A D Morero

H Procopiou

R Vargiolu

J Johns

H Zahouani

Carving techniques of Fatimid rock crystal ewers (10-12th cent

Keywords: Wear traces, Wavelength, Archaeology, Ancient technology, Rock crystal

Fatimid art is known for the production of luxury artifacts, particularly rock crystal vessels. The appearance in 2008 of the Francis Mills Ewer, which seemed to belong to a famous group of 6-8 rock crystal ewers attributed to Fatimid Egypt, prompted an investigation of the techniques used to carve them. A comparison of the carving technique of the Francis Mills Ewer with that of the other members of the group offers the best criterion for determining whether the new ewer belongs to this group. To this end, the traces of manufacture (mainly polishing and carving) were analysed on a group of fourteen artifacts. The topography of the surfaces has been measured with a confocal rugosimeter using silicon replicas. To identify and characterise the multi-scale wear signature and the traces left by the tools, a Fourier isotropic filtering technique was applied. Using these complementary methods, we were able to confirm that the Francis Mills Ewer belongs to the Fatimid group.

Introduction 1.The Fatimid rock crystal ewers

The Fatimid caliphs of Egypt (969-1173 A.D.) are well known for their patronage of luxury and prestige arts and particularly for the production of vessels in one of the harder and most difficult stones to work: rock crystal (7 on the Mohs hardness scale). Amongst these is an especially famous group of pearshaped vessels -now called the 'Magnificent Seven' -two of which bear Arabic inscriptions indicating that they were produced for the Fatimid court in Egypt in circa 1000 A.D. [1]. These seven masterpieces are plausibly thought to have been manufactured in Cairo in circa 1000 A.D., using similar methods of production. They have certain features in common, including their shape (piriform or globular), their size (H. 15.6-24 cm, D. ranging from 9.5-13.5 cm) and the organisation in three panels of the carved relief decoration (depicting animals and vegetal scrolling and stem work -'palmettetree'). Finally all the ewers were carved from a single block of rock crystal, including the integral handle.

The problem of the authentication of the Francis Mills Ewer

When the ewer (Fig. 1) appeared on the art market in 2008, its authenticity was questioned. Although doubts were initially allayed by the historical and stylistic arguments carefully advanced in the Christie's sale catalogue entry [1], further questions were subsequently raised when it emerged that both the Francis Mills Ewer, and the ewer in the Victoria and Albert Museum which it most closely resembles, were the only two ewers to be first attested in the mid 19th century and had both passed through the hands of the London dealer, John Webb, who is known to have 'improved' and even faked antiquities, including rock crystal objects [START_REF] Wainwright | The making of the South Kensington museum IV. Relationships with the trade: Webb and Bardini[END_REF]. Moreover, Webb had occasionally done so with the assistance of the Parisian jeweller Jean-Valentin Morel, who had added the enamelled gold mounts to the Francis Mills Ewer, and is celebrated for having revived the lapidary's art in 19th century France [START_REF] Lucas | Jean-Valentin Morel and the revival of the Lapidary's art[END_REF]. As already suggested, the ewer presents a number of features in common with the rest of the group (carved from a single block, dimensions, shape, style and organisation of the relief decoration, etc.). While the ewer was attributed to the Fatimid group on stylistic grounds, and was shown not to be a forgery on historical grounds, it remained highly desirable to test the authenticity of the new ewer by comparing the techniques of its manufacture with those of indisputably genuine Fatimid pieces.

The ancient techniques: wear traces identification

The manufacturing processes of Fatimid rock crystal vessels remain largely unknown. No tools or workshops of medieval Islamic rock crystal carving industries have so far been reported in Fatimid Egypt or elsewhere. Valuable information on the tools and techniques that might have been used by Fatimid craftsmen is provided by the sparse references in medieval Arabic sources (al-Bīırūnī, al-Tifāshī, etc.), and by the more circumstantial and detailed accounts of early modern and modern European scholars and travellers in Asia (e.g. Bernier, de Thevenot, Fryer [START_REF] Talboys-Wheller | European travellers in India. First series. India: past and present[END_REF]). Studies of traditional industries of Chinese jade carving [START_REF] Hansford | Chinese Carved Jades[END_REF] and Indian carnelian manufacture [6] are particularly useful. We have also used data obtained during our previous research on stone vase manufacturing techniques in the East Mediterranean Bronze Age [7,[START_REF] Morero | Artisanat lapidaire en Cr!ete Minoenne, les techniques de fabrication des vases en pierre[END_REF]. Indeed, despite variations over space and time, the ancient and traditional tools and techniques for hard stone carving have a wide range of common features. This study will focus on carving and polishing techniques.

The first tool to be used was certainly the saw. Probably made of steel, saws were employed with a mixture of water and abrasive for different tasks during the preliminary steps of manufacturing, such as the initial shaping of the raw crystal into the rough form of the finished vessel [START_REF] Hoyland | Medieval swords and swordmaking: kindi's treatise on swords and their kinds[END_REF]10]. Smaller bowsaws and files were also used for decoration in China [START_REF] Sax | The emperor's terrapin[END_REF]. The lathe was used from the Bronze Age in the Near East for glyptic production [START_REF] Gwinnett | Ancient lapidary. a study using scanning electron microscopy and functional analysis[END_REF], and also for gem and soft-stone vase manufacture in the Greek and Roman periods [START_REF] Boardman | Greek and Finger Rings: Early Bronze Age to Late Classical[END_REF]. A series of different tools could be attached to the horizontal axis of a lathe, which was certainly driven by a bow (Fig. 2a) [START_REF] Skelton | The relations between the Chinese and Indian jade carving traditions[END_REF].

Among these rotary tools, cutting discs, drill bits of various shapes, and grinding wheels (Fig. 2b), were presumably used as vectors for the abrasive powder that actually did the cutting, which was facilitated by a lubricant. They were employed for the shaping of the object and also, according to their size, for carving the decoration and finishing the surfaces. Cutting was done by discs of different diameters. Different sizes of grinding wheels were employed for grinding down the irregularities and polishing the surface. Decoration was cut using smaller discs of thin sheet steel [START_REF]The 'Magnificent Seven' comprise: (1) the Francis Mills Ewer, (Christie's. Art of the Islamic and Indian Worlds. Sale 7615[END_REF]. In addition, both Chinese and Indian craftsmen used a range of tubular drills and gouges for hollowing the vessel, and a variety of drill bits including diamond points, which seem to have been in use since the 1st Millennium BC in India for beadmaking [START_REF] Kenoyer | A new look at stone drills of the Indus Valley tradition[END_REF]. The tools employed for the traditional working of hard stone were generally made in soft steel or iron [START_REF] Sax | Methods of engraving Mesopotamian cylinder seals: experimental confirmation[END_REF] used in conjunction with an abrasive powder made of hard stone such as diamond, garnet (Mohs 7.5) and especially corundum (Mohs 9 -a naturally occurring crystalline form of aluminium oxide) and emery [18]. Such abrasives were used in the shaping and cutting processes; a finer and less effective abrasive must have been employed for final polishing [START_REF] Procopiou | The role of tactile perception during stone-polishing in Aegean prehistory (5th-4th millennium)[END_REF]. A powder such as hematite (Mohs 5.5-6.5, the mineral form of ferric oxide) might have been used [20]. Today, to polish quartz, some workshops use an ultra fine powder, such as jeweller's or tin putty, also known as putty powder, which consists chiefly of stannic oxide, mixed with water and applied by a tin wheel. A lubricant was used to contain the abrasive particles during rotation of the tool, and to ensure even distribution of the abrasive on the surface being cut or polished. The lubricant also served to remove loose particles of rock, thereby making the abrasive process more efficient. The viscosity of the lubricant is an important factor in the manufacturing process. In the Fatimid period, vegetable oil (olive or poorer quality 'lamp oils' such as sesame, turnip seed, radish seed etc.), mixed with water in different proportions, might well have been used as a lubricant.

