Planetary Dynamics: Shape, Gravity, Orbit, Tides, and Rotation from Observations and Models
Shape, gravity field, orbit, tidal deformation, and rotation state are fundamental geodetic parameters of any planetary object. Measurements of these parameters are prerequisites for e.g. spacecraft navigation and mapping from orbit, but also for modelling of the interior and evolution. This session welcomes contributions from all aspects of planetary geodesy, including the relevant theories, observations and models in application to planets, satellites, ring systems, asteroids, and comets.
Daniel Scheeres, Andrew French, Pasquale Tricarico, Steven Chesley, Yu Takahashi, Davide Farnocchia, Jay McMahon, Daniel Brack, Alex Davis, Ronald Ballouz, Erica Jawin, Benjamin Rozitis, Josh Emery, Andrew Ryan, Ryan Park, Brian Rush, Nick Mastrodemos, Brian Kennedy, Julie Bellerose, and Daniel Lubey and the OSIRIS-REx Team Members
Introduction: Estimates of asteroid (101955) Bennu’s gravity have been determined based on a series of independent solutions from different teams involved on the OSIRIS-REx mission. In addition to classical radio science techniques for estimating a body's gravity field coefficients, the discovery of particles ejected from Bennu that persist in orbit for multiple revolutions provides a unique opportunity to probe the gravity field to higher degree and order than possible by using conventional spacecraft tracking . However, the non-gravitational forces acting on these particles must also be characterized, and their impact on solution accuracy must be assessed, requiring the different gravity field estimates to be compared and reconciled.
Given the measured gravity field of Bennu, rigorous constraints on its internal density heterogeneity can be found by comparing the measured field with the constant density field computed from the asteroid shape. These results in turn provide unique insight into the global geophysical processes that drive the external and internal morphology of small rubble-pile asteroids such as Bennu.
Finally, definitive results on the surface and close-proximity force environment of Bennu can be derived and updated from the initial analysis based on the total mass and constant density shape. Several aspects of the environment are highly sensitive to the gravity field and have changed from earlier results [2, 3, 4].
We will present the current gravity field solutions and uncertainties, update the surface and proximity environment models, and provide the geophysical implications and interpretations of these measurements.
Geophysical Models: The estimated gravity field solutions are compared with the constant density shape model to constrain models of the internal density variation. We find that these differences are consistent with Bennu having an under-dense core and equatorial ridge. The degree to which these are under-dense cannot be specifically constrained, but feasible ranges for these values can be determined.
An under-dense equator could be consistent with transport of material to the equator without compaction. Given the slope transition at the Roche lobe, this would also be consistent with the ballistic transport of material into the equatorial region. Estimates of the rate of particle migration do not seem to be enough to account for the overall equatorial bulge of Bennu, however, implying that this feature could be older and not due to the more recent transport of material to the equator.
The lower-density interior is consistent with a period of rapid spin and failure of the interior of the body . This could also be consistent with the raised equatorial bulge. This interior failure could have occurred in an earlier epoch of YORP-induced rapid rotation or could trace to the initial formation of Bennu as a distinct rubble-pile body . Tests of this hypothesis require additional simulations of how rubble-pile asteroids coalesce after the catastrophic disruption of their parent body.
Acknowledgements: This material is based upon work supported by NASA under Contract NNM10AA11C issued through the New Frontiers Program. Part of this research was conducted at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. We are grateful to the entire OSIRIS-REx Team for making the encounter with Bennu possible.
References:  Lauretta D.S. & Hergenrother C.W. et al. (2019) Science 366, eaay3544.  Scheeres D.J. et al. (2019) Nature Astronomy 3, 352-361.  Barnouin O.S. et al. 2019. Nature Geoscience 12, 247-252.  Tricarico P. et al. (2019) EPSC-DPS Abstract #2019-547-1.  Scheeres D.J. et al. (2016) Icarus 276, 116-140.  Michel P. et al. (2018) AGU Fall Meeting 2018 Abstract #P33C-P33850.
