1. INTRODUCTION

1. EPSC2013-335Cavities in cometsSergei I. Ipatov1Space Research Institute, Moscow, Russia2Catholic University of America, Washington, DC, USA. (siipatov@hotmail.com, http://faculty.cua.edu/ipatov/)This presentation can be found on http://faculty.cua.edu/ipatov/EPSC2013-cavities.ppt.

2. 1. INTRODUCTION • On July 4, 2005 370 kg Deep Impact impactor collided with Comet 9P/Tempel 1 at velocity of 10.3 km/s (A’Hearn et al. 2005). It was an oblique impact, and the angle above the horizon was about 20-35o. Evolution of the cloud of ejected material was observed by Deep Impact (DI) cameras, by space telescopes (e.g. Rosetta, Hubble Space Telescope, Chandra, Spitzer), and by over 80 observatories on the Earth. • Outbursts from several comets were observed. • Objectives of our studies are to analyze the ejection of material after the DI collision, especially due to the triggered outburst, and to estimate the location of cavities with dust and gas under pressure that could cause the triggered outburst.

3. We analyzed several series of DI images. In each series, the integration time and the size of image were the same. For series Ma, Ha, and Hc, we analyzed the differences in brightness between a current image and that before the impact. These series are marked by “(dif)”. For other series, we analyzed the brightness in current images. Observed brightness of the cloud of ejected material was mainly due to particles with diameters d<3 micron, and we discuss mainly ejection of such particles.

4. 2. Triggered outburst from Comet Tempel 1 • The main points of our studies of the outburst are: Ipatov& A’Hearn (2011) analyzed images of the cloud of material ejected after this collision. Based on analysis of the images made by the DI cameras during the first 13 minutes, they studied the process of ejection of material and concluded that, besides the normal ejection, there was a triggered outburst of particles with diameters mainly less than a few microns. • Theoutburst was considerable at 8-60 s after the impact. It increased the duration of ejection and mean velocities of observed ejected particles (compared with the normal ejection). Mean velocities of small observed particles almost did not change with time te after the impact (and were about 100 m/s) at 10<te<20 s. • Ipatov& A’Hearn (2011) supposed that the outburst was caused by the ejection of material from cavities containing dust and gas under pressure. Velocities of such `fast` outburst particles could be mainly ~100 m/s (such velocities were obtained at various observations of the leading edge of the DI cloud). • Studies of the rate of ejection of material after the DI collision were based on analysis of contours of constant brightness in DI images.

5. Contours corresponding to CPSB (calibrated physical surface brightness) equal to 1, 0.3, 0.1, and 0.03.Our studies were based mainly on analysis of distances from the place of ejection to such contours. The bumps on the contours correspond to rays of excessive ejection.

6. Rays of ejected material • The excess ejection of material to a few directions (rays of ejected material) was considerable during the first 100 s, took place during several minutes, and was still observed in images at t~500-770 s even close to the place of ejection. It shows that the outburst could continue up to ~10 min. • Considerable excessive ejection to a few directions (some rays) began approximately at the same time te~10 s when the direction from the place of ejection to the brightest pixel changed, the peak brightness began to increase, and there was a local peak of the rate of ejection. These features could be caused by the outburst triggered by the impact. • The sharpest rays were caused by material ejected at te~20 s. • The upper-right excessive ejection (perpendicular to the direction of impact) began mainly at te~15 s (though there was some ejection at te~2 s), could reach maximum at te~25-50 s, could still be considerable at te~100 s, but then could decrease, though it still could be seen at te~400 s. The value of te~15 s is correlated with the changes of the direction to the brightest pixel at t~12-13 s. • The upper bump of the outer contours is clearly seen at 66<t<665 s, especially at t~200-350 s. The direction from the place of impact to this bump is not far from the direction opposite to the impact direction.

7. Time variations in sizes L (in km) of regions inside contours of CPSB=const. The number after a designation of the series in the figure legend shows the value of brightness of the considered contour. The curves have local minima and maxima that were used for estimates of velocities of particles at several moments in time. Based on the supposition that the same particles correspond to local maxima (or minima)of L at different CPSB(e.g., to values L1 and L2 on images made at t1 and t2),we calculated the characteristic velocitiesv=(L1-L2)/(t1-t2) of ejected particles at time of ejection te=t1-L1/v. For series Ma, we considered L as the distance from the place of impact to the contour down in y-direction. For other series, we considered the difference between maximum and minimum values of x for the contour.

