2015年年度度前期  第２回衝突勉強会
テーマ：氷天体におけるイジェクタサイズ分布

Secondary  craters  from  large  impacts  on  Europa  and  Ganymede
Ejecta  size-‐‑‒velocity  distributions  on  icy  world,
  and  the  scaling  of  ejected  blocks
Kelso  N.  Singer  ,  William  B.  McKinnon  ,  L.T.  Nowicki
Icarus  226  (2013)  865-‐‑‒884

神⼾戸⼤大学  理理学研究科
M2  松榮  ⼀一真Outline of this study² 本研究では、EuropaとGanymedeの巨⼤大クレーターの解析を⾏行行ったØ ２次クレーターを調べることで、イジェクタのsize-‐‑‒velocity  distribution(SVD)を詳細に調べることが可能Ø ejection  velocity,  ejection  position,  mass  of  material  ejected,  number  of  fragment(Alvarellos  er  al.2002,  Housen  and  Holsapple,  2011)² 本研究のアウトライン
1. イントロダクション
2. 今回調べた３つクレーターの２次クレーター場とカウンティング⽅方法
3. 重⼒力力⽀支配域における２次クレーターを形成したイジェクタ破⽚片サイズと速度度の⾒見見積もり⽅方法
4. SVDの上限を決定した結果とスケーリング則から求めることのできる関係式との⽐比較
5. 氷衛星の結果を岩⽯石天体との結果と⽐比較
6. 本研究の結果を踏まえ、氷衛星に存在する1.5次クレーターのサイズの⾒見見積もり
7. まとめIntroduction² 1.5次クレーター² 2次クレーター(Zahnle  et  al.2008)

ヒル圏

ヒル圏
ejecta² Europa上の直径＜1kmのクレーターの95%は２次クレーター  (Bierhaus  et  al.2005)Ø ２次クレーターの空間分布はランダムであるため、クレーターカウンティングによる地表⾯面年年代決定に影響がでる
Ø ⼩小天体or彗星の衝突と⾒見見分けることが難しくなるため² 2次クレーターを1次クレーターとカウントすると、クレーター年年代を過⼤大評価することがあるØ ⼩小さなクレーターの累累積個数が⼤大きくなるØ クレーターカウンティングによる年年代決定するた

累累
積
個
数

Ns ∝ Db

Ns ∝ Da

めには、１次クレーターと２次クレーターを⾒見見分ける必要がある

クレーター直径Introduction² １次クレーターと2次クレーターの⾒見見分け
1. １次クレーター近傍Ø ２次クレーターの数密度度は、１次クレーターからの距離離とともに減少
Ø 形状が不不規則Ø ２次クレーターの特徴

chain 　 　ｰ  clustersやradial  chainsを形成

cluster

2. １次クレーター遠⽅方Ø 形状は円形で、空間分布がランダムØ １次クレーターと区別することは難しい
１次² 氷衛星上でのクレーターのSVDについて調べた研究が少なくあまりわかっていないØ 破⽚片のSVDは⼩小さなクレーターの分布に寄与する  (Zahnle et al.2008, Bierhaus et al.2012)
Ø 岩⽯石天体(⽔水星・⽉月・⽕火星)でのSVDは調べられている

Introduction

Strom et al.1981

STROM ET AL.: CRATER POPULATION² 岩⽯石天体と氷天体の違い

I

[

I

[ I I [ •[II

I I I I ]

I

I

[

I

[ [ I [

⽉月の⾼高地Ø ⽉月とGanymedeを⽐比較

l

l

l

ｰ  Ganymedeでは⼤大きなクレーターが少ない
ｰ  50<D<100kmでベキ〜～­−３

I

I

•

-I - Lunar Highlands--.

（D>100km）Ø 天体表層の違い
ｰ  ⽉月：岩⽯石質地殻 　 　Ganymede：氷質地殻

Ø 氷の強度度は岩⽯石より⼩小さいため、氷のSpall破⽚片は岩⽯石に⽐比べ⼩小さくなるØ 形成される２次クレーターのサイズが異異なる可能性-

CallistoGanymede
cratered
Ganymeden-'----Callisto

0

-2-

%

ｰ  標的物質の強度度が異異なる² 衝突後の破⽚片サイズ

l

LunarPost-

-

t

/

⽉月の海

-:32.

J I0

I00

I000

CRATER DIAMETER-(km)Fig. l l. Curvesfor the craterpopulationsmeasuredon the heavily crateredterrainsof both Ganymedeand Callisto,with the lunar
curve for reference. The differences between the Moon, Callisto andGanymedeare muchgreaterthan are the similarities.The Ganymede
and Callistocurvesare similarbeyondabout 50 km diameter,but differ substantiallyat smaller diameters.

本研究：but only degradeslarger craters.Figure 14 showsan area ofgroovedterrainwherethe formationof new ice hasdestroyed
EuropaとGanymede上の巨⼤大クレーターのまわりの２次クレーターをカウン
a largeportionof the rims of severalcraters.Smallerones,ofcourse, would have been completely obliterated. The proposedpreferentialobliteration of small craterseven on the
ティングしたcratered terrain may have been the result of an ancient episodeof grooved-terrain
formation(now hiddenby the recraØ 岩⽯石天体の、⽔水星・⽉月・⽕火星の結果と⽐比較 tering)associatedwith theresurfacingor later crustalfreezing mentionedearlier. Alternatively, the formation of thearcuatetroughsmay have beenresponsiblefor the lossof theØ 氷天体におけるイジェクタ破⽚片サイズと速度度の関係を明らかにする
smaller craters.

