Atmospheric impacts, fragmentation, and small craters on Venus

Icarus 169 (2004) 287–299 www.elsevier.com/locate/icarus

Atmospheric impacts, fragmentation, and small craters on Venus D.G. Korycansky a,∗ and Kevin J. Zahnle b a CODEP, Department Earth Sciences, University of California, Santa Cruz, CA 95064, USA b 245-3 NASA Ames Research Center, Moffett Field, CA 94035, USA

Received 7 October 2003; revised 14 January 2004 Available online 12 April 2004

Abstract We use high-resolution three-dimensional numerical models of aerodynamically disrupted asteroids to predict the characteristic properties of small impact craters on Venus. We map the mass and kinetic energy of the impactor passing though a plane near the surface for each simulation, and find that the typical result is that mass and energy sort themselves into one to several strongly peaked regions, which we interpret as more-or-less discrete fragments. The fragments are sufficiently well separated as to imply the formation of irregular or multiple craters that are quite similar to those found on Venus. We estimate the diameters of the resulting craters using a scaling law derived from the experiments of Schultz and Gault (1985, J. Geophys. Res. 90 (B5), 3701–3732) of dispersed impactors into targets. We compare the spacings and sizes of our estimated craters with measured diameters tabulated in a Venus crater database (Herrick and Phillips, 1994a, Icarus 111, 387–416; Herrick et al., 1997, in: Venus II, Univ. of Arizona Press, Tucson, AZ, pp. 1015–1046; Herrick, 2003, http://www.lpi.usra.edu/research/vc/vchome.html) and find quite satisfactory agreement, despite the uncertainty in our crater diameter estimates. The comparison of the observed crater characteristics with the numerical results is an after-the-fact test of our model, namely the fluid-dynamical treatment of large impacts, which the model appears to pass successfully.  2004 Elsevier Inc. All rights reserved. Keywords: Impacts; Craters; Venus

1. Introduction All bodies in the Solar System have been subject over geologic time to impact by smaller objects. A subset of those bodies are planets (and the moon Titan) with atmospheres thick enough to substantially affect the impact and the resulting crater. Venus has the thickest atmosphere (∼ 100× that of Earth) and hence shows the strongest effects. The effects on venusian craters have been extensively studied (Phillips et al., 1991, 1992; Zahnle, 1992; Schaber et al., 1992; Ivanov et al., 1992; Schultz, 1992; Herrick and Phillips, 1994a, 1994b; Basilevsky et al., 1987; McKinnon et al., 1997). Earlier work on modeling the breakup and ablation of impacts was done by Baldwin and Schaeffer (1971), and an extensive discussion of the physics involved can be found in Bronshten (1983). Impactor fragmentation in the Earth’s atmosphere and resulting fields of multiple craters was investigated by Passey and Melosh (1980), who de* Corresponding author. Fax: 831-459-3074.

E-mail address: [email protected] (D.G. Korycansky). 0019-1035/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2004.01.010

scribed the known “strewn fields” (crater fields) on the Earth at the time of writing, and developed a fragmentation model that was the basis of much following work. In this model, an impactor is envisioned as splitting into two pieces (binary fragmentation). The interaction of bow shocks around separating fragments produces a velocity of separation due to the high pressures between the shocks. The separation velocity depends primarily on the downwards impactor velocity and the ratio of impactor density to that of the atmosphere. A coefficient of order unity in the expression for the velocity is calibrated to the observed crater fields on the Earth. Herrick and Phillips (1994b) developed a similar model for impactor fragmentation and crater formation specifically for Venus and used it to explain multiple craters and crater fields on that planet. Recently, detailed models for explicit treatment of impactor fragments were made by Artemieva and Shuvalov (2001), Bland and Artemieva (2003), and Artemieva and Bland (2003) and applied to the Earth, Venus, and Mars. In the first of these papers, numerical hydrodynamical calculations of gas flow around rigid (but separately-moving) blocks were made and the results related to the Passey–Melosh

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model, in particular, a calibration of the numerical factor in the expression for the transverse velocity of the fragments. The calculations extended to the study of multiple fragments (up to 27 in an initially cubic array), and the authors developed a stochastic fragmentation model (the “separated fragments” model) for general impacts by objects of non-zero strength that break up into smaller and typically stronger pieces due to dynamic pressure of the atmosphere. The model was applied to the Sikhote–Alin meteor shower, the Benešov meteor, and crater fields on Mars. Bland and Artemieva (2003) used the “separated fragments” model to derive impact rates for small asteroids by the atmosphere of the Earth, suggesting that stony objects are totally disrupted at much larger sizes than has been previously thought. Artemieva and Bland (2003) applied the same model to Venus in addition to the Earth and Mars. In their model, stony impactors make crater fields if their initial sizes fall in the diameter range 7–15 km, while iron objects in the range 0.5–1 km make fields. As a consequence of the relative numbers of impactors of different types in those size ranges, they propose that iron meteoroids are the source of crater fields on Venus, as they appear to be on the Earth (Passey and Melosh, 1980; Bland and Artemieva, 2003). Other calculations of meteoritic breakup in the atmosphere of the Earth and Venus include those by Ivanov et al. (1992) and Svetsov et al. (1995) who have done 2D numerical calculations by several methods. Ivanov et al. (1992) modeled Venus impacts and the resulting Venus crater sizefrequency distribution, using first a simple model in which impactors below a certain critical diameter are destroyed and make no craters; fits to the crater distribution yielded a critical diameter of 0.8 km/ sin θ , where θ is the impact zenith angle. Further calculations in that paper included Free–Lagrange (hybrid particle-grid) hydrodynamical calculations of icy bodies impacting vertically, for which it was found that a 1 km diameter body could reach the surface in a partially disrupted state and produce a crater of 22 km diameter. Svetsov et al. (1995) provide a discussion of the relevant physics, including ablation, material strength, and fragmentation. They discuss models of fragmentation, both progressive and catastrophic as well as hydrodynamic models. They give parameter ranges (impactor radius and altitude) for which various effects may be expected to predominate in the Earth’s atmosphere. Hydrodynamic calculations include those done by the Free–Lagrange method used by Ivanov et al. (1992) and an Eulerian method in which the impactor body is taken to be incompressible and its boundary tracked by a so-called volume-of-fluid method. For the hydrodynamic calculations, impacts in the Earth’s atmosphere of 100-m radius ice spheres and cylinders were made that showed the development of Rayleigh–Taylor instability in the front faces of the objects. Results were similar between the two calculation schemes. A 1 km-radius ice impactor was also studied using the Free–Lagrange method. In addition, meteoroid dispersion was studied using a “sandbag”

