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
Seismological
Laboratory,CaliforniaInstituteof Technology,Pasadena,California91125
Upon impactof a meteoritewith a planetarysurfacethe resultingshockwave both 'processes'
the material in the vicinity of the impact and setsa larger volumeof material than was subjectedto high pressureinto motion. Most of the volume which is excavatedby the impact leavesthe crater after the shock
wavehasdecayed.The kineticenergywhichhasbeendepositedin the planetarysurfaceis convertedinto
reversibleand irreversiblework, carriedout againstthe planetarygravityfield and againstthe strengthof
the impactedmaterial,respectively.By usingthe resultsof compressible
flow calculationsprescribingthe
initial stagesof the impact interaction(obtainedwith finite differencetechniques)the final stagesof crateringflow alongthe symmetryaxisare described,usingthe incompressible
flow formalismproposedby
Maxwell. The fundamentalassumptionin this descriptionis that the amplitude of the particle velocity
field decreases
with time as kineticenergyis convertedinto heat and gravitationalpotentialenergy.At a
given time in a sphericalcoordinatesystemthe radial velocity is proportionalto R-z, where R is the
radius(normalizedby projectilevelocity) and z is a constantshapefactor for the duration of flow and a
weak function of angle. The azimuthal velocity, as well as the streamlines,is prescribedby the incompressibilitycondition.The final crater depth (for fixed strength Y) is found to be proportional to
Ro[2(z+ l)Uor2/g]
l/(z+l),whereUoris the initial radialparticlevelocityat (projectilenormalized)radius
P,o,g is planetarygravity,and z (whichvaried from 2 to 3) is the shapefactor.The final craterdepth (for
fixedgravity)is alsofoundto be proportional
to [PUor2/Yz]
l/(z+•),wherep and Y are planetarydensity
and yield strength,respectively.By usinga Mohr-Coulomb yield criterionthe effectof varying strength
on transientcraterdepthand on craterformationtime in the gravity field of the moon is investigatedfor
5-km/s impactors
with radii in the 10-to 107-cmrange.Comparison
of craterformationtime and maximumtransientcraterdepthasa functionof gravityyieldsdependencies
proportionalto g-O.SS
and g-O.•9,
respectively,
compared
to g-O.61s
andg-O.165observed
by Gault andWedekindfor hypervelocity
impact
cratersin the 16- to 26-cm-diameterrange in a quartz sand(with Mohr-Coulomb type behavior) carried
out overan effectivegravityrangeof 72-980 cm/s2.
1.
INTRODUCTION
at sizesand time scalesfor which gravity does not affect the
flow. Theoretically basedscalinglaws for relating final crater
dimensionsto targetand projectiledensitiesand projectilekinetic energyor momentum(or a combinationof these),planetary gravity, sound speed,and material strengthhave been
developed on the basis of laboratory and field data [e.g.,
Chabai, 1965;Gault, 1973;Gault et al., 1975;O'Keefeand Ahrens, 1978].
In the presentpaper we examine how maximum transient
craterdepthis relatedto projectileenergyfor a rangeof planetary gravities and crustal strengthconditions.Moreover, we
available are limited, and theoretical estimates must be used examine how the scalinglaws themselvesdepend on gravity
and target strength.Although, in principle,sucha studycould
to understandcrater populations.
be
carried out by using only the finite difference numerical
There have been a number of studiesentailing the analytimethods
previouslyapplied to meteoriteimpact crateringon
cal and numerical predictionof crateringphenomenaand the
scalingof laboratory experimentaldata to planetary cratering. the earth and the terrestrialplanets[Bjork, 1961;O'Keefeand
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 aptional fields,centimeterscaleexplosioncrateringexperiments proachtaken is (1) to establishthe validity for impact crater-
In order to obtain the relative ages of cratered surfaces
formed on different terraneson a singleplanet [e.g., Neukum,
1977], as well as to obtain meaningful interplanetary comparisons[e.g., Gault et al., 1975;Neukum and Wise, 1976],the
effect of planetary crustal strengthand gravity on the dimensionsof cratersmust be understood.Becausethe largestmanmade impact craters(producedon the moon by empty rocket
casings)are only some40 m in diameter [Whitaker, 1972] and
the largestnuclear explosions(formed in wet coral sand) are
only 1.8 km in diameter [F'aile, 1961],the experimentaldata
have been carried
out under
conditions
both
less than
the
earth'sgravity field [Johnsonet al., 1969;Gault and Wedekind,
1977] and greater than the earth's gravity field [F'ictorovand
Stepenov,1960; Schmidtand Holsapple, 1980]. The effect of
rock or soil strengthon laboratorycrateringexperimentshas
been addressedby Moore et al. [1964], Gaffhey [1979], and
Holsapple and Schmidt [1979], and on the basis of extensive
numerical calculationsthe scaling of hypervelocity impact
craters is discussedby Dienes and Walsh [1970]. In the latter
paper the conditionsare describedfor similitudeof the hydrodynamicflowsinducedby hypervelocityimpactsin materials
Copyright¸ 1981 by the American GeophysicalUnion.
Paper number 80R 1189.
0034-6853/81/080R- 1189506.00
ing of the Maxwell z model,whichwaspreviouslyappliedt6
explosivecrateringby Maxwell [1973, 1977]and Maxwell and
Seifert [1976], (2) to compute the partitioning of impact energy into gravitationalpotential energyfor large-scalecratering and the effect of weakening of planetary material via
'shock processing,'and (3) to make a comparisonof calculated and laboratorycrater excavationtimesand depths.The
physicalbasisfor the incompressibleflow descriptionof late
stagecrateringprocessesis describedin detail in the next section. Specifically,we use the incompressible
formalismto examine the effectsof planetary crustalstrengthand gravity on
the moon and to comparecalculationalresultswith laboratory
crateringexperiments.
2
O'KEEFEAND AHRENS:CRATERING,STRENGTH,AND GRAVITY
Cavity
,-Trans•ent
/-Transient
/
i
/
ß
'n
• Velocity
Scales
D•ista
)
•//•
(a
Fig. 1. Incompressible
flow field crateringmodel.(a) Particlevelocityvectorsfor R0 -- 1 m, a -- 10 m/s, and z -- 3 for 0
= 0ø-110ø. (b) Streamlinesfor incompressible
flow for Ro = 1 m, a = 10 m/s, and z = 2.
2.
LATE
STAGE
CRATERING
FLOW
The z Model
Bjork et aL [ 1967] first pointed out that the impact of a meteorite with a solid half space,suchas a planetary surface,sets
a large region of the target into motion after the shockwave
has 'processed'a portion of target (planetary material) in the
immediatevicinity of the impact and that this large regionis
much greater in size than that which experiencedpeak dynamic pressures.The decaying shock wave propagatesaway
from the immediate region of impact and becomes'detached'
from the impactor.Thus a large low-stressregionwhich was
set into motion by the shockwave becomesinvolved in crater
excavation. Once the initial impact-inducedparticle velocity
flow in the target has been establishedand the stresseshave
decayed such that the kinetic energy per unit massis much
greater than recoverablestrain energy, the size of the crater
producedis solelycontrolledby conservationof energyfor a
fixed geometry, as was suggestedin an early model of cratering by Charters and Summers [1959]. We assumein the incompressiblelate stageflow z model that the kinetic energyof
the mass of target material along a given flow streamline is
equal to the irreversiblework that is performed against the
material due to its finite strengthplus the work done against
the planetary gravity field. Since the total kinetic energy imparted to the impacted target region is boundedas a result of
the initial shockand rarefactionwave interaction,the equality
of initial kinetic energy along a streamlinewith the irreversible work doneagainstthe materialstrengthand the planetary
gravity field limits the final crater dimensions.
