Chapter 4: Analysis and Results of 1989 (2D) Experiments

4.1 Objectives of the Analysis

The analysis stage of this project was initially geared toward measurement of four key parameters for as many individual ejected fragments as possible:

• Size and shape (by direct measurement using callipers upon the collected fragments, and 'optical' means from the digitised cine film);
• Fragment velocity projected onto the plane perpendicular to the camera axis;
• Initial ejection angle of the fragments;
• Fragment rotation rate and rotation sense;

Once these measurements had been made using the HV-2 program, the plan was to use these data to conduct a more detailed study of the disruption, including such parameters as:

• Angular momentum, and its relation to ejection parameters;
• Energy partitioning (E_rotational/E_{translational});
• Anisotropies in the fragment ejection field, and possible examples of collimated jetting;

Another essential objective, particularly during the analysis of the first shot, was the further development and completion of an adequate and powerful software package. This was done while bearing in mind the eventual need to solve the problems of stereo recordings and to study fragmentation of more complex targets in three dimensions - to this end, as much of the program code as possible was written to be 'portable', i.e. such that it could be used later in a different software package.

4.2 General Points on the Analysis

Some remarks can be made at this point concerning the analysis system in general. It is worth bearing in mind that, before this project, no software whatsoever existed for the task of analysing fast-framing film of this kind, either within the Sussex/Turin collaborative group or (to our knowledge) anywhere else. It is certainly true to say that no computer system of this complexity had been applied to the analysis of the problems which we were tackling, and that there was no available body of previous experience from which to learn prior to the start of the project.

A few surprises were in store within the first few weeks of the study of these digitised films. The most significant was the critical dependence of the program 'usability', at almost all stages of data collection, upon the image load and display time. The process of 'tracking' fragments, i.e. recording their 2D (projected) trajectories, was reliant upon reasonably paced automatic animation of sections of the digitised films, in order that the individual fragments could be seen by eye as separate shapes. For this process to be workable, the lower limit was found to be around 2 frames per second (sustained). Similarly a minimum frame rate was found for the fragment spin analysis, this being approximately 15 frames per second in most cases, although this was more dependent upon contrast and image background than was the normal animation used while tracking fragments. An upper limit of 50 frames per second was imposed, upon any animation, by the computer screen display rate of 50 Hz.

Another surprise was the very great amount of time required to make the necessary measurements on the NAC images. This applies to almost all measurements, from simple tracking to the measurement of fragment sizes. Some improvements were made in the form of better coded routines, new features added to the software and new ways to use the program, which were developed as the analysis progressed (these are discussed in Section 3.6). Typically, the time required to study an ejected fragment was of the order of 20 minutes during the early stages, and this time had, by the last shot, reduced to approximately 8 minutes including a full trajectory track and the fragment size and rotation rate measurements.

There are some basic limitations to the accuracy of this system. They arise from two main features:

• Framing rate of the NAC camera, limiting the time resolution of the data;
• Digital image resolution, affecting the spatial resolution of the digitised cine film;

In principle there is a third point, namely that of the focal resolution of the NAC film, but this is dominated by the issue of the digital image resolution, the latter being much lower (i.e. less detailed images). Even in cases where fragments approach the camera, thus losing focus, the human eye can provide very good estimates of the boundaries involved - this is harder with the digitised images because of the reduction in intrinsic resolution, equivalent to a larger grain size on the film.

These inaccuracies affect the various measurements by a varying amount - for instance, a time uncertainty of one field affects the rotation measurements more for short rotation times, though velocity data is measured over a fixed time and as such contains a fixed maximum error. Size measurements are of course independent of time but, similarly to the rotation time measurements, suffer from increased errors at small sizes. These points are covered in more detail in each relevant section.

4.3 Digitisation of the NAC film

The MultiGrab program, developed as a tool for multiple image grabs (see Section 3.4.3), was used with good effect as soon as video data was available. Before the major part of the digitisation was carried out however, some thought was given to the issue of precisely which frames of NAC film should be digitised from the video tape. The most dynamic part of the NAC high-speed film sequence is the moment immediately after the detonation of the contact charge. Even at 700 frames per second, the fragment envelope changes significantly between frames for at least 50 ms, and for this initial period all available frames were digitised. As the dust and fragment cloud expands, there is less need for all frames and the spacing between successive digitised images was increased to 2, then to 4 or 8 frames spacing. This resulted in a set of grabbed frames which were densest during the most interesting part of the disruption, but also allowed slow fragments (including the core, where present) to be followed for a significant length of time.

In the case of shot 1, which was transcribed and digitised first, the divisions between full and reduced digitisation were chosen at times of approximately 150 ms (for the transition to 1-in-2 grabbing) and 300 ms (for the transition to 1-in-4 grabbing) with frames being digitised up to a maximum of 406 ms after the impact. It became clear once some study had been made that full digitisation of approximately the first 100 ms is crucial for the analysis, and that data can easily be collected from video frames of much reduced density thereafter. No precise rules were used, except that the idiom 'too much is always better than not enough' was strictly enforced! Details of the digitisation pattern of all 1989 shots are given in Table 4.1.

Table 4.1: Video digitisation frequency for the four shots in 1989.

The total time taken to digitise and organise all these video sequences was approximately 2 working weeks, including some extra work on 'MultiGrab'. A significant fraction of that time was taken up in maintaining backup discs of all grabbed frames. Performance of the image compression system was good, with the average compression factor (i.e. ) around 2.5. The details of the data for all 4 shots are summarised in table 4.2, below.

Table 4.2: Summary of extent of the digital video data and average compression ratio in each case. Note that, following hard disc convention (as opposed to volatile RAM convention), 1 Mbyte º 10⁶ bytes in this context.

It was unrealistic, during the first year of the project, to keep all video image files on the hard disc of the Archimedes 440/1 which has only a 40Mb internal hard disc. A small number of frames were retained from each shot with only the shot currently under investigation being fully represented. The full complement of data has been available on hard disc since mid-1992, when a new disc with a capacity of 330Mb was added to the system.

4.4 Initial Calibration of the System

In order to be able to make any quantitative measurements, it was necessary to determine the geometry of the images and estimate the potential accuracy of the system. Although the fast-framing camera had been set up carefully and pointed directly at the targets using the cross-hair markers of the camera itself, the process of transcription to video followed by digitisation affected the absolute position of the image within the view. This was not a serious problem, since the software had been written with this in mind and uses a movable origin. By using the 'Callipers' option in HV-2 with the first image displayed (i.e. before any target disruption) it is straightforward to measure the width of the target in terms of screen pixels. This width was known to be 205 mm (± 2 mm) and allowed the angular width of the field of view to be calculated easily.

Figure 4.1: Schematics of the camera/target arrangement, seen from above and from the right. The broken lines represent the edges of the NAC camera view in each case.

The basic geometry of the target and camera arrangement used in 1989 is illustrated in Fig. 4.1. For the purposes of fragment tracking, the most useful indicator of the field of view is the largest angle at which fragments are visible on both sides of the view, i.e. the minimum of and . This parameter, called , is calculated as shown below.

Table 4.3 lists the values which result from the geometric analysis of the first images in each series. Note that the lens focal length, ¦, for shot 1 (28 mm) gave a view wider than that of other shots, but at the cost of absolute image resolution.

Shot	¦	DNAC	x1	x2		y
1	28mm	8.0m	1.10m	1.25m	15.7°	1.88m	13.2°
2	50mm	8.5m	0.57m	0.77m	7.7°	1.07m	7.2°
3	50mm	8.5m	0.70m	0.75m	9.5°	1.15m	7.7°
4	50mm	8.5m	0.62m	0.73m	8.4°	1.09m	7.3°

Table 4.3: Geometric parameters for each shot

Once the target diameter had been measured on the computer screen (in terms of screen pixels) it was used by the program in conjunction with the known value (in mm) entered for all distance calculations.

