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 satellites, 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 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/ R 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 dissipative, 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 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/ R 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. 1 for a set of loosely connected aggregate satellite models. –3,000 –2,500 –2,000 –1,500 –1,000 –500 0 Several important observations can be made. First, the evolution of Time from present (Myr)

Rp shows that the satellites’ orbits intersected, depending on their Fig. 1 | The orbital history of Phobos and Deimos. a, Tidal evolution 0 Kl , between 1 Gyr and 2.7 Gyr ago and that this intersection hap- (backward integrated over time) of the planetocentric distance for a set of penedI close to or above the synchronous radius (Fig. 1a). Second, loosely connected aggregate models of Phobos and Deimos (defined by both satellite orbits were initially eccentric and became gradually k2/Q2; Supplementary Table 1), with the tidal dissipation inside both Mars circularized by tidal dissipation in Mars and the moons (Fig. 1b). and the satellites included; k2/Q2 for the individual Phobos and Deimos Yet, throughout the integrations, the eccentricities remained small curves span the end-member range indicated in the legend linearly. enough (<0.35), reducing the possibility of chaotic tumbling24,25 b,c, Corresponding eccentricity (b) and semi-major axes (c) curves. or chaotic transitions between spin–orbit resonances17,24. Third, The eccentricity jumps are due to resonance interactions (see main text) Phobos’s and Deimos’s semi-major axes (Fig. 1c) remained below that result in rapid changes in planetocentric distances. Since Rp resides within the interval a 1 e Rp a 1 e , the curves in a for Phobos and above the synchronous radius, respectively. Although coun- ð� Þ  ðÞþ terintuitive, this fact agrees well with our scenario because of the and Deimos correspondI to the maximal and minimal planetocentric eccentricity values involved. Note that a common origin becomes distances, respectively. The point where the orbits intersect, that is, where Deimos Phobos possible when the maximal value of Phobos’s planetocentric dis- minimal Rp ≤ maximal Rp , is indicative of a common origin. Both I I tance becomes equal to the minimal value of Deimos’s distance. the planetocentric distance and semi-major axis are normalized to RMars. From the planetocentric inequality referred to earlier, we see that, The inset in c shows a plot of the backward-integrated tidal evolution of the although a must obey Rp/(1 + e) < a, it nevertheless can stay below inclinations relative to Mars’s equator of the moons for the end-member Rp. Thus, in the course of our backward integration, Phobos’s Rp can cases (for Deimos, the end-members are superimposed). The smallness become larger than the synchronous radius, with its a value remain- of the inclination over the entire lifetime is justified since in the course of ing less than this radius. Fourth, the changes in the orbital incli- uniform equinoctial precession of an oblate host planet, the inclination of nations are found to be small (<0.021 rad) throughout their entire a near-equatorial satellite follows the evolving equator, with very small history (Fig. 1c inset). oscillations about it (Supplementary Section 2). Because of the resonances Figure 1a shows that the orbits intersected close to or above the between Phobos and Mars, Phobos and the Sun, and Phobos and Deimos, synchronous radius (distance range 5.9–6.9 RMars) from as recently rapid eccentricity changes have occurred over the past ~650 Myr (b).

