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Ejecta Pattern of the Impact of Comet Shoemaker–Levy 9
Icarus 138, 157–163 (1999) Article ID icar.1998.6070, available online at http://www.idealibrary.com on
Ejecta Pattern of the Impact of Comet Shoemaker–Levy 9 Alexey A. Pankine and Andrew P. Ingersoll Division of Geological and Planetary Sciences, California Institute of Technology, MS 150-21, Pasadena, California 91125 E-mail: [email protected] Received June 24, 1996; revised November 20, 1998
The collision of Comet Shoemaker–Levy 9 (SL 9) with Jupiter created crescent-shaped ejecta patterns around impact sites. Although the observed impact plumes rose through a similar height of ∼3000 km, the radii of the created ejecta patterns differ from impact to impact and generally are larger for larger impacts. The azimuthal angle of the symmetry axis of the ejecta pattern is larger than that predicted by the models of oblique impacts, due to the action of the Coriolis force that rotates ejecta patterns counterclockwise from the south. We study the formation of ejecta patterns using a simple model of ballistic plume above a rotating plane. The ejected particles follow ballistic trajectories and slide horizontally for about an hour after reentry into the jovian atmosphere. The lateral expansion of the plume is stopped by the friction force, which is assumed to be proportional to the square of the horizontal velocity. Two different mass–velocity distributions used in the simulations produce qualitatively similar results. The simulated ejecta patterns fit very well the “crescents” observed at the impact sites. The sizes and azimuthal angles of symmetry axis of ejecta patterns depend on a parameter L, which has dimension of length and is related to the mass of the fragment. Thus more massive impacts produce larger ejecta patterns that are rotated through a wider angle. °c 1999 Academic Press Key Words: comets, S-L9; Jupiter; impact processes.
I. INTRODUCTION
The explosions of fragments of Comet Shoemaker–Levy 9 (SL 9) in the atmosphere of Jupiter created distinctive ejecta patterns around impact sites. The ejecta patterns are formed by the mixed comet–jovian material that was ejected through considerable heights and then reentered the atmosphere several thousand kilometers from the impact point. The ejecta patterns show consistent morphology: large crescent-shaped ejecta to the southeast of the impact site, circular ring and sometimes smaller faint ring inside the main ring, and a small triangle with its apex near the ring center and its base to the southeast (see Fig. 1). The “crescents” are probably made visible by the dark debris the core of which is composed of silicate grains that condense in the expanding plume (Friedson, 1998). Although all the impact plumes rose through a similar height of about 3000 km, the sizes of impact sites vary from about 6000 km for small impacts (Fragments S and W) to about
13000 km for large impacts (G, K, and L) (Hammel et al. 1995). The symmetry axis of the ejecta patterns (azimuth angle 35◦ ± 5◦ counterclockwise from the south for the G impact (Hammel et al. 1995)) does not line up with the fragment trajectory (16◦ counterclockwise from south) as predicted by the models of oblique impacts (Boslough et al. 1995, Crawford et al. 1994, Takata et al. 1994). It was suggested that the increase of the azimuth angle is due to the Coriolis force that rotates the plume of ejecta counterclockwise. To match the observations, the rotation must take place for 1 hr or more, which means the plume must continue its expansion after reentry into the atmosphere (Hammel et al. 1995). The radial expansion of the plume after the reentry also helps to explain the sizes of the largest impact sites (∼13000 km) since a plume of equal velocity ballistic particles will reach a height of 3000 km (particles going straight up) and a maximum range of only 6000 km (particles ejected at elevation angle of 45◦ ). The present paper is an attempt to test this hypothesis with a qualitative model. Numerous authors already addressed issues of the “rings,” the asymmetry of the plumes and ejecta patterns, the same height of the plumes associated with different impacts, and other issues (Shoemaker et al. 1995, Boslough et al. 1995, Takata and Ahrens 1997, Takata et al. 1994, Gryaznov et al. 1994, Zahnle and Mac Low 1995, Mac Low and Zahnle 1994, Ingersoll and Kanamori 1995). At the same time the shape of the ejecta patterns (crescent) have not been convincingly explained. Crawford et al. (1995) speculate that the crescent reflects the vertical distribution of the source of the visible ejecta in the jovian atmosphere. This argument implies that farther thrown ejecta originates from some particular layer in the atmosphere that contains dark material. This dark material also has the highest ejection velocity. The inner part of the ejecta pattern is made of transparent lowvelocity ejecta that originates from below the layer of dark material. The shape of the ejecta pattern (crescent) thus suggests a layered distribution of the dark material in the atmosphere that is preserved in the plume. Although such a distribution of ejecta in the plume is not improbable, the justification of it requires extensive numerical modeling of the explosion itself. Zahnle and Mac Low (1995) argue that the crescent is formed by carbonaceous dust generated from jovian air in the reentry shock. Reentry velocities must exceed 4 km/sec for shock
157 0019-1035/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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The model described below explains the bulk of the observations in a simple way. It is capable of producing the ejecta patterns that match all major features of the observed crescents—the shape, the size, and the azimuth angle of the symmetry axis. The model treats the ejecta as a cloud of ballistic particles above a rotating plane. Upon reentry onto the initial level the particles (ejecta) continue to move horizontally, slowing down under the influence of the friction force which is assumed to be proportional to the square of the velocity. II. MODEL
Ballistic Plume
FIG. 1. The D and G impact sites. North is up and east is to the right. Reprinted with permission from Fig. 1 West et al. (1995). Copyright 1995 American Association for the Advancement of Science.
heating to be strong enough to produce dust. Thus the crescent is produced by the ejecta with the highest velocities. This argument does not take into account the fact that high-velocity ejecta do not necessarily go the farthest unless all ejected material is launched at the same elevation angle. The model plume used by Zahnle and Mac Low is axisymmetric with a wide range of the launching angles. Hence the shock-generated carbonaceous dust is generated at all distances from impact site and the crescent does not form. The ejecta pattern presented by Takata and Ahrens (1997) resembles the observed patterns fairly well but does not match the observed azimuth angle of the symmetry axis. In their model, the descent of the plume is artificially prolonged to match the HST data at t > 1200 sec after the impact and the plume loses all its kinetic energy after returning onto the initial level, which contradicts the theoretical conclusion that the horizontal component of the velocity is preserved during the “splash back” (Zahnle and Mac Low 1995). Prolonging the descent time produces an ejecta pattern that is larger and that is rotated through a wider angle than would be expected for the equal velocity ballistic plume. Still, the angle is too small to match the observed azimuth angle of the symmetry.
