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1 Icaus 289 (217) Contents lsts avalable at ScenceDect Icaus jounal homepage: An analytcal model of cate count equlbum Masatosh Habayash a,, Davd A. Mnton a, Caleb I. Fassett b a Eath, Atmosphec and Planetay Scences, Pudue Unvesty, 55 Stadum Mall Dve, West Lafayette, IN , Unted States b NASA Mashall Space Flght Cente, Huntsvlle, AL 3585, Unted States a t c l e n f o a b s t a c t Atcle hstoy: Receved 5 August 216 Revsed 2 Novembe 216 Accepted 2 Decembe 216 Avalable onlne 27 Decembe 216 Keywods: Cateng Impact pocesses Regolths Cate count equlbum occus when new cates fom at the same ate that old cates ae eased, such that the total numbe of obsevable mpacts emans constant. Despte substantal effots to undestand ths pocess, thee eman many unsolved poblems. Hee, we popose an analytcal model that descbes how a heavly cateed suface eaches a state of cate count equlbum. The poposed model fomulates thee physcal pocesses contbutng to cate count equlbum: cooke-cuttng (smple, geometc ovelap), ejecta-blanketng, and sandblastng (dffusve eoson). These thee pocesses ae modeled usng a degadaton paamete that descbes the effcency fo a new cate to ease old cates. The flexblty of ou newly developed model allows us to epesent the pocesses that undele cate count equlbum poblems. The esults show that when the slope of the poducton functon s steepe than that of the equlbum state, the powe law of the equlbum slope s ndependent of that of the poducton functon slope. We apply ou model to the cateng condtons n the Snus Med egon and at the Apollo 15 landng ste on the Moon and demonstate that a consstent degadaton paametezaton can successfully be detemned based on the empcal esults of these egons. Futhe developments of ths model wll enable us to bette undestand the suface evoluton of aless bodes due to mpact bombadment. 217 Elseve Inc. All ghts eseved. 1. Intoducton A suface s cate populaton s sad to be n equlbum when the tean loses vsble cates at the same ate that cates ae newly geneated (e.g., Melosh, 1989, 211 ). Snce the Apollo ea, a numbe of studes have been wdely conducted that have povded us wth useful empcal nfomaton about cate count equlbum on planetay sufaces. In a cumulatve sze-fequency dstbuton (CSFD), fo many cases, the slope of the equlbum state n log-log space anges fom 1.8 to 2. f the slope of the cate poducton functon s steepe than 2 ( Gault, 197; Hatmann, 1984; Chapman and McKnnon, 1986; Xao and Wene, 215 ). When new cates fom, they ease old cates. Thee ae thee cate easue pocesses that pmaly contbute to cate count equlbum ( Fg. 1 ). The fst pocess s cooke-cuttng, n whch a new cate smply ovelaps old cates. Complete ovelap can ease the cates beneath the new cate. Howeve, f ovelappng s ncomplete, the old cates may be stll vsble because the m sze of the new cate stctly estcts the ange of cooke-cuttng. Cooke-cuttng s a geometc pocess and only depends on the aea occuped by new cates. Howeve, because cates ae not two-dmensonal ccles but thee-dmensonal depessons, cooke-cuttng s neffectve when a newly-fomed cate s smalle than a pe-exstng cate beneath t. The second pocess consdes ths thee-dmensonal effect. Ths pocess s sometmes called sandblastng, 1 whch happens when small cates collectvely eode a lage cate by nducng downslope dffuson, thus fllng n the lage depesson ove tme ( Ross, 1968; Sodeblom, 197; Fassett and Thomson, 214 ). Lastly, blanketng by ejecta deposts coves old cates outsde the new cate m (e.g., Fassett et al., 211 ). The thckness of ejecta blankets detemnes how ths pocess contbutes to cate count equlbum. Howeve, t has long been noted that they ae elatvely neffcent at easng old cates ( Woonow, 1977 ). Fo these easons, ths study fomulates the ejecta-blanketng pocess as a geometc ovelappng pocess (lke cooke-cuttng) and neglects t n the demonstaton execse of ou model. To ou knowledge, Gault (197) s the only eseache known to have conducted compehensve laboatoy-scale demonstatons fo the cate count equlbum poblem. In a 2.5-m squae box flled 3-cm deep wth quatz sand, he ceated sx szes of cates to geneate cate count equlbum. The photogaphs taken dung Coespondng autho. E-mal addess: thabayash@pudue.edu (M. Habayash). 1 Eale woks called ths pocess small mpact eoson ( Ross, 1968; Sodeblom, 197 ). Hee, we follow the temnology by Mnton et al. (215) / 217 Elseve Inc. All ghts eseved.

