Impact Cratering the Effect of Crustal Strength and Planetary Gravity

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 material 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 cratering 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: immediatevicinity 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 cratering 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 is that althoughthe shock-induced •0t particlevelocities in the flow field decreasein magnitudewith time, the geometryof the streamlinesremains nearly constant where (a) is the mean value. for the duration of the crater excavationprocess for a given explosion[Maxwell, 1973] and, as we will seein the next sec- b. Transientand Final Cavity tion, alsofor a givenimpact. The simpleassumption made by If the craterdepth is denotedby Rc,it followsfrom (6) that Maxwell [1973] and demonstratedin section3, which yields an approximatedescription of the late stageimpact-induced ac(t) • [(z + 1)(a)t] •/(•+•) (7) flow, is that for the sphericalgeometry defined in Figure la for Ro >> Re. The final crater depth, and hence total crater ex- the radial particle velocity Urat a given time t is given by cavation time, is specifiedby total energy consideration,for Ur : dR/dt -- a(t)/Rz (1) example,(49). Equations(6) and (7) yield a generalexpression O'KEEFE AND AHRENS:CRATERING, STRENGTH, AND GRAVITY 3 for the radial positionof a particle R(t) in terms of its initial radial positionand the currenttransient crater depth: R(t) • [Ro(to)z+' + Rc(t)z+']'/•+'• (8) R0 which may be expressedas [Maxwell, 1973] In [R(O/Ro]• -In [1 - (Rc/Ro)•:+'•]/(z+ 1) (9) c. VolumeIntegrals In order to compute(1) the lossin kinetic energyas a result of energydissipation due to finite strengthand (2) work done againstthe gravitationalfield (as is describedin section3) we Fig. 3. Radiusversus time diagramshowing the trajectoryof a require an expressionfor a volume elementalong a stream generalpoint along a radiusR, the craterdepth Ro and the elasticde- tube. A stream tube of initial area Ao at reference radius Ro formational boundary versustime. with velocity Uois shownwith an area A at a radiusR in Fig- ure 2. The incompressibleassumption implies that radial (dErldt),angular (dEo/dt), and azimuthal(d•ddt) strain Au ----AoUo (10) rates [Batchelor, 1967, p. 601]: and der/dt = OUr/OR= -z(dR/dt)/R = -az/R •+• (17) U '• (Ur2 '•' U02)1/2 (11) deo/dt = (Ouo/OO)/R+ ur/R = Ur[1+ (Z -- 2)/(1 + COSO)]/R which from (1) and (4) is given as (18) u = a {1 + [sinO(z - 2)/(1 + cos0)]:} '/:ZlR: (12) d%/dt = (Ou,/O4•)/(Rsin O) + ur/R + Uocot O/R Denotingvalues of A, u, R, and 0 at the referencesurface with = a[1 + (z + 2) cos0/(1 + cosO)]/R <•+'• (19) the subscriptzero, and substituting(l l) and (12) into (10) where from cylindricalsymmetry, 0u,/0•b = 0. yields We note, following Maxwell [1973], that for the on-axis A = Ao(R/Ro)• {1 + [sinOo(z - 2)/(1 + cos0o)]:} strain rate (0 = 0), {1 + [sinO(z - 2)/(1 + cos0)]:} '/: (13) deo/dt-- d%/dt-- -(d•r/dt)/2 (20) sincea volume elementalong a streamlineis given by and the principal stressdifference between the radial 'stressp, and azimuthal stressPo is lessthan or equal to the Tresea yield dV = A dS (14) stressY in the sphericalapproximation: where dS is the differentialalong the streamlinegiven by -- pol -< r (21) dS -- dR/cos • = u dR/ur (15) From (20) and Hooke's law it followsthat and • is defined in Figure 2. 2•[•r- (--•r/2)l --<--Y Substituting(12) and (13) into (15) and this expressioninto (14) yields,for the volumeelement along a streamline, or dV-- Ao(R/Ro)z {1 + [sinOo(z + 2)/(1 + cos0o)]:} '/: dR 3t•rIR_

Tnitial •r-- --Z dR/R = -z In (R/Ro) (23) AO-•C;avity yields,upon substitutionfrom (9), Er• Z In [1 - (Rc/Ro)•Z+')]/(z+ 1) (24) which upon substitutioninto (22) yields -r(z + 1)/(3/•z) • In [1 - (Rc/Ro)•:+'•l (25) which specifiesthe radius to the elastic behavior surfa• • te•s of the radius to the Mterface between the yielded and not yet yielded material versuscrater depth. SMce, • general,I•/• << l, (25) may • s•p•ed to Fig.2. Relationof surface element A t• lineelement dS and streamlines. (Rc/R*)•:+'• • Y(z + 1)/3• (26) 4 O'KEEFEAND AHRENS:CRATERING, STRENGTH, AND GRAVITY

