Thermal Regime

Disc thermal regime is addressed as a key of its original inner structure and first solids formation. The viscous disc evolution with radial temperature lapse rate is regarded as the most realistic scenario of the solar system bodies’ formation (Alexander, Boss, Keller, Nuth, & Weinberger, 2007; Dorofeeva & Makalkin, 2004; Makalkin & Dorofeeva, 1995; Lynden-Bell & Pringle, 1974; Marov et al., 2013).

A hot rather than cold model of planets accumulation from the matter similar to chondriticmeteorites seems most plausible. It is supported by the established differences in the abundances of many elements and their isotopic ratios between the Sun, undifferentiated meteorites, and Earth. Indeed, all chondrites (except CI) and our planet are depleted in moderately volatile (Na, K, Rb, Sn, etc.) and highly volatile (Cs, Pb, etc.) elements relative to the solar abundances that are similar to their abundances in carbonaceous CI chondrites. The depletion is most pronounced in such elements as Bi, Cd, Cs, Hg, In, Pb, Se, Te, Tl, Zn, S, etc. (Lodders, 2003; Palme & Boyton, 1993) that was found to be typical not only for various types of chondrites but also for the bulk composition of the terrestrial planets and some large planetesimals, for example, for the parent bodies of basaltic achondrites—eucrites. Hence one may conclude that the differentiation of moderately and highly volatile elements was an important large-scale process at the early evolutionary stages of the protoplanetary accreting disk under relatively high temperature (Marov et al., 2013).

Basically, the observed depletion of highly and moderately volatile elements could result from either partial evaporation or incomplete condensation of the proto-matter of planets and the parent bodies of chondrites, because the higher the volatility of an element, the greater the depletion. Thermodynamic calculations showed that temperatures of no lower than 1200–900 K are required in both cases (Petaev & Wood, 1998; Saxena & Eriksson, 1986). Under such temperatures only non-equilibrium partial evaporation may occur and indeed, some CAIs show clear chemical and isotopic evidence of that. It is of interest to note that an experimental study of the fractional evaporation of the material of CI chondrites aimed at obtaining the material of chondrites of other types showed that, irrespective of the redox evaporation conditions, the residue obtained through heating differs radically from the actual chondrites material in the abundances of highly and moderately volatile elements (Lipschutz, Biswas, & McSween, 1983). These differences were claimed to be particularly true for the abundances of such pairs of elements as Zn and Se; Sn and Pb; Rb and Cs; etc., with similar degrees of depletion relative to the abundances in chondrites. The results, however, are not unambiguous because all chondritic meteorites are cosmic sediments; that is, their components have formed at very different conditions within the nebula and experienced different thermal history before being accreted together later.

The condensation mechanism for the differentiation of moderately and highly volatile elements accompanied by their incomplete accumulation in the proto-matter of planets and the parent bodies of chondrites seems more justified. For most elements, there is a clear correlation between the degree of depletion and the temperature of its condensation from a solar-composition gas. It is most likely attributable to the influence of the kinetic constraints on the heterogeneous reactions in a gas-solid system associated with the reduction of the reaction surface of small dust particles during their accumulation and some other factors (Fegley, 2000). This allowed us to conclude that much of the chondrites material formed through the condensation of the gas phase of the protoplanetary disk. Temperature initially increased at the disk formation stage, reaching the evaporation temperature of magnesian silicates and metallic iron at a distance of 1 AU (1400 K) and then decreased in the course of viscous disk evolution. In the formation region of the terrestrial planets, it has always remained well above the H₂O condensation temperature—it did not dropped below 300–500 K.

Apart from mineral phases containing moderately volatile elements, there are crystalline Mg- and Fe-bearing silicates in chondrites that support the hot protoplanetary disk formation model. One may assume that these amorphous and crystalline silicates formed as a result of the condensation processes in the inner zone of the circumstellar disks. Amorphous silicates of the interstellar molecular cloud from the fragment of which the solar system formed are believed to have evaporated in its inner zone at the early evolutionary stages to be condensed as the nano-to-micrometer sized crystalline silicate grains during subsequent cooling of the disk gaseous phase (Arakawa & Nakamoto, 2016; Ciardi et al., 2005; Honda et al., 2003). One may suggest that the inclusions enriched in the most refractory elements (such as Са and Al composing CAIs, as well as the rare Hf, Sc, Lu, etc.) were the earliest condensates formed near the Sun (r < 0.5 AU) at Т ~2000–1700 K and partially carried outward through the radial drift up to the formation zone of the parent bodies of chondrites with refractory chondrules embedded in their matrix and entered rocky planetesimals. Obviously, a small fraction of them reached r ~5–10 AU, as evidenced by the findings of olivine crystals in the material of Halley and Hale-Bopp comets as well as in the Comet 9 P/Tempel nucleus investigated in the Deep Impact experiment (Wooden, Desch, Harker, Gail, & Keller, 2007).

