2014年年度度後期  第１回衝突勉強会
テーマ：観測に基づく推定

Craters  on  asteroids

Reconciling  diverse  impact  records  with  a  common  impacting  population
David  P.Oʼ’Brien  ,  Richard  Greenberg  ,  James  E.Richardson
Icarus  183  (2006)  79-‐‑‒92

神⼾戸⼤大学  理理学研究科
M1  松榮  ⼀一真Introduction² ⼩小惑星帯での衝突速度度は~5km/s (Opik 1951, Piotrowski 1952)Ø

カタストロフィック破壊(エネルギー⼤大)・クレーター形成(エネルギー⼩小)Ø   クレーター形成の⽅方が多く、⼩小惑星上のクレーターの記録は⼩小惑星のサイズ分布を推            測するのに⽤用いられる
Ø

クレーター消失過程を考慮する必要がある² ガリレオ探査機によって⼩小惑星GaspraとIda上のクレーター分布を調べた(Greenberg 1994,1996)Ø クレーターサイズ分布と衝突体のサイズ分布が類似Ø 衝突体サイズ分布  → ~100mのべき：ｰ2.95,100ｰ6mのべき：ｰ4 (Van Houten et al.1970)Ø Gaspra,Idaの⼩小さなクレーターの数が少ない

100m以下の衝突体のサイズ分布の傾きが⼤大きいこと、Gaspra,Ida上の⼩小さなクレー
ターが少ないこととの関係を説明する必要がある

Gaspraのクレー
ターサイズ分布(Belton et al.1992)

Introduction² 詳細なクレーター記録がわかっている⼩小惑星：Gaspra, Ida, Mathilde, ErosØ ⼩小惑星サイズ・形状とNEA・⼩小惑星帯のサイズ分布への理理解が広がったØ ⼩小惑星のサイズ分布は、collisionalとdynamical effectにより変化
collisional :  クレーター形成orカタストロフィック破壊により⼩小さな破⽚片が形成
dynamical : Yarkovsky効果と軌道共鳴によりNEAへ移動しNEA天体上にクレーター形成² ⼩小惑星帯とNEA分布のcollisional,dynamicalによる変化を数値計算  (O’Brien and Greenberg 2005)
1. ⼩小惑星帯の観測可能な天体のサイズ分布
2. NEAの観測可能な天体のサイズ分布3. 隕⽯石の宇宙線照射年年代によるm-サイズの天体の寿命
4. 200km以上の⺟母天体から形成された⼩小惑星群の数
5. Vesta上の⽞玄武岩質地殻と巨⼤大衝突盆地＜⽬目的＞

Gaspra,Ida,Mathilde,Erosの観測されたクレーター記録とモデルの結果を⽐比較すること→モデルをクレーター形成・消去過程を考慮し再構築Crater production~simulations~² Gaspraサイズの球天体に対し、衝突体とクレーターサイズの関係を計算(Nolan et al.1996)
Ø

衝撃波・掘削流流によって強度度が0になる領領域がトランジェントクレーター直径Dtrに等しいØ

strength,fractured regimeでは重⼒力力崩壊がないためDtrは最終クレーター直径と同じ

Ø

gravity regimeではリム直径はDtrより25%⼤大きい(Melosh 1989)² Asphaug et al.1996
Ø

Idaで同様のシミュレーションØ

Weibull flaw distributionを⼀一定

gravity(Dcrat>10km)
Dcrat= 161.4Dp0.78

strength(Dp<Dtr)² Greenberg et al.1996
Ø

Idaでの計算結果をNolan et alに適応Østrength,fractured regimeで正しくない

AsteroidcrAsteroDcrat = 35Dp

fractured(Dp>Dtr)
Dcrat = 26.61D1.159
p 　    かもしれない² Asphaug et al.2000
ØMathilde,Erosで数値計算

Ø

クレーターと弾丸サイズの関係を推定していない

Fig. Fig.
1. Crater
diametervs projectile
diameterfor impacts
into ainto
Gaspra-sized
1. Crater
diametervs projectile
diameterfor impacts
a Gaspra-si

Gaspraへの計算結果(Nolan et al.1996)の結果を、他のS-type⼩小惑星
spherical
bodybody
at 5300
m/s,m/s,
fromfromhydrocode
simulations
by Nolanet al.et(1996).
spherical
at 5300hydrocode
simulations
by Nolanal. (199
The solidline istheirleast-squares
fit tofitthetodata
pointsfor projectile
diame(Ida,Eros)・クレーター分布がわかっているC-type⼩小惑星(Mathilde)に適応する
The solid
line
is their
least-squares
the data
pointsfor projectile
diamters greaterthan than
5.6 m5.6anda slopeof 1 offor1projectiles
below
5.6 m,
ters greaterm follows
and follows
a slopefor projectiles
below
5.6
which
formform
craters
in theinstrength
regime.
The twolineslines
showshow
the gravwhichcraters
the strength
regime.
The dashed
two dashedthe grCrater production~scaling~² Holsapple and Schmidt(1982)に基づいたπｰscalingをIda,Erosに適応
規格化直径

規格化重⼒力力重⼒力力⽀支配域1!ρ $3
π D = Dat ## t &&" mp %

π D = CDπ 2− β
強度度⽀支配域

π D = CD! ππ2 =1.61gDp
vi2

規格化強度度π3 =−σ
3² C-type⼩小惑星Mathilde82Ø

密度度がS-typeの半分程度度→内部空隙の存在
D.P. O’Brien et al. / Icarus 183 (2006) 79–92

ØMathildeの半径程度度を直径とする巨⼤大クレーターの存在

Table 2
衝撃波が減衰し破壊された領領域が⼩小さい
Pi-groupØscaling
constants(Chapman et al.1999)
MaterialCDβCompetent rock

1.6

²
(右図)
Looseシミュレーション結果
sand
1.54

0.22
0.165

Notes. Pi-groupscaling constants from Melosh (1989). The values for looseØ GaspraとErosはほぼ同じ関係
sand are an average of the values for Ottawa sand and quartz sand.

