- Email: [email protected]
Energy partition into translational and rotational motion of fragments in catastrophic disruption by impact: An experiment and asteroid cases
ICARUS 70, 5 3 6 - 5 4 5 (1987)
Energy Partition into Translational and Rotational Motion of Fragments in Catastrophic Disruption by Impact: An Experiment and Asteroid Cases AKIRA FUJIWARA Department of Physics, Kyoto University, Kyoto 606, Japan
Translational kinetic energy Et and rotational kinetic energy Er of the fragments produced in the laboratory catastrophic disruption of a basalt sphere by the impact of a highspeed projectile were determined. It was found that the maximum value of Er/Et is of the order of 10 -z, which is not so different in the order of magnitude from the value estimated for family asteroids in spite of the great difference of the scale. However, the laboratory value is significantly higher than the value for the family. © 1987AcademicPress,Inc.
I. I N T R O D U C T I O N
Fragmental bodies produced in catastrophic disruption of minor planets by impact acquire spinning motion as well as the translational motion relative to the parent body. Knowing the amount of energy partitioned into rotational and translational motion is very useful in understanding the orbital and rotational properties of groups of minor bodies of collisional origin such as some asteroid families. Elementary processes on the catastrophic disruption have been studied on the small scale of laboratory experiments; typically centimeter- to millimeter-sized projectiles are impacted into 1- to 10-cm-sized targets. The velocity and kinetic energy (of translational motion) of some fragments ejected in catastrophic disruption have been determined by Fujiwara and Tsukamoto (1980), Mizutani et al. (1985), Waza et al. (1985), and the spin periods of the fragments by Fujiwara and Tsukamoto (1981). Experimental activities and the results in this field are reviewed elsewhere (Cerroni 1986, Fujiwara 1986). In this paper we present data on the translational and the rotational energies for the individual fragments produced by a labora-
tory catastrophic disruption of a small rock target by an impact. As shown later, in order to know the translational and rotational energies a set of many physical parameters such as mass, velocity, size, shape, and angular velocity of spin must be determined at the same time for the individual fragment. Since determining many parameters at the same time presents many difficulties, the data points presented here are rather sparse, and in the present stage we cannot give the energy distribution of the fragments as a function of size. However, a few new results are obtained on the order of magnitude of the translational and rotational energies, and on the ratio of these energies, which has an intriguing meaning for asteroid families. In the next section the experimental conditions are given, in Section 3 the procedures of analysis and some results are given, and in Section 4 the results are compared with some asteroid families. 2. E X P E R I M E N T A L
Experimental conditions are similar to those described in our previous series of experiments (Fujiwara et al. 1977, Fujiwara and Tsukamoto 1980), and we summarize 536
0019-1035/87 $3.00 Copyright ~c~1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
CONDITIONS
CATASTROPHIC DISRUPTION BY IMPACT them here only briefly. A basalt sphere of diameter 6.35 cm and mass 357 g was destroyed by a central impact of a cylindrical polycarbonate projectile of diameter 8 mm and mass 0.37 g launched at velocity 2.5 km/sec by a two-stage light-gas gun. The projectile kinetic energy E is 1.16 x 101° erg, and the impact energy per unit target mass E / M is 3.24 × 10 7 erg/g. The destruction type by the Fujiwara et al. (1977) classification is the "core t y p e " ; the surface and near-surface layers are peeled off and the central part is left intact. The core mass was 55 g, or 15% of the target mass. The motion of the fragments was pictured with a high-speed framing camera at a framing rate of 7,169 frames/sec through the side window of the chamber containing the target. Some selected frames from the film are shown in Fig. I. Characteristic behaviors of
537
the fragments ejected in a similar impact condition are described by Fujiwara and Tsukamoto (1980). 3. REDUCTION PROCEDURES AND RESULTS
We present the procedures used in obtaining the translational and rotational energies of the individual fragments from the movie film, and show some results.
(1) Translational Energy E1 The translational energy is expressed as Et = my2~2,
where m and v are the mass and translational velocity of the fragment, respectively. (a) Translational velocity v. The translational velocity v was obtained by tracing the translational displacement of the individual
FIG. 1. Selected frames from the high-speed film. The basalt target is 6.35 cm in diameter. The projectile comes from the right. The numbers show the time elapsed after the impact (in msec).
538
AKIRA FUJIWARA b
a
I FIG. 2. Example of tracing of some fragments. Numbers show the flame number counted from the " i m p a c t " flame (numbered zero). Arrows show the rotation of the fragments. An alphabetical letter is assigned to each fragment (also in Figs. 4, 5, 7, 8, and 9).
fragments one by one on the frames using a movie analyzer (a movie projector with variable framing speeds). Figure 2 shows examples of the movements of some fragments. In this work we consider only the fragments ejected from the target surface facing the camera, since only for those fragments could three-dimensional velocities be determined by the method discussed below. As shown in Fig. 3, the observer looks at the fragment images projected onto a plane 7r' (x-z plane in the figure). The observable quantities are projected velocity v/, and the projected ejection point P'/. The velocity v/ obtained directly from the movie film is shown in Fig. 4. In order to transform the projected velocities and ejection points to three-dimensional ones, we made the assumption that each velocity vector is on one of the sets of planes (in Fig. 3) which include the line connecting the center of the target with the impact point (the line of flight of the projectile). Very fine fragments near the impact point, whose images could not be resolved on the film, were excluded from the analysis. The total mass of the
fragments analyzed is estimated to be approximately 10-20% of the mass excluding the core. Translational velocity (and angular velocity) of the core could not be determined, because both were very small. The resultant velocity is shown in Fig. 5. The x 71-
"Jr"
Y FIG. 3. Configuration of the observer, target, and projectile trajectory, v" and r are velocity and initial position of a fragment, respectively, situated on the plane ~r. ~' and z" are projected onto the plane 7r' which includes x and z axes.
CATASTROPHIC DISRUPTION BY IMPACT
\
tl w
539
;
u
v
S
r
t
c
MPACT
."
:
*.
