Icarus 214 (2011) 1–20 Contents lists available at ScienceDirect Icarus journal homepage: www.elsevier.com/locate/icarusNonuniform cratering of the Moon and a revised crater chronology of the innerSolar SystemMathieu Le Feuvre ⇑, Mark A. WieczorekInstitut de Physique du Globe de Paris, Saint Maur des Fossés, Francea r t i c l e i n f o a b s t r a c tArticle history: We model the cratering of the Moon and terrestrial planets from the present knowledge of the orbital andReceived 18 August 2010 size distribution of asteroids and comets in the inner Solar System, in order to reﬁne the crater chronol-Revised 1 March 2011 ogy method. Impact occurrences, locations, velocities and incidence angles are calculated semi-analyti-Accepted 7 March 2011 cally, and scaling laws are used to convert impactor sizes into crater sizes. Our approach isAvailable online 31 March 2011 generalizable to other moons or planets. The lunar cratering rate varies with both latitude and longitude: with respect to the global average, it is about 25% lower at (±65°N, 90°E) and larger by the same amountKeywords: at the apex of motion (0°N, 90°W) for the present Earth–Moon separation. The measured size-frequencyCrateringMoon distributions of lunar craters are reconciled with the observed population of near-Earth objects under theTerrestrial planets assumption that craters smaller than a few kilometers in diameter form in a porous megaregolith. Vary-Impact processes ing depths of this megaregolith between the mare and highlands is a plausible partial explanation for dif- ferences in previously reported measured size-frequency distributions. We give a revised analytical relationship between the number of craters and the age of a lunar surface. For the inner planets, expected size-frequency crater distributions are calculated that account for differences in impact conditions, and the age of a few key geologic units is given. We estimate the Orientale and Caloris basins to be 3.73 Ga old, and the surface of Venus to be 240 Ma old. The terrestrial cratering record is consistent with the revised chronology and a constant impact rate over the last 400 Ma. Better knowledge of the orbital dynamics, crater scaling laws and megaregolith properties are needed to conﬁdently assess the net uncertainty of the model ages that result from the combination of numerous steps, from the observation of asteroids to the formation of craters. Our model may be inaccurate for periods prior to 3.5 Ga because of a different impactor population, or for craters smaller than a few kilometers on Mars and Mercury, due to the presence of subsurface ice and to the abundance of large secondaries, respectively. Standard parameter values allow for the ﬁrst time to naturally reproduce both the size distribution and absolute number of lunar craters up to 3.5 Ga ago, and give self-consistent estimates of the planetary cratering rates relative to the Moon. Ó 2011 Elsevier Inc. All rights reserved.1. Introduction 2001a; Stöfﬂer and Ryder, 2001, and references therein). For statis- tical robustness, craters are generally counted within a number of The counting of impact craters offers a simple method to esti- consecutive diameter ranges, allowing one to recognize certainmate the ages of geologic units on planetary surfaces when biases, such as erosion, resurfacing or crater saturation processes.in situ rock samples are lacking. The crater chronology method is Measurements over various geologic units has led to the postulatebased on the simple idea that old surfaces have accumulated more that the relative shape of the size-frequency distributions (SFD) ofimpact craters than more recent ones (Baldwin, 1949). The rela- lunar impact craters was similar for all surfaces, but unfortunately,tionship between geologic age and number of lunar craters, based the exact shape of this production function in the 2–20 km range ison the radiometric dating of existing lunar rock samples, is found still debated after decades of study. The total predicted size-fre-to be approximately linear from the present to about 3 Ga ago, quency distribution for any given time is obtained by multiplyingand approximately exponential beyond that time (Neukum et al., the production function, assumed independent of age, by a time- variable constant. The age of a geologic unit is estimated by ﬁnding the best ﬁt between the standardized and measured distributions. ⇑ Corresponding author. Present address: Laboratoire de Planétologie et Géody- The approach taken in this work models directly crater distribu-namique, Université de Nantes, France. tions from the current knowledge of the impactor population. This E-mail address: email@example.com (M. Le Feuvre). allows us to infer properties of the impact history of a planetary0019-1035/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2011.03.010
2 M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20body that the sole observation of craters could not reveal. In partic- orek, 2008), whereas longitudinal asymmetries result from syn-ular, whereas the crater chronology method assumes that craters chronous rotation since the relative encounter velocity, henceaccumulate uniformly on the surface of the planetary body, this impact rate, are maximized at the apex of motion (e.g., Morotamethod allows to quantify possible spatial variations in the impact et al., 2005). In addition, the Earth may focus low inclination andrate. Moreover, in the absence of dated samples with known geo- velocity projectiles onto the nearside of the satellite or act as alogical context from Mercury, Venus and Mars, the generalization shield at small separation distances.of the crater chronology to these planets requires an estimate of Several studies have attempted to give estimates of the lunarthe relative cratering rates with respect to the Moon: in the same cratering asymmetries. Wiesel (1971) used a simpliﬁed asteroidperiod of time, the number of craters of a given size that form on population, and Bandermann and Singer (1973) used analytical for-two planets differ according to both the different impact probabil- mulations based on strongly simplifying assumptions in order toities of the planet-crossing objects and the impact conditions (e.g., calculate impact locations on a planet. These formulations didimpact velocity, surface gravity). not allow to investigate any latitudinal effects. Wood (1973) Using this bottom-up approach, Neukum et al. (2001b) pro- numerically integrated the trajectories of ecliptic projectiles; Pinetposed age estimates of mercurian geologic features based on a (1985) numerically studied the asymmetries caused by geocentricMercury/Moon cratering rate ratio estimated from telescopic projectiles that were potentially present early in the lunar history.observations. Similarly, Hartmann and Neukum (2001) adapted Horedt and Neukum (1984), Shoemaker and Wolfe (1982), Zahnlethe lunar cratering chronology to Mars, using the Mars/Moon cra- et al. (1998) and Zahnle et al. (2001) all proposed analytical formu-tering rate ratio of Ivanov (2001). The venusian surface, which lations for the apex/antapex effect, but the range of predictedcontains a small number of craters that appear to be randomly amplitudes is very large (the ﬁrst, in particular, claimed that thisdistributed, has been the subject of several attempts of dating. effect is negligible for the Moon). Moreover, these four studiesThe most recent estimates can be found in Korycansky and Zahnle based their results on isotropic encounter inclinations. Gallant(2005) and McKinnon et al. (1997), and are both based on the et al. (2009) numerically modeled projectile trajectories in thepopulation of Venus crossers as