## DDA 2015 – Constraints on Titan’s rotation from Cassini mission radar data

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.

Bruce Bills (JPL)

#### Abstract

We present results of a new analysis of the rotational kinematics of Titan, as constrained by Cassini radar data, extending over the entire currently available set of flyby encounters. Our analysis provides a good constraint on the current orientation of the spin pole, but does not have sufficient accuracy and duration to clearly see the expected spin pole precession. In contrast, we do clearly see temporal variations in the spin rate, which are driven by gravitational torques which attempt to keep the prime meridian oriented toward Saturn.

Titan is a synchronous rotator. At lowest order, that means that the rotational and orbital motions are synchronized. At the level of accuracy required to fit the Cassini radar data, we can see that synchronous rotation and uniform rotation are not quite the same thing. Our best fibng model has a fixed pole, and a rotation rate which varies with time, so as to keep Titan’s prime meridian oriented towards Saturn, as the orbit varies.

A gravitational torque on the tri-axial figure of Titan attempts to keep the axis of least inertia oriented toward Saturn. The main effect is to synchronize the orbit and rotation periods, as seen in inertial space. The response of the rotation angle, to periodic changes in orbital mean longitude, is modeled as a damped, forced harmonic oscillator. This acts as a low-pass filter. The rotation angle accurately tracks orbital variations at periods longer than the free libration period, but is unable to follow higher frequency variations.

The mean longitude of Titan’s orbit varies on a wide range of time scales. The largest variations are at Saturn’s orbital period (29.46 years), and are due to solar torques. There are also variations at periods of 640 and 5800 days, due to resonant interaction with Hyperion.

For a rigid body, with moments of inertia estimated from observed gravity, the free libration period for Titan would be 850 days. The best fit to the radar data is obtained with a libration period of 645 days, and a damping time of 1000 years.

The principal deviation of Titan’s rotation from uniform angular rate, as seen in the Cassini radar data, is a periodic signal resonantly forced by Hyperion.

#### Notes

• Titan:
• hard to see surface
• Cassini’s radar intended for mapping surface
• didn’t get much by way of repeat observations (“tie points”), which are needed to constrain rotation
• most data near poles — not terribly helpful
• Rotation model from tie-point observations
• Stiles et al. 2008: 50 tie points over 2.8 yr
• Now: 2602 tie points over 10 yr
• solve for 3 params (RA & DEC of spin pole, angular rate)
• $P = 15.94547727 \pm 6.03 \times 10^{-7}$ d
• spin pole precession
• gravity model: ~250 yr
• not clearly seen in data
• spin rate variations
• seen in data
• dynamical model
• assume Titan in synch. rotation
• gravity torque
• dissipation
• $\rightarrow$ libration period ~850 d
• Hyperion has nontrivial influence
• fit: libration period = 645.4 d, damping time = 430 yr, rotation period slightly changed

## DDA 2015 – Recent Formation of Saturnian Moons: Constraints from Their Cratering Records

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 I

Henry C. (Luke) Dones (SWRI)

#### Abstract

Charnoz et al. (2010) proposed that Saturn’s small “ring moons” out to Janus and Epimetheus consist of ring material that viscously spread beyond the Roche limit and coagulated into moonlets. The moonlets evolve outward due to the torques they exert at resonances in the rings. More massive moonlets migrate faster; orbits can cross and bodies can merge, resulting in a steep trend of mass vs. distance from the planet. Canup (2010) theorized that Saturn’s rings are primordial and originated when a differentiated, Titan-like moon migrated inward when the planet was still surrounded by a gas disk. The satellite’s icy shell could have been tidally stripped, and would have given rise to today’s rings and the mid-sized moons out to Tethys. Charnoz et al. (2011) investigated the formation of satellites out to Rhea from a spreading massive ring, and Crida and Charnoz (2012) extended this scenario to other planets. Once the mid-sized moons recede far from the rings, tidal interaction with the planet determines the rate at which the satellites migrate. Charnoz et al. (2011) found that Mimas would have formed about 1 billion years more recently than Rhea. The cratering records of these moons (Kirchoff and Schenk 2010; Robbins et al. 2015) provide a test of this scenario. If the mid-sized moons are primordial, most of their craters were created through hypervelocity impacts by ecliptic comets from the Kuiper Belt/Scattered Disk (Zahnle et al. 2003; Dones et al. 2009). In the Charnoz et al. scenario, the oldest craters on the moons would result from low-speed accretionary impacts. We thank the Cassini Data Analysis program for support.

References
Canup, R. M. (2010). Nature 468, 943
Charnoz, S.; Salmon, J., Crida, A. (2010). Nature 465, 752
Charnoz, S., et al. (2011). Icarus 216, 535
Crida, A.; Charnoz, S. (2012). Science 338, 1196
Dones, L., et al. (2009). In Saturn from Cassini-Huygens, p. 613
Kirchoff, M. R.; Schenk, P. (2010). Icarus 206, 485
Robbins, S. J.; Bierhaus, E. B.; Dones, L. (2015). Lunar and Planetary Science Conference 46, abstract 1654
(http://www.hou.usra.edu/meetings/lpsc2015/eposter/1654.pdf)
Zahnle, K.; Schenk, P.; Levison, H.; Dones, L. (2003). Icarus 163, 263

