Abstract
All of the gas giants in our Solar System host ring systems, in contrast to the inner planets. One proposed mechanism of planetary ring formation is disruption or mass shedding of moons. The orbit of Phobos, the larger of Mars’s two moonlets, is gradually spiralling inwards towards Mars and the moon is experiencing increasing tidal stresses. Eventually, Phobos will either break apart to form a ring or it will crash into Mars. We evaluate these outcomes based on geologic, spectral and theoretical constraints, in conjunction with a geotechnical model that helps us determine the strength of Phobos. Our analysis suggests that much of Phobos is composed of weak, heavily damaged materials. We suggest that—with continued inward migration of the moon—the weakest material will disperse tidally in 20 to 40 million years to form a Martian ring. We predict that this ring will persist for 10^{6} to 10^{8} years and will initially have a comparable mass density to that of Saturn’s rings. Any large fragment of Phobos that is strong enough to escape tidal breakup will eventually collide with Mars in an oblique, lowvelocity impact. Our analysis of the evolution of Phobos underscores the potential orbital and topographic consequences of the growth and selfdestruction of other inwardly migrating moons, including those that met their demise early in our Solar System’s history.
Main
Tides play a key role in the orbital evolution of moons. The synchronous radius is the distance at which a satellite’s orbital period matches the spin period of its primary^{1}. As a result of tidal dissipation, moons that form beyond the synchronous radius will move gradually outwards with time, and moons within the synchronous radius will move inwards. Earth’s Moon is moving outwards; its orbit has probably expanded by an order of magnitude since the Moonforming event^{2}. In contrast, the small moon Phobos has a shorter orbital period than the rotational period of Mars, its central body^{3,4}. Therefore, over time the orbit of Phobos contracts and the moon edges inexorably towards Mars^{5,6,7,8,9}.
Inwardly migrating moons may have been more common in our solar system’s past^{10,11,12}. Saturn’s icerich rings—which are among the most distinctive planetary features in the solar system—may have originated through tidal stripping of a large, inwardly migrating, differentiated satellite^{12}. With Mars’s present spin, its synchronous radius is at ∼6.03 Mars radii^{4}. At 4.5 Ga, the satellite would have been at 5.8–6.0 Mars radii (depending on the assumed tidal parameters, see Methods), just inside the synchronous radius. In general, large, closein inwardly migrating moons should selfdestruct more quickly, whereas only a small moon such as Phobos (mean radius 11 km; ref. 13) with an initial position barely within the synchronous radius could survive until the present day. In this sense, it is not a coincidence that Phobos is the only surviving inwardly migrating Martian satellite, nor is it a coincidence that Phobos will expire imminently. Phobos thus offers the last possible glimpse of the signatures and processes that applied to inwardly migrating moons and the interplay with ring formation early in our solar system’s history.
Orbital evolution and tidal stresses on Phobos
The inward drift of Phobos results from dissipation associated with tides raised by Phobos on Mars. This dissipation is most sensitive to the Love number (k_{2}) and the tidal quality factor (Q) of Mars^{8,14}. Measurements of the rate at which Phobos is moving inwards^{8,9}, along with observations of periodic variations in the Martian gravity field in response to tides from Phobos and the Sun, have recently been used to deduce that k_{2} = 0.148 ± 0.017 and Q = 88 ± 16 for Mars (although depending on the treatment of higherdegree terms Q may be as high as 190 (refs 8,14)). In Fig. 1, we use these values to calculate Phobos’s future orbital semimajor axis, velocity relative to Mars, and eccentricity (see Methods), and we also illustrate the sensitivity of Phobos’s orbital evolution to higher and lower assumed values of k_{2} and Q (ref. 14). Barring tidal breakup, in at minimum 20 Myr and at most 70 Myr (and most likely in ∼30 Myr) the semimajor axis of Phobos’s orbit will diminish to 3,550 km, which is the top of the Martian atmosphere.
Phobos at present experiences tidal stresses (σ_{tidal}) and rotational stresses (σ_{centrifugal}), which are counterbalanced by selfgravitation (σ_{self−gravity}). The propagation of cracks leading to tidal failure depends on the distribution of strength and stress within a heterogeneous body. Because the internal structure of Phobos is unknown, we consider bodyaveraged stresses, which will provide an upper bound on the failure load^{15}. If the stresses exceed its yield strength (ϒ) at any time before atmospheric entry, Phobos will break apart (the stresses and failure criteria we consider in Fig. 2 are described in detail in the Methods). The gravitational, tidal and rotational stresses are well constrained as a function of orbital radius, mass and the shape of Phobos^{15}. The major uncertainty is the yield strength of Phobos.
Constraints on the strength of Phobos
To assess whether or not Phobos will in fact break apart before it reaches the top of the Martian atmosphere, we next consider the implications of compositional and density data and the presence of the Stickney impact crater for the probable strength of Phobos.
The visible and nearinfrared spectra of Phobos most closely resemble the spectra of CM carbonaceous chondrites (the unusual Tagish Lake meteorite provides a particularly good match)^{16}. CM carbonaceous chondrites typically exhibit extensive aqueous alteration^{17}, which degrades their strength (see Methods). Furthermore, the density of Phobos is only ∼1,860 kg m^{−3} (ref. 18). This low density implies a high porosity, consistent with gravitational aggregation or a high degree of damage^{19}.
The Stickney impact crater is ∼10 km in diameter^{20}, spanning approximately onesixth the circumference of Phobos. The preservation of the crater implies that the impact itself did not totally disrupt the satellite, despite significant fragmentation^{21}. Such a large impact would be sufficient to disperse a target that is either a rubble pile or an undamaged rigid object^{21,22}, indicating that at the time of the impact Phobos was already modestly fragmented and porous^{22}, but was stronger than a rubblepile asteroid of comparable size. The Stickney impact itself would have generated significant additional damage^{22,23}. Threedimensional simulations confirm that the Stickney impact caused probable heavy damage to a large region around the impact site, that much of this damaged mass did not escape, and that major fractures probably formed throughout the moon^{23}. Although uncertainties remain regarding the internal strength of Phobos, the density and compositional data and these hydrodynamic modelling results suggest that at present Phobos is a porous, heterogeneous (and possibly altered) composite of heavily damaged and more intact rock.
