Theory and equations for Craters from Impacts and Explosions. Keith A. Holsapple

Transcription

1 Theory and equations or Craters rom Impacts and Explosions Keith A. Holsapple 1. Dimensionless Forms As the primary example, the volume V o a crater ormed by a given impact can be expected to depend on the impactor radius a, its velocity U, and its mass density δ. Note that those three variables also deine the kinetic energy, momentum, and mass o the impactor. The target has some strength measure Y, a mass density ρ and the surace gravity is denoted as g. Then there is some unctional relationship: V = [{ a,u,δ}, { ρ,y}, g] (1) There are 7 variables in this relation, and three independent dimensions, so the standard tools o dimensional analysis can be used to obtain the reduced orm using 4 dimensionless groups: ρv m = ga U, Y ρu, ρ, m = 4π δ 3 δa3 () which is commonly written as π V = π,π 3, ρ δ (a) Generally most o the eects o the impact are in a region very much larger than the impactor. In that case a urther approximation is to suppose that the impactor acts as an instantaneous point-source, which has neither length nor time measures. Then it can have only one single measure o its magnitude, not three independent ones. Holsapple 1983 coined the term coupling parameter or that measure. It must be some single power-law variable o a, Uand δ o the orm C = au µ δ ν (3)

2 where µ and ν are exponents to be determined. This reduces the number o independent variables by two, and now the most general orm can be written in terms o only two dimensionless groups, either as or ρv m ρv m Y ρu ga U ρ +µ ρ 3ν 1 6 ν µ + µ = F ga ρu U Y = G Y ga ρu U + µ ν ρ + µ ρ A careul examination o these orms will divulge that indeed the only dependence on the impactor is indeed the combination o Eq. (3) in all places. The irst orm is the more useul when the event is small and dominated by the target strength. Then the gravity can be ignored, and the rhs o (4) is just a constant, so that V m ρu ρ Y ρ 1 3ν deining what is called the strength regime o cratering. In the other limit, the event is suiciently large and gravitational stresses dominate the strength measure. Then, dropping the strength term, the orm (5) gives or this gravity regime V m ga ρ U + µ ρ + µ 6ν + µ A general orm with those limits and that interpolates between these two regimes is taken as ν + µ (4) (5) (6) ρ π V = K 1 π 6ν µ π V = ρv m, π = ga U, π 3 = + K ρ π 3 δ Y ρu 6ν +µ + µ (7)

3 which is the starting point or the its here or impact cratering. (The last exponent o (7) is oten denoted as α.) The two constants K 1 and K and the two exponents µ and ν come rom experiments and the database. It is airly well established (see the reerences) that or relatively dissipative materials such as dry soils 1 the exponent µ is about 0.4, and or wet targets is about The exponent ν is 1/3 i it is the mass and velocity o the impactor that determine its measure, but experiments give uncertain values, ranging rom about 0. to Impact Cratering Volume In Holsapple 1993, igures are given or impacts into each o six target types, with deinite values o the two exponents, in the igures 4, 5, 6 and 7 Those were obtained by the same data as now being considered, and the impact data is augmented by the act that impacts give about the same volume as an explosive with the same speciic energy and mass density i the explosive is buried about 1.5 impactor radii. (see Holsapple, 1979) Here I take the same curves as those igures, and added two more: lunar regolith and cold ice. Cold ice was assumed to have the same gravity regime crater size as the other non-porous materials. Estimates or the strengths o the materials were made, and then the constant K was chosen to give the strength regime asymptote shown in the igures. The results used in this web application are then (units are all cgs): 1 That is, nominally dry : even desert alluvium can have a ew % moisture.. Note that there are errors in the Table 1 and; in the legends o the igures, the exponent on π 3 is incorrectly written as (+ µ)/ µ, it should be (+ µ)/. However, the curves are correct.

4 Material K 1 K µ ν Y (dynes/cm ) ρ (gm/cm 3 ) Water Dry Sand Dry Soil E6 1.7 Wet Soil E6.1 Sot Rock E7.1 (Hard Soil) Hard Rock E8 3. Lunar E5 1.5 Regolith Cold Ice E These strength and mass density values can be changed by the user i the choice Other Soil or Other Rock is made in the pull-down menu. So ar, the other two properties shown, the porosity and riction angle are not explicitly used in the estimates; their eects are implicitly included by the division o the targets into the material types. For the programmed impactors the mass density is needed. I use Impactor type Mass density δ Aluminum.7 Plastic 0.95 Steel 7.8 C-Type 1.8 S-Type 3.0 Comet 0.8 Any changes are made by choosing Other in the pull-down menu or the impactor type. Gravity is pre-set or Terrestrial, Lunar or two asteroid diameters, the input is arbitrary i Other is selected. The velocity can be set to any value, but a warning ensues or a values below 1 km/sec, where the data is sketchy and the point source assumption becomes iy. For non-vertical impacts, the vertical component U cos(θ) is used. The

5 energy and mass o the impactor are calculated and presented, as well as the Pi-groups and the crater volume..1 Impact Crater Shapes: Simple Craters The shapes o simple craters are calculated rom R=K r V 1/3 D=depth=K d V 1/3 The values indicated by the data and programmed are: Material K r K d Water Dry Sand Dry Soils (some cohesion) Sot Rock Cold Ice The rim diameter is assumed to be 1.3 times the excavation diameter and the lip height 0.36 times the rim diameter, consistent with the data and measured lunar simple craters. The ejecta volume is assumed to be 80% o the excavation volume. The crater ormation time is rom Schmidt and Housen 1987, and the Figure 1 in Holsapple, 1993a as T = 0.8 V1/3 g (8). Melt and Vapor Volumes Melt and vaporization o target material occurs when the initial impact pressure is high enough. For melt, I assume that the velocity threshold is U = 10E melt. I take a generic value or the melt energy or silicates as 5E10. I use the less than energy scaling rom Holsapple, 003 matched to some o the results rom Pierazzo et al 1997 and get U V melt = 0.5V projectile 5E (9)

