PTYS 554
Evolution of Planetary Surfaces
Impact Cratering III
PYTS 554 – Impact Cratering III
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Impact Cratering I
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Impact Cratering II
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Size-morphology progression
Propagation of shocks
Hugoniot
Ejecta blankets - Maxwell Z-model
Floor rebound, wall collapse
The population of impacting bodies
Rescaling the lunar cratering rate
Crater age dating
Surface saturation
Equilibrium crater populations
Impact Cratering III
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Strength vs. gravity regime
Scaling of impacts
Effects of material strength
Impact experiments in the lab
How hydrocodes work
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PYTS 554 – Impact Cratering III
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Scaling from experiments and weapons tests to planetary impacts
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PYTS 554 – Impact Cratering III
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Morphology progression with size…
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Transient diameters smaller than final diameters
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Simple ~20%
Complex ~30-70%
Simple
4
Moltke – 1km
Complex
Peak-ring
Euler – 28km
Schrödinger – 320km
Orientale – 970km
PYTS 554 – Impact Cratering III
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Scaling laws apply to the transient crater
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Apparent diameter (Dat), diameter at original surface, is most often used
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Target properties
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Density, strength, porosity, gravity
Projectile properties
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Size, density, velocity, angle
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PYTS 554 – Impact Cratering III
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Lampson’s law
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Length scales divided by cube-root of energy are constant
Crater size affected by burial depth as well
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D
E
Do
=
13
E 1o 3
or
D = æç E ö÷
Do è E o ø
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Very large craters (nuclear tests) show exponent closer to 1/3.4
1
3
PYTS 554 – Impact Cratering III
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Hydrodynamic similarity (Lab results vs. Nature)
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Conservation of mass, momentum & energy
(Mostly) invariant when distance and time are
rescaled x→αx and t →αt
i.e. x'= a x
t'= a t
r'= r
so : u'= u & P'= P & E'= E
Mass, Momentum and energy conservation
for compressible fluid flow
¶r ¶ rui
+
=0
¶t ¶x i
¶ rui ¶
+
( ruiu j + P - Sij ) = gir
¶t
¶x j
¶ rE t
¶
+
( ru j E t + u j P - uiSij ) = u j g j r where E t = 1 2 uiui + E
¶t
¶x j
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
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…but gravity is rescaled as g→g/α
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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
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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
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PYTS 554 – Impact Cratering III
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If g is fixed… (one crater vs another crater)
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If x→αx then D→αD and E ~ ½mv2 → α3E (mass proportional to x3)
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So D/Do= α and (E/Eo)⅓ = α
1
Lampson’s scaling law: D = æç E ö÷ 3
D
E
o
è
o
ø
exponent closer to 1/3.4 in ‘real life’ (nuclear explosions)
(
2 p D r g D so D = æç E ö÷
3
Do è E o ø
)
1
4
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In the gravity regime (large craters) energy is proportional to
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Experiments show that strength-less targets (impacts into liquid) have scaling exponents of 1/3.83
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PYTS 554 – Impact Cratering III
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PI group scaling
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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
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Cratering efficiency:
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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
(p H
at
8Dat
)D
3
at
- 13
æ pH at ö
pD = ç
÷
è 8Dat ø
1
1
pV 3 » 2pV 3
pV =
r tV
m
æ rt ö
p D = Dat ç ÷
è mø
1
3
PYTS 554 – Impact Cratering III
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Other PI groups are numbered
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Ratio of the lithostatic to inertial forces
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A measure of the importance of gravity
Inverse of the Froude number
Ratio of the material strength to inertial forces
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πD = F(π2, π3, π4)
A measure of the effect of target strength
Density ratio
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Usually taken to be 1 and ignored
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æ rt ö
p D = Dat ç ÷
è mø
1.61gL
p2 =
v i2
Y
p3 =
r p v i2
rt
p4 =
rp
1
3
PYTS 554 – Impact Cratering III
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When is gravity important?
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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
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Gravity regime
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Strength/gravity transition ≠
simple/complex crater transition
π3 can be neglected, also let π4 → 1
so πD = F(π2)
Strength regime
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π2 can be neglected, also let π4 → 1
so πD = F(π3)
Holsapple 1993
PYTS 554 – Impact Cratering III
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In the gravity regime strength is small
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so π3 can be neglected, also let π4 → 1
so πD = F’(π2)
-b
Experiments show: p D = CD p 2
or
p V = CV p -2 g
g
b
Incidentally
@3
If H/D is a constant… seems to be the case
- 13
æ pH at ö
pD = ç
÷
è 8Dat ø
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1
1
pV 3 » 2pV 3
In the strength regime gravity is small
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so π2 can be neglected, also let π4 → 1
so πD = F’(π3)
'