2 M. Habayash et al. / Icaus 289 (217) a. Cooke cuttng b. Sandblastng pocess c. Blanketng pocess Tme Fg. 1. A schematc plot of the pocesses that make cates nvsble. (a) The cookecuttng pocess, whee each new cate ovepnts olde cates. (b) The sandblastng pocess, whee multple small cates collectvely eode a lage cate. (c) The blanketng pocess, whee ejecta fom a new cate bues old cates. The bown ccle wth the sold lne shows ejecta blankets. The gay ccles wth the sold lnes ndcate fesh cates, and the gay ccles wth the dashed lnes descbe patally degaded cates. (Fo ntepetaton of the efeences to colo n ths fgue legend, the eade s efeed to the web veson of ths atcle.) the expements captued the natue of cate count equlbum (Fg. 5 n Gault (197) ). Hs expements successfully ecoveed the equlbum level of cate counts obseved n heavly cateed luna teans. Eale woks conducted analytcal modelng of cate count equlbum ( Macus, 1964, 1966, 197; Ross, 1968; Sodeblom, 197; Gault et al., 1974 ). Macus (1964, 1966, 197) theoetcally exploed the cate count equlbum mechansm by consdeng smple ccle emplacements. The ole of sandblastng n cate eoson was poposed by Ross (1968), followed by a study of sandblastng as an analog of the dffuson poblem ( Sodeblom, 197 ). Each of these models addessed the fact that the equlbum slope was found to be 2, whch does not fully captue the obseved slope descbed above. Gault et al. (1974) also developed a tmeevoluton model fo geometc satuaton fo sngle szed cates. Advances n computes have made Monte-Calo smulaton technques popula fo nvestgatng the evoluton of cate count equlbum. Eale woks showed how such technques could descbe the evoluton of a cateed suface ( Woonow, 1978 ). Snce then, the technques have become moe sophstcated and have been capable of descbng complcated cateng pocesses ( Hatmann and Gaskell, 1997; Mach et al., 214; Rchadson, 29; Mnton et al., 215 ). Many of these Monte Calo codes epesent cates on a suface as smple ccles o ponts on a m ( Woonow, 1977; 1978; Chapman and McKnnon, 1986; Mach et al., 214 ). The man dawback of the eale analytcal models and some Monte-Calo technques that smply emplaced ccles was that cate easue pocesses dd not account fo the thee-dmensonal natue of cate fomaton. As cateng poceeds, dffusve eoson becomes mpotant ( Fassett and Thomson, 214 ). Technques that model cates as thee-dmensonal to- pogaphc featues captue ths pocess natually ( Hatmann and Gaskell, 1997; Mnton et al., 215 ); howeve, these technques ae computatonally expensve. Hee, we develop a model that accounts fo the ovelappng pocess (cooke-cuttng and ejecta-blanketng) and the dffuson pocess (sandblastng) to descbe the evoluton of cate count equlbum whle avodng the dawbacks dscussed above. Smla to Macus (1964, 1966, 197), we appoach ths poblem analytcally. In the poposed model, we ovecome the computatonal uncetantes and dffcultes that hs model encounteed, such as hs complex geometcal fomulatons. Because the poposed model has an analytcal soluton, t s effcent and can be used to nvestgate a lage paamete space than numecal technques. Although a ecent study epoted a model of the topogaphc dstbuton of cateed teans, whch have also eached cate count equlbum at small cate szes, on the Moon ( Rosenbug et al., 215 ), the pesent pape only focuses on the populaton dstbuton of cates. We emphasze that the pesented model s a poweful tool fo consdeng the cate count equlbum poblem fo any aless planets. In the pesent execse, we consde a sze ange between and to bette undestand cate count equlbum. Also, although the cate count equlbum slope may potentally depend on the cate adus, we assume that t s constant. We oganze the pesent pape as follows. In Secton 2, we ntoduce a mathematcal fom that descbes the poduced cate CSFD deved fom the cate poducton functon. Secton 3 povdes the geneal fomulaton of the poposed model. Ths fom wll be elated to the fom deved by Macus (1964) but wll be moe flexble to consde the detaled easue pocesses. In Secton 4, we deve an analytcal soluton to the deved equaton. In Secton 5, we apply ths model to the cate count equlbum poblems of the Snus Med egon and the Apollo 15 landng ste on the Moon, and demonstate how we can use obsevatonal cate counts to nfe the natue of the cate easue pocesses. 2. The poduced cate CSFD To chaacteze cate count equlbum, we eque two populatons: the cate poducton functon and the poduced cates. The poducton functon s an dealzed model fo the populaton of cates that s expected to fom on a tean pe tme and aea. In ths study, we use a CSFD to descbe the poduced cates. The cate poducton functon n a fom of CSFD s gven by Cate Analyss Technques Wokng Goup (1979) as P ( ) = σ x ˆ η, (1) whee η s the slope, 2 σ s a constant paamete wth unts of m η 2 s 1, x ˆ s a cateng chonology functon, whch s defned to be dmensonless (e.g., fo the luna case, Neukum et al., 21 ), and s the cate adus. Fo notatonal smplfcaton, we choose to use the adus nstead of the damete. In contast, the poduced cates ae those that actually fomed on the suface ove fnte tme n a fnte aea. If all the poduced cates ae counted, the mean of the poduced cate CSFD ove many samplngs s obtaned by factozng the cate poducton functon by a gven aea, A, and by ntegatng t ove tme, t. We wte the poduced cate CSFD as t t C t( ) = Aσ η x ˆ dt = Aξ η xdt = Aξ X η, (2) whee ξ = σ t s x ˆ dt [m η 2 ], (3) 2 Fo notatonal smplfcaton, we defne the slopes as postve values.

