Scaling laws of impact induced shock pressure and particle velocity in planetary mantle
Introduction
Small planets are formed by accreting a huge number of planetesimals, a few km to a few tens of km in size, in the solar nebula (e.g. Wetherill and Stewart, 1989, Matsui, 1993, Chambers and Wetherill, 1998, Kokubo and Ida, 1995, Kokubo and Ida, 1996, Kokubo and Ida, 1998, Kokubo and Ida, 2000, Wetherill and Inaba, 2000, Rafikov, 2003, Chambers, 2004, Raymond et al., 2006). An accreting body may generate shock wave if the impact-induced pressure in the target exceeds the elastic Hugoniot pressure, ∼3 GPa, implying that collision of a planetesimal with a growing planetary embryo can generate shock waves when the embryo’s radius exceeds 150 km, assuming that impact occurs at the escape velocity of the embryo and taking the mean density of the embryo and projectile to be 3000 kg/m3. Hundreds of thousands of collisions must have occurred during the formation of small planets such as Mercury and Mars when they were orbiting the Sun inside a dense population of planetesimal. Such was also the case during the formation of embryos that later were accreted to produce Venus and Earth. Terrestrial planets have also experienced large high velocity impacts after their formation. Over 20 giant impact basins on Mars with diameters larger than 1000 km (Frey, 2008), the Caloris basin on Mercury with a 1550 km diameter, and the South Pole Aitken basin on Moon with a 2400 km diameter are likely created during catastrophic bombardment period at around 4 Ga. The overlapping Rheasilvia and Veneneia basins on 4-Vesta are probably created by projectiles with an impact velocity of about 5 km/s within the last 1–2 Gyr (Keil et al., 1997, Schenk et al., 2012).
The shock wave produced by an impact when the embryo is undifferentiated and completely solid propagates as a spherical wave centered at the impact site until it reaches the surface of the embryo in the opposite side. Each impact increases the temperature of the embryo within a region near the impact site. Because impacts during accretion occur from different directions, the mean temperature in the upper parts of the embryo increases almost globally. On the other hand, the shock wave produced by a large impact during the heavy bombardment period must have increased the temperature in the mantle and the core of the planets directly beneath the impact site, enhancing mantle convection (e.g. Watters et al., 2009, Roberts and Arkani-Hamed, 2012, Roberts and Arkani-Hamed, 2014), modifying the CMB heat flux which could in turn favor a hemispheric dynamo on Mars (Monteux et al., 2015), or crippling the core dynamo (e.g. Arkani-Hamed and Olson, 2010a).
The impact-induced shock pressure inside a planet has been investigated by numerically solving the shock dynamic equations using hydrocode simulations (e.g. Pierazzo et al., 1997, Wünnemann and Ivanov, 2003, Wünnemann et al., 2006, Barr and Citron, 2011, Kraus et al., 2011, Ivanov et al., 2010, Bierhaus et al., 2012) or finite difference techniques (e.g. Ahrens and O’Keefe, 1987, Mitani, 2003). However, these numerical solutions demand considerable computer capacity and time and are not practical for investigating the huge number of impacts that occur during the growth of a planet. Hence, the scaling laws derived from field experiments (e.g. Perret and Bass, 1975, Melosh, 1989) or especially from hydrocode simulations (Pierazzo et al., 1997) are of great interest when considering the full accretionary history of a planetary objects (e.g. Senshu et al., 2002, Monteux et al., 2014) or when measuring the influence of a single large impact on the long-term thermal evolution of deep planetary interiors (e.g. Monteux et al., 2007, Monteux et al., 2009, Monteux et al., 2013, Ricard et al., 2009, Roberts et al., 2009, Arkani-Hamed and Olson, 2010a, Arkani-Hamed and Ghods, 2011). Although the scaling laws provide approximate estimates of the shock pressure distribution, their simplicity and the small differences between their results and those obtained by the hydrocode simulations of the shock dynamic equations (that are likely within the numerical errors that could have been introduced due to the uncertainty of the physical parameters used in the hydrocode models) make them a powerful tool that can be combined with other geophysical approaches such as dynamo models (e.g. Monteux et al., 2015) or convection models (e.g. Watters et al., 2009, Roberts and Arkani-Hamed, 2012, Roberts and Arkani-Hamed, 2014).
