2015年度前期 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 神⼾⼤学 理学研究科 M2 松榮 ⼀真 Outline of this study
² 本研究では、EuropaとGanymedeの巨⼤クレーターの解析を⾏った 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法 3. 重⼒⽀配域における２次クレーターを形成したイジェクタ破⽚サイズと速度の⾒積も り⽅法 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較 5. 氷衛星の結果を岩⽯天体との結果と⽐較 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積もり 7. まとめ Introduction
² 2次クレーター ² 1.5次クレーター (Zahnle et al.2008)
ヒル圏 ヒル圏
ejecta
² 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 ⾯年代決定に影響がでる Ø ⼩天体or彗星の衝突と⾒分けることが難しくなるため
b ² 2次クレーターを1次クレーターとカウントすると、 Ns ∝ D クレーター年代を過⼤評価することがある 累 Ø ⼩さなクレーターの累積個数が⼤きくなる 積 a 個 Ns ∝ D Ø クレーターカウンティングによる年代決定するた 数 めには、１次クレーターと２次クレーターを⾒分 ける必要がある クレーター直径 Introduction
² １次クレーターと2次クレーターの⾒分け 1. １次クレーター近傍 Ø ２次クレーターの数密度は、１次クレーターからの距離とともに減少 Ø 形状が不規則 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 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 破⽚のSVDは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 岩⽯天体(⽔星・⽉・⽕星)でのSVDは調べられている
ntroduction I STROM StromET AL.: CRATER et al.1981 POPULATIONS ON GANYMEDE AND CALLISTO 8671
I [ I [ I I [ • [ I I I I I I I ] I I [ I [ [ I [ ets. We tend to favor the latter explanationfor the following ² 岩⽯天体と氷天体の違い reasons.Figures 11 through 12 show that sevendifferent cra- ⽉の⾼地 ter curves,representing vastly different densities,on different terrains and even on different satellitesall possessthis steep- Ø Ganymede l ⽉と を⽐較 l l slope index (•-4.7) distribution function. Furthermore, as I pointed out earlier, even though older cratersup to about 100 • l ｰ Ganymedeでは⼤きなクレーターが少ない km diameter have been degraded,i.e., fiat floors at about the -I - Lunar Highlands--. - level of the surroundingterrain, their rim sharpnessis more or （D>100km） ｰ 50 limit of about -3 for its slope.If an ancientepisode of obliter- Lunm ation, similar to that proposedfor Ganymede, operated on Callisto as well, then the production function could have an index more negativethan -3. In any case,it is far from the -•,2 I • • •,•1IO I I • i •j,Ioo , i i I I I IOOOI I -2.3 index observedon the terrestrialplanets. CRATER DIAMETER (km) All terrains on both Ganymede and Callisto show a de- Fig. 12. Comparisonof the crater populationson Ganymede's creasein slope index for cratersgreater than about 50 kin. groovedand heavily crateredterrains, with the lunar curve for refer- Two plausibleexplanations for this decreaseare: (1) a great ence.The groovedterrain is similar in slopeto the heavily cratered deal of crater obliteration due to crater relaxation in the icy terrainfor diametersabove 30 km, but at smallerdiameters a progres- crust,the vigor of the processincreasing with crater size [Par- sive loss of smaller craters has increasedthe slope index of the groovedterrain compared to that of the heavilycratered terrain. Also mentieret al., 1980],or (2) the curvesbasically represent the notice that the groovedterrains which were measuredare of at least productionfunction with a deficiencyof impactingbodies in two ages,one being about as denselycratered as the heavily cratered this crater sizerange compared to that for the terrestrialplan- terrain, the other much less cratered. Sites and mapping methods ² GalileoとVoyager2 missionで得られた画像を⽤いた 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次クレーターと⼤きく異なるクレーターは除外 K.N. Singer et al. / Icarus 226 (2013) 865–884 867 Table 1 Summary of primary and secondary crater field characteristics. Primary Primary Primary transient Diameter of primary Mosaic Number of Largest observed Fragment size for largest 1. Tyre-Europa a b c d crater diameter diameter (km) impactor (km) resolution secondaries secondary (km) secondary (m) 1 (km) (m pxÀ ) mapped Europa Tyre 38e 23 1.8 170 1,165 2.8 1160 e f (f Bierhaus et al.2005,2009) 直径が〜38kmで、⽐較的若くクレーターが少ない領域Pwyll 27 17 1.2 27, 21, 54 180 に位置する410 Ganymede Achelous 35g 21 1.9 180 630 2.7 1200 Ø Gilgamesh 585h 271 49.1 550 445 21.3 (18.6) 5760 (5000) ~175kma 中⼼から Sectionまで円形の溝が存在3. b 1 1 Assumes cometary impactors at 26 km sÀ (Europa) and 20 km sÀ (Ganymede) (Eq. (10)). c 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). -1 -1 Ø e Schenk and Turtle (2009). 170mpx (~30mpx ) 画像に限りがあるがf Alpert and、 Melosh① (1999)解像度が. ②⾼解像度 g Schenk (2010). h Ø 1165 0.5-2.8kmSchenk et al. (2004). 868 K.N. Singer et al. / Icarus 226 (2013) 865–884 ① 個： from the secondary diameter we can estimate the fragment’s size. The inset in Fig. 5 illustrates the quantities described in the follow- Ø 375 ~180m ing section, and Table 2 lists all symbols and associated parameters ② 個： が最⼩ utilized. Fragment velocities were calculated from the range equa- tion for a ballistic trajectory on a planetary sphere: 2 =R g 1 tej sin h cos h p R 2R tanÀ ; 1 b p 2 2 ¼ 1 t cos h=Rpg ð Þ À ej ! where Rb is the measured ballistic range from the primary to each secondary, Rp is the radius of the planet or moon (see Table 2 for values), g is surface gravity, h is the ejection angle, and tej is the ejection velocity we solve for. There are several assumptions that go into this calculation. We do not know the exact launch point or ejection angle of each frag- ment. The ejection angle is assumed to be 45°, as laboratory exper- iments in a granular media show this to be a practical average value with some scatter ±15° (Cintala et al., 1999; Durda et al., 2012). There is a subtle aspect to this assumption, however, which will be returned to in Section 5. An approximate value of one-half of the transient primary crater radius (0.25Dtr) was used as the starting point for measuring the ballistic range to each secondary (we justify this in a later section). The transient crater represents Fig. 1. Europan impact basin Tyre (centered at 34°N, 147°W) and outline of high resolution mosaic shown in Fig. 2. Dashed circle indicates the equivalent rim of Tyre (38 km a nominal cavity shape as the excavation flow ceases but before 1 1 in diameter). Mosaic is 170 m pxÀ , from Galileo imagery. Mapped secondaries (n = 1165) indicated in yellow, with lower density to the eastFig. possibly 2. Portion indicating of high an oblique resolution mosaic of Tyre’s secondary field ( 30 m pxÀ ). (a) gravitational collapse proceeds; thus Dtr/4 represents an average impact from the east. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) This high-Sun image also reveals dark material concentrated in secondary crater distance from which fragments might have been ejected. The tran- floors. (b) Mapping at higher resolution confirms craters mapped at lower sient diameter for each of the 3 primary craters was estimated resolution and permits mapping of considerably smaller secondaries (n = 375). are secondaries from Gula, but most appear to be from Achelous. 2.3. Gilgamesh – Ganymede To mitigate this we measured only to 63.5°N. This scene is again below: a total of 445 craters in the diameter range of 2.3–21.3 km. limited by the mosaic extent, to radial distances 45 – 85 km from Gilgamesh, Ganymede’s largest impact basin, has an equivalent the crater rim. rim diameter of 585 km (Schenk et al.,In 2004 some); this cases rim location this includes is ‘‘chains’’ where there is only one large Although Ganymede has many more primary craters than Euro- consistent with scaling from ejecta depositscrater in Schenk but there and Ridolfi is also a smaller crater, or elongation/gouging in pa, Achelous formed on younger grooved terrain, which helps elim- (2002) and is also marked by a prominent,the direction inward-facing radial scarp. to Gilgamesh (‘‘Gilgamesh sculpture’’). If a cra- inate some contamination from small primaries. A few craters Gilgamesh and the terrain to the east were imaged by Voyager 2 1 ter had a distinctly different degradation state than the majority of which were obviously out of place in terms of size or morphology at moderate resolution (550 m pxÀ ; see Fig. 4). Most identifiable when compared to the secondary field as a whole were excluded chains and secondaries are concentratedthe to secondaries the north and south(either much fresher, rayed, or much more de- from the analysis (unmarked craters in Fig. 3). Achelous on Gany- of the basin. Given the more heavily crateredgraded) surface it wasof Ganymede, excluded from the analysis. Additionally, if a crater mede makes a good comparison with Tyre on Europa because it is the Gilgamesh scene was the most difficultwas of particularly the three study large areas for its distance from Gilgamesh, compared to similar in terms of size and the resolution it was imaged at, and the in which to distinguish primaries from secondaries. Only craters gravities and surface compositions of the two moons are similar. that appear to be in a chain or cluster werethe included surrounding in the dataset secondaries, it was excluded (these craters ap- peared as obvious outliers in the dataset). The craters that were mapped or excluded can be seen in Fig. 4a and some specific exam- ples of the mapping decisions are given in Fig. 4b. Topography and example secondary crater profiles are shown in Fig. 4c and d, respectively. During the analysis described below, we also ex- plored an augmented dataset for Gilgamesh, which includes other craters that could be secondaries, but were not in an obvious chain or cluster. 2.4. Pwyll – Europa Alpert and Melosh (1999) measured 180 secondaries in high- 1 resolution Galileo mosaics (27, 21, 54 m pxÀ ) at three distances from Pwyll, a 27-km-diameter crater. One mosaic is centered over Pwyll, and the other two are almost due north, 900 km and 1,070 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 sam- pled, but a disadvantage to this site is that measurements only ex- tend in one radial direction away from Pwyll and mapping was only possible in small patches of surface area. Without broader ra- dial coverage the results may not reflect the largest secondaries at a given distance, due to stochastic variations with launch azimuth. Fig. 3. (a) Ganymede’s Achelous and Gula craters, 35 and 37 km in diameter, 3. Fragment size and velocity 1 respectively. Achelous is centered at 62°N, 12°W. Mosaic is 180 m pxÀ , from Galileo imagery. Mapped secondaries (n = 630) in yellow. (For interpretation of the From measured secondary crater ranges we can calculate the references to color in this figure legend, the reader is referred to the web version of velocity of the ejecta fragment that formed each secondary, and this article.) 868 K.N. Singer et al. / Icarus 226 (2013) 865–884 from the secondary diameter we can estimate the fragment’s size. The inset in Fig. 5 illustrates the quantities described in the follow- ing section, and Table 2 lists all symbols and associated parameters utilized. Fragment velocities were calculated from the range equa- tion for a ballistic trajectory on a planetary sphere: 2 =R g 1 tej sin h cos h p R 2R tanÀ ; 1 b p 2 2 ¼ 1 t cos h=Rpg ð Þ À ej ! where Rb is the measured ballistic range from the primary to each secondary, Rp is the radius of the planet or moon (see Table 2 for values), g is surface gravity, h is the ejection angle, and tej is the ejection velocity we solve for. There are several assumptions that go into this calculation. We do not know the exact launch point or ejection angle of each frag- ment. The ejection angle is assumed to be 45°, as laboratory exper- iments in a granular media show this to be a practical average value with some scatter ±15° (Cintala et al., 1999; Durda et al., 2012). There is a subtle aspect to this assumption, however, which will be returned to in Section 5. An approximate value of one-half of the transient primary crater radius (0.25Dtr) was used as the starting point for measuring the ballistic range to each secondary (we justify this in a later section). The transient crater represents Sites and mapping methodsa nominal cavity shape as the excavation flow ceases but before 1 Fig. 2. Portion of high resolution mosaic of Tyre’s secondary field ( 30 m pxÀ ). (a) gravitational collapse proceeds; thus D /4 represents an average tr This high-Sun image also reveals dark material concentrated in secondary crater distance from which fragments might have been ejected. The tran- floors. (b) Mapping at higher resolution confirms craters mapped at lower sient diameter for each of the 3 primary craters was estimated 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. 2. Achelous-GanymedeIn some cases this includes ‘‘chains’’ where there is only one large crater but there is also a smaller crater, or elongation/gouging in the direction radial to Gilgamesh (‘‘Gilgamesh sculpture’’). If a cra- 直径が〜35kmで、⽐較的新鮮なクレーターter had a distinctly different degradation state than the majority of the secondaries (either much fresher, rayed, or much more de- graded) it was excluded from the analysis. Additionally,-1 if a crater Ø 解像度はTyreの画像とほぼ同じwas particularly large for its distance180mpx from Gilgamesh, compared to the surrounding secondaries, it was excluded (these craters ap- Ø 630個：0.7-2.7kmpeared as obvious outliers in the dataset). The craters that were mapped or excluded can be seen in Fig. 4a and some specific exam- ples of the mapping decisions are given in Fig. 4b. Topography and Ø 明確なイジェクタ堆積があるexample secondary crater profiles are shown in Fig. 4c and d, respectively. During the analysis described below, we also ex- Ø ploredGula an augmented dataset for Gilgamesh, which includes other 北側に同サイズのcraters that couldクレーターがある be secondaries, but were not in an obvious chain or cluster. Ø Gulaはクレーター内部に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- 1 resolution Galileo mosaics (27, 21, 54 m pxÀ ) at three distances • AchelousはfromGula Pwyll,より年代が若い a 27-km-diameter crater. One mosaic is centered over Pwyll, and the other two are almost due north, 900 km and • 1,070Achelous km away, in Conamara Chaos and just north of Conamara, ほとんどがrespectively. Pwyll’sの２次クレーター bright rays are visible over 1000 km away, thus secondaries from Pwyll can be identified at large distances. Ø GulaとAchelousThisクレーターのイジェクタブランケッ allows for a greater range of fragment velocities to be sam- pled, but a disadvantage to this site is that measurements only ex- tend in one radial direction away from Pwyll and mapping was トが重なっている領域があるonly possible in small patches of surface area. Without broader ra- dial coverage the results may not reflect the largest secondaries at • 63.5度より北の部分のクレーターはカウントしa given distance, due to stochastic variations with launch azimuth. Fig. 3. (a) Ganymede’s Achelous and Gula craters, 35 and 37 km in diameter, 3. Fragment size and velocity 1 respectively. Achelous is centered at 62°N, 12°W. Mosaic is 180 m pxÀ , from ていない Galileo imagery. Mapped secondaries (n = 630) in yellow. (For interpretation of the From measured secondary crater ranges we can calculate the references to color in this figure legend, the reader is referred to the web version of velocity of the ejecta fragment that formed each secondary, and this article.) K.N. Singer et al. / Icarus 226 (2013) 865–884 869 K.N. Singer et al. / Icarus 226 (2013) 865–884 from an approximate scaling for complex869 craters on Ganymede D 1:176D1:108 (McKinnon and Schenk, 1995), where D and final tr final D are the final and transient crater diameters, respectively, given tr K.N. Singer et al. / Icarus 226 (2013) 865–884 869 from an approximate scaling forin complex km (Dfinal is craters measured on to rim-to-rim; Ganymede values of Dtr are given in 1:108 Table 1). Dfinal 1:176Dtr (McKinnon and Schenk, 1995), where Dfinal and Sites and mapping methodsTo derive the ejecta fragment size (dfrag) from the measured D are the final and transient crater diameters, respectively, given K.N. Singer et al. / Icarus 226 (2013) 865–884 869 tr diameter of each secondary crater (D ), we use the Schmidt–Hols- in km (D is measured to rim-to-rim; values of D are givensec in from an approximate scaling for complex craters on Ganymede final apple scaling relations fortr hypervelocity impacts (e.g., Holsapple, 1:108 3. Gilgamesh-Ganymede Table 1). 1993) with updated parameters (keith.aa.washington.edu/crater- fromD anfinal approximate1:176Dtr scaling(McKinnon for complex and craters Schenk, on Ganymede 1995), where Dfinal and To derive the ejecta fragmentdata/scaling/theory.pdf). size (d ) from the Thescaling measured equations relate the proper- 1:108 frag D Dtr1:176areD the final(McKinnon and transient and Schenk, crater 1995 diameters,), where D respectively,and given diameter of each secondary craterties (D of), the we impactor use the to Schmidt–Hols- the volume of the resulting transient final tr final 585km (Schenk et al.2004) cratersec on a given body (which can then be converted to a final 直径が〜 の最⼤盆地 Dtr arein the km final (Dfinal andis transient measured crater to diameters, rim-to-rim; respectively, values given of Dtr are given in apple scaling relations for hypervelocitydiameter for impacts complex (e.g., craters),Holsapple, or vice versa. For the purposes of Ø Voyager2の画像で解像度550mpx1993-1) with updated parametersthese (keith.aa.washington.edu/crater- calculations, we consider both solid, low-porosity ice and in kmTable (Dfinal 1is). measured to rim-to-rim; values of Dtr are given in data/scaling/theory.pdf). The scalingporous equations ice regolith asrelate models the for proper- the surfaces of Europa and Gany- Table 1). mede. For lower surface gravities and smaller impactors, the To derive the ejecta fragment size (dfrag) from the measured ties of the impactor to the volume of the resulting transient Ø 445個：2.3-21.3km strength of solid surface material will strongly control the size of To derive the ejecta fragment size (dfrag) from the measured crater on a given body (which canthe resulting then be crater converted (i.e., the ‘‘strength to a final regime’’). For larger gravities diameter of each secondary crater (Dsec), we use the Schmidt–Hols- diameter of each secondary crater (Dsec), we use the Schmidt–Hols- diameter for complex craters), orand vice impactor versa. sizes, For the the strength purposes of the material of is more easily over- Ø chain構造とcluster構造 appleapple scaling scaling relations relations for hypervelocity for hypervelocity impacts (e.g., impactsHolsapple, (e.g., Holsapple, these calculations, we considercome. both In solid, this ‘‘gravity low-porosity regime’’ the ice final and crater size is limited by conversion of flow field kinetic energy into gravitational potential 1993) with updated parameters (keith.aa.washington.edu/crater- porous ice regolith as models for the surfaces of Europa and Gany- 1993) with updated parameters (keith.aa.washington.edu/crater- Ø ２次クレーターがクレーターの北部と南部に集中 energy and its frictional dissipation as heat, both of which are di- mede. For lower surface gravitiesrect functions and smaller of gravity. impactors, A function the that interpolates between data/scaling/theory.pdf).data/scaling/theory.pdf). The scaling The equations scaling relate equations the proper- relate the proper- strength of solid surface materialthe will two strongly regimes is control given by the size of ties ofties the of impactor the impactor to the volume to the of volume the resulting of the transient resulting transient 3l the resulting crater (i.e., the ‘‘strength regime’’). ForÀ larger gravities ² 2 l 2 l crater on a given body (which can then be converted to a final þ １次クレーターと２次クレーターの判別が難しい 2 þ pV K1 p2 K2p ; 2 crater on a given body (which can then be converted to a final and impactor sizes, the strength of the¼ materialþ is3 more easily over- ð Þ diameter for complex craters), or vice versa. For the purposes of Ø cluster chain come. In this ‘‘gravity regime’’ the final crater size is limited by diameter for complex craters), or vice versa. For the purposes of と 構造をなしている⽐較的⼤きなク where K1 and K2 are scaling coefficients for a given surface material these calculations, we consider both solid, low-porosity ice and conversion of flow field kinetic energy into gravitational potential and l is the scaling exponent (see Table 2 for values). The scaling porousthese ice regolith calculations, as models we for consider the surfaces both of Europa solid, and low-porosity Gany- ice and レーターのみカウンティング energy and its frictional dissipationparameters as heat, are both empirically of which estimated are for di- different materials from rect functions of gravity. A functionlaboratory that experiments, interpolates numerical between computations, and comparison mede.porous For lower ice regolith surface gravities as models and for smaller the surfaces impactors, of Europa the and Gany- the two regimes is given by to explosion cratering results. The p-groups are non-dimensional: strengthmede. of solid For surface lower