8. Typical projections vmodel of velocities (in km/s) on the plane perpendicular to the line of sight at time te of ejection for the model for which velocities vmodel at te are the same as velocities vexpt=c×(t/0.26)-α of particles at the edge of observed bright region at time t. The values of vyobs and vxobs are based on analysis of local minima and maxima of plots presented on the previous slide. The distance from the place of ejection to the edge was used to find the dependence of t on te. As the first approximation, the characteristic velocity of particles at te>1 s can be considered to be proportional tote-0.75 or te-0.7 (i.e. α~0.7-0.75; 0.71 corresponds to sand; 0.75, to the ejection mainly governed by momentum). The values of vymin and vxmin show the minimum velocities of particles needed to reach the edge of the bright region (in an image considered at time t) from the place of ejection moving in y or x-direction, respectively.

9. Calculations of the relative rate of ejection Considering that the time needed for particles to travel a distance Lr to the edge of the bright region is equal to dt=Lr /vexpt (where vexpt(t)=c×(t/0.26)-α ), we found the timete=t−dtof ejection of material of the edge of the bright region considered at time t. Ipatov and A’Hearn (2011) showed that, as the first approximation, the characteristic velocity of particles at te>1 s can be considered to be proportional to te-0.75 or te-0.7(i.e. α~0.7-0.75; 0.71 corresponds to sand; 0.75, to the ejection mainly governed by momentum). The volume Vol of a spherical shell of radius Lr and width h is proportional to Lr2h, and the number of particles per unit of volume is proportional to rte∙(Vol∙v)-1, where v is the velocity of the material. Here rte corresponds to the material that was ejected at te and reached the shell with Lr at time t. The number of particles on a line of sight, and so the brightness Br, are approximately proportional to the number of particles per unit of volume multiplied by the length of the segment of the line of sight inside the DI cloud, which is proportional to Lr. Actually, the line of sight crosses many shells characterized by different rte, but as a first approximation we supposed that Br is proportional to rte(v∙Lr)-1. For the edge of the bright region, Br≈const. Considering v=vexpt, we calculated the relative rate of ejection as rte=c∙Lr∙t-α. 9

10. Relative rate of ejection at different times te of ejection for the model in which characteristic velocities of particles constituting the edge of the observed bright region at time t are equal to vexpt=c×(t/0.26)-α. At 1<te<3 and 8<te<60 s, the plot of time variation in the estimated rate rte of ejection of observed material was essentially greater than the exponential line connecting the values of rte at 1 and 300 s. (The exponential monotonic decrease in the rate is predicted by theoretical models.)At te~10 s therewas an increase in the rate of ejection of particles (with typical projections of velocities vp~100 m/s), and there was a sharp decrease in the raterte at te~60 s. Such variations in rte can be mainly due to the triggered outburst. Our studies do not contradict to a continuous ejection of material during at least 10 minutes after the collision.

11. Relative amount fev of observed material ejected with velocities greater than v vs. vfor the model VExp in which characteristic velocities of particles at the edge of the observed bright region in an image at time t are equal to v=vexpt=c×(t/0.26)-αfor five pairs of α and c. fev=1 for material ejected before the time te803 of ejection of particles located at the edge of the bright region in an image made at t=803 s. Compared to theoretical models, there was an excess of ejection of particles at velocities of several tens of meters per second. For theoretical models, exponents of the velocity dependence of the relative volume fev of material ejected with velocities greater than v, equal to -1, -1.23, -1.66, and -2, correspond to α equal to 0.75, 0.71, 0.644, and 0.6, respectively (α=0.71 is for sand).

