## DDA 2015 – Recent dynamical evolution of Mimas and Enceladus

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

### Session: Moon Formation and Dynamics III

Maja Cuk (SETI Institute)

#### Abstract

Mimas and Enceladus are the smallest and innermost mid-sized icy moons of Saturn. They are each caught in a 2:1 orbital resonance with an outer, larger moon: Mimas with Tethys, Enceladus with Dione. This is where the similarities end. Mimas is heavily cratered and appears geologically inactive, while Enceladus has a young surface and high tidal heat flow. Large free eccentricity of Mimas implies low tidal dissipation, while Enceladus appears very dissipative, likely due to an internal ocean. Their resonances are very different too. Mimas is caught in a 4:2 inclination type resonance with Tethys which involves inclinations of both moons. Enceladus is in a 2:1 resonance with Dione which affects only Enceladus’s eccentricity. The well-known controversy over the heat flow of Enceladus can be solved by invoking a faster tidal evolution rate than previously expected (Lainey et al. 2012), but other mysteries remain. It has been long known that Mimas has very low probability of being captured into the present resonance, assuming that the large resonant libration amplitude reflects sizable pre-capture inclination of Mimas. Furthermore, Enceladus seems to have avoided capture into a number of sub-resonances that should have preceded the present one. An order of magnitude increase in the rate of tidal evolution does not solve these problems. It may be the time to reconsider the dominance of tides in the establishment of these resonances, especially if the moons themselves may be relatively young. An even faster orbital evolution due to ring/disk torques can help avoid capture into smaller resonances. Additionally, past interaction of Mimas with Janus and Epimetheus produce some of the peculiarities of Mimas’ current orbit. At the meeting I will present numerical integrations that confirm the the existence of these problems, and demonstrate the proposed solutions.

#### Notes

• tidal rates $\dfrac{1}{a}\dfrac{d a}{d t}$: Mimas = 59, Enceladus = 23
• numerical integrations — brute force
• artificial migration
• slow
• the trouble with Mimas
• Mimas and Tethys in inclination-type 4:2 MMR
• inclination of both moons affected by the resonance
• libration amp. of resonance is large, ~100 deg $\rightarrow$ primordial Mimas inclination — doesn’t work
• eccentricity of Tethys has complex effects
• Mimas-Tethys evolution rate: $\dfrac{da_{moon}}{dt} \propto \dfrac{R^5_{planet}}{a^{3/2}}$
• introduce ad hoc ring torques — artificial torque on Prometheus
• gives Tethys resonance a kick
• $\therefore$ don’t take Mimas-Tethys resonance too seriously
• rings-Janus-Mimas-Enceladus-Dione system evolution is very complex

## DDA 2015 – Forced libration of tidally synchronized planets and moons

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

### Session: Moon Formation and Dynamics I

Valeri Makarov (for Bryan Dorland) (USNO)

#### Abstract

Tidal dissipation of kinetic energy, when it is strong enough, tends to synchronize the rotation of planets and moons with the mean orbital motion, or drive it into long-term stable spin-orbit resonances. As the actual orbital motion undergoes periodic acceleration due to a finite orbital eccentricity, the spin rate oscillates around the equilibrium mean value too, giving rise to the forced, or eccentricity-driven, librations. We contend that both the shape and amplitude of forced librations of synchronous viscoelastic planets and moons are defined by a combination of two different types of perturbative torque, the tidal torque and the triaxial torque, the latter related to a permanent deformation of the distribution of mass from a perfect rotational symmetry. Consequently, forced librations can be tidally dominated (e.g., Io and possibly Titan) or deformation-dominated (e.g., the Moon) depending on a set of orbital, rheological, and other physical parameters. With small eccentricities, for the former kind, the largest term in the libration angle can be minus cosine of the mean anomaly, whereas for the latter kind, it is minus sine of the mean anomaly. The shape and the amplitude of tidal forced librations determine the rate of orbital evolution of synchronous planets and moons, i.e., the rate of dissipative damping of the semimajor axis and eccentricity. The known super-Earth exoplanets can exhibit both kinds of libration, or a mixture of thereof, depending on, for example, the effective Maxwell time of their rigid mantles. Our approach can be extended to estimate the amplitudes of other libration harmonics, as well as the forced libration in non-synchronous spin-orbit resonances.

