- Email: [email protected]
Early-stage ejecta velocity distribution for vertical hypervelocity impacts into sand
Icarus 209 (2010) 866–870
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Note
Early-stage ejecta velocity distribution for vertical hypervelocity impacts into sand Brendan Hermalyn *, Peter H. Schultz Department of Geological Sciences, Brown University, Box 1846, Providence, RI 02912, United States
a r t i c l e
i n f o
Article history: Received 27 October 2009 Revised 26 May 2010 Accepted 27 May 2010 Available online 4 June 2010
a b s t r a c t The ejecta dynamics during main-stage excavation flow in a cratering event have previously been well characterized, particularly for vertical impacts. In this experimental study, we present new results addressing the early-time, lowangle, high-speed component of the ejecta velocity distribution as a function of time for hypervelocity vertical impacts into sand. Although this regime represents a very small portion of total ejected mass in laboratory experiments, it comprises a greater percentage of growth for larger craters.
Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Impact processes Cratering Collisional physics Geological processes
1. Introduction
2. Experimental procedure
The ejecta velocity distribution of an impact event determines the emplacement of material on planetary bodies. Main-stage excavation flow comprises the majority of ejected material and has been well studied for vertical impacts as a function of launch position (e.g., McGetchin et al., 1973; Schultz and Gault, 1979). Dimensional analysis (Housen et al., 1983) predicts a single power-law relationship between ejection velocity and time (or ejection position) for main-stage excavation. Numerical simulations (Wada et al., 2006) and experimental studies (Piekutowski, 1980; Oberbeck and Morrison, 1976; Piekutowski et al., 1977; Cintala et al., 1999; Anderson et al., 2003) follow the predicted main-stage trend fairly well. This agreement reflects the ‘‘late-stage equivalence” concept of crater growth (Dienes and Walsh, 1970), where the details of the impact, i.e., the energy and momentum transfer of the projectile to the target, can be replaced by a singular point in time and space (e.g., Chabai, 1959; Orphal, 1977; Housen et al., 1983). Implicit in the dimensional analysis is that this ‘‘point source” approximation can be defined by a coupling parameter, C, where C is a function of projectile radius, impact velocity, and projectile density, as detailed by Holsapple and Schmidt (1987). Thus, early-time coupling establishes the ejecta flow field behind the initial shock. Subsequent ejection of material is described by main-stage ejecta flow and is limited by properties of the target and/or environment (i.e., gravity and atmosphere). As the projectile transfers its energy and momentum to the target at early times, however, the single power-law description of ejecta velocity does not hold. Prior experimental studies of ejecta velocities have generally not extended to this early-time regime. Moreover, hydrocode modeling of granular materials is still poorly constrained and uncalibrated, particularly for this stage of growth (Brown et al., 2007; Pierazzo et al., 2008). The goal of this study is to investigate the ejecta velocity distribution as a function of time and how it departs from standard scaling laws early in the cratering process.
2.1. Strategy
* Corresponding author. E-mail address: [email protected] (B. Hermalyn). 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.05.025
A suite of experiments conducted at the NASA Ames Vertical Gun Range (AVGR) addressed the effect of impact variables on the early-stage ejecta velocity distribution. All experiments were vertical impacts (i.e., orthogonal to the target surface) into #20–30 Ottawa sand. Time, rather than launch position, is used as the independent parameter in this study. Previous studies (e.g., Anderson and Schultz, 2006) have shown that the impact point in oblique impacts does not correspond to either the flow-field center (which migrates throughout much of crater growth) or crater center (which requires complicated correction terms), thus making the launch position metric convoluted. A temporal reconstruction of the event is more physically meaningful for the early-stage and will allow comparison with future studies of the ejecta velocity distribution in oblique impacts. In order to explore the influence of different impact variables on the early-time regime, three separate factors were varied. First, identical aluminum projectiles (6.35 mm diameter) were launched at different velocities: 1.6 km/s, 2.5 km/s, and 5.6 km/s. Second, aluminum projectiles with a range of diameters were used: 6.35 mm, 9.525 mm, and 12.7 mm. And third, the projectile density was varied through the use of different impactor materials: aluminum (q = 2.8 g/cm3), polyethylene (q = 0.93 g/cm3), and titanium (q = 4.4 g/ cm3) at the 6.35 mm diameter. 2.2. Measurement technique A nonintrusive imaging technique called particle tracking velocimetry (PTV) was used to determine the ejecta velocity of the experiments. The PTV technique, which is widely used in experimental fluid dynamics research, tracks the motion of individual particles on the profile of the ejecta curtain using several high-speed cameras (both color and grayscale) operating at 8000–15,000 frames per second (fps). The high degree of temporal resolution afforded in this study allows the tracking of discrete sand grains in ballistic flight within a limited vertical plane (see Fig. 1). Successive image frames permit investigation of the dynamics of the ejecta flow field in a regime much earlier than that examined in previous contributions. The measured displacement of individual particles between frames yields the
B. Hermalyn, P.H. Schultz / Icarus 209 (2010) 866–870