4.5 Fragment Size and Shape Distribution

For the study of fragment sizes, shapes and axial ratios in our experiments, data were combined from both visual data (film) and in-hand measurements (that is, laboratory work on the collected fragments). Fragments were studied first in-hand, and their dimensions measured where possible using a pair of callipers as described in Section 2.8.

Using the 'callipers' facility in the 'HV-2' program, fragments were also measured as precisely as possible on the NAC fast-framing film. In the majority of cases, fragments rotated at least once before disappearing from the field of view - this gave a good idea of the three-dimensional geometry of the fragment under investigation. Where possible, the tri-axial dimensions were measured directly from the computer screen, as close as possible to the original target position in order to reduce errors which would arise from movement toward or away from the camera. While studying fragments using the 'Spin Analysis' facility, which allows fast animation of a small area of the images, it was possible to get an idea of which fragments were moving toward or away from the camera, and those which had velocity vectors close to the plane of view. Although not good enough for a quantitative analysis, this was helpful during the analysis to at least estimate the unknown velocity component.

In all cases the largest fragment axis was measured first, and labelled a; the second axis, b, was measured as the largest dimension perpendicular to this and the third, c, perpendicular to both. Where a fragment dimension could not be measured in flight, its value was estimated based upon the known average values measured from collected fragments, if appropriate.

4.5.1 Fragment Axial Ratios

It is possible to make a valid comparison between fragment and real asteroid shapes as inferred from lightcurves (e.g. Binzel et al. 1989) as well as with the results of previous experimenters (e.g. Fujiwara et al. 1978; Bianchi et al. 1984; Capaccioni et al. 1984, 1986).

The results of other laboratory investigations in this field have been in fairly good agreement that the distributions of the fragment axial ratios c/a and b/a in general peak at around 0.5 and 0.7 respectively, with standard deviations of between 0.12 and 0.15 in most cases. There has not been any significant dependence found upon fragment size.

When plotted as in Figs. 4.2(a) and (b), the scatter plot of the points of c/a vs. b/a for fragments studied in our experiments, and for previous ones by the same group (reported in Capaccioni et al., 1984) showed a very weak correlation between the values of b/a and c/a, and an almost entirely unpopulated region corresponding to c/a < 0.2, i.e. elongated or very flat fragments. It has been proposed (Giblin et al., 1994a) that this region of low population might correspond to fragments which have broken after the initial fragmentation event - i.e. fragments which have undergone secondary fragmentation when they hit the ground. Since we have been able to measure the shapes of many fragments in flight, we have avoided this uncertainty and can make an interesting comparison, not only between our results and those of other researchers, but also and more importantly between our own visual (in-flight) and ground (collected fragments) data.

Figure 4.2(a) and (b): Scatter diagrams (axial ratios b/a vs. c/a) resulting from laboratory in-hand measurement of collected fragments from shots 1 & 3.

Comparison can be made with Figs. 4.2(c) and (d), which show the visual (from the NAC film) measurements of the same experiments as Figs. 4.2(a) and (b).

Figure 4.2(c) and (d): Scatter diagrams (axial ratios b/a vs. c/a) resulting from NAC film measurement of fragments from shots 1 & 3 using program HV-2.

Table 4.4 summarises the axial ratio results for the four shots; note that no 'ground' data has been collected for shots 2 or 4. The comparison between our 'visual' and 'ground' data is good; values agree to within s in all cases, suggesting that the method of measuring fragment sizes from the NAC film is reliable, on average at least.

Table 4.4: Average fragment axial ratios and their standard deviations (in italics) for visual and ground data, where available.

There is no strong evidence to support the theory of widespread secondary fragmentation in our experiments. If secondary fragmentation were a significant effect, one would expect both c/a and b/a values to be higher in the case of 'ground' data, implying a move toward more rounded fragments - this is not the case (we would also have found a large number of broken fragments very close together on the ground, which we did not). Although statistically insignificant, a few fragments were found to have broken on landing, even on the soft floor of the quarry. Figure 4.3 shows a tracing of one particularly flat fragment which illustrates the tendency of fragment shapes to move toward more 'roundness' when undergoing secondary fragmentation.

Figure 4.3: A tracing of an elongated surface fragment (the lowest curve is the surface of a target) which broke, on hitting the ground, along the fractures shown.

It is interesting to note that higher specific energy impacts (increasing around the 10⁷ erg g^-1 level) often result in a greater degree of spallation around either an artificial or emergent core (Fujiwara 1985). Thus, in the case of a target with a harder core there could be a greater energetic contribution to the fragmentation of the 'mantle' for two reasons - firstly that the impact shock wave will be partially reflected at the first core/mantle interface and secondly that the core will consistently fragment less (regardless of any reflection considerations), i.e. it will absorb less energy, since energy is lost in the formation of fractures. If the result is more widespread spallation, there will be a greater proportion of long and flat fragments.

A comparison between shots 1 and 3 in Table 4.4 reveals a significantly lower value of c/a (representing more long and flat fragments) for shot 3, a cored target, compared to shot 1, which was uncored. If this is due to the reasons outlined in the previous paragraph, one would expect to see a correlation of some sort between the aspect ratios of fragments and their original positions within the target. In order to investigate this, I have generated scatter plots of the aspect ratios b/a and c/a of fragments from shots 1 and 3 versus ejection angle (which gives an indication of a fragment's original position within the target). These plots, which are based upon visual data only (since only fragments measured from the NAC film have associated ejection angle values), are shown in Fig. 4.4.

Figure 4.4: Scatter plots of fragment aspect ratios and ejection angles measured from shots 1 and 3.

Clearly there is no straightforward strong relationship between a fragment's ejection angle and its aspect ratios. However, a slight trend is apparent in that the cored target (shot 3, right hand plots in Fig. 4.4) exhibits a modest peak (representing 'rounder' fragments not typical of spallation) in both b/a and c/a values around the vertical direction, while the uncored target exhibits the opposite effect, particularly in the case of the c/a plots. Without a deeper statistical analysis it is impossible attach a quantitative significance to these features, but they could possibly be explained by the core of shot 3 partially reflecting the impact shock wave while at the same time fragmenting less, as described above. Furthermore, while the presence of the core might give rise to a higher energy density within the mantle on average, it might directly inhibit the transmission of energy through to the antipodal region, corresponding to fragments ejected near zero degrees. Clearly a more detailed study is in order here, and might be possible in the future using the data collected in 1992.

Across all targets, our average values of b/a and c/a are systematically lower both than those reported by other research teams, and than the estimates of asteroid shapes derived from lightcurves (e.g. Binzel et al., 1989). Average aspect ratios of asteroids are similar to those found by other experimenters, specifically values^[1] of around b/a = 0.7, c/a = 0.5; this agreement with asteroid data lends weight to the results of other researchers. The discrepancy is not highly statistically significant, since the difference between our results and the normally expected values remain within the standard ±s. If not due to a systematic error in our measurements or those by other research teams, this might have arisen from the use of a contact explosive charge in our experiments rather than a genuine hypervelocity impactor. Perhaps the overall estimate of delivered energy in our experiments was low - if more energy than expected had been delivered to the targets, the result would be an increased level of spallation and lower axial ratios as a result. The high level of agreement between other aspects of our results and the expected values (where comparison is possible with previous work), shows that this disagreement is exceptional.

