionis outburst, the entire radial belt from about 4 to 10 au could host significant preferential recondensation. By restricting the recondensation to a small subset of dust grains, those grains would grow to sizes associated with both planetesimal formation and pebble accretion (Johansen et al. 2007; Lambrechts & Johansen 2012; Carrera, Johansen & Davies 2015). This provides a pathway to rapidly triggering and promoting the formation of giant planets in the outer disc where densities are low and dust coagulation is unlikely to proceed apace.

2 MODEL

As might be expected, there is a vast literature discussing ice formation in the context of terrestrial cloud formation, far beyond the author's expertise and impossible to summarize (Cantrell & Heymsfield 2005; Hoose & Möhler 2012). Protoplanetary discs are, however, expected to be in a different regime from our atmosphere, with nearly unity IN-to-water density ratios, and correspondingly large effective IN number densities. Note that, excepting explicit densities of solid grains, any densities we refer to are densities per unit volume of protoplanetary disc gas plus solids. Protoplanetary discs also have sufficiently slow evolution time-scales, sufficient turbulent mixing, and sufficiently long molecular mean-free-paths that freeze-out is not expected to be diffusion limited: Turbulent mixing and molecular diffusion replenish the water vapour near a dust grain as fast as it is lost to condensation. Furthermore, grains cannot rapidly move out of condensation regions without first growing to a meaningful size.

Those large potential IN-to-water density ratios, combined with relatively slow temperature fluctuations, make homogeneous freezing a negligible phenomenon in protoplanetary discs. In the case of the terrestrial atmosphere, one is often in the situation where multiple condensation conditions are met, and which path is taken, such as continued homogeneous condensation or inhomogeneous condensation on to newly homogeneously formed INs, is a kinetic question (Eidhammer, DeMott & Kreidenweis 2009). We can instead assume that the condensation SRs are significantly separated for different INs, and that those SRs are, in turn, all significantly below the SRs required for homogeneous freezing.

In this section, we derive a basic model for the kinetics which uses those simplifying assumptions. By assuming that different INs have significantly different SR critical values to begin nucleating ice, we allow condensation to be restricted to a subset of INs even if the SR value temporarily rises. Thus, we only need to find the conditions required to avoid large SR fluctuations. More sophisticated models take the time derivation of the saturation ratio SR, e.g. Kärcher & Lohmann (2002). Making effective use of those models would, however, require possessing a detailed model of the distribution of the critical SR values for the potential IN actually present.

2.1 Comparing time-scales

The saturation pressure of water vapour over ice is approximately (Murphy & Koop 2005)
(1)
where pice and T are measured in Pa and K, respectively. We can use equation (1) to define
(2)
the time-scale for the vapour pressure to change as a function of the heating or cooling rates.
We can compare this time-scale τp with an equilibration time-scale between ice and vapour, which we will quantify through the time required for a vapour water molecule to encounter and expect to stick to an icy target:
(3)
where α is the accommodation coefficient, σ = πa2 is the collisional cross-section of the icy grains assumed to be spheres of radius a, ns is the number density of icy grains, and
(4)
is the thermal speed of a water molecule. The appropriate accommodation coefficient is unclear. Modelling observed cloud formation suggests low values potentially below 10−2, but recent laboratory studies have found α ≳ 0.5, which value we will use (Skrotzki et al. 2013).

Equation (3) estimates the time-scale on which water molecules freeze out on to ice grains, which means that τc is also the time-scale on which the ice partial pressure drops due to recondensation (any drop in partial pressure due to cooling is negligible for our purposes). In a cooling disc, recondensation is rapid enough to keep water ice in rough equilibrium with icy surfaces as long as τc < τp. If on the other hand τc > τp, then recondensation will lag and the SR will rise, triggering recondensation on to less favourable dust grains, causing τc to drop over time (more grains to recondense on). Once exactly enough dust grains begin condensing water that τc = τp, equilibrium can be maintained, and new grains will not join in.

Assuming ice–vapour equilibration (τc ≲ τp), it takes very little cooling for nearly complete freezing, allowing us to approximate
(5)
where ρs is the approximate solid density of our ice-mantled grains and ρw is the fluid density of water molecules in the disc. Note that we have assumed that all the ice-nucleating grains are of the same size, and have nucleated sufficient ice to dominate their mass and radius.
Combining equations (3) and (5), we arrive at
(6)
Denoting the water-to-gas mass ratio as ε = ρw/ρg, we can rewrite equation (6) as
(7)
where
(8)
is the Epstein regime drag time-scale of the dust. The thermal speed of the gas is
(9)
where mg ∼ 2 amu is the gas mean molecular mass.
We can use equations (2) and (7) to write the condition τc = τp as
(10)
where we have used ε ≃ 0.005, vth ≃ 3vw, and T = 160 K, and ▵TOrb is the change in temperature in K over a local orbital period. Equation (10) estimates the largest Stokes number St at which icy grains can recondense ice fast enough to maintain equilibrium with water vapour for a cooling rate defined through ▵TOrb. Alternatively, it defines the fastest cooling rate ▵TOrb at which icy grains with Stokes number St can maintain equilibrium with water vapour.
In equation (10), the drag time has been non-dimensionalized with the local orbital frequency Ω through the Stokes number of the dust:
(11)
If a region in the disc heats enough to evaporate the ice and then cools at a rate of about 1 K per orbit, the water vapour is expected to recondense on to a small number of INs, forming dust grains with St ∼ 0.01, large enough to have significant consequences for planet formation (Johansen et al. 2007; Lambrechts & Johansen 2012; Carrera et al. 2015).