3 136 M. Habayash et al. / Icaus 289 (217) x = X = x ˆ t s x ˆ dt [s 1 ], (4) t xdt. (5) Consde a specal case that the cateng chonology functon s constant, fo example, x ˆ = 1. Fo ths case, x = 1 /t s and X = t/t s. In these foms, t s can be chosen abtaly. Fo nstance, t s convenent to select t s such that Aξ η ecoves the poduced cate CSFD of the empcal data at X = 1. Also, we set the ntal tme as zeo wthout losng the genealty of ths poblem. Late, we use x and X n the fomulaton pocess below. Gven the poduced cate CSFD, C t ( ), the pesent pape consdes how the vsble cate CSFD, C c ( ), evolves ove tme. The followng sectons shall omt the subscpt, ( ), fom the vsble cate CSFD and the poduced cate CSFD to smplfy the notatonal expessons. 3. Development of an analytcal model 3.1. The concept of cate count equlbum We fst ntoduce how the obseved numbe of cates on a tean subject to mpact bombadment evolves ove tme. The most dect way to nvestgate ths evoluton would be to count newly geneated and eased cates ove some nteval of tme. Ths s what Monte Calo codes, lke CTEM, do ( Rchadson, 29; Mnton et al., 215 ). In the poposed model, the degadaton pocesses ae paametezed by a quantty that descbes how many cates ae eased by a new cate. We call ths quantty the degadaton paamete. Consde the numbe of vsble cates of sze on tme step s to be N s. We assume that the poducton ate of cates of ths sze s such that exactly one cate of ths sze s poduced n each tme step. In ou model, we teat patal degadaton of the cates by ntoducng a factonal numbe. Note that because we do not account fo the topologcal featues of the cateed suface n the pesent veson, ths paamete does not dstngush degadaton effects on shapes such as a m facton and change n a cate depth. Ths consdeaton s beyond ou scope hee. Fo nstance, f N 2 = 1. 5, ths means that at tme 2, the fst cate s halfway to beng uncounted. 3 If thee s no loss of cates, N s should be equal to s. When old cates ae degaded by new cates, N s becomes smalle than s. We descbe ths pocess as N s = N s k s N s 1, (6) whee k s s the degadaton paamete descbng how many cates of sze lose the denttes (ethe patal o n full) on tme step s, and s a constant facto that ncludes the poduced cate CSFD and the geometcal lmts fo the th-szed cates. In the analytcal model, we aveage k s ove the total numbe of the poduced cates. Ths opeaton s defned as k = s max s =1 k s N s 1 s max s =1 N s 1. (7) Eq. (7) povdes the followng fom: N s = N s k N s 1. (8) In the followng dscusson, we use ths aveaged value, k, and smply call t the degadaton paamete wthout confuson. 3 Because N =, at tme 1 thee s only one cate emplaced The case of a sngle szed cate poducton functon To help develop ou model, we fst consde a smplfed case of a tean that s bombaded only by cates wth a sngle adus. Ths case s smla to the analyss by Gault et al. (1974). We consde the numbe of cates, nstead of the facton of the cateed aea that was used by Gault et al. (1974). Consde a squae aea n whch sngle szed cates of adus,, ae geneated and eased ove tme. n s the numbe of the poduced cates, A s the aea of the doman, N s the numbe of vsble cates at a gven tme, and N, s the maxmum numbe of vsble cates of sze that s possbly vsble on the suface (geometc satuaton). In the followng dscusson, we defne the tme-devatve of n as n. N s the quantty that we wll solve, and N, s defned as N, =, (9) π 2 whee q s the geometc satuaton facto, whch descbes the hghest cate densty that could theoetcally be possble f the cates wee effcently emplaced onto the suface n a hexagonal confguaton ( Gault, 197 ). Fo sngle szed ccles q = π/ The numbe of vsble cates wll ncease lnealy wth tme at ate, n, at the begnnng of mpact cateng. Afte a cetan tme, the numbe of vsble cates wll be oblteated wth a ate of k n N /N,, whee N / N, means the pobablty that a newly geneated cate can ovelap old cates. Fo ths case, = n /N,, and k epesents the numbe of cates eased by one new cate at the geometcal satuaton condton. Ths pocess povdes the fst-ode odnal dffeental equaton, whch s gven as dn dt = n N k n. (1) N, The ntal condton s N = at t =. The soluton of Eq. (1) s gven as { ( )} N = N, 1 exp k n, k N, = k π 2 [ 1 exp { k π 2 n }]. (11) To poceed futhe, we must undestand the physcal meanng of the degadaton paamete. Eq. (1) ndcates that the degadaton paamete epesents how many old cates ae eased by a new cate. Fg. 2 shows two examples that descbe dffeent degadaton paametes n the case of a sngle cate-sze poducton functon. New cates ae epesented as gay ccles, old vsble cates ae whte ccles wth sold bodes, and lost old cates ae whte ccles wth dashed bodes. If k s less than 1, moe new cates ae necessay to be emplaced to ease old cates. If k s lage than 1, one new cate can ease moe than one old cate. Fom Eq. (11), as t, N eaches N, / k, not N,. Thus, snce N, / k N,, k 1. Ths means that the case descbed n Fg. 2 a, k < 1, does not happen The case of a multple cate sze poducton functon Ths secton extends the sngle cate-sze case to the multple cate-sze case. In the analytcal model by Macus (1964, 1966, 197), complex geometc consdeatons wee necessay, and thee wee many uncetantes. We wll see that the extenson of Secton 3.2 makes ou analytcal fomulaton clea and flexble so that the developed model can take nto account ealstc easue pocesses. We ceate a dffeental equaton fo ths case based on Eq. (1), whch states that fo a gven adus, the change ate of

