DOCSLIB.ORG

Dynamical Evidence for Phobos and Deimos As Remnants of a Disrupted Common Progenitor

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dynamical Evidence for Phobos and Deimos As Remnants of a Disrupted Common Progenitor LETTERS https://doi.org/10.1038/s41550-021-01306-2 Dynamical evidence for Phobos and Deimos as remnants of a disrupted common progenitor Amirhossein Bagheri 1 ✉ , Amir Khan 1,2, Michael Efroimsky 3, Mikhail Kruglyakov 1 and Domenico Giardini1 The origin of the Martian moons, Phobos and Deimos, remains on their dynamical figures, which enhances tidal dissipation16. elusive. While the morphology and their cratered surfaces This effect is more pronounced for Phobos than Deimos, owing to suggest an asteroidal origin1–3, capture has been questioned Phobos’s higher eccentricity and triaxiality. All properties are sum- because of potential dynamical difficulties in achieving the marized in Supplementary Table 1. current near-circular, near-equatorial orbits4,5. To circumvent For a non-librating planet hosting a satellite librating about a this, in situ formation models have been proposed as alterna- 1:1 spin–orbit resonance, the tidal rates of the semi-major axis and tives6–9. Yet, explaining the present location of the moons on eccentricity can be written in terms of the mean motion (n), satellite opposite sides of the synchronous radius, their small sizes and libration amplitude ( ), and planet and satellite quality functions 2 A apparent compositional differences with Mars has proved (K k /Q and K0 k0=I Q0), masses (M and M0), spin rates (θ_ and θ_ 0) l = l l l ¼ l l challenging. Here, we combine geophysical and tidal-evolution and radii (R andI R′): I I modelling of a Mars–satellite system to propose that Phobos and Deimos originated from disintegration of a common 5 da G M M0 M0 R progenitor that was possibly formed in situ. We show that ð þ Þ Kl; θ_; n; e dt ¼ na2 M a ½Fð Þ 1 tidal dissipation within a Mars–satellite system, enhanced ð Þ by the physical libration of the satellite, circularizes the K0; θ_ 0; n; e K0; n; e; ; þFð l Þ þ Gð l AÞ post-disrupted eccentric orbits in <2.7 Gyr and makes Phobos descend to its present orbit from its point of origin close to or above the synchronous orbit. Our estimate for Phobos’s 5 de G M M0 M0 R maximal tidal lifetime is considerably less than the age of ð þ Þ Kl; θ_; n; e dt ¼ na3 M a ½Lð Þ 2 Mars, indicating that it is unlikely to have originated alongside ð Þ _ 0 Mars. Deimos initially moved inwards, but never transcended Kl0; θ ; n; e Kl0; n; e; ; the co-rotation radius because of insufficient eccentricity and þLð Þ þ Hð AÞ therefore insufficient tidal dissipation. Whereas Deimos is where G is the gravitational constant, unprimed and primed vari- very slowly receding from Mars, Phobos will continue to spiral ables refer to those of the planet and the satellite, respectively, and towards and either impact with Mars or become tidally dis- the terms in the square brackets represent the dissipation due to the _ rupted on reaching the Roche limit in ≲39 Myr. main tides on the planet ( Kl; θ; n; e and Kl; θ_; n; e ), the main Fð_ Þ Lð_ Þ Tidal interactions between celestial bodies result in energy dis- tides on the satellite ( K0I; θ; n; e and K0I; θ0; n; e ), and satellite Fð l Þ Lð l Þ sipation and drive systems towards equilibrium states, in part by libration ( K0; n; e; I and K0; n; e; I ). Inclination (i) of a sat- Gð l AÞ Hð l AÞ pushing eccentricity and obliquity to zero and spin rates towards ellite orbit onI Mars’s equator isI governed by synchronization. This evolution is governed by the dissipative prop- 5 erties (including the frequency-scaling laws) of both the planet and di M0 R n sin i Kl; θ_; n; i; e ; 3 the moons10. To determine the orbital history of a Mars–satellite dt ¼ M a ½Ið Þ ð Þ system, we use up-to-date geophysical data for Mars and its satel- lites, including Martian seismic data from the currently operating where the term in square brackets refers to the main tides on the InSight mission11,12, laboratory-based viscoelastic models13 describ- planet. Detailed expressions for the functions , , , and F G L H I ing Mars’s rheology14, and a comprehensive tidal-evolution model are given in Methods. To ensure precision, theI functionsI I were based on the extended Darwin–Kaula theory of tides15, including expanded to order 18 in eccentricity. Equations (1)–(3) were inte- the contribution from a satellite’s physical libration in longitude. grated backwards in time using a Runge–Kutta explicit iterative We consider Mars–Phobos and Mars–Deimos as separate orbital solver. We also track the planetocentric distances (Rp) of the two 2 systems and integrate the semi-major axis (a), eccentricity (e), satellites, given by Rp a 1 e = 1 e cos f ; where f is the true ¼ ðÞ� ð þ Þ Mars’s spin rate (θ_) and inclinations of the satellites (i) backwards anomaly. R assumesI values in the interval a 1 e ≤Rp ≤a 1 e , p ðÞ� ðÞþ in time (t), starting from the current configuration. Our tidal model so a satellite always resides between the twoI circles. We shall com- includes degree-2 and -3 inputs, because of the proximity of the pare the minimal distance of Deimos with the maximal distance of moons to Mars. Although Phobos and Deimos are tidally locked, Phobos, and shall be particularly interested in the case where mini- Deimos Phobos their uniform rotational motion is modified by physical libration mal Rp ≤ maximal Rp (Supplementary Section 2), that is, arising from the time-varying gravitational torque exerted by Mars whereI the orbits of the moonsI intersect. 1Institute of Geophysics, ETH Zürich, Zurich, Switzerland. 2Physik-Institut, University of Zürich, Zurich, Switzerland. 3US Naval Observatory, Washington DC, USA. ✉e-mail: [email protected] NATURE ASTRONOMY | www.nature.com/natureastronomy LETTERS NATURE ASTRONOMY a During the orbital evolution, Phobos undergoes 2:1 and 3:1 spin– 7.5 orbit resonances with Mars’s figure at a = 3.8RMars and a = 2.9RMars, Mars 7.0 R respectively, where RMars is the mean radius of Mars, and a 1:1 reso- 6.5 nance with the Sun at a = 2.6RMars when its pericentre rate equals the 6.0 Martian mean motion. Deimos is affected by a 2:1 mean motion resonance with Phobos. These resonances result in rapid eccentric- 5.5 17 Mars Mars 5.0 ity changes Δe (ref. ). For Phobos, Δe2:1 0:032, Δe3:1 0:002 –4 Sun ¼Phobos ¼ Deimos k2/Q2 = 2.0 × 10 and Δe 0:0085, whereas for Deimos,I Δe I0:002. Finally, 4.5 –4 1:1 2:1 Deimos k2/Q2 = 1.2 × 10 ¼ ¼ –6 in theI course of our integrations, we assumeI that the system has not 4.0 Phobos k2/Q2 = 1.0 × 10 –7 been affected by any other planetary material. 3.5 Phobos k2/Q2 = 6.0 × 10 Synchronous radius To compute the quality functions, models of Mars, Phobos 3.0 and Deimos are required. For Mars, self-consistently computed Max/min planetocentric distance/ 2.5 interior-structure models are obtained by inversion of geophysi- –3,000 –2,500 –2,000 –1,500 –1,000 –500 0 cal data14 (Supplementary Sections 3 and 4 and Supplementary Fig. Time from present (Myr) b 1), which include the degree-2 tidal amplitude in the form of the 0.40 Love number (k2) and the phase response (Q2), mean density and mean moment of inertia (Supplementary Table 1). One of the main 0.35 parameters that controls the orbital history of Phobos is the fre- 0.30 quency dependence of tidal dissipation (through the exponent α)14. Current observations of a few of the largest low-frequency mars- 0.25 quakes are compatible with an effective mantle Martian seismic Q of 0.20 approximately 300 (refs. 11,12). These, together with the observation Eccentricity 0.15 of the Phobos-induced tidal Q around 95 ± 10, suggest an α value in the range of 0.25–0.35, in agreement with previous studies14,18. 