How to cite:
Scheeres, D., French, A., Tricarico, P., Chesley, S., Takahashi, Y., Farnocchia, D., McMahon, J., Brack, D., Davis, A., Ballouz, R., Jawin, E., Rozitis, B., Emery, J., Ryan, A., Park, R., Rush, B., Mastrodemos, N., Kennedy, B., Bellerose, J., and Lubey, D. and the OSIRIS-REx Team Members: The Measured Gravity and Global Geophysical Properties of (101955) Bennu, Europlanet Science Congress 2020, online, 21 Sep–9 Oct 2020, EPSC2020-929, https://doi.org/10.5194/epsc2020-929, 2020.
Yun Zhang, Patrick Michel, Olivier S. Barnouin, Michael G. Daly, Ronald-Louis Ballouz, Kevin J. Walsh, and Dante S. Lauretta
Previous studies have shown that material strength and density heterogeneity play important roles in asteroid reshaping processes through YORP spin-up (Holsapple 2010). The final shapes are also dependent on the magnitude and distribution of these intrinsic properties. In turn, material properties as well as the reshaping history of an asteroid could be revealed by examining its detailed morphology. The high-resolution shape, detailed surface characteristics, and internal density distribution of (101955) Bennu measured by the OSIRIS-REx mission (Barnouin et al. 2019; Walsh et al. 2019; Scheeres et al. 2019; Lauretta et al. 2019) now grant us an opportunity to decipher its material properties from its current state.
We use the Bennu shape model with a resolution of 1.68 m per facet derived from the data collected by the OSIRIS-REx Laser Altimeter (Seabrook et al. 2019; Barnouin et al. 2020) to construct rubble-pile models consisting of ~10,000 to ~100,000 spheres with different particle size distributions. The soft-sphere discrete element method is applied to simulate the spin-up process of these rubble piles (Schwartz et al., 2012; Zhang et al., 2017, 2018). The contact interactions between the constituent spheres are used to control the material shear and cohesive strengths. We study the behaviors of our simulated rubble piles against rotation as a function of frictional and cohesive properties.
In response to the rotational acceleration, we find that the contact-force networks adjust themselves to maintain the overall stability. The stress distributions in the Bennu rubble-pile models change with the spin rate. When no cohesion is included, the local regions subject to the highest shear stress are located near the surface at the slow spin state and shift to the interior during the subsequent spin-up. The critical spin period value of this transition decreases with a larger friction angle of the asteroid. For example, with a friction angle of ~20°, this transition occurs before achieving a spin period of ~ 5 hr and the Bennu rubble-pile model begins to fail internally and deform before achieving the current spin period of ~4.276 hr (Barnouin et al. 2019). With a friction angle of 30°, the rubble-pile Bennu is able to marginally keep its structure stable at 4.276 hr with an internal region subject to the highest shear stress. When the friction angle is larger than ~37°, the most sensitive region subject to the highest shear stress occurs at the surface at Bennu’s current spin rate. Since some recent surface mass movement is evidenced on Bennu (Jawin et al. 2020), Bennu’s material may have a high friction angle larger than 37° to promote surface movement. This friction-angle magnitude is common for terrestrial granular materials and is comparable to the maximum surface slope of Bennu (~40°).
The critical spin period to induce structural failure for Bennu modeled as a rubble pile with a friction angle of ~37° is ~3.4 hr, which is notably faster than its current spin period. If this is the case, Bennu might have been spun up to a spin period smaller than 3.4 hr in the past to induce some macroscopic reshaping effects. The critical spin period decreases to 2.6 hr if the rubble-pile material contains a small amount of cohesion ~ 3 Pa that is homogenously distributed. This fast critical spin state would lift any surface material that is not cohesively attached to the surface. This is inconsistent with the recent surface mass movement observed on Bennu. Furthermore, the structural failure is induced by surface cracking, preventing surface shedding from occurring. Therefore, our study suggests that Bennu should not have an overall cohesion larger than ~3 Pa.
However, if the cohesion distribution is highly heterogeneous, it is possible that some regions have a large material cohesion and some other regions are cohesionless. This inhomogeneous cohesion distribution is consistent with the estimated porosity and boulder distribution on Bennu, and could account for the formation of the observed features such as longitudinal ridges and internal heterogeneity.
