Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://star.arm.ac.uk/preprints/2014/660.pdf Äàòà èçìåíåíèÿ: Thu Nov 20 17:09:11 2014 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 02:41:48 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: ï ï ï ï ï ï ï ï ï ï ï ï

The meteoroid environment and impacts on Phobos
Apostolos A. Christoua,, Jurgen Oberstb,c , Valery Lupovkad , Vasily ¨ Dmitrievd , Maria Gritsevichg,f,e
Armagh Observatory, Col lege Hil l, Armagh BT61 9DG, Northern Ireland, UK German Aerospace Centre, Institute of Planetary Research, Rutherfordstrasse 2, D-12489 Berlin, Germany c Technische Universitat Berlin, Institute for Geodesy and Geoinformation Science, ¨ Planetary Geodesy, Strasse des 17. Juni 135, 10623 Berlin, Germany Moscow State University for Geodesy and Cartography, Extraterrestrial Laboratory, 4, Gorokchovsky pereulok, 105064, Moscow, Russia e Russian Academy of Sciences, Dorodnicyn Computer Centre, Department of Computational Physics, Vavilova ul. 40, 119333, Moscow, Russia f Lomonosov Moscow State University, Institute of Mechanics, Laboratory of General Aerodynamics, Michurinskii prt. 1, 119192, Moscow, Russia g Finnish Geodetic Institute, Department of Remote Sensing and Photogrammetry, Geodeetinrinne 2, P. O. Box 15, FI-02431 Masala, Finland
b a

d

Abstract We review current knowledge of the flux of meteoroids on Phobos, a key to interpreting its cratering record and understanding the origin of the Martian satellite system. Past observational attempts to estimate the flux of small (mm to cm) meteoroids as meteors in the Martian atmosphere highlight the need for customised instrumentation onboard future missions bound for Mars. The temporal distribution of cometary meteoroid streams as predicted by recent work is non-uniform; we advo cate emplacing seismic stations on Phobos or cameras optimised to monitor the Martian atmosphere for meteors
Corresponding author Email addresses: aac@arm.ac.uk (Apostolos A. Christou), Fax: +44 2837 522928 (Apostolos A. Christou), juergen.oberst@dlr.de (Jurgen Oberst), ¨ v.lupovka@miigaik.ru (Valery Lupovka), v.dmitriev@miigaik.ru (Vasily Dmitriev), gritsevich@list.ru (Maria Gritsevich)


Preprint submitted to Planetary and Space Science

July 11, 2013


as a means to elucidate this and other features of the meteoroid population. We construct a mo del of the sporadic flux of metre-sized or larger meteoroids and use it to predict a leading/trailing ratio in crater density of 4 if crater pro duction by asteroidal meteoroids dominates over that by cometary ones. It is found that the observed distribution of craters 100 m as determined from spacecraft images is consistent with a 50/50 contribution from the two meteoroid populations in our mo del. A need for more complete mo dels of the meteoroid flux is identified. Finally, the prospects for new observational constraints on the meteoroid environment are reviewed. Keywords: Meteoroids, Phobos, Impact Cratering, Comets, Asteroids 1. Introduction As with every other bo dy in the solar system, the Martian system is subject to a continuous influx of meteoroids ranging in size from under a µm to several km. Their effect on the Martian atmosphere is to produce ionisation layers (P¨tzold et al., 2005) and meteors (Adolfsson et al., 1996) with a the larger ones reaching the Martian surface and creating impact craters, crater clusters or meteorites (Flynn and McKay, 1989; Popova et al., 2003; Chappelow and Sharpton, 2006). Impacts of "large" (metre-sized or larger) meteoroids on the airless surface of Phobos is the primary pro cess for producing the craters seen in spacecraft images. In addition, seismic shaking and the pro duction of ejecta contribute to the displacement and transport of surface material. On the other hand, impacts by "small" (decimetre or smaller) meteoroids can launch ejecta in circum-martian space, contributing to the dust environment in the orbital vicinity of this mo on (eg Krivov and