4 M. Habayash et al. / Icaus 289 (217) Fg. 2. The physcal meanng of the degadaton paamete fo the case of a sngle szed cate poducton functon. The gay ccles ae newly emplaced cates, whle the whte ccles ae old cates. The sold ccles epesent vsble cates, and the dashed ccles epesent lost cates. (a) The case whee 2 old cates wee completely lost and seveal ones wee patally eased afte 1 new cates fomed ( k < 1). (b) The case whee 8 old cates wee completely lost and seveal ones wee patally eased afte 6 new cates fomed ( k 1). vsble cates s equal to the dffeence between the numbe of newly emplaced cates pe tme and that of newly eased cates pe tme. We fst develop a dscetzed model and then convet t nto a contnuous model. In the dscusson, we keep usng the notatons defned n Secton 3.2. That s, fo each th cate, the adus s, the numbe of vsble cates s N, the maxmum numbe of vsble cates n geometc satuaton s N,, the cate poducton ate s n, and the total numbe of the poduced cates s n. Cates of sze ae now affected by those of sze j, and we defne these quanttes fo cates of sze j n the same way. Modelng the degadaton pocess of dffeently szed cates stats fom fomulatng how the numbe of vsble cates of one sze changes due to cates of othe szes on each step. Fg. 3 shows how the numbe of vsble cates changes based on the degadaton pocesses that opeate dung the fomaton of new cates. In ths fgue, we llustate the cateng elaton between small cates and lage cates to vsualze the pocess clealy. Fo ths case, thee ae thee possbltes. Fst, new lage cates ease smalle, olde ones by ethe cooke-cuttng o ejecta-blanketng ( Fg. 3 a). Second, new cates elmnate the same-szed cates by ethe cooke-cuttng o ejecta-blanketng ( Fg. 3 b). Fnally, new small cates can degade lage ones though sandblastng o ejecta-blanketng ( Fg. 3 c). Fg. 3 d shows the total effect when all possble pemutatons ae consdeed. Consde the change n the numbe of vsble cates of sze. The th-szed cates ae geneated wth the ate, n, on evey tme step. Ths accumulaton ate s povded as dn dt acc = n. (12) The pesent case eques consdeaton of dffeently szed cates. The degadaton paamete should vay accodng to the degadaton pocesses of the j th-szed cates. To account fo them ( Fg. 3 ), we defne the degadaton paamete descbng the effect of the j th-szed cates on the th-szed cates as k j. Gven the th-szed cates, we gve the degadaton ate due to cates of sze j as dn dt deg, j N = k j n j 2 j N, 2. (13) In ths equaton, n Eq. (6) becomes dependent on cates of sze j. By defnng ths functon fo ths case as j, we wte 2 j 2 n j j =. (14) N, k j s a facto descbng how effectvely cates of sze ae eased by one cate of sze j. Fo example, when k j = 1, 2 j / 2 s the to- tal numbe of cates of sze eased by one cate of sze j at the geometc satuaton condton. In the followng dscusson, we constan k j to be contnuous ove the ange that ncludes = j to account fo the degadaton elatonshps among any dffeent szes. Based on Eqs. (12) and (13), we take nto account both the accumulaton pocess and the degadaton pocess. Consdeng the possble ange of the cate sze, we obtan the fst-ode dffeental equaton fo the tme evoluton of N as dn dt = dn dt acc = n N N, + max j= mn max j= mn dn dt deg, j, 2 j k j n j. (15) 2 The summaton opeaton on the ght hand sde means a sum fom the lagest cates to the smallest cates. Eq. (15) can descbe vsble cates that ovelap each othe ( Fg. 4 ). Smla to the sngle-szed case, we set the ntal condton such that N = at t =. The soluton of ths equaton s wtten as N = π max n j= mn k j 2 j n j [ 1 exp ( π max j= mn k j 2 j n j ) ]. (16) As dscussed n Secton 2, t s common to use the CSFDs to descbe the numbe of vsble cates. To enable ths model dectly to compae ts esults wth the empcal data, we convet Eq. (16) to a contnuous fom. k j s ewtten as a contnuous fom, k. We wte the contnuous fom of and that of j as and ř, espectvely. We defne C c as the CSFD of vsble cates and ewte N and n as N dc c d d, n dc t d, (17) d espectvely. Substtutons of these foms nto Eq. (16) yelds a dffeental fom of the CSFD, d C dc t [ ( c d = d π )] max dc t π 1 exp max d C t d ř k ř 2 d ř mn d ř k ř 2 d ř, (18) mn whee mn and max ae the smallest cate adus and the lagest cate adus, espectvely. The adus of the mn th-szed cates and that of the max th-szed cates coespond to mn and max, espectvely. Also, C t s the tme-devatve of C t. In the followng dscusson, we wll consde mn and max afte we model