Nature Astronomy | www.nature.com/natureastronomy Nature Astronomy Letters as 1 Gyr ago, for less consolidated aggregates, to 2.7 Gyr ago, for to contradict the geophysical observations11,12,14,21,22, would it be pos- more consolidated bodies. This suggests a common provenance (in sible to move the origin of Phobos beyond 4 Gyr ago (α is discussed space and time) in the form of a larger progenitor2,26 that disinte- in Supplementary Section 10 and Supplementary Fig. 7). grated to produce Phobos and Deimos. Different initial eccentrici- The effects of past increases in temperature in a previously hot- ties of the satellites (Fig. 1b) support an impact disruption, since ter and therefore more dissipative Mars30 are shown, for both loose post-collisional planetesimal fragments generally vary widely in and more consolidated satellites, with both low and high eccentrici- eccentricity8. The subsequent orbital evolution has separated the ties in Supplementary Section 11 and Supplementary Figs. 8 and 9. satellites in space, providing a natural explanation for their cur- They suggest earlier encounters relative to the nominal case. Any rent orbital configuration. The low initial orbital inclinations found dissipation not accounted for here (for example, a possible early here favour an equatorially orbiting parent body formed in situ6–9. presence of oceans on or melt inside Mars30, or chaotic tumbling of Although the details of the disruption process require more study, it the satellites24,25) would move the origin of the moons closer to the has already been demonstrated that subcatastrophic low-energy dis- present. Consequently, our results represent lower bounds on their ruptive events could result in two main fragments27; had more been orbital history. produced, the remaining debris could have fallen onto Mars28,29, Finally, forward integration to assess the future fate of the sat- contributing to what we observe as the Martian cratering record ellites (Supplementary Section 12 and Supplementary Fig. 10) (Supplementary Section 5). shows that, while Deimos very slowly continues to ascend, Phobos Contrary to popular belief4,17, the orbits of both bodies may have will impact on Mars in ~39 Myr (refs. 10,31) or tidally disintegrate started above the synchronous radius (Supplementary Section 6 into a ring on reaching the Roche limit. The results presented and Supplementary Figs. 2 and 3). The curves show that dissipation here stand to be improved with Mars InSight geophysical data, in inside Phobos is strong enough to drive it through the synchronous particular the dissipation in Mars and its frequency dependence limit on its descent towards Mars. This happens when the orbital that control the orbital history of Phobos. The upcoming Martian evolution is dominated by dissipation in the satellite, as the eccen- Moons Exploration mission will also provide crucial information tricity stays high enough. In contrast, dissipation inside Deimos is on the moons’ interiors, which will help to settle the question of initially only large enough to make it descend within the vicinity their origin. of its current orbit. Hence, Deimos’s distance to Mars has not been monotonically increasing with time, as is presently the case, but ini- Methods tially evolved inwards. As the eccentricities decreased, so did the Orbital evolution theory. Te time evolution of each orbital parameter of the dissipation