Several important formation stages of the gas-dust disk around the proto-Sun and its subsequent dynamical, thermal, and cosmochemical evolution, including the non-equilibrium partial evaporation, condensation, and compaction of protoplanetary matter were traced in the framework of quite comprehensive self-consistent numerical model (Dorofeeva & Makalkin, 2004; Marov et al., 2013; Morbidelli et al., 2009). In particular, it yielded the distributions of thermodynamic parameters (Т—Р) in the viscous disk around the young Sun passing through the T Tauri stage, as a function of radial-vertical distance (the rz coordinates). Radial temperature profiles in the disk’s mid-plane at the different stages of its evolution were derived, as it is shown in Figure 9.

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Figure 9. Radial temperature distribution T(r) in the midplane of a protoplanetary disk between 10 and 0.5 AU. 1–4—formation stage between 0.12 and 0.45 Ma (measured in millions of years) from the disk origin; 5, 6—viscous dissipation stage between 0.65 и 1.8 Ma; 7а–7в—stages of dust subdisk compaction and dust clusters formation at about 2 Ma. Temperatures are controlled by the processes of condensation/evaporation of forsterite, enstatite, and iron; maximum values T_si ≥ 1,600 К at r = 0,1 AU and T_si ≤ 1,400 К at r = 1,4 AU. Water ice is formed at T_w ~140–160 К under gas pressure Р = 10^‒7–10^‒4 bar at about 3 AU (break in the dot-dashed curve). The green zone shows the range of possible temperatures at the stage of dust clusters formation.

Source: Marov et al. (2009).

Variation of subdisk dust surface density ρ‎_d as a function of the radial distance r, the dust/gas densities ratio and size of dust particles is shown in Figure 10 as a sequence of subdisk dust density evolution, before critical surface density is achieved and the subdisc break up into dust clusters.

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Figure 10. Variation of subdisk dust density ρ‎_d depending on the radial distance r from 100 AU for particles of mean size 10 cm (left) and 1 cm (right). Broad diagonal band—critical dust density ρ‎_dcr corresponding to the disk gravity instability and its fragmentation into dust clusters. Curves 1–6 – time sequence 0, 1 × 10³, 5 × 10³, 2 × 10⁴, 5 × 10⁴ and 1 × 10⁵ (yr) from the beginning of subdisk evolution.

Source: Marov et al. (2009). Note that in the refined model, ρ‎_dcr approaches at larger r and the ρ‎_d change at the snow line is more pronounced.

Dynamics

It should be specially noticed that solar system formation was intrinsically related to the dynamical processes at all stages of origin and evolution including different kind of resonances defining the system’s structure, configuration, and stability. Complex nonlinear dynamical processes impose certain constraints on the models of disc evolution and planetary embryos origin. The role of dynamics is manifested, in particular, by observations of the formation of some peculiar configurations in the extrasolar planetary systems around the parent solar-type stars (see, e.g., Barucci, Boehnhardt, Cruikshank, Morbidelli, & Dotson, 2008; Ferraz-Mello, Michtchenko, Beaugé, & Callegari, 2005; Marov & Shevchenko, 2017; for further discussion).

The basic principles of the dynamical systems were derived by Anri Puankare and thoroughly developed by the numerous researchers (see Contopoulos, 2002, and citations therein). The main parameters defining planetary system configuration are resonances and dynamical chaos. The study is used to be performed as general N-body problem or in the framework of restricted three-body approximation with involvement of some constraints. Besides resonant and chaotic motions, this approach allowed us to describe periodic orbits, collisions, escape, and other processes of the long-term dynamics based on either analytical or (mostly) on the direct numerical integration of a set of ordinary differential equations. It was shown that orbits of the solar system planets with small eccentricities and inclinations are weakly chaotic and do not exhibit essential secular variations on the time scales comparable with the solar system age. Resonances (a near-exact relation between characteristic frequencies, e.g., orbital periods, of two bodies) and tidal oscillations are pronounced in the systems of small bodies and their tidal interactions with planets, which exerts strong influence on stability of their orbits being responsible for migration processes. The stability problems including interaction of nonlinear resonances in Hamiltonian systems were analyzed in detail in the framework of Kolmogorov-Аrnold-Мoser (КАМ-theory). Recently, these and some other studies (see Chirikov, 1979, 1982) laid the basis for investigation of the structure and dynamics of the systems of exoplanets around binary and multiple stars (Marov & Shevchenko, 2017).

Migration involving both forming planets and small bodies could play a major role in the history of early solar system affecting its configuration and planets composition. Great diversity of bodies revealed by compositional mapping of the main asteroid belt implies substantial mixing of bodies of different taxonomic classes through the various dynamical processes affected by planetary migration (DeMeo & Carry, 2014). The outer planets’ instability could cause massive influx of bodies to the inner solar system (e.g., Walsh, Morbidelli, Raymond, O’Brien, & Mandell, 2012). Hydrated C-type asteroids and comets transported from outer regions of the solar system would contribute to the above diversity and also be responsible for delivery of water and volatiles to the Earth and terrestrial planets deposited as a veneer at the final stage of these planets’ accumulation (Anders & Owen, 1977; Marov & Ipatov, 2005).