At diameters smaller than about 10 km the curve for theSites and mapping methods² GalileoとVoyager2 missionで得られた画像を⽤用いたØ

ejecta blancketの外側に位置する２次クレーターをカウンティングØ サイズと形状が周りの２次クレーターと⼤大きく異異なるクレーターは除外
867

K.N. Singer et al. / Icarus 226 (2013) 865–884

1. Tyre-Europa

Table 1
Summary of primary and secondary crater field characteristics.
PrimarycraterPrimary
diameter(km)

Primary transientdiameter (km)aEuropa
Tyre
Pwyll

38e
27e

23
17

GanymedeAchelous
Gilgamesh

35g
585h

Diameter of primary
impactor (km)b

Mosaicresolution(m px"1)

Number ofsecondaries
mappedLargest observed
secondary (km)c170
27, 21, 54

1,165
180f

2.8

180
550

630
445

Fragment size for largest
secondary (m)d

直径が〜～38kmで、⽐比較的若若くクレーターが少ない領領域に位置する(Bierhaus et al.2005,2009)1.8
1.2

Ø   中⼼心から~175kmまで円形の溝が存在
21
271

1.9
49.1

2.7
21.3 (18.6)

1160
410f

1200
5760 (5000)aSection 3.
Assumes cometary impactors at 26 km s"1 (Europa) and 20 km s"1 (Ganymede) (Eq. (10)).
For Gilgamesh, the value listed is the largest crater likely to be a secondary, and in parentheses the largest crater in an unquestionable radial chain (see Fig. 4b).
d
Fragment sizes assume non-porous ice surface and gravity-regime scaling (Section 4).
eSchenk and Turtle (2009).
f
Alpert and Melosh (1999).
g
Schenk (2010).
h
Schenk et al. (2004).
868bØ 画像に限りがあるが、①解像度度が170mpx-1 　②⾼高解像度度(~30mpx-1)cØ ①1165個：0.5-2.8km

Ø ②375個：~180mが最⼩小

K.N. Singer et2012). There is a subtle aspect to this assumption, however, which
will be returned to in Section 5. An approximate value of one-halfof the transient primary crater radius (0.25Dtr) was used as thestarting point for measuring the ballistic range to each secondary(we justify this in a later section). The transient crater represents
a nominal cavity shape as the excavation flow ceases but before
gravitational collapse proceeds; thus Dtr/4 represents an average
distance from which fragments might have been ejected. The transient diameter for each of the 3 primary craters was estimatedSites and mapping methods
Fig. 2. Portion of high resolution mosaic of Tyre’s secondary field ("30 m px!1). (a)This high-Sun image also reveals dark material concentrated in secondary crater
floors. (b) Mapping at higher resolution confirms craters mapped at lower
resolution and permits mapping of considerably smaller secondaries (n = 375).

below: a total of 445 craters in the diameter range of 2.3–21.3 km.

In some cases this includes ‘‘chains’’ where there is only one large
2. Achelous-Ganymede
crater but there is also a smaller crater, or elongation/gouging in
ter had a distinctly different degradation state than the majority of
直径が〜～35kmで、⽐比較的新鮮なクレーターthe direction radial to Gilgamesh (‘‘Gilgamesh sculpture’’). If a cra-the secondaries (either much fresher, rayed, or much more degraded) it was excluded from the analysis. Additionally,
if a crater-1
was particularly large for its distance from Gilgamesh, compared tothe surrounding secondaries, it was excluded (these craters appeared as obvious outliers in the dataset). The craters that were
mapped or excluded can be seen in Fig. 4a and some specific examples of the mapping decisions are given in Fig. 4b. Topography andexample secondary crater profiles are shown in Fig. 4c and d,respectively. During the analysis described below, we also explored an augmented dataset for Gilgamesh, which includes othercraters that could be secondaries, but were not in an obvious chain
or cluster.

Ø 解像度度はTyreの画像とほぼ同じ180mpx
Ø 630個：0.7-2.7km

Ø 明確なイジェクタ堆積がある

Ø 北北側に同サイズのGulaクレーターがあるØ Gulaはクレーター内部に⼩小さなクレーターがあるが2.4. Pwyll – Europa

Achelousには存在しない
Alpert and Melosh (1999) measured 180 secondaries in high••resolution Galileo mosaics (27, 21, 54 m px!1) at three distancesAchelousはGulaより年年代が若若いfrom Pwyll, a 27-km-diameter crater. One mosaic is centered overPwyll, and the other two are almost due north, "900 km and1,070 km away, in Conamara Chaos and just north of Conamara,respectively. Pwyll’s bright rays are visible over 1000 km away,
thus secondaries from Pwyll can be identified at large distances.
This allows for a greater range of fragment velocities to be sampled, but a disadvantage to this site is that measurements only extend in one radial direction away from Pwyll and mapping was
only possible in small patches of surface area. Without broader radial coverage the results may not reflect the largest secondaries at
a given distance, due to stochastic variations with launch azimuth.

ほとんどがAchelousの２次クレーターØ GulaとAchelousクレーターのイジェクタブランケッ
トが重なっている領領域がある•

63.5度度より北北の部分のクレーターはカウントしていない

3. Fragment size and velocity
From measured secondary crater ranges we can calculate thevelocity of the ejecta fragment that formed each secondary, andFig. 3. (a) Ganymede’s Achelous and Gula craters, "35 and 37 km in diameter,
respectively. Achelous is centered at "62!N, 12!W. Mosaic is 180 m px!1, from
Galileo imagery. Mapped secondaries (n = 630) in yellow. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)from an approximate scaling for complex
869 craters on Ganymede
Dfinal ! 1:176D1:108(McKinnon
andSchenk,1995), where Dfinal andtr
Dtr are the final and transient crater diameters, respectively, given
K.N.inS
from an approximate scaling forin complex
Ganymede
km (Dfinal iscraters
measuredon
to rim-to-rim;values of Dtr are given
1:108
Table
1).
Dfinal ! 1:176Dtr(McKinnon and
Schenk,1995), where Dfinal andderive the ejecta fragment size (dfrag) from the K.N.
measuredSinge
Dtr are the final and transient craterTodiameters,
respectively, given
diameter of each secondary crater (Dsec), we use the Schmidt–Holsin km (Dfinal is measured to rim-to-rim;valuesof Dforgiven inimpacts (e.g., Holsapple,
tr areapple scaling
relations
hypervelocity
Table 1).
1993) with updated parameters (keith.aa.washington.edu/craterdata/scaling/theory.pdf).
The scaling
equations relate the properTo derive the ejecta fragment
size (dfrag) from themeasuredties of the impactor to the volume of the resulting transientdiameter of each secondary crater (Dsec), we use the Schmidt–Holscrater on a given body (which can then be converted to a final
apple scaling relations for hypervelocity
impacts
Holsapple,
diameter forcomplex (e.g.,craters),
or vice versa. For the purposes of-1 with updated parameters thesecalculations, we consider both solid, low-porosity ice and1993)(keith.aa.washington.edu/craterporousequations
ice regolith asmodelsfor properthe surfaces of Europa and Ganydata/scaling/theory.pdf). The scaling
relatethemede. For lower surface gravities and smaller impactors, theties of the impactor to the volumeof the resulting transient
strength of solid surface material will strongly control the size ofcrater on a given body (which canthenconverted
to a final
the resultingbe
crater(i.e., the ‘‘strength
regime’’). For larger gravities
andimpactorsizes,
thestrengthof the material
is more easily overdiameter for complex craters), or vice versa. For the purposes
of
come.
In
this
‘‘gravity
regime’’the
finalcratersize is limited bythese calculations, we consider both solid, low-porosity ice andconversion of flow field kinetic energy into gravitational potential
porous ice regolith as models for energy
the surfacesof Europa
and Ganyand its frictional
dissipation
as heat, both of which are dimede. For lower surface gravities
and smaller
impactors,
rect functionsof gravity.
A functionthethat interpolates between
twostrongly
regimes is control
given by the size ofstrength of solid surface materialthewill