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Fig. 1. Early-stage velocity vs. time and experimental setup. The absolute magnitude of ejecta velocity (Ve) is scaled to the square root of gravity times the final crater radius R; the time of launch for individual particles is scaled to the time of crater formation, Tc. Subplots correspond to the parameters varied in each experiment: A covers a range of velocities, B is variable density, and C is variable size. The power-law slope predicted by Housen et al. (1983) of 0.28 t/Tc0.709 (using a = 0.51, as supported by our data) is superimposed as a dashed gray line. Although velocities scale as expected by around t/Tc = 0.05, they display a noticeable augmentation at early times. A running median filter was applied to the data for enhanced visibility. Note that the latest times represent <10% of the total crater formation time. D is a time series of high-speed PTV images of a vertical impact experiment (6.35 cm diameter Al into #20–30 sand at 2.5 km/s) at 0.06, 0.13, 0.20, and 20 ms after impact. Red arrow denotes impact point and direction. 1 and 2 are the early-stage, low-angle, high-speed component. The emergence of nominal sand grains comes in the bottom part of 3 (near the hinge point); this illustrates the transition to main-stage behavior. 4 Shows ejection during the main-stage regime, overlaid with the derived velocity vectors of the ejecta particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
instantaneous velocity. This measurement is accomplished by first finding the displacement of a group of particles through cross correlation between images, thereby yielding the generalized flow-field of the ejecta (a 2 dimensional velocimetry method). Next, the individual grains are identified by a size criteria (i.e., number of pixels) and, using the measured reference velocity field from the first step, located in successive frames with higher measurement accuracy. Along the leading edge of the curtain, the velocity of the ejecta is reduced to two components (i.e., there is little out-of-plane motion). This permits planar measurement of the ejecta, as has been applied in prior studies (Oberbeck and Morrison, 1976; Piekutowski et al., 1977; Cintala et al., 1999). The main advantage of this technique (over prior studies) is the temporal and spatial accuracy, which allow the measurement of ejecta in a time-resolved manner (rather than just launch position as determined from non-time resolved parabolic flight or discrete snapshots). The accuracy of the method depends on the resolution of the images, precision of timing, and the validity of the 2-dimensional simplification. Most PTV studies benefit from fairly dense seeding of tracer particles, which allows a high degree of statistical rigor in rejecting outliers. High spatial resolution (i.e., >5 pixels/sand grain) is used here to compensate for the restriction of the measurable area. Impact timing is known to within 1 ls for all experiments. The out-of-plane error can be constrained by the measurement plane, which was optically established by a narrow depth of field (DOF). The cameras were fitted with long focal length lenses at low f-stops, thus limiting the DOF and allowing only the particles on the profile of the curtain to be in-focus and measurable. The greatest out-of-plane velocity error introduced by this DOF would be 20% for the fastest ejecta. Since the particles
are measurable for many frames within the field of view, this error quickly diminishes at later times (slower velocities). After measurement, the vectors of the particles are ballistically regressed to the surface in order to establish ejection velocities as a function of time of launch or position. The addition of an out-of-plane component would only serve to increase the absolute magnitude of the ejection velocity; it would not affect the vertical component (hence, the ballistic regression) of the ejecta. The limitations on measurable velocities are dependent on the frame rate and exposure time of the cameras. The particles must be present in at least two successive frames for velocity measurement. The fastest measured particle velocity in these experiments is 600 m/s. On the lower end of the spectrum, the measurement terminates when the particles are no longer in the field of view of the cameras, or when they begin descent to the target surface; i.e., once they have reached maximum ballistic height (around a meter per second). 3. Results and analysis 3.1. Effects of impact variables During main-stage gravity-controlled growth, scaling considerations (Post, 1974; Schultz and Gault, 1979) and dimensional analysis (e.g., Housen et al., 1983) predict that ejection velocity pffiffiffiffiffiffi is related to the square root of gravity times the final crater radius, ðV e / gRÞ, whereas time of launch t goes as Tc, the time
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of crater formation. When scaled in this manner, the early-stage ejecta velocity distributions exceed the power-law characterizing late-stage excavation flow (see Fig. 1) and exhibit a different slope prior to reaching main-stage expectations by