2.2 Latent heat

There is a further complication to estimating ▵TOrb: The latent heat released by water freezing is significant. From Murphy & Koop (2005), we have
(12)
Writing
(13)
we find that the latent heat is sufficient to correspond to a temperature change of
(14)
At a background temperature of T ∼ 160 K, equation (1) implies that the 3 K temperature difference provided by condensing water vapour in a protoplanetary disc is sufficient to halve the saturation water vapour pressure. If cooling is sufficiently slow for equation (10) to have significant implications, latent heat could meaningfully further slow the cooling rate.

3 COOLING REGIMES

The details of ice deposition matter only when condensation occurs but is not total, i.e. near a water ice frost line. We examine two cases: static frost lines and evolving frost lines.

3.1 Static frost line

In the case of a static frost line, we have ▵TOrb = 0, which in conjunction with equation (10) would seem to imply the growth of very large ice-rimmed grains indeed. However, water vapour will recondense only after being transported across the frost line. Turbulence mixes the water vapour into regions with preexisting water ice-rimmed grains, moving radially at most a turbulent length-scale within a turbulent time-scale assumed to be approximately the orbital time-scale (Fromang & Papaloizou 2006).

Turbulence has a length-scale
(15)
where αSS is the Shakura–Sunyaev (Shakura & Sunyaev 1973) and H the local scaleheight. We expect the disc background temperature to scale as R−1/2 as in a Hayashi (1981) minimum mass solar nebula (MMSN), so moving one turbulent length-scale would correspond to about
(16)
where we have used αSS ∼ 10−3 and H/R ∼ 0.05. At a background temperature of T = 160 K, equation (16) implies
(17)
Thus, it is unlikely that water vapour would be able to freeze out, even in an inhomogeneous manner, except on to preexisting ice-mantled dust grains. Even in that case, the approximation that freeze-out or evaporation is total does not apply for such a small δT: While narrow, frost lines are clearly broader than turbulent length-scales.

Turbulently mixing two equal volumes of protoplanetary gas and dust outside a frost line, one with ice condensed and the other evaporated, will result in the water vapour recondensing only on the preexisting ice-mantled grains. Even complete recondensation would at most double the mass of the icy grains. From equation (8), we can see that this would increase the icy grain stopping time by only a factor of between 21/3 (solid grains growing at a constant density) and 2 (highly porous grains growing at a constant radius).

Ros & Johansen (2013) showed, however, that a small number of icy particles outside a frost line will remain there long enough to be mixed into water vapour rich parcels of gas turbulently transported outside the frost line several times, allowing them to grow to a significant size. Considerations of differing condensation nuclei qualities only strengthens this conclusion by arguing that the bare INs also carried in the water vapour rich parcels are unlikely to begin to recondense ice before the icy grains freshly mixed into the parcels can do so.

3.2 Evolving frost line

3.2.1 Cooling discs

A more interesting case from our perspective is a cooling disc whose frost line is contracting, causing large radial regions to experience freeze-out. As long as the frost line retreats sufficiently, it will eventually reveal completely dry grains. When the global scale cooling is rapid enough, equation (5) applies, and hence equation (10) holds. That will certainly occur when the frost line retreats faster than turbulence can mix material from both sides of the line. Taking advantage of the very strong temperature dependence of the vapour pressure, we quantify that limit by requiring the frost line to retreat more than a turbulent length-scale over a turbulent period, which we estimate as an orbit (Fromang & Papaloizou 2006).

To accurately determine the radial temperature profile of a disc, the full radiative transport equations must be taken into account, and discs possess vertically varying thermal structures (Dullemond, van Zadelhoff & Natta 2002). The strong dust dependence of radiative transport further complicates the issue in the case of preferential recondensation, when the size of the dust varies rapidly (Inoue, Oka & Nakamoto 2009). In the case of a cooling disc, we can even expect a dust wall just outside of the frost line, with a jump in the opacity. Given our uncertainties, we adopt the simplifying approximation that T ∝ R−1/2.