K.N. Singer et al. / Icarus 226 (2013) 865–884

Sites and mapping methods
3. Gilgamesh-Ganymede

直径が〜～585kmの最⼤大盆地  (Schenk et al.2004)
Ø Voyager2の画像で解像度度550mpx
Ø 445個：2.3-21.3km
Ø chain構造とcluster構造Ø ２次クレーターがクレーターの北北部と南部に集中!
" larger gravities
the resulting crater (i.e., the ‘‘strength regime’’).
For² １次クレーターと２次クレーターの判別が難しい
p ¼K p þK p
;
2þl
2

Ø
Ø

$3l
2þl

ð2Þ
1
2
2 is
3 more easily overand impactor sizes, the strength of Vthe material
come. In this ‘‘gravity regime’’ the final crater size is limited byclusterとchain構造をなしている⽐比較的⼤大きなク
where K1 and K2 are scaling coefficients for a given surface material
conversion of flow field kinetic energy
into gravitational potential
and l is the scaling exponent (see Table 2 for values). The scaling
energyanditsfrictional
dissipationasheat,of which
are fordi-different materials fromparameters areboth
empiricallyestimated
レーターのみカウンティング
laboratory
experiments,
numerical
computations,and comparison
rect functions of gravity. A function that interpolates between
to explosion cratering results. The p-groups are non-dimensional:
the two regimes is given by
リム構造の傾斜が⼤大きく、chain状に位置していな

!

l
"$3
2þl
2þl

く、周りより⼤大きなクレーターは除外(図b⻩黄⾊色)
pV ¼ K 1 p2 þ K 2 p3;
2pV describes the overall cratering efficiency, and is dependent on
the gravity-scaled size (p2) and the strength measure (p3). Generaldefinitions of the p-groups are

qV

ga

Y

ð2Þ

pV ¼ ; p2 ¼ 2 ; p3 ¼ 2 ;
ð3Þ
m
U
qU
where K1 and K2 are scaling coefficients for a given surface material
where
q is 2thefor
densityof theThetargetmaterial, V is the volume of theand l is the scaling exponent (seeTablevalues).
scaling
resulting crater, m is the mass of the impactor, a is the impactor raparameters are empirically estimated for different materials from
dius, U is impact velocity, and Y is a measure of target strengthlaboratory experiments, numericalcomputations,andcomparison(Holsapple,
1993). Thereis nosimple size division between craters
formedin the strength
versus the gravity regimes as illustrated in
to explosion cratering results. The
p-groups
are non-dimensional:
5, as the transitionan orderonof magnitude or more inpV describes the overall crateringFig.efficiency,
and is spans
dependent
p2 and appears to depend on impactor velocity. We note that thethe gravity-scaled size (p2) and thestrength measure (p3). Generalof the ratio of the target
cratering efficiency itself is independent
definitions of the p-groups are to impactor densities, but this ratio does appear on the right hand

pV ¼

qV
m

;

p2 ¼

ga
U

;
2

side of Eq. (2) in its most general form (see Holsapple, 1993). The

p3 ¼

Y density ratio is nominally taken as unity for secondaries, and is;
ð3Þ
qU 2not included for the ice-on-ice secondary impacts considered here.

In principle one could substitute Eq. (3) into Eq. (2) to solve for aFragment velocity of ejectau 破⽚片サイズ・破⽚片速度度の導出² ２次クレーターの存在領領域→エジェクタの破⽚片速度度² ２次クレーターサイズ→イジェクタ破⽚片サイズ
Ø イジェクタは弾道軌道を仮定² エジェクタ破⽚片速度度の推定Ø イジェクタ放出⾓角度度は45° (Cintala et al.1999)(Melosh,1989)Ø 0.25Dtrから破⽚片が放出

D final ~ 1.176D
Ø Dtrの⾒見見積もりは 　 　 　 　 　 　                を⽤用いる
1.108
tr

(McKinnon and Schenk,1995)" vej2 sin θ cosθ%$Rp g '−1'
Rb = 2Rp tan $
2
2
v
cos
θ
$
'
$ 1− ejRp g '&#

Rp:天体半径g:重⼒力力加速度度² クレータースケーリング則から破⽚片サイズと２次クレーターサイズの関係式を導く
サイズスケーリング則 (Hoslapple,1993)−3µ2+µ 2+µ
2
V
1
2
2 3

!
π = K #π + K π"

規格化クレーター体積

$
&
%

πV =

ρV
m

規格化重⼒力力

規格化強度度

π2 =

π3 =

ga
U2

Y
ρU 2

Dtr:トランジェントクレーター直径
Dfinal:最終クレーター直径   ρ:標的密度度
Fig. 5. Cratering efficiency ( V) as a function of gravity-scaled
size ( 2) for im
m:弾丸質量量 　 　g:重⼒力力加速度度※２次クレーターは標的物質の破⽚片が弾丸になるため
Y:標的物質強度度 　   U:衝突速度度
into cold, non-porous ice and ice regolith (http://keith.aa.washington.edu/c
π4=1 となる
a:弾丸直径 　 　   　 V:クレーター体積pp