t/Tc 102. Ejection angles (Fig. 2) also display a pronounced departure from main-stage excavation flow, evolving upward to the expected he 45° for impacts into sand. The transition to nominal flow regimes occurs at approximately the same time in both the velocity and the ejection angle for each experiment. The full range of experimental variables is presented in Fig. 1. 3.2. Discussion The lower ejection angles (and corresponding higher velocities) indicate a downward-migrating source or flow field center (Maxwell, 1977; Anderson and Schultz, 2006). Low ejection angles would, geometrically, indicate a shallower depth of penetration for a given ejection aperture or opening. Color cameras used during this study show that this component is thermally self-luminous; therefore, the high-speed material in question appears to be thoroughly heated. Ballistic return of this low-angle, high-speed component would scour the target surface out to many crater radii. The convergence on a constant ejection angle is consistent with the attainment of main-stage, gravity-controlled crater growth. Variations in impact velocity for the 6.35 mm projectiles yield a small change in the ejection velocity: lower speed impacts reach main-stage growth more quickly. The density and size of the projectile also cause differences in ejection velocity and angle. Ejecta from the low-density projectile retain characteristically lower ejection angles throughout the main-stage regime. This is attributed to the projectile coupling its energy/momentum very near the surface, thereby leading to a penetration depth shallower than is reached with denser projectiles. Larger projectiles, especially the low-speed 9.525 mm experiment, couple their energy and momentum deeper in the target and at later times, leading to higher ejection angles. The late-stage excavation flow regime follows expectations from previous studies and converges on a common ejecta-velocity distribution during main-stage growth. The early-stage flow, however, requires different dimensionless scaling criteria corresponding to the initial conditions of the impact event. This early-time regime can be scaled according to the shock, where time is non-dimensionalized by the penetration time scale (a/Vi) as s / t/(a/Vi) for a given time t, projectile radius a, and impact velocity Vi. The coupling parameter, C, is defined as:
C ¼ aV li dmp f
c ; Pm Vi
where m and l are constants for a given impact and f stands for some dimensionless function of c/Vi and the Pm dimensionless material set (Holsapple and Schmidt, 1987). C should be a constant for any given impact event. The point source assumption is that as a ? 0, Vi ? 1, and thus c/Vi ? 0 for fixed materials. Since the f(0, Pm) term is constant for most materials, it is usually omitted from the parameter. C therefore has dimensions of (energy)1/3 when m = 1/3, l = 2/3, and (momentum)1/3 when m = l = 1/3; representing the two endpoints and range of the coupling parameter. Very near the impact point, however, the projectile is still transferring its energy and momentum to the target, and this approximation does not hold. Instead, an impedance term, ‘‘I,” with functional dependence f Vc ; Pm , can be reinserted i (after Holsapple and Schmidt, 1987) with the working assumption of the form:
I¼f
c ; Pm Vi
l
¼
dmp V i
!
dm c l t
where dp and dt are the densities of the projectile and target respectively, and c is the sound speed (e.g., Schultz and Gault, 1990). Then l
l
C ¼ aV i dm
dmp V i dmt cl
! l
¼ aV i dm I 0
We can then write the early-time modified coupling length scale as a :
! l
a0 ¼ aI ¼ a
dmp V i
dmt cl
C ¼ l m Vi d 0
0
An adjusted dimensionless s now replaces s using a instead of a:
s0 ¼
Fig. 2. Ejection angle vs. time. As in Fig. 1, subplots A, B, and C represent variable velocity, density, and size, respectively. In all cases, ejection angles are initially lower than expected for vertical impacts into sand. The angle of ejection quickly rises upward before reaching a maximum and attaining the expected nominal angle of 45°. Ejection angles and ejection velocities both reach main-stage values at approximately the same time for each experiment. A running median filter was applied to the data for enhanced visibility.
t ða0 =V
iÞ
¼
t dmt cl ða=V i Þ dmp V li
!
This can be considered as a matching procedure between a set of independent variables incorporating impedance terms, and is consistent with other studies of early-stage scaling (Bjork and Olshaker, 1965) and dimensional scaling. In this form, the dimensionless time s0 is then be bounded by the same exponent endpoint values as C. The early-time component of the ejecta is best accommodated (Fig. 3) under energy scaling (l = 2/3). This phase is vanishingly short and represents only a small portion of excavation flow in laboratory-scale experiments. 4. Conclusions The PTV technique coupled with high-speed imaging of the ejecta curtain permits time-resolved measurement of the ejecta flow field. For the first time, our re-
B. Hermalyn, P.H. Schultz / Icarus 209 (2010) 866–870
869
0
Fig. 3. Scaled velocity vs. s and s . Ejection velocities are scaled to Vi, the impact velocity; time is presented as s = t/(a/Vi) for subgroup 1 (left). Note that the early-time 0 component, particularly for the velocity and density studies, is not accounted for in this scaling. For subgroup 2 on the right, time is represented by s as defined in the text. The early-time component is best scaled for l = 2/3, i.e., energy scaling (Schultz, 1988). At later times during the ballistic flow regime, the early-stage scaling does not hold. m is taken to be 1/3 for all impacts (Holsapple, 1993).
sults reveal the scaling relations controlling the earliest stages of excavation in granular material (without vaporization). As the impactor size or density increases, the early-time effects become more important in the overall formation of the crater: the lower the cratering efficiency, the greater the duration of early-time effects relative to the total time of crater growth (Schultz, 1988, 1992). Oblique impacts further deviate from the point-source approximation due to asymmetry of the shock wave (Dahl and Schultz, 2001) and ejecta velocity (Anderson et al., 2003), and will likely require similar adjustments to ejecta scaling relations to account for early-time processes. Understanding this stage of crater growth is integral for
interpreting ejecta deposits already emplaced on planetary surfaces. It is also essential for a more complete analysis of data from impact missions such as Deep Impact and the Lunar CRater Observation and Sensing Satellite (LCROSS).
