- Email: [email protected]
Dynamical evolution of Chelyabinsk-type bodies from sungrazing orbits to near-Earth space
Planetary and Space Science 118 (2015) 302–304
Contents lists available at ScienceDirect
Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
Dynamical evolution of Chelyabinsk-type bodies from sungrazing orbits to near-Earth space V.V. Emel’yanenko n Institute of Astronomy of the Russian Academy of Sciences, 48 Pyatnitskaya, Moscow 119017, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2015 Received in revised form 23 July 2015 Accepted 10 August 2015 Available online 21 August 2015
Studies of the orbital evolution show that, with a high probability, the Chelyabinsk object was near the Sun about 1 Myr ago. This is consistent with the estimate of a cosmic ray exposure age (Popova et al., 2013). Dynamical features of the flux of Chelyabinsk-type bodies have been studied by numerical integrations, assuming their origin near the Sun about 1 Myr ago. The most frequent fate of these bodies is solar collision. But about a quarter of all the original objects survive until the present epoch. The majority of the surviving bodies are typical near-Earth objects. The rate of their encounters with the Earth is almost constant for the last 0.5 Myr. About a third of the Chelyabinsk-type objects approach the Earth from the Sun direction. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Asteroid Meteorite Dynamics Hazard
1. Introduction After the Chelyabisk event it is evident that not only large asteroids but also 10 m size meteoroids pose a substantial hazard to the Earth civilisation. Although the number of known near-Earth objects has been growing rapidly in this century due to special surveys, there are large uncertainties in the population count, physical properties and dynamical characteristics of small asteroids. In particular, recent studies of bolides indicate that the number of impactors with diameters of 10 m may be an order of magnitude higher than estimates based on telescopic surveys (Silber et al., 2009; Brown et al., 2013). It is well known that near-Earth objects evolve frequently to orbits with small perihelion distances (Farinella et al., 1994; Gladman et al., 2000; Foschini et al., 2000; Marchi et al., 2009). In particular, there are large chances that the famous Chelyabinsk object was near the Sun in the past (Emel’yanenko et al., 2014). Asteroids undergo thermal and tidal destruction near the Sun. They may not maintain their physical integrity during this near-Sun period, creating many smaller bodies. Here we discuss the dynamical role of near-Sun states of asteroids in producing a flux of Chelyabinsk-type objects.
2. Approaches of observed meteoroids to the Sun in the past A study of the dynamical evolution of the Chelyabinsk object in the paper (Emel’yanenko et al., 2014) is based on integrations of n
Tel.: þ 7 495 9531624; fax: þ7 495 9515557. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.pss.2015.08.005 0032-0633/& 2015 Elsevier Ltd. All rights reserved.
860 orbits from the confidence region for 15 Myr. The six-dimensional vector q of orbital parameters from the confidence region is modelled ^ + C1/2η, where q ^ is the vector q at nominal by using the formula q = q values of orbital parameters, η is the six-dimensional standard normally distributed random vector, C1/2(the Cholesky matrix) is such a matrix that C1/2 (C1/2)T = C , and C is the covariance matrix. The orbital elements of the Chelyabinsk object have the following nominal values and standard deviations for the osculating epoch 2013.02.15.0 TT (Equinox J2000): semimajor axis a¼1.88070.068 AU, eccentricity e¼ 0.60970.017, argument of perihelion ω ¼108.92670.536°, longitude of the ascending node Ω ¼326.44670.002°, inclination i¼ 5.93870.427°, mean anomaly M0 ¼16.87371.069° (Emel’yanenko et al., 2014). This investigation shows that, with a high probability, this meteoroid was near the Sun in the past. The most probable time for the encounters of the Chelyabinsk object with the Sun is consistent with the estimate of a cosmic ray exposure age of 1.2 Myr (Popova et al., 2013). Fig. 1 shows typical examples of changes for perihelion distances q and inclinations of two objects from the confidence region that reach the near-Sun state about this time. The data are given for integrations from the present epoch until t ¼ 1.1 Myr for the first object and until t ¼ 1.2 Myr for the second object. The first object has q¼ 0.038 AU at t¼ 1.1 Myr, the second object has q ¼0.050 AU at t¼ 1.2 Myr. There are different dynamical mechanisms that lead near-Earth objects to the Sun (Farinella et al., 1994; Gladman et al., 2000; Foschini et al., 2000), but the decrease of q to very small values in the vicinity of the Chelyabinsk orbit is mainly associated with the secular resonance ν6 (Emel’yanenko et al., 2014).