The key to utilizing the conservationof energyconceptdescribedqualitativelyabove is that althoughthe shock-induced
particlevelocitiesin the flow field decreasein magnitudewith
time, the geometryof the streamlinesremainsnearly constant
for the duration of the crater excavationprocessfor a given
explosion[Maxwell, 1973] and, as we will seein the next section, alsofor a givenimpact.The simpleassumptionmade by
Maxwell [1973] and demonstratedin section3, which yields
an approximatedescriptionof the late stageimpact-induced
flow, is that for the sphericalgeometrydefined in Figure la
the radial particle velocity Urat a given time t is given by
Ur : dR/dt -- a(t)/Rz
(1)
whereR is the projectile-normalized
radius,a(t) is a parameter which dependson the strengthof the flow, and z is a constantfor a givenstreamlinewhichspecifies
the shapeof the
streamline.The validity of the latter statementcan be demonstratedby consideringthe divergenceof the flow (equal to
zero on accountof the incompressibility
assumption)
in twodimensionalsphericalcoordinates:
0 = SinO0(ReUr)/OR
+ RO(sin0 uo)/00
(2)
where0 and uoare definedin Figurel a. Substituting
(1) into
(2) and integratingover the angularrange 0-0 of the flow
f0Sin0 dO= fod(sin0 u0)
(2 -- 2)Ur
(3)
yields,for the azimuthalparticlevelocity,
uo-- R dO/dt= sin 0 (z - 2)Ur/(I q- COS
0)
(4)
Againsubstituting
dR/dt for Urin (4) and integratingfrom 0o
to 0 and Roto R, the trajectoryof a streamline
(Figurelb) is
given by
(a/ao)(•-2)= (1 - cos•)/(1 - cos00)
(5)
As is shownin Figure la, R0and 0orefer to a point on a referencesurfacewhichis takento be on the boundaryof the crater transientcavityat a givenreferencetime. The time dependence of the particle radial position can be obtained by
integrating(1) to yield
•0ta(t) dt-- (z +
R(t)•+'- Ro(to)
:+'= (z + 1)
(6)
where (a) is the mean value.
b.
Transientand Final Cavity
If the craterdepthis denotedby Rc,it followsfrom (6) that
ac(t) • [(z + 1)(a)t] •/(•+•)
(7)
for Ro >> Re.The final crater depth, and hencetotal crater excavation time, is specifiedby total energy consideration,for
example,(49). Equations(6) and (7) yield a generalexpression
O'KEEFEAND AHRENS:CRATERING,STRENGTH,AND GRAVITY
3
for the radial positionof a particle R(t) in terms of its initial
radial positionand the currenttransientcraterdepth:
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)
c.
(9)
VolumeIntegrals
In orderto compute(1) the lossin kinetic energyas a result
of energydissipationdue to finite strengthand (2) work done
againstthe gravitationalfield (asis describedin section3) we
require an expressionfor a volume elementalong a stream
Fig. 3. Radiusversustime diagramshowingthe trajectoryof a
generalpointalonga radiusR, thecraterdepthRo andthe elasticde-
tube. A stream tube of initial area Ao at reference radius Ro
formational boundary versustime.
with velocityUois shownwith an area A at a radiusR in Figure 2. The incompressible
assumptionimpliesthat
Au ----AoUo
(10)
radial (dErldt),angular(dEo/dt),and azimuthal(d•ddt) strain
rates [Batchelor, 1967, p. 601]:
der/dt = OUr/OR
= -z(dR/dt)/R = -az/R •+•
and
U '• (Ur2 '•' U02)
1/2
(11)
deo/dt = (Ouo/OO)/R
+ ur/R = Ur[1+ (Z -- 2)/(1 + COSO)]/R
(18)
which from (1) and (4) is given as
(12)
d%/dt = (Ou,/O4•)/(RsinO)+ ur/R + UocotO/R
Denotingvaluesof A, u, R, and 0 at the referencesurfacewith
the subscriptzero, and substituting(l l) and (12) into (10)
yields
= a[1 + (z + 2) cos0/(1 + cosO)]/R<•+'•
u = a {1 + [sinO(z- 2)/(1 + cos0)]:}'/:ZlR:
A = Ao(R/Ro)
• {1 + [sinOo(z- 2)/(1 + cos0o)]:}
{1 + [sinO(z- 2)/(1 + cos0)]:}'/:
dV = A dS
(14)
deo/dt-- d%/dt-- -(d•r/dt)/2
(15)
-- pol-< r
2•[•r-
3t•rIR_<R*
------ Y
(22)
at the positionwhere R _<R*. Here Eris the maximum elastic
radial strain and R* is the point beneath the crater where
transition
Strain Energy
(--•r/2)l --<--Y
or
dV-- Ao(R/Ro)z{1 + [sinOo(z+ 2)/(1 + cos0o)]:}'/: dR
(16)
(21)
From (20) and Hooke'slaw it followsthat
and • is defined in Figure 2.
Substituting
(12) and (13) into (15) and thisexpression
into
(14) yields,for the volumeelementalonga streamline,
(20)
and the principal stressdifferencebetweenthe radial 'stress
p,
and azimuthal stressPois lessthan or equal to the Treseayield
stressY in the sphericalapproximation:
wheredS is the differentialalongthe streamlinegiven by
dS -- dR/cos • = u dR/ur
(19)
where from cylindricalsymmetry,0u,/0•b= 0.
We note, following Maxwell [1973], that for the on-axis
strain rate (0 = 0),
(13)
sincea volume elementalong a streamlineis given by
d.
(17)
occurs between material
which
has and has not
Whereasthe decreasein kineticenergyand gain in gravita= yielded (Figure 3).
from (17) as
tionalenergyfor a volumeelementmay be simplyexpressed, Denoting erlR_<n-
the strainenergyrate mustbe givenin termsof the principal
Tnitial
AO-•
C;avity
•r-- --Z
dR/R = -z In (R/Ro)
(23)
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. Relation
ofsurface
element
A t• lineelement
dSandstreamlines.
(Rc/R*)•:+'•• Y(z + 1)/3•
(26)
4
O'KEEFEAND AHRENS:CRATERING,STRENGTH,AND GRAVITY
For an elasticincompressible
mediumthe (shear)strainen-
which may be approximatedby
ergy per unit volumeis given by
r•e-- t•
(Er&r/dt + •o&ddt + % &,/dt) dt
(27)
Ee-• 1.5pAo[Z/(Z
+ 1)]2Rc2(Z+!)/Ro
z . R-{•+2•
dR
(41)
ne-- t•
Integrating(41) and usingthe approximationof (26) yields
{(-z) In (R/Ro)(-z)d(inR)/dt + 2[(z/2)
ßIn (R/Ro)(Z/2) d In (R/Ro)/dt} dt
(28)
•e = 1.5#
•2 In2(R/Ro)
(29)
•e = 1.5].•[Z/(Z
"['1)]2 In2[1 -- (ac/Ro)(z+!)]
