REVIEWS OF GEOPHYSICS AND SPACE PHYSICS, VOL. 19, NO. 1, PAGES 1-12, FEBRUARY 1981 Impact Cratering: The Effect of Crustal Strengthand Planetary Gravity JOHN D. O'KEEFE AND THOMAS J. AHRENS SeismologicalLaboratory, California Institute of Technology,Pasadena, California 91125 Upon impactof a meteoritewith a planetarysurface the resultingshock wave both 'processes'the ma- terial in the vicinity of the impact and setsa larger volume of material than was subjectedto high pres- sureinto motion. Most of the volume which is excavatedby the impact leavesthe crater after the shock wave hasdecayed. The kineticenergy which hasbeen deposited in the planetarysurface is convertedinto reversibleand irreversiblework, carriedout againstthe planetarygravity field and againstthe strengthof the impactedmaterial, respectively. By usingthe resultsof compressibleflow calculationsprescribing the initial stagesof the impact interaction(obtained with finite differencetechniques) the final stagesof cra- tering flow along the symmetryaxis are described,using the incompressibleflow formalismproposed by Maxwell. The fundamentalassumption in this descriptionis that the amplitude of the particle velocity field decreaseswith time as kinetic energyis convertedinto heat and gravitationalpotential energy. At a given time in a sphericalcoordinate system the radial velocity is proportional to R -z, where R is the radius (normalizedby projectilevelocity) and z is a constantshape factor for the duration of flow and a weak function of angle. The azimuthal velocity, as well as the streamlines,is prescribedby the in- compressibilitycondition. The final crater depth (for fixed strength Y) is found to be proportional to Ro[2(z+ l)Uor2/g]l/(z+l), whereUor is the initial radialparticle velocity at (projectilenormalized) radius P,o,g is planetarygravity, and z (which varied from 2 to 3) is the shapefactor. The final craterdepth (for fixedgravity) is alsofound to be proportionalto [PUor2/Yz]l/(z+•), where p and Y are planetarydensity and yield strength,respectively. By usinga Mohr-Coulomb yield criterionthe effectof varying strength on transientcrater depth and on craterformation time in the gravity field of the moon is investigatedfor 5-km/s impactorswith radii in the 10-to 107-cmrange. Comparison of craterformation time and maxi- mumtransient crater depth as a functionof gravityyields dependencies proportional to g-O.SSand g-O.•9, respectively,compared to g-O.61sand g-O. 165 observed by Gault and Wedekindfor hypervelocityimpact cratersin the 16- to 26-cm-diameterrange in a quartz sand (with Mohr-Coulomb type behavior) carried out overan effectivegravity range of 72-980 cm/s2. 1. INTRODUCTION at sizesand time scalesfor which gravity does not affect the In order to obtain the relative ages of cratered surfaces flow. Theoretically basedscaling laws for relating final crater formed on different terraneson a singleplanet [e.g., Neukum, dimensionsto target and projectiledensities and projectileki- 1977], as well as to obtain meaningful interplanetary com- netic energyor momentum(or a combinationof these),plan- parisons[e.g., Gault et al., 1975;Neukum and Wise, 1976], the etary gravity, sound speed,and material strengthhave been effect of planetary crustal strengthand gravity on the dimen- developed on the basis of laboratory and field data [e.g., sionsof cratersmust be understood.Because the largestman- Chabai, 1965;Gault, 1973;Gault et al., 1975;O'Keefe and Ah- made impact craters(produced on the moon by empty rocket rens, 1978]. casings)are only some40 m in diameter [Whitaker, 1972] and In the presentpaper we examine how maximum transient the largestnuclear explosions(formed in wet coral sand) are crater depthis relatedto projectileenergy for a rangeof plan- only 1.8 km in diameter [F'aile, 1961], the experimental data etary gravities and crustal strengthconditions. Moreover, we available are limited, and theoretical estimates must be used examine how the scalinglaws themselvesdepend on gravity to understandcrater populations. and target strength.Although, in principle, sucha studycould There have been a number of studiesentailing the analyti- be carried out by using only the finite difference numerical cal and numerical prediction of crateringphenomena and the methodspreviously applied to meteoriteimpact crateringon scalingof laboratory experimentaldata to planetary cratering. the earth and the terrestrialplanets [Bjork, 1961;O'Keefe and In order to test theories [e.g., Chabai, 1965] of scaling of the Ahrens, 1975, 1976, 1977a, b, 1978; Swift, 1977; Bryan et al., sizesof explosive(and impact) cratersin a variety of gravita- 1978],this would be computationallyuneconomical. The ap- tional fields, centimeterscale explosion cratering experiments proach taken is (1) to establishthe validity for impact crater- have been carried out under conditions both less than the ing of the Maxwell z model,which was previously applied t6 earth's gravity field [Johnsonet al., 1969;Gault and Wedekind, explosivecratering by Maxwell [1973, 1977]and Maxwell and 1977] and greater than the earth's gravity field [F'ictorovand Seifert [1976], (2) to compute the partitioning of impact en- Stepenov,1960; Schmidtand Holsapple, 1980]. The effect of ergy into gravitationalpotential energy for large-scalecrater- rock or soil strengthon laboratory cratering experimentshas ing and the effect of weakening of planetary material via been addressedby Moore et al. [1964], Gaffhey [1979], and 'shock processing,'and (3) to make a comparisonof calcu- Holsapple and Schmidt [1979], and on the basis of extensive lated and laboratorycrater excavationtimes and depths.The numerical calculations the scaling of hypervelocity impact physicalbasis for the incompressibleflow descriptionof late craters is discussedby Dienes and Walsh [1970]. In the latter stagecratering processesis describedin detail in the next sec- paper the conditionsare describedfor similitudeof the hydro- tion. Specifically,we use the incompressibleformalism to ex- dynamicflows inducedby hypervelocityimpacts in materials amine the effectsof planetary crustal strengthand gravity on the moon and to comparecalculational results with laboratory Copyright¸ 1981 by the American GeophysicalUnion. cratering experiments. Paper number 80R 1189. 0034-6853/81/080R- 1189506.00 2 O'KEEFEAND AHRENS:CRATERING, STRENGTH, AND GRAVITY ,-Trans•entCavity / /-Transient i / ß • VelocityScales D•ista •//• (a'n) Fig. 1. Incompressibleflow field crateringmodel. (a) Particlevelocity vectors for R0 -- 1 m, a -- 10 m/s, and z -- 3 for 0 = 0ø-110ø. (b) Streamlinesfor incompressibleflow for Ro = 1 m, a = 10 m/s, and z = 2. 2. LATE STAGE CRATERING FLOW whereR is the projectile-normalizedradius, a(t) is a parame- The z Model ter which dependson the strengthof the flow, and z is a con- stantfor a givenstreamline which specifiesthe shapeof the Bjork et aL [ 1967] first pointed out that the impact of a me- streamline.The validity of the latter statementcan be demon- teorite with a solid half space,such as a planetary surface,sets stratedby consideringthe divergenceof the flow (equal to a large region of the target into motion after the shock wave zero on accountof the incompressibilityassumption) in two- has 'processed'a portion of target (planetary material) in the dimensionalspherical coordinates: immediate vicinity of the impact and that this large region is much greater in size than that which experiencedpeak dy- 0 = SinO0(ReUr)/OR + RO(sin 0 uo)/00 (2) namic pressures.The decaying shock wave propagatesaway from the immediate region of impact and becomes'detached' where0 and uoare definedin Figurel a. Substituting(1) into (2) and integratingover the angular range 0-0 of the flow from the impactor. Thus a large low-stressregion which was set into motion by the shockwave becomesinvolved in crater excavation. Once the initial impact-inducedparticle velocity (2 -- 2)Ur Sin0 dO= d(sin0 u0) (3) flow in the target has been establishedand the stresseshave f0 fo decayed such that the kinetic energy per unit mass is much yields,for the azimuthalparticle velocity, greater than recoverablestrain energy, the size of the crater producedis solelycontrolled by conservationof energyfor a uo-- R dO/dt= sin 0 (z - 2)Ur/(I q- COS0) (4) fixed geometry, as was suggestedin an early model of crater- ing by Charters and Summers [1959]. We assumein the in- Again substitutingdR/dt for Urin (4) and integratingfrom 0o compressiblelate stageflow z model that the kinetic energyof to 0 and Roto R, the trajectoryof a streamline(Figure lb) is the mass of target material along a given flow streamline is given by equal to the irreversiblework that is performed against the (a/ao)(•-2) = (1 - cos•)/(1 - cos00) (5) material due to its finite strengthplus the work done against the planetary gravity field. Since the total kinetic energy im- As is shownin Figure la, R0 and 0orefer to a point on a refer- parted to the impacted target region is bounded as a result of encesurface which is takento be on the boundaryof the cra- the initial shockand rarefactionwave interaction,the equality ter transientcavity at a givenreference time. The time depen- of initial kinetic energy along a streamlinewith the irrevers- dence of the particle radial position can be obtained by ible work done againstthe material strengthand the planetary integrating(1) to yield gravity field limits the final crater dimensions. The key to utilizing the conservationof energyconcept de- R(t)•+'- Ro(to):+'= (z + 1) a(t) dt -- (z + (6) scribedqualitatively above