Methodology

The Archaeometrical Study of the Early Islamic Rock Crystal Industry is based on an interdisciplinary research project. Archaeological, ethnographic, and historical sources provided us with an initial database of the tools and techniques that might have been used by early Islamic craftsmen. But artifacts such as the Fatimid ewers are fragile, rare, and of great artistic and monetary value, and therefore cannot readily be moved from their present locations, so we have adopted for this study the non-destructive method of taking impressions of details of the surfaces of the artifacts with Silflos (Flexico Developments Limited), a light bodied silicone rubber impression material with a standardised viscosity, which flows freely into the carved and polished recesses and, when removed, preserves its original shape. The traces of manufacture recorded on these silicone rubber impressions were then described and measured in the laboratory.

Archaeological samples

A sample of eleven artifacts was analysed for which the dates of production are either certain or generally accepted. This first assemblage consists of medieval Islamic vessels (mainly ovoidal and cylindrical flasks and jars) from the Edmund de Unger Collection and the Museum für Islamische Kunst (Berlin), and the ewer in the Victoria and Albert Museum (London). The second group includes a vase with cover in the form of a dragon ridden by an amorino, attributed to Jean-Valentin Morel (Victoria and Albert Museum, Fig. 3), and an imitation of Fatimid ewer, possibly made in Iran in the late 19 th or early 20 th century, and now held in a private collection (Table 1).

The traces of manufacture recovered from the Francis Mills Ewer were compared with both those obtained from the medieval objects and those observed on the vase attributed to Morel and the modern imitation. It was hoped that this would permit us both to establish whether or not the tools and techniques used for the Francis Mills Ewer were consistent with those used to produce the other medieval pieces, and to determine whether or not the Francis Mills Ewer bore traces of manufacture found on the vase attributed to Morel and on the modern imitation.

Different scales of observation, from macroscopic to microscopic analyses of wear traces

The manufacturing processes leave traces on the object's surface that may be observed at different scales. Some traces are visible by the naked eye, but other microscopic traces, measuring no more than a few nanometres, require high magnification. These wear patterns are indicative of the processes of carving and polishing, and of the craftsman's tools and techniques. Attention was focused on the reconstruction of techniques used to create the external relief decoration of the objects, which was done mainly by the creation of raised 'islets' into which the ornament was carved, and by different phases of polishing. In particular, we concentrated upon the techniques used to create a range of motifs common to all or most of the objects examined-including palmettes, leaf-scrolls, multiple circular dots, and the line-and-dot motif. After macroscopic examination of the artifact, a digital microscope was used to examine the surface, and to identify those abrading and cutting traces that were most important or representative of the object under examination. During this initial examination, close attention was paid to the organisation of the traces of production and, in particular, to the overlapping or superimposition of one set of traces over another, so as to reconstruct in part the sequence of the process of manufacture.

To further characterize the traces of manufacture, we measured surface topography with a confocal rugosimeter. The principle of this optical device is based on the chromatic aberration, which focuses different wavelengths of white light at different points. Only the wavelength focused on the surface to be measured passes through a filter located in front of the spectrometer while other wavelengths are removed. The optical system is associated with a confocal microscope, which selects the wavelengths focused on the surface to be measured. The detector consists of a spectrometer that analyses the wavelength with the highest intensity. Knowledge of the relationship between the wavelength and the distance of the focal point permits the measurement of roughness. In order to analyse different motifs with a good lateral resolution, we measured 10 x 10 mm 2 surfaces with a resolution of 4 mm. The surface of the motifs is composed of two families of wavelengths, medium and small. Medium wavelengths give information about the motifs, small wavelengths about their To identify wear traces of manufacture it is important to analyse small wavelengths. To that end, we applied the Fourier isotrope filtering technique [START_REF] Zahouani | Spectral and 3D motifs identification of anisotropic components: analysis and filtering of anisotropic patterns by morphological approach[END_REF][START_REF] Zahouani | The morphological tree transform of surface motifs. Incidence in tribology[END_REF][START_REF] Zahouani | Filtrage tridimensionnel des surfaces rugueuses[END_REF], which enabled us to obtain three-dimensional images z (x, y) of the wear traces, and to produce a detailed picture of the traces of the tools used by the craftsman. The morphology of the striations and carved relief could also be observed, as well as the profiles of the different traces, at different wavelengths. These precise measurements and detailed 2D and 3D views of the traces of manufacture allowed us to develop a series of working hypotheses on the tools and techniques used by the Fatimid rock crystal craftsmen.

Hypotheses and experimental reconstitution

In order to pursue further our research on the reconstruction of Fatimid rock crystal production, experiments will be designed to test our working hypotheses concerning ancient tools and techniques. The traces of manufacture observed on the Fatimid objects will be compared with the traces produced by experimental reconstruction. At this early stage, only a few preliminary experiments have been conducted, both for the carving process using a motorised mini-drill fitted with a steel disc and diamond powder, and for the polishing process, using corundum, clay, water and olive oil. We have also started the construction of a bow-lathe.

Results: the attribution of the Francis Mills Ewer to the group of Fatimid ewers

We demonstrated that the traces of manufacture observed on the Francis Mills Ewer were identical to those observed on the other Fatimid objects examined. same techniques and tools were employed for creation of the relief 'islets', for cutting incised lines, and for drilling ornamental dots. However, markedly different tools and techniques were used to carve both the vase attributed to Morel (Fig. 3) and the modern imitation of a Fatimid ewer (Table 1).