To translate these qualitative properties of Phobos into a quantitative estimate of strength, we employ the Hoek–Brown model for rock mass strength, which has a long history of successful application to tunnelbuilding and other underground construction projects on Earth^{24,25}. To permit direct comparison, we give all estimates of strength in terms of cohesion (also known as the cohesive strength), which is defined as the shear strength of a material under zero confining stress^{15}. Accounting for variable degrees of fragmentation, alteration and macroscale heterogeneity, we select inputs to our Hoek–Brown model appropriate for Phobos. We obtain estimates for the cohesion of rock masses on Phobos spanning 0.03–0.5 MPa (Fig. 2a; we split this range of cohesions to represent stronger and weaker parts of Phobos as described in the Methods).
For comparison, the maximum topographic relief on Phobos (∼1.5–3 km; ref. 18) requires a minimum cohesion of ∼0.005–0.015 MPa. The ram pressure at first breakup of the Tagish Lake meteoroid (which also had a similar porosity to Phobos) implies cohesion of ∼0.05–0.08 MPa (ref. 26). The good agreement between these independent estimates of strength and estimates from our Hoek–Brown model (Fig. 2a) decidedly supports the validity of the model and our chosen parameter ranges.
Finally, we consider two commonly applied failure criteria to relate the estimated Hoek–Brown strengths to the bodyaveraged stresses (see Methods). The Hoek–Brown criterion^{24,25} predicts failure at 2.4–2.0 Mars radii. The Drucker–Prager criterion^{15,27} gives the minimum cohesion required for Phobos to withstand increasing tidal stresses as its orbit contracts (Fig. 2a); it predicts that the weakest, most heavily damaged materials in Phobos should succumb to tidal stresses beginning at ∼1.6 Mars radii, whereas the stronger elements may survive intact (Fig. 2b).
The fate of Phobos and the consequences of tidal disruption
We conclude that Phobos’s inward migration will culminate with division of the satellite according to its internal distribution of strength and damage. The most damaged regions will disperse into a planetary ring. The timescale for full disruption of a homogeneous satellite is ∼5 orbital periods^{28}. In the case of a disintegrating Phobos, the timescale for a nascent ring to form around Mars is 50–130 orbital periods, or 10^{2}–10^{3} h (see Methods). In other words, ring formation will proceed very quickly. The orbits of the individual particles that comprise the ring will continue to contract, but because the tidal evolution timescale is proportional to mass, the timescale for these particles to collide with Mars will be much longer than for intact Phobos (Fig. 3a). Consequently the ring will be stable for 10^{6}–10^{8} years (assuming complete disruption at 1.2–2.0 Mars radii). Over this interval, collisional spreading causes the ring to gradually widen towards the top of the Martian atmosphere. The total mass of the ring will be relatively small (at most the mass of Phobos), but in the case of rapid, total fragmentation (in Fig. 3b we consider the aftermath of breakup at 1.2, 1.6, or 2.0 Mars radii with an equivalent particle radius of 1 m) the mass density will initially rival that of Saturn’s major rings^{29}. Mass density will gradually decline as collisional spreading pares the ring (see Methods).
On the other hand, any fraction of Phobos with a cohesion that exceeds ∼200 kPa will escape tidal breakup. Because tidal stresses depend on satellite size, if disruption is gradual (or if the outermost layers of Phobos are siphoned away first^{12}) even a damaged, weak interior may be prone to survive as one or several large fragments. The high mass of large intact remnants will cause them to continue a rapid inward evolution towards the surface of Mars (precise timescales may be sensitive to angular momentum exchange with the ring^{30}), and eventually they will enter the Martian atmosphere.
The impact velocity of large, intact fragments will be far lower than that of typical asteroid and comet impacts on Mars^{31}, and the impacts will be highly oblique. If ram pressure induces breakup in the Martian atmosphere, the fragments will be dispersed according to their mass^{32}, generating a strewn field with aligned elliptical craters (Fig. 3c). At least one such set of Martian elliptical craters has been attributed to atmospheric breakup of a past inwardly evolving moonlet^{11}, offering a possible template for the detection of vanished satellites (see Methods and Supplementary Fig. 1). In the case of small eccentricity orbits close to Mars, projectile velocities will be low (for example, see Fig. 1), and craters will be relatively modest in size (Fig. 3c). For comparison, collision of a Phobossized inwardly evolving moon with Earth would occur at much higher velocities owing to Earth’s greater mass; such a collision would generate a Chicxulubscale impact crater.
In general, the fate of inwardly migrating moons depends on their size and strength. The available constraints on the strength of Phobos suggest that a significant fraction of the satellite is heavily damaged and consequently weak, and will disintegrate to form a longlived ring. A future mission to Phobos, such as the proposed PADME (ref. 33), PANDORA (ref. 34) and MERLIN (ref. 35) Discoveryclass missions, will provide better constraints on the interior structure and strength of Phobos, and consequently will allow us to test this conclusion. We speculate that diminutive Phobos may be the last of many inwardly migrating prograde satellites in our solar system. Numerical simulations of planet formation suggest that Earthlike planets typically experience multiple giant impacts, which may lead to circumplanetary disk formation and a stochastic distribution of planetary spins (including a contingent of slowly spinning primaries)^{36,37}. Thus, inwardly migrating satellites—some of which may break up tidally, some of which may collide with their primaries—are likely to be an underappreciated and important component of planetary evolution.
Methods
The future orbital evolution of Phobos.
Semimajor axis evolution. Owing to tidal dissipation, the orbit of Phobos evolves inwards and the spin period of Phobos (ω_{Ph}) is approximately equal to the mean orbital angular velocity n. The timescales for the future contraction of Phobos’s orbit have been broadly understood for several decades (for example, refs 6,7). To calculate Phobos’s orbital evolution until it reaches the top of the Martian atmosphere (∼160 km above the Martian surface), we include higherorder terms in the expression for changes in the semimajor axis a_{Ph}, which can speed up the evolution of the system and decrease the tidal timescales^{40}. With R_{Ph}/R_{M} → 0: refs 40,41,42. Here κ = (ρ_{Ph}/ρ_{M})(R_{Ph}/R_{M})^{3}. For our calculations, we prefer values of Mars’s degree2 tidal Love number k_{2} = 0.148 ± 0.017 and Mars’s dissipation factor Q = 88 ± 16 (ref. 14), but we also consider an alternative value of Q = 190 (refs 8,14) as a conservative upper limit. Additional variable definitions and initial values are given in Supplementary Table 1. As the uncertainty in the tidal parameters (k_{2} and Q) for Mars is large, we do not consider a frequency dependence of the tidal quality factor Q in the calculations in this paper. With improved estimates of Martian Q, future models should include both the frequency dependence of tidal dissipation (see refs 43,44 for details) as well as resonances that Phobos might encounter as it evolves inwards^{45,46}.