estimated by Shoemaker et al. Sun–Earth–Moon system, and reported a signiﬁcant apex/antapex(1990). In this study, we use improved estimates of the orbital effect. In our study, we derive a semi-analytic approach for calcu-characteristics and size-frequency distribution of the impactor lating cratering rates of synchronously locked satellites, and applypopulation. it to the Moon. This method yields results that are nearly identical The early lunar cratering record indicates that impactors were to full numerical simulations, is computationally very rapid, andhundred of times more numerous than today, possibly due to the easily generalizable to other satellites.massive injection in the inner system of main-belt and/or Kuiper- The ‘‘asteroid to crater’’ modeling requires several steps to cre-belt objects about 700 Ma after the Moon formed, an event ate synthetic crater distributions. First, one needs to know theknown as the Late Heavy Bombardment (LHB) (see Gomes impactor population as a function of orbital elements, size andet al., 2005; Tera et al., 1974; Hartmann et al., 2000). More than time. Second, impact probabilities are calculated over precession3 Ga ago, the impactor population appears to have reached a rel- and revolution cycles of both the projectile and target, based onative state of equilibrium, being replenished both in size and orbit geometrical considerations (Öpik, 1951; Wetherill, 1967; Green-by, respectively, collisions inside the main asteroid belt and the berg, 1982; Bottke and Greenberg, 1993). As these impact probabil-ejection by resonances with the giant planets (Bottke et al., ities predict typically that a given planetary crosser will collide2002). Under the assumption of a steady-state distribution of with a given planet over timescales of about 10 Ga, whereas theimpactors, the distribution of craters on $3 Ga old surfaces typical lifetime of these objects is thought to be a thousand timesshould be consistent with the present astronomically inferred lower (Michel et al., 2005) as a result of ejection from the Solar Sys-cratering rates. tem or collision with another body, the calculated bombardment The lunar cratering record, in the form of the standardized pro- must be seen as the product of a steady-state population, whereduction functions, agrees reasonably well with telescopic observa- vacant orbital niches are continuously reoccupied (Ivanov et al.,tions of planet-crossing objects down to a few kilometers in 2007). Third, scaling laws derived from laboratory experimentsdiameter (Stuart, 2003; Werner et al., 2002). Smaller craters appear and dimensional analyses are used to calculate the ﬁnal crater sizeto be not numerous enough to have been formed by an impacting produced for given a impact condition (e.g., impactor size, velocitypopulation similar to the present one. But, in Ivanov et al. (2007), it and cohesion of the target material), for which several differentwas suggested that the presence of a porous lunar megaregolith scaling laws have been proposed (Schmidt and Housen, 1987;may reduce the predicted size of small craters, accounting for this Holsapple and Schmidt, 1987; Gault, 1974; Holsapple, 1993; Hols-observation. In Strom et al. (2005), the distinction was made be- apple and Housen, 2007). Subsequent gravitational modiﬁcation oftween pre- and post-LHB lunar crater distributions. Older distribu- the transient cavity that gives rise to the ﬁnal crater size have beentions, depleted in small craters, would reﬂect the SFD of Main Belt deduced from crater morphological studies (Pike, 1980; Croft,asteroids – massively provided by the LHB – rather than the SFD of 1985).near-Earth asteroids. Marchi et al. (2009) revised the crater chro- The rest of this paper is organized as follows. In the followingnology method by deriving a new production function from impact section, we describe the employed orbital and size distribution ofmodeling that differs signiﬁcantly from those based on measure- the impactor population. In Section 3, the necessary equationsments. In this study, we attempt to fully reconcile the measured lu- used in generating synthetic cratering rates are summarized; thenar production functions with the telescopic observations of full derivations are given in the appendix. In Section 4, we compareplanet-crossing objects. our synthetic lunar crater distribution with observations, we Modeling the impact bombardment also allows us to quantify describe the lunar spatial asymmetries, and give the predicted pla-spatial cratering asymmetries that are not accounted for in the tra- net/Moon cratering ratios. We also provide simple analytic equa-ditional crater chronology method. The presence of such asymme- tions that reproduce the predicted cratering asymmetries on thetries would bias the ages based on crater densities according to the Moon, as well as the latitudinal variations predicted for the terres-location of the geologic unit. The Moon is subject to nonuniform trial planets in Le Feuvre and Wieczorek (2008). Section 5 is dedi-cratering, since it is not massive enough to gravitationally homog- cated to revising the crater chronology method. New age estimatesenize encounter trajectories. Latitudinal asymmetries are produced are proposed for geologic units on the Moon, Earth, Venus andby the anisotropy of encounter inclinations (Le Feuvre and Wiecz- Mercury.
M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20 32. Impactor population 10 5 10 4 Let us denote a, e, i and d respectively the semi-major axis, 10 3eccentricity, inclination and diameter of those objects whose orbits Cumulative impact probability 10 2can intersect the inner planets. The entire population can be 10 1written 10 0 nð d; a; e; iÞ ¼ nð 1Þ Â oða; e; iÞ Â sð dÞ; ð1Þ 10 −1 10 −2where n can be expressed as the product of nðd 1Þ, the total num-ber of objects with a diameter greater than 1 km; o(a, e, i), the rela- 10 −3tive number of objects with a given set of orbital elements, 10 −4 Rnormalized so that oða; e; iÞ da de di ¼ 1; and s(d), the normal- 10 −5ized cumulative number of objects larger than a given diameter, 10 −6such that s(1 km) = 1. This formulation assumes that no correla- 10 −7tions exist between the size of the object and its orbit, which is con- 10 −8sistent with the observations of Stuart and Binzel (2004) for 10 −9diameters ranging from $10 m to $10 km. 10−10 0.0001 0.001 0.01 0.1 1 10 1002.1. Orbital distribution diameter (km) The orbital distribution of near-Earth objects (NEOs) is taken Fig. 1. Impact probability per year on Earth, for impactors larger than a givenfrom the model of Bottke et al. (2002), which provides a debiased diameter. Estimates come from atmospheric records or Öpik probabilities derivedestimate of the orbital distribution of bolides that can potentially from telescopic observations (black triangles: Halliday et al. (1996); black diamonds: ReVelle (2001); gray triangles: Brown et al. (2002); gray squares and white triangles:encounter the Earth–Moon system. This model assumes that the Rabinowitz et al. (2000); gray circles: Harris (2002); white circles: Morbidelli et al.NEO population is in steady-state, continuously replenished by (2002); black squares: Stuart and Binzel (2004)). Our compilation at large sizes isthe inﬂux coming from source regions associated with the main augmented by including the observed size-frequency distribution of Mars-crossingasteroid belt or the transneptunian disk. The model was deter- objects with sizes greater than 4 km, scaled to the terrestrial impact rates of Stuartmined through extensive numerical integrations of test particles and Binzel (2004) (gray diamonds). The red curve is the best ﬁt polynomial.from these