Ø Idaはクレーターが少し⼩小さい
crater diametercan be found fromØ Mathildeのみ傾向が異異なる−βπD = CD π2 ,

(6)

′ −σ
π3 ,
πD = CD(7)

where β and CD are constants (see Table 2). The strengthscaled crater diameter can be found fromY
ρ P vi2

Dat:トランジェントクレーター直径
ρp,ρt:弾丸・標的密度度
mp:弾丸質量量 　 　  g:重⼒力力加速度度
Y:標的物質強度度 　  vi:衝突速度度
Dp:弾丸直径

MathildeJedicke and Metcalfe, 1998). Likewise, for an NEA still coupled to the main belt, less than 1% of potential impactors will
be NEAs. The intrinsic collision probability and mean collisionvelocity is a few times larger for NEA–MB and NEA–NEA collisions than for MB–MB collisions, but this difference is far

Crater production~Impacting population~² 全ての⼩小惑星について共通の衝突体のサイズ分布を仮定
ØOʻ‘Brien and Greenberg 2005を⽤用いる² NEA,MBAの衝突現象例例)Eros(NEA)Ø クレーター分布がMBAsの衝突分布を反映Ø NEAへは⼩小惑星帯から離離れた天体が衝突し、⼩小惑星帯の天体に
⽐比べ~1000倍も衝突する可能性が⼩小さい(Rabinowitz et al.2000, Jedicke
andMetcalfe,1998)
Fig. 3. Plotof thebest-fit main-belt size distribution from O’Brien and Greenberg (2005), which we use as the impacting population in our cratering simulations. The size distribution has been smoothed somewhat in order to remove fluctuations due to the discrete binning in the O’Brien and Greenberg
simulations. Also shown is the impacting population from Greenberg et al.
(Bottke et al.2000,2002)(1994, 1996). The O’Brien and Greenberg population has a differential slopeof −3.85 below ∼200 m, and the Greenberg et al. population has a differential
slope of −4 below 100 m.

² ⼩小惑星帯からやってきたNEAsの平均寿命を計算

Ø 地球型惑星や太陽に衝突・太陽系から放出されるまで6.5MyrØ Eros軌道周りの天体の数から、Erosは安定軌道であることがわかり50Myr
Ø NEAに50Myr存在することは⼩小惑星帯に1Myr存在することとと同等
Ø NEAに多数のクレーターが存在することはMBAによる衝突で形成

ErosCrater erasure mechanisms² クレーターを消す過程  (Greenberg,1994,1996, Richardson et al.2004)
84

D.P. O’Brien et al. / Ic

1.   Grobal seismic jolting
巨⼤大衝突が⽣生じたとき、衝撃波が伝播し全球を振動させある
サイズより⼩小さなクレーターを全て消す過程
Ø Gaspra,Ida(Greenberg1994,1996)
Ø ErosはGaspraよりサイズが40%⼤大きいこと以外は同じとして計算Ø Mathildeでは⽣生じない現象
84

2.   Local seismic jolting

D.P. O’Brien et al. / Icarus 183 (2006) 79–92

⼤大きな衝突が⽣生じたとき、衝撃波が伝播し局所的にjoltが⽣生じてある範囲内に存在するクレーターがサイズに関係なく消える過程Ø Gaspra,Ida,Eros同様の傾向
Ø Mathildeでは⽣生じない現象

Fig. 4. The diameter of a crater needed to erase craters of a given diameter by
global jolting on the different asteroids. Global seismic jolting is assumed to beineffective on Mathilde, given its likely porous internal structure.

simplified crater scaling relationships that are linear fits to theGaspra, Ida, and Eros curves in Fig. 2. Other variables, such as
the seismic frequency of the disturbance causing the jolt, areassumed to be the same for the different asteroids, as a moredetailed analysis would require hydrocode and seismic modeling beyond the scope of this work. Fig. 4 shows the size of acrater needed to erase craters of a given diameter by global jolting on Gaspra and Ida (from Greenberg et al., 1996) and ourestimate of the Eros jolt relationship using the Gaspra and Ida
jolt relationships, the values in Table 1, and the assumptions
above. Given the fact that even its largest craters do not significantly destroy topography beyond their rims, we follow theCrater erasure mechanisms² クレーターを消す過程  (Greenberg,1994,1996, Richardson et al.2004)
3.   Cumulative effects of seismic shaking (Richardson et al.2004)
⽐比較的⼩小さな衝突によるjoltが複数回⽣生じることでレゴリス層が崩れクレーターを消す
Ø Erosでのseismic shakingをモデル化した

4.   Superposition of craters

クレーターが平衡状態となり、形成と消失の割合が⼀一定となりサイズ分布が⼀一定(Melosh 1989)² Greenberg 1994,1996によって平衡状態を２つに分類
1) Sandblasting：⼤大きいクレーターを⼩小さいクレーターで消失
2) Cookie-cutter：⼩小さなクレーターを⼤大きなクレーターで消失