5 138 M. Habayash et al. / Icaus 289 (217) a. The effect of lage cates on small cates (cooke-cuttng + blanketng) b. The effect of the same cates (cooke-cuttng + blanketng) c. The effect of the smalle cates on the lage cates (sandblastng + blanketng) d. The total effect Fg. 3. Schematc plot of degadaton pocesses fo a cate poducton functon wth two cate szes, lage and small. The model accounts fo thee dffeent cases to descbe the total effect. (a) The effect of lage cates on small cates. (b) The effect of the same-szed cates. (c) The effect of small cates on lage cates. (d) The total effect obtaned by summng the effects gven n (a) though (c). The gay ccles ae new cates, the whte ccles wth sold lnes descbe vsble old cates, and the whte ccles wth dashed lnes ndcate nvsble old cates. the k paamete. Eq. (18) s smla to Eq. (47) n Macus (1964), whch s the key equaton of hs sequental studes. The cate bth ate, λ, and the cate damagng ate, μ, ae elated to d C t /d and π max d C t mn d ř k ř 2 d ř, espectvely. Whle hs λ and μ ncluded a numbe of geometc uncetantes and dd not consde the effect of thee dmensonal depessons on cate count equlbum, ou fomulaton ovecomes the dawbacks of hs model and povdes much stonge constants on the equlbum state than hs model. 4. Analytcal solutons 4.1. Fomulaton of the degadaton paamete To detemne a useful fom of the degadaton paamete, k, we stat by dscussng how ths paamete vaes as a functon of ř. If cates wth a adus of ř ae lage than those wth a adus of, cooke-cuttng and ejecta-blanketng ae man contbutos to easng the -adus cates. In ths study, we consde that cookecuttng and ejecta-blanketng ae only elated to the geometcal elatonshp between cates wth a adus of ř and those wth a adus of. Cooke-cuttng only entals the geometc ovelap of the -adus cates; thus k should always be one. Ejecta-blanketng makes addtonal cates nvsble ( Pke, 1974; Fassett et al., 211; Xe and Zhu, 216 ), so k s descbed by some small constant, α eb. Addng these values, we obtan the degadaton paamete at ř as 1 + α eb. If cates wth a adus of ř ae smalle than those wth a adus of, the possble pocesses that degade the -adus cates ae ejecta-blanketng and sandblastng ( Fassett and Thomson, 214 ). Fo smplcty, we only consde the sze-dependence of sandblastng. It s easonable that as ř becomes smalle, the tmescale of degadng the -adus cate should become longe ( Mnton et al., 215 ). Ths means that wth a small adus, the effect of new cates on the degadaton pocess becomes small. To account fo ths fact, we assume that at ř <, k nceases as ř becomes lage. Hee, we model ths featue by ntoducng a sngle slope functon of ř / whose powe s a functon of. Combnng these condtons, we defne the degadaton paamete as ( Fg. 5 ) { ( k = (1 + α eb ) ř ) b() f ř <, 1 + α eb f ř, (19) whee b ( ) s a postve functon changng due to. We multpled 1 + α eb by the sze-dependent tem, ( ř /) b(), at < ř fo convenency. Ths opeaton satsfes the contnuty at = ř. We substtute the poduced cate CSFD defned by Eq. (2) and the degadaton paamete gven by Eq. (19) nto the ntegal tem

6 M. Habayash et al. / Icaus 289 (217) a. The numbe of countable cates b. The numbe of countable, small cates c. The numbe of countable, lage cates Fg. 4. Schematc plot fo how the analytcal model computes the numbe of vsble cates fo the multple cate-sze case. The model tacks the numbe of vsble cates fo each sze and sums up that of all the consdeed szes. In case small cates ae emplaced on lage cates (e.g., the sold squae), the model counts both szes and sums t up to compute the CSFD. (a) The numbe of vsble cates that the model s supposed to count. (b)and (c) The cate countng fo each case. The lght and dak gay ccles show small and lage cates, espectvely. [ η+2 η b() { + η+2 1 η 2 { ( max ( ) η+2+ b() } mn ) η+2 }] 1. (2) Fg. 5. Schematc plot of the degadaton paamete n log-log space. The x axs ndcates ř n a log scale, whle the y axs shows the value of the degadaton paamete n a log scale. If ř, k s always 1 + α eb because cooke-cuttng and ejectablanketng ae domnant. If ř <, sandblastng and ejecta-blanketng ae consdeed to be domnant. Fo ths case, k s descbed as (1 + α eb )( ř /) b(). The slope, b ( ), changes as a functon of. n Eq. (18). We ewte the ntegal tem of Eq. (18) as max mn dc t d ř k ř 2 d ř = ηaξ X (1 + α eb ) { ( ) b() ř ř η+1 d ř + mn = ηaξ X (1 + α eb ) max ř η+1 d ř }, Fo the case of C t that appeas n the denomnato of the facton tem n Eq. (18), we can use the devaton pocess above by eplacng X by x (see Eq. (5)). We have the smla opeatons below and only ntoduce the C t case wthout confuson. Note that the second opeaton n ths equaton s vald unde the assumpton that nethe η + 2 no η b() s zeo. Unde ths condton, we examne whethe o not Eq. (2) has a easonable value at mn and at max. Late, we wll show that an addtonal condton s necessay fo b ( ) fo mn. The tem n the last ow n Eq. (2) has ( max /) η+2. If the slope of the poduced cate CSFD satsfes η 2 >, we obtan ( ) η+2 max < 1. (21) Thus, when max, ths tem goes to zeo. The tem n the second to the last ow n Eq. (2) povdes constants on how the sandblastng pocess woks to ceate the equlbum states. Snce b ( ) > and η 2 >, the powe of mn /, η b(), can only take one of the followng cases: negatve ( η b() < ) o postve ( η b() > ). If η b() <, the tem, ( mn /) η+2+ b(), becomes at mn. Ths condton yelds C c at mn. We ule out ths condton by conductng the followng thought expement. We assume that ths case s tue. In natue, mcometeoods play sgnfcant oles n cate degadaton ( Melosh, 211 ). Because we assumed that ths case s tue, thee