Preliminary identification of Fatimid tools and techniques

Decoration of the exterior took place after the hollowing of the interior, and after preliminary shaping of the outer surface. As a first step, the craftsman cut the areas of raised relief or 'islets' into which the ornament would subsequently be carved, presumably following a pattern drawn onto the blank exterior of the vessel. Several tools were certainly used in this first stage. One type of trace is characterised by parallel striations of different sizes, which suggest the use of grinding wheels with vertical sides. At the same time, however, the angular shape of the 'islets' indicates that cutting discs, with an acute angled edge, of different sizes and shapes were also used to produce the islets, just as in Chinese traditional jade workshops. The abrasive used, at this stage of our work, is difficult to identify but we suggest the use of a coarse abrasive such as corundum or diamond such as it was used in ancient workshops in Mesopotamia and China [START_REF] Sax | Methods of engraving Mesopotamian cylinder seals: experimental confirmation[END_REF]18]. As to the carving of the detailed decoration, the cutting disc was used to create both straight and curved lines. Regardless of the desired width and length of the incision, the disc seems always to have been the same. Indeed, we observed cutting of similar shape and thickness on different Fatimid artifacts. Thus, to produce a narrow, straight line, the disc was applied to the surface of the object once and for a short time. Whereas, to produce a wide line, the same disc was passed several times along the same groove, enlarging the cutting trace. Similarly, to produce a curved line, the disc was applied repeatedly to cut a succession of lines at different angles, which ultimately formed a composite curved line (Fig. 4a,b).We found that for the entire Fatimid group, including the Francis Mills Ewer, the cutting discs always had a diameter of 2.3 cm-3.2 cm, had blades with a U-sectioned edge, and were always applied vertically to the surface of the object (Fig. 4c).

The diameter of the circular dots (Fig. 5), in the series of dots used to represent the fur of the cheetahs on the Francis Mills Ewer, and in other composite motifs, such as the line-and-dot pattern in the palmette-trees or the dots at the centre of some leaf-spirals, varies in size on the same object from 1-2 mm and larger 4.2 mm. However, the dots on the Fatimid group are always hemispherical, rather than conical, in section and have the same characteristics, suggesting that they were made by the same type of drill bits, albeit with different diameters. In many cases, especially on the ewer and ovoidal flask (R11) in the de Unger Collection, the base of the cavity is a horizontal plateau. Moreover, the dots are not always precisely circular in plan, but can be oval or even completely irregular. These observations suggest that the dots were made by a rotating drill bit fixed to a horizontal lathe, and that the surface of the rock crystal was applied to the drill bit by the craftsman. The drill bit itself was not precisely spherical but rather slightly ovoidal in section. Over prolonged use, the ovoidal point of the bit was flattened by wear, producing the horizontal plateau at the base of some cavities. The Fatimid craftsmen apparently presented the rock crystal at different angles to the drill bit. When applied perpendicular to the surface, the bit produced a dot that was perfectly circular in plan, but when the crystal was presented at an angle to the drill, the dots were oval or irregular in plan, and of shallower depth. It should be noted, however, that we have observed in the laboratory that dots with the irregular outer contours might, in some cases, be caused by lateral scaling, which took place after drilling and final polishing. It is suggested that, after the decoration was carved with the cutting disc, the contours of complex curves were occasionally regularised and finished by hand, although traces of this process appear to be scarce and relatively isolated. The deep grooves and straight striations at the base of the incisions appear to be produced by the cutting disc that was used to make a preliminary but very angular and irregular curved line. Subsequently, to harmonise and regularise the contours of the curve, the craftsman used a freehand tool, such as a diamond or corundum point mounted on a rod, which left a series of irregular lines incised or scratches in the bottom of grooves. These are different from the traces of cutting made by a disc, in that a curved, apparently continuous line can be produced in a single operation.

Preliminary polishing of the outer surface may well have taken place on several occasions during production but, at the end of the process, the crystal was polished for the last time in order to restore its natural transparency. For this final polishing, we suggest the use of a soft abrasive powder such as haematite or stannic oxide mixed with water or, more probably, with oil which added more gloss to the surface. This gives a perfectly transparent, glossy surface. The striations from the passage of the abrasive particles on the surface are almost invisible to the naked eye. Magnified under a digital microscope (x 50), fine crossing lines appear that are the product of the successive polishing sequences. This final work was certainly done by a specialised craftsman, as was the case in the traditional workshops of India and China. Indeed, the relatively homogeneity of the traces of production so far observed on the Fatimid group suggests that only a few highly skilled craftsmen in specialised workshops may have produced these luxury objects.

The modern imitation and the vase attributed to J.-V. Morel

When compared with the traces of manufacture on vessels from the Fatimid group, the incisions on the modern imitation are much straighter and more regular (Fig. 6a). This suggests that the tool was a steel cutting disc mounted on a modern, possibly motor-driven, rotary system, and that the abrasive was diamond powder. Moreover, the diameter of the cutting disc would seem to have been much smaller (<1 cm) than that used by the Fatimid craftsman (2.3 cm-3.2 cm). Again, the drill bit used to make dots on the modern imitation left very different traces (Fig. 5c) to the bits used on the Fatimid group (Fig. 5a,b). The diameter of the dots is smaller and the section of the cavity more sharply angled, almost conical. Moreover, while the bases of the dots on the Fatimid group are smooth, those on the modern imitation display clear striations, indicating that not only a different type of bit was used, but also a coarser abrasive. Again, it is likely that the drill was powered by a modern, perhaps motorised, device. Finally, the modern craftsman used a particular technique to finish and homogenise all of the lines of the incised ornament. This has left a continuous line in the centre of the base of all of the incised lines, even the curved ones (Fig. 6a). Exactly how such a line could have been made is yet to determine, but its constant and regular appearance is not a feature of the Fatimid group.

As to the vase attributed to J.-V. Morel, the cutting technique that he used for the decoration was identical to that used by the Fatimid craftsman, to the extent that he used cutting discs driven by a rotary lathe. However, the were applied in a very different manner, so that the disc was applied at an acute angle to the surface of the crystal (Fig. 6c), instead of perpendicularly (Fig. 6b), leading to very different effect. Moreover, the section of the edge of the disc used by Morel is sharply V-shaped, and very different from the U-shaped section of the blade of the Fatimid cutting disk.

Conclusion

Using these complementary methods, we were able to confirm that the traces of manufacture observed on the Francis Mills Ewer are wholly consistent with those present on the Fatimid group of rock crystal objects. Moreover, they are distinctly different from the traces observed on both a modern imitation and the vase attributed to Morel. On this basis, we conclude that the Francis Mills Ewer is exclusively the product of Fatimid craftsmanship, and that Morel's contribution was confined to the addition of the enamelled gold mounts; he did not 'improve' the carved ornament or polish, nor did he himself carve the decoration. As to the modern imitation, the traces of carving present on this vessel are markedly different from those common to the Fatimid group. More importantly for future research, we have established the utility and validity of the methods employed as a first step towards the reconstruction of the tools and techniques used in the medieval Islamic rock crystal industry.