Combining production and erasure² 全ての⼩小惑星の地表⾯面が、クレーターの平衡状態に達するほど⼗十分に古いわけではないØ 巨⼤大衝突によるjoltが⽣生じ⼩小さなクレーターが消されるため² クレーター形成と消失過程の関係

クレーター形成
衝突頻度度f(Dp)
Dp:弾丸サイズ   Pi:衝突確率率率
Np:天体数
rt:標的天体の平均半径2f (Dp ) = Pi rt N p (Dp )82

D.P. O’Brien et al. / Icarus 183 (2006) 79–92

時間texp

衝突数

Table 2
Pi-group scaling constants
Material

CD

Competent rock
Loose sand

1.6
1.54

f (Dp )texp
β

0.22
0.165

−β

πD = CD π2 ,Grobal jolt
Cumulative
seismic
shaking※Erosのみ適応

Notes. Pi-group scaling constants from Melosh (1989). The values for loose
sand are an average of the values for Ottawa sand and quartz sand.

crater diameter can be found from

クレーター消失※

superposition
local jolt

(6)

Dcrat
のクレー
where
β and
CD are constants (see Table 2). The strengthscaled crater diameter can be found from
ターを形成
′ −σ
N crat (Dcrat (7))π3 ,πD = CD′ is a constant and
した衝突数
where
CD
β
.
σ=
(8)
1−β
Melosh (1989) contains a misprinted version of Eq. (8) that is
the negative of the correct value given here (Melosh, personalcommunication). The fracture regime is not predicted explicitly
by scaling theory, which considers only the effects of material strength and gravity, not the effect of pre-fracturing due tothe passage of the shock wave through the target (Nolan et al.,1996).

Fig. 2. Crater diameter vs projectile diameter curves for impacts on Gaspra,
Ida, Mathilde, and Eros. Ida and Erospcurves are scaled from the Gaspra simulations shown in Fig. 1, as described in the text. The Gaspra and Eros curves
nearly overlap, given their similar sizes
and impact velocities. For Mathilde, thecratgravity scaling relationship for loose sand is assumed for all impacts.

※衝突体(D )で形成されるク
レーター(D )の関係を適応

where αs and αf are the exponents of the strength- and fracturescaling relationships from Eq. (1).
Equations (9)–(11), along with the impact velocities in Table 1 and Pi-scaling coefficients in Table 2, can be used to scale
the Gaspra crater relationships for the strength and fracture※巨⼤大クレーター
のみ適応