Acknowledgements: Y. Z. acknowledges funding from the Université Côte d’Azur “Individual grants for young researchers” program of IDEX JEDI. Y.Z. and P.M. acknowledge funding support from the French space agency CNES and from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 870377 (project NEO-MAPP). This material is based upon work supported by NASA under Contract NNM10AA11C issued through the New Frontiers Program. We are grateful to the entire OSIRIS-REx Team for making the encounter with Bennu possible.
Barnouin, O. S., Daly, M. G., Palmer, E. E., et al. 2019, Nat. Geosci., 12, 247.
Barnouin, O. S., Daly, M. G., Palmer, E. E., et al. 2020, Planet. Space Sci., 180, 104764.
Holsapple, K. A. 2010, Icarus 205, 430.
Jawin, E. R., Walsh, K. J., McCoy, T. J., et al. 2020, LPSC, 2326, 1201.
Lauretta, D. S., DellaGiustina, D. N., Bennett, C. A., et al. 2019, Nature, 568, 55.
Scheeres, D. J., McMahon, J. W., French, A. S., et al. 2019, Nat. Astron., 3, 352.
Seabrook, J. A., Daly, M. G., Barnouin, O. S., et al. 2019, Planet. Space Sci. 177, 104688.
Schwartz, S. R., Richardson, D. C., & Michel, P. 2012, Granular Matter, 14(3), 363–380.
Walsh, K. J., Jawin E. R., Ballouz, R.-L., et al. 2019, Nat. Geosci. 12, 242–246.
Zhang, Y., Richardson, D. C., Barnouin, O. S., et al. 2017, Icarus, 294, 98–123.
Zhang, Y., Richardson, D. C., Barnouin, O. S., et al. 2018, Astrophys. J., 857(1), 15.
How to cite:
Zhang, Y., Michel, P., Barnouin, O. S., Daly, M. G., Ballouz, R.-L., Walsh, K. J., and Lauretta, D. S.: Numerical modeling of Bennu’s structural stability and implications for its internal and surface properties, Europlanet Science Congress 2020, online, 21 Sep–9 Oct 2020, EPSC2020-722, https://doi.org/10.5194/epsc2020-722, 2020.
Stefania Soldini, Takanao Saiki, Hitoshi Ikeda, Koji Wada, Masahiko Arakawa, and Yuichi Tsuda
Abstract: A sphere cluster (SPH-Mas) based gravity model allows a semi-analytic expression of the linearised equations around the equilibrium points. Depending on the sphere packing distribution, the SPH-Mas model can retrieve the same dynamical objects common to others gravity models (i.e. spherical harmonics and polyhedron) or for non-uniform density objects. This model has the advantage to define the same particles mesh distribution for both astrophysical and astrodynamics tools. The Hayabusa2’s Small Carry-on Impactor operation is used as a scenario to study the ejecta particle dynamics around an irregular body. The goNEAR (gravitational orbit Near Earth Asteroid Regions) tool was used to simulate the impact operation in a non-linear sense when the effect of the solar radiation pressure perturbation is taken into account for particles size of 10 cm, 5 cm, 1 cm and 1 mm in diameter.
Introduction: In November 2019, the Japanese Hayabusa2 spacecraft completed an 18 months mission exploration around the asteroid Ryugu  and it is expected to return to Earth late this year (2020). JAXA’s Hayabusa2 and NASA’s OSIRIS-Rex missions  are contributing to answer fundamental questions related to the formation of our solar system and the origin of Life . After a successful touchdown in March 2019, Japan has set a new first when in April 2019 the Hayabusa2 spacecraft deployed and activated the explosive Small Carry-on Impactor (SCI) to successfully form an artificial crater .
We propose a genearlised methodology to study the dynamics around Equilibrium Points (EPs) of irregular bodies with application to the asteroid Ryugu . To the core of our study, we aim to gain a general insight on the dynamics around irregular shape bodies for studying the dynamics of ejecta particles. Moreover, we are looking into a generalised gravity model of celestial bodies that can be easily extended not only to any irregular shape bodies but also to arbitrary density distributions . The selected generalised gravity model provides a mass distribution that can be used for both hydrodynamics impact simulations and orbital dynamics around EPs.