7 14 M. Habayash et al. / Icaus 289 (217) should be no cates on the suface. Ths esult obvously contadcts what we have seen on the suface of aless bodes (we see cates!). The only possble case s the postve slope case, povdng the condton that the sandblastng effect leads to cate count equlbum as b() > η 2. Snce ths case satsfes ( ) η+2+ b() mn < 1, (22) Eq. (2) at mn and max s dc t d ř k ř 2 d ř = ηaξ X (1 + α eb ) η+2 ( ) 1 η b() + 1. (23) η 2 Hee, we also assume a constant slope of the equlbum state. To gve ths assumpton, we fnd b ( ) such that α sc β = 1 η b() + 1 η 2, (24) whee α sc and β ae constants. Ths fom yelds b( ) = a sc β (η 2) 2 α sc β. (25) (η 2) 1 Usng ths β value, we ewte Eq. (23) as dc t d ř k ř 2 d ř = ηaξ X (1 + α eb ) α sc η+2+ β. (26) 4.2. Equlbum state Ths secton ntoduces the equlbum slope at mn and max. Impact cateng acheves ts equlbum state on a suface when t. Usng Eqs. (18) and (26), we wte an odnal dffeental equaton of the equlbum state as dc c d = π d C t d, d C t d ř k ř 2 d ř = π (1 + α eb ) α 3 β. (27) sc Integatng Eq. (27) fom to, we deve the vsble cate CSFD at the equlbum condton as C c = = dc c d d, π (1 + α eb ) α sc (2 + β) 2 β. (28) Eq. (28) ndcates that the facton tem on the ght-hand sde s ndependent of ξ and η, and the equlbum slope s smply gven as 2 + β. If β =, the equlbum slope s exactly 2. Ths statement esults fom a constant value of b ( ), the case of whch was dscussed by Macus (197) and Sodeblom (197). These esults ndcate that the equlbum state s ndependent of the cate poducton functon and only dependent on the suface condton. Theefoe, a bette undestandng of the degadaton paamete may povde stong constants on the popetes of cateed sufaces, such as egonal slope effects, mateal condtons, and denstes. We befly explan the case of η 2 <. Ths would be a shallow-sloped CSFD, such as seen n lage cate populatons on heavly-cateed ancent sufaces. Fo ths case, cooke-cuttng s the pmay pocess that eases old cates ( Rchadson, 29 ). Consdeng that mn and max becomes qute lage ( ), we use Eq. (2) to appoxmately obtan max dc t d ř k ř 2 d ř η+2 max, (29) whch s constant. Thus, fom Eq. (28), we deve C c C t. (3) Ths equaton means that the slope of the equlbum state s popotonal to that of the poduced cate CSFD, whch s consstent wth the aguments by Chapman and McKnnon (1986) and Rchadson (29). We leave detaled modelng of ths case as a futue wok Tme evoluton of countable cates Ths secton calculates the tme evoluton of the vsble cate CSFD, C c. Substtutng Eq. (28) nto Eq. (18) yelds dc c d = π (1 + α eb ) α [ { sc 1 exp 3 β πηξx (1 + α q eb ) α sc }]. η+2+ β (31) Integatng Eq. (31) fom to, we obtan dc c C c = d d, 2 = β π (1 + α eb ) α sc (2 + β) 3 + β π (1 + α eb ) α sc { exp πηξx (1 + α q eb ) α sc }d. η+2+ β (32) Whle the second ow n ths equaton dectly esults fom Eq. (28), the thd ow needs addtonal opeatons. To deve the analytcal fom of the ntegal tem, we ntoduce an ncomplete fom of the gamma functon, whch s gven as Ɣ(a, Z) = Z Z a 1 exp ( Z) dz. (33) We focus on the ctcal ntegal pat of Eq. (32), whch s gven as f = whee 3 β exp ( χ η+2+ β ) d, (34) χ = πηξx (1 + α q eb ) α sc. (35) To apply Eq. (33) to Eq. (34), we consde the followng elatonshps, χ Z =, (36) η 2 β = ( χ Z ) 1 η 2 β. (37) Eq. (37) povdes d = 1 η 2 β ( χ Z Usng Eqs. (36) (38), we descbe f as ) 1 η 2 β dz Z. (38)

8 M. Habayash et al. / Icaus 289 (217) ( 1 1 f = η 2 β χ ( 1 1 = η 2 β χ ) 2+ β η 2 β ) 2+ β η 2 β γ Z 2+ β Z η 2 β 1 exp ( Z) dz, ( 2 + β η 2 β, Z ), (39) whee γ (, Z ) s called a lowe ncomplete gamma functon. Fo the cuent case, ths functon s defned as γ ( 2 + β η 2 β, Z ) = Ɣ ( ) ( ) 2 + β 2 + β Ɣ η 2 β η 2 β, Z, (4) whee Ɣ( ) = Ɣ(, ). We eventually obtan the fnal soluton as C c = + 1 π (1 + α eb ) α sc 1 π (1 + α eb ) α sc η 2 β 2 + β 2 β (41) ( 1 ) ( ) 2+ β η 2 β 2 + β γ χ η 2 β, Z. At an ealy stage, both the fst tem and the second tem play a ole n detemnng C c, whch should be close to the poduced cate CSFD. Howeve, as the tme nceases, X also becomes lage. Fom Eq. (34), when X 1, f 1, and thus the second tem becomes neglgble. Ths pocess causes C c to become close to C c. We ntoduce a specal case of β = and η = 3. Fom Eq. (25), b ( ) becomes constant and s gven as Fg. 6. Compason of the analytcal esults wth the empcal data of the Snus Med egon by Gault (197). The aea plotted s 1 km 2. The ed-edged ccles ae the empcal data. The blue and black lnes show the CSFDs of the poduced cates, C t, and that of the vsble cates, C c, espectvely. We plot the esults at thee dffeent tmes: X =. 