sources, and calibrated with the real population ob-served by the Spacewatch survey. The relative number of objects a constant mean geometric albedo of 0.1 in order to convert theiris discretized in orbital cells that spans 0.25 AU in semi-major axis, telescopic observations from magnitude to diameter at all sizes.0.1 in eccentricity, and 5° in inclination. Though small objects are expected to possess a larger albedo as In order to model the martian impact ﬂux, we have amended they are generally younger than large objects, we did not attemptthis model by including the known asteroids that cross the orbit to correct for this effect, since estimates of Rabinowitz et al. (2000)of Mars, but which are not part of the NEO population. These are show a general agreement with other studies. In atmospheric ﬂashtaken from the database provided by E. Bowell (Lowell Observa- estimates, masses have been deduced from kinetic energy, the lat-tory), that gives the orbit and absolute magnitude H of telescopic ter being estimated from luminous energy. We convert kineticdiscoveries. We consider the population of H 15 objects to be energies of Brown et al. (2002) into diameters using a mean den-the best compromise between a sufﬁcient number of objects (in or- sity of 2700 kg mÀ3 and a mean impact velocity of 20 km sÀ1 (Stu-der to avoid sparseness in the orbital space), and completeness. art and Binzel, 2004). In order to increase the range of sizes in ourTheir distribution as a function of perihelion is very similar in compilation, and to reduce the statistical uncertainties associatedshape to brighter (hence larger) H 13 objects (see Le Feuvre and with the larger objects, we have also included the size-frequencyWieczorek, 2008), the latter being large enough to ensure they distribution of Mars-crossing objects with sizes greater thando not suffer observational bias in the martian neighborhood. 4 km, and scaled these to the terrestrial impact rates of StuartThe H 15 population is therefore considered as complete, and is and Binzel (2004).scaled to match the modeled NEO population. From the relation- Various assumptions have led to all these estimates. Amongship between absolute magnitude and geometric albedo of Bowell them, the assumed impact velocity and bolide density are only ofet al. (1989), and a mean albedo of 0.13 (Stuart and Binzel, 2004), moderate inﬂuence. As an example, varying the density fromH 15 corresponds to d $4 km. 2700 to 2000 kg mÀ3, or the mean impact velocity from 20 to 17 km sÀ1, changes the estimates of Brown et al. (2002) only by2.2. Size distribution about 10%. Of major inﬂuence are the luminous efﬁciency used to obtain kinetic energy from ﬂashes (see Ortiz et al., 2006), and The size distribution of impactors is taken from a compilation of the debiasing process in the case of telescopic observations, whichvarious estimates of the size-dependent impact rate on Earth, as are difﬁcult to assign a statistical uncertainty to. Consequently, weshown in Fig. 1. This compilation gathers atmospheric recordings simply ﬁt a 10th-order polynomial to the entire dataset, assumingof meteoroids and impact probabilities calculated with Öpik each data is error free, and that the average combination of all esti-equations from debiased telescopic observations. Estimates from mates gives a good picture of the impactor population.telescopic observations are those of Rabinowitz et al. (2000), Mor- We express the resulting analytic size-frequency distribution asbidelli et al. (2002), Harris (2002) and Stuart and Binzel (2004). the product of two terms: the normalized size distribution PEstimates based on atmospheric recordings include Halliday et al. log sð dÞ ¼ 10 si ðlog dkm Þi , whose coefﬁcients are listed in Table i¼0(1996), ReVelle (2001) and Brown et al. (2002). Concerning the 1, and the Earth’s impact rate for objects larger than 1 km, À2LINEAR survey, estimates of Harris (2002) have been scaled at large /e ðd 1Þ ¼ 3:1 Â 10À6 GaÀ1 km . An overbar is appended to the sizes to the estimates of Stuart and Binzel (2004). In contrast to Earth’s impact rate symbol, denoting that this quantity is spatiallyMorbidelli et al. (2002) and Stuart and Binzel (2004), Rabinowitz averaged over the planet’s surface. The size-frequency distributionet al. (2000) did not use a debiased albedo distribution, but rather of impactors is here assumed to be the same for all bodies in the
4 M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20Table 1 PImpactor size distribution: log sð dÞ ¼ 10 si ðlog dkm Þi . i¼0 s0 s1 s2 s3 s4 s5 1.0 3.1656EÀ01 1.0393EÀ01 5.7091EÀ02 À8.1475EÀ02 À2.9864EÀ02 s6 s7 s8 s9 s10 1.3977EÀ02 5.8676EÀ03 À4.6476EÀ04 À3.8428EÀ04 À3.7825EÀ05inner Solar System. The relative impact rates for these bodies with the encounter probability are avoided following Dones et al.respect to Earth are calculated in Section 3 using the orbital distri- (1999). For a given orbital geometry, the encounter probabilitybution of the planet crossing objects. is proportional to the gravitational cross section, whose radius is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3. From asteroids to craters s ¼ R 1 þ C; ð2Þ Here we describe how is calculated the number of craters where R is the target’s radius andthat form on the Moon and planets, per unit area and unit time, GMas a function of the crater diameter and location on the surface. C¼2 ð3Þ RU 2We ﬁrst need to calculate the encounter conditions generated bythe impactor population, then the corresponding impact rate and is the Safronov parameter, with G the gravitational constant and Mconditions (impact velocity and incidence angle), in order to ﬁ- the target’s mass.nally obtain a cratering rate by the use of impact crater scaling Note that the calculated impact probabilities are long-termlaws. For the reader’s convenience, derivations are given in the averages over precession cycles of both the projectile and targetappendix. (i.e., the longitude of node and argument of pericenter can take any value between 0 and 2p). We account for secular variations3.1. Encounter probability of the planetary orbital elements using the probability distribu- tions given as a function of time in Laskar (2008). Our results are Following Öpik (1951), the assumptions under which an only sensitive to secular variations for Mars, and in the following,encounter is considered to occur can be summarized as follows: two values are quoted for this planet, that correspond to the pres- ent day value and to a long-term average (1 Ga and 4 Ga averages An encounter between the target (Moon or planet) and impac- yield nearly identical results). tor occurs at the geometrical point of crossing of the two orbits By calculating the encounter probability P and velocity U asso- (the mutual node). The geometry of encounter is given by the ciated with a given orbital element set (a, e, i), by weighting this relative velocity vector U at this point, which is expressed here probability with the relative number of objects o(a, e, i), and by in a frame where the X-axis points towards the central body summing over the entire planet-crossing population for each ter- (planet or Sun), (XY) deﬁnes the target’s orbital plane, and the restrial planet, we build the probability distribution of the encoun- Z-axis points upward. ter conditions, p(U), using bins of 1 km sÀ1 for each component of The relative encounter velocity does not account for the acceler- the encounter velocity. Analytically, we have ation generated by the mass of the target. p0 ðUÞ The impactor, as seen by the target, is treated as if it were pðUÞ ¼ R ; ð4Þ U p0 ðUÞ dU approaching from an inﬁnite distance, under only the gravita- tional inﬂuence of the