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Hamilton, 1997). Consequently, as can be read in the relevant chapters of this volume, an independent determination of the meteoroid flux is needed to compare with mo dels of the pro duction rate of craters and boulders on Phobos, the impact flux on Mars itself now and in the past as well as the dust torus that is postulated by several studies. A study of the Phobos meteoroid environment also has value in itself. It represents an opportunity to learn more about the population of meteoroids which do not intersect the Earth's orbit. The physical properties of meteoroids causing meteors in the Earth's atmosphere have been studied theoretically in a number of works (Lebedinets, 1987; Babadzhanov, 1994; Kikwaya et al., 2006). According to the classical physical theory of meteors, these are considered to be solid bo dies similar to meteorites of stony/iron composition with bulk densities ranging from 3.5 g cm-3 to 7.7 g cm
-3

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(Levin, 1956).

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Lebedinets (1987) and Babadzhanov (1994) obtained an average bulk density of 3.3 g cm-3 with values for individual cases in the range from 0.1 g cm-
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to 8.0 g cm-3 . Bellot Rubio et al. (2002) gave even lower estimates ranging from 0.1 g cm
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to 4.5 g cm-3 . In contrast, a wide range of the laboratory

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measurements for meteorites, micrometeorites and interplanetary dust particles did not confirm such a low limit for the bulk density. A selection effect may be at work here since fragile, low density material is likely to ablate and consume itself high in the Earth's atmosphere rather than land as a meteorite. The same is true for the Martian atmosphere (Christou and Beurle, 1999; McAuliffe, 2006) but not for airless Phobos where such ob jects would impact directly onto the surface. This work is concerned with the meteoroid environment and impact effects

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at Phobos, particularly crater pro duction. In the Section that follows we review past attempts to constrain the meteoroid flux in the Martian system while in Section 3 we consider the particular question of the existence and detectability of meteoroid streams. In Sections 4 we present a mo del of the "sporadic" (i.e. non-stream) flux of large meteoroids at Phobos. This is used to predict relative crater abundances over its surface through the pro cedure described in Section 5. Section 6 describes the results of this mo del and a comparison with observationally determined crater counts from in situ spacecraft data. Finally, Section 7 contains our main conclusions and outlines prospects for future development in the field. 2. Observations to-date No direct in situ measurements of the meteoroid flux at Mars exist todate. Since meteoroids obey a size distribution, the flux of particles larger than a few tens of microns across is to o low to be picked up by contemporary instrumentation for dust detection in situ. Rather, such particles can be detected as meteors in the Martian atmosphere (Adolfsson et al., 1996; McAuliffe and Christou, 2006). Adolfsson et al. (1996) estimated the flux of meteoroids pro ducing so-called "photographic" meteors - ie in the absolute visual magnitude range -1m - +4m - at Mars to be 50% of that at the Earth. Domokos et al. (2007) concluded that the negative result of their meteor search in nighttime images taken by the MER rovers, which they translated into an upper flux limit of 4.4 â 10 km-2 hr
-1 -6 -2 -1

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limit of 4g, was broadly consistent with the Adolfsson et al prediction of 4 â 10
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derived by their scaling of the Earth flux according 4