#### Notes

• Numerical sim of Lunarspinlibrations in longitude with polar torques exerted by Earth
• spectrum of harmonics
• dominant term: $\dfrac{\dot{\theta}}{n}\, – 1 \propto -\cos M$
• Geometry offorcedlibrations
• longest axis of planet tries to align with line of centers (but can’t)
• triaxial torque (Eckhardt 1981) $\ddot{\theta}\approx 6 n^2 \dfrac{B-A}{C^3}\left[e\, – \dfrac{31}{16}e^2 + O\left(e^5\right)\right] \sin M$
• Problem: Io
• striking difference with moon
• dominant term now $\dfrac{\dot{\theta}}{n}\, – 1 \propto +\sin M$
• Tidal torques
• torque = triaxial + tidal
• tidal = secular + periodic
• secular can dominate in the “linear” regime
• plot: $\ddot{\theta}$ vs. $\dfrac{\dot{\theta}}{n}\, – 1$
• It’s not the amplitude that’s important, but the gradient, which provides a subtle but powerful feedback.
• tiny interval of pert freqs where secular torque can be important
• Problem: doesn’t work for Moon
• because Moon is not sitting on that linear portion of $\dfrac{\dot{\theta}}{n}\, – 1$
• $\therefore$ need further harmonic terms

## DDA 2015 – Gravity and Tide Parameters Determined from Satellite and Spacecraft Orbits

This is one of a series of notes taken during the 2015 meeting of the AAS Division on Dynamical Astronomy, 3-7 May, at CalTech. An index to this series (all the papers presented at the meeting) is here.

### Session: Dynamics of Small Solar System Bodies II

Robert A. Jacobson (JPL)

#### Abstract

As part of our work on the development of the Jovian and Saturnian satellite ephemerides to support the Juno and Cassini missions, we determined a number of planetary system gravity parameters. This work did not take into account tidal forces. In fact, we saw no obvious observational evidence of tidal effects on the satellite or spacecraft orbits. However, Lainey et al. (2009 Nature 459, 957) and Lainey et. al (2012 Astrophys. J. 752, 14) have published investigations of tidal effects in the Jovian and Saturnian systems, respectively. Consequently, we have begun a re-examination of our ephemeris work that includes a model for tides raised on the planet by the satellites as well as tides raised on the satellites by the planet. In this paper we briefly review the observations used in our ephemeris production; they include astrometry from the late 1800s to 2014, mutual events, eclipses, occultatons, and data acquired by the Pioneer, Voyager, Ulysses, Cassini, Galileo, and New Horizons spacecraft. We summarize the gravity parameter values found from our original analyses. Next we discuss our tidal acceleration model and its impact on the gravity parameter determination. We conclude with preliminary results found when the reprocessing of the observations includes tidal forces acting on the satellites and spacecraft.

#### Notes

• Jupiter and Saturn gravity fields program at JPL
• started with Pioneer
• probably end with Juno (or proposed Europa) mission
• also Earth-based
• 1874-2014
• Saturnrigh stellar occultations
• pole orientation
• Saturn ring plane crossing times
• pole orientation
• spacecraft:
• imaging
• VLBI
• Saturn ring occultations
• But no tidal forces used in any analysis so far.
• But tidal effects are not zero
• Lainey et al. 2009, 2012
• Efroimsky & Lainey 2007 (JGR 112)
• $U_{jk} = k_2^k \left(\dfrac{\mu_j}{R_k}\right)^3 \left(\dfrac{R_k}{r}\right)^3 \left(\dfrac{R_k}{r^*_{jk}}\right)^3 P_2\left(\hat{r} \cdot \hat{r}^*_{jk}\right)$
• $r^*_{jk} = r_{jk} – \Delta t_j \left[\dot{r}_{jk} + \dot{W}_k\left(\hat{r}_{jk}\times\hat{h}_k\right)\right]$
• Tidal lag effects
• Put tides in fitting model
• $\rightarrow k_2$
• $\rightarrow$ gravity harmonic coefficients
• tidal lags: indeterminate from existing data
• tidal dissipation function $Q = \dfrac{2 \pi E}{\Delta E} = f(\Delta t)$
• $E$ = max energy stored in one tidal cycle
• $\Delta E$ = energy dissipated during that cycle
• $f(\Delta t) = \dfrac{1}{\omega^{\alpha} \Delta t}$
• comparison to Lainey for Jupiter:
• indeterminate
• comparison to Lainey for Saturn (common $Q$):
• $\Delta t$ and $\dfrac{k_2}{Q}$ successfully detected for Mimas, Enceladus, Tethys, Dione, and Rhea, $k_2 = 0.379 \pm 0.011$
• Lainey: $\dfrac{k_2}{Q} = 2.3\pm0.7 \times 10^{-4}$, $k_2 = 0.341$
• JPL:$\dfrac{k_2}{Q} = 1.0\pm0.2 \times 10^{-4}$, $k_2 = 0.381 \pm 0.011$