method in which a large number of individual particles interacting with the atmosphere were followed, and gas motion was calculated on a grid. Collective effects were present, so that the assemblage of particles moved as a gradually dispersing unit. In these calculations, large individual fragments did not tend to form, except for a case in which populations of different-sized particles were included from the outset. In this study we focus on the properties of craters made by aerodynamically disrupted bolides, treated as a continuum. The bolides are generated by detailed high resolution 3D numerical simulations. By contrast with the fragmentation models described above, our calculations assume zero-strength objects, modeling the impactors in a fluid-dynamical framework. In this model, large impactors fragment due to the growth of perturbations arising from hydrodynamic instabilities (Rayleigh–Taylor and Kelvin– Helmholtz). The model is best suited to the description of large (km-scale) objects that are capable of making craters on Venus. Aerodynamic forces will exceed the strength of competent rock at heights below ∼ 60 km in the atmosphere of Venus for impacts at 20 km s−1 , so that kilometer-scale impactors into the atmosphere of Venus (and Titan) are likely to break up into a large number of fragments (even if they are not already in “rubble-pile” state) for which a continuum fluid-dynamical treatment is likely to give a reasonable result. Likewise, for large objects, radiative effects, such as ablation (not included here) are likely to be small by comparison to dynamical instabilities (Svetsov et al., 1995; Korycansky et al., 2000, 2002). We find, as discussed below, that fragments hitting the ground are of roughly equal masses, or extend over a relatively small mass range, so that the resulting sub-craters are expected to be of similar size. A good deal of material (the majority, even) also reaches the ground in a more dispersed state, but probably cannot make discrete craters. Comparison of our results with the properties of venusian craters is a test of our model of the impact of large objects into the atmosphere of Venus. The model produces results that are compatible with the observed characteristics of small venusian craters, successfully passing the test. Databases of Venus craters can be found in the literature and online (Schaber et al., 1998; Herrick and Phillips, 1994a; Herrick et al., 1997; Herrick, 2003) including images and detailed lists of crater properties. Of particular interest to us for the present work are the classifications of craters by type and the characteristic statistical properties of each type. The two online databases are (as they should be) broadly compatible, though there are differences in detail. Schaber et al. (1998) divide craters into “irregular,” “multiple,” and other types of less concern to us, and Herrick’s database lists crater planforms including “irregular” and has a separate category for “multi-floored” and “crater fields.” Based on inspection of crater images, it would appear that Herrick’s distinction between multi-floored and crater field is made on the basis of whether the sub-craters overlap to any

Fragmentation and small craters on Venus

degree, with the “crater field” designation being assigned when the sub-craters are completely separate. Comparison of craters between the two databases shows that Herrick’s multi-floored and field categories are broadly equivalent to the multiple craters of Schaber et al. (1998), and that both put mostly the same craters into the irregular class. The distribution by size is compatible between the two listings, although there are ∼ 20 small features in Schaber et al. (1998) that are rejected by Herrick as being probably volcanic in origin rather than impact features. In addition, Herrick and Phillips (1994b) give more details about multiple craters. Table I in that paper lists the sizes and dispersions (downrange and crossrange) of individual components of multiple-floor craters and crater fields, as well as effective diameters of the features according to two different definitions. (The “effective single-object diameter” denoted SD by Herrick and Phillips (1994b) is the same as the diameter given in the online database of Herrick (2003) for crater fields only. For other categories, Herrick’s diameter is based on the area covered by the crater. There are also small differences in the listed diameters of some of the craters between Herrick and Phillips (1994b) and the database.) Other information is given, such as the probable zenith angle of the impact, which is also included in the online database. Multiple craters (both multi-floored and crater fields) are the clearest sign of fragmented impactors. Irregular craters are less so, since surface processes could have altered the shapes of small features after formation. Nevertheless, it is clear from the crater-size distributions that the objects that made small craters on Venus have been heavily affected by the planet’s atmosphere. Examples of small craters on Venus are shown in Fig. 1. Three kinds of crater are shown, as classified by Herrick (2003): irregular craters Vanessa and Urazbike (Figs. 1a, 1b), multi-floored craters Parvina and Oshalche (Figs. 1c, 1d), and the crater fields Khafiza and an unnamed feature (Figs. 1e, 1f). (The crater field in Fig. 1f is also shown as Fig. 10 of Phillips et al. (1992), which paper also includes the multi-floored crater Lilian as Fig. 9.) The features shown in Fig. 1 are typical members of their classes. The Venus crater fields differ from those on Earth in the specific respect in there are most often two (sometimes more, up to a maximum of seven) sub-craters of comparable sizes in the multi-floored craters and the fields. Crater fields on Earth such as the Sikhote–Alin field [containing over twenty craters (Krinov, 1966; Artemieva and Shuvalov, 2001)], or the fields discussed by Passey and Melosh (1980), are often dominated by one large crater, or have wider crater spacings relative to their diameters. √ Sorting craters by type and size into 2 bins shows that irregular, multiple-floor, and field craters make up the greater part in each bin up to the 8–11.3 km bin, and that ∼ 75% of craters smaller than 11.3 km are of these types. Actual multiple-floor craters plus crater fields are a minority of the small craters (≈ 90 out of ≈ 290). They are relatively more

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common at smaller sizes. A quantitative view of the number distribution of venusian craters and type is provided by Fig. 2, which is compiled from the databases. Figure 2 shows histograms of the number distribution for the various types of craters on Venus and demonstrates quite clearly how small craters tend to be irregular or multiple.