(30)
Ee• 0.5Yz•oRc(Z+!)/[(z
-{"1):Roq
(42)
A morecompleteexpression
for the energy• the axialstrem
Using (9) gives
tube of an elastic-plastic
targetmay • obta•ed by substitut•g•to thesumof •tegralsof theformsof (37)and(38)
us•g (31) and (35). This yields
Et. = Ee+ Ea• •Ao/[2•q
or using(22) and (23) in (29) yields
ne= y2/6•-•
dR'
• Rc
(31)
+r•a•c'•+"/[•o•(•
+•)1
[•/•
(43)
• Rc
In the caseof materialwhichhasyielded,the energydensity
deposited
by workcarriedouton themediumis givenby the wherethesecond
tern h (35)hasbeens•p•ed by theapsum 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 thetimerequiredfor the stress
at a depthR* to expotentialenergyEs •volved h excavathg
ceedthe materialyield strength-Y. Substituting
(23) into The gravRational
the axial streamtu• of areaAoh a planeta• surfaceof den-
(32) yields
sity p is
n•-- Yz tn (R/R*) = Yz[tn(Ro/R*) + tn (R/Ro)] (33)
gg'• •/•W
Inverting the argumentof the first term in (33) and substituting(9) for the secondterm yields
n•-= -Yz tn (R*/Ro)- Yz tn [• - (Rc/•){•+!q/(z+ •)
gRz+!dR
(45)
or, if g is assumedto be nearly independentof R,
Es = AWgRc(Z+2)/(z
q- 2)
(34)
(46)
Using(23) andthen(22) to substitute
in the firsttermof (34) Equations(42) and (46) werealsogivenby Maxwell [1973].
yields
n•-- YV3• - Yz tn [• - (Rc/R){•+!q/(z
+ •)
(35)
In the elastic-plastic
deformationregionthe total strain energy is given by
Finally, the initial kineticenergyin the axial streamtube at t
-- tois given, using(1), by
E•e
=[Aop/2]
fn•[a(to)/R•]2R•
dR
=-Aopot2(to)Ro{
- 2)
(47)
• = 7½"• •d
(36)
In orderto observethe effectof varyingRo,g, and Y on the
maximum transientdepth crater it is useful to substitutethe
e. EnergyIntegrals
valueof the initial particlevelocityUorin (47) to yield
Somesimpleequationsfor final craterdepthversusplanetary gravity and (constant)strengthin termsof initial crater
parametersmay be obtainedby consideringintegralsof the
form
Etcœ
= --pUor2Ro(Z+l)/(2Z2)
(48)
Equating the initial kinetic energy in the axial tube to the
strainenergygeneratedby producingthe crater,the latter,either specified
by (42) or (44) plusthe gravitational
energyof
excavationof the axial tube, yields,for the final to initial crater depth ratio,
Ed=
•lddV
(38)
c
Rc/Ro-• (OUor2)!/(z+!)/
{2(z+ 1)[pg/(z+ 1)+ 0.5Yz/Roq}
!/(z+l)
wherethevolumeelementfor the axialstreamtubefrom(16) If Y = 0, we obtain
for 19-- 0 is
dV = Ao(R/Ro)
• dR
(39)
Rc/Ro• [2(z+ 1)Uor2/g]!/(z+!)/2(Z
+ 1)
(49)
(50)
whereaswhen g-- 0,
Substituting(30) and (39) into (37) yields
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 resultsof section3, in which valuesof 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 changingplanetarygravity and strengthfor differ(00r/og)f 4- 020r/OR
2 •-'•'-pUr2q
t-- f2)
ently shapedcrateringflows.Maxwell [1977] showsthat for
cratersfor which z -- const,for the entire incompressibleflow
regime,
Rcocg-i/(2z+O
(5lb)
We demonstratein Figure 6, and more recently,Austin etal.
[1980] have also shown,that for impact-inducedflows,z increasesfrom a value between2 and 3 along the centerline of
the flow to values of •3.5-4
(59)
where f' -- Of/OR.
From (13), for 0 -- 0 we infer that
,4 = ,4o'Rz
(60)
Ao' ---Ao/Rd
(6 l)
f ---z/R
(62)
f' --- -z/R 2
(63)
where
near 0 -- 90 ø.
then
f.
Variable YieM Strength
In (31) and (32) we have treated- Y as a constantin the integrationover time. Whereason a laboratoryscalesuchan assumptionmay be adequateonly if the coefficientof internal
friction is zero, on a planetary scale,if it is assumedthat crustal strengthincreaseswith overburdenpressureor mean principal stressas
IY] = Y, + a•
(52)
where• is meanprincipalstressor pressure
P and I Y] is allowed to increaseuntil a maximum value Y3 is achieved(the
von Miseslimit), a more complicatedmodel shouldbe considered. In addition to this staticcontributionto the pressure,al-
Substituting(62) and (63) with (59) yields
020,/OR
2 + (00,/OR)g/R= -p[a(t)]2z(z+ 1)/R(2z+2) (64)
Assuminga general solutionof (64) of the form
o,-- [a(t)12•(R)
(65)
yields upon substitution
n" + z•'/R = -pz(z + 1)/o,2•+2
(66)
Assuminga form for f] suchthat
the flow givesrise to a dynamic pressurewhich will increase
• •' GRm
(67)
the effectivepressure(D. E. Maxwell, private communication,
1979).In order to describethe dynamicpressureP along an yields, upon substitutionand equating, exponentsand coefficients of R
axial stream tube we considerone-dimensionalflow along R
thoughthe late stageflow is incompressible,
the vorticityof
with variable area ,4. Conservation
of momentum
and mass is
then
rn -- -2z
(68)
(69)
dur/dt --- OUr/Or+ UrOUr/OR-----(0Or/OR)/p
(53)
G -- -p/2
A OUr/OR+ u, OA/OR = 0
(54)
respectively.
The homogeneousform of (66),
OUr/OR
-----ud
(55)
f -- (O,4/OR)/,4
(56)
hence
•t"+ z•t'/R -- 0
(70)
has a solution of the form
where
ft - A•R(•-• + A2
Operatingon (53) with O/ORyields
02u,/OROt+ u,(O2u,/OR
2) + (Ou,/OR)
2 -- -(02o,/dR2)/p
(57)
which, using(55), may be written as
-f(du/dO 4- u2(f2 - Of/OR)-- -(020,/OR2)/p
(58)
TABLE 1. Radial Stresseso, Derived From Velocity Gradients,
Lunar Gravity, and Vorticity for z Model Flow
Radius,*
Velocity
Lithostatic
Vorticity
Pressure,
Stress,
Stress,
Mbar
Mbar
cm
9.19•
28
66
142
180
Mbar
0
8.1 x
3.4 x
1.02 x
2.7 x
0
10-4
10-•
10-4
10-5
1.2 x
2.5 x
4.3 x
5.5 x
0
10-8
10-8
10-8
10-8
1.5 x
4.4 x
2.0 x
4.9 X
10-5
10-5
10-8
10-9
*Fromthe lunarsurface(2.94g/cm3),Y• = Y3= 10-7Mbar impactedby a 10-cm-diameter
projectile.
-•Theradiusof theinnerboundary
at t -- 0 was9 cm.After2.57
of flow the boundarymoved0.19 cm downward.
(71)
Adding the solutions of the particular and homogeneous
equationand taking into account(65) yields
Or= [AiR(l-z)4- ,42- pR-2•/2l[a(t)]2
(72)
Applying the boundary conditionsthat at R -, oo, Or----O, and
at R = Re,Or----0 yields
,42 = 0
(73)
,4, = o/[2R?+'•I
(74)
respectively;thus the final solutionof (64) is
Or= [a(t)]2{R('-')/R½
('+') - R-2z}/2
(75)
In calculatingthe pressure
from the increased
dynamicradial
stressto specifythe yield strengthalongthe axisstreamtube
accordingto (52) the pressure
calculatedfrom (75) mustbe
added to the lithostatic pressure.