1,.5, and 1.. The dashed lne ndcates the equlbum condton. (Fo ntepetaton of the efeences to colo n ths fgue legend, the eade s efeed to the web veson of ths atcle.) b = α sc α sc 1. (42) Eq. (41) s smplfed as C c = 2 π (1 + α eb ) α 2 + χ sc π (1 + α eb ) α 2 γ (2, Z), sc = 2 π (1 + α eb ) α 2 + χ sc π (1 + α eb ) α 2 { ( sc χ ) ( exp χ )}. (43) Eq. (42) shows that when α sc 1, b. Howeve, the followng execses show that α sc Sample applcatons We apply the developed model to the vsble cate CSFD of the Snus Med egon on the Moon (see the ed-edged ccles n Fg. 6 ) and that of the Apollo 15 landng ste (see the ededged ccles n Fg. 8 ). These locatons ae consdeed to have eached cate count equlbum. In these execses, we obtan C t by consdeng the cate szes that have not eseached equlbum, yet. Then, assumng that α eb =, we detemne α sc such that C c matches the empcal datasets. Fg. 7. Vaaton n b ( ) fo the Snus Med case. The sold lne shows b ( ), whch s gven n Eq. (25). The dotted lne epesents the mnmum value of b ( ), whch s η 2 = fo the Snus Med case The Snus Med egon Gault (197) obtaned ths CSFD (see Fg. 14 n hs pape), usng the so-called nestng countng method. Ths method accounts fo lage cates n a global egon, usually obtaned fom lowesoluton mages, and small cates n a small egon, gven fom hgh-esoluton mages. In the followng dscusson, cauton must be taken to deal wth the unts of the degadaton constants. Studes of the cate poducton functon (e.g. Neukum et al., 21) showed that a hgh slope egon, whch usually appeas at szes 1 m to 1 km mght be smla to the poduced cate CSFD. Hee, we obseved that such a steep slope appeas between 1 m and 4 m on the Snus Med suface. By fttng ths hgh slope, we obtan C t as C t = (44) Fg. 8. Compason of the analytcal esults wth the empcal data of the Apollo 15 landng egon ove an aea of 1 km 2. C.I.F. counted cates on ths egon n Robbns et al. (214). The ed-edged ccles ae the empcal data. The defntons of the lne fomats ae the same as those n Fg. 6. (Fo ntepetaton of the efeences to colo n ths fgue legend, the eade s efeed to the web veson of ths atcle.)

9 142 M. Habayash et al. / Icaus 289 (217) C t fo the Snus Med case s the poduced cate CSFD fo an aea of 1 km 2 (we consde an aea of 1 m 2 to be the unt aea), and ths quantty s dmensonless, and the facto has unts of m Snce η = > 2, ths case s the hgh-slope cate poducton functon. Based on ths fttng functon, we set X = 1, ξ = 2. 5 m 1.25, and A = 1 km 2. Also, we obtan the fttng functon of the equlbum slope as C c = (45) Smla to C t, the unts of the facto ae m 1.8. Fg. 6 compaes the empcal data wth the tme evoluton of C c that s gven by Eq. (41). We descbe dffeent tme ponts by vayng X wthout changng ξ and A. It s found that the model captues the equlbum evoluton popely. By fttng C c wth the empcal data, the pesent model can povde constants on the sandblastng exponent, b ( ), of the degadaton paamete. To obtan ths quantty, we detemne α sc and β. Snce α eb s assumed to be neglgble, we wte 1 + α eb 1. Fst, snce 2 β = 1. 8, we deve β =. 2. Second, opeatng the unts of the gven paametes, we have the followng elatonshp, [m 1. 8 ] = = 1 6 [m 2 ]. 97. (46) π (2 + β) α sc π (2. 2) α sc [m. 2 ] Then, we obtan α sc = [m. 2 ]. (47) Snce the unts of β ae m. 2, ths esult guaantees that Eq. (25) consstently povdes a dmensonless value of b ( ). Usng these quanttes, we obtan the vaaton n b ( ). Fg. 7 ndcates that the obtaned values of b ( ) fo the Snus Med satsfy the sand-blastng condton, b() > η 2. Fo ths case, whch has a constant slope ndex, b ( ) monotoncally nceases. As newly emplaced cates become smalle, they become less capable of easng a cate. These esults mply that fo a lage smple cate, t would take longe tme fo smalle cates to degade ts deep excavaton depth and ts hgh cate m. These esults could be used to constan models fo net downslope mateal dsplacement by cates; howeve, ths s beyond ou scope n ths pape The Apollo 15 landng ste We also consde the cate equlbum state on the Apollo 15 landng ste. Co-autho Fassett counted cates at ths aea n Robbns et al. (214), and we dectly use ths empcal esult. The used mage s a sub-egon of M L taken by Luna Reconnassance Obte Camea Naow-Angle Camea, the mage sze s pxels, and the sola ncdence angle s 77 ( Robbns et al., 214 ). The pxel sze of the used mage s.63 m/pxel. The total doman of the counted egon s 3.62 km 2, and the numbe of vsble cates s We plot the empcal data n Fg. 8. To make ths fgue consstent wth Fg. 6, we plot the vsble cate CSFD wth an aea of 1 km 2. In the followng dscusson, we wll show C t, C c, and C c by keepng ths aea,.e., A = 1 km 2, to make compasons of ou execses clea. Fo the Apollo 15 landng ste, the steep-slope egon anges fom 5 m to the maxmum cate adus, whch s 131 m. The fttng pocess yelds C t = (48) Ths fttng pocess shows that C t of the Apollo 15 landng ste s consstent wth that of the Snus Med case. Then, gven A = 1 km 2, we obtan ξ = 2. 2 m We also set X = 1 fo the condton that fts the empcal dataset. C c s gven as C c = (49) Fg. 9. Vaaton n b ( ) fo the Apollo 15 landng ste case. The unts of the facto ae m 1.8. Fg. 8 shows compasons of the analytcal model and the empcal data fo the Apollo 15 landng ste. We also obtan α sc and β. Agan, α eb s assumed to be zeo. Snce the slope of the equlbum state s 1.8, we deve β =. 