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to the Grun et al. (1985) mo del. No imaging system suitable for the routine ¨ detection of meteors in the Martian atmosphere has yet flown but many of the key technologies are now available (Oberst et al., 2011). A detection of a meteor asso ciated with a known Jupiter-family comet by the dual-eye Pancam imager onboard the Spirit rover on Mars was claimed by Selsis et al. (2005). Later work by Domokos et al. (2007) quantified the effects of cosmic ray hits (CRHs) on the Spirit Pancam as part of a dedicated meteor search and placed the 2005 detection in doubt. 3. Streams Many cometary meteoroids remain close to the orbit of the parent comet for up to 105 yr forming dense streams (McIntosh, 1991). These create meteor showers in the atmospheres of the Earth, where they dominate the bright (+0m - -4m) meteor flux (Hughes, 1987; Atreya and Christou, 2009). The ma jority of observational and theoretical work to-date concerns this class of meteoroids. Their orbits retain a memory of their birth, with the result that, if observed and recorded, they can be asso ciated with their specific comet of origin. In addition, the high particle density within trails, which results in shortlived outbursts (of duration a few hr) in meteor activity when these trails encounter the Earth's atmosphere (eg McNaught and Asher, 1999), pro duce a proportionate increase in meteoroid flux on the surfaces of airless bo dies. For example, the flux corresponding to a Leonid-type meteor storm translates into 400 meteoroids 1 mm in size impacting the surface of a bo dy with the dimensions of Phobos every hr (Christou and Beurle, 1999) exceeding the 5

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sporadic flux of same-sized ob jects at the Earth by 4 orders of magnitude. In the absence of observations, arguments for the existence of Marsintersecting meteoroid streams have been primarily based on orbital geometry considerations (Terentjeva, 1993; Christou and Beurle, 1999; Treiman and Treiman, 2000; Larson, 2001; Selsis et al., 2004; Neslusan, 2005; Jenniskens, 2006). Christou (2010) re-examined the problem of meteor shower parentho o d in light of the tendency for strong meteor showers at the Earth to be asso ciated with certain types of comets (eg Halley-type) rather than others. He identified 19 Intermediate Long Perio d (ILPCs) and Halley Type Comets (HTCs) that are potentially responsible for meteor showers at Mars. In addition, that author found that most of the meteoroid streams asso ciated with Encke-type comets and responsible for shower activity at the Earth are likely to also intersect the orbit of Mars. Expected characteristics of these showers such as speed and radiant lo cation - but not their intensity or particle population properties - can be determined in a straightforward way from the planet-approaching geometry and are illustrated in Fig. 1. Filled circles correspond to Halley-Type (HTCs) and Intermediate Long Perio d Comets (ILPCs) while open circles correspond to Encke-Type Comets (ETCs). The size of the circle is proportional to the impact speed in the Martian atmosphere while the colour indicates either a radiant near the Sun (bright red), along the terminator (dark red) or in the anti-Sun direction (black). The lo cation corresponding to LS = 0 is indicated by the grey arrow. It is noted that the temporal distribution of these putative streams is non-uniform with twice as many stream encounters o ccurring during the perio d 180 LS 3 6 0 .

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Christou and Vaubaillon (2011) refined the results of Christou by simu-

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lating numerically these meteoroid streams. They found that in 9 cases, a significant fraction of test meteoroids encounter the orbit of Mars and can pro duce annually-recurring showers, the intensity of which remains to be calibrated through observations. Potential meteoroid flux enhancements will o ccur as the Martian system passes through a stream. This may be detected by e.g. emplacing seismic stations on Phobos as was done previously for the Mo on (Oberst and Nakamura, 1991) or through the meteors pro duced in the Martian atmosphere (Adolfsson et al., 1996; Christou and Beurle, 1999; Christou et al., 2012). Their effect, if any, on the density and distribution of circum-martian dust is another open question. 4. A model of the sporadic meteoroid flux at Phobos Asteroidal and cometary meteoroids that have suffered significant dynamical evolution after ejection form the sporadic continuum (or sporadic background). Recently, Wiegert et al. (2009) pro duced a self-consistent mo del of the sporadic meteoroid complex at the orbit of the Earth and concluded that (a) most sporadic meteoroids are of cometary origin and (b) the bulk of the sporadic flux is provided by only a fraction of the comet population. No similar mo del exists for Mars, partly because there is no data on meteoroid flux (e.g. from meteor observations) that such a mo del can be fitted to. It can, however, be fitted to Phobos crater abundances. In what follows we describe a relatively simple mo del of the flux of crater-pro ducing meteoroids at Phobos that we use to predict the relative density of craters over different areas of its surface. The mo del is simple in that it assumes that the mete7