In the initial stagesof incompressible
flow,whenthe velocity gradientis high, Oras calculatedfrom (75) dominatesthe
principalstresses.
As is demonstrated
in Table 1, Or varied
from 1 to 10n timeseither the gravitationallyinducedstressor
that producedby the vorticityof the flow.
6
O'KEEFE AND AHRENS: CRATERING, STRENGTH,AND GRAVITY
TABLE 3. Material StrengthModel Used in CompressibleFlow
u')
L.i.I
Calculations
Y3
cr'
u•
Equation
Y
L.iJ
YI
z YI
E _<Em
Y=0
MEAN
o
Condition
Y= (.44-B# 4- C#2)(1- E/Era)
PRINCIPAL
E>Em
Parameter
STRESS
Value
A, kbar
B, kbar
C, kbar
i
I= -
PL7'+YL
•)
2.69
33.8
-901.8
Era,10!øergs/g
1.70
A, B, and C are from O'Keefeand Ahrens[1976].Em is the internal
energyfor incipientmelting under standardconditions,and E is internal energy.
o
YL
mental measurements.
We have previouslycomputedthe flow
fieldsdue to the impact of anorthositeand iron projectileson
an anorthositehalf spacefor impactvelocitiesrangingfrom 5
Fig. 4. Strengthmodelsusedin incompressible
flow calculations.
to
45 km/s [O'KeefeandAhrens,1977b].The material strength
(a) Mohr-Coulomb model; Y! representscohesivestrength,and Y3
representsmaximum or von Mises strength. (b) Shock-induced model was basedon Hugoniot elasticlimit measurementsby
strengthdegradationmodel describedby (76).
Ahrenset al. [1973] and is representativeof strongrocks. In
additionto thesefinite strengthcalculationswe 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 generalmaterialstrengthmodel,usedin this study,is a anorthositeprojecttiesimpacting an anorthositehalf spaceat
Mohr-Coulomb type with avon Mises limit, as already dis- velocitiesfrom 5 to 45 km/s under the assumptionof zero tarcussedin relation to (52). Referring to Figure 4a, the material get and projectfiestrength.The equationof stateand material
under zero mean stresshas a cohesion Y,, which increaseslinstrengthparametersused are listed in Tables 2 and 3. In Figearly with pressureas given by (51a), where a, the coefficient ure 5 we examine both the zero and the nonzero strength5of internal friction, is a material constant. The value of the
km/s impact of anorthositeupon anorthositeto determinethe
yield stressis limited to valueslessthan or equal to the von applicability of the incompressibleflow equationsoutlined in
P, PEAK SHOCK PRESSURE
Mises limit' YA.
section 2.
A number of researchers,including Knowles and Brode
[ 1977],Orphal[ 1977],and Swift[ 1977],haveproposedthat the
passageof the initial shockwave throughrock,precedingcrater excavation,weakensthe rock. Swift [1977]carriedout a series of explosiveexcavationcalculationswith a shock-induced
degradationof the material strengthmodel.
The model usedfor shockdegradationof material strength
is shown in Figure 4b. Referring to Figure 4b, if the peak
shock stressis lessthan a minimum stressfor failure, PL, then
there is no change.If the peak stressis greater than PL, then
the cohesionis given by
Y,' = (Y, - y,)(p,_.lp), +
(76)
where Y• is a lower limit on the shock alteration of the cohesion.
4. RELATING INCOMPRESSIBLE
FLOW (z) MODELS
TO COMPRESSIBLE FLOW
CALCULATIONS
The applicability of (1) was determined by plotting the
radial velocity field averagedover a given angular increment
at various times in the crater evolution for the 5-km/s case.
Referring to Figure 6, we seethat the velocity profile behind
the subkilobarshockis approximatelysatisfiedby (1). Values
shownfor the parametersa and z were obtainedby fitting the
flow regionbehindthe shockwave. Sincethe shockstrengthis
expectedto decreaseas 8 increases,
the averagevalueof a decreaseswith increasingangular coverage.The value of z near
the axis and averagedover 0ø-10 ø is 2.1 +_0.2, and the value
averagedover the whole flow field is 3.0 + 0.2 (seealsoAustin
et al. [1980]). This latter value is approximately equal to the
value z = 2.7 found by Maxwell and Seifert [1976] for calculations of cratering flow from buffed nuclear explosions.
Moreover, from the work of Oberbeck [1971] it has been
shownthat the flow fieldsfrom shallowbuffed explosivesimulate those found in impact events; thus general agreement
would be expected.
The results of the calculations of the evolution of the crater
As wasdiscussed
in section2e, the incompressible
(z) model
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,
b
Mbar
Mbar
1012ergs/g
2.936
0.145
0.705
0.751
3.965
0.128
2.357
1.258
g/cm3
Gabbroic
anorthosite
4.89
(low-pressurephase)
Gabbroic
anorthosite
(high-pressurephase)
The parametersare from the Tillotson equation of state [Ahrensand O'Keefe, 1977].
18.0
O'KEEFEANDAHRENS:
CRATERING,
STRENGTH,
ANDORAVlTY
projectilekinetic energyinto the initial targetvelocityfield is
insensitiveto materialstrength.At later timesthe solutionsdivergeand entera regimein whichthe growthrate is approximatelyconstant.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 stronglydependent
upon strength;however,the duration of this regimeis material strengthdependent.In the caseof strongmaterialsthe durationof thisregimeis shortandisdominatedby plasticwork.
In contrast,for weak materialsthe durationis long and is termMatedby gravitationalwork. It is the latter, or coasting,regime that resultsin very long computerrun times for compressibleflow calculationsfor low-strengthgeologicmaterials.
The compressibleflow code solutionswere not carried out to
the final crater depth becauseof this constraint. One of the
major advantagesof the z model is the efficiencyachievedin
' ' '1
103
7
........
I
""'""---,;/._
rl/
-
ml
....
o - •
9.0
2,5,
II
.....
0-9m
7.0
•.0,0.•
10-}
, , ,I
,
0,2
.....
I
10-}
100
RADIUS (M)
Fig. 6. Calculatedradial pa•iclc vcl•ity distributionfor 5-•/s
•pact of ano•bositcimo ano•bositc,60 •s after the impactof a 10cm-diamctcrano•bositc sphere,dcmonstrat•g averagevalues
and z which fit (1), upon averagingover diEcrcnt intc•als in 8. U•ts
of • are kilometersper second.
calculationsin the late time regime. The variable yield z
modelgivenby (52) and (75) wasusedto computethe crater
depth evolutionunder the assumptionof strengthvaluescorand gravitationalenergy.The initial projectilekinetic energy
respondingto the detailed code calculations(see Table 3).
can be convertedirreversibly by shock heating or deformaThe effect of reducingthe cohesivestrengthwas to increase
tional work into internal energy or storedreversiblyas elastic
or gravitational energy. Irreversible heating of the axial
streamtube material occursas a resultof 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 calculationof the computed by using the compressibleflow formulation deimpact of a 5-km/s anorthositesphereinto an anorthositehalf scribedby O'Keefe and Ahrens[1977b].Plasticwork occursat
spacedepictedin Figures5 and 6, the partitioningof energy both high and low stresslevels and was computedby using
in the axial streamtube was computedby usingthe valuesof both the compressibleand the incompressibleformulations.
a(to)and z obtainedfor the 0 = 0ø- 10ø sector.A shockdegra- The gravitational energy is important at late times and was
method.The partitiondation model for material strengthwas not assumedin these computedby usingthe incompressible
calculations.The z model parametersusedare summarizedin ing of energy as a function of meteorite radius is shown in
Table 4. There are severalpossibleforms that the impacting Figures 7a and 7b for von Mises and Mohr-Coulomb-von
energymay take. The partitioningof energyfor impactat var- Mises materials, respectively.