V.V. Emel’yanenko / Planetary and Space Science 118 (2015) 302–304
303
Fig. 1. Typical examples of changes for perihelion distances and inclinations of two objects from the confidence region: left – the object 1; right – the object 2 (data for orbits are plotted every 500 years, crosses show initial positions of the objects in 2013).
Fig. 2. Distributions of a and q for particles that survive until the present epoch. The models correspond to the objects 1 and 2 shown in Fig. 1.
This dynamical behaviour of the Chelyabinsk meteoroid is not unique. Frequent collisions with the Sun were found earlier in integrations of orbits for 20 bolides in the work (Foschini et al., 2000).
3. Dynamical features of the Chelyabinsk-type objects It is natural to assume that a parent body could be disrupted due to the strong solar tide, thermal stresses and interaction with the solar atmosphere at Sun-grazing conditions. The cluster of the observed Kracht and Marsden sungrazing objects is probably a pattern of such disruption near the present epoch (Sekanina and Chodas, 2005). But it is impossible to describe in detail dynamical features of Chelyabinsk-type bodies if they originate from a parent body near the Sun about 1.2 Myr ago. Only statistical and qualitative considerations are possible in this case. Therefore, we studied the most common features of the distribution of Chelyabinsktype bodies if they originate due to disruption of a parent body near the Sun. Two examples shown in Fig. 1 were considered. In each case we studied the evolution of 400 test particles starting from the epoch t¼ 1.1 Myr for the first object and from the epoch t¼ 1.2 Myr for the second object. The first object had orbital elements q ¼0.038 AU, a ¼2.49 AU, ω ¼ 265.4°, Ω ¼47.7°, i¼9.9°, the second object had orbital elements q ¼0.050 AU, a ¼2.30 AU, ω ¼252.4°, Ω ¼162.8°, i¼6.3°. Orbital elements of the test particles were equal to the corresponding values for two original objects except for q. Initial values of q for the test particles were distributed uniformly in the small interval of 10 8 AU (1.5 km) around the above values of q for each original object. The dynamical evolution
of the test particles under the planetary perturbations was calculated until the present epoch, using the symplectic integrator (Emel’yanenko, 2007). Particles were removed from integrations when qo 0.005 AU or semimajor axis a4 50 AU. The most frequent fate of these particles is solar collision. Fig. 2 shows the distributions of a and q for particles that survive until the present epoch. These distributions are similar in both examples. The majority of surviving particles (about a quarter of all the original particles) are concentrated in the region 0o qo 2.0 AU, 1.5 oa o3.0 AU. Fig. 3 shows the number of encounters with the Earth to minimum distances Δ o0.01 AU (the Hill radius of the Earth) during certain time intervals of the integrations. Although numbers are substantially different for the studied variants, approaches to the Earth are continuous on the whole interval of integrations in both cases. The number of encounters decreases insignificantly near the present epoch (approximately for 0.5 Myr). Thus the level of a hazard to the Earth associated with Chelyabinsk-type bodies originating in near-Sun conditions is almost constant for the last 0.5 Myr. We checked also the angle γ between the direction of motion relative to the Earth and the Sun–Earth direction for particles reaching the minimum geocentric distance Δo 0.01 AU on the interval ( 0.1, 0) Myr of our integrations. Fig. 4 shows that the distributions of γ depend on the chosen initial condition. Nevertheless, not less than a third of all the encounters take place at γ o60°. This stresses that discovery of dangerous Chelyabinsk-type objects approaching the Earth from the Sun direction is a very serious task in the asteroid hazard problem (it is known that the Chelyabinsk object came from the sunward direction).
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V.V. Emel’yanenko / Planetary and Space Science 118 (2015) 302–304
Fig. 3. Numbers of encounters with the Earth to minimum distances Δ o 0.01 during certain time intervals. The models correspond to the objects 1 and 2 shown in Fig. 1.
Fig. 4. Distributions of γ at close encounters with the Earth near the present epoch. The models correspond to the objects 1 and 2 shown in Fig.1.
