第三回衝突勉強セミナー 　2013/1/31

衝突クレーターのスケーリング解析とは？
参考論論⽂文Scaling  of  oblique  impacts  in  frictional  targets:  Implications  for  crater  size  and  formation  mechanisms
Dirk  Elbeshausen,  Kai  Wunnemann,  Gareth  S.  Collins
Icarus  204  (2009)  716-‐‑‒731

河本  泰成（神⼾戸⼤大M1）

• 太陽系天体の起源と進化にとって衝突過程は⾮非常に重要• 衝突天体、標的、クレーターサイズの関係→スケーリング則→実験室スケールでの衝突結果を惑星スケールに外挿する• 衝突には様々なパラメータが複雑に関わっている→それぞれの関わりを調べるのには次元解析が有効→最も成功しているものがπスケーリング

πスケーリング• クレーター体積Vに関係する量量→衝突速度度U、標的密度度ρ、弾丸密度度δ、強度度Y、重⼒力力加速度度g、弾丸質量量m→V=F(U,  ρ,  δ,  Y,  g,  m)：７つの物理理量量• [kg],  [m],  [s]の３つを基本単位とすると
７­−３＝４つの無次元数（πv=F(π2,  π3,  π4)）ができる

πv=ρV/m

規格化クレーター体積π2=1.61gL/U2

規格化弾丸直径π3=Y/(δU2)

規格化強度度

π4=ρ/δ

規格化密度度• π2とπ3を⽐比較する；π3/π2=Y/ρgL→Y>>ρgL：強度度⽀支配域→Y<<ρgL：重⼒力力⽀支配域

ste,yt• クレーター形成は弾丸の直径L、速度度U、密度度δに個別に依存するのでmelt production, material ejection, etc., arew
はなく、それらを組み合わせた量量に依存
power-laws in the various
non-dimensional qμν→カップリングパラメータ：C=LU δ   (Dienes  and  Walsh,  1970)
stance, the functional relation in Eq. (1) is well• この変数を導⼊入するとV=F(C,  ρ,  Y,  g)
a power-lawof the form (e.g. Schmidt and Hous
→重⼒力力⽀支配域(π3無視)で、弾丸標的密度度⽐比(π4)~∼1として次元解析

$c¼
C…(※)
VV 2and the open symbols refer to models

p7

D. Elbeshausen et al. / Icarus 204 (2009) 716–731

pALE. Following
an approach suggested
Cv：標的により異異なる量量communication), the dependence of D
γ=3μ/(2+μ) V
approximated by an exponential funcR)?0 the curve intersects the pD-axis at
Ø 衝突では様々なγの値
9.14 for p2 = 6.e!4, which can be conpD–value乾燥砂への衝突：γ=0.507
at infinitely high resolution
he scaled crater depth we obtain ‘‘true”= 1.2e!4湿った砂、⽔水：γ=0.65
and pH = 4.45 for p2 = 6.e!4.
ime for a single model run as small as• πスケーリングに組み込まれ
he numerical models in an acceptabley low resolutionof only eight CPPR forていない標的の性質でγをe, the resulting crater depth H, diameter
imately 変化させうるのは内部摩擦と2.8%, 6.3%, and 12% too small,s inaccuracyto beindependent of the
空隙率率率(Wunnemann,  2006)
he angle of impact.

107-0.646

water/water (2D): πV=1.1•π2numerical (3D): πV=1.2•π2-0.658
numerical (2D): πV=1.4•π2-0.636

πV = ρV/mwhere C and c are 10material dependent scaling p
related to the velocity-exponent
in the cou
10waterestim2/3),
c = 3l/(2 + l). For c 10= 0.75 (wherel=the coate
wet sandis exactly proportional
tothekineticenergy ofest
10
imateergy-scaling). The otherend-member
case, wh10Gault/Sonett (1982)
Schmidt/Housen (1987)
parameter is proportional
to the momentumq10
1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001
0.43 (l = 1/3). L(momentum scaling), occurs
at cπ == 1.61gL/Uy,ments (Schmidt andHousen,of vertical
impFig. 3. Cratering
efficiency as a1987)
function of gravity-scaledsize. Results forverti-ng Ø これまで、これらはスケーリング則で考慮されていない