Fragment size of ejecta
870

K.N. Singer et al. / Iu クレーター形成メカニズム² 氷のパラメータをスケーリング則に適応² ２次クレーターは“重⼒力力⽀支配域”で形成
Ø 観測したクレーターの条件(π2)が、右図 　  の灰⾊色の領領域になったため(衝突速度度は200-1000m/s)

Ø 破⽚片をm-scaleにしても、πVは遷移領領域
 　    に位置した

Fig. 5. Cratering efficiency (pV) as a function of gravity-scaled size (p2) for impact
into cold, non-porous ice and ice regolith (http://keith.aa.washington.edu/crater
data/scaling/theory.pdf). Shaded bar shows the range of p2 values for secondariedetermined in this study, and the inset illustrates quantities used in application oballistic range and crater scaling equations to calculate ejecta fragment velocity andsize. All of the secondaries in question form safely in the gravity regime, but thlargest and slowest reach the limit of point-source impactor scaling,
declineejas pV fragtowards unity. The cratering efficiency in the strength regime shown above is highcompared with that for laboratory experiments on intact ice samples (e.g., Langand Ahrens, 1987), but in actuality Y decreases with increasing scale; at thejgeological scales of interest here, secondary
formation
most likely occurs
θfrag in th
gravity regime (cf. Holsapple, 1993).

Ø EuropaとGanymedeには強度度⽀支配域の２次クレーターは観測されなかった² クレータースケーリング則は重⼒力力⽀支配域のみで考察をするØ 放出後の⼤大気による抵抗は無視する

π V = K1π

−3µ
2+µ
2

v =v

v

v

−3µ
2+µ image resolutions.2 For a porous, regolith-like materia
the current! g( f / 2) $
Vsec
frag
we &
use the parameters for ‘‘dry sand’’, which has no effective= K1 ##
strength;
will always be in the grav
2 & thus impacts into this material δ=ρV frag
(v
cos
θ
)
ity
regime,
also
shown
in
Fig.
5.
We
comment
on the low cratering" frag%efficiencies implied by Fig. 5, and other potential scaling limitations
at the end of this section and in Appendix A.

Fragment size of ejecta
Dsec² クレーター体積から直径へØ Europaの２次クレーターの直径深さ⽐比H/D=0.135

H(Bierhaus and Schenk 2010)Ø スケーリング則は表⾯面からの深さを⽤用いるので、H/D＝0.125を⽤用いる² イジェクタのスケーリング則２次クレーターサイズと破⽚片サイズの関係式−3µ! g(d / 2) $ 2+µ
0.05D
frag
&
= K1 ##
3
2&
(v
cos
θ
)
!
$
" frag%4 π # d frag &
3 "
2%
2
sec

( )

solid ice surface

! g $
1.277
d frag = 1.003Dsec ## 2 &&" v frag %※Gilgameshの２次クレーターは数kmより⼤大 　 　 　きく形状が複雑になっている※Schenk 2002によって、Ganymedeのク 　レーターの深さHは、直径Dの0.42乗に⽐比例例
HD(= 0.125 Dsec

3km

)

0.58

Dsec > 3km

0.277

icy regolith

! g $
1.205
d frag = 0.878Dsec ## 2 &&" v frag %0.205

Dsec>3km

! g $
1.03
d frag = 1.304Dsec ## 2 &&" v frag %

0.277

! g $
0.972
d frag = 1.126Dsec ## 2 &&" v frag %0.205

Size-distribution of secondary craters² ２次クレーターのサイズ分布Ø １次クレーターからの距離離と２次クレーターのサイズの関係  (右図)ー  ⿊黒線は１次クレーターのサイズ
ー  遠くになるにつれてクレーターサイズは⼩小さい
Ø ２次クレーターのサイズ頻度度分布  (下図) 　     　ー  １次クレーターと傾きが異異なる 　 　  ー  解像度度や観測場所によって変化する
K.N. Singer et al. / Icarus 226 (2013) 865–884

872

K.N. Singer et al. / IcaFragment diameter and velocity
872

K.N. Singer et al. / Icarus 226 (2013) 865–884² 破⽚片速度度vejとvejにおける最⼤大破⽚片サイズdfmax
Ø dfmaxとvejの関係

d f max = Avej− βØ 分位点回帰法でフィッテイング  (Koenker, 2005) 　 　ー  データの99%回帰線より下に位置する
1. 太線：すべてのデータ 　 　ー  破⽚片速度度が⼤大きいと破⽚片サイズは⼩小さい
2. 細線：最⼤大破⽚片の持つvejを下限値と設定

873

K.N. Singer et al. / Icarus 226 (2013) 865–884 　 　ー  最⼤大破⽚片が最も速度度が遅いわけではない
Table 3

Quantile regression fits to 99th quantile of size–velocity distributionsa. Solid and porous targets, gravity regime, all secondaries and subset.
Primary crater (diameter in km)

All secondariesb

Velocity-limited subset
ln(A)

A

ln(A)

b

Solid target
Pwyllb (27)
1.21
3.4 # 105
873
K.N. Singer et al. / Icarus 226 (2013) 865–884
K.N. Singer et al. / Icarus 226 (2013) 5865–884
Tyre (38)0.96 ± 0.13
12.3 ± 0.7
2.24 # 10
1.13 ± 0.16
13.3 ± 1.0
6Achelous (35)
1.41 ± 0.45
14.7 ± 2.6
2.52 # 10
2.44 ± 0.80
20.7 ± 4.6
Table
3
Gilgamesh(585)
2.07 ± 0.34
22.3 ± 2.3
4.73 # 109
2.55 ± 0.44
25.5 ± 3.0aa. Solidregression
and porous
gravityregime,all secondaries
and subset.
–velocity distributions. Solid and porous targets, gravity regime, all secondaries and subset.
Quantile
fitstargets,to 99thquantileof size–velocity
distributions
Porous target
b
Pwyll
1.02
1.5 # 105
Primary crater (diameter in km)
All secondaries subset
Velocity-limited subset
All secondaries
Velocity-limited
Tyre
0.79 ± 0.12
11.4 ± 0.7
8.94 # 104
0.95 ± 0.15
12.3 ± 0.9
5
b
ln(A)
A
b
ln(A)
b
ln(A)
A
b ± 0.41
ln(A)± 2.4
AAchelous
1.21
13.7
8.63 # 10
2.18 ± 0.72
19.3
± 4.2
9
Gilgamesh
1.84
±
0.34
20.8
±
2.3
1.11
#
10
2.29
±
0.41
23.9
± 2.8
Solid target