Fig. 1 .

 1 Fig. 1. (a) The Francis Mills Ewer and (b) detail of hunting cheetah.

Fig. 2 .

 2 Fig. 2. (a) Lapidary using a bow-lathe to polish a gemstone. From an album of the Mughal emperor Jahāngīr, dated 1610-11 AD. (Prague, Náprstek Museum, MS A.12182, f. 16); and (b) tool kit from a traditional jade-carving workshop in Peking (Hansford [5], Figure 11, p. 81).

Fig. 3 .

 3 Fig. 3. Vase attributed to Morel, Victoria and Albert Museum, inv. no. 1428-1870.

Fig. 4 .

 4 Fig. 4. (a) Cutting disc traces for the creation of curved lines on the jar (R9) in the de Unger Collection; (b) traces showing further cutting by disc to make a curved line on the ovoidal flask (R11), in the de Unger Collection; and (c) 3D view and profile of the cutting disc traces on the Francis Mills Ewer.

Fig. 4 .

 4 Fig. 4. (a) Cutting disc traces for the creation of curved lines on the jar (R9) in the de Unger Collection; (b) traces showing further cutting by disc to make a curved line on the ovoidal flask (R11), in the de Unger Collection; and (c) 3D view and profile of the cutting disc traces on the Francis Mills Ewer.

Fig. 5 .

 5 Fig. 5. 3D views and profiles of the dots on (a) the ovoidal flask (R11) in the de Unger Collection; (b) the Francis Mills Ewer; and (c) the modern imitation.

Fig. 6 .

 6 Fig. 6. (a) Views of the carving traces on the modern imitation. A continuous line appears on the base of all the grooves; (b) 3D view and profile of the cutting traces on the Francis Mills Ewer showing the perpendicular angle of the U-shaped disc to the surface; and (c) 3D view and profile of the cutting traces on the vase attributed to Morel showing the oblique angle of the V-shaped disc to the surface.

Table 1 .

 1 Groups of vessels analysed.

		Group 1	Fatimid or		Group 2	Fatimid or
			early Islamic			early Islamic
	de Unger	Ovoidal flask (R11)	Yes	Private	Imitation of Fatimid ewer	No
	Collection	Jar with 19th century Italian mounts (R9)	Yes	collection
		Toilet flask (R3)	Yes
		Cylindrical flask (R7)	Yes
	V & A	The ewer (inv. no. 7904-1862)	Yes	V & A	J.-V. Morel's dragon ridden	No
	Museum	Ovoidal flask (inv. no. 975-1050)	Yes	Museum	by an amorino
		3 Cylindrical flasks (inv. nos. A.11-1942,	Yes		(inv. no. 1428-1870, Fig. 3)
		A.45-1928, A.46-1928)

Acknowledgements

This study is part of the project An Archaeometrical Study of the Early Islamic Rock Crystal Industry: Phase One, directed by J. Johns and E. Morero, based at the Khalili Research Centre, University of Oxford. We would like to thank Ranros Universal SA for supporting this research, and Dr Stefan Weber and the staff of the Museum für Islamische Kunst (Berlin) and Dr. Mariam Rosser-Owen and the staff of the Victoria and Albert Museum (London) for granting us access to the rock crystal objects in their collections. We also thank Mr. Secretan, a lapidary and jeweller in Lyon, France, for information concerning modern jewellery techniques.


S Deboeuf

P Gondret

M Rabaud

Dynamics of grain ejection by sphere impact on a granular bed

Keywords: numbers: 45.70.-n, 45.50.-j, 83.80.Fg, 96.15.Qr

The dynamics of grain ejection consecutive to a sphere impacting a granular material is investigated experimentally and the variations of the characteristics of grain ejection with the control parameters are quantitatively studied. The time evolution of the corona formed by the ejected grains is reported, mainly in terms of its diameter and height, and favourably compared with a simple ballistic model. A key characteristic of the granular corona is that the angle formed by its edge with the horizontal granular surface remains constant during the ejection process, which again can be reproduced by the ballistic model. The number and the kinetic energy of the ejected grains is evaluated and allows for the calculation of an effective restitution coefficient characterizing the complex collision process between the impacting sphere and the fine granular target. The effective restitution coefficient is found to be constant when varying the control parameters.