4. Conclusions The estimated cosmic ray exposure age of the Chelyabinsk body is 1.2 Myr (Popova et al., 2013). The recovered fragments of the Chelyabinsk meteorite contain significant portions of shock blackened material and melt veins (Galimov et al., 2013; Kohout et al., 2014; Ozawa et al., 2014). Collisions with other solid bodies are usually considered for the origin of these features (e.g., Borovička et al., 2013; Galimov et al., 2013; Kohout et al., 2014; Ozawa et al., 2014). In particular, Borovička et al. (2013) suggest that the Chelyabinsk object originated recently from the 2 km diameter near-Earth asteroid 86039 (1999 NC43) via a collision. However, Reddy et al. (2015) show that the lack of spectral match between Chelyabinsk and 1999 NC43, and their different chemistries suggest that the asteroid is probably not the source object for this meteorite. Moreover, the cosmic ray exposure age of 1.2 Myr for the Chelyabinsk body appears incompatible with the very short separation timescales discussed in Borovička et al. (2013). But it is shown in the paper (Emel’yanenko et al., 2014) that, with high probability, the Chelyabinsk object was near the Sun about 1 Myr ago (Emel’yanenko et al., 2014). Therefore, while we do not have the expertise to reject the impact origin of the Chelyabinsk body, we consider another possibility that a parent body of the Chelyabinsk object could be disrupted due to strong thermal and tidal stresses near the Sun. We have studied dynamical features of the flux of Chelyabinsktype bodies assuming their origin near the Sun about 1 Myr ago. The most frequent fate of these bodies is solar collision. But about a quarter of all the original objects survive until the present epoch. The majority of the surviving bodies are typical near-Earth objects. The rate of their encounters with the Earth is almost constant for the last 0.5 Myr. This dynamical behaviour of the Chelyabinsk meteoroid is not unique. It was shown earlier (Foschini et al., 2000) that the near-Sun states are typical in the evolution of
bodies associated with bright bolides. Thus, there are expected to be objects, originating from Sun-grazing orbits, in the population of near-Earth asteroids that pose a long-term hazard to the Earth. About a third of the Chelyabinsk-type objects approach the Earth from the Sun direction.
Acknowledgements This work was supported by the Ministry of Education and Science of the Russian Federation as part of the project RFMEFIBBB14X0288 under the Federal Targeted Programme ‘Research and Development on Priority Directions of Scientific and Technological Complex of Russia for 2014–2020’. The calculations were carried out using the MVS 100K supercomputer of the Joint Supercomputer Center of the Russian Academy of Sciences.
References Borovička, J., Spurný, P., Brown, P., et al., 2013. Nature 503, 235. Brown, P.G., Assink, J.D., Astiz, L., et al., 2013. Nature 503, 238. Emel’yanenko, V.V., 2007. Celest. Mech. Dyn. Astron. 98, 191. Emel’yanenko, V.V., Naroenkov, S.A., Jenniskens, P., Popova, O.P., 2014. Meteorit. Planet. Sci. 49, 2169. Farinella, P., Froeschle, Ch., Froeschle, Cl., et al., 1994. Nature 371, 314. Foschini, L., Farinella, P., Froeschle, Ch., et al., 2000. Astron. Astrophys. 353, 797. Galimov, E.M., Kolotov, V.P., Nazarov, M.A., et al., 2013. Geochem. Int. 51, 522. Gladman, B., Michel, P., Froeschle, Ch., 2000. Icarus 146, 176. Kohout, T., Gritsevich, M., Grokhovsky, V.I., et al., 2014. Icarus 228, 78. Marchi, S., Delbo, M., Morbidelli, A., et al., 2009. Mon. Not. R. Astron. Soc. 400, 147. Ozawa, S., Miyahara, M., Ohtani, E., et al., 2014. Nat. Sci. Rep. 4, 5033, id. Popova, O.P., Jenniskens, P., Emel'yanenko, V., et al., 2013. Science 342, 1073. Reddy, V., Vokrouhlicky, D., Bottke, W.F., et al., 2015. Icarus 252, 129. Sekanina, Z., Chodas, P.W., 2005. Astrophys. J. 161, 551. Silber, E.A., ReVelle, D.O., Brown, P.G., Edwards, W.N., 2009. J. Geophys. Res. 114, E08006.