1.21
0.96 ± 0.13
1.41 ± 0.45
2.07 ± 0.34ab5Pwyllb (27)
1.21
3.4 #
dfmax ¼ At"b, where A andtej10are5 in m and m s"1.
ej
Tyre
(38)
0.96±±0.16
0.13
12.3
±
0.7
2.24
#
10
1.13
Alpert and Melosh, (1999).
Achelous
1.41±±0.80
0.4514.7 ± 2.6(35)2.52 # 106
2.44
Gilgamesh2.07±±0.440.3422.3 ± 2.3 (585)
4.73 # 109
2.55

12.3± ±1.0
0.7
13.3
14.7± ±4.6
2.6
20.7
22.3± ±3.0
2.3
25.5

3.4 # 105
5 5
2.24
6.12
##
1010
6
2.52
#
10
9.83 # 108
11 9
4.73
1.24
##
1010

1.13 ± 0.16
2.44 ± 0.80
2.55 ± 0.44

13.3 ± 1.0
20.7 ± 4.6
25.5 ± 3.0

A6.12 # 105
9.83 # 108
1.24 # 1011

873

2.30 # 105
A # 108
2.32
2.42 # 1010

6.12 # 105
9.83 # 108
1.24 # 1011

Fragment diameter and velocity
874

K.N. Singer et al. / Ica² 破⽚片速度度vejとvejにおける最⼤大破⽚片サイズdfmax
Ø ⿊黒：表⾯面がsolidØ ⻘青：表⾯面がicy regolith² ポーラスな標的に、同じ速度度で同サイズのクレーターを形成するためには⼤大きな破⽚片が必要になる
Ø 弾丸の運動エネルギーが標的物質の圧密に使われたり、散逸が⽣生じるため  (Holsapple 1993)
Ø 破⽚片サイズは⼤大きくなるがSVDのベキ乗数βは、標K.N. Singer et al. / Icarus 226 (2013) 865–884

873

K.N. Singer et al. / Icarus 226 (2013) 865–884

的物質によらず同じような値をとる

873

Table 3aa
. Solidandquantile
porous oftargets,
gravity regime,
all secondaries
subset.
to 99th quantile of size–velocity. Solid andandporoustargets, gravity regime, all secondaries and subset.
Quantile distributions
regression fits
to 99thsize–velocity
distributions

eter in km)

All Primary
secondaries
crater (diameter in km)

All secondaries

ln(A)

A b

Solid target
1.21Pwyllb (27)
0.96Tyre± 0.13(38)
1.41± 0.45 (35)
Achelous
2.07Gilgamesh± 0.34(585)

12.3 ± 0.7
14.7 ± 2.6
22.3 ± 2.3

3.4 1.21# 105
2.240.96# 10± 50.132.521.41# 10± 60.454.732.07# 10± 90.34

Porous target
1.02Pwyllb
0.79Tyre± 0.12
1.21Achelous± 0.41
1.84Gilgamesh± 0.34

11.4 ± 0.7
13.7 ± 2.4
20.8 ± 2.3

1.5 1.02# 105
8.940.79# 10± 40.128.631.21# 10± 50.41
1.111.84# 10± 90.34b

A and tej are in m and mas"1. ¼ At"bdfmax, where A and tej are in m and m s"1.
ejbVelocity-limited subset

Velocity-limited subset

ln(A)
A

A b

ln(A)

A

1.1312.3± 0.16± 0.7
2.4414.7± 0.80± 2.6
2.5522.3± 0.44± 2.3

3.4 # 105
13.3
± 1.0
2.24
# 105
20.7
± 4.6
2.52
# 106
25.5
± 3.0
4.73
# 109

6.121.13# 10± 50.16
9.832.44# 10± 80.80
1.242.55
# 10
± 11
0.44

13.3 ± 1.0
20.7 ± 4.6
25.5 ± 3.0

6.12 # 105
9.83 # 108
1.24 # 1011

0.9511.4± 0.15± 0.7
2.1813.7± 0.72± 2.4
2.2920.8± 0.41± 2.3

1.5 # 105
12.3
± 0.9
8.94
# 104
19.3
± 4.2
8.63
# 105
23.9
± 2.8
1.11
# 109

2.300.95# 10± 50.15
2.322.18# 10± 80.72
2.422.29
# 10
± 10
0.41

12.3 ± 0.9
19.3 ± 4.2
23.9 ± 2.8

2.30 # 105
2.32 # 108
2.42 # 1010

b

ln(A)

Fragment diameter and velocity
d f max = Avej− β² 分位点回帰法でフィッテイング  (Koenker, 2005) 　 　 　ー  データ範囲を99%,95%,90%に変更更K.N. Singer et al. / Ica 　 　 　ー  ⼀一般的に範囲を下げた⽅方が値は正確になる² Tyreのデータが最も信頼出来る² GilgameshのβがTyre,Achelousより⼤大きくなり、傾向が異異なることを⽰示唆² Gilgameshの２次クレーターはclusterとchainのみカウントØ 別の100個のクレーターをカウンティング

K.N. SingeØ サイズが⼩小さいため、βが減少する² クレーターの北北側と南側に２次クレーターが集中Ø 南側が多少⼤大きく、２次クレーターのチェーン構造が卓越している部分があるØ North:2.15、South:2.45
Fig. 10. Comparison of velocity exponents (b) for the three primary craters in thisØ チェーン状に位置する⼤大きな２次クレーターが多い
study. Above are given for (a) the full dataset and (b) the data subset (low velocity
limit). All are for cold, non-porous ice scaling in the gravity regime.