#### Notes

• Can cratering records constrain moon ages?
• see http://space.jpl.nasa.gov
• small inner moons (and Mimas) interact strongly with rings — the so-called “ring moons”
• migrated from outer edge of rings ~100 Myr
• regular moons (Mimas-Iapetus) are (assumed?) primordial
• transition is abrupt where tidal forces prevent formation
• formation of moons from spreading rings:Charnoz et al. 2010,Canup 2010,Charnoz et al. 2011,Crida &Charnoz 2012
• outside Roche limit, formation
• Lainey et al. 2012: dissipation stronger than thought
• decreases timescale considerably
• Impact rates
• $R_{moon} = R_J \dfrac{R_S}{R_J} \dfrac{R_{moon}}{R_S}$
• Crater scaling: diameter vs. velocity
• impacts/$10^9$ yr: Mimas 8.5, Rhea 48
• Mimas & Rhea counts: Robbins et al. 2015 (LPSC)
• plot: #craters larger than D vs. D
• Mimas: saturated up to $D \sim 20-45$ km
• Rhea: saturated up to $D \sim 25$ km
• Summary
• Mimas: Craters are near saturation for diameters < 20 km
• Rhea: saturation < 25 km
• Ages may be underestimated

## DDA 2015 – Rotational and interior models for Enceladus II

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.

#### Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

#### Notes

• Libration measurement
• 3D reconstruction of coords of a network of control point (fiducial satellite surface points — e.g. craters)
• most of Enceladus’s orbit was covered
• Thomas et al. 2015
• minimize RMS residual $\rightarrow 0.120 \pm 0.014$ deg
• Solid models
• core plus two-layer in hydro.equilib. plus south polar sea
• measured libration amplitude rules this out
• decoupled shell from the core (indep.librations)
• consistent with observed libration amplitude if shell thickness 21-26 km and ocean thickness 26-31 km
• Gravity data
• suggests a local mass anomaly — interpreted as ocean thicker under south pole

## DDA 2015 – Rotational and interior models for Enceladus I

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 II

Matthew S. Tiscareno (Cornell)

#### Abstract

We will discuss the underlying dynamical models and the consequent interior models that pertain to our discovery of a forced rotational libration for Saturn’s moon Enceladus (Thomas et al. 2015).

Despite orbital variations that change the mean motion on timescales of several years owing to mutual satellite interactions, the rotation state of Enceladus should remain synchronous with the varying mean motion, as long as damping is as expected (Tiscareno et al. 2009, Icarus). Taking that dynamically synchronous rotation as the ground state, we construct a model that naturally focuses on the physically interesting librations about the synchronous state that occur on orbital timescales. We will discuss the differences between the method used here and other dynamical methods (e.g., Rambaux et al. 2010, GRL; cf. Tajeddine et al. 2014, Science), and we will review the rotation states (whether known or predicted) of other moons of Saturn.

We will also describe our measurements of the control point network on the surface of Enceladus using Cassini images, which was then used to obtain its physical forced libration amplitude at the orbital frequency. The fit of Cassini data results in a libration amplitude too large to be consistent with a rigid connection between the surface and the core, ruling out the simplest interior models (e.g., homogeneous, two-layer, two-layer with south polar anomaly). Alternatively, we suggest an interior model of Enceladus involving a global ocean that decouples the shell from the core, with a thinner icy layer in the south polar region as an explanation for both the libration (Thomas et al. 2015) and the gravity (Iess et al. 2014, Science) measurements.

#### Notes

• 2nd largest Saturnian moon
• Plumes — salty jets — observed by Cassini
• What is under the surface?
• Rotational parameters $\rightarrow$ interior structure
• Forcedlibrations
• same period as orbital
• nat. freq. $\omega_0 \approx n \sqrt{3 (B-A)/C}$
• near-spherical: moon always points at empty focus (synchronous)
• elongated: moon would always point at Saturn
• Enceladus axis oscillates around empty focus (synchronous rotation)
• as $\dfrac{B-A}{C} \rightarrow \dfrac{1}{3}$, resonance (Tiscareno et al. 2009)
• but Enceladus $\dfrac{B-A}{C} \ll \dfrac{1}{3}$
• Enceladus libration $0.120\pm0.014$ deg
• rules out rigid connection between surface and core
• hence, some kind of global subsurface ocean
• Mean motion variations
• Enceladus resonant arguments from interaction with Dione:
$ILR_D = \lambda_E\, – 2 \lambda_D + \varpi_E$ (librating)
$CIR_D = \lambda_E\, – 2 \lambda_D + \Omega_D$ (circulating)
$CER_D = \lambda_E\, – 2 \lambda_D + \varpi_D$ (circulating)
• As long as damping is sufficiently strong, synchronous rotation maintained
• damping must be $\gamma_{\pi/2} = \dfrac{2 e}{1\, – \left(\dfrac{n}{\omega_0}\right)^2} \Rightarrow \tau \approx 1.0\,Q\ \mathrm{days}$
• but $10 \lt Q \lt 100$ days
• rot. rate varies with the CER and ILR freqs
• not really “librations”
• maintaining synch. rot., while the mean motion varies quasiperiodically
• Rotational models
• Global Fourier components have limited usefulness
• MM variation more complex than a few periodic terms
• Define rot.statewrt Saturn
• base state: synch rot (expected for low triaxiality)
• accounts for MM variation
• easy to generate a range of kernels for many vals of $\gamma$
• Tiscareno 2015
• deflect $\psi(t) = (2 e+\gamma)\sin M$
• generate kernels of $\psi(t)$ for a wide range of $\gamma$ values, check for best control-point resids
• dissipation?