Contents lists available at ScienceDirect
Planetary and Space Science journal homepage: www.elsevier.com/locate/pss
Dynamical evolution of Chelyabinsk-type bodies from sungrazing orbits to near-Earth space V.V. Emel’yanenko n Institute of Astronomy of the Russian Academy of Sciences, 48 Pyatnitskaya, Moscow 119017, Russia
art ic l e i nf o
a b s t r a c t
Article history: Received 20 January 2015 Received in revised form 23 July 2015 Accepted 10 August 2015 Available online 21 August 2015
Studies of the orbital evolution show that, with a high probability, the Chelyabinsk object was near the Sun about 1 Myr ago. This is consistent with the estimate of a cosmic ray exposure age (Popova et al., 2013). Dynamical features of the flux of Chelyabinsk-type bodies have been studied by numerical integrations, assuming their origin near the Sun about 1 Myr ago. The most frequent fate of these bodies is solar collision. But about a quarter of all the original objects survive until the present epoch. The majority of the surviving bodies are typical near-Earth objects. The rate of their encounters with the Earth is almost constant for the last 0.5 Myr. About a third of the Chelyabinsk-type objects approach the Earth from the Sun direction. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Asteroid Meteorite Dynamics Hazard
1. Introduction After the Chelyabisk event it is evident that not only large asteroids but also 10 m size meteoroids pose a substantial hazard to the Earth civilisation. Although the number of known near-Earth objects has been growing rapidly in this century due to special surveys, there are large uncertainties in the population count, physical properties and dynamical characteristics of small asteroids. In particular, recent studies of bolides indicate that the number of impactors with diameters of 10 m may be an order of magnitude higher than estimates based on telescopic surveys (Silber et al., 2009; Brown et al., 2013). It is well known that near-Earth objects evolve frequently to orbits with small perihelion distances (Farinella et al., 1994; Gladman et al., 2000; Foschini et al., 2000; Marchi et al., 2009). In particular, there are large chances that the famous Chelyabinsk object was near the Sun in the past (Emel’yanenko et al., 2014). Asteroids undergo thermal and tidal destruction near the Sun. They may not maintain their physical integrity during this near-Sun period, creating many smaller bodies. Here we discuss the dynamical role of near-Sun states of asteroids in producing a flux of Chelyabinsk-type objects.
2. Approaches of observed meteoroids to the Sun in the past A study of the dynamical evolution of the Chelyabinsk object in the paper (Emel’yanenko et al., 2014) is based on integrations of n
Tel.: þ 7 495 9531624; fax: þ7 495 9515557. E-mail address: [email protected]
http://dx.doi.org/10.1016/j.pss.2015.08.005 0032-0633/& 2015 Elsevier Ltd. All rights reserved.
860 orbits from the confidence region for 15 Myr. The six-dimensional vector q of orbital parameters from the confidence region is modelled ^ + C1/2η, where q ^ is the vector q at nominal by using the formula q = q values of orbital parameters, η is the six-dimensional standard normally distributed random vector, C1/2(the Cholesky matrix) is such a matrix that C1/2 (C1/2)T = C , and C is the covariance matrix. The orbital elements of the Chelyabinsk object have the following nominal values and standard deviations for the osculating epoch 2013.02.15.0 TT (Equinox J2000): semimajor axis a¼1.88070.068 AU, eccentricity e¼ 0.60970.017, argument of perihelion ω ¼108.92670.536°, longitude of the ascending node Ω ¼326.44670.002°, inclination i¼ 5.93870.427°, mean anomaly M0 ¼16.87371.069° (Emel’yanenko et al., 2014). This investigation shows that, with a high probability, this meteoroid was near the Sun in the past. The most probable time for the encounters of the Chelyabinsk object with the Sun is consistent with the estimate of a cosmic ray exposure age of 1.2 Myr (Popova et al., 2013). Fig. 1 shows typical examples of changes for perihelion distances q and inclinations of two objects from the confidence region that reach the near-Sun state about this time. The data are given for integrations from the present epoch until t ¼ 1.1 Myr for the first object and until t ¼ 1.2 Myr for the second object. The first object has q¼ 0.038 AU at t¼ 1.1 Myr, the second object has q ¼0.050 AU at t¼ 1.2 Myr. There are different dynamical mechanisms that lead near-Earth objects to the Sun (Farinella et al., 1994; Gladman et al., 2000; Foschini et al., 2000), but the decrease of q to very small values in the vicinity of the Chelyabinsk orbit is mainly associated with the secular resonance ν6 (Emel’yanenko et al., 2014).