During the decompression of shocked material much of the internal energy of the shock state is converted into heat leading to a temperature increase below the impact site. The present study focuses on deriving scaling laws of shock pressure and particle velocity distributions in silicate mantle of a planet on the basis of several hydrocode simulations. The scaling laws of Pierazzo et al. (1997) were derived using impact velocities of 10–80 km/s, hence may not be viable at low impact velocities. For example, at an impact velocity of 5 km/s, comparable to the escape velocity of Mars, the shock pressure scaling law provides an unrealistic shock pressure that increases with depth. Here we model shock pressure and particle velocity distributions in the mantle using hydrocode simulations for impact velocities of 4–10 km/s and projectile diameters ranging from 100 to 400 km, as an attempt to extend Pierazzo et al.’s (1997) scaling laws to low impact velocities and reasonable impactor radii occurring during the formation of terrestrial planets. Hence, on the basis of our scaling laws it is possible to estimate the temperature increase as a function of depth below the impact site for impact velocities compatible with the accretionary conditions of terrestrial protoplanets. These scaling laws can easily be implemented in a multi-impact approach (e.g. Senshu et al., 2002, Monteux et al., 2014) to monitor the temperature evolution inside a growing protoplanet whereas it is not yet possible to adopt hydrocode simulations for that purpose.
The hydrocode models we have calculated are described in the first section, while the second section presents the scaling laws derived from the hydrocode models. The concluding remarks are relegated to the third section.
Section snippets
Hydrocode models of shock pressure distribution
The huge number of impacts during accretion makes it impractical to consider oblique impacts. Not only it requires formidable computer time, but more importantly because of the lack of information about the impact direction, i.e. the impact angle relative to vertical and azimuth relative to north. Therefore, we consider only head-on collisions (vertical impact) to model the thermo-mechanical evolution during an impact between a differentiated Mars size body and a large impactor. We use the
Shock pressure and particle velocity scaling laws at low impact velocities
A given hydrocode simulation may take on the order of 48 h to determine a 2D shock pressure and particle velocity distributions in the mantle of our model planet. The impact velocity is about 4 km/s for a protoplanet with a radius of 2860 km and mean density of 3500 kg/m3, assuming that impacts occur at the escape velocity of the protoplanet. Mars is more likely a runaway planetary embryo formed by accreting small planetesimals and medium size neighboring planetary embryos. This indicates that the
Conclusions
We have modeled the shock pressure and particle velocity distributions in the mantle of a Mars size planet using hydrocode simulations (iSALE-2D) for impact velocities of 4–10 km/s and projectile diameters ranging from 100 to 400 km. We have extended Pierazzo et al.’s (1997) scaling laws to low impact velocities and also considered large impactor radii occurring during the formation of terrestrial planets. We propose three distinct regions in the mantle: a near field region, which extends to 1–3
Acknowledgments
This research was supported by Agence Nationale de la Recherche (Oxydeep decision No. ANR-13-BS06-0008) to JM, and by Natural Sciences and Engineering Research Council (NSERC) of Canada to JAH. We gratefully acknowledge the developers of iSALE (www.isale-code.de), particularly the help we have received from Gareth S. Collins. We are also grateful to the two reviewers for very helpful suggestions.
References (57)
- et al.
Impact on the Earth, ocean and atmosphere
Int. J. Impact Eng.
(1987) - et al.
Could giant impacts cripple core dynamos of small terrestrial planets?
Icarus
(2011) - et al.
Scaling of melt production in hypervelocity impacts from high-resolution numerical simulations
Icarus
(2011) - et al.
The origin of the Moon and the single-impact hypothesis III
Icarus
(1989) Planetary accretion in the inner Solar System
Earth Planet. Sci. Lett.