Unlike classical models of the terrestrial planets evolution with Jupiter–Saturn’s stable orbit positions (e.g., Raymond, O’Brien, Morbidelli, & Kaib, 2009; Wetherill, 1996), there were different scenarios developed in order to explain inner solar system’s water under action of the outer planets’ dynamical instability. Migration of bodies and dust from the disturbed trans-Neptunian regions in Jupiter Crossing Orbits (JCO) and then in the main asteroid belt to eventually become NEOs was studied in the numerical models (Marov & Ipatov, 2005, 2005). The mechanism involving unstable positions of the giant planets in the course of their formation was explored by several teams. Inward migration of Jupiter from ~ 20 to ~ 5AU and getting it captured in 2:3 or 1:2 resonance with growing Saturn (which could halt and reverse the inward migration of Jupiter) was considered in the model known as the Grand Tack scenario (Jacobson & Morbidelli, 2014; Mandell, Raymond, & Sigurdsson, 2007; Walsh, Morbidelli, Raymond, O’Brien, & Mandell, 2012). It was assumed that this process would shepherd outer belts’ bodies inward and exert influence on planetary embryos in the inner region. This would result in volatile implantation through planetesimals’ impacts during terrestrial planets formation and storage of remnant pristine bodies in the main asteroid belt, as well as the position and coexistence of asteroids of different compositions (Morbidelli, Lunine, O’Brien, & Walsh, 2009; Villeneuve, Chaussidon, & Liboured, 2009). Moreover, such a scenario was invoked to explain depletion of water in the zone of terrestrial planets’ formation in terms of a “fossilized” snow line position outside Jupiter’s precursor orbit beyond the snow line at ~3 AU preventing condensation of water and ice inside (Morbidelli et al., 2015) and/or mechanism of water delivery related to giant impacts (O’Brien, Walsh, Morbidelli, Raymond, & Mandell, 2014).

Protoplanets embedded in a gaseous disc would experience drift changing their orbital positions. The movement of Uranus and Neptune from the place of their initial formation close to the Jupiter–Saturn zone further away from the Sun was assumed to exert formation of the trans-Neptunian Kuiper belt. This scenario was justified by evaluating the time required for the accretion of Uranus and Neptune in their present-day orbits. Indeed, modeling showed that such a time would exceed the age of the solar system. The model, which is sometimes referred to as reconfiguration of giant planet orbits (“Nice model,” Morbidelli, Levison, Tsiganis, & Gomes, 2005; Tsiganis, Gomes, Morbidelli, & Levison, 2005) suggested an existence of the primordial disk of several tens of Earth masses made of comet-like objects residing just outside the initial orbits of the giant planets. The disk is thought to be scattered throughout the solar system by gravitational interactions between the giant planets leading also to their migration. It is worth noting that movement of Uranus–Neptune outward due to gravitational interactions with the planetesimals would ultimately eject them into hyperbolic orbits.

The dynamical models involving instability in the giant planets zone were associated with the dramatic event of Late Heavy Bombardment (LHB) of the Moon and terrestrial planets (Gomes, Levison, Tsiganis, & Morbidelli, 2005) and the Moon crater chronology. However, timing of the instability (well before the LHB or later) underlying these models remain unclear. Most recently (Morbidelli et al., 2017), instead of using the terminal cataclysm hypothesis that poorly explained the triggering flux of bodies from the outer asteroid belt, linked the source of LHB projectiles with leftovers of planetesimals (“tail end of planet accretion”) from the time of outer planets’ formation. In support of this idea, they brought some evidence of lunar craters’ records and matching density of craters with the Highly Siderophile Elements (HSE) in the lunar mantle. These “iron loving” elements were found in chondrite meteorites and presumably they were delivered with projectiles of similar composition after a major event of the lunar matter differentiation, with the core emergence and sinking of the original HSE in the core. Hence, geochemical constraints of large craters’ composition formed by projectiles and the timescale of mantle ocean crystallization (~ 4.35 Ma, with the account for Fe-S segregation), advocate for the imprint of LHB event at the tail end of planet accretion (Morbidelli et al., 2017). The modern model coupling LHB at ~ 3.9 Ma with the time of outer planets’ instability appears to accommodate the current knowledge with minimum assumptions.

The critical role of C-chondrites and water transport in the inner solar system could be played by the process of outer planets’ formation with no change of their orbital positions (Raymond & Izidoro, 2017). This is a kind of scattering model that postulates a slow accretion of gas in the broad outer belt rich of volatiles (~5–20 AU) by the giant’s cores until some threshold with the onset of Jupiter’s runaway growth. This process provides destabilization, gravitational scattering of nearby planetesimals by Jupiter and Saturn and their large-scale radial mixing, some of them moving inward and residing (resonance) orbits’ interior to that of Jupiter. As gas dissipates, orbits of the disc planetesimals become progressively eccentric, including those trapped in the main asteroid belt and crossing the region of growing Earth and terrestrial planets, thus delivering water with C-chondrites and other type of meteorites. Later implantation of these species by scattering triggered by growth and possible migration of Saturn and icy planets turns out to be non-efficient. This scenario with no or very limited key assumptions related to the Grand Tack or Nice models seems quite plausible and implies a close relationship of both inner and outer planets’ formation.