target. The encounter trajectory is there- with fore hyperbolic in the reference frame of the target. Z For a given U, there are an inﬁnite number of hyperbolic trajecto- p0 ðUÞ ¼ PðDÞoða; e; iÞdðUðDÞ À U0 Þ dD; ð5Þ D ries that can actually strike the target, that are distributed uni- formly on a circle perpendicular to U with a surface equal to the where the integration is performed over the eight-dimensional target’s gravitational cross section. domain À Á D ¼ U 0X ; U 0Y ; U 0Z ; a; e; i; w0 ; DX , with w0 the target’s argument of These approximations hold as long as the radius of the tar- perihelion, DX the difference between the target and projectile’sget’s Hill sphere is large enough with respect to the target size. longitudes of the ascending node (see Greenberg, 1982), and dWe performed three-body numerical simulations that show that the Kronecker function which equals 1 when U = U0 and 0 other-a factor of ten between the Hill sphere and target radii sufﬁces wise. The impact rate relative to the Earth isto ensure the validity of the above approximations. For the ter- R 0 p ðUÞdUrestrial planets, this condition is largely veriﬁed. For the Moon, it r¼RU 0 : ð6Þ U pe ðUÞ dUcorresponds to a minimum Earth–Moon separation of $17Earth’s radii. where p0e is calculated from Eq. (5) with the Earth being the target. In the case of planets in nearly circular orbits, the encounter The lunar case requires a speciﬁc treatment, which is detailed ingeometry U and probability P (providing the two orbits inter- Appendix A.2. For simplicity and without altering the results, it issect) are simply given by the well known Öpik equations assumed that the lunar orbit is circular about the Earth and possess(Öpik, 1951). In order to account for the eccentricity of the tar- a zero inclination with respect to the ecliptic. We ﬁrst calculateget (which is important for Mercury and Mars, but not for the encounters probabilities P0 and velocities V with the entireEarth), we use the improved formulation of Greenberg (1982) Earth–Moon system, whose expression is the same as for the Earth,and Bottke and Greenberg (1993). The probability is largest except that the gravitational cross section radius is replaced in thefor low inclination encounters, and for encounters occurring Öpik equation by what we call here the lunar orbit cross section,near the projectile’s pericenter and apocenter. Singularities of deﬁned as Zahnle et al. (1998)
M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20 5 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v2 from Eqs. (A.51) and (A.54). The impact velocity u is only dependent ms0 ¼ a m 1 þ 2 2 ; ð7Þ on U and C, while the incidence angle h further depends on position. Vwhere am and vm are respectively the lunar semi-major axis and 3.3. Cratering ratevelocity. On the cross sectional disk, the distance from the centeris denoted by the impact parameter b. Only when b 6 s0 is it possi- To obtain cratering rates from impact rates, we need to convertble for a hyperbolic orbit to impact the Moon (note that the gravi- the impactor diameters into crater diameters. For this purpose, wetational cross section of the Moon itself is accounted for later in use equations that have been derived in the framework of p-scal-the calculation, see Appendix A.2). The probability distribution of ing dimensional analysis (Holsapple and Schmidt, 1987; Holsapple,encounter conditions p(V) is ﬁrst calculated according to this new 1993), where the transient crater is given as a function of the pro-deﬁnition of the encounter probability. Then, the relative lunar jectile diameter, impact velocity, surface gravity and projectile/tar-encounter velocity U and probability Pm are derived analytically get density ratio (see Appendix A.4). It is assumed that only thefor each hyperbolic orbit crossing the Earth–Moon system (Eqs. vertical component of the impact velocity, whose value is obtained(A.19), (A.18), (A.24), (A.25), (A.30)–(A.33)). The probability distri- from the impact angle, contributes to the crater size (Pierazzobution of lunar encounter conditions, p(U), is then determined by et al., 1997), though other relations could be easily incorporatedintegrating numerically Pm for all possible hyperbolic orbits of each into this analysis. The scaling equation and parameters are takenencounter V, and for all encounter velocities. Similarly, the impact from the summary of Holsapple and Housen (2007) for the caseprobability with Earth Pe is determined for each hyperbola, and of porous and non-porous scaling. It will be seen that both forma-the Moon/Earth impact ratio r is calculated over all possible tion regimes are necessary to reconcile the impactor and craterencounters. Mathematically, p(U) and r are given by Eqs. (A.34)– SFDs. We only consider craters that form in the gravity regime,(A.36). where the tensile strength of rock is negligible, that is, craters lar- Note that there is a dependence of the lunar impact rate on the ger than a few hundred meters in competent rock, and larger thanEarth–Moon separation distance, am. This distance has evolved out- a few meters in consolidated soils. An increase of the transient cra-ward with time, and we test various separation distances in the ter diameter by wall slumping and rim formation is accounted forsimulations. A major difference between our approach and previ- as given in Melosh (1989). Finally, large craters collapse due toous investigations (Shoemaker and Wolfe, 1982; Zahnle et al., gravity, becoming complex craters, and the relationship between1998; Zahnle et al., 2001) is that the argument of pericenter of simple and complex crater diameters is taken from Holsapplethe hyperbolic orbits is not assumed to precess uniformly within (1993). Putting this altogether, we obtain the relation d(D, u, h) thatthe Earth–Moon system, but is explicitly given by the encounter gives the impactor diameter d as a function of the crater diametergeometry. Our formulation allows to calculate explicitly lateral D, impact velocity u and incidence angle h (Appendix A.4, Eqs.asymmetries in the lunar cratering rate. (A.63), (A.59)). The impactor diameter d required to create a crater of size D is ultimately a function of k, / and U (see Eqs. (A.51) and3.2. Impact rate (A.54)). The cratering rate, that is the number of craters larger than D Let us express the cumulative impact ﬂux, that is the number of that form at (k, u) per unit time and area, isobjects with diameters greater than d that hit the planet per unit Ztime and area, as Cð D; k; uÞ ¼ /ð d; k; /ÞpðUÞdU; ð12Þ U /ð d; k; uÞ ¼ /ð dÞ Â D/ðk; uÞ; ð8Þ where where k and u are respectively the latitude and longitude, /ð dÞ is d ¼ dðD; k; /; UÞ: ð13Þthe spatially averaged impact rate for projectiles larger than d, andD/(k, u) is the relative impact rate as a function of position, normal- For convenience, we separate the cratering rate intoized to the global average. Using our normalized impactor SFD, theaverage impact rate expresses as Cð D; k; uÞ ¼ Cð DÞ Â DCð D; k; uÞ: ð14Þ /ð dÞ ¼ /ð 1Þsð dÞ; ð9Þ where Cð DÞ is the spatially averaged rate and DC(D, k, u) is the relative spatial variation. Note that DC depends on D, though in and /ðd 1Þ is obtained from the impact ratio between the target practice, this dependence is moderate (see next section). and Earth, r, and the terrestrial impact rate /e ðd 1Þ as /ð 1Þ ¼ r /e ð 1Þ: ð10Þ 4. ResultsThe net spatial asymmetry D/(k, u) is found by integrating the spa- 4.1. Crater size-frequency distributionstial asymmetries d/(k, u, U) associated with each encountergeometry: We ﬁrst present our synthetic size-frequency distribution of lu- Z nar craters. Following the terminology of Marchi et al. (2009), weD/ðk; uÞ ¼ d/ðk; u; UÞpðUÞdU; ð11Þ refer to this as a model production function. We compare our mod- U el production function with the two standard measured productionwhere d/ is given in Appendix A.3 as a function of the Safronov functions of Neukum (1983, 1994) and Hartmann (Basaltic Volca-parameter C and obliquity of the target (Eqs. (A.47)–(A.49)). The nism Study Project, 1981; Hartmann, 1999). We note that theimpact ﬂux is homogeneous for C = 1, that is, for encounter veloc- two are in good agreement over the crater diameter range fromities negligible with respect to the target’s surface gravitational po- 300 m to 100 km, but differ between 2 and 20 km, with atential (Eq. (3)). On the other hand, for C = 0, encounter trajectories maximum discrepancy of a factor 3 at 5 km.are straight lines, and the impact ﬂux is a simple geometrical pro- Using the traditional non-porous scaling relations and ajection of the spatially uniform encounter ﬂux (on a plane perpen- standard target density of 2800 kg mÀ3, we calculate thatdicular to the radiant) onto the target’s spherical surface. The 2.88 Â 10À11 craters larger than 1 km would be created each yearassociated impact velocities and incidence angles are calculated on the lunar surface by the present impactor population. Using
6 M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20the time-dependence established by Neukum (1983) that predicts 1 dz ðzÞ ¼ ððT À zÞdp þ zdnp Þ for z 6 T;a quasi-constant impact ﬂux over the last $3 Ga, the Hartmann T ð17Þand Neukum production functions return respective values of dz ðzÞ ¼ dnp for z P T;7.0 Â 10À13 and 8.2 Â 10À13, which are about 10 times lower,implying that the present ﬂux must be considerably larger thanthe time averaged value. with dp and dnp the impactor diameters respectively required from However, by using the porous scaling law instead, in order to the porous and non-porous regimes. In the calculation of dnp, theaccount for the presence of megaregolith on the lunar surface, target density is set to 2800 kg mÀ3 (solid rocks), whereas we as-our calculated spatially averaged lunar cratering rate is sume in calculating dp that the density of the porous material is À2 2500 kg mÀ3, based on Bondarenko and Shkuratov (1999) who in-C m ðD 1Þ ¼ 7:5 Â 10À13 yrÀ1 km ; ð15Þ ferred an upper regolith density comprised between 2300 anda value in excellent agreement with the two measured production 2600 kg mÀ3 from correlations between the surface regolith thick-functions under the assumption of a constant impact ﬂux over the ness and the Soderblom’s crater parameter (Soderblom and Lebof-last $3 Ga. sky, 1972). We note that given the simplicity of our crater-scaling Let us now reconcile the entire shape of the measured produc- procedure in the transition zone, the correspondance between Ttion functions with the observed impactor population. As shown in and the actual megaregolith thickness should not be expected toFig. 2, the two measured distributions are very well ﬁtted by using be exact.the porous regime for small craters (D 2 km), and the non-porous As shown in Fig. 2, our model reproduces both the Hartmannregime for larger craters (D 20 km). We model a simple smooth and Neukum production functions within the 100 m – 300 kmtransition between the two regimes by considering that the impac- diameter range, for respective values of T equal to 250 and 700tor size d required to create a crater of diameter D is a linear com- m, respectively. For these diameter ranges, the maximum discrep-bination of the sizes required from the porous and non-porous ancy between our model and the Neukum production function isscaling relations, the inﬂuence of each regime depending on the only 30% at 200 m, and always less than 20% for craters larger thandepth of material excavated by the crater. The depth of excavation 500 m. Below 100 m, we note that our model is in reasonablezT is about 1/10 of the transient crater diameter DT, and does not agreement with the Neukum production function, and we leaveseem to depend on target properties (Melosh and Ivanov, 1999). the implications for the contribution of secondary craters to fur-The impactor size is averaged over the depth of excavation: ther investigations. The model production function proposed by Z zT Marchi et al. (2009) is also shown in Fig. 2. A detailed comparison 1 with this latter study is given in the discussion section.d¼ dz ðzÞ dz; ð16Þ zT 0 The use of porous scaling was ﬁrst suggested by Ivanov (2006)where dz is the impactor size required by the material at depth z, (see also Ivanov, 2008; Ivanov et al., 2007), and is a natural conse-given by the porous regime at the surface, by the non-porous re- quence of a highly fractured megaregolith on airless bodies. Alsogime at depths larger than a given ‘‘megaregolith thickness’’ T, natural is to expect that the thickness of the megaregolith will de-and by a linear combination between z = 0 and z = T: pend upon both age and local geology. We note that the need for a transition regime falls within the diameter range where the mea- sured production functions differ the most (excluding very large craters). We suspect that this is partially a result of the Hartmann production function being based on crater counts performed solely over mare units, whereas the Neukum production function also in- 10−5 cludes older highlands terrains (Neukum et al., 2001a). Estimated megaregolith thicknesses are roughly consistent with 10−6 seismic models of the lunar crust (e.g., Warren and Trice, 1977; 10−7 Lognonné et al., 2003) that generally predict reduced seismic Cumulative number (km−2 yr − 1) 10−8 velocities for the upper km, which is attributed to an increased 10−9 porosity and fractures. Furthermore, seismic data at the Apollo 17 landing site, overlaid by mare basalt, indicates that the upper 10−10 250 / 400 m show a very low P-wave velocity with respect to the 10−11 deeper basalt (Kovach and Watkins, 1973; Cooper et al., 1974), 10−12 the lower estimate being in agreement with our calculated megar- 10−13 egolith depth of 250 m for the Hartmann production function. Fi- nally, Thompson et al. (1979) show by analysis of radar and 10−14 infrared data (which are dependent on the amount of near surface 10−15 Neukum et al. (2001) rocks) that craters overlying highlands show different signatures 10−16 Hartmann (1999) for craters greater and less than 12 km, and that mare craters down 10−17 Marchi et al. (2009) to 4 km in diameter possess a similar signature to that of highlands Model, megaregolith 700 m craters greater than 12 km. They attribute this difference to the 10−18 Model, megaregolith 250 m presence of a pulverized megaregolith layer that is thicker in the 10−19 older highlands than the younger mare. 0.01 0.1 1 10 100 By the use of a porous regime dictated by the properties of a Crater diameter (km) megaregolith, our model production function reproduces the mea- sured crater distributions in shape and in the absolute number ofFig. 2. Model production function of lunar craters, for 1 yr, in comparison with the craters formed over the past 3 Ga, under the assumption of a con-Hartmann and Neukum measured production functions, and the model production stant impact ﬂux. We caution that our simple formulation of thefunction of Marchi et al. (2009). Respective megaregolith thicknesses of 700 and250 m allow to ﬁt either the Neukum or Hartmann production functions in the porous/non-porous transition does not account for the temporaldiameter range 2–20 km. The thin dotted red curve is obtained by using only the evolution of the megaregolith and that the inferred megaregolithnon-porous scaling relation. thicknesses are only qualitative estimates.