~ 3 0 m s -1
FIG. 4. Projected velocity vectors of the fragments. velocity d e c r e a s e s systematically with the size, which was m e a s u r e d as described below. (b) F r a g m e n t m a s s . Determination of the fragment size on the m o v i e film is not easy, b e c a u s e each fragment flies a w a y changing its shape f r o m f r a m e to f r a m e due to its spinning motion. The m i n i m u m size of any fragment was determined in the following way. The smallest size o f the fragment is m e a s u r e d on each f r a m e as the distance between a pair of parallel lines contacting the fragment image f r o m both sides. W h e n the smallest size takes a m i n i m u m value on one of the sequential frames, the minimum value is defined as the m i n i m u m size. The m a x i m u m sizes were also m e a s u r e d but they m a y contain m a n y errors. This is due to the following reason. S o m e typical frag-
_
I0
f
. . . . . . . . . .
0.01
'
'
'
0.I
' '
`% '~
I
SIZE
-,,
,c
;-,
cm
FIG. 5. Velocity and size relation. Filled circles show the minimum size, and open circles show the maximum size. The curves show the constant-energy line. The energy is calculated for the ellipsoidal fragment of three axis ratios 2s, V~s, and s (s is the size expressed on abscissa).
Fic. 6. Four typical motions of the fragment ejected from the surface of the spherical target are illustrated schematically. It is assumed that the fragments are rectangular shaped and that the longest, intermediate, and shortest axes are along the meridians, latitudes, and normal to the target surface, respectively. Arrows attached to each fragment show the spin vector. Notice that all the fragments manifest their shortest axes during rotation.
ment images flying a w a y are illustrated schematically in Fig. 6. In this figure the fragments are a s s u m e d to be rectangular shaped and initially the longest axis is along the meridian lines on the target surface (the N - S line is a s s u m e d to be along the z-axis), the shortest axis is along the normal to the surface, and the intermediate axis is along the latitude lines. (These a s s u m p t i o n s are verified later as a rough approximation.) As found in the figure, each fragment manifests its shortest axis once during a half-period of rotation while the longest or intermediate axes do not a p p e a r as real lengths. E v e n for the actual nonrectangular-shaped fragments the shortest axis can be o b s e r v e d in the film, but the longest and intermediate axes cannot. Therefore, only in Fig. 5 (and Fig. 8) w e r e the m a x i m u m sizes shown as a reference as well as the m i n i m u m sizes. The average three axes ratio o f the rock fragments was determined by m a n y authors as 2 : x/2 : 1 in a good a p p r o x i m a t i o n (Fujiwara et el. 1978, Matsui et el. 1982, Bianchi et el. 1984, Capaccioni et el. 1984). In the following analysis the v e r y simplified assumption was m a d e that all fragments have the rectangular shape with axial ratios given
540
AKIRA FUJIWARA
above. It should be noted that this assumption brings some arbitrariness into the determination of the fragment mass and that if we assume the ellipsoidal body of the same axial size the mass is reduced by a factor 7r/6 = 0.52. The translational energy Et is shown in Fig. 7. As found in this figure and in Fig. 5, most of the surface fragments have Et of order less than 10 6-.7 erg, or less t h a n 10 - 4 - - 3 times the projectile kinetic energy. It seems that E, values have some trend as a function of size, but in the present stage it is difficult to decide the functional dependence.
T 10 -2 b
m
~v
taJ
ut --~w
LO
10 -3
,
i
J ~,,,,1~/,
10-I SIZE
10 3
1 cm
FIG. 8. Rotation period (frequency) and size. The left and right ends of each horizontal error bar are the minimum and the maximum sizes, respectively. Tm~, shows the lower bound on the rotational period for the fragment to be able to sustain itself against the centrifugal force (see main text).
(2) Rotational Energy Rotational energy of a fragment is expressed by 1 Er = ~ lto 2, where 1 and to are moment of inertia and angular velocity, respectively. The angular velocity or rotational frequency is determined directly from the movie record. Resolution of the images seen on the still film are not so good, and determining the rotational frequency by following a fragment on the successive still frames is difficult. However, when the frag-
x rov 0o
6
ql~obj
~ °k
el
5
4 wo
2
i
0
i
I
i
i
i
0.5 M I N I M U M SIZE
,
,
'
•
1.0 cm
FIG. 7. Translational energy plotted against minimum size of the fragments.
ment is observed in a movie, the fragment image and its motion become very clear, because in our eyesight the vague images in every frame are overlapped and noises in the sequential frames are averaged and the signal to noise ratio is greatly enhanced. Observing the movie, when the fragment makes a full rotation or a certain fraction of the full rotation, the movie analyzer is stopped and the frame number is counted. In Fig. 8 the rotation frequency vs size diagram is shown. The accuracy in determination of the frequency depends on the fragment shape and attitude seen from the observer, and the errors are given in the figure. The data occupy approximately the same region in the diagram as in our previous report (Fujiwara and Tsukamoto 1981). In order to determine I, we must know the orientation of the spin axis in the coordinate system fixed to the fragment body as well as the shape of the fragment. It is difficult to determine the orientation of the spin axis in the coordinate system fixed to the fragment body by following the fragment image, but a general tendency of the orientation is obtained by noting the following. First, some other spherical targets which have been fragmented under similar impact conditions but at lower E/M values (less severe fragmentation) were reassembled from a considerable number of fragments
CATASTROPHIC DISRUPTION BY IMPACT like a three-dimensional jigsaw puzzle, and from these reassembled targets we learned in what attitude the individual fragments were originally packed in the target. We found that each surface fragment is a body having, as a first rough approximation, six fundamental facets as illustrated in Fig. 6, although many erratic facets appear due to the statistical nature common to the fracture phenomena. One of the facets is, of course, the target surface, and the second facet is the spallation plane, which is approximately parallel to the target surface. The other four facets are given by two sets of fracture planes appearing near the target surface; two approximately parallel facets along the planes which include the z-axis (F-plane; zr and ~-' in Fig. 3 belong to this plane group), and the other two approximately parallel facets along the spherical planes with their centers at the impact point (S-plane). We found that the longest axis of each fragment tends to be along the intersecting line of the F-plane and the target surface (meridional line), the intermediate axis along the intersecting line of the Splane with the target surface (latitude line), and hence the shortest axis along the normal to the target surface except for the fragments from the rear side of the target, whose longest and intermediate sizes are approximately comparable. This result is valid for spherical targets, and there is evidence that different target shapes give different results (P. Cerroni, private communication, 1985). On the other hand, there is a clear tendency in the orientation of the spin axis in the space coordinate system. The fragments tumble as shown in Figs. 2 and Fig. 6 (Fujiwara and Tsukamoto 1981). The spin vectors are oriented clockwise around the vector directed from the target center to the impact point. Therefore, we conclude that the spin axis is approximately parallel to the intermediate axis of the nearly rectangular shape of the fragments. For some fragments this property was clearly observed on the sequence of frames (see, for example, the fragment f i n Fig. 2).
