A. B. Makalkin, I. N. Ziglina, Schmidt Institute of Earth Physics Russian Academy of Sciences,

1. MODELING FORMATION OF SELF-GRAVITATING DUST CONDENSATIONS AND ORIGINAL PLANETESIMALS IN A PROTOPLANETARY DISK A. B. Makalkin, I. N. Ziglina, Schmidt Institute of Earth Physics Russian Academy of Sciences,

2. Obstacles to the growth of large dust particles and small bodies in a protoplanetary disk.Meter-sized barrier • Beyond millimeter-centimeter sizes, experiments show that particle collisions often result inbouncing or breaking instead of sticking (Blum & Wurm 2000; Zsom et al. 2010; Weidling et al.2012). • Bodies of decimeter to meter in size in the solar nebula (or another protoplanetary disk) quickly migrated to the sun (for ~ 103 years from 1 AU) and /or were destroyed in collisions with other particles. • This forms so called “meter-sized barrier” to planetesimal accretion.

3. Even in the absence of the global turbulence in the disk, dust particles could not settle to the disk midplane and reach high concentration there because of counteraction from local turbulence triggered by shear stresses at the boundaries between the dust layer (which is in Keplerian rotation) and the gas above and below, which rotates slower owing to radial pressure gradient. The shear turbulence prevents the thinning dust layer from reaching the critical density necessary for GI: cr0.5M*/r3, where M* is the stellar (solar) mass and r is the distance from the star (sun). Gravitational instability (GI) of the dense dust layer is the main way to jump over the barrier and first form self-gravitating dust condensations. After compaction and dust accretion they turn into initial planetesimals (with sizes no less than 1km to be self-gravitating).

4. Aerodynamic mechanisms for concentration of particles in a protoplanetary disk • Radial inward drift of particles (due to gas drag) leading to compaction of the layer. • Particle concentration in localized pressure maxima of any origin. • Particle concentration in long-lived turbulent vortices. • Streaming instability (caused by gas drag). In this modeling we study the first of the above mechanisms: global compaction of the particle layer through the particle inward drift due to gas drag

5. We consider the global redistribution of mass of the dust (particle) layer (also called subdisk) which in fact is the dust-gas layer in the gas-dust protoplanetary disk. Settling of particles (dust aggregates) towards the midplane and particle migration towards the sun is included. Evaporation of water ice at the “snowline” is taken into account. We calculate time variation of volume (spatial) density and surface (column) density of dust (particle) component of the layer (these densities present the total mass of particles in the unit volume and that per unit area respectively). The radial mass flux of particles in the layer is also computed. In the modeling we assume no global turbulence (0), but do account for the induced shear turbulence through calculation of shear stress acting on the layer and rms turbulent random velocities of particles (with suggestion of vanishing average turbulent velocities). Main features of this modeling

6. When computing particle movement we consider gas drag on the individual particles and also account for the “collective” drag on the layer which is the shear stress. Both drag mechanisms lead to loss of angular momentum. The total particle velocity in the inertial frame is the sum of two parts: the center-of-mass velocity of the narrow annular fragment of the layer and the particle velocity relative the center of mass of this annulus. The total velocity is substituted into the mass conservation (continuity) equation for the column density of dust component. Dividing column density of the layer by layer thickness yields the mean volume density to be compared with the critical one for the onset of GI. The calculated parameters of the layer are used in the derivation of dispersion equation, estimating the increment of perturbations and the initial parameters of dust condensations. We also consider interaction of the condensations with surrounding dust particles to trace the mass increase and compaction of condensations till their turning into planetesimals. For simplicity we assume all particles to be of the same size at any given distance at any time. Main features of this modeling(continuation)

7. The main stages of evolution of the protoplanetary disk (solar nebula). This scenario is presented in the book by V.S. Safronov (1969).The figure from the book of B.Y. Levin (1964) a– the gas-dust protoplanetary disk around the young sun; b – formation of the dense dust layer in the midplane of the disk due to dust settling; c – breakup of the dust layer on dust condensations; d – formation of initial self-gravitating planetasimals; e, f – рост этих тел при взаимных соударениях и увеличение их относительных скоростей при взаимных сближениях; g, h– превращение диска по мере удаления из него газа и роста планетезималей в рой крупных допланетных тел – зародышей планет (g), а затем и в систему планет (h).