Referring to Figure 7a, in avon Mises material (Y, = Y3 =
iousscales(assuming10-to (2 x 107)-cmdiameterprojectiles)
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 storedelasticenergy actial 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 casestudied, approximately than 2 x 105cm. For meteoriteradii greaterthan 2 x 105cm
102time stepswith severaliterationsper time stepwerecar- the storedelasticenergyis decreased,and gravitationalpotenried out, usingthe abovereferencedintegrals,to reduceall the tial energyincreased,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
meteorite radii lessthan 104cm. In the range of meteorite
IOOCALCULATION
STOPPED-FINAL
CRATER
•.•
radii between l04 and 107 cm the overburdenpressurein•
ZERO
•'•'
creasesthe yield stress,and part of the energy is partitioned
?•
STRENGTH'-•
I •..,.•'-/•,-.'/' '• Yi
=1.0
xI0-I0
I-__HYDRODYNAMIC •
Y•=2 69 x IO-3
into stored elastic energy. At meteorite radii greater than 2 x
105cm the gravitationalenergystartsto becomeimportant.
the crater depth by more than a factor of 4 and the crater formation time by more than 2 ordersof magnitude.
,-,
NOT
FINAL
CRATER
7 //--DEPTH
/.•'•
./
n,' IO
'
ß
x IO-3
TABLE 4.
Summary of z Model ParametersUsed in Cratering
Calculations
U
/
I
IO
•----Z
CODE
STARTING
•_• SHOCK-DOMINATED
REGIME
I
IO2
I
IO3
I
IO4
Case
IO5
TIME(p.SEC)
Fig. 5. Crater depthversustime calculatedby usingcompressible
flow calculationand linked incompressible
calculations(z code)for
the caseof zero strengthand variousMohr-Coulomb strengthcrusts
with lunarsurfacegravity.Asterisksindicatefinal craterdepths.Units
of yield strengthare megabars.
Material StrengthParameters
Yl, Mbar
Y2, Mbar
Y3, Mbar
1
10-•ø
1.0
2
3
l0 -lø
10-5
1.0
1.0
2.7 X 10-3
10-5
dry sand
hardclay
1.0
1.0
2.7 X 10-3
2.7 X 10-3
soft rock
hard rock
4
5
10-5
2.7 X 10-3
10-lø
Typical
Medium
fluid
The densityis 2.94g/cm3, andthe initial velocityis 0.075km/s.
8
O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY
and the final crater formation
•
m
IRREVERSIBLE
DEFORMATIONAL
WORK
Coulomb-dominated
growthis around l0 3 cm and is depen-
Z .4• (VONMISES)
t.d
o,;>
•: I01
10
2
10:5 10
4
10
5
10
6
t
10
7
METEORITE RADIUS, (CM)
dent upon the specificcombinationof Yi and Y3 assumed.In
the Mohr-Coulombregime (R,n -• 104-106)the overburden
stressincreasesthe yield stressof Mohr-Coulomb materials,
and the formationtime scalesas R,n/•,where 0 < fl < 1. In
terms of impacting meteorite radius the Mohr-Coulomb regime extendsto radii at which the overburdenstressincreases
the yield strengthof the bulk of the material involved in the
excavationprocessto the von Mises limit. In the von Mises re-
z
i-
('-')
n,LL 4
I01
IRREVER_S
IBL..E_•__ /__• ^CT,C \
DEFORMATIONAL
WORK
ELASTIC
••
GRAVITATIONAL
(MOHR-COULOMB)
gime(R,n'" l0 s cm) the magnitudeof the cohesivestrengthis
102
I03
104
105
106
METEORITE RADIUS, (CM)
lessimportant than the von Mises limit. Here the crater formation time again scaleslinearly with meteorite radius. Fi-
107
nally, in the gravityregime(R,n> •-106-107cm) the gravitational work of excavationdominatesthe work done against
the material strength.In this regime the crater formation time
Fig. 7. Energypartitioningversusmeteoriteradiusfor a 5-km/s
impactof anorthositeon anorthositein the gravityfield of the moon,
where the strengthof the planetarysurfaceis assumed(a) to be a constant2.7 kbar and (b) to increasefrom a cohesivestrengthof 10 bars
to a maximum strengthof 2.7 kbar as a Mohr-Coulomb solid.
is proportionalto R,nl/2.
The crater depth was also computed as a function of the
meteorite radius. As in the case of the crater formation times,
the calculational
6.
radius
R,n. For the moon the range of meteorite radii separatingthe
FLAST,C
••
cohesive strength-dominated crater growth and the MohrGRAVITATIONAL'--"•
)-'6•
--
time scales as meteorite
CRATER
FORMATION
results obtained
can be cast in terms of four
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 calculatingcrater depth formation times for lunar magnitudesof cohesivestrength. In the Mohr-Coulomb regravity for a 5-km/s impact of anorthositeon anorthositeas a gime the overburdenstressincreasesthe yield stresssuchthat
function of meteorite radius Rm (Figure 8) it was discovered the relative crater depth decreaseswith increasingmeteorite
that the resultsare approximatelydescribablein termsof dif- radius. In the von Mises regime the depth is independentof
ferent assumedmaterial strengthproperties(Table 4). These the cohesivestrengthand 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 threeregimesare determined material strength.For example,as is shownin Figure 9, in the
by the material strength.In the cohesivestrengthregime (Rrn case of the moon, for avon Mises limit of Y3 = 2.7 x l0 -3
< 104 cm) the cohesivestrengthof the material dominates, Mbar the Mohr-Coulomb regime extendsfrom meteorite radii
1000
COHESIVE
STRENGTH
......
,.•......
•
REGIME
._,oo-
1
i
....
MOHR-COULOMB
.....
REGIME
,,
•
T•
o
•
••
_
•/
I -
•
'3-'• .•/
•//
• I0-••
7'
10
-5
/
;-
REGIME'
=2.7xI0-•
Y3 27x10 3
_
_
Y•=I
u
m0-2 _
i0-3
D
m0
-
-
I
I
m0
2
I
m0
)
•ETE0•ITE
FiS. 8. Craterdepthfo•afion t•e for a 5-•/s
m0
4
•DIU•,
I
m0
5
m0
6
m0
7
(C•)
•pact on •e moonveBusano•hositemeteo•teradiusfor va•ous
Mohr-Coulomb strenSthmodels.
O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY
of--, 1• to l0 s cm, the von Misesregimeextendsfrom --,10s to
• 107cm, and the gravity regimeextendsfrom • 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 pressurereducesthe magnitude of the
cohesivestrength Y, and not the yon Mises limit Y3. The
shockwave amplitudesused to degradethe value of Y, were
obtainedfrom the pressuredecaycalculationsgiven by Ahrens
and O'Keefe [1977]. The resulting relative crater depth as a
function of meteoriteradius for a $-km/s impact is presented
in Figure 10. The example chosenhas an initially large cohesive strengthwhich was equal to the yon Mises limit, Y, = Y3
-- 2.7 x 10-3 Mbar. The strengthdegradationeffectwas expectedto be mostpronouncedin this case.The passageof the
shockwave reducedthe cohesivestrengthto 10-'ø Mbar for
shockpressures
greaterthan 10-4 Mbar in one caseand 10-3
Mbar in another. As is demonstrated in Figure 10, in the
I
9
I
I
I
I
I
Zyi-2.7
xI0-3
y3:2.7x I0-3
I
I100
I
I01
I
102
103
I
104
I
I
105
106
METEORITERADIUS,(CM)
Fig. 10. Normalized crater depth versus anorthosite meteorite
radiusfor a 5-km/s impacton the moonfor differentstrengthdegradation modelsof the type shownin Figure 4b..
the case of Mohr-Coulomb materials (cases2-5 in Table 4)
and large yon Miseslimits (e.g., Y3 = 2.7 x 10-3 Mbar) the
depthscalesas E•'/3'4.This intermediatetype of scalingwas
first proposedby Vaile [1961] on the basisof correlationsof
cohesiveregime(meteoriteradius<10s cm), shockdegrada- chemical and nuclear explosiondata.
tion could be an important mechanismin determiningcrater
depth. The craterdepth more than doubleswhen rock is processedby a thresholdshockpressureof 10-4 Mbar. The effect
diminishesin the Mohr-Coulomb regime and vanishesin the
yon Mises and gravity regimes.