541
Again, assuming that all fragments have a rectangular shape, we calculate the moment of inertia I and the rotational energy of each fragment. The resultant rotational energy of the individual fragments has a value of the order of 104 erg or less. In Fig. 9 the ratio Er/Et is shown as a function of size. Notice that Er/Et does not include the fragment mass, although it depends on the shape of the fragment slightly through the expression of the moment of inertia. This is a great merit, because accurate determination of the mass is very difficult, and the considerable amount of error is inevitable in Er or Et representation. This figure shows that the ratio has the order of 10-2 or less. The size dependence of Et, Er, and Er/Et is not clear because of the paucity of the amount of data. Also the correlation between Er and Et is not clear, as long as the data for the two small fragments named h and i, which have much smaller Er and Et compared with other larger fragments, are not included, since in the smaller size range the obtained data are only for these two fragments and may be accidental. 4. COMPARISONWITH ASTEROID FAMILIES It is intriguing to compare our ratio Er/Et with that of some asteroids belonging to families, some of which, for example, Themis, Eos, and Koronis, are considered to be of collisional origin. The formation of each family as an outcome of a catastrophic -1
,
i
,
. . . .
i
i
i
Iw f==J g ~ b
Ik
., oi , Io i ~ -~
I,
rZ~.
o,
MINIMUM
0.5 SIZE
1.0 cm
FIG. 9. Ratio of rotational energy to translational energy of the fragments.
542
AKIRA FUJIWARA
disruption by an impact was discussed by Fujiwara (1982) and Zappal~ et al. (1984). Translational kinetic energy of the asteroid members belonging to a family, which is to be compared with the laboratory result, is the sum of the energy of motion after escape from the parent body and the escape energy from the parent body. Velocity component Vf,p of each asteroid member relative to and projected to the velocity vector of the largest member in the family is calculated from the difference Aa between semimajor axes of the member and the largest member, respectively, with the assumption that the orbital element of the largest member is unchanged from that of the parent body before disruption (Fujiwara 1982). F o l l o w i n g Ip (1979), and Zappal~_ et al. (1984), Vf,p = nAa/2, where n is the orbital mean motion of the largest asteroid. We assume that the three-
dimensional velocity vectors vr'S are approximately uniformly oriented in the velocity space and that v~ = 30~,p is statistically most probable. This assumption is not strictly justified, but the minor change in the multiplication factor does not seriously affect the translational energy and will suffice for the order of magnitude estimate in the present purpose. Coupled with the mass, we calculate the energy Er = (1/ 2)mv~ for each family member. Adding the escape energy from the largest asteroid we obtain the translational energy Et for each asteroid. Rotational energies of the asteroids are calculated from the rotational periods and the size data in TRIAD (1979) assuming that the asteroids have a spherical shape. Figure 10 shows E, vs Er for Themis, Eos, and Koronis members. The ratio Er/Et distributes in the region of the order of 10-2 or less. The contribution of the escape energy part to Et is so great that if the escape energy part was not included in Et the upper
30
28
O .J
26
22
24
26
28
LOGIo E r erg
FIG. 10. Translationalenergyplotted against rotationalenergyfor Themis, Eos, and Koronisfamily members. Asteroid numbers are indicated. Marks with arrows show upper limits.
CATASTROPHIC DISRUPTION BY IMPACT limit of Er/Et for the families would be higher than or as high as an order of 1. It should be noted that the highest Er/Et values for the three families are comparable in order in spite of the fact that Themis members have higher Er and Et values compared with the other two family members. It is remarkable that the upper bounds on the ratios Er/Et obtained for the rock targets in the laboratory and the asteroid family members have orders of magnitude that are not so different in spite of the great difference in scale between the laboratory target and the asteroids. We consider the possibility that the upper bound of Er/Et obtained in the laboratory may be insensitive to the scale. A brief consideration of the acquisition of the rotation in the catastrophic disruption is given by Fujiwara (1981). When the strain energy density in the target material reaches a certain value, a fragment is cut out by fracturing. The rotation energy of the fragment is supplied by the release of strain energy stored in the fragment before the fracture. If the maximum ratio of the strain energy converted to the rotation energy of the fragment is k, the maximum rotation energy is given by Er,max = keV,
where e is strain energy per unit volume of the target material before the fracture and V is the volume of the fragment. The upper bound on Er/Et is Er,max _
Et
ke
lpV2'
where p is the material density. The parameter k depends on several parameters specifying the fracture process, for example, dynamical properties of the material. As long as the material properties are the same and the stress distribution in the material is similar, the parameter k will be approximately similar for the differing scale of events, and therefore the maximum Er/Et will be insensitive to the scale. An alternate explanation may be given in
543
terms of strength (D. Davis, private communication, 1986). The fracturing occurs in such a way that the centrifugal force induced due to the rotation of the fragment does not exceed the stress inside the newly formed fragment. A rotating sphere has nonhydrostatic stresses which vary as pto2r2 (Weidenschilling 1981), where p is the density, to the angular velocity, and r is the radius. The limiting spin rate for a body is considered to occur when these stresses equal the tensile strength, S. Hence an upper bound on the rotational energy is approximately given by
1
o~m__~s
Er = ~ otmr2to 2 < 2p '
where a is a constant and S is the tensile strength. Hence the energy ratio is Er,max
Et
o~S
p V 2"
Here, again, we obtain an expression for the upper bound to Er/Et, which is independent of the scale. This explanation is based on the assumption that the maximum angular velocity should be given by ptoZaxr2 = S.