Background: The mascons (“mas”s “con” centrations) has been mainly used for explaining the Lunar gravity anomalies originally detected in 1968 . Conversely, Smooth Particles Hydrodynamics (SPH) codes are often used to simulate asteroid impact events and share the problem to handle the transition between a SPH simulation and N-body simulations . Since the SPH and Mascons make use of the same mass conservation law and we are interested to interface the SPH simulations with the N-Body simulations, we will rename the selected gravity model as the SPH-Mascons (SPH-Mas) model.
Figure 1: Sphere packing and the equilibrium points .
SPH-Mas Gravity Model: The gravity of an irregular shape body is modelled with a cluster of spheres, SPH-Mas. Each spherical particle contributes in the overall gravity field of the body. The exterior gravity potential of each sphere behaves as a single point mass. The potential of the irregular body is the result of the summation of each point mass’s potential that contributes to the overall potential field such that:
where mi (i = 1, ..., Nsph) is the mass of each SPH-Mas for a total of Nsph masses. r is the distance from the field point and the center of the asteroid. ri is the distance of each masses with respect to the center of the asteroid. The total mass of the asteroid is conserved and given by mb = ∑(i = 1,.., Nsph) mi.
Figure 2: Shape model  and the equilibrium points.
Sphere Packing: We consider Ryugu’s polyhedron model published in  as our “high fidelity” gravity model. We distribute the SPH-Mas within the asteroid shape such that we can approximate Ryugu’s “high fidelity” gravity field. For the scope of testing our semi-analytic formula, we compared a uniform sphere packing approach with a random packing approach for different numbers of SPH-Mas. Fig. 1 shows the comparison between the uniform distribution in the left panel and the random distribution in the right panel for Nsph = 19, 58, 1,605 and 1,406,146. By comparing the location of the EPs between Fig 1 and Fig 2, it is clear that under the assumption of uniform density polyhedron, the uniform sphere packing is preferable to the random sphere packing even for the case of Nsph major to the order of million spheres. Indeed, the random sphere packing does not necessarily preserve the geometry of the EPs that affects the ejecta dynamics.
Effect of SPH-Mas Packing onto Particles Dynamics: The derived semi-analytical formula based on an SPH-Mas gravity model is a direct function of the sphere packing distribution (density), their position (ri) and the asteroid’s spin axis angular velocity (7.6 h for Ryugu) which allows to find families of periodic orbits for ejecta particles around an non-uniform irregular shaped asteroid as shown in Fig 3.
Fate of Ryugu’s Ejecta: We made use of goNEAR tool to simulate the dynamics of 10 cm, 5 cm, 1 cm and 1mm in diameter size particles under the effect of the solar radiation pressure perturbation. In the numerical experiment, few particles seemed to survive in orbit for diameter of 5–10 cm (Fig 4). The search for evidence of particles in Ryugu orbit is still unconfirmed however the stability of EPs can be linked to long survival particles in orbit.
Figure 3: Family of periodic orbits as function of sphere packing .
Figure 4: SCI’s ejecta dynamics with the goNEAR tool .
References:  Watanabe et al. (2019) Science, 364, 268–272.  Lauretta et al. (2015) Meteoritics & Planet. Sci., 50, 834–849.  Sugita et al. (2019) Science, 364, 6437.  Arakawa et al. (2019) Science, under review  Soldini et al, (2019) PSS, (2020) 180  Melosh et al., (2013) Science, 340,1552–1555  Ballouz et al., (2018) 49th LPSC.
How to cite:
Soldini, S., Saiki, T., Ikeda, H., Wada, K., Arakawa, M., and Tsuda, Y.: The effect of "MASCONS" Sphere Packing onto the Dynamical Environment around Rubble-Pile Asteroids: Application to Ryugu, Europlanet Science Congress 2020, online, 21 Sep–9 Oct 2020, EPSC2020-808, https://doi.org/10.5194/epsc2020-808, 2020.
Marin Ferrais, Pierre Vernazza, Laurent Jorda, Nicolas Rambaux, and Josef Hanuš and the HARISSA team
Asteroid (16) Psyche is the largest M-type asteroid in the main belt and the only metal rich asteroid of this size (D > 200 km). It has been proposed that Psyche could be an exposed planetary core [S17,D18]. This hypothesis and the uniqueness of Psyche's characteristics are the main reasons for its selection as the rendez-vous target of a NASA Discovery mission that is scheduled to launch in 2022 [E17].