M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20 7 The present Earth–Moon distance has been used in the above For illustrative purpose, Rc is shown for the inner planets in Fig. 3 bycalculation of the lunar cratering rate, and temporal variations in assuming that craters with diameters less than 10 km form in a por-the lunar semi-major axis could, in principle, modify the Earth/ ous soil on both the planet and Moon, while craters with greaterMoon impact ratio and encounter velocity distribution with the sizes form in solid rocks (except for the Earth and Venus where onlyMoon. Nevertheless, it is found in our simulations that, for a lunar the non-porous regime is used). Note that Rc can be easily calcu-semi-major axis as low as 20 Earth radii, the average lunar crater- lated from Eq. (18) and Table 2 for a different (and more realistic)ing rate is changed by less than 3%. This implies that both the transition between porous and non-porous regimes.shielding and gravitational focusing of projectiles by the Earth The mean impact velocity on the Moon is calculated to beare of very moderate effects, especially since the Moon is believed um ¼ 19:7 km sÀ1 . The full probability distribution of impact veloc- to have spent the vast majority of its history beyond 40 Earth radii. ities for each planet is shown in Fig. 4. The quantities /=/m ; u=um Our globally averaged planetary cratering rates Cð DÞ are ﬁttedby 10th-order polynomials for the Moon and inner planets. Thecoefﬁcients (with units of yrÀ1 km2) are listed in Table 2. Since 4the megaregolith thickness is not necessarily the same on each pla-net, and may depend on the age and geology of the surface, coefﬁ-cients are given for the two scaling regimes (T = 1 and T = 0 km)for diameters between 0.1 and 1000 km (except for the Earth and 3Venus, where only non-porous scaling is given). A linear transitionsimpler than ours can be used by deﬁning two threshold diame-ters, Dp and Dnp, such that the porous and non-porous regime ap- Mercury R c (D)plies alone respectively below Dp and above Dnp. The cratering Venusrate in the transition regime is then calculated as Cð DÞ ¼ 2 Earth C ðD ÞÀC ðD ÞC p ð Dp Þ þ np Dnp ÀDpp p ðD À Dp Þ, where Cp and Cnp are given in Ta- Mars npble 2 in the porous and non-porous columns, respectively. Notethat the martian cratering rate is sensitive to the eccentricity ofthe planet, since the number of potential impactors increases dra- 1matically as this planets gets closer to the Main Asteroid Belt (LeFeuvre and Wieczorek, 2008; Ivanov, 2001). In addition to calculat-ing the present day martian cratering rate, we also used the prob-ability distribution of the martian eccentricity provided by Laskar(2008) to calculate an average over the past 1 Ga (note that this va- 0 0.1 1 10 100 1000lue is nearly insensitive for longer averages). Planetary size-frequency distributions are generally expressed D (km)with respect to the lunar one. This is done by deﬁning the size- Fig. 3. Planetary cratering ratios with respect to the Moon, for craters larger than adependent quantity Rc, which is the cratering ratio with respect given diameter D. For illustrative purposes, craters with D 10 km and D 10 kmto the Moon, are respectively assumed to form in the porous and non-porous regimes. Curves are not shown for Venus and the Earth for D 10 km, since the porous regime is not Cð DÞ expected, and erosion or atmospheric shielding are known to be of signiﬁcantRc ð DÞ ¼ : ð18Þ C m ð DÞ inﬂuence at these sizes.Table 2 PPlanetary crater size-frequency distributions for 1 yr (kmÀ2): log Cð DÞ ¼ 10 C i ðlog Dkm Þi ; D 2 ½0:1—1000 km. i¼0 Moon Moon Mercury Mercury Venus Non-porous Porous Non-porous Porous Non-porous C0 À0.1049E+02 À0.1206E+02 À0.9939E+01 À0.1159E+02 À0.1073E+02 C1 À0.4106E+01 À0.3578E+01 À0.3994E+01 À0.3673E+01 À0.4024E+01 C2 À0.8715E+00 0.9917E+00 À0.1116E+01 0.9002E+00 À0.4503EÀ01 C3 0.1440E+01 0.7884E+00 0.1269E+01 0.9609E+00 0.1374E+01 C4 0.1000E+01 À0.5988E+00 0.1272E+01 À0.5239E+00 0.2433E+00 C5 À0.8733E+00 À0.2805E+00 À0.8276E+00 À0.3622E+00 À0.7040E+00 C6 À0.2725E+00 0.1665E+00 À0.3718E+00 0.1508E+00 À0.3962EÀ01 C7 0.2373E+00 0.3732EÀ01 0.2463E+00 0.5224EÀ01 0.1541E+00 C8 0.9500EÀ02 À0.1880EÀ01 0.2091EÀ01 À0.1843EÀ01 À0.8944EÀ02 C9 À0.2438EÀ01 À0.1529EÀ02 À0.2756EÀ01 À0.2510EÀ02 À0.1289EÀ01 C10 0.3430EÀ02 0.7058EÀ03 0.3659EÀ02 0.8053EÀ03 0.2102EÀ02 Earth Mars (long term) Mars (long term) Mars (today) Mars (today) Non-porous Non-porous Porous Non-porous Porous C0 À0.1099E+02 À0.1089E+02 À0.1213E+02 À0.1082E+02 À0.1207E+02 C1 À0.3996E+01 À0.4068E+01 À0.3124E+01 À0.4072E+01 À0.3134E+01 C2 0.2334E+00 0.2279E+00 0.1295E+01 0.2157E+00 0.1293E+01 C3 0.1333E+01 0.1422E+01 0.1542E+00 0.1426E+01 0.1713E+00 C4 0.2286EÀ02 0.2470EÀ01 À0.7519E+00 0.3330EÀ01 À0.7518E+00 C5 À0.6476E+00 À0.7150E+00 0.3125EÀ01 À0.7171E+00 0.2232EÀ01 C6 0.2875EÀ01 0.2056EÀ01 0.1779E+00 0.1856EÀ01 0.1786E+00 C7 0.1313E+00 0.1535E+00 À0.2311EÀ01 0.1540E+00 À0.2129EÀ01 C8 À0.1410EÀ01 À0.1526EÀ01 À0.1369EÀ01 À0.1510EÀ01 À0.1396EÀ01 C9 À0.1006EÀ01 À0.1252EÀ01 0.2623EÀ02 À0.1255EÀ01 0.2492EÀ02 C10 0.1810EÀ02 0.2247EÀ02 0.5521EÀ05 0.2246EÀ02 0.3212EÀ04
8 M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20 0.20 velocity is added to the projectile velocity for impacts at the apex, whereas it is subtracted at the antapex. The apex/antapex ratio is Mercury 1.37. We note that there is a negligible nearside/farside effect: Venus the nearside experiences the formation of about 0.1% more craters Earth that the farside. The Earth does indeed concentrate very low incli- 0.15 Moon nation (and moderate velocity) projectiles onto the lunar nearside, Mars but these are not numerous enough to inﬂuence the global distribution. Probability The lateral cratering variations depend on the crater size since 0.10 the size-frequency distribution of impactors s(d) is not a simple power law, and the impact conditions are not everywhere the same. Nevertheless, the maximum/minimum cratering ratio varies only by about 5% for D ranging from 30 m to 300 km. Consequently, 0.05 for the following discussion, we shall consider that DC(D, k, u) ’ DC(1, k, u). For smaller Earth–Moon distances am, the apex/antapex effect increases as the lunar velocity increases. Between 20 and 60 Earth radii, this dependency is ﬁt by the simple equation 0.00 10 20 30 40 50 60 70 80 90 am CðapexÞ=CðantapexÞ ¼ 1:12eÀ0:0529 Re þ 1:32; ð19Þ Impact velocity (km/s) where Re is the Earth radius. Over this range of Earth–Moon separa-Fig. 4. Probability distribution of impact velocities for the Moon and terrestrial tions, latitudinal variations and the nearside/farside effect are