material surface will gravities strongly control and smaller the size of impactors, the Ø リム構造の傾斜が⼤きく、chain状に位置していな pV describes the overall cratering efficiency, and is dependent on 3l the gravity-scaled size ( ) and the strength measure ( ). General À p2 p3 the resulting crater (i.e., the ‘‘strength regime’’). For larger gravities 2 l 2 l strength of solid surface material will strongly control the size of þ þ definitions of the -groups are ( b 2 ) p く、周りより⼤きなクレーターは除外pV K1 p図2 K⻩⾊2p3 ; 2 and impactor sizes, the strength of the material is more easily over- ¼ þ qV ga Y ð Þ the resulting crater (i.e., the ‘‘strength regime’’). For larger gravities pV ; p2 ; p3 ; 3 come. In this ‘‘gravity regime’’ the final crater size is limited by ¼ m ¼ U2 ¼ qU2 ð Þ and impactor sizes, the strength of the material is more easily over- 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 (seewhereTableq is 2 thefor density values). of the The target scaling material, V is the volume of the energycome. and its In frictional this ‘‘gravity dissipation regime’’ as heat, the both final of which crater are size di- is limited by parameters are empirically estimatedresulting for crater, differentm is the materials mass of the fromimpactor, a is the impactor ra- dius, U is impact velocity, and Y is a measure of target strength rect functionsconversion of gravity. of flow A field function kinetic that energy interpolates into gravitational between potential laboratory experiments, numerical computations, and comparison (Holsapple, 1993). There is no simple size division between craters the two regimes is given by to explosion cratering results. Theformedp-groups in the strength are non-dimensional: versus the gravity regimes as illustrated in energy and its frictional dissipation as heat, both of which are di- Fig. 5, as the transition spans an order of magnitude or more in pV describes the overall cratering efficiency, and is dependent on 3l rect functionsÀ of gravity. A function that interpolates between p2 and appears to depend on impactor velocity. We note that the 2 l 2 l þ þ the gravity-scaled size (p2) and thecratering strength efficiency measure itself is (p independent3). General of the ratio of the target 2 pV theK1 p two2 K regimes2p3 ; is given by 2 definitions of the p-groups are to impactor densities, but this ratio does appear on the right hand ¼ þ ð Þ side of Eq. (2) in its most general form (see Holsapple, 1993). The 3l À qV ga Y density ratio is nominally taken as unity for secondaries, and is 2 l 2 l pV ; p2 ; p3 ; 3 where K and K are scaling coefficientsþ þ for a given surface material m 2 2not included for the ice-on-ice secondary impacts considered here. 1 2 2 ¼ ¼ U ¼ qU ð Þ pV K1 p2 K2p ; 2 In principle one could substitute Eq. (3) into Eq. (2) to solve for a and l is the¼ scalingþ exponent3 (see Table 2 for values). The scaling ð Þ Fig. 4. (a) Ganymede’s largest basin, Gilgamesh (585 km in diameter; dashed circle in a given cratering situation, but there is no closed-form solution. indicates the equivalent rim, centeredwhere at 62°S,q 125is°W), the with density mapped secondaries of the target material, V is the volume of the parameters are empirically estimated for different materials from Instead we follow the example of Holsapple (1993) and plot pV as a (n = 445) in yellow. Large, white outlineresulting shows the crater, extent ofm higheris the resolution mass of the impactor, a is the impactor2 ra- 1 function of p2, and insert Y/qU for p3. Fig. 5 shows this curve for laboratory experiments, numerical computations, and comparison Voyager 2 imagery (180 m pxÀ ) and smaller white rectangle is inset of secondary where K and K are scaling coefficients for a given surface material dius, U is impact velocity, and Y is a measure of target strength 1 2 chains in b. (b) Red solid circle is centered on the largest measured secondary cold ice parameters and various impact velocities. The values of to explosion cratering results. The p-groups are non-dimensional: (21.3 km in diameter); blue, dotted circle(Holsapple, is centered on 1993 the largest). There secondary is in no an simplep2 for size the fragments division thatbetween made the craters secondary craters measured and l is the scaling exponent (see Table 2 for values). The scaling obvious radial chain (18.6 km); and yellow dashed circle shows an example of a around Tyre, Achelous, and Gilgamesh are in the range of pV describes the overall cratering efficiency, and is dependent on crater that was marked as a possibleformed primary only in due the to (1) strength its somewhat versus fresher the gravity4 regimes2 as illustrated in 4 10À to 1 10À (fragment sizes are determined below), thus parameters are empirically estimated for different materials from and sharper rim compared with surroundingFig. 5, secondaries, as the transition(2) its large size spans at this an order of magnitude or more in the gravity-scaled size (p2) and the strength measure (p3). General radial distance (24-km diameter), and (3) that it is not in an obvious radial chain. these impacts are predicted to be in the gravity regime for the 1 1 laboratory experiments, numerical computations, and comparison The latter crater also showed up as an obvious2 and outlier appears in the SVD to and depend thus was onnot impactorassociated velocity. range of impact We note velocities that ( the200 m sÀ to 1 km sÀ ). In definitions of the p-groups are p included in the main analysis. Alternatively, all three circled craters are later cratering efficiency itself is independentfact, only meter-sized of the ratio impactors of the would target shift pV into the transition to explosion cratering results. The p-groups are non-dimensional: primary impacts. (c) Topography of portion of scene in (b) derived from stereo- region between the two regimes as a function of impact speed controlled photoclinometry (courtesyto of impactor P.M. Schenk). (d) densities, Topographic but profiles this of ratio does appear on the right hand qV ga Y (lower speeds for secondaries, intermediate speeds for sesqui- pV pV ;describesp2 ; thep3 overall; cratering efficiency, and is3 dependent on largest measured secondary (a–a0) andside nearby of chain Eq. secondaries(2) in its (b–b most0) and (c–c general0). form (see Holsapple, 1993). The 2 2 Crater depths (rim-to-floor) vary from 0.8 to 1 km; floors appear flat with central naries, and higher, cometary speeds for primaries); thus, strength ¼ m ¼ U ¼ qU ð Þ peaks. Note vertical exaggeration. density ratio is nominally takenregime as unity craters for are secondaries, nearly invisible andon Europa is and Ganymede given the gravity-scaled size (p2) and the strength measure (p3). General not included for the ice-on-ice secondary impacts considered here. wheredefinitionsq is the density of the of thep-groups target material, are V is the volume of the In principle one could substitute Eq. (3) into Eq. (2) to solve for a resulting crater, m is the mass of the impactor, a is the impactor ra- Fig. 4. (a) Ganymede’s largest basin, Gilgamesh (585 km in diameter; dashed circle in a given cratering situation, but there is no closed-form solution. qV ga Y indicates the equivalent rim, centered at 62°S, 125°W), with mapped secondaries dius, U is impact velocity, and Y is a measure of target strength Instead we follow the example of Holsapple (1993) and plot pV as a pV ; p2 2 ; p3 2 ; 3 (n = 445) in yellow. Large, white outline shows the extent of higher resolution 2 (Holsapple,¼ 1993m ). There¼ is no simple¼ size division between craters ð Þ 1 function of p2, and insert Y/qU for p3. Fig. 5 shows this curve for U qU Voyager 2 imagery (180 m pxÀ ) and smaller white rectangle is inset of secondary formed in the strength versus the gravity regimes as illustrated in chains in b. (b) Red solid circle is centered on the largest measured secondary cold ice parameters and various impact velocities. The values of (21.3 km in diameter); blue, dotted circle is centered on the largest secondary in an p2 for the fragments that made the secondary craters measured Fig. 5where, as theq transitionis the density spans an of order the target of magnitude material, orV moreis the in volume of the obvious radial chain (18.6 km); and yellow dashed circle shows an example of a around Tyre, Achelous, and Gilgamesh are in the range of p2 and appears to depend on impactor velocity. We note that the crater that was marked as a possible primary only due to (1) its somewhat fresher 4 2 resulting crater, m is the mass of the impactor, a is the impactor ra- 4 10À to 1 10À (fragment sizes are determined below), thus and sharper rim compared with surrounding secondaries, (2) its large size at this   crateringdius, efficiencyU is impact itself is velocity, independent and ofY theis ratio a measure of the target of target strength radial distance (24-km diameter), and (3) that it is not in an obvious radial chain. these impacts are predicted to be in the gravity regime for the 1 1 to impactor densities, but this ratio does appear on the right hand The latter crater also showed up as an obvious outlier in the SVD and thus was not associated range of impact velocities ( 200 m sÀ to 1 km sÀ ). In (Holsapple, 1993). There is no simple size division between craters included in the main analysis. Alternatively, all three circled craters are later side of Eq. (2) in its most general form (see Holsapple, 1993). The fact, only meter-sized impactors would shift pV into the transition formed in the strength versus the gravity regimes as illustrated in primary impacts. (c) Topography of portion of scene in (b) derived from stereo- region between the two regimes as a function of impact speed density ratio is nominally taken as unity for secondaries, and is controlled photoclinometry (courtesy of P.M. Schenk). (d) Topographic profiles of (lower speeds for secondaries, intermediate speeds for sesqui- not includedFig. 5, for as the the ice-on-ice transition secondary spans impacts an order considered of magnitude here. or more in largest measured secondary (a–a0) and nearby chain secondaries (b–b0) and (c–c0). naries, and higher, cometary speeds for primaries); thus, strength Crater depths (rim-to-floor) vary from 0.8 to 1 km; floors appear flat with central Inp principle2 and appears one could to substitute depend Eq. on(3) impactorinto Eq. (2) velocity.to solve for Wea note that the peaks. Note vertical exaggeration. regime craters areFig. nearly 4. (a) invisible Ganymede’s on Europa largest and basin, Ganymede Gilgamesh given (585 km in diameter; dashed circle in a givencratering cratering efficiency situation, itself but there is independent is no closed-form of the solution. ratio of the target indicates the equivalent rim, centered at 62°S, 125°W), with mapped secondaries Instead we follow the example of Holsapple (1993) and plot pV as a (n = 445) in yellow. Large, white outline shows the extent of higher resolution to impactor densities,2 but this ratio does appear on the right hand 1 function of p2, and insert Y/qU for p3. Fig. 5 shows this curve for Voyager 2 imagery (180 m pxÀ ) and smaller white rectangle is inset of secondary side of Eq. (2) in its most general form (see Holsapple, 1993). The chains in b. (b) Red solid circle is centered on the largest measured secondary cold ice parameters and various impact velocities. The values of density ratio is nominally taken as unity for secondaries, and is (21.3 km in diameter); blue, dotted circle is centered on the largest secondary in an p2 for the fragments that made the secondary craters measured obvious radial chain (18.6 km); and yellow dashed circle shows an example of a aroundnot Tyre, included Achelous, for the and ice-on-ice Gilgamesh secondary are in the impacts range considered of here. crater that was marked as a possible primary only due to (1) its somewhat fresher 4 2 4 10ÀInto principle 1 10À (fragment one could sizes substitute are determined Eq. (3) below),into Eq. thus(2) to solve for a and sharper rim compared with surrounding secondaries, (2) its large size at this   Fig. 4. radial(a) Ganymede’s distance (24-km largest diameter), basin, andGilgamesh (3) that it (585 is not km in in an diameter; obvious radial dashed chain. circlethesein impacts a given are cratering predicted situation, to be in the but gravity there regime is no closed-form for the solution. 1 1 indicatesThe the latter equivalent crater also rim, showed centered up as an at obvious62°S, outlier 125°W), in the with SVD mapped and thus secondarieswas not associated range of impact velocities ( 200 m sÀ to 1 km sÀ ). In Instead we follow the example of Holsapple (1993) and plot pV as a included in the main analysis. Alternatively, all three circled craters are later (n = 445) in yellow. Large, white outline shows the extent of higher resolutionfact, only meter-sized impactors would shift2 pV into the transition primary impacts. (c) Topography1 of portion of scene in (b) derived from stereo- function of p2, and insert Y/qU for p3. Fig. 5 shows this curve for Voyager 2 imagery (180 m pxÀ ) and smaller white rectangle is inset of secondaryregion between the two regimes as a function of impact speed controlled photoclinometry (courtesy of P.M. Schenk). (d) Topographic profiles of chains in b. (b) Red solid circle is centered on the largest measured secondary(lowercold speeds ice parametersfor secondaries, and intermediate various impact speeds velocities. for sesqui- The values of largest measured secondary (a–a0) and nearby chain secondaries (b–b0) and (c–c0). (21.3 kmCrater in diameter); depths (rim-to-floor) blue, dotted vary circle from 0.8 is centered to 1 km; floors on the appear largest flat secondary with central in annaries,p2 andfor higher, the fragments cometary speeds that formade primaries); the secondary thus, strength craters measured obviouspeaks. radial Note chain vertical (18.6 exaggeration. km); and yellow dashed circle shows an example of aregimearound craters are Tyre, nearly Achelous, invisible on and Europa Gilgamesh and Ganymede are given in the range of crater that was marked as a possible primary only due to (1) its somewhat fresher 4 2 4 10À to 1 10À (fragment sizes are determined below), thus and sharper rim compared with surrounding secondaries, (2) its large size at this   radial distance (24-km diameter), and (3) that it is not in an obvious radial chain. these impacts are predicted to be in the gravity regime for the 1 1 The latter crater also showed up as an obvious outlier in the SVD and thus was not associated range of impact velocities ( 200 m sÀ to 1 km sÀ ). In included in the main analysis. Alternatively, all three circled craters are later fact, only meter-sized impactors would shift pV into the transition primary impacts. (c) Topography of portion of scene in (b) derived from stereo- region between the two regimes as a function of impact speed controlled photoclinometry (courtesy of P.M. Schenk). (d) Topographic profiles of (lower speeds for secondaries, intermediate speeds for sesqui- largest measured secondary (a–a0) and nearby chain secondaries (b–b0) and (c–c0). Crater depths (rim-to-floor) vary from 0.8 to 1 km; floors appear flat with central naries, and higher, cometary speeds for primaries); thus, strength peaks. Note vertical exaggeration. regime craters are nearly invisible on Europa and Ganymede given 870 K.N. Singer et al. / Icarus 226 (2013) 865–884 Fragment velocity of ejecta Finally, we must make some assumptions about the crater shape to convert crater volume into diameter. We use a depth- u 破⽚サイズ・破⽚速度の導出 to-diameter (H/D) ratio of 0.125, lower than the canonical ratio ² ２次クレーターの存在領域→エジェクタの破⽚速度 of 0.2 for simple, primary craters (e.g., as on the Moon), but a better ² ２次クレーターサイズ→イジェクタ破⽚サイズ approximation for fresh secondary craters. Our H/D value is slightly lower than the mean or mode found in the recent study of Europa Ø イジェクタは弾道軌道を仮定 secondaries by Bierhaus and Schenk (2010), H/D 0.135. Our Ø イジェクタ放出⾓度は45° (Cintala et al.1999) ² エジェクタ破⽚速度の推定 (Melosh,1989) slightly lower value offsets the fact that the secondary depths in Ø 0.25Dtrから破⽚が放出 2 1.108 " % Bierhaus and Schenk were measured from the rim crest, whereas vej sinθ cosθ Ø Dtrの⾒積もりは Dfinal ~ 1.176 D tr を⽤いる $ R g' R 2R tan−1 $ p ' the Schmidt–Holsapple equations apply to the apparent depth of (McKinnon and Schenk,1995) b = p 2 2 $ vej cos θ ' R : $ 1− ' p 天体半径 the transient crater, or the depth measured from the ground plane. # Rpg & g:重⼒加速度 A value of 0.125 is also the same H/D assumed by Vickery (1986, ² クレータースケーリング則から破⽚サイズと２次クレーターサイズの関係式を導く 1987), and so aids in comparisons. Using a paraboloid shape for サイズスケーリング則 (Hoslapple,1993) the ‘‘excavated’’ crater (meaning the excavated and displaced vol- −3µ 規格化クレーター体積 規格化重⼒ 規格化強度 3 3 2 V ga ume), this H/D ratio yields V 0:4 D =2 0:05D and thus ! +µ $2+µ ρ Y sec sec sec 2 πV = π 2 = 2 π 3 = 2 ¼ ð Þ ¼ πV = K1#π 2 + K2π 3 & m U ρU the equation becomes " % Dtr:トランジェントクレーター直径 : : 3l Dfinal 最終クレーター直径 ρ 標的密度 À Fig. 5. Cratering efficiency (pV) as a function of gravity-scaledm: g: size (p2) for impacts 3 2 l 弾丸質量 重⼒加速度 þ ※２次クレーターは標的物質の破⽚が弾丸になるためinto cold, non-porous ice and ice regolith (http://keith.aa.washington.edu/crater-Y:標的物質強度 U:衝突速度 0:05Dsec g dfrag =2 π4=1 となる a:弾丸直径 V:クレーター体積 K1 ð Þ : 6 data/scaling/theory.pdf). Shaded bar shows the range of values for secondaries 3 2 p2 4=3 p dfrag =2 ¼ tfrag cos h ! ð Þ determined in this study, and the inset illustrates quantities used in application of ð Þ ð Þ ð Þ ballistic range and crater scaling equations to calculate ejecta fragment velocity and With the above assumptions and parameters we solve for the frag- size. All of the secondaries in question form safely in the gravity regime, but the ment diameter (dfrag). In the case of a solid surface the resulting largest and slowest reach the limit of point-source impactor scaling, as pV declines towards unity. The cratering efficiency in the strength regime shown above is high equation is compared with that for laboratory experiments on intact ice samples (e.g., Lange 0:277 and Ahrens, 1987), but in actuality Y decreases with increasing scale; at the 1:277 2 dfrag 1:003Dsec g=tfrag : 7a geological scales of interest here, secondary formation most likely occurs in the ¼ ð Þ ð Þ gravity regime (cf. Holsapple, 1993). Europa and Ganymede have both been extensively resurfaced (Ganymede to a lesser extent overall, although Achelous formed the current image resolutions.2 For a porous, regolith-like material on younger grooved terrain). This suggests regolith layers from im- we use the parameters for ‘‘dry sand’’, which has no effective pact gardening are relatively thin (ostensibly, only a few meters for strength; thus impacts into this material will always be in the grav- Europa; Moore et al., 2009), although the surface may be or have ity regime, also shown in Fig. 5. We comment on the low cratering been fractured to a greater depth by tectonics and locally by larger efficiencies implied by Fig. 5, and other potential scaling limitations, primary impacts. Many of the secondary craters we mapped are at the end of this section and in Appendix A. fairly large, especially for Gilgamesh on Ganymede, and thus For the gravity regime we are concerned with the relation should be mostly forming in low-porosity ice below any regolith. 