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oroid flux is solely due to the known Mars-approaching asteroids and comets serving as the parent bo dies. It do es not take into account the orbital evolution of either the parent bo dies or their meteoroids. Although the mo del encompasses, by definition, both the stream and sporadic flux we assume, as is the case for the Earth (Jones and Brown, 1993; Brown et al., 2008), that the meteoroid flux within streams is a minor contributor to the overall flux. Finally, we note that the crater-pro ducing meteoroids we consider here are metre-sized or larger as opposed to the sub-mm - cm meteoroids considered in the work by Wiegert et al. They are, however, generally smaller than the ob jects considered in studies of the impact flux and crater pro duction on the Mo on (Gallant et al., 2009; Ito and Malhotra, 2010; Le Feuvre and Wieczorewk, 2011). Non-gravitational forces such as Poynting-Robertson drag as included in the work by Wiegert et al. will be ineffective in mo difying the orbits of the meteoroids considered in the present work. Instead, the Yarkovsky effect, not included in our mo del, will likely result in significant orbital changes over time (Farinella et al., 1998). 4.1. The Cometary component From a database of 1037 perio dic comets within JPL's HORIZONS service (http://ssd.jpl.nasa.gov/?horizons) we found 137 comets in orbits that approach Mars' orbit to within 0.15AU. Of those, 54 are Jupiter Family Comets (JFCs) and the remainder are Long Perio d Comets (LPCs). Both types have been included as they are thought to contribute to different components of the sporadic flux at the Earth. The meteoroid flux for each of these comets at Mars was calculated using the following assumptions:

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1. The orbits of the comets change very slowly over time; the theoretical stream radiant positions are therefore fixed. 2. The density of meteoroid orbits in the stream tube is distributed symmetrically with respect to the parent comet orbit, and maximum flux o ccurs at the comet orbit. 3. Meteoroids are uniformly distributed along the stream orbit, and their distribution do es not depend on the position of the parent comet. Specifically, the cumulative particle flux is mo delled as an exponentially decaying function of the distance from the stream centre (Dmitriev et al., 2012a,b). 4.2. The Asteroidal component Asteroids on Mars-approaching orbits are another potentially important contributor to the Phobos meteoroid environment due to Mars' proximity to the asteroid belt. Using HORIZONS, we identified 5957 asteroids in orbits that approach Mars' orbit to within 0.1 AU. This sample is composed of Apollo and Amor group asteroids, as well as inner main belt members. Mo delling of asteroid impact probability was performed using Opik's metho d (Opik, 1976). Formulas from Chesley et al. (2002) were used in order to convert asteroid magnitudes into diameters and masses. The number of asteroids with H> 18 suffers from significant observational bias; this was estimated from the distribution of brighter asteroids by extrapolating the Hartmann function (Ivanov et al., 2002) applicable to this latter population.

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5. Model of crater production and distribution For cometary impactors, the cumulative impactor flux was calculated for 60 days before and 60 days after the orbit-crossing point with a time step of one day. The number of potential impacts during this time interval is then
t1

N =S

Phob t0

(r )dt

(1 )

where t0 and t1 correspond to the boundaries of the aforementioned 120day perio d, S
Phob

being the surface area of Phobos' disk. For asteroids the

probability used for the random generation of impacts was calculated after Opik (1976): p=
2 fS U 2 + 0.625e2 + sin2 i0 0

2 sin2 i + sin2 i0

1/2

2 (Ux + 0.5e2 ) 0

1/2

(2 )

where s = (2/3) s, s2 = q 2 (1 - 2m/q U 2 ) and a is the orbital helio centric semima jor axis of the impactor in units of that of Mars. Impact directions and velo cities were calculated in a Phobos-centric, Phobos-fixed co ordinate system.The velo city V of the impactor relative to Phobos is given by V =V where V
impactor impactor

where f = L/ with L given by the expression a(1-e)-s -(1-e2 ) 0 if a > 1, (a(1-e)-s )e0 cos L = (1-e2 )-s -a(1+e) 0 if a < 1 (a(1+e)+s )e0

