1. PTYS 554 Evolution of Planetary Surfaces Impact Cratering III

2. PYTS 554 – Impact Cratering III 2 l Impact Cratering I n Size-morphology progression n Propagation of shocks n Hugoniot n Ejecta blankets - Maxwell Z-model n Floor rebound, wall collapse l Impact Cratering II n The population of impacting bodies n Rescaling the lunar cratering rate n Crater age dating n Surface saturation n Equilibrium crater populations l Impact Cratering III n Strength vs. gravity regime n Scaling of impacts n Effects of material strength n Impact experiments in the lab n How hydrocodes work

3. PYTS 554 – Impact Cratering III 3 l Scaling from experiments and weapons tests to planetary impacts

4. PYTS 554 – Impact Cratering III 4 l Morphology progression with size… l Transient diameters smaller than final diameters n Simple ~20% n Complex ~30-70% Moltke – 1km Euler – 28km Schrödinger – 320km Orientale – 970km Simple Complex Peak-ring

5. PYTS 554 – Impact Cratering III 5 l Scaling laws apply to the transient crater l Apparent diameter (D at ), diameter at original surface, is most often used l Target properties n Density, strength, porosity, gravity l Projectile properties n Size, density, velocity, angle

6. PYTS 554 – Impact Cratering III 6 l Lampson’s law n Length scales divided by cube-root of energy are constant n Crater size affected by burial depth as well n Very large craters (nuclear tests) show exponent closer to 1/3.4

7. PYTS 554 – Impact Cratering III 7 l Hydrodynamic similarity (Lab results vs. Nature) n Conservation of mass, momentum & energy n (Mostly) invariant when distance and time are rescaled x→αx and t →αt n i.e. n Lab experiments at small scales and fast times = large-scale impacts over longer times w1cm lab projectile can be scaled up to 10km projectile (α = 10 6 ) wEvents that take 0.2ms in the lab take 200 seconds for the 10km projectile wShock pressures & energy densities are equivalent at the same scaled distances and times l …but gravity is rescaled as g→g/α n Lab experiments at 1g correspond to bodies with very low g n In the above example… the results would be accurate on a body with g~10 -5 ms -2 l Workaround… increase g n Centrifuges in lab can generate ~3000 g moon n So α up to 3000 can be investigated… wA 30cm lab crater can be scaled to a 1km lunar crater Mass, Momentum and energy conservation for compressible fluid flow

8. PYTS 554 – Impact Cratering III 8 l If g is fixed… (one crater vs another crater) l If x→αx then D→αD and E ~ ½mv 2 → α 3 E (mass proportional to x 3 ) n So D/D o = α and (E/E o ) ⅓ = α n Lampson’s scaling law: exponent closer to 1/3.4 in ‘real life’ (nuclear explosions) n In the gravity regime (large craters) energy is proportional to n Experiments show that strength-less targets (impacts into liquid) have scaling exponents of 1/3.83

9. PYTS 554 – Impact Cratering III 9 l PI group scaling n Buckingham, 1914 n Dimensional analysis technique n Crater size D at function of projectile parameters {L, v i, ρ i }, and target parameters {g, Y, ρ t } n Seven parameters with three dimensions (length, mass and time) n So there are relationships between four dimensionless quantities wPI groups l Cratering efficiency: n Mass of material displaced from the crater relative to projectile mass n Popular with experimentalists as volume is measured n An alternative measure n Popular with studies of planetary surfaces as diameter is measured n Close to the ratio of crater and projectile sizes n Crater volume (parabolic) is ~ n If H at /D at is constant then

10. PYTS 554 – Impact Cratering III 10 l Other PI groups are numbered n π D = F(π 2, π 3, π 4 ) l Ratio of the lithostatic to inertial forces n A measure of the importance of gravity n Inverse of the Froude number l Ratio of the material strength to inertial forces n A measure of the effect of target strength l Density ratio n Usually taken to be 1 and ignored

11. PYTS 554 – Impact Cratering III 11 l When is gravity important? n ρgL > Y gravity regime n ρgL < Y strength regime n Gravity is increasingly important for larger craters n If Y~2MPa (for breccia) wTransition scales as 1/g wAt D~70m on the Earth, 400m on the Moon n Strength/gravity transition ≠ simple/complex crater transition l Gravity regime n π 3 can be neglected, also let π 4 → 1 so π D = F(π 2 ) l Strength regime n π 2 can be neglected, also let π 4 → 1 so π D = F(π 3 ) Holsapple 1993

12. PYTS 554 – Impact Cratering III 12 l In the gravity regime strength is small n so π 3 can be neglected, also let π 4 → 1 so π D = F’(π 2 ) Experiments show: Incidentally If H/D is a constant… seems to be the case l In the strength regime gravity is small n so π 2 can be neglected, also let π 4 → 1 so π D = F’(π 3 ) Experiments show:

13. PYTS 554 – Impact Cratering III 13 l Combining results for gravity regime… (competent rock) l Crater size scales as: l Combining results for strength regime… (competent rock)