0.10 For our nominal cases, we employ 0.27. Densities of Phobos and α= 0.05 Deimos are <2 g cm−3, implying porous and therefore highly dissi- pative, yet weakly bonded, aggregates19,20. This assumption is based 0 –3,000 –2,500 –2,000 –1,500 –1,000 –500 0 on the moons’ ability to sustain sharp features (such as ubiquitous Time from present (Myr) 21 16 grooves and fractures) , their ability to wobble and the presence c of the Stickney crater, an event that would have shattered Phobos 8 completely if it had been a monolith or a complete rubble pile, but would have left a weakly connected Phobos intact22. For Phobos, 7 we use Q2 values based on viscosity estimates and granular friction Mars 23 R 6 studies of loose aggregates, whereas k2 is computed numerically for a two-layer model comprising a consolidated core and a porous 5 0.021 outer layer, each of which is half the satellite radius. Since μ ≫ ρgR Inclination (rad) (where is the shear rigidity modulus, is the mean density and g μ ρ 4 0.019 is the surface gravity) for both satellites, k2 and Q2 of Deimos can be Semi-major axis/ approximated by size-scaling it to Phobos23. 3 The evolution of planetocentric distances, eccentricities, 0.015 –3,000 –2,000 –1,000 0 semi-major axes and inclinations of the two satellites is shown 2 in Fig.
Load more

Recommended publications
  • Viscoelastic Dissipation Models and Geophysical Data
    Confidential manuscript submitted to JGR-Planets 1 Tidal Response of Mars Constrained from Laboratory-based 2 Viscoelastic Dissipation Models and Geophysical Data 1 2;1 3 3 1 3 A. Bagheri , A. Khan , D. Al-Attar , O. Crawford , D. Giardini 1 4 Institute of Geophysics, ETH Zürich, Zürich, Switzerland 2 5 Institute of Theoretical Physics, University of Zürich, Zürich, Switzerland 3 6 Bullard Laboratories, Department of Earth Sciences, University of Cambridge, Cambridge, UK 7 Key Points: • 8 We present a method for determining the planetary tidal response using laboratory- 9 based viscoelastic models and apply it to Mars. • 10 Maxwellian rheology results in considerably biased (low) viscosities and should be 11 used with caution when studying tidal dissipation. • 12 Mars’ rheology and interior structure will be further constrained from InSight mea- 13 surements of tidal phase lags at distinct periods. Corresponding author: Amirhossein Bagheri, [email protected] –1– Confidential manuscript submitted to JGR-Planets 14 Abstract 15 We employ laboratory-based grain-size- and temperature-sensitive rheological models to 16 describe the viscoelastic behavior of terrestrial bodies with focus on Mars. Shear modulus 17 reduction and attenuation related to viscoelastic relaxation occur as a result of diffusion- 18 and dislocation-related creep and grain-boundary processes. We consider five rheological 19 models, including extended Burgers, Andrade, Sundberg-Cooper, a power-law approxima- 20 tion, and Maxwell, and determine Martian tidal response. However, the question of which 21 model provides the most appropriate description of dissipation in planetary bodies, re- 22 mains an open issue. To examine this, crust and mantle models (density and elasticity) are 23 computed self-consistently through phase equilibrium calculations as a function of pres- 24 sure, temperature, and bulk composition, whereas core properties are based on an Fe-FeS 25 parameterisation.