rate in the satellites, and the orbital evolution became two-body system can be cast as governed mainly by dissipation in Mars. Consequently, the inward dx dx main dx main dx libration motion of Deimos changed to outward migration, while dissipation ; 4 dt ¼ dt þ dt þ dt ð Þ in Phobos was and still is intensive enough that it keeps descend- planet satellite satellite 5 ing. The case of crossing satellite orbits was considered earlier , but where x is either a or e. The three terms refer, respectively, to the tides in the planet, was ruled out, partly due to the difficulty of circularizing Deimos’s the tides in the satellite (with no libration taken into account) and the input from orbit within the lifetime of the Solar System. This difficulty resulted the satellite’s libration. In the following, we provide only the main formulae for from: the application of a simplistic tidal model (inappropriate the orbital evolution. For the full derivations, see Supplementary Section 13. The for e > 0.15) that ignores libration (Supplementary Section 7 and semi-major axis rate is Supplementary Fig. 4) and resonance interactions; the use of ad hoc da l l m ! l 2an l12 m 0 ð � Þ 2 δ0m p 0 q1 s1 j1 viscoelastic rheologies based on the limited geophysical data then dt ¼� ¼ ¼ l m !ð � Þ ¼ ¼�1 ¼�1 ¼�1 ð þ Þ P P 2l 1 P P P 2l 1P available; and the application of too small Kl0 values. R þ M R þ G e G e l 2p q s 0F2 i K β 0 5 To test the variation of Phobos’s tidal lifetimeI with initial condi- lpqð Þ lpjð Þð � þ � Þ a M lmpð Þ lð Þþ a ð Þ     tions, we considered low (e = 0.015), medium (e = 0.15) and high M 2 Jj q s m Js m Flmp i0 Kl β0 ; (e = 0.3) starting eccentricities, and integrated its orbit forwards M � þ ð AÞ ð AÞ ð Þ ð Þ 0  in time from the synchronous radius. The results (Supplementary Section 8 and Supplementary Fig. 5) indicate that a satellite with where G(e) are the eccentricity functions (Supplementary Table 2), F(i) are the inclination functions, Js is the order-s Bessel function, β is the tidal mode and all an initial eccentricity <0.2 would crash into Mars in <3.1 Gyr. For of the other variables are as defined for equations (1)–(3). Since the inclinations of any higher initial eccentricity, Phobos’s lifetime would be <2 Gyr. A the orbits remain small, only F201 and F220 are relevant and are equal to 1/2 and 3, short-lived Phobos presents an obstacle to it having formed along- respectively. Similarly to da/dt, the general expression for the eccentricity rate is side Mars. The progenitor, conversely, could have been billions of 2 de 1 e n l l m ! years old before breaking up, provided that its eccentricity was low, � l12 m 0 ð � Þ dt ¼� e MM ¼ ¼ l m ! 0 ð þ Þ since tidal evolution in the vicinity of the co-rotation radius is slow. l P P ´ 2 δ0m p 0 q1 s1 j1 Those of its remnants which were born with a sufficient eccentricity ð � Þ ¼ ¼�1 ¼�1 ¼�1 2l 1 2l 1 P P P R þPM0 R0 þ (such as Phobos) were dissipative enough to descend and cross the G e G e l 2p q s F2 i K β lpqð Þ lpjð Þð � þ � Þ a M lmpð Þ lð Þþ a synchronicity radius.     p 2 These results provide additional support for the assertion that the M 2 1 e n Jj q s m Js m Flmp i0 Kl0 β0 � l12 M � þ ð AÞ ð AÞ ð Þ ð Þ � e MM ¼ satellites cannot be monoliths. Indeed, had the moons been mono- 0  ffiffiffiffiffiffiffiffiffiffiffiffi 0 lithic, the dissipation in them would have not been sufficient to l l m ! l P m 0 ð � Þ 2 δ0m p 0 q1 s1 j1 Glpq e Glpj e l 2p ¼ l m !