For an elasticincompressible medium the (shear)strain en- which may be approximatedby ergy per unit volume is given by Ee-• 1.5pAo[Z/(Z+ 1)]2Rc2(Z+!)/Ro z . R-{•+2• dR r•e-- t• (Er&r/dt + •o&ddt + % &,/dt) dt (27) (41) Integrating(41) and usingthe approximationof (26) yields ne-- t• {(-z) In (R/Ro)(-z) d(inR)/dt + 2[(z/2) Ee• 0.5Yz•oRc(Z+!)/[(z -{"1):Roq (42) ßIn (R/Ro)(Z/2) d In (R/Ro)/dt} dt (28) A morecomplete expression for the energy• the axialstrem •e = 1.5#•2 In2(R/Ro) (29) tube of an elastic-plastictarget may • obta•ed by sub- stitut•g•to thesum of •tegralsof theforms of (37)and (38) Using (9) gives us•g (31) and (35). This yields •e = 1.5].•[Z/(Z"[' 1)]2 In2[1 -- (ac/Ro)(z+!)] (30) Et. = Ee+ Ea • •Ao/[2•q dR' or using(22) and (23) in (29) yields • R c ne= y2/6•-• (31) +r•a•c'•+"/[•o•(• +•)1 [•/• (43) In the caseof materialwhich has yielded, the energydensity • R c depositedby workcarried out on themedium is givenby the wherethe second tern h (35)has been s•p•ed by the ap- sum of the elasticwork plus the plasticwork. The latter is prox•ation of (26). Ushg (26) to s•pfify the resultsof both given by htegrals h (43) yields %-- -Y ft,terdt (32) E, • •oYR?+!V(z + •)Rd {•.Sz- Y/2• + z • [3•/Y(z + •)]/(z + •)} (•) wheret' is the timerequired for the stressat a depthR* to ex- ceedthe materialyield strength-Y. Substituting(23) into The gravRationalpotential energy E s •volved h excavathg (32) yields the axial streamtu• of areaAo h a planeta• surfaceof density p is n•-- Yz tn (R/R*) = Yz[tn(Ro/R*) + tn (R/Ro)] (33) Inverting the argumentof the first term in (33) and sub- gg'• •/•W gRz+! dR (45) stituting(9) for the secondterm yields or, if g is assumedto be nearly independentof R, n•-= -Yz tn (R*/Ro)- Yz tn [• - (Rc/•){•+!q/(z+ •) (34) Es = AWgRc(Z+2)/(zq- 2) (46) Using(23) and then(22) to substitutein the firstterm of (34) Equations(42) and (46) were alsogiven by Maxwell [1973]. yields Finally, the initial kinetic energyin the axial streamtube at t -- to is given, using(1), by n•-- YV3• - Yz tn [• - (Rc/R){•+!q/(z+ •) (35) In the elastic-plasticdeformation region the total strain energy is given by E•e=[Aop/2] fn•[a(to)/R•]2R• dR=-Aopot2(to)Ro{!-•'/(2z - 2) (47) • = 7½"• •d (36) In orderto observethe effectof varyingRo, g, and Y on the maximum transient depth crater it is useful to substitutethe e. Energy Integrals valueof the initial particlevelocity Uor in (47) to yield Somesimple equations for final craterdepth versus planetary gravity and (constant)strength in termsof initial crater Etcœ= --pUor2Ro(Z+l)/(2Z-2) (48) parametersmay be obtained by consideringintegrals of the form Equating the initial kinetic energy in the axial tube to the strainenergy generated by producingthe crater,the latter, ei- ther specifiedby (42) or (44) plusthe gravitationalenergy of excavationof the axial tube, yields,for the final to initial crater depth ratio,

Ed= •lddV (38) Rc/Ro-• (OUor2)!/(z+!)/{2(z+ 1)[pg/(z+ 1)+ 0.5Yz/Roq} !/(z+l) c (49) wherethe volumeelement for the axialstream tube from (16) If Y = 0, we obtain for 19-- 0 is Rc/Ro• [2(z+ 1)Uor2/g]!/(z+!)/2(Z+ 1) (50) dV = Ao(R/Ro)• dR (39) Substituting(30) and (39) into (37) yields whereaswhen g-- 0, Rc= (2pUor2/Y•)'/(z+')/2(•+ 1) (51a) Ee• 1.5p, Ao[z/(z+ 1)]2/Ro z/R.øø1I!2 [1- (Rc/Ro)(Z+!)]R zdR Using the results of section3, in which values of z between 2 (40) and 3 are obtainedfor the centerlinestreamline, (50) and O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY

(5 la) are usefulto predictthe effecton maximumexcavation Substitutingfor du/dt from (53) yields depth of changingplanetary gravity and strengthfor differ- ently shapedcratering flows. Maxwell [1977] showsthat for (00r/og)f 4- 020r/OR2 •-'•'-- pUr2q t-- f2) (59) cratersfor which z -- const,for the entire incompressibleflow regime, where f' -- Of/OR. From (13), for 0 -- 0 we infer that Rc ocg-i/(2z+O (5lb) ,4 = ,4o'Rz (60) We demonstratein Figure 6, and more recently, Austin etal. [1980] have also shown,that for impact-inducedflows, z in- where creasesfrom a value between 2 and 3 along the centerline of Ao' --- Ao/Rd (6 l) the flow to values of •3.5-4 near 0 -- 90 ø. then f. Variable YieM Strength f ---z/R (62) In (31) and (32) we have treated- Y as a constantin the in- tegrationover time. Whereason a laboratoryscale such an as- f' --- -z/R 2 (63) sumptionmay be adequateonly if the coefficientof internal friction is zero, on a planetary scale,if it is assumedthat crus- Substituting(62) and (63) with (59) yields tal strengthincreases with overburdenpressure or mean prin- 020,/OR2 + (00,/OR)g/R= -p[a(t)]2z(z+ 1)/R(2z+2) (64) cipal stressas Assuminga general solutionof (64) of the form I Y] = Y, + a• (52) o,-- [a(t)12•(R) (65) where• is meanprincipal stress or pressureP and I Y] is al- lowed to increaseuntil a maximum value Y3 is achieved (the yields upon substitution von Miseslimit), a more complicatedmodel shouldbe consid- n" + z•'/R = -pz(z + 1)/o,2•+2 (66) ered. In addition to this staticcontribution to the pressure,althoughthe late stageflow is incompressible,the vorticity of Assuminga form for f] suchthat the flow gives rise to a dynamic pressurewhich will increase • •' GRm (67) the effectivepressure (D. E. Maxwell, private communication, 1979).In order to describethe dynamicpressure P along an yields, upon substitutionand equating, exponentsand coeffi- axial stream tube we consider one-dimensionalflow along R cients of R with variable area ,4. Conservation of momentum and mass is then rn -- -2z (68)

