SB9

   The problem of the delivery of water to the terrestrial planets was studied by several scientists (see references e.g. in [1]). Below I present the results of migration of planetesimals under the gravitational influence of 7 planets (from Venus to Neptune). In each variant of the calculations, the initial values of semi-major axes of orbits of 250 planetesimals varied from a_min to a_min+d_a with d_a=0.1 AU, their initial eccentricities equaled to e_o, and the initial inclinations equaled to e_o/2 rad. a_min varied from 3 to 4.9 AU with a step of 0.1 AU, e_o=0.02 or e_o=0.15. The calculations of the probability p_E of a collision of a planetesimal with the Earth were made similar to those in [1-3]. In [1-3], it was concluded that the total mass of water delivered to the Earth from outside Jupiter’s orbit could be about the mass of Earth’s oceans for the total mass of planetesimals beyond that orbit about 200 Earth’s masses.

   In Figs. 1-2 the values of 10⁶p_E are presented for several calculations at a_min from 3 to 4.9 AU, e_o=0.02 or e_o=0.15, and for time interval T equal to 1, 10, 100, 200, 500, and 1000 Myr. Results of calculations with T>1000 Myr (up to 5000 Myr) are presented as “end” runs in the figures. Similar plots were presented in [4] for calculations at a_min from 5 to 40 AU and d_a=2.5 AU.

   For some values of a_min and e_o, the values of p_E calculated for 250 planetesimals can differ by a factor up to several hundreds for different calculations with almost the same initial orbits. This difference can be caused by that one of thousands of planetesimals could move in an Earth-crossing orbit for millions of years. Such difference was also noted earlier in [2-3] for initially Jupiter-crossing objects. The mean time of motion of a former Jupiter-crossing object in Earth-crossing orbit was about 30 Kyr.

   For some calculations at a_min from 3 to 3.8 AU, there was a considerable growth of p_E after 10 Myr. At T=100 Myr and 3≤a_min≤4.9 AU, the values of p_E vary from values less than 10^-6 to values of the order of 10^-3 (and of 0.01 at T=1000 Myr), but they are often between 10^-6 and 10^-5, as for many our previous calculations with a_min≥5 AU. There were more runs with p_E>2×10^-5 for 3.2≤a_min≤3.3 AU, a_min=3.5 AU and 3.7≤a_min≤4.1 AU than for other values of a_min, including the values of a_min≥5 AU. The delivery of water and volatiles from such distances from the Sun was more effective (up to hundreds of times for some initial data) than from other distances. It is not clear how much material was at distances from 3 to 4 AU from the Sun, compared to that in the zone of the giant planets. If we suppose that the density of a protoplanetary disk is proportional to R^-0.5, then the ratio of the mass of material with a distance R from the Sun between 4 and 15 AU is greater by a factor of 209/7≈30 than that with R between 3 and 4 AU. For such a model, the amount of material delivered to the Earth from the outer asteroid belt can be comparable with the amount of material delivered from the zone of Jupiter and Saturn.

   For T=100 Myr, at 3.0≤a_min≤3.6 AU or a_min=4.2 AU and e_o=0.02, and also at 3.0≤a_min≤3.1 AU and e_o=0.15, more than a half of planetesimals were left in elliptical orbits. At a_min=4.2 AU planetesimals were close to the Hilda family asteroids. Planetesimals that originally crossed the orbit of Jupiter may have come to the Earth's orbit mostly within the first million years. Most of collisions with the Earth of bodies, originally located at a distance of 4 to 5 AU from the Sun, occurred during the first 10 million years. The times elapsed before the collisions of some bodies from the Uranus and Neptune zones with the Earth could exceed 20 million years. For 3≤a_min≤3.5 AU and e_o≤0.15, individual bodies could fall onto the Earth and the Moon in a few billion years. For example, p_E=4×10^-5 for a_min=3.3 AU, e_o=0.02 at 0.5≤t≤0.8 Myr (the time of the late-heavy bombardment) and p_E=6×10^-6 at 2≤t≤2.5 Myr. For a_min=3.2 AU and e_o=0.15, p_E=0.015 at 0.5≤t≤1 Myr, and p_E=6×10^-4 at 1≤t≤2 Myr. The zone of the outer asteroid belt can be one of the sources of the late heavy bombardment.  

   The ratio of the number of bodies colliding with the Earth to that with the Moon varied mainly from 20 to 40 for planetesimals from the feeding zone of the terrestrial planets [5]. For bodies arriving from distances from the Sun greater than 3 AU, this ratio was mainly in the interval from 16.4 to 17.4. So more planetesimals per mass of a celestial body collided with the Moon than with the Earth. However, at collisions of planetesimals with the Moon the fraction of ejected material was greater than that for the Earth.

   The characteristic velocities of collisions with the Moon and the Earth of bodies in calculations with a_min from 3 to 15 AU were mainly from 20 to 23 km/s and from 23 to 26 km/s, respectively. The characteristic velocities of collisions of planetesimals from the feeding zone of the terrestrial planets with the Moon varied from 8 to 16 km/s depending on initial semi-major axes and eccentricities of planetesimals.

   Studies of collisions of bodies with the Earth were carried out within the framework of the state assignment of the GEOKHI RAS No. 0137-2021-0004. Studies of collisions of bodies with the Moon were carried out with the support of the Russian Science Foundation grant No. 21-17-00120, https://rscf.ru/project/21-17-00120/.