V.V. Emel’yanenko / Planetary and Space Science 118 (2015) 302–304
303
Fig. 1. Typical examples of changes for perihelion distances and inclinations of two objects from the confidence region: left – the object 1; right – the object 2 (data for orbits are plotted every 500 years, crosses show initial positions of the objects in 2013).
Fig. 2. Distributions of a and q for particles that survive until the present epoch. The models correspond to the objects 1 and 2 shown in Fig. 1.
This dynamical behaviour of the Chelyabinsk meteoroid is not unique. Frequent collisions with the Sun were found earlier in integrations of orbits for 20 bolides in the work (Foschini et al., 2000).
3. Dynamical features of the Chelyabinsk-type objects It is natural to assume that a parent body could be disrupted due to the strong solar tide, thermal stresses and interaction with the solar atmosphere at Sun-grazing conditions. The cluster of the observed Kracht and Marsden sungrazing objects is probably a pattern of such disruption near the present epoch (Sekanina and Chodas, 2005). But it is impossible to describe in detail dynamical features of Chelyabinsk-type bodies if they originate from a parent body near the Sun about 1.2 Myr ago. Only statistical and qualitative considerations are possible in this case. Therefore, we studied the most common features of the distribution of Chelyabinsktype bodies if they originate due to disruption of a parent body near the Sun. Two examples shown in Fig. 1 were considered. In each case we studied the evolution of 400 test particles starting from the epoch t¼ 1.1 Myr for the first object and from the epoch t¼ 1.2 Myr for the second object. The first object had orbital elements q ¼0.038 AU, a ¼2.49 AU, ω ¼ 265.4°, Ω ¼47.7°, i¼9.9°, the second object had orbital elements q ¼0.050 AU, a ¼2.30 AU, ω ¼252.4°, Ω ¼162.8°, i¼6.3°. Orbital elements of the test particles were equal to the corresponding values for two original objects except for q. Initial values of q for the test particles were distributed uniformly in the small interval of 10 8 AU (1.5 km) around the above values of q for each original object. The dynamical evolution
of the test particles under the planetary perturbations was calculated until the present epoch, using the symplectic integrator (Emel’yanenko, 2007). Particles were removed from integrations when qo 0.005 AU or semimajor axis a4 50 AU. The most frequent fate of these particles is solar collision. Fig. 2 shows the distributions of a and q for particles that survive until the present epoch. These distributions are similar in both examples. The majority of surviving particles (about a quarter of all the original particles) are concentrated in the region 0o qo 2.0 AU, 1.5 oa o3.0 AU. Fig. 3 shows the number of encounters with the Earth to minimum distances Δ o0.01 AU (the Hill radius of the Earth) during certain time intervals of the integrations. Although numbers are substantially different for the studied variants, approaches to the Earth are continuous on the whole interval of integrations in both cases. The number of encounters decreases insignificantly near the present epoch (approximately for 0.5 Myr). Thus the level of a hazard to the Earth associated with Chelyabinsk-type bodies originating in near-Sun conditions is almost constant for the last 0.5 Myr. We checked also the angle γ between the direction of motion relative to the Earth and the Sun–Earth direction for particles reaching the minimum geocentric distance Δo 0.01 AU on the interval ( 0.1, 0) Myr of our integrations. Fig. 4 shows that the distributions of γ depend on the chosen initial condition. Nevertheless, not less than a third of all the encounters take place at γ o60°. This stresses that discovery of dangerous Chelyabinsk-type objects approaching the Earth from the Sun direction is a very serious task in the asteroid hazard problem (it is known that the Chelyabinsk object came from the sunward direction).
304
V.V. Emel’yanenko / Planetary and Space Science 118 (2015) 302–304
Fig. 3. Numbers of encounters with the Earth to minimum distances Δ o 0.01 during certain time intervals. The models correspond to the objects 1 and 2 shown in Fig. 1.
Fig. 4. Distributions of γ at close encounters with the Earth near the present epoch. The models correspond to the objects 1 and 2 shown in Fig.1.