Particles Growth and Solid Bodies Formation

The key problem of the solar system origin is how the solar system bodies (original dust and condensates, as well as those produced in coagulation) progressively grew on scales ranging from nano- and micron-size particles to planetesimals and planets thus ranging over dozens orders of magnitude in mass. As mentioned above, time-dependent modeling with the different particle size distribution functions and limited lab experiments is the only tool to gain insight into the problem. Numerous attempts to reconstruct the process taking into account nebula thermal structure; evaporation fronts (EFs’) position for different components depending on radial temperature lapse rates in the disc’s midplane; particle growth by sticking limited by bouncing; fragmentation and radial drift due to nebula headwind drag of growing bodies influencing mass redistribution; etc., have been undertaken (see, e.g., Birnstiel, Dullemon, & Brauer, 2010; Dominik, Blum, Cuzzi, & Wurm, 2007; Estrada, Cuzzi, & Morgan, 2016; Nakagawa, Sekiya, & Hayashi, 1986; Ormel, Spaans, & Tielens, 2007; Wada, Tanaka, Suyama, Kimura, & Yamamoto, 2008, 2009; Weidenschilling, 1980). Also, lab experiments with silicate and ice particles collisions in microgravity conditions revealed some important patterns of particles and particle aggregates formation. Some important constraints were deduced for particles bouncing and sticking, including translational energy and coefficients of restitution estimates depending on impact parameters (see, e.g., Beitz et al., 2011; Blum, 2004; Blum, Schrapler, Davidson, & Trigo-Rodriguez, 2006; Brisset et al., 2013; Chiang & Youdin, 2010; Güttler, Blum, Zsom, Ormel, & Dullemond, 2010; Hill, Heißelmann, Blum, & Fraser, 2015; Ida, Guilot, & Morbidelli, 2016; Lankowski, Teiser, & Blum, 2008; Schrapler, Blum, Seizinger, & Kley, 2012; Weidling, Güttler Blum, & Brauer, 2009; Weidling, Güttler, & Hium, 2011).

According to the modern views, the process started owing to the above-mentioned streaming/gravitational instability development in the dense subdisk formed out of the dust component settled down to the midplane of the gas-dust disk (Dorofeeva & Makalkin, 2004; Kolesnichenko & Marov, 2014; Makalkin & Dorofeeva, 1996; Marov et al., 2013; Youdin & Shu, 2002). This was followed by the primary porous dust clusters formation from which the first solid bodies of pebble-boulder size and eventually planetesimals of asteroid size have emerged. Some of these cm-size and larger pebbles were possibly assembled into porous clumps giving birth to comets; this approach accommodates the primordial rubble pile theory though other theory postulates that comets were made out of debris left over from the main planet-building phase, thus preserving remnants of the protosolar nebula matter. Nonetheless, collisional rubble pile theory better explains a rather small size of comet’s nucleus (~5–10 km) and even the bi-lobed shape of some of them, because of their gradual formation at low speed of leftover pebbles/grains after larger bodies have accumulated and gas in disc has disappeared, followed by gentle collisions owing to stirring the cometary orbits, specifically at the skirt of planetary system.

The sequence of solids growth and timescale of such a scenario is shown schematically in Figure 11;

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Figure 11. Sequence of the of protosolar disk evolution. (a) Disk formation due to accretion gas and dust from the protosolar nebula and protosun emergence in the center. (b) Disk flatteniand dust particle sedimentation toward the midplane and dense dust subdisc formation; particle growth. (c) Subdisc fragmentation into dust clusters due to streaming and gravity instability development, cluster collisional interaction and solid growth, including first proto-planetesimal accumulation with gravity domain. (d) Planetesimals and planetary embryo formation, gas dissipation, and original solar system architecture setup evolving ultimately to the contemporary configuration. Time span: (a)–(b) 10⁵–10⁶ yr; (c)–(d) 10⁶–10⁷ yr.

Source: Dorofeeva and Makalkin (2004).

the initial process is thought to last less than 10⁵–10⁶ years of the overall ~10⁸ years of planets formation. Note that the same short timescale is assumed for comets’ formation as well as for initial growth phase of the TNOs probably influencing the cometary-like bodies origin and evolution. The shear turbulence mechanism would promote ring-like contraction of dust in protoplanetary cloud into a thin disc (h << r) of non-regular shape widening toward the periphery. Solid bodies becoming planet embryos are formed from initially “loose” gas-dust (porous) clumps filling the main part of their sphere of attraction (Hill’s sphere) and slowly contracting due to internal gravitational forces. Let us recall that whereas the growth of particles during their collisions is hampered in chaotic turbulence, their coalescence and enlargement can occur inside turbulent eddies’ coherent structures promoting dust clusters set up in vortexes of baroclinic nature in a broad range of Stokes (St) number defined by the ratio of particle drag in an ambient gas to characteristic time of the system (defined as inverse Kepler frequency Ω‎⁻¹) and dependent on size and density of a particle. Solid particles may be also concentrated in the disks with quite weak turbulence where denser than average matter structures (with large dust-to-gas ratio) would create a large population of aggregates and trigger streaming instability. This could be the case in outer zone of massive discs where rapid growth of aggregates to planetesimals was suggested (Krijt, Ormel, Dominik, & Tielens, 2016).