2. Calculations and analysis The calculations discussed in this paper are a subset of those described in previous papers (Korycansky et al., 2002; Korycansky and Zahnle, 2003) using the ZEUS3D and ZEUSMP hydrodynamic codes. The subset consists of highresolution runs (27 out of 34 calculations in those papers) that happened to have detailed output saved for additional analysis and are listed in Table 1. The calculations were simulations of the impacts of asteroids (shaped like 4769 Castalia) at various velocities and angles into the venusian atmosphere (except for one calculation from Korycansky et al. (2002) that was a sphere). The dimensions of the model object were 1×, 2×, or 3× that of the actual asteroid, with corresponding masses, as listed in Table 1. For a number of cases, initial velocities were varied by ±0.01 km s−1 to sample sensitivity to initial conditions. Densities of the objects were 2.7 gm cm−3 in most cases, except for several cases from Korycansky et al. (2002) with 2.4 gm cm−3 . Impactor equations of state were of the Tillotson type, except as noted. A small number (five) of the calculations involved porous objects with 50% initial porosity. The calculations are more fully described in the cited work. The fifth column of the table gives the orientation of the impactor in terms of angles ψ, χ relative to its incoming velocity vector. Column six of the table contains an estimate of the type of crater (or crater field) that might result, based on the position of fragments crossing a fiducial plane, and estimates of resulting craters. Columns seven and eight give the maximum separation Lmax [which is the same as Mt in the notation of Herrick (2003)] of the centers of the fragments that crossed a fiducial plane near the ground, and the equivalent crater diameter D that corresponds to the area covered by craters. The last column gives the maximum value for each calculation of the kinetic energy per unit area max crossing a plane near the surface. These findings are the basis of this study and will be discussed in detail in following sections. One point requiring additional description is the coordinate system of the computational grid. The grid is Cartesian and oriented at the zenith angle θ of the impactor’s trajectory. In terms of coordinates x, y, z oriented with respect to the surface, the grid coordinates x  , y  , z are given by x  = x cos θ − z sin θ, z = x sin θ + z cos θ.

y  = y, (1)

The detailed numerical output from the simulations consisted of density, material tracer, and downward velocity data

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(a) Vanessa

(b) Urazbike

(c) Parvina

(d) Oshalche

(e) Khafiza

(f) (unnamed)

Fig. 1. Examples of small craters on Venus. The cropped images are taken from the Venus crater database of Herrick (2003), with contrast enhanced for reproduction. The classifications follow that of Herrick and Phillips (1994b). Panels (a) and (b): two craters classed as single, irregular craters, Vanessa (lat = −6◦ , long = 1.9◦ ), 12.5 km diameter, and Urazbike (lat = −9◦ , long = 202.5◦ ), 6.7 km diameter. Panel (c): multiple-floor crater Parvina (lat = −62.1◦ , long = 153◦ ), D = 7.4 km. The two sub-craters are 6.6 and 3 km in diameter. Panel (d): multiple-floor crater Oshalche (lat = 29.7◦ , long = 155.5◦ ), D = 9.6 km. The two sub-craters are 8.3 and 6.7 km in diameter. Panel (e): crater field Khafiza (lat = 29.7◦ , long = 155.5◦ ), D = 7.2 km. The two sub-craters are 6.3 and 5.5 km in diameter. Panel (f): an unnamed group at (lat = −46.3◦ , long = 126◦ ), classified as a crater field. The individual members have diameters of 3.8, 3.3, 2.8, and 2 km.

(in the grid coordinate system) saved in time-slices at intervals of 0.1 sec (usually). We have used the data to compute time-integrated fluxes of mass and momentum, as functions of the “horizontal” coordinates (x  , y  ) on the grid, for vari-

ous values of the (grid) “vertical” coordinate z . Not all the impactors were massive enough to deliver material to the ground (or the z = 0 level that we take as a proxy). We have no examples of objects impacting at angles greater than 45◦

Fragmentation and small craters on Venus

Fig. 2. Size distributions of craters on Venus, from the databases of Schaber et al. (1998) and Herrick and collaborators (Herrick and Phillips, 1994b; Herrick et al., 1997; Herrick, 2003). Top panel: Histograms of crater types. Solid triangles are data from the database of Schaber et al. (1998), and open squares refer to data from Herrick (2003). Solid lines: total numbers √ of craters of all types, plotted as number of craters N vs. diameter D in 2 bins. Dotted lines: histogram of multiple + irregular craters (Schaber) and crater fields + multiple-floor + irregular craters (Herrick). Dashed lines: multiple craters only (Schaber) and crater fields + multiple-floor craters (Herrick). Bottom panel: fractions of total craters in each bin, from Herrick’s database only. Solid line: fraction of total that are crater fields + multiple-floor + irregular craters. Dotted line: fraction of total that are crater fields + multiple-floor craters. Dashed line: fraction total that are crater fields.

that also strike the ground (our sole 60◦ example stops at z ≈ 10 km). The mass flux µ(z0 ; x  , y  ) and momentum flux λ(z0 ; x ,  y ) at given grid height z0 are given by the integrals
 µ z0 ; x , y   


= − ρ z0 ; x  , y  C z0 ; x  , y  vz z0 ; x , y  dt,
 λ z0 ; x  , y   


= ρ z0 ; x , y  C z0 ; x  , y  vz2 z0 ; x  , y  dt, (2) where the integration over time is done at a fixed gridcoordinate height z0 . The quantities inside the integrals are the material density ρ, a tracer 0 < C < 1 that gives the fraction of matter that is due to the impactor, and the