foundplanets. to vary by less than 1%. These calculations assume that the lunarand g/gm, that give the relative impact ﬂux, impact velocity and obliquity stayed equal to its present value in the past.surface gravity with respect to the Moon, are given in Table 3 for Fig. 6 plots the relative cratering rate DC as a function of thethe inner planets. Mars experiences a high impact rate with respect angular distance from the apex of motion for the present Earth–to the Moon (about 3) due to its proximity to the main asteroid Moon distance, and compares this with the counts of rayed cratersbelt. In comparison, the martian cratering ratio is reduced (be- with diameters greater than 5 km given in Morota et al. (2005).tween about 0.5 and 2.5) because the impact velocity on Mars is Rayed craters are younger than about 1 Ga (Wilhelms et al.,signiﬁcantly lower than on the Moon, requiring larger (and hence 1987), which should corresponds to an Earth–Moon separation dis-less numerous) impactors to create a crater of a given size. Mercury tance very close to the present one (Sonett and Chan, 1998; Eriks-exhibits also a high impact rate, and the impact velocity is about son and Simpson, 2000). As is seen, the model compares favorablytwice as large as on the Moon, resulting in a high value of the cra- to the data.tering ratio Rc, comprised between 2 and 4. The impact rate is lar- We note that the impact rate exhibits nearly the same behaviorger on the Earth and Venus than on the Moon, as these planets as the cratering rate, but with a reduced amplitude. The pole/equa-possess a higher gravitational attraction. Their higher surface grav- tor ratio is 0.90, whereas the apex/antapex ratio is 1.29. The latitu-ities compensate the differences in impact velocities with the dinal cratering variations are enhanced with respect to the impactMoon, and Rc is comprised between 0.5 and 1.5 for the Earth and rate variations, as the mean impact angle and impact velocity arebetween 1 and 2 for Venus. smaller at the poles than at the equator (respectively by 2.5° and 500 m/s), requiring a larger projectile to create the same crater size. As large projectiles are less numerous than small ones, the im-4.2. Spatial variations pact rate at the poles is smaller than the cratering rate (Le Feuvre and Wieczorek, 2008). This is also true for the apex/antapex asym- The relative spatial cratering variations on the Moon, metry, as the average impact velocity is 500 m/s higher at the apexDC(D, k, u), are shown in Fig. 5 for the present Earth–Moon dis- than at the antapex. However, the increase is moderate, as thetance of about 60 Earth radii, and for crater diameters larger than mean impact angle is only about 1.5° smaller.1 km. The cratering rate varies from approximately À20% to +25% It is seen in Fig. 7 that our predicted apex/antapex effect differswith respect to the global average. It is minimized at about from that of Zahnle et al. (2001). These authors describe their vari-(±65°N, 90°E), whereas the maximum, which is a factor 1.5 higher, ations of the impact rate as a function of c, the angular distance tois located at the apex of motion (0°N, 90°W). the apex, as Two effects conjugate to give such a distribution. First, a latitu- 0 12dinal effect, detailed in Le Feuvre and Wieczorek (2008), comes B vm Cfrom the higher proportion of low inclination asteroids associated D/ðcÞ ¼ @1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ cos cA ; ð20Þwith the higher probabilities of low inclination encounters. The 2v 2 þ V 2 mpole/equator ratio is 0.80. Second, a longitudinal apex/antapex ef-fect comes from the synchronous rotation of the Moon and the where V ’ 19 km sÀ1 is the mean encounter velocity with thehigher relative encounter velocities at the apex. The lunar orbital Earth–Moon system. We are able to reproduce their analytical solu-Table 3Impact rate, mean impact velocity and surface gravity for the inner planets, normalized to the Moon’s. Moon Mercury Venus Earth Mars (long term) Mars (today) Impact rate ratio 1 1.82 1.75 1.58 2.76 3.20 Mean velocity ratio 1 2.16 1.28 1.04 0.53 0.54 Surface gravity ratio 1 2.2 5.3 5.9 2.2 2.2
M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20 9 90° 60° 30° 180° 210° 240° 270° 300° 330° 0° 30° 60° 90° 120° 150° 180° −30° −60° −90° 0.85 0.90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 Relative cratering rateFig. 5. Relative cratering rate on the Moon for the current Earth–Moon distance and for crater diameters larger than 1 km. This image is plotted in a Mollweide projectioncentered on the sub-Earth point. 1.2 1.4 1.1 Relative impact rate Relative cratering rate 1.2 1 1.0 0.8 0.9 0.6 0 30 60 90 120 150 180 0 30 60 90 120 150 180 Angular distance from apex (º) Angular distance to apex (°)Fig. 6. Relative lunar cratering rate as a function of angular distance from the apexof motion. Rayed crater data are for crater diameters greater than 5 km from Morota Fig. 7. Relative lunar impact rate as a function of the angular distance from theet al. (2005) that were counted between approximately [70°E, 290°E] in longitude apex (solid black), in comparison with the equation from Zahnle et al. (2001) (solidand [À42°N, 42°N] in latitude. In comparison, the predicted apex/antapex cratering gray). The latter is reproduced if the model forces latitudinal isotropy of encountereffect is shown over the same count area (solid black) and for the entire Moon inclinations (dotted black).(dotted black). 2008). Spatial variations in the impact ﬂux and cratering rate are parameterized by a sum a spherical harmonic functionstion, but only under the condition where we force the encounterinclinations with respect to the lunar orbit plane to be isotropic in X X 1 lspace. These authors used Öpik equations (Shoemaker and Wolfe, DCðk; uÞ ¼ C lm Y lm ðk; uÞ; ð21Þ l¼0 m¼Àl1982) for hyperbolic orbits that were assumed to precess uniformlyinside the planet–Moon system. We nevertheless point out that where Ylm is the spherical harmonic function of degree l and orderZahnle et al. (2001) applied Eq. (20) to the moons of Jupiter, where m, Clm is the corresponding expansion coefﬁcient, and (k, u) repre-this approximation might be valid. sents position on the sphere in terms of latitude k and longitude We next provide analytical solutions for the relative variations u, respectively. The real spherical harmonics are deﬁned asof both the impact and cratering rates on the Moon. We also give (solutions for the latitudinal variations presented on the terrestrial Plm ðsin kÞ cos mu if m P 0 Y lm ðk; uÞ ¼ ð22Þplanets in Le Feuvre and Wieczorek (2008). Two values are quoted Pljmj ðsin kÞ sin jmju if m 0;for Mars, one that corresponds to its present obliquity and eccen-tricity and the other to averaged results using variations over and the corresponding unnormalized Legendre functions are listed3 Ga as given in Laskar et al. (2004) (see Le Feuvre and Wieczorek, in Table 4. Many of the expansion coefﬁcients are nearly zero since