Acknowledgments We gratefully acknowledge the technical crew at the NASA Ames Vertical Gun Range: D. Holt, R. Smythe, and D. Bowling. Without their considerable efforts, this
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study would not have been possible. We also wish to thank J.T. Heineck and E. T. Schairer for critical advice and expertise with the techniques used. This work has been supported by NASA Planetary Geology and Geophysics Grant #NNX08AM45G and a NASA Rhode Island Space Grant Fellowship. References Anderson, J.L.B., Schultz, P.H., 2006. Flow-field center migration during vertical and oblique impacts. Int. J. Impact Eng. 33, 35–44. Anderson, J.L.B., Schultz, P.H., Heineck, J.T., 2003. Asymmetry of ejecta flow during oblique impacts using three-dimensional particle image velocimetry. J. Geophys. Res. (Planets) 108, 5094. Bjork, R.L., Olshaker, A.E., 1965. A Proposed Scaling Law for Hypervelocity Impacts between a Projectile and a Target of Dissimilar Material. Memorandum RM2926-PR, United States Air Force Project Rand. Brown, J.L., Vogler, T.J., Chhabildas, L.C., Reinhart, W.D., Thornhill, T.F., 2007. Shock Response of Dry Sand. Tech. Rep. SAND2007-3524, Sandia National Laboratories. Chabai, A., 1959. Crater Scaling Laws for Desert Alluvium. Tech. Rep., Sandia Corp., Albuquerque, N. Mex. Cintala, M.J., Berthoud, L., Hörz, F., 1999. Ejection-velocity distributions from impacts into coarse-grained sand. Meteorit. Planet. Sci. 34 (July), 605– 623. Dahl, J.M., Schultz, P.H., 2001. Measurement of stress wave asymmetries in hypervelocity projectile impact experiments. Int. J. Impact Eng. 26, 145–155. Dienes, J.K., Walsh, J.M., 1970. Theory of impact: Some general principles and the method of Eulerian codes. In: Kinslow, R. (Ed.), High-Velocity Impact Phenomena. Academic Press, pp. 45–104 (Chapter 3). Holsapple, K.A., 1993. The scaling of impact processes in planetary sciences. Annu. Rev. Earth Planet. Sci. 21, 333–373. Holsapple, K.A., Schmidt, R.M., 1987. Point source solutions and coupling parameters in cratering mechanics. J. Geophys. Res. 92, 6350–6376. Housen, K.R., Schmidt, R.M., Holsapple, K.A., 1983. Crater ejecta scaling lawsfundamental forms based on dimensional analysis. J. Geophys. Res. 88 (March), 2485–2499.
Maxwell, D., 1977. Simple z model of cratering, ejection and the overturned flap. In: Roddy, D.J., Pepin, R.O., Merrill, R.B. (Eds.), Impact and Explosion Cratering. Pergamon, NY, pp. 907–917. McGetchin, T.R., Settle, M., Head, J.W., 1973. Radial thickness variation in impact crater ejecta: Implications for lunar basin deposits. Earth Planet. Sci. Lett. 20, 226–236. Oberbeck, V.R., Morrison, R.H., 1976. Candidate areas for in situ ancient lunar materials. In: Merrill, R.B. (Ed.), Lunar and Planetary Science Conference Proceedings, vol. 7. pp. 2983–3005. Orphal, D.L., 1977. Calculations of explosion cratering-II cratering mechanics and phenomenology. In: Roddy, D.J., Pepin, R.O., Merrill, R.B. (Eds.), Impact and Explosion Cratering. Pergamon, NY, pp. 907–917. Piekutowski, A.J., 1980. Formation of bowl-shaped craters. Lunar Planet. Inst. Sci. 11, 877–878 (abstract). Piekutowski, A.J., Andrews, R.J., Swift, H.F., 1977. Studying small-scale explosive cratering phenomnea photographically. In: Richardson, M.C. (Ed.), Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 97. pp. 177–183. Pierazzo, E., and 13 colleagues, 2008. Validation of numerical codes for impact and explosion cratering: Impacts on strengthless and metal targets. Meteorit. Planet. Sci. 43(12), 1917–1938. Post, R., 1974. Ejecta Distributions from Near-surface Nuclear and HE Burst. Tech. Rep. AFWL-TR-74-51, Air Force Weapons Lab, Kirtland Air Force Base. Schultz, P.H., 1988. Cratering on Mercury - A relook. In: Vilas, F., Chapman, C.R., Matthews, M.S. (Eds.), Mercury. University of Arizona Press, Tucson, pp. 274– 335. Schultz, P.H., 1992. Atmospheric effects on ejecta emplacement and crater formation on Venus from Magellan. J. Geophys. Res. 97, 11623. Schultz, P.H., Gault, D.E., 1979. Atmospheric effects on martian ejecta emplacement. J. Geophys. Res. 84, 7669–7687. Schultz, P.H., Gault, D.E., 1990. Prolonged global catastrophes from oblique impacts. In: Sharpton, V.L., Ward, P.D. (Eds.), Earth History: An Interdisciplinary Conference on Impacts, Volcanism, and Mass Mortality, Geological Society of America Special Paper, vol. 247. pp. 239–261. Wada, K., Senshu, H., Matsui, T., 2006. Numerical simulation of impact cratering on granular material. Icarus 180 (2), 528–545.