However, the true nature of Psyche remains enigmatic which leads to the formulation of several distinct formation scenarios. Psyche’s density appears compatible with that of stony-iron meteorites such as mesosiderites [Vi18] as well as that of pallasites and CB chondrites [E20]. It is also unknown if its interior is intact or a is a rubble pile and if it is differentiated.
We obtained 35 images of Psyche at 7 epochs in July and August 2019 using VLT/SPHERE/ZIMPOL. They complement the first 25 images obtained in 2018 that were already presented in [Vi18], for a total of 60 images taken at 12 epochs. Psyche was observed near opposition with a pixel size corresponding to ~6 km/px. The first apparition in 2018 was limited to the northern hemisphere of Psyche but the second apparition in 2019 covered well the equatorial region and allowed us to achieve a complete coverage of the surface.
First, we generated an updated shape model of Psyche with the ADAM inversion algorithm [Vi15]. We used the same procedure as in [Vi18] and added the new SPHERE images and a new stellar occultation recorded in October 2019.
We then applied our Multi-resolution PhotoClinometry by Deformation (MPCD; [C13]) method on a selection of the SPHERE images to reconstruct the 3D shape of Psyche. The MPCD software gradually deforms the vertices of an initial mesh to minimize the difference between the observed images and realistic images of the surface. The ADAM model was used as input for the initial mesh and the spin parameters.
Results and conclusions
The ADAM and MPCD shape models are remarkably similar with a small volume difference and radial differences mean. The comparison between the SPHERE images and the corresponding synthetic images is given in Fig. 1. The densities derived from the volumes of both shape models combined with the average of available mass estimates are close to ~4 g/cm³ which is in agreement with other recent estimates [S17,D18,Vi18].
A shape analysis was performed by computing the radial differences between Psyche’s shape model and its best-fitting ellipsoid to obtain the average residuals relative to the mean radius. We then computed the sphericity index of Psyche using the same approach as in [V19]. We repeated the process for other large main-belt asteroids, the terrestrial planets and smaller asteroids visited in-situ by space missions. It revealed that Psyche’s shape appears intermediate between those of larger asteroids and those of smaller or similarly sized bodies. Psyche’s appearance is close to an ellipsoid with flat regions at the poles even though we identified three depression regions along its equator.
Finally, we investigated whether the shape of Psyche may be close to the equipotential shape of an hydrostatic body. The flatness and density of Psyche are compatible with a formation at hydrostatic equilibrium as a Jacobi ellipsoid with a shorter rotation period of ~3 h. Later impacts may have slowed down Psyche’s rotation, which is currently ~4.2 h, while also creating the imaged depressions. Our results open the possibility that Psyche acquired its primordial shape either after a giant impact while its interior was already frozen or while its interior was still molten owing to the decay of the short-lived radionuclide 26Al.
Figure 1: Comparison between VLT/SPHERE/ZIMPOL deconvolved images of Psyche (top row) and the corresponding synthetic images of our MPCD (second row) and ADAM (third row) shape models. The red arrows indicate the direction of the spin axis.
[C13] Capanna, C., Gesquière, G., Jorda, L., Lamy, P., & Vibert, D. 2013, The Visual Computer, 29, 825
[D18] Drummond, J. D., Merline, W. J., Carry, B., et al. 2018, Icarus, 305, 174
[E17] Elkins-Tanton, L. T., Asphaug, E., Bell, J. F., et al. 2017, in Lunar and Planetary Science Conference, Lunar and Planetary Science Conference, 1718
[E20] Elkins-Tanton, L., Asphaug, E., Bell, J., et al. 2020, Journal of Geophysical Research: Planets
[S17] Shepard, M. K., Richardson, J., Taylor, P. A. et al. 2017, Icarus, 281, 388
[Vi15] Viikinkoski, M., Kaasalainen, M., & ˇ Durech, J. 2015, A&A, 576, A8
[Vi18] Viikinkoski, M., Vernazza, P., Hanuš, J., et al. 2018, A&A, 619, L3
How to cite:
Ferrais, M., Vernazza, P., Jorda, L., Rambaux, N., and Hanuš, J. and the HARISSA team: Asteroid (16) Psyche's primordial shape: A possible Jacobi ellipsoid, Europlanet Science Congress 2020, online, 21 Sep–9 Oct 2020, EPSC2020-289, https://doi.org/10.5194/epsc2020-289, 2020.