クレーター形成とクレーター消失過程を考慮して、クレーターのサイズ分布を
数値計算

5.
Resultsis applied. For each crater bin centered at D , ourblasting)
craters quite well but produces too few large craters. At 1 Gyr, our simulat
blasting) is applied. For each crater bin centered at Dcrat ,crat
ourcode calculates the total effective area Atot of craters between
more closely matches the large craters but produces too many small craters
ofSectioncraters 4between
code calculates
the total
effective area
Atot in
Weof
now
apply
procedure
outlinedto the four
addition, the slope of the model population is shallower than the small cra
andthethelargest crater
that hasaccumulated
per
1/10
Dcrat
andhavethe largest
crater thathas
accumulatedper
1/10
of Dcratthat
population.
asteroidsbeen
observedbyspacecraft.
The
impact86 unit area since global jolt or cumulative seismic
D.P. O’Brienet al. / last
Icarus 183(2006) 79–92
shaking
unitingarea
sinceglobaljolt
orcumulative
seismic
shaking
last
population
we
use
is
the
best-fit
main-belt
populationfromerased craters of diameter Dcrat (i.e., over the lesser of tjolt andover thelesserof tjolt thisandimerased
cratersofGreenberg
diameter D(2005),
crat (i.e.,
O’Brien
and
shown
in
3. Using, or overtseismic
globaljolt effective
norFig.
cumulative
seismic
tsscumulative
exp if neither
thenshaking
is
more
than
global² exposure  time=65Myrでの⼤大きなクレーター 　
or overpopulation
texp if neither
globaljolt noroutlined
cumulative
seismic
tss , pacting
andAs
the
procedurein3.4,
Sectionweshakingareapplied).
described
in Section
when4,. cal(D)
isdecreasedbyt/tinstead
of
t
jolt
and
N
shakingareAs
described
inSection3.4,
whencrat applied).
crat
ss
exp
jolt /tcalexp
calculatepopulations
the different
asteroids.
Inculatingerasure
Athethebyareasof craters on
smallerthan Dcrat
aresandreduced
1Gyrでの⼩小さなクレーターで観測結果と合わない
tot , crater
Finally,superposition(cookie-cutter
andreduced
culating
A
,
the
areas
of
craters
smaller
than
D
are
tot
crat
all
unlessotherwise
noted,
apply
globalby asimulations,factorof 3 toaccount
for thebin
fact
thatwesmall
are jolt
notblasting)
is ofapplied.
For
eachcratercentered
at craters
Dcrat
, our
by aerasure,factor3
toaccount
for
the
fact
that
small
craters
are
not
local
jolt
erasure,anderasurebysuperposition(cookieperfectly
efficient
at erasinglarger craters,
the diametersofof andcraters
betweencode
calculates
theattotal
effective
Atotandperfectly
efficienterasinglarger area
craters,
thediametersof ofcutterand sandblasting).
Cumulative
seismic
shaking
erasurelarge
cratersareincreased
by
local
jolt
effects
(cf.
Fig.
5)
if
loand
the largest
craterjolt
that
has accumulatedper
1/10
of Dcratarelarge
cratersincreasedby localeffects(cf.
Fig.
5) ifasteroid
lo65Myrより後でクレーター消失過程が重要
small
craters
is only
appliedto Eros,
asis itthen
is the
only
cal
jolt
is
being
applied.
Equation
(17)
used
to
calculateunit
areais since
globaljolt
or cumulative
seismic
shaking
last
cal forjolt
beingtheapplied.
Equation(17) is thenusedcalculatewhich
necessaryparameterization[Eq. to(13)]
has beenFig. 6. Plot showing results of our simulations for Gaspra using the bes
thenumberofcratersthatstillremainvisible.
(i.e., overthe lesser of tjolt anderasedcratersofdiameter
Dcratremain
Fig. 6. Plot showing results of our simulations for Gaspra using the best-fit
the derived
numberofcratersthat
stillvisible.
main-belt population from O’Brien and Greenberg (2005), shown in Fig(Richardson et al., 2004). In addition, the other aster- main-belt
population from O’Brien and Greenberg (2005), shown in Fig. 3,
tss , or over texp if neither global jolt nor cumulative seismic
the impacting population. At 65 Myr, our simulation matches the smaoidshave not been imaged at small enough scales that the effect as theas impacting
5.
Results
population. At 65 Myr, our simulation matches the smaller
shaking
are applied). As described in Section 3.4, when calcraters quite well but produces too few large craters. At 1 Gyr, our simulat
5. Results² Gaspraが経験したシナリオ
of cumulative seismic shaking was detected in their cratering craters
quite well but produces too few large craters. At 1 Gyr, our simulation
more closely matches the large craters but produces too many small craters
culating Atot , the areas of craters smaller than Dcrat are reduced
more closely matches the large craters but produces too many small craters. In
records(althoughit isprocedureprobablyoutlined
acting on
craters
smaller
than
We
now
apply
the
in
Section
4
to
the
four
addition, the slope of the model population is shallower than the small cr
by1.
aWe
factorof
3 totheaccount
for thefact thatcraters
arefour
not
now
apply
procedure
outlinedin small
Section4ourto the巨⼤大クレーター(≧3km)の数を増やすくらい
addition, the slope of the model population is shallower than the small cratertheresolution
limit).
Sections
5.1–5.4
describe
resultsforpopulation.
asteroidsthat athavebeenlarger
observedby andspacecraft.
The impactperfectly
efficienterasing
craters,
the diametersof
population.
asteroids
that
have
been
observedbyspacecraft.
The
impacteachof the fourwe
observed