(3 )

-V

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P hobos

(4 )

and V

M ar s

are the inertial helio centric velo cities of the comet/asteroid
P hobos

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is that mo on's orbital velo city relative to

Mars. Applying a uniform random-event generator on the disk perpendicular to the arrival velo city vector (Press et al., 2007) followed by orthographic 10


pro jection onto a sphere pro duces the impact lo cation of meteoroids in the inertial frame. These are then transformed into a Phobos-fixed co ordinate system using the IAU-recommended rotation mo del for that moon (Archinal and 15 co-authors, 2009). At the same time, the effect of meteoroid screening by Mars is calculated and - for the slower asteroidal impactors only - the effect of Mars' gravitational attraction is taken into account. The crater size is estimated using the formula by Bottke et al. (2000): Dc = 1.25(M /)
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1/3

1.61g d/V

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(5 )

where Dc is the crater diameter, M the meteoroid mass, d the meteoroid diameter, g and the gravity acceleration and density at Phobos surface respectively, Vn the component of the impact velo city that is normal to Phobos surface and an experimentally determined constant. The values assumed in this work are: g = 5.1 â 10
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m s-2 and = 1900 kg m-3 . For we used

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the value 0.22 for competent ro ck or saturated soil from Richardson et al. (2005). The meteoroid diameter is calculated assuming a density of 1200 kg m-3 for the cometary case and 2000 kg m-3 for the asteroidal one. 6. Results of the sporadic flux model The results of the mo del described in Section 4 for both asteroidal and cometary sources were used by the pro cedure of Section 5 to generate expected relative densities of craters 100m in size. These are shown as percentages of the total for the entire surface of Phobos in columns 2-5 of Table 1. Column 6 shows observed relative crater densities on Phobos (see paper by Kharachevtseva et al in this issue) for a completeness diameter limit 11

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of 100 m. In what follows we discuss some features observed in the resulting mo del distributions and compare it with the observed crater densities. A potential consequence of Phobos' proximity to Mars is a decrease of the impactor flux on the Mars-facing hemisphere of Phobos due to screening by the planet. In our mo del such a feature exists at the levels of 11% and 2% for the mo del cometary and asteroidal particles respectively. The difference is likely due to the lower degree of gravitational fo cusing for the faster-moving cometary particles (see also later discussion on velo city). Also, both northern and southern hemispheres of Phobos are expected to receive the same number of impacts to within 10%. Interestingly, cometary meteoroids in our mo del - mainly those from Jupiter-family comets - appear to pro duce 30% more craters near the equator than near the poles while this trend is reversed for asteroidal meteoroids. The cause of this feature is not immediately obvious since the typical orbital inclinations for the two populations are similar. We believe that a partial explanation may be found in the perihelion distance distributions of the mo del impactors. Fig. 2 shows these distributions with the perihelion and aphelion distances of Mars indicated by the red bars. Asteroidal and JFC perihelia mainly o ccur near Mars' distance from the sun. In the case of the highly eccentric JFC orbits, the helio centric velo city at pericentre is high enough that the velo city relative to Mars is dominated by the ecliptic component. For the asteroidal case however, perihelic velo cities of these low-to-mo derate eccentricity orbits, are similar to the orbital velo city of Mars and the small - in absolute terms - vertical component of the relative velocity becomes significant compared to the ecliptic component. Consequently, asteroidal