Generally, both instability mechanisms support the basic Safronov and Goldreich-Ward ideas about disc viscous accretion and subdisk matter gravitational collapse. As we have seen, in the modern models, the focus is also given to the competing processes of gas and dust photo-evaporation by the solar EUV-X-ray radiation and gas condensation/condesate growth, as well as to dust trapping, clustering, and coagulation including particles size distribution and stratified turbulence. These processes are accompanied by dust aggregates formation/restructuring through particles sticking, gas-dust coupling decrease, solids growth, and radial drift resulting ultimately in gravitationally bound planetesimals and planet embryos origin (Armitage, 2007; Birnstiel, Fang, & Johansen, 2016; Carrera, Gorti, Johansen, & Davies, 2017; Johansen et al., 2014; Marov & Kolesnichenko, 2013; Raettig, Klahr, & Lyra, 2015; Schaefer, Yang, & Johansen, 2017).

Although the basic scenario of ongoing particle growth is generally understood, many details of the processes involved at the different stages remain unclear. First, still uncertain are the details of physical mechanisms responsible for primary small dust particles growth before gravitational interactions of hundred meters-kilometer size bodies become effective. The process of particles’ mutual collisions is usually invoked as the factor that presumably gave rise to the aggregation of small particles to yield either dense particle clumps (Carrera, Johansen, & Davies, 2015) or pebble/boulder-size compact aggregates migrating in the protoplanetary disc and controlling planetesimals’ growth (Güttler, Blum, Zsom, Ormel, & Dullemond, 2010; Ida, Guilot, & Morbidelli, 2016; Krijt, Ormel, Dominik, & Tielens, 2016; Nakagawa, Nakazawa, & Hayashi, 1981; Zsom, Ormel, Güttler, Blum, & Dullemond, 2010; Ormel, Spaans, & Tielens, 2007). However, a probability of destruction rather than growth of forming bodies may be higher if collisional relative velocities and bouncing barriers for particle aggregates are significant, until gravitational attraction of planetesimals becomes dominant. Hence, an efficiency of sticking of dust particles into aggregates through collisions is a rather delicate mechanism, specifically if equal size particles of ~mm size are considered (Blum & Wurm, 2000). The formation of larger objects limited by bouncing is thought to be augmented by the streaming instability, gravitational collapse, collective particle behavior, and/or by static compression of fluffy dust aggregates (Kataoka, Tanaka, Okuzumi, & Wada, 2013; Ward, 2000; Yang, Johansen, & Carrera, 2016).

As a compromise ensuring higher efficiency of agglomerates creation we advocate for collisional integration of fluffy clusters made up by micron-sized particles rather than individual particles themselves, in support of the lab experiments mentioned above. It is further assumed in the developed models that clusters and their individual particles have fractal structure. Such an approach was thoroughly explored and validated theoretically (Kolesnichenko & Marov, 2013) involving concurrence of gravitational and brownian coagulation of dust monomers, aggregates growth, and porous bodies interaction/growth.

The bottom line of our numerical model is the agglomeration of small particles in the primary fluffy dust aggregates of low-bulk density and fractal nature of the latter, giving rise to their compression and formation of first dense bodies of progressively larger size. Unlike an approach used in the previous simulations (e.g., Dominik, Blum, Cuzzi, & Wurm, 2007; Wada, Tanaka, Suyama, Kimura, & Yamamoto, 2008, 2009), the mode of particle on-head and offset collisions inside dust clusters was more strictly assessed in numerical N-body models (Marov & Rusol, 2011, 2015a, 2015b). Method of permeable particles and modified Newton model of collisions were utilized in terms of restitution coefficient Kr dependent on the distance between centers of particles and their relative velocity, taking into account internal structure of particles in fluffy clusters and complicated patterns of their interactions, specifically in the contact zone. This enabled us to examine collisional evolution of fluffy clusters beginning from nano-particles sticking together through electrostatic forces and growing up to compressed aggregates of larger size and different patterns. Example of 3D modeling of evolution of fractal dusty clusters with different numbers of populated submicron particles is shown in Figures 12 and 13.

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Figure 12. 3D-numerical modeling of submicron particles, collisional interaction, and growing within a grid in the primordial dusty cloud (density d = 1.5 × 10^‒21 g/nm³. (a) An example of the original 3D structure of a selected grid. (b) Examples of agglomerate formation in the process of collisional evolution of particles in some grids.

Source: Marov and Rusol (2015a).

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Figure 13. 3D structure of fractal dusty clusters with different numbers of populated particles evolution (a–d) Red and blue colors denote positively and negatively charged particles of about 20-nm characteristic size in a quasi-neutral medium.

Source: Marov and Rusol (2015b).

Figure 14 is a snapshot of a cross-section of randomly selected grids for fractal dusty clusters with different numbers of populated particles in Figure 13.

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Figure 14. Snapshot of a cross section of a randomly selected grid when getting through fractal dusty clusters with different numbers of populated particles (clusters a and b, respectively).

Source: Marov and Rusol (2015b).

Of special interest is the evolution of fluffy dust clusters in mutual collisions that is critical for dusty agglomerates formation, as shown in Figure 15.

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Figure 15. Evolution of the structures of fluffy dust clusters; computer modeling of collisional interactions. (a) Fluffy clusters of characteristic size 50 nm, fractal dimension 2.55, and number of particles = 8,192. (b) Fluffy clusters of characteristic size 75 nm, fractal dimension 2.15, and number of particles = 3,072.