asteroids,
and Section
5.5 describes
ing
population
use byis thebest-fit
main-belt
population
⼗十分⻑⾧長い時間exposeされた
large
cratersarewe
increased
local
jolt
effects(cf.
Fig. 5) iffrom
lo-froming thepopulation
use
is
the
best-fit
main-belt
population
sensitivity
of
our
model
to
the
main
input
parameters.
O’Brienandapplied.
Greenberg(2005),(17)shownthenin Fig. 3.toUsing
this imcal
jolt
isand
being
Equation
O’Brien
Greenberg (2005),
shown isin Fig. used
3. Usingcalculate
this im 　 　→Grobal  joltで3kmより⼩小さいクレーターは消去
pacting
population
andtheprocedure
outlinedinSection4, we Fig. 6. Plot showing results of our simulations for Gaspra using the best-fit
thenumberof craters
that
still
remain outlined
visible. in Section 4, we
pacting
population
andtheprocedureFig. 7. Results of a simulation in which Gaspra is exposed to the impact
5.1.
Gaspracalculatethe crater populations on the different asteroids. In main-belt
population from O’Brien and Greenberg (2005), shown in Fig. 3,
calculatethe crater populations on the different asteroids. In
population for ∼1 Gyr and is then impacted by a projectile that is large enou 　 　  →Grobal  joltは弾丸が200mの衝突
as the impacting population. At 65 Myr, our simulation matches the smallerallsimulations,
unlessotherwise
noted,
weapply
globaljolt
to erase all craters below 3 km in diameter. 65 Myr of exposure following
5.
all Results
simulations,
unlessotherwise
noted, we
apply
globaljolt
craters quite well but produces too few large craters. At 1 Gyr, our simulationFig.
6localshows
craterand
population
insuperposition
our
model
at(cookieseveral
erasure,jolttheerasure,erasure by
resetting impact allows the small crater population to fill back in and ma
2.
観測された数だけ⼩小さなクレーターが1)の後
erasure,local
jolt erasure,and erasureby superposition(cookiemore closely matches the large craters but produces too many small craters. In
pointsin time.
Crater counts
are from
Table I shaking
in Chapman
et al.
the current population of small craters. This type of scenario was described
cutterandsandblasting).
Cumulative
seismic
erasureof
We andnowsandblasting).
apply the procedure
outlinedin Section
4 to
the fouraddition, the slope of the model population is shallower than the small crater
cutterCumulative
seismic
shaking
erasureofdetail by Greenberg et al. (1994).
(1996a),augmented
by
the
large
concavities
andcraters
identiに形成
small
craters
is been
only applied
toby
Eros,
as it is theasteroid population.
asteroidsthathave
observed
spacecraft.
Theonly
impactsmall
cratersis
only
applied
to
Eros,
as
it
is
the
only
asteroid
fied
by Greenberg
et al. (1994).
We have added
bars
for whichtheparameterization[Eq. ±1σ(13)]error
has
been
ing
population
we necessary
use is thebest-fit
main-belt
populationfromfor based
whichonthe√necessary
parameterization
[Eq.
(13)]
has
been
N counting
statistics(e.g.,
Woronowal., 1978).
(!3 km) craters. It then experienced a relatively recent impa
derivedet al.,
2004).
In
addition,
theetthisotherasterO’Brien
and(Richardson
Greenberg(2005),
shown
in Fig.
3.theUsingimderived(Richardsonetal.,2004).
In
addition,otherasterAt65
Myr,
oursimulation
matchesthesmallercratersquite
oidspopulation
have not beenat smalloutlined
enough inscalesthat 4,theweeffectby a projectile large enough that it erased all craters smaller th
Gaspraは1億年年exposeされていて、65Myrに約
pacting
andimaged
the atprocedure
Sectionoids
havebut
notproduces
been imaged
small
enoughscalesthat theeffect
well
too
few
large
craters.
After
only
65
Myr,
eraof cumulative
seismic
shaking
detectedintheir cratering∼3 km by global jolting. Finally, subsequent cratering filledcalculatethe crater
populations
onwas
the
different
asteroids.
Inof200mの天体の衝突が⽣生じ3kmより⼩小さなクレー
cumulative
seismic
shaking
was
detectedin theircrateringsureeffects
have
yet
to
be
importantandthecraterpopulation
records (although
it otherwise
is probably
actingweonapply
cratersglobal
smallerthe small crater population observed today. Fig. 7 shows the rallsimulations,
unlessnoted,
joltthanrecords(althoughit isjoltによって消失した
probably
actingoncraters smallerthan
ターがGrobal
isa directreflectionofSections
the impacting
population.
At 1results
Gyr,
our
the
resolution
limit).
5.1–5.4
describe
ourforsult from our model for such a scenario, with 1 Gyr of exposu
erasure,local jolt
erasure,and erasureby superposition(cookiethe simulation
resolution
limit).
Sections
5.1–5.4
our resultsformoreclosely
matchesthedescribelargecraters
but produces
each
of
the
four
observed
asteroids,
andSection
5.5
describes
cutterandCumulative
seismic
shaking
erasureofand a resetting event 65 Myr ago that erased all craters smal
eachtooof
thesandblasting).
fourobserved
asteroids,
and due
Section
5.5competing
describes
many
small
craters.
In addition,
to theefthesensitivity
of
our
model
to
the
main
input
parameters.
smallcraters isofonly
appliedtotothe
Eros, asinput
it is the only asteroid
than 3 km, followed by continued accumulation of craters fro
the fects
sensitivity
modelof craterour
production
and mainerasure, theparameters.
slope of the model