14. PYTS 554 – Impact Cratering III 14 l Pi scaling continued n How does projectile size affect crater size n If velocity is constant, ratio of π D ’s will give diameter scaling for projectile size: For competent rock β~0.22 so D/D o = (E/E o ) 1/3.84 n (verified experimentally) n Pi scaling can be used for lots of crater properties wCrater formation time wEjecta scaling Gravity regime Strength regime

15. PYTS 554 – Impact Cratering III 15 l More recent formulations just combine these two regimes into one scaling law l Simplify with: l Into: Holsapple 1993

16. PYTS 554 – Impact Cratering III 16 l Mass of melt and vapor (relative to projectile mass) n Increases as velocity squared n Melt-mass/displaced-mass α (gD at ) 0.83 v i 0.33 n Very large craters dominated by melt Earth, 35 km s -1

17. PYTS 554 – Impact Cratering III 17 l Impacting bodies can explode or be slowed in the atmosphere l Significant drag when the projectile encounters its own mass in atmospheric gas: n Where P s is the surface gas pressure, g is gravity and ρ i is projectile density n If impact speed is reduced below elastic wave speed then there’s no shockwave – projectile survives l Ram pressure from atmospheric shock Crater-less impacts? n If P ram exceeds the yield strength then projectile fragments n If fragments drift apart enough then they develop their own shockfronts – fragments separate explosively n Weak bodies at high velocities (comets) are susceptible n Tunguska event on Earth n Crater-less ‘powder burns’ on venus n Crater clusters on Mars

18. PYTS 554 – Impact Cratering III 18 l ‘Powder burns’ on Venus l Crater clusters on Mars n Atmospheric breakup allows clusters to form here wScreened out on Earth and Venus wNo breakup on Moon or Mercury Mars Venus

19. PYTS 554 – Impact Cratering III 19 l Impact Cratering I n Size-morphology progression n Propagation of shocks n Hugoniot n Ejecta blankets - Maxwell Z-model n Floor rebound, wall collapse l Impact Cratering II n The population of impacting bodies n Rescaling the lunar cratering rate n Crater age dating n Surface saturation n Equilibrium crater populations l Impact Cratering III n Strength vs. gravity regime n Scaling of impacts n Effects of material strength n Impact experiments in the lab n How hydrocodes work

20. PYTS 554 – Impact Cratering III 20 l Hydrocode simulations n Commonly used simulate impacts n Computationally expensive Total number of timesteps in a simulation, M, depends on: 1) the duration of the simulation, T 2) the size of the timestep,  t Smallest timestep:  t  Δx/c s (Stability Rule) (Δx is the shortest dimension)  Overall: M = T/  t  N and run time = N r  M  N r+1 Oslo University, Physics Dept. Courtesy of Betty Pierazzo

21. PYTS 554 – Impact Cratering III 21 Example: problem with N=1000 10 double-precision numbers are stored for each cell (i.e., 80 Bytes/cell) For 1D Storage: 80 kBytes (trivial!) Runtime: 1 million operations (secs) For 2D Storage: 80 MBytes (a laptop can do it easily!) Runtime: 1 billion operations (hrs) For 3D Storage: 80 GBytes (large computers) Runtime: 1 trillion operations (days) (and N=1000 isn’t very much) Courtesy of Betty Pierazzo

22. PYTS 554 – Impact Cratering III 22 l Problem… n Some results depend on resolution n Need several model cells per projectile radius n Ironically small impacts take more computational power to simulate than longer ones n Adaptive Mesh Refinement (AMR) used (somewhat) to get around this Crawford & Barnouin-Jha, 2002 Courtesy of Betty Pierazzo

23. PYTS 554 – Impact Cratering III 23 There are two basic types of hydrocode simulation Lagrangian and Eulerian Cells follow the material - the mesh itself moves Cell volume changes (material compression or expansion) Cell mass is constant  Free surfaces and interfaces are well defined  Mesh distortion can end the simulation very early Courtesy of Betty Pierazzo

24. PYTS 554 – Impact Cratering III 24 There are two basic types of hydrocode simulations Lagrangian and Eulerian Material flows through a static mesh Cell volume is constant Cell mass changes with time  Cells contain mixtures of material  Material interfaces are blurred  Time evolution limited only by total mesh size Courtesy of Betty Pierazzo

25. PYTS 554 – Impact Cratering III 25 Equations of State account for compressibility effects and irreversible thermodynamic processes (e.g., shock heating) Deviatoric Models relate stress to strain and strain rate, internal energy and damage in the material Change of volume Change of shape COMPRESSIBILITY STRENGTH Artificial Viscosity Artificial term used to ‘smooth’ shock discontinuities over more than one cell to stabilize the numerical description of the shock (avoiding unwanted oscillations at shock discontinuities) Courtesy of Betty Pierazzo

26. PYTS 554 – Impact Cratering III 26 l Given all that… models differences should be expected n Compare results from impact into water Courtesy of Betty Pierazzo