4.5.2 Fragment Mass Distribution

Several representations have been used to describe the fragment size (and therefore mass) distribution resulting from a fragmentation event. The simplest of these is the mass of the largest resulting fragment, normalised to the total target mass, . The resulting fraction is conventionally called ¦_ð. General guidelines have been developed (e.g. Fujiwara 1986) which relate the value of ¦_ð to the type of fragmentation, specifically concerning the classification of the outcome as either non-catastrophic (conventionally ¦_ð > 0.5) or catastrophic (¦_ð £ 0.5). In our experiments, there is no doubt of the catastrophic nature of the fragmentation, since ¦_ð was found to be less than 0.2 in all four experiments. Table 4.5, below, summarises this aspect of the results.

Shot	Target mass,	Largest frag. mass,	¦ð
1	8.724 kg	0.28 kg	0.033
2	8.642 kg	0.79 kg	0.092
3	8.632 kg	1.62 kg	0.188
4	8.518 kg	0.44 kg	0.051

Table 4.5: Absolute and relative mass of largest fragments seen after impact.

The estimated error in these ¦_ð figures is ~30% (a lower error than that associated with small fragments because the fragments are correspondingly larger and can therefore be measured more accurately). The relatively large values of ¦_ð in shots 2 and 3 result primarily from the fact that the core itself did not fragment as extensively as the outer mantle. This is not surprising - the core material is harder than than that used to form the mantle and would therefore be expected to fragment less (i.e. into larger fragments) for a given specific energy. It is interesting to note that most of these relatively large core fragments did not survive the trauma of 'landing' - they shattered when they hit the table or ground, and were obviously very much weakened by the disruption process. For this reason, these data presented in Table 4.5 are derived from measurements made upon the NAC film alone,

Another worthwhile way to study distribution of fragment sizes resulting from a catastrophic disruption is the cumulative mass distribution. The cumulative mass is defined as follows:

where, for we use

What emerges when log(M_c) is plotted against log() is a set of points, one corresponding to each of the measured fragments, with these points together forming a line or curve. The gradient (or that of the most linear section) of the cumulative mass distribution is conventionally labelled s. This exponent is connected (see e.g. Paolicchi et al. 1989) to the exponent q of the differential mass distribution describing the number of fragments in a mass interval (m, m+dm):

by the expression

The value of q is generally between 5/3 and 2 for asteroids and previous laboratory studies (Zappalà et al., 1984, Cappacioni et al., 1986, Fujiwara 1986, Fujiwara et al., 1988), leading to values of s between 0 and 1.

Figure 4.5 shows an example, taken directly from Di Martino et al., 1990. The example in this Figure shows a 'knee' around millimetre sizes. This effect is thought by the authors to be due to two different processes producing the fragments - an initial cratering process producing dust and fine fragments, plus a slower and more extensive shattering process giving rise to a distribution of fragments above approximately millimetre sizes. Any feature present below sizes of a few tens of millimetres would not be apparent in our experiments, since the smallest fragments measured were of the order of 10 mm. On either side of this 'knee', it can be seen that the points form reasonably straight lines, with the larger fragment regime (which can be compared to our own data) exhibiting a gradient of ~1.

Figure 4.5: An example of a cumulative mass distribution. This has been directly reproduced from Di Martino et al., 1990. The different symbols represent different types of target material, but the general form of the plot is similar across all target types. Note that the vertical axis appears to have been incorrectly labelled, saying where it should clearly say .

For the study of our experiments, the primary and most significant plots are taken from the 'ground' data - i.e. fragments collected and measured by hand. These data are naturally more accurate than those collected from 'in-flight' measurements of fragment sizes, but data are currently available only for shots 1 and 3 in the 1989 experiments.

An important question arose during the processing of data for the cumulative mass distribution plots, concerning the way in which the 'unmeasured' mass is incorporated. Clearly the value of the total mass, M_t (of the original target) is reliable, but the sum of individual fragment masses, m_i, is not equal to this. This is not surprising - not only would a small number of fragments have been missed on the ground after the experiment (the term M_lost below), but some of the target material was immediately turned to dust in the simulated impact and would not have been visible on the ground (M_dust below). A further proportion of the fragments were collected but not measured for size because they had crumbled before or after collection (called "unsized fragments" - mass measurements were still possible on these). This contribution is labelled M_tiny below, and is significant particularly because it represents some initially large fragments which shattered on landing. The total mass can now be expressed simply as:

Three types of plot are available in the HV-2 program, each handling the data differently:

• Unsized or uncollected fragments ignored (type A) - the cumulative mass is expressed in terms of only fragments collected and measured (m_i above). All fragments not measured for size are ignored, so M_c(0) º 0;
• Unsized fragments assumed small (type B) - dust and 'lost' fragments are ignored but all fragments which were collected and not measured for size are included at the bottom end of the distribution, so M_c(0) º M_tiny;
• All unsized mass assumed small (type C) - in this case the dust is represented as if it were small fragments, and any fragments which were collected but not measured for size are also included at the bottom end of the distribution. Formally, i.e. .

Examples of these three possible types of cumulative plot, all derived from essentially the same data (from shot 1, 'ground' data) are shown in Figs. 4.6(a) to 4.6(c) on the following page. If the explanation for the knee in the plots of Di Martino et al., 1990 is to be accepted, it would make sense to generally use plot type B, where dust and lost fragments (meaning that they were too small to be collected) are effectively excluded from the plot. Whilst plots A and C are also of interest, the type 'B' plots are the most useful for direct comparison. It can be seen that this plot, 4.6(b) shows a good linear distribution with a gradient of ~2.

Figure 4.6(a): The log-log cumulative mass distribution plot (type A) from shot 1.

Figure 4.6(b): The log-log cumulative mass distribution plot (type B) from shot 1.

Figure 4.6(c): The log-log cumulative mass distribution plot (type C) from shot 1.

Given that the largest fragments often broke when they landed, it is not surprising that the distribution shows some strange features at the top end - this feature is also visible in the equivalent plot from shot 3, shown in Fig. 4.7. The shot 3 plot also shows a linear distribution very similar to that from shot 1, with a gradient of ~2.

Figure 4.7: The log-log cumulative mass distribution (type 'B') from shot 3.

These values of s » 2 for the cumulative mass distribution suggest q » , a result which is at odds with previous experimental work and, despite agreeing with Paolicchi et al.'s results suggests some error in the generation of the cumulative mass distributions.

If, on the other hand, we take the type C plots, which both exhibit gradients of s » 1, we do find agreement with previous experimental results though the mass distribution is not linear, as already shown in Fig. 4.6(c). This aspect of our experimental results remains somewhat unexplained, and clearly deserves further work.

^[1]  These values do not hold for large asteroids, which are more rounded (with b/a » 0.85 for 27 examples larger than 200 km), probably due to gravitational accretion and settling.

4.6 Velocity of Ejected Fragments

This aspect of collisional disruption has been fairly well studied, in general terms at least, by other teams as well as ourselves. As in our 1989 research programme, most other experiments have been carried out using single-camera filmed records of a catastrophic disruption, (see e.g. Gault and Wedekind 1978; Fujiwara and Tsukamoto 1980; Davis and Ryan 1990; Di Martino et al. 1990; Martelli et al. 1991; Giblin et al. 1994a,b). Nakamura and Fujiwara, 1991, and Nakamura et al., 1992, on the other hand, used a pair of fast framing cameras placed at a separation of 90° to record the 3D velocities of some fragments and evolution of the ejection field.

