Elsevier

Icarus

Volume 264, 15 January 2016, Pages 246-256
Icarus

Scaling laws of impact induced shock pressure and particle velocity in planetary mantle

https://doi.org/10.1016/j.icarus.2015.09.040Get rights and content

Highlights

  • Used hydrocode simulations to monitor shock wave propagation in a Mars type body.

  • Derived scaling laws of shock pressure and particle velocity versus distance.

  • 100–400 km diameter impactors with 4–10 km/s impact velocities are investigated.

  • Using our power law distributions, we identify three regions from the impact site.

  • The scaling law parameters are impact velocity dependent.

Abstract

While major impacting bodies during accretion of a Mars type planet have very low velocities (<10 km/s), the characteristics of the shockwave propagation and, hence, the derived scaling laws are poorly known for these low velocity impacts. Here, we use iSALE-2D hydrocode simulations to calculate shock pressure and particle velocity in a Mars type body for impact velocities ranging from 4 to 10 km/s. Large impactors of 100–400 km in diameter, comparable to those impacted on Mars and created giant impact basins, are examined. To better represent the power law distribution of shock pressure and particle velocity as functions of distance from the impact site at the surface, we propose three distinct regions in the mantle: a near field regime, which extends to 1–3 times the projectile radius into the target, where the peak shock pressure and particle velocity decay very slowly with increasing distance, a mid field region, which extends to ∼4.5 times the impactor radius, where the pressure and particle velocity decay exponentially but moderately, and a more distant far field region where the pressure and particle velocity decay strongly with distance. These scaling laws are useful to determine impact heating of a growing proto-planet by numerous accreting bodies.

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.

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      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.

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