dur/dt --- OUr/Or+ Ur OUr/OR---- -(0Or/OR)/p (53) G -- -p/2 (69) A OUr/OR+ u, OA/OR = 0 (54) respectively. The homogeneousform of (66), hence •t"+ z•t'/R -- 0 (70) OUr/OR---- -ud (55) has a solution of the form where ft - A•R (•-• + A2 (71) f -- (O,4/OR)/,4 (56) Adding the solutions of the particular and homogeneous Operatingon (53) with O/ORyields equationand taking into account(65) yields 02u,/OROt + u,(O2u,/OR2) + (Ou,/OR)2 -- -(02o,/dR2)/p (57) Or= [AiR(l-z) 4- ,42- pR-2•/2l[a(t)]2 (72) which, using(55), may be written as Applying the boundary conditionsthat at R -, oo, Or---- O, and -f(du/dO 4- u2(f2 - Of/OR)-- -(020,/OR2)/p (58) at R = Re, Or---- 0 yields

,42 = 0 (73) ,4, = o/[2R?+'•I (74) TABLE 1. Radial Stresseso, Derived From Velocity Gradients, Lunar Gravity, and Vorticity for z Model Flow respectively;thus the final solutionof (64) is Velocity Lithostatic Vorticity Radius,* Pressure, Stress, Stress, Or= [a(t)]2 {R('-')/R½ ('+') - R-2z} /2 (75) cm Mbar Mbar Mbar In calculatingthe pressurefrom the increaseddynamic radial 9.19• 0 0 0 stressto specifythe yield strengthalong the axisstream tube 28 8.1 x 10-4 1.2 x 10-8 1.5 x 10-5 accordingto (52) the pressurecalculated from (75) mustbe 66 3.4 x 10-• 2.5 x 10-8 4.4 x 10-5 142 1.02 x 10-4 4.3 x 10-8 2.0 x 10-8 added to the lithostatic pressure. 180 2.7 x 10-5 5.5 x 10-8 4.9 X 10-9 In the initial stagesof incompressibleflow, when the velocity gradientis high, Oras calculatedfrom (75) dominatesthe *Fromthe lunarsurface (2.94 g/cm3), Y• = Y3= 10-7Mbar im- principalstresses. As is demonstratedin Table 1, Or varied pactedby a 10-cm-diameterprojectile. -•Theradius of the innerboundary at t -- 0 was9 cm.After 2.57 from 1 to 10n timeseither the gravitationallyinduced stress or of flow the boundarymoved 0.19 cm downward. that producedby the vorticityof the flow. 6 O'KEEFE AND AHRENS: CRATERING, STRENGTH,AND GRAVITY

TABLE 3. Material StrengthModel Used in CompressibleFlow

L.i.Iu') Y3 Calculations cr' u• Y Equation Condition Y= (.4 4- B# 4- C#2)(1- E/Era) E _Em

MEAN PRINCIPAL STRESS Parameter Value

A, kbar 2.69 B, kbar 33.8 z YI C, kbar -901.8 o Era,10 !ø ergs/g 1.70 i I = - •)PL 7' +YL

o A, B, and C are from O'Keefeand Ahrens[1976]. Em is the internal energyfor incipientmelting under standardconditions, and E is internal energy. YL mental measurements.We have previouslycomputed the flow P, PEAK SHOCK PRESSURE fieldsdue to the impact of anorthositeand iron projectileson an anorthositehalf spacefor impactvelocities ranging from 5 Fig. 4. Strengthmodels used in incompressibleflow calculations. (a) Mohr-Coulomb model; Y! representscohesive strength, and Y3 to 45 km/s [O'Keefe and Ahrens,1977b]. The material strength representsmaximum or von Mises strength. (b) Shock-induced model was based on Hugoniot elasticlimit measurementsby strengthdegradation model describedby (76). Ahrens et al. [1973] and is representativeof strong rocks. In addition to thesefinite strengthcalculations we have,by using the formalism describedby O'Keefe and Ahrens [1977b] and 3. MATERIAL STRENGTH MODELS Ahrens and O'Keefe [1977], computed the flow fields for The generalmaterial strengthmodel, used in this study,is a anorthositeprojectties impacting an anorthositehalf spaceat Mohr-Coulomb type with avon Mises limit, as already dis- velocitiesfrom 5 to 45 km/s under the assumptionof zero tar- cussedin relation to (52). Referring to Figure 4a, the material get and projectfiestrength. The equationof stateand material under zero mean stresshas a cohesion Y,, which increaseslin- strengthparameters used are listed in Tables 2 and 3. In Fig- early with pressureas given by (51a), where a, the coefficient ure 5 we examine both the zero and the nonzero strength5- of internal friction, is a material constant. The value of the km/s impact of anorthositeupon anorthositeto determinethe yield stressis limited to valuesless than or equal to the von applicability of the incompressibleflow equationsoutlined in Mises limit' YA. section 2. A number of researchers,including Knowles and Brode The applicability of (1) was determined by plotting the [ 1977],Orphal [ 1977],and Swift[ 1977],have proposed that the radial velocity field averagedover a given angular increment passageof the initial shockwave throughrock, preceding cra- at various times in the crater evolution for the 5-km/s case. ter excavation,weakens the rock. Swift [1977] carried out a se- Referring to Figure 6, we seethat the velocity profile behind ries of explosiveexcavation calculations with a shock-induced the subkilobarshock is approximatelysatisfied by (1). Values degradationof the material strengthmodel. shownfor the parametersa and z were obtainedby fitting the The model usedfor shockdegradation of material strength flow regionbehind the shockwave. Sincethe shockstrength is is shown in Figure 4b. Referring to Figure 4b, if the peak expectedto decreaseas 8 increases,the averagevalue of a de- shock stressis lessthan a minimum stressfor failure, PL, then creaseswith increasingangular coverage.The value of z near there is no change. If the peak stressis greater than PL, then the axis and averaged over 0ø-10 ø is 2.1 +_0.2, and the value the cohesionis given by averagedover the whole flow field is 3.0 + 0.2 (seealso Austin et al. [1980]). This latter value is approximately equal to the Y,' = (Y, - y,)(p,_.lp), + (76) value z = 2.7 found by Maxwell and Seifert [1976] for calcu- where Y• is a lower limit on the shock alteration of the cohe- lations of cratering flow from buffed nuclear explosions. sion. Moreover, from the work of Oberbeck [1971] it has been shownthat the flow fields from shallowbuffed explosivesimu- 4. RELATING INCOMPRESSIBLEFLOW (z) MODELS late those found in impact events; thus general agreement TO COMPRESSIBLE FLOW CALCULATIONS would be expected. As was discussedin section2e, the incompressible(z) model The results of the calculations of the evolution of the crater requiresinitial valuesfor the parametersa(to) and z at Ro(to). radius are shown in Figure 5. At early times, in the shock These have been obtained from compressibleflow calcu- wave-dominated regime the crater growth rate is identical for lations but could, in principle, also be derived from experi- the zero and nonzero strengthcases. Thus the transfer of the