This body of research has revealed the following general properties of the ejecta velocity field for spherical or spheroidal targets:

• Hypervelocity impacts using 'laboratory size' (10-100 cm) targets generally result in velocities of values of < 30 m s^-1 for macroscopic (> 1 mm) fragments, and the velocity distribution depends little upon the impact generating technique, projectile type or target type;
• A fine spray of ejected dust travelling at relatively high velocity (>100 m s^-1) can sometimes be observed very near the impact point;
• Fragment velocities are highest for material originating near the impact point and decrease with increasing initial distance from it;
• Surface fragments, by the time they become visibly separated, have velocities higher than those originating inside the target - as a consequence fragments are well dispersed upon ejection and do not normally undergo secondary collisions in flight;
• When the impact results in core-type fragmentation (see section 4.5.1), the core fragment usually travels at very low velocity.

Di Martino et al. (1990) report results of experiments carried out by the collaborative Sussex/Turin group (but not covered by this thesis) which involved impacting, in vacuo, leucititic basalt targets with 1g aluminium projectiles travelling at 9.4 (± 0.02) km s^-1. These authors conclusively observed not only a fine spray of high-velocity (>100 m s^-1) ejecta originating from very close to the impact point but also the outer edge of a dust cloud expanding at a few km s^-1. This effect, if present, would not have been observable in our 1989 experiments because (i) our experiments were carried out in air - the atmosphere would rapidly slow down the dust cloud, and (ii) the positioning of the target on the table and the distance from which filmed records were made would have made any such effects almost unresolvable.

In experiments of this type, other authors (e.g. Fujiwara and Tsukamoto 1980, 1981; Takagi and Mizutani 1984; Waza et al. 1985; Fujiwara 1987) have generally measured fragment velocity and rotation for only a few of the largest fragments from each shot, whilst we have been able to successfully measure these parameters for most of the fragments in our experiments.

Also, axial symmetry has been assumed by most researchers, and sometimes a 3-D velocity field has been estimated from the 2-D data collected. Nakamura and Fujiwara, 1991, and Nakamura et al., 1992, have carried out the most extensive analysis to date, and these results will be compared to our own whenever possible.

4.6.1 General Properties of the Ejection Velocity Field

The first analysis of any kind carried out on the visual data from these experiments was a simple study of the CCD camera records, as part of a 3rd year project by Martin Wettstein, in June 1990. This part of the project involved studying the digitised images from the CCD and estimating the rate of expansion of the fragment envelope. This was done by measuring the position of the envelope in two adjacent frames, using a program which displayed the co-ordinates of a cursor which could be moved about the image. Although the time of detonation was known only to an accuracy of 20 ms (i.e. 1 video field) the average rate of expansion of the fragment envelope could be estimated accurately using the position in the two subsequent adjacent frames. Wettstein's results, which apply to shot 1 only, are shown in Fig. 4.8.

Figure 4.8: Plot of fragment envelope speed against angle from CCD data. Triangles and circles indicate the left and right hand sides of the target respectively. Data are from Wettstein, 1990.

Three features of Fig. 4.8 are immediately worth noting: Firstly, the velocities involved are of the order of tens of metres per second. Secondly, the velocity of fragments is higher at low angles, i.e. closer to the impact point (where target thickness is at a minimum). Thirdly, the velocity distribution is quite symmetrical about the vertical. These three features are exactly as expected from other hypervelocity impact experiments and, although not yielding any new qualitative information, strengthen further the case for the contact charge being equivalent to a hypervelocity impact by a solid projectile.

A second type of plot, illustrating the general shape of the velocity distribution, is shown in Fig. 4.9. This data has been collected from the NAC film using the HV-2 program. The analysis was carried out on a fragment-by-fragment basis and has provided a large amount of data on individual fragment velocities from all four shots. With the peak at 8 - 12 m s^-1, the plot shown in Fig. 4.9 confirms that the general velocity distribution data agree with the findings of other authors (e.g. Fujiwara and Tsukamoto, 1980, Nakmura et al., 1992).

Figure 4.9: The observed 2D velocity distributions from each shot, measured from the NAC film. Note that the vertical axes are not identical between plots.

As can be seen from the four plots shown in Fig. 4.9, the approximate peak of the 2D velocity distributions is consistent in being around 8 - 12 m s^-1 for all four shots. The marked broadening in the shot 1 and 2 plots is not explained by any consideration of the experimental set-up since the shot 1 target was homogeneous and the shot 2 target had a harder core.

The only noticeable difference between the general velocity distributions for the cored and homogeneous targets is that a few unexpectedly fast fragments were observed in the homogeneous cases - no fragments were observed to be travelling faster than 30 m s^-1 in shots 2 and 3, while totals of 5 fragments (3.4% of total) and 6 fragments (4.7% of total) above this value were observed in shots 1 and 4 respectively. One possible explanation for this is that the core, when present, interferes with the propagation of the impact-generated shock wave, reducing the likelihood of fragments being spalled off at high velocity in the antipodal region.

The low-velocity end of the distribution does not seem to be significantly affected by the presence of a core. The ejection velocity value at which the contribution of the core, in shifting the spectrum to high velocity, becomes important, is difficult to assess without further detailed analysis of wave propagation within the target. However, if we choose (rather arbitrarily) a lower velocity cut-off (say 20 m s^-1) we still find that the percentage of high-velocity fragments is still significantly greater in the 'no-core' case - 18.5% and 10% for shots 1 and 4 compared to 3.0% and 4.4% for shots 2 and 3 respectively.

4.6.2 Specific Fragment Velocities

In parallel with a general analysis of the form of the velocity distribution, as described in the previous section, some researchers (e.g. Nakamura et al. 1992, Nakamura and Fujiwara 1991) have used the velocity of characteristic fragments to classify the outcome of disruption experiments.

When a target has originally contained a core, or when fragmentation has resulted in the formation of a core-type fragment (see section 4.5.1), the velocity of this core has been measured and found to characterise the experimental outcome. Two of our 1989 targets (shots 2 and 3) were provided with a core. Both cores were visible in the NAC high-speed film, and a (possibly) core-type fragment was observed emerging from the dust cloud in shot 4. No core fragment could be observed in shot 1.

Together with the velocity of a core or core-type fragment, the antipodal fragment velocity represents an important measurable parameter of the impact, particularly in the context of the various scaling laws since it is expected to scale with collisional specific energy (see e.g. Nakamura, 1993).

The antipodal velocity, v_antipodal, has been measured for all four 1989 shots and is shown in Table 4.6 along with the velocity of the core(-type) fragments, v_core, and limits (highest, v_max, and lowest, v_min, measured velocities) for each shot. It is worth noting that the values of v_antipodal and v_core do not suffer any loss of accuracy through the use of 2D single-camera analysis: In our geometry the antipodal velocity, by definition, has only a vertical component, and the core and core-type fragments invariably travel in a near-vertical direction, even in the case of the shot 2 core which broke into two pieces during the initial stages of the disruption. From a statistical viewpoint the value of v_max is also highly reliable, given the general symmetry of the velocity field, whilst v_min does not represent any directly meaningful measure, since a fragment travelling directly toward the camera has zero observed (2D) velocity but an unknown velocity component toward or away from the camera.

Table 4.6: Measured 2D velocities (all in m s^-1) of characteristic individual fragments from the 1989 experiments. The shot 2 core broke into two pieces within the first few ms of the disruption - hence two data points. Subscripts are the (arbitrary) indices of the fragments in HV-2.

The first feature which is apparent from the four sets of figures is the internal consistency of the measurements. Some variation between the v_max values from shot 1 to shot 4 is to be expected, due to the change in optics (see section 4.4), and in this sense the shot 4 data is likely to be more reliable (the time resolution is identical between these two sets of experimental data, but the spatial resolution is better for shot 4).

A number of researchers (e.g. Fujiwara and Tsukamoto, 1980, Davis and Ryan, 1991) have reported results on the antipodal velocity found in impact experiments, though each set of experiments represents only a few data points. Nakamura (1993) has carried out the most extensive study of target types. Figure 4.10, which has been reproduced directly from Nakamura (1993) summarises these data. Our results, which agree very well with those of previous workers, are indicated by two pairs of dashed lines (at 4.0 ± 1.0 ´ 10⁷ erg g^-1 and 9.0 ± 1.0 m s^-1).