I. INTRODUCTION

Impact cratering has been recognized as an important geologic process for the last decades when the lunar craters have been finally attributed to impact structures rather than giant volcanoes as believed until 1950's [START_REF] Melosh | Impact cratering: A geologic process[END_REF]. The planetary impact craters such as the ones observed commonly on the Moon or the Earth result from very high energy impacts of meteorites and thus involve numerous and very complex phenomena such as shock and rarefaction wave propagation, melt and vaporization of the projectile and target materials, together with excavation by displacement and ejection of the target material [START_REF] Melosh | Impact cratering: A geologic process[END_REF]. In light of evidences of the discrete and sandy nature of planet surface, laboratory scale experiments of high energy impacts on granular materials were conducted [START_REF] Cook | Impact cratering in granular materials[END_REF][START_REF] Yamamoto | Transient crater growth in granular targets: An experimental study of low velocity impacts into glass sphere targets[END_REF]. Since a few years, physicists have conducted laboratory scale experiments with rather low impact energy on granular matter, interesting in the crater morphology and searching for scaling laws for the crater size [START_REF] Walsh | Morphology and scaling of impact craters in granular media[END_REF][START_REF] Uehara | Low-speed impact craters in loose granular media[END_REF][START_REF] Zheng | Impact craters in loose granular media[END_REF][START_REF] Vet | Shape of impact craters in granular media[END_REF]. Even though their energies are typically many orders of magnitude smaller than those of meteorite impacts, these small scale experiments on granular impacts may be relevant to planetary impact processes, as the progression of crater morphologies as a function of impact energy has been shown to mirror that seen in lunar craters [START_REF] Walsh | Morphology and scaling of impact craters in granular media[END_REF]. In these impact experiments, physicists have also been interested in the penetration of the impacting sphere in the granular target [START_REF] De Bruyn | Penetration of spheres into loose granular media[END_REF][START_REF] Ciamarra | Dynamics of drag and force distributions for projectile impact in a granular medium[END_REF][START_REF] Ambroso | Penetration depth for shallow impact cratering[END_REF][START_REF] Hou | Dynamics of a projectile penetrating in granular systems[END_REF][START_REF] Katsuragi | Unified force law for granular impact cratering[END_REF][START_REF] Seguin | Influence of confinement on granular penetration by impact[END_REF][START_REF] Goldman | Scaling and dynamics of sphere and disk impact into granular media[END_REF]. Indeed, despite recent progress on the complex rheology of granular matter [START_REF] Midi | On dense granular flows[END_REF], the penetration dynamics of a solid sphere into a granular medium is still difficult to understand well as it involves both the complex drag resulting from frictional and collisional processes, and the final stop involving the complex "liquid/solid" transition exhibited by granular matter [START_REF] Mills | Rheology and structure of granular materials near the jamming transition[END_REF]. The penetration dynamics of the impacting sphere and the grain ejection have been shown to be very different when the granular material is not dense but loose: A spectacular thin granular jet can raise very high after the impact as first demonstrated in Refs [START_REF] Cook | Impact cratering in granular materials[END_REF][START_REF] Thoroddsen | Granular jets[END_REF][START_REF] Mikkelsen | Granular Eruptions: Void Collapse and Jet Formation[END_REF]. The effects on this granular jet of the interstitial fluid [START_REF] Royer | Gasmediated impact dynamics in fine-grained granular materials[END_REF][START_REF] Caballero | Role of Air in Granular Jet Formation[END_REF] and of the initial packing fraction of the target [START_REF] Marston | Effect of packing fraction on granular jetting from solid sphere entry into aerated and fluidized beds[END_REF] have then been studied. In the dense case, no granular jet but a growing granular corona is seen after the impact. These different kinds of grain ejection can be related to similar kinds of liquid ejection consecutive to the impact of a droplet into a deep or thin layer of liquid that have been first filmed by [START_REF] Worthington | A study of splashes[END_REF] and then studied extensively [START_REF] Fedorchenko | The formation and dynamics of a blob on free and wall sheets induced by a drop impact on surfaces[END_REF][START_REF] Yarin | Drop impact dynamics: Splashing, spreading, receding, bouncing[END_REF]. Much less studies focusing on the grain ejection have been performed in the dense granular case: Ogale et al. [START_REF] Ogale | Impact-induced splash and spill in a quasi-confined granular medium[END_REF] have measured the mass of the spilled-over grains and Boudet et al. [START_REF] Boudet | Dynamics of impact cratering in shallow sand layers[END_REF] have proposed a model of ejection from the crater growth in a layer of thickness small compared to the size of both projectile and target grains at low impact velocities (∼ 1 m/s), while Cintala et al. [START_REF] Cintala | Ejection-velocity distribution from impact into coarse-grained sand[END_REF] measured ejection speeds and angles of grains at high impact velocities (∼ 1 km/s). Another type of experiments concern the impact of one bead with a granular target made of the same beads [START_REF] Beladjine | Collision process between an incident bead and a threedimensional granular packing[END_REF]. Such impacts and grain ejections by an impacting projectile have been recently simulated in limiting cases, for which the impact energy is very low or very high [START_REF] Tsimring | Modeling of impact cratering in granular media[END_REF][START_REF] Wada | Numerical simulation of impact cratering on granular material[END_REF] or for which the projectile size is about the grain size [START_REF] Oger | Discrete element method studies of the collision of one rapid sphere on 2D and 3D packings[END_REF][START_REF] Crassous | Impact of a projectile on a granular medium described by a collision model[END_REF][START_REF] Bourrier | Physical processes within a 2D granuler layer during an impact[END_REF]. We focus in the present paper on the dynamics of the granular corona formed by the ejected grains from a dense and deep pile upon low speed (∼ 1 m/s) impacts. In section II, the experimental setup is described together with the measurements. A simple ballistic model is then presented in section III and compared to the experimental data. The results are discussed in section IV and section V ends this paper with a conclusion.

II. EXPERIMENTS

Each experimental run consists of dropping a solid sphere into a granular medium. Four different steel spheres of density ρ s =7800 kg/m 3 are used as impactors, with different radius R = 5.15, 6.75, 7.55, 9.50 mm and masses M ranging thus from 4.5 g to 30 g. The steel sphere is initially hold by a magnet through a semi-spherical hole, so that the sphere can be dropped without any translational nor rotational velocity by pulling up the magnet. The sphere is released directly above the center of a container and falls along its axis. Being dropped from the height h above the granular surface which is varied between 8 cm and 60 cm, the sphere thus impacts the granular material with the velocity U c varying from 1 m/s to 4 m/s and energy E c = M gh ranging from 3.10 -3 J to 2.10 -1 J. The target material consists in sieved glass beads of density ρ g = 2500 kg/m 3 (ρ s /ρ g = 3.1) and mean diameter 2r = 0.4 ± 0.1 mm (R/r ∼ 25 -50), thus of mass m = 8.10 -5 g (M/m ∼ 10 4 -10 5 ). The granular material fills the cylindrical container of diameter 19 cm and height 26 cm. The size ratio of the container diameter over the sphere diameter is always larger than 10, so that there is no influence of the radial confinement by the lateral walls of the container neither of the bottom wall [START_REF] Seguin | Influence of confinement on granular penetration by impact[END_REF] as the height of the packing is large. Before each drop, the granular medium is prepared by gently stirring the grains with a thin rod. The container is then overfilled and the surface levelled using a straightedge. The typical value of the solid volume fraction of the packing is 60%. Each impact experiment is lighted from the front to enhance contrast between the grains and the black background, and side view images are recorded by a high speed camera at the rate of 500 images/s and resolution of 0.16 mm/pixel. The 256 gray level images are thresholded to identify grains. Figure 1 shows a sequence of side view images separated by 30 ms illustrating the ejection dynamics of the grains after the sphere impact defined as the time t = 0. One can see few isolated ejected grains above a dense corona of grains that expands radially and vertically. The amount of ejected grains can be quantified by measuring the apparent surface area A tot of all the grains in each image. The evolution of A tot as a function of time is reported in Fig. 2 for several experiments corresponding to different dropping heights h of the same sphere. After the impact at t = 0 defined as the first contact between the bottom of the sphere and the granular surface, A tot increases up to a maximal value denoted A tot max at the time t Atot max before decreasing. Each curve corresponds to a single impact experiment, thus without any ensemble averaging, but with a little smoothing by a slide-average over a time window of 10 ms so that no data appears before t = 4 ms in the reported figures. Increasing the dropping height h of a given sphere, the values A tot max and t Atot max increase, accounting for the increase of both the number of ejected grains and the dynamics duration. The same goes when keeping constant the dropping height h and increasing the mass M of the impacting sphere. It is worthnoting that the exact relation between the measured apparent surface area A tot of the ejected grains and the real total number of ejected grains N ej is not straightforward. For dilute zones of ejected grains, all the grains can be seen in the 2D images, but there is still the problem of focusing: A far grain appears smaller than a close grain so that the grains do not have the same apparent area. The most problematic case concerns however the dense zones where some grains can be hidden by other grains. Using the fact that the ejection process is axisymmetric, the number of ejected grains N ej is related to the measured apparent surface area A tot by:

N ej = nwπA tot max , (1)
where w is the unknown corona thickness and n the unknown number of grains per unit volume. The unknown product nw will be deduced in the following by compar- ing our scaling laws for the grain ejection with scaling laws already known in the literature for the crater size.