3l À For completeness, however, we also calculate the results for sec- 2 l pV K1p þ : 4 ¼ 2 ð Þ ondary impacts with porous material parameters representing an icy regolith: The cratering efficiency, pV, is the ratio of the mass of the material that is ejected or displaced when making the crater (secondary cra- 1:205 2 0:205 dfrag 0:878Dsec g=tfrag : 7b ter in this case) to the mass of the ejecta fragment. This relation can ¼ ð Þ ð Þ Many of Gilgamesh’s secondary craters are larger than a few km be written as Vsecq/Vfragd, where Vsec, Vfrag, q, and d are the volumes and densities of the surface material and the impacting fragment, across, and are thus complex rather than simple in shape. Appro- respectively. As noted, for secondary craters on icy bodies q and d priately, we must modify our crater volume formula. The depths are logically assumed to be the same. For secondaries, the impactor (H) of smaller complex craters on Ganymede (primaries) were found by Schenk (2002) to be proportional to D0.42. We adopt this radius afrag = dfrag/2. For the above equations, t refers to the vertical velocity component, so a factor of cosh is introduced for non-verti- slope, and extrapolate from Schenk’s simple-to-complex transition cal impactors (e.g., Chapman and McKinnon, 1986; Holsapple, depth of 0.4 km to arrive at a self-consistent formula of H/ 0.58 1993), and again h is assumed to be 45° (this angular dependence, D = 0.125(Dsec/3 km)À , for Dsec > 3 km. Actual depth measure- based on numerical calculations, appears reasonably accurate for ments for Gilgamesh (Fig. 4c and d) are well-matched by this for- frictional materials [Elbeshausen et al., 2009]). Additionally, the im- mula. The resulting scaling equation for fragment diameters for pact velocity for the secondary craters is taken to be the same as the solid ice is ejection velocity ( = ), as there is no atmospheric resistance to tfrag tej 1:030 2 0:277 dfrag 1:304D g=t ; 8a slow the fragments after ejection on the icy satellites in question. ¼ sec ð frag Þ ð Þ The resulting equation is and for porous icy regolith is 3l À 2 l 0:972 0:205 gd =2 þ 2 V sec frag dfrag 1:126Dsec g=tfrag ; 8b K1 2 : 5 ¼ ð Þ ð Þ V frag ¼ tfrag cos h ! ð Þ ð Þ where all variables are in MKS units. Rigorously, Eqs. (7) and (8) should refer to the transient cavity, which is less wide but deeper We use the K1 and l most suitable for cold, non-porous ice, as well as a porous material (dry sand) representing icy regolith. than the apparent crater we use. Unfortunately we have little guid- ance on the specific geometric details for secondary craters, even for those on the Moon (this in contrast to the situation for primary cra- 2 From another perspective, the transition crater diameter between the strength ters (Grieve and Garvin, 1984)). Hence, while excavated volumes and gravity regimes only depends on Y and g, for scale- and rate-independent strength, and is predicted to be of order 100 m on Europa and Ganymede (Chapman are likely to be similar to our estimates, a systematic uncertainty and McKinnon, 1986). of 50% would not be unwarranted. Fragment size870 of ejecta K.N. Singer et al. / Icarus 226 (2013) 865–884 Finally, we must make some assumptions about the crater shape to convert crater volume into diameter. We use a depth- u クレーター形成メカニズム to-diameter (H/D) ratio of 0.125, lower than the canonical ratio of 0.2 for simple, primary craters (e.g., as on the Moon), but a better ² 氷のパラメータをスケーリング則に適応 approximation for fresh secondary craters. Our H/D value is slightly lower than the mean or mode found in the recent study of Europa secondaries by Bierhaus and Schenk (2010), H/D 0.135. Our ² ２次クレーターは“重⼒⽀配域”で形成 slightly lower value offsets the fact that the secondary depths in Bierhaus and Schenk were measured from the rim crest, whereas Ø 観測したクレーターの条件(π2)が、右図 the Schmidt–Holsapple equations apply to the apparent depth of the transient crater, or the depth measured from the ground plane. の灰⾊の領域になったため A value of 0.125 is also the same H/D assumed by Vickery (1986, (衝突速度は200-1000m/s) 1987), and so aids in comparisons. Using a paraboloid shape for the ‘‘excavated’’ crater (meaning the excavated and displaced vol- ume), this H/D ratio yields V 0:4 D =2 3 0:05D3 and thus Ø 破⽚をm-scaleにしても、πVは遷移領域 sec ¼ ð sec Þ ¼ sec the equation becomes 3l À に位置した Fig. 5. Cratering efficiency (pV) as a function of gravity-scaled size (p2) for impacts 2 l 3 þ into cold, non-porous ice and ice regolith (http://keith.aa.washington.edu/crater- 0:05Dsec g dfrag =2 3 K1 ð Þ 2 : 6 data/scaling/theory.pdf). Shaded bar shows the range of p2 values for secondaries ¼ ð Þ Ø EuropaとGanymedeには強度⽀配域の２次クレーターは観測されなかった 4=3 p dfrag =2 tfrag cos h ! determined in this study, and the inset illustrates quantities used in application of ð Þ ð Þ ð Þ ballistic range and crater scaling equations to calculate ejecta fragment velocity and With the above assumptions and parameters we solve for the frag- ² size. All of the secondaries in question form safely in the gravity regime, but the クレータースケーリング則は重⼒⽀配域のみで考察をする v =v ment diameter (dfrag). In the case of a solid surface the resulting largest and slowest reach the limit of point-source impactor scaling,ej as pV fragdeclines Ø towards unity. The cratering efficiency in the strength regime shown above is high equation is 放出後の⼤気による抵抗は無視する compared with that for laboratory experiments on intact ice samples (e.g., Lange 0:277 and Ahrens, 1987), but in actuality Y decreases with increasing scale; at the 1:277 2 dfrag 1:003D g=t : 7a vej ¼ sec ð frag Þ ð Þ geological scales of interest here, secondaryθ formation most likelyv occursfrag in the gravity− regime3µ (cf. Holsapple, 1993). Europa and Ganymede have both been extensively resurfaced (Ganymede to a lesser extent overall, although Achelous formed −3µ ! the$ current2+µ image resolutions.2 For a porous, regolith-like material g( f / 2) on younger grooved terrain). This suggests regolith layers from im- Vsec frag we use the parameters for ‘‘dry sand’’, which has no effective 2+µ pact gardening are relatively thin (ostensibly, only a few meters for = K1# strength;2 & thus impacts into this material will= always be in the grav- πV = K1π 2 # & δ ρ Europa; Moore et al., 2009), although the surface may be or have V (v cosθ)ity regime, also shown in Fig. 5. We comment on the low cratering frag " frag % been fractured to a greater depth by tectonics and locally by larger efficiencies implied by Fig. 5, and other potential scaling limitations, primary impacts. Many of the secondary craters we mapped are at the end of this section and in Appendix A. fairly large, especially for Gilgamesh on Ganymede, and thus For the gravity regime we are concerned with the relation should be mostly forming in low-porosity ice below any regolith. 3l À For completeness, however, we also calculate the results for sec- 2 l pV K1p þ : 4 ¼ 2 ð Þ ondary impacts with porous material parameters representing an icy regolith: The cratering efficiency, pV, is the ratio of the mass of the material that is ejected or displaced when making the crater (secondary cra- 1:205 2 0:205 dfrag 0:878Dsec g=tfrag : 7b ter in this case) to the mass of the ejecta fragment. This relation can ¼ ð Þ ð Þ Many of Gilgamesh’s secondary craters are larger than a few km be written as Vsecq/Vfragd, where Vsec, Vfrag, q, and d are the volumes and densities of the surface material and the impacting fragment, across, and are thus complex rather than simple in shape. Appro- respectively. As noted, for secondary craters on icy bodies q and d priately, we must modify our crater volume formula. The depths are logically assumed to be the same. For secondaries, the impactor (H) of smaller complex craters on Ganymede (primaries) were found by Schenk (2002) to be proportional to D0.42. We adopt this radius afrag = dfrag/2. For the above equations, t refers to the vertical velocity component, so a factor of cosh is introduced for non-verti- slope, and extrapolate from Schenk’s simple-to-complex transition cal impactors (e.g., Chapman and McKinnon, 1986; Holsapple, depth of 0.4 km to arrive at a self-consistent formula of H/ 0.58 1993), and again h is assumed to be 45° (this angular dependence, D = 0.125(Dsec/3 km)À , for Dsec > 3 km. Actual depth measure- based on numerical calculations, appears reasonably accurate for ments for Gilgamesh (Fig. 4c and d) are well-matched by this for- frictional materials [Elbeshausen et al., 2009]). Additionally, the im- mula. The resulting scaling equation for fragment diameters for pact velocity for the secondary craters is taken to be the same as the solid ice is ejection velocity ( = ), as there is no atmospheric resistance to tfrag tej 1:030 2 0:277 dfrag 1:304D g=t ; 8a slow the fragments after ejection on the icy satellites in question. ¼ sec ð frag Þ ð Þ The resulting equation is and for porous icy regolith is 3l À 2 l 0:972 0:205 gd =2 þ 2 V sec frag dfrag 1:126Dsec g=tfrag ; 8b K1 2 : 5 ¼ ð Þ ð Þ V frag ¼ tfrag cos h ! ð Þ ð Þ where all variables are in MKS units. Rigorously, Eqs. (7) and (8) should refer to the transient cavity, which is less wide but deeper We use the K1 and l most suitable for cold, non-porous ice, as well as a porous material (dry sand) representing icy regolith. than the apparent crater we use. Unfortunately we have little guid- ance on the specific geometric details for secondary craters, even for those on the Moon (this in contrast to the situation for primary cra- 2 From another perspective, the transition crater diameter between the strength ters (Grieve and Garvin, 1984)). Hence, while excavated volumes and gravity regimes only depends on Y and g, for scale- and rate-independent strength, and is predicted to be of order 100 m on Europa and Ganymede (Chapman are likely to be similar to our estimates, a systematic uncertainty and McKinnon, 1986). of 50% would not be unwarranted. Fragment size of ejecta D ² クレーター体積から直径へ sec Ø Europaの２次クレーターの直径深さ⽐H/D=0.135 (Bierhaus and Schenk 2010) H Ø スケーリング則は表⾯からの深さを⽤いるので、 H/D＝0.125を⽤いる 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サイズの関係式 −3µ 0.05D2 ! g(d / 2) $2+µ sec K # frag & solid ice surface icy regolith 3 = 1# 2 & !d frag $ "(v frag cosθ) % 0.277 0.205 4 π # & ! $ ! $ 3 2 1.277 g 1.205 g ( ) " % d =1.003D # & d 0.878D # & frag sec # 2 & frag = sec # 2 & " v frag % " v frag % ※Gilgameshの２次クレーターは数kmより⼤ きく形状が複雑になっている Dsec>3km ※Schenk 2002によって、Ganymedeのク レーターの深さHは、直径Dの0.42乗に⽐例 0.277 0.205 0.58 ! $ ! $ D 1.03 g 0.972 g H 0.125 sec D 3km d =1.304D # & d =1.126D # & D = 3km sec > frag sec # 2 & frag sec # 2 & ( ) " v frag % " v frag % Size-distribution of secondary craters 872 K.N. Singer et al. / Icarus 226 (2013) 865–884 ² 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と２次クレーターのサイズ の関係 (右図) ー ⿊線は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さい Ø ２次クレーターのサイズ頻度分布 (下図) ー １次クレーターと傾きが異なる 882 ー 解像度や観測場所によって変化するK.N. Singer et al. / Icarus 226 (2013) 865–884 Fig. 7. Size–velocity distributions (SVDs) for ejecta fragments at the three primary Fig. 6. Secondary crater measurements for the three primary craters studied. Black Fig. D.20. Cumulative size–frequency plots for secondary crater fields around the three primary craters in this study. Gray points indicate diameter range use for cumulative craters studied (scaling for a solid target in the gravity regime). Quantile regression 4 6.3±0.2 bars indicate primary radius. Striping in Achelous data is due to measuring curve fitting to largest craters (solid red line). A fit to the size–frequency distribution for the Tyre high resolution inset (Fig. 2b), N(>D) = 1.4 10 DÀ , is similar to the fit (QR) was used to fit the maximum fragment size for a given ejection/impact  secondary diameters to whole numbers of pixels. Note changes in axes. Size– for Tyre as a whole. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) velocity. Fits to the 99th quantile are shown (fit parameters listed in Table 3). Two frequency distributions for these secondary populations are given in Appendix D. different fits are shown for each primary, one for the full dataset and one for a subset where the largest secondary sets a low velocity cutoff. Note change in axes 1/3 and Holsapple, 2011). Non-porous materials in particular appear to (gtr/gtr,Tyre) , based on the scaling above. We do not explicitly con- for Gilgamesh. converge along a well-defined trend (Fig. 14 in Housen and Holsap- sider q or T in the scaling in Fig. 14 because (1) we do not know T, presumably due to the transition from the ejecta block and blanket ple, 2011), with tej= gR = constant at the apparent crater rim. and (2) we expect q and T to be similar for the four icy satellite cra- Defining dà as the largest ejected block diameter, Eq. (A.17) thus ters regardless. This normalization does not make thestyle secondary of emplacement to the field of secondary craters (or possibly fmax pffiffiffiffiffiffi becomes distributions of all four craters (Tyre, Achelous, Gilgamesh,due to and the overestimated cratering efficiencies as previously dis- or change in power-law slope at small sizes, lest the total ejected cussed); therefore, anchoring our curve fitting in this manner mass become infinite. l b Pwyll) perfectly coincident, but goes a long way. The secondary À à 2 dfmax qgR data may still be subject to azimuthal biases and uncertaintiesmay give in us a better measure, particularly for Tyre and Gilgamesh. To explore the robustness of the above results, we examined ; A:18 R / T ð Þ proper transient crater diameter, in some instances. NoteThe also velocity that exponents (b) for the two different fits are consistent several variations of the scaling or fitting parameters. Fragment we still fit for b in an overall quantile regression, althoughwithin formally the formal errors, for each crater respectively (Table 3), scaling for a porous target material (regolith) is compared with or 2=3 b = 1.2 is determined by assuming dà D . It is debatablealthough which only barely so for Achelous. The Achelous mosaic, how- that for the solid in Fig. 8. In the gravity regime, a larger fragment fmax / 2 l b 2 l b þ À þ À exponent is actually better determined, that for maximumever, only frag- allows for measurements over a fairly limited velocity is necessary to form the same size secondary crater in a porous tar- dà R 2 D 2 ; A:19 fmax / / ð Þ ment size or velocity! After all, icy ejecta blocks may notrange scalewith (already a problem), so the subset may not be as representa- get because more of the energy goes into compaction of the mate- for a given q, g, and T. crater diameter exactly as rocky ejecta blocks do. tive as the full dataset (b is in particular poorly constrained). rial and dissipation (e.g., Holsapple, 1993). This effect is not as 2=3 à Finally, if we do assume b = l + 2/3 and non-porousNote scaling that an average or median fragment size is not particularly pronounced for larger secondary craters (the large secondaries of From Moore (1971) and Bart and Melosh (2007), dfmax D , / ( = 0.55), then Eq. (A.17) explicitly becomes which yields b = l + 2/3 (for rock). The scaling exponent l is con- l meaningful for these datasets because they would be strongly con- Gilgamesh being the prime example), as the gravity-scaled strained to the interval [1/3,2/3], corresponding to the momentum 1:2 trolled by the smallest measured secondaries, which are controlled cratering efficiencies for cold ice and ice regolith converge at large 1=3 À dfmax qgR À tej and energy scaling limits (Holsapple and Schmidt, 1982), so like- : by ourA:20 mosaic resolutions, and unlikely to be an actual physical p2 (Fig. 5). Overall, the distributions and fits are quite similar R / T ð Þ wise b is constrained to the interval [1,4/3]. It is interesting to note gR! characteristic of the ejecta formation process. The size–frequency (Table 3), with fits to the porous scaling predicting slightly larger that many of the b values reported here for Europa and Ganymede distributions of such secondary crater populations are quite steep, fragments sizes. The velocity exponents (b) are mostly within the Compared with Eq. p(9)ffiffiffiffiffiffi(or Eq. (15) in Melosh (1984)), we see that (Fig. 10), and for the terrestrial planets (Fig. 16), fall within this 1=3 however (McEwen and Bierhaus, 2006), which implies a ‘‘cut-off’’ error bars of each other. dà T=q , not T/q. By this logic, maximal ice ejecta fragments, range, when errors are considered. For non-porous rock or ice, l fmax / ð Þ that is, spalls, should be no more than 20–35% smaller than their is well determined to be 0.55, which implies b = 1.2 in particular. We tentatively conclude that the exponent reported by Bart and rocky counterparts, for similar gravity and crater size. If so, the ‘‘dis- Melosh (2007), and by Moore (1971) before them, is consistent crepancies’’ in Figs. 17 and 18 may be more illusory than real. Of course, point-source (coupling parameter) scaling does not rule with the b values reported in this paper. out d T, but such would imply a steeper velocity dependence The astute reader may note that whereas above derivation best fmax / applies to simple (or transient) craters, the D2/3 ejecta-block scaling than in Eq. (A.20). More work on secondary and ejecta block distri- of Moore, Bart, and Melosh incorporates data from complex craters butions for terrestrial planet craters would be very valuable in this as well. The scatter in the block/boulder measurements is such, regard, and as well as high-resolution numerical modeling of the and the range in crater diameter covered so great, however, that ejection and spallation process. adjustments to transient diameter for the handful of complex cra- ters measured do not affect the log–log slope (see figure in Moore, Appendix D. Secondary crater size–frequency distributions 1971). 2=3 2/3 Continuing, if dà D , then (l b)/2 = 1/3 in Eq. (A.18). D The secondary craters mapped for the three craters in this study fmax / À À ejecta-block scaling motivated our original normalization of scaled generally exhibit steep size–frequency distribution (SFD) slopes at 1/3 ejecta fragment size in Fig. 14, i.e., a factor of (Rtr/Rtr,Tyre) on the the large size end (Fig. D.20 and Table D.7), typical of secondary ordinate (and we referenced an argument for D0.8 scaling as crater SFDs (McEwen and Bierhaus, 2006). Other SFD slopes well). Upon review, we multiplied an additional factor of reported for secondary populations on rocky and icy bodies are Fragment diameter and velocity 872 K.N. Singer et al. / Icarus 226 (2013) 865–884 ² 破⽚速度vejとvejにおける最⼤破⽚サイズdfmax Ø d v −β fmaxと ejの関係 d = Av f max ej Ø 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 破⽚速度が⼤きいと破⽚サイズは⼩さい 2. 細線：最⼤破⽚の持つvejを下限値と設定 K.N. Singer et al. / Icarus 226 (2013) 865–884 873 ーTable 3 最⼤破⽚が最も速度が遅いわけではない 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 secondaries Velocity-limited subset b ln(A) A b ln(A) A Solid target Pwyllb (27) 1.21 3.4 105 K.N. Singer et al. / Icarus 226 (2013) 865–884 K.N. Singer et al. / Icarus 226 (2013) 5 865–884873 5 873 Tyre (38) 0.96 ± 0.13 12.3 ± 0.7 2.24 10 1.13 ± 0.16 13.3 ± 1.0 6.12 10   Achelous (35) 1.41 ± 0.45 14.7 ± 2.6 2.52 106 2.44 ± 0.80 20.7 ± 4.6 9.83 108   Table 3 TableGilgamesh 3 (585) 2.07 ± 0.34 22.3 ± 2.3 4.73 109 2.55 ± 0.44 25.5 ± 3.0 1.24 1011   Quantile regression fits to 99th quantile of size–velocity distributionsQuantilea. Solid regression and porous fits targets, to 99th gravity quantile regime, of size–velocity all secondaries distributions and subset.a. Solid and porous targets, gravity regime, all secondaries and subset. Porous target Pwyllb 1.02 1.5 105 Primary crater (diameter in km) All secondariesPrimary crater (diameter in km) Velocity-limited All secondaries subset  Velocity-limited subset Tyre 0.79 ± 0.12 11.4 ± 0.7 8.94 104 0.95 ± 0.15 12.3 ± 0.9 2.30 105   b Achelousln(A) A 1.21bb ± 0.41 13.7ln(ln(AA) ±) 2.4A 8.63A 105 2.18b ± 0.72 19.3ln(A ±) 4.2 2.32A 108   Gilgamesh 1.84 ± 0.34 20.8 ± 2.3 1.11 109 2.29 ± 0.41 23.9 ± 2.8 2.42 1010 Solid target Solid target   b b 5 5 Pwyll (27) 1.21a Pwyll (27)b 3.4 10 1.211 3.4 10 dfmax A À , where A and tej are in m and m sÀ . tej  5  5 5 5 Tyre (38) 0.96 ± 0.13b Tyre 12.3¼ (38) ± 0.7 2.24 10 1.13 0.96 ± ± 0.16 0.13 13.3 12.3 ± ± 1.0 0.7 6.12 2.24 1010 1.13 ± 0.16 13.3 ± 1.0 6.12 10 Alpert and Melosh, (1999).    6  8 Achelous (35) 1.41 ± 0.45Achelous 14.7 ± 2.6 (35) 2.52 106 2.44 1.41 ± ± 0.80 0.45 20.7 14.7 ± ± 4.6 2.6 9.83 2.52 10108 2.44 ± 0.80 20.7 ± 4.6 9.83 10    9  11 Gilgamesh (585) 2.07 ± 0.34Gilgamesh 22.3 ± 2.3 (585) 4.73 109 2.55 2.07 ± ± 0.44 0.34 25.5 22.3 ± ± 3.0 2.3 1.24 4.73 101011 2.55 ± 0.44 25.5 ± 3.0 1.24 10     Porous target AdditionalPorous target fits to the 95th and 90th quantiles for cold, non-por- especially for larger secondaries, supports the results presented in b 5 Fig. 7. Size–velocity distributions (SVDs) for ejecta fragments at the three primary b Pwyll Fig. 6.5 Secondary crater1.02 measurements for the three primary craters 1.5 studied.10 Black Pwyll 1.02ous ice are plotted with 1.5 the10 data in Fig. 9, the fitted parameters are this paper for Tyre. craters studied (scaling for a solid target in the gravity regime). Quantile regression bars indicate primary radius. Striping in Achelous data is due to measuring4 5 Tyre 0.79 ± 0.12Tyre 11.4 ± 0.7 8.94 104 0.95 0.79 ± ± 0.15 0.12 12.3 11.4 ± ± 0.9 0.7 2.30 8.94 10105 (QR)0.95 ± was 0.15 used to fit the 12.3 maximum ± 0.9 fragment 2.30 size10 for a given ejection/impact given in Tables B.5 andsecondary B.6Â, and adiameters graphical to whole comparison numbers of of the pixels. var- Note changesGanymede in  axes.5 Size– and Europa have similar gravities, and the similarly- 8 Achelous 1.21 ± 0.41Achelous 13.7 ± 2.4 8.63 105 2.18 1.21 ± ± 0.72 0.41 19.3 13.7 ± ± 4.2 2.4 2.32 8.63 10108 velocity.2.18 ± 0.72 Fits to the 99th 19.3 quantile ± 4.2 are shown (fit 2.32 parameters10 listed in Table 3). Two frequency distributions for these secondary populations are given in Appendix 9 D.  10 iousGilgamesh velocity exponents (bÂ) is9 given in Fig. 1.84 10 ±. 