The plume of ejecta is treated as a cloud of ballistic particles above a rotating plane. The motion of the particles occurs above the plane that coincides with the 0.1 bar level in the jovian atmosphere. Each particle represents a parcel of mixed jovian and cometary material in the plume. The plume is axisymmetric and is initially ejected from a cone with opening angle θ 0 . The plume can be visualized as a conic section of the radially expanding cloud centered on the impact point (see Fig. 2). The opening angle θ 0 is left to be a free parameter of the model, although simulations show that it must be about 70◦ to yield the crescent that extends 180◦ around the impact site. The axis of the cone is aligned with the incoming trajectory of the bolide (elevation angle of 45◦ and azimuth angle 16◦ counterclockwise from south) as predicted by numerical models (Boslough et al. 1995, Crawford et al. 1994, Takata et al. 1994) and as possibly seen on some HST images of rising plumes (Hammel et al. 1995). The particles are ejected with initial velocities that are governed by one of two mass–velocity distributions (see below) and with ejection angles that follow a uniform angular distribution. Particles ejected downward are not included in the calculations. This corresponds to a horizontal “cutoff” of the plume, which is consistent with numerical modeling (Takata et al. 1994, Boslough et al. 1995, Takata and Ahrens 1997). The particles follow ballistic trajectories bent by the Coriolis force. The equations that describe trajectories of the particles above the rotating flat Jupiter are u˙ = 2Ä(v sin ϕ − w cos ϕ) v˙ = −2Äu sin ϕ w˙ = 2Äu cos ϕ − g, where u, v, and w are the east, north, up components of velocity vector V¯ , ϕ is jovian latitude, and Ä is the rotation rate. This system of equations is solved assuming that ϕ is constant. Mass–Velocity Distributions We use two different mass–velocity distributions to model the plume. These mass–velocity distributions are the well-known solutions to the key gas-dynamic problems and are widely used in impact modeling. Since the initial conditions of the SL 9 explosions are very uncertain it is not possible to discriminate
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FIG. 2. The plume geometry. The plume is axisymmetric and is initially ejected from a cone with opening angle θ 0 . The axis of the cone is aligned with the incoming trajectory of the fragment (elevation angle of 45◦ and azimuth angle 16◦ counterclockwise from the south). The plume can be visualized as a conic section of the radially expanding cloud centered on the impact point. The ejection angles of the individual particles follow a uniform angular distribution. Particles ejected with negative vertical velocities are not included into simulations.
between these two distributions a priori. Although the distributions are quite different they produce qualitatively similar results in our simulations. The first distribution is that of the gas cloud isentropically expanding into a vacuum. This distribution is assumed to be valid for rather shallow explosions when the explosion fireball forms in the upper part of the atmosphere. According to Zel’dovich and Raizer (1967) the differential mass–velocity distribution of such a cloud evolves toward the form d M ∼ (1 − x 2 )α x 2 d x,
(1)
√ where x = v/vmax is a normalized velocity, vmax = 2γ /(γ − 1) √ 2E/M is the maximum velocity in the cloud, E/M is the thermal energy of the gas in the cloud, γ is the ratio of the specific heats, and the constant α is determined by conservation of mass and energy α = (γ + 5)/2/(γ − 1) (Zahnle 1990). For the fragment G fireball γ ≈ 1.2 (Carlson et al. 1997), thus α ≈ 15.5. Figure 3 shows the plot of the normalized mass–velocity distribution (1). Instead of mass–velocity distribution we use number– velocity distribution, assuming that all particles have the same mass, so the number of particles in an interval d x is given by (1). Parameter vmax is constrained by comparing the height reached by the simulated plume at different times with the HST data
(Hammel et al. 1995). Figure 4 shows a simulated plume with mass–velocity distribution (1), vmax = 25 km/sec, at t = 500 sec. At this time the plumes observed by HST reached their maximum height of ∼3200 km. The plume shown in Fig. 4 does not have a clearly defined edge. Moreover, the particle cross section is largely unknown. Thus, to compare the heights of the simulated and observed plumes, we assume that the edge of the plumes corresponds to a specific contour of the column density along the direction of the observation. The fact that the edges look sharp on the HST images thus suggests rapid density decrease toward outer parts of the plume. The procedure employed to calculate the height of the simulated plumes is as follows. First, we project the numerical plume at t = 500 sec on the south–north plane, calculate column density contours, and choose the contour that has maximum height of ∼3200 km. This contour of column density corresponds now to the edge of the plume. Second, we repeat this calculation at different times and determine the maximum height of the previously chosen column density contour (the edge) at these different times (the edge is not a material surface). This gives us the height–time dependence for the plume with that particular vmax . Finally, we repeat the whole procedure for the plumes with different vmax and compare the height–time dependencies of the simulated plumes with
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FIG. 3. Normalized mass–velocity distribution (1) for α ≈ 15.5. Note that the bulk of the mass is ejected with the velocity that is just a fraction of vmax .
the HST observations. We estimate the maximum velocity to be in the range of 25 ± 10 km/sec in order for the height of the simulated plume to be consistent with the HST observations (Fig. 3 of Hammel et al. 1995). Plumes with higher vmax rise and fall faster than suggested by HST observations and reach heights greater then ∼3200 km. Plumes with lower vmax rise slower than suggested by HST observations. This estimate will be further constrained later when we determine the sliding parameter L for the plume with mass–velocity distribution (1). The bulk of the mass of the plume with mass–velocity distribution (1) is ejected with the velocity which is just a fraction of vmax , thus our estimate differs from that of Hammel et al. (1995) (∼13 km/sec).