4. Conclusions The estimated cosmic ray exposure age of the Chelyabinsk body is 1.2 Myr (Popova et al., 2013). The recovered fragments of the Chelyabinsk meteorite contain significant portions of shock blackened material and melt veins (Galimov et al., 2013; Kohout et al., 2014; Ozawa et al., 2014). Collisions with other solid bodies are usually considered for the origin of these features (e.g., Borovička et al., 2013; Galimov et al., 2013; Kohout et al., 2014; Ozawa et al., 2014). In particular, Borovička et al. (2013) suggest that the Chelyabinsk object originated recently from the 2 km diameter near-Earth asteroid 86039 (1999 NC43) via a collision. However, Reddy et al. (2015) show that the lack of spectral match between Chelyabinsk and 1999 NC43, and their different chemistries suggest that the asteroid is probably not the source object for this meteorite. Moreover, the cosmic ray exposure age of 1.2 Myr for the Chelyabinsk body appears incompatible with the very short separation timescales discussed in Borovička et al. (2013). But it is shown in the paper (Emel’yanenko et al., 2014) that, with high probability, the Chelyabinsk object was near the Sun about 1 Myr ago (Emel’yanenko et al., 2014). Therefore, while we do not have the expertise to reject the impact origin of the Chelyabinsk body, we consider another possibility that a parent body of the Chelyabinsk object could be disrupted due to strong thermal and tidal stresses near the Sun. We have studied dynamical features of the flux of Chelyabinsktype bodies assuming their origin near the Sun about 1 Myr ago. The most frequent fate of these bodies is solar collision. But about a quarter of all the original objects survive until the present epoch. The majority of the surviving bodies are typical near-Earth objects. The rate of their encounters with the Earth is almost constant for the last 0.5 Myr. This dynamical behaviour of the Chelyabinsk meteoroid is not unique. It was shown earlier (Foschini et al., 2000) that the near-Sun states are typical in the evolution of
bodies associated with bright bolides. Thus, there are expected to be objects, originating from Sun-grazing orbits, in the population of near-Earth asteroids that pose a long-term hazard to the Earth. About a third of the Chelyabinsk-type objects approach the Earth from the Sun direction.
Acknowledgements This work was supported by the Ministry of Education and Science of the Russian Federation as part of the project RFMEFIBBB14X0288 under the Federal Targeted Programme ‘Research and Development on Priority Directions of Scientific and Technological Complex of Russia for 2014–2020’. The calculations were carried out using the MVS 100K supercomputer of the Joint Supercomputer Center of the Russian Academy of Sciences.
References Borovička, J., Spurný, P., Brown, P., et al., 2013. Nature 503, 235. Brown, P.G., Assink, J.D., Astiz, L., et al., 2013. Nature 503, 238. Emel’yanenko, V.V., 2007. Celest. Mech. Dyn. Astron. 98, 191. Emel’yanenko, V.V., Naroenkov, S.A., Jenniskens, P., Popova, O.P., 2014. Meteorit. Planet. Sci. 49, 2169. Farinella, P., Froeschle, Ch., Froeschle, Cl., et al., 1994. Nature 371, 314. Foschini, L., Farinella, P., Froeschle, Ch., et al., 2000. Astron. Astrophys. 353, 797. Galimov, E.M., Kolotov, V.P., Nazarov, M.A., et al., 2013. Geochem. Int. 51, 522. Gladman, B., Michel, P., Froeschle, Ch., 2000. Icarus 146, 176. Kohout, T., Gritsevich, M., Grokhovsky, V.I., et al., 2014. Icarus 228, 78. Marchi, S., Delbo, M., Morbidelli, A., et al., 2009. Mon. Not. R. Astron. Soc. 400, 147. Ozawa, S., Miyahara, M., Ohtani, E., et al., 2014. Nat. Sci. Rep. 4, 5033, id. Popova, O.P., Jenniskens, P., Emel'yanenko, V., et al., 2013. Science 342, 1073. Reddy, V., Vokrouhlicky, D., Bottke, W.F., et al., 2015. Icarus 252, 129. Sekanina, Z., Chodas, P.W., 2005. Astrophys. J. 161, 551. Silber, E.A., ReVelle, D.O., Brown, P.G., Edwards, W.N., 2009. J. Geophys. Res. 114, E08006.