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Note
Early-stage ejecta velocity distribution for vertical hypervelocity impacts into sand Brendan Hermalyn *, Peter H. Schultz Department of Geological Sciences, Brown University, Box 1846, Providence, RI 02912, United States
a r t i c l e
i n f o
Article history: Received 27 October 2009 Revised 26 May 2010 Accepted 27 May 2010 Available online 4 June 2010
a b s t r a c t The ejecta dynamics during main-stage excavation flow in a cratering event have previously been well characterized, particularly for vertical impacts. In this experimental study, we present new results addressing the early-time, lowangle, high-speed component of the ejecta velocity distribution as a function of time for hypervelocity vertical impacts into sand. Although this regime represents a very small portion of total ejected mass in laboratory experiments, it comprises a greater percentage of growth for larger craters.
Ó 2010 Elsevier Inc. All rights reserved.
Keywords: Impact processes Cratering Collisional physics Geological processes
1. Introduction
2. Experimental procedure
The ejecta velocity distribution of an impact event determines the emplacement of material on planetary bodies. Main-stage excavation flow comprises the majority of ejected material and has been well studied for vertical impacts as a function of launch position (e.g., McGetchin et al., 1973; Schultz and Gault, 1979). Dimensional analysis (Housen et al., 1983) predicts a single power-law relationship between ejection velocity and time (or ejection position) for main-stage excavation. Numerical simulations (Wada et al., 2006) and experimental studies (Piekutowski, 1980; Oberbeck and Morrison, 1976; Piekutowski et al., 1977; Cintala et al., 1999; Anderson et al., 2003) follow the predicted main-stage trend fairly well. This agreement reflects the ‘‘late-stage equivalence” concept of crater growth (Dienes and Walsh, 1970), where the details of the impact, i.e., the energy and momentum transfer of the projectile to the target, can be replaced by a singular point in time and space (e.g., Chabai, 1959; Orphal, 1977; Housen et al., 1983). Implicit in the dimensional analysis is that this ‘‘point source” approximation can be defined by a coupling parameter, C, where C is a function of projectile radius, impact velocity, and projectile density, as detailed by Holsapple and Schmidt (1987). Thus, early-time coupling establishes the ejecta flow field behind the initial shock. Subsequent ejection of material is described by main-stage ejecta flow and is limited by properties of the target and/or environment (i.e., gravity and atmosphere). As the projectile transfers its energy and momentum to the target at early times, however, the single power-law description of ejecta velocity does not hold. Prior experimental studies of ejecta velocities have generally not extended to this early-time regime. Moreover, hydrocode modeling of granular materials is still poorly constrained and uncalibrated, particularly for this stage of growth (Brown et al., 2007; Pierazzo et al., 2008). The goal of this study is to investigate the ejecta velocity distribution as a function of time and how it departs from standard scaling laws early in the cratering process.
2.1. Strategy
* Corresponding author. E-mail address: [email protected] (B. Hermalyn). 0019-1035/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2010.05.025
A suite of experiments conducted at the NASA Ames Vertical Gun Range (AVGR) addressed the effect of impact variables on the early-stage ejecta velocity distribution. All experiments were vertical impacts (i.e., orthogonal to the target surface) into #20–30 Ottawa sand. Time, rather than launch position, is used as the independent parameter in this study. Previous studies (e.g., Anderson and Schultz, 2006) have shown that the impact point in oblique impacts does not correspond to either the flow-field center (which migrates throughout much of crater growth) or crater center (which requires complicated correction terms), thus making the launch position metric convoluted. A temporal reconstruction of the event is more physically meaningful for the early-stage and will allow comparison with future studies of the ejecta velocity distribution in oblique impacts. In order to explore the influence of different impact variables on the early-time regime, three separate factors were varied. First, identical aluminum projectiles (6.35 mm diameter) were launched at different velocities: 1.6 km/s, 2.5 km/s, and 5.6 km/s. Second, aluminum projectiles with a range of diameters were used: 6.35 mm, 9.525 mm, and 12.7 mm. And third, the projectile density was varied through the use of different impactor materials: aluminum (q = 2.8 g/cm3), polyethylene (q = 0.93 g/cm3), and titanium (q = 4.4 g/ cm3) at the 6.35 mm diameter. 2.2. Measurement technique A nonintrusive imaging technique called particle tracking velocimetry (PTV) was used to determine the ejecta velocity of the experiments. The PTV technique, which is widely used in experimental fluid dynamics research, tracks the motion of individual particles on the profile of the ejecta curtain using several high-speed cameras (both color and grayscale) operating at 8000–15,000 frames per second (fps). The high degree of temporal resolution afforded in this study allows the tracking of discrete sand grains in ballistic flight within a limited vertical plane (see Fig. 1). Successive image frames permit investigation of the dynamics of the ejecta flow field in a regime much earlier than that examined in previous contributions. The measured displacement of individual particles between frames yields the
B. Hermalyn, P.H. Schultz / Icarus 209 (2010) 866–870
867