te,n,
fs

65
4
3
2
122

impacts in a hydrodynamic target derived by 3D-calculations (iSALE-3D), 2• 実際には斜め衝突は⽀支配的→クレーター体積は衝突⾓角度度とともに正弦関数的に減少(Gault  and  Wedekind,  1978)→この関係はある⼀一定の弾丸標的組み合わせでしか 　成り⽴立立たないかもしれない• 衝突速度度の垂直成分のみを使⽤用して、衝突⾓角度度の影響をスケーリング則に組み込む(Chapman  and  Mckinnon,  1986)→砂への室内実験を除いて、適切切に試されていない
Ø 衝突⾓角度度とクレーターサイズの関係はまだ確⽴立立されていない
Ø 特に、斜め衝突での摩擦係数の影響がよくわかっていない

数値モデル• 斜め衝突のモデル化には３つの空間座標が必要• クレーター形成過程全体をシミュレーションするため、広範囲でのかなりの数のシミュレーションが必要Ø ⾮非常に効率率率的な計算コードが必要→iSALE-‐‑‒3Dを開発→多様な物質をシミュレーションできる
数値実験条件• 地球のような重⼒力力条件• 衝突速度度6.5km/s• 衝突⾓角度度30-‐‑‒90°• 結合強度度０• 摩擦係数0,  0.2,  0.7(典型的な砂の値)• 弾丸直径430m-‐‑‒3km

クレーター形状への衝突⾓角度度の影響
D. Elbeshausen et al. / Icarus 204 (2009) 716–731• f=0：30°でもほぼ対称• f  増加：衝突⾓角度度の影響 　 　 　    が⾼高まる• 明らかに⾮非対称だが、すべてのモデルでクレーターの楕円率率率は1.01以下→円形クレーター

g. 4. Snapshots of the transient crater (for our definition see Section 2.1) of akm-diameter projectile impacting with 6.5 km/s for different angles of incidencend target weakness. Both target and projectile material are composed of granite,
ut cohesion is neglected to show the effect of friction coefficient for different

クレーター体積への衝突⾓角度度の影響
D. Elbeshausen et al. / Icarus 204 (2009) 716–731

摩擦係数f=0

摩擦係数f=0.7

異異なるサイズの弾丸でπge of thecratering efficiency with the impact
angle for hydrodynamic (f = 0) and frictional (f = 0.7) targets. Both show results for a n2の値を変えた、衝突⾓角度度と規格化クレーター体積の関係
izes: p2 = 1.12e!3, 8.6e!4, 6.47e!4, 5.01e!4, 4.04e!4, 3.29e!4, 2.62e!4, 2.09e!4, and 1.62e!4. Bold dashed line shows normalize
an and McKinnon (1986)).

• どちらの場合も、クレーター体積は衝突⾓角度度のsinにおおよそ⽐比例例するas was the case for hydrodynamic targets. Plottin(Fig. 7) illustrates that cratering efficiency is muchtional targets than in hydrodynamic targets. In add
f=0.7の場合、衝突速度度の垂直成分だけがクレーター体積を決める
it is hard to see in this diagram, the slope of the plines for targets
with a friction coefficient of 0.7 (c2γに⽐比例例）に近いとした場合（規格化クレーター体積がsin(α)
ler than in the case of hydrodynamic targets (c = 0• π2にはほとんど依存しない•cratering efficiency πV = ρV/mrocess of a meteorite impact is (Fig. 7) illustrates that cratering efficiency is muchtional targets than in hydrodynamic targets. In add
inof thethp
is hard power-laws
to see in this diagram, the slope
n of 100anクレーター体積への衝突⾓角度度の影響
explosive
source, except itlines
pfor targets with a friction coefficient of 0.7 (c =the functio
than instance,
the case of hydrodynamictargets (c = 0.6e) is moving. As a consequence, ler Fig.
6b illustrates that for impacts into targets wiof asfriction
f power-law= case0.7 thedecrease in crateringeP
wasathefor hydrodynamictargets.
oftheays1000a10 mixture of pure ‘‘energyモデル結果を(※)式でフィッティ
a)/V(90!) =issdecreasing
angle
is best-fitby V(efficiency(Fig. 7) impact
illustrates
thatcratering
is closethe idealized
casewhere only theverticalI
tionaltotargets
than inhydrodynamictargets.
ング$cthe cratervolume:, which considers the impact theit impact
is hard velocity
diagram,the slopeof
p to see¼incontrols
Cthis pcratering efficiency πV = ρV/mg. 6. Relative change of the cratering efficiency with the impact angle for hydrodynamic (f = 0) and frictional (f = 0.7) targets. Both show results
ifferent projectile sizes: 2 = 1.12e!3, 8.6e!4, 6.47e!4, 5.01e!4, 4.04e!4, 3.29e!4, 2.62e!4, 2.09e!4, and 1.62e!4. Bold dashed line shows nor
uggested by Chapman and McKinnon (1986)).