ð � Þ ¼ ¼�1 ¼�1 ¼�1 ð Þ ð Þð � Þ dampen the eccentricity jumps associated with the above-mentioned ð þ Þ P 2l 1 P P2l 1 P P resonances (Supplementary Section 9 and Supplementary Fig. 6). R þ M0 2 R0 þ M 2 Flmp i Kl β Jj q s m Js m Flmp i0 Kl0 β0 : The backward integration of the orbits indicates that for the eccen- a M ð Þ ð Þþ a M � þ ð AÞ ð AÞ ð Þ ð Þ     0  tricity excursions to become efficiently damped, k2/Q2 needs to be 6 at least ~10−7 and ~10−4 for Phobos and Deimos, respectively, which ð Þ is achievable only in the case of sufficiently fractured and there- Due to very slow convergence of the series and the relatively high eccentricities 6–9,30 found in this study, we have to include higher-order terms to ensure precision of fore dissipative moons. Thus, although early accretion of the our results and stability of integration for high eccentricities. Martian moons cannot entirely be ruled out, our results indicate that only in the extreme case of a monolithic Phobos and a very low Contribution from tides. In equation (4), the contribution of tides raised by the frequency exponent (α ≈ 0.1) for dissipation in Mars, which appears satellite in the planet is

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da main R5 M da 3 R 7M 0 ´ θ_ 0 2 θ_ n 4 Kl; ; n; e : 7 an 5 1 12e K3 3n 3 dt planet¼ a M Fð Þ ð Þ dt l 3 ¼�4 a M ð � Þ ð � Þ 17 ¼ ð Þ 2 Â 4 2 1 4e K3 n θ_ O e O i ; The ‘main’ (unrelated to libration) contribution due to the tides raised by the planet þð þ Þ ð � Þ þ ð Þþ ð Þ Ã in the satellite looks similar: where O refers to the order of the truncation error. However, we have explored main 5 whole groups of terms, those with lmpq = 330q and lmpq = 311q. A direct da R0 M n ´ K 0 ; θ_ 0; n; e : 8 calculation has shown that in both groups, the important terms are those with dt ¼ a4 M Fð l Þ ð Þ satellite 0 q = −1, 0, 1: 7 Here, is a function of the eccentricity, the mean motion, the satellite spin rate (θ_ 0) de M R 5 F 0 2 _ and its Itidal response n ´ e 1 2e K3 2n 3θ dt l 3 ¼ M a 8 ð � Þ ð � Þ ¼ 9 1 15 49 2 _ 125 113 2 _ 0 _ 2i � 0 _ 2i e 1 e K3 3n 3θ e 1 e K3 4n 3θ Kl ; θ0; n; e e Kl jn 2θ0 φj þ16 � 4 ð � Þ� 8 � 10 ð � Þ F ¼ i 0 j 7 � ¼ ¼� �P P � 9 16;129 3 3 11 2 3 15 2 11 9 ð Þ e K3 5n 3θ_ e 1 e K3 θ_ e 1 e K3 n θ_ 0 _ 2i 0 2i � 256 ð � Þþ8 � 2 ð� Þþ8 þ 16 ð � Þ Kl jn 2θ0 φj Kl jn φ^j : þ j 1 � þ j 1 ð Þ ! ¼ ¼ 27 1 2 _ 8;427 3 _ 5 2 P �P e 1 e K3 2n θ ne K3 3n θ O e O i : � 8 þ 3 ð � Þ 256 ð � Þ þ ð Þþ ð Þ The coefficients φ2i and φ^2i of the series are tabulated in Supplementary Tables 3 j j 18 and 4. Note that inI these tables,I the terms of the series that are not mentioned are ð Þ equal to zero. The above equations have been derived for the general case, that is, Note that here we do not include terms of higher order in the eccentricities, with neither of the bodies assumed to be synchronous. In the specific situation because the overall effect of degree-3 tides is much less than that of degree-2, so of a synchronized moon, we have θ_ 0 n, and therefore the semi-diurnal term in ¼ such terms can be neglected. equation (8) vanishes: Kl 2n 2θ_ 0I 0. In the contribution from the planet, the I ð � Þ¼ semi-diurnal term vanishes when the satellite is at the synchronous orbit, that is, Inclination. Finally, we compute the rate of the orbit inclination on the Martian when n θ_ 0. I ¼ equator. Given the smallness of both i and