The minimum rotation period is, therefore, Tmin =
"n'D ~/~,
where D is the diameter of the spherical fragment. The Tmi,vs D line is drawn in Fig. 8 for the present experimental conditions (p = 2.7 g c m - 3 , S = 3 x 107 dyn cm-2). The Tmi, line is an order of magnitude lower than the apparent lower bound to the domain of the data points. It seems that the actual maximum rotation rate cannot attain the theoretical "centrifugal" limit, although the above explanation may be oversimplified and more sophisticated theory is needed. Although the upper bounds o n Er/Et are not so unsimilar, those for the laboratory experiments have significantly higher values than those for the family asteroids.
544
AKIRA FUJIWARA
Some possible causes for the difference must be considered. The first possible cause is that the constant upper bound to Er/Et obtained from the laboratory experiment may not be scaled strictly up to the asteroid sizes due to the difference in the material properties and/or the shock profiles or due to some other unknown reasons. The second possibility is that the family members had, at first, approximately the same limiting E/EI value as in the laboratory experiments immediately after the catastrophic breakup, but the rotation has been gradually slowed down to the current state due to many repeated small impacts which act as drags to the spinning motion (Harris 1979). This possibility is plausible, but more detailed study is needed. 5. CONCLUSIONS In our laboratory simulation of catastrophic disruption the following results were obtained. (1) The maximum ratio of rotational kinetic energy to translational kinetic energy of the fragment is of the order of 10-2. (2) The maximum value of Er/Et for the asteroid family members is not so different from the one obtained in the laboratory experiments in spite of the great difference in scale. However, the value for the asteroid is significantly lower than that in the laboratory. This may be due to the fact that the laboratory result cannot strictly be scaled or that the family members have been slowed down by repeated collisions after the catastrophic disruption of the parent bodies. Finally, our experimental approach to investigating the translational and rotational energies is still in the preliminary stage in many respects. The distributions of Er/Et, Er, and Et, as the function of fragment size and other parameters, are not clear in this experiment. The fragments measured in our analysis are only those from the surface of the target. The possibility cannot be corn-
pletely excluded that there may exist fragments of Er/Et higher than 10 -1 among the fragments from the surface or undersurface positions neighboring the impact point. More extensive and systematic experimental studies as well as theoretical work are needed. ACKNOWLEDGMENTS I thank A. Tsukamoto for assistance, and M. Yoneda for using the movie analyzer. Critical reviewing by D. Davis, P. Cerroni, and P. Farinella for the first version is gratefully acknowledged. Part of this work was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, and Culture of Japan (No. 59390007).
REFERENCES B1ANCHI, R., F. CAPACCION1, P. CERRONI, M. CORADINI, E. FLAMINI, P. HURREN, G. MARTELLI, AND P, N. SMITH 1984. Experimental simulation of asteroidal fragmentation by macroscopic hypervelocity impacts against free falling bodies. Astron. Astrophys. 139, 1-6. CAPACCIONI, F., P. CERRON[, M. CORADINI, P. FARINELLA, E. FLAMINI, G. MARTELLI, P. PAOLICCHI, P. N. SMITH, AND V. ZAPPAL)t 1984. Shapes of asteroids compared with fragments from hypervelocity impact experiments. N a t u r e 308, 832-834. CERRONI, P. 1986. Simulation of catastrophic fragmentation events: Review of the accelerating techniques and a resume of the experiments performed until now. Mere. Soc. Astron. ltal. 57, 13-45. FUJIWARA, A. 1982. Complete fragmentation of the parent bodies of Themis, Eos, and Koronis families. Icarus 52, 434-443. FUJIWARA, A. 1986. Results obtained by laboratory simulations of catastrophic impact. Mere. Soc. Astron. ltal. 57, 47-64. FUJIWARA, A., G, KAMIMOTO, AND A. TSUKAMOTO 1977. Destruction of basaltic bodies by high-velocity impact. Icarus 31, 277-288. FUJIWARA, A., G. KAMIMOTO, AND A. TSUKAMOTO 1978. Expected shape distribution of asteroids obtained from laboratory impact experiments. Nature 272, 602-603. FUJIWARA, A., AND A. TSUKAMOTO 1980. Experimental study on the velocity of fragments in collisional breakup. Icarus 44, 142-153. FUJIWARA, A., AND m. TSUKAMOTO 1981. Rotation of fragments in catastrophic impact. Icarus 48, 329334. HARRIS, A. W. 1979. Asteroid rotation rates. I1. A theory for the collisional evolution of rotation rates. Icarus 40, 145-153.
CATASTROPHIC
DISRUPTION BY IMPACT
IP, W, -H. 1979. On three types of fragmentation processes observed in the asteroid belt. Icarus 40, 418422. MATSUI, T., T. WAZA, K. KANI, AND S. SUZUKI 1982. Laboratory simulation of planetesimal collisions. J. Geophys. Res. 87, 10,968-10,982. MIZUTANI, H., Y. TAKAGI, S. KAWAKAMI 1985. New Scaling Law on Impact Fragmentation. Preprint. TRIAD (TUCSON REVISED INDEX OF ASTEROID DATA)
545
1979. IN Asteroids (T. Gehrels, Ed.), pp. 1011-1154. Univ. of Arizona Press, Tucson. WAZA, T., T. MATSUI, AND K. KANI 1985. Laboratory simulation of planetesimal collision. 2. Ejecta velocity distribution. J. Geophys. Res. 90, 19952011. ZAPPALA, V., P. FARINEELA, Z. KNEZEVIC, AND P. PAOLICCHI 1984. Collisional origin of the asteroid families: Mass and velocity distributions. Icarus 59, 261-285.