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velocity vz in the z direction. From these, we can calculate the kinetic energy flux at z0 = 0, (z0 ; x  , y  ) = 1/2λ(z0 ; x  , y  )2 /µ(z0 ; x  , y  ). Figure 3 shows (z = 0; x , y  ) for a selection of runs, along with our estimation of possible resulting craters. The crater locations and sizes were calculated by the method described below. Most of the simulated impactors were of 2X Castalia size (dimensions ∼ 3.2 × 2.0× 1.6 km, mass 1.45 × 1016 gm), which is roughly the critical mass and dimensions of an object that just reaches the ground; such objects are just right for examining break-up and multiple crater formation. Contours of  are shown on a linear scale keyed to the maximum value max for each calculation, listed in Table 1. Larger objects tend to break up less, and also deliver so much mass and energy to the ground, that the resulting craters are much larger than the size of the “footprint.” Thus, any traces of fragmentation that might have occurred are obscured. Smaller objects are stopped or completely disrupted, so little material reaches the ground. However, they may be responsible for the radar-dark “splotches” that are seen in the data (Zahnle, 1992). As discussed at length in Korycansky et al. (2000, 2002), the inherent chaos (sensitivity to initial conditions) of the hydrodynamics produces a wide variety of results. This variety is enhanced by the greater number of degrees of freedom, so to speak, for craters: sizes and placement of craters are additional quantities that can vary to produce widely divergent outcomes. At this writing we have not yet developed a systematic way to characterize crater patterns, so we concentrate on the most basic and salient points. The main result is that the impactor does indeed tend to fragment into several lumps of comparable size, at least as far as ground impact is concerned. In our previous work (Korycansky and Zahnle, 2003) we did not apply this particular analysis and therefore missed this feature of the results. We looked for separated fragments on the grid, but found the mass of ejecta largely formed a connected single cloud of debris, albeit one with high-density concentrations inside it. The new results are compatible with that finding. However, this analysis makes it clear that within such a cloud, separated lumps are present, and are in fact sufficiently distinct to form multiple or irregular craters on the ground. The energy deposition peaks have a typical spacing of a few kilometers, comparable to the separations of multiple craters as listed in the table and analysis of Herrick and Phillips (1994b). There are typically one to half a dozen such peaks. 2.1. Mass and energy across the impact plane Having calculated µ and , we can characterize the resulting crater field. In the absence of a rigorous formulation for results of a dispersed impact, we adopt the following algorithm: (1) Find the maximum peak of (z = 0; x  , y  ).

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Table 1 3D calculations discussed in this paper Run

Size

θ

v0

χ, ψ

Type

Lmax

D

max

A1 A2 A3 A4 B1 B2 B3 B4 B5 B1p B2p B3p B4p B5p C1 D1 D2 E1 G2 H1 J1 J2 J3 J4 J5 M1 N1

2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 2X 1X 1X 1X 1X 1X 3X 3X

0 0 0 0 45 45 45 45 45 45 45 45 45 45 60 0 0 45 0 45 0 0 0 0 0 0 45

20 20 20 20 20 20.01 20.02 19.99 19.98 20 20.01 20.02 19.99 19.98 20 40 40 40 10 10 20 20 20 20 20 20 20

50, 140 50, 140 0, 0 150, 140 50, 140 50, 140 50, 140 50, 140 50, 140 50, 140 50, 140 50, 140 50, 140 50, 140 50, 140 0, 90 0, 90 50, 140 90, 0 90, 0 0, 0 – 0, 0 0, 90 90, 0 50, 140 50, 140

I I I I F S M I S F F I F I – I I M M F – F F F – I I

5.8 4.4 5.0 2.2 9.2 0.0 6.6 5.1 0.0 5.9 10.9 2.0 11.1 1.6 – 5.7 4.6 8.0 5.0 6.9 – 5.4 3.0 6.9 – 8.8 9.6

10.6 14.7 13.9 9.4 7.2 3.9 7.8 6.9 7.8 4.8 5.6 6.6 4.3 4.3 – 16.3 17.7 7.8 8.5 4.9 – 3.1 2.6 2.1 – 20.6 17.5

3.4 × 1017 1.0 × 1018 8.7 × 1017 5.7 × 1017 2.3 × 1016 1.0 × 1016 2.9 × 1016 1.1 × 1017 1.8 × 1017 9.5 × 1015 5.0 × 1015 1.0 × 1017 3.8 × 1015 2.8 × 1016 0.0 1.6 × 1018 2.6 × 1018 2.6 × 1016 8.0 × 1016 5.2 × 1016 4.3 × 109 5.2 × 1015 4.9 × 1014 2.9 × 1014 7.8 × 1012 2.0 × 1018 9.4 × 1017

Calculations B1p–B5p were done with object porosity  = 0.5. Calculations J2–J5 are from Korycansky et al. (2000); ρ0 = 2.4 gm cm−3 with Murnaghan EOS. Kinetic energy/unit area “footprints” for runs A1, B3, E1, G2, J2, and N1 are shown in Fig. 3. Column definitions: θ : impact zenith angle (degrees); size: 1X, 2X, 3X dimensions of asteroid 4769 Castalia (≈ 1.6 × 1.0 × 0.8 km); v0 : impactor velocity (km s−1 ); χ , ψ : orientation angles of impactor relative to head-on (degrees); type: I = irregular, F = field, M = multiple, S = single, − = no crater; Lmax : maximum separation of crater centers (km); equivalent to Mt in Herrick (2003); D: equivalent diameter of crater of crater group (km); max : maximum kinetic energy per unit area passing through z = 0 plane (erg cm−2 ).