The thickness of the edge of the plume is defined as the length on which the value of the column density contour decreases by half from the value of the column density contour chosen as the edge (see above). The thickness of the edge is larger for the plumes with higher vmax and increases with time. For the plumes with vmax = 25 ± 5 km/sec it comes to ∼400 ± 200 and ∼1100 ± 300 km at times t = 500 and 900 sec, respectively. The resolution of the instrument (PC1) that observed rising SL9 plumes was 170 km per pixel, according to Ref. 6 of Hammel et al. (1995). We conclude that the edges of the simulated plumes would look sharp during the early stages of plume evolution, consistent with the HST observations.
FIG. 4. Projection of the simulated plume with mass–velocity distribution (1) on the south–north plane. vmax is 25 km/sec and t is 500 sec after the impact. The viewing geometry approximately corresponds to the geometry of the HST observations (the plume is viewed from the east) and the time corresponds to the time when the plumes observed by HST reached their maximum height. The axis of the plume is to the left (south) and slightly toward the observer.
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Another mass–velocity distribution is based on the results of numerical simulations (Shoemaker et al. 1995, Boslough et al. 1995, Takata et al. 1994, Crawford et al. 1994) that suggest that the ejecta exit the atmosphere through a wake left by the incoming cometary fragment. The mass–velocity distribution in this case differs considerably from (1). As shown by Carlson et al. (1997), the bulk of the material is ejected in a narrow range of velocities as in the case of strong explosion (Zel’dovich and Raizer 1967). It is then reasonable to assume that all the mass exited with the same velocity vmax and that the resultant mass– velocity distribution has the form d M ∼ δ(1 − x) d x,
(2)
where, as before, x = v/vmax is a normalized velocity and δ(x) is a δ-function. The maximum velocity vmax in this case is immediately defined by the maximum height reached by the plume to be ∼13 km/sec. Sliding Phase When the plume rains back onto the atmosphere (or, in other words, it returns to the initial level), the kinetic energy associated with its vertical motion is dissipated in a reentry shock. The energy goes into the thermal energy of the ejecta and of the ambient atmosphere, and eventually it is radiated into space and the lower atmosphere (Zahnle and Mac Low 1995). The collapsed plume continues to expand horizontally and rotate counterclockwise due to the Coriolis force (Hammel et al. 1995) until the plume is stopped by friction force. Friction in our model is proportional to the square of the horizontal velocity of particles and is governed by the parameter L, which has dimension of length. Variations of the parameter L affect both the size and the azimuthal angle of the ejecta pattern. The equations that govern horizontal sliding of the ejecta are u˙ = 2Äv sin ϕ − uV /L v˙ = −2Äu sin ϕ − vV /L w = w˙ = 0,
(3)
√ where V is the total horizontal velocity, V = u 2 + v 2 . The rationale behind the friction term in (3) is as follows. Let us approximate the spreading of the plume as a horizontal flow of one fluid over another. The surface stress per unit area τ is then τ = C D ρV 2 , where C D is a drag coefficient (or skin friction coefficient) and ρ and V are the density and the velocity of the upper fluid (ejecta), respectively. The mass of the upper fluid per unit area is ρh, where h is the thickness of the layer of the fluid, so that the deceleration is −τ/ρh or V˙ = −C D V 2 / h = −V 2 /L .
The parameter L is then equal to the thickness of the layer of the ejecta divided by the drag coefficient. We found L to be of the order of several thousand kilometers in our simulations. If the thickness of the layer of ejecta is of the order of tens of kilometers, the drag coefficient is then ∼0.01–0.001, which is consistent with the values of the skin friction coefficient for the turbulent flow over a flat surface (see, for example, Schlichting 1968). Parameter L is chosen in such a way that the radial extent and azimuthal angle of the symmetry axis agree with the observation. In the case of ejecta pattern produced by the plume with the mass–velocity distribution (1) this means that we pick a density contour that outlines the crescent with the outer radius of ∼13000 km and vary L so that the inner radius of this contour becomes equal to ∼8000 km (the inner and outer radius are for the G impact site (Hammel et al. 1995)). It is not always possible to match both radii and the azimuth angle of the symmetry axis for a given vmax . This circumstance further constrains the range of possible maximum velocities for the plume with the mass–velocity distribution (1) to vmax = 30 ± 5 km/sec, for which L = 4000 ± 500 km. In the case of the ejecta pattern produced by the plume with the mass–velocity distribution (2) the radius and the angle are matched simultaneously. Given the uncertainty of the azimuthal angle measurement (35◦ ± 5◦ ), L = 2500 ± 500 km for the mass–velocity distribution (2). III. RESULTS
Figure 5 shows simulated ejecta patterns 90 min after the impact, when horizontal spreading of the plume has slowed down significantly. The HST observations were made at approximately the same time. The ejecta pattern shown in Fig. 5a is formed by the plume with the mass–velocity distribution (1) and parameters vmax = 30 km/sec and L = 4000 km. The plume particles concentrate to the southeast of the impact point and form a feature that resembles the observed crescent of the G impact. The azimuth angle of the symmetry axis is 38◦ ± 3◦ counterclockwise from south, in agreement with the observations. The simulated feature does not have a sharp boundary, but the number density of the particles increases drastically at the location of the observed crescent. The inner and outer radius of the feature, determined as described above in Section II, are at ∼8000 and ∼13000 km. Figure 5b shows the ejecta pattern formed by the plume with the mass–velocity distribution (2) and parameters vmax = 13 km/sec and L = 2500 km. The sharp edges of the simulated crescent are at ∼8000 and ∼13000 km, and the azimuth angle of the symmetry axis is 35◦ ± 2◦ counterclockwise from south. The outer edge of the crescent is formed by the particles that were ejected at elevation angle of 45◦ . The inner edge of the crescent is formed by the particles that were ejected along the surface and started sliding right after the impact. The thinly populated area around the impact point and toward the northwest of the impact point is filled with particles that were ejected at elevation angles >45◦ and to the right of the z axis in Fig. 2.