12. Acceleration of particles • Hosapple & Housen (2007) supposed that ejected particles could be accelerated by the dust-gas interaction. Such acceleration could be important during time intervals of about several hours. • Ipatov & A’Hearn (2011) made their conclusions basing on analysis of variations in brightness in images made during the first 13 minutes and considering distances R from a place of ejection between 1 and 10 km. • They concluded that particles could not increase their velocities by more than a few m/s during not more than a few minutes, when the particles moved at R between 1 and 10 km. • Ipatov and A’Hearn analysed the brightness of the DI cloud at such distances and did not consider a plume base at R<1 km as Richardson et al. (2007). In many DI images, the region corresponding to R<1 km consisted mainly of saturated pixels (in 8-bit images received from the DI spacecraft, the brightness of most saturated pixels equaled to 255).

13. Triggered outburst vs. a model of a layered target • In our opinion, it is difficult to explain the time variations in the brightness of the DI cloud at 1<R<10 km without consideration of the triggered outburst with velocities of about 100 m/s. Several authors (e.g. Richardson et al. 2007) suppose that variation in brightness of the DI cloud was caused by variation in particle size distribution due to striking a layered target. • In our opinion, the model of a layered target plays some role in explanation of the variation in brightness of the DI cloud, but it cannot explain all details of such variation (e.g., why at the same time ~10 s there was the jump in the direction from the place of ejection to the brightest pixel of the cloud image by 50 deg, the increase in the rate of ejection of small particles, and the increase in brightness of the brightest pixel; why at the same time of ejection te~60 s, there was the sharp decrease in the rate of ejection of small particles, and the direction from the place of ejection to the brightest pixel in an image returned to that at 1<t<12 s; why mean ejection velocities of observed particles were almost the same at te~10-20 s, etc.).

14. 3. Cavities in cometsSize of the Deep Impact crater • On 2011 February 14, the Stardust spacecraft made photos of the crater formed after the DI impact. Peter Schultz et al. (2013) concluded that the diameter of the transient (кратковременный) DI crater was about 150-200 m. Smaller estimates of the diameter were made by several other scientists. For example, Melosh and Richardson supposed that there is an approximately 50 meter diameter depression that most likely represents the DI crater. • The crater size observed by Stardust spacecraft could be different from the transient crater size just after its formation. This is because the sublimation process as well as various modification processes changed the crater size after its formation. • On the image made by the Stardust spacecraft (http://stardustnext.jpl.nasa.gov/Multimedia/popups/Tempel1ImageSite.html), the diameter of the brightest part of the ring zone of ejected material around the crater is about 90-100 m. The diameters of the inner and outer edges of the ring zone are ~60-70 m and ~130-140 m, respectively. The ring zone may correspond to ejected material, and the diameter of an excavation zone might not exceed 100 m.

15. Growth of a forming crater • It is considered (e.g. Croft 1980; Melosh 1989) thatejected material originates from an excavation cavity which has a geometry distinct from a transient crater. The excavation cavity and the transient crater have the same diameter dtc, but the depth dhe of the excavation cavity is ~0.1dtc, or about one-third of the transient crater depth, and, in the case of simple bowl-shaped craters, about one-half of the depth of the final apparent crater. For example, dhe~10 m at dtc=100 m. For some craters, Croft (1980) considered the ratio dhe/dtc to be in the range [0.09, 0.17]. • For theoretical models (e.g. Holsapple & Housen 2007), during the most time of crater formation (except for initial and final stages), the diameter dc of a craterat time te elapsed after the impact is proportional to teγ, where γ is about 0.25-0.4. In their table 4, Holsapple & Housen (2007) considered models at 0.29≤γ≤0.36.

16. Estimates of the depth of the DI crater at the beginning of excavation of the main cavity According to fig. 12 in (Holsapple 1993), diameter dc of a cratergrows faster at the initial stage (which duration is usually less than 0.1Te, where Te corresponds to the end of the stage of proportionality of dc to teγ). Ipatov & A’Hearn (2011) concluded that outburst and excavation of a large cavity began at ~8 s after the DI collision. Supposing the diameter dc of a craterto be proportional to teγ, Ipatov (2012b) estimated the lower limit of the depth, dcavDI, of the DI crater at the time of the beginning of excavation of the main cavity teb=8 s as dcavmin=dhe×(teb/Te)γ. At dhe=0.1dtc, the value of dcavmin equals dcmn=0.1dtc×(teb/Te)γ. At teb=8 s, the values of dcmn for three values of γ are presented in Table 1. The real value of dcavDIis greater by some value dh1 than dcmn because, during the short initial stage, the real growth of the crater is greater than that for the model used for calculation of the lower limit dcmn. We suppose that the difference dh1 can be about 1 m. In Table 1 we consider that dcavDI=0.1dtc×(teb/Te)0.3+dh1, where dh1=1 m.