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impactors will appear to approach Mars from high ecliptic latitudes. Irrespective of the cause of the feature in the mo del population, there is no clear evidence for it in the observed crater abundances. A higher by 25% - crater abundance is observed near the South pole relative to the equatorial region but not in the North. Its origin is not clear but may be related to the presence of large craters such as Stickney rather than e.g. a higher proportion of asteroidal impactors on Phobos. Probably the most significant feature in our mo del is the difference between the number of mo del craters on the eastern and western hemispheres of Phobos. These correspond to the regions of its surface that trail and lead respectively in Phobos' motion around Mars; we will also make use of the notation "leading" and trailing" in what follows. The number of cometary meteoroids is 5% larger on the western hemisphere but for asteroidal meteoroids this difference amounts to 45%. In fact, as Table 1 shows, Phobos' leading quadrant - in other words, the centre half of its western hemisphere should receive four times as many asteroidal impacts as its trailing quadrant. A clue as to the cause of this asymmetry may be found in the velocity distribution of the impactors (Fig. 3). The ma jority of cometary impacts o ccur at velo cities ranging from 20 to 50 km sec ulation and 15 to 30 km sec
-1 -1

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other hand, asteroidal impacts o ccur at velo cities between 2 to 5 km sec-1 , comparable to the orbital velo city of Phobos around Mars (2.1 km sec-1 ) as well as the escape velo city at Phobos' distance from Mars (3 km sec-1 ). Under the circumstances applicable to asteroidal impacts, a satellite in a synchronous lo ck with its primary will receive a significantly higher flux of

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impactors on its leading side than on the trailing side (see Kawamura et al., 2011, and references therein). Following Gallant et al. (2009) (see also Zahnle et al., 2001) this asymmetry can be expressed as ¯ ( ) = (1 + cos )g

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(6 )

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where is the elongation angle from the apex of Phobos' orbital motion ¯ around Mars and represents the crater density at = 90 . The quantity is given by the ratio between vorb , Phobos' orbital velo city, and vimp , the velo city of the pro jectile with hyperbolic velo city v at the areo centric distance of that mo on, in other words = vorb /
2 2 2vorb + v . The exponent g

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depends on the size distribution of the impactors. Here we adopt the value g = 2.81 also used by Gallant et al. when investigating the flux asymmetry on the Earth's Mo on. If we adopt typical v values for LPC (JFC) impactors of 40 (20) km sec
-1

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respectively (assuming v vimp if either is much larger than the escape ¯ velo city), we obtain relative crater densities (i.e. /) of 1.12 (1.24) and 0.89 (0.79) for = 0 and = 180 respectively or apex-antapex differences of 7% and 14% respectively. Adopting < v >= 4 km sec
-1

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for asteroidal

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impacts yields values of 2.15 and 0.30, a difference by a factor of 7 between apex and antapex crater density. The statistical output of our mo del (Table 1) for the Leading and Trailing sectors predicts a difference of 7% for both LPCs and JFCs and a factor of 4 difference for asteroidal impactors. As can be seen in column 6 of Table 1, the observed crater densities on the leading and trailing sides of Phobos show an asymmetry of a factor of 2 that, if taken at face value, can be interpreted as a vindication of 14

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our mo del for an approximately equal ratio of cometary/asteroidal impacts (actually 43/57). We prefer to treat this result with some caution. Some of the reasons, to be addressed in future work, include: (i) our mo del do es not provide us with a self-consistent steady state, either in the Mars-approaching population of impactors or the pro duction of craters (ii) consideration of the efficiency of meteoroid delivery from different sources was not done as rigorously as in previous works dealing with the sporadic flux at the Earth (iii) the observed craters likely formed > 1 Gyr ago when Phobos' orbit would have been wider and its orbital velo city lower owing to tidal evolution (iv) Phobos' shape (including its departure from sphericity) may result in preferential shielding of some areas over others and thus skew the statistics, and (v) the possibility that Phobos broke its rotational lo ck to Mars when one or more of its craters formed. Nevertheless, our results demonstrate the potential of mo dels of this type for constraining the relative contributions to the impact flux in the Martian system from different source populations when combined with accurate crater inventories for Phobos that are now available. 7. Conclusions and Future Prospects Despite the current intense scientific interest on Mars, direct measurements of its meteoroid environment have not been forthcoming. In terms of Phobos science, it would be important to feed these into mo dels that attempt to repro duce the main features of its crater population as well as predictions of its dust torus. In the foreseeable future, some progress is expected through the ongoing detection campaign of new impact craters caused by large me15