Eccentricity evolution. We model the evolution of Phobos’s orbital eccentricity using the following expression, which is valid in the limit of e ≪ 1 (refs 3,41): The first term represents tides raised on Mars whereas the second term represents tides raised on Phobos. k_{2, Ph} and Q_{Ph} are the tidal parameters for Phobos.
Because the orbital eccentricity of Phobos is very small (∼0.0151), the temperature increase of Phobos due to tidal dissipation should be minor (at most ∼1–2 K)^{47,48,49}.
Orbital evolution. The density of Phobos (∼1,860 kg m^{−3}) implies large porosities broadly similar to those of rubblepile asteroids. Correspondingly, we use the internal structure model proposed by ref. 50 to estimate Phobos’s rigidity and its k_{2, Ph} value: where is the dimensionless rigidity (see ref. 50). For Phobos, a reasonable range for is 2.5 × 10^{3} to 2.5 × 10^{4}, whereas Q_{Ph} ∼ 50–100 (see ref. 51).
Using the above equations, we integrate Phobos’s orbit into the future and calculate the evolution of a_{Ph} and e_{Ph} over time (Fig. 1 in the main text).
Tidal stresses and the strength of Phobos.
Tidal stress model. The volumeaveraged stress fields on a rotating, selfgravitating homogeneous ellipsoidal satellite can be expressed analytically^{15,52,53} give analytical expressions for the volumeaveraged stress fields on a rotating, selfgravitating homogeneous ellipsoidal satellite. The principal stresses (accounting for rotation, selfgravity and tides, with semimajor axes a, b and c) are^{15}: where A_{x}, A_{y}, A_{z} are dimensionless coefficients accounting for the object’s shape. For Phobos, the values of A_{x}, A_{y}, A_{z} are respectively (bc/a^{2}) × (0.879,1.037,1.343).
Strength models for Phobos. In this subsection, we develop the available constraints on the actual strength of Phobos from an adaptation of the Hoek–Brown geotechnical model, the strength required to support Phobos’s topography, the atmospheric breakup of the Tagish Lake meteorite, and the strength of a comparably sized rubblepile asteroid. To make a direct comparison in Fig. 2a to the Drucker–Prager failure criterion (described in detail in the following subsection), in all cases we calculate the cohesive strength (also known as cohesion; see ref. 54 for definition of cohesive strength both in terms of Drucker–Prager and Hoek–Brown failure criterion parameters).
Hoek–Brown model for the bulk strength of Phobos. For fractured or disturbed rock, Hoek and Brown^{23,24} have developed an empirical methodology to estimate rock strength that has been successfully employed during the construction of underground power stations, tunnels, and other largescale engineering projects over the course of several decades^{55,56}. The Hoek–Brown criterion has also been employed previously to estimate the strength of Mercury^{57}, asteroid 433 Eros^{58} and the lunar crust^{59}.
A Hoek–Brown approach to estimating strength has several advantages. It gives the strength of a largescale rock mass, rather than a smallscale sample^{55,56}. The Hoek–Brown methodology also accounts for largescale blasting (through a disturbance factor D, which is not equivalent to the damage factor employed in other rock strength models^{60}), offering one of the only available analogues for impact modification of planetary materials^{55}. Finally, the Hoek–Brown approach incorporates a large quantity of laboratory and field data to relate qualitative rock properties to strength^{23}.
The Hoek–Brown approach can be used to determine σ_{cm}, the total rock mass strength, from the uniaxial compressive strength of an intact rock (σ_{ci}) and the following Hoek–Brown parameters: a_{h}, s, m_{i} (the Hoek–Brown constant for intact rock), D and GSI (the Geologic Strength Index)^{23,24}: with and The cohesive strength (Y_{t}, which is a required parameter for the Drucker–Prager failure criterion described below) can then be calculated from the total rock mass strength σ_{cm} (ref. 24): Here, φ is the internal friction angle (which is typically taken to be 40° for lunar regolith^{61} and ranges from 25° to 45° for granular material^{62}).
For the results shown in Fig. 2a, we consider values for m_{i} from 5 to 15 (ref. 63). For comparison, the experimentally determined value of m_{i} for marble is 9 ± 3, for diabase is 15 ± 5, and for schists is 12 ± 3 (refs 63,64). For weak and porous rock, a m_{i} value of 5 has been suggested^{64}.
We consider values of GSI from 5 to 25, following published tabulations of GSI values^{63,65}. The GSI depends on three qualitative properties of a rock mass: the extent to which rock pieces are interlocking (from massive to poorly interlocked or foliated), the quality and extent of alteration of the surfaces of those rock pieces, and the degree of heterogeneity. The constraints on each of these properties for Phobos are summarized below.
Interlocking: As discussed in the main text, the presence of Stickney and other impact craters implies that at least some sectors of Phobos have suffered heavy damage^{20,21,22}.
Surface alteration: The carbonaceous chondrites (including the Tagish Lake meteorite) that are most spectrally similar to Phobos^{16,66} have been pervasively aqueously altered. The original wellformed silicate minerals have degraded to much weaker, poorly formed fibres including phyllosilicates^{17,67}. The presence of phyllosilicates may also explain some of the observed spectral features on Phobos^{68}.
Heterogeneity: The bestfit porosity of Phobos is ∼30% (ref. 13). Its high porosity (for example, ref. 13) and hypothesized origin as a captured asteroid^{69} or gravitational aggregate^{51,70,71,72} imply Phobos is likely to be highly heterogeneous (possibly including ice in pore spaces in the subsurface^{73}, although at present no surface ice has been detected from spacecraft data^{16}).
Accordingly, we expect GSI values at the lower end of the range we consider to be most representative of the weakest regions of Phobos, where altered surfaces may combine with high porosity and impact damage. We define the weakest elements of Phobos (orange shaded regions in Fig. 2) as the subset of material that is weaker than the critical Drucker–Prager cohesive strength (see section 2.3.1) for 25°–45° friction angles. Stronger fractions of Phobos will probably still qualify as poorquality rock masses, but will have GSI and m_{i} values at the higher end of the ranges we consider. We define the stronger elements of Phobos (green shaded regions in Fig. 2) as the subset of material that is stronger than the critical Drucker–Prager cohesive strength for 25°–45° friction angles.