17. Table 1.The distance, dcavDI, between the pre-impact surface of Comet Tempel 1 and the upper border of the cavity excavated at teb=8 s for several values of the diameter, dtc, of the DI transient crater and the time, Te, of formation of the crater. The minimum value, dcmn=0.1dtc(teb/Te)γ, of the distance at three values of γ is also given. The value of dcmn at γ=0.3 was used for estimates of dcavDI. The distance between the pre-impact surface of the comet and the upper border of the main excavated cavity is about 5-6 m for the diameter of the DI transient crater of about 150-200 m.

18. The depth of the DI crater at the beginning of excavation of the main cavity • Based on the above estimates, we can suppose that dcavmin was probably not less than 3 m at dtc=100 m. During the intermediate stage of crater formation (when diameter dc of a crater is proportional to teγ), time usually increases by more than a factor of 10 (see fig. 12 in Holsapple 1993). Therefore, during the time interval [0.1Te, Te ] (we consider Te as the time that corresponds to the end of the stage of proportionality of dc to teγ) dc increases by a factor of 10γ, where 10γ is 1.8-2.5 at 0.25≤γ≤0.4. • These estimates show that at teb=8 s and Te>80 s (this inequality is fulfilled for the DI crater) the maximum value, dcavmax, of dcavDI does not exceed dhe/10γ+dh1, which is in the range [0.4dhe+dh1, 0.56dhe+dh1], that is dcavmax≤0.056dtc+dh1 at dhe/dtc=0.1 (e.g. dcavmax≤0.05dtc+dh1 for γ=0.3). At Te=80 s and γ=0.3, the value of 0.05dtc is greater by about 2 and 5 m than the values of dcmn presented in Table 1 for dtc=100 m and dtc=200 m, respectively. • Based on the values of dcavDI presented in Table 1, we concluded that the distance between the pre-impact surface of the comet and the upper border of the main excavated cavity is about 5-6 m for the estimates (150-200 m) of the diameter of the DI transient crater presented by Schultz et al. (2012). The estimated distance is 4 m for the diameter of 100 m.

19. Cavities in different comets The distance between the pre-impact surface of Comet 9P/Tempel 1 and the upper border of the largest excavated cavity equal to ~5 m, and sizes of particles inside the cavities of a few microns are in a good agreement with the results obtained by Kossacki & Szutowicz (2011). • These authors made calculations for several models of the explosion of Comet 17P/Holmes. They concluded that the nonuniform crystallization of amorphous water ice itself is probably not sufficient for an explosion, which could be caused by a rapid sublimation of the CO ice (моно-оксид углерода, угарный газ) leading to the rise of gas pressure above the tensile strength of the nucleus. In their models, the initial sublimation front of the CO ice was located at a depth of 4, 10, or 20 m, and calculations were finished when the CO pressure exceeded the threshold value 10 kPa. It was shown that the pressure of CO vapor can rise to the thresholdvalue only when the nucleus is composed of very fine grains, a few microns in radius. • Our estimates of location of cavities in Comet 9P/Tempel 1 and the studies of the initial sublimation front of the CO ice in Comet 17P/Holmes made by Kossacki & Szutowicz show that the upper borders of relatively large cavities containing dust and gas under pressure can be located at a depth of 3-20 m below surfaces of different comets.