Result~Gaspra~

5. Results
craters quite well but produces too few large craters. At 1 Gyr, our simul
blasting) is applied. For each crater bin centered at Dcrat , ourmore closely matches the large craters but produces too many small crate
code calculates the total effective area Atot of craters between
We now apply the procedure outlined in Section 4 to the four
addition, the slope of the model population is shallower than the small c
1/10 of Dcrat and the largest crater that has accumulated per
population.
asteroids that have been observed by spacecraft. The
impactAsteroid cratering records
87
unit area since global jolt or cumulative seismic shaking
last
ing population we use is the best-fit main-belt population fromerased craters of diameter Dcrat (i.e., over the lesser of tjolt andO’Brien and Greenberg (2005), shown in Fig. 3. Using this imOur
thus
confirms
the global
plausibilityof thescenario seismic
proor overtexp
if neither
jolt nor
cumulative
tss ,model² 250Myrより⻑⾧長い時間でのモデル
pacting
population
andtheprocedure
outlinedinSectionwe
posedby Greenberg
et al.As described in Section 3.4, when4,calshaking
are applied).
calculatethe crateron necessary
the different
asteroids.
Ø culating
クレーター分布は10kmより⼩小さいと飽和して
From
Figs.
andareas
4,populations
thetocratcausesuch In
Atot ,2theof projectile
craters smaller
than Dare reduced
all simulations,
unless otherwise
noted,and
wegivenapplytheglobaljolt
a resetting
event
m for
in diameter,
numby a factorof 3 isto ∼200
account
the fact that
small craters
are not
いて良良く合っている
jolt erasure,and erasureby superposition(cookiebererasure,ofsuchlocal
objects
impacting(Fig.
andthe ofperfectly
efficientinatthe
erasinglargerpopulationcraters, andthe 3)
diameters
cutterandsandblasting).
Cumulative
seismic
shaking
erasureofandcraters
intrinsic
probabilityof Gaspra
Table
1, lolarge
arecollision
increasedby local jolt
effects from(cf. Fig.
5) ifØsizeIdaの地表はこれくらい古いとされているが、
craters
is onlyfor
appliedimpacts
to Eros,isas∼1
it Gyr.
is theThus,
only itasteroid
thesmallaveragetimescaleapcal
jolt is being
applied.such
Equation (17)is thenused to calculate
for
which
the
necessaryparameterization
[Eq.
(13)]
has
been
pears
that Gaspra
experienced
once-in-a-billion-years
event
Fig. 6. Plot showing results of our simulations for Gaspra using the bethe10kmより⼤大きなクレーター分布の変動でばら
numberof cratersthat stilla remain
visible.
derived(Richardsonetal.,2004).
In
addition,theotherastermain-belt population from O’Brien and Greenberg (2005), shown in F
about 65 million years ago. Using Poisson statistics, such an
oids
have
not
been
imagedatsmall
enoughscalesthattheeffectas the impacting population. At 65 Myr, our simulation matches the sm
event
has about a 6% chance of occurring on a given asteroid.
5. つきが⽣生じる
Results
craters quite well but produces too few large craters. At 1 Gyr, our simulofcumulative
seismic
shaking
was
detectedintheircratering
A more relevant statistic is that the chance of an event with that
more closely matches the large craters but produces too many small crate
records (although
it is exactly
probablyone
actingon thecraters
than
probability
occurringout offour smaller
We now
apply theonprocedure
outlinedin Section
4asteroids
to the four
addition, the slope of the model population is shallower than the small cthehave
resolution
limit).by
Sections
5.1–5.4
describe
ourresults
for
that
been
imagedspacecraft
is
about
20%.
Hence,
itispopulation.
asteroids that have been observed by spacecraft. The impact² Idaの年年代で注⽬目すべき点
of the four
observed
asteroids,
and Sectiona5.5
describes
noteachsurprisingthat
Gaspracould
haveexperienced
relatively
ing population we use is the best-fit main-belt population fromthesensitivityof our(asmodelto the inmainparameters.
event,
while
explained
theininput
following
sections)(a)Øunlikely
直径が~∼1.5kmの衛星のDactyl
O’Brienand Greenberg(2005), shown
Fig. 3. Using
this imthepacting
other threeasteroids
didnot.
population and the procedure outlined in Section 4, weFig. 7. Results of a simulation in which Gaspra is exposed to the impa
5.1. Gaspra
calculatethe crater populations on the different asteroids. InØ5.2.サイズよりcollisional  lifetime~∼100Myr
population for ∼1 Gyr and is then impacted by a projectile that is large en
all Idasimulations, unless otherwise noted, (Greenberg  et  al.1996)
we apply global jolt
to erase all craters below 3 km in diameter. 65 Myr of exposure followin
Fig. 6 shows the crater population in our model at several
resetting impact allows the small crater population to fill back in and merasure, local jolt erasure, and erasure by superposition (cookieinshows
time. Cratercountsarefrom
Table I for
in Chapman
et al.
Fig.
8
theresultsof
oursimulation
Ida
forexØ points
モデルよりcollisional  lifetime~∼800Myr
the current population of small craters. This type of scenario was describ
cutter and sandblasting). Cumulative seismic shaking erasure ofdetail by Greenberg et al. (1994).
(1996a),augmented
bythetolarge
concavities
andcraters
identiposure
times
from
100
Myr
4
Gyr.
Crater
data
are
the
total
(Oʼ’Brien  and  Greenberg  2005)
small craters is only applied to Eros,
as it is the only asteroid
fied by
Greenberg(1994).
We(1994)
have added±1σerrorofbars
countsfromFig. 5b etin al.
Beltonet al.
and thecountsforwhich√the necessary parameterization [Eq. (13)] has beenØlarge
Idaを含むKoronis族<1.5Gyrcraters
Table IIstatistics
in Chapmanetal. (1996b).
N counting
Woronow
et al.,±1σ
1978).
basedon from(!3 km) craters. It then experienced a relatively recent im√ (e.g.,derived (Richardson et al., 2004).
In addition, (Durda  et  al.1993)
the other asterN counting
statistics
and inerror
wereour
added
based onmatches
At bars
65 Myr,simulation
the smaller
cratersquite
by a projectile large enough that it erased all craters smaller
oids have not been imaged at small enough scales that the effectthe samplingareascraters.
for different
size 65
rangeswell
but aboutproduces
too few large
After only
Myr,oferaØformation
クレーターが多いことから地表年年代は古い∼3 km by global jolting. Finally, subsequent cratering filleof cumulative seismic shaking was detected in their cratering(Belton  et  al.1994)
craters
given inhave
BeltonAt and100 theMyrcraterof exposure,
sure effects
yet etto al.
be (1994).
importantpopulation
the small crater population observed today. Fig. 7 shows therecords (although it is probably acting on craters smaller thantheismodel
population
below thepopulation.
observed population.
a directcrater
reflectionof theisimpacting
At 1 Gyr, ourthe resolution limit). Sections 5.1–5.4 describe our results for
sult from our model for such a scenario, with 1 Gyr of expo
After
250 Myrmore
of exposure,the crateris essentiallyIdaの年年代は、~500Myrｰ1Gyrと推測される
simulation
closely matches
thepopulation
large craters
but produceseach of the four observed asteroids, and Section 5.5 describes
and a resetting event 65 Myr ago that erased all craters sma
saturatedbelow
10 craters.
km at a In
level
that matches
pop- eftoo manysmall
addition,
due tothe
thecurrent
competingthesensitivity
of
our
model
to
the
main
input
parameters.
than 3 km, followed by continued accumulation of craters fulation.
Ida’s
surfaceis at leastold, butvariationsinfects ofcraterproduction
and thaterasure,the theslopeof the model