The effect of strengthon the transientcrater depth versus
kinetic energy of impact (E) is presentedin Figure 11 on the
basisof variousstrengthslisted in Table 4. We emphasizethe
word transient, becauseobservationally,it has been found
that the crater depthsdo not exceed•$ km, regardlessof diameter, on the moon, Mercury, and probably other planets.
We believethat the limit observedin crater depth for larger
cratersresultsfrom later stagesof crater developmentthan
those describedhere [e.g., Melosh, 1977; McKinnon, 1978].
Like the relationshipof the transientcrater growth times and
crater depth versusmeteorite radius, the impact energy can
also be describedas occurringwithin severalregimeswhich
are relatedto the assumedstrengthmodel. In the caseof very
weak materials(case 1 in Table 4) the crater depth is dominated by gravity, and the depth scalesas E '/4 [e.g., Chabai,
1965;Gaultet al., 1975;Sauer,1978].In the caseof very strong
7.
"b.%
•
I
•
I
•o
I
I
•
•y•lo-,O
•
r•
Yi='O
-IO ••,
i0-
o
•4•0
Io3
o• I
I_
(77)
-CHEMiCAl-iNUCLEAR
'I•"
I
--
ß•...__E4.o
_
-
I
/// 'E•-•.
MODIFIED
REGIME
_
kDOMINATED/
%
E$.4
bJ
•
MODEL
EXPERIMENTS
--
8
.
1
• ,r'-
OF INCOMPRESSIBLE
CRATERING
Y (Mbar)= 25 x 10-'6 + 33 x 10-'4 P
3 -y
SAND
The comparisonof impact cratering experimentswith detailed calculationshas to date, surprisingly,only been carried
out in the case of metals [Rosenblattet al., 1970] and compositestructures[Gehring,1970].
Our objectivewas to find a setof well-controlledimpact experimentsin a medium of geologicinterestand compare the
incompressibleflow model to those resultsso that we could
have confidencein computingcrateringover a broad range of
conditions.The sand impact experimentsof Gault and Wedekind[1977] providean appropriatedata base.The propertiesof
the sandusedin theseexperimentswas measuredand demonstrated a Mohr-Coulomb type behavior for which the yield
stressin shear can be approximatedby
materialsthe craterdepthscalesas E '/3 at impactenergiesof
_<10
•ø ergs,whereaswhen gravitationalforcesdominate,the
craterdepth scalesas E '/4 as energiesapproach103øergs.In
ioo
COMPARISON
WITH
10
or'
3-
/
BOWL- SHAPED
HARD ROCK
•
_
o 2
ix
STRENGTH
DOMINATED] •
o
Y•:•.•,o-•
_.1
YON•,S•S
--
CRATER
REGIME
_
2/,
•OHR-COULO•B
AND YON •ISES
u
I100
i01
i0•'
103
104
105
106
I07
METEORITE RADIUS, (CM)
Fig. 9. Normalized crater depth versus anorthositc meteorite
radiusfor a 5-km/s impact on the moon for different Mohr-Coulomb
strengthmodels.
o
io
I
I
I
I
I
15
20
25
30
35
LOGIo IMPACT ENERGY (ergs)
Fig. 11. Log,0 of transientcrater depth versusanorthositeimpactor energy for a 5-km/s impact on the moon for different strength
models specifiedin Table 4.
10
O'KEEFE AND AHRENS: CRATERING, STRENGTH, AND GRAVITY
TABLE 5. Parametersof z Model Used for ComparisonWith
Experimentsof Gaultand Wedekind[ 1977]
Material StrengthParameter
Value
Y•, Mbar
Y•_,tan q•
2.5 x 10-•6
0.33
Y3,Mbar
2.7 x 10-3
The material strengthparameterswere calculatedfrom a graph of
thedataof Gaultand Wedekind
[1977].The densityis 1.65g/cm3, and
the initial velocity is 0.075 km?s.
where1ø is the overburden
stress,
whichis assumed
to be equal
to the averageprincipal stressas usedin (76). Gault and Wedekind's[1977] experimentswere carried out on a size scale
between that of cohesion-and that of gravity-dominated scaling and thusprovidea stringenttestof presentmodelingcapability in the Mohr-Coulomb regime.
These experimentsconsistedof impacting aluminum pro-
jectilesinto sandat velocitiesin the rangeup to 6 km/s. Measurements of the crater diameter as a function of time for vari-
ous values of acceleration, simulating various surface
gravities,were carried out. The depth to diameterratios remained nearly constantover the range of test conditions,so
that the scaling of characteristiccrater diameter and depth
formation times should be proportional (i.e., have the same
scaling exponent).
We have made a comparisonof the scalingof the crater formation time as a function of the size of impacting projectile
and surfacegravity. The objectivesof thesecalculationswere
to show a correlation betweentheory and experimentsof the
scalingof crater depth with time and gravity. This scalingis
insensitiveto the exact magnitude of the initial velocity field.
Becauseof this insensitivitywe did not use the detailed com-
means of coupling the resultsof compressibleflow finite difference calculations
carried out to stress levels below the com-
pressionaldynamic yield point (Hugoniot elasticlimit) with
the incompressiblefluid flow model of Maxwell [1973]. The
incompressibleformulation is applicableto the flow after the
stresslevelshad decayedto -• 1.5 kbar, resultingfrom the impact of a 5-km/s anorthosite sphere on a half space of
anorthositein the gravity field of the moon. We found that,
like the case of explosions,the radial particle velocity at a
given time field can be representedas being proportionalto
(I/R) z, wherez • 2 along the axisstreamtube and z • 3 when
averagedover a larger portion of the half space.We outlined
the physical and mathematical framework of the Maxwell
model. This model assumesthat the kinetic energywithin the
incompressibleflow generatedby the passageof the shock
wave in the crater excavationregion is convertedinto elastic,
plastic, and potential energy by doing work againstthe target'smaterial strengthand body forces(in this case,planetary
gravity); thus the final crater depth and volume is determined
from conservationlaws. We developedexpressionsfor the final crater depth in the strength-dominatedcase, for fixed
gravity g, in terms of target densityp and yield strengthY,
which is proportionalto [pUor•/Yz]
•/(•+•), where Uoris the
radial particlevelocityon the surfaceof the early time cavity
of radius Ro. In addition, we developed comparable expressionsfor the gravity-dominatedcasefor a fixed material
strength,in which the final crater depth is proportional to
Ro[2(z+ 1)Uofi/g]
•/•+•>.When Y is specified
by a Mohr-Coulombrelationof theform Y-- Y, + a• upto a vanMiseslimit
Y3, analytical solutions were not found, and the effect of
strengthon the crater depth wascomputednumericallyby using the lithostaticpressurein additionto the dynamicpressure
pressibleflow code to computethe initial flow field in sand resultingfrom the vorticity of the flow.