In impact experiments of a drop onto a liquid layer, a beautiful corona is classically observed [START_REF] Worthington | A study of splashes[END_REF][START_REF] Yarin | Drop impact dynamics: Splashing, spreading, receding, bouncing[END_REF]. In such a liquid case, the corona is easy to define and extract as the liquid is a continuous medium. In the granular case, the corona is less easy to define as the ejected grains are individual entities. The granular corona is here defined as the largest connected part of grains in each side view image. We have checked that the size of the corona does not depend significantly on the lighting and contrast of the images. To investigate the time evolution of the corona and characterize its shape which is basically axisymmetrical, its contour is extracted for each image. An example of such a contour is drawn in Fig. 3. Note that this corresponds to the external contour of the ejecta curtain. This allows to define the bottom diameter D b and the top diameter D t as the minimal and maximal diameter of the corona. The height H of the corona is measured as the distance between the mean vertical position of the corona top contour and the granular surface level before impact. As the corona lateral edge appears quite straight except in a small zone at the base of the corona, we extract also the angle θ formed by the corona edge with the horizontal, by a linear fit of the straight portion. The time evolution of the corona, in terms of its height H, its top and bottom diameters D t and D b , and its edge slope θ, is displayed in Fig. 4 for the same experiments as in Fig. 2. The expansion of the corona is demonstrated in Fig. 4a by the increase of its height H up to a maximal value denoted H max at time t Hmax before decreasing. For a given sphere, H max and time t Hmax increase monotically with the dropping height h, so that the different curves of Fig. 4a appear in order. Note that for large impact energies (large impact heights h), H decreases to a significant non-zero final value, because of the final crater rims lying above the initial free surface [START_REF] Walsh | Morphology and scaling of impact craters in granular media[END_REF]. In the same time, the top diameter D t and the bottom diameter D b increase with time, as shown in Fig. 4b andc, up to their maximal values, D t max and D b max , when the corona disappears and its height vanishes. The evolutions of D t and D b are different: About linear for D t but parabolic for D b . Besides, D t increases significantly with the dropping height h whereas D b does not vary so much. Note that at the nearly end of the corona life, for vanishing height H, the values of D t and D b become very noisy and are thus not shown here. The angle θ the corona edge forms with the horizontal, shown in Fig. 4d, is roughly constant as a function of time and whatever the dropping height h, and equal to about 56 • with relative variations of ∼ 5%. The same kind of evolutions of all these parameters are observed for the different tested spheres. The corona evolution reported here for granular impacts can be compared to the corona evolution for the liquid case [START_REF] Yarin | Drop impact dynamics: Splashing, spreading, receding, bouncing[END_REF]. For the liquid case, the corona base spreads radially as the square root of time whereas it seems more close to a quadratic evolution for the granular case (see D b (t) in Fig. 4c). The scaling laws for spreading is thus different in the granular and liquid cases, with inverse curvatures in the D b (t) plot. Besides, the angle of the granular corona is found independent of the impact velocities, as found for the liquid corona, with a value closer to the case of shallow liquid layers than the case of deep liquid layers [START_REF] Fedorchenko | The formation and dynamics of a blob on free and wall sheets induced by a drop impact on surfaces[END_REF]. This may be related to the collisional chains that redirected the impact velocity within a few grains layer only [START_REF] Crassous | Impact of a projectile on a granular medium described by a collision model[END_REF]. With the measurements of the geometrical parameters of the corona, we can now define the apparent area of the corona A cor , that is related to the amount of ejected grains contained in the corona, and compare it to the total apparent area of ejected grains A tot , related to the total amount of the ejected grains, both in the corona and isolated. A cor is measured as the area included inside the corona contour, which is not far from the area H(D t + D b )/2 corresponding to the approximate corona trapezium shape in the images. The time evolution of A cor is reported in Fig. 5 for the same experimental parameters as in Figs. 2 and4. Even if A cor is always smaller than A tot , the evolution of A cor is qualitatively the same as the one of A tot (Fig. 2). This is confirmed by the quantitative comparison of the corresponding coordinates of the curve maxima, (t Atot max , A tot max ) and (t Acor max , A cor max ), that are reported in Fig. 6 for all the experiments (all dropping heights, all different impacting spheres). The evolutions of A cor and A tot are synchronized in time as illustrated by the equality t Acor max ≃ t Atot max (Fig. 6a). Besides, A cor max is found proportional to A tot max with the same ratio for all experiments: A cor max ≃ 0.75A tot max (Fig. 6b). All this suggests that the investigation of the corona dynam-ics is a good first order for the study of the dynamics of grain ejection due to an impact. Let us now interpret these experimental results by a simple ballistic model.

III. BALLISTIC MODEL FOR GRAIN EJECTION

The dynamics of the grains is found axisymmetric and rapidly contact forces between grains play no role (Fig. 1). Furthermore air friction can be estimated and is negligible compared to the grain weight. Thus rapidly after impact, each grain trajectory corresponds to a parabolic free flight under the action of gravity alone. Using these preliminary remarks, we now build a very simple axisymmetric model that reproduces the observed corona dynamics in the (r, z) plane of cylindrical coordinates.

As a first attempt, we assume that all grains start at the same time t = 0, from the same position (r 0 , 0), in the same initial direction making an angle α with the horizontal, but with different velocity amplitudes. It is thus easy to show (Fig. 7a) that at any time t all the grains will be located on a cone making the same angle α with a bottom diameter 2r 1 increasing quadratically with time as 2r 1 (t) = 2r 0 + gt 2 / tan α. Furthermore, if u 0 is the maximal amplitude of initial velocity, the top diameter of the cone 2r 2 increases linearly with time as 2r 2 (t) = 2r 0 + 2u 0 cos α t and the height of the cone z 2 evolves quadratically in time as z 2 (t) = u 0 sin α tgt 2 /2. These analytic results are quite similar to the dynamics observed in Fig. 1 and to the time evolution of the corona parameters H, D t , D b and θ presented in Fig. 4. The hypothesis of an instantaneous release of all the grains at the same place and the same time t = 0 is however clearly oversimplified. Indeed grains are ejected during a time interval δt of the order of the penetration time R/U c ∼ 5 ms and, as the projectile decelerates the ejection velocity of the grains should be a decreasing function of time. In Fig. 7b, we plot the apparent cone angle θ(t) of the moving grains when released with the same ejection angle α but with various models of time decreasing velocities u 0 (t) during the time interval δt = 5 ms: A linear decrease [u ej (t) = u 0 (1 -t/δt)], a quadratic decrease [u ej (t) = u 0 (1 -t 2 /δt 2 )], or an exponential decrease [u ej (t) = u 0 exp(-t/δt)] from the initial value u 0 = 1 m/s. Except at very short times the edge angle θ remains constant and always close to the ejection angle α. While the time evolutions for the top diameter 2r 2 and the height z 2 of the apparent cone are obviously the same than for an instantaneous release, it appears to be also the same but with a small delay time δt for the bottom diameter 2r 1 . As this delayed grain ejection and initial velocity decrease have no visible effect in the corona dynamics, we will use in the following the first simpler model where all the grains are ejected at the same time with ejection velocities 0 ≤ u ej ≤ u 0 .