(2) Choose a radius q for the size of a region around the peak (e.g., 1 km) and a trial centroid x0 , y0 . We chose q on a basis of inspection of the plots of µ and  for each run like those shown in Fig. 3. (3) Integrate the total mass m = µ dA and energy E =    dA over the circular region A = πq 2 centered on x0 , y0 . The radius of this region will be held fixed during the iteration of the integrations done in the next step. (4) Likewise, calculate a characteristic radius for the region  the radial moment r = (3/2)E −1 × A  by calculating 2  esti[(x − x0) + (y − y0 )2 ]1/2  dA and an improved  −1 x   dA , = E mate for the centroid of the region: x 0  y0 = E −1 y   dA . (The factor 3/2 in the radial moment is chosen so that a uniform“top-hat” distribution of  over a circle of radius r0 would give r = r0 .) (5) Iterate the previous two steps until convergence, particularly for x0 , y0 . (6) Repeat steps 2–5 for other peaks of . After finishing, the result is a number of centers in the x  , y  plane with associated values of m, E, and impact region

diameters d = 2r. We use these to estimate the sizes of craters using the following crater scaling rule. 2.2. Crater scaling for dispersed impactors In this section we describe a simple scaling relation for craters made by dispersed impactors. In this we are guided by the results found by Schultz and Gault (1985) for dispersed clusters impacting pumice and sand targets. We begin with a relation for the mass M excavated by an impactor of mass m, velocity v, impact angle θ , and diameter d, in the gravity regime:  2 α v M =C cos θ. (3) m gd In this relation, g is the gravitational acceleration and C and α are constants that are determined empirically. Schultz and Gault (1985) find that experimental results for a cluster of small fragments are fit quite well by a relation of this type, if d is taken to be the diameter of the dispersed cluster of fragments. Writing M = ρt V , where ρt is the target density

Fragmentation and small craters on Venus

293

Fig. 3. “Footprints” of energy/unit area  passing through the x  , y  plane perpendicular to the impact for runs A1, B3, E1, G2, J2, and N1, as listed in Table 1, plus inferred craters (shown as circles) as estimated from the scheme given in Section 2 and the crater scaling Eqs. (6) and (7). The x and y scales are in km. Ten contour levels are shown for , running linearly from 0.05max to 0.95max , where max is listed in Table 1. Note the difference in xy scale from panel to panel. The calculation in run A1 is a 2X Castalia model (mass 1.45 × 1016 gm) impacting at zenith angle 0◦ at 20 km s−1 . Run B3 is a 2X Castalia model impacting at zenith angle 45◦ at 20 km s−1 . Run E1 is a 2X Castalia model impacting at zenith angle 45◦ at 40 km s−1 . Run H1 is the same mass Castalia model impacting at 10 km s−1 and zenith angle 45◦ . Run J2 is the calculation of a sphere of density ρ = 2.4 gm cm−3 , impacting at zenith angle 0◦ at 20 km s−1 . Run N1 is a 3X Castalia model (mass 4.89×1016 gm) impacting at 20 km s−1 , and zenith angle 45◦ .

and V the crater volume, and denoting E = 1/2mv 2 we have V =C

  m1−α 2E α cos θ. ρt gd

(4)

We take the crater to be a shallow paraboloid of diameter D and depth-to-diameter ratio B, so that V ∝ BD 3 . Schultz and Gault (1985) find B ∼ (ρi /ρt )β , where ρi is the cluster (dispersed impactor) density. Writing ρi ∼ m/d 3 gives   m[1−(α+β)]/3 2E α/3 β−α/3 D=C d (cos θ )1/3 . (1−β)/3 g ρt

(5)

Scaling relations derived from Schmidt and Housen (1987) give α = 0.651, C = 1.30 (in cgs units) for the transient crater diameter. Schultz and Gault (1985) find β = 0.04 for sand and 0.38 for pumice. Inserting the sand value for β and the normalization from Schmidt and Housen (1987), we arrive at the following crater scaling rule: D = 1.3

  m0.103 2E 0.217 −0.177 d (cos θ )0.333 . g ρt0.32

(6)

In the above rule, all the units are cgs. We also include a correction for relaxation of transient craters to a final diameter

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Fig. 4. Top: “Footprints” of energy/unit area  passing through the x  , y  plane perpendicular to the impact for runs B1, B2, and B5, shown for reference. Scales and contour levels as the same as in Fig. 3 Bottom: crater diameter Df plotted against integration parameter q, showing the sensitivity of Df as a function of q. Curves labeled B1a, B1b, and B1c correspond to the peaks labeled a, b, and c in the B1 plot at top. Curves labeled B2 and B5 correspond to the peaks in the corresponding plots at top. “×” symbols show the choices for q and the corresponding crater diameters that were chosen for the entries in Table 1 and the ensuing analysis in the paper.

(Zahnle et al., 2003):  D, D < Dc , Df = D(D/Dc )ξ , D > Dc ,

(7)

where ξ = 0.13 and Dc = 2.5 km for Venus. 2.3. Some results As mentioned above, Fig. 3 also shows craters from a representative selection of the impact simulations. Figure 4 shows three additional examples in connection with tests of the sensitivity of the craters diameters with respect to choices of q. We see what would probably result in single, irregular, multi-floored craters, and crater fields. A key point is the pattern of craters and their sizes look quite a lot like the multi-floored craters and crater fields on Venus, as seen in the examples of Fig. 1. Whether a given body would create a single or a multiple crater is a complicated function of its mass (and other parameters) and its breakup history. Sufficiently massive objects would not break up at all, and so create classical single circular craters (unless impacting at very oblique angles). Somewhat smaller objects would break up before impact, but still deliver enough mass and energy to the ground to

make a single crater. If the separations of the impact points were much smaller than the sizes of individual sub-craters, the appearance of the final crater would be little changed. As the sub-crater sizes decrease (for smaller impactors), or the separations increase, we would see increasingly irregular craters. Panels A1 and N1 of Fig. 3 show examples in which we might expect largely circular final craters. Transitional cases might occur for impacts like those depicted in panels E1, and G2 of Fig. 3. For smaller objects we would start to see crater fields in panels B3, J2, and B1 of Fig. 4. (Smaller impacts might also fortuitously make single craters, as seen in panels B2 and B5 of the same figure.) Finally, as we consider smaller objects, the amount of mass and energy hitting the surface is probably not enough to make craters that are larger than the “footprints” of the mass/energy deposition, and would probably not result in recognizable craters. The mass and energy contained in the peaks are considerably less than the total quantities crossing the plane. That is, the greater part of the mass, and a significant part of the energy, are distributed outside the obvious peaks that we have identified in Fig. 3. For the calculations listed in Table 1, the average fractions of mass and energy in the peaks (compared to the total) are ∼ 0.39 and 0.62, respectively. As