20. Observed outbursts from several comets Astronomers observed outbursts from several comets. Cometary activity was observed even for some comets (e.g. Comet Hale-Bopp) moved outside of Jupiter’s orbit and for some objects moving in typical asteroid orbits. Natural outbursts from comets could last for weeks or months. The total mass of material ejected at the 2007 October 24 outburst of Comet 17P/Holmes (~1-4% of the nucleus mass of the comet, i.e. (1-3)×1011 kg) was much greater than that at the DI collision (~107 kg). Paganini et al. (2010) observed the outburst activity of Comet 73P/Schwassmann-Wachmann 3 in May 2006. They obtained a decrease in gaseous productivity of this comet by a factor of 2 in about a week. Prialnik et al. (2004) concluded that the outburst of Comet 1P/Halley could take place during a few months when the comet moved at a distance greater than 5 AU from the Sun. Results obtained by Ivanova et al. (2011) support the idea that the observed activity of Comet 29P/Schwassmann-Wachmann 1 at distance >5.5 AU requires a permanent demolition of the upper surface layers. A few Main Belt Comets, or “icy-asteroids”, were found in the asteroid belt (Bertini 2011). In our opinion, cometary activity of asteroid 7968 Elst-Pizarro, also known as Comet 133P/Elst-Pizarro, (a=3.16 AU, e=0.16, and i=1.39o) could be caused by the same internal processes as natural and triggered outbursts from other comets (including Comet Tempel 1), but its solid crust could be thicker than that of Comet Temple 1. In 1996, 2002, and 2007, the object Elst-Pizarro had a comet tail for several months.

21. Sources of gas pressure in cavities • The more time a comet has spent close to the Sun, the greater are distances from the surface of the comet to the upper borders of cavities containing dust and gas under considerable pressure. • It is usually considered that the main sources of gas pressure are the water ice sublimation, the sublimation of a more volatile ice such as CO or CO2 at a lower temperature than required for water ice, and the crystallization of amorphous ice in the interior of porous nucleus. Other discussed potential mechanisms of outbursts include the polymerization of hydrogen cyanide HCN, the thermal stresses, the anneling of the amorphous water ice, and meteoritic impacts (see, e.g., Gronkowski & Sacharczuk 2010; more references are presented by Ipatov 2012a). The porous structure of comets provides enough space for sublimation and testifies in favor of existence of cavities. • The projection of velocity of the leading edge of the DI cloud (onto the plane perpendicular to the line of sight) was about 100-200 m/s (see references in Ipatov & A’Hearn 2011) and is typical for outburst particles ejected from comets. According to Feldman et al. (2007), at the 2005 June 14 natural outburst from Comet Tempel 1, velocities of ejection were 60-145 m/s. Sarugaku et al. (2010) obtained that the dust cloud caused by the outburst from Comet 217P/LINEAR expanded at a velocity of 120-140 m/s. The similarity of velocities of particles ejected at the triggered and natural outbursts (~100 m/s) shows that these outbursts could be caused by similar internal processes in comets.

22. 4. Conclusions • At time of ejection te~10 s, there was a local maximum of the rate of ejection of observed particles (mainly with diameter d<3 μm)with typical projections of velocities vp~100 m/s. At 1<te<3 s and 8<te<60 s the estimated rate of ejection of observed material was essentially greater than that for theoretical monotonic exponential decrease. Such difference could be caused by that the impact was a trigger of an outburst. • At te~55-72 s, the ejection rate sharply decreased and the direction from the place of ejection to the brightest pixel quickly returned to the direction that was before 10 s. It could be caused by a sharp decrease of the outburst that began at te~10 s. • Analysis of observations of the DI cloud and of outbursts from different comets testifies in favor of that there can be large cavities with material under gas pressure a few meters below a considerable fraction of comet’s surface.

23. Conclusions • The upper border of the largest cavity excavated during ejection of material after the collision of the impact module of the Deep Impact spacecraft with Comet 9P/Tempel 1 could be located at a depth of about 5-6 metres below the pre-impact surface of the comet if the diameter of the DI transient crater was about 150-200 m, as presented by Schultz et al. (2013). The estimated depth was 4 m at the diameter equaled 100 m. The largest cavity excavated after the DI collision could be relatively deep because a considerable excess ejection lasted during ~50 s. • These estimates of the depth are in accordance with the depth (4-20 m) of the initial sublimation front of the CO ice in the models of the explosion of Comet 17P/Holmes considered by Kossacki & Szutowicz (2011). Our studies testify in favour of that cavities with dust and gas under pressure located a few metres below surfaces of comets can be common.

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