In the experiments we measured the time evolution of the corona parameters, i.e. the optically opaque zone formed by ejected grains, when viewed from the side. The relation between this corona that corresponds to a 2D projection of the 3D real dynamics with this latter is not straightforward. But the bottom diameter D b and the edge angle θ are clearly the same. Furthermore, Fig. 6 shows that most of the grains are inside the corona. In the following we will then assume that the height H and the top diameter D t of the corona in the experiments are identical to the height z 2 and the top diameter 2r 2 of the cone in the ballistic model. Assuming this, the time evolution of H, D t , D b and θ are given by the following set of equations:

H(t) = u 0 sin α t -gt 2 /2 + H(0), (2a) D t (t) = D t (0) + 2u 0 cos α t, (2b) D b (t) = D b (0) + gt 2 / tan α, (2c) θ = α, (2d)
An initial height H(0) has been introduced in the dynamics of H(t) as we observe in the experiment an initial swelling of the granular bed before any grain ejection.

This swelling corresponds to a deformation of the substrate when the projectile starts to penetrate the bed. This deformation process is clearly not contained in our ballistic description. The horizontal position of ejection r 0 is related to the projectile size: We checked that D t (0) and D b (0) increase with the impacting sphere diameter 2R. The set of equations ( 2a)-(2c) can be written in dimensionless form as:

H(t) -H(0) H max -H(0) = 2 t t Hmax - t t Hmax 2 , (3a) D t (t) D t (0) = 1 + 2u 0 cos α D t (0) t, (3b)
D b (t) D b (0) = 1 + g D b (0) tan α t 2 . ( 3c
)
with the two scaling parameters: t Hmax = u 0 sin α/g and H max -H(0) = gt 2 Hmax /2. For each of the 65 experiments, our experimental data for the height H(t) and the top diameter D t (t) of the corona are well fitted by the ballistic model equations as shown in the non-dimensional plots of Fig. 8a andb. The evolution of H is basically quadratic, while D t is rather linear in time. The fitting parameters u 0 sin α for H(t) and u 0 cos α for D t give access to the typical grain ejection velocity u 0 and to the typical ejection angle α for each impact experiment. The time evolution of the bottom diameter of the corona D b (t) is shown in the nondimensional plot of Fig. 8c, with the already deduced fitting parameters: The predicted quadratic increase of D b is rather well followed by the experimental data. The angle of ejection α, extracted from the fits of Fig. 8, is found to be constant for all experiments: α ≃ 53 • ± 3 • , which corresponds well to the corona edge angle measured previously (Fig. 4d): θ ≃ 56 • ± 3 • . We have checked that α and θ are robustly constant for all experiments and do not depend on any impact parameter, as shown in Fig. 9: No clear correlation of α with the impact energy E c can be seen. The equality found between the fitted ejection angle α and the measured corona edge angle θ is again in favour of the present grain ejection scenario. From eqs. (2a)-(2c), we can calculate the area of the corona A cor (t): The predicted evolution is in good agreement with experimental results and gives t Acor max = 1.36t Hmax very close to the coefficient 1.38 measured experimentally. Although we do not measure directly the grain ejection velocities, the fits of Fig. 8 give a characteristic ejection velocity u 0 . Figure 10a shows the slow increase of u 0 with the impact velocity U c , and its dependence on the sphere radius, where we can see that U c is not the relevant parameter to account for the variations of u 0 . The kinetic impact energy E c = M U 2 c /2 is the relevant parameter, as shown in Fig. 10b where the kinetic energy of one ejected grain mu 2 0 /2 is plotted as a function of the impact energy E c in log-log axes. The best power law fit is: 0 /2 as a function of impact energy Ec and the best power law fit Eq. ( 4). Same symbols as in Fig. 6.

mu 2 0 /2 ≃ 0.56 10 -9 E 0.37±0.05 c , (4)
showing a moderate effect of the impact energy on the kinetic energy of one ejected grain. From the ballistic model presented above and this experimental result of the scaling of the ejection velocity u 0 with the impact energy E c , we can infer the scaling for the total duration of the corona ballistic dynamics T , for its maximal height H max and its maximal top diameter D t max with the impact energy E c as:

T = 2t Hmax = 2u 0 sin α/g ∝ E 0.19 c , (5)
H max = gt 2 Hmax /2 ∝ E 0.37 c , (6)
D t max = u 2 0 sin 2α/g ∝ E 0.37 c , (7)
allowing to characterize the duration and the extension of deposits upon an impact. These results show that the corona duration T depends only slightly on the impact energy E c , whereas the corona expansion depends moderatly on E c . In Eq. ( 1), the characteristic number of ejected grains N ej has been related to the maximal value of the total apparent surface of all of the ejected grains A tot max , through the unknown product nw. We will estimate in section IV that nw ≃ 12 mm -2 and we use from now this value to 8). b) Kinetic energy of ejecta Eej as a function of impact energy Ec and best power law fit given in Eq. ( 9). Same symbols as in Fig. 6.

calculate the number of ejected grains N ej according to Eq. ( 1). In Fig. 11a N ej is plotted as a function of the impact energy E c in log-log axes. We see that whatever the projectile size all the experiments collapse close to the power law:

N ej ≃ 7.2 10 5 E 0.70±0.05 c . (8)
We can now estimate the kinetic energy transmitted to all the ejected grains as E ej ≃ N ej mu 2 0 /2. Figure 11b shows the evolution of E ej as a function of E c . The total kinetic energy of the ejected grains is found proportional to the impact energy as:

E ej ≃ 0.031E c . (9)
This unexpected result allows for the definition of an effective restitution coefficient ρ = E ej /E c ≈ 0.18. The small value of ρ confirms the well known and well used fact that granular beds are very efficient dissipating systems.