Fig. 1. Early-stage velocity vs. time and experimental setup. The absolute magnitude of ejecta velocity (Ve) is scaled to the square root of gravity times the final crater radius R; the time of launch for individual particles is scaled to the time of crater formation, Tc. Subplots correspond to the parameters varied in each experiment: A covers a range of velocities, B is variable density, and C is variable size. The power-law slope predicted by Housen et al. (1983) of 0.28 t/Tc0.709 (using a = 0.51, as supported by our data) is superimposed as a dashed gray line. Although velocities scale as expected by around t/Tc = 0.05, they display a noticeable augmentation at early times. A running median filter was applied to the data for enhanced visibility. Note that the latest times represent <10% of the total crater formation time. D is a time series of high-speed PTV images of a vertical impact experiment (6.35 cm diameter Al into #20–30 sand at 2.5 km/s) at 0.06, 0.13, 0.20, and 20 ms after impact. Red arrow denotes impact point and direction. 1 and 2 are the early-stage, low-angle, high-speed component. The emergence of nominal sand grains comes in the bottom part of 3 (near the hinge point); this illustrates the transition to main-stage behavior. 4 Shows ejection during the main-stage regime, overlaid with the derived velocity vectors of the ejecta particles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
instantaneous velocity. This measurement is accomplished by first finding the displacement of a group of particles through cross correlation between images, thereby yielding the generalized flow-field of the ejecta (a 2 dimensional velocimetry method). Next, the individual grains are identified by a size criteria (i.e., number of pixels) and, using the measured reference velocity field from the first step, located in successive frames with higher measurement accuracy. Along the leading edge of the curtain, the velocity of the ejecta is reduced to two components (i.e., there is little out-of-plane motion). This permits planar measurement of the ejecta, as has been applied in prior studies (Oberbeck and Morrison, 1976; Piekutowski et al., 1977; Cintala et al., 1999). The main advantage of this technique (over prior studies) is the temporal and spatial accuracy, which allow the measurement of ejecta in a time-resolved manner (rather than just launch position as determined from non-time resolved parabolic flight or discrete snapshots). The accuracy of the method depends on the resolution of the images, precision of timing, and the validity of the 2-dimensional simplification. Most PTV studies benefit from fairly dense seeding of tracer particles, which allows a high degree of statistical rigor in rejecting outliers. High spatial resolution (i.e., >5 pixels/sand grain) is used here to compensate for the restriction of the measurable area. Impact timing is known to within 1 ls for all experiments. The out-of-plane error can be constrained by the measurement plane, which was optically established by a narrow depth of field (DOF). The cameras were fitted with long focal length lenses at low f-stops, thus limiting the DOF and allowing only the particles on the profile of the curtain to be in-focus and measurable. The greatest out-of-plane velocity error introduced by this DOF would be 20% for the fastest ejecta. Since the particles
are measurable for many frames within the field of view, this error quickly diminishes at later times (slower velocities). After measurement, the vectors of the particles are ballistically regressed to the surface in order to establish ejection velocities as a function of time of launch or position. The addition of an out-of-plane component would only serve to increase the absolute magnitude of the ejection velocity; it would not affect the vertical component (hence, the ballistic regression) of the ejecta. The limitations on measurable velocities are dependent on the frame rate and exposure time of the cameras. The particles must be present in at least two successive frames for velocity measurement. The fastest measured particle velocity in these experiments is 600 m/s. On the lower end of the spectrum, the measurement terminates when the particles are no longer in the field of view of the cameras, or when they begin descent to the target surface; i.e., once they have reached maximum ballistic height (around a meter per second). 3. Results and analysis 3.1. Effects of impact variables During main-stage gravity-controlled growth, scaling considerations (Post, 1974; Schultz and Gault, 1979) and dimensional analysis (e.g., Housen et al., 1983) predict that ejection velocity pffiffiffiffiffiffi is related to the square root of gravity times the final crater radius, ðV e / gRÞ, whereas time of launch t goes as Tc, the time
868
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of crater formation. When scaled in this manner, the early-stage ejecta velocity distributions exceed the power-law characterizing late-stage excavation flow (see Fig. 1) and exhibit a different slope prior to reaching main-stage expectations by