0.34 In general, these 20.8 ± low- 2.3sized 1.11 Tyre1010 and Achelous2.29 were ± 0.41 mapped at 23.9 comparable ± 2.8 resolutions. 2.42 10 In Gilgamesh 1.84 ± 0.34 20.8 ± 2.3 1.11 10 2.29 ± 0.41 23.9 ± 2.8 2.42 Â10 different fits are shown for each primary, one for the full dataset and one for a er quantiles are increasingly statistically more robust, although Fig. 11 we plot the SVDs for both craters together, assuming non- a b 1 subset where the largest secondary sets a low velocity cutoff. Note change in axes a b 1 dfmax AtejÀ , where A and tej are in m and m sÀ . dfmax AtÀ , where A and tej are in m and m sÀ . they cannot¼ fully capture the largest secondaries at any radial dis- porous ice in the gravityfor Gilgamesh. regime for scaling. The merged datasets ¼ ej b Alpert and Melosh, (1999). b Alpert and Melosh, (1999). tance or ejection velocity. The 99th quantile is particularly suscep- clearly overlap, and could easily be considered to be drawn from tible to statistical outliers.presumably Of the three due to craters the transition studied from here, the Tyre ejecta blockthe same and blanket parent population. For the Achelous data, the upper enve- style of emplacement to the field of secondary craters (or possibly givesAdditional the most consistent fits to the velocity95th and exponents, 90th quantiles full dataset for cold, or non-por- subset. lopeespecially (99th quantile) for larger drops secondaries, off more supports rapidly at the higher results velocities presented than in Additional fits to the 95th and 90th quantiles for cold, non-por- especiallydue to for the larger overestimated secondaries, cratering supports efficiencies the results as presented previously in dis- or change in power-law slope at small sizes, lest the total ejected Theous magnitude ice are plotted of the with pre-exponential the data in Fig. factor 9, theA fittedgenerally parameters scales to are thethis Tyre paper fit, for but Tyre. we suspect this is due to the more limited image ous ice are plotted with the data in Fig. 9, the fitted parameters are thiscussed); paper for therefore, Tyre. anchoring our curve fitting in this manner mass become infinite. lowergiven values in Tables with B.5 lower and B.6 quantiles,, and a graphical as expected, comparison but the of the slopes var- coverageGanymede of the secondaryand Europa field have at similar Achelous. gravities, If one wereand the to truncatesimilarly- given in Tables B.5 and B.6, and a graphical comparison of the var- Ganymedemay give us and a better Europa measure, have similar particularly gravities, for and Tyre the and similarly- Gilgamesh. To explore the robustness of the above results, we examined areious comparable velocity exponents across quantiles, (b) is given again in Fig. often 10 within. In general, the error these bars. low- thesized Tyre Tyre distribution and Achelous to the were same mapped velocity at range comparable that is resolutions. available for In ious velocity exponents (b) is given in Fig. 10. In general, these low- sizedThe Tyre velocity and Achelous exponents were (b mapped) for the at two comparable different resolutions.fits are consistent In several1 variations of the scaling or fitting parameters. Fragment Nevertheless,er quantiles it are is apparent increasingly that statisticallythe slope of the more SVD robust, for Gilgamesh although AchelousFig. 11 we (200–375 plot the m SVDs sÀ ), for the both quantile craters regression together, fit assuming to the Tyre non- er quantiles are increasingly statistically more robust, although Fig.within 11 we plot the theformal SVDs errors, for both for craters each crater together, respectively assuming (Table non- 3), scaling for a porous target material (regolith) is compared with isthey considerably cannot fully steeper capture than the that largest of secondariesTyre or Achelous at any ( radialFig. 10 dis-), dataporous would ice bein the much gravity steeper regime and morefor scaling. uncertain The merged(b = 2.90 datasets ± 0.70 they cannot fully capture the largest secondaries at any radial dis- porousalthough ice in only the gravity barely soregime for Achelous. for scaling. The The Achelous merged mosaic, datasets how- that for the solid in Fig. 8. In the gravity regime, a larger fragment regardlesstance or ejection of scaling velocity. or fitting The approach. 99th quantile is particularly suscep- forclearly Tyre overlap, in this case). andis could Thusnecessary easily while to be the form considered full the dataset same tosize and be secondary drawn subset from for crater in a porous tar- tance or ejection velocity. The 99th quantile is particularly suscep- clearlyever, overlap, only allows and could for measurements easily be considered over a fairlyto be limiteddrawn from velocity tibleWe to also statistical investigated outliers.range an alternative Of (already the three a set problem), craters of secondary sostudied the subset craters here, may Tyre for not beAchelousthe as same representa- yield parentb values population.get consistent because For more the with Achelous of the the uncertainties, energy data, goesthe upper into we compactioncon- enve- of the mate- tible to statistical outliers. Of the three craters studied here, Tyre the same parent population. For the Achelous data, the upper enve- Gilgamesh,gives the most including consistent 100tive velocityadditional as the full exponents, datasetcraters (inb fullis thedataset insize particular range or subset. poorly of constrained).cludelope (99th that the quantile) full datasetrial drops and off for dissipation more Achelous rapidly (e.g., is at more higherHolsapple, representative, velocities 1993). than This effect is not as gives the most consistent velocity exponents, full dataset4-to-24 or subset. km in diameterlope (99th that quantile) could be drops secondaries off more but rapidly are not at higher in velocitiesand that than the Tyre and Achelous distributions are very similar over- The magnitude of the pre-exponentialNote that an average factor orA generally median fragment scales to size isthe not particularlyTyre fit, but wepronounced suspect this for is duelarger to secondary the more craterslimited (the image large secondaries of The magnitude of the pre-exponential factor A generallyradiallower chains scales values to or with clustersthe lowermeaningful Tyre (not quantiles, fit, shown but for we here). these as suspect expected, Only datasets this a few is because but due of these the to they the slopes cra- more would limitedall. becoverage Indeed,strongly image of a con- the composite secondaryGilgamesh fit (99th field at being quantile) Achelous. the yields prime If one a wereb example),= 1.02 to truncate ± 0.12, as the gravity-scaled lower values with lower quantiles, as expected,ters butare fell the comparable within slopes the across uppercoveragetrolled quantiles, diameter of by the the again secondary range smallest often of the field measured within SVD, at Achelous. the and secondaries, error thus If bars. are one which werestatisticallythe to are truncate Tyre controlled distribution similar tocratering theto the Tyre same efficiencies fit alone velocity (b for= range 0.96 cold that ± ice 0.13, and is availableTable ice regolith 3). for converge at large 1 are comparable across quantiles, again often withincapable theNevertheless, error of bars. affecting it is apparentthe the Tyreby quantile our distribution that mosaic regression the slope resolutions, to the of fits. the same A SVD fit and velocity to for unlikely 99th Gilgamesh range quan- to that be an is availableAchelous actual physical for (200–375p m2 s(ÀFig.), the 5). quantileOverall, the regression distributions fit to the and Tyre fits are quite similar 1 Nevertheless, it is apparent that the slope of the SVDtileis for of considerably Gilgamesh this augmented steeperAchelous SVDcharacteristic than resulted (200–375 that in of of a m the Tyreb sofÀ ejecta 2.53), or the Achelous ± formation quantile 0.48 (cold, ( regressionFig. process. non- 10), The fit4.1.data tosize–frequency Ejecta the would Tyre fragment be much scaling(Table steeper – 3), preliminary with and fitsmore to considerationsuncertain the porous (b scaling= 2.90 predicting ± 0.70 slightly larger is considerably steeper than that of Tyre or Achelousporousregardless (Fig. ice), 10 whichof), scaling isdata in ordistributions the would fitting range be approach. of much the of such fits steeper in secondaryTables and 3, more crater B.5 and uncertain populations B.6. (b =for are 2.90 Tyrequite ± 0.70 in steep, this case).fragments Thus while sizes. the The full velocity dataset exponents and subset (b) are for mostly within the regardless of scaling or fitting approach. GilgameshWe also ejecta investigated appearsfor Tyrehowever an most in alternative this prominent (McEwen case). set Thus and to of the Bierhaus,secondary while north the 2006 andcraters full south), dataset which for impliesandAchelous subsetTo acompare ‘‘cut-off’’ for yield theb values SVDserror consistentacross bars of primaries each with other. the more uncertainties, generally, we we ap-con- We also investigated an alternative set of secondary(Fig.Gilgamesh, 4 cratersa), no doubtfor including stronglyAchelous 100 influenced additional yield b values by craters resolution consistent in the and with size Sun the range angle, uncertainties, of pealclude towe the that con- ejecta the full model dataset of Housen for Achelous and Holsapple is more (2011) representative,. For grav- Gilgamesh, including 100 additional craters in theand size4-to-24 fits range to the km of northern in diameterclude and that thatsouthern the could full sections be dataset secondaries of for the Achelous secondary but are is not field more in representative,ity-dominatedand that the Tyreprimary and Achelousimpacts we distributions scale the secondaryare very similar fragment over- (as originally mapped) yield b values of 2.15 ± 0.87 and 2.45 ± 0.34, 4-to-24 km in diameter that could be secondariesradial but are chains not in or clustersand that (not the shown Tyre and here). Achelous Only a distributions few of these are cra- veryvelocities similarall. Indeed, over- by a gravity composite and fit the (99th primary quantile) crater yields transient a b = radius, 1.02 ± 0.12,tej/ respectively. These are similar to the b for all Gilgamesh secondar- 1/2 radial chains or clusters (not shown here). Only a fewters of these fell within cra- theall. upper Indeed, diameter a composite range offit the(99th SVD, quantile) and thus yields are a b (=gRstatistically 1.02tr) ±, 0.12, and scale similar the to fragment the Tyre diameters fit alone (b also= 0.96 by the ± 0.13, transientTable ra-3). ies (2.07 ± 0.34), considering the uncertainties, but the steeper va- ters fell within the upper diameter range of the SVD,capable and thus of areaffectingstatistically the quantile similar regression to the fits.Tyre A fit fit alone to 99th (b = quan- 0.96 ± 0.13,dius,Tabledfrag 3/).Rtr (Fig. 12), recognizing that additional factors may be lue to the south reflects the influence of the prominent chains of capable of affecting the quantile regression fits. A fittile to 99th of this quan- augmented SVD resulted in a b of 2.53 ± 0.48 (cold, non- involved for the larger secondaries (e.g., spallation fragment size large secondaries there (the actual regression fit for all secondaries 4.1. Ejecta fragment scaling – preliminary considerations tile of this augmented SVD resulted in a b of 2.53 ± 0.48porous (cold, ice), non- which is in the range of the fits in Tables 3, B.5 and B.6. nominally depends on surface tensile strength (Melosh, 1984), lies between the two,4.1. south Ejecta above fragment and scaling north below). – preliminary We conclude considerations porous ice), which is in the range of the fits in TablesGilgamesh 3, B.5 and B.6 ejecta. appears most prominent to the north and south and see below). As expected, Tyre and Achelous are still similar gi- that the steepness of the SVD fits for Gilgamesh derives, at least To compare the SVDs across primaries more generally, we ap- Gilgamesh ejecta appears most prominent to the north(Fig. and 4a), south no doubt strongly influenced by resolution and Sun angle, ven their similar SVDs and similar Rtr. The Gilgamesh secondaries, in part, from these spectacular,To compare large the secondary SVDs across crater primaries chains. more generally,peal weto the ap- ejecta model of Housen and Holsapple (2011). For grav- (Fig. 4a), no doubt strongly influenced by resolution andand Sun fits to angle, the northern and southern sections of the secondary field however, are significantly offset, with generally smaller scaled Among the threepeal secondary to the ejecta fields model we mapped, of Housen the and results Holsapple for (2011)ity-dominated. For grav- primary impacts we scale the secondary fragment (as originally mapped) yield b values of 2.15 ± 0.87 and 2.45 ± 0.34, fragment sizes and scaled ejection velocities. The scaled distribu- and fits to the northern and southern sections of the secondary field velocities by gravity and the primary crater transient radius, tej/ Tyre are likely theity-dominated most robust. Imageprimary coverageimpacts extends we scale around the secondarytion fragment of Gilgamesh’s secondary craters suggests that Gilgamesh’s (as originally mapped) yield b values of 2.15 ± 0.87 andrespectively. 2.45 ± 0.34, These are similar to the b for all Gilgamesh secondar- (gR )1/2, and scale the fragment diameters also by the transient ra- the entire crater andvelocities yields the by largest gravity number and the of primary measurable crater sec- transienttransient radius,tr diametertej/ may be overestimated in comparison with Tyre respectively. These are similar to the b for all Gilgameshies (2.07 secondar- ± 0.34), considering1/2 the uncertainties, but the steeper va- dius, d /R (Fig. 12), recognizing that additional factors may be ondaries with the largest(gRtr) range, and scale of calculated the fragment velocities diameters (from also 225 by theand transient Achelous.frag ra-tr Certainly, the transient diameter for Gilgamesh is ies (2.07 ± 0.34), considering the uncertainties, but thelue steeper to the1 va-south reflects the influence of the prominent chains of to 580 m sÀ ), in eitherdius, absoluted /R or(Fig. gravity-scaled 12), recognizing terms that (discussed additional factorsinvolved may be for the larger secondaries (e.g., spallation fragment size large secondaries there (thefrag actualtr regression fit for all secondaries the least constrained of the three. If we take the point-of-view that lue to the south reflects the influence of the prominentbelow). chains Additionally, of this scene is the least likely to be contami- nominally depends on surface tensile strength (Melosh, 1984), lies between the two,involved south for above the andlarger north secondaries below). We (e.g., conclude spallation fragmentthe transition size from ejecta blanket to secondary field should gravity large secondaries there (the actual regression fit fornated all secondaries by primariesnominally or secondaries depends from on other surface craters tensile given strength Euro- (Melosh,and see 1984 below).), As expected, Tyre and Achelous are still similar gi- that the steepness of the SVD fits for Gilgamesh derives, at least scale (Housen et al., 1983), then Dtr for Gilgamesh should be re- lies between the two, south above and north below).pa’s We young conclude surface, estimated to average between 20 and 200 Ma ven their similar SVDs and similar Rtr. The Gilgamesh secondaries, in part, from theseand spectacular, see below). large As expected, secondary Tyre crater and chains. Achelous are stillduced similar by gi-1.5. This by itself would not make the upper envelope that the steepness of the SVD fits for Gilgamesh derives,(Bierhaus at et least al., 2009). The results for Achelous have a much more however, are significantly offset, with generally smaller scaled Among the threeven secondary their similar fields SVDs we and mapped, similar theRtr. The results Gilgamesh for of secondaries, the Gilgamesh SVD match or parallel the other distributions in in part, from these spectacular, large secondary craterlimited chains. velocity range and the Gilgamesh secondaries are the most fragment sizes and 4scaled ejection velocities. The scaled distribu- Tyre are likely thehowever, most robust. are significantly Image coverage offset, extends with generally around smallerterms ofscaledb, however. Possibly, ejecta/spallation physics may limit Among the three secondary fields we mapped,difficult the results to tell for apart from potential primaries of similar size. Our tion of Gilgamesh’s secondary craters suggests that Gilgamesh’s the entire crater andfragment yields thesizes largestand scaled number ejection of measurable velocities. sec- The scaled distribu- Tyre are likely the most robust. Image coverage extendssecondary around size distribution versus distance for Tyre (Fig. 6a) is transient diameter may be overestimated in comparison with Tyre ondaries with thetion largest of Gilgamesh’srange of calculated secondary velocities craters (from suggests 225 that Gilgamesh’s4 Alternatively, the transient crater sizes for Tyre and Achelous could be increased quite similar to that in Bierhaus and Schenk (2010; their Fig. 7). and Achelous. Certainly, the transient diameter for Gilgamesh is the entire crater and yields the largest number of measurable sec-1 transient diameter may be overestimated in comparisonby with1.5. Although Tyre Tyre’s equivalent and transient crater size could be underesti- Bierhausto 580 m and sÀ Schenk), in either (2010) absoluteuseda or clustering gravity-scaled algorithm terms to (discussed identify the least constrained of the three. If we take the point-of-view that ondaries with the largest range of calculated velocities (from 225 and Achelous. Certainly, the transient diameter formated, Gilgamesh a 33% underestimationis seems unlikely (cf. Turtle et al., 1999). Achelous’ rim 1 below). Additionally, this scene is the least likely to be contami- to 580 m sÀ ), in either absolute or gravity-scaled termssecondaries (discussed (and more of them); the similarity of the two datasets, diameterthe transition is secure, offrom course. ejecta blanket to secondary field should gravity nated by primariesthe or least secondaries constrained from of other the three. craters If we given takeEuro- the point-of-view that below). Additionally, this scene is the least likely to be contami- scale (Housen et al., 1983), then Dtr for Gilgamesh should be re- pa’s young surface,the estimated transition to from average ejecta between blanket to20 secondary and 200 Ma field should gravity nated by primaries or secondaries from other craters given Euro- duced by 1.5. This by itself would not make the upper envelope (Bierhaus et al., 2009scale). The (Housen results et for al., Achelous 1983), then haveDtr afor much Gilgamesh more should be re- pa’s young surface, estimated to average between 20 and 200 Ma of the Gilgamesh SVD match or parallel the other distributions in limited velocity rangeduced and by the1.5. Gilgamesh This byitself secondaries wouldnot are themake most the upper envelope 4 terms of b, however. Possibly, ejecta/spallation physics may limit (Bierhaus et al., 2009). The results for Achelous havedifficult a much tomore tell apartof the from Gilgamesh potential SVD primaries match ofor similarparallel size. the other Our distributions in limited velocity range and the Gilgamesh secondariessecondary are the most size distributionterms of b, versus however. distance4 Possibly, for ejecta/spallation Tyre (Fig. 6a) is physics may limit 4 Alternatively, the transient crater sizes for Tyre and Achelous could be increased difficult to tell apart from potential primaries of similarquite size. similar Our to that in Bierhaus and Schenk (2010; their Fig. 7). by 1.5. Although Tyre’s equivalent and transient crater size could be underesti- secondary size distribution versus distance for Tyre (Fig. 6a) is Bierhaus and Schenk4 (2010)Alternatively,used the a transient clustering crater algorithm sizes for Tyre to and identify Achelous couldmated, be increased a 33% underestimation seems unlikely (cf. Turtle et al., 1999). Achelous’ rim quite similar to that in Bierhaus and Schenk (2010; their Fig. 7). secondaries (and moreby 1.5. of Althoughthem); the Tyre’s similarity equivalent of and the transient two datasets, crater size coulddiameter be underesti- is secure, of course. Bierhaus and Schenk (2010) used a clustering algorithm to identify mated, a 33% underestimation seems unlikely (cf. Turtle et al., 1999). Achelous’ rim secondaries (and more of them); the similarity of the two datasets, diameter is secure, of course. Fragment diameter and874 velocityK.N. Singer et al. / Icarus 226 (2013) 865–884 position (such as we have), validating our use of Dtr/4 as a nominal ² launch position of the secondary fragments. 