Asteroid mass determination is performed by analyzing an asteroid's gravitational interaction with another object, such as a spacecraft, Mars, a companion in the case of binary asteroids, or a separate asteroid during a close encounter. During asteroid-asteroid close encounters, perturbations caused by the masses of larger asteroids can be detected in the post-encounter orbits of the smaller test asteroid involved in such an encounter. This can be described as an inverse problem where the aim is to fit six orbital elements for each asteroid and mass(es) for the perturbing asteroid(s), for a total of 13 parameters at minimum unless more asteroid-asteroid encounters are modeled simultaneously.
To solve this inverse problem, which is traditionally done with least-squares methods, we have implemented a Markov-chain Monte Carlo (MCMC) based solution and recently (Siltala & Granvik 2020) reported, among others, significantly lower than expected masses and densities for the asteroid (16) Psyche in particular. Psyche is an interesting, and topical, object as it is the target of NASA's eponymous Psyche mission and is commonly thought to be of metallic or stony-iron composition, which our previous density estimates disagreed with. In our previous work our two separate mass estimates for Psyche were based on modeling encounters with two separate test asteroids in both cases. Since then we have further refined our mass estimate for Psyche by simultaneously using eight separate test asteroids for this object, significantly increasing the amount of observational data included on the model which, in turn, will narrow down the uncertainties of our results at the cost of additional model complexity. Here we report and discuss our latest results for the mass of Psyche based on this case and compute corresponding densities based on existing literature values for the volume. We obtain a mass of (0.972 ± 0.148) * 10^-11 solar masses for Psyche corresponding to a bulk density of (3.37 ± 0.58) g/cm³ which is higher than our previous results while remaining consistent with them considering the uncertainties involved. It still remains lower than other previous literature values. We compare our results to these previous literature values and briefly discuss possible physical implications of these results.
In addition, due to previous interest from the scientific community, we have also computed mass estimates for Ceres and Vesta, both of which already have very precisely known masses from the Dawn mission. As such, our results for these two asteroids are not of direct scientific interest but they serve as an useful benchmark to verify that our algorithm provides realistic results as we have 'ground truth' values to compare our results to. We find that for both cases, our results are in line with those of Dawn.
How to cite:
Siltala, L. and Granvik, M.: New estimates of the mass and density of asteroid (16) Psyche, Europlanet Science Congress 2020, online, 21 Sep–9 Oct 2020, EPSC2020-752, https://doi.org/10.5194/epsc2020-752, 2020.
Julia Marín-Yaseli de la Parra, Michael Kueppers, Jacint Roger Pérez, and the Osiris Team
Comet 67P/C-G is a dusty object. As it neared its closest approach to the Sun in late July and August 2015, instruments on Rosetta recorded a huge amount of dust enshrouding the comet. This is tied to the comet’s proximity to our parent star, its heat causing the comet’s nucleus to release gases into space, lifting the dust along . Spectacular jets were also observed, blasting more dust away from the comet. This disturbed, ejected material forms the ‘coma’, the gaseous envelope encasing the comet’s nucleus, and can create a beautiful and distinctive tail. A single image from Rosetta’s OSIRIS instrument can contain hundreds of dust particles and grains surrounding the 4 km-wide comet nucleus.
The study of the dust behaviour is vital for understanding the global evolution of the comet and has direct consecuences in the research of the origins of the solar system. 
That helps for calculating the average mass loss rate per period. Calculations are not so simple since two situations may occur. The diurnal thermal cycle plus the irregularities in the shape of the comet produces a flow of particles from the southern hemisphere to the northern and most particles are redeposited. However some areas, like Hapi region, are eroded around 1.0± 0.5 m per orbit. Some other areas are growing with deposits of dust creating moving shifting dunes. It is believed the 95% of the ejected dust particles are falling back into the comet.
A simple image of the OSIRIS instrument can contain hundreds of dust and grain particles around a 4km sphere around the core. The images above show the level of complexity when processing an image. Partly, most image sequences are processed manually