IV. DISCUSSION

In this paper, we have shown that the number of ejected grains N ej obeys a scaling law with the impact energy E c (see Eq. ( 8)). In the literature, the impact energy has been shown to be also the relevant parameter that governs the crater size, in terms of its typical radius and depth [START_REF] Walsh | Morphology and scaling of impact craters in granular media[END_REF][START_REF] Uehara | Low-speed impact craters in loose granular media[END_REF][START_REF] Zheng | Impact craters in loose granular media[END_REF][START_REF] Vet | Shape of impact craters in granular media[END_REF]. Let us now relate these different scaling laws. The scaling laws known from Ref. [START_REF] Vet | Shape of impact craters in granular media[END_REF] for the typical radius R crat and depth H crat of craters are R crat ∝ E 0.23 c and H crat ∝ E 0.21 c . The volume V of craters can thus be also related to the impact energy as

V ∼ H crat R 2 crat ∝ E 0.67 c
, with a numerical prefactor that depends on the precise shape of the crater and on the grain and projectile properties. From the crater profile measurements of Ref. [START_REF] Vet | Shape of impact craters in granular media[END_REF] for glass beads, we deduce the precise scaling law for the crater volume:

V ≃ 3.6 10 4 E 0.67 c . (10)
From this expression for the crater volume, we can estimate the number of grains that have been ejected as N crat = V Φ/v where v = 4πr 3 /3 is the volume of one grain and Φ is the solid volume fraction of the granular packing (Φ ≃ 0.6). This number of excavated grains follows thus a power law with the impact energy with the exponent 0.67, that is very close to the exponent 0.70 measured here for the ejected grains in Eq. ( 8). This definitely confirms that the geometrical measurements made on the corona give relevant and quantitative information on the grain ejection dynamics. Writing that the number of ejected grains N ej measured here is equal to the number of excavated grains N crat from [START_REF] Vet | Shape of impact craters in granular media[END_REF] allows to bypass the unknow prefactor wn in Eq. ( 1) and to accurately calculate N ej and E ej shown in Fig. 11: Imposing the equality N ej = N crat gives wn ≃ 12 mm -2 . The scaling laws u 0 ∝ E 0.18 c and N ej ∝ E 0.70 c obtained here for a large projectile over grain size ratio (R/r ∼ 25-50) are close to the results obtained for a small size ratio (R/r = 1) in [START_REF] Beladjine | Collision process between an incident bead and a threedimensional granular packing[END_REF]: u 0 ∝ E 0.12 c and N ej ∝ E 0.75 c , where u 0 and N ej refer to ejection velocity and number of ejected grains averaged over many experiments. This suggests that the same scaling laws hold whatever the ratio R/r. The constant value of the ejection angle equal to about 55 • ± 5 • , is in agreement with previous values measured between 40 • and 60 • in the case of high speed impacts [START_REF] Cintala | Ejection-velocity distribution from impact into coarse-grained sand[END_REF][START_REF] Wada | Numerical simulation of impact cratering on granular material[END_REF]. Let us recall that the energy of the ejected grains is shown to correspond to 3% of the impact energy (Eq. ( 9)), that is much larger than the tiny fraction (0.3% according to [START_REF] Vet | Shape of impact craters in granular media[END_REF]) required to excavate the crater, i.e. for the grains to move just above the granular surface with zero velocity. This is illustrated by the larger values of the maximal height of the corona H max (between 10 mm and 40 mm in Fig. 4a) than the values of crater depth (between 4 mm and 10 mm in [START_REF] Walsh | Morphology and scaling of impact craters in granular media[END_REF][START_REF] Vet | Shape of impact craters in granular media[END_REF]) for the same material properties and impact conditions. This suggests to take into account the kinetic energy of the ejected grains when considering balance of energies. However the essential part (∼ 97%) of the impact energy is dissipated in the granular target through frictional contacts and inelastic collisions.

V. CONCLUSION

The dynamics of grain ejection due to a large sphere impacting a granular material has been experimentally investigated through the time evolution of the corona, mainly in terms of its geometry. Whereas the dimensions of the corona -its height, top and bottom diameterschange with time, the angle formed by its edge with the horizontal granular surface remains constant during the ejection process. All these geometrical properties are well described by a simple ballistic model, supporting that grains are quasi-instantaneously ejected at time of impact from the same position in one direction. This direction appears to be constant when varying the experimental parameters and equal to about 55 • . One may wonder how this angle changes with the shape of the impacting projectile. By contrast, the typical ejection velocities and number of ejected grains change when varying experimental parameters and are controlled by the impact energy through power laws. The evaluation of the energy of the ejected grains allows finally for the calculation of an effective coefficient of restitution characterizing the complex collision process between the impacting sphere and the fine granular target. An important result is that this effective restitution coefficient is constant when varying the experimental parameters and equal to about 0.2.

FIG. 1 :

 1 FIG. 1: Sequence of side view images separated by 30 ms, showing the typical time evolution of ejected grains after the impact of a steel sphere of radius R = 6.75 mm dropped from the height h = 30 cm (Uc = 2.45 m/s, Ec = 30 mJ).

2 )FIG. 2 :FIG. 3 :

 223 FIG. 2: (Color online) Time evolution of the total apparent surface of ejected grains Atot for an impacting steel sphere of radius R = 7.55 mm dropped from different heights h = 13, 23, 33, 43, 48 and 58 cm (from bottom to top).

FIG. 4 :

 4 FIG. 4: (Color online) Time evolution of a) the height H, b) the top diameter Dt, c) the bottom diameter D b , and d) the angle of the corona θ for the same experimental parameters as in Fig. 2. The horizontal line in d) is the time and ensemble average of θ(t) for the 65 experiments.

2 )FIG. 5 :FIG. 6 :

 256 FIG. 5: (Color online) Time evolution of the apparent surface area of the corona Acor for the same experimental parameters as in Fig. 2.

FIG. 7 :

 7 FIG. 7: (Color online) a) Ballistic model for grains ejected instantaneously from the same position r0 in the same direction making the angle α with the horizontal, with different velocity amplitudes. Trajectories are drawn in dashed lines. At any time grains align along a straight line forming the corona edge of angle θ = α drawn in continuous lines. b) Time evolution of the angle θ of the cone edge if grains are ejected during a finite time δt = 5 ms in the direction α = 53 • , with a linear, quadratic or exponential decrease of ejection velocity from the initial value u0 = 1 m/s.

FIG. 8 :

 8 FIG. 8: (Color online) Rescaled data of a) heights (H -H(0))/(Hmax -H(0)), b) top diameter Dt/Dt(0) and c) bottom diameter D b /D b (0) as a function of rescaled times appearing in Eqs. 3 for all of the 65 experiments.

FIG. 9 :

 9 FIG. 9: (Color online) Fitted ejection angle α as a function of the impact energy Ec for the 65 experiments (same symbols as in Fig.6).

FIG. 10

 10 FIG.10:(Color online) Variations of ejection parameters with the impact parameters: a) Ejection velocity u0 as a function of impact velocity Uc; b) Kinetic energy of one grain mu 2 0 /2 as a function of impact energy Ec and the best power law fit Eq. (4). Same symbols as in Fig.6.

FIG. 11 :

 11 FIG.11:(Color online) a) Number of ejected grains Nej as a function of impact energy Ec for all experiments, and best power law fit given in Eq. (8). b) Kinetic energy of ejecta Eej as a function of impact energy Ec and best power law fit given in Eq. (9). Same symbols as in Fig.6.