Experiments show:  D  CD 3
with   
 1
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PYTS 554 – Impact Cratering III
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Combining results for gravity regime… (competent rock)
ær ö
p D = Dat ç t ÷
è mø
1.61gL
p2 =
v i2
1
3
p D = CDp -2 b
-0.33 -0.22 0.13
Dat =1.8 r 0.11
g L W 0.22
p rt
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Crater size scales as:
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Combining results for strength regime… (competent rock)
ær ö
p D = Dat ç t ÷
è mø
Y
p3 =
r p v i2
1
3
 D  CD'  3
with   
 1
-0.33 -0.28 0.16
0.28
Dat µr 0.33
r
Y
L
W
p
t
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PYTS 554 – Impact Cratering III
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Pi scaling continued
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How does projectile size affect crater size
If velocity is constant, ratio of πD’s will give diameter scaling for projectile size:
pD
pD
o
D Lo
=
Do L
æp 2
ö- b æ Lo ö b
and ç p ÷ = ç ÷
èLø
2o ø
è
1- b
1- b
æEö
D æLö
Þ
=ç ÷ =ç ÷
Do è Lo ø
è Eo ø
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Gravity regime
For competent rock β~0.22 so D/Do= (E/Eo)1/3.84
 (verified experimentally)
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Pi scaling can be used for lots of crater properties
 Crater formation time
 Ejecta scaling
pD
pD
o
D Lo
=
Do L
-s
æp
ö
and ç 3
÷
è p 3o ø
1
D æLö æEö
Þ
=ç ÷ =ç ÷
Do è Lo ø è E o ø
1
3
Strength regime
=1
PYTS 554 – Impact Cratering III
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More recent formulations just combine these two regimes into one scaling law
p4 =
r
d
Holsapple 1993
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Simplify with:
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Into:
n=
1
and K 2Y = Y
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PYTS 554 – Impact Cratering III
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Mass of melt and vapor (relative to projectile mass)
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Increases as velocity squared
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Melt-mass/displaced-mass α (gDat)0.83 vi0.33
Very large craters dominated by melt
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Earth, 35 km s-1
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PYTS 554 – Impact Cratering III
Crater-less impacts?
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Impacting bodies can explode or be slowed in the atmosphere
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Significant drag when the projectile encounters its own mass in
atmospheric gas: i.e. Di  3PS 2 g P i
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Where Ps is the surface gas pressure, g is gravity and ρi is projectile density
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If impact speed is reduced below elastic wave speed then there’s no
shockwave – projectile survives
Ram pressure from atmospheric shock
Pram  v 2  atmosphere
if
T  const. Pram  v
where H  kT
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2
g ATM
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
 ATM
v 2 PS  z H
Pz  
e
kT
gH
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PYTS 554 – Impact Cratering III
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‘Powder burns’ on Venus
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Crater clusters on Mars
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Atmospheric breakup allows clusters to form here
 Screened out on Earth and Venus
 No breakup on Moon or Mercury
Mars
Venus
PYTS 554 – Impact Cratering III
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Impact Cratering I
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



Impact Cratering II






Size-morphology progression
Propagation of shocks
Hugoniot
Ejecta blankets - Maxwell Z-model
Floor rebound, wall collapse
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
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PYTS 554 – Impact Cratering III
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Courtesy of Betty Pierazzo
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Hydrocode simulations
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Commonly used simulate impacts
Computationally expensive
Oslo University, Physics Dept.
Total number of timesteps in a simulation, M, depends on:
1) the duration of the simulation, T
2) the size of the timestep,
Smallest timestep:
Dt
Dt  Δx/cs
(Stability Rule)
(Δx is the shortest dimension)
 Overall:
and
M = T/ Dt  N
run time = NrM  Nr+1
PYTS 554 – Impact Cratering III
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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)
PYTS 554 – Impact Cratering III
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Courtesy of Betty Pierazzo
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Problem…
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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
PYTS 554 – Impact Cratering III
There are two basic types of hydrocode simulation
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Courtesy of Betty Pierazzo
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
PYTS 554 – Impact Cratering III
There are two basic types of hydrocode simulations
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Courtesy of Betty Pierazzo
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
PYTS 554 – Impact Cratering III
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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)
Change of volume
COMPRESSIBILITY
Deviatoric Models
relate stress to strain and
strain rate, internal energy
and damage in the material
Change of shape
STRENGTH
PYTS 554 – Impact Cratering III
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Courtesy of Betty Pierazzo
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Given all that… models differences should be expected
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Compare results from impact into water