Source: Marov and Rusol (2016).

Examples of the respective numerical experiments carried out with clusters composed of various number of submicron particles of different fractal dimensions D_β‎ ranging from 2.025 (very fluffy structure) to 2.975 (well-packed structure) are shown in Figure 16 (Marov & Rusol, 2016, in press). The results of modeling of cluster interactions reveals how K_r (restitution coefficient or its equivalent—collision recovery ratio) changes for clusters of different D_β‎ depending on the relative distance—between colliding i-j particles (0–1), and collision energy E_ij/E_int (E_ij is kinetic energy of collision; E_int—the internal energy of molecular coupling preserving structure) ranging from 0.01 to 0.99. The strong dependence of K_r on the parameters involved and are bound threshold were found, for example, for dense colliding clusters with relative velocity less than the critical value(corresponding to the energy of plastic deformation)bouncing occurs at K_r> 0.87.

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Figure 16. Restitution coefficient K_r (collision recovery ratio) for clusters with various fractal dimensions $D_{\beta }$ as shown here. Note: r_i, r_j — radii of i and j model particles; r_ij — distance between i and j colliding model particles; r_col = r_i +r_j — collision distance; $\frac{r_{ij}}{r_{col}}$ — relative distance between i and j colliding particles; $E_{int}$ — cluster’s internal energy; $E_{ij}$ — kinetic energy of collision; $\frac{E_{ij}}{E_{int}}$ — relative collision energy.

The results are regarded as an important milestone toward in-depth study of “cluster-cluster coagulation” at the initial stage of the protoplanetary gas-dust disc evolution and results are addressed as precursors of primary objects growing. They are in accord with the conclusion (Wada et al., 2008) that a cluster with a larger number of particles is harder to destroy (easier to survive) in energetic collisions. Nonetheless, the problem is still far from being solved, especially if one keeps in mind that the global qualitative effect of disk gravity further increases collision/impact velocities and adds additional jitter to the orbital evolution of the primary bodies.

The next stage of growth of primordial seeds of bodies to planetesimals, with the allowance for mechanism of self-gravitation, seems more conceivable. At the first approximation, the mass distribution of particle agglomerates and proto-planetesimals obeys the known Smoluhovsky coagulation equation (a sort of the Boltzmann kinetic equation when dealing with coagulation process) with the account for gravity mutual attraction and fragmentation in non-gentle collisions. This interaction along with the dust sedimentation onto the formed bodies residing on intersected orbits (with chaotic velocities superimposed on quasi-circular orbital velocities) results in further growth of these protoplanetary embryos followed by the gradual scooping up of smaller bodies in due course of the swarm evolution. It is supposed, however, that there is a sort of barrier at around one meter size-range (see, e.g., Nakagawa, Hayashi, & Nakazawa, 1983; Weidenschilling, 1977; Youdin & Kenyon, 2012) because of the effects of body destruction and their inward drift toward the central star. The latter is caused by aerodynamic braking of these bodies in the remaining gas whose rotation velocity becomes lower than the Keplerian one. Primordial bodies beyond the meter-size barrier to proto-planetesimals growth through “coagulation” after more complete gas evacuation seems more feasible.

Obviously, competitive processes of destruction and particle trapping during radial matter exchange, along with particle concentration via streaming and/or gravitational instability, eventually results in bodies’ growth. Based on the results of modeling supported by lab simulation (Jansson, Johansen, Syed, & Blum, 2017) millimeter- to centimeter-size pebbles in a gravitationally bound collapsed cloud (due to, for example, disk streaming instability) would experience numerous collisions with different outcomes depending on their relative speeds and the cloud density, resulting eventually in a quite intense formation of planetesimals. As far as a planet’s accumulation is concerned, gravitationally focused mutual collisions of asteroids and planetary embryos are thought to be responsible for the emergence of progressively growing proto-planets possessing larger gravitational potentials (oligarchs), such an oligarchic growth being accompanied by catastrophic fragmentation of other bodies in a swarm of planetesimals (Chambers, 2008; Greenberg, Hartman, Chapman, & Wacker, 1978; Weidenschilling, 2010;Youdin & Kenyon, 2012). A system of satellites around a proto-planet could be explained invoking the same mechanism (Canup & Esposito, 1996).

A possibility of the formation of bodies from dust clusters in mass range from 10²⁰ to 10²² g (corresponding to asteroids of tens to hundred kilometers in size) was shown in the numerical model (Marov et al., 2013). A mechanism of gravitational collapse of massive swarms of small particles and their pile-up without passing through the intermediate phases of growing was proposed to explain early formation of the ~100 km bodies, a source of very old iron meteorites, thereby circumventing the meter-size barrier problem (Cuzzi, Hogan, & Shariff, 2008; Johansen et al., 2007; Morbidelli, Bottke, Nesvorny, & Levison, 2009). Similarly, a prompt planetesimal’s formation in regions beyond the snow line due to streaming instability-induced gravity collapse of radially drifting millimeter-size dust particles to ensure material deposition for the giant planet cores was suggested (Armitage, Eisner, & Simon, 2016). Anyway, whatever particular solution of the problem, further growth occurs through interaction of planetesimals with solid and gaseous components under gravity domain and sporadic mutual collisions (Chambers, 2010).

