Constraints on Titan rotation from Cassini radar data

46th Lunar and Planetary Science Conference (2015)
Constraints on Titan rotation from Cassini radar data
B.G. Bills1, B.W. Stiles1, R.L. Kirk2, A. Hayes3, S. Birch3, P. Corlies3
Jet Propulsion Laboratory, Caltech, Pasadena, CA, 91109, 2Astrogeology, USGS, Flagstaff, AZ, 86011
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
= 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)
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
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
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 [4].
A decoupled shell would also allow the spin pole to
closely approximate a Cassini state, or fully damped
configruation [5].
ଶ 3ଶ References:
[1] Stiles, B.W. et al. (2008), A.J. 135, 1669-1680
[2] Stiles, B.W. et al. (2010), A.J. 139, 311-311.
[3] Iess, L. et al. (2012), Science, 327, 1367-1369.
[4] van Hoolst, T. et al. (2013), Icarus, 226, 299-315.
[5] Bills, B.G. et al. (2011), Icarus, 214, 351-354.