8. Sufficient compaction of the dust layer for attainment of GI is found to be possible not for all radial distributions of surface density and temperature of the gas in the protoplanetary disk. The slope of power distributions of gas surface density and temperature g=1(r/1AU)p and T=T1(r/1AU)q should be sufficiently low to satisfy the inequality p + 1.5q < 2, e.g. we obtained the effective compaction for the case p=1 and q=1/2 , but not for the cases of p=3/2 and q=1. The exponent p=1 is confirmed by observations of circumstellar disks, and q0.5 is characteristic for thin passive disks. Modeling results

9. Time variation of the dust layer half-thickness (in AU) Curve 1 corresponds to the initial half-thickness of the dust layer, equal to the half-thickness of the homogeneous gas-dust protoplanetary disk; curve 2is half–thickness at the time 1103years; curve 3 –5103 yr; curve 4 – 2104 yr ; curve 5 – 5104yr ; curve 6 –1105yr . Vertical segments indicate the outer boundary of the layer, moving inward.

10. Evolution of dust density of the layer , g/сm3 d The volume (spatial) density of the dust component of the dust-gas layer, averaged over its thicknessat the following time moments: 0 years (curve 1), 1103yr (curve 2), 5103yr (curve 3), 210 4 yr (curve 4), 5104 yr(curve 5), 1105yr (curve 6). The blue band running diagonally –shows the range of values of critical density for GI. Red points show places where density becomes greater than average critical value. Density jump at 4 AU ти на 4 а.е. соответствует фронту испарения льда. AU

11. Evolution of dust density of the layer for different particle sizes Time variation of mean density distribution in the dust layer of the protoplanetary disk with initial radius rd=100 AU. (The density is averaged over the thickness of the layer.) Curves 1, 2, …6 correspond to the following time instants from the onset of layer formation (in years): 0, 1103, 5103, 2104, 5104 and 1105. The diagonal band (in gray) shows the critical density for the onset of GI. The band width indicates the uncertainty in the determination of critical density. The bold dots mark the intersections of curves with the middle of the band. The adopted resultant particle size (diameter) in the inner, ice-free region of the layer is 10 cm (left) and 1 cm (right).

12. GI first occurred in the outer part of the disk (at 50-70 AU for the 100-AU disk) in 5103 yr from the beginning of the layer formation, if particles in the layer reach decimeter size, and GI in 5104 yr if particles are cm-sized. (For simplicity we assume all particles to be of the same size at any given distance at any time). GI covers the formation zone of giant planets in 5104 yr (at any particle size), and at the same time GI reaches the region near 1 AU, but comes to this region from the inner (not outer) zone, close to the metal-silicate evaporation boundary of the layer. (In the inner zone GI happens in 2104 yr.) Finally, in 1105 yr GI occurs in the region about 3–4 AU. Modeling dust evolution in the disk with radius 50 AU shows similar times of GI, but at lower distances. First GI occurs in 5103 yr at ~40 AU. Almost simultaneously it happens at 1 AU and 8 AU. But in the 50-AU disk GI does not occur at 3–4 AU, because the outer boundary of the dust layer passes through this region before the dust layer undergoes sufficient compaction for GI. Results and discussion

13. The modeling shows that during radial contraction and compaction of the dust layer particles pile up and the dust surface density d rises drastically. At 1 AU we obtained increase of d from 11 g cm2 (at the onset of dust layer formation) to 150 g cm2 (at GI). At Jupiter and Saturn distances the enrichment in solid material is triple – double. It follows from this result that most of dust particles would never take part in subsequent processes of planetesimal formation and planet accretion inner part of the disk. The particles probably were evaporated at the inner disk boundary and were consequently accreted by the sun. In addition to density, another barrier for GI is particle random velocity which should be rather low to prevent dispersion of newly formed condensations. When modeling GI, we derived dispersion relation and estimated perturbation increment. We found that the turbulent random velocities of particles are equal to about a half – one third of the critical velocity for GI. With account for parameter uncertainties (e.g., in critical Reynolds number) this result suggests a rather low probability of formation of dust condensation from the environment which is very abundant in dust. Thus we get one more argument for the rather low probability of formation of dust condensations in addition to high dust surface density obtained by the modeling.