Fragmentation and small craters on Venus

Fig. 5. Plot of maximum separation Lmax [Mt in Herrick (2003)] of craters (i.e., fragments at z = 0) vs. single-crater diameter D for real Venus craters [as classified by Herrick and Phillips, Herrick (1994b, 2003)] and the numerical hydrodynamical calculations. Solid squares: Venus multi-floored craters as classified by Herrick (2003). Solid triangles: Venus crater fields as classified by Herrick (2003). Open symbols: Hydrodynamical results. assuming that the For the calculations, the separation Lmax is calculated √ x-positions of the fragments have been scaled by 2 for the 45◦ cases, and the maximum distance among pairs is equated to Lmax . The single-crater equivalent diameter D = (4A/π )1/2 , where A is the total area covered by craters whose centers are the fragment positions (peaks of ) and whose individual diameters are calculated according to the integration scheme and scaling rules given in the text. The three craters listed as “single” (S) are not plotted, as their values of Lmax = 0. Open squares: craters listed as “irregular” (I) in Table 1. Crosses: craters listed as “multiple”(M) in Table 1. Circles: craters listed as “fields” (F) in Table 1.

noted above, we chose the radii q of regions that included the peaks by inspecting the plots analogous to Fig. 3. This step of the analysis thus contains a somewhat subjective element that we plan to eliminate in future work when we have a more concrete formulation of how dispersed impactors create craters. We find that crater diameters Df are not highly sensitive to the choice of q, provided q is chosen to be large enough to encompass the entire peak of mass/energy deposition. We have checked this result with several cases for which the peaks are well separated. Test cases are displayed in Fig. 4. Runs B2, and B5 all had single peaks (which would create a single crater), and B1 had three well separated peaks (labeled a, b, and c in Fig. 5). We followed the procedure outlined in Section 2.1 for those five cases (B1a, B1b, B1c, B2, and B5) with a range of values of q to see how the resulting values of Df changed. As can be seen in the lower panel of Fig. 4, Df reaches an asymptotic value as q increases, as might be expected. The values that we initially chose for q by inspection yield reasonable crater diameters. Numerical details for the results depicted in Fig. 3 are given in Table 2. Each mass/energy peak has a line in the table. Entries are the radii q (in km), the mass and kinetic

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energy m and E (in gm and erg, respectively), and the effective diameter d of the region from the moment integration described above. We also list (for comparison with solid impacts) measures of density and velocity of material associated with the peaks: ρ ≈ m/d 3 , v = (2E/m)1/2. These rough values give an indication of the dispersion of the impactor fragments. We also list quantities that would result from the  combining the peaks into a single impact (the rows labeled “ ”) and the total mass and energy integrated over the entire grid (rows labeled “total”). For the peak-summed rows, we calculated an effective diameter d by summing the  approximate volumes dj3 of the fragments: d = ( j dj3 )1/3 , which value we use for the density and crater diameter estimations. For the rows containing the total mass and energy, d is calculated using the same radial-moment formula given above, calculated about the center of mass of all the material crossing the z = 0 plane. It is easy to show, using crater scaling relations like Eq. (6), that the diameter of a crater made by a uniformly dispersed “pancake” impactor increases more slowly than the diameter of the impactor itself, so that a point is reached at which the crater is smaller than the impactor. Obviously the ground trace of such an event would not be a normal crater. We do not have a very good notion at this point about what sort of craters are made by dispersed impactors that strike the ground in a peak + envelope distribution. Our calculations produce highly peaked distributions; the experiments of Schultz and Gault (1985) were done with impactors that were broken up into more-or-less uniform-density objects. Our numerical results resemble more closely the situation depicted in Fig. 6c of Schultz (1992). To investigate this question will require numerical simulations and experiments.

3. Discussion In Table 1, we have classified the resulting craters (or crater fields), based on their appearance. “Single” craters (type S) are produced in cases where only one peak reached the z = 0 plane. “Irregular” craters (type I) are cases in which the potential craters overlap by distances equal to their radii or more, so that the resulting object might look like a single, possibly irregular, crater. “Multiple” craters (type M) are cases in which individual sub-craters overlap, but by smaller distances. Finally, “crater fields” (type F) are assigned to the cases where significant non-overlapping craters might form. As noted above, the z = 0 planes are perpendicular to the impactor velocity vector, √ so that ground tracks might be elongated by a factor of 2 for the 45◦ impacts, such as cases B1–B5. We have taken that into account in making Fig. 3, in which the footprints √ have been stretched by the appropriate factor (sec θ = 2 ) for the oblique impact cases. In addition, the angular factor (cos θ )1/3 has been introduced into the crater scaling Eq. (6) for those cases.

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Table 2 Details of selected calculations Run A1

 Total B3

 Total E1

 Total G2

 Total J2  Total N1



q

m

E

d

ρ

v

Df

1.4 1 1 1 – – 1 1 1 1 1 – – 1.5 1 1 0.5 1.0 – – 1 0.5 0.5 0.5 0.5 – – 1 1 – – 1.5 1 0.5 0.5 1 – –

2.37 × 1015

1.57 × 1027

1.01 × 1015 8.63 × 1014 7.27 × 1014 4.97 × 1015 1.16 × 1016 5.10 × 1014 5.27 × 1014 1.41 × 1014 4.85 × 1014 5.92 × 1014 2.25 × 1015 6.84 × 1016 5.61 × 1014 3.16 × 1014 3.49 × 1014 7.57 × 1013 2.93 × 1014 1.59 × 1015 6.60 × 1016 1.38 × 1015 6.46 × 1014 3.21 × 1014 2.19 × 1014 3.78 × 1014 2.94 × 1015 1.02 × 1016 1.11 × 1014 9.32 × 1013 2.04 × 1014 3.84 × 1014 7.93 × 1015 4.26 × 1015 9.45 × 1014 5.82 × 1014 1.46 × 1015 1.52 × 1016 3.96 × 1016