Result~Ida~as the impacting population. At 65 Myr, our simulation matches the
5. Finally,
Results erasure by superposition (cookie-cutter and sandcraters quite well but produces too few large craters. At 1 Gyr, our sim
blasting) is applied. For each crater bin centered at Dcrat , ourmore closely matches the large craters but produces too many small cr
Wecalculates
now applythe
thetotalprocedure
outlined
4 tobetween
the fourof craterscode
effective
area AintotSection
addition, the slope of the model population is shallower than the sma
population.
asteroidsbeen
observed
spacecraft.
The impactand thelargestcraterbythat
has accumulatedper
1/10
of Dthatcrat have
88
D.P. O’Brien et al. /ing population
use istheorbest-fit
main-belt
population
unit
area since weglobaljolt
cumulative
seismic
shakingfrom
last
O’Briencraters
and Greenberg(2005),
in Fig.
3. Using
im(i.e., overthe lesserof tthis
erasedof diameter
Dcrat shownjolt and² S-‐‑‒type⼩小惑星より低密度度であるため、内部に空隙がpopulation
the procedurein Section
4, we
texp if andneither
global joltoutlined
nor cumulative
seismic
tpacting
ss , or over
ありラブルパイル構造かもしれない
calculatearetheapplied).
crater populations
on in
theSection
different
asteroids.
Inshaking
As described
3.4,
when calall
simulations,
unlessotherwise
noted,
weapply
globaljolt
culating Atot , the areas of craters smaller than Dcrat are reduced
Ø ⼤大きなクレーターがあることからjoltは⽣生じていerasure,local
superpositionbya factorof jolt
3 toerasure,accountand
for erasure
the factby
that
small craters(cookieare notcutter andefficient
sandblasting).
Cumulative
seismic
shaking
erasure of
perfectlyat erasinglarger craters,
andthe diameters
るはず
small craters
cratersare
is only
appliedEros,
it is thelarge
increased
by to
local
joltaseffects(cf.onlyFig. asteroid
5) if lofor jolt
which
the necessary
parameterization[Eq.used
(13)]
been
cal
is being
applied. Equation(17) is thento hascalculate² NEAR探査によりMathildeでは地震エネルギーは伝播
Fig. 6. Plot showing results of our simulations for Gaspra using thederived(Richardsonal.,still2004).
In addition,thenumberof cratersetthat
remainvisible. the other asterしていないことを⽰示唆
main-belt population from O’Brien and Greenberg (2005), shown inoids have not been imaged at small enough scales that the effectas the impacting
population. At 65 Myr, our simulation matches the
⾼高解像度度of cumulative
seismic shaking was detected in their cratering
5.
Results
craters quite well but produces too few large craters. At 1 Gyr, our sim
records (although it is probably acting on craters smaller thanmore closely matches the large craters but produces too many small cr
theWe
resolution
limit).
Sections 5.1–5.4
our4 results
for
now
apply
the procedure
outlineddescribein Section
to the four
addition, the slope of the model population is shallower than the sma
Joltの影響は除外している
population.
each
of
the
four
observed
asteroids,
andSection
5.5
describes
asteroids that have been observed by spacecraft. The impactthe sensitivity
the main
input parameters.
ing
populationof
weour
usemodelis thetobest-fit
main-belt
population from(a)O’Brien and Greenberg (2005), shown in Fig. 3. Using this im² km-‐‑‒scaleと⼩小さなクレーターでは⼀一定に飽和Fig. 7. Results of a simulation in which Gaspra is exposed to the im5.1. Gaspra
pacting
population and the procedure outlined in Section 4, we
population for ∼1 Gyr and is then impacted by a projectile that is large
calculate the crater populations on the different asteroids. In² exposureが⻑⾧長くなると⼤大きなクレーターが増加
to erase all craters below 3 km in diameter. 65 Myr of exposure follow
6 shows unlessthe crater
populationin we
our apply
modelglobal
at several
all Fig.
simulations,otherwise
noted,
jolt
resetting impact allows the small crater population to fill back in andpoints inlocal
time.jolt
Cratercountsfrom byTable
I in Chapman
et al.
erasure,erasure,andareerasuresuperposition(cookieØ ⼤大きなクレーターで結果があるのは4Gyr
the current population of small craters. This type of scenario was desc
detail by Greenberg et al. (1994).
(1996a),by thelarge concavities
and craters
identicutter
andaugmented
sandblasting).
Cumulative
seismic shaking
erasureof
fied bycraters
Greenberg
et al.
(1994).
have
bars
small
appliedto We
Eros,
as added
it is the±1σ
onlyerror
asteroid√ is only
N necessary
counting statistics(e.g., Woronow
et al.,
basedon theforwhichparameterization[Eq. (13)]has1978).
been(!3 km) craters. It then experienced a relatively recent i
At
65
Myr,
our
simulation
matchesthesmallercraters
quite
derived (Richardson et al., 2004). In addition, the other aster内部空隙があるためJoltは⽣生じておらず数kmよりby a projectile large enough that it erased all craters smalle
well
but
produces
too
few
large
craters.
After
only
65
Myr,
eraoids
have
not
been
imagedatsmall
enoughscalesthattheeffect∼3 km by global jolting. Finally, subsequent cratering fi
⼩小さなクレーターでは飽和に達しており、⼤大きな
低解像度度
surecumulative
effects have
yet toshaking
be importantand the incraterofseismic
was detected
theirpopulationcratering
the small crater population observed today. Fig. 7 shows
クレーターから地表年年代は~4Gyrと推定
is a directreflection
the impacting
population.
1 Gyr,thanourrecords(althoughit ofis probably
actingon cratersAt
smallersult from our model for such a scenario, with 1 Gyr of exp
simulation
moreclosely
matchesthe large
cratersour
butresults
producestheresolution
limit).
Sections
5.1–5.4
describeforand a resetting event 65 Myr ago that erased all craters s
too many
addition, anddue Section
to the competing
efeachof thesmall
four craters.
observedInasteroids,
5.5 describes
than 3 km, followed by continued accumulation of crater

