impact cratering iii
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PTYS 554
Evolution of Planetary Surfaces
Impact Cratering IIIImpact Cratering III
PYTS 554 – 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
PYTS 554 – Impact Cratering III 3
Scaling from experiments and weapons tests to planetary impacts
PYTS 554 – Impact Cratering III 4
Morphology progression with size…
Transient diameters smaller than final diameters Simple ~20% Complex ~30-70%
Moltke – 1km
Euler – 28km
Schrödinger – 320km
Orientale – 970km
Simple Complex Peak-ring
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 7
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 Velocities (u), Shock pressures (P) & energy densities(E) 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
Mass, Momentum and energy conservation for compressible fluid flow
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 12
In the gravity regime strength is small so π3 can be neglected, also let π4 → 1
so πD = F’(π2)
Experiments show:
If H/D is a constant… seems to be the case
So:
In the strength regime gravity is small so π2 can be neglected, also let π4 → 1
so πD = F’(π3)
Experiments show: 13
'
withCDD
PYTS 554 – Impact Cratering III 13
Combining results for gravity regime… (competent rock)
Crater size scales as:
Combining results for strength regime… (competent rock)
13'
withCDD
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 15
More recent formulations just combine these two regimes into one scaling law
Simplify with:
Into:
Holsapple 1993
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 17
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
Crater-less impacts?
iPSi gPDei 23..
ATM
Hz
SATMram
atmosphereram
gkTHwhere
eHg
PvzP
TkvPconstTif
vP
22
2
.
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
PYTS 554 – Impact Cratering III 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
MarsVenus
PYTS 554 – Impact Cratering III 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
PYTS 554 – Impact Cratering III 20
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, t
Smallest timestep: t Δx/cs (Stability Rule)
(Δx is the shortest dimension)
Overall: M = T/ t N
and run time = NrM Nr+1
Oslo University, Physics Dept.
Courtesy of Betty Pierazzo
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 1DStorage: 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 3DStorage: 80 GBytes (large computers) Runtime: 1 trillion operations (days)
(and N=1000 isn’t very much)
Courtesy of Betty Pierazzo
PYTS 554 – Impact Cratering III 22
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
Courtesy of Betty Pierazzo
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
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
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 ViscosityArtificial 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
PYTS 554 – Impact Cratering III 26
Given all that… models differences should be expected Compare results from impact into water
Courtesy of Betty Pierazzo
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