46th Lunar and Planetary Science Conference (2015) 1636.pdf Constraints on Titan rotation from Cassini radar data B.G. Bills1, B.W. Stiles1, R.L. Kirk2, A. Hayes3, S. Birch3, P. Corlies3 1 Jet Propulsion Laboratory, Caltech, Pasadena, CA, 91109, 2Astrogeology, USGS, Flagstaff, AZ, 86011 3 Astronomy, Cornell University, Ithaca, NY, 14850 Introduction: 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. Previous published analyses of this data type included only 6.7 years of data [1,2]. 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. Data description: The current data set includes 670 tie-point pairs. Each tie-point comprises two observations of the position of a fixed feature on the surface of Titan. There are thus two observation times, and two position vectors for each surface point. The position is given relative to the center of mass of Titan, in an inertial reference frame. The expected measurement error of these positions is roughly 1 km in each component. A lower limit on the error is due to the finite resolution of the radar data. Additional errors can arise from misidentification of pixel locations of matching points on the surface. In the complex geometry associated with Titan’s surface, features can appear quite different when viewed from different directions. Simple solution: Our simplest model has a fixed spin pole orientation, and a fixed rate of rotation. This model has an RMS misfit to the data of 2.29 km. The spin pole orientation is given by enced to different fiducial points. The rotation period is measured relative to an inertially fixed direction, while the orbital period is measured relative to the location of periapse, which is itself precessing. Motion of the spin pole: The spin and orbit poles of Titan both change their orientations, but do so on time scales which are long compared to the time span of our data. The orbit pole precesses about Saturn’s spin pole, in response to torques from Saturn’s oblate mass distribution, and from the Sun. These yield periods of motion of ~700 years, for the main precession, and 29.47 years for the solar-driven nutation. The rate at which the spin pole ̂ precesses about the orbit pole , is given by ̂ = ∙ ̂ + × ̂ The rate parameters are given by 3 ଶ + ଶ,ଶ = 2 ଶ,ଶ /2 where and , are gravitational potential harmonic coefficients, n is orbital mean motion, and c is dimensionless polar moment of inertia. Based upon current estimates of these parameters [1,2,3], the spin pole precession rate will be such that2⁄ + ≈ 250. Figure 1 shows the motion of orbit and spin poles of Titan, for a 700 year time span, centered on J2000. = 39.3935 ± 0.0063 = 83.4432 ± 0.0007 where is right ascension, and is declination. The rate of roation is = 22.576938 ± 1.6 × 10ି / which corresponds to a rotation period of = 15.945474 ± 1.7 × 10ି which is quite close to the mean orbital period. The osculating period of orbital motion for Titan varies on a wide range of time scales. The mean and standard deviation of the osculating orbital period, over the 400 year interval (1800-2200) AD is = 15.947586 ± 0.00032 The main cause of the observed difference between rotation period and orbital period is that they are refer- Figure 1. Precessional trajectories of orbit pole (heavy line) and spin pole (light line) of Titan, for 700 years. Origin is Saturn’s spin pole. 46th Lunar and Planetary Science Conference (2015) 1636.pdf Orbit rate variations: The mean longitude of Titan’s orbit varies on a wide range of time scales. That variation is dominated by a linear trend, which makes it cycle around the orbit in a mean period just under 16 days. The longitude of periapse circulates, with a mean period of roughly 700 years, but also has periodic variations due to perturbations from Jupiter and the Sun, and smaller perturbations from interactions with other satellites, principally Iapetus and Hyperion. Figure 2 shows variations in the mean orbital longitude, with a linear trend removed, over a 400 year time span. where A < B < C are the principal moments of inertia. This acts as a low-pass filter. The rotation angle accurately tracks orbital variations at frequencies below , but is unable to follow higher frequency variations. 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. Figure 3 shows variations in the rotation angle of the prime meridian of Titan, over a 30 year time span, as given by our best fitting model. Figure 2. Variations in orbital mean longitude of Titan, minus linear trend. Dominant effects are 700 year precession, due to oblate figure of Saturn, and ~29 year period oscillations, due to solar torques. Figure 3. Time variations in orientation of Titan’s prime meridian, compared to uniform rotation case An angle of 0.05 deg, on the equator, is a distance of 2.24 km.. Rotation rate variations: 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 fitting 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. The RMS misfit of this model to the radar data is 0.98 km. 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 inertia space. If the orbital mean longitude is L, and the inertial frame orientation of Titan’s prime meridian is Λ, the torque balance can be written in the form ଶ Λ ଶ Λ Λ ଶ where is a viscous damping rate, and is the free libration rate, which is given by Summary and Conclusions: We have found a kinematic model of the rotation of Titan which is dynamically plausible, and fits the Cassini radar data at the expected level. Titan is a synchronous rotator, but not quite a uniform rate rotator. The best fitting model of Titan’s rotation has a free libration period of 2⁄ 645 days, whereas the rigid body solution predicts a period of 850 days. A plausible interpretation of this difference is that Titan contains an internal fluid layer, which mechanically decouples the surface from the deeper interior, and allows it to librate more rapidly. The effective moments of inertia would be those of the decoupled shell. That interpretation is consistent with theoretical models of libration of decoupled shells . A decoupled shell would also allow the spin pole to closely approximate a Cassini state, or fully damped configruation . ଶ 3ଶ References:  Stiles, B.W. et al. (2008), A.J. 135, 1669-1680  Stiles, B.W. et al. (2010), A.J. 139, 311-311.  Iess, L. et al. (2012), Science, 327, 1367-1369.  van Hoolst, T. et al. (2013), Icarus, 226, 299-315.  Bills, B.G. et al. (2011), Icarus, 214, 351-354.
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