Thus, assuming a disc locally experiencing external irradiation (from the protostar or in the case of an FU Orionis event, the innermost rapidly accreting disc), we can write
(18)
for some constant A, at an orbital position R with external luminosity L. From equation (1) the frost line temperature varies only slowly with the local gas density. Assuming a frost temperature of Tf ∼ 160 K, quasi-constant as a function of radius, the frost radius is given by
(19)
and the speed of its retreat is
(20)
The distance ▵ROrb retreated in an orbit is then simply
(21)
where ▵LOrb is the chance in luminosity in one local orbit. Requiring ▵ROrb > lt, we arrive at the constraint that cooling must be faster than
(22)
At constant R, we can use equation (18) to write
(23)
We can use equation (22) to further determine that cooling outpacing turbulent mixing requires a dimming rate of
(24)
The condition for cooling to outpace mixing is that the external (inner disc or protostar) luminosity drops more than 0.3 per cent per local orbit.

3.2.2 Applications to preferential recondensation

Combining equations (10), (22), and (23), we arrive at
(25)
where we used T ≃ 160 K. For reasonable estimates of H/R = 0.05 and α = 10−3, equation (25) becomes
(26)
implying that in the slow cooling limit, preferential recondensation can create quite large icy grains indeed.
We can also combine equations (10) and (23), estimating T = 160 K, to write
(27)
We are interested in preferential recondensation if it generates large grains. Arbitrarily setting the lower limit for large at St ≥ 10−3, equation (27) implies ▵LOrb/L < 0.25. Even extreme dimming rates can result in the condensation of respectably large grains. Equation (25) implies that discs that are cooling sufficiently fast that the frost line outpaces turbulent mixing, but not utterly outclasses it, are expected to see inhomogeneous freeze-out on a small enough fraction of the ambient potential INs so as to condense into large grains.
The range of dimming rates for which we expect preferential condensation on to favoured INs to result in large (here St > 10−3) grains is therefore
(28)
although variations in αSS or H/R would adjust these dimming rate bounds. Modest differences in the thermal profile (T ∝ R−1/2) will adjust, but not qualitatively alter, equation (28). The rates in equation (28) have potential astrophysical implications but, especially at the lower end, will require long-term monitoring surveys to fully explore (one orbit at 4 au taking 8 yr for a 1-M⊙ star). In particular, the bounds match dimming rates associated with FU Orionis, the namesake for FU Orinis type objects (Hartmann & Kenyon 1996). FU Orionis objects undergo violent accretion events, increasing in luminosity by around 6 mag, before dimming on a time-scale of about a century.
Recent observations have found that FU Orionis’ continuum dimmed by 12 per cent over 12 yr, although there is a yet uncertain difference in the dimming rate between shorter and longer wavelengths, similar to previous estimates for BBW 76 and slower than the dimming of V1057 Cyg by a factor of about 2 (Clarke et al. 2005; Green et al. 2006, 2016). Increasing the luminosity of a Hayashi MMSN by 6 mag would move the water frost line to approximately 45 au, while during quiescence, the frost line is closer to 4 au. This estimate has been recently confirmed by Cieza et al. (2016). At
(29)
the corresponding dimming rates in local orbits would be approximately
(30)
Out to 10 au, those rates fall within the estimated bounds of equation (28), suggesting that preferential recondensation was significant from Jupiter to Saturn, and possibly well beyond once the latent heat of water is taken into account. That suggests that as FU Orionis, or a similar object, fades, significant preferential water ice recondensation occurs generating icy pebbles.

4 DISCUSSION AND CONCLUSIONS

The aerodynamics of dust grains, as measured through their Stokes number, plays into nearly every aspect of the formation of and potentially also the growth of planetesimals (Lambrechts & Johansen 2012). Preferential recondensation naturally occurs in the aftermath of powerful accretion events such as FU Orionis type events, providing a mechanism to create grains with thick enough icy mantles to be moderately decoupled from the gas (St ≳ 0.01), a potential observable. Furthermore, different FU Orionis type objects, with differing cooling rates, will have ice-mantled dust grains of differing sizes in their recently cooled regions.

Massive accretion events, FU Orionis outbursts occur early in the life cycle of a protoplanetary disc with lots of gas left to play with, and are believed to be a common phenomenon with most protostars undergoing such outbursts (Hartmann & Kenyon 1996). While the radial extent of the accretion flow associated with the outburst is unclear, most of the energy is released at the disc's inner edge, and it is reasonable to assume a localized engine. We have shown that FU Orionis outbursts naturally combine with preferential recondensation to provide a very rapid (orbital time-scale) pathway to creating large ice-mantled dust grains. These pebbles can be of an appropriate size to trigger the Streaming Instability, leading to planetesimal formation very early in the protostar's life potentially at a large orbital separation (Johansen et al. 2007). The pebbles could also supply pebble accretion (Carrera et al. 2015), allowing those early planetesimals to grow to become the cores of gas giants. Thus, evaporation and recondensation could easily have played a major role in the formation of the gas giants in our own Solar system, and could play major roles in other forming planetary systems. This reinforces the concept of intermittent thermal processing of solids in protoplanetary discs playing an important role in the process of planet formation (Hubbard & Ebel 2014).

Acknowledgments

The research leading to these results was funded by NASA Origin of Solar Systems grant NNX14AJ56G.

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