Figure 4.10: Summary of results on antipodal velocity (from Nakamura, 1993) with our own results added as two pairs of dashed lines, indicating the maxima and minima.

4.7 Velocity and Ejection Angle: Collimated Jetting?

The possible existence of preferential planes and groupings among the ejected fragments from a catastrophic disruption is interesting because it may account for the formation of binary asteroids and self-gravitating asteroidal aggregates ("rubble piles") - these phenomena both indicate that the fragment ejection velocity field after target break-up may be strongly an-isotropic. Following the accurate measurement of the positions of the collected fragments in our experiments, it has been possible to carry a considerable analysis of the distribution of ejected fragments in terms of numerical population as a function of angle. Before the fragment positions were even measured, however, they were examined on the ground. By standing next to the target table, it was often possible to spot lines of fragments before they were collected. In several cases these lines turned out to be statistically significant, as explained later in this chapter. Figure 4.11 is a scanned photograph of one of these lines of fragments, where each separate object has been marked by a small plastic flag.

Figure 4.11: A line of fragments, photographed before any measurement of fragment positions. The apparent alignment of many fragments suggested that a proper analysis might well reveal statistically significant 'jets'. This picture shows a line of pieces found during the 1992 campaign.

With the experimental procedure used in 1989, where fragment positions (relative to the target position) were accurately recorded on the ground, anisotropic fragment ejection will only be detectable if it takes the form of vertical planes passing through the target centre^[1]. Figures 4.12(a) to (d) show the simplest possible representation of the fragment distributions for shots 1 to 4 respectively, with each fragment just represented by a dot.

Figure 4.12 (a) to (d): Positions of fragments after each shot. Spacing between successive concentric circles is 5 m, and angular divisions are 15°.

The type of plot shown in figure 4.12 was used as a first attempt to identify any possible jets. Although some lines and groups of fragments are apparent, any interpretation of this graphical format is somewhat subjective, and a more quantitative assessment of fragment angular distribution is clearly required.

4.7.1 Some Comments on the Statistical Analysis

The angular frequency distribution of fragments from an impact is not random, since there are restrictions placed upon the distribution by the physical nature of the experiment. With the simulated impact made from below the target, as in our experiment, the horizontal linear momentum has no preferential component. Thus, conservation of momentum requires the total linear momentum of the target/impactor system after the impact, in the horizontal plane, to be zero.

From an analytical viewpoint, this really only restricts the momentum of one fragment, since the first fragments can exhibit any random distribution with the fragment cancelling the momentum of all others. This therefore excludes the possibility of, for example, all fragments being ejected to one side of the target, and brings the expected distribution closer to an even distribution - i.e. jets are less likely in reality (albeit by a very small factor) than in a truly random and independent model. Clearly any standard analysis, based upon such a random model, which seems to suggest significant anisotropic ejection (jetting of fragments) will be underestimating the significance of these jets.

Several types of analysis can be used to quantitatively investigate the isotropy or anisotropy of the fragment angular distribution. The two used by ourselves are the Poisson and c² (chi-squared) tests.

4.7.2 Statistical Significance 1: The Poisson Test

A Poisson distribution is a discrete distribution involving small numbers of independent events within an interval of some variable. The probability distribution of a Poisson random variable with mean (expected value) m is given by

This is applicable in the case of our experiments if we study the number of fragments in each small azimuthal angular interval of (arbitrarily) 3°. Since there are then 120 bins, the expected number of fragments per bin will be for each shot, where is the total number of fragments found, and the probability of finding r fragments in any one bin is .

In order to apply this analysis effectively to the fragment data from the 1989 experiments, we first plotted the fragment frequency distributions from these shots, as shown in Fig. 4.13.

Fig 4.13: The fragment frequency distributions from the four 1989 experiments. 120 angular bins of 3° width have been used.

The plots in Fig. 4.13 highlight the fact that, even after the averaging effect of using 'bins', none of the distributions are isotropic. The two homogeneous (uncored) targets exhibit the strongest anisotropies in the form of pronounced 'spikes' in these plots. Table 4.7 summarises the results of applying the Poisson test to the most prominent spikes (with heights of N_spike) in each plot.

Shot	Core?	Nspike	Mean m	´ 120
1	No	9	1.08	0.023 %
2	Yes	4	0.78	86 %
3	Yes	5	0.81	15 %
4	No	6	1.15	12 %

Table 4.7: Results of the Poisson statistical analysis of possible jets

The conclusion which can be drawn from Table 4.7 is that the largest spike found in shot 1 is extremely unlikely to be the result of a normal (i.e. isotropic) distribution. The distributions from shots 3 and 4 are also improbable in an isotropic model, though not to the same extent as shot 1.

Another notable feature of the data in Fig. 4.11 is the fact that two fairly large spikes can be seen almost 180° apart (shot 4, lower left plot). This suggests that a possible origin of these fragments is a common vertical plane extending on both sides of the target. It is possible to calculate an approximate figure for the probability of this happening by chance - that is, two spikes to be matched across the target as they are in this case. If we label the spikes by A and B and their respective heights and fragments, and choose an angular window for B of Dj = 24° (i.e. 8 bins), then the probability of finding B within Dj, given A, is

So, although the probability of the largest spike (of height 6 fragments) in shot 4 is 12 %, the probability of also finding another jet of height 5 at the approximate opposite point (or rather, within any 24° section, the position of which is fixed) is only 0.5 %. This contributes further evidence to the case for jetting.

4.7.3 Statistical Significance 2: The c² Test

Within a set of samples, some variation is to be expected in their values and statistics (mean and variance) relative to the true population statistics. The c² test is used to measure the consistency of a set of samples which all come from the same expected distribution. If n randomly chosen samples to , with mean , are taken from a normally distributed set of points expected variance s², the c² statistic is given by

This c² result is then looked up in a probability distribution function (p.d.f.) table and a corresponding probability value found, which represents the likelihood that the overall distribution is normal.

The necessary code was incorporated into the HV-2 program and this test applied to the angular distribution of fragments from our experiments, with the aim of estimating the probability that our distribution is normal. Figure 4.14 shows an example of the fragment distribution plot produced - note that the c² value is not 'looked up' automatically because the p.d.f. table is too large to be included within the program, and cannot be easily calculated.

It is also possible, within the program, to change the angular divisions (thus changing the number of samples) and the offset from which the first is measured (thus allowing the same data set to be sampled more than once). These settings are displayed alongside the fragment distribution plot. Generally the c² test is not applied to more than 100 samples, and indeed the tables available to us do not cover the region above . For this reason, 60 bins of 6° each were used.

Fig 4.14: An example screen shot from the HV-2 program, showing the fragment angular distribution from shot 4. The c² test parameter must be referenced against a p.d.f. table together with the number of 'bars' (that is, samples) in our test.

We applied the c² test to each of the four 1989 experiments. The results are summarised in Table 4.8, below, where P(normal) is the probability that the distribution is normal, given c² .

Table 4.8: Results of the c² analysis of possible jets using 60 bins of 6° each in all cases (the remarkably similar values for shots 2 and 3 arise by coincidence from two different distributions). This table can be compared with Table 4.7, which shows the results of the Poisson statistical analysis.

The result of the c² analysis is clear - the distributions of fragments from shots 1 and 4 are highly unlikely to be normal distributions. It is worth remarking, with respect to these probability values, that the c² statistics from the p.d.f. table only reach a value of 99.6, corresponding to a probability P(normal) = 0.1%, with the previous value being 91.9 for P(normal) = 0.5 %, so values of 115 and 124 in the c² statistic represent probability values considerably below 0.1%.