5.95 × 1026 4.27 × 1026 3.76 × 1026 2.97 × 1027 4.30 × 1027 1.00 × 1026 1.12 × 1026 3.55 × 1025 1.06 × 1026 1.55 × 1026 5.09 × 1026 8.72 × 1026 2.14 × 1026 1.32 × 1026 1.36 × 1026 3.05 × 1025 1.39 × 1026 6.52 × 1026 1.13 × 1027 2.52 × 1026 1.29 × 1026 5.32 × 1025 4.28 × 1025 6.56 × 1025 5.43 × 1026 9.24 × 1026 8.70 × 1024 1.03 × 1025 1.90 × 1025 2.21 × 1025 7.60 × 1027 4.59 × 1027 8.66 × 1026 4.91 × 1026 7.84 × 1026 1.43 × 1028 2.19 × 1028

2.20 1.48 1.76 1.43 2.81 8.25 1.53 1.58 0.91 1.73 1.66 2.62 8.02 2.14 1.65 1.73 0.86 1.64 2.91 9.59 1.67 0.80 0.88 0.75 0.86 1.91 8.26 1.29 1.20 1.57 8.64 2.14 1.64 0.96 0.93 1.77 2.78 8.26

0.22 0.31 0.16 0.25 0.22 0.020 0.14 0.13 0.19 0.094 0.130 0.13 0.13 0.057 0.070 0.067 0.12 0.066 0.064 0.074 0.29 1.28 0.47 0.52 0.59 0.42 0.018 0.051 0.054 0.054 0.0006 0.81 0.97 1.08 0.73 0.26 0.71 0.070

11.5 10.9 9.9 10.2 10.9 8.6 6.3 6.5 7.5 6.6 7.2 6.7 1.6 8.7 9.1 8.8 9.0 9.7 9.1 1.9 6.0 6.3 5.8 6.3 5.9 6.1 4.3 4.0 4.7 4.3 3.4 13.8 14.7 13.5 12.9 10.4 13.7 10.5

9.9 7.6 6.7 6.6 12.0 11.6 4.0 4.1 2.9 3.9 4.4 6.2 6.6 4.5 4.0 4.0 2.7 4.0 6.3 6.7 6.2 5.8 4.1 3.8 4.2 8.0 7.9 2.1 2.2 2.7 2.1 14.7 13.9 8.0 6.6 7.2 17.7 17.6

Total  Rows labeled “ ” refer to mass and energy summed over the peak-values given in the preceding rows. Rows labeled “total” refer to mass and energy integrated  over the z = 0 plane. Other quantities (d, ρ, v, D) are calculated as defined in the text. Column definitions: q: radius of footprint integration region for each peak (km); m: mass of peak crossing the z = 0 plane(gm); E: kinetic energy of peak crossing the z = 0 plane(erg); d: effective diameter of peak (km); ρ: effective density = m/d 3 (gm cm−3 ); v: effective velocity = (2E/m)1/2 (km s−1 ); D: estimated diameter of crater (km).

We find that impactors of order 1–2 km in diameter (1015–1016 gm) suffer disruption and make irregular and multiple craters. This contrasts with the results of Artemieva and Bland (2003) for stony impactors, whose model evidently made crater fields only for stony impactors a thousand times more massive (1018–1019 gm, or diameters 7–15 km). Some of our inferred crater diameters are smaller than the minimum crater diameter identified on Venus (1.3 or 1.5 km), for example cases J1 and J5. In those cases, the footprint is larger than the diameter of the crater. It is likely that such cases would lead to features not identifiable as craters. When making comparisons, or assessing the populations, we need to separate different parameter sets in the runs. To

first order, the outcome of the impact of an object is controlled by the amount of atmospheric mass that it encounters. The largest homogeneous population in our runs are the groups B1–B5 and B1p–B5p, all representing a body shaped like the Asteroid 4769 Castalia, with twice the dimensions of the real object, entering at 45◦ at a speed of 20 km s−1 . The five runs in each group differed in initial velocity by increments of ±0.01 km s−1 in order to sample the sensitivity to initial conditions. The group B1p–B5p were calculations of a body of the same mass but 50% porosity (and thus twice the volume). That particular configuration is optimal for examining the possibility of multiple small-crater formation. (The fact that objects of such a mass just make it to the surface was part of our motivation for the original calculations.)

Fragmentation and small craters on Venus

We would like to make quantitative assessments of our results, compared to the craters observed on Venus. While we do that here, it should be remembered that several limitations hamper such a comparison. First, there is the question of small-number statistics and the uncontrolled nature of the calculations. We cannot be sure that our results are representative of what a large set of simulations would produce. More important, the calculations discussed here are not a representative sample of the impactor population. Probably the most important missing factor is the lack of high-zenith-angle calculations. According to Herrick and Phillips (1994b), the majority of the multiple impacts have inferred zenith angles larger than 45◦ . Our single example of a 60◦ impact produced no material that made it below z = 10 km, so we have no real idea of the craters that would be produced by large bodies impacting at high angles. We find that 2X Castalias impacting from the zenith produce mostly “irregular” craters (type I). More scatter, and some crater fields, (type F craters) are found in oblique impact cases (e.g., cases B1, B1p, B2p, and B5p). Also producing crater fields are cases in which the impactor has barely made it to z = 0, so that craters are very small, as seen in cases J2, J3, and J4. Truly separate craters are found in 8 cases out of 26, which is consistent with the observed proportion. If anything, the proportion of our crater fields is higher than actually found on Venus. However, it must be remembered that our calculations are not a complete sample of impactors, either by impact mass, velocity, or material type, so that the similarity might be fortuitous. A comparison of our results can be seen in Fig. 5, which is comparable to Fig. 7 of Herrick and Phillips (1994b). In Fig. 5 we plot the maximum separation Lmax of craters, both real and modeled, against their diameter D. The solid symbols are data from Herrick’s database (Herrick, 2003). The triangles are the crater fields, while the squares are the multifloored craters. The open symbols (squares, crosses, circles) are from our calculations. The maximum separation of the crater centers (as given by the locations of the impactor fragments, i.e., peaks of ) is plotted against effective crater diameter D. We calculated D by integrating the area covered by the craters whose sizes are determined by scheme given above. The total area A yields D from D = (4A/π)1/2 . This definition is consistent with that used for the real craters plotted in the figure. Crater type is correlated with placement on Fig. 5 by simple geometrical considerations. Crater groups for which Lmax < D will appear and be classified as multiple-floor or irregular single craters. Our intent in showing the figure is to see how the numerical results compare with real venusian craters: do the numerical craters and groups thereof fall in the same region of the diagram as the data? On the whole the overlap of the results is quite satisfactory. As we note above, multi-floored craters together with crater fields make up about 30% of small craters, with the bulk of the rest in the irregular category. We find a similar proportion of multiple craters among our calculations. The calculated