TABLE 2. Equation of State ParametersUsed in CompressibleFlow Calculations Density, A, B, Eo, g/cm3 b Mbar Mbar 1012ergs/g

Gabbroic anorthosite 2.936 0.145 0.705 0.751 4.89 (low-pressurephase) Gabbroic anorthosite 3.965 0.128 2.357 1.258 18.0 (high-pressurephase) The parametersare from the Tillotson equation of state [Ahrensand O'Keefe, 1977]. O'KEEFEAND AHRENS: CRATERING, STRENGTH, AND ORAVlTY 7 projectilekinetic energyinto the initial targetvelocity field is ' ' '1 ...... I insensitiveto materialstrength. At later timesthe solutionsdi- 103 vergeand enter a regimein whichthe growthrate is approximately constant.Maxwell and Selferr[1976] found that for explosion cratering in porous and nonporousmaterials the ""'""---,;/._ velocity and stressfields behind the initial shock wave were the same. The crater growth rate is not strongly dependent upon strength;however, the duration of this regime is mate- rl/ - rial strengthdependent. In the caseof strongmaterials the du- rationof thisregime is shortand is dominatedby plasticwork. In contrast,for weak materialsthe durationis long and is ter- ml .... o - • 9.0 2,5, 0,2 II ..... 0-9m 7.0 •.0,0.• mMatedby gravitationalwork. It is the latter, or coasting,regime that resultsin very long computerrun times for com- 10-} , , ,I , ..... I pressibleflow calculationsfor low-strengthgeologic materials. 10-} 100 The compressibleflow code solutionswere not carried out to RADIUS (M) the final crater depth becauseof this constraint. One of the Fig. 6. Calculatedradial pa•iclc vcl•ity distributionfor 5-•/s •pact of ano•bositc imo ano•bositc,60 •s after the impact of a 10- major advantagesof the z model is the efficiencyachieved in cm-diamctcr ano•bositc sphere,dcmonstrat•g averagevalues calculationsin the late time regime. The variable yield z and z which fit (1), upon averagingover diEcrcnt intc•als in 8. U•ts modelgiven by (52) and (75) wasused to computethe crater of • are kilometersper second. depth evolutionunder the assumptionof strengthvalues cor- and gravitationalenergy. The initial projectilekinetic energy respondingto the detailed code calculations(see Table 3). can be converted irreversibly by shock heating or deforma- The effect of reducingthe cohesivestrength was to increase tional work into internal energy or storedreversibly as elastic the crater depth by more than a factor of 4 and the crater for- or gravitational energy. Irreversible heating of the axial mation time by more than 2 ordersof magnitude. streamtube material occursas a result of hydrodynamicshock and is independentof scalefor fixed impact velocity and is a 5. PARTITIONING OF ENERGY constantfraction (-0.3) of the total kinetic energy. This was Using the resultsof a compressibleflow calculation of the computed by using the compressibleflow formulation de- impact of a 5-km/s anorthositesphere into an anorthositehalf scribedby O'Keefe and Ahrens[1977b]. Plasticwork occursat spacedepicted in Figures 5 and 6, the partitioning of energy both high and low stresslevels and was computedby using in the axial streamtube was computedby using the values of both the compressibleand the incompressibleformulations. a(to) and z obtainedfor the 0 = 0 ø- 10ø sector.A shockdegra- The gravitational energy is important at late times and was dation model for material strengthwas not assumedin these computedby usingthe incompressiblemethod. The partition- calculations.The z model parametersused are summarizedin ing of energy as a function of meteorite radius is shown in Table 4. There are severalpossible forms that the impacting Figures 7a and 7b for von Mises and Mohr-Coulomb-von energymay take. The partitioningof energyfor impact at var- Mises materials, respectively. iousscales (assuming 10- to (2 x 107)-cmdiameter projectiles) Referring to Figure 7a, in avon Mises material (Y, = Y3 = was studied by calculatingthe total plastic work (equation 2.7 x 10-3 Mbar) the shockand plasticwork accountfor 75% (35)), elasticenergy (equation (31)), and gravitationalpoten- of the initial kinetic energy, and the storedelastic energy ac- tial energy (equation (46)) in the axial flow tube for the grav- counts for 25% for anorthosite meteorites having radii less ity field of the moon. For each case studied, approximately than 2 x 105cm. For meteoriteradii greaterthan 2 x 105cm 102time stepswith severaliterations per time stepwere car- the storedelastic energy is decreased,and gravitationalpoten- ried out, usingthe abovereferenced integrals, to reduceall the tial energy increased,the latter being completelydominant at initial kinetic energyin the flow tube to elastic,deformational, radii greaterthan 107cm. In the caseof Mohr-Coulomb-von Mises materials (Y, _< 1 x 10-5, Y3 = 2.7 x 10-3 Mbar) the shockand deformational I I I work account for nearly all of the initial kinetic energy for COMPRESSIBLE FLOW IOO- CALCULATIONSTOPPED-- FINALCRATER •.• meteorite radii less than 104cm. In the range of meteorite radii between l0 4 and 107 cm the overburdenpressure in- ,-,• NOTZEROFINAL CRATER 7 //--DEPTH •'•' /.•'• ' ?• STRENGTH'-•I /•,-.'/' '• Yi=1.0 xI0 -I0 creasesthe yield stress,and part of the energy is partitioned I-- __HYDRODYNAMIC • •..,.•'- Y•=./ 2 ß69 x IO-3 into stored elastic energy. At meteorite radii greater than 2 x 105cm the gravitationalenergy starts to becomeimportant. n,' IO x IO-3 TABLE 4. Summary of z Model ParametersUsed in Cratering Calculations U / •----ZCODE STARTING Material StrengthParameters Typical •_• SHOCK-DOMINATEDREGIME Case Yl, Mbar Y2, Mbar Y3, Mbar Medium I I I I IO IO2 IO3 IO4 IO5 1 10-•ø 1.0 10-lø fluid TIME(p. SEC) 2 l0 -lø 1.0 2.7 X 10-3 dry sand Fig. 5. Crater depth versustime calculatedby usingcompressible 3 10-5 1.0 10-5 hardclay flow calculationand linked incompressiblecalculations (z code)for 4 10-5 1.0 2.7 X 10-3 soft rock the caseof zero strengthand variousMohr-Coulomb strengthcrusts 5 2.7 X 10-3 1.0 2.7 X 10-3 hard rock with lunar surfacegravity. Asterisks indicate final craterdepths. Units of yield strength are megabars. The densityis 2.94 g/cm3, andthe initial velocityis 0.075km/s. 8 O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY

and the final crater formation time scales as meteorite radius • FLAST,C •• R,n. For the moon the range of meteorite radii separatingthe )-'6 • GRAVITATIONAL'--"•cohesive strength-dominated crater growth and the Mohr- m IRREVERSIBLEDEFORMATIONAL WORK Coulomb-dominatedgrowth is around l0 3 cm and is depen- Z .4• (VON MISES) dent upon the specificcombination of Yi and Y3 assumed.In t.do,;> t the Mohr-Coulomb regime (R,n -• 104-106)the overburden •: I01 102 10:5 104 105 106 107 stressincreases the yield stressof Mohr-Coulomb materials, -- METEORITE RADIUS, (CM) and the formation time scalesas R,n/•,where 0 < fl < 1. In terms of impacting meteorite radius the Mohr-Coulomb re- z gime extendsto radii at which the overburdenstress increases i- I RREVER_SIBL..E_•__ /__• ^CT,C \ the yield strengthof the bulk of the material involved in the ('-') DEFORMATIONALWORK ELASTIC • n,- GRAVITATIONAL • excavationprocess to the von Mises limit. In the von Mises re- LL 4 (MOHR-COULOMB) gime (R,n'" l0 s cm) the magnitudeof the cohesivestrength is less important than the von Mises limit. Here the crater for- I01 102 I03 104 105 106 107 mation time again scaleslinearly with meteorite radius. Fi- METEORITE RADIUS, (CM) nally, in the gravityregime (R,n > •-106-107cm) the gravita- Fig. 7. Energy partitioningversus meteorite radius for a 5-km/s tional work of excavationdominates the work done against impact of anorthositeon anorthositein the gravityfield of the moon, the material strength.In this regime the crater formation time where the strengthof the planetarysurface is assumed(a) to be a con- is proportionalto R,nl/2. stant2.7 kbar and (b) to increasefrom a cohesivestrength of 10 bars The crater depth was also computed as a function of the to a maximum strengthof 2.7 kbar as a Mohr-Coulomb solid. meteorite radius. As in the case of the crater formation times, the calculational results obtained can be cast in terms of four 6. CRATER FORMATION AS A FUNCTION regimes.In the cohesiveregime the crater depth for a given OF STRENGTH AND GRAVITY radius can vary by over an order of magnitude for differing Upon calculating crater depth formation times for lunar magnitudesof cohesivestrength. In the Mohr-Coulomb re- gravity for a 5-km/s impact of anorthositeon anorthositeas a gime the overburdenstress increases the yield stresssuch that function of meteorite radius Rm (Figure 8) it was discovered the relative crater depth decreaseswith increasingmeteorite that the resultsare approximatelydescribable in termsof dif- radius. In the von Mises regime the depth is independentof ferent assumedmaterial strengthproperties (Table 4). These the cohesivestrength and dependsonly on the von Mises regimesare (1) cohesivestrength, (2) Mohr-Coulomb, (3) von limit. In the gravity regime, crater depth is independentof the Mises,and (4) gravity.The first three regimesare determined material strength.For example, as is shownin Figure 9, in the by the material strength.In the cohesivestrength regime (Rrn case of the moon, for avon Mises limit of Y3 = 2.7 x l0 -3 < 104 cm) the cohesivestrength of the material dominates, Mbar the Mohr-Coulomb regime extendsfrom meteorite radii