破⽚速度vejとvejにおける最⼤破⽚サイズdfmax We emphasize that although the ejection velocities in Figs. 12 and 13 appear to gravity scale, ejecta are launched from positions Ø ⿊：表⾯がsolid where gravity does not influence (or only modestly influences) the cratering flow field. We are relating the secondaries to primary cra- Ø icy regolith ter size, however, which is something that (in most cases) we can ⻘：表⾯が measure, as opposed to primary impactor size and velocity (which we do not know), and primary crater size does gravity scale (Hou- sen and Holsapple, 2011). Hence, the gravity scaling is embedded in Figs. 12 and 13, and especially in Fig. 13, which illustrates ejec- tion velocities for the bulk of the ejecta. As will be discussed in the ² ポーラスな標的に、同じ速度で同サイズのクレー next section, however, secondary crater formation is thought to be strongly influenced by surface spallation and tensile fragmentation (Melosh, 1984), and this is especially true for the largest secondar- ターを形成するためには⼤きな破⽚が必要になる ies. In spallation theory, tensile strength (T) and longitudinal 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、散逸が⽣じるため (Holsapple 1993) Ø 破⽚サイズは⼤きくなるがSVDのベキ乗数βは、標 K.N. Singer et al. / Icarus 226 (2013) 865–884 K.N. Singer et al. / Icarus 226 (2013) 865–884 873 873 的物質によらず同じような値をとる Table 3 Table 3 Quantile regression fits to 99th quantile of size–velocityQuantile distributions regression fitsa. Solid to 99th and quantile porousof targets, size–velocity gravity regime,distributions all secondariesa. Solid and and porous subset. targets, gravity regime, all secondaries and subset. Primary crater (diameter in km) AllPrimary secondaries crater (diameter in km) All secondaries Velocity-limited subset Velocity-limited subset b ln(A) A b b ln(A) ln(AA) A b ln(A) A Solid target Solid target Pwyllb (27) 1.21Pwyllb (27) 3.4 1.21105 3.4 105   Tyre (38) 0.96Tyre ± 0.13 (38) 12.3 ± 0.7 2.24 0.9610 ±5 0.131.13 12.3 ± 0.16 ± 0.7 13.3 2.24 ± 1.0105 6.121.1310 ±5 0.16 13.3 ± 1.0 6.12 105     Achelous (35) 1.41Achelous ± 0.45 (35) 14.7 ± 2.6 2.52 1.4110 ±6 0.452.44 14.7 ± 0.80 ± 2.6 20.7 2.52 ± 4.6106 9.832.4410 ±8 0.80 20.7 ± 4.6 9.83 108     Gilgamesh (585) 2.07Gilgamesh ± 0.34 (585) 22.3 ± 2.3 4.73 2.0710 ±9 0.342.55 22.3 ± 0.44 ± 2.3 25.5 4.73 ± 3.0109 1.242.5510 ±11 0.44 25.5 ± 3.0 1.24 1011     Porous target Porous target Pwyllb 1.02Pwyllb 1.51.02105 1.5 105   Tyre 0.79Tyre ± 0.12 11.4 ± 0.7 8.94 0.7910 ±4 0.120.95 11.4 ± 0.15 ± 0.7 12.3 8.94 ± 0.9104 2.300.9510 ±5 0.15 12.3 ± 0.9 2.30 105     Achelous 1.21Achelous ± 0.41 13.7 ± 2.4 8.63 1.2110 ±5 0.412.18 13.7 ± 0.72 ± 2.4 19.3 8.63 ± 4.2105 2.322.1810 ±8 0.72 19.3 ± 4.2 2.32 108     Gilgamesh 1.84Gilgamesh ± 0.34 20.8 ± 2.3 1.11 1.8410 ±9 0.342.29 20.8 ± 0.41 ± 2.3 23.9 1.11 ± 2.8109 2.422.2910 ±10 0.41 23.9 ± 2.8 2.42 1010     a b a 1 b 1 dfmax AtÀ , where A and tej are in m and m sÀdfmax. AtÀ , where A and tej are in m and m sÀ . ¼ ej ¼ ej b Alpert and Melosh, (1999). b Alpert and Melosh, (1999). Fig. 8. Comparison of fragment sizes impied by scaling for a porous versus a solid surface material (both in the gravity regime). The fits for the upper envelope (99th Additional fits to the 95th and 90th quantilesAdditional for fitscold, to non-por- the 95th andespecially 90th quantiles for larger for cold, secondaries, non-por- supportsespecially the results for larger presentedquantile) secondaries, in of the supportsSVDs are similar, the results but the presented porous scaling in predicts slightly larger fragments (fits shown are for the full dataset; parameters listed in Table 3). Note ous ice are plotted with the data in Fig.ous 9, the ice arefitted plotted parameters with the are data inthisFig. paper 9, the for fitted Tyre. parameters are this paper for Tyre. change in axes for Gilgamesh. given in Tables B.5 and B.6, and a graphicalgiven comparison in Tables B.5 of and the B.6 var-, and a graphicalGanymede comparison and Europa of the have var- similar gravities,Ganymede and and the Europasimilarly- have similar gravities, and the similarly- ious velocity exponents (b) is given in Fig.ious 10 velocity. In general, exponents these (low-b) is givensized in Fig. Tyre 10 and. In general,Achelous these were low- mapped atsized comparable Tyre and resolutions.Achelous were In mapped at comparable resolutions. In er quantiles are increasingly statisticallyer quantiles more robust, are increasingly although statisticallyFig. 11 we more plot the robust, SVDs although for both cratersFig. together, 11 we plot assuming the SVDs non- for both craters together, assuming non- 1 ice fragment sizes as ejection velocities reach and exceed 1 km sÀ they cannot fully capture the largest secondariesthey cannot at fully any capture radial dis- the largestporous secondaries ice in the at gravity any radial regime dis- for scaling.porous Theice in merged the gravity datasets regime for scaling. The merged datasets (truly hypervelocity for ice; Gaffney, 1985), accounting for the rela- tance or ejection velocity. The 99th quantiletance or is ejectionparticularly velocity. suscep- The 99thclearly quantile overlap, is particularly and could suscep- easily be consideredclearly overlap, to be anddrawntively could from sharp easily decline be considered in fragment to be size drawn with velocity from for Gilgamesh. tible to statistical outliers. Of the threetible craters to statistical studied here,outliers. Tyre Of thethe three same craters parent studied population. here, For Tyre the Achelousthe same data, parent the upperpopulation. enve-The scaled For the velocities Achelous can data, also the be upper compared enve- to a compilation of gives the most consistent velocity exponents,gives the full most dataset consistent or subset. velocitylope exponents, (99th quantile) full dataset drops or subset.off more rapidlylope at(99th higher quantile) velocitiesdata drops than related off more to rapidly ejection at velocity higher velocities versus fragment than launch position The magnitude of the pre-exponentialThe factor magnitudeA generally of the scales pre-exponential to the Tyre factor fit, butA generally we suspect scales this to is duethe to Tyre the more fit, but limited wefrom suspect image within this the is due transient to the crater more (x limited) given imageby Housen and Holsapple lower values with lower quantiles, aslower expected, values but with the lower slopes quantiles,coverage as expected, of the secondary but the field slopes at Achelous.coverage If one of thewere secondary to(2011) truncate field. In their at Achelous.Fig. 14 Ifthe one authors were to plot truncate both their ejecta model are comparable across quantiles, againare often comparable within the across error quantiles, bars. the again Tyre often distribution within the to error the same bars. velocitythe range Tyre distribution that is availabledescribing to the for same the velocity scaled velocities range that as is a available function for of scaled launch posi- 1 tion (1x/R, where R is the apparent radius, thus 1.1x/R (see Table 2 Nevertheless, it is apparent that the slopeNevertheless, of the SVD it for is Gilgameshapparent that theAchelous slope of (200–375 the SVD for m sGilgameshÀ ), the quantileAchelous regression (200–375 fit to them sÀ Tyre), the quantile regression fit to the Tyre tr is considerably steeper than that ofis Tyre considerably or Achelous steeper (Fig.10 than), thatdata ofwould Tyre or be Achelous much steeper (Fig. 10 and), moredata uncertain would be (b much= 2.90in steeper ± Grieve 0.70 and and more Garvin, uncertain 1984)) and(b = 2.90 data ± from 0.70 gravity-controlled regardless of scaling or fitting approach.regardless of scaling or fitting approach.for Tyre in this case). Thus while thefor full Tyre dataset in this and case). subsetexperiments. Thus for while We the replot full dataset their ejecta and modelsubset fits for in our Fig. 13. The scaled velocities for the three primary craters considered here We also investigated an alternative setWe of alsosecondary investigated craters an for alternativeAchelous set ofyield secondaryb values craters consistent for withAchelous the uncertainties, yield b values we con-consistent with the uncertainties, we con- are in the range of 1.6 to 4.7 (Fig. 12). The corresponding scaled Gilgamesh, including 100 additionalGilgamesh, craters in the including size range 100 of additionalclude craters that the in thefull size dataset range for of Achelousclude is that more the representative, fulllaunch dataset positions for Achelous (x/R) isclosely more overlap representative, 0.5 for all three primaries, 4-to-24 km in diameter that could be4-to-24 secondaries km in but diameter are not that in couldand be that secondaries the Tyre and but Achelous are not in distributionsand that are the very Tyre similar and Achelous over- distributions are very similar over- Fig. 9. Quantile regression fits to the 99th, 95th and 90th percentiles of the SVDs the value we take as the launch position (Dtr/4), for both porous radial chains or clusters (not shown here).radial Only chains a few or clustersof these (not cra- shownall. here). Indeed, Only a composite a few of these fit (99th cra- quantile)all. Indeed, yields a composite= 1.02 ± 0.12, fit (99th quantile) yields a = 1.02 ± 0.12, (shown for the full dataset, non-porous ice scaling, gravity regime). Velocity b and non-porous scaling. This rangeb of values is thus in good agree- exponents (Tables 3, B.5 and B.6) are similar within formal errors in all cases. Note ters fell within the upper diameter rangeters of fell the within SVD, the and upper thus are diameterstatistically range of the similar SVD, to and the thus Tyre are fit alonestatistically (b = 0.96 ± similar 0.13, Table toment the 3). with Tyre theoryfit alone and (b = data 0.96 about ± 0.13, ejectionTable 3 velocity). versus launch change in axes for Gilgamesh. capable of affecting the quantile regressioncapable fits. of A affecting fit to 99th the quan- quantile regression fits. A fit to 99th quan- tile of this augmented SVD resulted intile a b of this2.53 augmented ± 0.48 (cold, SVD non- resulted4.1. in Ejecta a b of fragment 2.53 ± 0.48 scaling (cold, – non- preliminary4.1. considerations Ejecta fragment scaling – preliminary considerations porous ice), which is in the range of theporous fits in ice),Tables which 3, B.5 is and in the B.6 range. of the fits in Tables 3, B.5 and B.6. Gilgamesh ejecta appears most prominentGilgamesh to the ejecta north appears and south most prominentTo compare to the the north SVDs and across south primariesTo more compare generally, the SVDs we ap- across primaries more generally, we ap- (Fig. 4a), no doubt strongly influenced( byFig. resolution 4a), no doubt and Sun strongly angle, influencedpeal to by the resolution ejecta model and Sun of Housen angle, andpeal Holsapple to the ejecta (2011) model. For grav- of Housen and Holsapple (2011). For grav- and fits to the northern and southern sectionsand fits of to the the secondary northern and field southernity-dominated sections of theprimary secondaryimpacts field we scaleity-dominated the secondaryprimary fragmentimpacts we scale the secondary fragment (as originally mapped) yield b values of(as 2.15 originally ± 0.87 and mapped) 2.45 ± yield 0.34,b values of 2.15 ± 0.87 and 2.45 ± 0.34, velocities by gravity and the primaryvelocities crater transient by gravity radius, andt theej/ primary crater transient radius, tej/ respectively. These are similar to the brespectively.for all Gilgamesh These secondar- are similar to the b1/2for all Gilgamesh secondar- 1/2 (gRtr) , and scale the fragment diameters(gRtr) also, and by the scale transient the fragment ra- diameters also by the transient ra- ies (2.07 ± 0.34), considering the uncertainties,ies (2.07 ± but 0.34), the considering steeper va- the uncertainties, but the steeper va- dius, dfrag/Rtr (Fig. 12), recognizing thatdius, additionaldfrag/Rtr factors(Fig. 12 may), recognizing be that additional factors may be lue to the south reflects the influencelue of tothe the prominent south reflects chains the of influenceinvolved of the for prominent the larger secondaries chains of (e.g.,involved spallation for the fragment larger secondaries size (e.g., spallation fragment size large secondaries there (the actual regressionlarge secondaries fit for all secondaries there (the actualnominally regression depends fit for all on secondaries surface tensilenominally strength depends (Melosh, on 1984 surface), tensile strength (Melosh, 1984), lies between the two, south above andlies north between below). the We two, conclude south aboveand and see north below). below). As expected, We conclude Tyre andand Achelous see below). are still As similar expected, gi- Tyre and Achelous are still similar gi- that the steepness of the SVD fits forthat Gilgamesh the steepness derives, of at the least SVD fits for Gilgamesh derives, at least ven their similar SVDs and similar Rtr.ven The their Gilgamesh similar secondaries, SVDs and similar Rtr. The Gilgamesh secondaries, in part, from these spectacular, large secondaryin part, from crater these chains. spectacular, largehowever, secondary aresignificantly crater chains. offset, withhowever, generally are smaller significantly scaled offset, with generally smaller scaled Among the three secondary fields weAmong mapped, the the three results secondary for fieldsfragment we mapped, sizes and thescaled results ejection for velocities.fragment The sizes scaledand distribu-scaled ejection velocities. The scaled distribu- Tyre are likely the most robust. ImageTyre coverage are likely extends the most around robust.tion Image of coverage Gilgamesh’s extends secondary around craterstion suggests of Gilgamesh’s that Gilgamesh’s secondary craters suggests that Gilgamesh’s the entire crater and yields the largestthe number entire of crater measurable and yields sec- the largesttransient number diameter of measurable may be overestimated sec- transient in comparison diameter with may Tyrebe overestimated in comparison with Tyre ondaries with the largest range of calculatedondaries velocities with the (from largest 225 range ofand calculated Achelous. velocities Certainly, (from the 225 transientand diameter Achelous. for Certainly, Gilgamesh the is transient diameter for Gilgamesh is 1 1 to 580 m sÀ ), in either absolute or gravity-scaledto 580 m sÀ terms), in either (discussed absolute orthe gravity-scaled least constrained terms of (discussed the three. If wethe take least the constrained point-of-view of the that three. If we take the point-of-view that below). Additionally, this scene is thebelow). least likely Additionally, to be contami- this scene isthe the transition least likely from to ejecta be contami- blanket to secondarythe transition field fromshould ejecta gravity blanket to secondary field should gravity nated by primaries or secondaries fromnated other by craters primaries given or secondariesEuro- from other craters given Euro- scale (Housen et al., 1983), then Dtr forscale Gilgamesh (Housen should et al., 1983 be re-), then Dtr for Gilgamesh should be re- pa’s young surface, estimated to averagepa’s between young surface, 20 and estimated 200 Ma toduced average by between1.5. This 20 by and itself 200 would Ma notduced make by the1.5. upper This envelope by itself would not make the upper envelope (Bierhaus et al., 2009). The results for( AchelousBierhaus have et al., a 2009 much). The more resultsof for the Achelous Gilgamesh have SVD a much match more or parallelof the the Gilgamesh other distributions SVD match in or parallel the other distributions in limited velocity range and the Gilgamesh secondaries are the most 4 4 limited velocity range and the Gilgameshterms of secondariesb, however. arePossibly, the most ejecta/spallationterms of b, physics however. mayPossibly, limit ejecta/spallation physics may limit difficult to tell apart from potential primariesdifficult to of tell similar apart size. from Our potential primaries of similar size. Our secondary size distribution versus distancesecondary for size Tyre distribution (Fig. 6a) is versus distance for Tyre (Fig. 6a) is 4 Alternatively, the transient crater sizes for Tyre4 andAlternatively, Achelous could the transient be increased crater sizes for Tyre and Achelous could be increased quite similar to that in Bierhaus andquite Schenk similar (2010 to; their that Fig.in Bierhaus 7). and Schenk (2010; their Fig. 7). by 1.5. Although Tyre’s equivalent and transientby crater1.5. Although size could Tyre’s be underesti- equivalent and transient crater size could be underesti- Bierhaus and Schenk (2010) used a clusteringBierhaus algorithm and Schenk to identify (2010) usedmated, a clustering a 33% underestimation algorithm to seems identify unlikely (cf.mated,Turtle a et33% al., underestimation 1999). Achelous’ seems rim unlikely (cf. Turtle et al., 1999). Achelous’ rim secondaries (and more of them); the similaritysecondaries of the (and two more datasets, of them); thediameter similarity is secure, of of the course. two datasets, diameter is secure, of course. Fragment diameter and velocity −β ² 分位点回帰法でフィッテイング (Koenker, 2005) d = Av f max ej K.N. Singer et al. / Icarus 226 (2013) 865–884 875 ー データ範囲を99%,95%,90%に変更 ー ⼀般的に範囲を下げた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より⼤きくなり、 傾向が異なることを⽰唆 Fig. 11. Size–velocity distributions for the two similarly sized craters Tyre (Europa), ² Gilgameshの２次クレーターはclusterとchainのみカウント in black and Achelous (Ganymede), in red. The two distributions are very similar K.N. Singer et al.over / Icarus their 226 shared (2013) 865–884 velocity range. The limited velocity range for the Achelous data869 is Ø 別の100個のクレーターをカウンティング likely behind the significantly steeper velocity exponent found for the subset of the Achelousfrom data an(99th approximate percentile in scaling all cases; for see complexTable 3). Thecraters 95th on and Ganymede 90th QR fits for Achelous are not so divergent. (For interpretation of the references to color in D 1:176D1:108 (McKinnon and Schenk, 1995), where D and Ø サイズが⼩さいため、βが減少する this figurefinal legend, the readertr is referred to the web version of this article.)final Dtr are the final and transient crater diameters, respectively, given in km (Dfinal is measured to rim-to-rim; values of Dtr are given in ² クレーターの北側と南側に２次クレーターが集中 velocity).Table This 1). normalization sidesteps the implicit gravity scaling of the transientTo derive primary the ejecta crater fragment size. size The (d threefrag) from secondary the measured fields Ø 南側が多少⼤きく、２次クレーターのチェーン構造が卓越し measureddiameter by us of each would secondary appear crater to sample, (Dsec), we for use thedifferent Schmidt–Hols- velocity intervals,apple a more scaling universal relations forSVD hypervelocity for icy impacts, impacts but (e.g., for theHolsapple, excep- tion of1993 Pwyll.) with Quantile updated regression parameters to (keith.aa.washington.edu/crater- the 99th percentile of the ている部分がある data/scaling/theory.pdf). The scaling equations relate the proper- combined SFDs in Fig. 15 yields a master power law of ties of the impactor to the volume of the resulting transient 1:17 0:06 1 dfmax=Dfinal 19:9tÀ Æ , where tej is in m sÀ . Of course, the data Ø North:2.15、South:2.45 crater¼ on a givenej body (which can then be converted to a final from Pwylldiameter (Alpert for complex and Melosh, craters), 1999 or), vice if accepted versa. For as the representative purposes of Fig. 10. Comparison of velocity exponents ( ) for the three primary craters in this b of thatthese crater, calculations, imply this we power consider law both is not solid, universal. low-porosity ice and Ø study. Above are given for (a) the full dataset and (b) the data subset (low velocity チェーン状に位置する⼤きな２次クレーターが多い porous ice regolith as models for the surfaces of Europa and Gany- limit). All are for cold, non-porous ice scaling in the gravity regime. 