-0.66-0.57

α = 90° : πV(f=0.) = 1.19•π2-0.66 | πV(f=0.7) = 0.32•π2-0.57
α = 75° : πV(f=0.) = 1.17•π2-0.65 | πV(f=0.7) = 0.31•π2-0.57
α = 60° : πV(f=0.) = 1.14•π2-0.64 | πV(f=0.7) = 0.29•π2-0.57
α = 45° : πV(f=0.) = 1.03•π2-0.62 | πV(f=0.7) = 0.24•π2-0.58
α = 30° : πV(f=0.) = 0.77•π2| πV(f=0.7) = 0.14•π2100 10.0001

V2 coefficient of 0lines for targets
with Va friction
ler than in the case of hydrodynamic targets
4.3. The
combined
effect ofthat
impactangle andinto
frictionFig.
6b illustrates
for impacts
targc
cratering efficiencyof friction f = 0.7 the Vdecrease in crater
decreasing impact angle is best-fit by V(a)/V(9The model results presented so far demonstrate
is closev to the idealized case where only the v
efficiency depends significantly on the coefficientthe impact
velocity controls the crater vo
the target.
This becomes even more evident by plot
2c.
p sin
for a aseriesof vertical impacts (a = 90!) into targe
sin2ca.

0.001

π = 1.61•g•L/Uwhere C and c are
dimensional
analysisof impact• 衝突⾓角度度減少
10減少
to the v
group scaling (e.g. Holsapple, →π relatedc = 3l/(2 + l). For
2, 1987;
Schmidt and Housen,
1friction coefficient f = 0.0–0.7. Fig. 8 shows crate• ent摩擦係数増加
0.0001
0.001
proportiof exactly
gravity-scaledfor vertical
impac
Theis
combined
effect of size
impactangle andfriater volume V as a function of aswitha4.3.function
different
friction coefficients. These data arecratering
efficiency
π = 1.61•g•L/U→π減少v
mated byergy-scaling).
power-laws (Eq. (4)). The scaling
expon
The
least
six (or more) parametersacdecrease
withpresented
increasing so
coefficientofintercept
TheCmodelresults
far demon
To test whether the effect of the impact angle chaefficiency
depends significantlyison thecoef
近似直線はほぼ平⾏行行
parameter
prop
where U is impact velocity, q• tioncoefficient, we conducted additional simulatio
the target. This becomes even more evident b
projectile size of 1 km (p = 4.e!4) and varied bo
p for a(momentumseries of vertical impactsscaling(a = 90!) int
nsity, Y is strength, g is gravity, →クレーター体積へのent friction coefficient f = 0.0–0.7. Fig. 8 show
as a functionof gravity-scaled
size for verticalments(Schmidt
anscaling introduces dimension-  　衝突⾓角度度の影響はwith different friction coefficients. These da1000
mated by power-laws (Eq. (4)). The scaling
revealeda scaling 　πによらない
ndependent
variables to three:ac2
intercept
C decrease with increasing coefficie
22Fig. 7. Cratering efficiency as a function of gravity-scaled size for different impact-0.66and frictional (f = 0.7,-0.57
angles and both αhydrodynamic(dashedlines)
solid lines)= 90° : πV(f=0.)= 1.19•π2-0.66 | πV(f=0.7) = 0.32•π2-0.57targets. For discussion
see: Sections
and 24.2. | πV(f=0.7) = 0.31•π2
α = 75°πV(f=0.) =4.11.17•πα = 60° : πV(f=0.) = 1.14•π2-0.65| πV(f=0.7) = 0.29•π2-0.57-0.64-0.57
α = 45° : πV(f=0.) = 1.03•π2-0.62 | πV(f=0.7) = 0.24•π2-0.58
α = 30° : πV(f=0.) = 0.77•π2| πV(f=0.7) = 0.14•π2VTable 4Power-law constants for crater volume derived for different
impact angles and2friction coefficients.
2fCV0.00.0
0.0