1000 COHESIVE 1 ,.•...... STRENGTH...... • .... MOHR-COULOMB..... REGIME i REGIME

._• ,oo- ,, T•

• _ • •/ '3-'• .•/ / ;- REGIME' • I -

• •// 10-5 =2.7 x I0 -• • I0-• - 7' _ Y3 27x10 3 _

• Y•=I

u m0-2 _ D - -

i0-3 I I I I m0 m02 m0) m04 m05 m06 m07 •ETE0•ITE •DIU•, (C•) FiS. 8. Craterdepth fo•afion t•e for a 5-•/s •pact on •e moonveBus ano•hosite meteo•te radius for va•ous Mohr-Coulomb strenSthmodels. O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY 9 of--, 1• to l0 s cm, the von Misesregime extends from --,10 s to I I I I I I • 107cm, and the gravity regime extendsfrom • 107cm to greater values. A seriesof calculationswere carried out using the shock degradationmodel specifiedby (76) with y -- 1. In this model the peak shock wave pressurereduces the magnitude of the Zyi--2.7x I0 -3 cohesive strength Y, and not the yon Mises limit Y3. The y3:2.7x I0-3 shockwave amplitudesused to degrade the value of Y, were obtained from the pressuredecay calculations given by Ahrens I I I I I I and O'Keefe [1977]. The resulting relative crater depth as a I100 I01 102 103 104 105 106 METEORITERADIUS, (CM) function of meteorite radius for a $-km/s impact is presented in Figure 10. The example chosenhas an initially large cohe- Fig. 10. Normalized crater depth versus anorthosite meteorite sive strengthwhich was equal to the yon Mises limit, Y, = Y3 radiusfor a 5-km/s impacton the moon for differentstrength degra- dation modelsof the type shown in Figure 4b.. -- 2.7 x 10-3 Mbar. The strengthdegradation effect was expectedto be mostpronounced in this case.The passageof the the case of Mohr-Coulomb materials (cases2-5 in Table 4) shockwave reducedthe cohesivestrength to 10-'ø Mbar for and large yon Miseslimits (e.g., Y3 = 2.7 x 10-3 Mbar) the shockpressures greater than 10-4 Mbar in one caseand 10-3 depth scalesas E•'/3'4. This intermediatetype of scalingwas Mbar in another. As is demonstrated in Figure 10, in the first proposedby Vaile [1961] on the basisof correlationsof cohesiveregime (meteorite radius <10 s cm), shockdegrada- chemical and nuclear explosion data. tion could be an important mechanismin determiningcrater depth. The crater depth more than doubleswhen rock is proc- 7. COMPARISON OF INCOMPRESSIBLE MODEL essedby a thresholdshock pressure of 10-4 Mbar. The effect WITH SAND CRATERING EXPERIMENTS diminishesin the Mohr-Coulomb regime and vanishesin the The comparisonof impact cratering experimentswith de- yon Mises and gravity regimes. tailed calculationshas to date, surprisingly,only been carried The effect of strengthon the transient crater depth versus out in the case of metals [Rosenblattet al., 1970] and com- kinetic energy of impact (E) is presentedin Figure 11 on the positestructures [Gehring, 1970]. basisof variousstrengths listed in Table 4. We emphasizethe Our objectivewas to find a set of well-controlledimpact ex- word transient, becauseobservationally, it has been found perimentsin a medium of geologicinterest and compare the that the crater depthsdo not exceed•$ km, regardlessof di- incompressibleflow model to those resultsso that we could ameter, on the moon, Mercury, and probably other planets. have confidencein computingcratering over a broad range of We believethat the limit observedin crater depth for larger conditions.The sand impact experimentsof Gault and Wede- craters resultsfrom later stagesof crater developmentthan kind[1977] provide an appropriatedata base.The propertiesof those describedhere [e.g., Melosh, 1977; McKinnon, 1978]. the sand usedin theseexperiments was measuredand demon- Like the relationshipof the transientcrater growth times and strated a Mohr-Coulomb type behavior for which the yield crater depth versusmeteorite radius, the impact energy can stressin shear can be approximated by also be describedas occurringwithin severalregimes which are related to the assumedstrength model. In the caseof very Y (Mbar)= 25 x 10-'6 + 33 x 10-'4 P (77) weak materials(case 1 in Table 4) the crater depth is dominated by gravity, and the depth scalesas E '/4 [e.g., Chabai, 1965;Gault et al., 1975;Sauer, 1978]. In the caseof very strong -CHEMiCAl-i-NUCLEAR 'I•" materialsthe craterdepth scales as E '/3 at impactenergies of 8 -- I -- _<10•ø ergs,whereas when gravitationalforces dominate, the ß•...__ E 4. o craterdepth scalesas E '/4 as energiesapproach 103ø ergs. In

ioo "b.% I I I Io3 I_ . • •o o• I 1 • I • 3 -- _ /// 'E•-•.MODIFIED REGIME_ •y•lo-,O •r• y - Yi='O-IO ••, • ,r'- I kDOMINATED/

o i0- •4•0 % E$.4 bJ • 10 3- or' / BOWL- SHAPED o 2 i x HARD ROCK • _ STRENGTHDOMINATED] • CRATERREGIME _ o Y•:•.•,o-• YON•,S•S _.1 2/, -- •OHR-COULO•B AND YON •ISES u o I I I I I I100 i01 i0•' 103 104 105 106 I07 io 15 20 25 30 35 METEORITE RADIUS, (CM) LOGIo IMPACT ENERGY (ergs) Fig. 9. Normalized crater depth versus anorthositc meteorite Fig. 11. Log,0 of transientcrater depth versusanorthosite impac- radius for a 5-km/s impact on the moon for different Mohr-Coulomb tor energy for a 5-km/s impact on the moon for different strength strengthmodels. models specifiedin Table 4. 10 O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY

TABLE 5. Parametersof z Model Used for ComparisonWith means of coupling the resultsof compressibleflow finite dif- Experimentsof Gaultand Wedekind[ 1977] ference calculations carried out to stress levels below the com- Material StrengthParameter Value pressionaldynamic yield point (Hugoniot elastic limit) with the incompressiblefluid flow model of Maxwell [1973]. The Y•, Mbar 2.5 x 10-•6 incompressibleformulation is applicable to the flow after the Y•_,tan q• 0.33 Y3,Mbar 2.7 x 10-3 stresslevels had decayedto -• 1.5 kbar, resultingfrom the impact of a 5-km/s anorthosite sphere on a half space of The material strengthparameters were calculatedfrom a graph of anorthositein the gravity field of the moon. We found that, the dataof Gaultand Wedekind[1977]. The densityis 1.65g/cm 3, and the initial velocity is 0.075 km?s. like the case of explosions,the radial particle velocity at a given time field can be representedas being proportional to where1 ø is the overburdenstress, which is assumedto be equal (I/R) z, where z • 2 along the axis streamtube and z • 3 when to the averageprincipal stressas usedin (76). Gault and We- averagedover a larger portion of the half space.We outlined dekind's[1977] experimentswere carried out on a size scale the physical and mathematical framework of the Maxwell between that of cohesion-and that of gravity-dominated scal- model. This model assumesthat the kinetic energywithin the ing and thusprovide a stringenttest of presentmodeling capa- incompressibleflow generated by the passageof the shock bility in the Mohr-Coulomb regime. wave in the crater excavationregion is convertedinto elastic, These experimentsconsisted of impacting aluminum pro- plastic, and potential energy by doing work against the tar- jectilesinto sandat velocitiesin the rangeup to 6 km/s. Mea- get'smaterial strengthand body forces(in this case,planetary surements of the crater diameter as a function of time for vari- gravity); thus the final crater depth and volume is determined ous values of acceleration, simulating various surface from conservationlaws. We developedexpressions for the fi- gravities,were carried out. The depth to diameter ratios re- nal crater depth in the strength-dominatedcase, for fixed mained nearly constantover the range of test conditions,so gravity g, in terms of target densityp and yield strength Y, that the scaling of characteristiccrater diameter and depth which is proportional to [pUor•/Yz]•/(•+•), where Uoris the formation times should be proportional (i.e., have the same radial particle velocity on the surfaceof the early time cavity scaling exponent). of radius Ro. In addition, we developed comparable ex- We have made a comparisonof the scalingof the crater for- pressionsfor the gravity-dominatedcase for a fixed material mation time as a function of the size of impacting projectile strength,in which the final crater depth is proportional to and surfacegravity. The objectivesof thesecalculations were Ro[2(z+ 1)Uofi/g]•/•+•>. When Y is specifiedby a Mohr-Cou- to show a correlation between theory and experimentsof the lombrelation of theform Y-- Y, + a• up to a vanMises limit scalingof crater depth with time and gravity. This scalingis Y3, analytical solutions were not found, and the effect of insensitiveto the exact magnitude of the initial velocity field. strengthon the crater depth was computednumerically by us- Becauseof this insensitivitywe did not use the detailed com- ing the lithostaticpressure in additionto the dynamicpressure pressibleflow code to computethe initial flow field in sand resultingfrom the vorticity of the flow. but rather assumedthe same flow field that we calculatedpre- The effect of shockwave degradationof material strength viously [O'Keefeand/lhrens, 1977b],associated with impact of on crater depth was examined.The cohesivestrength Y• was anorthositeupon an anorthositehalf spaceat 5 km/s. This im- modeled as decreasingwith the exposureof rock to increased pact flow field was usedto determinethe initial transientcav- shockpressure and was found to have the effect of increasing ity size Ro, z, and velocity magnitude a (e.g., (1) and (7)). A strongrock crater excavationdepths by as much as a factor of summary of the z model parametersis given in Table 5. 2 for impactorradii of • 103 cm. The crater depth formation time was computedby usingthe The scaling of transient crater depth with impact energy z model as a function of crater depth ranging from 0.4 cm to 4 was found in the caseof the moon to be strengthdominated x 103 cm. The associated formation time varied from 3.5 x for high-strengthsurfaces at impactenergies E • 102ø ergs and 10-3 to 0.4 s. Referring to Figure 12, the slopeof the calcu- hencescale as E '/3, whereasfor weakmaterials and for strong lated crater formation time or the scaling exponent is very surfaceswhere the impact energy is •10 3øergs, the crater closeto the value found by Gault and Wedekind over the experimental range. I0 O, The crater depth formation time was also computedover a range of surfaceaccelerations less than lg. Referring to Figure Experiments 13, the slopeor scalingexponent in that casealso agreedquite Gault & Wedekind (1977) well with the value found by Gault and Wedekind (t oc g-O.6,8).Note that for gravity-dominatedcraters the craterfor- •---Rangeof-• • Slopeo:D0-477 / mation time scalesas g-O.6•_5;thus the small shearstrength of Gault and Wedek•nd ..•.•' sand decreasesthe formation time in relation to strengthless gravity scaling.Finally, transientcrater depth versusg is cal- •,•,•" •-•Present culated over the range studiedby Gault and Wedekind[1977] •,•,•.•---••, CaJculahons (exceptthat they determinedthe diameterD). In Figure 14 we show that the calculated exponent of depth (or diameter) is I i I i again close to the value obtained by Gault and Wedekind ioo ioI io2 io3 0 4 [1977].Both are lessthan the D ocg-O.•_5 expected for pure Crater Depth (cm) gravity scaling. Fig. 12. Variation of crater depth formation time with crater depth usingthe presentincompressible flow formulation with the as- 8. CONCLUSIONS sumptionof a 5-km/s anorthosite-anorthositevelocity field. Shown also is the range of crater depths(diameters) observed in the sand ex- The effect of varying planetary crustalstrength and surface perimentsof Gaultand I•edekind[ 1977]and the associatedslope and gravity on the depth of impact craters was investigatedby scalingrelationship. O'KEEFE AND AHRENS: CRATERING, STRENGTH,AND GRAVITY 11 depthsscale as E 1/4.However, for a broadrange of observable io z crater sizesand expectedstrength parameters the depth scales as -E •/3'4,which is consistentwith chemical,nuclear, and impact crater statistics. The transientcrater depth and formationtime was shown to varygreatly depending upon planetary strength. For im- pactingmeteorite radii lessthan 103 cm the craterformation time varied by about 5 ordersof magnitudefrom 10-3 to 10 S, and the crater depth to meteorite radius ratio varied from 4 I- to 32; this behavior is characteristicof what we call the cohe- z sive strengthregime. For meteoriteradii greaterthan 103 cm the crater depth decreasedsharply with increasingmeteorite z radius for Mohr-Coulomb materials, and its rate was un- changed for von Mises materials. We denote this behavior i0 I I I I I i with the terms Mohr-Coulomb and von Mises regimes.At suf- 0.1 1.0 ficiently large meteorite radii (e.g., -10 s to -106 cm) the ACCELERATION g strengthof Mohr-Coulomb materials is limited by the von Fig. 14. Variationof transientcrater depth D withgravity using Miseslimit. Finally, for impactorradii of •10 ? cm the crater thepresent incompressible flow formulation with the assumption of a 5-km/sanorthosite-anorthosite velocity field. Shown also is theslope depth and formation time are dominatedby gravity. andscaling relationship found by Gaultand Werekind[1977] in sand The partitioning of energy in the axial stream tube was ex- experiments. amined for impactson the moon in which the surfacestrength was constantand high (2.7 kbar) versusone whose strength kind's reported shear yield strengths,crater formation times increasedwith confiningpressure from -0.01 to 2.7 kbar. For were found to vary with gravity as g-o.ss,whereas they ob- the former (von Mises type) material some75% of the initial serveda g-O.6•8dependence over a rangefrom 0.03 to 0.12 s. kinetic energy goes into reversibleelastic strain energy, for Similarly, a maximum crater depth dependencewas calcu- meteorite radii less than 2 x l0 s cm, whereas for the latter latedwhich varied as g-o.•9, whereas the diameterdependence (Mohr-Coulomb type) material, virtually all the impact en- of Gault and Wedekind's experimentsfollowed a g-o.•6sde- ergy is depositedas shock,or deformational, energy for im- pendence.The comparisonof the presentincompressible flow pactorshaving a diameterless than -104 cm. Moreover,it was calculationwith experimentsdemonstrates the need for data demonstratedthat some 25% of the impact energy goesinto describingthe strengthproperties of planetary surfacesin or- work againstthe moon'sgravity field for meteoriteradii ap- der to relate cratersize populations to impacthistory on plan- proaching-10 ? cm. ets with differing surfacematerials and gravities. The experimental data of Gault and Wedekind [1977] for crateringin sandwith energiesof 6 x 10? to 10•ø ergs under Acknowledgments.We are grateful for both the help and the ac- effectivegravities of 0.07g-lg provide a test of both strength cessto unpublishedmaterials so generouslyproffered by D. E. Max- well of ScienceApplications, Incorporated. The final manuscripthas effectsand gravity on the presentincompressible ttow calcu- benefitedfrom the rigorousreviews of R. Schmidtof the BoeingCom- lations. The starting condition for the calculations was the pany and H. Moore of the USGS. Technical discussionswith B. Min- scaled ttow field for an anorthositesphere impacting an ster and J. Melosh and computationalassistance by M. Lainhart are anorthositehalf space at 5 km/s. Using Gault and Wede- also appreciated.The researchwas supportedunder NASA grant NSG 7129. Contribution3300, Division of Geologicaland Planetary