The effect of shockwave degradationof material strength
but rather assumedthe sameflow field that we calculatedpreviously[O'Keefeand/lhrens, 1977b],associatedwith 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- shockpressureand was found to have the effect of increasing
ity size Ro, z, and velocity magnitude a (e.g., (1) and (7)). A strongrock crater excavationdepthsby as much as a factor of
2 for impactorradii of • 103 cm.
summary of the z model parametersis given in Table 5.
The scalingof transient crater depth with impact energy
The craterdepth formation time wascomputedby usingthe
z model as a function of crater depth rangingfrom 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-strength
surfacesat impactenergiesE • 102øergsand
10-3 to 0.4 s. Referring to Figure 12, the slopeof the calcu- hencescaleasE '/3, whereasfor weakmaterialsand for strong
lated crater formation time or the scaling exponent is very surfaceswhere the impact energyis •10 3øergs, the crater
closeto the value found by Gault and Wedekind over the exI0 O,
perimental range.
The crater depth formation time was also computedover a
rangeof surfaceaccelerationslessthan lg. Referringto Figure
Experiments
Gault & Wedekind
13, the slopeor scalingexponentin that casealsoagreedquite
(1977)
well with the value found by Gault and Wedekind (t oc
g-O.6,8).
Note that for gravity-dominated
cratersthe craterformation time scalesas g-O.6•_5;
thusthe small shearstrengthof
sand decreasesthe formation time in relation to strengthless
gravity scaling.Finally, transientcrater depth versusg is calculated over the range studiedby Gault and Wedekind[1977]
(exceptthat they determinedthe diameterD). In Figure 14 we
show that the calculated exponent of depth (or diameter) is
again close to the value obtained by Gault and Wedekind
[1977].Both are lessthan the D ocg-O.•_5
expectedfor pure
gravity scaling.
8.
CONCLUSIONS
The effectof varying planetary crustalstrengthand surface
gravity on the depth of impact craters was investigatedby
•--Range
of
-• •
Slope
o:D
0-477
Gault and Wedek•nd
/
..•.•'
•,•,•" •-•Present
•,
•,•.•---•
•,
CaJculahons
I
i
I
i
ioo
ioI
io2
io3
04
Crater Depth(cm)
Fig. 12. Variation of crater depth formation time with crater
depth usingthe presentincompressible
flow formulationwith the assumptionof a 5-km/s anorthosite-anorthosite
velocity field. Shown
alsois the range of crater depths(diameters)observedin the sand experimentsof Gaultand I•edekind[ 1977]and the associated
slopeand
scalingrelationship.
O'KEEFE AND AHRENS: CRATERING, STRENGTH,AND GRAVITY
11
io z
depthsscaleasE 1/4.However,for a broadrangeof observable
cratersizesand expectedstrengthparametersthe depth scales
as -E •/3'4,which is consistentwith chemical,nuclear, and impact crater statistics.
The transientcrater depth and formationtime was shown
to varygreatlydepending
uponplanetarystrength.
For impactingmeteoriteradii 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
to 32; this behavior is characteristicof what we call the cohe-
Iz
sivestrengthregime.For meteoriteradii greaterthan 103 cm
the crater depth decreasedsharply with increasingmeteorite
radius for Mohr-Coulomb
z
materials, and its rate was un-
changed for von Mises materials. We denote this behavior
with the terms Mohr-Coulomb and von Misesregimes.At suf-
i0 I
I
I
I
I
0.1
i
1.0
ACCELERATION
g
ficiently large meteoriteradii (e.g., -10 s to -106 cm) the
Fig. 14. Variationof transient
craterdepthD withgravityusing
strengthof Mohr-Coulomb materials is limited by the von
incompressible
flowformulation
withtheassumption
of a
Miseslimit. Finally, for impactorradii of •10 ? cm the crater thepresent
5-km/sanorthosite-anorthosite
velocity
field.Shownalsois theslope
depth and formation time are dominatedby gravity.
andscaling
relationship
foundby GaultandWerekind
[1977]in sand
The partitioning of energy in the axial streamtube was examined for impactson the moon in which the surfacestrength
was constantand high (2.7 kbar) versusone whose strength
increasedwith confiningpressurefrom -0.01 to 2.7 kbar. For
the former (von Mises type) material some75% of the initial
kinetic energy goes into reversibleelastic strain energy, for
meteorite radii less than 2 x l0 s cm, whereas for the latter
(Mohr-Coulomb type) material, virtually all the impact energy is depositedas shock,or deformational,energy for impactorshavinga diameterlessthan -104 cm. Moreover,it was
demonstratedthat some 25% of the impact energy goesinto
work againstthe moon'sgravity field for meteoriteradii approaching-10 ? cm.
The experimental data of Gault and Wedekind[1977] for
experiments.
kind's reported shear yield strengths,crater formation times
were found to vary with gravity as g-o.ss,whereasthey observeda g-O.6•8
dependence
over a rangefrom 0.03 to 0.12 s.
Similarly, a maximum crater depth dependencewas calcu-
latedwhichvariedasg-o.•9,whereasthe diameterdependence
of Gault and Wedekind'sexperimentsfollowed a g-o.•6sdependence.The comparisonof the presentincompressibleflow
calculationwith experimentsdemonstratesthe need for data
describingthe strengthpropertiesof planetary surfacesin order to relatecratersizepopulationsto impacthistoryon planets with differing surfacematerialsand gravities.
Acknowledgments.We are grateful for both the help and the accrateringin sandwith energiesof 6 x 10? to 10•øergsunder
effectivegravitiesof 0.07g-lg providea test of both strength cessto unpublishedmaterialsso generouslyprofferedby D. E. Maxwell of ScienceApplications,Incorporated.The final manuscripthas
effectsand gravity on the presentincompressiblettow calcu- benefitedfrom the rigorousreviewsof R. Schmidtof the BoeingCom-
lations. The starting condition for the calculations was the
scaled ttow field for an anorthositesphere impacting an
anorthositehalf space at 5 km/s. Using Gault and WedeI
.,,,
•;•
I
i
i
.,
t ([ g-0.618
JGAULT
AND
WEDEKIND
(1977)
pany and H. Moore of the USGS. Technical discussions
with B. Minster and J. Melosh and computationalassistance
by M. Lainhart are
also appreciated.The researchwas supportedunder NASA grant
NSG 7129. Contribution3300, Division of Geologicaland Planetary
Sciences,California Institute of Technology.
REFERENCES
Ahrens,
T. J.,andJ. D. O'Keefe,
Equations
of stateandimpact-inRESENT
RESULTS
•....
t o::•0.58
duced shock-waveattenuation on the moon, in Impact and ExplosionCratering,edited by D. J. Roddy, R. O. Pepin, and R. B.
Merrill, pp. 639-656, Pergamon,New York, 1977.
Ahrens, T. J., J. D. O'Keefe, and R. V. Gibbons, Shock compression
of a recrystallizedanorthositicrock from Apollo 15, Geochim.Cosmochim. Acta, 3, 2575-2590, 1973.
Austin, M. G., J. M. Thomsen, and S. F. Ruhl, Calculational investi-
gation of impact crateringdynamics:Material motionsduring the
cratergrowthperiod(abstract),Lunar Planet.$ci., XI, 46-48, 1980.