Note that in the numerical experiments, some constraints were placed on the value of angular momentum of the porous (low density) gas-dust clusters (pre-planetesimals) of about Hill sphere in size and their moving before collision along heliocentric orbits. The model aimed to assist dynamical formation of planetary/satellite bodies with application to trans-Neptunian system. Arguments were drawn for consistency of the properties of wide binaries in the Kuiper Belt with a primordial origin during gravitational collapse (Nesvorny, Youdin, & Richardson, 2010). As the outcome, origin of the Earth-Moon system was attempted to explain in terms of the compression of a low-density cluster of 0.1 mass of the contemporary Earth formed from the collision of two original clusters on the close heliocentric orbits, which allowed the system to acquire the necessary angular momentum (Ipatov, 2014).

As it was said above, significant role at the final stage of planet’s growth could play migration and resonances in the forming planetary system. The commonly adopted scenario of the solar system origin involving planetesimals growing to planet embryos together with asteroid-size and comet-like bodies as remnants of planets is illustrated in Figure 17.

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Figure 17. Artist’s concept of a planetary system formation, as seen from the outer edge of the gas-dust disc around an emerged central protostar. Planetesimals growing to planet embryos, together with asteroid-size and cometlike bodies as remnants of planets’ formation, are seen. The picture resembles the commonly adopted scenario of the solar system origins.

Source: NASA.

It reflects our current general understanding of how original bodies and then planets formed though we are still far from resolving this mysterious subject. Some fundamental bottlenecks in planets formation challenging future studies were most recently summarized in Morbidelli and Raymond (2016).

Inner Planets Evolution

The key question intrinsically related to the origin of the terrestrial planets is why the three neighbor planets—Earth, Venus, and Mars—inheriting disc composition in a close neighborhood and basically, exhibiting similar chemistry in the formation (Saxena & Eriksson, 1986) took different evolutionary paths? Namely, what caused, unlike the pleasant climate of Earth, hostile thermal regime of Venus, and why the assumed original clement climate on Mars has drastically changed?

To answer this question, many complicated problems closely related to planetary cosmogony should be addressed. Some of them relevant to this important topic were considered in previous chapters of the paper and they are discussed in more detail in some reviews and monographs (see, e.g., Morbidelli, Lunine, O’Brien, & Walsh, 2009; Raymond, O’Brien, Morbidelli, & Kaib, 2009; Lissauer & de Pater, 2013; Armitage, 2007; Marov & Grinspoon, 1998). Here, we focus only on limited aspects of the problems involved that concerns a rather simple approach in the treatment of thermal regime formation on the inner planets and the fate of water as the key climatic factor impacting on their evolution.

Let us first note that Earth is located in a comparatively narrow zone of circumsolar space where the development of favorable (for life’s existence) climatic conditions is possible. This is the so-called habitable zone in the solar system whose inner boundary is located 10–15 million km closer to the Sun than the Earth’s orbit while its outer boundary extends to approximately the orbit of Mars (Figure 18). The orbit of Venus turns out to be outside this zone, at a distance that is almost triple the critical value. Obviously, if the Earth were moved to the place of Venus (actually, even less) then it would probably evolve according to the Venusians’ scenario.

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Figure 18. The habitable zone in the vicinity of Earth (adapted from Wikipedia). The zone is limited by only about 10–15 million km toward the Sun and extends outward to nearly the Mars orbit. The planet shifting closer to the Sun would meet runaway greenhouse similar to that on Venus (T_s = 735 K; P_s = 92 atm), whereas at Mars’s position, greenhouse gases would condense out and the atmosphere would collapse. Unlike Earth, the atmospheres of Venus and Mars consist mainly of carbon dioxide; their abundance is controlled by the carbonate-vollastonite equilibrium between atmosphere and crust, which is strongly dependent on surface temperature T_s . CO₂ storage in the Venus atmosphere is the same as what is locked in the Earth’s crust, and it would release into the atmosphere amounting to P_s ~ 90 atm if the clement climate on the home planet irreversibly changed (temperature dramatically raised) and the CO₂ balance maintenance ceased.

Based on the average annual balance of the solar radiation absorbed by the terrestrial surface and atmosphere with an account of its albedo the effective (equilibrium) temperature of Earth is T_e~255 K, which is even below the freezing point of salt water. This means that favorable climate conditions could hardly have arose on the home planet unless the greenhouse effect caused by absorption of solar radiation in its relatively thick atmosphere would raise the mean temperature near the surface to the pleasant 288 K. This atmosphere of secondary origin, unlike the primary atmosphere captured from the protosolar nebula and composed of light atmophile elements that were lost, formed owing to degassing of volatiles from the interior and from collisions with remnant planetesimals and migrating bodies. It presumably contained carbon dioxide, nitrogen, water vapor, and some minor species including such greenhouse gas like methane.

Earth could therefore retain its water on the surface, which was mainly concentrated in the oceans. In turn, carbon dioxide would accumulate incarbonates of sedimentary rocks through its binding with metal oxides incorporated into the ocean crust and upper mantle (with the formation of aqueous silicates) and biogenically, through the deposits of lime skeletons of sea organisms. Dramatic change of the Earth’s atmosphere with its modern composition occurred much later, about 2.8 Ma, due to bacterial production of oxygen accumulated in the planet’s gas envelope, which made possible the life forms rich diversity and proliferation.