Ejecta fragment scaling
876

876

K.N. Singer et al. / Icarus

K.N. Singer et al. / I² Ejectaのスケール則  (Housen and Hoslapple,2011)Ø 規格化したイジェクタ破⽚片速度度：vej/(gRtr)1/2
Ø 規格化したイジェクタ破⽚片サイズ：dfrag/Rtr² Gilgameshのみデータにオフセット
Fig. 12. Fragment size and ejection velocity scaled by primary size and gravity,Ø Gilgameshのトランジェントクレーターサイズを⼤大きく⾒見見積もったため

following Housen and Holsapple (2011). The maximum fragment size derived for
Pwyll by Alpert and Melosh (1999), appropriately scaled, is also shown (solid curve
where data exist; dashed where interpolated). Fragment sizes calculated assuming
non-porous
scaling in size
the gravity
regimevelocity
in all cases.
Fig. 12. Fragment
and ejection
scaled by primary size and gravityØ イジェクタブランケットから推測すると、1.5倍⼩小さくなる  (Housen et al.1983)² 放出位置と速度度の関係  (Housen and Holsapple2011)Ø 規格化放出速度度のデータ範囲:1.6-4.7

following Housen and Holsapple (2011). The maximum fragment size derived fo
Pwyll by Alpert and Melosh (1999), appropriately scaled, is also shown (solid curv
where data exist; dashed where interpolated). Fragment sizes calculated assumin
non-porous scaling in the gravity regime in all cases.

イジェクタの放出位置

Ø 放出位置の範囲の中に、x/Rtr=Dtr/4=0.5を含む
Ø イジェクタ速度度と放出位置についての理理論論とデータを⽐比較することが可能Ø 放出位置がDtr/4であることは妥当
Fig. 13. Scaled launch positions and ejection velocities from Housen and Holsapple(2011; their Fig. 14 and Eq. 14). Our study uses an approximate fragment launch
position (x) of Dtr/4 (or Rtr/2, indicated). This scaled launch position is within therange of positions inferred by the Housen and Holsapple ejecta model for thesecondary fragments measured in our study (Fig. 12), although some fragments

2013), and not in layers of structurally coherent mare basalt.
Bart and Melosh (2010) examined the SVDs of ejected boulders
around 18 lunar craters. These were mostly small primaries, 0.2to-4 km in diameter, but they also looked at two larger craters withdiameters of 27.4 and 41.2 km. Power law fits for the boulders gave
exponents in the range of !0.3 to !3.7 (for the largest crater), andalthough there is some scatter in the data, they determined an² 新たなファクターを導⼊入
overall increasing trend in b with increasing crater size, meaningØ ３つのクレーターを⽐比較することが可能larger craters had a steeper fragment size dependence on velocity.
The b values for the km-scale craters were particularly shallow
2/3
Ø positions
⽉月のクレーターリム上にあるイジェクタブロックの最⼤大値がD(Moore1971)
Scaled launchand ejection velocities from Housen and Holsapple(0 < b < 1). When combined withで変化する  the results from
Vickery(1987)
heir Fig. 14 and Eq. 14). Our study uses an approximate fragment launch(thus, boulders and secondary forming fragments) this trend was(x) of Dtr/4Ø(or R最⼤大イジェクタブロックの質量量はイジェクタ総質量量の0.8乗に⽐比例例  (O’Keefe and Ahrens 1987)
tr/2, indicated). This scaled launch position is within thereinforced, albeit with scatter for larger craters. Our results (b valpositions inferred by the Housen and Holsapple ejecta model for theues) and that of Alpert and Melosh (1999) further strengthen thisy fragments
in our study (Fig. 12), although some fragmentsØ measured
速度度のベキ乗数βは0.91±0.09
ve been launched from locations nearer or farther from the transient rim
trend, such as it is. We note that rim boulders and the largest secejectaスケーリング則). The exact
position is only important for calculating ranges andØlaunch
Gilgameshのクレーターサイズより、βの値が⼤大きくなる可能性
ondary-forming fragments might come from
(Housen different
and Holsappledistinct
2011)
s of nearby secondaries. For distant secondaries and sesquinary fragments,
ejecta components (Melosh, 1989), but this is not entirely clear.
osition uncertainties of even 0.5Rtr are an insignificant fraction of the totalβα² 放出速度度と２次クレーターを形成する破⽚片サイズ
It is possible that the largest rim boulders
representα＝ｰ1/3
the ‘‘last
traveledby the fragment.
f max
ej
spalls’’ (or at least the last surviving spalls, prior to any rimØ Pwylで得た結果より⼤大きな破⽚片が放出している
collapse).
tr
226(2013)
865–884size of fragments, it is important to controlortheabsoluteØ 先⾏行行研究との結果とは合わない
mary crater size. For modest sized craters, it still appears
gment size on icy bodies is generally less than that on rockyd f max
at equivalent ejection velocities, but the difference is far less= 19.9vej−1.17±0.06
D finalnced than for Pwyll alone, whose fragments are signifismaller in comparison (Figs. 14 and 15). Gilgamesh secondgments (or fragment clusters) can be quite large, however,atch or exceed the fragment sizes derived in the Vickery. This result is not surprising, as the proximal secondaries
icy satellite basin are up to twice as large as the largest seccraters identified by Wilhelms et al. (1978) for Orientale.
other, more recent studies have used Vickery’s approach to
craters on the Moon and Mars. Hirase et al. (2004) examhat were judged to be secondary craters around Kepler
istarchus on the moon (32- and 40-km diameter, respec-

Ejecta fragment scalingdR! ρ gR $ ! v $
∝#
&&
& ##
" T % " gR %Comparison with terrestrial planet secondaries

K.N. Singer et al. / Icarus² 速度度のベキβを⽐比較
Ø 岩⽯石天体に⽐比べ氷衛星のイジェクタ破⽚片のほうが⼩小さいØ 氷衛星のβは岩⽯石天体に⽐比べて⼩小さいØ Gilgameshのみβが⼤大きい 　－    ⽉月のOrientaleの２次クレーターより⼤大きい

Vickery 1986,1987

Fig. 16. Velocity exponents for ejecta fragments (as measured from secondary
fields) on terrestrial bodies compared with icy satellites. Crater names anddiameters in km are given. Exponents displayed for icy satellites are the preferredfits: the data subset for Tyre and Gilgamesh, and the full dataset for Achelous (all for
99th quantile, solid surface, gravity regime). Exponent for Pwyll is from Alpert andMelosh (1999). Exponents for Mercury, the Moon and Mars are from Vickery (1987).