14. Modeling showed that formation of dust condensations after the GI in the dust layer is very rapid process requiring no more than 100 yr. Subsequent evolution of condensations is determined by their interaction with surrounding dust particles. Our calculations also showed that initial masses of condensations in the region near 1 AU are ~ 1020–1021 g. After the 103–104 years of absorption of dust particles and aggregates the condensations compact to densities of solids, tripling their masses and turning into bodies of asteroid mass and size of the order of 1021 g and 50 km respectively. The subsequent growth of these bodies to sizes of a few hundred km requires no more than 105 yr. Therefore the whole period of formation of 100-km-sized bodies from dust aggregates would not be longer than ~ 105 yr. However, the onset of planetesimal formation could be delayed for ~106 yr (after CAI). The latter time interval may correspond to the duration of the active turbulent phase of the solar nebula. This interval is consistent with observations of disks around young stars and with 182Hf–182W systematics of some magmatic iron meteorites, which provide data on timing of accretion and differentiation of parent bodies of these meteorites in the solar nebula (~1 Myr after CAI formation) (Kleine et al. 2009).

15. Modeling showed that formation of dust condensations after the GI in the dust layer is very rapid process requiring no more than 100 yr. Subsequent evolution of condensations is determined by their interaction with surrounding dust particles. Our calculations also showed that initial masses of condensations in the region near 1 AU are ~ 1020–1021 g. After the 103–104 years of absorption of dust particles and aggregates the condensations compact to densities of solids, tripling their masses and turning into bodies of asteroid mass and size of the order of 1021 g and 50 km respectively. The subsequent growth of these bodies to sizes of a few hundred km requires no more than 105 yr. Therefore the whole period of formation of 100-km-sized bodies from dust aggregates would not be longer than ~ 105 yr. However, the onset of planetesimal formation could be delayed for ~106 yr (after CAI). The latter time interval may correspond to the duration of the active turbulent phase of the solar nebula. This interval is consistent with observations of disks around young stars and with 182Hf–182W systematics of some magmatic iron meteorites, which provide data on timing of accretion and differentiation of parent bodies of these meteorites in the solar nebula (~1 Myr after CAI formation) (Kleine et al. 2009).

16. Estimations of masses of initial dust condensations at the distances 5–10 AU give ~(0.36)1022 г. These values correspond to radii of initial planetesimals from 80 to 200 km (at densities 1 – 1.5 g cm-3 ). Such large planetesimals, if formed in the first 1 Myr after CAI, would be sufficiently heated by 26Al to melt water ice and form internal water ocean and silicate core. This would result in ice/rock fractionation at subsequent collisional evolution of planetesimals during formation of giant planets. The larger bodies would be depleted in ice. The lower mass bodies captured by proto-satellite disks of these planets would be enriched in ice.

17. Modeling of radial contraction of the dust layer shows a possibility of rapid (during 0.1 Myr) formation of about 50-km-sized planetesimals at the Earth distance and 100-200-km-sized planetesimals at Jupiter-Saturn distances. If this formation was delayed for not more than ~1 Myr, heating by 26Al could significantly affect evolution of the planetesimals. Dust condensation and planetesimal formation in the inner-planet zone were processes with rather low efficiencies. A lot of particles probably were evaporated near the inner disk boundary and were consequently accreted by the sun. More calculations and improvements of the model are needed. In particular, it is necessary to improve modeling of the particle growth at collisions and their fragmentation at ice sublimation in order to account for specific features of evolution of the dust layer near the snowline. Summary and future work