We select σ_{ci} = 50 MPa for the uniaxial compressive strength^{74}. We also assume that the effects of large impact craters on Phobos are similar to those of bulk blasts, implying D ∼ 1 (ref. 23), following Watters and colleagues^{58}.
In conjunction with equation (11), these values for GSI, D, φ and m_{i} predict a cohesive strength for the stronger and weaker fractions of Phobos as shown in Fig. 2a of the main text. We use the failure criteria discussed below to compare the predicted strength with the strength required to avert failure.
Strength required to support topographic relief. If we assume that Phobos’s topography is largely uncompensated, the strength required to support the observed relief gives an approximate lower bound on the satellite’s strength. Estimates for the maximum topographic relief on Phobos vary depending on the reference surface, and range from ∼1.5–3 km (ref. 18). The true extent of compensation is uncertain. The gravitational stress P = ρ_{Ph}g_{Ph}h gives an estimated minimum strength to support the observed topography of ∼0.02–0.05 MPa. If we take this strength as the minimum total rock mass strength, for friction angles between 25°–45°, equation (11) gives a lower limit on cohesive strength of ∼0.005–0.015 MPa, which is less than the lowest estimates from the Hoek–Brown model (as illustrated in Fig. 2a in the main text), supporting the validity of our Hoek–Brown calculations.
Strength of the Tagish Lake meteoroid. Among known meteorites, the spectral characteristics and porosity of Tagish Lake most closely resemble those of Phobos^{25,75,76}. Observations of the fireball yield an estimate of the ram pressure when the meteoroid first began to break up^{25}, which in turn provides an estimate for the global binding strength of ∼0.25 MPa. Because of the socalled ‘size effect’, in which larger bodies are more likely to incorporate zones of weakness and will therefore fail under smaller stresses^{77}, strength inferred from Tagish Lake may in fact exceed the true strength of Phobos. Taking ∼0.25 MPa as the total rock mass strength of Tagish Lake, we again apply equation (11) with friction angles between 25°–45° to obtain an estimated cohesive strength of ∼0.05–0.08 MPa, which is indistinguishable from our independent Hoek–Brown estimates for the weakest fractions of Phobos. This agreement further supports the validity of our Hoek–Brown model (and the GSI, m_{i}, and other parameters we selected to represent Phobos).
Strength of a Phobossized rubblepile asteroid. The observation that the extant asteroid population occupies a distinct spin perioddiameter envelope yields a model for the strength of a rubblepile asteroid (radius ‘r’) with bulk density ρ_{b} (ref. 15): For a Phobossized body, this model estimates Y_{t} ∼ 800 Pa, much weaker than we calculate from our Hoek–Brown model. This result is expected because hydrodynamic simulations show that if Phobos were a fully damaged rubble pile it would probably have broken apart at the time of the Stickney impact^{20,21}.
Failure criteria. In the previous two subsections, we defined volumeaveraged stresses on Phobos and developed several independent estimates of the strength of Phobos. Finally, to compare the stresses with our strength estimates and determine whether breakup will occur, we consider two failure criteria. In detail, stress and strength on small bodies are nuanced, and landslides and other local failures may occur (for example, refs 78,79). Here, because we are considering volumeaveraged stresses, if failure occurs it will be global in extent^{15}.
Drucker–Prager failure criterion. We use equation (11) to translate the Hoek–Brown rock mass strength σ_{cm} to a cohesive strength Y_{t}, which can then be related to the volumeaveraged stresses given in equations (4)–(6) via the Drucker–Prager failure criterion^{52,78}:
Here we use the inscribed version of the Drucker–Prager criterion because it more closely matches results from triaxial extension tests^{26,80}.
Hoek–Brown failure criterion. In two dimensions, the generalized Hoek–Brown failure criterion is^{24}: where σ_{1} and σ_{3} are the major and minor effective principal stresses at failure.
Using the Hoek–Brown parameters given above, this criterion predicts imminent breakup of Phobos at 2.4–2.0 Mars radii (these bounds are shown along with the predictions from the Drucker–Prager criterion in Fig. 2b).
Although the Drucker–Prager failure criterion has been frequently applied to planetary objects (for example, refs 15,55), it is known to overestimate rock strength under tension^{80}. The Hoek–Brown failure criterion (which predicts that a larger fraction of Phobos should disrupt sooner) may therefore be more robust for tensile regimes such as the tidal breakup we consider here.
Orbital evolution of a Martian ring.
Timescale for ring formation. When tidal breakup occurs (at a_{Ph} = a_{r}), the disrupting material will rapidly shear out into a coherent, needlelike ellipsoid. Secondary gravitational instabilities will then lead to the formation of a collisional belt. Numerical simulations indicate that the timescale for this progression spans only ∼100 orbits, with the newly formed belt having an initial width ∼0.02a_{r} (ref. 27). For an assumed disruption location of Phobos at 1.6 R_{M}, this implies an initial ring width (Δa_{r}) of ∼10 R_{Ph}. The timescale for the debris to spread out to encircle the entire orbital azimuth is given by: (for approximately circular orbits of the satellite before disruption)^{81}.
Because the ring filling timescale is much shorter than the timescale for tidal evolution of ring particles (∼60–120 orbital periods for breakup at 1.2–2.6 R_{M}), we expect a complete azimuthal ring to form quickly after disruption of Phobos.
Viscous spreading of the ring. Because tidal evolution is very slow for small particles (Fig. 3a), viscous spreading will control the evolution and decay of the ring.
We use the dynamical optical depth τ of a ring as a measure of the ring particle number density. The dynamical optical depth of a circumplanetary ring is directly related to the surface mass density Σ_{r} and is: for an equivalent particle radius ‘d’, particle mass m_{r} and density ρ_{r}. In equation (19), N is the average particle surface density (particles m^{−2}) and R_{out} and R_{in} are the outer and inner radius of the ring respectively. Equation (19) makes the upperlimit assumption that the total ring mass is equal to Phobos’s mass.
To first order, viscous spreading due to interparticle collisions or mutual gravitational encounters^{82,83} will control the evolution of a future circumplanetary Martian ring.
The viscous timescale is defined as: and represents the time needed for a ring with constant kinematic viscosity ν to reach the width L. To find this viscous spreading timescale (equation (20)), we first need an estimate of ν in the ring which in turn depends on the ring particles’ velocity dispersion (σ_{r}, the random velocities of the particles due to collisions and gravitational excitation^{84}).
