A new chronology for the Moon and Mercury Simone Marchi German Aerospace Center (DLR), Institute of Planetary Research, Rutherfordstr. 2, D-12489 Berlin Dipartimento di Astronomia, Universit`adi Padova, Vicolo dell’Osservatorio 2, I-35122 Padova [email protected] Stefano Mottola German Aerospace Center (DLR), Institute of Planetary Research, Rutherfordstr. 2, D-12489 Berlin Gabriele Cremonese INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova Matteo Massironi Dipartimento di Geoscienze, Universit`adi Padova, via Giotto 1, I-35137, Padova and Elena Martellato INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova ABSTRACT In this paper we present a new method for dating the surface of the Moon, obtained by modeling the incoming flux of impactors and converting it into a size distribution of resulting craters. We compare the results from this model with the standard chronology for the Moon showing their similarities and discrepancies. In particular, we find indications of a non-constant impactor flux in the last 500 Myr and also discuss the implications of our findings for the Late Heavy Bombardment hypothesis. We also show the potential of our model for accurate dating of other inner Solar System bodies, by applying it to Mercury. arXiv:0903.5137v1 [astro-ph.EP] 30 Mar 2009 Subject headings: solar system: general — planets and satellites: Earth, Mercury, Moon 1. Introduction the evolution of the Solar System and in particu- lar of our own planet, the Earth. Recently, thanks Craters are among the most spectacular surface to a fleet of new space missions (Mars Express features of the solid bodies of the Solar System. to Mars, MESSENGER to Mercury, and Kaguya Cratering studies provide a fundamental tool for to the Moon, only to name a few), this field of the age determination of planetary and asteroidal research entered a new exciting phase, where ac- terrains. Since the beginning of the lunar explo- curate age estimates provide means for detailed ration, age estimates for the lunar terrains were small-scales geological studies. derived, followed by chronology models for the The method currently used for dating purposes other terrestrial planets. The development of the defines a chronology (crater surface density vs. lunar chronology greatly helped in interpreting absolute age) for a reference body for which ra- 1 diometric ages are available for different terrains. impactor sizes and velocities for any body in the Then, on the basis of models predicting the im- inner Solar System. Thus, the lunar calibration pactor flux ratio between the reference body and can be exported with precision to any other body. another generic body, it is possible to estimate A drawback of the method, however, is that the the age of the latter. This method can be defined chronology of the reference body depends -through experimental, since it develops a chronology of a the scaling law- on the physics of the cratering reference body for which two measurable quanti- process, which is as yet not fully understood. Fur- ties are available: absolute ages and number of thermore, the precision of the age estimate relies craters for selected areas. The Moon is the only on the accuracy of the dynamical model. Encour- body of the Solar System that could fulfill both agingly, the adopted dynamical model is capable requirements so far. This method has been de- of providing a good representation of the present veloped and refined in the last 30 years by many asteroid population and size distribution both in researchers and represents the reference for dat- the near-Earth space and in the main belt, with ing purposes (e.g. Hartmann et al. 1981; Neukum a maximum deviation amounting to less than a 1983). In this approach, the development of the factor of 2 (Bottke et al. 2005b). chronology for the reference body does not depend explicitly on the physics of the cratering process. 2. Modeling the cratering on the Earth- However, such information becomes necessary -via Moon system a cratering scaling law- in order to apply the Moon chronology to other bodies or to infer the flux of The present inner Solar System is continu- impactors from the observed cratering record. ously reached by a flux of small bodies having A possible alternative -purely theoretical- ap- sizes smaller than few tens of km. A fraction proach could be based on the accurate estimation of these bodies (namely those having perihelion of the time-dependent impactor flux for each tar- < 1.3 AU) are called near-Earth objects (NEOs). get body, through the history of the Solar System. Contributors to this flux are represented by main With this method, a direct comparison between belt asteroids (MBAs) and Jupiter family comets the observed distribution of craters on a given ter- (JFCs). Both contributions have been modeled rain and the isochrones produced by the model by a number of authors and it is generally ac- would give its age. Although the knowledge of cepted that the MBAs constitute the main source the formation and evolution of the Solar System for the flux presently observed in the inner Solar greatly improved in recent years, such ambitious System (e.g. Morbidelli et al. 2002), therefore we goal is presently far beyond the capabilities of the neglect the cometary contribution. This flux is available models, due to the large uncertainties in sustained by a few fast escape tracks, which are the early stages of its formation. continuously replenished with new bodies as a re- In this paper we introduce a third approach for sult of collisional processes and semimajor axis determining the chronology of objects in the inner mobility. The main gateways to the inner Solar Solar System. This method, which can be con- System are the ν6 secular resonance with Saturn sidered as a hybrid of the first two approaches, and the 3:1 mean motion resonance with Jupiter is based on the dynamical model by Bottke et al. (Morbidelli & Gladman 1998; Morbidelli et al. (2000, 2002, 2005a,b) which describes the forma- 2002). One of the most recent and accurate models tion and evolution of asteroids in the inner Solar concerning NEOs formation has been developed System. In this framework, first the flux of im- by Bottke et al. (2000, 2002), while Bottke et al. pactors on the Moon is derived, then the impactor (2005a,b) modeled the main belt dynamical evo- size distribution is converted into a cumulative lution. These models have been adopted in order crater distribution via an appropriate scaling law, to estimate the properties of the impactor flux at and finally the radiometric ages of different regions the Moon. -the Apollo and Luna landing sites- are used for the calibration of the lunar chronology. The main advantage of our approach is that the adopted model naturally describes the distributions of the 2 2.1. Modeling the impactor flux: the sta- The impactor size distribution for the Earth (he) tionary case can be written as follows: The flux of impactors may be described as a differential distribution, φ(d, v), which represents he(d)= Pehn(d) (2) the number of incoming bodies per unit of im- where hn is the NEO differential size distri- pactor size (d) and impact velocity (v). An im- bution and Pe is the intrinsic collision probabil- portant quantity -since it can be constrained from ity. Following Bottke et al. (2005b), we adopted observations- is the size distribution of the incom- −9 −1 Pe =2.8 10 yr . The distribution hn has been ing bodies, namely the number of impactors per derived from· Bottke et al. (2002), and it can be unit impactor size. Let h(d) be such distribution, scaled to the Moon and other bodies. In the case we then have: h(d) φ(d, v) dv. Therefore, ≡ of the Moon, this is done considering the Earth- without loss of generality,R we can write: Moon ratio of the impact probability as a function of impactor size (Marchi et al. 2005). Let Σ(d) be φ(d, v)= h(d)f(d, v) (1) this ratio, then the Moon impactor size distribu- tion becomes: where f(d, v) is the distribution of the impact velocities (i.e. the impact probability per unit im- pact velocity) for any dimension d, and which ful- hm(d)= Pehn(d)Σ(d) (3) fills the normalization constraint f(d, v) dv 1. ≡ In figure 2 the cumulative impactor size dis- We used the NEO orbital distributionR model com- tribution for both the Moon and the Earth are puted by Bottke et al. (2000, 2002), generated by shown. A similar approach can also be used for integrating two sets of test particles placed into the evaluating the flux on other targets, like Mercury ν and 3:1 resonances (with 2 106 and 7 105 par- 6 × × (see section 6). In equation 1 the dependence of h ticles, respectively). For each particle they com- and f over time is neglected since we are consider- puted the impact probability and the impact ve- ing a stationary problem, while eq. 2 and 3 strictly locity relative to the target body. Following the describe the present flux. Arguments about possi- work of Marchi et al. (2005), we computed the im- ble variations in the flux intensity and shape have pact velocity distribution (corrected for the gravi- been discussed since the late 70s, and this still re- tational cross-section of the targets). We assumed mains one of the most debated topics in the field that the Moon revolves around the Sun with an of lunar cratering. In their work, Neukum and orbit identical to that of the Earth.