² ⽉月のKeplerとAristarchusクレーターと⽔水星上の３つのクレーター周りの２次クレーターについて調べた(Hirase et al.2004)Ø SVDの上限を決定しておらず、距離離と破⽚片サイズの減少の傾向が⾒見見られなかった
5.2. Secondaries and spallation theory² ⽉月のTychoクレーターで同様の研究を⾏行行った(Hirata
and Nakamura 2006)
Spall fragments are those ejected at high velocities (but low
shock pressures) from coherent, near surface material. Spalls areØ SVDの上限は決めていないが、Copernicusクレーターのデータとほぼ⼀一致
generally the highest velocity fragments ejected from a given
launch point within the crater. The majority of ejected fragments,Ø ⽞玄武岩層ではなくMegaregolithを深くまで掘削したことを⽰示唆  (Wieczorek
et al.2013)
however, are ejected as partof the main excavation
flow and are
generally considered to be Grady–Kipp fragments, after the theory
predicts
the breakup(Bartthat
and
Melosh2010)of coherent rock (or ice) into roughly
equal-sized blocks (Melosh, 1989, Chapter 6). As the generally
slower moving and much larger fraction of the ejecta (at a givenx), Grady–Kipp fragments presumably result in many near-fieldsecondaries and most of the continuous ejecta (though fragments
ejected from below the surface spall layer are expected to travel
at close to spall speeds; Melosh, 1987).
Melosh (1984) provides an estimate of spall plate thickness, for² 18個の⽉月クレーター周りのSVDを求めた  Ø クレーターサイズが⼤大きくなるに従いβが⼤大きくなる(β：ｰ0.3~ｰ3.7)
Ø ⼤大きなクレーターほど、破⽚片サイズ分布の傾きが⼤大きくなるSecondaries and spallation theory² クレーター形成時に発⽣生する標的物質の破⽚片
Ø Spall破⽚片が地表⾯面付近において⾼高速で放出される
Ø Grady-Kipp fragments：掘削流流による破⽚片の⼤大部分² hydrodynamic ejection model
lspall:スポール破⽚片厚さ
ai:衝突体半径 　ρ:標的密度度
δ:衝突体密度度 　  T:引っ張り強度度
CL:P波速度度 　 　ai:弾丸直径

122a T (δ ρ )lspall = i
ρCL vej
H/D=0.2

1.277
tr2ai = 1.215D

!g$
# 2&
" vi %

0.277² Spall破⽚片は低速イジェクタでは破壊されないと仮定Ø Spall破⽚片のサイズの予測は、観測された２次クレーターから推測した破⽚片サイズより⼩小さい² この違いが何で起きているのか？
Ø H/D=0.1では破⽚片サイズが10%減少
Ø 速度度、衝突⾓角度度で説明できない

878

K.N. Singer et al. /Secondaries and spallation theory² 氷と岩⽯石の違いØ ポアソン⽐比が液体の0.5により近いため  “fluid material”   (岩⽯石:0.25、氷:0.325)
Ø ejectionの⼒力力学がより流流体⼒力力学的になり、放出⾓角度度がより⼤大きくなる  (Melosh,1984)Ø 弾丸半径で規格化した距離離(x/ai)が5-10に増えるに従って放出⾓角度度が70°-50°と変化Ø 氷衛星の２次クレーターから推測したvej(θ=45°)は過⼩小評価² 衝突⾓角度度を70°で計算をした

Spall theoryØ 破⽚片サイズは30%ほど減少
Ø 破⽚片に与えられる速度度が⼤大きくなるためあるサイズの２次 　    クレーターを形成するために必要な破⽚片サイズは⼩小さくなる

122aiT (δ ρ )lspall =ρCL vej

K.N. Singer et al. / Ic

² Spall破⽚片サイズ
Ø 破⽚片の⼤大きさは標的の強度度によって決まる
Ø 氷の強度度Tは岩⽯石より⼩小さいため、氷のSpall破⽚片は岩⽯石に⽐比べ⼩小さくなる

Spall理理論論だけでは岩⽯石天体・氷天体の⼤大きな
２次クレーターについて完全に説明できない

Fig. 18. Effect of changing the assumed ejection angle. Smaller fragments areGrady-‐‑‒Kipp fragments² Grady-Kipp fragmentsのサイズ  (Melosh 1989)Ø 2次クレーター領領域の内側での最⼤大破⽚片サイズはとても⼩小さい

lGK =Ø Gilgamesh~20m、Tyre~1m
Ø 観測された２次クレーターを形成するために必要な破⽚片サイズ
        より⼩小さい

2aiTρ vej2 3vi4 3

lGK:Grady-kipp破⽚片厚さ
ai:衝突体半径 　ρ:標的密度度   　   　T:引っ張り強度度  vej:破⽚片放出速度度
vi:１次クレーター形成衝突速度度² Grady-Kipp fragmentsの問題点
Ø この理理論論はcracksがランダムな流流れによって活発もしくは成⻑⾧長することを想定
Ø 地表⾯面構造を想定していない  (断層、Megaregolith、層構造など)
Ø EuropaやGanymedeの表層も破砕されていて、構造的に⼀一様ではないØ ある速度度で同じサイズの破⽚片が放出されると予想
Ø １次クレーターからの距離離に伴い２次クレーターのサイズが変化することを説明できない

