Dynamics and Fate of the Past Martian Rings

The two small satellites of Mars, Phobos and Deimos, have low-eccentricity obits close to the plane of the Martian equator, implying their in-situ formation (Burns, 1992). The most common proposed hypothesis for their origin is that the moons formed from debris ejected by a giant impact onto early Mars (Craddock, 2011; Rosenblatt and Charnoz, 2012; Citron et al., 2015; Rosenblatt et al., 2016; Hesselbrock and Minton, 2017; Hyodo et al., 2017; Canup and Salmon, 2018). Much is unknown about how the initial giant impact gave rise to the current moons, as Phobos (and any other more massive moons present in the past) likely experienced large-scale tidal evolution over the age of the Solar System. As Mars is a relatively slow rotator, all moons orbiting interior to the synchronous orbit at about 6 Mars radii (RM) orbit faster than Mars spins and tidally migrate inward (Deimos is just outside synchronous orbit, at 6.92 RM).

 

Hesselbrock and Minton (2017) (HM17) have proposed that Phobos is not primordial, but a product of ring-moon cycles over the age of the Solar System. According to HM17, successive generations of Martian moons formed from successive generations of rings. After forming at the outer edge of the rings (near the Fluid Roche Limit at 3.2 RM ), the moons migrate outward through interactions with the ring until the ring dissipates. When the ring torques become too weak to counter Martian tides, the moons migrate inward, until the Martian tides disrupt them at the rigid Roche Limit (at 1.6 RM ), forming the next generation of the Martian ring (cf. Black and Mittal, 2015). Each generation of moons is several times less massive than the previous, as inward tidal evolution drains angular momentum from the system, and the rest of the mass is accreted onto Mars.

 

One of the conclusions of HM17 was that, over several Gyr, mass many times larger than that of Phobos was deposited over the equatorial regions of Mars due to ring infall. So far, there have not been any claims of geological or geochemical features on Mars that would be related to this mass infall. Here we revisit the dynamics of the putative past ring of Mars and show that its evolution would have been complex and largely shaped by solar resonances.

 

During its past and future evolution Phobos crosses several solar semi-secular resonances (SSRs), in which a moon’s precession period is a multiple or a simple fraction of Mars’s heliocentric orbital period. Unlike mean-motion resonances in which capture can happen only during convergent evolution (Murray and Dermott, 1999), capture into eccentricity and inclination-type solar SSRs requires different directions of migration. When the planet’s oblateness is the dominant source of perturbation, a satellite’s orbit precesses faster as eccentricity grows (Danby, 1992), making capture into constant-precession-rate resonances possible during outward migration (e.g. evection resonance; Touma and Wisdom, 1998; Cuk and Stewart, 2012). However, inclined orbits precess more slowly than planar (Danby, 1992), making inward migration a requirement for capture into an inclination-type SSR. This has previously been confirmed numerically for the 2:3 SSR (Touma and Wisdom, 1998) and 2:1 SSR (Yokoyama, 2002; Yokoyama et al., 2005). The 2:1 SSR is of particular interest for modeling the future dynamics of inward migrating moons and rings resulting from their disruption.

Figure 1: The capture of a ring particle (or a moon) into the 2:1 solar semi-secular resonance, at about 2.15 RM. As the particle's orbit is shrinking, it is moving right to left in the plot. At this distance the apsidal and nodal precession periods are 0.5 martian years. The particle's orbit is captured into the resonance and the inclination keeps increasing as the particle drifts closer to Mars. This resonance was first described by Yokoyama et al. (2005), so we refer to it as the Yokoyama resonance.

 

Yokoyama et al. (2005) find that Phobos is very likely to be captured into the 2λ_M +Ω−3Ω_Eq harmonic of the 2:1 SSR at 2.15 RM (“Yokoyama resonance”), where λ_{M i}s the mean longitude of Mars,Ω is the longitude of the node of the moon (or a dust particle) and Ω_Eq is the longitude of Mars’s equinox. We confirm the general process using our integrator SIMPL in the simulation shown in Fig. 1. A particle migrating inward relatively rapidly (on sub-Myr timescales) is captured into the Yokoyama resonance, and its inclination grows over time as it maintains a constant nodal precession of half a Martian year as its semimajor axis continues shrinking. The bottom panel in Fig. 1 shows the equivalent semimajor axis, if the particle were to damp onto a planar orbit while conserving orbital angular momentum (as in collisions within a ring). But the equatorial angular momentum is proportional to the cosine of inclination and the square root of semimajor axis, so we can see that by the time inclination within the resonance reaches about 45 deg, the equivalent planar semimajor axis is within the planet. The physical meaning of this result is that, if the resonance is broken and particles formerly in the resonance are allowed to collide, they will impact Mars before managing to form a thin ring. For the angular momentum seen at the end of the simulation in Fig. 1, particles will have inclinations of about 26 deg when impacting Mars.

 

We will present more in-depth explorations of both the exact distance at which we expect the disruption of Phobos (and its putative forebears) and the fate of martian ring particles under the influence of solar resonances.