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Impact Cratering III

Impact Cratering III. Impact Cratering I Size-morphology progression Propagation of shocks Hugoniot Ejecta blankets - Maxwell Z-model Floor rebound, wall collapse Impact Cratering II The population of impacting bodies Rescaling the lunar cratering rate Crater age dating

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Impact Cratering III

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  1. Impact Cratering III

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

  3. Scaling from experiments and weapons tests to planetary impacts

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

  5. Scaling laws apply to the transient crater • Apparent diameter (Dat), diameter at original surface, is most often used • Target properties • Density, strength, porosity, gravity • Projectile properties • Size, density, velocity, angle

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

  7. Mass, Momentum and energy conservation for compressible fluid flow • Hydrodynamic similarity (Lab results vs. Nature) • Conservation of mass, momentum & energy • (Mostly) invariant when distance and time are rescaled x→αx and t →αt • i.e. • Lab experiments at small scales and fast times = large-scale impacts over longer times • 1cm lab projectile can be scaled up to 10km projectile (α = 106) • Events that take 0.2ms in the lab take 200 seconds for the 10km projectile • Shock pressures & energy densities are equivalent at the same scaled distances and times • …but gravity is rescaled as g→g/α • Lab experiments at 1g correspond to bodies with very low g • In the above example… the results would be accurate on a body with g~10-5 ms-2 • Workaround… increase g • Centrifuges in lab can generate ~3000 gmoon • So α up to 3000 can be investigated… • A 30cm lab crater can be scaled to a 1km lunar crater

  8. If g is fixed… (one crater vs another crater) • If x→αx then D→αD and E ~ ½mv2 → α3E (mass proportional to x3) • So D/Do= α and (E/Eo)⅓ = α • Lampson’s scaling law: exponent closer to 1/3.4 in ‘real life’ (nuclear explosions) • In the gravity regime (large craters) energy is proportional to • Experiments show that strength-less targets (impacts into liquid) have scaling exponents of 1/3.83

  9. PI group scaling • Buckingham, 1914 • Dimensional analysis technique • Crater size Dat function of projectile parameters {L, vi, ρi}, and target parameters {g, Y, ρt} • Seven parameters with three dimensions (length, mass and time) • So there are relationships between four dimensionless quantities • PI groups • Cratering efficiency: • Mass of material displaced from the crater relative to projectile mass • Popular with experimentalists as volume is measured • An alternative measure • Popular with studies of planetary surfaces as diameter is measured • Close to the ratio of crater and projectile sizes • Crater volume (parabolic) is ~ • If Hat/Dat is constant then

  10. Other PI groups are numbered • πD = F(π2, π3, π4) • Ratio of the lithostatic to inertial forces • A measure of the importance of gravity • Inverse of the Froude number • Ratio of the material strength to inertial forces • A measure of the effect of target strength • Density ratio • Usually taken to be 1 and ignored

  11. When is gravity important? • ρgL > Y gravity regime • ρgL < Y strength regime • Gravity is increasingly important for larger craters • If Y~2MPa (for breccia) • Transition scales as 1/g • At D~70m on the Earth, 400m on the Moon • Strength/gravity transition ≠ simple/complex crater transition • Gravity regime • π3 can be neglected, also let π4 → 1 so πD = F(π2) • Strength regime • π2 can be neglected, also let π4 → 1 so πD = F(π3) Holsapple 1993

  12. In the gravity regime strength is small • so π3 can be neglected, also let π4 → 1 so πD = F’(π2) Experiments show: Incidentally If H/D is a constant… seems to be the case • In the strength regime gravity is small • so π2 can be neglected, also let π4 → 1 so πD = F’(π3) Experiments show:

  13. Combining results for gravity regime… (competent rock) • Crater size scales as: • Combining results for strength regime… (competent rock)

  14. Pi scaling continued • How does projectile size affect crater size • If velocity is constant, ratio of πD’s will give diameter scaling for projectile size: For competent rock β~0.22 so D/Do= (E/Eo)1/3.84 • (verified experimentally) • Pi scaling can be used for lots of crater properties • Crater formation time • Ejecta scaling Gravity regime Strength regime

  15. More recent formulations just combine these two regimes into one scaling law • Simplify with: • Into: Holsapple 1993

  16. Mass of melt and vapor (relative to projectile mass) • Increases as velocity squared • Melt-mass/displaced-mass α (gDat)0.83 vi0.33 • Very large craters dominated by melt Earth, 35 km s-1

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

  18. ‘Powder burns’ on Venus • Crater clusters on Mars • Atmospheric breakup allows clusters to form here • Screened out on Earth and Venus • No breakup on Moon or Mercury Mars Venus

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

  20. Courtesy of Betty Pierazzo • Hydrocode simulations • Commonly used simulate impacts • 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,Dt Smallest timestep: Dt Δx/cs (Stability Rule) (Δx is the shortest dimension)  Overall: M = T/ Dt  N and run time = NrM  Nr+1 Oslo University, Physics Dept.

  21. Courtesy of Betty Pierazzo 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)

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

  23. Courtesy of Betty Pierazzo 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

  24. Courtesy of Betty Pierazzo There are two basic types of hydrocode simulations Lagrangian andEulerian 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

  25. Change of volume Change of shape Courtesy of Betty Pierazzo 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) 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 COMPRESSIBILITY STRENGTH

  26. Courtesy of Betty Pierazzo • Given all that… models differences should be expected • Compare results from impact into water

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