The minimum velocity dispersion for an optically thin selfgravitating ring is either the relative Keplerian velocity between particles (∼2dΩ_{r}, where Ω_{r} is the local angular velocity) or the escape velocity of a particle (). However, if the disk is gravitationally unstable, as measured by the Toomre Q parameter^{85}, selfgravity wakes significantly enhance the velocity dispersion. The Toomre Q parameter is defined as:
Previous studies have shown that selfgravity wakes start to form for Q ≤ 2, and subsequently the Toomre Q parameter always remains about 1–2 (refs 86,87,88). Therefore, the typical velocity dispersion is given by fixing Q = 2 and using equation (21). In practice, we choose the largest value for the velocity dispersion among relative Keplerian velocity, escape velocity and selfgravitationally excited velocity (which is relevant only if Q ≤ 2).
The total effective kinematic viscosity of the ring (ν_{eff}) has three components: ν_{l}, the local kinematic shear viscosity^{83,89}, ν_{nl}, the nonlocal shear viscosity^{90}, and ν_{grav}, the gravitational viscosity^{87,89}. The total viscosity for a ring is: where C is a correction factor defined as min{53r_{h}^{5}, 30} (ref. 89) and r_{h} is the ratio of the mutual Hill radius R_{Hill} = a_{r}(2m_{r}/3M_{M})^{1/3} of two equal sized colliding particles to the sum of their radii (2d). If the Toomre Q ≤ 2, ν_{l} = ν_{grav} (ref. 87).
Following ref. 91, we represent the particle size distribution in the ring with a single equivalent particle radius ‘d’. We adopt d = 1 m, which is commonly applied to simulations of Saturn’s rings^{29,92,93}.
We can then estimate the ring spreading timescale from equation (20), solving iteratively for surface mass density, velocity dispersion and viscosity as the ring spreads. We cease our calculations when the ring spreads to the top of the Martian atmosphere.
We confirmed the validity of our ring spreading timescale estimates by calculating the onedimensional viscous evolution of a Keplerian pressureless disk^{93} (with a timedependent and spacedependent viscosity model as in equation (22)).
The time for a ring to spread to the top of the Martian atmosphere is ∼10^{6}–10^{8} years, depending on the initial breakup location (see Fig. 3b in the main text). The total lifetime of a future Martian ring could be significantly longer as hundreds of viscous timescales are required to completely accrete the disk onto the planet.
Assuming no significant interaction with small satellites at disk edges, the time evolution of the surface density of a circumMartian ring with total mass equivalent to Phobos’s mass is shown in Fig. 3b of the main text. For comparison, the surface density of Saturn’s A ring is ∼40–60 g cm^{−2} (refs 28,94), that of the C ring is^{95} ∼1–5 g cm^{−2} and that of the B ring is^{96,97} ∼140–260 g cm^{−2}.
Projectile dispersion in the Martian atmosphere and impact crater scaling.
If some fraction of Phobos reaches the top of the Martian atmosphere without succumbing to tidal breakup, it will experience friction and ablation in the atmosphere. Ram pressure can be estimated as The entry velocity V will be ∼3.5 km s^{−1}, implying a peak ram pressure of ∼0.25 MPa as a projectile approaches the Martian surface, where atmospheric density ρ_{atmosphere} ∼ 0.02 kg m^{−3}. This ram pressure corresponds to a critical cohesive strength of ∼0.05–0.08 MPa to avoid breakup during atmospheric entry. These values are comparable to the cohesive strength we estimate for the weaker regions of Phobos. Consequently, breakup in the atmosphere is unlikely to be catastrophic, but because tidal stresses are inversely proportional to mass, whereas ram pressures are independent of projectile size, we expect some remnants to survive tidal disruption only to break apart and disperse when they enter the Martian atmosphere.
The final velocities and masses will determine the size of the craters that result from the fragments of Phobos when they impact on Mars; the atmospheric dispersion will determine the distribution of those craters. To characterize the probable size and spatial distribution of impact craters from a Phoboslike body (Fig. 3c in the main text), we solve the equations of motion for projectiles in the Martian atmosphere^{31,98}: with variables and initial values as defined in Supplementary Table 1. We employ a density structure for the Martian atmosphere from Mars Pathfinder^{37}.
To calculate the size of the resulting elliptical impact craters, we rely on impact crater scaling relationships^{38}: where D is the effective crater diameter shown in Fig. 3. V is the velocity when the projectile hits the Martian surface and θ is the angle from horizontal; we obtain these values from the atmospheric ablation and dispersion model described above. L is the diameter of the object and g is gravitational acceleration (see Supplementary Table 1 for a summary of variables).
The initial position of Phobos.
With Mars’s present spin, its synchronous distance is at ∼6.03 Mars radii (for example, ref. 4). This distance may have decreased by at most ∼10% or so during Martian differentiation^{99}. Exchange of angular momentum with the satellites or solar tides has not significantly affected the rotation period of Mars^{100}.
Assuming zero eccentricity and values of k_{2} = 0.148 ± 0.017 and Q = 88 ± 16 for Mars^{14}, backwards solution of equation (1) implies that at 4.5 Ga, the orbital semi major axis of Phobos would have been ∼5.8–6 Mars radii (for k_{2} = 0.1463 and Q = 190 for Mars, at 4.5 Ga Phobos would have been at 5.15 Mars radii). In other words, because any satellites forming much beyond this distance would evolve outwards, ∼5 Gyr may be the maximum lifetime for an inwardly evolving Martian satellite. Thus, although we suggest that other inwardly evolving satellites may have existed in our solar system in the past, it may not be surprising that at present there is only one inwardly evolving satellite in the Martian system. Only a body that began as far out as Phobos could have survived until the present day.
The topographic signature of vanished inwardly evolving moons.
If other inwardly evolving moons once existed in the Martian system, they may likewise have formed rings or (if they were sufficiently strong) they may have impacted on the surface of Mars^{11,101}. In the latter case, the resulting elongated strewn field of elliptical impact craters could have left a topographic signature. The most recent survey of elliptical impact craters tabulated 8,724 elliptical craters (ellipticity > 1.2) on Mars larger than ∼2 km diameter^{102}. It has been argued^{11} that atmospheric breakup of an inwardly migrating moonlet is the most plausible explanation for an aligned pair of highly elliptical craters located at 40.5° N, 222.5° E (illustrated in Supplementary Fig. 1). Future work to distinguish elliptical craters from hypervelocity asteroidal impacts versus elliptical craters from lowervelocity impacts could enable the detection of the topographic aftermath of vanished inwardly evolving moons.