Energy Partition into Translational and Rotational Motion of Fragments in Catastrophic Disruption by Impact: An Experiment and Asteroid Cases AKIRA FUJIWARA Department of Physics, Kyoto University, Kyoto 606, Japan
Translational kinetic energy Et and rotational kinetic energy Er of the fragments produced in the laboratory catastrophic disruption of a basalt sphere by the impact of a highspeed projectile were determined. It was found that the maximum value of Er/Et is of the order of 10 -z, which is not so different in the order of magnitude from the value estimated for family asteroids in spite of the great difference of the scale. However, the laboratory value is significantly higher than the value for the family. © 1987AcademicPress,Inc.
I. I N T R O D U C T I O N
Fragmental bodies produced in catastrophic disruption of minor planets by impact acquire spinning motion as well as the translational motion relative to the parent body. Knowing the amount of energy partitioned into rotational and translational motion is very useful in understanding the orbital and rotational properties of groups of minor bodies of collisional origin such as some asteroid families. Elementary processes on the catastrophic disruption have been studied on the small scale of laboratory experiments; typically centimeter- to millimeter-sized projectiles are impacted into 1- to 10-cm-sized targets. The velocity and kinetic energy (of translational motion) of some fragments ejected in catastrophic disruption have been determined by Fujiwara and Tsukamoto (1980), Mizutani et al. (1985), Waza et al. (1985), and the spin periods of the fragments by Fujiwara and Tsukamoto (1981). Experimental activities and the results in this field are reviewed elsewhere (Cerroni 1986, Fujiwara 1986). In this paper we present data on the translational and the rotational energies for the individual fragments produced by a labora-
tory catastrophic disruption of a small rock target by an impact. As shown later, in order to know the translational and rotational energies a set of many physical parameters such as mass, velocity, size, shape, and angular velocity of spin must be determined at the same time for the individual fragment. Since determining many parameters at the same time presents many difficulties, the data points presented here are rather sparse, and in the present stage we cannot give the energy distribution of the fragments as a function of size. However, a few new results are obtained on the order of magnitude of the translational and rotational energies, and on the ratio of these energies, which has an intriguing meaning for asteroid families. In the next section the experimental conditions are given, in Section 3 the procedures of analysis and some results are given, and in Section 4 the results are compared with some asteroid families. 2. E X P E R I M E N T A L
Experimental conditions are similar to those described in our previous series of experiments (Fujiwara et al. 1977, Fujiwara and Tsukamoto 1980), and we summarize 536
0019-1035/87 $3.00 Copyright ~c~1987 by Academic Press, Inc. All rights of reproduction in any form reserved.
CONDITIONS
CATASTROPHIC DISRUPTION BY IMPACT them here only briefly. A basalt sphere of diameter 6.35 cm and mass 357 g was destroyed by a central impact of a cylindrical polycarbonate projectile of diameter 8 mm and mass 0.37 g launched at velocity 2.5 km/sec by a two-stage light-gas gun. The projectile kinetic energy E is 1.16 x 101° erg, and the impact energy per unit target mass E / M is 3.24 × 10 7 erg/g. The destruction type by the Fujiwara et al. (1977) classification is the "core t y p e " ; the surface and near-surface layers are peeled off and the central part is left intact. The core mass was 55 g, or 15% of the target mass. The motion of the fragments was pictured with a high-speed framing camera at a framing rate of 7,169 frames/sec through the side window of the chamber containing the target. Some selected frames from the film are shown in Fig. I. Characteristic behaviors of
537
the fragments ejected in a similar impact condition are described by Fujiwara and Tsukamoto (1980). 3. REDUCTION PROCEDURES AND RESULTS
We present the procedures used in obtaining the translational and rotational energies of the individual fragments from the movie film, and show some results.
(1) Translational Energy E1 The translational energy is expressed as Et = my2~2,
where m and v are the mass and translational velocity of the fragment, respectively. (a) Translational velocity v. The translational velocity v was obtained by tracing the translational displacement of the individual
FIG. 1. Selected frames from the high-speed film. The basalt target is 6.35 cm in diameter. The projectile comes from the right. The numbers show the time elapsed after the impact (in msec).
538
AKIRA FUJIWARA b
a
I FIG. 2. Example of tracing of some fragments. Numbers show the flame number counted from the " i m p a c t " flame (numbered zero). Arrows show the rotation of the fragments. An alphabetical letter is assigned to each fragment (also in Figs. 4, 5, 7, 8, and 9).
fragments one by one on the frames using a movie analyzer (a movie projector with variable framing speeds). Figure 2 shows examples of the movements of some fragments. In this work we consider only the fragments ejected from the target surface facing the camera, since only for those fragments could three-dimensional velocities be determined by the method discussed below. As shown in Fig. 3, the observer looks at the fragment images projected onto a plane 7r' (x-z plane in the figure). The observable quantities are projected velocity v/, and the projected ejection point P'/. The velocity v/ obtained directly from the movie film is shown in Fig. 4. In order to transform the projected velocities and ejection points to three-dimensional ones, we made the assumption that each velocity vector is on one of the sets of planes (in Fig. 3) which include the line connecting the center of the target with the impact point (the line of flight of the projectile). Very fine fragments near the impact point, whose images could not be resolved on the film, were excluded from the analysis. The total mass of the
fragments analyzed is estimated to be approximately 10-20% of the mass excluding the core. Translational velocity (and angular velocity) of the core could not be determined, because both were very small. The resultant velocity is shown in Fig. 5. The x 71-
"Jr"
Y FIG. 3. Configuration of the observer, target, and projectile trajectory, v" and r are velocity and initial position of a fragment, respectively, situated on the plane ~r. ~' and z" are projected onto the plane 7r' which includes x and z axes.
CATASTROPHIC DISRUPTION BY IMPACT
\
tl w
539
;
u
v
S
r
t
c
MPACT
."
:
*.