Result~Mathilde~

1/10 of Dcrat and the largest crater that has accumulated perunit area since global jolt or cumulative seismic shaking lasterased craters of diameter Dcrat (i.e., over the lesser of tjolt and
tss , or over texp if neither global jolt nor cumulative seismic
shaking are applied). As described in Section 3.4, when calAtot , the areas of craters smaller than Dcrat are reduced (Richardson  2004,2005)²culating
Erosは、モデルが作られている唯⼀一の⼩小惑星by a factor of 3 to account for the fact that small craters are not²perfectly
⾼高解像度度の画像がある
efficient at erasing larger craters, and the diameters oflarge craters are increased by local jolt effects (cf. Fig. 5) if local jolt is being applied. Equation (17) is then used to calculate
Fig. 6. Plot showing results of our simulations for Gaspra using the best-fit
number of craters that still remain visible.
²the 200mより⼩小さなクレーターではどの年年代でもモデルが⼀一致

Result~Eros~main-belt population from O’Brien and Greenberg (2005), shown in Fig. 3,
as the impacting population. At 65 Myr, our simulation matches the smaller
5. ResultsØ   Cumulative  seismic  eﬀectを考慮しているため
craters quite well but produces too few large craters. At 1 Gyr, our simulation
more closely matches the large craters but produces too many small craters. In
We now apply the procedure outlined in Section 4 to the four
addition, the slope of the model population is shallower than the small crater
population.
asteroids that have been observed by spacecraft. The impact-

D.P. O’Brien et al. / Icarus 183 (2006) 79–92

population we use is the best-fit main-belt population from²ing 観測結果と最も整合性が良良いのは、t=120Myr

O’Brien and Greenberg (2005), shown in Fig. 3. Using this imØ population
Richardsonのモデルでは、400±200Myr
pacting
and the procedure outlined in Section 4, we
calculate the crater populations on the different asteroids. InØ 計算に⽤用いた衝突体のサイズとクレー
all simulations, unless otherwise noted, we apply global jolt
erasure, local
jolt erasure, and erasure by superposition (cookieターサイズの関係が異異なるため
cutter and sandblasting). Cumulative seismic shaking erasure of
正しい関係を⽤用いた数値シミュレーショ
smallØcraters
is only applied to Eros, as it is the only asteroid
for which the necessary parameterization [Eq. (13)] has beenンが必要となるderived (Richardson
et al., 2004). In addition, the other asteroids have not been imaged at small enough scales that the effectof cumulative seismic shaking was detected in their cratering
records(although it is probably acting on craters smaller than
⼩小さなクレーターは、Cumulative  seismic  the resolution limit). Sections 5.1–5.4 describe our results foreﬀectより消失していて、地表年年代はeach of the four observed asteroids, and Section 5.5 describes~120Myrと最も整合性が良良い
the sensitivityof our model to the main input parameters.
5.1. Gaspra

Fig. 7. Results of a simulation in which Gaspra is exposed to the impacting
population for ∼1 Gyr and is then impacted by a projectile that is large enoughSummary² Gaspra,Ida,Eros,Mathildeのクレーター分布は、共通の衝突分布で表せることがわかった² クレーター形成と消失過程を考慮したモデルと観測結果を⽐比較
1. GaspraØ

1億年年exposeされていて、65Myrに約200mの天体の衝突が⽣生じ3kmより⼩小さなクレーターがGrobal joltによって消失した

2. Ida3.

4.

Ø

10kmより⼩小さなクレーターは飽和状態に近いため、クレーター消失過程が⼤大きく影響しているØ

年年代は、~500Myrｰ1Gyrと推測

Asteroid
crateringrecords
Asteroid
crateringrecords

Mathilde
At Myr,50 Myr,populationof craters
betweenabout
At 50
the thepopulationof craters
betweenabout
200200m m
1 in
kmdiameterin diameterin oursimulation
is sufficient
not sufficient
to matchand
1 km
in oursimulation
is notto matchØ and内部空隙があることからgrobal,local
joltは⽣生じていない
observations.
Similarly,
at 200populationof craters
observations.
Similarly,
at 200
Myr,Myr,
the thepopulationof cratersØ
NEARによる探査結果と調和的!1 km
in diameterin oursimulationis larger
the observed!1 km
in diameterin oursimulationis larger
thanthanthe observed
at 120At of
all ofpopulation.
We We
findfindthatthat
the thebestbest
fit isfitatis120
Myr.Myr.
At allØ population.
⼤大きいクレーター(D>10km)より~4Gyrと推定
the times
shown,
the flatteningof size
the size
distributionof craters
the times
shown,
the flatteningof thedistributionof craters
Eros
"200
in diameteris dueto erasureby cumulative
seismic"200
m inmdiameteris dueto erasureby cumulative
seismic
shaking.
Ø shaking.
直径が200mより⼩小さなクレーターの分布を説明する
Using
same
impactingpopulation
Using
the thesame
impactingpopulation
thatthatwe we
use usehere,here, 　 　ためには、cumulative
seismic
shakingが必要
Richardson
al. (2004)
found
that
best
fit wasachieved
Richardson
et al.et (2004)
found
that
theirtheir
best
fit wasachieved400± 200
of exposure.
primaryreasonfor the
after
400
± 200
MyrMyrof exposure.
TheTheprimaryreasonfor theØ after
若若い地表年年代は~120Myr
discrepancy
betweenestimatesis thatRichardson
al. used
discrepancy
between
our our
estimatesis thatRichardson
et al.etuseda somewhat
different
relationshipbetween
impactordiametera somewhat
different
relationshipbetween
impactordiametercraterthe relationshipwe use2). Forand andcratersizesizethanthanthe relationshipwe useherehere(cf. (cf.
Fig.Fig.
2). For89