Code Availability.
The Python scripts we used to compute the orbital evolution of Phobos, its strength and its failure envelope, and the spreading of a ring can be accessed at https://github.com/PhobosMars/Phobos.
References
 1.
Murray, C. D. & Dermott, S. F. Solar System Dynamics (Cambridge Univ. Press, 1999).
 2.
Canup, R. M. & Asphaug, E. Origin of the Moon in a giant impact near the end of the Earth’s formation. Nature 412, 708–712 (2001).
 3.
Goldreich, P. & Soter, S. Q in the Solar System. Icarus 5, 375–389 (1966).
 4.
Veverka, J. & Burns, J. A. The moons of Mars. Annu. Rev. Earth Planet. Sci. 8, 527–558 (1980).
 5.
Soter, S. & Harris, A. Are striations on Phobos evidence for tidal stress? Nature 268, 421–422 (1977).
 6.
Mignard, F. Evolution of the Martian satellites. Mon. Not. R. Astron. Soc. 194, 365–379 (1981).
 7.
Szeto, A. M. Orbital evolution and origin of the Martian satellites. Icarus 55, 133–168 (1983).
 8.
Bills, B. G., Neumann, G. A., Smith, D. E. & Zuber, M. T. Improved estimate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos. J. Geophys. Res. 110, E07004 (2005).
 9.
Jacobson, R. A. The orbits and masses of the Martian satellites and the libration of Phobos. Astron. J. 139, 668–679 (2010).
 10.
Burns, J. A. Where are the satellites of the inner planets? Nature 242, 23–25 (1973).
 11.
Chappelow, J. & Herrick, R. On the origin of a double, oblique impact on Mars. Icarus 197, 452–457 (2008).
 12.
Canup, R. M. Origin of Saturn’s rings and inner moons by mass removal from a lost Titansized satellite. Nature 468, 943–926 (2010).
 13.
Pätzold, M., Andert, T., Jacobson, R., Rosenblatt, P. & Dehant, V. Phobos: Observed bulk properties. Planet. Space Sci. 102, 86–94 (2014).
 14.
Nimmo, F. & Faul, U. Dissipation at tidal and seismic frequencies in a meltfree, anhydrous Mars. J. Geophys. Res. 118, 2558–2569 (2013).
 15.
Holsapple, K. A. & Michel, P. Tidal Disruptions: II. A continuum theory for solid bodies with strength, with applications to the solar system. Icarus 193, 283–301 (2008).
 16.
Fraeman, A. et al. Analysis of diskresolved OMEGA and CRISM spectral observations of Phobos and Deimos. J. Geophys. Res. 117, E00J15 (2012).
 17.
Rubin, A. E., TrigoRodríguez, J. M., Huber, H. & Wasson, J. T. Progressive aqueous alteration of CM carbonaceous chondrites. Geochim. Cosmochim. Acta 71, 2361–2382 (2007).
 18.
Willner, K., Shi, X. & Oberst, J. Phobos’ shape and topography models. Planet. Space Sci. 102, 51–59 (2014).
 19.
Richardson, D., Leinhardt, Z., Melosh, H., Bottke, W. Jr & Asphaug, E. Gravitational aggregates: Evidence and evolution. Asteroids III 1, 501–515 (2002).
 20.
Willner, K. et al. Phobos control point network, rotation, and shape. Earth Planet. Sci. Lett. 294, 541–546 (2010).
 21.
Asphaug, E. & Melosh, H. The Stickney impact of Phobos: A dynamical model. Icarus 101, 144–164 (1993).
 22.
Asphaug, E., Ostro, S., Hudson, R., Scheeres, D. & Benz, W. Disruption of kilometresized asteroids by energetic collisions. Nature 393, 437–440 (1998).
 23.
Rovny, J., Owen, J. M., Howley, K. & Wasem, J. in Proc. 44th Lunar Planet. Sci. Conf. 2013 1076 (LPI, 2013).
 24.
Hoek, E. & Brown, E. Practical estimates of rock mass strength. Int. J. Rock Mech. Min. Sci. 34, 1165–1186 (1997).
 25.
Hoek, E., CarranzaTorres, C. & Corkum, B. Hoek–Brown failure criterion2002 edition. Proc. NARMSTac 1, 267–273 (2002).
 26.
Brown, P. G., Revelle, D. O., Tagliaferri, E. & Hildebrand, A. R. An entry model for the Tagish Lake fireball using seismic, satellite and infrasound records. Meteorit. Planet. Sci. 37, 661–675 (2002).
 27.
Colmenares, L. & Zoback, M. A statistical evaluation of intact rock failure criteria constrained by polyaxial test data for five different rocks. Int. J. Rock Mech. Min. Sci. 39, 695–729 (2002).
 28.
Leinhardt, Z. M., Ogilvie, G. I., Latter, H. N. & Kokubo, E. Tidal disruption of satellites and formation of narrow rings. Mon. Not. R. Astron. Soc. 424, 1419–1431 (2012).
 29.
Colwell, J. et al. Saturn from Cassini–Huygens (eds Dougherty, M. K., Esposito, L. W. & Krimigis, S. M.) 375–412 (Springer, 2009).
 30.
Goldreich, P. & Tremaine, S. Disk–satellite interactions. Astrophys. J. 241, 425–441 (1980).
 31.
JeongAhn, Y. & Malhotra, R. The current impact flux on Mars and its seasonal variation. Icarus 262, 140–153 (2015).
 32.
Beech, M. Towards an understanding of the fall circumstances of the Hoba meteorite. Earth Moon Planets 111, 15–30 (2013).
 33.
Lee, P. et al. in Proc. 46th Lunar Planet. Sci. Conf. 2015 2856 (LPI, 2015).
 34.
Raymond, C., Prettyman, T. & Diniega, S. in Proc. 46th Lunar Planet. Sci. Conf. 2015 2792 (LPI, 2015).
 35.
Murchie, S. et al. in Proc. 46th Lunar Planet. Sci. Conf. 2015 2047 (LPI, 2015).
 36.
Kokubo, E. & Genda, H. Formation of terrestrial planets from protoplanets under a realistic accretion condition. Astrophys. J. Lett. 714, L21 (2010).
 37.
Raymond, S. N., Kokubo, E., Morbidelli, A., Morishima, R. & Walsh, K. J. in Protostars and Planets VI (eds Beuther, H. et al.) 595–618 (Univ. Arizona, Tucson, 2014).