4.7.4 Summary of Evidence of Jetting

The results presented here provide a very strong case for the presence of genuine anisotropies in the fragment ejection fields of shots 1 and 4, both of which are homogeneous targets. There is weaker but significant evidence of anisotropic ejection in shots 2 and 3, both of which involved cored targets. By applying both the Poisson and c² tests, we have tested the fragment angular distributions from a local and more general standpoint, and the good agreement reduces any chance of the apparent jetting being a result of error on our parts, perhaps during the measurement of fragment positions.

Since our data has only consisted of fragment positions after they have come to rest, a large amount of information about the initial ejection field has been lost. We have still found statistically significant anisotropies, and this lends further weight to our findings. The fact that fragments were not generally closely grouped on the ground, but that preferential directions seem to be present suggests that these jets emerged from the disruption in vertical planes. We cannot exclude the possibility of a number of fragments being ejected in closely aligned initial directions with a range of speeds, though the existence two possible jets at approximately opposite angular positions in shot 4 does not support such a hypothesis.

This study, and the resulting successful quantitative measurement of the anisotropies of the fragment ejection field, have been the primary subject of a recent paper, of which I am a co-author, entitled "Fragment Jets from Catastrophic Break-Up Events and the Formation of Asteroid Binaries and Families" (Martelli et al., 1993).

4.8 The Relationship Between Fragment Size and Fragment Velocity

Only a small amount of detailed data is available from previous work on this aspect of collisional disruption. Previous researchers (e.g. Fujiwara and Tsukamoto 1980, 1981, Takagi and Mizutani 1984, Fujiwara 1987, Davis and Ryan 1990) have tended to concentrate on the study and analysis of a small number of fragments from a selection of experiments and little attention has been paid to the general trends in the size (hence mass) versus velocity data.

Nakamura and Fujiwara, 1991, carried out an extensive study of fragment velocity and ejection angle using alumina and basalt targets. The experimental conditions were comparable to our own, with targets being 6 cm in diameter and specific energy of the impacts ranging from 3.2 to 5.3 ´ 10⁷ erg g^-1. Assuming a general form for the velocity v as a function of fragment mass, m, as

where M_t is the total mass. They found from a weighted least-squares fit to data from experiments using both target types.

We can apply this analysis to our own data - Fig. 4.13 (following page) shows the log-log plots of fragment velocity against ), our standard size measure. The measured gradients (using a linear least-squares algorithm) from these data are summarised above each plot along with the number of points considered.

The values of b indicated by our results are summarised in Table 4.9 in a form allowing direct comparison to the Nakamura and Fujiwara, 1991 result.

Shot	Core?	b	r
1	No		-0.34
2	Yes		-0.05
3	Yes		-0.10
4	No		-0.10

Table 4.9: Measured values of the b exponent and correlation coefficient, r, from Fig. 4.15.

Figure 4.15: Plots (log-log) of velocity versus size for each of the shots in 1989. The gradient and straight line resulting from the least squares fit is also shown. The label boxes have been positioned such that they are not covering any data points.

As can be seen from the plots in Fig. 4.15, the errors in these values are likely to be large - we can assess the level of significance by looking at the correlation coefficient, conventionally labelled r, for each plot. The correlation coefficient is calculated^[2] for these data as

and will always have a value between -1 and +1, where the former indicates a 100% negative linear correlation and the latter a 100% positive linear correlation (that is to say, the points lie precisely on a line with positive gradient). The correlation coefficients, which have been included in Table 4.9, indicate very little linear correlation for shots 2, 3 and 4. Shot 1 represents the best (by far) linear form and indeed the corresponding b value is the closest to that found by Nakamura and Fujiwara, 1991. It should be borne in mind, however, that (i) no data is available concerning the correlation coefficients from the Nakamura and Fujiwara experiments, and (ii) this single agreement with previous results does not support any theory for a functional relationship between velocity and size. In fact, from Table 4.9 it is clear that the results of our group indicate that there is no straightforward correlation.

^[1]  Preferential planes not passing through the target centre could, in principle, be detected in the fragment ejection field, though the analysis applied here would not reveal any such features.

^[2]  This formula, which is more amenable to computer calculation, is used in preference over the conventional though the two are equivalent.

4.9 The Fragment Rotation Field

The rotational properties of fragments from these disruption experiments is of interest for a number of reasons; firstly the general distribution can be compared to known parameters of real asteroids; secondly, some insight can be gained as to the partitioning between rotational energy E_R and kinetic (translational) energy E_T which occurs during break-up and, thirdly, a direct comparison can be made with the rotation field implied by numerical models, specifically with that presented by Paolicchi et al., 1989.

With such a wide-ranging study of the fragments observed in our experiments, we are in a good position to examine correlations between the various parameters. Size, ejection angle and velocity can all be considered in a study of fragment rotation. An interesting and somewhat unexpected result has been the significant differences found between fragment rotation fields from the homogeneous and cored targets, discussed in Section 4.9.2.

4.9.1 General Form of the Rotation Distribution

The histogram distribution of asteroid rotation rates is often compared to a Maxwellian distribution (e.g. Harris and Burns, 1979, Farinella et al., 1981, Binzel et al., 1989), which would arise if the rotational axes were normally distributed with zero mean and equal dispersions. This would be the expected distribution of objects which had undergone a large number of collisions and reached some approximate equilibrium state.

Although considered very suitable for a highly evolved asteroid population, the Maxwell distribution is not expected to give a good fit to young primary fragments from disruption experiments such as our own - fragments observed in our experiments have undergone no collisional evolution themselves (as distinct from their collisional origin). If any significant Maxwellian distribution could be detected in the rotation field of fragments from laboratory impact experiments it would suggest that Maxwellian distributions amongst the asteroid population did not necessarily arise, as normally assumed, from a long-term collisional evolution but (at least partially) from primary fragmentation events.

The presence of a Maxwellian form in the rotation data from our fragments, known to have resulted from primary fragmentations, would imply certain characteristic energy distributions normally associated with systems which have reached some equilibrium state - in this case the interactions between fragments would have to have occurred during the early stages of target fragmentation and separation, and probably on a smaller scale than any phenomena studied in this series of experiments.

Figure 4.16 shows the measured rotation distributions of the four 1989 shots, with single Maxwellian fits to the data.

Figure 4.16: Rotation rate distributions from each shot in the 1989 simulated impact experiments. Solid lines indicate the best fit Maxwellians in each case; the results of the c² analysis are shown in Table 4.10.

By applying a c² analysis to the data in Fig. 4.16 it is possible to assess the degree to which the distribution is Maxwellian. The results of this analysis are shown in Table 4.10, where the statistical analysis has been applied with respect to the hypothesis H₀ º "The distribution is Maxwellian" and the confidence level represents the probability that H₀ is false. In calculating the c² parameter, empty bins have been ignored but small counts have not been grouped together (as they were by Binzel et al., 1989). This has the effect that, in our analysis, small discrepancies in these bins (that is, departures from the best-fit Maxwellian) contribute significantly to the c² parameter, calculated as

where the expected value for some Maxwell parameter s and rotation rate w is given by

which tends to zero for large w and thus contributes disproportionately to the c² result.

Table 4.10: Results of the rotation rate distribution analysis.

By inspection, the curve fits in Fig. 4.16 are good in the middle of the rotation range but exhibit an excess of slow and fast rotators. This is in agreement with the comments made by (e.g.) Farinella, 1981 and Binzel et al., 1989, the latter in their discussion of the plot shown in Fig. 4.18, which shows the best-fit Maxwellian to the histogram of observed asteroid rotation rates across all size ranges, with rotation rates below 10 revolutions per day. A further plot, covering all data from our experiments, is shown in Fig. 4.17 - it is acceptable to group our distributions in this way since there are no significant differences between those from the cored and uncored targets.