its rate, we here keep only the quadrupole (degree-2) terms32 Contribution from libration. The contribution from the longitudinal libration of 5 a synchronized satellite is di M0 R n sin i ´ Kl; θ_; n; i; e ; 19 libration 5 dt ¼ M a Ið Þ ð Þ da R0 M ´ 0 10 n 4 Kl ; n; e; ; dt ¼ a M0 Gð AÞ ð Þ where satellite 243 27 11 9 being a function of the eccentricity, the mean motion, the libration amplitude and 1 ρ e4K 2n θ_ e2 1 ρ ρ e2 K n θ_ G I¼ 64 ð þ Þ 2ð� � Þþ16 þ þ 4 þ 4 2ð� � Þ the tidal response 3 7 63 3 1 17 18 i 9 ρ ρ 2 ρ 4 θ_ 2 ρ ρ 2 � 1 3 e 6 e K2 e 1 1 e 0 i 0 j;s þ4 þ þ 2 þ þ 8 þ ð� Þþ16 � þ 4ð þ Þ Kl ; n; e; e Kl jn γi : 11 Gð AÞ ¼ i 0 s 1 j 1 A ð Þ ð Þ 3 49 41 3 9 X¼ X¼ X¼ K n 2θ_ e2 1 ρ ρ e2 K n θ_ 1 ρ 5ρ e2 2ð � Þþ2 þ þ 16 þ 16 2ð � Þþ4 � � 2 � j;s The coefficients γi are tabulated in Supplementary Table 5. Similarly, to compute 23 63 4 3 9 2 11 45 4 I ρ e K2 2n 2θ_ 1 ρ 5ρ e ρ e the eccentricity evolution, we write down all the inputs entering equation (6). The þ 4 � 8 ð � Þ�4 þ � 2 þ þ 16 þ 16 input generated by the tides in the planet is 147 2 109 123 2 147 2 main 5 K2 2n θ_ e 1 ρ ρ e K2 3n 2θ_ e de M0 R ð � Þþ 16 � � 28 � 28 ð � Þ� 16 n ´ Kl; θ_; n; e ; 12 dt ¼� M a Lð Þ ð Þ 109 123 867 planet 1 ρ ρ e2 K 3n θ_ 1 ρ e4K 4n 2θ_ þ � 28 þ 28 2ð � Þþ 16 ð � Þ 2ð � Þ while the ‘main’ (unrelated to libration) input from the tides in the satellite is 867 4 3 6 1 ρ e K2 4n θ_ O i O e ; main 5 � 16 ð þ Þ ð � Þ þ ð Þþ ð Þ de M R0 n ´ K 0 ; θ_ 0; n; e ; 13 dt ¼� M a Lð l Þ ð Þ 20 satellite 0 ð Þ with where the function is defined as L 2 MM0na 9 1 ρ ; 21 _ 2i 1 � _ 2i 1 _ Kl; θ; n; e e � Kl jn 2θ λj � ¼ M M0 Cθ ð Þ Lð Þ¼i 1 j 7 � ð þ Þ ¼ ¼� P P � 14 11 9 2i 1 ð Þ where C is the polar moment of inertia of the planet. _ 2i 1 ^ � Kl jn 2θ λj � Kl jn λj : þ j 1 � þ j 1 ð Þ ¼ ¼ ! P �P Data availability 2i 1 2i 1 The data that support the findings of this study are available from the The coefficients λ À and ^λ À are tabulated in Supplementary Tables 6 and 7. j j corresponding author on request. Note that, similarlyI to da/dIt, the expression is general, in that neither of the bodies is a priori assumed to be synchronous. For the synchronized Martian moons, the term associated with the semi-diurnal tide in equation (12) vanishes. The input Code availability generated by the satellite’s longitudinal libration about the 1:1 spin–orbit resonance The code for computing the orbital evolution is available on request from the is corresponding author.

libration 5 de M R0 n ´ K 0 ; n; e; ; 15 Received: 28 July 2020; Accepted: 8 January 2021; dt ¼ M a Hð l AÞ ð Þ satellite 0 Published: xx xx xxxx where the function is given by HI References 16 18 i 9 1. Rosenblatt, P. Te origin of the Martian moons revisited. Astron. Astrophys. � j;s K 0 ; n; e; ei sK 0 jn η : 16 Rev. 19, 44 (2011). Hð l AÞ ¼ A l ð Þ i i 1 s 1 j 1 ð Þ 2. Pajola, M. et al. Phobos as a D-type captured asteroid, spectral modeling X¼� X¼ X¼ from 0.25 to 4.0 m. Astron. J. 777, 127 (2013). j;s μ The coefficients ηi are tabulated in Supplementary Table 8. 3. Fraeman, A. et al. Spectral absorptions on Phobos and Deimos in the visible/ Owing to the Isatellites’ proximity to Mars, degree-3 terms have also been taken near infrared wavelengths and their compositional constraints. Icarus 229, into account. Of these, leading are those with {lmpq} = {3300} and {3110}: 196–205 (2014).

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