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FIG. 5. Simulated ejecta patterns 90 min after impact. North is to the top and east is to the right. The axes are labeled in thousands of kilometers from the origin (the impact point), which is marked by a cross. (a) Ejecta pattern of the G impact generated by numerical plume with mass–velocity distribution (1), vmax = 30 km/sec, L = 4000 km. (b) Ejecta pattern of the G impact generated by numerical plume with mass–velocity distribution (2), vmax = 13 km/sec, L = 2500 km. (c) Simulated ejecta pattern of the A impact, same plume as in (a), L = 1500 km. (d) Simulated ejecta pattern of the A impact, same plume as in (b), L = 500 km.
Both mass–velocity distributions (1) and (2) produce qualitatively similar results. The “real” mass–velocity distribution of the ejected material probably lies somewhere in between these two extreme cases so that the edges of the resultant ejecta pattern is softer than the edges in Fig. 5b but not so diffuse as the edges in Fig. 5a. A small triangle—or streak—that is common for major impacts forms probably independently of the crescent. HST images of rising plumes E and G show faint emission that follows the plume. Numerical simulations (Yabe et al. 1995) suggest that two groups of material were ejected. Penetration of the comet must be rather deep in this case (300 km below 1 bar level). Be it as it may, the streak can be reproduced by our model only if the mass–velocity distribution of ejecta differs significantly from (1) and (2) in its low-velocity part.
The ejecta patterns of other impact sites can be reproduced by our model by changing the parameter L. For smaller L, the rotation of the ejecta pattern is less and the radius of the resultant crescent is smaller. In the previous section we speculated that L is related to the thickness of the sliding layer of ejecta. It seems reasonable to assume that parameter L is hence related to the mass of the plume, since more massive plumes create a thicker layer of ejecta. Thus, more massive plumes produce larger ejecta patterns and rotate through a wider angle, although the initial mass–velocity distribution of the plumes are the same since all plumes go to the same height. Figures 5c and 5d show simulated ejecta patterns of the smaller impact A produced by the same plumes that create ejecta patterns in Figs. 5a and 5b, respectively. The only difference is that the parameter L was reduced by a factor of several. The resultant
SL 9 EJECTA PATTERNS
ejecta patterns are smaller (outer radius ∼8000 km) and were rotated through a smaller angle (∼30◦ ), which is consistent with the observations of the fragment A impact site (K.-L. Jessup, personal communication, 1998).
IV. CONCLUSION
We developed a simple model of dynamics of ejecta from impacts of Comet SL 9 that can adequately reproduce the crescentshaped ejecta patterns observed by HST at the impact sites. The model assumes that ejecta follow ballistic trajectories and slide horizontally upon reentry into the atmosphere until stopped by the friction force. We use two different mass–velocity distributions, both of which produced qualitatively similar results. Mass–velocity distribution (2) seems to be closer to reality because it is backed by the numerical calculations of Carlson et al. (1997) and the plume and the resultant crescent have sharp boundaries, which is more consistent with the observations. The friction force during the sliding phase is assumed to be proportional to the square of the velocity. This assumption represents the notion that the ejecta forms a layer that slides over the surface of the underlying fluid (jovian atmosphere). The thickness of the layer and the resultant frictional deceleration vary with the mass of the plume. This explains why the plumes rose to similar heights but produced ejecta patterns of different sizes. Obviously, the real process that governs the horizontal expansion of the plume is much more complex. Nevertheless, our model seems to capture the essence of the phenomenon. We hope the results will be of use to future modelers.
ACKNOWLEDGMENTS Helpful suggestions have been received from N. Schneider and T. Takata. This work was supported by the NASA Planetary Atmospheres Program under Grant NAGW-1956.
REFERENCES Boslough, M. B., D. A. Crawford, T. G. Trucano, and A. C. Robinson 1995. Numerical modeling of Shoemaker–Levy 9 impact as a framework for interpreting observations. Geophys. Res. Lett. 22, 1821–1824.