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teoroids on Mars by the HiRISE camera onboard the Mars Reconnaissance Orbiter (Daubar et al., 2011) and the detection of impacts by a seismometer to be placed on Mars in 2016 as part of the InSight mission (Lognonne and InSight SEIS VBB Team, 2012). The possibility and value of coordinated observations has also been recognised (Daubar et al., 2012). The detection of smaller meteoroids, on the other hand, must await the arrival of the first meteor camera in the vicinity of Mars (Christou et al., 2012). Acknowledgements This paper has been prepared following the 7th European Strategic Meteor workshop sponsored by the Europlanet Research Infrastructure (RI) and held at the Moscow State University of Geo desy and Cartography in 5-6 July 2012. This work has been supported by Grant 11.G34.31.0021 from the Ministry of Education and Science of the Russian Federation. MG was supported by the Academy of Finland. Astronomical research at the Armagh Observatory is funded by the Northern Ireland Department of Culture, Arts and Leisure (DCAL). References Adolfsson, L., Gustafson, B. A. S., Murray, C. D., 1996. The Martian atmosphere as a meteoroid detector. Icarus 119, 144­152. Archinal, B. A., 15 co-authors, 2009. Division I-III / Working Group Cartographic Co ordinates and Rotational Elements. Trans. IAU 4, 68.

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List of Figures 1 Top: Areo centric latitudes referenced to the Martian equator as a function of LS for the radiants of the shower candidates given in Christou (2010). Bottom: Lo cation of the shower candidates along the Martian orbit. See text for details. . . . . 26 2 Perihelion distance distribution for the sample of Long Perio d comets (top), Jupiter Family comets (middle) and asteroids (bottom) used in constructing the sporadic meteoroid flux mo del as explained in the text. . . . . . . . . . . . . . . . . . 27 3 Impact velo city distribution for the sample of Long Perio d comets (top), Jupiter Family comets (middle) and asteroids (bottom) used in constructing the sporadic meteoroid flux mo del as explained in the text. . . . . . . . . . . . . . . . . . 28

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Table 1: Statistics of model and observed meteoroid impact craters on Phobos.

Lo cation on Phobos Sub-Mars (-45 < < 45 ) Trailing (45 < < 135) Anti-Mars (135 < < 225 ) Leading (225 < < 315 ) North Polar ( > 30 ) South Polar ( < -30 ) South Equatorial (-30 < < 0 ) North Equatorial (0 < < 30 ) Northern Hemisphere Southern Hemisphere Western Hemisphere Eastern Hemisphere

Long Perio d Comets 23.1±1.3 23.4±1.3 26.1±1.4 27.4±1.5 25.4±1.4 25.3±1.4 24.2±1.4 25.2±1.4 50.6±2.0 49.4±1.9 52.7±2.0 47.4±1.9

Jupiter Family Comets 21.7±1.3 23.9±1.4 26.7±1.4 27.7±1.5 19.5±1.2 21.7±1.3 28.9±1.5 29.0±1.5 48.5±1.9 51.5±2.0 52.7±2.0 47.3±1.9

All Comets Asteroids 21.4±1.3 22.5±1.3 24.2±1.4 10.8±0.9 26.6±1.4 24.4±1.4 27.8±1.5 42.3±1.8 19.5±1.2 28.7±1.5 21.7±1.3 26.1±1.4 29.9±1.5 23.3±1.3 28.9±1.5 21.9±1.3 48.4±1.9 50.6±2.0 51.6±1.9 49.4±1.9 52.7±2.0 72.5±2.5 47.3±1.9 27.5±1.5

Observed crater Abundance 26.0 15.9 24.9 33.2 22.6 30.1 24.9 22.4 47.3 52.7 57.3 42.7

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