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. 2014 Sep 13;372(2024):20130174.
doi: 10.1098/rsta.2013.0174.

Lunar and terrestrial planet formation in the Grand Tack scenario

S A Jacobson  1 , A Morbidelli  2
Affiliations

Lunar and terrestrial planet formation in the Grand Tack scenario

S A Jacobson et al. Philos Trans A Math Phys Eng Sci. .

Abstract

We present conclusions from a large number of N-body simulations of the giant impact phase of terrestrial planet formation. We focus on new results obtained from the recently proposed Grand Tack model, which couples the gas-driven migration of giant planets to the accretion of the terrestrial planets. The giant impact phase follows the oligarchic growth phase, which builds a bi-modal mass distribution within the disc of embryos and planetesimals. By varying the ratio of the total mass in the embryo population to the total mass in the planetesimal population and the mass of the individual embryos, we explore how different disc conditions control the final planets. The total mass ratio of embryos to planetesimals controls the timing of the last giant (Moon-forming) impact and its violence. The initial embryo mass sets the size of the lunar impactor and the growth rate of Mars. After comparing our simulated outcomes with the actual orbits of the terrestrial planets (angular momentum deficit, mass concentration) and taking into account independent geochemical constraints on the mass accreted by the Earth after the Moon-forming event and on the time scale for the growth of Mars, we conclude that the protoplanetary disc at the beginning of the giant impact phase must have had most of its mass in Mars-sized embryos and only a small fraction of the total disc mass in the planetesimal population. From this, we infer that the Moon-forming event occurred between approximately 60 and approximately 130 Myr after the formation of the first solids and was caused most likely by an object with a mass similar to that of Mars.

Keywords: Moon formation; Solar System formation; accretion.

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Figures

Figure 1.
Figure 1.
Each sub-panel shows planets from simulations that started from a different suite of initial conditions. The sub-panels are arranged according to initial embryo mass Me and the initial ratio between the total masses of embryos and planetesimals (ΣMe:ΣMp). Within each sub-panel, the mass of every planet from each simulation suite is shown as a function of semi-major axis. Each black dot locates the planet at its semi-major axis, and the black line shows its perihelion-to-aphelion excursion. Red (grey) dots show the real masses of Mercury, Venus, Earth and Mars, and the red (grey) lines show the width of their radial excursions. The shaded boxes highlight the regions that correspond to Mercury, Earth and Mars analogues. (Online version in colour.)
Figure 2.
Figure 2.
The angular momentum deficit Sd and concentration statistic Sc for each simulated Solar System in that suite. The dashed lines are the angular momentum deficit Sd=0.0018 and concentration statistic Sc=89.9 for the terrestrial planets in the Solar System. The sub-panels are arranged the same as in figure 1.
Figure 3.
Figure 3.
The relative late accreted mass for each Earth-like planet as a function of angular momentum deficit Sd (b) and concentration statistic Sc (a) for simulated Solar Systems. The points are from all of the Earth-like planets in the Grand Tack simulations and triangles are the values from published classical simulations [23,22]. The vertical dashed lines are the angular momentum deficit Sd=0.0018 and concentration statistic Sc=89.9 for the terrestrial planets in the Solar System [18]. The horizontal dashed line is the estimated late accreted mass from the highly siderophile elements and the uncertainty is shown in grey: MLA=1.8±0.6×10−3M⊕ [52].
Figure 4.
Figure 4.
The ratio of the impactor to total mass ratio for each last giant (Moon-forming) impact on each Earth analogue as a function of the time of that impact. The horizontal bands highlight the hypothesized lunar formation impactor to total mass ratios : equal sized colliders (top) [63], standard hit-and-run (middle) [64,65] and rapidly rotating Earth (bottom)[66]. The vertical bands highlight predicted lunar formation times from geochemical evidence: early (left) [–46] and late (right) [–52]. The sub-panels are arranged the same as in figure 1. (Online version in colour.)
Figure 5.
Figure 5.
The late accreted mass (relative to the final mass of the planet) and the time of the last giant impact is shown for every Earth analogue as a dot for Grand Tack simulations and a triangle for classical simulations [23,22]. The downward arrow represents three Earth analogues that did not accrete any planetesimals after their final giant impact at nearly 150 Myr. Restricting ourselves solely to the Grand Tack simulations, the solid staircase is the running geometric mean over time of the late accreted mass and the grey region shows its 1σ uncertainty assuming that the recorded values are distributed lognormally. The dashed staircase is the same, but including also the results from classical simulations. The horizontal dashed line and grey region indicate the late accreted mass on the Earth MLA= 4.8×10−2M⊕ determined from the highly siderophile element abundance in the Earth's mantle.
Figure 6.
Figure 6.
Each dot shows the impact velocity scaled by the escape velocity of the total mass and the impact parameter b of the last giant (Moon-forming) impact on each Earth analogue. The dots are coloured according to the impactor to total mass ratio colour code used in figure 4: equal sized colliders (orange stars) [63], standard hit-and run (magenta triangles) [64,65], rapidly rotating Earth (cyan circles) [66] and between these regions (grey squares). The polygonal regions are coloured according to the same scheme to indicate the impact characteristics postulated for each mass ratio hypothesis. The standard scenario is the lower magenta box, whereas the hit-and-run scenario is the upper magenta box. The sub-panels are arranged the same as in figure 1. (Online version in colour.)
Figure 7.
Figure 7.
The scaled growth history is shown for each Mars analogue. From Hf–W evidence, Nimmo & Kleine [71] predict with 90% confidence that Mars growth history should not pass through the dark square region and with 63% confidence that it should not pass through the light red (grey) rectangular region. Using the same evidence but assuming an exponential growth model, Dauphas & Pourmand [72] predict that Mars accretion history should evolve inside the green top left triangular region. (Online version in colour.)
Figure 8.
Figure 8.
The normalized angular momentum deficit of the entire population of protoplanetary disc objects (green or light grey curves) and the separate embryo population (black curves) and planetesimal population for each simulated Solar System. The planetesimal population is shown normalized to the circular orbits of the entire population of disc objects (red or lower set of dark grey curves) as are the embryo and entire population curves, naturally. However, the planetesimal population is also shown normalized only to the angular momentum deficit of the planetesimal population on circular orbits (blue or upper set of dark grey curves). The normalized angular momentum deficit of the terrestrial planets of the Solar System Sd=0.0018 is shown as a blue dashed line and blue or dark grey zone representing a factor of 2 about that line is also included. The green or lighter grey region underneath corresponds to a prediction made by Brasser et al. [60] for the inner Solar System before the giant planet instability, i.e. Nice model. We only have angular momentum evolution data for the Walsh et al. [31] simulations and not for the O'Brien et al. [11] simulations, so there are fewer 1:1-0.025 and 1:1-0.05 simulations. The sub-panels are arranged the same as in figure 1. (Online version in colour.)
Figure 9.
Figure 9.
The relative angular momentum deficit of the embryo to the planetesimal populations for each simulated Solar System. We only have angular momentum evolution data for the Walsh et al. [31] simulations and not for the O'Brien et al. [11] simulations, so there are fewer 1:1-0.025 and 1:1-0.05 simulations. The primary plot and the inset share identical data; however the time is shown logarithmically in the primary plot but linearly in the inset. The sub-panels are arranged the same as in figure 1.
Figure 10.
Figure 10.
The mass in the terrestrial disc (black curves), embryo population (blue or upper set of grey curves eventually joining the black curves) and planetesimal (red or lower set of grey curves) population. The sub-panels are arranged the same as in figure 1. (Online version in colour.)
See this image and copyright information in PMC

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