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Carlson, R. W., P. Drossart, Th. Encrenaz, P. R. Weissman, J. Hui, and M. Segura 1997. Temperature, size, and energy of the Shoemaker–Levy 9 G-impact fireball. Icarus 128, 251–274. Crawford, D. A., M. B. Boslough, T. G. Trucano, and A. C. Robinson 1994. The impact of Comet Shoemaker–Levy 9 on Jupiter. Shock Waves 4, 47–50. Crawford, D. A., M. B. Boslough, T. G. Trucano, and A. C. Robinson 1995. Comet Shoemaker–Levy 9 plumes and ejecta: Constraints on impactor size and penetration depth. In Abstracts Presented at the 27th Annual DPS Meeting, Hawaii, Oct. 9–13, 1995, p. 60. Friedson, A. J. 1998. Formation of refractory grains in Shoemaker–Levy 9 fireballs. Icarus 131, 179–197. Gryaznov, V. K., B. A. Ivanov, A. B. Ivlev, B. A. Klumov, S. V. Utyuzhnikov, and V. E. Fortov 1994. Collision of the Comet Shoemaker–Levy 9 with Jupiter: Interpretation of observed data. Earth, Moon, Planets 66, 99–128. Hammel, H. B., R. F. Beebe, A. P. Ingersoll, G. S. Orton, J. R. Mills, A. A. Simon, P. Chodas, J. T. Clarke, E. De Jong, T. E. Dowling, J. Harrington, L. F. Huber, E. Karkoschka, C. M. Santori, A. Toigo, D. Yeomans, and R. A. West 1995. HST imaging of atmospheric phenomena created by the impact of Comet Shoemaker–Levy 9. Science 267, 1288–1296. Ingersoll, A. P., and H. Kanamori 1995. Waves from the impact of Shoemaker– Levy 9 with Jupiter. Nature 374, 706–708. Mac Low, M.-M., and K. Zahnle 1994. Explosion of Comet Shoemaker–Levy 9 on entry into the jovian atmosphere. Astrophys. J. 434, L33–L36. Schlichting, H. 1968. Boundary Layer Theory. McGraw Hill, New York. Shoemaker, E. M., P. J. Hassing, and D. J. Roddy 1995. Numerical simulations of the Shoemaker–Levy 9 impact plumes and clouds: A progress report. Geophys. Res. Lett. 22, 1825–1828. Takata, T., and T. J. Ahrens 1997. Impact of Comet Shoemaker–Levy 9— Size, origin, and plumes: Comparison of numerical analysis and observations. Icarus 125, 317–330. Takata, T., J. D. O’Keefe, and T. J. Ahrens 1994. Comet Shoemaker–Levy 9: Impact and plume evolution. Icarus 109, 3–19. West, R. A., E. Karkoschka, A. J. Friedson, M. Seymour, K. H. Baines, and H. B. Hammel 1995. Impact debris particles in Jupiter’s stratosphere. Science 267, 1296–1301. Yabe, T., T. Aaki, M. Tajima, F. Xiao, Sh. Sasaki, Y. Abe, and J. Watanabe 1995. Possible explanation of the secondary flash and strong flare on IR light curves upon impact of Shoemaker–Levy 9. Geophys. Res. Lett. 22, 2429–2432. Zahnle, K. 1990. Atmospheric chemistry by large impacts. In Global Catastrophes in Earth History: An Interdisciplinary Conference on Impacts, Volcanism and Mass Mortality (V. L. Sharpton and P. D. Ward, Eds.), Vol. 247, pp. 271–287. Geological Society of America Special Paper. Zahnle, K., and M.-M. Mac Low 1995. A simple model for the light curve generated by a Shoemaker–Levy 9 impact. J. Geophys. Res. 100, 16,885– 16,894. Zel’dovich, Ya. B., and Yu. P. Raizer 1967. Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena. Academic Press, New York.
Ejecta Pattern of the Impact of Comet Shoemaker–Levy 9 Alexey A. Pankine and Andrew P. Ingersoll Division of Geological and Planetary Sciences, California Institute of Technology, MS 150-21, Pasadena, California 91125 E-mail: [email protected] Received June 24, 1996; revised November 20, 1998
The collision of Comet Shoemaker–Levy 9 (SL 9) with Jupiter created crescent-shaped ejecta patterns around impact sites. Although the observed impact plumes rose through a similar height of ∼3000 km, the radii of the created ejecta patterns differ from impact to impact and generally are larger for larger impacts. The azimuthal angle of the symmetry axis of the ejecta pattern is larger than that predicted by the models of oblique impacts, due to the action of the Coriolis force that rotates ejecta patterns counterclockwise from the south. We study the formation of ejecta patterns using a simple model of ballistic plume above a rotating plane. The ejected particles follow ballistic trajectories and slide horizontally for about an hour after reentry into the jovian atmosphere. The lateral expansion of the plume is stopped by the friction force, which is assumed to be proportional to the square of the horizontal velocity. Two different mass–velocity distributions used in the simulations produce qualitatively similar results. The simulated ejecta patterns fit very well the “crescents” observed at the impact sites. The sizes and azimuthal angles of symmetry axis of ejecta patterns depend on a parameter L, which has dimension of length and is related to the mass of the fragment. Thus more massive impacts produce larger ejecta patterns that are rotated through a wider angle. °c 1999 Academic Press Key Words: comets, S-L9; Jupiter; impact processes.
I. INTRODUCTION
The explosions of fragments of Comet Shoemaker–Levy 9 (SL 9) in the atmosphere of Jupiter created distinctive ejecta patterns around impact sites. The ejecta patterns are formed by the mixed comet–jovian material that was ejected through considerable heights and then reentered the atmosphere several thousand kilometers from the impact point. The ejecta patterns show consistent morphology: large crescent-shaped ejecta to the southeast of the impact site, circular ring and sometimes smaller faint ring inside the main ring, and a small triangle with its apex near the ring center and its base to the southeast (see Fig. 1). The “crescents” are probably made visible by the dark debris the core of which is composed of silicate grains that condense in the expanding plume (Friedson, 1998). Although all the impact plumes rose through a similar height of about 3000 km, the sizes of impact sites vary from about 6000 km for small impacts (Fragments S and W) to about
13000 km for large impacts (G, K, and L) (Hammel et al. 1995). The symmetry axis of the ejecta patterns (azimuth angle 35◦ ± 5◦ counterclockwise from the south for the G impact (Hammel et al. 1995)) does not line up with the fragment trajectory (16◦ counterclockwise from south) as predicted by the models of oblique impacts (Boslough et al. 1995, Crawford et al. 1994, Takata et al. 1994). It was suggested that the increase of the azimuth angle is due to the Coriolis force that rotates the plume of ejecta counterclockwise. To match the observations, the rotation must take place for 1 hr or more, which means the plume must continue its expansion after reentry into the atmosphere (Hammel et al. 1995). The radial expansion of the plume after the reentry also helps to explain the sizes of the largest impact sites (∼13000 km) since a plume of equal velocity ballistic particles will reach a height of 3000 km (particles going straight up) and a maximum range of only 6000 km (particles ejected at elevation angle of 45◦ ). The present paper is an attempt to test this hypothesis with a qualitative model. Numerous authors already addressed issues of the “rings,” the asymmetry of the plumes and ejecta patterns, the same height of the plumes associated with different impacts, and other issues (Shoemaker et al. 1995, Boslough et al. 1995, Takata and Ahrens 1997, Takata et al. 1994, Gryaznov et al. 1994, Zahnle and Mac Low 1995, Mac Low and Zahnle 1994, Ingersoll and Kanamori 1995). At the same time the shape of the ejecta patterns (crescent) have not been convincingly explained. Crawford et al. (1995) speculate that the crescent reflects the vertical distribution of the source of the visible ejecta in the jovian atmosphere. This argument implies that farther thrown ejecta originates from some particular layer in the atmosphere that contains dark material. This dark material also has the highest ejection velocity. The inner part of the ejecta pattern is made of transparent lowvelocity ejecta that originates from below the layer of dark material. The shape of the ejecta pattern (crescent) thus suggests a layered distribution of the dark material in the atmosphere that is preserved in the plume. Although such a distribution of ejecta in the plume is not improbable, the justification of it requires extensive numerical modeling of the explosion itself. Zahnle and Mac Low (1995) argue that the crescent is formed by carbonaceous dust generated from jovian air in the reentry shock. Reentry velocities must exceed 4 km/sec for shock
157 0019-1035/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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The model described below explains the bulk of the observations in a simple way. It is capable of producing the ejecta patterns that match all major features of the observed crescents—the shape, the size, and the azimuth angle of the symmetry axis. The model treats the ejecta as a cloud of ballistic particles above a rotating plane. Upon reentry onto the initial level the particles (ejecta) continue to move horizontally, slowing down under the influence of the friction force which is assumed to be proportional to the square of the velocity. II. MODEL
Ballistic Plume
FIG. 1. The D and G impact sites. North is up and east is to the right. Reprinted with permission from Fig. 1 West et al. (1995). Copyright 1995 American Association for the Advancement of Science.