10 M. Le Feuvre, M.A. Wieczorek / Icarus 214 (2011) 1–20Table 4 the following relationship between the number of craters withUnnormalized associated Legendre functions. diameters greater than 1 km and the age of the geologic unit: l, m Plm(sin k) NðD 1; tÞ ¼ aðebt À 1Þ þ ct; ð23Þ 0, 0 1 1, 1 cosk where NðD 1; tÞ is given per 106 km2, t is the age expressed in Ga, 2, 0 1 2 2 ð3 sin k À 1Þ and a = 5.44 Â 10À14, b = 6.93 and c = 8.38 Â 10À4. This relationship 2 2, 2 3 cos k 3, 1 3 2 is essentially linear over the last 3.3 Ga (constant cratering rate in 2 ð5 sin k À 1Þ cos k 4, 0 1 4 2 time) and approximately exponential beyond. The data used to con- 8 ð35 sin k À 30 sin k þ 3Þ struct this empirical curve are obtained from radiometric ages of the Apollo and Luna rock samples, compared to the crater density covering the associated geologic unit. We emphasize that no agreedTable 5 upon calibration data exist between 1 and 3 Ga and beyond 3.9 GaSpherical harmonic coefﬁcients of the lunar relative impact ﬂux D/(k, u) for Earth–Moon separations of 30, 45, and 60 Earth radii. (Stöfﬂer and Ryder, 2001). We also note that Eq. (23) was originally obtained using age estimates of the highland crust and Nectaris im- Clm 30 Earth radii 45 Earth radii 60 Earth radii pact basin, both of which are disputed and considerably older than C0,0 1 1 1 3.9 Ga. C1,À1 À1.7779020 Â 10À1 À1.4591830 Â 10À1 À1.2670400 Â 10À1 Accounting for our calculated spatial variations, we ﬁrst convert C2,0 À6.3209891 Â 10À2 À6.2923420 Â 10À2 À6.2755592 Â 10À2 the measured crater density at a given site, N(D 1, k, u), into the C2,2 À1.7937283 Â 10À3 À1.2524090 Â 10À3 À9.8998262 Â 10À4 C4,0 À9.9735381 Â 10À3 À1.0141450 Â 10À2 À1.0222370 Â 10À2 corresponding spatially averaged quantity: NðD 1Þ ¼ NðD 1; k; uÞ=DCðD 1; k; uÞ: ð24ÞTable 6 The cratering asymmetry DC is given in Table 2, and we use theSpherical harmonic coefﬁcients of the lunar relative cratering rate DC(D 1,k, u) for function that corresponds to the present Earth–Moon separationEarth–Moon separations of 30, 45, and 60 Earth radii. for cratered surfaces that are less than 1 Ga (consistent with the ti- Clm 30 Earth radii 45 Earth radii 60 Earth radii dal deposit data of Sonett and Chan (1998)). We choose a Earth– C0,0 1 1 1 Moon separation of 40 Earth radii for units that are older than C1,À1 À2.2715950 Â 10À1 À1.8571530 Â 10À1 À1.6092760 Â 10À1 3 Ga (based on the tidal deposits data of Eriksson and Simpson C2,0 À1.3954110 Â 10À1 À1.3874170 Â 10À1 À1.3831914 Â 10À1 (2000)). This lunar semi-major axis value corresponds to a lunar C2,2 À3.3499412 Â 10À3 À2.2729362 Â 10À3 À1.7822560 Â 10À3 orbital velocity twice as large as the present one. We further as- C3,À1 2.9040180 Â 10À3 2.3263713 Â 10À3 1.9948010 Â 10À3 C4,0 2.7412530 Â 10À3 2.7863080 Â 10À3 2.8083000 Â 10À3 sume that the lunar obliquity was equal to its present value (nearly zero) for the entire time between 3.9 Ga and the present. The data used to calibrate the crater density versus age relation- ship are listed in Table 9, along with their corrections accountingTable 7 for spatial variations in the cratering rate. We use the crater den-Spherical harmonic coefﬁcients of the relative impact ﬂux D/(k) for the terrestrialplanets. sity and ages values quoted in Stöfﬂer and Ryder (2001). We did not attempt to ﬁt crater distributions with our model production Planet C00 C20 C40 function to re-estimate N(D 1) values, since our model already Mercury 1 3.7395410 Â 10À2 À7.9170623 Â 10À3 reproduces very well the Neukum production function that was Venus 1 7.3546990 Â 10À3 À6.0267052 Â 10À3 used to estimate this quantity. For the case of very young calibra- Earth 1 À2.6165971 Â 10À2 À1.8682412 Â 10À3 tion surfaces, we suspect that a thinner megaregolith might change Mars (today) 1 1.6254980 Â 10À1 À1.0738801 Â 10À2 Mars (long-term average) 1 8.7425552 Â 10À2 4.7442493 Â 10À3 the crater distribution with respect to the Neukum production function at sizes larger than one kilometer, but, for small exposure times, the largest observed craters are below this diameter value. Based on the data and interpretations of Norman (2009), we ex-Table 8 clude the Nectaris basin from consideration, as it is possible thatSpherical harmonic coefﬁcients of the relative cratering rate DC(D 1, k) for the samples assigned to this basin have instead an Imbrium prove-terrestrial planets. nance. We thus assume that the Descartes formation is not Necta- Planet C00 C20 C40 ris ejecta, but rather Imbrium ejecta with an age of 3.85 Ga. We Mercury 1 4.5850560 Â 10À2 2.4749320 Â 10À3 also exclude the Crisium basin from the ﬁt, due to the uncertain Venus 1 À1.8545722 Â 10À3 3.5865970 Â 10À4 provenance of samples dated at the Luna 20 site. Finally, we use Earth 1 À7.6586370 Â 10À2 2.4353234 Â 10À4 the recent crater counts performed on Copernicus deposits by Mars (today) 1 3.3900970 Â 10À1 2.2655340 Â 10À3 Mars (long-term average) 1 1.7986312 Â 10À1 À4.3484250 Â 10À4 Hiesinger et al. (2010) with high-resolution Lunar Reconnaissance Orbiter images, which agree with a constant impact ﬂux during the last 800 Ma, in contrast to previous studies.the cratering rate is symmetric about both the equator and the axis As seen in Table 9, the spatial correction is in general moderate,connecting the apex and antapex of motion. Only the most signiﬁ- since the calibration terrains are located in the central portion ofcant coefﬁcients are listed in Tables 5 and 6, which for most cases the nearside hemisphere, far from the extrema of the spatially-reproduce the data to better than 0.2%. The coefﬁcients for the lat- dependent cratering rate. Nevertheless, we point out that after cor-itudinal variations on the terrestrial planets are listed in Tables 7 rection, the Apennines, Fra Mauro and Descartes formations, whichand 8. are all Imbrian in age, exhibit nearly the same globally averaged crater density, which is consistent with our assignment of an5. Crater chronology Imbrium origin to the Descartes formation, as recently suggested by Norman (2009) (see also Haskin et al., 1998). Neukum et al. (2001a) (see also Neukum, 1983; Strom and We perform a new ﬁt of the calibration point, using the sameNeukum, 1988; Neukum and Ivanov, 1994) established empirically functional form as in Eq. (23). Our resulting best parameters are