Batchelor,G. K., An Introductionto Fluid Mechanics,615 pp., CamLLI
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 appliesto the meteoroidhazard, NASA Con10-2
I
I
I
I
tract. Rep., CR- 757, 1967.
O. I
1.0
Bryan, J. B., D. E. Burton, M. E. Cunningham,and L. A. Lettis, Jr., A
ACCELERATION g
two-dimensionalcomputersimulationof hypervelocityimpact craFig. 13. Variationof craterdepthformationtime with gravityustering: 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-anorthosite
velocity field. Shown also is the Chabai, A. J., On scalingdimensionsof cratersproducedby buried
slopeand scalingrelationshipfoundby Gaultand Wedekind[1977]in
explosions,J. Geophys.Res., 70, 5075-5098, 1965.
sand experiments.
Charters,A. C., and J. L. Summers,Somecommentson the phenomb.I
12
O'KEEFE AND AHRENS:CRATERING,STRENGTH,AND GRAVITY
Nav. Ordnance Lab., White Oak, Silver Spring, Md., 1959.
Dienes,J. K., and J. M. Walsh, Theory of impact:Somegeneralprin-
targetstrengthwith increasingsizeof cratersfor hypervelocityimpact craters,Proc. HypervelocityImpact Symp. 7th, 4, 35-45, 1964.
Neukum, G., Lunar cratering,Philos. Trans. R. Soc. London, Ser. •4,
ciplesof the method of Eulerian codes,in High VelocityImpact
Phenomena,edited by R. Kinslow, pp. 45-156, Academic, New
Neukum, G., and D. V. Wise, Mars: A standardcratercurveand pos-
ena of high speedimpact, Rep. NOLTR 1238, pp. 200-201, U.S.
York, 1970.
Gaffhey,E. S., Effectsof gravityon explosive-craters
(abstract),Lunar
Planet. Sci. Conf. 9th, 3831-3842, 1979.
Gault, D. E., Displacedmass,depth,diameterand effectsof oblique
trajectoriesfor impact cratersformed in densecrystallinerocks,
Moon, 6, 32-44, 1973.
Gault, D. E., and J. A. Wedekind, Experimentalhypervelocityimpact
285, 267-272, 1977.
sible new time scale, Science, 194, 1381-1387, 1976.
Oberbeck,V. R., Laboratorysimulationof impactcrateringwith high
explosives,J. Geophys.Res., 76, 5732-5749, 1971.
O'Keefe, J. D., and T. J. Ahrens,Shockeffectsfrom a largeimpacton
the moon, Proc. Lunar Sci. Conf. 6th, 2831-2844, 1975.
O'Keefe, J. D., and T. J. Ahrens, Impact ejecta on the moon, Proc.
Lunar Sci. Conf. 7th, 3007-3025, 1976.
O'Keefe, J. D., and T. J. Ahrens,Meteorite impact ejecta:Dependenceof massand energylost on planetaryescapevelocity,Science,
in quartzsand,II, Effectsof gravitationalacceleration,
Impactand
ExplosionCratering,editedby D. J. Roddy,R. O. Pepin,and R. B.
198, 1249-1251, 1977a.
Merrill, pp. 1231-1234,Pergamon,New York, 1977.
Gault, D. E., J. E. Guest,J. B. Murray, D. Dzurisin,and M. C. Malin,
O'Keefe,J. D., and T. J. Ahrens,Impact-inducedenergypartitioning,
melting and vaporization on terrestrial planets, Proc. Lunar Sci.
Somecomparisons
of impactcraterson Mercuryand the moon,J.
Conf. 8th, 3357-3374, 1977b.
Geophys.
Res.,80, 2444-2460, 1975.
Gehring,J. W., Jr., Theoryof impacton correlationwith experiment, O'Keefe, J. D., and T. J. Ahrens,Impact flowsand craterscalingon
the moon, Phys. Earth Planet. Inter., 16, 341-351, 1978.
in High VelocityImpactPhenomena,
editedby R. Kinslow,pp. 105156, Academic, New York, 1970.
Orphal, D. L., Calculationsof explosioncratering,II, Crateringmechanicsand phenomenology,in Impact and ExplosionCratering,
Holsapple,K. A., and R. M. Schmidt,A material-strength
modelfor
editedby D. J. Roddy, R. O. Pepin,and R. B. Merrill, pp. 907-917,
apparentcratervolume,Proc.LunarPlanet.Sci. Conf loth, 27572777, 1979.
Johnson,S. W., J. A. Smith, E.G. Franklin, L. K. Morashi, and D. J.
Teal, Gravity and atmospheric
pressureeffectson craterformation
in sand,J. Geophys.Res., 74, 4838-4850, 1969.
Knowles, C. P., and H. L. Brode, The theory of crateringphenomena:
An overview, in Impact and ExplosionCratering,edited by D. J.
'Roddy, R. O. Pepin, and R. B. Merrill, pp. 869-895, Pergamon,
New York, 1977.
Maxwell, D. E., Cratering flow and crater predictionmethods,Rep.
TC•4M 73-17,50 pp., Phys.Int., San Leandro,Calif., 1973.
Maxwell, D. E., Simple Z model of cratering,ejection,and the overturned flap, in Impact and ExplosionCratering,edited by D. J.
Roddy, R. O. Pepin,and R. B. Merrill, pp. 1003-1008,Pergamon,
New York, 1977.
Maxwell, D. E., and R. Seifert, Modeling of cratering,close-in displacementsand ejecta,Rep. DN•4 3628F, p. 111, Phys. Int., San
Pergamon,New York, 1977.
Rosenblatt,M., J. J. Piechocki,and K. N. Kreyenhagen,Cratering
and surfacewavescausedby detonationof a small explosivecharge
in aluminum, D•4SA Rep. 2495, Shock Hydrodynamics,Sherman
Oaks, Calif., 1970.
Sauer,F. M., The influenceof targetmaterialstrengthon the scaling
of crateringand ejectadata, Lunar Planet. Sci., IX, 993-995, 1978.
Schmidt, R. M., and K. A. Holsapple, Theory and experimentson
centrifugecratering,J. Geophys.Res., 85(B1), 235-252, 1980.
Swift, R. P., Material strengthdegradationeffecton crateringdynamics, in Impact and ExplosionCratering,edited by D. J. Roddy, R. O.
Pepin, and R. B. Merrill, pp. 1025-1042, Pergamon,New York,
1977.
Vaile, R. B., Jr., Pacificcratersand scalinglaws, J. Geophys.Res., 66,
3413-3438, 1961.
Viktorov, V. V., and R. D. Stepenov,Modeling of the action of an exLeandro, Calif., 1976.
plosionwith concentratedchargein homogeneous
ground(in Russian), Inzh. Sb., 28, 87-96, 1960. (English translation,Rep. SCL-TMcKinnon, W. B., An investigationinto the role of plasticfailure in
392, SandiaLab., Albuquerque,N.M., 1961.)
crater modification,Proc. Lunar Planet. Sci. Conf 9th, 3965-3973,
1978.
Whitaker, E. A., Artificial lunar impact craters:Four new identifications, N•4S•4 Spec.Publ., SP-315, 29-39-29-45, 1972.
Melosh, H. J., Crater modificationby gravity:A mechanicalanalysis
of slumping,in Impact and ExplosionCratering,edited by D. J.
(Received September 1, 1979;
Roddy, R. O. Pepin, and R. B. Merrill, pp. 1245-1260,Pergamon,
New York, 1977.
revisedJuly 1, 1980;
Moore, H. J., D. E. Gault, and E. D. Heitowit, Change of effective
acceptedJuly 21, 1980.)