heating to be strong enough to produce dust. Thus the crescent is produced by the ejecta with the highest velocities. This argument does not take into account the fact that high-velocity ejecta do not necessarily go the farthest unless all ejected material is launched at the same elevation angle. The model plume used by Zahnle and Mac Low is axisymmetric with a wide range of the launching angles. Hence the shock-generated carbonaceous dust is generated at all distances from impact site and the crescent does not form. The ejecta pattern presented by Takata and Ahrens (1997) resembles the observed patterns fairly well but does not match the observed azimuth angle of the symmetry axis. In their model, the descent of the plume is artificially prolonged to match the HST data at t > 1200 sec after the impact and the plume loses all its kinetic energy after returning onto the initial level, which contradicts the theoretical conclusion that the horizontal component of the velocity is preserved during the “splash back” (Zahnle and Mac Low 1995). Prolonging the descent time produces an ejecta pattern that is larger and that is rotated through a wider angle than would be expected for the equal velocity ballistic plume. Still, the angle is too small to match the observed azimuth angle of the symmetry.
The plume of ejecta is treated as a cloud of ballistic particles above a rotating plane. The motion of the particles occurs above the plane that coincides with the 0.1 bar level in the jovian atmosphere. Each particle represents a parcel of mixed jovian and cometary material in the plume. The plume is axisymmetric and is initially ejected from a cone with opening angle θ 0 . The plume can be visualized as a conic section of the radially expanding cloud centered on the impact point (see Fig. 2). The opening angle θ 0 is left to be a free parameter of the model, although simulations show that it must be about 70◦ to yield the crescent that extends 180◦ around the impact site. The axis of the cone is aligned with the incoming trajectory of the bolide (elevation angle of 45◦ and azimuth angle 16◦ counterclockwise from south) as predicted by numerical models (Boslough et al. 1995, Crawford et al. 1994, Takata et al. 1994) and as possibly seen on some HST images of rising plumes (Hammel et al. 1995). The particles are ejected with initial velocities that are governed by one of two mass–velocity distributions (see below) and with ejection angles that follow a uniform angular distribution. Particles ejected downward are not included in the calculations. This corresponds to a horizontal “cutoff” of the plume, which is consistent with numerical modeling (Takata et al. 1994, Boslough et al. 1995, Takata and Ahrens 1997). The particles follow ballistic trajectories bent by the Coriolis force. The equations that describe trajectories of the particles above the rotating flat Jupiter are u˙ = 2Ä(v sin ϕ − w cos ϕ) v˙ = −2Äu sin ϕ w˙ = 2Äu cos ϕ − g, where u, v, and w are the east, north, up components of velocity vector V¯ , ϕ is jovian latitude, and Ä is the rotation rate. This system of equations is solved assuming that ϕ is constant. Mass–Velocity Distributions We use two different mass–velocity distributions to model the plume. These mass–velocity distributions are the well-known solutions to the key gas-dynamic problems and are widely used in impact modeling. Since the initial conditions of the SL 9 explosions are very uncertain it is not possible to discriminate
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FIG. 2. The plume geometry. The plume is axisymmetric and is initially ejected from a cone with opening angle θ 0 . The axis of the cone is aligned with the incoming trajectory of the fragment (elevation angle of 45◦ and azimuth angle 16◦ counterclockwise from the south). The plume can be visualized as a conic section of the radially expanding cloud centered on the impact point. The ejection angles of the individual particles follow a uniform angular distribution. Particles ejected with negative vertical velocities are not included into simulations.