Grady-Kipp fragmentsを求める式は、観測された２次クレーターを形成す
る破⽚片サイズの⾒見見積もりには使えない

5-km-diameter craters. On the other hand,
5-km-diameterthe highest speed
craters.
GilgaOn the other
hand,thethehighestsp
bodies.
Tyre,largestmesh ejecta blocks are almost all less thanmeshhalfejecta
the sizeblocksof thearelargalmost allnaryless craters
than half
the
size
o
about 1 km
est ejecta blocks at those velocities (Fig.est
7c).
ejectaIt is blocks
also implausibleat those velocities(Fig. 7c).
It is alsoFoim
re-impact
on Europa).
that very-high-speed, sesquinary-formingthat tabularvery-high-speed,
ice spalls sesquinary-forming
cantabularicethson Ganymede,
we findsurvive intact. By this logic, the largest survive
sesquinary
intact.
craters
By thisformedlogic, theimpact
largestonsesquinary
Ganymedecrate
woby Gilgamesh are likely to be no larger bythanGilgamesh
2–3 km inarediameter.
likely to be noGilgamesh
larger than
2–3
kma in
SVD
has
verd

Sesquinary craters on Europa and Ganymede² 1.5次クレーター

cating fragment size develocity. In this study
6. Final remarks
6. Final remarks
et al., 2001; Alvarellos etfromGilgamesh(Ganyme
Mapping of secondary craters providesMapping
a windowof secondary
into ejectacraters
providesa windowin
ヒル圏
enoughtoaccount
forfitt
fragment sizes and velocities, and quantile
fragment
regression
sizes fitting
and velocities,
char- and quantile regression
observed
on Ganymede
acterizes the decline in fragment sizesacterizes
with increasing
the decline
velocity.
in fragment
sizes withincreasinga

Based on the physics of cratering andBasedejectaonwetheexpect
physics
to ofseecratering and ejecta we expeØ SVDの速度度をヒル圏脱出速度度まで外挿して、放出される最trends when comparing the SVDs of different
trendssecondary
when comparing
fields with
the SVDsof different secondary fi
Acknowledgments

⼤大破⽚片を⾒見見積もった

the size of the primary, surface gravity of
the
the
size
body,
of theor composition
primary, surface gravity of the body, or coof the target/impactor material, etc. Forof
the
the
two
target/impactor
large icy satellites
material, etc.We
Forthank
the twoicy
P.M.large
Schenkconsidered here, power law fits to the consideredmaximum ejecta
here, power
size aslaw
a fitsthree
to theregions
maximum
ejectamappedØ 再衝突速度度：Europa,2.5km/s (Zahnle et al.2008) 　

Table 4Largest sesquinary diameters.

 　 　 　 　 　 　 　    Ganymede,3.5km/s (Alvarellos et al.2002)
PrimarycraterLargest fragment at
escape velocity –full dataset (m)a² Gilgameshからの1.5次クレーターのサイズを⾒見見積もった
EuropaTyre
PwyllØ 氷破⽚片のスケーリング則で、破⽚片サイズは1km

GanymedeAchelous
Gilgamesh

Table 4Largest sesquinary diameters.
Largest fragment
Primary Largest
Largestfragment
fragment
at at
at escape velocity
crater escapeescapevelocity
velocity– master–a– subset (m)datasetfullofdataset
Fig. 15(m)(m)a b

150
34

120

36
360

5
220

Ø 最も速く放出された破⽚片は、最⼤大破⽚片の半分以下のサイズ

EuropaTyre
Pwyll

10015034

GanymedeAchelous 64 36
Gilgamesh
1070360

LargestLargest fragment
sesquinary Largest
Larg
fra
at
craterescapeon thevelocitysame bodyescapecrateveac–– subset
full dataset(m) (km) dataset
body
of
120
1.1
0.3

100 0.9

0.45 (0.3)d220
2.4

64 0.071070 1.6aaCalculated by extrapolation of equations in Table
Calculated
3 (for non-porousby extrapolation
ice scaling)of equations
to the Hillinsphere
Table escape3 (for non-porou
velocity gbCalculated with the quantile regression fit (99thCalculated
quantile) forwith
thethecombinedquantiledataset
regression
includingfit (99th
all threequantile)
primaries
for theinccCalculated using gravity regime parameters into
Calculated
a non-porous
using
target,gravity
withregime
typicalparameters
re-impact speed
into a of
non-porous
2.5 km s"1targ
foddAssumes porous, ice regolith target.
Assumes porous, ice regolith target.
eeComplex crater.
Complex crater.
bØ 氷のSpallは⽣生き残ったまま衝突し1.5次クレーターを形成するとする

Ø Gilgameshで形成された最⼤大の1.5次クレーターサイズは2­−3kmより⼤大きくない² 2,1.5次クレーターのサイズを⾒見見積もった先⾏行行研究Ø Europa:100mの破⽚片が衝突して0.5~1kmのクレーターが形成  (Alvarellos et al.2002)
Ø Gilgamesh:1kmより⼩小さな破⽚片が5km/sで衝突して~5kmほどのクレーターを形成(Zahnle et al.2001)

D>30kmにおけるクレーターサイズ分布に影響はないSummary² イジェクタサイズと速度度の関係を調べるために２次クレーターをカウンティングØ 速度度が⼤大きくなるにつれてイジェクタサイズは⼩小さくなるØ ２次クレーターのSVDは、天体重⼒力力や標的物質が異異なることにより⽣生じるクレーター形成、
イジェクタの物理理過程が異異なることによるØ 25-40kmのクレーターではβ〜～ｰ1、585kmの巨⼤大盆地ではβ〜～ｰ2-ｰ3
Ø 氷衛星でのSVDの傾きは、岩⽯石天体の同サイズクレーターに⽐比べ⼩小さくなる傾向Ø 岩⽯石天体ではspall theoryで説明できるが、氷衛星では説明できない² 本研究の結果をもちいて、氷衛星における1.5次クレーター分布に制約ができる
Ø EuropaとGnymedeの巨⼤大クレーターから求めたイジェクタSVDを外挿することで算出
Ø 1.5次クレーターはサイズが⼩小さいことがわかった  Ø EuropaのTyreクレーターの1.5次クレーターは1kmより⼩小さいØ Ganymedeの巨⼤大盆地Gilgameshでも2-3kmより⼩小さいØ βが⼤大きいため、衝突速度度が⼤大きいと破⽚片サイズが⼩小さくなるため
Ø 先⾏行行研究で予測したGilgameshの２次、1.5次クレーターのサイズは、Ganymede上の直径30kmより⼤大きなクレーター分布に影響が出るほど⼤大きいものではないことを確かめた