 38.
Schofield, J. T. et al. The Mars Pathfinder atmospheric structure investigation/meteorology (ASI/MET) experiment. Science 278, 1752–1758 (1997).
 39.
Collins, G., Elbeshausen, D., Davison, T., Robbins, S. & Hynek, B. The sizefrequency distribution of elliptical impact craters. Earth Planet. Sci. Lett. 310, 1–8 (2011).
 40.
Taylor, P. A. & Margot, J.L. Tidal evolution of close binary asteroid systems. Celest. Mech. Dyn. Astron. 108, 315–338 (2010).
 41.
Goldreichand, P. & Gold, T. On the eccentricity of satellite orbits in the solar system. Mon. Not. R. Astron. Soc. 126, 257–268 (1963).
 42.
Rosenblatt, P. & Charnoz, S. On the formation of the Martian moons from a circumMartian accretion disk. Icarus 221, 806–815 (2012).
 43.
Efroimsky, M. & Lainey, V. Physics of bodily tides in terrestrial planets and the appropriate scales of dynamical evolution. J. Geophys. Res. 112, E12003 (2007).
 44.
AuclairDesrotour, P., Le PoncinLafitte, C. & Mathis, S. Impact of the frequency dependence of tidal Q on the evolution of planetary systems. Astron. Astrophys. 561, L7 (2014).
 45.
Yokoyama, T., Mana, M. R., Do Nascimento, C., Santos, M. T. & Callegari, N. Jr On the dynamics of some resonances of Phobos in the future. Astron. Astrophys. 429, 731–738 (2005).
 46.
Yokoyama, T. Possible effects of secular resonances in Phobos and Triton. Planet. Space Sci. 50, 63–77 (2002).
 47.
Peale, S. J. & Cassen, P. Contribution of tidal dissipation to lunar thermal history. Icarus 36, 245–269 (1978).
 48.
Peale, S. J., Cassen, P. & Reynolds, R. T. Melting of Io by tidal dissipation. Science 203, 892–894 (1979).
 49.
Wisdom, J. Spin–orbit secondary resonance dynamics of Enceladus. Astron. J. 128, 484–491 (2004).
 50.
Goldreich, P. & Sari, R. Tidal evolution of rubble piles. Astrophys. J. 691, 54–60 (2009).
 51.
Le Maistre, S. et al. Phobos interior from librations determination using Doppler and star tracker measurements. Planet. Space Sci. 85, 106–122 (2013).
 52.
Holsapple, K. A. Equilibrium configurations of solid cohesionless bodies. Icarus 154, 432–448 (2001).
 53.
Holsapple, K. A. & Michel, P. Tidal disruptions: A continuum theory for solid bodies. Icarus 183, 331–348 (2006).
 54.
Hoek, E. Estimating Mohr–Coulomb friction and cohesion values from the Hoek–Brown failure criterion. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 27, 227–229 (1990).
 55.
Holsapple, K. A. On the “strength” of the small bodies of the solar system: A review of strength theories and their implementation for analyses of impact disruptions. Planet. Space Sci. 57, 127–141 (2009).
 56.
Eberhardt, E. The Hoek–Brown Failure Criterion. The ISRM Suggested Methods for Rock Characterization, Testing and Monitoring 2007–2014 233–240 (Springer, 2015).
 57.
Klimczak, C., Byrne, P. K. & Solomon, S. C. A rockmechanical assessment of Mercury’s global tectonic fabric. Earth Planet. Sci. Lett. 416, 82–90 (2015).
 58.
Watters, T. R., Thomas, P. C. & Robinson, M. S. Thrust faults and the nearsurface strength of asteroid 433 Eros. Geophys. Res. Lett. 38, L02202 (2011).
 59.
Williams, N., Watters, T, Pritchard, M., Banks, M. & Bell, J. Faults dislocation modeled structure of lobate scarps from Lunar Reconnaissance Orbiter Camera digital terrain models. J. Geophys. Res.: Planets 118, 224–233 (2013).
 60.
Jutzi, M. Sph calculations of asteroid disruptions: The role of pressure dependent failure models. Planet. Space Sci. 107, 3–9 (2015).
 61.
Mitchell, J. K., Bromwell, L. G., Carrier III, W. D., Costes, N. C. & Scott, J. F. Soil mechanical properties at the Apollo 14 site. J. Geophys. Res. 77, 5641–5664 (1972).
 62.
Lambe, T. W. & Whitman, R. V. Soil Mechanics SI Version (John Wiley, 2008).
 63.
Hoek, E. Practical Rock Engineering (2000); www.rocscience.com
 64.
Cai, M. Practical estimates of tensile strength and Hoek–Brown strength parameter m_{i} of brittle rocks. Rock Mech. Rock Eng. 43, 167–184 (2010).
 65.
Marinos, V., Marinos, P. & Hoek, E. The geological strength index: Applications and limitations. Bull. Eng. Geol. Environ. 64, 55–65 (2005).
 66.
Pajola, M. et al. Phobos as a dtype captured asteroid, spectral modeling from 0.25 to 4.0 μm. Astrophys. J. 777, 127 (2013).
 67.
Tomeoka, K. & Buseck, P. R. Indicators of aqueous alteration in cm carbonaceous chondrites: Microtextures of a layered mineral containing Fe, S, O and Ni. Geochim. Cosmochim. Acta 49, 2149–2163 (1985).
 68.
Fraeman, A. A. et al. Spectral absorptions on Phobos and Deimos in the visible/near infrared wavelengths and their compositional constraints. Icarus 229, 196–205 (2014).
 69.
Hunten, D. M. Capture of Phobos and Deimos by photoatmospheric drag. Icarus 37, 113–123 (1979).
 70.
Burns, J. A. Contradictory clues as to the origin of the Martian moons. Mars 1, 1283–1301 (1992).
 71.
Andert, T. P. et al. Precise mass determination and the nature of Phobos. Geophys. Res. Lett. 37, L09202 (2010).
 72.
Citron, R. I., Genda, H. & Ida, S. Formation of Phobos and Deimos via a giant impact. Icarus 252, 334–338 (2015).
 73.
Fanale, F. P. & Salvail, J. R. The influence of CO ice on the activity and nearsurface differentiation of comet nuclei. Icarus 84, 403–413 (1990).
 74.
Popova, O. et al. Very low strengths of interplanetary meteoroids and small asteroids. Meteorit. Planet. Sci. 46, 1525–1550 (2011).