between these two distributions a priori. Although the distributions are quite different they produce qualitatively similar results in our simulations. The first distribution is that of the gas cloud isentropically expanding into a vacuum. This distribution is assumed to be valid for rather shallow explosions when the explosion fireball forms in the upper part of the atmosphere. According to Zel’dovich and Raizer (1967) the differential mass–velocity distribution of such a cloud evolves toward the form d M ∼ (1 − x 2 )α x 2 d x,
(1)
√ where x = v/vmax is a normalized velocity, vmax = 2γ /(γ − 1) √ 2E/M is the maximum velocity in the cloud, E/M is the thermal energy of the gas in the cloud, γ is the ratio of the specific heats, and the constant α is determined by conservation of mass and energy α = (γ + 5)/2/(γ − 1) (Zahnle 1990). For the fragment G fireball γ ≈ 1.2 (Carlson et al. 1997), thus α ≈ 15.5. Figure 3 shows the plot of the normalized mass–velocity distribution (1). Instead of mass–velocity distribution we use number– velocity distribution, assuming that all particles have the same mass, so the number of particles in an interval d x is given by (1). Parameter vmax is constrained by comparing the height reached by the simulated plume at different times with the HST data
(Hammel et al. 1995). Figure 4 shows a simulated plume with mass–velocity distribution (1), vmax = 25 km/sec, at t = 500 sec. At this time the plumes observed by HST reached their maximum height of ∼3200 km. The plume shown in Fig. 4 does not have a clearly defined edge. Moreover, the particle cross section is largely unknown. Thus, to compare the heights of the simulated and observed plumes, we assume that the edge of the plumes corresponds to a specific contour of the column density along the direction of the observation. The fact that the edges look sharp on the HST images thus suggests rapid density decrease toward outer parts of the plume. The procedure employed to calculate the height of the simulated plumes is as follows. First, we project the numerical plume at t = 500 sec on the south–north plane, calculate column density contours, and choose the contour that has maximum height of ∼3200 km. This contour of column density corresponds now to the edge of the plume. Second, we repeat this calculation at different times and determine the maximum height of the previously chosen column density contour (the edge) at these different times (the edge is not a material surface). This gives us the height–time dependence for the plume with that particular vmax . Finally, we repeat the whole procedure for the plumes with different vmax and compare the height–time dependencies of the simulated plumes with
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FIG. 3. Normalized mass–velocity distribution (1) for α ≈ 15.5. Note that the bulk of the mass is ejected with the velocity that is just a fraction of vmax .
the HST observations. We estimate the maximum velocity to be in the range of 25 ± 10 km/sec in order for the height of the simulated plume to be consistent with the HST observations (Fig. 3 of Hammel et al. 1995). Plumes with higher vmax rise and fall faster than suggested by HST observations and reach heights greater then ∼3200 km. Plumes with lower vmax rise slower than suggested by HST observations. This estimate will be further constrained later when we determine the sliding parameter L for the plume with mass–velocity distribution (1). The bulk of the mass of the plume with mass–velocity distribution (1) is ejected with the velocity which is just a fraction of vmax , thus our estimate differs from that of Hammel et al. (1995) (∼13 km/sec).
The thickness of the edge of the plume is defined as the length on which the value of the column density contour decreases by half from the value of the column density contour chosen as the edge (see above). The thickness of the edge is larger for the plumes with higher vmax and increases with time. For the plumes with vmax = 25 ± 5 km/sec it comes to ∼400 ± 200 and ∼1100 ± 300 km at times t = 500 and 900 sec, respectively. The resolution of the instrument (PC1) that observed rising SL9 plumes was 170 km per pixel, according to Ref. 6 of Hammel et al. (1995). We conclude that the edges of the simulated plumes would look sharp during the early stages of plume evolution, consistent with the HST observations.
FIG. 4. Projection of the simulated plume with mass–velocity distribution (1) on the south–north plane. vmax is 25 km/sec and t is 500 sec after the impact. The viewing geometry approximately corresponds to the geometry of the HST observations (the plume is viewed from the east) and the time corresponds to the time when the plumes observed by HST reached their maximum height. The axis of the plume is to the left (south) and slightly toward the observer.
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Another mass–velocity distribution is based on the results of numerical simulations (Shoemaker et al. 1995, Boslough et al. 1995, Takata et al. 1994, Crawford et al. 1994) that suggest that the ejecta exit the atmosphere through a wake left by the incoming cometary fragment. The mass–velocity distribution in this case differs considerably from (1). As shown by Carlson et al. (1997), the bulk of the material is ejected in a narrow range of velocities as in the case of strong explosion (Zel’dovich and Raizer 1967). It is then reasonable to assume that all the mass exited with the same velocity vmax and that the resultant mass– velocity distribution has the form d M ∼ δ(1 − x) d x,
(2)
where, as before, x = v/vmax is a normalized velocity and δ(x) is a δ-function. The maximum velocity vmax in this case is immediately defined by the maximum height reached by the plume to be ∼13 km/sec. Sliding Phase When the plume rains back onto the atmosphere (or, in other words, it returns to the initial level), the kinetic energy associated with its vertical motion is dissipated in a reentry shock. The energy goes into the thermal energy of the ejecta and of the ambient atmosphere, and eventually it is radiated into space and the lower atmosphere (Zahnle and Mac Low 1995). The collapsed plume continues to expand horizontally and rotate counterclockwise due to the Coriolis force (Hammel et al. 1995) until the plume is stopped by friction force. Friction in our model is proportional to the square of the horizontal velocity of particles and is governed by the parameter L, which has dimension of length. Variations of the parameter L affect both the size and the azimuthal angle of the ejecta pattern. The equations that govern horizontal sliding of the ejecta are u˙ = 2Äv sin ϕ − uV /L v˙ = −2Äu sin ϕ − vV /L w = w˙ = 0,
(3)
√ where V is the total horizontal velocity, V = u 2 + v 2 . The rationale behind the friction term in (3) is as follows. Let us approximate the spreading of the plume as a horizontal flow of one fluid over another. The surface stress per unit area τ is then τ = C D ρV 2 , where C D is a drag coefficient (or skin friction coefficient) and ρ and V are the density and the velocity of the upper fluid (ejecta), respectively. The mass of the upper fluid per unit area is ρh, where h is the thickness of the layer of the fluid, so that the deceleration is −τ/ρh or V˙ = −C D V 2 / h = −V 2 /L .