~ 3 0 m s -1
FIG. 4. Projected velocity vectors of the fragments. velocity d e c r e a s e s systematically with the size, which was m e a s u r e d as described below. (b) F r a g m e n t m a s s . Determination of the fragment size on the m o v i e film is not easy, b e c a u s e each fragment flies a w a y changing its shape f r o m f r a m e to f r a m e due to its spinning motion. The m i n i m u m size of any fragment was determined in the following way. The smallest size o f the fragment is m e a s u r e d on each f r a m e as the distance between a pair of parallel lines contacting the fragment image f r o m both sides. W h e n the smallest size takes a m i n i m u m value on one of the sequential frames, the minimum value is defined as the m i n i m u m size. The m a x i m u m sizes were also m e a s u r e d but they m a y contain m a n y errors. This is due to the following reason. S o m e typical frag-
_
I0
f
. . . . . . . . . .
0.01
'
'
'
0.I
' '
`% '~
I
SIZE
-,,
,c
;-,
cm
FIG. 5. Velocity and size relation. Filled circles show the minimum size, and open circles show the maximum size. The curves show the constant-energy line. The energy is calculated for the ellipsoidal fragment of three axis ratios 2s, V~s, and s (s is the size expressed on abscissa).
Fic. 6. Four typical motions of the fragment ejected from the surface of the spherical target are illustrated schematically. It is assumed that the fragments are rectangular shaped and that the longest, intermediate, and shortest axes are along the meridians, latitudes, and normal to the target surface, respectively. Arrows attached to each fragment show the spin vector. Notice that all the fragments manifest their shortest axes during rotation.
ment images flying a w a y are illustrated schematically in Fig. 6. In this figure the fragments are a s s u m e d to be rectangular shaped and initially the longest axis is along the meridian lines on the target surface (the N - S line is a s s u m e d to be along the z-axis), the shortest axis is along the normal to the surface, and the intermediate axis is along the latitude lines. (These a s s u m p t i o n s are verified later as a rough approximation.) As found in the figure, each fragment manifests its shortest axis once during a half-period of rotation while the longest or intermediate axes do not a p p e a r as real lengths. E v e n for the actual nonrectangular-shaped fragments the shortest axis can be o b s e r v e d in the film, but the longest and intermediate axes cannot. Therefore, only in Fig. 5 (and Fig. 8) w e r e the m a x i m u m sizes shown as a reference as well as the m i n i m u m sizes. The average three axes ratio o f the rock fragments was determined by m a n y authors as 2 : x/2 : 1 in a good a p p r o x i m a t i o n (Fujiwara et el. 1978, Matsui et el. 1982, Bianchi et el. 1984, Capaccioni et el. 1984). In the following analysis the v e r y simplified assumption was m a d e that all fragments have the rectangular shape with axial ratios given
540
AKIRA FUJIWARA
above. It should be noted that this assumption brings some arbitrariness into the determination of the fragment mass and that if we assume the ellipsoidal body of the same axial size the mass is reduced by a factor 7r/6 = 0.52. The translational energy Et is shown in Fig. 7. As found in this figure and in Fig. 5, most of the surface fragments have Et of order less than 10 6-.7 erg, or less t h a n 10 - 4 - - 3 times the projectile kinetic energy. It seems that E, values have some trend as a function of size, but in the present stage it is difficult to decide the functional dependence.
T 10 -2 b
m
~v
taJ
ut --~w
LO
10 -3
,
i
J ~,,,,1~/,
10-I SIZE
10 3
1 cm
FIG. 8. Rotation period (frequency) and size. The left and right ends of each horizontal error bar are the minimum and the maximum sizes, respectively. Tm~, shows the lower bound on the rotational period for the fragment to be able to sustain itself against the centrifugal force (see main text).
(2) Rotational Energy Rotational energy of a fragment is expressed by 1 Er = ~ lto 2, where 1 and to are moment of inertia and angular velocity, respectively. The angular velocity or rotational frequency is determined directly from the movie record. Resolution of the images seen on the still film are not so good, and determining the rotational frequency by following a fragment on the successive still frames is difficult. However, when the frag-
x rov 0o
6
ql~obj
~ °k
el
5
4 wo
2
i
0
i
I
i
i
i
0.5 M I N I M U M SIZE
,
,
'
•
1.0 cm
FIG. 7. Translational energy plotted against minimum size of the fragments.
ment is observed in a movie, the fragment image and its motion become very clear, because in our eyesight the vague images in every frame are overlapped and noises in the sequential frames are averaged and the signal to noise ratio is greatly enhanced. Observing the movie, when the fragment makes a full rotation or a certain fraction of the full rotation, the movie analyzer is stopped and the frame number is counted. In Fig. 8 the rotation frequency vs size diagram is shown. The accuracy in determination of the frequency depends on the fragment shape and attitude seen from the observer, and the errors are given in the figure. The data occupy approximately the same region in the diagram as in our previous report (Fujiwara and Tsukamoto 1981). In order to determine I, we must know the orientation of the spin axis in the coordinate system fixed to the fragment body as well as the shape of the fragment. It is difficult to determine the orientation of the spin axis in the coordinate system fixed to the fragment body by following the fragment image, but a general tendency of the orientation is obtained by noting the following. First, some other spherical targets which have been fragmented under similar impact conditions but at lower E/M values (less severe fragmentation) were reassembled from a considerable number of fragments
CATASTROPHIC DISRUPTION BY IMPACT like a three-dimensional jigsaw puzzle, and from these reassembled targets we learned in what attitude the individual fragments were originally packed in the target. We found that each surface fragment is a body having, as a first rough approximation, six fundamental facets as illustrated in Fig. 6, although many erratic facets appear due to the statistical nature common to the fracture phenomena. One of the facets is, of course, the target surface, and the second facet is the spallation plane, which is approximately parallel to the target surface. The other four facets are given by two sets of fracture planes appearing near the target surface; two approximately parallel facets along the planes which include the z-axis (F-plane; zr and ~-' in Fig. 3 belong to this plane group), and the other two approximately parallel facets along the spherical planes with their centers at the impact point (S-plane). We found that the longest axis of each fragment tends to be along the intersecting line of the F-plane and the target surface (meridional line), the intermediate axis along the intersecting line of the Splane with the target surface (latitude line), and hence the shortest axis along the normal to the target surface except for the fragments from the rear side of the target, whose longest and intermediate sizes are approximately comparable. This result is valid for spherical targets, and there is evidence that different target shapes give different results (P. Cerroni, private communication, 1985). On the other hand, there is a clear tendency in the orientation of the spin axis in the space coordinate system. The fragments tumble as shown in Figs. 2 and Fig. 6 (Fujiwara and Tsukamoto 1981). The spin vectors are oriented clockwise around the vector directed from the target center to the impact point. Therefore, we conclude that the spin axis is approximately parallel to the intermediate axis of the nearly rectangular shape of the fragments. For some fragments this property was clearly observed on the sequence of frames (see, for example, the fragment f i n Fig. 2).