2. Smla to the Snus Med case, we calculate α sc fo the Apollo 15 case as 34.9 m. 2. These quanttes yeld the vaaton n b ( ) ( Fg. 9 ). The esults ae consstent wth those fo the Snus Med case. 6. Necessay mpovements Futhe nvestgatons and mpovements wll be necessay as we made fve assumptons n the pesent study. Fst, we smply consdeed the lmt condton of the cate adus,.e., mn and max. Howeve, ths assumpton neglects consdeaton of a cut-off effect on cate countng. Such an effect may happen when a local aea s chosen to count cates on a tean that eaches cate count equlbum. Due to ths effect, la ge cates n the aea may be accdentally tuncated, and the cates counted thee may not follow the typcal slope featue. Second, we gnoed the effect of ejecta-blanketng n ou execses. Howeve, t s necessay to nvestgate the detals fo t to gve stonge constants on the cate count equlbum poblem. Thd, we assumed that the equlbum state s chaactezed by a sngle slope. Howeve, an eale study has shown that the equlbum slope could vay at dffeent cate szes fom case to case (e.g. Robbns et al., 214). To adapt such complex equlbum slopes, we eque moe sophstcated foms of Eq. (24). Fouth, the cuent veson of ths model does not dstngush cate degadaton wth cate oblteaton. Fo example, f t 1 n Eq. (41), thee s a chance that cates would be degaded due to sandblastng but not oblteated. At ths condton, they all would be vsble, whle Eq. (41) pedcts some oblteaton. Ffth, the measued cate adus can ncease due to sandblastng, whle the cuent model does not account fo ths effect. We wll attempt to solve these poblems n ou futue woks. We fnally addess that although we took nto account cookecuttng, ejecta-blanketng, and sandblastng as the physcal pocesses contbutng to cate count equlbum n ths study, we have not mplemented the effect of cate countng on the degadaton paamete. Accodng to Robbns et al. (214), the vsblty of degaded cates could depend on seveal dffeent factos: shapness of cates, suface condtons, and mage qualtes (such as mage esoluton and Sun angles). Also, puposes that a cate counte has also played a sgnfcant ole n cate countng. A bette undestandng of ths mechansm wll shed lght on the effect of human cate countng pocesses on cate count equlbum. We wll conduct detaled nvestgatons and constuct a bette methodology fo chaactezng ths effect.

10 M. Habayash et al. / Icaus 289 (217) Concluson We developed an analytcal model fo addessng the cate count equlbum poblem. We fomulated a balance condton between cate accumulaton and cate degadaton and deved the analytcal soluton that descbed how the cate count equlbum evolves ove tme. The degadaton pocess was modeled by usng the degadaton paamete that gave an effcency fo a new cate to ease old cates. Ths model fomulated cooke-cuttng, ejectablanketng, and sandblastng to model cate count equlbum. To fomulate the degadaton paamete, we consdeed the slope functons of the ato of one cate to the othe fo the followng cases: f the sze of newly emplaced cates was smalle than that of old cates, ejecta-blanketng and sandblastng wee domnant; othewse, ejecta-blanketng and cooke-cuttng manly eased old cates. Based on ou fomulaton of the degadaton paamete, we deved the elatonshp between ths paamete and a fttng functon obtaned by the empcal data. If the slope of the cate poducton functon was hghe than 2, the equlbum state was ndependent of the cate poducton functon. We ecoveed the esults by eale studes that the slope of the equlbum state was always ndependent of the poduced cate CSFD. If the physcal pocesses wee scale-dependent, the slope devated fom the slope of 2. Usng the empcal esults of the Snus Med egon and the Apollo 15 landng ste on the Moon, we dscussed how ou model constaned the degadaton paametes fom obseved cate counts of equlbum sufaces. We assumed that the ejectablanketng pocess was neglgble. Ths execse showed that ths model popely descbed the natue of cate count equlbum. Futhe wok wll be conducted to bette undestand the slope functons of the degadaton paametes, whch wll help us do valdaton and vefcaton pocesses fo both ou analytcal model and the numecal cateed tean model CTEM ( Rchadson, 29; Mnton et al., 215 ). Acknowledgements M.H. s suppoted by NASA s GRAIL msson and NASA Sola System Wokngs NNX15AL41G. The authos acknowledge D. Keslavsky and the anonymous evewe fo detaled and useful comments that substantally mpoved ou manuscpt. The authos also thank D. H. Jay Melosh at Pudue Unvesty, D. Jason M. Sodeblom at MIT, Ms. Ya-Hue Huang at Pudue Unvesty, and D. Colleen Mlbuy at West Vgna Wesleyan College fo useful advce to ths poject. Cate Analyss Technques Wokng Goup, Standad technques fo pesentaton and analyss of cate sze-fequency data. Icaus 37 (2), Fassett, C.I., Head, J.W., Smth, D.E., Zube, M.T., Neumann, G.A., 211. Thckness of poxmal ejecta fom the oentale basn fom luna obte lase altmete (lola) data: mplcatons fo mult-ng basn fomaton. Geophys. Res. Lett. 38 (17). Fassett, C.I., Thomson, B.J., 214. Cate degadaton on the luna maa: topogaphc dffuson and the ate of eoson on the moon. J. Geophys. Res. Planets 119 (1), Gault, D., Höz, F., Bownlee, D., Hatung, J., Mxng of the luna egolth. In: Luna and Planetay Scence Confeence Poceedngs, vol. 5, pp Gault, D.E., 197. 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