The parameter L is then equal to the thickness of the layer of the ejecta divided by the drag coefficient. We found L to be of the order of several thousand kilometers in our simulations. If the thickness of the layer of ejecta is of the order of tens of kilometers, the drag coefficient is then ∼0.01–0.001, which is consistent with the values of the skin friction coefficient for the turbulent flow over a flat surface (see, for example, Schlichting 1968). Parameter L is chosen in such a way that the radial extent and azimuthal angle of the symmetry axis agree with the observation. In the case of ejecta pattern produced by the plume with the mass–velocity distribution (1) this means that we pick a density contour that outlines the crescent with the outer radius of ∼13000 km and vary L so that the inner radius of this contour becomes equal to ∼8000 km (the inner and outer radius are for the G impact site (Hammel et al. 1995)). It is not always possible to match both radii and the azimuth angle of the symmetry axis for a given vmax . This circumstance further constrains the range of possible maximum velocities for the plume with the mass–velocity distribution (1) to vmax = 30 ± 5 km/sec, for which L = 4000 ± 500 km. In the case of the ejecta pattern produced by the plume with the mass–velocity distribution (2) the radius and the angle are matched simultaneously. Given the uncertainty of the azimuthal angle measurement (35◦ ± 5◦ ), L = 2500 ± 500 km for the mass–velocity distribution (2). III. RESULTS
Figure 5 shows simulated ejecta patterns 90 min after the impact, when horizontal spreading of the plume has slowed down significantly. The HST observations were made at approximately the same time. The ejecta pattern shown in Fig. 5a is formed by the plume with the mass–velocity distribution (1) and parameters vmax = 30 km/sec and L = 4000 km. The plume particles concentrate to the southeast of the impact point and form a feature that resembles the observed crescent of the G impact. The azimuth angle of the symmetry axis is 38◦ ± 3◦ counterclockwise from south, in agreement with the observations. The simulated feature does not have a sharp boundary, but the number density of the particles increases drastically at the location of the observed crescent. The inner and outer radius of the feature, determined as described above in Section II, are at ∼8000 and ∼13000 km. Figure 5b shows the ejecta pattern formed by the plume with the mass–velocity distribution (2) and parameters vmax = 13 km/sec and L = 2500 km. The sharp edges of the simulated crescent are at ∼8000 and ∼13000 km, and the azimuth angle of the symmetry axis is 35◦ ± 2◦ counterclockwise from south. The outer edge of the crescent is formed by the particles that were ejected at elevation angle of 45◦ . The inner edge of the crescent is formed by the particles that were ejected along the surface and started sliding right after the impact. The thinly populated area around the impact point and toward the northwest of the impact point is filled with particles that were ejected at elevation angles >45◦ and to the right of the z axis in Fig. 2.
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FIG. 5. Simulated ejecta patterns 90 min after impact. North is to the top and east is to the right. The axes are labeled in thousands of kilometers from the origin (the impact point), which is marked by a cross. (a) Ejecta pattern of the G impact generated by numerical plume with mass–velocity distribution (1), vmax = 30 km/sec, L = 4000 km. (b) Ejecta pattern of the G impact generated by numerical plume with mass–velocity distribution (2), vmax = 13 km/sec, L = 2500 km. (c) Simulated ejecta pattern of the A impact, same plume as in (a), L = 1500 km. (d) Simulated ejecta pattern of the A impact, same plume as in (b), L = 500 km.
Both mass–velocity distributions (1) and (2) produce qualitatively similar results. The “real” mass–velocity distribution of the ejected material probably lies somewhere in between these two extreme cases so that the edges of the resultant ejecta pattern is softer than the edges in Fig. 5b but not so diffuse as the edges in Fig. 5a. A small triangle—or streak—that is common for major impacts forms probably independently of the crescent. HST images of rising plumes E and G show faint emission that follows the plume. Numerical simulations (Yabe et al. 1995) suggest that two groups of material were ejected. Penetration of the comet must be rather deep in this case (300 km below 1 bar level). Be it as it may, the streak can be reproduced by our model only if the mass–velocity distribution of ejecta differs significantly from (1) and (2) in its low-velocity part.
The ejecta patterns of other impact sites can be reproduced by our model by changing the parameter L. For smaller L, the rotation of the ejecta pattern is less and the radius of the resultant crescent is smaller. In the previous section we speculated that L is related to the thickness of the sliding layer of ejecta. It seems reasonable to assume that parameter L is hence related to the mass of the plume, since more massive plumes create a thicker layer of ejecta. Thus, more massive plumes produce larger ejecta patterns and rotate through a wider angle, although the initial mass–velocity distribution of the plumes are the same since all plumes go to the same height. Figures 5c and 5d show simulated ejecta patterns of the smaller impact A produced by the same plumes that create ejecta patterns in Figs. 5a and 5b, respectively. The only difference is that the parameter L was reduced by a factor of several. The resultant
SL 9 EJECTA PATTERNS
ejecta patterns are smaller (outer radius ∼8000 km) and were rotated through a smaller angle (∼30◦ ), which is consistent with the observations of the fragment A impact site (K.-L. Jessup, personal communication, 1998).
IV. CONCLUSION
We developed a simple model of dynamics of ejecta from impacts of Comet SL 9 that can adequately reproduce the crescentshaped ejecta patterns observed by HST at the impact sites. The model assumes that ejecta follow ballistic trajectories and slide horizontally upon reentry into the atmosphere until stopped by the friction force. We use two different mass–velocity distributions, both of which produced qualitatively similar results. Mass–velocity distribution (2) seems to be closer to reality because it is backed by the numerical calculations of Carlson et al. (1997) and the plume and the resultant crescent have sharp boundaries, which is more consistent with the observations. The friction force during the sliding phase is assumed to be proportional to the square of the velocity. This assumption represents the notion that the ejecta forms a layer that slides over the surface of the underlying fluid (jovian atmosphere). The thickness of the layer and the resultant frictional deceleration vary with the mass of the plume. This explains why the plumes rose to similar heights but produced ejecta patterns of different sizes. Obviously, the real process that governs the horizontal expansion of the plume is much more complex. Nevertheless, our model seems to capture the essence of the phenomenon. We hope the results will be of use to future modelers.
ACKNOWLEDGMENTS Helpful suggestions have been received from N. Schneider and T. Takata. This work was supported by the NASA Planetary Atmospheres Program under Grant NAGW-1956.
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