Transcript Impact Processes III - Lunar and Planetary Laboratory
PTYS 554 Evolution of Planetary Surfaces
Impact Cratering III
PYTS 554 – 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 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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Scaling from experiments and weapons tests to planetary impacts
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Morphology progression with size…
Transient diameters smaller than final diameters
Simple ~20% Complex ~30-70% Simple Complex Peak-ring Moltke – 1km
4
Euler – 28km Schrödinger – 320km Orientale – 970km
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Scaling laws apply to the transient crater
Apparent diameter (D at ), diameter at original surface, is most often used
Target properties
Density, strength, porosity, gravity
Projectile properties
Size, density, velocity, angle
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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
D E
1 3 =
D o E
1 3
o or D D o
= ç
E E o
ö ø 1 3 6
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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.
x
' = a
t
' = r ' = a
t
r
x so
:
u
' =
u
&
P
' =
P
&
E
' =
E
Mass, Momentum and energy conservation for compressible fluid flow
¶r ¶
t
+ ¶ r
u i
¶
t
¶ r
E t
¶
t
¶ r
u i
¶
x i
+ ¶ ¶
x j
+ ¶ ¶
x j
( ( = 0 r
u i u j
r
u j E t
+ +
P
- S
ij u j P
) =
g i
r
u i
S
ij
) =
u j g j
r
where E t
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 = 10 6 ) 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
= 1 2
u i u i
+
E
…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 g
moon
So α up to 3000 can be investigated…
A 30cm lab crater can be scaled to a 1km lunar crater
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If g is fixed… (one crater vs another crater) If x →αx then D→αD and E ~ ½mv 2
→ α 3 E (mass proportional to x 3 ) So D/D o = α and (E/E o ) ⅓ = α Lampson
’
s scaling law: exponent closer to 1/3.4 in
D o
= ç
E E o
ö ø 1 3 ‘
real life
’
(nuclear explosions)
In the gravity regime (large craters) energy is proportional to
( 2 3 p
D
3 ) r
g D so D D o
= ç
E E o
ö ø 1 4
Experiments show that strength-less targets (impacts into liquid) have scaling exponents of 1/3.83
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PI group scaling
Buckingham, 1914 Dimensional analysis technique
Crater size D at function of projectile parameters {L, v i , ρ i }, and target parameters {g, Y, ρ t } 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 ~
( p
H at
If H at /D at is constant then
8
D at
)
D at
3 p
D
= æ è p
H at
8
D at
ö ø 1 3 p
V
1 3 » 1 3
V
p
V
= r
t V m
p
D
=
D at
ç r
m t
ö ø 1 3 9
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Other PI groups are numbered
π D = F( π 2 , π 3 , π 4 )
p
D
=
D at
r
t
è
m
ø 1 3 10
Ratio of the lithostatic to inertial forces
A measure of the importance of gravity Inverse of the Froude number
p 2 = 1.61
gL v i
2
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
p 3 =
Y
r
p v i
2 p 4 = r
t
r
p
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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
π 3 can be neglected, also let π 4 so π D = F( π 2 ) → 1
Strength regime
π 2 can be neglected, also let π 4 so π D = F( π 3 ) → 1 Holsapple 1993
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PYTS 554 – Impact Cratering III
In the gravity regime strength is small
so π 3 so π D can be neglected, also let π 4 = F
’
( π 2 ) Experiments show:
p
D
=
C D
p 2 b
→ 1
or
p
V
=
C V
p 2 g
Incidentally
g @ 3 b
If H/D is a constant… seems to be the case
p
D
= è p
H at
8
D at
ö ø 1 3 p
V
1 3 » 2 p
V
1 3
In the strength regime gravity is small
so π 2 so π D can be neglected, also let π 4 = F
’
( π 3 ) Experiments show:
D
C D
' 3
→ 1
with
1 12
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Combining results for gravity regime… (competent rock)
p
D
=
D at
ç r
m t
ö ø 1 3 p 2 = 1.61
gL v i
2
Crater size scales as:
p
D
=
C D
p 2 b
D at
= 1.8
r
p
0.11
r
t
0.33
g
0.22
L
0.13
W
0.22
Combining results for strength regime… (competent rock)
p
D
p 3 = =
D at
ç r
m t
ö ø 1 3
Y
r
p v i
2
D
C D
' 3
with
1
D at
µ r
p
0.33
r
t
0.33
Y
0.28
L
0.16
W
0.28
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Pi scaling continued
How does projectile size affect crater size If velocity is constant, ratio of π D
’
s will give diameter scaling for projectile size:
p
D
p
D o
=
D D o L o L
Þ
D D o
= æ è
L L o
ö ø 1 b
and
è p 2 ö b p 2
o
ø = æ è
E E o
ö ø 1 b 3 = ç
L o L
ö ø b
Gravity regime For competent rock β~0.22 so D/D o = (E/E o ) 1/3.84
(verified experimentally)
p
D
p
D o
=
D D o L o L and
Þ
D D o
= æ è
L L o
ö ø 1 = æ è
E E o
ö ø 1 3 æ è p 3 p 3
o
ö ø s = 1
Strength regime
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Pi scaling can be used for lots of crater properties
Crater formation time Ejecta scaling
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More recent formulations just combine these two regimes into one scaling law
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Simplify with:
n = 1 3
and K
2
Y
=
Y
Into:
p 4 = r d
Holsapple 1993
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Mass of melt and vapor (relative to projectile mass)
Increases as velocity squared
Melt-mass/displaced mass α (gD at ) 0.83
v i 0.33
Very large craters dominated by melt
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Earth, 35 km s -1
PYTS 554 – Impact Cratering III 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:
i
.
e
.
D i
3
P S
2
g P
i
Where P s is the surface gas pressure, g is gravity and ρ i is projectile density
If impact speed is reduced below elastic wave speed then there
’
s no shockwave – projectile survives
Ram pressure from atmospheric shock
P ram
v
2
atmosphere if T
const
.
P ram
v
2
ATM k T P where H
kT g
ATM
v
2
P S e
z H g H
If P ram 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
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‘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
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Mars Venus
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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
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Hydrocode simulations PYTS 554 – Impact Cratering III
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Courtesy of Betty Pierazzo
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,
D
t
Smallest timestep:
D
t
Δx/c
s ( Δx is the shortest dimension)
Overall:
M = T/
D
t
N
and (Stability Rule)
run time = N
r
M
N
r+1
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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)
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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
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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
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Courtesy of Betty Pierazzo 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
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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)
Deviatoric Models
relate stress to strain and strain rate, internal energy and damage in the material Change of volume Change of shape COMPRESSIBILITY STRENGTH
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Courtesy of Betty Pierazzo Given all that… models differences should be expected
Compare results from impact into water