., I I i i Sciences,California Institute of Technology.

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b.I gation of impact crateringdynamics: Material motionsduring the cratergrowth period (abstract),Lunar Planet.$ci., XI, 46-48, 1980.

LLI Batchelor,G. K., An Introductionto Fluid Mechanics,615 pp., Cam- bridge University Press,New York, 1967. Q:: Bjork, R. L., Analysisof the formation of Meteor Crater, Arizona: A U preliminary report, J. Geophys.Res., 66, 3379-3388, 1961. Bjork, R. L., K. N. Kreyenhagen,and M. H. Wagner, Analytic study of impact effectsas applies to the meteoroidhazard, NASA Con- 10-2 I I I I O. I 1.0 tract. Rep., CR- 757, 1967. ACCELERATION g Bryan, J. B., D. E. Burton, M. E. Cunningham,and L. A. Lettis, Jr., A two-dimensionalcomputer simulation of hypervelocityimpact cra- Fig. 13. Variationof craterdepth formation time with gravityus- tering: Some preliminary resultsfor Meteor Crater, Arizona, Proc. ing the presentincompressible flow formulationwith the assumption Lunar $ci. Conf. 9th, 3931-3964, 1978. of a 5-km/s anorthosite-anorthositevelocity field. Shown also is the Chabai, A. J., On scalingdimensions of cratersproduced by buried slopeand scalingrelationship found by Gaultand Wedekind[1977] in explosions,J. Geophys.Res., 70, 5075-5098, 1965. sand experiments. Charters,A. C., and J. L. Summers,Some comments on the phenom- 12 O'KEEFE AND AHRENS:CRATERING, STRENGTH, AND GRAVITY

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