(2004)- et al.
Making the terrestrial planets: N-body integrations of planetary embryos in three dimensions
Icarus
(1998) - et al.
Numerical modeling of heating in porous planetesimal collisions
Icarus
(2010) - et al.
Orbital evolution of protoplanets embedded in a swarm of planetesimals
Icarus
(1995) - et al.
On runaway growth of planetesimals
Icarus
(1996) - et al.
Oligarchic growth of protoplanets
Icarus
(1998)
Formation of protoplanets from planetesimals in the solar nebula
Icarus
Impacts onto H2O ice: Scaling laws for melting, vaporization, excavation, and final crater size
Icarus
A model of metal-silicate separation on growing planets
Earth Planet. Sci. Lett.
Dynamics of core merging after a martian mega-impact
Icarus
Can large icy Moons accrete undifferentiated?
Icarus
Giant impacts, heterogeneous mantle heating and a past hemispheric dynamo on Mars
Phys. Earth Planet. Int.
A re evaluation of impact melt production
Icarus
High-resolution simulations of the final assembly of Earth-like planets, Terrestrial accretion and dynamics
Icarus
A multi-phase model of runaway core-mantle segregation in planetary embryos
Earth Planet. Sci. Lett.
Impact-induced mantle dynamics on Mars
Icarus
Some recent advances in the scaling of impact an explosion cratering
Int. J. Impact Eng.
Accumulation of a swarm of small planetesimals
Icarus
Numerical modelling of impact crater depth-diameter dependence in an acoustically fluidized target
Planet. Space Sci.
Magnetic crust of Mars
J. Geophys. Res.
Giant impact stratification of the martian core
Geophys. Res. Lett.
Giant impacts, core stratification, and failure of the martian dynamo
J. Geophys. Res.
The effect of target properties on crater morphology: Comparison of central peak craters on the Moon and Ganymede
Meteorit. Planet. Sci.
Cited by (11)
Thermal state of earth's mantle during accretion
2022, Physics of the Earth and Planetary InteriorsCitation Excerpt :Decompression behind the shock wave converts the energy into heat. The shock is strongly attenuated as it propagates in the core and becomes extremely weak when it re-emerges into the mantle antipodal to the impact site (Monteux and Arkani-Hamed, 2016). The core thus provides a “shadowing” effect and the mantle on the far side is not heated significantly (Arkani-Hamed and Ivanov, 2014).
Shock wave propagation in layered planetary interiors: Revisited
2019, IcarusCitation Excerpt :The ANEOS type equations of state were used for the dunitic mantle rocks and the iron core. In the present study, we adopt a new technique based on hydrocode models by Monteux and Arkani-Hamed [2016] while deriving a new scaling law for the iron core to assess shock pressure and the shock-related temperature increase in the entire Mars type planet. A given shock front propagates from the impact site at the surface of the planet down to the CMB.
Globally smooth approximations for shock pressure decay in impacts
2017, IcarusCitation Excerpt :The more promising fitting models for p(r), i.e., the inverse-r and the arc cotangent models, can be combined with these parameter fits into a tentative general expression in which the corresponding coefficients a, b, and n in Eqs. (2a), (2c) and (3c) are functions of v of the form Eqs. (5b) and (5c), respectively. These fits are shown in Fig. 4, along with the general fitting formulae by Monteux and Arkani-Hamed (2016) and by Pierazzo et al. (1997) (in modified form). On a general note, all datasets indicate that the numerical models approach the impedance-match solution at r → 0 the better the higher the velocity of the impactor is; this was already noticed by Ahrens et al. (1977), who suggested that it may be a numerical effect, namely a consequence of the shorter timesteps in models with greater v. Another conspicuous feature is the fact that the transition between the near and the far field seems to become sharper and the far-field slope becomes steeper as v increases; this fact is reflected by the v dependence of the exponent n in the model functions.
Large Impacts onto the Early Earth: Planetary Sterilization and Iron Delivery
2022, Planetary Science Journal