    [Show full text]
  • A. C. Quillen – Research Narrative (2021)
    A. C. Quillen { Research Narrative (2021) Contents 1 Celestial Mechanics { multiple exoplanet and satellite systems 1 2 Soft-astronomy: Spin dynamics using viscoelastic mass-spring model simulations 2 3 Robotic Locomotion, collective phenomena and biophysics 3 4 Laboratory investigations of low velocity impacts into granular materials 4 5 Galaxy Dynamics 5 6 Infrared extragalactic studies and warped disks 5 7 Infrared imaging and constraints on dark matter in galaxies 6 8 Observations of Active Galaxies 7 9 Light curves and the transient dimming sky 8 10 Turbulence, Winds and Feedback 8 1 Celestial Mechanics { multiple exoplanet and satellite sys- tems In 2006 I worked on general theory on orbital resonance capture [1] to cover the non-adiabatic regime, extending Peter Goldreich and Nicole Borderie's work that was restricted to adiabatic drift only. A dimensional analysis argument facilitates estimating whether pairs of drifting bodies can capture into orbital resonance. My estimate has since been applied to interpret migrating multi- ple exo-planet systems, orbital dust dynamics, satellite systems and early evolution of planetary embryos. In 2011 I developed a theory, based on a three-body resonance overlap condition for chaotic behavior, to predict numerically measured scaling relations for stability times in closely packed satellite or planetary systems [2]. This is a first attempt to theoretically explain numerically mea- sured scaling relations. I and Rob French corrected and extended the theory [3] to cover more types of three-body resonances with a study of the inner Uranian satellite system in 2014, where I have tried to better understand the chaotic evolution and stability of resonant chains or consecutive pairs of bodies in first order resonances.
    [Show full text]
  • Natural Moons Dynamics and Astrometry
    OBSERVATOIRE DE PARIS MEMOIRE D’HABILITATION A DIRIGER DES RECHERCES Présenté par Valéry LAINEY Natural moons dynamics and astrometry Soutenue le 29 mai 2017 Jury: Anne LEMAITRE (Rapporteur) François MIGNARD (Rapporteur) William FOLKNER (Rapporteur) Doris BREUER (Examinatrice) Athéna COUSTENIS (Examinatrice) Carl MURRAY (Examinateur) Francis NIMMO (Examinateur) Bruno SICARDY (Président) 2 Index Curriculum Vitae 5 Dossier de synthèse 11 Note d’accompagnement 31 Coordination d’équipes et de réseaux internationaux 31 Encadrement de thèses 34 Tâches de service 36 Enseignement 38 Résumé 41 Annexe 43 Lainey et al. 2007 : First numerical ephemerides of the Martian moons Lainey et al. 2009 : Strong tidal dissipation in Io and Jupiter from astrometric observations Lainey et al. 2012 : Strong Tidal Dissipation in Saturn and Constraints on Enceladus' Thermal State from Astrometry Lainey et al. 2017 : New constraints on Saturn's interior from Cassini astrometric data Lainey 2008 : A new dynamical model for the Uranian satellites 3 4 Curriculum Vitae Name: Valéry Lainey Address: IMCCE/Observatoire de Paris, 77 Avenue Denfert-Rochereau, 75014 Paris, France Email: [email protected]; Tel.: (+33) (0)1 40 51 22 69 Birth: 19/08/1974 Qualifications: Ph.D. Observatoire de Paris, December 2002 Title of thesis: “Théorie dynamique des satellites galiléens" Master degree “Astronomie fondamentale, mécanique céleste et géodésie” Observatoire de Paris 1998. Academic Career: Sep. 2006 - Present Astronomer at the Paris Observatory Apr. 2004 - Aug. 2006 Post-doc position at the Royal Observatory of Belgium Jun. 2003 - Mar. 2004 Post-doc position at the Paris Observatory Mar. - May 2003 Invited fellow at the Indian Institute of Astrophysics Dec. 2002 - Feb.