5. Discussionmede. For lower surface gravities and smaller impactors, the strength of solid surface material will strongly control the size of 5.1. Comparisonthe resulting with crater terrestrial (i.e., the planet‘‘strength secondaries regime’’). For larger gravities (P-wave) sound speed (CL) are important (Melosh, 1984), so we and impactor sizes, the strength of the material is more easily over- 2 might expect such dimensionless quantities as T/qU or CL/U to come. In this ‘‘gravity regime’’ the final crater size is limited by The velocity exponents (b) found for the craters on icy satellites show up in the scaling. These quantities are not expected to vary conversion of flow field kinetic energy into gravitational potential span the range of exponents found by Vickery (1986, 1987) for ter- strongly between Europa and Ganymede; nor do we have individ- energy and its frictional dissipation as heat, both of which are di- restrial planets, but the three ‘‘smaller’’ icy craters (Pwyll, Ache- ual estimates of primary impact speed (U) for the craters in rect functions of gravity. A function that interpolates between lous, and Tyre) have velocity exponents at the lowest end question. the two regimes is given by whereas Gilgamesh on Ganymede3l is towards the higher end of À Given overall scaling uncertainties, we present a version of 2 l 2 l þ the terrestrial range (Fig.2 16þ ). Vickery (1986, 1987) mapped the Fig. 12 in which the vertical axis is normalized by an additional fac- pV K1 p2 K2p3 ; 2 secondary¼ fields forþ 5 craters on the Moon, 4 on Mars and 3ð onÞ tor of [(D g)/(D g )]1/3 (Fig. 14). This was initially motivated tr tr,Tyre Tyre Mercury, in the diameter range of 26-to-227 km, and found by the empirical observation that maximum ejecta block size on where K1 and K2 are scaling coefficients for a given surface material 2/3 b 1.4–2.8.and l is The the pre-exponential scaling exponent (see factorTable (A) 2 generallyfor values). scales The scaling with lunar crater rims varies as D (Moore, 1971; Bart and Melosh, the sizeparameters of the primary. are empiricallyVickery estimated (1986, 1987) for differentfound materials values in from the 2007). The inclusion of g in the normalization stems from consid- range oflaboratory6 10 experiments,7-to-7 10 numerical10. Our three computations, smaller craters and comparison have or- eration of point-source scaling, as is discussed at length in Appen-   der of magnitudeto explosion lower cratering values results. of A The, andp-groups Gilgamesh’s are non-dimensional:A is somewhat dix C. With this normalization, the SVDs for Tyre, Achelous, and higher.p5VGilgameshdescribes the is by overall far the cratering largest efficiency, crater in anyand ofis dependentthe studies on so Gilgamesh are made much more compatible (though not perfectly the gravity-scaled size (p2) and the strength measure (p3). General it may be less comparable with the available terrestrial examples (in so, which as discussed above, may result from an overestimate of definitions of the p-groups are the sense that a better comparison may be with Orientale on the Dtr for Gilgamesh). It is especially worth noting, however, that in qV ga Y moon (Wilhelms,; 1976; Wilhelms; et; al., 1978)). neither Fig. 12 nor Fig. 14 does the secondary distribution from Al- pV p2 2 p3 2 3 Alpert¼ andm Melosh¼ U (1999) concluded¼ qU from their study of Pwyllð Þ pert and Melosh (1999) align with the others (the upper limit to the secondaries that icy bodies ejected smaller fragments than impacts Pwyll SVD is plotted). Also, if we were to adopt a sometimes where q is the density of the target material, V is the volume of the on silicate bodies at equivalent ejection velocities, and that frag- quoted relationship in which maximum ejected block mass is pro- resulting crater, m is the mass of the impactor, a is the impactor ra- ment sizedius, decreasesU is impact more velocity, gradually and Y is as a velocitymeasure of increases target strength (lower portional to the total ejected mass to the 0.8 power (i.e., b). Our(Holsapple, results for 1993 Tyre). There and is Achelous no simple support size division the between latter, as craters the b d D0:8 ) derived from older explosion crater data (see O’Keefe fmax / final valuesformed close to in 1 the are strength distinctive versus for the all gravity 3 icycraters regimes ( asFig. illustrated 16). Gilga- in and Ahrens, 1987), our scaled fragment sizes would be intermedi- mesh doesFig. 5 not, asfollow the transition this trend, spans of an course, order of so magnitude caution is or warranted more in ate between Figs. 12 and 14. A fit to the 99th quantile of the com- and appears to depend on impactor velocity. We note that the with respectp2 to this conclusion, although as noted above there may bined dataset in Fig. 14 yields a velocity exponent of 0.91 ± 0.09, a cratering efficiency itself is independent of the ratio of the target be a scale or strength limit effect (Gilgamesh is more than an order value consistent with the individual fits for Tyre and Achelous, to impactor densities, but this ratio does appear on the right hand of magnitude larger than the other 3 craters). although on the low side of the overall range. If Gilgamesh’s size side of Eq. (2) in its most general form (see Holsapple, 1993). The has been overestimated, this joint b value will steepen. density ratio is nominally taken as unity for secondaries, and is 5 not included for the ice-on-ice secondary impacts considered here. In Fig. 15 we plot secondary-forming fragment size for all three We note for completeness that the ‘‘least-squares’’ and ‘‘best-fit’’ methods used by VickeryIn (1987) principleand oneAlpert could and substitute Melosh (1999) Eq. (3),into respectively, Eq. (2) to to solve determine for a primaries simply as a function of ejection velocity and instead nor- Fig. 4. (a) Ganymede’s largest basin, Gilgamesh (585 km in diameter; dashed circlemaximumin a fragment given cratering size as a situation, function of but velocity there is were no closed-form not specified. solution. Nor were indicates the equivalent rim, centered at 62°S, 125°W), with mapped secondaries malize fragment size to the final crater radius, as this is the closest secondaryInstead or fragment we follow size data the examplepresented of forHolsapple 9 of the 12 (1993) cratersand studied plot inpVickeryV as a (n = 445) in yellow. Large, white outline shows the extent0:92 of higher resolution 2 direct measure of primary impactor1 size (Dfinal ai at fixed (1987). function of p2, and insert Y/qU for p3. Fig. 5 shows this curve for Voyager 2 imagery (180 m pxÀ ) and smaller white/ rectangle is inset of secondary chains in b. (b) Red solid circle is centered on the largest measured secondary cold ice parameters and various impact velocities. The values of (21.3 km in diameter); blue, dotted circle is centered on the largest secondary in an p2 for the fragments that made the secondary craters measured obvious radial chain (18.6 km); and yellow dashed circle shows an example of a around Tyre, Achelous, and Gilgamesh are in the range of crater that was marked as a possible primary only due to (1) its somewhat fresher 4 2 4 10À to 1 10À (fragment sizes are determined below), thus and sharper rim compared with surrounding secondaries, (2) its large size at this   radial distance (24-km diameter), and (3) that it is not in an obvious radial chain. these impacts are predicted to be in the gravity regime for the 1 1 The latter crater also showed up as an obvious outlier in the SVD and thus was not associated range of impact velocities ( 200 m sÀ to 1 km sÀ ). In included in the main analysis. Alternatively, all three circled craters are later fact, only meter-sized impactors would shift pV into the transition primary impacts. (c) Topography of portion of scene in (b) derived from stereo- region between the two regimes as a function of impact speed controlled photoclinometry (courtesy of P.M. Schenk). (d) Topographic profiles of (lower speeds for secondaries, intermediate speeds for sesqui- largest measured secondary (a–a0) and nearby chain secondaries (b–b0) and (c–c0). Crater depths (rim-to-floor) vary from 0.8 to 1 km; floors appear flat with central naries, and higher, cometary speeds for primaries); thus, strength peaks. Note vertical exaggeration. regime craters are nearly invisible on Europa and Ganymede given Ejecta fragment876 scaling K.N. Singer et al. / Icarus 226 (2013) 865–884 876 K.N. Singer et al. / Icarus 226 (2013) 865–884 ² Ejectaのスケール則 (Housen and Hoslapple,2011) 1/2 Ø 規格化したイジェクタ破⽚速度：vej/(gRtr) Ø 規格化したイジェクタ破⽚サイズ：dfrag/Rtr ² Gilgameshのみデータにオフセット Fig. 14. Scaled fragment velocities and normalized fragment diameters. This Ø Gilgameshのトランジェントクレーターサイズを⼤きく⾒積もったためFig. 12. Fragment size and ejection velocity scaled by primary size and gravity, additional normalization (compared with Fig. 12) follows from the empirical following Housen and Holsapple (2011). The maximum fragment size derived for observation that lunar ejecta boulders are related to their parent crater size by D2/3 Pwyll by Alpert and Melosh (1999), appropriately scaled, is also shown (solid curve (see text and Appendix C). Curve for Pwyll as in Fig. 12. Ø where data exist; dashed( whereHousen interpolated). et al.1983) Fragment sizes calculated assuming イジェクタブランケットから推測すると、1.5倍⼩さくなる Fig. 14. Scaled fragment velocities and normalized fragment diameters. This non-porous scaling in the gravity regime in all cases. Fig. 12. Fragment size and ejection velocity scaled by primary size and gravity, additional normalization (compared with Fig. 12) follows from the empirical following Housen and Holsapple (2011). The maximum fragment size derived for theyobservation did not perform that lunar their ejecta own boulders fits, are but related do show to their that parent the crater power- size by D2/3 Pwyll by Alpert and Melosh (1999), appropriately scaled, is also shown (solid curve law(see fit textfor andthe Appendix 93-km-diameter C). Curve for Copernicus Pwyll as in Fig. crater 12. from Vickery ² 放出位置と速度の関係 (Housen and Holsapple2011) where data exist; dashed where interpolated). Fragment sizes calculated assuming non-porous scaling in the gravity regime in all cases. (1987) matches well with their Tycho data. This result is notable as the Tycho impact occurred in the lunar nearside southern high- Ø they did not perform their own fits, but do show that the power- 規格化放出速度のデータ範囲:1.6-4.7 イジェクタの放出位置 lands, presumably deeply fractured megaregolith (Wieczorek et al., law fit for the 93-km-diameter Copernicus crater from Vickery 2013), and not in layers of structurally coherent mare basalt. Ø 放出位置の範囲の中に、x/Rtr=Dtr/4=0.5を含む Bart(1987) andmatches Melosh (2010) well withexamined their Tychothe SVDs data. of ejected This result boulders is notable aroundas the 18 Tycho lunar craters. impact occurredThese were in themostly lunar small nearside primaries, southern 0.2- high- Ø イジェクタ速度と放出位置についての理論とデータ to-4lands, km in presumably diameter, but deeply they also fractured looked megaregolith at two larger craters (Wieczorek with et al., diameters2013), of and 27.4 not and in 41.2 layers km. of Power structurally law fits coherentfor the boulders mare basalt.gave exponentsBart in and the Melosh range of (2010)0.3 toexamined3.7 (for the the SVDs largest of crater), ejected and boulders を⽐較することが可能 À À althougharound there 18 lunar is some craters. scatter These in the were data, mostly they small determined primaries, an 0.2- overallto-4 increasing km in diameter, trend in butb with they also increasing looked crater at two size, larger meaning craters with Ø tr/4 放出位置がD であることは妥当 largerdiameters craters had of 27.4 a steeper and 41.2 fragment km. Power size dependence law fits for on the velocity. boulders gave The exponentsb values for in the range km-scale of craters0.3 to were3.7 (for particularly the largest shallow crater), and À À Fig. 13. Scaled launch positions and ejection velocities from Housen and Holsapple (0 < althoughb < 1). When there combined is some with scatter the in results the data, from Vickery they determined (1987) an (2011; their Fig. 14 and Eq. 14). Our study uses an approximate fragment launch (thus,overall boulders increasingand secondary trend in formingb with fragments)increasing crater this trend size, was meaning position (x) of Dtr/4 (or Rtr/2, indicated). This scaled launch position is within the range of positions inferred by the Housen and Holsapple ejecta model for the reinforced,larger craters albeit with had scattera steeper for fragment larger craters. size dependence Our results ( onb val- velocity. secondary fragments measured in our study (Fig. 12), although some fragments ues)The andb thatvalues of Alpert for the and km-scale Melosh (1999) cratersfurther were strengthen particularly this shallow couldFig. have 13. Scaled been launched launch positions from locations and ejection nearer velocities or farther from fromHousen the transient and Holsapple rim trend,(0 such < b < as 1). it When is. We combined note that rim with boulders the results and thefrom largestVickery sec- (1987) (x/(2011Rtr = 1).; their The exactFig. 14 launchand Eq. position 14). Our is only study important uses an for approximate calculating fragment ranges and launch ondary-forming(thus, boulders fragmentsand secondary might come forming from fragments) different this distinct trend was velocitiesposition of (x nearby) of D secondaries./4 (or R /2, For indicated). distant secondaries This scaled and launch sesquinary position fragments, is within the tr tr ejectareinforced, components albeit (Melosh, with scatter 1989), for but larger this craters.is not entirely Our results clear. ( val- launchrange position of positions uncertainties inferred of byeven the 0.5 HousenRtr are an and insignificant Holsapple fraction ejecta of model the total for the b distancesecondary traveled fragments by thefragment. measured in our study (Fig. 12), although some fragments It isues) possible and that that of theAlpert largest and rim Melosh boulders (1999) representfurther strengthen the ‘‘last this could have been launched from locations nearer or farther from the transient rim spalls’’trend, (or such at least as it theis. We last note surviving that rim spalls, boulders prior and to the any largest rim sec- (x/Rtr = 1). The exact launch position is only important for calculating ranges and collapse).ondary-forming fragments might come from different distinct velocities of nearby secondaries. For distant secondaries and sesquinary fragments, As for the absolute size of fragments, it is important to control ejecta components (Melosh, 1989), but this is not entirely clear. forlaunch primary position crater uncertainties size. For of even modest 0.5Rtr sizedare an craters, insignificant it still fraction appears of the total distance traveled by the fragment. It is possible that the largest rim boulders represent the ‘‘last that fragment size on icy bodies is generally less than that on rocky spalls’’ (or at least the last surviving spalls, prior to any rim bodies at equivalent ejection velocities, but the difference is far less collapse). pronounced than for Pwyll alone, whose fragments are signifi- As for the absolute size of fragments, it is important to control cantly smaller in comparison (Figs. 14 and 15). Gilgamesh second- for primary crater size. For modest sized craters, it still appears ary fragments (or fragment clusters) can be quite large, however, that fragment size on icy bodies is generally less than that on rocky and match or exceed the fragment sizes derived in the Vickery bodies at equivalent ejection velocities, but the difference is far less studies. This result is not surprising, as the proximal secondaries pronounced than for Pwyll alone, whose fragments are signifi- of this icy satellite basin are up to twice as large as the largest sec- ondarycantly craters smaller identified in comparison by Wilhelms (Figs. 14 et al.and (1978) 15). Gilgameshfor Orientale. second- aryTwo fragments other, more (or recent fragment studies clusters) have used can Vickery’sbe quite large,approach however, to lookand at match craters or on exceed the Moon the and fragment Mars. Hirase sizes derived et al. (2004) in theexam- Vickery inedstudies. what This were result judged is not to be surprising, secondary as craters the proximal around secondaries Kepler andof thisAristarchus icy satellite on the basin moon are up(32- to and twice 40-km as large diameter, as the largest respec- sec- tively)ondary and craters also around identified 3 smaller by Wilhelms unnamed et al. martian (1978) craters,for Orientale. 2.0, 3.1, andTwo 3.5 other, km morein diameter. recent studies These authors have used did Vickery’s not fit curves approach to to Fig. 15. Fragment sizes normalized by final primary crater diameter, plotted as a thelook upper at craters envelope on ofthe the Moon SVDs, and and Mars. noneHirase of their et al. distributions (2004) exam- function of ejection velocity. A joint fit to the maximum fragment size (99th quantile), excluding Pwyll, is shown for illustrative purposes showined a what strong were or clear judged decrease to be in secondary fragment size craters with around increasing Kepler 1:17 0:06 1 (dfrag =Dfinal 19:9tÀ Æ , with tej in m sÀ ). Maximum fragment sizes for Pwyll ¼ ej distance.and AristarchusHirata and on Nakamura the moon (2006) (32- andconducted 40-km a diameter, similar study respec- (lower solid and dashed curve; Alpert and Melosh, 1999) are considerably smaller fortively) the young and also lunar around crater 3 Tycho smaller (85 unnamed km in diameter). martian Similarly craters, 2.0, than those of the three primaries considered in this work. 3.1, and 3.5 km in diameter. These authors did not fit curves to Fig. 15. Fragment sizes normalized by final primary crater diameter, plotted as a the upper envelope of the SVDs, and none of their distributions function of ejection velocity. A joint fit to the maximum fragment size (99th quantile), excluding Pwyll, is shown for illustrative purposes show a strong or clear decrease in fragment size with increasing 1:17 0:06 1 (dfrag =Dfinal 19:9tÀ Æ , with tej in m sÀ ). Maximum fragment sizes for Pwyll ¼ ej distance. Hirata and Nakamura (2006) conducted a similar study (lower solid and dashed curve; Alpert and Melosh, 1999) are considerably smaller for the young lunar crater Tycho (85 km in diameter). Similarly than those of the three primaries considered in this work. 876 K.N. Singer et al. / Icarus 226 (2013) 865–884 Fig. 14. Scaled fragment velocities and normalized fragment diameters. This Fig. 12. Fragment size and ejection velocity scaled by primary size and gravity, additional normalization (compared with Fig. 12) follows from the empirical following Housen and Holsapple (2011). The maximum fragment size derived for observation that lunar ejecta boulders are related to their parent crater size by D2/3 Pwyll by Alpert and Melosh (1999), appropriately scaled, is also shown (solid curve (see text and Appendix C). Curve for Pwyll as in Fig. 12. where data exist; dashed where interpolated). Fragment sizes calculated assuming non-porous scaling in the gravity regime in all cases. they did not perform their own fits, but do show that the power- law fit for the 93-km-diameter Copernicus crater from Vickery (1987) matches well with their Tycho data. This result is notable as the Tycho impact occurred in the lunar nearside southern high- lands, presumably deeply fractured megaregolith (Wieczorek et al., 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, 0.2- Ejecta fragmentto-4 km in diameter, scaling but they also looked at two larger craters with diameters 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), and À À ² 新たなファクターを導⼊ although there is some scatter in the data, they determined an overall increasing trend in b with increasing crater size, meaning Ø ３つのクレーターを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 Fig. 13. Scaled launchØ positions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値が and ejection velocities from Housen and Holsapple (0 < b < 1). When combinedD withで変化する the results from (MooreVickery 1971) (1987) (2011; their Fig. 14 and Eq. 14). Our study uses an approximate fragment launch (thus, boulders and secondary forming fragments) this trend was position (x) of Dtr/4Ø (or R最⼤イジェクタブロックの質量はイジェクタ総質量のtr/2, indicated). This scaled launch position is within the 0.8乗に⽐例 (O’Keefe and Ahrens 1987) range of positions inferred by the Housen and Holsapple ejecta model for the reinforced, albeit with scatter for larger craters. Our results (b val- secondary fragmentsØ measured速度のベキ乗数 in our study (Fig.