Figure 4.17: Fragment rotation rates from all four 1989 experiments carried out by this group, with a Maxwellian best fit to the combined data (statistics are reported in Table 4.10).

Figure 4.18: Distribution of asteroid rotation rates with 20 equal bins between 0 and 10 rev/day. The solid curve shows the best fit Maxwellian, which is formally rejected under a c² test. This figure has been taken directly from Binzel et al., 1989.

Binzel et al., 1989, have made a further set of plots, with each covering a different size range in the asteroid population. The objective was to highlight any differences arising from the different size/strength regimes, particularly with respect to gravitational binding. As discussed by Binzel et al., the larger asteroidal bodies may have experienced (and survived) more collisions than smaller ones. Whilst no such division is meaningful in our data, we would expect the smaller objects to more closely resemble our own fragments. The four separate plots resulting from this analysis are shown in Fig. 4.19.

Figure 4.19: Histograms of asteroid rotation rates broken down into four size groups as indicated (units are kilometres), with solid lines showing the best-fit Maxwellians in each case. Only the two largest groups exhibit acceptable Maxwellian fits. (Figure from Binzel et al., 1989.)

The fact that the rotations of the smaller asteroid groups do not exhibit a significantly Maxwellian distribution agrees with the suggestion above that these smaller bodies are less evolved, that is to say their rotations have not undergone significant collisional alteration since the primary impact which produced them, although we cannot rule out the possibility that some other (unknown) process has affected these asteroids.

4.9.2 Rotation Rate and Fragment Size

The correlation between rotation rate and size for asteroids and fragments from impact experiments has been discussed by Paolicchi et al. (1989), Binzel et al. (1989), amongst others, and is covered in the paper recently co-authored by our group (Giblin et al., 1994a). Figure 4.20 shows two typical plots of rotation rate and fragment size, with a running box analysis applied using a box size of 10.

Figure 4.20: Fragment rotation rate vs. size for each fragment and an overlaid running box analysis using a box size of 10 points - dashed lines indicate the ±2s boundaries. These data are from shot 2 (left) and 4 but are typical of all shots.

The average rotational rate decreases only slightly for larger fragments. No strong dependence of w upon (as suggested by Fujiwara and Tsukamoto, 1981 and Paolicchi et al., 1989) is apparent, but the large dispersion of w at small fragment sizes - a feature also noted in the spin rate distributions of the Eos and Koronis families (Binzel et al., 1989) - agrees with these previous results.

4.9.3 Rotation Rate and Fragment Ejection Angle

Fragments ejected along the line of impact generally have the lowest rotational rates, and those at high angles relative to the impact (i.e. near the horizontal in the case of our experiments) generally rotate fastest. Experimentally measured rotation rates are fairly well correlated with ejection angle between these two extremes, as shown in Fig. 4.21 below. This relationship is as predicted by Paolicchi et al., 1989 and can be explained by considering the velocity field's variation with angle; the curl of this velocity field, essentially the origin of the rotation field, is highest at low angles and hence gives rise to the fastest rotators in this region.

Figure 4.21: Measured rotation rates for the fragments from the four 1989 shots. The plots have been arranged to allow comparison between the un-cored (1,4) and cored shots (2,3). The number of fragments in each plot are 106, 114, 100 and 109 for shots 1 to 4 respectively. A few points were off-scale in the shot 2 and shot 4 plots, and their co-ordinates are indicated. Note that the error bars on lower points are present, but are smaller than the point markers.

Figure 4.21 illustrates several interesting aspects of the fragment rotation field. The points are fairly well grouped around a curve which gives higher rotation rates at higher angles from the impact direction. The dispersion is much greater in shots 2 and 3, both cored targets, with a number of fast rotators close to the impact angle not present in the shot 1 and 4 plots. Clearly the core, when present, has affected the rotation field in some way, and the very good agreement between data from the two experiments of each type (with and without core) adds to the significance of the results. One possible explanation for this is that the reflected shock wave, having been modified by the core (by reflection, refraction, or most likely a combination of the two) reaches the antipodal surface obliquely and gives rise to a complex velocity (and hence rotation) field amongst the fragments ejected at this surface. A more detailed examination of the wave propagation, perhaps including a study of the shock wave arriving at the surface using fast piezoelectric detectors, would undoubtedly reveal some very interesting data concerning this aspect of the disruptions.

4.9.4 Fragment Angular Momentum

Since the impactor/target system is effectively isolated in our experiments and has no initial angular momentum, we expect the total angular momentum of the system after the impact to be zero.

The rotation sense (positive, meaning clockwise, or negative, meaning anti-clockwise) was recorded with the fragment rotation rate in all cases. Figure 4.22 shows typical results, taken from the shot 2 analysis, with different symbols which represent positive and negative rotations. Any fragments which were unmeasurable (i.e. unknown rotation) have been omitted from this Figure, while fragments visibly not rotating, or rotating too slowly to be measured, have been recorded as having a rotational period of 999 ms (as described in Section 3.6.9). These latter points appear very close to the horizontal axis in Fig. 4.22, since they have an effective rotation rate of 0.001 Hz.

Figure 4.22: A typical plot of rotation rate vs. ejection angle, with triangles and circles representing clockwise and anti-clockwise rotation respectively. The rather crowded region around the origin (marked by a dashed box) is enlarged in the right hand figure. This data is in fact from shot 2 - the error bars, which have been omitted here for clarity, can be seen on the top right plot in Fig. 4.16.

There is a very clear division between fragments on either side of the target; those to the left (from our viewpoint) generally rotate clockwise, while those to the right generally rotate anti-clockwise. This is easily explained in terms of the fragment velocity field (Fujiwara and Tsukamoto, 1981) - since the ejection velocity decreases as one moves away from the impact point, fragments on either side have opposite angular momenta about their centre of mass. It should be noted, however, (see Fig. 4.22 and enlargement) that fragments close to the vertical do not quite conform to this - some fragments are slowly rotating in the opposite direction to that expected. This may be due to some subtle interaction of fragments in the first stages of fragmentation, or may be an example of the counter-rotation of core and core-type fragments observed in experiments where artificial rock targets were impacted off-centre. The latter explanation would indicate some asymmetry in our simulated impact, but since these fragments are rotating very slowly they do not contribute significantly to the distribution of angular momentum - Nakamura and Fujiwara, 1991, indeed remark that cores were "minor carriers of energy and momentum" even in their experiments carried out under similar conditions but using real projectiles and impacted significantly off-centre.

In order to estimate the angular momentum of the fragments under investigation, we need to know the size, shape and rotation rate of each as well as, in principle, their respective axes of rotation. It was not possible, during the analysis, to measure the orientation of the rotational axis of each fragment. For the study of fragment angular momentum using HV-2, we make a number of simplifications. Since we do not know the orientation of the spin axes with respect to the fragment triaxial dimensions, we assume that they are evenly distributed. We then estimate the mass of the fragments based upon their known dimensions and the volume of an equivalent ellipsoid, using only the measured a values and incorporating estimates of b and c based upon the observed average axial ratios from collected fragments. From this volume and mass estimate we calculate the moment of inertia of an equivalent sphere, and this allows us to estimate the angular momentum of the original fragment. Although the errors in moments of inertia of individual fragments will be large, the considerable volume of data allows a worthwhile statistically significant study to be conducted.

The following lines lead to the formula used in the HV-2 program. Note that the conventional algebraic size measure of a sphere is the radius (and equivalent measures are used for an ellipsoid), rather than the diameter-like measures used for our fragments, hence the factors of which appear though the expressions below.