1.14
1.03
0.77

g. 7. Cratering efficiency as a function of gravity-scaled size for different impact
0.0
1.19frictional (f = 0.7, solid0.66lines)
ngles90and both hydrodynamic(dashed lines) and
75
0.0
1.17
0.66
rgets. For discussion see Sections 4.1 and 4.2.
60
4530

0.65
0.64
0.62

V2V

0.7
0.32
0.57
able90475
0.7
0.57 andower-law
constants for cratervolume derived 0.31
for different impact angles
60
0.7
0.29
0.57
ction coefficients.
4530

0.7

f 0.70.24CV0.14

0.570.58VπV(f=0.0) = 1πV(f=0.1) = 0πV(f=0.2) = 0πV(f=0.3) = 0πV(f=0.4) = 0πV(f=0.5) = 0

ated by power-laws (Eq. (4)). The scaling exponent c and thetercept CV decrease with increasing coefficient of friction.
To test whether the effect of the impact angle changes with fricon coefficient, we conducted additional simulations for a fixedrojectile size of 1 km (p2 = 4.e!4) and varied both the impact

衝突⾓角度度と摩擦係数の組み合わせ
1πV = ρV/m

1000

100

100.0001π2 = 1.61•g•L/U2

異異なる摩擦係数に対する垂直衝突0.001

normalized volume, V(α)/V(90°)-0.66πV(f=0.0) = 1.19•π2-0.65
πV(f=0.1) = 0.82•π2-0.64
πV(f=0.2) = 0.62•π2
πV(f=0.3) = 0.50•π2-0.63-0.61
πV(f=0.4) = 0.42•π2-0.60
πV(f=0.5) = 0.36•π2πV(f=0.6) = 0.32•π2-0.59-0.57
πV(f=0.7) = 0.32•π2

D. Elbeshausen et a0.90.8

π2⼀一定	0.70.6

f=0.0
f=0.2
f=0.4
f=0.70.50.430405060708090impact angle α

異異なる摩擦係数に対する衝突⾓角度度とクレーター体積の関係

Fig. 9. Relative change of the crater volume with the impact angle for differen
g. 8. Cratering efficiency as a function of gravity-scaled size for different friction
friction coefficients. Projectile size is 1 km (p2 = 4.e!4). Bold dashed line show
efficients.
normalized crater volume as suggested by Chapman and McKinnon (1986).

and the friction coefficient systematically. By plotting agai
• 左図：摩擦係数増加 　→ 　Canglevとγ減少

the normalized crater volume V(a)/V(90!) but now for differencoefficients of friction, Fig. 9 indicates that the effect of the impac
angle on cratering efficiency is slightly dependent on the frictio
coefficient.

• 右図：衝突⾓角度度のクレーター体積への影響は 　 　 　摩擦係数にわずかに依存
5. Discussion

er floor in the uprange-direction than in
ontrast to experimental observations by978) who found more symmetric craters as low as 30!. However, their observa-

value given by pure energy scaling (c = 0.75). Moreover, for a friction coefficient f = 0.7 c is constant and does not depend on the angle of impact (c = 0.57; see Table 4). This is in contrast to thehydrodynamic target case, where the scaling exponent c decreases

摩擦係数と衝突⾓角度度のスケーリング則への組み込み• 図a：f=0.7
衝突⾓角度度によらずγ⼀一定• f=0わずかに減少• この結果が重要かどうかはまだわからない•

図c：f=0.7

Cv∝Cv(90)sin(α)2γ→衝突⾓角度度の垂直成分仮定に⼀一致•

低いfでは⼀一致しない

Relationship between scaling parameters (c and CV, see Eq. (4)) and both the impact angle and friction coefficient.

• 摩擦係数に伴うCvとγの減少（図b,d）→摩擦係数はクレータースケーリングに影響する重要なパラメータ

まとめ
クレーター形成における衝突⾓角度度と標的摩擦係数の影響を調査した• 標的表⾯面への30°以上での衝突は円形クレーターを作る• 衝突⾓角度度と摩擦係数はπvをはっきりと減少させる• f=0.7の標的への衝突では、クレーター形成に衝突速度度の垂直成分だけが影響するØ πスケーリングはf=0.7で結合をほとんど持たない砂に似た性質を持つ物質への、惑星スケール斜め衝突に適⽤用できる