 75.
Brown, P. G. et al. The fall, recovery, orbit, and composition of the Tagish Lake meteorite: A new type of carbonaceous chondrite. Science 290, 320–325 (2000).
 76.
Hildebrand, A. R. et al. The fall and recovery of the Tagish Lake meteorite. Meteorit. Planet. Sci. 41, 407–431 (2006).
 77.
Bažant, Z. P, Kim, J.J. H., Daniel, I. M., BecqGiraudon, E. & Zi, G. Size effect on compression strength of fiber composites failing by kink band propagation. Int. J. Fract. 95, 103–141 (1999).
 78.
Sharma, I. The equilibrium of rubblepile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates. Icarus 200, 636–654 (2009).
 79.
Dobrovolskis, A. R. Internal stresses in Phobos and other triaxial bodies. Icarus 52, 136–148 (1982).
 80.
Alejano, L. R. & Bobet, A. Drucker–Prager criterion. Rock Mech. Rock Eng. 45, 995–999 (2012).
 81.
Veras, D., Leinhardt, Z. M., Bonsor, A. & Gänsicke, B. T. Formation of planetary debris discs around white dwarfs—i. Tidal disruption of an extremely eccentric asteroid. Mon. Not. R. Astron. Soc. 445, 2244–2255 (2014).
 82.
LyndenBell, D. & Pringle, J. E. The evolution of viscous discs and the origin of the nebular variables. Mon. Not. R. Astron. Soc. 168, 603–637 (1974).
 83.
Goldreich, P. & Tremaine, S. The velocity dispersion in Saturn’s rings. Icarus 34, 227–239 (1978).
 84.
Esposito, L. W. Composition, structure, dynamics, and evolution of Saturn’s rings. Annu. Rev. Earth Planet. Sci. 38, 383–410 (2010).
 85.
Toomre, A. On the gravitational stability of a disk of stars. Astrophys. J. 139, 1217–1238 (1964).
 86.
Salo, H. Simulations of dense planetary rings III. Selfgravitating identical particles. Icarus 117, 287–312 (1995).
 87.
Daisaka, H. & Ida, S. Spatial structure and coherent motion in dense planetary rings induced by selfgravitational instability. Earth Planets Space 51, 1195–1213 (1999).
 88.
Salo, H., Schmidt, J. & Spahn, F. Viscous overstability in Saturn’s B ring: I. Direct simulations and measurement of transport coefficients. Icarus 153, 295–315 (2001).
 89.
Yasui, Y., Ohtsuki, K. & Daisaka, H. Viscosity in planetary rings with spinning selfgravitating particles. Astron. J. 143, 110 (2012).
 90.
Shukhman, I. G. Collisional dynamics of particles in Saturn’s rings. Sov. Astron. 28, 574–585 (1984).
 91.
Shu, F. H. & Stewart, G. R. The collisional dynamics of particulate disks. Icarus 62, 360–383 (1985).
 92.
Morishima, R. & Salo, H. Simulations of dense planetary rings: IV. Spinning selfgravitating particles with size distribution. Icarus 181, 272–291 (2006).
 93.
Salmon, J., Charnoz, S., Crida, A. & Brahic, A. Longterm and largescale viscous evolution of dense planetary rings. Icarus 209, 771–785 (2010).
 94.
Dougherty, M., Esposito, L. & Krimigis, S. Saturn from Cassini–Huygens (Springer Science and Business Media, 2009).
 95.
Hedman, M. M. & Nicholson, P. D. Kronoseismology: Using density waves in Saturn’s C ring to probe the planet’s interior. Astron. J. 146, 12 (2013).
 96.
Hahn, J. M. & Spitale, J. N. An nbody integrator for gravitating planetary rings, and the outer edge of Saturn’s B ring. Astrophys. J. 772, 122 (2013).
 97.
Nicholson, P. D., French, R. G., Hedman, M. M., Marouf, E. A. & Colwell, J. E. Noncircular features in Saturn’s rings I: The edge of the B ring. Icarus 227, 152–175 (2014).
 98.
Beech, M. Electrophonic sound generation by the Chelyabinsk fireball. Earth Moon Planets 113, 33–41 (2014).
 99.
Pollack, J. B. & Burns, J. A. An origin by capture for the Martian satellites? Bull. Am. Astron. Soc. 9, 518 (1977).
 100.
Peale, S. J. & Canup, R. M. Treatise on Geophysics 2nd edn, 559–604 (Elsevier, 2015).
 101.
Schultz, P. H. & LutzGarihan, A. B. Grazing impacts on Mars: A record of lost satellites. J. Geophys. Res. 87, A84–A96 (1982).
 102.
Robbins, S. J. & Hynek, B. M. A new global database of Mars impact craters ≤ 1 km: 1. Database creation, properties, and parameters. J. Geophys. Res. 117, E06001 (2012).
Acknowledgements
B.A.B. thanks the Open Earth Systems project funded by National Science Foundation grant EAR1135382. K. Ferrier, B. Marsh, H. Lenferink, B. Buffett, R. Citron and M. Manga provided valuable feedback on earlier versions of this work. The authors gratefully acknowledge the support of M. Manga.
Author information
Affiliations
Department of Earth and Planetary Science, University of California, Berkeley, Berkeley, California 94720, USA
 Benjamin A. Black
 & Tushar Mittal
Department of Earth and Atmospheric Science, The City College of New York, City University of New York, 160 Convent Avenue, New York City, New York 10031, USA
 Benjamin A. Black
Authors
Search for Benjamin A. Black in:
Search for Tushar Mittal in:
Contributions
B.A.B. and T.M. jointly planned the research, performed the calculations, and wrote and revised the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Benjamin A. Black or Tushar Mittal.
Supplementary information
PDF files
 1.
Supplementary Information
Supplementary Information
Rights and permissions
To obtain permission to reuse content from this article visit RightsLink.
About this article
Further reading

Three Dynamical Evolution Regimes for Coupled Ringsatellite Systems and Implications for the Formation of the Uranian Satellite Miranda
The Astronomical Journal (2019)

The rheology and thermal history of Mars revealed by the orbital evolution of Phobos
Nature (2019)

A dynamical context for the origin of Phobos and Deimos
Monthly Notices of the Royal Astronomical Society (2018)

The ABCs of Viewing Ringed Planets
The Mathematical Intelligencer (2018)

Can Rocky Exoplanets with Rings Pose as SubNeptunes?
The Astronomical Journal (2018)