Consider an ellipsoidal fragment with characteristic dimensions a, b, c, and density r, rotating at angular velocity w, where the average observed axial ratios are k_ab and k_ac. We approximate the fragment to a sphere of radius r :

The volume of the ellipsoid is

This is calculated within the HV-2 program using the measured average k_ab and k_ac values, providing more accurate data over the whole data set. The constants in the last expression above do not play a part in any relative comparisons within each shot, but actual values of L are used in the analysis of energy partitioning (next section).

The most revealing representation of the angular momentum distribution is that within angular sectors of various widths Df. Initially, just two angular sectors are taken, giving results for the 90° on either side of the vertical. We would expect these two sectors to contain equal amounts of angular momentum and indeed this is the case, within errors, for shots 1 to 3. Another worthwhile choice of Df is 45° - this reveals the distribution of angular momentum as we move from the vertical to the horizontal. Figure 4.23 shows both these plots, together with further versions with Df = 30° and 15°. As can be seen from this Figure, the usefulness of the plot decreases as the angular divisions become too narrow to have any significant averaging effect.

Fig. 4.23: Rosette diagrams from all four shots indicating the sign (light grey = positive, dark grey = negative) and total angular momentum (represented by the area of a slice) within Df. Note that the scale is arbitrary, and only relative sizes of angular slices within each plot are significant. Maximum errors are indicated by the inner and outer sets of circular arcs.

Shot 4 exhibits a notable deficit of angular momentum on the left hand side; this does not arise from any physical process but from an observational systematic error introduced as a result of the very dark background on the left of all shot 4 images. This experiment was carried out late in the day and a lack of bright background makes fragment rotation measurements difficult - only 38 fragment rotations were measured on the left compared to 70 on the right of the vertical in the shot 4 analysis. Equivalent proportions for shots 1, 2 and 3 were 54:51, 53:60 and 52:47 respectively.

While not providing any remarkable new results, this analysis of the angular momentum distribution has shown that the rotation measurements are generally reliable and conform to expectations, including the explainable imbalance in shot 4. As such this represents an important check on our experimental method.

4.9.5 Energy Partitioning

The level of energy partitioning which takes place during a catastrophic disruption is a fundamental physical property of the fragmentation process. Division of impact energy between translational and rotational energy of fragments, comminution and heat has been studied by a small number of researchers in the past, e.g. Fujiwara, 1987, Nakamura and Fujiwara, 1991 and Nakamura et al., 1992. They found that the largest proportion of impact energy goes into heat, comminution and the relatively high velocity of fine fragments (see Section 4.6) - this resulted in the larger fragments carrying 0.3% to 3% of the impact energy. Fujiwara (1987) found that the rotational energy of fragments is generally less than 1% of their kinetic energy.

We are in a good position, having measured size, velocity and rotational rate of so many fragments, to investigate these parameters in our experiments and make some comparisons to this previous work. The total kinetic energy carried by large (that is, measurable) fragments can easily be estimated and compared to the nominal impact energy as well as the rotational energy In order to use best possible estimates of velocity in the calculation of kinetic energy, it is necessary to correct for the 2D nature of our data. This is done by assuming that the true horizontal velocity components are, on average, evenly distributed, and applying a correction to the horizontal velocity component while leaving the vertical velocity unchanged as follows:

This technique has been used in the past by our group (Giblin et al., 1994), to provide better estimates of the real velocity field in our experiments. It is strictly not suitable for any consideration of specific fragment velocities as discussed in Section 4.6 since it wrongly modifies the individual values - the maximum observed velocity (for example) is believed to be reliable, due to the large number of samples, and would be significantly changed by this 'correction'.

Combining these calculations within HV-2 yields the average values summarised in Table 4.11. Note that the values of in column 5 were calculated by averaging the energy ratios of all fragments - i.e. not by applying the formulae once to averages of v, a and w.

Shot	Ÿw / rad sec-1‰‰	Ÿa / mm	Ÿv / m s-1
1	ˆŠ‹ŠŠŠŠŠŠ1‰‰‰‰‰‰ˆˆˆˆˆˆ‰ˆ‰ˆ63 (239ˆˆˆˆˆˆˆˆˆˆˆ‰ˆ‰ˆˆ‰ˆˆ‰‰ˆ)	58 (14)	17.5 (9.30)	0.021 (0.045)
2	170 (283)Š	57 (25)	12.3 (5.83)	0.021 (0.036)
3	ˆŠ‹ŠŠŠŠŠŠ163 (239)	62 (23)	13.9 (5.89)	0.023 (0.043)
4	176 (364)	57 (23)	12.3 (8.56)	0.038 (0.110)

Table 4.11: Averages of rotation rate, major axis and rotational to translational energy ratio for each 1989 shot. Figures in italics are standard deviations in each case - these numbers are included for Ÿw, äa and Ÿv only to indicate the spread of values within the sample.

The relatively high average value of Ÿv found in shot 1 is not explained by any consideration of the experimental conditions, but could have been predicted from the velocity distribution in Fig. 4.9, which peaks at a higher velocity than those of the other targets, as well as being wider. Perhaps this is due to some bias in the sampling at the time when fragments were chosen for tracking, but the consistent values for other measurements suggest that only the velocity distribution was affected. Also, the random error in this value is large - the standard deviation is more than 50% in this case.

The averages of the translational to rotational energy ratio E_r/E_t are remarkably similar despite their widely differing (and large) standard deviations, and despite the fact that they each arise from measurement of different numbers of fragments. These figures are slightly higher than those quoted by Fujiwara, 1987, Nakamura and Fujiwara, 1991 and Nakamura et al., 1992, who found that the ratio E_r/E_t is generally below 1%. These previous authors approximated the fragments to tri-axial parallepipeds rotating along their major axes - this would have the effect of maximising the apparent angular momentum and might lead us to expect lower ratios in our own calculations. Our results, on the other hand, arise from a significantly larger sample than that used by the previous authors - of the order of 100 fragments have been considered for each shot in our study, compared with less than 20 (Fujiwara, 1991) and 26 (Nakamura et al., 1992) in the previous studies for which figures are available.

It is interesting to consider the implied rotational rate corresponding to the estimated ejection velocity of asteroid family members. Typical estimated ejection velocity for family members (see e.g. Chapman et al., 1989) is of the order of 100 m s^-1 with representative size, R, being 100 - 1000 m. Based upon a material density similar to that used in our experiments, and an initial energy partitioning of E_r/E_t » 0.01, this suggests rotational rates of approximately 5000 rev/day (for a 100 m body) and 500 rev/day (for a 1 km body). These figures are much higher than the observed values, which are of the order of 10 rev/day. Two possible explanations present themselves. The first is that the small number of collisions experienced by the family members in their lifetime have nevertheless significantly reduced their rotational rates. This is entirely possible but may or may not account for such a large discrepancy, according to the nature and outcome of these impacts. Secondly, it may well be the case that moving from the laboratory to the real asteroid population requires a consideration of the phenomenon of rotational bursting - that is, fragments breaking up as a result of internal stresses shortly after formation. While the rotational rate of our example fragments scales inversely with size R, the centripetal acceleration of a point (and thus the rotational stress experienced) at some given radius r varies with w², i.e. 1/R², for a given ejection velocity. Thus smaller fragments are more prone to rotational bursting. This is most applicable in the region where R is small on an asteroidal scale but large on a laboratory scale, since the static strength of the rock material does not change and is quite adequate to hold fragments together on our small laboratory scale (particularly because the radius decreases for small fragments, thus decreasing the maximum stress) but may not be so for bodies a few orders of magnitude larger.