6 Vapor production is in a volume much closer to the impactor, so I use strict energy scaling with a generic vapor energy o 1.5E11: U V vapor = 0.4V impactor I have not yet added the melt and vapor or impacts into ice, there are signiicant questions abut its many phases at cold temperatures. 1.0 (10).3 Complex Craters For craters with a simple radius greater than some value R *, the simple excavation crater with the radius R e undergoes a late-time readjustment into a much broader and shallower complex crater. The data or lunar craters by Pike 1977 gives a transition to complex shapes beginning at 10.6 km rim diameter. The transition in rim heights begins at the larger size,.8 km diameter. The onset o lat loors is gradual, but is ully developed at 0 km diameter. Let R r t denote the inal rim radius, and R r the transient (simple) rim radius. Holsapple, 1993b presented an analysis o the transormation rom simple to complex craters. It is based on an incompressible readjustment rom the details o simple crater shapes measured in laboratory experiments and those observed or lunar craters, using primarily the data o Pike The approach is outlined in Holsapple, 1993a. The primary result is an expression or the ratio o the inal to transient rim radius: R r R r t =1.0 R r R * which gives that, using the ratio 1.3 or the transient rim to excavation rim radii, (11) R r =1.33R e ( R * ) (1) For the transition radius I assume that R * Y ρg and, or lunar craters D* = R * is 10.6 km. The Pike data or lunar craters gives or the depth o complex craters d =1.044( D r ) in km units. This matches the simple crater result, d = 0.D r transition onset using the dimensionally consistent orm at the

7 d = 0.D * D r D * (13) For the rim height, Pike gives h = 0.36( D r ) or complex craters and h = 0.036D or simple. With the transition at.8 km diameter, that gives the equation h = 0.036D * D r D * (14) The lat loor diameter is given or lunar complex as D = 0.187( D r ) 1.49 or diameters greater than 0 km. Assuming this begins at zero at the 10.6 km onset o complex craters, the it used was D = 0.9 D * ( ) 0.49 ( D r D * ) 1.49 (15) Finally, the volume below the rim uses a proile with a lat loor, and a uniorm slope rom the loor diameter to the rim diameter with the rim height. It is given as vol = πd 4 D D ( r D )( D r + D ) (16) Note that the section o the output or complex craters only appears when the crater sizes are larger than the transition diameter. 3. Explosive Cratering Explosive cratering is much like impact cratering, but there is one additional independent variable: the depth o burial o the explosive. The speciic energy o the impactor, which is 1 U is replaced by the speciic energy Q o the explosive material.

8 Also, it is more common to use the explosive mass (weight) W rather than its radius. Thus, the dimensional orm or the crater volume becomes ρv m = g Q W 1 3 Y, ρq, ρ δ, d a (17) Note that the relation between the explosive gravity π group and that o the explosive is π = g Q W 1 3 = 3. ga U (18) I the point-source assumption is added, then again the explosive radius a, its speciic energy Q and its mass density d can only occur in a power-law group. Speciically, This equation (17) must then have the restricted orm I I told you anything more, I would have to shoot you..(to be completed) 4. Reerences: (Note: many o the Holsapple reerences can be downloaded rom ) Holsapple, K. A., 003 "Does melt volume give the signature o the impactor?" Proceedings, Workshop on Bridging the Gap between Modeling and Observations. Houston TX, February, 003. Holsapple, K. A., 1993a "The Scaling o Impact Processes in Planetary Sciences", Annual Reviews o Earth and Planetary Sciences 1 pp Holsapple, K.A. 1993b. "The Size o Complex Craters", Lunar and Planetary Science XXIV. Holsapple, K.A., 1987 "The Scaling o Impact Phenomenon", Intl. J. Impact Engr., 5, Holsapple, K.A., "The Equivalent Depth o Burst or Impact Cratering", Lunar and Planetary Science XI. Holsapple, K.A. and S. Peyton, "The Scaling o Nuclear Weapons Eects or Near Surace Bursts", Rpt. DNA 6543F, Deense Nuclear Agency, Washington, D.C.

9 Holsapple, K.A. and R.M. Schmidt, 198, "On the Scaling o Crater Dimensions. Impact Processes", J. Geophy. Res., 87, No. B3, Holsapple, K.A. and R.M. Schmidt, "On the Scaling o Crater Dimensions 1. Explosive Processes" J. Geophys. Res. 85, No. B1, Housen, K.R., and K.A. Holsapple, 003. "Impact Cratering on Porous Asteroids", Icarus Housen, K.R., R.M. Schmidt, and K.A. Holsapple, "Crater Ejection Scaling Laws: Fundamental Forms Based on Dimensional Analysis", J. Geophys. Res. 88, No. B3, Schmidt R.M., Housen K.R. and K.A. Holsapple, "Gravity Eects in Cratering", DNA-TR-86-18, Deense Nuclear Agency, Washington D.C. Schmidt, R.M. and K.A. Holsapple, 198. "Estimates o Crater Size or Large-Body Impact: Gravity-Scaling Results", Geo. Soc. Am., Special Paper No Schmidt, R.M. and K.A. Holsapple, "Theory and Experiments on Centriuge Cratering", J. Geophys. Res. 85, 35-5.