    [Show full text]
  • A Geophysical Perspective on the Bulk Composition of Mars A
    JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/, A geophysical perspective on the bulk composition of Mars A. Khan1, C. Liebske2, A. Rozel1, A. Rivoldini3, F. Nimmo4, J.A.D. Connolly2, A.-C. Plesa5, D. Giardini1 Abstract. We invert the Martian tidal response and mean mass and moment of iner- tia for chemical composition, thermal state, and interior structure. The inversion com- bines phase equilibrium computations with a laboratory-based viscoelastic dissipation model. The rheological model, which is based on measurements of anhydrous and melt- free olivine, is both temperature and grain size sensitive and imposes strong constraints on interior structure. The bottom of the lithosphere, defined as the location where the conductive geotherm meets the mantle adiabat, occurs deep within the upper mantle (∼250{ 500 km depth) resulting in apparent upper mantle low-velocity zones. Assuming an Fe- FeS core, our results indicate: 1) a Mantle with a Mg# (molar Mg/Mg+Fe) of ∼0.75 in agreement with earlier geochemical estimates based on analysis of Martian meteorites; 2) absence of bridgmanite- and ferropericlase-dominated basal layer; 3) core composi- tions (13.5{16 wt% S), core radii (1640{1740 km), and core-mantle-boundary temper- atures (1560{1660 ◦C) that, together with the eutectic-like core compositions, suggest the core is liquid; and 4) bulk Martian compositions that are overall chondritic with a Fe/Si (wt ratio) of 1.63{1.68. We show that the inversion results can be used in tandem with geodynamic simulations to identify plausible geodynamic scenarios and parameters. Specifically, we find that the inversion results are reproduced by stagnant lid convection models for a range of initial viscosities (∼1019{1020 Pa·s) and radioactive element par- titioning between crust and mantle around 0.001.
    [Show full text]
  • The Dynamical Evolution of Close-In Binary Systems Formed by a Super-Earth and Its Host Star Case of the Kepler-21 System S
    A&A 641, A109 (2020) Astronomy https://doi.org/10.1051/0004-6361/201936551 & © ESO 2020 Astrophysics The dynamical evolution of close-in binary systems formed by a super-Earth and its host star Case of the Kepler-21 system S. H. Luna1,2, H. D. Navone1,3, and M. D. Melita2,4 1 Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Av. Pellegrini 250, Rosario, Argentina 2 Instituto de Astronomía y Física del Espacio (IAFE), CONICET-Universidad de Buenos Aires, Buenos Aires, Argentina e-mail: [email protected] 3 Instituto de Física de Rosario (IFIR), CONICET-Universidad Nacional de Rosario, Rosario, Argentina 4 Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, La Plata, Argentina Received 22 August 2019 / Accepted 24 July 2020 ABSTRACT Aims. The aim of this work is to develop a formalism for the study of the secular evolution of a binary system which includes interaction due to the tides that each body imparts on the other. We also consider the influence of the J2-related secular terms on the orbital evolution and the torque, caused by the triaxiality, on the rotational evolution, both of which are associated only to one of the bodies. We apply these set of equations to the study of the orbital and rotational evolution of a binary system composed of a rocky planet and its host star in order to characterize the dynamical evolution at work, particularly near spin-orbit resonances. Methods. We used the equations of motion that give the time evolution of the orbital elements and the spin rates of each body to study the dynamical evolution of the Kepler-21 system as an example of how the formalism that we have developed can be applied.
    [Show full text]
  • Arxiv:1907.10575V4 [Astro-Ph.EP] 3 Aug 2020
    Astronomy & Astrophysics manuscript no. evrotorbST_final c ESO 2020 August 4, 2020 The dynamical evolution of close-in binary systems formed bya super-Earth and its host star Case of the Kepler-21 system S. H. Luna1, 2⋆, H. D. Navone1, 3, and M. D. Melita2, 4 1 Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, Av. Pellegrini 250, Rosario, Argentina. 2 Instituto de Astronomía y Física del Espacio (IAFE). CONICET-Universidad de Buenos Aires. Buenos Aires, Argentina. 3 Instituto de Física de Rosario (IFIR). CONICET-Universidad Nacional de Rosario. Rosario, Argentina. 4 Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata. Paseo del Bosque s/n. La Plata. Argentina. Received ...; accepted ... ABSTRACT Aims. The aim of this work is to develop a formalism for the study of the secular evolution of a binary system which includes interaction due to the tides that each body imparts on the other. We also consider the influence of the J2-related secular terms on the orbital evolution and the torque, caused by the triaxiality, on the rotational evolution, both of which are associated only to one of the bodies. We apply these set of equations to the study of the orbital and rotational evolution of a binary system composed of a rocky planet and its host star in order to characterize the dynamical evolution at work, particularly near spin-orbit resonances. Methods. We used the equations of motion that give the time evolution of the orbital elements and the spin rates of each body to study the time evolution of the Kepler-21 system as an example of how the formalism that we have developed can be applied.
    [Show full text]