t/Tc 102. Ejection angles (Fig. 2) also display a pronounced departure from main-stage excavation flow, evolving upward to the expected he 45° for impacts into sand. The transition to nominal flow regimes occurs at approximately the same time in both the velocity and the ejection angle for each experiment. The full range of experimental variables is presented in Fig. 1. 3.2. Discussion The lower ejection angles (and corresponding higher velocities) indicate a downward-migrating source or flow field center (Maxwell, 1977; Anderson and Schultz, 2006). Low ejection angles would, geometrically, indicate a shallower depth of penetration for a given ejection aperture or opening. Color cameras used during this study show that this component is thermally self-luminous; therefore, the high-speed material in question appears to be thoroughly heated. Ballistic return of this low-angle, high-speed component would scour the target surface out to many crater radii. The convergence on a constant ejection angle is consistent with the attainment of main-stage, gravity-controlled crater growth. Variations in impact velocity for the 6.35 mm projectiles yield a small change in the ejection velocity: lower speed impacts reach main-stage growth more quickly. The density and size of the projectile also cause differences in ejection velocity and angle. Ejecta from the low-density projectile retain characteristically lower ejection angles throughout the main-stage regime. This is attributed to the projectile coupling its energy/momentum very near the surface, thereby leading to a penetration depth shallower than is reached with denser projectiles. Larger projectiles, especially the low-speed 9.525 mm experiment, couple their energy and momentum deeper in the target and at later times, leading to higher ejection angles. The late-stage excavation flow regime follows expectations from previous studies and converges on a common ejecta-velocity distribution during main-stage growth. The early-stage flow, however, requires different dimensionless scaling criteria corresponding to the initial conditions of the impact event. This early-time regime can be scaled according to the shock, where time is non-dimensionalized by the penetration time scale (a/Vi) as s / t/(a/Vi) for a given time t, projectile radius a, and impact velocity Vi. The coupling parameter, C, is defined as:
C ¼ aV li dmp f
c ; Pm Vi
where m and l are constants for a given impact and f stands for some dimensionless function of c/Vi and the Pm dimensionless material set (Holsapple and Schmidt, 1987). C should be a constant for any given impact event. The point source assumption is that as a ? 0, Vi ? 1, and thus c/Vi ? 0 for fixed materials. Since the f(0, Pm) term is constant for most materials, it is usually omitted from the parameter. C therefore has dimensions of (energy)1/3 when m = 1/3, l = 2/3, and (momentum)1/3 when m = l = 1/3; representing the two endpoints and range of the coupling parameter. Very near the impact point, however, the projectile is still transferring its energy and momentum to the target, and this approximation does not hold. Instead, an impedance term, ‘‘I,” with functional dependence f Vc ; Pm , can be reinserted i (after Holsapple and Schmidt, 1987) with the working assumption of the form:
I¼f
c ; Pm Vi
l
¼
dmp V i
!
dm c l t
where dp and dt are the densities of the projectile and target respectively, and c is the sound speed (e.g., Schultz and Gault, 1990). Then l
l
C ¼ aV i dm
dmp V i dmt cl
! l
¼ aV i dm I 0
We can then write the early-time modified coupling length scale as a :
! l
a0 ¼ aI ¼ a
dmp V i
dmt cl
C ¼ l m Vi d 0
0
An adjusted dimensionless s now replaces s using a instead of a:
s0 ¼
Fig. 2. Ejection angle vs. time. As in Fig. 1, subplots A, B, and C represent variable velocity, density, and size, respectively. In all cases, ejection angles are initially lower than expected for vertical impacts into sand. The angle of ejection quickly rises upward before reaching a maximum and attaining the expected nominal angle of 45°. Ejection angles and ejection velocities both reach main-stage values at approximately the same time for each experiment. A running median filter was applied to the data for enhanced visibility.
t ða0 =V
iÞ
¼
t dmt cl ða=V i Þ dmp V li
!
This can be considered as a matching procedure between a set of independent variables incorporating impedance terms, and is consistent with other studies of early-stage scaling (Bjork and Olshaker, 1965) and dimensional scaling. In this form, the dimensionless time s0 is then be bounded by the same exponent endpoint values as C. The early-time component of the ejecta is best accommodated (Fig. 3) under energy scaling (l = 2/3). This phase is vanishingly short and represents only a small portion of excavation flow in laboratory-scale experiments. 4. Conclusions The PTV technique coupled with high-speed imaging of the ejecta curtain permits time-resolved measurement of the ejecta flow field. For the first time, our re-
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0
Fig. 3. Scaled velocity vs. s and s . Ejection velocities are scaled to Vi, the impact velocity; time is presented as s = t/(a/Vi) for subgroup 1 (left). Note that the early-time 0 component, particularly for the velocity and density studies, is not accounted for in this scaling. For subgroup 2 on the right, time is represented by s as defined in the text. The early-time component is best scaled for l = 2/3, i.e., energy scaling (Schultz, 1988). At later times during the ballistic flow regime, the early-stage scaling does not hold. m is taken to be 1/3 for all impacts (Holsapple, 1993).
sults reveal the scaling relations controlling the earliest stages of excavation in granular material (without vaporization). As the impactor size or density increases, the early-time effects become more important in the overall formation of the crater: the lower the cratering efficiency, the greater the duration of early-time effects relative to the total time of crater growth (Schultz, 1988, 1992). Oblique impacts further deviate from the point-source approximation due to asymmetry of the shock wave (Dahl and Schultz, 2001) and ejecta velocity (Anderson et al., 2003), and will likely require similar adjustments to ejecta scaling relations to account for early-time processes. Understanding this stage of crater growth is integral for
interpreting ejecta deposits already emplaced on planetary surfaces. It is also essential for a more complete analysis of data from impact missions such as Deep Impact and the Lunar CRater Observation and Sensing Satellite (LCROSS).