In contrast, equilibrium temperature of the early Venus is supposed to be not lower than T_e~325 K, which is above the water boiling temperature in a low pressure environment. One may reasonably expect surface pressure on ancient Venus as an Earth-like plane to be substantially lower than 1 bar. Under these conditions, carbon dioxide would gradually accumulate in the atmosphere together with water vapors, which in turn, have contributed to a further rise in surface temperature. The runaway greenhouse eventually developed resulting in the transport of increasingly large amounts of СО₂ and Н₂О into the atmosphere, up to some equilibrium state corresponding temperature 735 K and pressure 92 atm. At such a temperature, carbon dioxide turned out to be not bound in carbonates of sedimentary rocks, as on Earth but was released into the atmosphere giving rise to very high pressure. According to the model estimates, the amount of carbon dioxide locked in the Earth’s sedimentary rocks is comparable to the content of СО₂ in the Venus atmosphere and it would constitute nearly the same pressure had the Earth’s temperature raised to similar temperature and СО₂ released in the atmosphere. The latter is characterized by the relationships between mineral phases and volatiles on the surface, with the carbonate-silicate interaction in the upper layer of the planetary crust being the most important. The above scenario is characteristic of a system with positive feedback and internal instability when initial perturbation is not suppressed but, on the contrary, fairly rapidly enhanced (see Marov & Grinspoon, 1998).

As far as water is concerned, its accumulation on Venus with the involvement of both endogenic and exogenic sources, likewise on Earth and Mars, would result in the initial amounts of water nearly equal to the volume of the terrestrial hydrosphere (Marov & Ipatov, 2005; Morbidelli, Lunine, O’Brien, & Walsh, 2009). However, water could not be preserved on the Venus surface at the temperature above the critical point (647 K), even as salty aqueous solutions (brines) with a slightly higher critical temperature (675–700 K). As to the atmosphere, the amount of water in it is negligible. Thus, the key problems of the planet evolution are whether water was on early Venus, where it was stored, and, if so, when and how was it lost. The deuterium-to-hydrogen ratio measured in the Venusian atmosphere, which turned out to be larger than that in the Earth’s atmosphere by two orders of magnitude, confirms the concept of ancient Venus ocean because high deuterium enrichment can be explained by an efficient thermal escape of the lighter hydrogen isotope and leaving behind heavier deuterium (Donahue, Hoffman, Hodges, & Watson, 1982). Note that adoption of this mechanism requires an assumption that enormous masses of liberated hydrogen and oxygen through dissociation by the solar EUV radiation were removed from the atmosphere by efficient dissipation and bound by surface rocks, respectively, which seems, however, hardly realistic (Marov & Grinspoon, 1998).

There is also a different viewpoint that Venus was initially formed as a “dry” planet, though the introduction of volatiles through the mechanism of heterogeneous accretion is assumed to play an important role as well. In this scenario, the water content was essentially constant throughout the geological history of Venus, remaining approximately at the current level while the efficiencies of heterogeneous accretion and hydrogen dissipation from the atmosphere were considerably lower. In such a case, the observed enrichment of the Venusian atmosphere in deuterium during the isotope fractionation in the process of their dissipation from the atmosphere can be also explained.

Addressing now the problem of Mars evolution, we start from a well-supported concept that the quite pleasant natural conditions presumably were on Mars at early epoch, which is backed by the morphological features preserved on the planet’s surface as well as by some other evidence. Since equilibrium temperature of early Mars would not exceed T_e~220 K it would be able to retain only frozen water on its surface. However, there is widespread evidence of the presence of liquid water flows meaning considerably denser atmosphere and much more favorable climate ~3.8–3.5 billion years ago. Obviously a catastrophic drought suddenly occurred on Mars rather than a long-term evolution has resulted in the present-day natural conditions. This dramatic change cannot be explained by the fact that the planet is 0.5 AU farther away from the Sun than Earth. More likely it should be associated with the fact that the mass of Mars is almost an order of magnitude lower than that of the Earth implying a lower storage and an early depletion of the former in the long-lived radiogenic isotopes, which served as the main energy sources responsible for the thermal history of a rocky planet. Mars began to cool, volcanism ceased, atmosphere diminished. In other words, a limited inventory of radiogenic isotopes rather than the distance from the Sun has dramatically impacted the Mars evolution changing its geology and causing collapse of the atmosphere. Likewise Earth and Venus, Mars probably possessed an ocean, currently its significant fraction being preserved beneath the surface as underground water (Lissauer & de Pater, 2013; Marov, 2015).

It is important to emphasize that synergy of the Earth and planetary sciences is aimed at a better understanding of the present, past, and future of the Earth based on comparative planetology approach. It is intended to answer the key questions about feedback for the existing regulation mechanisms to maintain clement climate. Concurrently, this contributes to the cardinal problems of planetary cosmogony allowing us to put more stringent constraints on the range of parameters used in modeling of the origin and evolution of the solar and other planetary systems.