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results fall in the region where crater fields exist, although large crater fields (for which Lmax > 10 km) were not produced in our calculations. We expect that large crater fields would result from large impactors (masses of ∼ 5 × 1016 gm and larger) coming in at highly oblique angles. (One example is the crater field Marianne, for which the largest crater is ∼ 9 km in diameter, and Lmax = 27 km; according to Herrick (2003), the zenith angle of the impact fell in the range 75◦ to 90◦ .) We do not have any such highly oblique impacts in our data set, but we believe that such a calculation would produce a similar result. It is also worth noting that we assumed that the impactors were uniform in composition. Internal structure is likely to lead to greater separation among fragments and greater dynamic range in what hits the ground. As for the largest cases of multifloored craters (Wollstonecraft, Heloise, and possibly Nina), it is arguable that they were produced by binary asteroids, i.e., incoming objects that were already multiple, given the large separations and circular shape of the smaller components. One problem in our results is that there seems to be little sign of the downrange/crossrange differences found by Herrick and Phillips (1994b), who find that the downrange separation is two to four times that of the crossrange separations. For our calculations, “downrange” is the x-direction in Fig. 3, and there is no indication of systematic trends in crater placement either by crater position or size. Once again, however, we expect that simulations of highly oblique impacts would produce results that would match this particular observation. Finally, we discuss the relation of our results to fragmentation models like those of Passey and Melosh (1980) and Artemieva and Shuvalov (2001). Passey and Melosh (1980) derived the transverse velocity vt of two fragments in terms of the ratio ρa /ρi of atmospheric density to impactor density, the radii of the two fragments R1 and R2 , and the impactor’s downward velocity vz :   3 R1 ρa 1/2 C vt = (8) vz . 2 R2 ρi The coefficient C can be interpreted as the number of meteoroid radii beyond which the bow-shock interaction no longer takes place. If we assume that fragmentation occurs at a height h and leads to a spread of impacts on the ground of size δ, then (assuming constant velocities vz and vt ) we have   δ cos θ vt 3 ρa 1/2 (9) = C = , h vz 2 ρi where we have neglected systematic variations of R1 /R2 . The coefficient C is expected to be in the range 0.1–1 from numerical modeling of solid fragments (Artemieva and Shuvalov, 2001). Terrestrial crater fields yield values between ∼ 0.02 and 1.5 (Passey and Melosh, 1980). Inspection of our runs shows breakups beginning to take place at heights in the range 20 to 40 km. Taking the average Lmax = 4.5 km value

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of the group B1–B5 and B1p–B5p, and substituting for ρa at the slant heights z sec θ = 20 or 40 km, we find equivalent values of C ≈ 3 and 2.3, respectively. 4. Conclusion We have presented some further analysis of results from a number of high-resolution hydrodynamical simulations of the impacts of asteroids into the atmosphere of Venus. Our main result is that impactors evidently break up into a small number of fragments of comparable mass that separate into distinct “lumps,” whose masses, velocities, sizes, and separations are compatible with the objects that have made small irregular and multiple craters on Venus. The resulting craters have multiplicities, sizes, and separations that are similar to those that are observed on the planet’s surface. In effect, the comparisons we have made are an after-the-fact test of our model of atmospheric impact against observations. Subject to the qualifications given below, we believe the model has passed this particular test. We do not wish to push the analysis presented here too far. We have (deliberately) over-simplified many aspects of this problem, in order to elucidate a central point. In order to make further progress, a number of questions need to be specifically addressed. In particular, crater formation by dispersed impactors needs to be studied. This is a good problem for numerical simulation, and one which has not to our knowledge been studied. The results of Schultz and Gault (1985) are a good starting point for validation of numerical calculations. Direct comparison of our results with venusian craters is difficult for two reasons. First, we do not have a very good idea of how our plots of µ and  (mass and energy per unit area) translate into actual craters. This is due to a lack of knowledge about crater-making by dispersed impactors, and also because we have not fully taken into account the effects of impact obliquity with respect to the surface. Second, a comparison between impact and crater-making models requires a more systematic coverage of parameter space. In particular, calculations need to be carried out with this problem in mind rather than relying on analysis of serendipitous data left over from previous studies. The central question is: how can we turn hydrodynamic calculations into a useful statistical model? Doing this requires additional answers to questions such as: What governs the properties of the spectrum of fragments? How much mass remains in lumps that are distinct enough to make separate craters? (Some must, as the radar images show that such craters exist on Venus.) How much, by contrast, spreads out into an amorphous cloud of material that does not make craters? What effect does such material have on the ground and what sort of traces does it leave? These questions can be answered by a combination of a close examination of the Venus crater fields, (and those on Titan, when the data become available), and further experimental and numerical work.

Acknowledgments Primary support for the work in this paper was provided by NSF Planetary Astronomy grant #0098787. Additional support came from the NASA Planetary Atmospheres and Exobiology programs through Cooperative agreement NCC 2-5428. ZEUS-MP was developed at the Laboratory for Computational Astrophysics of the National Computational Science Alliance. This work was partially supported by the National Computational Science Alliance under grant numbers AST010013N. Some of the high-resolution calculations utilized the NCSA SGI/CRAY Power Challenge array at the University of Illinois, Urbana-Champaign. Other high-resolution calculations were done on the 32-node Beowulf cluster of the IGPP at UC Santa Cruz. We thank N. Artemieva and P. Thomas for helpful referee comments and suggestions about the organization of the paper.

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