Acknowledgments We gratefully acknowledge the technical crew at the NASA Ames Vertical Gun Range: D. Holt, R. Smythe, and D. Bowling. Without their considerable efforts, this
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study would not have been possible. We also wish to thank J.T. Heineck and E. T. Schairer for critical advice and expertise with the techniques used. This work has been supported by NASA Planetary Geology and Geophysics Grant #NNX08AM45G and a NASA Rhode Island Space Grant Fellowship. References Anderson, J.L.B., Schultz, P.H., 2006. Flow-field center migration during vertical and oblique impacts. Int. J. Impact Eng. 33, 35–44. Anderson, J.L.B., Schultz, P.H., Heineck, J.T., 2003. Asymmetry of ejecta flow during oblique impacts using three-dimensional particle image velocimetry. J. Geophys. Res. (Planets) 108, 5094. Bjork, R.L., Olshaker, A.E., 1965. A Proposed Scaling Law for Hypervelocity Impacts between a Projectile and a Target of Dissimilar Material. Memorandum RM2926-PR, United States Air Force Project Rand. Brown, J.L., Vogler, T.J., Chhabildas, L.C., Reinhart, W.D., Thornhill, T.F., 2007. Shock Response of Dry Sand. Tech. Rep. SAND2007-3524, Sandia National Laboratories. Chabai, A., 1959. Crater Scaling Laws for Desert Alluvium. Tech. Rep., Sandia Corp., Albuquerque, N. Mex. Cintala, M.J., Berthoud, L., Hörz, F., 1999. Ejection-velocity distributions from impacts into coarse-grained sand. Meteorit. Planet. Sci. 34 (July), 605– 623. Dahl, J.M., Schultz, P.H., 2001. Measurement of stress wave asymmetries in hypervelocity projectile impact experiments. Int. J. Impact Eng. 26, 145–155. Dienes, J.K., Walsh, J.M., 1970. Theory of impact: Some general principles and the method of Eulerian codes. In: Kinslow, R. (Ed.), High-Velocity Impact Phenomena. Academic Press, pp. 45–104 (Chapter 3). Holsapple, K.A., 1993. The scaling of impact processes in planetary sciences. Annu. Rev. Earth Planet. Sci. 21, 333–373. Holsapple, K.A., Schmidt, R.M., 1987. Point source solutions and coupling parameters in cratering mechanics. J. Geophys. Res. 92, 6350–6376. Housen, K.R., Schmidt, R.M., Holsapple, K.A., 1983. Crater ejecta scaling lawsfundamental forms based on dimensional analysis. J. Geophys. Res. 88 (March), 2485–2499.
Maxwell, D., 1977. Simple z model of cratering, ejection and the overturned flap. In: Roddy, D.J., Pepin, R.O., Merrill, R.B. (Eds.), Impact and Explosion Cratering. Pergamon, NY, pp. 907–917. McGetchin, T.R., Settle, M., Head, J.W., 1973. Radial thickness variation in impact crater ejecta: Implications for lunar basin deposits. Earth Planet. Sci. Lett. 20, 226–236. Oberbeck, V.R., Morrison, R.H., 1976. Candidate areas for in situ ancient lunar materials. In: Merrill, R.B. (Ed.), Lunar and Planetary Science Conference Proceedings, vol. 7. pp. 2983–3005. Orphal, D.L., 1977. Calculations of explosion cratering-II cratering mechanics and phenomenology. In: Roddy, D.J., Pepin, R.O., Merrill, R.B. (Eds.), Impact and Explosion Cratering. Pergamon, NY, pp. 907–917. Piekutowski, A.J., 1980. Formation of bowl-shaped craters. Lunar Planet. Inst. Sci. 11, 877–878 (abstract). Piekutowski, A.J., Andrews, R.J., Swift, H.F., 1977. Studying small-scale explosive cratering phenomnea photographically. In: Richardson, M.C. (Ed.), Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, vol. 97. pp. 177–183. Pierazzo, E., and 13 colleagues, 2008. Validation of numerical codes for impact and explosion cratering: Impacts on strengthless and metal targets. Meteorit. Planet. Sci. 43(12), 1917–1938. Post, R., 1974. Ejecta Distributions from Near-surface Nuclear and HE Burst. Tech. Rep. AFWL-TR-74-51, Air Force Weapons Lab, Kirtland Air Force Base. Schultz, P.H., 1988. Cratering on Mercury - A relook. In: Vilas, F., Chapman, C.R., Matthews, M.S. (Eds.), Mercury. University of Arizona Press, Tucson, pp. 274– 335. Schultz, P.H., 1992. Atmospheric effects on ejecta emplacement and crater formation on Venus from Magellan. J. Geophys. Res. 97, 11623. Schultz, P.H., Gault, D.E., 1979. Atmospheric effects on martian ejecta emplacement. J. Geophys. Res. 84, 7669–7687. Schultz, P.H., Gault, D.E., 1990. Prolonged global catastrophes from oblique impacts. In: Sharpton, V.L., Ward, P.D. (Eds.), Earth History: An Interdisciplinary Conference on Impacts, Volcanism, and Mass Mortality, Geological Society of America Special Paper, vol. 247. pp. 239–261. Wada, K., Senshu, H., Matsui, T., 2006. Numerical simulation of impact cratering on granular material. Icarus 180 (2), 528–545.