541
Again, assuming that all fragments have a rectangular shape, we calculate the moment of inertia I and the rotational energy of each fragment. The resultant rotational energy of the individual fragments has a value of the order of 104 erg or less. In Fig. 9 the ratio Er/Et is shown as a function of size. Notice that Er/Et does not include the fragment mass, although it depends on the shape of the fragment slightly through the expression of the moment of inertia. This is a great merit, because accurate determination of the mass is very difficult, and the considerable amount of error is inevitable in Er or Et representation. This figure shows that the ratio has the order of 10-2 or less. The size dependence of Et, Er, and Er/Et is not clear because of the paucity of the amount of data. Also the correlation between Er and Et is not clear, as long as the data for the two small fragments named h and i, which have much smaller Er and Et compared with other larger fragments, are not included, since in the smaller size range the obtained data are only for these two fragments and may be accidental. 4. COMPARISONWITH ASTEROID FAMILIES It is intriguing to compare our ratio Er/Et with that of some asteroids belonging to families, some of which, for example, Themis, Eos, and Koronis, are considered to be of collisional origin. The formation of each family as an outcome of a catastrophic -1
,
i
,
. . . .
i
i
i
Iw f==J g ~ b
Ik
., oi , Io i ~ -~
I,
rZ~.
o,
MINIMUM
0.5 SIZE
1.0 cm
FIG. 9. Ratio of rotational energy to translational energy of the fragments.
542
AKIRA FUJIWARA
disruption by an impact was discussed by Fujiwara (1982) and Zappal~ et al. (1984). Translational kinetic energy of the asteroid members belonging to a family, which is to be compared with the laboratory result, is the sum of the energy of motion after escape from the parent body and the escape energy from the parent body. Velocity component Vf,p of each asteroid member relative to and projected to the velocity vector of the largest member in the family is calculated from the difference Aa between semimajor axes of the member and the largest member, respectively, with the assumption that the orbital element of the largest member is unchanged from that of the parent body before disruption (Fujiwara 1982). F o l l o w i n g Ip (1979), and Zappal~_ et al. (1984), Vf,p = nAa/2, where n is the orbital mean motion of the largest asteroid. We assume that the three-
dimensional velocity vectors vr'S are approximately uniformly oriented in the velocity space and that v~ = 30~,p is statistically most probable. This assumption is not strictly justified, but the minor change in the multiplication factor does not seriously affect the translational energy and will suffice for the order of magnitude estimate in the present purpose. Coupled with the mass, we calculate the energy Er = (1/ 2)mv~ for each family member. Adding the escape energy from the largest asteroid we obtain the translational energy Et for each asteroid. Rotational energies of the asteroids are calculated from the rotational periods and the size data in TRIAD (1979) assuming that the asteroids have a spherical shape. Figure 10 shows E, vs Er for Themis, Eos, and Koronis members. The ratio Er/Et distributes in the region of the order of 10-2 or less. The contribution of the escape energy part to Et is so great that if the escape energy part was not included in Et the upper
30
28
O .J
26
22
24
26
28
LOGIo E r erg
FIG. 10. Translationalenergyplotted against rotationalenergyfor Themis, Eos, and Koronisfamily members. Asteroid numbers are indicated. Marks with arrows show upper limits.
CATASTROPHIC DISRUPTION BY IMPACT limit of Er/Et for the families would be higher than or as high as an order of 1. It should be noted that the highest Er/Et values for the three families are comparable in order in spite of the fact that Themis members have higher Er and Et values compared with the other two family members. It is remarkable that the upper bounds on the ratios Er/Et obtained for the rock targets in the laboratory and the asteroid family members have orders of magnitude that are not so different in spite of the great difference in scale between the laboratory target and the asteroids. We consider the possibility that the upper bound of Er/Et obtained in the laboratory may be insensitive to the scale. A brief consideration of the acquisition of the rotation in the catastrophic disruption is given by Fujiwara (1981). When the strain energy density in the target material reaches a certain value, a fragment is cut out by fracturing. The rotation energy of the fragment is supplied by the release of strain energy stored in the fragment before the fracture. If the maximum ratio of the strain energy converted to the rotation energy of the fragment is k, the maximum rotation energy is given by Er,max = keV,
where e is strain energy per unit volume of the target material before the fracture and V is the volume of the fragment. The upper bound on Er/Et is Er,max _
Et
ke
lpV2'
where p is the material density. The parameter k depends on several parameters specifying the fracture process, for example, dynamical properties of the material. As long as the material properties are the same and the stress distribution in the material is similar, the parameter k will be approximately similar for the differing scale of events, and therefore the maximum Er/Et will be insensitive to the scale. An alternate explanation may be given in
543
terms of strength (D. Davis, private communication, 1986). The fracturing occurs in such a way that the centrifugal force induced due to the rotation of the fragment does not exceed the stress inside the newly formed fragment. A rotating sphere has nonhydrostatic stresses which vary as pto2r2 (Weidenschilling 1981), where p is the density, to the angular velocity, and r is the radius. The limiting spin rate for a body is considered to occur when these stresses equal the tensile strength, S. Hence an upper bound on the rotational energy is approximately given by
1
o~m__~s
Er = ~ otmr2to 2 < 2p '
where a is a constant and S is the tensile strength. Hence the energy ratio is Er,max
Et
o~S
p V 2"
Here, again, we obtain an expression for the upper bound to Er/Et, which is independent of the scale. This explanation is based on the assumption that the maximum angular velocity should be given by ptoZaxr2 = S.