[1] Marov M.Ya., Ipatov S.I. Solar System Research, 2018, 52, 392-400. https://arxiv.org/ftp/arxiv/papers/2003/2003.09982.pdf

[2] Ipatov S.I., Mather J.C. Annals of the NYAS, 2004, 1017, 46-65. http://arXiv.org/format/astro-ph/0308448

[3] Ipatov S.I., Mather J.C. Advances in Space Research, 2004, 33, 1524-1533. http://arXiv.org/format/astro-ph/0212177

[4] Ipatov S.I. EPSC2020-71, 2020. https://meetingorganizer.copernicus.org/EPSC2020/EPSC2020-71.html.

[5] Ipatov S.I. Solar System Research, 2019, 53, 332-361. http://arxiv.org/abs/2003.11301

How to cite: Ipatov, S.: Migration of planetesimals to the Earth and the Moon from the region of the outer asteroid belt, Europlanet Science Congress 2021, online, 13–24 Sep 2021, EPSC2021-100, https://doi.org/10.5194/epsc2021-100, 2021.

We present a semianalytical model that describes the resonance strength, width in au, location and stability of fixed points, and periods of small-amplitude librations [1]. The model is valid for any two gravitationally interacting massive bodies, and is thus applicable to planets around single or binary stars with arbitrary eccentricities and inclinations. The model assumes some simplifications that limit its validity in some specific cases, mainly for first order resonances at very small eccentricities. In spite of these simplifications the model shows as very helpful to understand the properties of the planetary resonances and its potential is revealed at high eccentricities and/or high inclinations. In particular, it is very simple to generate atlas of resonances for a given planetary system, that means, the domains of the resonances in the plane (a,e) or (a,i). In Fig. 1 we show an example for the system HD31527, where the atlas calculated with the model predicts that the two outher planets are in the resonance 16/3. When one of the bodies has negligible mass, the model reproduces the results of the restricted, asteroidal, case [2]. In this talk we also show how to apply the model and the use of the codes which are available at http://www.fisica.edu.uy/~gallardo/atlas/plares.html.

Fig. 1. Atlas of resonances (white lines) near the outher planet of the system HD31527 and corresponding MEGNO map of the region (colored background). Dark blue corresponds to more regular orbits, and lighter tones of green indicate increasingly chaotic motion. The location of the outer planet is shown with a filled yellow circle. The libration widths of the most relevant MMRs determined with the model are plotted as white lines showing that the outher planet is well placed inside the resonance 16/3 with the middle planet [1].

References

[1] Gallardo, T. et al. “Semianalytical model for planetary resonances: Application to planets around single and binary stars", Astronomy and Astrophysics, 646, A148 (2021).

[2] Gallardo, T. “Three dimensional structure of mean motion resonances beyond Neptune", CMDA 132:9 (2020).

How to cite: Gallardo, T., Beauge, C., and Giuppone, C.: Semianalytical model for planetary resonances, Europlanet Science Congress 2021, online, 13–24 Sep 2021, EPSC2021-147, https://doi.org/10.5194/epsc2021-147, 2021.

1. Introduction

We are interested in systems where two phases coexist: a solid phase with a high effective viscosity, and a low-viscosity liquid phase which fills the porosity of the solid phase. Two-phase systems, or "mush", are common in planetary sciences: magma extraction, Earth's inner core, icy moons... On terrestrial or icy bodies, such biphasic layers can compact under their own weight, which leads to the extraction of the liquid phase from the solid phase, and compaction of this solid phase. We present here a new model of two-phase flows with phase change (melting or freezing), and focus on 2 applications:

• Earth's inner core compaction: Whether the inner core crystallises dendritically or by sedimentation, it is likely that some liquid iron is trapped within the inner core, which may explain its low rigidity [1]. We extend here the studies of Sumita et al., 1996 [2] and Lasbleis et al., 2019 [3] by taking into account phase change within the inner core. 

• Transneptunian objects (TNOs): Ices may contain antifreezes like ammonia or methanol which could depress the melting point by up to 155 K [4]. Due to their presence, the melting temperature depends on their concentration. Then it is very likely that there would not be any clear border between solid and liquid if the melting point was reached, but rather a mushy layer. Several authors have studied the thermal evolution of TNOs without these considerations (for instance [5], [6], [7]).

2. Two-phase flow model

The model is based on the two-phase formalism developed by Bercovici et al., 2001 [8] for a non-reacting two-phase medium, which we generalise to allow for non-congruent phase change. This requires solving equations of conservation of mass and momentum for each phases, energy, and solute, with appropriate boundary conditions. In our model, the solid and liquid phases are assumed to be in thermodynamic equilibrium, which allows to link temperature and composition through the phase diagram. To simplify the problem, the liquidus temperature is taken to be linear (Fig. 1), which implies that the temperature in the two-phase region is a linear function of solute concentration. We further assume that the solute is considered to be totally contained in the melt (solid/liquid partition coefficient equal to 0). The set of equations is written in 1D spherical geometry. These assumptions allow us to reduce the system of equations to a system of only three equations (conservation of momentum, energy, and solute), which we solve to obtain the temperature (directly connected to composition), radial velocity of liquid (linked to velocity of solid) and porosity (the proportion of liquid).

The code solves equations in a 1D sphere, which size can evolve with time, either due to accretion (TNOs during their formation), or solidification (inner core). This is taken into account thanks to an adaptive grid. The code has been developed from the code written by Lasbleis et al., 2019 [3].

3. Preliminary results

Application to Earth's inner core

In the inner core, 'light elements', and in particular oxygen, act as antifreezes [9]. Within the inner core, the effect of pressure on the liquidus temperature cannot be neglected, and we thus take this factor into account in the model. The temperature-composition relation thus depends on pressure, or radius; the slope of the liquidus with respect to solute concentration is take to be constant but T_L decreases with decreasing pressure and increasing radius. At the inner core boundary, the temperature is set to T_L(χ₀, r). Fig. 2 and Fig. 3 show the evolution of porosity and temperature in the inner core as functions of time and radius.