Impact Cratering III

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


Scaling from experiments and weapons tests to planetary impacts


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


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


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


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


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


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


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


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


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


Combining results for gravity regime… (competent rock)

Crater size scales as:

Combining results for strength regime… (competent rock)

13'

withCDD


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


More recent formulations just combine these two regimes into one scaling law

Simplify with:

Into:

Holsapple 1993


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


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


‘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


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


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


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)



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



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



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



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)



Given all that… models differences should be expected Compare results from impact into water


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Impact Cratering III