The minimum rotation period is, therefore, Tmin =
"n'D ~/~,
where D is the diameter of the spherical fragment. The Tmi,vs D line is drawn in Fig. 8 for the present experimental conditions (p = 2.7 g c m - 3 , S = 3 x 107 dyn cm-2). The Tmi, line is an order of magnitude lower than the apparent lower bound to the domain of the data points. It seems that the actual maximum rotation rate cannot attain the theoretical "centrifugal" limit, although the above explanation may be oversimplified and more sophisticated theory is needed. Although the upper bounds o n Er/Et are not so unsimilar, those for the laboratory experiments have significantly higher values than those for the family asteroids.
544
AKIRA FUJIWARA
Some possible causes for the difference must be considered. The first possible cause is that the constant upper bound to Er/Et obtained from the laboratory experiment may not be scaled strictly up to the asteroid sizes due to the difference in the material properties and/or the shock profiles or due to some other unknown reasons. The second possibility is that the family members had, at first, approximately the same limiting E/EI value as in the laboratory experiments immediately after the catastrophic breakup, but the rotation has been gradually slowed down to the current state due to many repeated small impacts which act as drags to the spinning motion (Harris 1979). This possibility is plausible, but more detailed study is needed. 5. CONCLUSIONS In our laboratory simulation of catastrophic disruption the following results were obtained. (1) The maximum ratio of rotational kinetic energy to translational kinetic energy of the fragment is of the order of 10-2. (2) The maximum value of Er/Et for the asteroid family members is not so different from the one obtained in the laboratory experiments in spite of the great difference in scale. However, the value for the asteroid is significantly lower than that in the laboratory. This may be due to the fact that the laboratory result cannot strictly be scaled or that the family members have been slowed down by repeated collisions after the catastrophic disruption of the parent bodies. Finally, our experimental approach to investigating the translational and rotational energies is still in the preliminary stage in many respects. The distributions of Er/Et, Er, and Et, as the function of fragment size and other parameters, are not clear in this experiment. The fragments measured in our analysis are only those from the surface of the target. The possibility cannot be corn-
pletely excluded that there may exist fragments of Er/Et higher than 10 -1 among the fragments from the surface or undersurface positions neighboring the impact point. More extensive and systematic experimental studies as well as theoretical work are needed. ACKNOWLEDGMENTS I thank A. Tsukamoto for assistance, and M. Yoneda for using the movie analyzer. Critical reviewing by D. Davis, P. Cerroni, and P. Farinella for the first version is gratefully acknowledged. Part of this work was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Science, and Culture of Japan (No. 59390007).
REFERENCES B1ANCHI, R., F. CAPACCION1, P. CERRONI, M. CORADINI, E. FLAMINI, P. HURREN, G. MARTELLI, AND P, N. SMITH 1984. Experimental simulation of asteroidal fragmentation by macroscopic hypervelocity impacts against free falling bodies. Astron. Astrophys. 139, 1-6. CAPACCIONI, F., P. CERRON[, M. CORADINI, P. FARINELLA, E. FLAMINI, G. MARTELLI, P. PAOLICCHI, P. N. SMITH, AND V. ZAPPAL)t 1984. Shapes of asteroids compared with fragments from hypervelocity impact experiments. N a t u r e 308, 832-834. CERRONI, P. 1986. Simulation of catastrophic fragmentation events: Review of the accelerating techniques and a resume of the experiments performed until now. Mere. Soc. Astron. ltal. 57, 13-45. FUJIWARA, A. 1982. Complete fragmentation of the parent bodies of Themis, Eos, and Koronis families. Icarus 52, 434-443. FUJIWARA, A. 1986. Results obtained by laboratory simulations of catastrophic impact. Mere. Soc. Astron. ltal. 57, 47-64. FUJIWARA, A., G, KAMIMOTO, AND A. TSUKAMOTO 1977. Destruction of basaltic bodies by high-velocity impact. Icarus 31, 277-288. FUJIWARA, A., G. KAMIMOTO, AND A. TSUKAMOTO 1978. Expected shape distribution of asteroids obtained from laboratory impact experiments. Nature 272, 602-603. FUJIWARA, A., AND A. TSUKAMOTO 1980. Experimental study on the velocity of fragments in collisional breakup. Icarus 44, 142-153. FUJIWARA, A., AND m. TSUKAMOTO 1981. Rotation of fragments in catastrophic impact. Icarus 48, 329334. HARRIS, A. W. 1979. Asteroid rotation rates. I1. A theory for the collisional evolution of rotation rates. Icarus 40, 145-153.
CATASTROPHIC
DISRUPTION BY IMPACT
IP, W, -H. 1979. On three types of fragmentation processes observed in the asteroid belt. Icarus 40, 418422. MATSUI, T., T. WAZA, K. KANI, AND S. SUZUKI 1982. Laboratory simulation of planetesimal collisions. J. Geophys. Res. 87, 10,968-10,982. MIZUTANI, H., Y. TAKAGI, S. KAWAKAMI 1985. New Scaling Law on Impact Fragmentation. Preprint. TRIAD (TUCSON REVISED INDEX OF ASTEROID DATA)
545
1979. IN Asteroids (T. Gehrels, Ed.), pp. 1011-1154. Univ. of Arizona Press, Tucson. WAZA, T., T. MATSUI, AND K. KANI 1985. Laboratory simulation of planetesimal collision. 2. Ejecta velocity distribution. J. Geophys. Res. 90, 19952011. ZAPPALA, V., P. FARINEELA, Z. KNEZEVIC, AND P. PAOLICCHI 1984. Collisional origin of the asteroid families: Mass and velocity distributions. Icarus 59, 261-285.