β 12は),0.91±0.09 although some fragments ues) and that of Alpert and Melosh (1999) further strengthen this could have been launched from locations nearer or farther from the transient rim trend, such as it is. We note that rim boulders and the largest sec- ejectaスケーリング則 (x/Rtr = 1). The exact launch position is only important for calculating ranges and Ø Gilgamesh ondary-forming fragments might come from(Housen differentand Holsapple distinct2011) velocities of nearby secondaries. For distantのクレーターサイズより secondaries and sesquinary fragments,、βの値が⼤きくなる可能性 ejecta components (Melosh, 1989), but this is not entirely clear. launch position uncertainties of even 0.5Rtr are an insignificant fraction of the total β α 1/3 distance traveled² by the fragment. It is possible that the largest rim boulders! represent$ α＝ｰ the ‘‘last 放出速度と２次クレーターを形成する破⽚サイズ d f max ! ρgR $ vej spalls’’ (or at least the last∝# surviving& spalls,# prior& to any rim Ø Pwylで得た結果より⼤きな破⽚が放出しているcollapse). # & Rtr " T % " gR % 876 K.N. Singer et al. / IcarusAs for 226 the (2013) absoluteØ 865–884 size of fragments, it is important to control for primary crater size.先⾏研究との結果とは合わない For modest sized craters, it still appears that fragment size on icy bodies is generally less than that on rocky d bodies at equivalent ejection velocities, but the difference is far less f max −1.17±0.06 =19.9vej pronounced than for Pwyll alone, whose fragments are signifi- Dfinal cantly smaller in comparison (Figs. 14 and 15). Gilgamesh second- ary fragments (or fragment clusters) can be quite large, however, and match or exceed the fragment sizes derived in the Vickery studies. This result is not surprising, as the proximal secondaries of this icy satellite basin are up to twice as large as the largest sec- ondary craters identified by Wilhelms et al. (1978) for Orientale. Two other, more recent studies have used Vickery’s approach to look at craters on the Moon and Mars. Hirase et al. (2004) exam- ined what were judged to be secondary craters around Kepler and Aristarchus on the moon (32- and 40-km diameter, respec- tively) and also around 3 smaller unnamed martian craters, 2.0, Fig. 15. Fragment sizes normalized by final primary crater diameter, plotted as a 3.1, andFig. 3.5 14.kmScaled in diameter. fragment Thesevelocities authors and normalized did not fragment fit curves diameters. to This function of ejection velocity. A joint fit to the maximum fragment size (99th Fig. 12. Fragment size and ejection velocity scaled by primary size and gravity,the upperadditional envelope normalization of the SVDs, (compared and nonewith Fig. of 12 their) follows distributions from the empirical following Housen and Holsapple (2011). The maximum fragment size derived for observation that lunar ejecta boulders are related to their parent crater size by Dquantile),2/3 excluding Pwyll, is shown for illustrative purposes show a strong or clear decrease in fragment size with increasing 1:17 0:06 1 Pwyll by Alpert and Melosh (1999), appropriately scaled, is also shown (solid curve (dfrag =Dfinal 19:9tÀ Æ , with tej in m sÀ ). Maximum fragment sizes for Pwyll (see text and Appendix C). Curve for Pwyll as in Fig. 12. ¼ ej where data exist; dashed where interpolated). Fragment sizes calculated assumingdistance. Hirata and Nakamura (2006) conducted a similar study (lower solid and dashed curve; Alpert and Melosh, 1999) are considerably smaller non-porous scaling in the gravity regime in all cases. for the young lunar crater Tycho (85 km in diameter). Similarly than those of the three primaries considered in this work. they did not perform their own fits, but do show that the power- law fit for the 93-km-diameter Copernicus crater from Vickery (1987) matches well with their Tycho data. This result is notable as the Tycho impact occurred in the lunar nearside southern high- lands, presumably deeply fractured megaregolith (Wieczorek et al., 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, 0.2- to-4 km in diameter, but they also looked at two larger craters with diameters 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), and À À although there is some scatter in the data, they determined an overall increasing trend in b with increasing crater size, meaning larger craters had a steeper fragment size dependence on velocity. The b values for the km-scale craters were particularly shallow Fig. 13. Scaled launch positions and ejection velocities from Housen and Holsapple (0 < b < 1). When combined with the results from Vickery (1987) (2011; their Fig. 14 and Eq. 14). Our study uses an approximate fragment launch (thus, boulders and secondary forming fragments) this trend was position (x) of Dtr/4 (or Rtr/2, indicated). This scaled launch position is within the range of positions inferred by the Housen and Holsapple ejecta model for the reinforced, albeit with scatter for larger craters. Our results (b val- secondary fragments measured in our study (Fig. 12), although some fragments ues) and that of Alpert and Melosh (1999) further strengthen this could have been launched from locations nearer or farther from the transient rim trend, such as it is. We note that rim boulders and the largest sec- (x/Rtr = 1). The exact launch position is only important for calculating ranges and ondary-forming fragments might come from different distinct velocities of nearby secondaries. For distant secondaries and sesquinary fragments, ejecta components (Melosh, 1989), but this is not entirely clear. launch position uncertainties of even 0.5Rtr are an insignificant fraction of the total distance traveled by the fragment. It is possible that the largest rim boulders represent the ‘‘last spalls’’ (or at least the last surviving spalls, prior to any rim collapse). As for the absolute size of fragments, it is important to control for primary crater size. For modest sized craters, it still appears that fragment size on icy bodies is generally less than that on rocky bodies at equivalent ejection velocities, but the difference is far less pronounced than for Pwyll alone, whose fragments are signifi- cantly smaller in comparison (Figs. 14 and 15). Gilgamesh second- ary fragments (or fragment clusters) can be quite large, however, and match or exceed the fragment sizes derived in the Vickery studies. This result is not surprising, as the proximal secondaries of this icy satellite basin are up to twice as large as the largest sec- ondary craters identified by Wilhelms et al. (1978) for Orientale. Two other, more recent studies have used Vickery’s approach to look at craters on the Moon and Mars. Hirase et al. (2004) exam- ined what were judged to be secondary craters around Kepler and Aristarchus on the moon (32- and 40-km diameter, respec- tively) and also around 3 smaller unnamed martian craters, 2.0, 3.1, and 3.5 km in diameter. These authors did not fit curves to Fig. 15. Fragment sizes normalized by final primary crater diameter, plotted as a the upper envelope of the SVDs, and none of their distributions function of ejection velocity. A joint fit to the maximum fragment size (99th quantile), excluding Pwyll, is shown for illustrative purposes show a strong or clear decrease in fragment size with increasing 1:17 0:06 1 (dfrag =Dfinal 19:9tÀ Æ , with tej in m sÀ ). Maximum fragment sizes for Pwyll ¼ ej distance. Hirata and Nakamura (2006) conducted a similar study (lower solid and dashed curve; Alpert and Melosh, 1999) are considerably smaller for the young lunar crater Tycho (85 km in diameter). Similarly than those of the three primaries considered in this work. Comparison with terrestrial planet secondaries K.N. Singer et al. / Icarus 226 (2013) 865–884 877 (Melosh, 1984; Vickery, 1986; Hirata and Nakamura, 2006) and ² 速度のベキβを⽐較 Mars (McEwen et al., 2005; Zahnle et al., 2008)? The discrepancy is too large (an order of magnitude in the case of Tyre) to be ex- Ø 岩⽯天体に⽐べ氷衛星のイジェクタ破⽚のほ plained by an even slower and/or more oblique than average primary impactor. Also, spallation theory systematically underpre- うが⼩さい dicts fragment size for all three of the craters studied here (Fig. 17). We may have overestimated secondary crater depths, but even a Ø 氷衛星のβは岩⽯天体に⽐べて⼩さい H/D of 0.1 (the lowest likely limit from Bierhaus and Schenk (2010)), would decrease fragment size estimates by less than 10%. Ø Gilgameshのみβが⼤きい We investigated whether a neglected aspect of ejection physics may be responsible. Ice is a rather ‘‘fluid’’ material compared with － ⽉のOrientaleの２次クレーターより⼤きい rock, in that its Poisson’s ratio of 0.325 is much closer to the fluid value of 0.5 than that typical of rocks (0.25). As such, ice ejection Vickery 1986,1987 Fig. 16. Velocity exponents for ejecta fragments (as measured from secondary physics is more hydrodynamic and ejection angles are steeper (Me- fields) on terrestrial bodies compared with icy satellites. Crater names and losh, 1984). Detailed calculations by Melosh (1984) for ice spalla- ² Kepler Aristarchus diameters in km are given. Exponents displayed for icy satellites are the preferred ⽉の と クレーターと⽔星上の３つのクレーター周りの２次クレー tion (his Fig. 6, case 7) show the ejection angle declining from fits: the data subset for Tyre and Gilgamesh, and the full dataset for Achelous (all for 70 to 50 as the distance from the impact site increases from 5 (Hirase et al.2004) 99th quantile, solid surface, gravity regime). Exponent for Pwyll is from Alpert and ° ° ターについて調べた Melosh (1999). Exponents for Mercury, the Moon and Mars are from Vickery (1987). to 10 projectile radii. Although the analytic stress-wave ejection model is itself a simplification of the real situation, it illustrates Ø SVDの上限を決定しておらず、距離と破⽚サイズの減少の傾向が⾒られなかった the possibility that we are underestimating the ejection angle for 5.2. Secondaries and spallation theory ice at 45°. If so, the ejection velocity necessary to reach a given ² ⽉のTychoクレーターで同様の研究を⾏った(Hirata and Nakamura 2006) range (Eq. (1)) is underestimated. More importantly, the impact Spall fragments are those ejected at high velocities (but low velocity of the ejected fragments becomes less oblique and closer Ø SVD Copernicus shock pressures) from coherent, near surface material. Spalls are to normal. This is the most important aspect and the total kinetic の上限は決めていないが、 クレーターのデータとほぼ⼀致generally the highest velocity fragments ejected from a given energy of a given fragment, available for cratering, may easily more launch point within the crater. The majority of ejected fragments, Ø Megaregolith (Wieczorek et al.2013) than double. Thus for a given secondary crater, the estimated frag- ⽞武岩層ではなく を深くまで掘削したことを⽰唆however, are ejected as part of the main excavation flow and are ment mass is correspondingly reduced. Fig. 18 illustrates the effect generally considered to be Grady–Kipp fragments, after the theory of changing the impact angle uniformly to 70°: fragment sizes are ² 18個の⽉クレーター周りのSVDを求めた (Bartthat and predicts Melosh the breakup2010) of coherent rock (or ice) into roughly reduced by up to 30%. However, fragment velocities are also in- equal-sized blocks (Melosh, 1989, Chapter 6). As the generally creased (shifted to the right), so although the difference between Ø クレーターサイズが⼤きくなるに従いβが⼤きくなるslower moving(β：ｰ and0.3~ much largerｰ3.7) fraction of the ejecta (at a given data and theory is reduced, it is not resolved.6 x), Grady–Kipp fragments presumably result in many near-field Ice spalls are predicted to be substantially smaller than rock Ø ⼤きなクレーターほど、破⽚サイズ分布の傾きが⼤きくなるsecondaries and most of the continuous ejecta (though fragments spalls from Eq. (9), all other things being equal, principally because ejected from below the surface spall layer are expected to travel the tensile strength of even cold ice is much less than that of rock. at close to spall speeds; Melosh, 1987). This physical fact seems inescapable. The relatively large second- Melosh (1984) provides an estimate of spall plate thickness, for ary craters on Euorpa and Ganymede are unexplained by conven- the so-called ‘‘hydrodynamic’’ ejection model: tional spallation theory. We could argue ice fragments cluster, 1=2 but why would ice fragments as opposed to rock fragments prefer- 2aiT d=q lspall ð Þ ; 9 entially cluster? Our understanding of fragment scaling has ¼ qCLtej ð Þ evolved over the course of the is project, however. Point-source impactor scaling offers the possibility that dfmax depends only where as a reminder, T and CL are the tensile strength and P-wave weakly on tensile strength. Full details are given in Appendix C, speed of the target material, respectively. The nominal diameter but the implication is that a comprehensive physical theory of of the primary impactor (2ai) is calculated with an equation similar spallation may ultimately explain the relatively large secondary to Eq. (7a), but with a H/D ratio of 0.2 assumed: craters on both rocky and icy surfaces. 1:277 2 0:277 For completeness we also calculate the characteristic or ‘‘pre- 2ai 1:215Dtr g=ti ; 10 ¼ ð Þ ð Þ ferred’’ size of Grady–Kipp fragments, from Melosh (1989; his Eq. where ti is the primary impactor velocity (and again using an aver- 6.4.2): age 45° impact angle). Impactor radii are given in Table 1 and were calculated using typical cometary impact speeds of 26 and 2aiT 1 lGK : 11 20 km sÀ for Europa and Ganymede, respectively (Zahnle et al., ¼ 2=3 4=3 ð Þ 3 qtej ti 2003; and d = q for simplicity). Using q = 920 kg mÀ , T=17 MPa (Melosh, 1984; based on Lange and Ahrens, 1983), and Even the largest of fragments calculated by Eq. (11), for dis- 1 C 4 km sÀ (Vogt et al., 2008), we can estimate the thickness of tances to the inner edge of the secondary field and with the speeds L % spall plates for the ejection velocities (tej) associated with the sec- given above, are quite small: only 20 m in size for the largest frag- ondaries in this study. Fig. 17 compares our fragment data with ment from Gilgamesh, and only 1 m for Tyre. These fragment $ the spall thickness predicted by Eq. (9), along with an estimate of sizes are several orders of magnitude smaller than the sizes neces- an effective diameter assuming the plates do not break up and are sary to form the observed secondaries. Grady–Kipp theory was a factor of several to 10 times larger in the two other, horizontal developed for a material with random flaws, which activate and dimensions (i.e., multiplying Eq. (9) by 1001/3). Even assuming the grow as cracks, but does not explicitly account for preexisting plates do not break up for the lower velocity portion of the ejecta, structure (faults, megaregolith layers and blocks, etc.) that may the prediction for the size of spall plates falls well below the frag- ‘‘predetermine’’ fragment size. Although we do not necessarily ex- ment sizes inferred for the secondary craters around the three pect a thick regolith or deep megaregolith such as that found on primaries in Fig. 17. What can account for this discrepancy, especially as spallation 6 A side benefit is that smaller, faster secondary-forming fragments are more likely theory appears to do a credible job for craters on the Moon to adhere to established point-source crater scaling relations. Secondaries and spallation theory 878 K.N. Singer et al. / Icarus 226 (2013) 865–884 ² クレーター形成時に発⽣する標的物質の破⽚ are appropriately small percentages, given we only measure por- tions of the secondary field (limited by image coverage), and not Ø Spall破⽚が地表⾯付近において⾼速で放出される the ejecta blanket mass, distant secondaries, or many small sec- ondaries (limited by image resolution). These percentages are also consistent with a similar calculation by Vickery (1986) for Coperni- Ø Grady-Kipp fragments：掘削流による破⽚の⼤部分 cus (0.7%), and which again point towards similar spallation effi- ciencies in ice and rock. ² hydrodynamic ejection model 5.3. Sesquinary craters on Europa and Ganymede In this subsection we use our size–velocity fits to extrapolate to 1 2 lspall:スポール破⽚厚さ the escape speeds of Europa and Ganymede, recognizing that this 2a T δ ρ is an extrapolation, and that we do not have the image coverage i ( ) ai:衝突体半径 ρ:標的密度 l to detect such large velocity secondaries. We extrapolate the fitted spall = δ:衝突体密度 T:引っ張り強度 ρC v upper-envelope equations to the Hill sphere escape velocity (tH, for L ej CL:P波速度 ai:弾丸直径 a fragment to escape to the Hill sphere for a particular moon rather than to infinity; see Table 2) and derive the maximum size frag- 0.277 ment that would be ejected from each primary crater. The diame- ter of the largest sesquinary fragments ejected and the size of a 1.277 ! g $ H/D=0.2 2a =1.215D resulting crater after impact on the parent moon are given in Ta- i tr # 2 & ble 4. Resulting crater sizes are calculated using Eq. (10) and with 1 " vi % typical re-impact speed of 2.5 km sÀ for Europa (Zahnle et al., 1 2008) and 3.5 km sÀ for Ganymede (Alvarellos et al., 2002). Each primary would also produce an array of smaller sesquinaries. We also include the (extrapolated) maximum fragment size for our ² Spall破⽚は低速イジェクタでは破壊されないと仮定 ‘‘master’’ ice secondary SVD from Fig. 15. Using Eq. (9) we calculate the largest spall thicknesses at escape Ø Spall velocity would have been 4, 3, and 86 m in size for Tyre, Achelous, 破⽚のサイズの予測は、観測された２次クレーター and Gilgamesh, respectively. These dimensions are considerably smaller than extrapolation of our measured SVDs (Table 4), except から推測した破⽚サイズより⼩さい for that of the Achelous subset, which is likely not the best charac- terization of the SVD to begin with. Eq. (9) calculates the thickness of the plates, and were they to remain intact, the other dimensions could be substantially larger. However, at high speeds spall plates ² この違いが何で起きているのか？ are predicted to break up (Melosh, 1984), and in that case the thickness of the spalls may be representative of fragment diame- Ø H/D=0.1では破⽚サイズが10%減少 ters, if not upper limits to fragment sizes. Zahnle et al. (2008) also use Eq. (9) to estimate the sesquinary fragments produced by an impact resulting in a 20-km-diameter crater and from a hypothet- Ø 速度、衝突⾓度で説明できない ical 100-km crater as well (although none are known on Europa). Accordingly, they find spall thicknesses that are much smaller than those predicted by our empirical extrapolations (e.g., they calculate Fig. 17. Comparison of secondary fragment data to predicted spall plate sizes from a spall thickness of 3 m for a sesquinary fragment from the 20-km Melosh (1984). Shown are derived fragment sizes and velocities (non-porous, gravity-regime scaling), spall plate thicknesses according to Eq. (9) (thin curve), an crater, similar to the small fragments at escape speed predicted by approximation for mean spall diameter assuming the other dimensions are 10 Eq. (9) for Tyre). times larger than the thickness (thick curve), and a more detailed elastic stress- Eq. (9) would predict very small fragment sizes for even a 100- wave estimate (dashed curve). At high ejection velocity spall plates are not likely to km impact on Europa: spall thickness of 430 m for near-field, remain intact. Thus, these spall sizes are upper limits, with spall fragments likely 1 much smaller with the exception of perhaps the slowest velocity fragments. All of secondary-forming fragments (using the same tej = 150 m sÀ as the predicted spall plate sizes are considerably smaller than the sizes of fragments Zahnle et al., 2008) or 20 m at escape speed. Of course, Tyre, at necessary to form the observed secondaries. only 40 km in diameter, already produces large ( 1000-m-diam- eter) secondary-forming fragments and by our empirical, data-dri- ven scaling could have produced sesquinary fragments as large as the Moon (Wieczorek et al., 2013), the surfaces of Europa and Gan- 150 m. Thus, Tyre alone could be an important if not dominant ymede are still fractured and not structurally homogenous. More- contributor to the globally distributed secondary population on over, we do not know the terrain that Gilgamesh formed in, and at Europa between 100-m and 1-km diameter (Bierhaus et al., 2001, least a few kilometers of structurally disturbed and non-thermally- 2009). But, one cannot come to this conclusion using Eq. (9) alone. annealed crust is possible (cf. Nimmo et al., 2003). Grady–Kipp the- On the other hand, we cannot be categorical about Tyre’s contribu- ory also predicts that all fragments will be about the same size for a tion to Europa’s global cratering record, because we do not have 1 given ejection velocity, whereas clearly there is a range of second- empirical data on ice fragment sizes for tej > 1.1 km sÀ (Fig. 15). ary sizes at any given distance from the primary. We conclude that At these greater speeds all ‘‘tabular’’ spalls may breakup and even Eq. (11) does not yield a useful prediction of ejecta fragment sizes equant spalls may be crushed by horizontal stresses during launch for our measured secondary craters. (Melosh, 1984, 1989). As one additional consistency check, we calculated the total In their study of Ganymede cratering, Alvarellos et al. (2002) mass of the fragments for each of the three mapped sites. The mass scaled from Vesta and the 5-to-10-km size asteroids thought to of the fragments is 0.7%, 0.8%, and 0.1% of the excavated transient be ejected from Vesta (the vestoids) to arrive at an estimate for cavity mass for Tyre, Achelous, and Gilgamesh, respectively. These the size of sesquinaries from Gilgamesh. They predicted the largest Secondaries and spallation theory ² 氷と岩⽯の違い vejの推定式 Ø ポアソン⽐が液体の0.5により近いため “fluid material” (岩⽯:0.25、氷:0.325) Ø ejectionの⼒学がより流体⼒学的になり、放出⾓度がより⼤きくなる (Melosh,1984) Ø 弾丸半径で規格化した距離(x/ai)が5-10に増えるに従って放出⾓度が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°)は過⼩評価 ² 衝突⾓度を70°で計算をした Spall theory Ø 破⽚サイズは30%ほど減少 2a T 1 2 Ø 破⽚に与えられる速度が⼤きくなるためあるサイズの２次 i (δ ρ) lspall = クレーターを形成するために必要な破⽚サイズは⼩さくなる ρCLvej K.N. Singer et al. / Icarus 226 (2013) 865–884 879 lspall:スポール破⽚厚さ function of velocity yield exponents ( b) near 1 for the mid-sized À À ai:衝突体半径 ρ:標的密度 craters (25–40 km in diameter), while the largest basin ( 585 km ² Spall : T: 破⽚サイズ in diameter) has a steeper exponent between 2 and 3. With δ 衝突体密度 引っ張り強度 À À two secondary fields measured on each of two moons, one must CL:P波速度 ai:弾丸直径 Ø 破⽚の⼤きさは標的の強度によって決まる be cautious when drawing strong (definitive) inferences about a trend in SVDs with primary size. It is clear that larger craters on Ø 氷の強度Tは岩⽯より⼩さいため、氷のSpall破 the same body eject larger fragments (as described by the pre- exponential factor A). Although the velocity exponents for the SVDs ⽚は岩⽯に⽐べ⼩さくなる of the mid-sized craters are consistently lower than those found for terrestrial craters of similar size, these exponents are consistent with spallation theory and with empirical observations of ejecta Spall理論だけでは岩⽯天体・氷天体の⼤きな block sizes (Bart and Melosh, 2010). Theories for spall sizes, how- ever, predict considerably smaller (and for Grady–Kipp theory, ２次クレーターについて完全に説明できない much smaller) fragments than are required to produce the ob- served secondary craters. The largest impact, Gilgamesh, has a con- Fig. 18. Effect of changing the assumed ejection angle. Smaller fragments are siderably steeper power-law exponent (b value), but this trend for predicted for more vertical impacts (angles given above and in the text refer to a steeper velocity dependence with larger craters is not apparent angle from the ground-plane). However, even changing the angle uniformly to 70° does not predict small enough fragment sizes to match those predicted by for all bodies (see Mercury and the Moon in Fig. 16). Fits for the ter- spallation theory. restrial planets should be re-explored with more primaries and modern, high-resolution imaging to see whether larger craters consistently have more negative velocity exponents or not, and Gilgamesh sesquinary fragments should have been 0.9-to-1.8 km to test the point-source scaling prediction that the velocity depen- in size. According to our study this would be an over-prediction dence and scale dependence of maximum block/boulder/spall size by a factor of 3–8, based on our measurements at Gilgamesh are related. alone (Table 4). Using our ‘‘universal’’ ice fragment scaling, in con- Our empirical approach also helps constrain the contribution of trast, would predict fragments as large as 1 km, more similar to the sesquinary ejecta fragments to the population of craters on icy sat- Alvarellos et al. estimate. It is worth remembering, also, that some ellites. By extrapolating from the size–velocity distributions of Gilgamesh secondary craters are quite large, 10-km diameter at mapped secondary crater populations around large impacts on 1 tej 1 km sÀ (Fig. 6c). It is not too much of a stretch to imagine Europa and Ganymede, we conclude that sesquinary craters would 4-km-diameter distant secondaries (if b 1, as in current spalla- be restricted to smaller craters, although not completely insignifi- tion theory) and therefore returning sesquinary fragments making cant when considering the very largest primary impacts on these 5-km-diameter craters. On the other hand, the highest speed Gilga- bodies. Tyre, the largest basin on Europa, would produce sesqui- mesh ejecta blocks are almost all less than half the size of the larg- nary craters about 1 km in diameter or smaller (from eventual est ejecta blocks at those velocities (Fig. 7c). It is also implausible re-impact on Europa). For the 585-km-diameter basin Gilgamesh that very-high-speed, sesquinary-forming tabular ice spalls can on Ganymede, we find the maximum sesquinary crater size for re- survive intact. By this logic, the largest sesquinary craters formed impact on Ganymede would be approximately 2–3 km at most. The by Gilgamesh are likely to be no larger than 2–3 km in diameter. Gilgamesh SVD has a very steep slope (approximately 2.5), indi- À cating fragment size decreases rapidly with increasing ejection 6. Final remarks velocity. In this study we confirm earlier estimations (Zahnle et al., 2001; Alvarellos et al., 2002) that secondaries or sesquinaries Mapping of secondary craters provides a window into ejecta from Gilgamesh (Ganymede’s largest surviving basin) are not large fragment sizes and velocities, and quantile regression fitting char- enough to account for the closer-to-isotropic crater distribution acterizes the decline in fragment sizes with increasing velocity. observed on Ganymede at crater diameters greater than 30 km. Based on the physics of cratering and ejecta we expect to see trends when comparing the SVDs of different secondary fields with Acknowledgments the size of the primary, surface gravity of the body, or composition of the target/impactor material, etc. For the two large icy satellites We thank P.M. Schenk for providing base image mosaics of the considered here, power law fits to the maximum ejecta size as a three regions mapped in this study, and the NASA Planetary Table 4 Largest sesquinary diameters. Primary Largest fragment at Largest fragment Largest fragment at Largest sesquinary Largest sesquinary Largest sesquinary crater on crater escape velocity – at escape velocity escape velocity – master crater on the same body crater on the same the same body – master full dataset (m)a – subset (m)a dataset of Fig. 15 (m)b – full dataset (km)c body – subset (km)c dataset of Fig. 15 (m)c Europa Tyre 150 120 100 1.1 0.9 0.8 Pwyll 34 0.3 Ganymede Achelous 36 5 64 0.4 (0.3)d 0.07 (0.04)d 0.6 Gilgamesh 360 220 1070 2.4 1.6 6.4e a Calculated by extrapolation of equations in Table 3 (for non-porous ice scaling) to the Hill sphere escape velocity given in Table 2. b Calculated with the quantile regression fit (99th quantile) for the combined dataset including all three primaries in our study (see Section 4.1 and Fig. 15). c 1 1 Calculated using gravity regime parameters into a non-porous target, with typical re-impact speed of 2.5 km sÀ for Europa and 3.5 km sÀ for Ganymede (Eq. (10)). d Assumes porous, ice regolith target. e Complex crater. Grady-Kipp fragments ² Grady-Kipp fragmentsのサイズ (Melosh 1989) 2aiT Ø 2次クレーター領域の内側での最⼤破⽚サイズはとても⼩さい l = GK ρv2 3v4 3 Ø Gilgamesh~20m、Tyre~1m ej i 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サイズ lGK:Grady-kipp破⽚厚さ a : : i 衝突体半径 ρ 標的密度 より⼩さい T:引っ張り強度 vej:破⽚放出速度 v : i １次クレーター形成衝突速度 ² Grady-Kipp fragmentsの問題点 Ø この理論はcracksがランダムな流れによって活発もしくは成⻑することを想定 Ø 地表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、構造的に⼀様ではない Ø ある速度で同じサイズの破⽚が放出されると予想 Ø １次クレーターからの距離に伴い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 る破⽚サイズの⾒積もりには使えない K.N. Singer et al. / Icarus 226 (2013) 865–884 K.N. Singer et al. / Icarus 226 (2013) 865–884 879 879 function of velocity yield exponents ( bfunction) near 1 of for velocity the mid-sized yield exponents ( b) near 1 for the mid-sized À À À À craters (25–40 km in diameter), while thecraters largest (25–40 basin km ( in585 diameter), km while the largest basin ( 585 km in diameter) has a steeper exponent betweenin diameter)2 and has a3. steeper With exponent between 2 and 3. With À À À À two secondary fields measured on eachtwo of twosecondary moons, fields one mustmeasured on each of two moons, one must be cautious when drawing strong (definitive)be cautious inferences when drawingabout a strong (definitive) inferences about a trend in SVDs with primary size. It is cleartrend that in SVDs larger with craters primary on size. It is clear that larger craters on the same body eject larger fragmentsthe (as same described body by eject the larger pre- fragments (as described by the pre- exponential factor A). Although the velocityexponential exponents factor for theA). AlthoughSVDs the velocity exponents for the SVDs of the mid-sized craters are consistentlyof lower the mid-sized than those craters found arefor consistently lower than those found for terrestrial craters of similar size, theseterrestrial exponents craters are consistent of similar size, these exponents are consistent with spallation theory and with empiricalwith observations spallation theory of ejecta and with empirical observations of ejecta block sizes (Bart and Melosh, 2010). Theoriesblock sizes for spall (Bart sizes, and Melosh,how- 2010). Theories for spall sizes, how- ever, predict considerably smaller (andever, for predict Grady–Kipp considerably theory, smaller (and for Grady–Kipp theory, much smaller) fragments than are requiredmuch smaller)to produce fragments the ob- than are required to produce the ob- served secondary craters. The largest impact,served Gilgamesh, secondary has craters. a con- The largest impact, Gilgamesh, has a con- Fig. 18. Effect of changing the assumed ejectionFig. angle. 18. Effect Smaller of changing fragments the are assumedsiderably ejection steeper angle. Smaller power-law fragments exponent are (siderablyb value), butsteeper this power-law trend for exponent (b value), but this trend for predicted for more vertical impacts (angles givenpredicted above and for inmore the vertical text refer impacts to (angles given above and in the text refer to a steeper velocity dependence with largera steeper craters velocity is not apparent dependence with larger craters is not apparent angle from the ground-plane). However, even changingangle from the angle the ground-plane). uniformly to 70 However,° even changing the angle uniformly to 70° does not predict small enough fragment sizesdoes to match not predict those small predicted enough by fragmentfor all sizes bodies to match (see Mercury those predicted and the by Moonfor in allFig. bodies 16). Fits (see for Mercury the ter- and the Moon in Fig. 16). Fits for the ter- spallation theory. spallation theory. restrial planets should be re-exploredrestrial with more planets primaries should and be re-explored with more primaries and modern, high-resolution imaging to seemodern, whether high-resolution larger craters imaging to see whether larger craters consistently have more negative velocityconsistently exponents have or not, more and negative velocity exponents or not, and Gilgamesh sesquinary fragments shouldGilgamesh have been sesquinary 0.9-to-1.8 fragments km toshould test the have point-source been 0.9-to-1.8 scaling kmpredictionto test that the the point-source velocity depen- scaling prediction that the velocity depen- in size. According to our study this wouldin size. be Accordingan over-prediction to our studydence this would and scale be an dependence over-prediction of maximumdence block/boulder/spall and scale dependence size of maximum block/boulder/spall size by a factor of 3–8, based on our measurementsby a factor of at Gilgamesh3–8, based on our measurements at Gilgamesh are related. are related. alone (Table 4). Using our ‘‘universal’’ icealone fragment (Table scaling, 4). Using in our con- ‘‘universal’’Our ice empirical fragment approach scaling, also in con- helps constrainOur empirical the contribution approach of also helps constrain the contribution of trast, would predict fragments as large astrast, 1 km, would more predict similar fragments to the assesquinary large as 1ejecta km, more fragments similar to to the the populationsesquinary of craters ejecta on fragments icy sat- to the population of craters on icy sat- Alvarellos et al. estimate. It is worth remembering,Alvarellos et also, al. estimate. that some It is worthellites. remembering, By extrapolating also, that from some the size–velocityellites. By distributions extrapolating of from the size–velocity distributions of Gilgamesh secondary craters are quiteGilgamesh large, 10-km secondary diameter craters at aremapped quite large, secondary10-km crater diameter populations at mapped around secondarylarge impacts crater on populations around large impacts on 1 1 tej 1 km sÀ (Fig. 6c). It is not too muchtej of1 a km stretch sÀ (Fig. to imagine 6c). It is not too much of a stretch to imagine Europa and Ganymede, we conclude thatEuropa sesquinary and Ganymede, craters would we conclude that sesquinary craters would 4-km-diameter distant secondaries (if b4-km-diameter1, as in current distant spalla- secondaries (if b 1, as in current spalla- be restricted to smaller craters, althoughbe not restricted completely to smaller insignifi- craters, although not completely insignifi- tion theory) and therefore returning sesquinarytion theory) fragments and therefore making returningcant whensesquinary considering fragments the making very largestcant primary when impacts considering on these the very largest primary impacts on these 5-km-diameter craters. On the other hand,5-km-diameter the highest speed craters. Gilga- On the otherbodies. hand, Tyre, the the highest largest speed basin Gilga- on Europa,bodies. would Tyre, produce the largest sesqui- basin on Europa, would produce sesqui- mesh ejecta blocks are almost all less thanmesh half ejecta the size blocks of the are larg- almost allnary less craters than half about the size 1 km of thein diameter larg- ornary smaller craters (from about eventual 1 km in diameter or smaller (from eventual est ejecta blocks at those velocities (Fig.est 7c). ejecta It is blocks also implausible at those velocitiesre-impact (Fig. 7 onc). Europa). It is also For implausible the 585-km-diameterre-impact onbasin Europa). Gilgamesh For the 585-km-diameter basin Gilgamesh that very-high-speed, sesquinary-formingthat tabular very-high-speed, ice spalls sesquinary-forming can on Ganymede, tabular we find ice the spalls maximum can sesquinaryon Ganymede, crater we size find for the re- maximum sesquinary crater size for re- survive intact. By this logic, the largestsurvive sesquinary intact. craters By this formed logic, theimpact largest on sesquinary Ganymede craters would formed be approximatelyimpact 2–3 on Ganymede km at most. would The be approximately 2–3 km at most. The Sesquinary craters on Europaby Gilgamesh are likelyand to be no larger by thanGanymede Gilgamesh 2–3 km in are diameter. likely to be noGilgamesh larger than SVD 2–3 has km a invery diameter. steep slope (approximatelyGilgamesh SVD has2.5), a veryindi- steep slope (approximately 2.5), indi- À À cating fragment size decreases rapidly²cating with fragment increasing size ejection decreases rapidly with increasingSpall ejection 6. Final remarks 6. Final remarks velocity. In this study we confirm earliervelocity. estimationsエジェクタ破⽚と In this study (Zahnle we confirm earlier estimations (Zahnle破⽚ et al., 2001; Alvarellos et al., 2002) that secondarieset al., 2001; orAlvarellos sesquinaries et al., 2002) that secondaries or sesquinaries Mapping of secondary craters providesMapping a window of secondaryinto ejecta cratersfrom provides Gilgamesh a window (Ganymede’s into ejecta largest survivingfrom Gilgamesh basin)Ø are (Ganymede’sSpall not large サイズは largest surviving basin)、 areTyre:4m not large ,Achelous:3m,Gilgamesh:86m ² 1.5次クレーター fragment sizes and velocities, and quantilefragment regression sizes fitting and velocities, char- andenough quantile to account regressionヒル圏 for fitting the closer-to-isotropic char- enough craterto account distribution for the closer-to-isotropic crater distribution acterizes the decline in fragment sizesacterizes with increasing the decline velocity. in fragmentobserved sizes withon Ganymede increasing at velocity. crater diametersobserved greaterØ on than Ganymede SVD 30 km. at crater diameters greater than 30 km. Based on the physics of cratering andBased ejecta on we the expect physics to of see cratering and ejecta we expect to see の外挿で求めた最⼤イジェクタ破⽚より⼩さい Ø SVDの速度をヒル圏脱出速度まで外挿して、放出される最trends when comparing the SVDs of differenttrends secondary when comparing fields with the SVDsAcknowledgments of different secondary fields with Acknowledgments the size of the primary, surface gravity ofthe the size body, of the or compositionprimary, surface gravity of the body, or composition Ø Spall theoryのでの破⽚は破砕を受けてはいないが、⾼速で放出されると破砕されること ⼤破⽚を⾒積もった of the target/impactor material, etc. Forof the the two target/impactor large icy satellites material, etc.We For thank the two P.M. large Schenk icy satellites for providing baseWe image thank mosaics P.M. Schenk of the for providing base image mosaics of the considered here, power law fits to theconsidered maximum ejectahere, power size as law a fitsthree to the regions maximum mapped ejecta in size this as study, a three and the regions NASA mapped Planetaryがある in this study,(Melosh,1984) and the NASA Planetary (Zahnle et al.2008) Ø 再衝突速度：Europa,2.5km/s Table 4 Table 4 Largest sesquinary diameters. Largest sesquinary diameters. Ganymede,3.5km/s (Alvarellos et al.2002)Primary Largest fragment at Largest fragmentPrimary LargestLargest fragment fragment at at LargestLargest fragment sesquinary LargestLargest fragment sesquinary at LargestLargest sesquinary sesquinary craterLargest on sesquinary Largest sesquinary crater on crater escape velocity – at escape velocitycrater escapeescape velocity velocity – master – atcrater escape on the velocity same bodyescapecrater velocity on the – master same craterthe sameon the body same – body mastercrater on the same the same body – master full dataset (m)a – subset (m)a datasetfull of datasetFig. 15 (m)(m)a b –– subset full dataset (m)a (km)c datasetbody of Fig. – subset 15 (m) (km)b c – fulldataset dataset of Fig. (km) 15c (m)c body – subset (km)c dataset of Fig. 15 (m)c Europa Europa ² Gilgamesh 1.5 Tyre 150 120Tyre 100 150 120 1.1 100 0.9 1.1 0.8 0.9 0.8 からの 次クレーターのサイズを⾒積もったPwyll 34Pwyll 34 0.3 0.3 Ganymede Ganymede Ø 氷破⽚のスケーリング則で、破⽚サイズは1kmAchelous 36 5Achelous 64 36 0.4 5 (0.3)d 640.07 (0.04)d 0.40.6 (0.3)d 0.07 (0.04)d 0.6 Gilgamesh 360 220Gilgamesh 1070 360 2202.4 1070 1.6 2.4 6.4e 1.6 6.4e a Calculated by extrapolation of equations in Tablea Calculated 3 (for non-porous by extrapolation ice scaling) of equations to the Hill in sphereTable escape 3 (for non-porous velocity given ice in scaling)Table2 to. the Hill sphere escape velocity given in Table 2. Ø b b 最も速く放出された破⽚は、最⼤破⽚の半分以下のサイズCalculated with the quantile regression fit (99thCalculated quantile) for with the the combined quantile dataset regression including fit (99th all three quantile) primaries for the in combined our study dataset (see Section including4.1 and all threeFig. 15 primaries). in our study (see Section 4.1 and Fig. 15). c c 1 1 1 1 Calculated using gravity regime parameters intoCalculated a non-porous using target, gravity with regime typical parameters re-impact speedinto a ofnon-porous 2.5 km sÀ target,for Europa with typicaland 3.5 re-impact km sÀ for speed Ganymede of 2.5 (Eq.km s(10)À for). Europa and 3.5 km sÀ for Ganymede (Eq. (10)). d Assumes porous, ice regolith target. d Assumes porous, ice regolith target. Ø 氷のSpallは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e Complex crater. e Complex crater. Ø Gilgameshで形成された最⼤の1.5次クレーターサイズは23kmより⼤きくない ² 2,1.5次クレーターのサイズを⾒積もった先⾏研究 Ø Europa:100mの破⽚が衝突して0.5~1kmのクレーターが形成 (Alvarellos et al.2002) Ø Gilgamesh:1kmより⼩さな破⽚が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 Summary ² イジェクタサイズと速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 速度が⼤きくなるにつれてイジェクタサイズは⼩さくなる Ø ２次クレーターのSVDは、天体重⼒や標的物質が異なることにより⽣じるクレーター形成、 イジェクタの物理過程が異なることによる Ø 25-40kmのクレーターではβ〜ｰ1、585kmの巨⼤盆地ではβ〜ｰ2-ｰ3 Ø 氷衛星でのSVDの傾きは、岩⽯天体の同サイズクレーターに⽐べ⼩さくなる傾向 Ø 岩⽯天体では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より⼩さい Ø Ganymedeの巨⼤盆地Gilgameshでも2-3kmより⼩さい Ø βが⼤きいため、衝突速度が⼤きいと破⽚サイズが⼩さくなるため Ø 先⾏研究で予測したGilgameshの２次、1.5次クレーターのサイズは、Ganymede上の直 径30kmより⼤きなクレーター分布に影響が出るほど⼤きいものではないことを確かめた